Dynamic obstacle avoidance of unmanned surface vehicle based on improved speed obstacle method

基于改进速度障碍法的水面无人艇动态避障 Dynamic obstacle avoidance of unmanned surface vehicle based on improved speed obstacle method 1 corresp first-author E-mail： mexbhong@scut.edu.cn E-mail： mexbhong@scut.edu.cn 洪晓斌（1979-），男，广东揭阳人，教授，博士生导师，2002年于武汉理工大学获得学士学位，2007年于华南理工大学获得博士学位，现为华南理工大学机械与汽车工程学院副院长，主要从事无损检测技术与装备、网络化测控技术及仪器及海上智能化无人装备等方面的研究。E-mail：mexbhong@scut.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932624&type= 22.01330566 32.00395203 洪晓斌 （1979-），男，广东揭阳人，教授，博士生导师，2002年于武汉理工大学获得学士学位，2007年于华南理工大学获得博士学位，现为华南理工大学机械与汽车工程学院副院长，主要从事无损检测技术与装备、网络化测控技术及仪器及海上智能化无人装备等方面的研究。E-mail： mexbhong@scut.edu.cn mexbhong@scut.edu.cn 1 徐郑攀（1997-），男，湖北荆州人，硕士研究生，主要从事水面无人艇自主航行控制、路径规划和避障技术等方面的研究。E-mail：243636483@qq.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932640&type= 22.01330566 32.00373840 徐郑攀 （1997-），男，湖北荆州人，硕士研究生，主要从事水面无人艇自主航行控制、路径规划和避障技术等方面的研究。E-mail： 243636483@qq.com 243636483@qq.com 1 1 2 华南理工大学 机械工程学院，广东 广州 510640 华南理工大学 机械工程学院，广东 广州 510640 College of Mechanical and Automotive Engineering， South China University of Technology， Guangzhou 510640， China College of Mechanical and Automotive Engineering， South China University of Technology， Guangzhou 510640， China 华南理工大学 土木与交通学院，广东 广州 510641 华南理工大学 土木与交通学院，广东 广州 510641 College of Civil Engineering and Transportation， South China University of Technology， Guangzhou 510641， China College of Civil Engineering and Transportation， South China University of Technology， Guangzhou 510641， China 针对水面无人艇在复杂海况下的动态避障问题，构建了障碍物模型和碰撞风险模型，通过引入多个约束条件的改进速度障碍法，实现了复杂海况下水面无人艇动态避障的航迹重规划。针对无人艇长宽比较大的几何特点，提出了矩形包围框的障碍物模型描述方法，建立无人艇相对运动模型，并在船体坐标系下计算相对运动参数。通过计算碰撞危险度来确定开始避障的时机。最后，在速度障碍法的基础上引入障碍物模型约束和水面无人艇自主局部避障系统特性约束对避障路径进行优化，并通过实验获得动态避障环境下的航迹重规划结果。基于V-REP（Virtual Robot Experimentation Platform）平台仿真实验以及实船避障实验表明，在航速12节的情况下，避碰过程中的路径重规划时间在15 s以下，满足在实际航行多约束条件下水面无人艇的动态避障要求。 Aiming at solving the dynamic collision avoidance problem for unmanned surface vehicle under complex marine status， this study constructs the obstacle model and the collision risk model to achieve the track re-planning of dynamic collision avoidance in complex sea conditions with the improved velocity obstacle method that introduced multiple constraints.Firstly， an obstacle model description method for rectangular bounding box was proposed for USV's geometric characteristics of large aspect ratio.A relative motion model of USV in hull coordinate system was established with the relative motion parameters calculated. Secondly，the collision risk was calculated to determine the time to start avoiding the obstacle.Lastly， improved obstacle avoidance path was constructed by introducing obstacle model constraints and characteristic constraints of USV’s autonomous local obstacle avoidance system based on velocity obstacle method. The track re-planning results were obtained through experiments.The simulated experiment on the physical simulation platform of V-REP （Virtual Robot Experimentation Platform） and the actual ship obstacle avoidance experiment show that the path re-planning time during collision avoidance is less than 15 s at a speed of 12 knots， meeting the dynamic collision avoidance requirements of unmanned surface vehicle under multiple constraints of actual navigation. 无人艇 碰撞危险度模型 改进速度障碍法 动态避障方法 unmanned surface vehicle collision risk model improved velocity obstacle method remote collision avoidance method 1 引　言 1 水面无人艇（Unmanned Surface Vehicle， USV）作为一种水上新型智能无人航行载体，常被用来执行危险系数高、作业环境恶劣等系列任务。在复杂多变的动态海洋环境下，高度智能化避障水平是USV能够安全航行的重要保障。相对于静态障碍物，海面上的动态障碍物更易导致碰撞危险，因此根据感知信息进行动态的局部路径规划具有重要意义。 在路径规划和避障方面，常用的算法有蚁群算法 ［ 1 1 ］ 、粒子群算法 ［ 2 2 ］ 和单目视觉避障算法 ［ 3 3 ］ 等。USV由于其运动环境的特殊性，需要对传统的避障算法进行针对性的改进以适应复杂的海况。针对USV的动态避障问题，国内外学者取得了多项研究成果。