{"id":686,"date":"2023-01-11T09:25:06","date_gmt":"2023-01-11T01:25:06","guid":{"rendered":"http:\/\/192.168.10.115\/wordpress\/index.php\/2023\/01\/11\/686\/"},"modified":"2023-01-15T17:14:43","modified_gmt":"2023-01-15T09:14:43","slug":"%e7%bb%bc%e8%bf%b0%ef%bc%9a%e5%9f%8e%e5%b8%82%e9%81%93%e8%b7%af%e8%87%aa%e5%8a%a8%e9%a9%be%e9%a9%b6%e8%bd%a6%e8%be%86%e8%a7%84%e5%88%92%e4%b8%8e%e6%8e%a7%e5%88%b6%e6%8a%80%e6%9c%af","status":"publish","type":"post","link":"http:\/\/222.128.65.89:18086\/index.php\/2023\/01\/11\/686\/","title":{"rendered":"\u7efc\u8ff0\uff1a\u57ce\u5e02\u9053\u8def\u81ea\u52a8\u9a7e\u9a76\u8f66\u8f86\u89c4\u5212\u4e0e\u63a7\u5236\u6280\u672f"},"content":{"rendered":"

\u7f16\u8f91\u4e8e 2023-01-10 19:00\u30fbIP \u5c5e\u5730\u5317\u4eac<\/p>\n

\u200b\u76ee\u5f55<\/p>\n

\u6536\u8d77<\/p>\n

\u6458\u8981<\/p>\n

1. \u5f15\u8a00<\/p>\n

2. \u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u7684\u51b3\u7b56\u5c42\u7efc\u8ff0<\/p>\n

A. \u8def\u7ebf\u89c4\u5212<\/p>\n

B. \u884c\u4e3a\u51b3\u7b56<\/p>\n

C. \u8fd0\u52a8\u89c4\u5212<\/p>\n

D. \u8f66\u8f86\u63a7\u5236<\/p>\n

3. \u89c4\u5212\u4e0e\u63a7\u5236\u7684\u5efa\u6a21<\/p>\n

A. \u5355\u8f68\uff08Single-Track\uff09\u8fd0\u52a8\u5b66\u6a21\u578b<\/p>\n

B. \u60ef\u6027\u6548\u5e94<\/p>\n

4. \u8fd0\u52a8\u89c4\u5212<\/p>\n

A. \u8def\u5f84\u89c4\u5212<\/p>\n

B. \u8f68\u8ff9\u89c4\u5212<\/p>\n

C. \u53d8\u5206\u6cd5<\/p>\n

D. \u56fe\u641c\u7d22\u7b97\u6cd5<\/p>\n

E. \u589e\u91cf\u641c\u7d22\u6280\u672f<\/p>\n

F. \u5b9e\u9645\u90e8\u7f72<\/p>\n

5. \u8f66\u8f86\u63a7\u5236<\/p>\n

A. \u57fa\u4e8e\u8fd0\u52a8\u5b66\u6a21\u578b\u7684\u8def\u5f84\u7a33\u5b9a<\/p>\n

B. \u57fa\u4e8e\u8fd0\u52a8\u5b66\u6a21\u578b\u7684\u8f68\u8ff9\u8ddf\u8e2a\u63a7\u5236<\/p>\n

C. \u9884\u6d4b\u63a7\u5236\u65b9\u6cd5<\/p>\n

D. \u7ebf\u6027\u53d8\u53c2\u6570\u63a7\u5236<\/p>\n

6. \u7ed3\u8bba<\/p>\n

7. \u53c2\u8003\u6587\u732e,<\/p>\n

\u672c\u6587\u662f2016\u5e74\u7684\u7efc\u8ff0\u8bba\u6587\u300aA Survey of Motion Planning and Control Techniques for Self-driving Urban Vehicles\u300b\u7684\u4e2d\u6587\u7a3f\u3002\u8be5\u7bc7\u6587\u7ae0\u51713\u4e071\u5343\u5b57\uff0c\u6700\u521d\u4f5c\u4e3a\u6df1\u84dd\u5b66\u9662\u81ea\u52a8\u9a7e\u9a76\u63a7\u5236\u4e0e\u89c4\u5212\u5728\u7ebf\u8bfe\u7a0b\u7684\u5185\u90e8\u8d44\u6599\uff0c\u53d7\u5230\u540c\u5b66\u4eec\u7684\u5e7f\u6cdb\u597d\u8bc4\uff0c\u73b0\u5c06\u5176\u516c\u5f00\uff0c\u5e0c\u671b\u5e2e\u52a9\u66f4\u591a\u540c\u884c\u3002\u7279\u522b\u611f\u8c22\u6df1\u84dd\u5b66\u9662\u7684\u77f3\u5353\u51e1\u540c\u5b66\uff08\u5c31\u8bfb\u4e8e\u82cf\u9ece\u4e16\u8054\u90a6\u7406\u5de5\u5b66\u9662\uff09\u5bf9\u8bba\u6587\u7684\u7ffb\u8bd1\u53ca\u89e3\u8bfb\u3002<\/p>\n

\u6458\u8981<\/p>\n

\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u662f\u4e00\u9879\u6210\u719f\u7684\u6280\u672f\uff0c\u6709\u6f5c\u529b\u901a\u8fc7\u63d0\u9ad8\u6c7d\u8f66\u8fd0\u8f93\u7684\u5b89\u5168\u6027\u3001\u53ef\u8fbe\u6027\u3001\u6548\u7387\u548c\u4fbf\u6377\u6027\u6765\u91cd\u5851\u5176\u673a\u52a8\u6027\u3002\u5fc5\u987b\u7531\u81ea\u52a8\u9a7e\u9a76\u8f66\u8f86\u6267\u884c\u7684\u5b89\u5168\u5173\u952e\u4efb\u52a1\u5305\u62ec\u5728\u4e0e\u5176\u4ed6\u8f66\u8f86\u548c\u884c\u4eba\u5171\u4eab\u7684\u52a8\u6001\u73af\u5883\u4e2d\u89c4\u5212\u8fd0\u52a8\uff0c\u4ee5\u53ca\u901a\u8fc7\u53cd\u9988\u63a7\u5236\u5b9e\u73b0\u9c81\u68d2\u7684\u6267\u884c\u529b\u3002\u672c\u6587\u7684\u76ee\u7684\u662f\u7efc\u8ff0\u57ce\u5e02\u73af\u5883\u4e2d\u8f66\u8f86\u89c4\u5212\u548c\u63a7\u5236\u7b97\u6cd5\u7684\u73b0\u72b6\u3002\u6211\u4eec\u5c06\u56de\u987e\u4e00\u4e9b\u516c\u5f00\u7684\u6280\u672f\uff0c\u5e76\u5bf9\u5176\u6709\u6548\u6027\u8fdb\u884c\u8ba8\u8bba\u3002\u6839\u636e\u4f7f\u7528\u7684\u8f66\u8f86\u79fb\u52a8\u6a21\u578b\u3001\u5bf9\u73af\u5883\u7ed3\u6784\u7684\u5047\u8bbe\u4ee5\u53ca\u8ba1\u7b97\u8981\u6c42\uff0c\u6211\u4eec\u5206\u522b\u91c7\u7528\u4e86\u4e0d\u540c\u7684\u7efc\u8ff0\u65b9\u6cd5\u3002\u672c\u7efc\u8ff0\u901a\u8fc7\u5e76\u6392\u6bd4\u8f83\u6765\u8ba8\u8bba\u4e0d\u540c\u7684\u6280\u672f\u65b9\u6cd5\uff0c\u8fd9\u6709\u52a9\u4e8e\u8bfb\u8005\u6df1\u5165\u4e86\u89e3\u5404\u79cd\u65b9\u6cd5\u7684\u4f18\u52bf\u548c\u5c40\u9650\u6027\uff0c\u540c\u65f6\u6709\u52a9\u4e8e\u7cfb\u7edf\u7ea7\u7684\u8bbe\u8ba1\u9009\u62e9\u3002<\/p>\n

1. \u5f15\u8a00<\/p>\n

\u8fc7\u53bb\u4e09\u5341\u5e74\uff0c\u5b66\u672f\u754c\u548c\u5de5\u4e1a\u754c\u5bf9\u5f00\u53d1\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u6280\u672f\u7684\u7814\u7a76\u529b\u5ea6\u7a33\u6b65\u589e\u52a0\u3002\u8fd9\u4e9b\u53d1\u5c55\u5f97\u76ca\u4e8e\u4f20\u611f\u548c\u8ba1\u7b97\u6280\u672f\u7684\u6700\u65b0\u8fdb\u5c55\uff0c\u4ee5\u53ca\u5bf9\u6c7d\u8f66\u8fd0\u8f93\u7684\u6f5c\u5728\u53d8\u9769\u6027\u5f71\u54cd\u548c\u5df2\u7ecf\u88ab\u89c1\u8bc1\u7684\u793e\u4f1a\u6548\u76ca\uff1a2014\u5e74\uff0c32675\u4eba\u56e0\u4ea4\u901a\u4e8b\u6545\u6b7b\u4ea1\uff0c230\u4e07\u4eba\u53d7\u4f24\uff0c610\u4e07\u4eba\u906d\u9047\u78b0\u649e\u4e8b\u6545[1]\u3002\u5176\u4e2d\u4f30\u8ba1\u6709 94% \u7684\u4e8b\u6545\u662f\u56e0\u4e3a\u9a7e\u9a76\u5458\u7684\u9519\u8bef\uff0c\u4f8b\u5982\u6709 31% \u7684\u4e8b\u6545\u6d89\u53ca\u6cd5\u5f8b\u4e0a\u88ab\u5224\u5b9a\u4e3a\u9189\u9152\u7684\u9a7e\u9a76\u5458\uff0c\u6709 10% \u7684\u4e8b\u6545\u6d89\u53ca\u9a7e\u9a76\u5458\u7684\u5206\u5fc3[2]\u3002<\/p>\n

\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u6709\u671b\u5927\u5927\u51cf\u5c11\u7531\u4e8e\u9a7e\u9a76\u5458\u9519\u8bef\u548c\u758f\u5ffd\u800c\u5bfc\u81f4\u7684\u8f66\u8f86\u78b0\u649e\u4e8b\u6545\uff0c\u540c\u65f6\u4e3a\u7531\u4e8e\u8eab\u4f53\u6216\u89c6\u89c9\u6b8b\u75be\u800c\u65e0\u6cd5\u9a7e\u9a76\u7684\u4eba\u63d0\u4f9b\u4e2a\u4eba\u79fb\u52a8\u7684\u65b9\u5f0f\u3002\u6700\u540e\uff0c\u8003\u8651\u5230 86% \u7684\u7f8e\u56fd\u52b3\u52a8\u529b\u6bcf\u5929\u5e73\u5747\u9700\u8981\u82b1\u8d3925\u5206\u949f\u5728\u9a7e\u9a76\u901a\u52e4\uff08\u5355\u7a0b\uff09\u4e0a[3]\uff0c\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u5c06\u6709\u52a9\u4e8e\u66f4\u6709\u6548\u5730\u5229\u7528\u4ea4\u901a\u65f6\u95f4\uff0c\u6216\u8005\u81f3\u5c11\u51cf\u5c11\u9a7e\u9a76\u538b\u529b\u5e26\u6765\u7684\u53ef\u8861\u91cf\u7684\u4e0d\u826f\u5f71\u54cd[4]\u3002\u8003\u8651\u5230\u8fd9\u9879\u65b0\u6280\u672f\u7684\u6f5c\u5728\u5f71\u54cd\uff0c\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u62e5\u6709\u957f\u4e45\u7684\u5386\u53f2\u5e76\u4e0d\u5947\u602a\u3002<\/p>\n

\u65e9\u572820\u4e16\u7eaa20\u5e74\u4ee3\uff0c\u8fd9\u79cd\u60f3\u6cd5\u5c31\u5df2\u7ecf\u5b58\u5728\uff0c\u4f46\u662f\u76f4\u523020\u4e16\u7eaa80\u5e74\u4ee3\uff0c\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u4f3c\u4e4e\u624d\u771f\u6b63\u6210\u4e3a\u53ef\u80fd\u3002Ernst Dickmanns\u572820\u4e16\u7eaa80\u5e74\u4ee3\u9886\u5bfc\u7684\u5f00\u521b\u6027\u5de5\u4f5c\uff08\u4f8b\u5982[5]\uff09\u4e3a\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u7684\u53d1\u5c55\u94fa\u5e73\u4e86\u9053\u8def\u3002\u5f53\u65f6\u4e00\u9879\u5927\u89c4\u6a21\u7684\u7814\u7a76\uff08PROMETHEUS\u9879\u76ee\uff09\u88ab\u8d44\u52a9\u4e8e\u5f00\u53d1\u4e00\u79cd\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u30021994\u5e74\uff0c\u8fd9\u9879\u5de5\u4f5c\u53d1\u5e03\u4e86\u4e00\u4e2a\u5f15\u4eba\u6ce8\u76ee\u7684\u6f14\u793a\uff0c\u5176\u4e2dVaMP\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u884c\u9a76\u4e861600\u516c\u91cc\uff0c\u5e76\u4e14\u6709 95% \u7684\u8def\u7a0b\u662f\u81ea\u52a8\u9a7e\u9a76\u7684[6]\u3002\u4e0e\u6b64\u540c\u65f6\uff0c\u5361\u5185\u57fa\u6885\u9686\u5927\u5b66NAVLAB\u4e5f\u53d6\u5f97\u4e86\u8fdb\u5c55\uff0c\u5b83\u4e8e1995\u5e74\u5728\u7f8e\u56fd\u5883\u5185\u5b8c\u6210\u4e865000\u516c\u91cc\u7684\u884c\u9a76\uff0c\u5176\u4e2d 98% \u7684\u8def\u7a0b\u662f\u81ea\u52a8\u9a7e\u9a76\u7684[7]\u3002<\/p>\n

\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u6280\u672f\u7684\u4e0b\u4e00\u4e2a\u91cd\u5927\u91cc\u7a0b\u7891\u662f2004\u5e74\u7684\u7b2c\u4e00\u6b21DARPA\u5927\u6311\u6218\uff08DARPA Grand Challenge\uff09\u3002\u7ade\u8d5b\u76ee\u6807\u662f\u8ba9\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u5728150\u82f1\u91cc\u7684\u8d8a\u91ce\u9053\u8def\u4e0a\u5c3d\u53ef\u80fd\u5feb\u901f\u5730\u884c\u9a76\u3002\u4e0e\u4e4b\u524d\u7684\u6f14\u793a\u76f8\u6bd4\uff0c\u8fd9\u662f\u4e00\u4e2a\u91cd\u5927\u6311\u6218\uff0c\u56e0\u4e3a\u5728\u6bd4\u8d5b\u671f\u95f4\u6ca1\u6709\u4eba\u4e3a\u5e72\u9884\u3002\u5c3d\u7ba1\u5148\u524d\u7684\u7814\u7a76\u5de5\u4f5c\u6f14\u793a\u4e86\u51e0\u4e4e\u5b8c\u5168\u7684\u81ea\u52a8\u9a7e\u9a76\uff0c\u4f46\u5728\u5173\u952e\u65f6\u523b\u4e0d\u8fdb\u884c\u4eba\u4e3a\u5e72\u9884\u88ab\u8bc1\u660e\u662f\u4e00\u9879\u91cd\u5927\u6311\u6218\u3002\u53c2\u52a0\u6bd4\u8d5b\u768415\u8f86\u8f66\u4e2d\u6ca1\u6709\u4e00\u8f86\u5b8c\u6210\u6bd4\u8d5b\u3002<\/p>\n

2005\u5e74\u4e3e\u884c\u4e86\u7c7b\u4f3c\u7684\u6d3b\u52a8\uff0c\u8fd9\u4e00\u6b2123\u652f\u8f66\u961f\u4e2d\u67095\u652f\u5230\u8fbe\u4e86\u7ec8\u70b9[8]\u3002\u540e\u6765\u57282007\u5e74\uff0cDARPA\u57ce\u5e02\u6311\u6218\u8d5b\uff08DARPA Urban Challenge\uff09\u4e3e\u884c\uff0c\u8981\u6c42\u8f66\u8f86\u5728\u6a21\u62df\u7684\u57ce\u5e02\u73af\u5883\u4e2d\u81ea\u52a8\u9a7e\u9a76\u3002\u516d\u652f\u961f\u4f0d\u5b8c\u6210\u4e86\u6bd4\u8d5b\uff0c\u8fd9\u8bc1\u660e\u5b8c\u5168\u81ea\u4e3b\u7684\u57ce\u5e02\u9a7e\u9a76\u662f\u53ef\u80fd\u7684[9]\u3002\u81eaDARPA\u6311\u6218\u4ee5\u6765\uff0c\u5df2\u7ecf\u5f00\u5c55\u4e86\u8bb8\u591a\u6d3b\u52a8\u548c\u5927\u578b\u81ea\u52a8\u9a7e\u9a76\u8f66\u8f86\u7cfb\u7edf\u6d4b\u8bd5\u3002<\/p>\n

\u503c\u5f97\u6ce8\u610f\u7684\u4f8b\u5b50\u5305\u62ec2009\u5e74\u81f32013\u5e74\u7684\u667a\u80fd\u6c7d\u8f66\u672a\u6765\u6311\u6218\u8d5b\uff08Intelligent Vehicle Future Challenges\uff09[10]\u30012010\u5e74\u7684\u73b0\u4ee3\u81ea\u52a8\u9a7e\u9a76\u6311\u6218\u8d5b\uff08Hyundai Autonomous Challenge\uff09[11]\u30012010\u5e74\u7684VisLab\u6d32\u9645\u81ea\u52a8\u9a7e\u9a76\u6311\u6218\u8d5b\uff08VisLab Intercontinental Autonomous Challenge\uff09[12]\u30012013\u5e74\u7684\u516c\u5171\u9053\u8def\u57ce\u5e02\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u6d4b\u8bd5\uff08Public Road Urban Driverless-Car Test\uff09[13]\uff0c\u4ee5\u53ca\u4f2f\u838e-\u5954\u9a70\u5386\u53f2\u8def\u7ebf\uff08Bertha-Benz historic route\uff09\u7684\u81ea\u52a8\u9a7e\u9a76[14]\u3002\u540c\u65f6\uff0c\u5b66\u672f\u754c\u548c\u5de5\u4e1a\u754c\u7684\u7814\u7a76\u90fd\u5728\u52a0\u901f\u8fdb\u884c\u3002<\/p>\n

\u8c37\u6b4c\u7684\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66[15]\u548c\u7279\u65af\u62c9\u7684\u81ea\u52a8\u9a7e\u9a76\u7cfb\u7edf\uff08Autopilot system\uff09[16]\u662f\u4e24\u4e2a\u5546\u4e1a\u6210\u679c\u7684\u4f8b\u5b50\uff0c\u5b83\u4eec\u53d7\u5230\u4e86\u5a92\u4f53\u7684\u5e7f\u6cdb\u5173\u6ce8\u3002\u6c7d\u8f66\u7684\u81ea\u52a8\u5316\u7a0b\u5ea6\u4ece\u5b8c\u5168\u4eba\u5de5\u64cd\u4f5c\u5230\u5b8c\u5168\u81ea\u4e3b\u3002SAE J3016\u6807\u51c6[17]\u5f15\u5165\u4e86\u4ece0\u52305\u7684\u7b49\u7ea7\uff0c\u7528\u4e8e\u5bf9\u8f66\u8f86\u81ea\u52a8\u5316\u8fdb\u884c\u5206\u7ea7\u3002\u5728\u8be5\u6807\u51c6\u4e2d\uff0c0\u7ea7\u8868\u793a\u6240\u6709\u9a7e\u9a76\u4efb\u52a1\u5747\u7531\u4eba\u7c7b\u9a7e\u9a76\u5458\u8d1f\u8d23\u7684\u8f66\u8f86\u3002<\/p>\n

1\u7ea7\u5305\u62ec\u57fa\u672c\u9a7e\u9a76\u8f85\u52a9\uff0c\u5982\u81ea\u9002\u5e94\u5de1\u822a\u63a7\u5236\u3001\u9632\u62b1\u6b7b\u5236\u52a8\u7cfb\u7edf\u548c\u7535\u5b50\u7a33\u5b9a\u63a7\u5236[18]\u3002<\/p>\n

\u7b2c2\u7ea7\u5305\u62ec\u5148\u8fdb\u7684\u8f85\u52a9\u63aa\u65bd\uff0c\u5982\u5371\u9669\u6700\u5c0f\u5316\u7eb5\u5411\/\u6a2a\u5411\u63a7\u5236[19]\u6216\u7d27\u6025\u5236\u52a8[20][21]\u3002\u8fd9\u4e9b\u8f85\u52a9\u63aa\u65bd\u7684\u57fa\u7840\u901a\u5e38\u662f\u57fa\u4e8e\u96c6\u7684\u5f62\u5f0f\u63a7\u5236\u7406\u8bba\u65b9\u6cd5\uff0c\u7528\u6765\u8ba1\u7b97\u53ef\u8bc1\u660e\u7684\u65e0\u78b0\u649e\uff08\u5b89\u5168\uff09\u72b6\u6001\u7684\u201c\u6700\u574f\u60c5\u51b5\u201d\u96c6[22\u201324]\u3002<\/p>\n

\u5728\u7b2c3\u7ea7\uff0c\u7cfb\u7edf\u76d1\u63a7\u73af\u5883\uff0c\u5e76\u53ef\u4ee5\u5728\u7279\u5b9a\u6761\u4ef6\u4e0b\u5b8c\u5168\u81ea\u4e3b\u9a7e\u9a76\uff0c\u4f46\u5982\u679c\u9a7e\u9a76\u4efb\u52a1\u8d85\u51fa\u81ea\u4e3b\u7cfb\u7edf\u7684\u64cd\u4f5c\u8303\u56f4\uff0c\u5219\u4ecd\u9700\u8981\u64cd\u4f5c\u5458\u8fdb\u884c\u63a7\u5236\u3002<\/p>\n

\u5177\u67094\u7ea7\u81ea\u52a8\u5316\u7684\u8f66\u8f86\u80fd\u591f\u5728\u67d0\u4e9b\u6761\u4ef6\u4e0b\u5b8c\u5168\u81ea\u52a8\u9a7e\u9a76\uff0c\u5e76\u4e14\u5982\u679c\u64cd\u4f5c\u5458\u672a\u80fd\u5728\u8bf7\u6c42\u5e72\u9884\u65f6\u91c7\u53d6\u63a7\u5236\u63aa\u65bd\uff0c\u7cfb\u7edf\u4ecd\u7136\u53ef\u4ee5\u5b89\u5168\u5730\u63a7\u5236\u8f66\u8f86\u3002<\/p>\n

5\u7ea7\u7cfb\u7edf\u5728\u6240\u6709\u9a7e\u9a76\u6a21\u5f0f\u4e0b\u90fd\u662f\u5b8c\u5168\u81ea\u4e3b\u7684\u3002<\/p>\n

\u8f66\u8f7d\u8ba1\u7b97\u548c\u65e0\u7ebf\u901a\u4fe1\u6280\u672f\u7684\u53ef\u7528\u6027\u4f7f\u6c7d\u8f66\u80fd\u591f\u4e0e\u5176\u4ed6\u6c7d\u8f66\u548c\u9053\u8def\u57fa\u7840\u8bbe\u65bd\u4ea4\u6362\u4fe1\u606f\uff0c\u4ece\u800c\u4ea7\u751f\u4e86\u4e00\u4e2a\u4e0e\u8054\u7f51\u667a\u80fd\u6c7d\u8f66\u5bc6\u5207\u76f8\u5173\u7684\u7814\u7a76\u9886\u57df[25]\u3002<\/p>\n

\u8be5\u7814\u7a76\u65e8\u5728\u901a\u8fc7\u5404\u8f66\u8f86\u4e4b\u95f4\u7684\u4fe1\u606f\u5171\u4eab\u548c\u534f\u8c03\uff0c\u63d0\u9ad8\u9053\u8def\u8fd0\u8f93\u7684\u5b89\u5168\u6027\u548c\u6027\u80fd\u3002\u4f8b\u5982\uff0c\u8054\u7f51\u8f66\u8f86\u6280\u672f\u6709\u53ef\u80fd\u63d0\u9ad8\u4ea4\u53c9\u8def\u53e3\u7684\u541e\u5410\u91cf[26]\u6216\u9632\u6b62\u8f66\u6d41\u51b2\u51fb\u6ce2\u7684\u5f62\u6210[27]\u3002\u4e3a\u4e86\u9650\u5236\u8fd9\u7bc7\u7efc\u8ff0\u7684\u8303\u56f4\uff0c\u6211\u4eec\u5c06\u91cd\u70b9\u653e\u5728\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u7684\u51b3\u7b56\u3001\u8fd0\u52a8\u89c4\u5212\u548c\u63a7\u5236\u65b9\u9762\uff0c\u7279\u522b\u662f\u81ea\u52a8\u5316\u7ea7\u522b\u4e3a3\u7ea7\u53ca\u4ee5\u4e0a\u7684\u7cfb\u7edf\u3002\u51fa\u4e8e\u540c\u6837\u7684\u539f\u56e0\uff0c\u8fd9\u7bc7\u7efc\u8ff0\u4e0d\u8ba8\u8bba\u81ea\u52a8\u9a7e\u9a76\u7684\u611f\u77e5\u9886\u57df\uff1b\u8bfb\u8005\u53ef\u4ee5\u53c2\u8003\u8bb8\u591a\u8be5\u65b9\u9762\u7684\u7efc\u8ff0\u548c\u6700\u8fd1\u7684\u4e3b\u8981\u6210\u679c[28\u201331]\u3002\u73b0\u4ee3\u81ea\u52a8\u9a7e\u9a76\u7cfb\u7edf\u4e2d\u7684\u51b3\u7b56\u901a\u5e38\u5206\u4e3a\u8def\u7ebf\u89c4\u5212\u3001\u884c\u4e3a\u51b3\u7b56\u3001\u5c40\u90e8\u8fd0\u52a8\u89c4\u5212\u548c\u53cd\u9988\u63a7\u5236\u3002<\/p>\n

\u7136\u800c\uff0c\u7531\u4e8e\u6587\u732e\u4e2d\u62a5\u9053\u4e86\u51b3\u7b56\u65b9\u6848\u7684\u4e0d\u540c\u53d8\u4f53\uff0c\u56e0\u6b64\u8fd9\u4e9b\u7ea7\u522b\u7684\u5212\u5206\u76f8\u5f53\u6a21\u7cca\u3002\u672c\u6587\u6982\u8ff0\u4e86\u4e3a\u89e3\u51b3\u81ea\u52a8\u9a7e\u9a76\u6838\u5fc3\u95ee\u9898\u800c\u5df2\u7ecf\u63d0\u51fa\u7684\u65b9\u6cd5\uff0c\u5e76\u4e14\u5c06\u4fa7\u91cd\u4e8e\u5c40\u90e8\u8fd0\u52a8\u89c4\u5212\u548c\u63a7\u5236\u7684\u65b9\u6cd5\u3002<\/p>\n

\u672c\u6587\u7684\u5176\u4f59\u90e8\u5206\u7ed3\u6784\u5982\u4e0b\uff1a<\/p>\n

\u5728\u7b2c2\u8282\u4e2d\uff0c\u5bf9\u51b3\u7b56\u8fc7\u7a0b\u7684\u5c42\u6b21\u7ed3\u6784\u53ca\u5176\u8bbe\u8ba1\u65b9\u6cd5\u8fdb\u884c\u4e86\u9ad8\u5c42\u6b21\u7684\u6982\u8ff0\u3002\u7b2c3\u8282\u56de\u987e\u4e86\u4e3a\u4e86\u5b9e\u73b0\u8fd0\u52a8\u89c4\u5212\u548c\u53cd\u9988\u63a7\u5236\u800c\u63d0\u51fa\u7684\u7528\u4e8e\u8fd1\u4f3c\u57ce\u5e02\u73af\u5883\u4e2d\u6c7d\u8f66\u7684\u673a\u52a8\u6027\u7684\u6a21\u578b\u3002\u7b2c4\u8282\u7efc\u8ff0\u4e86\u8fd0\u52a8\u89c4\u5212\u7684\u5927\u91cf\u6587\u732e\uff0c\u5e76\u8ba8\u8bba\u4e86\u8fd0\u52a8\u89c4\u5212\u5bf9\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u7684\u53ef\u7528\u6027\u3002\u540c\u6837\uff0c\u7b2c5\u8282\u8ba8\u8bba\u4e86\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u7684\u8def\u5f84\u548c\u8f68\u8ff9\u7a33\u5b9a\u95ee\u9898\u4ee5\u53ca\u5177\u4f53\u7684\u53cd\u9988\u63a7\u5236\u65b9\u6cd5\u3002\u6700\u540e\uff0c\u7b2c6\u8282\u603b\u7ed3\u4e86\u6700\u65b0\u6280\u672f\u548c\u672a\u6765\u7814\u7a76\u7684\u6f5c\u5728\u9886\u57df\u3002<\/p>\n

2. \u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u7684\u51b3\u7b56\u5c42\u7efc\u8ff0<\/p>\n

\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u63cf\u8ff0\u5178\u578b\u7684\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u7684\u51b3\u7b56\u5c42\u67b6\u6784\uff0c\u5e76\u8ba8\u8bba\u5176\u4e2d\u6bcf\u4e2a\u7ec4\u6210\u90e8\u5206\u7684\u4f5c\u7528\u3002\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u81ea\u4e3b\u51b3\u7b56\u7cfb\u7edf\uff0c\u5b83\u5904\u7406\u6765\u81ea\u96f7\u8fbe\u3001\u6fc0\u5149\u96f7\u8fbe\uff08LIDARs\uff09\u3001\u76f8\u673a\u3001GPS\/INS\uff08\u5168\u7403\u5b9a\u4f4d\u7cfb\u7edf\/\u60ef\u6027\u5bfc\u822a\u7cfb\u7edf\uff09\u5355\u5143\u548c\u91cc\u7a0b\u8ba1\u7b49\u8f66\u8f7d\u4f20\u611f\u5668\u7684\u89c2\u6d4b\u6d41\u3002\u8fd9\u4e9b\u89c2\u6d4b\u7ed3\u679c\u4e0e\u5173\u4e8e\u9053\u8def\u7f51\u7edc\u3001\u9053\u8def\u89c4\u5219\u3001\u8f66\u8f86\u52a8\u529b\u5b66\u548c\u4f20\u611f\u5668\u6a21\u578b\u7684\u5148\u9a8c\u77e5\u8bc6\u4e00\u8d77\uff0c\u7528\u4e8e\u81ea\u52a8\u9009\u62e9\u63a7\u5236\u8f66\u8f86\u8fd0\u52a8\u7684\u53d7\u63a7\u53d8\u91cf\u7684\u53d6\u503c\u3002\u667a\u80fd\u6c7d\u8f66\u7814\u7a76\u65e8\u5728\u5c3d\u53ef\u80fd\u591a\u5730\u5b9e\u73b0\u9a7e\u9a76\u4efb\u52a1\u7684\u81ea\u52a8\u5316\u3002\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u7684\u5e38\u7528\u65b9\u6cd5\u662f\u5c06\u611f\u77e5\u548c\u51b3\u7b56\u4efb\u52a1\u5212\u5206\u5e76\u7ec4\u7ec7\u6210\u4e00\u4e2a\u5c42\u6b21\u7ed3\u6784\u3002\u611f\u77e5\u7cfb\u7edf\u4f7f\u7528\u5148\u9a8c\u4fe1\u606f\u548c\u6536\u96c6\u7684\u89c2\u6d4b\u6570\u636e\uff0c\u6765\u63d0\u4f9b\u8f66\u8f86\u53ca\u5176\u5468\u56f4\u73af\u5883\u7684\u72b6\u6001\u4f30\u8ba1\u3002<\/p>\n

\u51b3\u7b56\u7cfb\u7edf\u968f\u540e\u4f7f\u7528\u8fd9\u4e9b\u4f30\u8ba1\u6765\u63a7\u5236\u8f66\u8f86\uff0c\u4ece\u800c\u5b9e\u73b0\u9a7e\u9a76\u76ee\u6807\u3002\u5178\u578b\u7684\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u7684\u51b3\u7b56\u7cfb\u7edf\u4f9d\u636e\u5c42\u7ea7\u5206\u89e3\u4e3a\u56db\u4e2a\u7ec4\u6210\u90e8\u5206\uff08\u53c2\u8003\u56fe2.1\uff09\uff1a\u5728\u6700\u9ad8\u5c42\uff0c\u7cfb\u7edf\u901a\u8fc7\u9053\u8def\u7f51\u7edc\u89c4\u5212\u51fa\u4e00\u6761\u8def\u7ebf\uff1b\u968f\u540e\u7684\u884c\u4e3a\u5c42\u6307\u5b9a\u4e00\u4e2a\u5c40\u90e8\u9a7e\u9a76\u4efb\u52a1\uff0c\u4f7f\u6c7d\u8f66\u671d\u7740\u76ee\u7684\u5730\u524d\u8fdb\u540c\u65f6\u9075\u5b88\u9053\u8def\u89c4\u5219\uff1b\u968f\u540e\uff0c\u8fd0\u52a8\u89c4\u5212\u6a21\u5757\u9009\u62e9\u4e00\u6761\u901a\u8fc7\u73af\u5883\u7684\u8fde\u7eed\u8def\u5f84\uff0c\u4ee5\u5b8c\u6210\u5c40\u90e8\u5bfc\u822a\u4efb\u52a1\uff1b\u6700\u540e\uff0c\u63a7\u5236\u7cfb\u7edf\u53cd\u5e94\u6027\u5730\u4fee\u6b63\u6267\u884c\u89c4\u5212\u8fd0\u52a8\u65f6\u7684\u8bef\u5dee\u3002\u672c\u8282\u7684\u5176\u4f59\u90e8\u5206\u5c06\u66f4\u8be6\u7ec6\u5730\u8ba8\u8bba\u6bcf\u4e2a\u7ec4\u6210\u90e8\u5206\u7684\u4f5c\u7528\u3002<\/p>\n

\u56fe2.1\uff1a\u51b3\u7b56\u8fc7\u7a0b\u7684\u5c42\u7ea7<\/p>\n

\u6839\u636e\u76ee\u7684\u5730\uff0c\u8def\u7ebf\u89c4\u5212\u5668\u901a\u8fc7\u9053\u8def\u7f51\u7edc\u751f\u6210\u4e00\u6761\u8def\u7ebf\u3002\u884c\u4e3a\u5c42\u5bf9\u73af\u5883\u8fdb\u884c\u63a8\u7406\uff0c\u5e76\u751f\u6210\u6cbf\u7740\u6240\u9009\u8def\u7ebf\u524d\u8fdb\u7684\u8fd0\u52a8\u89c4\u8303\u3002\u8fd0\u52a8\u89c4\u5212\u5668\u968f\u540e\u6c42\u89e3\u5b9e\u73b0\u89c4\u8303\u7684\u53ef\u884c\u8fd0\u52a8\u3002\u53cd\u9988\u63a7\u5236\u5bf9\u81f4\u52a8\u53d8\u91cf\u8fdb\u884c\u8c03\u53c2\uff0c\u4ee5\u4fee\u6b63\u6267\u884c\u53c2\u8003\u8def\u5f84\u65f6\u7684\u8bef\u5dee\u3002<\/p>\n

A. \u8def\u7ebf\u89c4\u5212\u5728\u6700\u9ad8\u5c42\u4e0a\uff0c\u8f66\u8f86\u7684\u51b3\u7b56\u7cfb\u7edf\u5fc5\u987b\u9009\u62e9\u4e00\u6761\u4ece\u5f53\u524d\u4f4d\u7f6e\u5230\u8bf7\u6c42\u7684\u76ee\u7684\u5730\u7684\u3001\u901a\u8fc7\u9053\u8def\u7f51\u7edc\u7684\u8def\u7ebf\u3002\u901a\u8fc7\u5c06\u9053\u8def\u7f51\u7edc\u8868\u793a\u4e3a\u4e00\u4e2a\u6709\u5411\u56fe\uff08\u6709\u5411\u56fe\u7684\u8fb9\u7684\u6743\u91cd\u5bf9\u5e94\u6240\u7a7f\u8fc7\u7684\u9053\u8def\u6bb5\u7684\u6210\u672c\uff09\uff0c\u53ef\u4ee5\u5c06\u8def\u7ebf\u89c4\u5212\u95ee\u9898\u6784\u9020\u4e3a\u5728\u9053\u8def\u7f51\u7edc\u56fe\u4e0a\u627e\u5230\u4e00\u6761\u6700\u5c0f\u6210\u672c\u8def\u5f84\u7684\u95ee\u9898\u3002\u7136\u800c\uff0c\u8868\u793a\u9053\u8def\u7f51\u7edc\u7684\u56fe\u53ef\u4ee5\u5305\u542b\u6570\u767e\u4e07\u6761\u8fb9\uff0c\u4f7f\u5f97\u7ecf\u5178\u7684\u6700\u77ed\u8def\u5f84\u7b97\u6cd5\uff08\u5982Dijkstra [32]\u6216A* [33]\uff09\u4e0d\u518d\u5b9e\u7528\u3002\u4ea4\u901a\u7f51\u7edc\u4e2d\u7684\u9ad8\u6548\u8def\u7ebf\u89c4\u5212\u95ee\u9898\u5f15\u8d77\u4e86\u4ea4\u901a\u79d1\u5b66\u754c\u7684\u6781\u5927\u5174\u8da3\uff0c\u50ac\u751f\u4e86\u4e00\u7cfb\u5217\u7b97\u6cd5\u3002\u8fd9\u4e9b\u7b97\u6cd5\u7ecf\u8fc7\u4e00\u6b21\u6027\u7684\u9884\u5904\u7406\u6b65\u9aa4\u540e\uff0c\u80fd\u591f\u5728\u6beb\u79d2\u65f6\u95f4\u5185\u8fd4\u56de\u5927\u9646\u89c4\u6a21\u7684\u9053\u8def\u7f51\u7edc\u4e0a\u7684\u4e00\u6761\u6700\u4f18\u8def\u7ebf[34][35]\u3002\u6709\u5173\u53ef\u7528\u4e8e\u9ad8\u6548\u7684\u4eba\u7c7b\u9a7e\u9a76\u548c\u81ea\u52a8\u9a7e\u9a76\u8f66\u8f86\u8def\u7ebf\u89c4\u5212\u7684\u5b9e\u7528\u7b97\u6cd5\u7684\u5168\u9762\u7efc\u8ff0\u548c\u6bd4\u8f83\uff0c\u8bf7\u53c2\u8003[36]\u3002<\/p>\n