Hu提出了一种新颖的分层多目标粒子群优化（H-MOPSO）智能算法，该算法使用多目标优化方法生成安全且符合海事规则的路径 ［ 4 4 ］ 。Park使用概率方法计算碰撞风险，并在预测的轨迹上配置碰撞风险区来避开动态障碍物 ［ 5 5 ］ 。Johansen结合船舶航行动力学提出了一种基于模型预测控制的船舶动态避障方法 ［ 6 6 ］ 。Chen设计了一种结合动态窗口算法和A*算法的混合算法来提高避障过程中的可靠性 ［ 7 7 ］ 。Lin将强化学习引入USV避障过程中，提出了一种基于策略优化的Q学习算法（SO-QL）来加快USV路径规划的速度 ［ 8 8 ］ 。Zhang提出了一种结合遗传算法和模拟退火算法的路径规划方法来优化避障路径 ［ 9 9 ］ 。Joohyun提出了一种基于深度强化学习（DRL）的动态避障方法 ［ 10 10 ］ 。 目前，国内外学者通过对USV避障技术的研究提出了多种可行的避碰方法，然而其中大部分方法建立的模型是将无人艇的几何形状特性简化，忽略了USV在水面运动时的动力学特性和航行参数的不确定性，因此在具体航行实验中难以达到理想效果。针对水面USV在实际航行中面临的静态障碍物实时避障问题，作者前期研制了一种面向单一静态障碍物的水面USV融合图像识别和VFH+避障方法 ［ 11 11 ］ 。该方法通过图像识别技术来获取环境感知信息，并利用VFH+避障算法来求取当前环境下代价函数值最小的静态障碍物的最佳避碰角度。然而，VFH+算法对动态障碍物的栅格化处理容易导致速度等动态信息的缺失。为了改善USV的动态避障性能，本文在文献［ 11 11 ］的环境感知方法以及多种避障算法的基础上，将速度障碍区域的几何模型通过矩形包围框进行描述，同时将速度障碍区域的运动模型以速度不确定度进行描述，从而实现对可行驶速度区域的模型建立，将国际海上避碰规则（International Regulations for Preventing Collisions at Sea，COLREGS）与USV自身的动力学特征相结合，将USV可行驶速度区域集合进一步缩小，最终通过代价函数的计算选择最优航行速度并且对航迹进行实时重规划。 2 相对运动模型与碰撞危险度计算 1 2.1　相对运动模型 2 分析USV与水面船舶动态障碍物的相对运动过程，需要先建立相对运动模型，即确定运动坐标系并在该坐标系下计算USV与障碍物之间的相对运动参数。在水面动态避障情景下，动态障碍物大多为船舶类障碍物，传统的速度障碍法使用的外接圆描述法不足以准确描述动态障碍物的外形特征，因此本文在相对运动模型中采用矩形包围框描述动态障碍物，障碍物模型如 图1 图1 所示。 使用船体坐标系作为相对运动模型的基本坐标系，水面USV与障碍物在船体坐标系下的相对运动参数关系如 图2 图2 所示。设USV的位置坐标为 <math id="M1"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932660&type= 11.09133339 4.74133301 ，速度在 <math id="M2"><mi>x</mi><mo>,</mo><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932897&type= 5.67266655 3.72533321 轴的分量为 <math id="M3"><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>x</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932675&type= 12.69999981 4.48733330 ，则USV的速度矢量 <math id="M4"><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932683&type= 11.09133339 4.65666676 可以用式（1）~ 式（3） 式（3） 表示；障碍物的位置坐标为 <math id="M5"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932692&type= 11.09133339 4.74133301 ，速度在 <math id="M6"><mi>x</mi><mo>,</mo><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932897&type= 5.67266655 3.72533321 轴的分量为 <math id="M7"><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>x</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932712&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932708&type= 12.69999981 4.48733330 ，障碍物的速度矢量 <math id="M8"><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932718&type= 11.09133339 4.65666676 可以用式（3）~ 式（5） 式（5） 表示。根据上述运动参数可以在船体坐标系中计算USV和障碍物的相对运动参数，包括船速比、相对速度矢量、相对距离、相对方位、最近会遇距离和最小会遇时间。 USV的速度矢量为 <math id="M9"><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932744&type= 11.09133339 4.65666676 ，其中 <math id="M10"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932751&type= 8.29733276 4.57200003 分别表示USV速度的大小和方向，数学公式计算如下： <math display="block" id="M11"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>=</mo><mroot><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932829&type= 25.14599991 5.24933338 ， <math display="block" id="M12"><msub><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfrac><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>x</mi></mrow></msub></mrow></mfrac><mo>+</mo><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932767&type= 30.31066895 8.46666718 ， <math display="block" id="M13"><mi>α</mi><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msup><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>∘</mo></mrow></msup><mtext>         </mtext><mi>i</mi><mi>f</mi><mtext> </mtext><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>x</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msup><mrow><mn mathvariant="normal">180</mn></mrow><mrow><mo>∘</mo></mrow></msup><mtext>     </mtext><mi>i</mi><mi>f</mi><mtext> </mtext><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>x</mi></mrow></msub><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub><mo><</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msup><mrow><mn