B. \u884c\u4e3a\u51b3\u7b56\u627e\u5230\u4e00\u6761\u8def\u7ebf\u8ba1\u5212\u540e\uff0c\u81ea\u52a8\u9a7e\u9a76\u8f66\u8f86\u5fc5\u987b\u80fd\u591f\u6839\u636e\u9a7e\u9a76\u60ef\u4f8b\u548c\u9053\u8def\u89c4\u5219\u6765\u5bfc\u822a\u9009\u5b9a\u7684\u8def\u7ebf\u5e76\u4e0e\u5176\u4ed6\u4ea4\u901a\u53c2\u4e0e\u8005\u8fdb\u884c\u4ea4\u4e92\u3002\u5f53\u7ed9\u5b9a\u4e86\u6307\u5b9a\u6240\u9009\u8def\u7ebf\u7684\u4e00\u7cfb\u5217\u8def\u6bb5\u65f6\uff0c\u884c\u4e3a\u5c42\u57fa\u4e8e\u611f\u77e5\u5230\u7684\u5176\u4ed6\u4ea4\u901a\u53c2\u4e0e\u8005\u7684\u884c\u4e3a\u3001\u9053\u8def\u72b6\u51b5\u548c\u57fa\u7840\u8bbe\u65bd\u4fe1\u53f7\uff0c\u8d1f\u8d23\u5728\u4efb\u4f55\u65f6\u95f4\u70b9\u9009\u62e9\u9002\u5f53\u7684\u9a7e\u9a76\u884c\u4e3a\u3002\u4f8b\u5982\uff0c\u5f53\u8f66\u8f86\u5728\u4ea4\u53c9\u8def\u53e3\u524d\u5230\u8fbe\u505c\u8f66\u7ebf\u65f6\uff0c\u884c\u4e3a\u5c42\u5c06\u547d\u4ee4\u8f66\u8f86\u505c\u8f66\uff0c\u89c2\u5bdf\u4ea4\u53c9\u8def\u53e3\u7684\u5176\u4ed6\u8f66\u8f86\u3001\u81ea\u884c\u8f66\u548c\u884c\u4eba\u7684\u884c\u4e3a\uff0c\u5e76\u5728\u8f6e\u5230\u8f66\u8f86\u884c\u9a76\u65f6\u8ba9\u8f66\u8f86\u7ee7\u7eed\u524d\u8fdb\u3002\u9a7e\u9a76\u624b\u518c\u89c4\u5b9a\u4e86\u7279\u5b9a\u9a7e\u9a76\u73af\u5883\u4e0b\u7684\u5b9a\u6027\u884c\u52a8\u3002\u7531\u4e8e\u9a7e\u9a76\u73af\u5883\u548c\u6bcf\u4e2a\u73af\u5883\u4e2d\u53ef\u7528\u7684\u884c\u4e3a\u90fd\u53ef\u4ee5\u88ab\u5efa\u6a21\u4e3a\u6709\u9650\u96c6\uff0c\u56e0\u6b64\u81ea\u52a8\u5316\u8fd9\u79cd\u51b3\u7b56\u7684\u4e00\u79cd\u81ea\u7136\u65b9\u6cd5\u662f\u5c06\u6bcf\u4e2a\u884c\u4e3a\u5efa\u6a21\u4e3a\u6709\u9650\u72b6\u6001\u673a\u4e2d\u7684\u72b6\u6001\uff0c\u5176\u4e2d\u72b6\u6001\u7684\u8f6c\u79fb\u53d7\u611f\u77e5\u5230\u7684\u9a7e\u9a76\u73af\u5883\u7684\u63a7\u5236\uff0c\u8fd9\u4e9b\u9a7e\u9a76\u73af\u5883\u5305\u62ec\u76f8\u5bf9\u4e8e\u89c4\u5212\u7684\u8def\u7ebf\u548c\u9644\u8fd1\u8f66\u8f86\u7684\u76f8\u5bf9\u4f4d\u7f6e\u3002<\/p>\n

\u4e8b\u5b9e\u4e0a\uff0c\u5728DARPA\u57ce\u5e02\u6311\u6218\u8d5b\u4e2d\uff0c\u5927\u591a\u6570\u56e2\u961f\u91c7\u7528\u4e86\u6709\u9650\u72b6\u6001\u673a\u548c\u7279\u5b9a\u4e8e\u6240\u8003\u8651\u9a7e\u9a76\u573a\u666f\u7684\u4e0d\u540c\u542f\u53d1\u5f0f\uff0c\u4f5c\u4e3a\u884c\u4e3a\u63a7\u5236\u7684\u673a\u5236[9]\u3002\u7136\u800c\uff0c\u73b0\u5b9e\u4e16\u754c\uff08\u5c24\u5176\u662f\u57ce\u5e02\u73af\u5883\uff09\u7684\u9a7e\u9a76\u7684\u7279\u70b9\u662f\uff0c\u5176\u4ed6\u4ea4\u901a\u53c2\u4e0e\u8005\u7684\u610f\u56fe\uff08intentions\uff09\u5177\u6709\u4e0d\u786e\u5b9a\u6027\u3002\u56e0\u6b64\uff0c\u5b66\u754c\u8fd8\u7814\u7a76\u4e86\u5bf9\u5176\u4ed6\u8f66\u8f86\u3001\u81ea\u884c\u8f66\u548c\u884c\u4eba\u7684\u672a\u6765\u8f68\u8ff9\u7684\u610f\u56fe\u9884\u6d4b\u548c\u4f30\u8ba1\u3002\u63d0\u51fa\u7684\u89e3\u51b3\u65b9\u6848\u5305\u62ec\u57fa\u4e8e\u673a\u5668\u5b66\u4e60\u7684\u6280\u672f\uff0c\u4f8b\u5982\u9ad8\u65af\u6df7\u5408\u6a21\u578b[37]\u3001\u9ad8\u65af\u8fc7\u7a0b\u56de\u5f52[38]\u3001\u88ab\u62a5\u9053\u7684\u5728\u8c37\u6b4c\u81ea\u52a8\u9a7e\u9a76\u7cfb\u7edf\u4e2d\u7528\u4e8e\u610f\u56fe\u9884\u6d4b\u7684\u5b66\u4e60\u6280\u672f[39]\uff0c\u4ee5\u53ca\u7528\u4e8e\u4ece\u4f20\u611f\u5668\u6d4b\u91cf\u503c\u4e2d\u76f4\u63a5\u4f30\u8ba1\u610f\u56fe\u7684\u57fa\u4e8e\u6a21\u578b\u7684\u65b9\u6cd5[40][41]\u3002\u5176\u4ed6\u4ea4\u901a\u53c2\u4e0e\u8005\u884c\u4e3a\u7684\u8fd9\u79cd\u4e0d\u786e\u5b9a\u6027\u901a\u5e38\u5728\u4f7f\u7528\u6982\u7387\u89c4\u5212\u5f62\u5f0f\uff08\u5982\u9a6c\u5c14\u53ef\u592b\u51b3\u7b56\u8fc7\u7a0b\uff08MDPs\uff09\u53ca\u5176\u63a8\u5e7f\u5f62\u5f0f\uff09\u8fdb\u884c\u51b3\u7b56\u7684\u884c\u4e3a\u5c42\u4e2d\u88ab\u8003\u8651\u3002\u4f8b\u5982\uff0c\u6587\u7ae0[42]\u5728MDP\u6846\u67b6\u4e2d\u6784\u9020\u4e86\u884c\u4e3a\u51b3\u7b56\u95ee\u9898\u3002\u4e00\u4e9b\u5de5\u4f5c[43\u201346]\u4f7f\u7528\u90e8\u5206\u53ef\u89c2\u5bdf\u9a6c\u5c14\u53ef\u592b\u51b3\u7b56\u8fc7\u7a0b\uff08POMDP\uff09\u6846\u67b6\uff0c\u4ee5\u663e\u5f0f\u5730\u6a21\u62df\u672a\u89c2\u5bdf\u5230\u7684\u9a7e\u9a76\u573a\u666f\u548c\u884c\u4eba\u610f\u56fe\uff0c\u5e76\u63d0\u51fa\u4e86\u5177\u4f53\u7684\u8fd1\u4f3c\u89e3\u51b3\u7b56\u7565\u3002<\/p>\n

C. \u8fd0\u52a8\u89c4\u5212\u884c\u4e3a\u5c42\u51b3\u5b9a\u7684\u8981\u5728\u5f53\u524d\u73af\u5883\u4e2d\u6267\u884c\u7684\u9a7e\u9a76\u884c\u4e3a\uff08\u4f8b\u5982\u8f66\u9053\u5de1\u822a\u3001\u53d8\u9053\u6216\u53f3\u8f6c\uff09\u5fc5\u987b\u88ab\u8f6c\u6362\u4e3a\u4e00\u6761\u53ef\u7531\u4f4e\u7ea7\u53cd\u9988\u63a7\u5236\u5668\u8ddf\u8e2a\u7684\u8def\u5f84\u6216\u8f68\u8ff9\u3002\u751f\u6210\u7684\u8def\u5f84\u6216\u8f68\u8ff9\u5bf9\u4e8e\u8f66\u8f86\u5fc5\u987b\u662f\u52a8\u6001\u53ef\u884c\u7684\uff0c\u540c\u65f6\u53ef\u4ee5\u4fdd\u8bc1\u4e58\u5ba2\u7684\u8212\u9002\uff0c\u5e76\u907f\u514d\u4e0e\u8f66\u8f7d\u4f20\u611f\u5668\u68c0\u6d4b\u5230\u7684\u969c\u788d\u7269\u53d1\u751f\u78b0\u649e\u3002\u627e\u5230\u8fd9\u6837\u7684\u4e00\u6761\u8def\u5f84\u6216\u8f68\u8ff9\u662f\u8fd0\u52a8\u89c4\u5212\u7cfb\u7edf\u7684\u5de5\u4f5c\u3002\u81ea\u52a8\u9a7e\u9a76\u8f66\u8f86\u7684\u8fd0\u52a8\u89c4\u5212\u4efb\u52a1\u5bf9\u5e94\u4e8e\u6c42\u89e3\u673a\u5668\u4eba\u6587\u732e\u4e2d\u8ba8\u8bba\u7684\u6807\u51c6\u8fd0\u52a8\u89c4\u5212\u95ee\u9898\u3002\u8fd0\u52a8\u89c4\u5212\u95ee\u9898\u7684\u7cbe\u786e\u89e3\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\u90fd\u662f\u96be\u4ee5\u8ba1\u7b97\u7684\u3002<\/p>\n

\u56e0\u6b64\uff0c\u5b9e\u8df5\u4e2d\u901a\u5e38\u4f7f\u7528\u6570\u503c\u8fd1\u4f3c\u65b9\u6cd5\uff0c\u6700\u6d41\u884c\u7684\u5305\u62ec\uff1a\uff081\uff09\u53d8\u5206\u6cd5\uff0c\u5176\u5c06\u95ee\u9898\u89c6\u4e3a\u5728\u51fd\u6570\u7a7a\u95f4\u4e2d\u8fdb\u884c\u975e\u7ebf\u6027\u4f18\u5316\uff1b\uff082\uff09\u56fe\u641c\u7d22\u65b9\u6cd5\uff0c\u5176\u6784\u9020\u8f66\u8f86\u72b6\u6001\u7a7a\u95f4\u7684\u56fe\u7684\u79bb\u6563\u5316\uff0c\u5e76\u7528\u56fe\u641c\u7d22\u7b97\u6cd5\u6765\u641c\u7d22\u6700\u77ed\u8def\u5f84\uff1b\uff083\uff09\u57fa\u4e8e\u589e\u91cf\u6811\uff08incremental tree\uff09\u7684\u65b9\u6cd5\uff0c\u5176\u4ece\u8f66\u8f86\u7684\u521d\u59cb\u72b6\u6001\u5f00\u59cb\uff0c\u589e\u91cf\u5730\u6784\u9020\u4e00\u4e2a\u53ef\u5230\u8fbe\u72b6\u6001\u6811\uff0c\u5e76\u9009\u62e9\u6811\u7684\u6700\u4f73\u5206\u652f\u3002\u7b2c4\u8282\u5c06\u66f4\u8be6\u7ec6\u5730\u8ba8\u8bba\u4e0e\u81ea\u52a8\u9a7e\u9a76\u76f8\u5173\u7684\u8fd0\u52a8\u89c4\u5212\u65b9\u6cd5\u3002<\/p>\n

D. \u8f66\u8f86\u63a7\u5236\u4e3a\u4e86\u6267\u884c\u6765\u81ea\u8fd0\u52a8\u89c4\u5212\u7cfb\u7edf\u7684\u53c2\u8003\u8def\u5f84\u6216\u8f68\u8ff9\uff0c\u4f7f\u7528\u53cd\u9988\u63a7\u5236\u5668\u6765\u9009\u62e9\u9002\u5f53\u7684\u81f4\u52a8\u5668\u8f93\u5165\u3002\u6267\u884c\u89c4\u5212\u8fd0\u52a8\u671f\u95f4\u4f1a\u4ea7\u751f\u8ddf\u8e2a\u8bef\u5dee\u7684\u90e8\u5206\u539f\u56e0\u662f\u8f66\u8f86\u6a21\u578b\u4e0d\u591f\u51c6\u786e\u3002\u56e0\u6b64\uff0c\u95ed\u73af\u7cfb\u7edf\u7684\u9c81\u68d2\u6027\u548c\u7a33\u5b9a\u6027\u53d7\u5230\u4e86\u6781\u9ad8\u7684\u91cd\u89c6\u3002\u5f53\u524d\uff0c\u5df2\u7ecf\u63d0\u51fa\u4e86\u8bb8\u591a\u6709\u6548\u7684\u53cd\u9988\u63a7\u5236\u5668\uff0c\u7528\u4e8e\u6267\u884c\u7531\u8fd0\u52a8\u89c4\u5212\u7cfb\u7edf\u63d0\u4f9b\u7684\u53c2\u8003\u8fd0\u52a8\u3002\u7b2c5\u8282\u8be6\u7ec6\u8ba8\u8bba\u4e86\u76f8\u5173\u6280\u672f\u7684\u6982\u51b5\u3002<\/p>\n

3. \u89c4\u5212\u4e0e\u63a7\u5236\u7684\u5efa\u6a21<\/p>\n

\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u7efc\u8ff0\u6700\u5e38\u7528\u7684\u7c7b\u8f66\u8f66\u8f86\uff08car-like vehicles\uff09\u7684\u79fb\u52a8\u6a21\u578b\u3002\u6b64\u7c7b\u6a21\u578b\u88ab\u5e7f\u6cdb\u7528\u4e8e\u63a7\u5236\u548c\u8fd0\u52a8\u89c4\u5212\u7b97\u6cd5\uff0c\u6765\u8fd1\u4f3c\u5728\u76f8\u5173\u64cd\u4f5c\u6761\u4ef6\u4e0b\u54cd\u5e94\u4e8e\u63a7\u5236\u52a8\u4f5c\u7684\u8f66\u8f86\u884c\u4e3a\u3002\u9ad8\u4fdd\u771f\u6a21\u578b\u53ef\u4ee5\u51c6\u786e\u5730\u53cd\u6620\u8f66\u8f86\u7684\u54cd\u5e94\uff0c\u4f46\u589e\u52a0\u7684\u7ec6\u8282\u53ef\u80fd\u4f1a\u4f7f\u89c4\u5212\u548c\u63a7\u5236\u95ee\u9898\u590d\u6742\u5316\u3002\u8fd9\u8981\u6c42\u5728\u6240\u9009\u6a21\u578b\u7684\u51c6\u786e\u6027\u548c\u51b3\u7b56\u95ee\u9898\u7684\u96be\u5ea6\u4e4b\u95f4\u505a\u51fa\u6743\u8861\u3002\u672c\u8282\u5c06\u5bf9\u901a\u7528\u7684\u5efa\u6a21\u6982\u5ff5\u8fdb\u884c\u6982\u8ff0\uff0c\u540c\u65f6\u5c06\u7efc\u8ff0\u7528\u4e8e\u8fd0\u52a8\u89c4\u5212\u548c\u63a7\u5236\u7684\u6a21\u578b\u3002\u5efa\u6a21\u7684\u7b2c\u4e00\u6b65\u662f\u8f66\u8f86\u6784\u578b\/\u914d\u7f6e\uff08configuration\uff09\u7684\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u8f66\u8f86\u5728\u4e16\u754c\u4e2d\u7684\u4f4d\u59ff\u6216\u4f4d\u7f6e\u3002\u4f8b\u5982\u53ef\u4ee5\u7528\u6c7d\u8f66\u4e0a\u67d0\u4e00\u70b9\u7684\u5e73\u9762\u5750\u6807\u4ee5\u53ca\u6c7d\u8f66\u7684\u65b9\u5411\u6765\u8868\u793a\u5176\u6784\u578b\uff0c\u8fd9\u4fbf\u662f\u6c7d\u8f66\u7684\u6784\u578b\u7a7a\u95f4\/\u914d\u7f6e\u7a7a\u95f4\uff08configuration space \/ C-space\uff09\u7684\u5750\u6807\u7cfb\u3002\u8be5\u5750\u6807\u7cfb\u63cf\u8ff0\u5e73\u9762\u521a\u4f53\u8fd0\u52a8\uff08\u8868\u793a\u4e3a\u4e8c\u7ef4\u7684\u7279\u6b8a\u6b27\u6c0f\u7fa4 SE(2) \uff09\uff0c\u662f\u5e38\u7528\u7684\u6784\u578b\u7a7a\u95f4[47-49]\u3002\u968f\u540e\uff0c\u5bf9\u8f66\u8f86\u8fd0\u52a8\u8fdb\u884c\u89c4\u5212\u548c\u63a7\u5236\uff0c\u4ee5\u5b8c\u6210\u9a7e\u9a76\u4efb\u52a1\uff0c\u540c\u65f6\u5fc5\u987b\u6ee1\u8db3\u6240\u9009\u6a21\u578b\u5f15\u5165\u7684\u7ea6\u675f\u3002<\/p>\n

A. \u5355\u8f68\uff08Single-Track\uff09\u8fd0\u52a8\u5b66\u6a21\u578b\u5b9e\u8df5\u4f7f\u7528\u7684\u6700\u57fa\u7840\u7684\u6c7d\u8f66\u6a21\u578b\u7531\u4e24\u4e2a\u901a\u8fc7\u521a\u6027\u8fde\u6746\u8fde\u63a5\u7684\u8f66\u8f6e\u7ec4\u6210\uff0c\u5e76\u4e14\u88ab\u9650\u5236\u5728\u5e73\u9762\u5185\u79fb\u52a8[48-52]\u3002\u8be5\u6a21\u578b\u5047\u8bbe\u8f66\u8f6e\u5728\u5176\u4e0e\u5730\u9762\u7684\u63a5\u89e6\u70b9\u5904\u4e0d\u4f1a\u6ed1\u52a8\uff0c\u4f46\u53ef\u4ee5\u7ed5\u5176\u65cb\u8f6c\u8f74\u81ea\u7531\u65cb\u8f6c\u3002\u524d\u8f6e\u5177\u6709\u989d\u5916\u7684\u81ea\u7531\u5ea6\uff0c\u5141\u8bb8\u5176\u7ed5\u5782\u76f4\u4e8e\u8fd0\u52a8\u5e73\u9762\u7684\u8f74\u65cb\u8f6c\uff0c\u4ee5\u6a21\u62df\u8f6c\u5411\u3002\u8fd9\u4e24\u4e2a\u5efa\u6a21\u7279\u5f81\u53cd\u6620\u4e86\u5927\u591a\u6570\u4e58\u5ba2\u7684\u4f53\u9a8c\uff0c\u5373\u6c7d\u8f66\u65e0\u6cd5\u5728\u4e0d\u5411\u524d\u79fb\u52a8\u7684\u540c\u65f6\u5b9e\u73b0\u6a2a\u5411\u4f4d\u79fb\u3002\u66f4\u6b63\u5f0f\u5730\uff0c\u8fd9\u79cd\u673a\u52a8\u6027\u9650\u5236\u88ab\u79f0\u4e3a\u975e\u5b8c\u6574\u7ea6\u675f[47][53]\u3002\u975e\u5b8c\u6574\u7ea6\u675f\u8868\u793a\u4e3a\u6c7d\u8f66\u8fd0\u52a8\u7684\u5fae\u5206\u7ea6\u675f\uff0c\u7ea6\u675f\u7684\u8868\u8fbe\u5f0f\u968f\u5750\u6807\u7cfb\u7684\u9009\u62e9\u800c\u53d8\u5316\u3002\u8fd9\u79cd\u6c7d\u8f66\u6a21\u578b\u7684\u53d8\u4f53\u88ab\u79f0\u4e3a\u7c7b\u8f66\u673a\u5668\u4eba\u3001\u81ea\u884c\u8f66\u6a21\u578b\u3001\u8fd0\u52a8\u5b66\u6a21\u578b\u6216\u5355\u8f68\u6a21\u578b\u3002\u63a5\u4e0b\u6765\u6211\u4eec\u5c06\u63a8\u5bfc\u8fd9\u79cd\u6784\u578b\u5728\u51e0\u79cd\u5e38\u7528\u5750\u6807\u7cfb\u4e2d\u7684\u5fae\u5206\u7ea6\u675f\u3002<\/p>\n

\u53c2\u8003\u56fe3.1\uff0c\u5411\u91cf p_r \u548c p_f \u5206\u522b\u8868\u793a\u9759\u6b62\u5750\u6807\u7cfb\u6216\u60ef\u6027\u5750\u6807\u7cfb\u4e2d\u540e\u8f6e\u548c\u524d\u8f6e\u7684\u4f4d\u7f6e\uff0c\u5750\u6807\u7cfb\u7531\u4e09\u4e2a\u57fa\u5411\u91cf left(hat{e}_x, hat{e}_y, hat{e}_zright) \u6240\u51b3\u5b9a\u3002\u822a\u5411 theta \u662f\u63cf\u8ff0\u8f66\u8f86\u671d\u5411\u7684\u89d2\u5ea6\uff0c\u5b9a\u4e49\u4e3a\u5411\u91cf hat{e}_x \u548c p_f-p_r \u7684\u5939\u89d2\u3002\u4e00\u4e2a\u5750\u6807\u7cfb\u4e0b\u7684\u5fae\u5206\u7ea6\u675f\u7684\u8868\u8fbe\u5f0f\u5305\u62ec\u8be5\u5750\u6807\u7cfb\u4e0b\u7684\u89d2\u5ea6 theta \u4ee5\u53ca\u8f66\u8f86\u4e0a\u4e00\u70b9\u7684\u8fd0\u52a8\uff0c\u5982\u5411\u91cf p_r\uff08\u53c2\u8003[54]\uff09\u6216 p_f\uff08\u53c2\u8003[55]\uff09\u3002<\/p>\n

\u56fe3.1\uff1a\u5355\u8f68\u6a21\u578b\u7684\u8fd0\u52a8\u5b66<\/p>\n

\u56fe3.1\uff1a\u5355\u8f68\u6a21\u578b\u7684\u8fd0\u52a8\u5b66\uff0c\u5176\u4e2d p_r \u548c p_f\u5206\u522b\u662f\u540e\u8f6e\u548c\u524d\u8f6e\u4e0e\u5730\u9762\u7684\u63a5\u89e6\u70b9\uff0c theta \u662f\u8f66\u8f86\u822a\u5411\uff0c delta \u662f\u524d\u8f6e\u7684\u8f6c\u5411\u89d2\u3002\u7531\u4e8e\u975e\u5b8c\u6574\u7ea6\u675f\uff0cp_r \u548c p_f \u7684\u65f6\u95f4\u5bfc\u6570\u88ab\u9650\u5236\u5728\u84dd\u8272\u7bad\u5934\u6307\u793a\u7684\u65b9\u5411\u4e0a\u3002\u70b9p_r \u548c p_f \u7684\u8fd0\u52a8\u5fc5\u987b\u4e0e\u8f66\u8f6e\u7684\u671d\u5411\u5171\u7ebf\uff0c\u4ee5\u6ee1\u8db3\u65e0\u6ed1\u79fb\u5047\u8bbe\u3002<\/p>\n

\u5bf9\u4e8e\u540e\u8f6e\u7684\u8be5\u7ea6\u675f\u53ef\u4ee5\u8868\u793a\u4e3aleft(dot{p}_r cdot hat{e}_yright) cos theta-left(dot{p}_r cdot hat{e}_xright) sin theta=0 (3.1)<\/p>\n

\u5bf9\u4e8e\u524d\u8f6e\u7684\u8be5\u7ea6\u675f\u53ef\u4ee5\u8868\u793a\u4e3aleft(dot{p}_f cdot hat{e}_yright) cos (theta+delta)-left(dot{p}_f cdot hat{e}_xright) sin (theta+delta)=0 (3.2)<\/p>\n

\u5229\u7528\u6bcf\u4e2a\u70b9\u6cbf\u57fa\u5411\u91cf\u7684\u8fd0\u52a8\u5206\u91cf\uff0c\u53ef\u4ee5\u91cd\u5199\u4e0a\u5f0f\u3002<\/p>\n

\u540e\u8f6e\u6cbf hat{e}_x \u65b9\u5411\u7684\u8fd0\u52a8\u53ef\u4ee5\u5199\u4f5c x_r:=p_r cdot hat{e}_x \uff0c\u76f8\u4f3c\u5730\uff0c\u6cbfhat{e}_y \u65b9\u5411\u7684\u8fd0\u52a8\u5199\u4f5c y_r:=p_r cdot hat{e}_y \u3002\u524d\u5411\u901f\u5ea6 v_r:=dot{p}_r cdotleft(p_f-p_rright) \/left|left(p_f-p_rright)right| \uff0c\u8868\u793a dot{p}_r \u7684\u5927\u5c0f\uff0c\u540c\u65f6\u5e26\u6709\u6b63\u786e\u7684\u6b63\u8d1f\u53f7\uff0c\u8be5\u6b63\u8d1f\u53f7\u7528\u4e8e\u63cf\u8ff0\u5f53\u524d\u7684\u884c\u9a76\u662f\u6b63\u5411\u6216\u53cd\u5411\u9a7e\u9a76\u3002\u5229\u7528 x_r, y_r, theta \u8fd9\u51e0\u4e2a\u6807\u91cf\uff0c\u5fae\u5206\u7ea6\u675f\u53ef\u4ee5\u5199\u4f5cbegin{aligned} dot{x}_r =v_r cos theta \\ dot{y}_r =v_r sin theta \\ dot{theta} =frac{v_r}{l} tan delta end{aligned} (3.3)<\/p>\n

\u7c7b\u4f3c\u5730\uff0c\u4e5f\u53ef\u4ee5\u5229\u7528 p_f \u7684\u8fd0\u52a8\u5c06\u5fae\u5206\u7ea6\u675f\u5199\u4f5cbegin{aligned} &dot{x}_f=v_f cos (theta+delta) \\ &dot{y}_f=v_f sin (theta+delta) \\ &dot{theta}=frac{v_f}{l} sin delta end{aligned} (3.4)<\/p>\n

\u5176\u4e2d v_f \u662f\u524d\u8f6e\u7684\u524d\u5411\u901f\u5ea6\uff0c\u5b83\u4e0e\u540e\u8f6e\u901f\u5ea6 v_r \u6ee1\u8db3\u5173\u7cfbfrac{v_r}{v_f}=cos delta (3.5)\u5229\u7528\u8fd9\u4e2a\u6a21\u578b\u6c42\u89e3\u89c4\u5212\u548c\u63a7\u5236\u95ee\u9898\u6d89\u53ca\u5728\u8f66\u8f86\u7684\u673a\u68b0\u9650\u5236\u8303\u56f4\u5185\u9009\u62e9\u8f6c\u5411\u89d2 delta inleft[delta_{min }, delta_{max }right] \uff0c\u4ee5\u53ca\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u9009\u62e9\u524d\u5411\u901f\u5ea6 v_r inleft[v_{min }, v_{max }right] \u3002<\/p>\n

\u4e00\u79cd\u7ecf\u5e38\u91c7\u7528\u7684\u7b80\u5316\u65b9\u5f0f\u662f\u9009\u62e9\u4e00\u4e2a\u822a\u5411\u7387 omega \u800c\u975e\u9009\u62e9\u4e00\u4e2a\u8f6c\u5411\u89d2 delta \u3002\u8fd9\u4e9b\u53d8\u91cf\u6ee1\u8db3\u5173\u7cfbdelta=arctan frac{l omega}{v_r} (3.6)\u5e76\u968f\u4e4b\u5f97\u4ee5\u5c06\u8f6c\u5411\u52a8\u529b\u5b66\u7b80\u5316\u4e3adot{theta}=omega, quad omega inleft[frac{v_r}{l} tan delta_{min }, frac{v_r}{l} tan delta_{max }right] (3.7)<\/p>\n

\u6b64\u65f6\u7684\u6a21\u578b\u7ecf\u5e38\u88ab\u79f0\u4e3a\u5355\u8f6e\uff08unicycle\uff09\u6a21\u578b\uff0c\u56e0\u4e3a\u5b83\u7684\u63a8\u5bfc\u76f8\u5f53\u4e8e\u53ea\u8003\u8651\u4e86\u4e00\u4e2a\u8f6e\u5b50\u7684\u8fd0\u52a8\u3002\u8be5\u6a21\u578b\u7684\u4e00\u4e2a\u91cd\u8981\u53d8\u4f53\u662f v_r \u56fa\u5b9a\u65f6\u7684\u60c5\u51b5\uff0c\u6b64\u65f6\u7684\u6a21\u578b\u53c8\u88ab\u79f0\u4e3aDubins\u8f66\uff08Dubins Car\uff09\uff0c\u4ee5\u7eaa\u5ff5Lester Dubins\u63a8\u5bfc\u51fa\u4e86\u5177\u6709\u6307\u5b9a\u5207\u7ebf\u7684\u4e24\u70b9\u4e4b\u95f4\u7684\u6700\u5c0f\u65f6\u95f4\u8fd0\u52a8[57]\u3002\u53e6\u4e00\u4e2a\u503c\u5f97\u6ce8\u610f\u7684\u53d8\u4f53\u662fReeds-Shepp\u8f66\uff0c\u6b64\u65f6\u5bf9\u4e8e\u6700\u77ed\u8def\u5f84\u7684\u6c42\u89e3\u5141\u8bb8 v_r \u662f\u6b63\u5411\u6216\u8005\u53cd\u5411\u884c\u9a76\u7684\u901f\u5ea6[58]\u3002\u8fd9\u4e24\u79cd\u6a21\u578b\u5df2\u88ab\u8bc1\u660e\u5bf9\u8fd0\u52a8\u89c4\u5212\u5177\u6709\u4e00\u5b9a\u7684\u91cd\u8981\u6027\uff0c\u56e0\u6b64\u5c06\u5728\u7b2c\u56db\u8282\u4e2d\u4f5c\u8fdb\u4e00\u6b65\u8ba8\u8bba\u3002\u8fd0\u52a8\u5b66\u6a21\u578b\u9002\u7528\u4e8e\u4f4e\u901f\uff08\u4f8b\u5982\u6cca\u8f66\u548c\u57ce\u5e02\u9a7e\u9a76\uff09\u4e0b\u7684\u8def\u5f84\u89c4\u5212\u3002\u5728\u8fd9\u4e9b\u4f4e\u901f\u884c\u9a76\u7684\u573a\u666f\u4e2d\uff0c\u76f8\u6bd4\u4e8e\u65e0\u6ed1\u79fb\u5047\u8bbe\u5bf9\u79fb\u52a8\u6027\u7684\u9650\u5236\uff0c\u60ef\u6027\u6548\u5e94\u8f83\u5c0f\u3002\u8be5\u6a21\u578b\u7684\u4e3b\u8981\u7f3a\u70b9\u662f\u5b83\u5141\u8bb8\u77ac\u65f6\u7684\u8f6c\u5411\u89d2\u53d8\u5316\uff1a\u5982\u679c\u8fd0\u52a8\u89c4\u5212\u6a21\u5757\u751f\u6210\u5177\u6709\u8fd9\u79cd\u77ac\u65f6\u53d8\u5316\u7684\u89e3\uff0c\u53ef\u80fd\u5c06\u5e26\u6765\u5f88\u591a\u9ebb\u70e6\u3002\u8f6c\u5411\u89d2\u7684\u8fde\u7eed\u6027\u53ef\u4ee5\u901a\u8fc7\u589e\u5f3a\u65b9\u7a0b(3.4)\u6765\u5b9e\u73b0\uff0c\u5176\u4e2d\u8f6c\u5411\u89d2\u88ab\u5904\u7406\u4e3a\u6307\u5b9a\u901f\u7387\u7684\u79ef\u5206\uff0c\u53c2\u8003[49]\u3002<\/p>\n

\u65b9\u7a0b(3.4)\u8fdb\u800c\u53d8\u4e3abegin{aligned} &dot{x}_f=v_f cos (theta+delta) \\ &dot{y}_f=v_f sin (theta+delta) \\ &dot{theta}=frac{v_f}{l} sin delta \\ & dot{delta}=v_delta end{aligned} (3.8)<\/p>\n

\u6b64\u65f6\u9664\u4e86\u9650\u5236\u8f6c\u5411\u89d2\uff0c\u8fd8\u8981\u7ea6\u675f\u8f6c\u5411\u7387 v_delta inleft[dot{delta}_{min }, dot{delta}_{max }right] \u3002\u540c\u6837\u7684\u95ee\u9898\u4e5f\u53ef\u80fd\u53d1\u751f\u5728\u8f66\u901f v_r\u4e0a\uff0c\u4f46\u53ef\u4ee5\u901a\u8fc7\u4e0e\u8f6c\u5411\u89d2\u76f8\u540c\u7684\u65b9\u5f0f\u6765\u89e3\u51b3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u7f3a\u70b9\u662f\u589e\u52a0\u4e86\u6a21\u578b\u7684\u7ef4\u5ea6\uff0c\u4f7f\u5f97\u8fd0\u52a8\u89c4\u5212\u4e0e\u63a7\u5236\u95ee\u9898\u88ab\u590d\u6742\u5316\u4e86\u3002 \u5750\u6807\u7cfb\u7684\u9009\u62e9\u5e76\u4e0d\u9650\u5b9a\u5fc5\u987b\u9009\u62e9\u4e00\u4e2a\u8f6e\u5b50\u7684\u4f4d\u7f6e\u4f5c\u4e3a\u8f66\u8f86\u7684\u4f4d\u7f6e\u5750\u6807\u3002\u5bf9\u4e8e\u901a\u8fc7\u7ecf\u5178\u529b\u5b66\u539f\u7406\u63a8\u5bfc\u5f97\u5230\u7684\u6a21\u578b\uff0c\u4f7f\u7528\u8d28\u5fc3\u4f5c\u4e3a\u4f4d\u7f6e\u5750\u6807\u53ef\u80fd\u66f4\u52a0\u65b9\u4fbf\uff0c\u53c2\u8003[59][60]\uff0c\u6216\u8005\u4f7f\u7528\u6446\u5fc3\uff0c\u53c2\u8003[61][62]\u3002B. \u60ef\u6027\u6548\u5e94\u5f53\u8f66\u8f86\u7684\u52a0\u901f\u5ea6\u8db3\u591f\u5927\u65f6\uff0c\u8f6e\u80ce\u4e0e\u5730\u9762\u4e4b\u95f4\u7684\u65e0\u6ed1\u79fb\u5047\u8bbe\u4e0d\u518d\u6210\u7acb\u3002\u6b64\u65f6\uff0c\u4e00\u4e2a\u66f4\u52a0\u51c6\u786e\u7684\u6a21\u578b\u662f\u5c06\u8f66\u8f86\u89c6\u4e3a\u4e00\u4e2a\u6ee1\u8db3\u52a8\u91cf\u5b9a\u7406\u7684\u521a\u4f53\uff0c\u5373\uff0c\u52a0\u901f\u5ea6\u6b63\u6bd4\u4e8e\u5730\u9762\u65bd\u52a0\u5728\u8f6e\u80ce\u4e0a\u7684\u529b\u3002<\/p>\n

\u53d6 p_c \u4f5c\u4e3a\u8f66\u8f86\u7684\u8d28\u5fc3\u4ee5\u53ca\u6784\u578b\u5750\u6807\uff08\u53c2\u8003\u56fe3.2\uff09\uff0c\u8f66\u8f86\u7684\u8fd0\u52a8\u670d\u4ece\u4ee5\u4e0b\u5173\u7cfbbegin{aligned} &m ddot{p}_c=F_f+F_r \\ & I_{z z} ddot{theta}=left(p_c-p_fright) times F_f+left(p_c-p_rright) times F_r end{aligned} (3.9)<\/p>\n

\u5176\u4e2d F_r \u548c F_f \u662f\u5730\u9762\u901a\u8fc7\u4e0e\u8f6e\u80ce\u7684\u63a5\u89e6\u800c\u65bd\u52a0\u4e8e\u8f66\u8f86\u7684\u529b\uff0cm \u662f\u8f66\u8f86\u7684\u603b\u8d28\u91cf\uff0c I_{Z Z} \u662f\u5728 hat{e}_z \u65b9\u5411\u4e0a\u5173\u4e8e\u8d28\u5fc3\u7684\u6781\u60ef\u6027\u77e9\u3002\u5728\u63a5\u4e0b\u6765\u7684\u63a8\u5bfc\u4e2d\uff0c\u6211\u4eec\u5c06\u57fa\u4e8e\u5e73\u5766\u5730\u9762\u3001\u521a\u4f53\u60ac\u67b6\u4ee5\u53ca\u8f66\u8f86\u4fdd\u6301\u5728\u9053\u8def\u4e0a\u7684\u5047\u8bbe\uff0c\u9ed8\u8ba4\u5ffd\u7565 p_c\u5728hat{e}_z\u65b9\u5411\u4e0a\u7684\u8fd0\u52a8\u3002<\/p>\n

<\/p>\n

\u56fe3.2\uff1a\u6ca1\u6709\u65e0\u6ed1\u79fb\u5047\u8bbe\u7684\u5355\u8f68\u6a21\u578b\u7684\u8fd0\u52a8\u5b66\u56fe\u89e3\u5176\u4e2d omega_{{r, f}} \u662f\u8f66\u8f6e\u76f8\u5bf9\u4e8e\u8f66\u8f86\u7684\u76f8\u5bf9\u89d2\u901f\u5ea6\uff08relative angular velocities\uff09\u3002F_r \u548c F_f \u7684\u8868\u8fbe\u5f0f\u968f\u5efa\u6a21\u5047\u8bbe[18][59][60][62]\u800c\u53d8\u5316\uff0c\u4f46\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\uff0c\u63a8\u5bfc\u8be5\u8868\u8fbe\u5f0f\u90fd\u53ef\u80fd\u5f88\u7e41\u7410\u3002\u56e0\u6b64\uff0c\u65b9\u7a0b(3.10)-(3.15)\u63d0\u4f9b\u4e86\u8be6\u7ec6\u7684\u63a8\u5bfc\u4f5c\u4e3a\u53c2\u8003\u3002\u5730\u9762\u4e0e\u8f6e\u80ce\u4e4b\u95f4\u7684\u529b\u88ab\u5efa\u6a21\u4e3a\u4e0e\u8f6e\u80ce\u5728\u5730\u9762\u4e0a\u6ed1\u52a8\u7684\u901f\u7387\u6709\u5173\u3002<\/p>\n

\u5c3d\u7ba1\u9009\u62e9\u8d28\u5fc3\u4f5c\u4e3a\u6784\u578b\u7684\u5750\u6807\uff0c\u4f46\u4ecd\u7136\u9700\u8981\u6bcf\u4e2a\u8f66\u8f6e\u76f8\u5bf9\u4e8e\u5730\u9762\u7684\u901f\u5ea6\uff0c\u4ee5\u786e\u5b9a\u8be5\u76f8\u5bf9\u901f\u5ea6\u3002\u8fd9\u4e09\u4e2a\u70b9\u4e4b\u95f4\u7684\u8fd0\u52a8\u5b66\u5173\u7cfb\u4e3abegin{aligned} &p_r=p_c+left(begin{array}{c} -l_r cos theta \\ -l_r sin theta \\ 0 end{array}right) \\ & dot{p}_r=dot{p}_c+left(begin{array}{c} 0 \\ 0 \\ & dot{theta} end{array}right) timesleft(begin{array}{c} -l_r cos theta \\ -l_r sin theta \\ 0 end{array}right) \\ &p_f=p_c+left(begin{array}{c} l_f cos theta \\ l_f sin theta \\ 0 end{array}right) \\ &dot{p}_f=dot{p}_c+left(begin{array}{c} 0 \\ 0 \\ dot{theta} end{array}right) timesleft(begin{array}{c} l_f cos theta \\ l_f sin theta \\ 0 end{array}right) end{aligned} (3.10)<\/p>\n