mathvariant="normal">180</mn></mrow><mrow><mo>∘</mo></mrow></msup><mtext>     </mtext><mi>i</mi><mi>f</mi><mtext> </mtext><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>x</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub><mo><</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msup><mrow><mn mathvariant="normal">360</mn></mrow><mrow><mo>∘</mo></mrow></msup><mtext>     </mtext><mi>i</mi><mi>f</mi><mtext> </mtext><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>x</mi></mrow></msub><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932778&type= 46.82066727 21.92866707 . 在坐标系的建立上规定 <math id="M14"><mi>x</mi><mo>,</mo><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932897&type= 5.67266655 3.72533321 轴正向为航行正方向，因此计算USV航行方向时需要加入偏置角度 <math id="M15"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932800&type= 1.94733346 2.79399991 。 障碍物的速度矢量为 <math id="M16"><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>,</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932813&type= 11.17599964 4.65666676 ，其中 <math id="M17"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>,</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932823&type= 8.38199997 4.57200003 分别表示障碍物速度的大小和方向，数学公式计算如下： <math display="block" id="M18"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>=</mo><mroot><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932829&type= 25.14599991 5.24933338 ， <math display="block" id="M19"><msub><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfrac><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow></msub></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>x</mi></mrow></msub></mrow></mfrac><mo>+</mo><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932839&type= 30.31066895 8.46666718 . 障碍物与USV之间的船速比为： <math display="block" id="M20"><mi>K</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932847&type= 12.10733318 8.46666718 . 障碍物相对于USV的速度矢量为 <math id="M21"><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932856&type= 10.41399956 4.65666676 ，其中 <math id="M22"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932865&type= 7.61999989 4.57200003 分别表示相对速度的大小和方向，计算公式如下： <math display="block" id="M23"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>=</mo><mroot><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mi>r</mi><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>r</mi><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932877&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932873&type= 24.38400078 5.24933338 ， <math display="block" id="M24"><msub><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfrac><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi><mi>y</mi></mrow></msub></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi><mi>x</mi></mrow></msub></mrow></mfrac><mo>+</mo><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932882&type= 29.80266571 8.46666718 ， 式中： <math id="M25"><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi><mi>x</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932891&type= 9.48266602 4.14866638 为相对速度在 <math id="M26"><mi>x</mi><mo>,</mo><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932897&type= 5.67266655 3.72533321 轴的分量，即： <math display="block" id="M27"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>x</mi></mrow></msub><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi><mi>y</mi></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow></msub><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932906&type= 23.02933502 9.99066734 . 障碍物与USV之间的相对距离 <math id="M28"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932913&type= 4.31799984 3.72533321 和相对真方位 <math id="M29"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932920&type= 3.64066648 3.72533321 为： <math display="block" id="M30"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>=</mo><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932929&type= 48.25999832 6.85799980 ， <math display="block" id="M31"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow></mfrac><mo>+</mo><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932936&type= 37.50733566 8.46666718 . 障碍物的相对方位角为： <math display="block" id="M32"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932944&type= 20.31999969 4.57200003 . 将USV等效为质点，该点与会遇船舶运动路径之间的最小距离即为最近会遇距离（DCPA），其计算公式如下： <math display="block" id="M33"><mi>D</mi><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>-</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">π</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932950&type= 38.77733231 4.82600021 . D 的正负号表示USV与会遇船舶的相对位置，当会遇船位于USV左舷时，会遇船舶过USV艇首为负，过USV艇尾为正；当会遇船舶位于USV舷时，会遇船舶过USV艇首为正，过USV艇尾为负。 最小会遇时间（TCPA）为会遇船舶从当前位置运动到DCPA垂足点所花费的时间，其计算公式如下： <math display="block" id="M34"><mi>T</mi><mo>=</mo><mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>-</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">π</mi></mrow></mfenced></mrow><mo>/</mo><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932961&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932958&type= 45.12733078 5.33400011 . T 的正负号表示会遇船舶是否航行至会遇点CPA，若会遇船舶已航行至会遇点，则碰撞不会发生，TCPA为负；反之，USV与会遇船舶存在碰撞的可能性，TCPA为正。 2.2　碰撞危险度计算 2 “碰撞危险度”描述的是船舶之间在某一时刻某一位置时某段时间内发生碰撞的可能性。在动态避障算法中，为了确定USV开始避障的时机，需要对碰撞危险度做定量分析，构建相应的计算模型。现阶段对其做定量分析计算模型一般依据船舶避碰危险度（Collision Risk Index， CRI）来构建。在实际情况中，影响CRI的因素有很多，主观因素有航行水域、能见度等；客观因素有DCPA、TCPA等。本文考虑到局部避障的自主性，主要采用TCPA和DCPA对应隶属度函数的加权组合来计算碰撞危险度。 2 . 2 . 1　DCPA隶属度函数 3 DCPA的隶属度函数 <math id="M35"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932964&type= 7.70466709 4.57200003 为： <math display="block" id="M36"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">1</mn><mtext>   </mtext><mi>D</mi><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">π</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mfenced separators="|"><mrow><mi>D</mi><mo>-</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow></mfenced><mtext>   </mtext><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi>D</mi><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mtext>   </mtext><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi>D</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932970&type= 91.60932922 21.59000015 . 式中： <math id="M37"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933042&type= 3.30200005 3.72533321 表示两船能够安全会遇的最小距离， <math id="M38"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933051&type= 3.30200005 3.72533321 表示两船能够安全通过的最小距离。当两船的DCPA值小于 <math id="M39"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933042&type= 3.30200005 3.72533321 时， <math id="M40"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933002&type= 14.56266594 4.57200003 ，此时USV与会遇船舶的碰撞危险度很高，需要改变航行状态避开该障碍物；当两船的DCPA值大于 <math id="M41"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933051&type= 3.30200005 3.72533321 时， <math id="M42"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933109&type= 14.56266594 4.57200003 ，此时USV与会遇船舶没有碰撞的可能性，可以保持目前的航行状态；当两船的DCPA值介于 <math id="M43"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933042&type= 3.30200005 3.72533321 和 <math id="M44"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933051&type= 3.30200005 3.72533321 之间时，此时需要计算CRI的值来判断两船是否有碰撞危险。 <math id="M45"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933042&type= 3.30200005 3.72533321 和 <math id="M46"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933051&type= 3.30200005 3.72533321 的计算公式如下： <math display="block" id="M47"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">1.1</mn><mo>-</mo><mn mathvariant="normal">0.2</mn><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></mrow><mo>/</mo><mrow><mn mathvariant="normal">180</mn><mo>°</mo><mtext>   </mtext><msup><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>∘</mo></mrow></msup><mo>≤</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo><</mo><mn mathvariant="normal">112.5</mn><mo>°</mo></mrow></mrow></mtd></mtr><mtr><mtd><mn mathvariant="normal">1.0</mn><mo>-</mo><mn mathvariant="normal">0.4</mn><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></mrow><mo>/</mo><mrow><mn mathvariant="normal">180</mn><mo>°</mo><mtext>   </mtext><mn mathvariant="normal">112.5</mn><mo>°</mo><mo>≤</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo><</mo><mn mathvariant="normal">180</mn><mo>°</mo></mrow></mrow></mtd></mtr><mtr><mtd><mn mathvariant="normal">1.0</mn><mo>-</mo><mn mathvariant="normal">0.4</mn><mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">360</mn><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></mrow></mfenced></mrow><mo>/</mo><mrow><mn mathvariant="normal">180</mn><mo>°</mo><mtext>   </mtext><mn mathvariant="normal">180</mn><mo>°</mo><mo>≤</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo><</mo><mn mathvariant="normal">247.5</mn><mo>°</mo></mrow></mrow></mtd></mtr><mtr><mtd><mn mathvariant="normal">1.1</mn><mo>-</mo><mn mathvariant="normal">0.2</mn><mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">360</mn><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></mrow></mfenced></mrow><mo>/</mo><mrow><mn mathvariant="normal">180</mn><mo>°</mo><mtext>   </mtext><mn mathvariant="normal">247.5</mn><mo>°</mo><mo>≤</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>≤</mo><mn mathvariant="normal">360</mn><mo>°</mo></mrow></mrow></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933061&type= 80.17933655 25.56933212 ， <math display="block" id="M48"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933068&type= 13.54666710 3.72533321 . 2 . 2 . 2　TCPA隶属度函数 3 TCPA的隶属度函数 <math id="M49"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933079&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933077&type= 7.70466709 4.57200003 为： <math display="block" id="M50"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">1</mn><mtext>   </mtext><mi>T</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi>T</mi></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mtext>   </mtext><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi>T</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mtext>   </mtext><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi>T</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933081&type= 49.44533539 22.18266678 ， 式中： <math display="block" id="M51"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mroot><mrow><msup><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933090&type= 26.58533096 9.31333351 ， <math display="block" id="M52"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mroot><mrow><msup><mrow><mn mathvariant="normal">12</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933094&type= 25.82333374 8.46666718 . 其中 <math id="M53"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933100&type= 3.97933316 3.72533321 表示最小避碰距离，一般由其本身的性质以及动力学特性来决定。 2 . 2 . 3　碰撞危险度综合评判 3 根据以上的描述，USV的碰撞危险度综合评判指标 <math id="M54"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933106&type= 1.94733346 3.64066648 存在3种情况，由DCPA的隶属度函数和TCPA的隶属度函数来决定。