\u8fd9\u4e9b\u8fd0\u52a8\u5b66\u5173\u7cfb\u7528\u4e8e\u786e\u5b9a\u6bcf\u4e2a\u8f6e\u80ce\u4e0e\u5730\u9762\u7684\u63a5\u89e6\u70b9\u7684\u901f\u5ea6 s_r \u548c s_f \u3002\u8fd9\u4e9b\u70b9\u7684\u901f\u5ea6\u53c8\u88ab\u79f0\u4e3a\u8f6e\u80ce\u6ed1\u52a8\u901f\u5ea6\u3002\u901a\u5e38\uff0c\u7531\u4e8e\u5b58\u5728\u8f6e\u80ce\u7684\u89d2\u901f\u5ea6\uff0c\u56e0\u6b64s_r \u548c s_f\u4e0d\u540c\u4e8e dot{p}_r \u548c dot{p}_f \u3002\u5b83\u4eec\u4e4b\u95f4\u7684\u8fd0\u52a8\u5b66\u5173\u7cfb\u5982\u4e0b\uff1abegin{aligned} & s_r=dot{p}_r+omega_r times R \\ & s_f=dot{p}_f+omega_f times R end{aligned} (3.11)<\/p>\n

\u5176\u4e2d\u8f66\u8f6e\u7684\u89d2\u901f\u5ea6\u7531\u4e0b\u5f0f\u7ed9\u51fa\uff1aomega_r=left(begin{array}{c} Omega_r sin theta \\ -Omega_r cos theta \\ 0 end{array}right), quad omega_f=left(begin{array}{c} Omega_f sin (theta+delta) \\ -Omega_f cos (theta+delta) \\ 0 end{array}right) (3.12)\u4e14 R=(0,0,-r)^T \u3002<\/p>\n

\u8f66\u8f6e\u534a\u5f84\u7528\u6807\u91cf r \u8868\u793a\uff0c\u6bcf\u4e2a\u8f6e\u80ce\u76f8\u5bf9\u4e8e\u6c7d\u8f66\u7684\u89d2\u901f\u7387\u7528 Omega_{{r, f}} \u8868\u793a\u3002\u56fe3.3\u63cf\u7ed8\u4e86\u540e\u8f6e\u4e0a\u7684\u5404\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/p>\n

<\/p>\n

\u56fe3.3:\u4e8c\u7ef4\u7684\u540e\u8f6e\u8fd0\u52a8\u5b66\u56fe\u89e3<\/p>\n

\u56fe3.3\uff1a\u4e8c\u7ef4\u7684\u540e\u8f6e\u8fd0\u52a8\u5b66\u56fe\u89e3\u8868\u660e\u4e86\u8f66\u8f6e\u6ed1\u79fb S_r \u4e0e\u540e\u8f6e\u901f\u5ea6 dot{p}_r \u4ee5\u53ca\u89d2\u901f\u7387\uff08angular speed\uff09Omega_r \u5747\u5b58\u5728\u5173\u8054\u3002\u901a\u5e38\uff0c S_r\u548c dot{p}_r\u4e0d\u5171\u7ebf\uff0c\u5e76\u4e14\u53ef\u80fd\u5728\u5782\u76f4\u4e8e\u6240\u793a\u5e73\u9762\u7684\u65b9\u5411\u4e0a\u5177\u6709\u975e\u96f6\u5206\u91cf\u3002 \u5728\u9759\u6001\u6761\u4ef6\u4e0b\uff0c\u6216\u8005\u5f53\u8d28\u5fc3\u7684\u9ad8\u5ea6\u53ef\u4ee5\u8fd1\u4f3c\u6ee1\u8db3 p_c cdot hat{e}_z approx 0 \u65f6\uff0c\u5730\u9762\u65bd\u52a0\u7ed9\u8f6e\u80ce\u7684\u529b\u5728\u5782\u76f4\u4e8e\u5730\u9762\u7684\u65b9\u5411\u4e0a\u7684\u5206\u91cf F_{{r, f}} cdot hat{e}_Z \u53ef\u4ee5\u901a\u8fc7\u9759\u529b\u5b66\u7684\u529b\u77e9\u5e73\u8861\u800c\u7ed9\u51fa\uff1aF_f cdot hat{e}_z=frac{l_r m g}{l_f+l_r}, quad F_r cdot hat{e}_z=frac{l_f m g}{l_f+l_r} (3.13)\u6cd5\u5411\u529b\u968f\u540e\u540c\u6ed1\u79fb\u4ee5\u53ca\u4e00\u4e2a\u63cf\u8ff0\u8f6e\u80ce\u7279\u6027\u7684\u6469\u64e6\u7cfb\u6570\u6a21\u578b mu \u4e00\u8d77\uff0c\u7528\u4e8e\u8ba1\u7b97\u6bcf\u4e2a\u8f6e\u80ce\u4e0a\u7684\u7275\u5f15\u529b\u3002<\/p>\n

\u540e\u8f6e\u4e0a\u7684\u7275\u5f15\u529b\u7684\u5206\u91cf\u7ed9\u51fa\u5982\u4e0b\uff1a<\/p>\n

begin{aligned} & F_r cdot hat{e}_x=-frac{left(F_r cdot hat{e}_zright) muleft(frac{left|s_rright|}{Omega_r r}right) s_r}{left|s_rright|} cdot hat{e}_x \\ & F_r cdot hat{e}_y=-frac{left(F_r cdot hat{e}_zright) muleft(frac{left|s_rright|}{Omega_r r}right) s_r}{left|s_rright|} cdot hat{e}_y end{aligned} (3.14)<\/p>\n

\u5c06\u4e0a\u5f0f\u4e2d\u7684\u4e0b\u6807r \u66ff\u6362\u4e3a f \uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u524d\u8f6e\u4e0a\u7684\u7275\u5f15\u529b\u7684\u8868\u8fbe\u5f0f\u3002\u4e0a\u5f0f\u5c06\u7275\u5f15\u529b\u5efa\u6a21\u4e3a\u53cd\u5411\u5e73\u884c\u4e8e\u6ed1\u79fb\uff0c\u4e14\u5176\u5927\u5c0f\u6b63\u6bd4\u4e8e\u6cd5\u5411\u529b\uff0c\u540c\u65f6\u4e0e\u6ed1\u79fb\u7387\u975e\u7ebf\u6027\u76f8\u5173\uff08\u6ed1\u79fb\u7387\u7531\u6ed1\u79fb\u7ecf\u8fc7\u540e\u8f6e\u7684 Omega_r r \u6216\u524d\u8f6e\u7684 Omega_f r \u7684\u5f52\u4e00\u5316\u800c\u7ed9\u51fa\uff09\u3002\u7ed3\u5408\u516c\u5f0f(3.10)-(3.15)\uff0c\u53ef\u4ee5\u5f97\u5230\u5173\u4e8e\u63a7\u5236\u53d8\u91cf\u3001\u5e7f\u4e49\u5750\u6807\u4ee5\u53ca\u8f6e\u80ce\u901f\u5ea6\u7684\u6bcf\u4e2a\u8f66\u8f6e\u4e0a\u7684\u5408\u529b\u7684\u8868\u8fbe\u5f0f\u3002\u5c06\u65b9\u7a0b(3.14)\u4ee5\u53ca\u5982\u4e0b\u6709\u5173 mu \u7684\u6a21\u578b\u7ed3\u5408\u8d77\u6765\uff0c\u662f\u4e00\u79cd\u5e38\u7528\u7684\u3001\u5904\u7406\u8f66\u8f6e\u548c\u5730\u9762\u63a5\u89e6\u7684\u60c5\u51b5\u7684\u6a21\u578b\u3002\u65b9\u7a0b(3.15)\u662fPacejka\u63d0\u51fa\u7684\u77e5\u540d\u6a21\u578b\u7684\u4e00\u79cd\u7b80\u5316\u7248\u672c\u3002<\/p>\n

muleft(frac{left|s_rright|}{Omega_r r}right)=D sin left(C arctan left(B frac{left|s_rright|}{Omega_r r}right)right) (3.15)<\/p>\n

\u65b9\u7a0b(3.14)\u7684\u65cb\u8f6c\u5bf9\u79f0\u6027\u4ee5\u53ca\u65b9\u7a0b(3.15)\u5b58\u5728\u7684\u5cf0\u503c\u5c06\u5bfc\u81f4\u8f6e\u80ce\u4ea7\u751f\u4e00\u4e2a\u80fd\u5728\u4efb\u4f55\u65b9\u5411\u4e0a\u65bd\u52a0\u7684\u8303\u6570\u6700\u5927\u7684\u529b\u3002\u8be5\u5cf0\u503c\u53c8\u88ab\u79f0\u4e3a\u6469\u64e6\u5706\uff08friction circle\uff09\uff0c\u5982\u56fe3.4\u6240\u793a\u3002<\/p>\n

\u56fe3.4<\/p>\n

\u56fe3.4\uff1a\u9876\u90e8\u7684\u56fe\u662f\u653e\u5927\u89c6\u56fe\uff0c\u5176\u63cf\u7ed8\u4e86\u6bcf\u4e2a\u8f6e\u80ce\u4e0a\u7684\u8f66\u8f6e\u6ed1\u79fb\u5230\u7275\u5f15\u529b\u7684\u6620\u5c04\uff1b\u5e95\u90e8\u7684\u56fe\u662f\u7f29\u5c0f\u89c6\u56fe\uff0c\u5176\u91cd\u70b9\u63cf\u7ed8\u4e86\u5b9a\u4e49\u6469\u64e6\u5708\u7684\u5cf0\u503c\u3002\u53c2\u8003\u65b9\u7a0b(3.15)\u3002\u672c\u8282\u8ba8\u8bba\u7684\u6a21\u578b\u7ecf\u5e38\u51fa\u73b0\u5728\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u7684\u8fd0\u52a8\u89c4\u5212\u548c\u63a7\u5236\u7684\u6587\u732e\u4e2d\u3002\u8fd9\u4e9b\u6a21\u578b\u9002\u7528\u4e8e\u672c\u7efc\u8ff0\u6240\u8ba8\u8bba\u7684\u8fd0\u52a8\u89c4\u5212\u548c\u63a7\u5236\u4efb\u52a1\u3002\u7136\u800c\uff0c\u8bf8\u5982\u7535\u5b50\u7a33\u5b9a\u63a7\u5236\u548c\u4e3b\u52a8\u60ac\u67b6\u7cfb\u7edf\u7b49\u8f83\u4f4e\u7ea7\u522b\u7684\u63a7\u5236\u4efb\u52a1\u901a\u5e38\u4f7f\u7528\u66f4\u590d\u6742\u7684\u6a21\u578b\u6765\u5bf9\u5e95\u76d8\u3001\u8f6c\u5411\u548c\u4f20\u52a8\u7cfb\u7edf\u8fdb\u884c\u5efa\u6a21\u3002<\/p>\n

4. \u8fd0\u52a8\u89c4\u5212\u6839\u636e\u51b3\u7b56\u5c42\u7684\u884c\u4e3a\u5c42\u7ed9\u51fa\u7684\u76ee\u6807\u6784\u578b\uff0c\u8fd0\u52a8\u89c4\u5212\u5c42\u8d1f\u8d23\u8ba1\u7b97\u51fa\u4e00\u6761\u4ece\u8f66\u8f86\u7684\u5f53\u524d\u6784\u578b\u5230\u76ee\u6807\u6784\u578b\u7684\u5b89\u5168\u3001\u8212\u9002\u4e14\u52a8\u529b\u5b66\u53ef\u884c\u7684\u8f68\u8ff9\u3002\u5728\u4e0d\u540c\u7684\u60c5\u5883\u4e2d\uff0c\u76ee\u6807\u6784\u578b\u53ef\u80fd\u6709\u6240\u4e0d\u540c\u3002\u4f8b\u5982\uff0c\u76ee\u6807\u4f4d\u7f6e\u53ef\u80fd\u662f\u884c\u9a76\u65b9\u5411\u4e0a\u524d\u65b9\u6570\u7c73\u5904\u7684\u5f53\u524d\u8f66\u9053\u7684\u4e2d\u5fc3\u70b9\u3001\u4e0b\u4e00\u4e2a\u5341\u5b57\u8def\u53e3\u5904\u7684\u505c\u8f66\u7ebf\u7684\u4e2d\u5fc3\u6216\u8005\u4e0b\u4e00\u4e2a\u671f\u671b\u7684\u6cca\u8f66\u70b9\u3002\u8fd0\u52a8\u89c4\u5212\u7ec4\u4ef6\u63a5\u53d7\u5173\u4e8e\u8f66\u8f86\u5468\u56f4\u7684\u9759\u6001\u548c\u52a8\u6001\u969c\u788d\u7269\u7684\u4fe1\u606f\uff0c\u5e76\u751f\u6210\u6ee1\u8db3\u8f66\u8f86\u8fd0\u52a8\u7684\u52a8\u529b\u5b66\u548c\u8fd0\u52a8\u5b66\u7ea6\u675f\u7684\u65e0\u78b0\u649e\u8f68\u8ff9\u3002\u901a\u5e38\uff0c\u8fd0\u52a8\u89c4\u5212\u5668\u6700\u5c0f\u5316\u4e00\u4e2a\u7ed9\u5b9a\u7684\u76ee\u6807\u51fd\u6570\u3002\u9664\u4e86\u884c\u7a0b\u65f6\u95f4\u5916\uff0c\u76ee\u6807\u51fd\u6570\u8fd8\u53ef\u80fd\u5305\u542b\u6709\u5173\u5371\u9669\u8fd0\u52a8\u6216\u5bfc\u81f4\u4e58\u5ba2\u4e0d\u9002\u7684\u8fd0\u52a8\u7684\u60e9\u7f5a\u9879\u3002\u5728\u5178\u578b\u7684\u8bbe\u7f6e\u4e2d\uff0c\u8fd0\u52a8\u89c4\u5212\u5668\u7684\u8f93\u51fa\u4f20\u9012\u7ed9\u5c40\u90e8\u53cd\u9988\u63a7\u5236\u5c42\u3002\u53cd\u9988\u63a7\u5236\u5668\u968f\u540e\u4ea7\u751f\u4e00\u4e2a\u8f93\u5165\u4fe1\u53f7\u4ee5\u63a7\u5236\u8f66\u8f86\u8ddf\u968f\u8fd9\u4e2a\u7ed9\u5b9a\u7684\u8fd0\u52a8\u8ba1\u5212\u3002\u8f66\u8f86\u7684\u8fd0\u52a8\u8ba1\u5212\u53ef\u80fd\u662f\u8def\u5f84\u5f62\u5f0f\u6216\u8f68\u8ff9\u5f62\u5f0f\u3002<\/p>\n

\u5728\u8def\u5f84\u89c4\u5212\u7684\u6846\u67b6\u5185\uff0c\u89e3\u8def\u5f84\u8868\u793a\u4e3a\u51fd\u6570 sigma(alpha):[0,1] rightarrow mathcal{X} \uff0c\u5176\u4e2d chi \u662f\u8f66\u8f86\u7684\u6784\u578b\u7a7a\u95f4\u3002\u6ce8\u610f\u8fd9\u79cd\u89e3\u5e76\u672a\u89c4\u5b9a\u5e94\u8be5\u5982\u4f55\u8ddf\u968f\u8be5\u8def\u5f84\uff0c\u56e0\u6b64\u8ddf\u968f\u7684\u65b9\u6848\u53ef\u4ee5\u662f\u9009\u62e9\u8def\u5f84\u7684\u901f\u5ea6\u5206\u5e03\uff0c\u6216\u8005\u662f\u5c06\u8be5\u4efb\u52a1\u79fb\u4ea4\u51b3\u7b56\u5c42\u4e2d\u7684\u4f4e\u7ea7\u5c42\u3002\u800c\u8f68\u8ff9\u89c4\u5212\u7684\u6846\u67b6\u660e\u786e\u8003\u8651\u5230\u4e86\u63a7\u5236\u6267\u884c\u65f6\u95f4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5bf9\u8f66\u8f86\u52a8\u529b\u5b66\u548c\u52a8\u6001\u969c\u788d\u7269\u8fdb\u884c\u76f4\u63a5\u5efa\u6a21\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u89e3\u8f68\u8ff9\u8868\u793a\u4e3a\u4e00\u4e2a\u65f6\u95f4\u53c2\u6570\u5316\u7684\u51fd\u6570 pi(t):[0, T] rightarrow mathcal{X} \uff0c\u5176\u4e2d T \u662f\u89c4\u5212\u8303\u56f4\/\u533a\u95f4\u3002\u4e0e\u8def\u5f84\u4e0d\u540c\uff0c\u8f68\u8ff9\u89c4\u5b9a\u4e86\u8f66\u8f86\u7684\u6784\u578b\u5982\u4f55\u968f\u65f6\u95f4\u53d8\u5316\u3002<\/p>\n

\u5728\u63a5\u4e0b\u6765\u7684\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u5bf9\u8def\u5f84\u89c4\u5212\u548c\u8f68\u8ff9\u89c4\u5212\u95ee\u9898\u8fdb\u884c\u6b63\u5f0f\u7684\u5b9a\u4e49\uff0c\u5e76\u7efc\u8ff0\u8fd9\u4e24\u79cd\u95ee\u9898\u5f62\u5f0f\u7684\u4e3b\u8981\u590d\u6742\u5ea6\u548c\u7b97\u6cd5\u7ed3\u679c\u3002<\/p>\n

A. \u8def\u5f84\u89c4\u5212\u8def\u5f84\u89c4\u5212\u95ee\u9898\u662f\u5728\u8f66\u8f86\uff08\u6216\u8005\u66f4\u4e00\u822c\u5730\u8bf4\uff0c\u673a\u5668\u4eba\uff09\u7684\u6784\u578b\u7a7a\u95f4 chi \u4e2d\u5bfb\u627e\u4e00\u6761\u4ece\u521d\u59cb\u6784\u578b\u5230\u76ee\u6807\u533a\u57df\u7684\u8def\u5f84 sigma(alpha):[0,1] rightarrow mathcal{X} \uff0c\u540c\u65f6\u6ee1\u8db3\u7ed9\u5b9a\u7684\u5168\u5c40\u548c\u5c40\u90e8\u7ea6\u675f\u3002\u6839\u636e\u662f\u5426\u8003\u8651\u89e3\u8def\u5f84\u7684\u8d28\u91cf\uff0c\u5206\u522b\u7528\u672f\u8bed\u201c\u53ef\u884c\uff08feasible\uff09\u201d\u548c\u201c\u6700\u4f18\uff08optimal\uff09\u201d\u6765\u63cf\u8ff0\u8be5\u8def\u5f84\u3002<\/p>\n

\u53ef\u884c\u8def\u5f84\u89c4\u5212\u662f\u6307\u5728\u4e0d\u5173\u6ce8\u89e3\u7684\u8d28\u91cf\u7684\u60c5\u51b5\u4e0b\uff0c\u786e\u5b9a\u4e00\u6761\u6ee1\u8db3\u67d0\u4e9b\u7ed9\u5b9a\u95ee\u9898\u7ea6\u675f\u7684\u8def\u5f84\u7684\u95ee\u9898\uff1b\u800c\u6700\u4f18\u8def\u5f84\u89c4\u5212\u662f\u6307\u5728\u7ed9\u5b9a\u7ea6\u675f\u6761\u4ef6\u4e0b\u627e\u5230\u4e00\u6761\u4f18\u5316\u67d0\u4e2a\u8d28\u91cf\u6807\u51c6\u7684\u8def\u5f84\u7684\u95ee\u9898\u3002\u6700\u4f18\u8def\u5f84\u89c4\u5212\u95ee\u9898\u7684\u4e00\u79cd\u6b63\u5f0f\u8868\u8ff0\u5982\u4e0b\u3002<\/p>\n

\u8bbe chi \u662f\u8f66\u8f86\u7684\u6784\u578b\u7a7a\u95f4\uff0c\u8bbe sum(mathcal{X}) \u8868\u793a\u6240\u6709\u8fde\u7eed\u51fd\u6570 [0,1] rightarrow mathcal{X} \u7684\u96c6\u5408\uff1b\u8f66\u8f86\u7684\u521d\u59cb\u6784\u578b\u4e3a mathbf{x}_{text {init }} in mathcal{X} \uff1b\u8def\u5f84\u9700\u8981\u5728\u76ee\u6807\u533a\u57df X_{text {goal }} subseteq mathcal{X} \u5185\u7ed3\u675f\u3002\u8f66\u8f86\u7684\u6240\u6709\u5141\u8bb8\u7684\u6784\u578b\u7684\u96c6\u5408\u79f0\u4e3a\u81ea\u7531\u6784\u578b\u7a7a\u95f4\/\u81ea\u7531\u914d\u7f6e\u7a7a\u95f4\uff0c\u8bb0\u4e3a chi_{text {free }} \u3002<\/p>\n

\u901a\u5e38\uff0c\u81ea\u7531\u6784\u578b\u662f\u90a3\u4e9b\u4e0d\u4f1a\u5bfc\u81f4\u4e0e\u969c\u788d\u7269\u53d1\u751f\u78b0\u649e\u7684\u6784\u578b\uff0c\u4f46\u81ea\u7531\u6784\u578b\u96c6\u4e5f\u53ef\u4ee5\u8868\u793a\u8def\u5f84\u4e0a\u7684\u5176\u4ed6\u5b8c\u6574\u7ea6\u675f\u3002\u8def\u5f84\u4e0a\u7684\u5fae\u5206\u7ea6\u675f\u7531\u8c13\u8bcd Dleft(mathrm{x}, mathrm{x}^{prime}, mathrm{x}^{prime prime}, ldotsright) \u8868\u793a\uff0c\u5e76\u4e14\u53ef\u4ee5\u7528\u4e8e\u5f3a\u5236\u8f66\u8f86\u5b9e\u73b0\u4e00\u5b9a\u7a0b\u5ea6\u7684\u8def\u5f84\u5e73\u6ed1\u5ea6\uff0c\u4f8b\u5982\u8def\u5f84\u66f2\u7387\u7684\u754c\u9650\u4ee5\u53ca\/\u6216\u8005\u66f2\u7387\u53d8\u5316\u7387\u3002<\/p>\n

\u4f8b\u5982\uff0c\u5728 mathcal{X} subseteq mathbb{R}^2 \u7684\u60c5\u51b5\u4e0b\uff0c\u5fae\u5206\u7ea6\u675f\u53ef\u4ee5\u4f7f\u7528\u5982\u4e0b\u7684Frenet-Serret\u516c\u5f0f\u6765\u9650\u5236\u8def\u5f84\u7684\u6700\u5927\u66f2\u7387<\/p>\n

kappa :Dleft(mathrm{x}, mathrm{x}^{prime}, mathrm{x}^{prime prime}, ldotsright) Leftrightarrow frac{left|mathrm{x}^{prime} times mathrm{x}^{prime prime}right|}{left|mathrm{x}^{prime}right|^3} leq kappa<\/p>\n

\u5982\u679c\u518d\u8fdb\u4e00\u6b65\u5730\u8bbe<\/p>\n

J(sigma): sum(mathcal{X}) rightarrow mathbb{R} \u4e3a\u6210\u672c\u6cdb\u51fd\uff0c\u90a3\u4e48\u6700\u4f18\u8def\u5f84\u89c4\u5212\u95ee\u9898\u53ef\u4ee5\u4e00\u822c\u5730\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n

\u95ee\u9898 4.1 \u6700\u4f18\u8def\u5f84\u89c4\u5212\u7ed9\u5b9a\u4e00\u4e2a\u4e94\u5143\u7ec4 left(mathcal{X}_{text {free }}, mathrm{x}_{text {init }}, X_{text {goal }}, D, Jright) \uff0c\u627e\u5230 sigma^*=arg min _{sigma in sum(X)} J(sigma)<\/p>\n

\u7ea6\u675f\u6761\u4ef6\uff1abegin{aligned} &sigma(0)=mathrm{x}_{text {init }}, quad sigma(1) in X_{text {goal }} \\ &sigma(alpha) in mathcal{X}_{text {free }}, quad forall alpha in[0,1] \\ & Dleft(sigma(alpha), sigma^{prime}(alpha), sigma^{prime prime}(alpha), ldotsright), quad forall alpha in[0,1] end{aligned}<\/p>\n

\u53ef\u884c\u548c\u6700\u4f18\u8def\u5f84\u89c4\u5212\u95ee\u9898\u5728\u8fc7\u53bb\u51e0\u5341\u5e74\u4e2d\u5f97\u5230\u4e86\u5e7f\u6cdb\u7684\u7814\u7a76\u3002\u6b64\u95ee\u9898\u7684\u590d\u6742\u5ea6\u5df2\u7ecf\u88ab\u5f88\u597d\u5730\u7406\u89e3\uff0c\u5e76\u4e14\u5df2\u7ecf\u5f00\u53d1\u4e86\u8bb8\u591a\u5b9e\u7528\u7684\u7b97\u6cd5\u3002<\/p>\n

\u590d\u6742\u5ea6\uff1a\u5927\u91cf\u6587\u732e\u81f4\u529b\u4e8e\u7814\u7a76\u8fd0\u52a8\u89c4\u5212\u95ee\u9898\u7684\u590d\u6742\u5ea6\u3002\u63a5\u4e0b\u6765\u672c\u6587\u5c06\u7efc\u8ff0\u5173\u4e8e\u8fd9\u4e9b\u95ee\u9898\u7684\u8ba1\u7b97\u590d\u6742\u5ea6\u7684\u4e00\u4e9b\u4e3b\u8981\u7ed3\u679c\u3002<\/p>\n

\u95ee\u98984.1\u9648\u8ff0\u4e86\u5728\u5b8c\u6574\u548c\u5fae\u5206\u7ea6\u675f\u4e0b\u627e\u5230\u4e00\u6761\u6700\u4f18\u8def\u5f84\u7684\u95ee\u9898\uff0c\u540e\u8005\u88ab\u8ba4\u4e3a\u662fPSPACE\u56f0\u96be\u7684\uff08PSPACE-hard\uff09[64]\u3002\u8fd9\u610f\u5473\u7740\u8be5\u95ee\u9898\u81f3\u5c11\u4e0e\u89e3\u51b3\u4efb\u4f55NP\u5b8c\u5168\u95ee\u9898\u4e00\u6837\u56f0\u96be\u3002\u56e0\u6b64\uff0c\u5982\u679c\u5047\u8bbe mathrm{P} neq mathrm{NP} \uff0c\u90a3\u4e48\u5c31\u4e0d\u5b58\u5728\u6709\u6548\u7684\uff08\u591a\u9879\u5f0f\u65f6\u95f4\u7684\uff09\u7b97\u6cd5\u80fd\u591f\u89e3\u51b3\u6b64\u95ee\u9898\u7684\u6240\u6709\u5b9e\u4f8b\u3002\u56e0\u6b64\uff0c\u7814\u7a76\u8005\u6b64\u540e\u4fbf\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u7814\u7a76\u8fd1\u4f3c\u65b9\u6cd5\uff0c\u6216\u7814\u7a76\u4e00\u822c\u8def\u5f84\u89c4\u5212\u95ee\u9898\u7684\u5b50\u96c6\u7684\u65b9\u6cd5\u4e0a\u3002<\/p>\n

\u6700\u521d\u7684\u7814\u7a76\u4e3b\u8981\u96c6\u4e2d\u4e8e\u5728\u591a\u8fb9\u5f62\/\u591a\u9762\u4f53\u73af\u5883\u4e2d\u7684\u5b8c\u6574\uff08holonomic\uff09\u8f66\u8f86\u6a21\u578b\u7684\u53ef\u884c\uff08\u975e\u6700\u4f18\uff09\u8def\u5f84\u89c4\u5212\u3002\u5373\uff0c\u5047\u8bbe\u969c\u788d\u7269\u662f\u591a\u8fb9\u5f62\/\u591a\u9762\u4f53\uff0c\u4e14\u5047\u8bbe\u5728\u751f\u6210\u7684\u8def\u5f84\u4e0a\u6ca1\u6709\u5fae\u5206\u7ea6\u675f\u3002<\/p>\n

1970\u5e74\uff0cReif\u53d1\u73b0\uff0c\u5728\u4e8c\u7ef4\u548c\u4e09\u7ef4\u73af\u5883\u4e2d\uff0c\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u627e\u5230\u5b8c\u6574\u8f66\u8f86\u7684\u65e0\u969c\u788d\u8def\u5f84\uff0c\u5176\u8db3\u8ff9\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u5355\u4e2a\u591a\u9762\u4f53[64]\u3002Canny\u5df2\u7ecf\u8bc1\u660e\uff0c\u4f7f\u7528\u591a\u9879\u5f0f\u8868\u793a\u7684\u81ea\u7531\u7a7a\u95f4\u4e2d\u7684\u53ef\u884c\u8def\u5f84\u89c4\u5212\u95ee\u9898\u662fPSPACE\u95ee\u9898\uff0c\u56e0\u6b64\uff0c\u6ca1\u6709\u5fae\u5206\u7ea6\u675f\u7684\u53ef\u884c\u8def\u5f84\u89c4\u5212\u7684\u51b3\u7b56\u95ee\u9898\u662fPSPACE\u5b8c\u5168\u95ee\u9898[65]\u3002<\/p>\n

\u6700\u4f18\u8def\u5f84\u89c4\u5212\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u4e00\u6761\u6700\u77ed\u7684\u65e0\u969c\u788d\u8def\u5f84\u3002\u5b66\u754c\u5f88\u65e9\u5c31\u610f\u8bc6\u5230\uff0c\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u627e\u5230\u5b8c\u6574\u8f66\u8f86\u5728\u5177\u6709\u591a\u8fb9\u5f62\u969c\u788d\u7269\u7684\u4e8c\u7ef4\u73af\u5883\u4e2d\u7684\u6700\u77ed\u8def\u5f84[66][67]\u3002\u66f4\u51c6\u786e\u5730\u8bf4\uff0c\u53ef\u4ee5\u5728 Oleft(n^2right) \u65f6\u95f4\u5185\u8ba1\u7b97\u5f97\u5230\uff0c\u5176\u4e2d n \u662f\u591a\u8fb9\u5f62\u969c\u788d\u7269\u7684\u9876\u70b9\u6570[68]\u3002<\/p>\n

\u6c42\u89e3\u7684\u65b9\u6cd5\u53ef\u4ee5\u662f\u6784\u9020\u548c\u641c\u7d22\u6240\u8c13\u7684\u201c\u53ef\u89c1\u56fe\uff08visibility graph\uff09\u201d[69]\u3002\u76f8\u5bf9\u5730\uff0cLazard\u3001Reif\u548cWang\u53d1\u73b0\uff0c\u5728\u591a\u8fb9\u5f62\u969c\u788d\u7269\u4e2d\u627e\u5230\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u66f2\u7387\u6709\u754c\u7684\u6700\u77ed\u8def\u5f84\uff08\u5373\u7c7b\u8f66\u673a\u5668\u4eba\u7684\u8def\u5f84\uff09\u7684\u95ee\u9898\u662fNP\u96be\u9898[70]\uff0c\u8fd9\u8868\u660e\u6ca1\u6709\u5df2\u77e5\u7684\u591a\u9879\u5f0f\u65f6\u95f4\u7b97\u6cd5\u53ef\u4ee5\u5728\u591a\u8fb9\u5f62\u969c\u788d\u4e2d\u627e\u5230\u7c7b\u8f66\u673a\u5668\u4eba\u7684\u6700\u77ed\u8def\u5f84\u3002\u76f8\u5173\u7684\u7814\u7a76\u6210\u679c\u662f\uff0c\u53ef\u4ee5\u5728\u6307\u6570\u65f6\u95f4\uff08EXPTIME\uff09\u5185\u786e\u5b9a\u4e00\u6761\u5728\u591a\u8fb9\u5f62\u73af\u5883\u4e2d\u7684\u5b58\u5728\u66f2\u7387\u7ea6\u675f\u7684\u8def\u5f84[71]\u3002\u4e00\u79cd\u80fd\u591f\u6709\u6548\u8ba1\u7b97\u89e3\u7684\u7279\u6b8a\u60c5\u51b5\u662f\u65e0\u969c\u788d\u73af\u5883\u4e2d\u7684\u66f2\u7387\u6709\u754c\u7684\u6700\u77ed\u8def\u5f84\u3002<\/p>\n

Dubins\u5df2\u7ecf\u8bc1\u660e[57]\uff0c\u7ed9\u5b9a\u4e24\u4e2a\u70b9 p_1, p_2 \uff0c\u5e76\u4e14\u7ed9\u5b9a\u4e24\u4e2a\u70b9\u5904\u7684\u5207\u7ebf theta_1, theta_2 \uff0c\u5219\u4e24\u70b9\u4e4b\u95f4\u7684\u3001\u9650\u5236\u6700\u5927\u66f2\u7387 k \u7684\u6700\u77ed\u8def\u5f84\u662f\u4e00\u6761\u7531\u6700\u591a\u4e09\u4e2a\u6bb5\u7ec4\u6210\u7684\u66f2\u7ebf\uff0c\u6bcf\u4e2a\u6bb5\u662f\u5706\u5f27\u6bb5\u6216\u76f4\u7ebf\u3002Reeds\u548cShepp\u5c06\u8be5\u65b9\u6cd5\u6269\u5c55\u5230\u80fd\u591f\u524d\u540e\u79fb\u52a8\u7684\u6c7d\u8f66\u4e0a[58]\u3002<\/p>\n

Agraval\u7b49\u4eba[72]\u63d0\u51fa\u7684\u53e6\u4e00\u79cd\u503c\u5f97\u6ce8\u610f\u7684\u60c5\u51b5\u662f Oleft(n^2 log nright) \u7b97\u6cd5\uff0c\u5176\u7528\u4e8e\u5728\u51f8\u591a\u8fb9\u5f62\u5185\u627e\u5230\u4e00\u6761\u66f2\u7387\u6709\u754c\u7684\u6700\u77ed\u8def\u5f84\u3002<\/p>\n

\u7c7b\u4f3c\u5730\uff0cBoissonnat\u548cLazard\u63d0\u51fa\u4e86\u4e00\u79cd\u591a\u9879\u5f0f\u65f6\u95f4\u7b97\u6cd5[73]\uff0c\u7528\u4e8e\u5728\u4e00\u79cd\u7279\u5b9a\u7684\u73af\u5883\u4e2d\u627e\u5230\u4e00\u6761\u7cbe\u786e\u7684\u66f2\u7387\u6709\u754c\u7684\u8def\u5f84\uff0c\u8fd9\u79cd\u73af\u5883\u4e2d\u7684\u969c\u788d\u7269\u7684\u8fb9\u754c\u90fd\u5177\u6709\u6709\u754c\u66f2\u7387\u3002\u7531\u4e8e\u81ea\u52a8\u9a7e\u9a76\u4e2d\u53d7\u5230\u5173\u6ce8\u7684\u5927\u591a\u6570\u95ee\u9898\u90fd\u4e0d\u5b58\u5728\u5177\u6709\u5b9e\u7528\u7684\u8ba1\u7b97\u590d\u6742\u5ea6\u7684\u7cbe\u786e\u7b97\u6cd5[70]\uff0c\u56e0\u6b64\u5fc5\u987b\u8f6c\u5411\u66f4\u4e00\u822c\u7684\u6570\u503c\u6c42\u89e3\u65b9\u6cd5\u3002\u8fd9\u4e9b\u65b9\u6cd5\u901a\u5e38\u4e0d\u4f1a\u627e\u5230\u4e00\u4e2a\u7cbe\u786e\u7684\u89e3\uff0c\u800c\u662f\u8bd5\u56fe\u627e\u5230\u4e00\u4e2a\u6ee1\u610f\u7684\u89e3\u6216\u4e00\u7cfb\u5217\u6536\u655b\u5230\u6700\u4f18\u89e3\u7684\u53ef\u884c\u89e3\u3002<\/p>\n

\u8fd9\u4e9b\u65b9\u6cd5\u7684\u6548\u7528\u548c\u6027\u80fd\u901a\u5e38\u901a\u8fc7\u5b83\u4eec\u9002\u7528\u7684\u95ee\u9898\u7c7b\u522b\u4ee5\u53ca\u5b83\u4eec\u662f\u5426\u80fd\u4fdd\u8bc1\u6536\u655b\u5230\u6700\u4f18\u89e3\u6765\u91cf\u5316\u3002\u8def\u5f84\u89c4\u5212\u7684\u6570\u503c\u65b9\u6cd5\u53ef\u4ee5\u5927\u81f4\u5206\u4e3a\u4e09\u5927\u7c7b\uff1a<\/p>\n

\uff081\uff09\u53d8\u5206\u6cd5\uff08Variational methods\uff09\u3002\u5176\u5c06\u8def\u5f84\u8868\u793a\u4e3a\u4e00\u4e2a\u7531\u6709\u9650\u7ef4\u5411\u91cf\u53c2\u6570\u5316\u7684\u51fd\u6570\uff0c\u5e76\u901a\u8fc7\u975e\u7ebf\u6027\u8fde\u7eed\u4f18\u5316\u6280\u672f\u5bf9\u5411\u91cf\u53c2\u6570\u8fdb\u884c\u4f18\u5316\uff0c\u4ee5\u5bfb\u627e\u6700\u4f18\u8def\u5f84\u3002\u7531\u4e8e\u80fd\u5feb\u901f\u6536\u655b\u5230\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\u56e0\u6b64\u5907\u53d7\u5173\u6ce8\uff1b\u7136\u800c\uff0c\u9664\u975e\u63d0\u4f9b\u4e00\u4e2a\u9002\u5f53\u7684\u521d\u59cb\u731c\u6d4b\uff0c\u5426\u5219\u53d8\u5206\u6cd5\u901a\u5e38\u96be\u4ee5\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\u3002\u6709\u5173\u53d8\u5206\u6cd5\u7684\u8be6\u7ec6\u8ba8\u8bba\uff0c\u8bf7\u89c1\u7b2c4-C\u8282\u3002<\/p>\n

\uff082\uff09\u56fe\u641c\u7d22\u65b9\u6cd5\uff08Graph-search methods\uff09\u3002\u5176\u5c06\u8f66\u8f86\u7684\u6784\u578b\u7a7a\u95f4\u79bb\u6563\u5316\u4e3a\u4e00\u4e2a\u56fe\u3002\u56fe\u7684\u9876\u70b9\u8868\u793a\u8f66\u8f86\u6784\u578b\u7684\u6709\u9650\u96c6\u5408\uff0c\u56fe\u7684\u8fb9\u8868\u793a\u9876\u70b9\u4e4b\u95f4\u7684\u8f6c\u79fb\u3002\u8be5\u65b9\u6cd5\u901a\u8fc7\u5728\u8fd9\u6837\u7684\u56fe\u4e2d\u641c\u7d22\u4e00\u6761\u6700\u5c0f\u6210\u672c\u8def\u5f84\uff0c\u6765\u627e\u5230\u671f\u671b\u7684\u8def\u5f84\u3002\u56fe\u641c\u7d22\u65b9\u6cd5\u4e0d\u5bb9\u6613\u9677\u5165\u5c40\u90e8\u6781\u5c0f\u503c\uff0c\u4f46\u662f\u5b83\u4eec\u4ec5\u9650\u4e8e\u5728\u6709\u9650\u7684\u8def\u5f84\u96c6\u4e0a\u8fdb\u884c\u4f18\u5316\uff0c\u8fd9\u79cd\u8def\u5f84\u96c6\u5305\u542b\u7684\u8def\u5f84\u90fd\u53ef\u4ee5\u7531\u56fe\u4e2d\u7684\u8fd0\u52a8\u57fa\u5143\uff08motion primitives\uff09\u6784\u5efa\u5f97\u5230\u3002\u6709\u5173\u56fe\u641c\u7d22\u65b9\u6cd5\u7684\u8be6\u7ec6\u8ba8\u8bba\uff0c\u8bf7\u89c1\u7b2c4-D\u8282\u3002<\/p>\n