如果 <math id="M55"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933109&type= 14.56266594 4.57200003 ，那么 <math id="M56"><mi>γ</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933135&type= 8.63599968 3.64066648 ；如果 <math id="M57"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933141&type= 14.56266594 4.65666676 且 <math id="M58"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933129&type= 14.64733315 4.57200003 ，那么 <math id="M59"><mi>γ</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933135&type= 8.63599968 3.64066648 ；如果 <math id="M60"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933141&type= 14.56266594 4.65666676 且 <math id="M61"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933150&type= 14.56266594 4.65666676 ，那么以CRI评价标准来计算，其公式如下： <math display="block" id="M62"><msub><mrow><mi>γ</mi></mrow><mrow/></msub><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>⋅</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>⋅</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933157&type= 48.00600052 4.57200003 ， 其中 <math id="M63"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933162&type= 7.70466709 3.72533321 和 <math id="M64"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933168&type= 7.70466709 3.72533321 均为权重值，且有 <math id="M65"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933176&type= 27.09333420 3.80999994 。 3 基于改进速度障碍法的航迹重规划 1 传统的速度障碍法是一种在理想状态条件下的动态避障方法，该方法忽略了USV在水面运动时的动力学特性（速度约束、角速度约束等）和航行参数的不确定性，也没有考虑海上避碰规则，因此在具体航行实验中难以达到理想效果。本文通过引入障碍物速度不确定度，结合水面USV动力学特性约束和海上避碰规则约束，提出了一种多约束条件下的改进速度障碍法。该方法可以获得一个最优避障速度，在该速度下可以实现水面USV的航迹重规划。 3.1　速度障碍区域构造 2 速度障碍区域是指：当USV的速度矢量在该区域时，在将来的一段时间内USV会与动态障碍物发生碰撞。在传统的速度障碍法中，该区域的构造仅由USV与动态障碍物的相对速度和相对尺寸决定，在实际航行过程中存在着很大的局限性。因此，本文提出的改进速度障碍法通过增加多种约束条件来构造更为合理的速度障碍区域，从而保证避障过程的可靠性与安全性。 考虑到海上障碍物大多为船舶，因此在构造速度障碍区域时障碍物考虑为船舶。构建几何模型时USV以及会遇船舶通过矩形包围框来表示，如 图3 图3 所示（彩图见期刊电子版）。USV使用蓝色矩形包围框 <math id="M66"><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933182&type= 3.04800010 2.79399991 来表示，会遇船舶使用黄色矩形包围框 <math id="M67"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933190&type= 2.96333337 2.87866688 来表示，其中 <math id="M68"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933933&type= 3.64066648 3.72533321 表示USV的位置， <math id="M69"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933327&type= 3.04800010 3.72533321 表示USV的速度； <math id="M70"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933207&type= 3.64066648 3.72533321 表示会遇船舶的位置， <math id="M71"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933314&type= 3.04800010 3.72533321 表示会遇船舶的速度。 定义一条从USV几何中心 <math id="M72"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933933&type= 3.64066648 3.72533321 沿相对速度 <math id="M73"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933259&type= 2.70933342 3.72533321 出发的射线为： <math display="block" id="M74"><mi>λ</mi><mfenced separators="|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mi>t</mi><mfenced open="|" close="" separators="|"><mrow><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933231&type= 43.94200134 5.16466665 ， 式中 <math id="M75"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933237&type= 1.10066664 2.79399991 代表时间，因此射线 <math id="M76"><mi>λ</mi><mfenced separators="|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933241&type= 13.03866673 4.82600021 实际表示的是在两船运动状态不发生改变的情况下，USV与会遇船舶的实时相对位置。当射线 <math id="M77"><mi>λ</mi><mfenced separators="|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933254&type= 13.03866673 4.82600021 随时间的增加与会遇船舶相交时，即射线 <math id="M78"><mi>λ</mi><mfenced separators="|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933254&type= 13.03866673 4.82600021 落在速度障碍区域包围框内时，两船在未来有碰撞危险。