\uff083\uff09\u589e\u91cf\u641c\u7d22\u65b9\u6cd5\uff08Incremental search methods\uff09\u3002\u5176\u5bf9\u6784\u578b\u7a7a\u95f4\u8fdb\u884c\u91c7\u6837\uff0c\u5e76\u9010\u6b65\u6784\u9020\u4e00\u4e2a\u53ef\u8fbe\u56fe\uff08\u901a\u5e38\u662f\u6811\uff09\u3002\u53ef\u8fbe\u56fe\uff08reachability graph\uff09\u5305\u542b\u53ef\u8fbe\u6784\u578b\u7684\u4e00\u4e2a\u79bb\u6563\u96c6\u5408\u4ee5\u53ca\u53ef\u8fbe\u6784\u578b\u4e4b\u95f4\u7684\u53ef\u884c\u8f6c\u79fb\u3002\u4e00\u65e6\u56fe\u5df2\u7ecf\u6784\u9020\u5f97\u8db3\u591f\u5927\uff0c\u4f7f\u5f97\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a\u8282\u70b9\u4f4d\u4e8e\u76ee\u6807\u533a\u57df\u4e2d\uff0c\u5219\u7b97\u6cd5\u901a\u8fc7\u8ddf\u8e2a\u4ece\u8d77\u59cb\u6784\u578b\u51fa\u53d1\u7684\u3001\u6700\u7ec8\u6307\u5411\u8be5\u8282\u70b9\u7684\u8fb9\uff0c\u6765\u83b7\u5f97\u671f\u671b\u8def\u5f84\u3002\u76f8\u6bd4\u66f4\u57fa\u672c\u7684\u56fe\u641c\u7d22\u65b9\u6cd5\uff0c\u57fa\u4e8e\u91c7\u6837\u7684\u65b9\u6cd5\u9010\u6e10\u589e\u52a0\u56fe\u7684\u5927\u5c0f\uff0c\u76f4\u5230\u5728\u56fe\u4e2d\u627e\u5230\u6ee1\u610f\u7684\u89e3\u3002\u6709\u5173\u589e\u91cf\u641c\u7d22\u65b9\u6cd5\u7684\u8be6\u7ec6\u8ba8\u8bba\uff0c\u8bf7\u89c1\u7b2c4-E\u8282\u3002<\/p>\n

\u663e\u7136\uff0c\u901a\u8fc7\u5c06\u8fd9\u4e9b\u65b9\u6cd5\u7ed3\u5408\u8d77\u6765\uff0c\u6211\u4eec\u6709\u671b\u5145\u5206\u5229\u7528\u6bcf\u79cd\u65b9\u6cd5\u7684\u4f18\u70b9\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u7c97\u7565\u7684\u56fe\u641c\u7d22\u6765\u83b7\u5f97[74]\u548c[75]\u4e2d\u62a5\u544a\u7684\u53d8\u5206\u6cd5\u7684\u521d\u59cb\u731c\u6d4b\u3002\u88681\u4e2d\u7ed9\u51fa\u4e86\u8def\u5f84\u89c4\u5212\u65b9\u6cd5\u7684\u5173\u952e\u6027\u8d28\u7684\u6bd4\u8f83\u3002\u672c\u8282\u7684\u5269\u4f59\u90e8\u5206\u5c06\u8be6\u7ec6\u8ba8\u8bba\u8def\u5f84\u89c4\u5212\u7b97\u6cd5\u4ee5\u53ca\u5b83\u4eec\u7684\u6027\u8d28\u3002<\/p>\n

B. \u8f68\u8ff9\u89c4\u5212\u52a8\u6001\u73af\u5883\u7684\u6216\u5177\u6709\u52a8\u529b\u5b66\u7ea6\u675f\u7684\u8fd0\u52a8\u89c4\u5212\u95ee\u9898\u53ef\u80fd\u66f4\u9002\u5408\u8868\u8ff0\u4e8e\u8f68\u8ff9\u89c4\u5212\u7684\u6846\u67b6\u4e2d\uff0c\u5176\u4e2d\u95ee\u9898\u7684\u89e3\u662f\u4e00\u6761\u8f68\u8ff9\uff0c\u5373\u65f6\u95f4\u53c2\u6570\u5316\u51fd\u6570 pi(t):[0, T] rightarrow mathcal{X} \uff0c\u5176\u89c4\u5b9a\u4e86\u8f66\u8f86\u6784\u578b\u968f\u65f6\u95f4\u7684\u53d8\u5316\u3002<\/p>\n

\u8bb0 Pi(x, T) \u4e3a\u6240\u6709\u8fde\u7eed\u51fd\u6570 [0, T] rightarrow mathcal{X} \u7684\u96c6\u5408\uff0c\u540c\u65f6\u8bb0 x_{text {init }} in mathcal{X} \u4e3a\u8f66\u8f86\u7684\u521d\u59cb\u6784\u578b\u3002\u76ee\u6807\u533a\u57df\u662f X_{text {goal }} subseteq mathcal{X} \u3002<\/p>\n

\u5728\u65f6\u95f4 t in[0, T] \u5185\u7684\u6240\u6709\u5141\u8bb8\u6784\u578b\u7684\u96c6\u5408\u8bb0\u4e3a x_{text {free }}(t) \uff0c\u5e76\u88ab\u7528\u4e8e\u7f16\u7801\u5b8c\u6574\u7ea6\u675f\uff0c\u4f8b\u5982\u8def\u5f84\u4e0a\u7684\u907f\u514d\u4e0e\u9759\u6001\u4ee5\u53ca\u53ef\u80fd\u7684\u52a8\u6001\u969c\u788d\u7269\u53d1\u751f\u78b0\u649e\u7684\u8981\u6c42\u3002\u8f68\u8ff9\u4e0a\u7684\u5fae\u5206\u7ea6\u675f\u8868\u793a\u4e3a\u8c13\u8bcd Dleft(mathrm{x}, mathrm{x}^{prime}, mathrm{x}^{prime prime}, ldotsright) \uff0c\u5e76\u53ef\u7528\u4e8e\u5bf9\u8f68\u8ff9\u65bd\u52a0\u52a8\u529b\u5b66\u7ea6\u675f\u3002\u6b64\u5916\uff0c\u518d\u8bbe J(pi): Pi(mathcal{X}, T) rightarrow mathbb{R} \u4e3a\u6210\u672c\u6cdb\u51fd\u3002<\/p>\n

\u5728\u8fd9\u4e9b\u5047\u8bbe\u4e0b\uff0c\u6700\u4f18\u8f68\u8ff9\u89c4\u5212\u95ee\u9898\u53ef\u4ee5\u975e\u5e38\u4e00\u822c\u5730\u8868\u8ff0\u4e3a\uff1a<\/p>\n

\u95ee\u9898 4.2 \u6700\u4f18\u8f68\u8ff9\u89c4\u5212\u7ed9\u5b9a\u4e00\u4e2a\u516d\u5143\u7ec4 left(X_{text {free }}, mathrm{x}_{text {init }}, X_{text {goal }}, D, J, Tright) \uff0c\u627e\u5230 pi^*=arg min _{pi in Pi(X, T)} J(pi)<\/p>\n

\u7ea6\u675f\u6761\u4ef6\uff1abegin{aligned} &pi(0)=mathrm{x}_{text {init }}, quad pi(T) in X_{text {goal }} \\ &pi(t) in X_{text {free }}, quad forall t in[0, T] \\ &Dleft(pi(t), pi^{prime}(t), pi^{prime prime}(t), ldotsright), quad forall t in[0, T] end{aligned}<\/p>\n

\u590d\u6742\u5ea6\uff1a\u7531\u4e8e\u52a8\u6001\u73af\u5883\u7684\u8f68\u8ff9\u89c4\u5212\u662f\u9759\u6001\u73af\u5883\u7684\u8def\u5f84\u89c4\u5212\u7684\u63a8\u5e7f\uff0c\u56e0\u6b64\u95ee\u9898\u4ecd\u7136\u662fPSPACE\u56f0\u96be\u7684\u3002\u6b64\u5916\uff0c\u52a8\u6001\u73af\u5883\u7684\u8f68\u8ff9\u89c4\u5212\u5df2\u88ab\u8bc1\u660e\u6bd4\u8def\u5f84\u89c4\u5212\u66f4\u96be\uff0c\u56e0\u4e3a\u5f53\u5728\u52a8\u6001\u73af\u5883\u4e2d\u8003\u8651\u4e00\u4e2a\u7c7b\u6bd4\u95ee\u9898\u65f6\uff0c\u5728\u9759\u6001\u73af\u5883\u4e2d\u6613\u4e8e\u5904\u7406\u7684\u95ee\u9898\u7684\u67d0\u4e9b\u53d8\u91cf\u4f1a\u53d8\u5f97\u96be\u4ee5\u5904\u7406\u3002<\/p>\n

\u5178\u578b\u5730\uff0c\u56de\u60f3\u4e00\u4e0b\uff0c\u5728\u9759\u6001\u7684\u4e8c\u7ef4\u591a\u8fb9\u5f62\u73af\u5883\u4e2d\u7684\u70b9\u673a\u5668\u4eba\u7684\u6700\u77ed\u8def\u5f84\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u88ab\u6709\u6548\u5730\u627e\u5230\uff1b\u4e0e\u6b64\u76f8\u5bf9\uff0cCanny\u548cReif\u7684\u7ed3\u679c[76]\u8868\u660e\uff0c\u5728\u79fb\u52a8\u7684\u591a\u8fb9\u5f62\u969c\u788d\u7269[1]\u4e2d\u627e\u5230\u5b8c\u6574\u70b9\u673a\u5668\u4eba\u7684\u4e00\u6761\u901f\u5ea6\u6709\u754c\u7684\u65e0\u78b0\u649e\u8f68\u8ff9\u662fNP\u56f0\u96be\u7684\u3002\u7c7b\u4f3c\u5730\uff0c\u5c3d\u7ba1\u5728\u4e09\u7ef4\u591a\u9762\u4f53\u73af\u5883\u4e2d\u5bf9\u5177\u6709\u56fa\u5b9a\u81ea\u7531\u5ea6\u7684\u673a\u5668\u4eba\u8fdb\u884c\u8def\u5f84\u89c4\u5212\u662f\u5bb9\u6613\u7684\uff0c\u4f46\u662fReif\u548cSharir\u6307\u51fa\uff0c\u5728\u5e73\u79fb\u548c\u65cb\u8f6c\u4e09\u7ef4\u591a\u9762\u4f53\u969c\u788d\u7269\u4e2d\u5bf9\u5177\u67092\u4e2a\u81ea\u7531\u5ea6\u7684\u673a\u5668\u4eba\u8fdb\u884c\u8f68\u8ff9\u89c4\u5212\u662fPSPACE\u56f0\u96be\u7684[85]\u3002\u5bf9\u4e8e\u81ea\u52a8\u9a7e\u9a76\u4e2d\u51fa\u73b0\u7684\u5173\u952e\u8f68\u8ff9\u89c4\u5212\u95ee\u9898\uff0c\u4e0d\u5b58\u5728\u6613\u4e8e\u5904\u7406\u7684\u7cbe\u786e\u7b97\u6cd5\uff0c\u8fd9\u4f7f\u5f97\u6570\u503c\u65b9\u6cd5\u6210\u4e3a\u5904\u4e8e\u8be5\u4efb\u52a1\u7684\u6d41\u884c\u9009\u62e9\u3002\u8f68\u8ff9\u89c4\u5212\u95ee\u9898\u7684\u6c42\u89e3\u53ef\u4ee5\u662f\u76f4\u63a5\u5728\u65f6\u57df\u4e2d\u4f7f\u7528\u4e00\u4e9b\u53d8\u5206\u65b9\u6cd5\uff0c\u6216\u8005\u662f\u5c06\u8f68\u8ff9\u89c4\u5212\u95ee\u9898\u8f6c\u6362\u4e3a\u9644\u52a0\u4e86\u65f6\u95f4\u7ef4\u5ea6\u7684\u6784\u578b\u7a7a\u95f4\u4e2d\u7684\u8def\u5f84\u89c4\u5212\u95ee\u9898[86]\u3002<\/p>\n

\u8f68\u8ff9\u89c4\u5212\u95ee\u9898 left(X_{text {free }}^T, mathrm{x}_{text {init }}^T, X_{text {goal }}^T, D^T, J^T, Tright) \u5230\u8def\u5f84\u89c4\u5212\u95ee\u9898 left(X_{text {free }}^P, x_{text {init }}^P, X_{text {goal }}^P, D^P, J^Pright) \u7684\u8f6c\u5316\u901a\u5e38\u5b9e\u73b0\u5982\u4e0b\u3002\u5b9a\u4e49 X^P:=X^T times[0, T] \u4e3a\u8fdb\u884c\u8def\u5f84\u89c4\u5212\u7684\u6784\u578b\u7a7a\u95f4\u3002<\/p>\n

\u5bf9\u4e8e\u4efb\u610f y in mathcal{X}^P \uff0c\u8bb0 t(mathrm{y}) in[0, T] \u4e3a\u65f6\u95f4\u5206\u91cf\uff0c\u8bb0 c(mathrm{y}) in mathcal{X}^T \u4e3a\u70b9 y \u7684\u201c\u6784\u578b\/\u914d\u7f6e\u201d\u5206\u91cf\u3002<\/p>\n

\u5bf9\u4e8e\u4e00\u6761\u8def\u5f84 sigma(alpha):[0,1] rightarrow mathcal{X}^P \u4ee5\u53ca\u4e00\u6761\u8f68\u8ff9 pi(t):[0, T] rightarrow mathcal{X}^T \uff0c\u5982\u679c\u8be5\u8def\u5f84\u5728\u8d77\u70b9\u548c\u7ec8\u70b9\u5904\u7684\u65f6\u95f4\u5206\u91cf\u6ee1\u8db3begin{aligned} & t(sigma(0))=0 \\ & t(sigma(1))=T end{aligned}<\/p>\n

\u5e76\u4e14\u8fd9\u6761\u8def\u5f84\u662f\u5355\u8c03\u9012\u589e\u7684\uff0c\u5373\u53ef\u4ee5\u8868\u793a\u4e3a\u5fae\u5206\u7ea6\u675ftleft(sigma^{prime}(alpha)right)>0, quad forall alpha in[0,1]<\/p>\n

\u90a3\u4e48\u8fd9\u6761\u8def\u5f84 sigma(alpha) \u5c31\u80fd\u88ab\u8f6c\u5316\u4e3a\u8fd9\u6761\u8f68\u8ff9 pi(t) \u3002<\/p>\n

\u518d\u8fdb\u4e00\u6b65\u5730\u8bf4\uff0c\u81ea\u7531\u6784\u578b\u7a7a\u95f4\u3001\u521d\u59cb\u6784\u578b\u3001\u76ee\u6807\u533a\u57df\u4ee5\u53ca\u5fae\u5206\u7ea6\u675f\u4ee5\u5982\u4e0b\u7684\u5f62\u5f0f\u88ab\u6620\u5c04\u5230\u8def\u5f84\u89c4\u5212\u4e2d\uff1abegin{aligned} & x_{text {free }}^P=left{(x, t): x in X_{text {free }}^T(t) wedge t in[0, T]right} \\ & mathrm{x}_{text {init }}^P=left(mathrm{x}_{text {init }}^T, 0right) \\ & X_{text {goal }}^P=left{(x, T): x in X_{text {goal }}^Tright} \\ &D^Pleft(mathrm{y}, mathrm{y}^{prime}, mathrm{y}^{prime prime}, ldotsright)=D^Tleft(c(mathrm{y}), frac{cleft(mathrm{y}^{prime}right)}{tleft(mathrm{y}^{prime}right)}, frac{cleft(mathrm{y}^{prime prime}right)}{tleft(mathrm{y}^{prime prime}right)}, ldotsright) end{aligned}<\/p>\n

\u968f\u540e\uff0c\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u80fd\u591f\u5904\u7406\u5fae\u5206\u7ea6\u675f\u7684\u8def\u5f84\u89c4\u5212\u7b97\u6cd5\u6765\u5f97\u5230\u8be5\u8def\u5f84\u89c4\u5212\u95ee\u9898\u7684\u89e3\uff0c\u6700\u7ec8\u5c06\u5176\u8f6c\u6362\u56de\u8f68\u8ff9\u7684\u5f62\u5f0f\u3002<\/p>\n

\u88681\uff1a\u4e0d\u540c\u8def\u5f84\u89c4\u5212\u65b9\u6cd5\u7684\u6bd4\u8f83\u8bf4\u660e\uff1a a: \u6700\u77ed\u8def\u5f84\u95ee\u9898\uff1b b:n \u662f\u7528\u4e8e\u5b9a\u4e49\u969c\u788d\u7269\u7684\u70b9\u7684\u4e2a\u6570\uff1b c: \u5b8c\u5907\uff08complete\uff09\u7684\u6982\u5ff5\u662f\u4ec5\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u56fe\u6240\u4ea7\u751f\u7684\u8def\u5f84\u96c6\u800c\u8a00\u7684\uff1b d: \u6700\u4f18\u7684\u6982\u5ff5\u662f\u4ec5\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u56fe\u6240\u4ea7\u751f\u7684\u8def\u5f84\u96c6\u800c\u8a00\u7684\uff1b e: n \u548c m \u5206\u522b\u662f\u56fe\u4e2d\u8fb9\u548c\u9876\u70b9\u7684\u6570\u91cf\uff1b f: \u5047\u8bbe\u78b0\u649e\u68c0\u6d4b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(1) \uff1b g: \u5177\u6709\u56fa\u5b9a\u534a\u5f84\u7684\u8fde\u63a5\u7b56\u7565\u7684\u6279\u7248\u672c\uff1b h:n \u662f\u91c7\u6837\u70b9\u7684\u6570\u91cf\u6216\u7b97\u6cd5\u7684\u8fed\u4ee3\u6b21\u6570\uff1b i: \u5bf9\u4e8e\u7b97\u6cd5\u7684\u67d0\u4e9b\u53d8\u4f53\u800c\u8a00\uff1b j: \u672a\u88ab\u660e\u786e\u5206\u6790\uff1b k: \u03f5 \u662f\u8ddd\u79bb\u6700\u4f73\u6210\u672c\u7684\u6240\u9700\u8ddd\u79bb\uff1b l: \u5728\u989d\u5916\u5047\u8bbe\u7684\u60c5\u51b5\u4e0b\uff0c\u53ef\u80fd\u5177\u6709\u66f4\u5feb\u7684\u901f\u7387\uff1b *: \u5bf9\u4e8e\u4e0d\u5177\u6709\u5fae\u5206\u7ea6\u675f\u7684\u7cfb\u7edf\uff0c\u5df2\u88ab\u8bc1\u660e\u6210\u7acb\uff1b \u271d: \u5bf9\u4e8e\u67d0\u4e9b\u7c7b\u522b\u7684\u975e\u5b8c\u6574\u7cfb\u7edf\uff0c\u5df2\u88ab\u8bc1\u660e\u6210\u7acb\u3002<\/p>\n

C. \u53d8\u5206\u6cd5<\/p>\n

\u6211\u4eec\u5c06\u9996\u5148\u5728\u975e\u7ebf\u6027\u8fde\u7eed\u4f18\u5316\u7684\u6846\u67b6\u4e0b\u89e3\u51b3\u8f68\u8ff9\u89c4\u5212\u95ee\u9898\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8be5\u95ee\u9898\u901a\u5e38\u88ab\u79f0\u4e3a\u8f68\u8ff9\u4f18\u5316\u3002\u5728\u672c\u5c0f\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u91c7\u7528\u8f68\u8ff9\u89c4\u5212\u7684\u5f62\u5f0f\u2014\u2014\u6ce8\u610f\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u4f1a\u5f71\u54cd\u4e00\u822c\u6027\uff0c\u56e0\u4e3a\u8def\u5f84\u89c4\u5212\u53ef\u4ee5\u8868\u793a\u4e3a\u5355\u4f4d\u65f6\u95f4\u95f4\u9694\u5185\u7684\u8f68\u8ff9\u4f18\u5316\u3002\u4e3a\u4e86\u5229\u7528\u73b0\u6709\u7684\u975e\u7ebf\u6027\u4f18\u5316\u65b9\u6cd5\uff0c\u6709\u5fc5\u8981\u5c06\u8f68\u8ff9\u7684\u65e0\u9650\u7ef4\u51fd\u6570\u7a7a\u95f4\u6295\u5f71\u5230\u4e00\u4e2a\u6709\u9650\u7ef4\u5411\u91cf\u7a7a\u95f4\u3002<\/p>\n

\u6b64\u5916\uff0c\u5927\u591a\u6570\u975e\u7ebf\u6027\u89c4\u5212\u6280\u672f\u8981\u6c42\u5c06\u95ee\u98984.2\u5f62\u5f0f\u7684\u8f68\u8ff9\u4f18\u5316\u95ee\u9898\u8f6c\u5316\u4e3a\u5982\u4e0b\u5f62\u5f0farg min _{pi in Pi(x, T)} J(pi)<\/p>\n

\u7ea6\u675f\u6761\u4ef6\uff1abegin{aligned} &pi(0)=mathrm{x}_{text {init }}, pi(T) in X_{g o a l} \\ &fleft(pi(t), pi^{prime}(t), ldotsright)=0, forall t in[0, T] \\ &gleft(pi(t), pi^{prime}(t), ldotsright) leq 0, forall t in[0, T] end{aligned}<\/p>\n

\u5176\u4e2d\u5b8c\u6574\u7ea6\u675f\u548c\u5fae\u5206\u7ea6\u675f\u88ab\u8868\u793a\u4e3a\u7cfb\u7edf\u7684\u7b49\u5f0f\u7ea6\u675f\u548c\u4e0d\u7b49\u5f0f\u7ea6\u675f\u3002\u4e00\u4e9b\u5e94\u7528\u573a\u666f\u901a\u8fc7\u4f7f\u7528\u60e9\u7f5a\u51fd\u6570\u6216\u969c\u788d\u51fd\u6570\uff0c\u5c06\u5e26\u7ea6\u675f\u7684\u4f18\u5316\u95ee\u9898\u653e\u677e\u4e3a\u65e0\u7ea6\u675f\u7684\u4f18\u5316\u95ee\u9898\u3002\u65e0\u8bba\u662f\u4f7f\u7528\u60e9\u7f5a\u51fd\u6570\u8fd8\u662f\u969c\u788d\u51fd\u6570\uff0c\u90fd\u662f\u5229\u7528\u4e00\u4e2a\u589e\u5f3a\u7684\u6210\u672c\u6cdb\u51fd\u4ee3\u66ff\u7ea6\u675f\u3002<\/p>\n

\u4f7f\u7528\u60e9\u7f5a\u65b9\u6cd5\u65f6\uff0c\u6210\u672c\u6cdb\u51fd\u7684\u5f62\u5f0f\u5982\u4e0btilde{J}(pi)=J(pi)+frac{1}{epsilon} int_0^Tleft[left|fleft(pi, pi^{prime}, ldotsright)right|^2+left|max left(0, gleft(pi, pi^{prime}, ldotsright)right)right|^2right] d t<\/p>\n

\u7c7b\u4f3c\u5730\uff0c\u969c\u788d\u51fd\u6570\u53ef\u4ee5\u7528\u4e8e\u66ff\u4ee3\u4e0d\u7b49\u5f0f\u7ea6\u675f\u3002\u6b64\u65f6\u7684\u589e\u5f3a\u6210\u672c\u6cdb\u51fd\u7684\u5f62\u5f0f\u4e3atilde{J}(pi)=J(pi)+epsilon int_0^T h(pi(t)) d t<\/p>\n

\u5176\u4e2d\u969c\u788d\u51fd\u6570\u6ee1\u8db3 g(pi)<0 Longrightarrow h(pi)<\/p>\n

\u56fe4.1\uff1a\u624b\u5de5\u7ed8\u5236\u7684\u56fe\uff0c\u8868\u793a\u6b63\u5e38\u60c5\u51b5\u4e0b\u7684\u671f\u671b\u884c\u9a76\u8def\u5f84\u3002\u5c3d\u7ba1\u5927\u591a\u6570\u65f6\u5019\u8ba9\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u6cbf\u7740\u9053\u8def\u8f66\u9053\u56fe\u4e2d\u7f16\u7801\u7684\u8def\u5f84\u884c\u9a76\u5c31\u8db3\u591f\u4e86\uff0c\u4f46\u662f\u6709\u65f6\u5b83\u5fc5\u987b\u80fd\u5728\u8bbe\u8ba1\u9053\u8def\u7f51\u7edc\u56fe\u65f6\u672a\u8003\u8651\u7684\u969c\u788d\u7269\u5468\u56f4\u5bfc\u822a\uff0c\u6216\u8005\u80fd\u5728\u56fe\u4e2d\u672a\u6db5\u76d6\u7684\u73af\u5883\u4e2d\u5bfc\u822a\u3002\u4f8b\u5982\uff0c\u8003\u8651\u4e00\u8f86\u6545\u969c\u8f66\u8f86\u963b\u585e\u4e86\u6211\u4eec\u7684\u8f66\u8f86\u8ba1\u5212\u7a7f\u8fc7\u7684\u8f66\u9053\u2014\u2014\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5fc5\u987b\u4f7f\u7528\u66f4\u4e00\u822c\u7684\u8fd0\u52a8\u89c4\u5212\u65b9\u6cd5\uff0c\u4ee5\u627e\u5230\u88ab\u68c0\u6d4b\u5230\u7684\u969c\u788d\u7269\u5468\u56f4\u7684\u4e00\u6761\u65e0\u78b0\u649e\u8def\u5f84\u3002\u6839\u636e\u73af\u5883\u4e2d\u969c\u788d\u7269\u7684\u8868\u793a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5c06\u4e00\u822c\u7684\u8def\u5f84\u89c4\u5212\u65b9\u6cd5\u5927\u81f4\u5206\u4e3a\u4e24\u7c7b\u3002\u6240\u8c13\u7684\u51e0\u4f55\u6216\u7ec4\u5408\u65b9\u6cd5\u9002\u7528\u4e8e\u969c\u788d\u7269\u7684\u51e0\u4f55\u8868\u793a\u2014\u2014\u5728\u5b9e\u8df5\u4e2d\uff0c\u969c\u788d\u7269\u6700\u5e38\u7528\u591a\u8fb9\u5f62\u6216\u591a\u9762\u4f53\u6765\u63cf\u8ff0\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u6240\u8c13\u7684\u57fa\u4e8e\u91c7\u6837\u7684\u65b9\u6cd5\u62bd\u8c61\u4e86\u969c\u788d\u7269\u7684\u5185\u90e8\u8868\u793a\u65b9\u5f0f\uff0c\u5e76\u4e14\u4ec5\u5047\u8bbe\u5bf9\u4e8e\u4e00\u4e2a\u51fd\u6570\u7684\u8bbf\u95ee\uff0c\u8be5\u51fd\u6570\u7528\u4e8e\u786e\u5b9a\u4efb\u4f55\u7ed9\u5b9a\u7684\u8def\u5f84\u6bb5\u662f\u5426\u4e0e\u4efb\u4f55\u969c\u788d\u7269\u53d1\u751f\u4e86\u78b0\u649e\u3002<\/p>\n

2) \u51e0\u4f55\u65b9\u6cd5\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u91cd\u70b9\u4ecb\u7ecd\u4f7f\u7528\u969c\u788d\u7269\u51e0\u4f55\u8868\u793a\u7684\u8def\u5f84\u89c4\u5212\u65b9\u6cd5\u3002\u6211\u4eec\u5c06\u9996\u5148\u5173\u6ce8\u65e0\u5fae\u5206\u7ea6\u675f\u7684\u8def\u5f84\u89c4\u5212\uff0c\u56e0\u4e3a\u5bf9\u4e8e\u6b64\u60c5\u51b5\uff0c\u5b58\u5728\u6709\u6548\u7684\u7cbe\u786e\u8def\u5f84\u89c4\u5212\u7b97\u6cd5\u3002\u5c3d\u7ba1\u7531\u4e8e\u65e0\u6cd5\u8003\u8651\u5bf9\u6700\u5c0f\u8f6c\u5f2f\u534a\u5f84\u7684\u7ea6\u675f\uff0c\u56e0\u6b64\u65e0\u6cd5\u5f3a\u5236\u5730\u65bd\u52a0\u5fae\u5206\u7ea6\u675f\uff0c\u8fdb\u800c\u5bf9\u4f20\u7edf\u8f6c\u5411\u8f66\u8f86\u7684\u8def\u5f84\u89c4\u5212\u4ea7\u751f\u4e86\u9650\u5236\uff0c\u4f46\u662f\u8fd9\u4e9b\u65b9\u6cd5\u53ef\u7528\u4e8e\u6c42\u89e3\u5177\u6709\u66f2\u7387\u7ea6\u675f\u7684\u8def\u5f84\u7684\u957f\u5ea6\u4e0b\u9650\u548c\u4e0a\u9650[1]\uff0c\u5e76\u4e14\u8fd8\u53ef\u7528\u4e8e\u80fd\u591f\u539f\u5730\u8f6c\u5f2f\u7684\u3001\u66f4\u7279\u6b8a\u7684\u8f66\u8f86\u7ed3\u6784\u7684\u8def\u5f84\u89c4\u5212\u3002<\/p>\n

\u5728\u8def\u5f84\u89c4\u5212\u4e2d\uff0c\u672f\u8bed\u201c\u8def\u7ebf\u56fe\uff08roadmap\uff09\u201d\u7528\u4e8e\u63cf\u8ff0 mathcal{X}_{text {free }}\u7684\u56fe\u79bb\u6563\u5316\u3002\u8fd9\u79cd\u79bb\u6563\u5316\u80fd\u591f\u5f88\u597d\u5730\u63cf\u8ff0\u81ea\u7531\u6784\u578b\u7a7a\u95f4\u7684\u8fde\u901a\u6027\uff0c\u5e76\u4e14 mathcal{X}_{text {free }}\u4e2d\u7684\u4efb\u610f\u70b9\u90fd\u53ef\u4ee5\u4ece\u8def\u7ebf\u56fe\u7684\u67d0\u4e9b\u9876\u70b9\u51fa\u53d1\u800c\u8f7b\u677e\u5230\u8fbe\u3002\u5f53\u96c6\u5408 mathcal{X}_{text {free }}\u53ef\u4ee5\u901a\u8fc7\u7ebf\u6027\u6216\u534a\u4ee3\u6570\u6a21\u578b\u88ab\u51e0\u4f55\u5f62\u5f0f\u5730\u63cf\u8ff0\u65f6\uff0c\u6211\u4eec\u80fd\u5728\u7b97\u6cd5\u4e0a\u6784\u9020\u7528\u4e8e mathcal{X}_{text {free }}\u7684\u4e0d\u540c\u7c7b\u578b\u7684\u8def\u7ebf\u56fe\uff0c\u8fdb\u800c\u5c06\u5176\u7528\u4e8e\u83b7\u5f97\u5b8c\u5907\u7684\u8def\u5f84\u89c4\u5212\u7b97\u6cd5\u3002<\/p>\n

\u6700\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u4e8e mathcal{X}_{text {free }} subset mathbb{R}^2 \u548c\u6784\u578b\u7a7a\u95f4\u7684\u591a\u8fb9\u5f62\u6a21\u578b\uff0c\u5b58\u5728\u51e0\u79cd\u7528\u4e8e\u6784\u9020\u8fd9\u79cd\u8def\u7ebf\u56fe\u7684\u6709\u6548\u7b97\u6cd5\uff0c\u5982\u5782\u76f4\u5355\u5143\u5206\u89e3\uff08vertical cell decomposition\uff09[93]\u3001\u5e7f\u4e49\u51af\u6d1b\u8bfa\u4f0a\u56fe\uff08generalized Voronoi diagrams\uff09[94][95]\u548c\u53ef\u89c1\u56fe[33][96]\u3002<\/p>\n

\u5bf9\u4e8e\u7531\u4e00\u822c\u534a\u4ee3\u6570\u6a21\u578b\u63cf\u8ff0\u7684\u9ad8\u7ef4\u6784\u578b\u7a7a\u95f4\uff0c\u4e00\u79cd\u79f0\u4e3a\u67f1\u5f62\u4ee3\u6570\u5206\u89e3\uff08cylindrical algebraic decomposition\uff09\u7684\u6280\u672f\u53ef\u7528\u4e8e\u5728\u6784\u578b\u7a7a\u95f4\u4e2d\u6784\u9020\u8def\u7ebf\u56fe[47][97]\uff0c\u4ece\u800c\u4e3a\u975e\u5e38\u4e00\u822c\u7684\u8def\u5f84\u89c4\u5212\u95ee\u9898\u63d0\u4f9b\u5b8c\u5907\u7684\u7b97\u6cd5\u3002\u8fd9\u7c7b\u7b97\u6cd5\u4e2d\u6700\u5feb\u7684\u662fCanny\u5f00\u53d1\u7684\u7b97\u6cd5[65]\uff0c\u5176\u5728\u6784\u578b\u7a7a\u95f4\u7684\u7ef4\u5ea6\u4e0a\u5177\u6709\uff08\u5355\u4e2a\uff09\u6307\u6570\u65f6\u95f4\u7684\u590d\u6742\u5ea6\u3002\u7136\u800c\uff0c\u5176\u7ed3\u8bba\u4e3b\u8981\u662f\u7406\u8bba\u6027\u8d28\u7684\uff0c\u8fc4\u4eca\u4e3a\u6b62\u6ca1\u6709\u4efb\u4f55\u5df2\u77e5\u7684\u5b9e\u73b0\u3002\u5bf9\u4e8e\u5177\u6709\u6700\u5927\u66f2\u7387\u7ea6\u675f\u7684\u8fd0\u52a8\u89c4\u5212\u95ee\u9898\uff0c\u76ee\u524d\u4e5f\u5b58\u5728\u4e00\u4e9b\u7814\u7a76\u6210\u679c\uff0c\u8fd9\u662f\u56e0\u4e3a\u8be5\u95ee\u9898\u4e0e\u7c7b\u8f66\u8f66\u8f86\u7684\u8fd0\u52a8\u89c4\u5212\u5b58\u5728\u5173\u8054\u3002<\/p>\n

Backer\u548cKirkpatrick\u63d0\u51fa\u4e86\u4e00\u79cd\u7b97\u6cd5[98]\uff0c\u7528\u4e8e\u6784\u9020\u4e00\u6761\u5177\u6709\u6709\u754c\u66f2\u7387\u7684\u591a\u9879\u5f0f\u8def\u5f84\uff0c\u4e14\u8be5\u591a\u9879\u5f0f\u662f\u76f8\u5bf9\u4e8e\u4ee5\u4e0b\u53d8\u91cf\u800c\u8a00\u7684\uff1a\u57df\u7684\u7279\u5f81\u7684\u6570\u91cf\u3001\u8f93\u5165\u7684\u7cbe\u5ea6\u3001\u8fde\u63a5\u7279\u5b9a\u6784\u578b\u7684\u6700\u7b80\u5355\u7684\u65e0\u969c\u788dDubins\u8def\u5f84\u4e0a\u7684\u8def\u5f84\u6bb5\u4e2a\u6570\u3002\u7531\u4e8e\u5728\u591a\u8fb9\u5f62\u969c\u788d\u7269\u4e2d\u627e\u5230\u4e00\u6761\u5177\u6709\u6709\u754c\u66f2\u7387\u7684\u6700\u77ed\u8def\u5f84\u7684\u95ee\u9898\u662fNP\u56f0\u96be\u7684\uff0c\u56e0\u6b64\u76ee\u524d\u5e76\u4e0d\u5b58\u5728\u7cbe\u786e\u7684\u591a\u9879\u5f0f\u89e3\u7b97\u6cd5\u3002<\/p>\n

Jacobs\u548cCanny\u9996\u5148\u63d0\u51fa\u4e86\u4e00\u79cd\u5728\u591a\u8fb9\u5f62\u969c\u788d\u7269\u4e2d\u5bfb\u627e\u66f2\u7387\u6709\u754c\u7684\u6700\u77ed\u8def\u5f84\u7684\u8fd1\u4f3c\u7b97\u6cd5[99]\uff0c\u540e\u6765Wang\u548cAgarwal\u8fdb\u884c\u4e86\u6539\u8fdb[100]\uff0c\u6539\u8fdb\u540e\u7684\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a Oleft(frac{n^2}{epsilon^4} log nright) \uff0c\u5176\u4e2d n \u662f\u969c\u788d\u7269\u7684\u9876\u70b9\u6570\uff0c epsilon \u662f\u8fd1\u4f3c\u56e0\u5b50\u3002\u6240\u8c13\u7684\u4e2d\u7b49\u969c\u788d\u7269\uff08moderate obstacles\uff09\u5177\u6709\u5149\u6ed1\u7684\u8fb9\u754c\uff0c\u4e14\u8fb9\u754c\u7684\u66f2\u7387\u5927\u5c0f\u4ee5 k \u4e3a\u754c\u3002\u5728\u8fd9\u79cd\u4e2d\u7b49\u969c\u788d\u7269\u7684\u7279\u6b8a\u60c5\u51b5\u4e0b\uff0cBoissonnat\u548cLazard\u5f00\u53d1\u4e86\u4e00\u79cd\u7cbe\u786e\u7684\u591a\u9879\u5f0f\u7b97\u6cd5\uff0c\u4ee5\u627e\u5230\u66f2\u7387\u6700\u5927\u4ee5 k \u4e3a\u754c\u7684\u4e00\u6761\u8def\u5f84[73]\u3002<\/p>\n

3) \u57fa\u4e8e\u91c7\u6837\u7684\u65b9\u6cd5 \u5728\u81ea\u52a8\u9a7e\u9a76\u4e2d\uff0c mathcal{X}_{text {free }}\u7684\u51e0\u4f55\u6a21\u578b\u901a\u5e38\u4e0d\u662f\u76f4\u63a5\u53ef\u7528\u7684\uff0c\u800c\u901a\u8fc7\u539f\u59cb\u4f20\u611f\u6570\u636e\u6784\u9020\u6a21\u578b\u7684\u6210\u672c\u53c8\u592a\u9ad8\u3002\u6b64\u5916\uff0c\u5bf9\u751f\u6210\u8def\u5f84\u7684\u8981\u6c42\u901a\u5e38\u6bd4\u7b80\u5355\u7684\u6700\u5927\u66f2\u7387\u7ea6\u675f\u66f4\u590d\u6742\u3002\u8fd9\u53ef\u4ee5\u89e3\u91ca\u57fa\u4e8e\u91c7\u6837\u7684\u65b9\u6cd5\u7684\u6d41\u884c\uff0c\u56e0\u4e3a\u540e\u8005\u5e76\u672a\u5f3a\u5236\u5bf9\u81ea\u7531\u6784\u578b\u96c6\u548c\u52a8\u529b\u5b66\u7ea6\u675f\u91c7\u7528\u7279\u5b9a\u7684\u8868\u793a\u65b9\u6cd5\u3002\u57fa\u4e8e\u91c7\u6837\u7684\u65b9\u6cd5\u4e0d\u662f\u901a\u8fc7\u51e0\u4f55\u8868\u793a\u8fdb\u884c\u63a8\u7406\uff0c\u800c\u662f\u4f7f\u7528\u8f6c\u5411\/\u5f15\u5bfc\uff08steering\uff09\u548c\u78b0\u649e\u68c0\u6d4b\uff08collision checking\uff09\u7684\u63a8\u7406\u65b9\u5f0f\u6765\u63a2\u7d22\u81ea\u7531\u6784\u578b\u7a7a\u95f4\u7684\u53ef\u8fbe\u6027\uff1a \u8f6c\u5411\/\u5f15\u5bfc\u51fd\u6570 text { steer }(mathrm{x}, mathrm{y}) \u8fd4\u56de\u4e00\u6761\u4ece\u6784\u578b x \u51fa\u53d1\u3001\u671d\u5411\u6784\u578b y\uff08\u4f46\u662f\u4e0d\u5fc5\u987b\u8981\u5230\u8fbe y\uff09\u7684\u8def\u5f84\u6bb5\uff0c\u5e76\u786e\u4fdd\u6ee1\u8db3\u5fae\u5206\u7ea6\u675f\uff0c\u5373\uff0c\u751f\u6210\u7684\u8fd0\u52a8\u5bf9\u4e8e\u6240\u8003\u8651\u7684\u8f66\u8f86\u6a21\u578b\u662f\u53ef\u884c\u7684\u3002\u8f6c\u5411\u51fd\u6570\u7684\u5177\u4f53\u5b9e\u73b0\u65b9\u5f0f\u53d6\u51b3\u4e8e\u5176\u6240\u5904\u7684\u73af\u5883\u3002<\/p>\n

\u6587\u732e\u4e2d\u62a5\u544a\u8fc7\u7684\u4e00\u4e9b\u5178\u578b\u9009\u62e9\u5305\u62ec\uff1a<\/p>\n

1) \u968f\u673a\u8f6c\u5411\uff08Random steering\uff09\uff1a\u8f6c\u5411\u51fd\u6570\u8fd4\u56de\u7684\u8def\u5f84\u662f\u5728\u72b6\u6001x\u65f6\uff0c\u5c06\u4e00\u4e2a\u968f\u673a\u63a7\u5236\u8f93\u5165\u5e94\u7528\u5230\u8f66\u8f86\u7684\u524d\u5411\u6a21\u578b\u3001\u5e76\u7ecf\u8fc7\u4e00\u4e2a\u56fa\u5b9a\u6216\u53ef\u53d8\u65f6\u95f4\u6b65\u957f\u800c\u5f97\u5230\u7684[101]\u3002<\/p>\n

2) \u542f\u53d1\u5f0f\u8f6c\u5411\uff08Heuristic steering\uff09\uff1a\u8f6c\u5411\u51fd\u6570\u8fd4\u56de\u7684\u8def\u5f84\u662f\u5e94\u7528\u4e00\u4e2a\u542f\u53d1\u5f0f\u6784\u9020\u7684\u63a7\u5236\u800c\u5f97\u5230\u7684\uff0c\u8fd9\u7c7b\u542f\u53d1\u5f0f\u7684\u6784\u9020\u65b9\u6cd5\u7528\u4e8e\u5f15\u5bfc\u7cfb\u7edf\u4ece x \u671d\u5411 y [102-104]\uff0c\u5305\u62ec\u4ece\u9884\u5148\u8bbe\u8ba1\u7684\u79bb\u6563\u7b56\u7565\u96c6\u5408\uff08\u5e93\uff09\u4e2d\u9009\u62e9\u4e00\u4e2a\u7b56\u7565\u3002<\/p>\n

3) \u7cbe\u786e\u8f6c\u5411\uff08Exact steering\uff09\uff1a\u8f6c\u5411\u51fd\u6570\u8fd4\u56de\u4e00\u6761\u7528\u4e8e\u5f15\u5bfc\u7cfb\u7edf\u4ece x \u5230 y \u7684\u53ef\u884c\u8def\u5f84\u3002\u8fd9\u6837\u4e00\u6761\u8def\u5f84\u5bf9\u5e94\u4e00\u4e2a\u4e24\u70b9\u8fb9\u503c\u95ee\u9898\u7684\u4e00\u4e2a\u89e3\u3002\u5bf9\u4e8e\u4e00\u4e9b\u7cfb\u7edf\u548c\u6210\u672c\u6cdb\u51fd\uff0c\u53ef\u4ee5\u89e3\u6790\u5730\u6c42\u51fa\u8fd9\u6837\u7684\u8def\u5f84\uff0c\u4f8b\u5982\u5b8c\u6574\u7cfb\u7edf\u7684\u4e00\u6761\u76f4\u7ebf\u3001\u524d\u5411\u79fb\u52a8\u7684\u5355\u8f6e\u8f66\u7684\u4e00\u6761Dubins\u66f2\u7ebf[57]\u3001\u6216\u53cc\u5411\u5355\u8f6e\u8f66[58]\u7684\u4e00\u6761Reeds-Shepp\u66f2\u7ebf\u3002\u5fae\u5206\u5e73\u5766\uff08differentially flat\uff09\u7cfb\u7edf\u540c\u6837\u5b58\u5728\u89e3\u6790\u89e3[61]\uff0c\u800c\u5bf9\u4e8e\u66f4\u590d\u6742\u7684\u6a21\u578b\uff0c\u53ef\u4ee5\u901a\u8fc7\u6c42\u89e3\u4e24\u70b9\u8fb9\u503c\u95ee\u9898\u6765\u83b7\u5f97\u7cbe\u786e\u7684\u8f6c\u5411\u89e3\u3002<\/p>\n

4) \u6700\u4f18\u7cbe\u786e\u8f6c\u5411\uff08Optimal exact steering\uff09\uff1a\u8f6c\u5411\u51fd\u6570\u8fd4\u56de\u76f8\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u6210\u672c\u6cdb\u51fd\u7684\u4e00\u6761\u6700\u4f18\u7cbe\u786e\u8f6c\u5411\u8def\u5f84\u3002\u4e8b\u5b9e\u4e0a\uff0c\u82e5\u5047\u8bbe\u6210\u672c\u6cdb\u51fd\u662f\u8def\u5f84\u7684\u5f27\u957f[57][58]\uff0c\u5219\u4e0a\u4e00\u6bb5\u63d0\u5230\u7684\u76f4\u7ebf\u3001Dubins\u66f2\u7ebf\u548cReeds-Shepp\u66f2\u7ebf\u90fd\u662f\u6700\u4f18\u89e3\u3002\u5982\u679c\u8def\u5f84\u6bb5 sigma \u5b8c\u5168\u5904\u4e8e mathcal{X}_{text {free }}\u4e2d\uff0c\u90a3\u4e48\u78b0\u649e\u68c0\u6d4b\u51fd\u6570 operatorname{col} text {-free }(sigma) \u5c06\u8fd4\u56de\u771f\u503c\uff08\u5373true\uff09\uff0c\u7528\u4e8e\u786e\u4fdd\u751f\u6210\u7684\u8def\u5f84\u4e0d\u4f1a\u4e0e\u4efb\u4f55\u969c\u788d\u7269\u53d1\u751f\u78b0\u649e\u3002<\/p>\n

\u5177\u5907\u4e86\u8f6c\u5411\u51fd\u6570\u548c\u78b0\u649e\u68c0\u6d4b\u51fd\u6570\u4e4b\u540e\uff0c\u4e3b\u8981\u7684\u6311\u6218\u5c31\u662f\u5982\u4f55\u6784\u9020\u4e00\u4e2a\u80fd\u5f88\u597d\u5730\u8fd1\u4f3c mathcal{X}_{text {free }}\u7684\u8fde\u63a5\u6027\u7684\u79bb\u6563\u5316\uff0c\u800c\u4e0d\u9700\u8981\u62e5\u6709\u5176\u51e0\u4f55\u7ed3\u6784\u7684\u663e\u5f0f\u6a21\u578b\u3002\u63a5\u4e0b\u6765\u6211\u4eec\u5c06\u4ece\u6587\u732e\u4e2d\u56de\u987e\u57fa\u4e8e\u91c7\u6837\u7684\u79bb\u6563\u5316\u7b56\u7565\u3002\u4e00\u79cd\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u9009\u62e9\u4e00\u7ec4\u8fd0\u52a8\u57fa\u5143\uff08\u56fa\u5b9a\u7b56\u7565\uff09\uff0c\u7136\u540e\u4ece\u8f66\u8f86\u7684\u521d\u59cb\u6784\u578b mathrm{x}_{text {init }}\u51fa\u53d1\uff0c\u9012\u5f52\u5730\u5e94\u7528\u8fd9\u4e9b\u57fa\u5143\uff0c\u6765\u751f\u6210\u641c\u7d22\u56fe\uff0c\u4f8b\u5982\u4f7f\u7528\u7b97\u6cd51\u7684\u65b9\u6cd5\u3002\u5bf9\u4e8e\u6ca1\u6709\u5fae\u5206\u7ea6\u675f\u7684\u8def\u5f84\u89c4\u5212\uff0c\u8fd0\u52a8\u57fa\u5143\u53ef\u4ee5\u7b80\u5355\u5730\u662f\u4e00\u7ec4\u5177\u6709\u4e0d\u540c\u65b9\u5411\u548c\u957f\u5ea6\u7684\u76f4\u7ebf\u3002\u5bf9\u4e8e\u7c7b\u8f66\u8f66\u8f86\uff0c\u8fd0\u52a8\u57fa\u5143\u53ef\u4ee5\u662f\u4e00\u7ec4\u5f27\uff0c\u8868\u793a\u6c7d\u8f66\u4ee5\u4e0d\u540c\u7684\u8f6c\u5411\u503c\u8ddf\u968f\u7684\u8def\u5f84\u3002\u591a\u79cd\u6280\u672f\u90fd\u53ef\u7528\u4e8e\u751f\u6210\u65e0\u4eba\u9a7e\u9a76\u8f66\u8f86\u7684\u8fd0\u52a8\u57fa\u5143\u3002\u4e00\u79cd\u7b80\u5355\u7684\u65b9\u6cd5\u662f\u91c7\u6837\u591a\u4e2a\u63a7\u5236\u8f93\u5165\uff0c\u5e76\u4f7f\u7528\u8f66\u8f86\u6a21\u578b\u5b9e\u65f6\u6a21\u62df\u524d\u8fdb\uff0c\u4ee5\u83b7\u5f97\u53ef\u884c\u7684\u8fd0\u52a8\u3002\u4e3a\u4e86\u83b7\u5f97\u5177\u6709\u8fde\u7eed\u66f2\u7387\u7684\u8def\u5f84\uff0c\u6709\u65f6\u4e5f\u4f7f\u7528\u56de\u65cb\u66f2\u7ebf\u6bb5[105]\u3002<\/p>\n

\u6b64\u5916\uff0c\u8fd8\u53ef\u4ee5\u901a\u8fc7\u8bb0\u5f55\u7531\u4e13\u5bb6\u9a7e\u9a76\u5458\u9a7e\u9a76\u7684\u8f66\u8f86\u7684\u8fd0\u52a8\u6765\u83b7\u5f97\u8fd0\u52a8\u57fa\u5143[106]\u3002<\/p>\n

<\/p>\n

\u56fe4.2\uff1a\u6805\u683c\u56fe\u548c\u975e\u6805\u683c\u56fe\uff0c\u5747\u5177\u67095000\u6761\u8fb9<\/p>\n

(a) \u901a\u8fc7\u9012\u5f52\u5730\u5e94\u7528 90\u2218 \u5de6\u5706\u5f27\u3001 90\u2218 \u53f3\u5706\u5f27\u548c\u76f4\u7ebf\u800c\u751f\u6210\u7684\u56fe\u3002(b) \u901a\u8fc7\u9012\u5f52\u5730\u5e94\u7528 89\u2218 \u5de6\u5706\u5f27\u3001 89\u2218 \u53f3\u5706\u5f27\u548c\u76f4\u7ebf\u5f97\u5230\u7684\u56fe\u5f62\u3002<\/p>\n

\u5bf9\u4e8e(b)\uff0c\u8fd9\u4e9b\u57fa\u5143\u7684\u9012\u5f52\u5e94\u7528\u786e\u5b9e\u6ca1\u6709\u751f\u6210\u6805\u683c\uff0c\u800c\u662f\u751f\u6210\u4e86\u6811\uff0c\u4e14\u6811\u7684\u5f88\u591a\u5206\u652f\u5728\u539f\u70b9\u9644\u8fd1\u73af\u7ed5\u3002\u56e0\u6b64\uff0c\u5176\u751f\u6210\u7684\u56fe\uff08\u53f3\u4fa7\u7684\u56fe\uff09\u6240\u8986\u76d6\u7684\u533a\u57df\u8f83\u5c0f\u3002<\/p>\n

\u6211\u4eec\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u9012\u5f52\u5730\u5e94\u7528\u8fd0\u52a8\u57fa\u5143\u80fd\u751f\u6210\u6811\u56fe\uff0c\u8fd9\u79cd\u56fe\u5728\u6700\u574f\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5b58\u5728\u4e24\u6761\u80fd\u5230\u8fbe\u76f8\u540c\u6784\u578b\u7684\u8fb9\u3002\u7136\u800c\uff0c\u79f0\u4e3a\u201c\u6805\u683c\u751f\u6210\uff08lattice-generating\uff09\u201d\u7684\u8fd0\u52a8\u57fa\u5143\u96c6\u80fd\u751f\u6210\u5f62\u4f3c\u6805\u683c\uff08lattice\uff09\u7684\u89c4\u5219\u56fe\u5f62\u3002<\/p>\n

\u89c1\u56fe4.2a\uff0c\u6805\u683c\u751f\u6210\u57fa\u5143\u7684\u4f18\u70b9\u662f\u641c\u7d22\u56fe\u7684\u9876\u70b9\u80fd\u591f\u5747\u5300\u5730\u8986\u76d6\u6784\u578b\u7a7a\u95f4\uff0c\u800c\u6811\u901a\u5e38\u53ef\u80fd\u5728\u6839\u9876\u70b9\u7684\u5468\u56f4\u51fa\u73b0\u9ad8\u5bc6\u5ea6\u5206\u5e03\u7684\u9876\u70b9\u3002Pivtoraiko\u7b49\u4eba\u4f7f\u7528\u672f\u8bed\u201c\u72b6\u6001\u6805\u683c\uff08state lattice\uff09\u201d\u6765\u63cf\u8ff0[107]\u4e2d\u7684\u6b64\u7c7b\u56fe\uff0c\u5e76\u6307\u51fa\uff0c\u5982\u679c\u9996\u5148\u5728\u539f\u70b9\u5468\u56f4\u751f\u6210\u89c4\u5219\u95f4\u9694\u7684\u6784\u578b\uff0c\u7136\u540e\u901a\u8fc7\u8868\u793a\u4e24\u70b9\u8fb9\u503c\u95ee\u9898\u7684\u89e3\u7684\u8def\u5f84\u5c06\u539f\u70b9\u8fde\u63a5\u5230\u8fd9\u79cd\u6784\u578b\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u5f53\u524d\u7cfb\u7edf\u7684\u4e00\u7ec4\u6805\u683c\u751f\u6210\u8fd0\u52a8\u57fa\u5143\u3002<\/p>\n

<\/p>\n

\u901a\u8fc7\u751f\u6210\u8986\u76d6\uff08\u81ea\u7531\uff09\u6784\u578b\u7a7a\u95f4\u7684\u79bb\u6563\u6837\u672c\u96c6\uff0c\u5e76\u901a\u8fc7\u4f7f\u7528\u7cbe\u786e\u7684\u8f6c\u5411\u7a0b\u5e8f\u83b7\u5f97\u7684\u53ef\u884c\u8def\u5f84\u6bb5\u5c06\u5b83\u4eec\u8fde\u63a5\u8d77\u6765\uff0c\u53ef\u4ee5\u5b9e\u73b0\u7c7b\u4f3c\u4e8e\u4ece\u521d\u59cb\u6784\u578b\u51fa\u53d1\u3001\u9012\u5f52\u5730\u5e94\u7528\u6805\u683c\u751f\u6210\u8fd0\u52a8\u57fa\u5143\u7684\u6240\u4ea7\u751f\u7684\u6548\u679c\u3002<\/p>\n

<\/p>\n

\u5927\u591a\u6570\u57fa\u4e8e\u91c7\u6837\u7684\u8def\u7ebf\u56fe\u7684\u6784\u9020\u65b9\u6cd5\u90fd\u9075\u5faa\u7b97\u6cd52\u6240\u793a\u7684\u65b9\u6848\uff0c\u4f46\u662f\u5728\u91c7\u6837\u70b9\u51fd\u6570 text { sample-points }(mathcal{X}, n) \u548c\u90bb\u70b9\u51fd\u6570 text { neighbors }(mathrm{x}, V) \u7684\u5b9e\u73b0\u4e0a\u6709\u6240\u4e0d\u540c\u3002\u91c7\u6837\u70b9\u51fd\u6570text { sample-points }(mathcal{X}, n)\u8868\u793a\u4ece\u6784\u578b\u7a7a\u95f4 chi \u4e2d\u9009\u62e9 n \u4e2a\u70b9\u7684\u7b56\u7565\uff0c\u800c\u90bb\u70b9\u51fd\u6570text { neighbors }(mathrm{x}, V)\u8868\u793a\u4e3a\u9876\u70b9 x \u9009\u62e9\u4e00\u7ec4\u90bb\u9876\u70b9 N subseteq V \u7684\u7b56\u7565\u3002\u5bf9\u4e8e\u90bb\u9876\u70b9 y\uff0c\u8be5\u7b97\u6cd5\u5c06\u5c1d\u8bd5\u901a\u8fc7\u7cbe\u786e\u8f6c\u5411\u51fd\u6570 text { steer }_{text {exact }}(mathrm{x}, mathrm{y}) \u6765\u627e\u5230\u4e00\u6761\u4ece y \u8fde\u63a5\u5230 x \u7684\u8def\u5f84\u6bb5\u3002<\/p>\n

\u51fd\u6570 text { sample-points }(mathcal{X}, n) \u7684\u6700\u5e38\u89c1\u7684\u4e24\u79cd\u5b9e\u73b0\u65b9\u5f0f\u662f\uff1a\uff081\uff09\u8fd4\u56de\u6392\u5217\u5728\u4e00\u4e2a\u89c4\u5219\u7f51\u683c\u4e2d\u7684 n \u4e2a\u70b9\uff0c\uff082\uff09\u8fd4\u56de chi \u4e2d\u7684 n \u4e2a\u968f\u673a\u91c7\u6837\u70b9\u3002\u5c3d\u7ba1\u968f\u673a\u62bd\u6837\u5177\u6709\u666e\u904d\u9002\u7528\u548c\u6613\u4e8e\u5b9e\u65bd\u7684\u4f18\u70b9\uff0c\u4f46\u6240\u8c13\u7684Sukharev\u7f51\u683c\u5df2\u7ecf\u88ab\u8bc1\u660e\u5728\u5355\u4f4d\u8d85\u7acb\u65b9\u4f53\u4e2d\u5b9e\u73b0\u4e86\u6700\u4f18 L_{infty} \u6563\u5ea6\uff0c\u5373\u5176\u6700\u5c0f\u5316\u4e86\u5185\u90e8\u6ca1\u6709\u91c7\u6837\u70b9\u7684\u6700\u5927\u7a7a\u7403\u7684\u534a\u5f84\u3002\u4e3a\u4e86\u6df1\u5165\u8ba8\u8bba\u57fa\u4e8e\u91c7\u6837\u7684\u8def\u5f84\u89c4\u5212\u73af\u5883\u4e2d\u968f\u673a\u91c7\u6837\u548c\u786e\u5b9a\u6027\u91c7\u6837\u7684\u76f8\u5bf9\u4f18\u70b9\uff0c\u6211\u4eec\u8bf7\u8bfb\u8005\u53c2\u8003[108]\u3002<\/p>\n

\u5b9e\u73b0\u51fd\u6570 text { neighbors }(mathrm{x}, V)\u7684\u4e24\u4e2a\u6700\u5e38\u7528\u7b56\u7565\u662f\uff1a\uff081\uff09\u4e0e x \u6700\u8fd1\u7684 k \u8fd1\u90bb\u7684\u96c6\u5408\uff0c\uff082\uff09\u4f4d\u4e8e\u4ee5 x \u4e3a\u4e2d\u5fc3\u7684\u3001\u534a\u5f84\u4e3a r \u7684\u7403\u5185\u7684\u70b9\u7684\u96c6\u5408\u3002\u7279\u522b\u5730\uff0c\u5728 d \u7ef4\u7f51\u683c\u4e2d\u4ee5\u786e\u5b9a\u6027\u65b9\u5f0f\u6392\u5217\u7684\u6837\u672c\uff0c\u82e5\u5176\u90bb\u57df\u5728\u4e8c\u7ef4\u4e2d\u53d6\u4e3a 4 \u4e2a\u6216 8\u4e2a\u6700\u8fd1\u90bb\u3001\u5728\u66f4\u9ad8\u7ef4\u5ea6\u4e2d\u53d6\u4e3a\u4e0e\u4e8c\u7ef4\u60c5\u51b5\u76f8\u7c7b\u4f3c\u7684\u6a21\u5f0f\uff0c\u90a3\u4e48\u8fd9\u4e9b\u6837\u672c\u4ee3\u8868\u4e86\u81ea\u7531\u6784\u578b\u7a7a\u95f4\u7684\u7b80\u5355\u786e\u5b9a\u6027\u79bb\u6563\u5316\uff0c\u90e8\u5206\u662f\u56e0\u4e3a\u5b83\u4eec\u81ea\u7136\u4ea7\u751f\u4e8e\u673a\u5668\u4eba\u6784\u578b\u7a7a\u95f4\u7684\u81ea\u7531\u548c\u5360\u7528\u533a\u57df\u6240\u5e7f\u6cdb\u4f7f\u7528\u7684\u4f4d\u56fe\u8868\u793a[109]\u3002<\/p>\n

Kavraki\u7b49\u4eba[110]\u4e3b\u5f20\u5728\u6982\u7387\u8def\u7ebf\u56fe\uff08PRM\uff09\u6846\u67b6\u5185\u4f7f\u7528\u968f\u673a\u62bd\u6837\uff0c\u4ee5\u4fbf\u5728\u9ad8\u7ef4\u6784\u578b\u7a7a\u95f4\u4e2d\u6784\u9020\u8def\u7ebf\u56fe\uff0c\u56e0\u4e3a\u4e0e\u7f51\u683c\u4e0d\u540c\uff0c\u5b83\u4eec\u968f\u65f6\u53ef\u4ee5\u81ea\u7136\u8fd0\u884c\u3002\u6279\u5904\u7406\u7248\u672c\u7684PRM\u9075\u5faa\u7b97\u6cd52\u7684\u65b9\u6848\uff0c\u5176\u91c7\u7528\u968f\u673a\u62bd\u6837\uff0c\u5e76\u4e14\u5728\u56fa\u5b9a\u534a\u5f84 r \u7684\u7403\u5185\u9009\u62e9\u90bb\u70b9[79]\u3002\u7531\u4e8ePRM\u5177\u6709\u901a\u7528\u7684\u5f62\u5f0f\uff0c\u5b83\u4eec\u5df2\u7528\u4e8e\u5404\u79cd\u7cfb\u7edf\u7684\u8def\u5f84\u89c4\u5212\uff0c\u5305\u62ec\u5177\u6709\u5fae\u5206\u7ea6\u675f\u7684\u7cfb\u7edf\u3002\u7136\u800c\uff0c\u8be5\u7b97\u6cd5\u7684\u7406\u8bba\u5206\u6790\u4e3b\u8981\u96c6\u4e2d\u4e8e\u65e0\u5fae\u5206\u7ea6\u675f\u7cfb\u7edf\u7684\u7b97\u6cd5\u6027\u80fd\uff0c\u5373\uff0c\u5f53\u4f7f\u7528\u76f4\u7ebf\u8fde\u63a5\u4e24\u79cd\u6784\u578b\u65f6\u3002\u5728\u8fd9\u6837\u7684\u5047\u8bbe\u4e0b\uff0c\u6587\u7ae0[80]\u4e2d\u7684PRM\u88ab\u8bc1\u660e\u662f\u6982\u7387\u5b8c\u5907\u4e14\u6e10\u8fd1\u6700\u4f18\u7684\u3002<\/p>\n

\u4e5f\u5c31\u662f\u8bf4\uff0c\u5176\u751f\u6210\u7684\u56fe\u5305\u542b\u6709\u6548\u89e3\uff08\u5982\u679c\u5b58\u5728\uff09\u7684\u6982\u7387\u968f\u7740\u56fe\u7684\u5927\u5c0f\u7684\u589e\u52a0\u800c\u6536\u655b\u5230 1\uff0c\u4e14\u56fe\u4e2d\u6700\u77ed\u8def\u5f84\u7684\u6210\u672c\u968f\u4e4b\u6536\u655b\u5230\u6700\u4f18\u6210\u672c\u3002Karaman\u548cFrazzoli\u63d0\u51fa\u4e86\u4e00\u79cd\u79f0\u4e3aPRM*\u7684\u6279\u5904\u7406PRM\u7684\u81ea\u9002\u5e94\u7248\u672c[80]\uff0c\u5176\u4ec5\u8fde\u63a5\u4e00\u4e2a\u7403\u5185\u7684\u90bb\u9876\u70b9\u3002\u8be5\u7403\u7684\u534a\u5f84\u968f\u7740\u6837\u672c\u6570\u91cf\u7684\u589e\u52a0\u800c\u5bf9\u6570\u5927\u5c0f\u5730\u51cf\u5c0f\u3002\u8fd9\u4e48\u505a\u7684\u76ee\u7684\u662f\u4fdd\u6301\u7b97\u6cd5\u7684\u6e10\u8fd1\u6700\u4f18\u6027\u4ee5\u53ca\u8ba1\u7b97\u6548\u7387\u3002<\/p>\n

\u5728\u540c\u4e00\u7bc7\u8bba\u6587\u4e2d\uff0c\u4f5c\u8005\u540c\u65f6\u63d0\u51fa\u4e86\u5feb\u901f\u63a2\u7d22\u968f\u673a\u56fe\uff08RRG*\uff09\u3002\u8fd9\u662f\u4e00\u79cd\u589e\u91cf\u79bb\u6563\u5316\u7b56\u7565\uff0c\u5728\u4fdd\u6301\u6e10\u8fd1\u6700\u4f18\u6027\u7684\u540c\u65f6\u80fd\u591f\u968f\u65f6\u7ec8\u6b62\u3002\u6700\u8fd1\uff0c\u5feb\u901f\u884c\u8fdb\u6811\uff08FMT*\uff09[111]\u88ab\u63d0\u51fa\u4f5c\u4e3aPRM*\u7684\u6e10\u8fd1\u6700\u4f18\u66ff\u4ee3\u65b9\u6848\u3002\u8be5\u7b97\u6cd5\u901a\u8fc7\u5bf9\u4e00\u7ec4\u91c7\u6837\u9876\u70b9\u6267\u884c\u60f0\u6027\u52a8\u6001\u89c4\u5212\u9012\u5f52\uff0c\u5c06\u79bb\u6563\u5316\u548c\u641c\u7d22\u7ed3\u5408\u5230\u4e00\u4e2a\u8fc7\u7a0b\u4e2d\u3002\u8fd9\u4e9b\u91c7\u6837\u9876\u70b9\u968f\u540e\u53ef\u4ee5\u88ab\u7528\u4e8e\u5feb\u901f\u786e\u5b9a\u4ece\u521d\u59cb\u6784\u578b\u5230\u76ee\u6807\u533a\u57df\u7684\u8def\u5f84\u3002<\/p>\n

\u6700\u8fd1\uff0c\u7406\u8bba\u5206\u6790\u4e5f\u6269\u5c55\u5230\u5fae\u5206\u7ea6\u675f\u7cfb\u7edf\u3002<\/p>\n

Schmerling\u7b49\u4eba[81]\u63d0\u51fa\u4e86PRM*\u548cFMT*\u7684\u5fae\u5206\u7248\u672c\uff0c\u5e76\u8bc1\u660e\u4e86\u8fd9\u4e9b\u7b97\u6cd5\u5bf9\u4e8e\u65e0\u6f02\u79fb\u7684\u4eff\u5c04\u63a7\u5236\u7684\u52a8\u529b\u7cfb\u7edf\u7684\u6e10\u8fd1\u6700\u4f18\u6027\u3002\u8fd9\u4e00\u7c7b\u52a8\u529b\u7cfb\u7edf\u5305\u62ec\u65e0\u6ed1\u79fb\u7684\u8f6e\u5f0f\u8f66\u8f86\u6a21\u578b\u3002<\/p>\n

4) \u56fe\u641c\u7d22\u7b56\u7565\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u8ba8\u8bba\u4e86\u5c06\u81ea\u7531\u6784\u578b\u7a7a\u95f4\u79bb\u6563\u5316\u4e3a\u56fe\u7684\u5f62\u5f0f\u7684\u6280\u672f\u3002\u4e3a\u4e86\u5728\u8fd9\u79cd\u79bb\u6563\u5316\u4e2d\u83b7\u5f97\u4e00\u6761\u5b9e\u9645\u7684\u6700\u4f18\u8def\u5f84\uff0c\u5fc5\u987b\u4f7f\u7528\u56fe\u641c\u7d22\u7b97\u6cd5\u3002<\/p>\n

\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u56de\u987e\u4e0e\u8def\u5f84\u89c4\u5212\u76f8\u5173\u7684\u56fe\u641c\u7d22\u7b97\u6cd5\u3002\u6700\u53d7\u5e7f\u6cdb\u8ba4\u53ef\u7684\u3001\u7528\u4e8e\u5728\u56fe\u4e2d\u627e\u5230\u6700\u77ed\u8def\u5f84\u7684\u7b97\u6cd5\u53ef\u80fd\u662fDijkstra\u7b97\u6cd5[32]\u3002\u5b83\u6267\u884c\u6700\u4f73\u4f18\u5148\u641c\u7d22\uff08best first search\uff09\uff0c\u4ee5\u6784\u5efa\u8868\u793a\u4ece\u7ed9\u5b9a\u6e90\u9876\u70b9\u5230\u56fe\u4e2d\u6240\u6709\u5176\u4ed6\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u7684\u6811\u3002\u5f53\u53ea\u9700\u8981\u5230\u5355\u4e2a\u9876\u70b9\u7684\u8def\u5f84\u65f6\uff0c\u53ef\u4ee5\u4f7f\u7528\u542f\u53d1\u5f0f\u65b9\u6cd5\u6765\u6307\u5bfc\u641c\u7d22\u3002\u6700\u8457\u540d\u7684\u542f\u53d1\u5f0f\u641c\u7d22\u7b97\u6cd5\u662fHart\u3001Nilsson\u548cRaphael\u5f00\u53d1\u7684A*\u7b97\u6cd5[112]\u3002<\/p>\n

\u5982\u679c\u7ed9\u5b9a\u7684\u542f\u53d1\u5f0f\u51fd\u6570\u662f\u53ef\u63a5\u53d7\u7684\uff08admissible\uff09\uff0c\u5373\uff0c\u51fd\u6570\u4ece\u4e0d\u4f1a\u9ad8\u4f30\u672a\u6765\u8981\u82b1\u8d39\u7684\u6210\u672c\uff08cost-to-go\uff09\uff0c\u5219\u6b64\u65f6A*\u662f\u6700\u4f18\u6709\u6548\u7684\uff0c\u5e76\u4fdd\u8bc1\u8fd4\u56de\u6700\u4f18\u89e3\u3002\u8bb8\u591a\u95ee\u9898\u53ef\u4ee5\u4f7f\u7528\u52a0\u6743A*\u7b97\u6cd5[113]\uff0c\u4ee5\u8f83\u5c11\u7684\u8ba1\u7b97\u91cf\u83b7\u5f97\u4e00\u4e2a\u6709\u754c\u7684\u6b21\u4f18\u89e3\uff0c\u8fd9\u76f8\u5f53\u4e8e\u7b80\u5355\u5730\u5c06\u542f\u53d1\u5f0f\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\u56e0\u5b50 epsilon>1 \u3002<\/p>\n

\u53ef\u4ee5\u8bc1\u660e\uff0c\u5229\u7528\u8fd9\u79cd\u81a8\u80c0\u7684\u542f\u53d1\u5f0f\u65b9\u6cd5\uff0cA*\u4fdd\u8bc1\u8fd4\u56de\u7684\u89e3\u8def\u5f84\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\u53ea\u6709\u6700\u4f18\u8def\u5f84\u7684\u6210\u672c\u7684 1+epsilon \u500d\u3002\u901a\u5e38\uff0c\u6bcf\u6b21\u4f7f\u7528\u4f20\u611f\u6570\u636e\u66f4\u65b0\u4e16\u754c\u6a21\u578b\u65f6\uff0c\u7b97\u6cd5\u4f1a\u518d\u6b21\u5bfb\u627e\u4ece\u8f66\u8f86\u5f53\u524d\u6784\u578b\u5230\u76ee\u6807\u533a\u57df\u7684\u6700\u77ed\u8def\u5f84\u3002\u7531\u4e8e\u6bcf\u6b21\u8fd9\u6837\u7684\u66f4\u65b0\u901a\u5e38\u53ea\u5f71\u54cd\u56fe\u7684\u5c0f\u90e8\u5206\u533a\u57df\uff0c\u6240\u4ee5\u6bcf\u6b21\u5b8c\u5168\u4ece\u5934\u5f00\u59cb\u641c\u7d22\u53ef\u80fd\u4f1a\u6d6a\u8d39\u65f6\u95f4\u3002\u5b9e\u65f6\u7684\u91cd\u65b0\u89c4\u5212\uff08replanning\uff09\u641c\u7d22\u7b97\u6cd5\uff0c\u4f8b\u5982D*\u7b97\u6cd5[114]\u3001\u805a\u7126D*\u7b97\u6cd5\uff08Focussed D*\uff09[115]\u548c\u8f7b\u91cfD*\u7b97\u6cd5\uff08D* Lite\uff09[116]\uff0c\u80fd\u5728\u6bcf\u6b21\u56fe\u53d1\u751f\u6539\u53d8\u65f6\u6709\u6548\u5730\u91cd\u65b0\u8ba1\u7b97\u6700\u77ed\u8def\u5f84\uff0c\u540c\u65f6\u5229\u7528\u6765\u81ea\u5148\u524d\u641c\u7d22\u7684\u4fe1\u606f\u3002<\/p>\n

\u968f\u65f6\u641c\u7d22\uff08anytime search\uff09\u7b97\u6cd5\u8bd5\u56fe\u9996\u5148\u5feb\u901f\u627e\u5230\u4e00\u4e2a\u6b21\u4f18\u8def\u5f84\uff0c\u518d\u7528\u66f4\u591a\u7684\u8ba1\u7b97\u65f6\u95f4\u4e0d\u65ad\u6539\u8fdb\u89e3\u3002\u968f\u65f6A*\u7b97\u6cd5\uff08anytime A*\uff09[117]\u4f7f\u7528\u52a0\u6743\u542f\u53d1\u5f0f\u51fd\u6570\u6765\u627e\u5230\u7b2c\u4e00\u4e2a\u89e3\uff0c\u518d\u4ee5\u7b2c\u4e00\u4e2a\u89e3\u8def\u5f84\u7684\u6210\u672c\u4e3a\u4e0a\u9650\u3001\u4ee5\u53ef\u63a5\u53d7\u7684\u542f\u53d1\u5f0f\u4e3a\u4e0b\u9650\uff0c\u6765\u7ee7\u7eed\u641c\u7d22\uff0c\u4ee5\u5b9e\u73b0\u968f\u65f6\u641c\u7d22\u7684\u529f\u80fd\u3002<\/p>\n

\u800c\u968f\u65f6\u4fee\u590dA*\u7b97\u6cd5\uff08Anytime Repair A* \/ ARA*\uff09[118]\u4f7f\u7528\u6743\u91cd\u964d\u4f4e\u7684\u81a8\u80c0\u542f\u53d1\u5f0f\u6765\u6267\u884c\u4e00\u7cfb\u5217\u641c\u7d22\uff0c\u5e76\u91cd\u590d\u5229\u7528\u6765\u81ea\u5148\u524d\u8fed\u4ee3\u7684\u4fe1\u606f\u3002<\/p>\n

\u53e6\u4e00\u65b9\u9762\uff0c\u968f\u65f6\u52a8\u6001A*\u7b97\u6cd5\uff08Anytime Dynamic A* \/ ADA*\uff09[119]\u7ed3\u5408\u4e86D* Lite\u548cARA*\u7684\u601d\u60f3\uff0c\u4e3a\u52a8\u6001\u73af\u5883\u4e2d\u7684\u5b9e\u65f6\u91cd\u65b0\u89c4\u5212\u63d0\u4f9b\u4e86\u4e00\u79cd\u968f\u65f6\u641c\u7d22\u7b97\u6cd5\u3002\u7528\u4e8e\u5728\u6784\u578b\u7a7a\u95f4\u7684\u79bb\u6563\u5316\u56fe\u4e0a\u641c\u7d22\u8def\u5f84\u7684\u7b97\u6cd5\u5b58\u5728\u7684\u660e\u663e\u9650\u5236\u662f\uff0c\u5728\u8fd9\u6837\u7684\u56fe\u4e0a\u5f97\u5230\u7684\u6700\u4f18\u8def\u5f84\u53ef\u80fd\u663e\u8457\u957f\u4e8e\u6784\u578b\u7a7a\u95f4\u4e2d\u7684\u771f\u5b9e\u6700\u77ed\u8def\u5f84\u3002\u4efb\u610f\u89d2\u5ea6\u8def\u5f84\u89c4\u5212\u7b97\u6cd5\uff08any-angle path planning algorithms\uff09[120-122]\u5728\u7f51\u683c\u4e0a\u8fd0\u884c\uff0c\u6216\u8005\u66f4\u4e00\u822c\u5730\uff0c\u5728\u8868\u793a\u81ea\u7531\u6784\u578b\u7a7a\u95f4\u7684\u5355\u5143\u5206\u89e3\uff08cell decomposition\uff09\u7684\u56fe\u4e0a\u8fd0\u884c\uff0c\u5e76\u901a\u8fc7\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u8003\u8651\u56fe\u4e2d\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u6377\u5f84\uff08shortcuts\uff09\u201d\u6765\u5c1d\u8bd5\u51cf\u8f7b\u8fd9\u4e00\u7f3a\u70b9\u3002<\/p>\n

\u6b64\u5916\uff0c\u771f\u5b9e\u573a\u666fD*\u7b97\u6cd5\uff08Field D*\uff09\u901a\u8fc7\u5c06\u7ebf\u6027\u63d2\u503c\u5f15\u5165\u641c\u7d22\u8fc7\u7a0b\u6765\u4ea7\u751f\u5e73\u6ed1\u8def\u5f84[123]\u3002E. \u589e\u91cf\u641c\u7d22\u6280\u672f\u5728\u56fa\u5b9a\u7684\u79bb\u6563\u5316\u56fe\u4e0a\u8fdb\u884c\u641c\u7d22\u7684\u6280\u672f\u7684\u7f3a\u70b9\u662f\uff0c\u5176\u53ea\u641c\u7d22\u53ef\u4ee5\u7531\u57fa\u5143\u6784\u9020\u7684\u4e00\u7ec4\u8def\u5f84\uff0c\u56e0\u6b64\u8fd9\u4e9b\u6280\u672f\u53ef\u80fd\u65e0\u6cd5\u8fd4\u56de\u53ef\u884c\u7684\u8def\u5f84\u6216\u8fd4\u56de\u660e\u663e\u6b21\u4f18\u7684\u8def\u5f84\u3002\u589e\u91cf\u53ef\u884c\u8fd0\u52a8\u89c4\u5212\u5668\u52aa\u529b\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u5e76\u5728\u7ed9\u5b9a\u8db3\u591f\u7684\u8ba1\u7b97\u65f6\u95f4\u7684\u60c5\u51b5\u4e0b\uff0c\u4e3a\u4efb\u4f55\u8fd0\u52a8\u89c4\u5212\u95ee\u9898\u5b9e\u4f8b\u63d0\u4f9b\u53ef\u884c\u8def\u5f84\uff08\u5982\u679c\u5b58\u5728\u53ef\u884c\u8def\u5f84\uff09\u3002<\/p>\n

\u901a\u5e38\uff0c\u8fd9\u4e9b\u65b9\u6cd5\u9010\u6b65\u6784\u9020\u5bf9\u6784\u578b\u7a7a\u95f4\u7684\u8d8a\u6765\u8d8a\u7cbe\u7ec6\u7684\u79bb\u6563\u5316\uff0c\u540c\u65f6\u5c1d\u8bd5\u786e\u5b9a\u5728\u6bcf\u4e00\u6b65\u79bb\u6563\u5316\u4e2d\u662f\u5426\u5b58\u5728\u4ece\u521d\u59cb\u6784\u578b\u5230\u76ee\u6807\u533a\u57df\u7684\u8def\u5f84\u3002\u5982\u679c\u5b9e\u4f8b\u662f\u201c\u7b80\u5355\u201d\u7684\uff0c\u5219\u53ef\u4ee5\u5feb\u901f\u5f97\u5230\u89e3\uff0c\u4f46\u662f\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8ba1\u7b97\u65f6\u95f4\u53ef\u80fd\u662f\u65e0\u9650\u957f\u7684\u3002\u7c7b\u4f3c\u5730\uff0c\u5728\u5feb\u901f\u627e\u5230\u53ef\u884c\u8def\u5f84\u7684\u57fa\u7840\u4e0a\uff0c\u589e\u91cf\u6700\u4f18\u8fd0\u52a8\u89c4\u5212\u65b9\u6cd5\u8bd5\u56fe\u63d0\u4f9b\u4e00\u7cfb\u5217\u8d28\u91cf\u4e0d\u65ad\u63d0\u9ad8\u7684\u89e3\uff0c\u4ece\u800c\u6536\u655b\u5230\u6700\u4f18\u8def\u5f84\u3002\u6587\u732e\u4e2d\u4f7f\u7528\u201c\u6982\u7387\u5b8c\u5907\uff08probabilistically complete\uff09\u201d\u4e00\u8bcd\u6765\u63cf\u8ff0\u4e00\u79cd\u7b97\u6cd5\u3002\u8fd9\u7c7b\u7b97\u6cd5\u627e\u5230\u89e3\uff08\u5982\u679c\u5b58\u5728\u89e3\uff09\u7684\u6982\u7387\u968f\u7740\u8ba1\u7b97\u65f6\u95f4\u7684\u589e\u52a0\u800c\u63a5\u8fd1 1\u3002<\/p>\n

\u8bf7\u6ce8\u610f\uff0c\u5982\u679c\u89e3\u5e76\u4e0d\u5b58\u5728\uff0c\u5219\u6982\u7387\u5b8c\u5907\u7684\u7b97\u6cd5\u53ef\u80fd\u4e0d\u4f1a\u7ec8\u6b62\u3002<\/p>\n

\u7c7b\u4f3c\u5730\uff0c\u6e10\u8fd1\u6700\u4f18\uff08asymptotically optimal\uff09\u4e00\u8bcd\u7528\u4e8e\u63cf\u8ff0\u4ee5\u6982\u7387 1 \u6536\u655b\u5230\u6700\u4f18\u89e3\u7684\u7b97\u6cd5\u3002\u6781\u9650\u60c5\u51b5\u4e0b\u83b7\u5f97\u5b8c\u5907\u6027\u548c\u6700\u4f18\u6027\u7684\u4e00\u79cd\u7b80\u5355\u7b56\u7565\u662f\u5728\u6784\u578b\u7a7a\u95f4\u7684\u56fa\u5b9a\u79bb\u6563\u5316\u4e0a\u89e3\u51b3\u4e00\u7cfb\u5217\u8def\u5f84\u89c4\u5212\u95ee\u9898\uff0c\u4e14\u6bcf\u6b21\u90fd\u91c7\u53d6\u66f4\u9ad8\u7684\u79bb\u6563\u5316\u5206\u8fa8\u7387\u3002\u6b64\u65b9\u6cd5\u7684\u4e00\u4e2a\u7f3a\u70b9\u662f\uff0c\u5404\u4e2a\u5206\u8fa8\u7387\u7ea7\u522b\u4e0a\u7684\u8def\u5f84\u89c4\u5212\u8fc7\u7a0b\u662f\u72ec\u7acb\u7684\uff0c\u65e0\u6cd5\u91cd\u590d\u5229\u7528\u4efb\u4f55\u4fe1\u606f\u3002\u6b64\u5916\uff0c\u5728\u5f00\u59cb\u65b0\u4e00\u8f6e\u56fe\u641c\u7d22\u524d\uff0c\u6211\u4eec\u5e76\u4e0d\u660e\u663e\u5730\u77e5\u9053\u5e94\u8be5\u591a\u5feb\u5730\u589e\u52a0\u79bb\u6563\u5316\u7684\u5206\u8fa8\u7387\uff0c\u5373\uff0c\u5230\u5e95\u662f\u6dfb\u52a0\u5355\u4e2a\u65b0\u6784\u578b\u3001\u8fd8\u662f\u589e\u52a0\u4e00\u500d\u6570\u91cf\u7684\u6784\u578b\u3001\u8fd8\u662f\u6cbf\u7740\u6784\u578b\u7a7a\u95f4\u7684\u6bcf\u4e2a\u7ef4\u5ea6\u589e\u52a0\u4e00\u500d\u6570\u91cf\u7684\u79bb\u6563\u503c\u66f4\u52a0\u5408\u9002\u3002\u4e3a\u4e86\u514b\u670d\u8fd9\u4e9b\u95ee\u9898\uff0c\u589e\u91cf\u8fd0\u52a8\u89c4\u5212\u65b9\u6cd5\u5c06\u6784\u578b\u7a7a\u95f4\u7684\u589e\u91cf\u79bb\u6563\u5316\u4e0e\u8def\u5f84\u641c\u7d22\u7ed3\u5408\u5230\u4e00\u4e2a\u6b65\u9aa4\u4e2d\u3002<\/p>\n

\u589e\u91cf\u8def\u5f84\u89c4\u5212\u7684\u4e00\u7c7b\u91cd\u8981\u65b9\u6cd5\u7684\u601d\u8def\u662f\uff0c\u5728\u8f66\u8f86\u521d\u59cb\u6784\u578b\u5904\u5411\u5916\u589e\u91cf\u751f\u957f\u6811\uff0c\u4ee5\u63a2\u7d22\u53ef\u5230\u8fbe\u7684\u6784\u578b\u7a7a\u95f4\u3002\u201c\u63a2\u7d22\u6027\u201d\u884c\u4e3a\u8fed\u4ee3\u5730\u9009\u62e9\u6811\u7684\u4e00\u4e2a\u968f\u673a\u9876\u70b9\uff0c\u5e76\u5728\u6811\u4e2d\u5e94\u7528\u8f6c\u5411\u51fd\u6570\uff0c\u4ee5\u6269\u5c55\uff08expand\uff09\u6240\u9009\u9876\u70b9\u3002\u4e00\u65e6\u6811\u751f\u957f\u5230\u8db3\u4ee5\u5230\u8fbe\u76ee\u6807\u533a\u57df\uff0c\u5219\u53ef\u4ee5\u4ece\u76ee\u6807\u533a\u57df\u5185\u7684\u9876\u70b9\u51fa\u53d1\uff0c\u53cd\u5411\u8ddf\u8e2a\u80fd\u5230\u8fbe\u521d\u59cb\u6784\u578b\u7684\u8fb9\uff0c\u6765\u83b7\u5f97\u751f\u6210\u7684\u8def\u5f84\u3002\u7b97\u6cd53\u4e2d\u63cf\u8ff0\u4e86\u57fa\u4e8e\u589e\u91cf\u6811\u7684\u7b97\u6cd5\u7684\u4e00\u822c\u65b9\u6848\u3002<\/p>\n

<\/p>\n

\u7b2c\u4e00\u4e2a\u57fa\u4e8e\u968f\u673a\u6811\u7684\u589e\u91cf\u89c4\u5212\u5668\u662fHsu\u7b49\u4eba\u63d0\u51fa\u7684\u6269\u5c55\u7a7a\u95f4\u6811\uff08EST\uff09\u89c4\u5212\u5668[124]\u3002\u5176\u4ee5\u7279\u5b9a\u7684\u6982\u7387\u4ece V \u4e2d\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u7528\u4e8e\u6269\u5c55\u7684\u9876\u70b9 mathrm{X}_{text {selected }}\u3002\u6982\u7387\u4e0e\u5176\u90bb\u57df\u4e2d\u9876\u70b9\u7684\u6570\u91cf\u6210\u53cd\u6bd4\uff0c\u4ece\u800c\u9f13\u52b1\u5411\u672a\u63a2\u7d22\u533a\u57df\u7684\u589e\u957f\u3002<\/p>\n

\u5728\u6269\u5c55\u8fc7\u7a0b\u4e2d\uff0c\u7b97\u6cd5\u5728 mathrm{X}_{text {selected }}\u5468\u56f4\u7684\u56fa\u5b9a\u534a\u5f84\u7684\u90bb\u57df\u5185\u91c7\u6837\u5f97\u5230\u4e00\u4e2a\u65b0\u9876\u70b9 y\u3002\u6211\u4eec\u4f7f\u7528\u76f8\u540c\u7684\u6280\u672f\u6765\u504f\u7f6e\u8fd9\u4e2a\u91c7\u6837\u8fc7\u7a0b\uff0c\u4f7f\u5176\u80fd\u5728\u76f8\u5bf9\u8f83\u5c11\u63a2\u7d22\u7684\u533a\u57df\u9009\u62e9\u9876\u70b9\u3002\u968f\u540e\uff0c\u7b97\u6cd5\u8fd4\u56de mathrm{X}_{text {selected }}\u548c y \u4e4b\u95f4\u7684\u4e00\u6761\u76f4\u7ebf\u8def\u5f84\u3002<\/p>\n

\u6587\u7ae0[125]\u5f15\u5165\u4e86\u5728\u52a8\u6001\u73af\u5883\u4e2d\u8fdb\u884c\u5177\u6709\u52a8\u529b\u5b66\u7ea6\u675f\u7684\u89c4\u5212\u7684\u63a8\u5e7f\u601d\u60f3\uff0c\u5176\u4e2d\u5728\u4e0d\u540c\u7684\u975e\u5b8c\u6574\u673a\u5668\u4eba\u7cfb\u7edf\u4e0a\u8bc1\u660e\u4e86\u7b97\u6cd5\u7684\u80fd\u529b\u3002\u6b64\u5916\uff0c\u4f5c\u8005\u8fd8\u4f7f\u7528\u7406\u60f3\u5316\u7248\u672c\u7684\u7b97\u6cd5\uff0c\u8bc1\u660e\u4e86\u65e0\u6cd5\u627e\u5230\u53ef\u884c\u8def\u5f84\u7684\u6982\u7387\u53d6\u51b3\u4e8e\u72b6\u6001\u7a7a\u95f4\u7684\u53ef\u6269\u5c55\u6027\uff0c\u5e76\u4e14\u8fd9\u4e2a\u6982\u7387\u968f\u7740\u6837\u672c\u7684\u6570\u91cf\u589e\u957f\u800c\u5448\u6307\u6570\u8870\u51cf\u3002<\/p>\n

La Valle\u63d0\u51fa\u4e86\u5feb\u901f\u63a2\u7d22\u968f\u673a\u6811\uff08RRT\uff09[101]\uff0c\u4f5c\u4e3a\u5bfb\u627e\u9ad8\u7ef4\u975e\u5b8c\u6574\u7cfb\u7edf\u7684\u53ef\u884c\u8f68\u8ff9\u7684\u4e00\u79cd\u6709\u6548\u65b9\u6cd5\u3002\u8be5\u7b97\u6cd5\u4ece\u81ea\u7531\u6784\u578b\u7a7a\u95f4\u4e2d\u968f\u673a\u91c7\u6837 x_{r n d} \uff0c\u5e76\u5728\u968f\u673a\u6837\u672c\u7684\u65b9\u5411\u4e0a\u6269\u5c55\u6811\uff0c\u6765\u5b9e\u73b0\u5feb\u901f\u63a2\u7d22\u3002\u5728RRT\u4e2d\uff0c\u9876\u70b9\u9009\u62e9\u51fd\u6570 text { select }(V) \u6839\u636e\u4e24\u4e2a\u6784\u578b\u4e4b\u95f4\u7684\u7ed9\u5b9a\u7684\u8ddd\u79bb\u5ea6\u91cf\uff0c\u8fd4\u56de\u968f\u673a\u6837\u672c x_{r n d} \u7684\u6700\u8fd1\u90bb\u3002\u6269\u5c55\u51fd\u6570 text { extend() } \u968f\u540e\u901a\u8fc7\u5728\u56fa\u5b9a\u65f6\u95f4\u6b65\u957f\u4e0a\u5e94\u7528\u4e00\u4e2a\u80fd\u6700\u5c0f\u5316\u5230 x_{r n d} \u7684\u8ddd\u79bb\u7684\u63a7\u5236\u8f93\u5165\uff0c\u4ee5\u5728\u6784\u578b\u7a7a\u95f4\u4e2d\u751f\u6210\u4e00\u6761\u8def\u5f84\u3002\u5728\u67d0\u4e9b\u7b80\u5316\u7684\u5047\u8bbe\uff08\u968f\u673a\u8f6c\u5411\u7528\u4e8e\u6269\u5c55\uff09\u4e0b\uff0cRRT\u7b97\u6cd5\u5df2\u88ab\u8bc1\u660e\u662f\u6982\u7387\u5b8c\u5907\u7684[83]\u3002\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u8fd9\u4e2a\u6982\u7387\u5b8c\u5907\u6027\u7684\u7ed3\u8bba\u5e76\u4e0d\u80fd\u8f7b\u6613\u63a8\u5e7f\u5230\u8bb8\u591a\u5b9e\u9645\u5b9e\u73b0\u7684RRT\u7248\u672c\uff0c\u540e\u8005\u901a\u5e38\u4f7f\u7528\u542f\u53d1\u5f0f\u8f6c\u5411\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6700\u8fd1[126]\u4e2d\u5df2\u7ecf\u8868\u660e\uff0c\u4f7f\u7528\u5177\u6709\u56fa\u5b9a\u65f6\u95f4\u6b65\u957f\u7684\u542f\u53d1\u5f0f\u8f6c\u5411\u7684RRT\u4e0d\u662f\u6982\u7387\u5b8c\u5907\u7684\u3002<\/p>\n

\u6b64\u5916\uff0cKaraman\u548cFrazzoli\u8bc1\u660e\u4e86RRT\u4ee5\u6982\u73871\u6536\u655b\u5230\u6b21\u4f18\u89e3\uff0c\u5e76\u8bbe\u8ba1\u4e86RRT\u7684\u6e10\u8fd1\u6700\u4f18\u7248\u672c\uff0c\u79f0\u4e3aRRT*\u7b97\u6cd5[127]\u3002\u5982\u7b97\u6cd54\u6240\u793a\uff0c\u5728\u6bcf\u6b21\u8fed\u4ee3\u65f6\uff0cRRT*\u8003\u8651\u4f4d\u4e8e\u65b0\u6dfb\u52a0\u7684\u9876\u70b9 x_{n e w} \u7684\u90bb\u57df\u5185\u7684\u4e00\u7ec4\u9876\u70b9\uff0c\u968f\u540e\uff08a\uff09\u5c06 x_{n e w} \u8fde\u63a5\u5230\u90bb\u57df\u4e2d\u7684\u4e00\u4e2a\u9876\u70b9\uff0c\u4ee5\u6700\u5c0f\u5316\u4ece mathrm{x}_{text {init }} \u5230 x_{n e w} \u7684\u8def\u5f84\u6210\u672c\uff0c\uff08b\uff09\u5982\u679c\u5c06\u8be5\u90bb\u57df\u5185\u7684\u67d0\u4e2a\u9876\u70b9\u91cd\u65b0\u8fde\u63a5\u5230 x_{n e w} \u540e\uff0c\u4ece mathrm{x}_{text {init }}\u5230\u8be5\u9876\u70b9\u7684\u8def\u5f84\u6210\u672c\u964d\u4f4e\u4e86\uff0c\u90a3\u4e48\u6267\u884c\u8fd9\u4e2a\u91cd\u8fde\u63a5\uff08rewire\uff09\u64cd\u4f5c\u3002\u8be5\u7b97\u6cd5\u7684\u4e00\u4e2a\u91cd\u8981\u7279\u5f81\u662f\uff0c\u90bb\u57df\u88ab\u5b9a\u4e49\u4e3a\u4ee5 x_{n e w} \u4e3a\u4e2d\u5fc3\u7684\u7403\uff0c\u5176\u534a\u5f84\u662f\u6811\u7684\u5927\u5c0f\u7684\u51fd\u6570\uff1a r=gamma sqrt[d]{(log n) \/ n} \uff0c\u5176\u4e2d n \u662f\u6811\u7684\u9876\u70b9\u7684\u6570\u91cf\uff0c d \u662f\u6784\u578b\u7a7a\u95f4\u7684\u7ef4\u5ea6\uff0c gamma \u662f\u4f9d\u8d56\u4e8e\u5b9e\u4f8b\u7684\u5e38\u6570\u3002\u7ed3\u679c\u8868\u660e\uff0c\u5bf9\u4e8e\u8fd9\u6837\u7684\u51fd\u6570\uff0c\u7403\u5185\u9876\u70b9\u7684\u671f\u671b\u4e2a\u6570\u662f\u6811\u7684\u5927\u5c0f\u7684\u5bf9\u6570\uff0c\u8fd9\u5bf9\u4e8e\u786e\u4fdd\u7b97\u6cd5\u51e0\u4e4e\u80af\u5b9a\u5730\u6536\u655b\u5230\u6700\u4f18\u8def\u5f84\u3001\u540c\u65f6\u4fdd\u6301\u4e0e\u6b21\u4f18RRT\u76f8\u540c\u7684\u6e10\u8fd1\u590d\u6742\u6027\u6765\u8bf4\u662f\u5fc5\u8981\u7684\u3002<\/p>\n

<\/p>\n

\u6587\u7ae0[80]\u4e2d\u9648\u8ff0\u4e86\u5fae\u5206\u7ea6\u675f\u4e0bRRT*\u7684\u6e10\u8fd1\u6700\u4f18\u6027\u7684\u5145\u5206\u6761\u4ef6\uff0c\u5e76\u4e14\u8bc1\u660e\uff0c\u8fd9\u5bf9\u4e8eDubins\u8f66\u8f86\u548c\u53cc\u79ef\u5206\u5668\u7cfb\u7edf\u6765\u8bf4\u662f\u53ef\u6ee1\u8db3\u7684\u3002\u5728\u968f\u540e\u7684\u5de5\u4f5c\u4e2d\uff0c\u4f5c\u8005\u8fdb\u4e00\u6b65\u8868\u660e\uff0c\u5728\u5c0f\u65f6\u95f4\u5c40\u90e8\u53ef\u8fbe\u7cfb\u7edf\uff08small-time locally attainable systems\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u8be5\u7b97\u6cd5\u4e0d\u4ec5\u53ef\u4ee5\u4fdd\u6301\u6e10\u8fd1\u6700\u4f18\u6027\uff0c\u800c\u4e14\u53ef\u4ee5\u4fdd\u6301\u8ba1\u7b97\u6548\u7387[82]\u3002\u5176\u4ed6\u76f8\u5173\u7684\u5de5\u4f5c\u5c06\u6ce8\u610f\u529b\u653e\u5728\u901a\u8fc7\u5c40\u90e8\u7ebf\u6027\u5316\u7cfb\u7edf\u52a8\u529b\u5b66[128]\u6216\u901a\u8fc7\u63a8\u5bfc\u5f97\u5230\u5177\u6709\u7ebf\u6027\u52a8\u529b\u5b66\u7684\u7cfb\u7edf\u7684\u95ed\u5f0f\u89e3[129]\uff0c\u6765\u5f97\u5230\u975e\u5b8c\u6574\u7cfb\u7edf\u7684\u8ddd\u79bb\u548c\u8f6c\u5411\u51fd\u6570\u3002\u53e6\u4e00\u65b9\u9762\uff0cRRTX\u662f\u4e00\u79cdRRT*\u7684\u6269\u5c55\u7b97\u6cd5\uff0c\u5176\u5141\u8bb8\u5728\u969c\u788d\u7269\u533a\u57df\u53d1\u751f\u53d8\u5316\u65f6\uff08\u4f8b\u5982\u9762\u5bf9\u6765\u81ea\u4f20\u611f\u5668\u7684\u65b0\u6570\u636e\u65f6\uff09\u8fdb\u884c\u5b9e\u65f6\u589e\u91cf\u91cd\u65b0\u89c4\u5212[130]\u3002\u57fa\u4e8e\u91c7\u6837\u7684\u7b97\u6cd5\u9886\u57df\u7684\u65b0\u8fdb\u5c55\u5305\u62ec\u63d0\u51fa\u4e86\u5728\u7f3a\u5c11\u7cbe\u786e\u8f6c\u5411\u7a0b\u5e8f\u7684\u60c5\u51b5\u4e0b\u5b9e\u73b0\u6e10\u8fd1\u6700\u4f18\u7684\u7b97\u6cd5\u3002\u7279\u522b\u662f\uff0cLi\u7b49\u4eba[84]\u6700\u8fd1\u63d0\u51fa\u4e86\u9762\u5411\u6e10\u8fd1\uff08\u8fd1\u4f3c\uff09\u6700\u4f18\u8def\u5f84\u89c4\u5212\u7684\u7a33\u5b9a\u7a00\u758f\u6811\uff08SST\uff09\u3002<\/p>\n

\u8be5\u65b9\u6cd5\u57fa\u4e8e\u901a\u8fc7\u7cfb\u7edf\u52a8\u529b\u5b66\u7684\u524d\u5411\u6a21\u578b\u6765\u4f20\u64ad\u7684\u968f\u673a\u91c7\u6837\u63a7\u5236\u6811\uff0c\u4ece\u800c\u4fee\u526a\u51fa\uff08prune out\uff09\u5c40\u90e8\u6b21\u4f18\u7684\u5206\u652f\uff0c\u4ee5\u786e\u4fdd\u6811\u4fdd\u6301\u7a00\u758f\u3002<\/p>\n

F. \u5b9e\u9645\u90e8\u7f72\u6211\u4eec\u5df2\u7ecf\u8ba8\u8bba\u4e86\u81ea\u52a8\u9a7e\u9a76\u8f66\u8f86\u7684\u4e09\u7c7b\u8def\u5f84\u89c4\u5212\u65b9\u6cd5\uff1a\u53d8\u5206\u6cd5\u3001\u56fe\u641c\u7d22\u6cd5\u548c\u57fa\u4e8e\u589e\u91cf\u6811\u7684\u65b9\u6cd5\u3002\u81ea\u52a8\u9a7e\u9a76\u7cfb\u7edf\u7684\u5b9e\u9645\u73b0\u573a\u90e8\u7f72\u7b97\u6cd5\u6765\u81ea\u4e0a\u8ff0\u7684\u6240\u6709\u7c7b\u522b\u3002<\/p>\n

\u4f8b\u5982\uff0c\u5373\u4f7f\u662fDARPA\u57ce\u5e02\u6311\u6218\u8d5b\u7684\u524d\u56db\u4f4d\u6210\u529f\u53c2\u4e0e\u8005\uff0c\u91c7\u7528\u7684\u8fd0\u52a8\u89c4\u5212\u65b9\u6cd5\u4e5f\u5b58\u5728\u663e\u8457\u5dee\u5f02\u3002\u4f5c\u4e3a\u6311\u6218\u7684\u80dc\u5229\u8005\uff0c\u5361\u5185\u57fa\u6885\u9686\u5927\u5b66\u7684Boss\u8f66\u8f86\u4f7f\u7528\u53d8\u5206\u6280\u672f\u6765\u5728\u7ed3\u6784\u5316\u7684\u73af\u5883\u4e2d\u751f\u6210\u5c40\u90e8\u8f68\u8ff9\uff0c\u5e76\u5728\u56db\u7ef4\u6784\u578b\u7a7a\u95f4\uff08\u7531\u4f4d\u7f6e\u3001\u65b9\u5411\u548c\u901f\u5ea6\u7ec4\u6210\uff09\u4e2d\u4f7f\u7528\u6805\u683c\u56fe\u4ee5\u53caAnytime D*\uff0c\u4ee5\u5728\u505c\u8f66\u573a\u4e2d\u627e\u5230\u65e0\u78b0\u649e\u8def\u5f84[131]\u3002\u636e\u62a5\u9053\uff0c\u65af\u5766\u798f\u5927\u5b66\u56e2\u961f\u5f00\u53d1\u7684\u4e9a\u519b\u8f66\u8f86\u4f7f\u7528\u4e86\u4e00\u79cd\u88ab\u79f0\u4e3a\u6df7\u5408A*\uff08Hybrid A*\uff09\u7684\u641c\u7d22\u7b56\u7565\u3002<\/p>\n

\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u5176\u901a\u8fc7\u9012\u5f52\u5730\u5e94\u7528\u6709\u9650\u7684\u4e00\u7ec4\u7b56\u7565\uff0c\u60f0\u6027\u5730\u6784\u5efa\u4e00\u4e2a\u8fd0\u52a8\u57fa\u5143\u6811\u3002\u8be5\u641c\u7d22\u7531\u7cbe\u5fc3\u8bbe\u8ba1\u7684\u542f\u53d1\u5f0f\u65b9\u6cd5\u6307\u5bfc\uff0c\u5e76\u4e14\u5728\u6784\u578b\u7a7a\u95f4\u7684\u7ed9\u5b9a\u533a\u57df\u4e2d\u4ec5\u4fdd\u7559\u5355\u4e2a\u8282\u70b9\uff0c\u6765\u786e\u4fdd\u6811\u7684\u7a00\u758f\u6027[132]\u3002\u7c7b\u4f3c\u5730\uff0c\u5f17\u5409\u5c3c\u4e9a\u7406\u5de5\u5927\u5b66VictorTango\u56e2\u961f\u5f00\u53d1\u7684\u5b63\u519b\u8f66\u8f86\u6784\u9020\u4e86\u4e00\u4e2a\u5305\u542b\u53ef\u80fd\u7684\u7b56\u7565\u7684\u79bb\u6563\u5316\u56fe\uff0c\u5e76\u4f7f\u7528A*\u7b97\u6cd5\u641c\u7d22\u6b64\u56fe[133]\u3002\u6700\u540e\uff0c\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u5f00\u53d1\u7684\u8f66\u8f86\u4f7f\u7528\u4e86\u4e00\u79cdRRT\u7b97\u6cd5\u7684\u53d8\u4f53\uff0c\u79f0\u4e3a\u5177\u6709\u504f\u7f6e\u91c7\u6837\uff08biased sampling\uff09\u7684\u95ed\u73afRRT\u7b97\u6cd5[134]\u3002<\/p>\n

5. \u8f66\u8f86\u63a7\u5236\u8fd0\u52a8\u89c4\u5212\u8fc7\u7a0b\u63d0\u4f9b\u4e86\u95ee\u98984.1\u62164.2\u7684\u89e3\u3002<\/p>\n

\u53cd\u9988\u63a7\u5236\u5668\u7684\u4f5c\u7528\u662f\uff0c\u5728\u5b58\u5728\u5efa\u6a21\u8bef\u5dee\u548c\u5176\u4ed6\u5f62\u5f0f\u7684\u4e0d\u786e\u5b9a\u6027\u7684\u60c5\u51b5\u4e0b\uff0c\u7a33\u5b9a\u4e8e\u53c2\u8003\u8def\u5f84\u6216\u8f68\u8ff9\u3002\u6839\u636e\u8fd0\u52a8\u89c4\u5212\u5668\u63d0\u4f9b\u7684\u53c2\u8003\uff0c\u63a7\u5236\u76ee\u6807\u53ef\u4ee5\u662f\u8def\u5f84\u7a33\u5b9a\u6216\u8f68\u8ff9\u7a33\u5b9a\u3002\u66f4\u6b63\u5f0f\u5730\u8bf4\uff0c\u8def\u5f84\u7a33\u5b9a\u95ee\u9898\u8868\u8ff0\u5982\u4e0b\uff1a\u95ee\u9898 5.1 \u8def\u5f84\u7a33\u5b9a\uff08Path Stabilization\uff09\u7ed9\u5b9a\u53d7\u63a7\u5fae\u5206\u65b9\u7a0b dot{x}=f(x, u) \uff0c\u53c2\u8003\u8def\u5f84 x_{text {ref }}: mathbb{R} rightarrow mathbb{R}^n \uff0c\u4ee5\u53ca\u53c2\u8003\u901f\u5ea6 v_{text {ref }}: mathbb{R} rightarrow mathbb{R} \uff0c\u627e\u5230\u4e00\u4e2a\u53cd\u9988\u5f8b\uff0c\u4f7f\u5f97 dot{x}=f(x, u(x)) \u7684\u89e3\u6ee1\u8db3\u5982\u4e0b\u547d\u9898\uff1a\u5bf9\u4e8e forall epsilon>0, t_10 \u4ee5\u53ca\u53ef\u5fae\u5206\u7684 S: mathbb{R} rightarrow mathbb{R} \uff0c\u4f7f\u5f97begin{aligned} & 1) rightarrowleft|xleft(t_1right)-x_{text {ref }}left(sleft(t_1right)right)right| leq delta quad Longrightarrowleft|xleft(t_2right)-x_{text {ref }}left(sleft(t_2right)right)right| leq epsilon \\ & 2) rightarrow lim _{t rightarrow infty}left|x(t)-x_{text {ref }}(s(t))right|=0 \\ & 3) rightarrow lim _{t rightarrow infty} dot{s}(t)=v_{text {ref }}(s(t))end{aligned}<\/p>\n

\u5b9a\u6027\u5730\u8bf4\uff0c\u4e0a\u8ff0\u7684\u6761\u4ef6\u8868\u793a\uff1a(1) \u5c0f\u7684\u521d\u59cb\u8ddf\u8e2a\u8bef\u5dee\u4ecd\u5c06\u4fdd\u6301\u5c0f\uff0c(2) \u8ddf\u8e2a\u8bef\u5dee\u5fc5\u987b\u6536\u655b\u5230\u96f6\uff0c(3) \u6cbf\u7740\u53c2\u8003\u8def\u5f84\u7684\u884c\u9a76\u8d8b\u8fd1\u4e8e\u89c4\u5b9a\u901f\u7387\uff08nominal rate\uff09\u3002\u8bb8\u591a\u5df2\u88ab\u63d0\u51fa\u7684\u8f66\u8f86\u63a7\u5236\u5f8b\uff0c\u5305\u62ec\u672c\u8282\u5c06\u8981\u8ba8\u8bba\u7684\u51e0\u4e2a\uff0c\u90fd\u4f7f\u7528\u4ee5\u4e0b\u5f62\u5f0f\u7684\u53cd\u9988\u5f8b\uff1au(x)=fleft(arg min _gammaleft|x-x_{r e f}(gamma)right|right) (5.1)\u5176\u4e2d\u53cd\u9988\u662f\u53c2\u8003\u8def\u5f84\u4e0a\u8ddd\u8f66\u8f86\u6700\u8fd1\u7684\u70b9\u7684\u51fd\u6570\u3002\u8be5\u5f62\u5f0f\u7684\u63a7\u5236\u7684\u4e00\u4e2a\u91cd\u8981\u95ee\u9898\u662f\u95ed\u73af\u5411\u91cf\u573a f(x, u(x)) \u975e\u8fde\u7eed\u3002\u5982\u679c\u8def\u5f84\u662f\u81ea\u76f8\u4ea4\u7684\u6216\u5728\u67d0\u4e00\u70b9\u4e0d\u53ef\u5fae\u5206\uff0c\u5219 f(x, u(x)) \u5b58\u5728\u7684\u4e0d\u8fde\u7eed\u6027\u5c06\u76f4\u63a5\u51fa\u73b0\u5728\u8def\u5f84\u4e0a\u3002\u82e5\u6267\u884c\u7684\u8f68\u8ff9\u5b58\u5728\u4e0d\u8fde\u7eed\u6027\uff0c\u5219\u5c06\u5bfc\u81f4\u4e0d\u53ef\u9884\u6d4b\u7684\u884c\u4e3a\u3002\u8fd9\u79cd\u4e0d\u8fde\u7eed\u6027\u5982\u56fe5.1\u6240\u793a\u3002<\/p>\n

\u6587\u7ae0[135]\u63d0\u51fa\u4e86\u4e00\u79cd\u53cd\u63a8\uff08backstepping\uff09\u63a7\u5236\u8bbe\u8ba1\uff0c\u5176\u4e0d\u4f7f\u7528\u516c\u5f0f(5.1)\u7684\u53cd\u9988\u5f8b\u3002\u8f68\u8ff9\u7a33\u5b9a\u95ee\u9898\u66f4\u52a0\u7b80\u5355\uff0c\u4f46\u662f\u8fd9\u4e9b\u63a7\u5236\u5668\u5bb9\u6613\u53d7\u5230\u6027\u80fd\u9650\u5236[136]\u3002<\/p>\n

\u95ee\u9898 5.2 \u8f68\u8ff9\u7a33\u5b9a\uff08Trajectory Stabilization\uff09\u7ed9\u5b9a\u53d7\u63a7\u5fae\u5206\u65b9\u7a0b dot{x}=f(x, u) \u548c\u53c2\u8003\u8f68\u8ff9 x_{r e f}(t) \uff0c\u627e\u5230 pi(x) \u4f7f\u5f97 dot{x}=f(x, pi(x)) \u7684\u89e3\u6ee1\u8db3\u5982\u4e0b\u547d\u9898\uff1a\u5bf9\u4e8e forall epsilon>0, t_10 \uff0c\u4f7f\u5f97begin{aligned} & text { 1) } rightarrowleft|xleft(t_1right)-x_{text {ref }}left(t_1right)right| leq delta \\ & quad Rightarrowleft|xleft(t_2right)-x_{text {ref }}left(t_2right)right| leq epsilon \\ & text { 2) } rightarrow lim _{t rightarrow infty}left|x(t)-x_{text {ref }}(t)right|=0 end{aligned}<\/p>\n

\u5728\u8bb8\u591a\u60c5\u51b5\u4e0b\uff0c\u5206\u6790\u8f68\u8ff9\u7684\u7a33\u5b9a\u6027\u53ef\u4ee5\u7b80\u5316\u4e3a\u786e\u5b9a\u65f6\u53d8\u7cfb\u7edf\u7684\u539f\u70b9\u7684\u7a33\u5b9a\u6027\u3002\u674e\u96c5\u666e\u8bfa\u592b\u5b9a\u7406\u7684\u57fa\u672c\u5f62\u5f0f\u4ec5\u9002\u7528\u4e8e\u65f6\u4e0d\u53d8\u7cfb\u7edf\u3002\u7136\u800c\uff0c\u65f6\u53d8\u7cfb\u7edf\u7684\u7a33\u5b9a\u6027\u7406\u8bba\u4e5f\u5f97\u5230\u4e86\u5f88\u597d\u7684\u5efa\u7acb\uff08\u4f8b\u5982[137]\u7684\u5b9a\u74064.9\uff09\u3002\u5bf9\u4e8e\u5404\u79cd\u7c7b\u578b\u7684\u7a33\u5b9a\u6027\uff0c\u4e00\u4e9b\u6709\u7528\u7684\u672f\u8bed\u5305\u62ec\uff1a\u65f6\u53d8\u7cfb\u7edf\u7684\u4e00\u81f4\u6e10\u8fd1\u7a33\u5b9a\u6027\uff08uniformly \/ uniform asymptotic stability\uff09\uff0c\u5176\u65ad\u8a00\u4e0a\u8ff0\u95ee\u9898\u7684\u6761\u4ef6 1 \u4e2d\u7684 delta \u72ec\u7acb\u4e8e t_{1} \u3002\u6307\u6570\u7a33\u5b9a\u6027\uff08exponential stability\uff09\uff0c\u5176\u65ad\u8a00\u6536\u655b\u901f\u7387\u4ee5\u6307\u6570\u8870\u51cf\u4e3a\u754c\u3002<\/p>\n

\u9700\u8981\u6ce8\u610f\u7684\u4e00\u4e2a\u5fae\u5999\u95ee\u9898\u662f\uff0c\u5f53\u65f6\u95f4\u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u63a7\u5236\u5668\u7684\u89c4\u8303\u901a\u5e38\u7528\u6e10\u8fd1\u8ddf\u8e2a\u8bef\u5dee\u6765\u8868\u793a\u3002\u5728\u5b9e\u8df5\u4e2d\uff0c\u53c2\u8003\u8f68\u8ff9\u662f\u6709\u9650\u7684\uff0c\u56e0\u6b64\u8fd8\u5e94\u5f53\u8003\u8651\u7cfb\u7edf\u7684\u77ac\u6001\u54cd\u5e94\u3002\u672c\u8282\u7684\u5269\u4f59\u90e8\u5206\u5c06\u9009\u62e9\u4e00\u4e9b\u9002\u7528\u4e8e\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u7684\u63a7\u5236\u8bbe\u8ba1\uff0c\u5e76\u5bf9\u5176\u8fdb\u884c\u7efc\u8ff0\u3002\u88682\u63d0\u4f9b\u4e86\u8fd9\u4e9b\u63a7\u5236\u5668\u7684\u6982\u8ff0\u3002\u7b2c5-A\u5c0f\u8282\u8be6\u7ec6\u4ecb\u7ecd\u4e86\u7528\u4e8e\u8fd0\u52a8\u5b66\u6a21\u578b\u7684\u8def\u5f84\u7a33\u5b9a\u7684\u4e00\u4e9b\u6709\u6548\u63a7\u5236\u7b56\u7565\u3002\u7b2c5-B2\u5c0f\u8282\u8ba8\u8bba\u4e86\u8f68\u8ff9\u7a33\u5b9a\u6280\u672f\u3002\u7b2c5-C\u5c0f\u8282\u5c06\u8ba8\u8bba\u7684\u9884\u6d4b\u63a7\u5236\u7b56\u7565\u5bf9\u4e8e\u66f4\u590d\u6742\u7684\u8f66\u8f86\u6a21\u578b\u662f\u6709\u6548\u7684\uff0c\u53ef\u4ee5\u5e94\u7528\u4e8e\u8def\u5f84\u548c\u8f68\u8ff9\u7a33\u5b9a\u3002<\/p>\n

<\/p>\n

\u56fe5.1\uff1a(5.5)\u7684\u53ef\u89c6\u5316\u9ed1\u7ebf\u8868\u793a\u53c2\u8003\u8def\u5f84\u7684\u6837\u4f8b\u3002\u5e73\u9762\u4e0a\u6bcf\u4e2a\u70b9\u7684 s \u503c\u7528\u989c\u8272\u8868\u793a\uff0c\u8fd9\u63cf\u7ed8\u4e86(5.5)\u4e2d\u7684\u4e0d\u8fde\u7eed\u6027\u3002<\/p>\n

<\/p>\n

\u88682\uff1a\u672c\u8282\u8ba8\u8bba\u7684\u63a7\u5236\u5668\u6982\u8ff0\u8bf4\u660e\uff1a a: \u5c40\u90e8\u6307\u6570\u7a33\u5b9a\u6027\uff08LES\uff09\uff1b b: \u5047\u8bbe(5.1)\u662f\u901a\u8fc7\u5728\u8def\u5f84\u6216\u8f68\u8ff9\u7684 n \u70b9\u79bb\u6563\u5316\u4e0a\u7684\u7ebf\u6027\u641c\u7d22\u6765\u8bc4\u4f30\u7684\uff1b c: \u5047\u8bbe\u4f7f\u7528\u5185\u70b9\u6cd5\u6c42\u89e3(5.28)\uff0c\u5176\u4e2d\u65f6\u95f4\u8303\u56f4\/\u533a\u95f4\uff08horizon\uff09\u4e3a n \uff0c\u89e3\u7684\u7cbe\u786e\u5ea6\u4e3a epsilon \uff1b d: \u57fa\u4e8e\u4f7f\u7528\u6700\u9661\u4e0b\u964d\u7684\u3001\u5411(5.25)\u7684\u5c40\u90e8\u6700\u5c0f\u503c\u6536\u655b\u7684\u6e10\u8fd1\u6536\u655b\u901f\u7387\uff0c\u4e0d\u4fdd\u8bc1\u8fd4\u56de\u89e3\u6216\u627e\u5230\u5168\u5c40\u6700\u5c0f\u503c\uff1b e: \u7531 mathbb{R}^m \u4e2d\u7684\u5404\u4e2a\u8f93\u5165\u5b9a\u4e49\u7684\u72b6\u6001\u7a7a\u95f4 mathbb{R}^n \u4e0a\u7684\u5411\u91cf\u573a\u662f\u8fde\u7eed\u53ef\u5fae\u5206\u51fd\u6570\uff0c\u4ece\u800c\u5b9a\u4e49\u5173\u4e8e\u53c2\u8003\u7684\u6210\u672c\u68af\u5ea6\u6216\u7ebf\u6027\u5316\u3002<\/p>\n

A. \u57fa\u4e8e\u8fd0\u52a8\u5b66\u6a21\u578b\u7684\u8def\u5f84\u7a33\u5b9a<\/p>\n

1) \u7eaf\u8ffd\u8e2a\uff08Pure Pursuit\uff09\u7eaf\u8ffd\u8e2a\u7b97\u6cd5\u662f\u6700\u65e9\u63d0\u51fa\u7684\u8def\u5f84\u8ddf\u8e2a\uff08path tracking\uff09\u7b56\u7565\u4e4b\u4e00\u3002\u6587\u7ae0[138]\u4e2d\u7b2c\u4e00\u6b21\u8ba8\u8bba\u4e86\u5b83\uff0c\u5e76\u5728[52][139]\u4e2d\u8fdb\u884c\u4e86\u8be6\u7ec6\u9610\u8ff0\u3002\u8be5\u7b56\u7565\u53ca\u5176\u53d8\u4f53\uff08\u4f8b\u5982[140][141]\uff09\u5df2\u88ab\u8bc1\u660e\u662f\u8f66\u8f86\u63a7\u5236\u4e0d\u53ef\u6216\u7f3a\u7684\u5de5\u5177\uff0c\u56e0\u4e3a\u5176\u5b9e\u65bd\u7b80\u5355\u4e14\u6027\u80fd\u4ee4\u4eba\u6ee1\u610f\u3002\u8bb8\u591a\u76f8\u5173\u6210\u679c\uff0c\u5305\u62ecDARPA\u5927\u6311\u6218\u4e2d\u7684\u4e24\u8f86\u8f66[8]\u548cDARPA\u57ce\u5e02\u6311\u6218\u4e2d\u7684\u4e09\u8f86\u8f66[9]\u90fd\u62a5\u9053\u4f7f\u7528\u4e86\u7eaf\u8ffd\u8e2a\u5668\u3002<\/p>\n

<\/p>\n

\u56fe5.2\uff1a\u7eaf\u8ffd\u8e2a\u63a7\u5236\u5668\u7684\u51e0\u4f55\u7ed3\u6784<\/p>\n

\u4e00\u4e2a\u5706\uff08\u84dd\u8272\uff09\u4f4d\u4e8e\u540e\u8f6e\u4f4d\u7f6e\u548c\u53c2\u8003\u8def\u5f84\uff08\u68d5\u8272\uff09\u4e4b\u95f4\uff0c\u4f7f\u5f97\u5f26\u957f\uff08\u7eff\u8272\uff09\u4e3a\u524d\u77bb\u8ddd\u79bb L \uff0c\u5e76\u4e14\u8be5\u5706\u4e0e\u5f53\u524d\u8f66\u8f86\u7684\u671d\u5411\u65b9\u5411\u76f8\u5207\u3002\u7eaf\u8ffd\u8e2a\u7684\u63a7\u5236\u5f8b\u7684\u57fa\u672c\u601d\u60f3\u662f\uff0c\u5229\u7528\u534a\u5706\u5f27\u6765\u62df\u5408\u4ece\u8f66\u8f86\u5f53\u524d\u6784\u578b\u5230\u53c2\u8003\u8def\u5f84\u4e0a\u7279\u5b9a\u4e00\u70b9\uff0c\u8fd9\u4e2a\u7279\u5b9a\u70b9\u5728\u8f66\u8f86\u524d\u65b9\u3001\u8ddd\u79bb\u4e3a L \uff0c\u79f0\u4e3a\u524d\u77bb\u8ddd\u79bb\uff08lookahead distance\uff09\u3002<\/p>\n

\u56fe5.2\u63cf\u8ff0\u4e86\u8fd9\u4e2a\u51e0\u4f55\u7ed3\u6784\u3002\u901a\u8fc7\u4e24\u4e2a\u70b9\u548c\u4e00\u4e2a\u5207\u7ebf\u6765\u5b9a\u4e49\u8fd9\u4e2a\u5706\uff1a\u4ee3\u8868\u6c7d\u8f66\u6240\u5904\u4f4d\u7f6e\u7684\u70b9\u3001\u5728\u8be5\u70b9\u5904\u4e0e\u8f66\u8f86\u671d\u5411\u65b9\u5411\u76f8\u540c\u7684\u5207\u7ebf\u3001\u8f66\u8f86\u524d\u65b9\u76f8\u8ddd\u4e00\u4e2a\u524d\u77bb\u8ddd\u79bb\u7684\u53c2\u8003\u8def\u5f84\u4e0a\u7684\u70b9\u3002\u5706\u7684\u66f2\u7387\u7531\u4e0b\u5f0f\u7ed9\u51fa\uff1akappa=frac{2 sin alpha}{L} (5.2)\u5bf9\u4e8e\u8f66\u8f86\u901f\u5ea6 v_r \uff0c\u6307\u4ee4\u7684\u822a\u5411\u901f\u7387\u4e3aomega=frac{2 v_r sin alpha}{L} (5.3)\u5728\u8fd9\u4e2a\u63a7\u5236\u5668\u7684\u6700\u65e9\u62a5\u9053[138]\u4e2d\uff0c\u4ece\u76f8\u673a\u7684\u8f93\u51fa\u6570\u636e\u4e2d\u76f4\u63a5\u8ba1\u7b97\u89d2\u5ea6 alpha \u3002<\/p>\n

\u7136\u800c\uff0c alpha \u53ef\u4ee5\u7528\u60ef\u6027\u5750\u6807\u7cfb\u8868\u793a\uff0c\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u72b6\u6001\u53cd\u9988\u63a7\u5236\u3002\u8003\u8651\u6784\u578b left(x_r, y_r, thetaright)^T \u4ee5\u53ca\u8def\u5f84\u70b9 left(x_{r e f}(s), y_{r e f}(s)right) \uff0c\u4f7f\u5f97left|left(x_{text {ref }}(s), y_{text {ref }}(s)right)-left(x_r, y_rright)right|=L \u3002\u7531\u4e8e\u8def\u5f84\u4e0a\u901a\u5e38\u4e0d\u6b62\u6709\u4e00\u4e2a\u8fd9\u6837\u7684\u70b9\uff0c\u6211\u4eec\u9009\u62e9\u53c2\u6570 s \u7684\u503c\u6700\u5927\u7684\u70b9\uff0c\u4ee5\u552f\u4e00\u5730\u5b9a\u4e49\u63a7\u5236\u3002\u8fdb\u800c\uff0c alpha \u7531\u4e0b\u5f0f\u7ed9\u51fa\uff1aalpha=arctan left(frac{y_{text {ref }}-y_r}{x_{text {ref }}-x_r}right)-theta (5.4)<\/p>\n

\u5047\u8bbe\u8def\u5f84\u6ca1\u6709\u66f2\u7387\u4e14\u8f66\u8f86\u901f\u5ea6\u6052\u5b9a\uff08\u53ef\u80fd\u4e3a\u8d1f\uff09\uff0c\u5219\u7eaf\u8ffd\u8e2a\u63a7\u5236\u5668\u53ef\u4ee5\u7ed9\u51fa\u95ee\u98985.1\u7684\u89e3\u3002\u5bf9\u4e8e\u56fa\u5b9a\u7684\u975e\u96f6\u66f2\u7387\uff0c\u7eaf\u8ffd\u8e2a\u5177\u6709\u8f83\u5c0f\u7684\u7a33\u6001\u8ddf\u8e2a\u8bef\u5dee\u3002\u5728\u8f66\u8f86\u5230\u8def\u5f84\u7684\u8ddd\u79bb\u5927\u4e8e L \u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u5668\u8f93\u51fa\u5e76\u672a\u88ab\u5b9a\u4e49\u3002\u53e6\u4e00\u4e2a\u8003\u8651\u662f\uff0c\u53c2\u8003\u8def\u5f84\u7684\u66f2\u7387\u7684\u53d8\u5316\u4f1a\u5bfc\u81f4\u6c7d\u8f66\u504f\u79bb\u53c2\u8003\u8f68\u8ff9\uff0c\u8fd9\u5bf9\u4e8e\u6cbf\u9053\u8def\u884c\u9a76\u7684\u4efb\u52a1\u53ef\u80fd\u53ef\u4ee5\u63a5\u53d7\uff0c\u4f46\u662f\u5bf9\u4e8e\u8ddf\u8e2a\u6cca\u8f66\u7b56\u7565\u53ef\u80fd\u5b58\u5728\u95ee\u9898\u3002\u6700\u540e\uff0c\u968f\u7740\u8f66\u8f86\u901f\u5ea6\u7684\u589e\u52a0\uff0c\u822a\u5411\u901f\u7387\u6307\u4ee4 omega \u5bf9\u53cd\u9988\u89d2 alpha \u66f4\u52a0\u654f\u611f\u3002\u89e3\u51b3\u6b64\u95ee\u9898\u7684\u4e00\u4e2a\u5e38\u89c1\u65b9\u6cd5\u662f\u6839\u636e\u8f66\u901f\u6765\u7f29\u653e L \u3002<\/p>\n

2) \u57fa\u4e8e\u540e\u8f6e\u4f4d\u7f6e\u7684\u53cd\u9988\u540e\u8f6e\u53cd\u9988\u63a7\u5236\u4f7f\u7528\u540e\u8f6e\u4f4d\u7f6e\u4f5c\u4e3a\u8f93\u51fa\uff0c\u4f7f\u4e4b\u7a33\u5b9a\u4e8e\u4e00\u6761\u6807\u51c6\u540e\u8f6e\u8def\u5f84[56]\u3002\u63a7\u5236\u5668\u89c4\u5b9as(t)=arg min _gammaleft|left(x_r(t), y_r(t)right)-left(x_{text {ref }}(gamma), y_{text {ref }}(gamma)right)right| (5.5)\u6587\u7ae0[56]\u4e2d\u63cf\u8ff0\u4e86\u5173\u4e8e\u53c2\u8003\u8def\u5f84\u548c\u5305\u542b\u53c2\u8003\u8def\u5f84\u7684\u6709\u9650\u57df\u7684\u8be6\u7ec6\u5047\u8bbe\uff0c\u5728\u8fd9\u4e2a\u6709\u9650\u57df\u5185\uff0c(5.5)\u662f\u8fde\u7eed\u51fd\u6570\u3002<\/p>\n

s(t) \u5904\u4e0e\u8def\u5f84\u76f8\u5207\u7684\u5355\u4f4d\u6b63\u5207\u7531\u4e0b\u5f0f\u7ed9\u51fahat{t}=frac{left(left.frac{partial x_{r e f}}{partial s}right|_{s(t)},left.frac{partial y_{r e f}}{partial s}right|_{s(t)}right)}{left|left(frac{partial x_{r e f}(s(t))}{partial s}, frac{partial y_{r e f}(s(t))}{partial s}right)right|} (5.6)\u4e14\u8ddf\u8e2a\u8bef\u5dee\u5411\u91cf\u4e3ad(t):=left(x_r(t), y_r(t)right)-left(x_{text {ref }}(s(t)), y_{text {ref }}(s(t))right) (5.7)<\/p>\n

\u8fd9\u4e9b\u503c\u7528\u4e8e\u6839\u636e\u8def\u5f84 e \u8ba1\u7b97\u6a2a\u5411\u8bef\u5dee\u5750\u6807\uff0c\u5176\u4e2d\u8def\u5f84 e \u662f\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u5411\u91cf\u79efe=d_x hat{t}_y-d_y hat{t}_x (5.8)\u5176\u4e2d\u4e0b\u6807\u8868\u793a\u5411\u91cf\u7684\u5206\u91cf\u7d22\u5f15\u3002\u8fd9\u79cd\u63a7\u5236\u7b56\u7565\u4f7f\u7528\u8f66\u8f86\u7684\u822a\u5411\u5411\u91cf\u548c\u8def\u5f84\u7684\u5207\u7ebf\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6 theta_e \u3002theta_e(t)=theta-arctan _2left(frac{partial y_{text {ref }}(s(t))}{partial s}, frac{partial x_{text {ref }}(s(t))}{partial s}right) (5.9)\u8fd9\u4e2a\u51e0\u4f55\u5173\u7cfb\u5982\u56fe5.3\u6240\u793a\u3002<\/p>\n

<\/p>\n

\u56fe5.3\uff1a\u57fa\u4e8e\u540e\u8f6e\u7684\u53cd\u9988\u63a7\u5236\u7684\u53cd\u9988\u53d8\u91cftheta_e \u662f\u8def\u5f84\u4e0a\u8ddd\u79bb\u540e\u8f6e\u6700\u8fd1\u7684\u70b9\u5904\u7684\u5207\u7ebf\u4e0e\u6c7d\u8f66\u822a\u5411\u4e4b\u95f4\u7684\u5dee\u503c\u3002\u6807\u91cf e \u7684\u5927\u5c0f\uff08\u7edd\u5bf9\u503c\uff09\u7528\u7ea2\u8272\u8868\u793a\u3002<\/p>\n

\u5982\u56fe\u6240\u793a\uff0c e>0 \uff0c\u800c\u5bf9\u4e8e\u6c7d\u8f66\u4f4d\u4e8e\u8def\u5f84\u5de6\u4fa7\u7684\u60c5\u51b5\uff0ce<0\u3002\u5750\u6807 left(s, e, theta_eright) \u7684\u53d8\u5316\u7387\u6ee1\u8db3\uff1abegin{aligned} & dot{s}=frac{v_r cos theta_e}{1-kappa(s) e} \\ &dot{e}=v_r sin theta_e \\ & theta_e=omega-frac{v_r kappa(s) cos theta_e}{1-kappa(s) e} end{aligned} (5.10)<\/p>\n

\u5176\u4e2d kappa(s) \u8868\u793a\u8def\u5f84\u5728 s \u5904\u7684\u66f2\u7387\u3002\u4e0b\u5f0f\u4e2d\u7684\u8f6c\u5411\u7387\u6307\u4ee4\u4e3a\u5c40\u90e8\u6e10\u8fd1\u6536\u655b\u63d0\u4f9b\u4e86\u8fde\u7eed\u4e8c\u6b21\u53ef\u5fae\u5206\u7684\u8def\u5f84\uff1aomega=frac{v_r kappa(s) cos theta_e}{1-kappa(s) e}-g_1left(e, theta_e, tright) theta_e-k_2 v_r frac{sin theta_e}{theta_e} e (5.11)<\/p>\n

\u5176\u4e2d g_1left(e, theta_e, tright)>0, k_2>0, v_r neq 0 \uff0c\u53ef\u4ee5\u901a\u8fc7[56]\u4e2d\u5229\u7528\u4e86\u5750\u6807\u7cfb(5.10)\u7684\u674e\u96c5\u666e\u8bfa\u592b\u51fd\u6570Vleft(e, theta_eright)=e^2+theta_e^2 \/ k_2 \u6765\u8bc1\u660e\u3002\u4e4b\u6240\u4ee5\u8981\u6c42\u8def\u5f84\u662f\u4e8c\u6b21\u53ef\u5fae\u5206\u7684\uff0c\u662f\u56e0\u4e3a\u53cd\u9988\u5b9a\u5f8b\u4e2d\u51fa\u73b0\u4e86\u66f2\u7387\u3002<\/p>\n

\u8fd9\u79cd\u63a7\u5236\u5f8b\u7684\u4e00\u4e2a\u4f18\u70b9\u662f\uff0c\u7a33\u5b9a\u6027\u4e0d\u53d7 v_r \u7684\u6b63\u8d1f\u53f7\u7684\u5f71\u54cd\uff0c\u4ece\u800c\u4f7f\u5176\u9002\u7528\u4e8e\u53cd\u5411\u9a7e\u9a76\u3002\u5f53 k_e>0 \u65f6\uff0c\u8bbe\u7f6e g_1left(v_r, theta_e, tright)=k_thetaleft|v_rright| \u53ef\u4ee5\u5b9e\u73b0\u5c40\u90e8\u6307\u6570\u6536\u655b\uff0c\u4e14\u6536\u655b\u901f\u7387\u4e0e\u8f66\u901f\u65e0\u5173\uff08\u53ea\u8981 v_r neq 0\uff09\u3002<\/p>\n

\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684\u63a7\u5236\u5f8b\u662fomega=frac{v_r kappa(s) cos theta_e}{1-kappa(s) e}-left(k_thetaleft|v_rright|right) theta_e-left(k_e v_r frac{sin theta_e}{theta_e}right) e (5.12)3)<\/p>\n

\u57fa\u4e8e\u524d\u8f6e\u4f4d\u7f6e\u7684\u53cd\u9988\u524d\u8f6e\u53cd\u9988\u63a7\u5236\u662f\u65af\u5766\u798f\u5927\u5b66\u57282005\u5e74DARPA\u5927\u6311\u6218\u4e2d\u63d0\u51fa\u5e76\u4f7f\u7528\u7684\u65b9\u6cd5[55][8]\uff0c\u5176\u5c06\u524d\u8f6e\u4f4d\u7f6e\u4f5c\u4e3a\u63a7\u5236\u53d8\u91cf\u3002\u5b83\u540c\u6837\u4f7f\u7528\u4e0a\u4e00\u8282\u4e2d\u7684\u53d8\u91cf s(t), e(t), theta_e(t) \uff0c\u4e0d\u540c\u7684\u662f\u5176\u6839\u636e\u524d\u8f6e\u4f4d\u7f6e\u800c\u4e0d\u662f\u540e\u8f6e\u4f4d\u7f6e\u6765\u8ba1\u7b97 e(t) \u3002\u53d6\u6a2a\u5411\u8bef\u5dee\u7684\u65f6\u95f4\u5bfc\u6570dot{e}=v_f sin left(theta_e+deltaright) (5.13)\u5bf9\u4e8e\u4e0a\u5f0f\u4e2d\u7684\u8bef\u5dee\u53d8\u5316\u7387\uff0c\u5982\u679c\u5927\u5c0f\u5c0f\u4e8e v_f \uff0c\u5219\u53ef\u76f4\u63a5\u7531\u8f6c\u5411\u89d2\u63a7\u3002\u8f6c\u5411\u89d2\u7684\u89e3\u5e94\u4f7f\u5f97 dot{e}=-k e \u80fd\u4ee5\u6307\u6570\u901f\u5ea6\u5c06 e(t) \u63a7\u5236\u5230\u96f6\u3002<\/p>\n

begin{aligned} v_f sin left(delta+theta_eright)=-k e Rightarrow & delta=arcsin left(-frac{k e}{v_f}right)-theta_e end{aligned} (5.14)<\/p>\n

\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u672f\u8bed theta_e \u4e0d\u88ab\u89e3\u91ca\u4e3a\u822a\u5411\u8bef\u5dee\uff0c\u56e0\u4e3a\u5373\u4f7f\u5728\u5b8c\u7f8e\u8ddf\u8e2a\u7684\u60c5\u51b5\u4e0b\uff0c\u5b83\u4e5f\u5c06\u662f\u975e\u96f6\u7684\u3002\u66f4\u6070\u5f53\u7684\u89e3\u91ca\u5e94\u8be5\u662f\uff0c theta_e \u662f\u4e3a\u7528\u4e8e\u8ffd\u8e2a\u53c2\u8003\u8def\u5f84\u7684\u6807\u51c6\u8f6c\u5411\u89d2\u7684\u524d\u9988\u9879\u548c\u822a\u5411\u8bef\u5dee\u9879\u7684\u7ec4\u5408\u3002\u8fd9\u79cd\u63a7\u5236\u5f8b\u7684\u7f3a\u70b9\u662f\uff0c\u5f53 left|frac{k e}{v_f}right|>1 \u65f6\uff0c\u5b83\u672a\u88ab\u5b9a\u4e49\u3002<\/p>\n

\u901a\u8fc7\u5982\u4e0b\u7684\u53cd\u9988\u5f8b\uff0c\u6709\u9650\u57df\u4e0a\u7684\u6307\u6570\u6536\u655b\u53ef\u4ee5\u88ab\u653e\u677e\u4e3a\u5c40\u90e8\u6307\u6570\u6536\u655b\u3002delta=arctan left(-frac{k e}{v_f}right)-theta_e (5.15)\u8003\u8651 e \u7684\u4e00\u9636\u5bfc\uff0c\u8be5\u5f0f\u4e0e\u524d\u9762\u7684\u516c\u5f0f\u76f8\u540c\uff0c\u5982\u56fe5.4\u6240\u793a\u3002\u4e0e(5.14)\u4e2d\u7684\u63a7\u5236\u5f8b\u4e00\u6837\uff0c\u6b64\u63a7\u5236\u5668\u5728\u8def\u5f84\u8fde\u7eed\u53ef\u5fae\u5206\u7684\u6761\u4ef6\u4e0b\uff0c\u5c40\u90e8\u6307\u6570\u5730\u5c06\u8f66\u8f86\u7a33\u5b9a\u4e8e\u5177\u6709\u53ef\u53d8\u66f2\u7387\u7684\u8def\u5f84\u3002\u4e4b\u6240\u4ee5\u8981\u6c42\u8def\u5f84\u8fde\u7eed\u53ef\u5fae\u5206\uff0c\u662f\u56e0\u4e3a\u53cd\u9988\u7b56\u7565\u4e2d\u5bf9 theta_e \u7684\u5b9a\u4e49\u3002\u6b64\u63a7\u5236\u5668\u7684\u7f3a\u70b9\u662f\u5b83\u5728\u5012\u8f66\u65f6\u4e0d\u7a33\u5b9a\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5904\u7406\u6cca\u8f66\u4efb\u52a1\u3002<\/p>\n

<\/p>\n

\u56fe5.4\uff1a\u57fa\u4e8e\u524d\u8f6e\u8f93\u51fa\u7684\u63a7\u5236\u8fd9\u79cd\u63a7\u5236\u7b56\u7565\u5c06\u524d\u8f6e\u6307\u5411\u8def\u5f84\uff0c\u4ee5\u4f7f\u524d\u8f6e\u5782\u76f4\u4e8e\u8def\u5f84\u7684\u901f\u5ea6\u5206\u91cf\u6b63\u6bd4\u4e8e\u524d\u8f6e\u5230\u8def\u5f84\u7684\u8ddd\u79bb\u3002\u8fd9\u79cd\u63a7\u5236\u5728\u5c40\u90e8\u5b9e\u73b0\uff0c\u5e76\u5bfc\u81f4\u5c40\u90e8\u6307\u6570\u6536\u655b\u3002<\/p>\n

<\/p>\n

\u56fe5.5\uff1a\u672c\u8282\u4e2d\u7684\u4e09\u79cd\u8def\u5f84\u7a33\u5b9a\u63a7\u5236\u5f8b\u7684\u8ddf\u8e2a\u6027\u80fd\u6bd4\u8f83\u3002<\/p>\n

(a) \u5f53\u66f2\u7387\u975e\u96f6\u65f6\uff0c\u7eaf\u8ffd\u8e2a\u63a7\u5236\u7b56\u7565\u504f\u79bb\u53c2\u8003\u8def\u5f84\u3002(b) \u57fa\u4e8e\u540e\u8f6e\u8f93\u51fa\u7684\u63a7\u5236\u5668\u5c06\u540e\u8f6e\u9a71\u52a8\u81f3\u540e\u8f6e\u53c2\u8003\u8def\u5f84\u3002\u7cfb\u7edf\u7684\u4e8c\u9636\u54cd\u5e94\u5bfc\u81f4\u4e86\u8fc7\u51b2\u3002(c) \u57fa\u4e8e\u524d\u8f6e\u8f93\u51fa\u7684\u63a7\u5236\u5668\u4ee5\u4e00\u9636\u54cd\u5e94\u5c06\u524d\u8f6e\u9a71\u52a8\u81f3\u53c2\u8003\u8def\u5f84\u5e76\u8ddf\u8e2a\u8def\u5f84\u3002<\/p>\n

\u9762\u5411\u8fd0\u52a8\u5b66\u6a21\u578b\u7684\u8def\u5f84\u8ddf\u8e2a\u63a7\u5236\u5668\u7684\u6bd4\u8f83\uff1a\u57fa\u4e8e\u5177\u6709\u8f66\u8f6e\u65e0\u6ed1\u79fb\u7ea6\u675f\u7684\u8fd0\u52a8\u5b66\u6a21\u578b\u7684\u63a7\u5236\u5668\u7684\u4f18\u70b9\u5728\u4e8e\uff0c\u5b83\u4eec\u7684\u8ba1\u7b97\u8981\u6c42\u4f4e\uff0c\u6613\u4e8e\u5b9e\u73b0\uff0c\u5e76\u4e14\u5728\u4e2d\u7b49\u901f\u5ea6\u4e0b\u5177\u6709\u826f\u597d\u7684\u6027\u80fd\u3002<\/p>\n

\u56fe5.5\u5b9a\u6027\u6bd4\u8f83\u4e86\u672c\u8282\u8ba8\u8bba\u7684\u8def\u5f84\u7a33\u5b9a\u63a7\u5236\u5668\uff0c\u5176\u4e2d\u5404\u4e2a\u63a7\u5236\u5668\u57fa\u4e8e(3.3)\u8fdb\u884c\u4e86\u53d8\u9053\u7b56\u7565\u7684\u6a21\u62df\u3002\u5728\u57fa\u4e8e\u524d\u8f6e\u8f93\u51fa\u7684\u63a7\u5236\u5668\u7684\u4eff\u771f\u4e2d\uff0c\u540e\u8f6e\u53c2\u8003\u8def\u5f84\u88ab\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u7684\u524d\u8f6e\u53c2\u8003\u8def\u5f84\u4ee3\u66ffbegin{aligned} & x_{r e f}(s) longmapsto x_{r e f}(s)+l cos theta \\ & y_{r e f}(s) longmapsto y_{r e f}(s)+l sin theta end{aligned} (5.16)\u6a21\u62df\u4e2d\u6d89\u53ca\u7684\u53c2\u6570\u5982\u88683\u6240\u793a\u3002<\/p>\n

\u53c2\u8003\u56fe5.5\uff0c\u7eaf\u8ffd\u8e2a\u63a7\u5236\u5728\u6ca1\u6709\u66f2\u7387\u7684\u5468\u671f\u5185\u80fd\u591f\u8ddf\u8e2a\u53c2\u8003\u8def\u5f84\u3002\u5728\u5bf9\u5e94\u9ad8\u66f2\u7387\u8def\u5f84\u7684\u533a\u57df\u4e2d\uff0c\u7eaf\u8ffd\u8e2a\u63a7\u5236\u5c06\u5bfc\u81f4\u7cfb\u7edf\u504f\u79bb\u53c2\u8003\u8def\u5f84\u3002\u4e0e\u4e4b\u76f8\u53cd\uff0c\u540e\u4e24\u79cd\u63a7\u5236\u5668\u6536\u655b\u4e8e\u53c2\u8003\u8def\u5f84\uff0c\u5e76\u4ecd\u80fd\u5728\u9ad8\u66f2\u7387\u7684\u533a\u57df\u5185\u8ddf\u8e2a\u53c2\u8003\u8def\u5f84\u3002\u5bf9\u4e8e\u5747\u4f7f\u7528(5.5)\u4f5c\u4e3a\u63a7\u5236\u7b56\u7565\u7684\u4e24\u79cd\u63a7\u5236\u5668\uff0c\u4ec5\u5f53\u5b58\u5728\u8def\u5f84\u7684\u90bb\u57df\u4f7f\u5f97(5.5)\u662f\u8fde\u7eed\u7684\uff0c\u5c40\u90e8\u6307\u6570\u7a33\u5b9a\u6027\u624d\u80fd\u88ab\u8bc1\u660e\u3002\u76f4\u89c9\u4e0a\uff0c\u8fd9\u610f\u5473\u7740\u8def\u5f84\u4e0d\u80fd\u81ea\u76f8\u4ea4\uff0c\u4e14\u5fc5\u987b\u662f\u53ef\u5fae\u5206\u7684\u3002<\/p>\n

<\/p>\n

\u88683\uff1a\u7528\u4e8e\u751f\u6210\u56fe5.5\u7684\u6a21\u62df\u548c\u63a7\u5236\u5668\u53c2\u6570B. \u57fa\u4e8e\u8fd0\u52a8\u5b66\u6a21\u578b\u7684\u8f68\u8ff9\u8ddf\u8e2a\u63a7\u52361) \u57fa\u4e8e\u63a7\u5236\u674e\u96c5\u666e\u8bfa\u592b\u51fd\u6570\u7684\u8bbe\u8ba1\u6587\u7ae0[142]\u63cf\u8ff0\u4e86\u57fa\u4e8e\u63a7\u5236\u674e\u96c5\u666e\u8bfa\u592b\u51fd\u6570\u7684\u63a7\u5236\u8bbe\u8ba1\uff0c\u5176\u5728\u6c7d\u8f66\u5750\u6807\u7cfb\u4e2d\u5b9a\u4e49\u8ddf\u8e2a\u8bef\u5dee\u3002<\/p>\n

\u5229\u7528\u53c2\u8003\u8f68\u8ff9\u548c\u53c2\u8003\u901f\u5ea6 left(x_{text {ref }}, y_{text {ref }}, theta_{text {ref }}, v_{text {ref }}, omega_{text {ref }}right) \uff0c\u6784\u578b\u8bef\u5dee\u53ef\u4ee5\u8868\u793a\u4e3a\u60ef\u6027\u5750\u6807\u7cfb\u7684\u57fa\u7684\u53d8\u5316\uff1aleft(begin{array}{l} x_e \\ y_e \\ theta_e end{array}right)=left(begin{array}{ccc} cos theta & sin theta & 0 \\ -sin theta & cos theta & 0 \\ 0 & 0 & 1 end{array}right)left(begin{array}{c} x_{text {ref }}-x_r \\ y_{text {ref }}-y_r \\ theta_{text {ref }}-theta end{array}right)<\/p>\n

\u6784\u578b\u8bef\u5dee\u7684\u53d8\u5316\u7ee7\u800c\u6ee1\u8db3\uff1abegin{aligned} & dot{x}_e=omega y_e-v_r+v_{r e f} cos theta_e \\ & dot{y}_e=-omega x_e+v_{r e f} sin theta_e \\ &dot{theta}_e=omega_{r e f}-omega end{aligned}<\/p>\n

\u901a\u8fc7\u5982\u4e0b\u7684\u63a7\u5236\u5206\u914dbegin{aligned} &v_r=v_{r e f} cos theta_e+k_1 x_e \\ &omega=omega_{r e f}+v_{r e f}left(k_2 y_e+k_3 sin theta_eright) end{aligned}<\/p>\n

\u53ef\u4ee5\u5f97\u5230\u5982\u4e0b\u7684\u95ed\u73af\u8bef\u5dee\u52a8\u529b\u5b66begin{aligned} &dot{x}_e=left(omega_{text {ref }}+v_{text {ref }}left(k_2 y_e+k_3 sin theta_eright)right) y_e-k_1 x_e \\ &dot{y}_e=-left(omega_{text {ref }}+v_{text {ref }}left(k_2 y_e+k_3 sin theta_eright)right) x_e+v_{text {ref }} sin theta_e \\ &theta_e=omega_{text {ref }}-omega end{aligned}<\/p>\n

\u5bf9\u4e8e k_{1,2,3}>0, dot{omega}_{r e f}=0, dot{v}_{text {ref }}=0 \uff0c\u5176\u7a33\u5b9a\u6027\u53ef\u4ee5\u7531\u5982\u4e0b\u7684\u674e\u96c5\u666e\u8bfa\u592b\u51fd\u6570\u5f97\u8bc1V=frac{1}{2}left(x_e^2+y_e^2right)+frac{1-cos theta_e}{k_2}<\/p>\n

\u4e14\u674e\u96c5\u666e\u8bfa\u592b\u51fd\u6570\u5177\u6709\u534a\u8d1f\u5b9a\u7684\u65f6\u95f4\u5bfc\u6570dot{V}=-k_1 x_e^2-frac{v_{r e f} k_3 sin ^2 theta_e}{k_2} \u5c40\u90e8\u7684\u5206\u6790\u8868\u660e\uff0c\u6b64\u63a7\u5236\u5f8b\u5b9e\u73b0\u4e86\u5c40\u90e8\u6307\u6570\u7a33\u5b9a\u6027\u3002<\/p>\n

\u7136\u800c\uff0c\u5bf9\u4e8e\u65f6\u4e0d\u53d8\u7cfb\u7edf\uff0c\u8981\u6c42 omega_{r e f} \u548c v_{text {ref }} \u662f\u5e38\u6570\u3002\u6587\u7ae0[143]\u63d0\u51fa\u4e86\u4e00\u79cd\u76f8\u5173\u7684\u63a7\u5236\u5668\uff0c\u5176\u5229\u7528\u53cd\u63a8\u7684\u8bbe\u8ba1\u65b9\u6cd5\uff0c\u4ee5\u8fbe\u5230\u5728\u5177\u6709\u65f6\u53d8\u53c2\u8003\u7684\u6709\u9650\u57df\u4e0a\u7684\u4e00\u81f4\u5c40\u90e8\u6307\u6570\u7a33\u5b9a\u6027\u30022) \u8f93\u51fa\u53cd\u9988\u7ebf\u6027\u5316\u5bf9\u4e8e\u8f83\u9ad8\u7684\u8f66\u8f86\u901f\u5ea6\uff0c\u540c\u7406(3.8)\uff0c\u5e94\u5f53\u7ea6\u675f\u8f6c\u5411\u89d2\u4ee5\u5b9e\u73b0\u8fde\u7eed\u8f6c\u5411\u3002\u7531\u4e8e\u9644\u52a0\u4e86\u4e00\u4e2a\u72b6\u6001\uff0c\u4ece\u7b80\u5355\u7684\u51e0\u4f55\u89d2\u5ea6\u6765\u8003\u8651\u5982\u4f55\u8bbe\u8ba1\u63a7\u5236\u5668\u53d8\u5f97\u66f4\u52a0\u56f0\u96be\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8f93\u51fa\u7ebf\u6027\u5316\u7684\u7cfb\u7edf\u662f\u4e00\u79cd\u597d\u7684\u9009\u62e9\u3002\u4f7f\u7528\u524d\u8f6e\u6216\u540e\u8f6e\u7684\u4f4d\u7f6e\u5e76\u4e0d\u5bb9\u6613\u5b9e\u73b0\u8fd9\u4e00\u70b9\u3002\u6587\u7ae0[144]\u4e2d\u63d0\u51fa\u4e86\u7b80\u5316\u53cd\u9988\u7ebf\u6027\u5316\u7684\u8f93\u51fa\uff0c\u5176\u4e2d\u9009\u62e9\u4e86\u4e0e\u8f6c\u5411\u89d2\u5bf9\u9f50\u7684\u3001\u8ddd\u79bb\u8f66\u8f86\u524d\u65b9\u4efb\u610f d neq 0 \u7684\u70b9\u3002\u8bbe x_p=x_f+d cos (theta+delta) \u548c y_p=y_f+d sin (theta+delta) \u4e3a\u7cfb\u7edf\u7684\u8f93\u51fa\u3002<\/p>\n

\u53d6\u8f93\u51fa\u7684\u5bfc\u6570\uff0c\u4ee3\u5165(3.8)\u7684\u52a8\u529b\u5b66\u5f97begin{aligned} left(begin{array}{l} dot{x}_p \\ dot{y}_p end{array}right)=left(begin{array}{cc} cos (theta+delta)-frac{d}{l} sin (theta+delta) sin delta & -d sin (theta+delta) \\ sin (theta+delta)+frac{d}{l} cos (theta+delta) sin delta & d cos (theta+delta) end{array}right)left(begin{array}{l} v_f \\ v_delta end{array}right) =: A(theta, delta)left(begin{array}{l} v_f \\ v_delta end{array}right) end{aligned} (5.17)<\/p>\n

\u518d\u5b9a\u4e49(5.17)\u7b49\u5f0f\u53f3\u4fa7\u4e3a\u8f85\u52a9\u63a7\u5236\u53d8\u91cf u_x \u548c u_y \uff0c\u6709left(begin{array}{l} dot{x}_p \\ dot{y}_p end{array}right)=left(begin{array}{l} u_x \\ u_y end{array}right) (5.18)<\/p>\n

\u8fd9\u4f7f\u5f97\u63a7\u5236\u53d8\u5f97\u7b80\u5355\u3002\u6839\u636e u_x \u548c u_y \uff0c\u4f7f\u7528(5.17)\u4e2d\u7684\u77e9\u9635\u7684\u9006\uff0c\u6765\u91cd\u65b0\u5f97\u5230\u539f\u59cb\u63a7\u5236 v_f \u548c v_delta \uff0c\u5982\u4e0b\u6240\u793a\uff1abegin{aligned} {[A(theta, delta)]^{-1}} & =left(begin{array}{cc} cos (theta+delta) & sin (theta+delta) \\ -frac{1}{d} sin (theta+delta)-frac{1}{l} cos (theta+delta) sin delta & frac{1}{d} cos (theta+delta)-frac{1}{l} sin (theta+delta) sin delta end{array}right) end{aligned} (5.19)<\/p>\n

\u6839\u636e\u8f93\u5165-\u8f93\u51fa\u7ebf\u6027\u7cfb\u7edf\uff0c\u53ef\u4ee5\u901a\u8fc7\u5982\u4e0b\u7684\u63a7\u5236\uff0c\u6765\u5b9e\u73b0\u5c40\u90e8\u8f68\u8ff9\u7a33\u5b9abegin{aligned} & u_x=dot{x}_{p, text { ref }}+k_xleft(x_{p, r e f}-x_pright) \\ & u_y=y_{p, text { ref }}+k_yleft(y_{p, r e f}-y_pright) end{aligned} (5.20)<\/p>\n

\u4e3a\u907f\u514d\u6df7\u6dc6\uff0c\u6ce8\u610f\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8f93\u51fa\u4f4d\u7f6e left(x_p, y_pright) \u548c\u53d7\u63a7\u901f\u5ea6 v_f \u5e76\u4e0d\u50cf\u524d\u9762\u8ba8\u8bba\u7684\u63a7\u5236\u5668\u90a3\u6837\u914d\u7f6e\u3002<\/p>\n

C. \u9884\u6d4b\u63a7\u5236\u65b9\u6cd5<\/p>\n

\u4e0a\u9762\u8ba8\u8bba\u7684\u7b80\u5355\u63a7\u5236\u5f8b\u9002\u7528\u4e8e\u4e2d\u7b49\u9a7e\u9a76\u6761\u4ef6\u3002\u7136\u800c\uff0c\u6e7f\u6ed1\u9053\u8def\u6216\u7d27\u6025\u673a\u52a8\u53ef\u80fd\u9700\u8981\u66f4\u7cbe\u786e\u7684\u6a21\u578b\uff0c\u4f8b\u5982\u7b2c3-B\u8282\u4e2d\u4ecb\u7ecd\u7684\u6a21\u578b\u3002\u66f4\u590d\u6742\u7684\u6a21\u578b\u6240\u589e\u52a0\u7684\u7ec6\u8282\u4f7f\u63a7\u5236\u8bbe\u8ba1\u590d\u6742\u5316\uff0c\u4f7f\u5f97\u96be\u4ee5\u6839\u636e\u76f4\u89c9\u548c\u6784\u578b\u7a7a\u95f4\u7684\u51e0\u4f55\u7ed3\u6784\u6765\u6784\u9020\u63a7\u5236\u5668\u3002\u6a21\u578b\u9884\u6d4b\u63a7\u5236[145]\u662f\u4e00\u79cd\u901a\u7528\u7684\u63a7\u5236\u8bbe\u8ba1\u65b9\u6cd5\uff0c\u5bf9\u8fd9\u79cd\u95ee\u9898\u975e\u5e38\u6709\u6548\u3002\u4ece\u6982\u5ff5\u4e0a\u8bb2\uff0c\u6b64\u65b9\u6cd5\u5728\u77ed\u65f6\u95f4\u8303\u56f4\/\u533a\u95f4\u5185\u6c42\u89e3\u8fd0\u52a8\u89c4\u5212\u95ee\u9898\uff0c\u53d6\u77ed\u65f6\u95f4\u95f4\u9694\u5185\u7684\u5f00\u73af\u63a7\u5236\uff0c\u5e76\u5c06\u5176\u5e94\u7528\u4e8e\u7cfb\u7edf\u3002<\/p>\n

\u5728\u6267\u884c\u65f6\uff0c\u91cd\u65b0\u6c42\u89e3\u8fd0\u52a8\u89c4\u5212\u95ee\u9898\uff0c\u4ee5\u627e\u5230\u4e0b\u4e00\u65f6\u95f4\u95f4\u9694\u7684\u9002\u5f53\u63a7\u5236\u3002\u8ba1\u7b97\u786c\u4ef6\u548c\u6570\u5b66\u7f16\u7a0b\u7b97\u6cd5\u7684\u8fdb\u6b65\u4f7f\u5f97\u9884\u6d4b\u63a7\u5236\u5728\u65e0\u4eba\u9a7e\u9a76\u8f66\u8f86\u4e2d\u7684\u5b9e\u65f6\u5e94\u7528\u6210\u4e3a\u53ef\u80fd\u3002MPC\u672c\u8eab\u662f\u4e00\u4e2a\u4e3b\u8981\u7684\u7814\u7a76\u9886\u57df\uff0c\u672c\u8282\u4ec5\u65e8\u5728\u7b80\u8981\u63cf\u8ff0\u8be5\u6280\u672f\uff0c\u5e76\u7efc\u8ff0\u5176\u5728\u65e0\u4eba\u9a7e\u9a76\u8f66\u8f86\u63a7\u5236\u4e2d\u7684\u5e94\u7528\u7ed3\u679c\u3002\u7531\u4e8e\u6a21\u578b\u9884\u6d4b\u63a7\u5236\u662f\u4e00\u79cd\u975e\u5e38\u901a\u7528\u7684\u63a7\u5236\u6280\u672f\uff0c\u56e0\u6b64\u5176\u6a21\u578b\u53ef\u8868\u793a\u4e3a\u5177\u6709\u63a7\u5236 u(t) in mathbb{R}^m \u548c\u72b6\u6001 x(t) in mathbb{R}^n \u7684\u4e00\u822c\u8fde\u7eed\u65f6\u95f4\u63a7\u5236\u7cfb\u7edf\uff1adot{x}=f(x, u, t) (5.21)\u95ee\u9898\u7ed9\u5b9a\u4e00\u6761\u53ef\u884c\u7684\u53c2\u8003\u8f68\u8ff9 x_{text {ref }}(t) \uff0c\u4ee5\u53ca\u5bf9\u4e8e\u4e00\u4e9b\u8fd0\u52a8\u89c4\u5212\u5668\uff0c\u8fd8\u7ed9\u5b9a u_{r e f}(t) \u3002\u8fd9\u4e9b\u7ed9\u5b9a\u6761\u4ef6\u6ee1\u8db3(5.21)\u3002<\/p>\n

\u968f\u540e\uff0c\u901a\u8fc7\u4e00\u79cd\u5408\u9002\u7684\u6570\u503c\u8fd1\u4f3c\u6765\u79bb\u6563\u5316\u7cfb\u7edf\uff0c\u4f7f\u5f97(5.21)\u5728\u79bb\u6563\u65f6\u95f4\u60c5\u51b5\u4e0b\u8868\u793a\u4e3ax_{k+1}=F_kleft(x_k, u_kright), quad k in mathbb{N} (5.22)\u6b27\u62c9\u6cd5\u662f\u6700\u7b80\u5355\u7684\u79bb\u6563\u5316\u7b56\u7565\u4e4b\u4e00\uff0c\u5176\u5bf9\u63a7\u5236\u65bd\u52a0\u96f6\u9636\u4fdd\u6301begin{aligned} F_k(x(k cdot Delta t), u(k cdot Delta t)) =x(k cdot Delta t)+Delta t & cdot fleft(x(k cdot Delta t), u(k cdot Delta t), t_kright), quad k in mathbb{N} end{aligned} (5.23)<\/p>\n

\u72b6\u6001\u548c\u63a7\u5236\u5728\u65f6\u95f4 t_k=k cdot Delta t \u65f6\u901a\u8fc7\u5176\u8fd1\u4f3c\u503c\u79bb\u6563\u5316\u3002\u79bb\u6563\u7cfb\u7edf\u7684\u89e3\u662f\u8fd1\u4f3c\u7684\uff0c\u56e0\u6b64\u4e0d\u4f1a\u7cbe\u786e\u5730\u5339\u914d\u8fde\u7eed\u65f6\u95f4\u65b9\u7a0b\u3002\u7c7b\u4f3c\u5730\uff0c\u5728\u79bb\u6563\u65f6\u95f4 t_k \u91c7\u6837\u7684\u53c2\u8003\u8f68\u8ff9\u548c\u63a7\u5236\u5c06\u4e0d\u6ee1\u8db3\u79bb\u6563\u65f6\u95f4\u65b9\u7a0b\u3002\u4f8b\u5982\uff0c(5.23)\u548c(5.21)\u7684\u89e3\u4e4b\u95f4\u7684\u8bef\u5dee\u5c06\u662f O(Delta t) \uff0c\u4e14\u5728\u65f6\u95f4 t_k \u91c7\u6837\u7684\u53c2\u8003\u8f68\u8ff9\u5c06\u5bfc\u81f4F_kleft(x_{r e f}left(t_kright), u_{r e f}left(t_kright)right)-x_{r e f}left(t_{k+1}right)=Oleft(Delta t^2right) (5.24)<\/p>\n

\u4e3a\u4e86\u907f\u514d\u8fc7\u5ea6\u590d\u6742\u5316\u63a5\u4e0b\u6765\u7684\u8ba8\u8bba\uff0c\u6211\u4eec\u5728\u672c\u8282\u7684\u5269\u4f59\u90e8\u5206\u5047\u8bbe\u79bb\u6563\u5316\u662f\u7cbe\u786e\u7684\u3002<\/p>\n

\u901a\u5e38\uff0c\u63a7\u5236\u5f8b\u6709\u5982\u4e0b\u5f62\u5f0fbegin{array}{r} u_kleft(x_{m e a s .}right)=arg min _{substack{x_n in X_n, u_n in U_n}}left{hleft(x_N-x_{r e f, N}, u_N-u_{r e f, N}right)right. \\ left.+sum_{n=k} g_nleft(x_n-x_{r e f, n}, u_n-u_{r e f, n}right)right} end{array} (5.25)\u7ea6\u675f\u6761\u4ef6\uff1abegin{aligned} &x_k=x_{text {meas. }} \\ &x_{n+1}=Fleft(x_n, u_nright) \\ &n in{k, ldots, k+N-1} end{aligned}<\/p>\n

\u51fd\u6570 g_n \u5c06\u6bcf\u4e2a\u65f6\u95f4\u6b65\u957f\u5904\u7684\u3001\u4e0e\u53c2\u8003\u8f68\u8ff9\u548c\u53c2\u8003\u63a7\u5236\u7684\u504f\u5dee\u4f5c\u4e3a\u60e9\u7f5a\u9879\uff0c\u800c\u51fd\u6570 h \u662f\u65f6\u95f4\u533a\u95f4\u7ed3\u675f\u65f6\u7684\u7ec8\u6b62\u60e9\u7f5a\u9879\u3002\u96c6\u5408 x_n \u662f\u53ef\u5141\u8bb8\u72b6\u6001\u7684\u96c6\u5408\uff0c\u4ee5\u9650\u5236\u975e\u671f\u671b\u7684\u4f4d\u7f6e\u6216\u901f\u5ea6\uff0c\u4f8b\u5982\u8fc7\u5ea6\u7684\u8f6e\u80ce\u6253\u6ed1\u6216\u969c\u788d\u7269\u3002\u96c6\u5408 mathcal{U}_n \u5bf9\u8f93\u5165\u4fe1\u53f7\u7684\u5e45\u5ea6\u9650\u5236\u8fdb\u884c\u7f16\u7801\u3002\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u662f\uff0c(5.25)\u7684\u7b49\u5f0f\u53f3\u4fa7\u7684\u89e3\u662f\u5426\u5b58\u5728\u3001\u4f55\u65f6\u5b58\u5728\uff0c\u4ee5\u53ca\u95ed\u73af\u7cfb\u7edf\u7684\u7a33\u5b9a\u6027\u548c\u9c81\u68d2\u6027\u3002\u9884\u6d4b\u63a7\u5236\u6587\u732e[146][147]\u7814\u7a76\u4e86\u8fd9\u4e9b\u95ee\u9898\u3002\u4e3a\u4e86\u5728\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u4e0a\u5b9e\u73b0MPC\uff0c\u5fc5\u987b\u6bcf\u79d2\u5185\u82e5\u5e72\u6b21\u5730\u6c42\u89e3\u51fa(5.25)\uff0c\u8fd9\u662f\u4f7f\u7528MPC\u7684\u4e3b\u8981\u969c\u788d\u3002\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\u4e0b\uff0c h \u548c g_n \u662f\u4e8c\u6b21\u7684\uff0c mathcal{U}_n \u548c x_n \u662f\u591a\u9762\u4f53\uff0c F \u662f\u7ebf\u6027\u7684\uff0c\u6b64\u65f6\u95ee\u9898\u53d8\u6210\u4e86\u4e8c\u6b21\u89c4\u5212\u3002\u4e0e\u4e00\u822c\u7684\u975e\u7ebf\u6027\u89c4\u5212\u4e0d\u540c\uff0c\u5185\u70b9\u7b97\u6cd5\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u6c42\u89e3\u4e8c\u6b21\u89c4\u5212\u3002\u4e3a\u4e86\u5229\u7528\u8fd9\u4e00\u70b9\uff0c\u590d\u6742\u7684\u8f66\u8f86\u6a21\u578b\u901a\u5e38\u88ab\u7ebf\u6027\u5316\u4ee5\u83b7\u5f97\u8fd1\u4f3c\u7684\u7ebf\u6027\u6a21\u578b\u3002\u7ebf\u6027\u5316\u65b9\u6cd5\u901a\u5e38\u5728\u8ba1\u7b97\u7ebf\u6027\u5316\u7684\u53c2\u8003\u4e0a\u6709\u6240\u4e0d\u540c\uff0c\u5305\u62ec\u5f53\u524d\u64cd\u4f5c\u70b9[148-150]\u3001\u53c2\u8003\u8def\u5f84[151]\u3001\u6216\u8005\uff0c\u66f4\u4e00\u822c\u5730\uff0c\u53c2\u8003\u8f68\u8ff9\u3002\u91c7\u7528\u53c2\u8003\u8f68\u8ff9\u4f5c\u4e3a\u7ebf\u6027\u5316\u7684\u53c2\u8003\uff0c\u5c06\u5f97\u5230\u5982\u4e0b\u7684\u8fd1\u4f3c\u7ebf\u6027\u6a21\u578b\uff1abegin{aligned} & x_{r e f, k+1}+xi_{k+1}\\ &= F_kleft(x_{r e f, k}, u_{r e f, k}right) +nabla_x F_kleft(x_{r e f, k}, u_{r e f, k}right) xi_k \\ &+nabla_u F_kleft(x_{r e f, k}, u_{r e f, k}right) eta_k+Oleft(left|xi_kright|^2right) + Oleft(|| eta_k |^2right) \\ &=: F_kleft(x_{r e f, k}, u_{r e f, k}right)+A_k xi_k+B_k eta_k+Oleft(left|xi_kright|^2right) +Oleft(|| eta_k |^2right) end{aligned} (5.26)<\/p>\n

\u5176\u4e2d xi:=x-x_{text {ref }} \u548c eta:=u-u_{r e f} \u5206\u522b\u662f\u72b6\u6001\u548c\u63a7\u5236\u4e0e\u53c2\u8003\u8f68\u8ff9\u95f4\u7684\u504f\u5dee\u3002\u6270\u52a8\u52a8\u529b\u5b66\u7684\u8fd9\u79cd\u4e00\u9636\u6269\u5c55\u5f97\u5230\u4e86\u4e00\u4e2a\u7ebf\u6027\u65f6\u53d8\uff08LTV\uff09\u7cfb\u7edf\uff0cxi_{k+1}=A_k xi_k+B_k eta_k (5.27)<\/p>\n

\u63a5\u7740\u5229\u7528\u4e8c\u6b21\u76ee\u6807\u51fd\u6570\uff0c\u5e76\u4ee3\u6570\u5f62\u5f0f\u5730\u8868\u793a\u591a\u9762\u4f53\u7ea6\u675f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230begin{aligned} u_kleft(x_{text {meas. }}right)= & arg min _{xi_k, eta_k}left{xi_N^T H xi_Nright. left.+sum_{n=k}^{k+N-1} xi_k^T Q_k xi_k+eta_k^T R_k eta_kright} end{aligned} (5.28)\u7ea6\u675f\u6761\u4ef6\uff1abegin{aligned} &xi_k=x_{text {meas. }}-x_{r e f} \\ &C_n xi_n leq 0, quad D_n eta_n leq 0 \\ &xi_{k+1}=A_k xi_k+B_n eta_n, quad n in{k, ldots, k+N-1} end{aligned}<\/p>\n

\u5176\u4e2d R_k \u548c Q_k \u662f\u534a\u6b63\u5b9a\u7684\u3002\u5982\u679c\u72b6\u6001\u548c\u8f93\u5165\u662f\u65e0\u7ea6\u675f\u7684\uff0c\u5373 mathcal{U}_n=mathbb{R}^m \uff0c x_n=mathbb{R}^n \u5219\u53ef\u4ee5\u901a\u8fc7\u52a8\u6001\u89c4\u5212\u6765\u83b7\u5f97\u534a\u5c01\u95ed\u5f62\u5f0f\u7684\u89e3\uff0c\u8fd9\u53ea\u9700\u8981\u8ba1\u7b97 N \u6b65\u7684\u77e9\u9635\u9012\u5f52[152]\u3002\u5f53\u8f66\u8f86\u6a21\u578b\u8868\u793a\u4e3a\u65e0\u72b6\u6001\u548c\u8f93\u5165\u7ea6\u675f\u7684\u53d7\u63a7\u81ea\u56de\u5f52\u79ef\u5206\u6ed1\u52a8\u5e73\u5747\u6a21\u578b\uff08controlled auto-regressive integrated moving average\uff0cCARIMA\uff09[153][154]\u65f6\uff0c\u7c7b\u4f3c\u7684\u95ed\u5f0f\u9012\u5f52\u89e3\u4e5f\u5f97\u5230\u4e86\u63a2\u7d22\u3002\u6a21\u578b\u9884\u6d4b\u63a7\u5236\u65b9\u6cd5\u7684\u4e00\u79cd\u8fdb\u9636\u53d8\u4f53\u662f\u7528\u6027\u80fd\u6cdb\u51fd\u4e2d\u7684\u60e9\u7f5a\u51fd\u6570\u6765\u4ee3\u66ff\u72b6\u6001\u7ea6\u675f\uff08\u4f8b\u5982\u969c\u788d\u7269\uff09\u548c\u8f93\u5165\u7ea6\u675f\u3002\u6587\u7ae0[155]\u63d0\u51fa\u4e86\u8fd9\u79cd\u57fa\u4e8e\u5177\u6709\u975e\u7ebf\u6027\u8f6e\u80ce\u884c\u4e3a\u7684\u52a8\u529b\u5b66\u6a21\u578b\u7684\u9884\u6d4b\u63a7\u5236\u65b9\u6cd5\u3002<\/p>\n

\u9664\u4e86\u60e9\u7f5a\u63a7\u5236\u8f93\u5165\u4ee5\u53ca\u60e9\u7f5a\u4ece\u53c2\u8003\u8def\u5f84\uff08\u4e0d\u8003\u8651\u969c\u788d\u7269\u4e14\u901a\u5f80\u76ee\u6807\uff09\u7684\u504f\u79bb\uff0c\u6027\u80fd\u51fd\u6570\u8fd8\u60e9\u7f5a\u5bf9\u8f93\u5165\u7ea6\u675f\u7684\u8fdd\u53cd\u4ee5\u53ca\u5728\u6709\u9650\u63a7\u5236\u533a\u95f4\u5185\u4e0e\u969c\u788d\u7269\u7684\u78b0\u649e\u3002\u5728\u8fd9\u4e2a\u610f\u4e49\u4e0a\uff0c\u5b83\u7c7b\u4f3c\u4e8e\u57fa\u4e8e\u52bf\u573a\u7684\u8fd0\u52a8\u89c4\u5212\uff0c\u4f46\u662f\u88ab\u8bc1\u660e\u5177\u6709\u66f4\u597d\u7684\u6027\u80fd\u3002<\/p>\n

\u4ee5\u4e0b\u662f\u5728\u6c7d\u8f66\u63a7\u5236\u5668\u6587\u732e\u4e2d\u53d1\u73b0\u7684\u6a21\u578b\u9884\u6d4b\u63a7\u5236\u6846\u67b6\u7684\u4e00\u4e9b\u53d8\u4f53\uff1a<\/p>\n

1) \u5229\u7528\u8fd0\u52a8\u5b66\u6a21\u578b\u7684\u65e0\u7ea6\u675f\u7684\u6a21\u578b\u9884\u6d4b\u63a7\u5236\u6587\u7ae0[153]\u63d0\u51fa\u7684\u6700\u65e9\u7684\u9884\u6d4b\u63a7\u5236\u5668\u5c5e\u4e8e\u6b64\u7c7b\uff0c\u5176\u4e2d\u4f7f\u7528CARIMA\u6a21\u578b\uff0c\u5728\u6ca1\u6709\u8f93\u5165\u6216\u72b6\u6001\u7ea6\u675f\u7684\u60c5\u51b5\u4e0b\u5e94\u7528\u6a21\u578b\u9884\u6d4b\u63a7\u5236\u6846\u67b6\u3002\u6240\u5f97\u5230\u7684\u534a\u5c01\u95ed\u5f62\u5f0f\u7684\u89e3\u5177\u6709\u6700\u5c0f\u7684\u8ba1\u7b97\u8981\u6c42\uff0c\u5e76\u5df2\u5728[154]\u4e2d\u91c7\u7528\u3002\u6b64\u5916\uff0c[154]\u4e2d\u4f7f\u7528\u7ebf\u6027\u5316\u7684\u8fd0\u52a8\u5b66\u6a21\u578b\uff0c\u8003\u8651\u4e86\u65e0\u8f93\u5165\u6216\u72b6\u6001\u7ea6\u675f\u7684\u65f6\u53d8\u7ebf\u6027\u4e8c\u6b21\u89c4\u5212\u65b9\u6cd5\u3002<\/p>\n

2) \u8def\u5f84\u8ddf\u8e2a\u63a7\u5236\u5668\u5728[151]\u4e2d\uff0c\u4f7f\u7528\u57fa\u4e8e\u8d28\u5fc3\u7684\u7ebf\u6027\u52a8\u529b\u5b66\u6a21\u578b\uff08\u5047\u8bbe\u6052\u5b9a\u901f\u5ea6\uff09\u548c\u8fd1\u4f3c\u8f6c\u5411\u6a21\u578b\u7814\u7a76\u4e86\u9884\u6d4b\u63a7\u5236\u3002\u901a\u8fc7\u8be6\u7ec6\u7684\u81ea\u52a8\u8f6c\u5411\u6a21\u578b\u548c\u4e00\u4e2a\u5177\u670927\u81ea\u7531\u5ea6\u7684CarSim\u8f66\u8f86\u6a21\u578b\uff0c\u751f\u6210\u7684\u96c6\u6210\u6a21\u578b\u5f97\u5230\u4e86\u9a8c\u8bc1\u3002<\/p>\n

3) \u8f68\u8ff9\u8ddf\u8e2a\u63a7\u5236\u5668\u6587\u7ae0[148]\u7814\u7a76\u4e86\u4f7f\u7528\u7c7b\u4f3c\u4e8e\u7b2c3\u8282\u7684\u8f6e\u80ce\u6a21\u578b\u7684\u9884\u6d4b\u63a7\u5236\u5668\u3002\u5728\u4eff\u771f\u4e2d\u6267\u884c\u4e86\u5b8c\u5168\u975e\u7ebf\u6027\u9884\u6d4b\u63a7\u5236\u7b56\u7565\uff0c\u5e76\u5728\u7ed3\u51b0\u6761\u4ef6\u4e0b\u4ee520\u8d6b\u5179\u7684\u63a7\u5236\u9891\u7387\u5b9e\u73b0\u4e86\u6a21\u62df\u7d27\u6025\u673a\u52a8\u7684\u7a33\u5b9a\u3002\u7136\u800c\uff0c\u7531\u4e8e\u63a7\u5236\u533a\u95f4\u53ea\u6709\u4e24\u4e2a\u65f6\u95f4\u6b65\u957f\uff0c\u7136\u800c\u8ba1\u7b97\u65f6\u95f4\u662f\u63a7\u5236\u5668\u91c7\u6837\u65f6\u95f4\u7684\u4e09\u500d\uff0c\u56e0\u6b64\u65e0\u6cd5\u8fdb\u884c\u5b9e\u9a8c\u9a8c\u8bc1\u3002\u5728[148-150]\u4e2d\u8fd8\u7814\u7a76\u4e86\u57fa\u4e8e\u7ebf\u6027\u5316\u7684\u65b9\u6cd5\uff0c\u5176\u57fa\u4e8e\u5f53\u524d\u65f6\u95f4\u6b65\u957f\u4e0b\u8f66\u8f86\u72b6\u6001\u7684\u5355\u4e00\u7ebf\u6027\u5316\u3002\u6b64\u65f6\uff0c\u6c42\u89e3\u4e8c\u6b21\u89c4\u5212\u7684\u590d\u6742\u5ea6\u964d\u4f4e\uff0c\u4f7f\u5f97\u8ba1\u7b97\u65f6\u95f4\u662f\u53ef\u63a5\u53d7\u7684\uff0c\u540c\u65f6\u8fd8\u62a5\u544a\u4e86\u5728\u7ed3\u51b0\u6761\u4ef6\u4e0b\u4ee5\u9ad8\u8fbe21 m\/s\u7684\u901f\u5ea6\u884c\u9a76\u7684\u6210\u529f\u5b9e\u9a8c\u7ed3\u679c\u3002\u6587\u7ae0[150]\u901a\u8fc7\u4e3a\u65f6\u53d8\u7cfb\u7edf\u7684\u4e00\u81f4\u5c40\u90e8\u6e10\u8fd1\u7a33\u5b9a\u6027\u63d0\u4f9b\u6761\u4ef6\uff0c\u6539\u8fdb\u4e86\u8be5\u65b9\u6cd5\u7684\u6a21\u62df\u548c\u5b9e\u9a8c\u7ed3\u679c\u3002<\/p>\n

D. \u7ebf\u6027\u53d8\u53c2\u6570\u63a7\u5236<\/p>\n

\u8bb8\u591a\u63a7\u5236\u5668\u8bbe\u8ba1\u6280\u672f\u53ef\u7528\u4e8e\u7ebf\u6027\u7cfb\u7edf\uff0c\u4f7f\u5f97\u4eba\u4eec\u901a\u5e38\u671f\u671b\u4f7f\u7528\u7ebf\u6027\u6a21\u578b\u3002\u7136\u800c\uff0c\u6b63\u5e38\u9a7e\u9a76\u6761\u4ef6\u4e0b\u9047\u5230\u7684\u64cd\u4f5c\u70b9\u8303\u56f4\u5f88\u5e7f\uff0c\u56e0\u6b64\u5f88\u96be\u4f9d\u8d56\u4e8e\u5728\u5355\u4e2a\u64cd\u4f5c\u70b9\u4e0a\u7ebf\u6027\u5316\u7684\u6a21\u578b\u3002\u4e3a\u4e86\u8bf4\u660e\u8fd9\u4e00\u70b9\uff0c\u8003\u8651(5.10)\u4e2d\u7684\u6a2a\u5411\u8bef\u5dee\u52a8\u529b\u5b66\u3002\u5982\u679c\u5047\u8bbe\u8ddf\u8e2a\u8bef\u5dee\u4fdd\u6301\u8f83\u5c0f\uff0c\u5219\u5728\u64cd\u4f5c\u70b9 theta_e=0 \u548c e=0 \u5468\u56f4\u7684\u52a8\u529b\u5b66\u7684\u7ebf\u6027\u5316\u4e3a\uff1aleft(begin{array}{c} dot{e} \\ dot{theta}_e end{array}right)=left(begin{array}{cc} 0 & v_r \\ 0 & 0 end{array}right)left(begin{array}{l} e \\ theta_e end{array}right)+left(begin{array}{l} 0 \\ 1 end{array}right) omega+left(begin{array}{c} 0 \\ -v_r kappa(s) end{array}right) (5.29)<\/p>\n

\u5f15\u5165\u4e00\u4e2a\u65b0\u7684\u63a7\u5236\u53d8\u91cf\u6765\u5408\u5e76\u524d\u9988 u=omega+v_r kappa(s) \u4ee5\u7b80\u5316\u4e0a\u5f0f\uff0c\u5f97\u5230\u52a8\u529b\u5b66\uff1aleft(begin{array}{c} dot{e} \\ dot{theta}_e end{array}right)=left(begin{array}{cc} 0 & v_r \\ 0 & 0 end{array}right)left(begin{array}{l} e \\ theta_e end{array}right)+left(begin{array}{l} 0 \\ 1 end{array}right) u (5.30)<\/p>\n

\u6211\u4eec\u89c2\u5bdf\u5230\u6a21\u578b\u786e\u5b9e\u662f\u7ebf\u6027\u7684\uff0c\u4f46\u662f\u524d\u8fdb\u901f\u5ea6 v_r \u51fa\u73b0\u5728\u7ebf\u6027\u6a21\u578b\u4e2d\u3002\u5177\u6709\u589e\u76ca k_p \u548c k_d \u7684\u7b80\u5355\u6bd4\u4f8b\u52a0\u5fae\u5206\u63a7\u5236\u53ef\u4ee5\u7a33\u5b9a\u6a2a\u5411\u52a8\u529b\u5b66\uff0c\u4f46\u662f\u95ed\u73af\u7cfb\u7edf\u7684\u6781\u70b9\u4e3a left(-k_d pm sqrt{k_d^2-4 k_p v_r}right) \/ 2 \u3002\u5728\u8f83\u9ad8\u901f\u5ea6\u4e0b\uff0c\u6781\u70b9\u5c06\u79fb\u52a8\u5230\u590d\u5e73\u9762\uff0c\u8fd9\u5c06\u5bfc\u81f4\u632f\u8361\u54cd\u5e94\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5c0f\u7684 k_p \u589e\u76ca\u5c06\u5bfc\u81f4\u4f4e\u901f\u65f6\u7684\u54cd\u5e94\u8f83\u5dee\u3002\u9488\u5bf9\u8fd9\u4e2a\u95ee\u9898\u7684\u4e00\u79cd\u975e\u5e38\u76f4\u89c2\u4e14\u5e7f\u6cdb\u4f7f\u7528\u7684\u89e3\u51b3\u65b9\u6848\u662f\u589e\u76ca\u8c03\u5ea6\u3002\u5728\u672c\u4f8b\u4e2d\uff0c\u5c06 k_p \u53c2\u6570\u5316\u4e3a v_r \u7684\u51fd\u6570\uff0c\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u901f\u5ea6\u4e0b\u7684\u6781\u70b9\u56fa\u5b9a\u4e3a\u4e00\u4e2a\u503c\u3002\u8be5\u6280\u672f\u5c5e\u4e8e\u7ebf\u6027\u53c2\u6570\u53d8\u5316\uff08LPV\uff09\u6a21\u578b\u7684\u63a7\u5236\u8bbe\u8ba1\u8303\u7574[156]\u3002\u589e\u76ca\u8c03\u5ea6\u662f\u8fd9\u7c7b\u63a7\u5236\u5668\u8bbe\u8ba1\u7684\u7ecf\u5178\u65b9\u6cd5\u3002\u9c81\u68d2\u63a7\u5236\u548c\u51f8\u4f18\u5316\u5de5\u5177\u5f88\u5bb9\u6613\u5e94\u7528\u4e8e\u89e3\u51b3\u66f4\u590d\u6742\u7684\u6a21\u578b\u3002\u5728[157-159]\u4e2d\u4ecb\u7ecd\u4e86\u6a2a\u5411\u63a7\u5236\u7684LPV\u63a7\u5236\u8bbe\u8ba1\u3002\u5728[160][161]\u4e2d\uff0c\u540c\u65f6\u4f7f\u7528\u4e86LPV\u6a21\u578b\u4ee5\u53ca\u9884\u6d4b\u63a7\u5236\u65b9\u6cd5\uff0c\u7528\u4e8e\u8def\u5f84\u548c\u8f68\u8ff9\u7a33\u5b9a\u3002<\/p>\n

\u5728\u8f83\u4f4e\u7684\u81ea\u52a8\u5316\u6c34\u5e73\u4e0a\uff0cLPV\u63a7\u5236\u6280\u672f\u5df2\u88ab\u63d0\u51fa\u7528\u4e8e\u96c6\u6210\u7cfb\u7edf\u63a7\u5236\u3002\u5728\u8fd9\u4e9b\u8bbe\u8ba1\u4e2d\uff0c\u591a\u4e2a\u5b50\u7cfb\u7edf\u88ab\u7ec4\u5408\u5728\u4e00\u4e2a\u63a7\u5236\u5668\u4e0b\uff0c\u4ee5\u5b9e\u73b0\u6539\u8fdb\u7684\u64cd\u7eb5\u6027\u80fd\u3002\u5728[162][163]\u4e2d\u5f00\u53d1\u4e86\u7528\u4e8e\u9a71\u52a8\u4e3b\u52a8\u548c\u534a\u4e3b\u52a8\u60ac\u67b6\u7cfb\u7edf\u7684LPV\u63a7\u5236\u7b56\u7565\uff0c\u800c\u5728[164][165]\u4e2d\u5f00\u53d1\u4e86\u96c6\u6210\u60ac\u67b6\u548c\u5236\u52a8\u63a7\u5236\u7cfb\u7edf\u3002<\/p>\n

6. \u7ed3\u8bba<\/p>\n

\u5728\u8fc7\u53bb\u4e09\u5341\u5e74\u4e2d\uff0c\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u6280\u672f\u7684\u8fdb\u6b65\u8d8a\u6765\u8d8a\u5feb\u3002\u9664\u4e86\u8ba1\u7b97\u548c\u611f\u77e5\u786c\u4ef6\u7684\u8fdb\u6b65\u4e4b\u5916\uff0c\u79fb\u52a8\u673a\u5668\u4eba\u8fd0\u52a8\u89c4\u5212\u548c\u53cd\u9988\u63a7\u5236\u7406\u8bba\u5728\u8ba1\u7b97\u65b9\u9762\u7684\u91cd\u5927\u7406\u8bba\u8fdb\u6b65\u4e5f\u63a8\u52a8\u4e86\u8fd9\u4e00\u5feb\u901f\u8fdb\u6b65\u3002\u6beb\u65e0\u7591\u95ee\uff0c\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u6240\u63d0\u4f9b\u7684\u9053\u8def\u7f51\u7edc\u7684\u5229\u7528\u7387\u548c\u5b89\u5168\u6027\u7684\u63d0\u9ad8\u63a8\u52a8\u4e86\u7814\u7a76\u5de5\u4f5c\u3002\u65e0\u4eba\u9a7e\u9a76\u6c7d\u8f66\u662f\u590d\u6742\u7cfb\u7edf\uff0c\u5b83\u5df2\u7ecf\u88ab\u5206\u89e3\u5230\u51b3\u7b56\u95ee\u9898\u7684\u5c42\u6b21\uff0c\u5176\u4e2d\u4e00\u4e2a\u95ee\u9898\u7684\u89e3\u5c31\u662f\u4e0b\u4e00\u4e2a\u95ee\u9898\u7684\u8f93\u5165\u3002\u5c06\u51b3\u7b56\u95ee\u9898\u5206\u89e3\u4e3a\u5404\u4e2a\u72ec\u7acb\u7684\u95ee\u9898\uff0c\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u4f7f\u7528\u6765\u81ea\u5404\u4e2a\u7814\u7a76\u9886\u57df\u7684\u6210\u719f\u65b9\u6cd5\u548c\u6280\u672f\u3002\u968f\u540e\u7684\u4efb\u52a1\u4fbf\u662f\u96c6\u6210\u8fd9\u4e9b\u65b9\u6cd5\uff0c\u4ee5\u4f7f\u5b83\u4eec\u7684\u4ea4\u4e92\u8bed\u4e49\u6709\u6548\uff0c\u5e76\u4f7f\u7ec4\u5408\u7684\u7cfb\u7edf\u5728\u8ba1\u7b97\u4e0a\u9ad8\u6548\u3002\u66f4\u6709\u6548\u7684\u8fd0\u52a8\u89c4\u5212\u7b97\u6cd5\u53ef\u80fd\u4ec5\u4e0e\u8ba1\u7b97\u5bc6\u96c6\u578b\u7684\u53cd\u9988\u63a7\u5236\u5668\uff08\u4f8b\u5982\u6a21\u578b\u9884\u6d4b\u63a7\u5236\uff09\u517c\u5bb9\u3002<\/p>\n

\u76f8\u53cd\uff0c\u4e00\u4e2a\u7b80\u5355\u7684\u63a7\u5236\u5f8b\u53ef\u80fd\u53ea\u9700\u8981\u66f4\u5c11\u7684\u8ba1\u7b97\uff0c\u4f46\u662f\u4e5f\u4e0d\u90a3\u4e48\u9c81\u68d2\uff0c\u5e76\u4e14\u9700\u8981\u4f7f\u7528\u66f4\u8be6\u7ec6\u7684\u8fd0\u52a8\u89c4\u5212\u6a21\u578b\u3002\u672c\u6587\u7efc\u8ff0\u4e86\u65e0\u4eba\u9a7e\u9a76\u8f66\u8f86\u51b3\u7b56\u95ee\u9898\u7684\u5404\u4e2a\u65b9\u9762\uff0c\u4e14\u5c06\u91cd\u70b9\u653e\u5728\u4e86\u8fd0\u52a8\u89c4\u5212\u548c\u53cd\u9988\u63a7\u5236\u3002\u5bf9\u5404\u79cd\u8fd0\u52a8\u89c4\u5212\u548c\u63a7\u5236\u6280\u672f\u7684\u6027\u80fd\u548c\u8ba1\u7b97\u8981\u6c42\u7684\u7efc\u8ff0\u53ef\u4f5c\u4e3a\u53c2\u8003\uff0c\u6765\u5e2e\u52a9\u8bc4\u4f30\u7cfb\u7edf\u7ea7\u8bbe\u8ba1\u7684\u4e0d\u540c\u9009\u62e9\u7684\u517c\u5bb9\u6027\u548c\u8ba1\u7b97\u6743\u8861\u30027. \u53c2\u8003\u6587\u732e\u8be6\u89c1\u539f\u59cbpaper\uff0cpaper\u94fe\u63a5\uff1aA Survey of Motion Planning and Control Techniques for Self-driving Urban Vehicles\u3002\u5bf9\u81ea\u52a8\u9a7e\u9a76\u7b97\u6cd5\u4e0e\u5de5\u7a0b\u5316\u611f\u5174\u8da3\u7684\u4f19\u4f34\uff0c\u6b22\u8fce\u8054\u7cfb\u6211\u4eec\uff0c\u6709\u66f4\u591a\u9ad8\u8d28\u91cf\u7684paper\u63a8\u8350\u7ed9\u60a8\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u7f16\u8f91\u4e8e 2023-01-10 19:00\u30fbIP […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,1],"tags":[],"class_list":["post-686","post","type-post","status-publish","format-standard","hentry","category-technology-frontier","category-home"],"views":5,"_links":{"self":[{"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/posts\/686"}],"collection":[{"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/comments?post=686"}],"version-history":[{"count":2,"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/posts\/686\/revisions"}],"predecessor-version":[{"id":813,"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/posts\/686\/revisions\/813"}],"wp:attachment":[{"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/media?parent=686"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/categories?post=686"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/222.128.65.89:18086\/index.php\/wp-json\/wp\/v2\/tags?post=686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}