定义集合 <math id="M79"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933259&type= 2.70933342 3.72533321 为相对碰撞区域（Relative Collisin Cone） <math id="M80"><mi>R</mi><mi>C</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933342&type= 11.09133339 3.72533321 ，若USV以集合内的速度行驶，则存在与会遇船舶的碰撞危险，该集合的数学表示如下： <math display="block" id="M81"><mi>R</mi><mi>C</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>U</mi><mi>O</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><mi>λ</mi><mfenced separators="|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced><mo>⋂</mo><mi>O</mi><mo>≠</mo><mi mathvariant="normal">∅</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933272&type= 52.66266251 6.34999990 . 根据前文所述，相对碰撞区域是以相对速度空间为基准的。如果多个船舶会遇就会出现多个基准，无法统一描述，为了应对这个情况，这里考虑将USV的绝对速度作为基准空间，对其他对象的速度进行转换，从而统一对各个对象的描述。此时，可以将集合 <math id="M82"><mi>R</mi><mi>C</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933342&type= 11.09133339 3.72533321 根据绝对速度集合 <math id="M83"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933327&type= 3.04800010 3.72533321 转化为新的速度障碍区域 <math id="M84"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933319&type= 7.87400007 3.80999994 ： <math display="block" id="M85"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub><mo>=</mo><mi>R</mi><mi>C</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>U</mi><mi>O</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933289&type= 30.81866837 4.40266657 ， 式中 <math id="M86"><mo>⊕</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933293&type= 3.72533321 3.47133350 为闵可夫斯基和运算，目的是求取两个欧几里得空间的点集合，即求取相对碰撞区域 <math id="M87"><mi>R</mi><mi>C</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933342&type= 11.09133339 3.72533321 与矢量空间 <math id="M88"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933314&type= 3.04800010 3.72533321 的点集和。如 图3 图3 所示，将相对碰撞区域 <math id="M89"><mi>R</mi><mi>C</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933342&type= 11.09133339 3.72533321 沿矢量 <math id="M90"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933314&type= 3.04800010 3.72533321 平移，获得的新区域 <math id="M91"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933319&type= 7.87400007 3.80999994 即为速度障碍区域。那么 式（24） 式（24） 可写成： <math display="block" id="M92"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><mi>λ</mi><mfenced separators="|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced><mo>⋂</mo><mi>O</mi><mo>≠</mo><mi mathvariant="normal">∅</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933323&type= 49.78400040 6.34999990 . 此时若USV的速度矢量 <math id="M93"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933327&type= 3.04800010 3.72533321 落在平移后的平面 <math id="M94"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933336&type= 7.87400007 3.80999994 内，那么判定两船存在碰撞危险；相反，USV可以保持原来的运动状态继续航行。当面对多船舶会遇的动态避障时，同样可以将USV的绝对速度作为基准空间，将每一艘会遇船舶的速度根据基准进行转换，求取所有转换后的速度与相对碰撞区域 <math id="M95"><mi>R</mi><mi>C</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933342&type= 11.09133339 3.72533321 的闵可夫斯基和，即可获得每艘会遇船舶的速度障碍区域 <math id="M96"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933351&type= 5.33400011 3.80999994 ，求所有速度障碍区域 <math id="M97"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933351&type= 5.33400011 3.80999994 的并集就能得到总的 <math id="M98"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933400&type= 4.74133301 3.80999994 集合： <math display="block" id="M99"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi></mrow></msub><mo>=</mo><mo>⋃</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933362&type= 17.61066628 4.23333359 . USV在航行过程中实时调整自身的速度矢量 <math id="M100"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933369&type= 3.04800010 3.72533321 ，使它偏离速度障碍区域 <math id="M101"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933400&type= 4.74133301 3.80999994 就可以实现USV的避碰。 3.2　障碍物速度不确定度约束 2 USV在海上航行过程中主要通过艇载摄像头、激光雷达、声呐和惯导模块等传感器探测障碍物的尺寸信息和运动信息，因此传感器的测量误差会影响速度障碍区域模型的构建。本文通过引入障碍物的速度不确定度 <math id="M102"><msub><mrow><mi>δ</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933383&type= 3.64066648 3.72533321 参量对集合 <math id="M103"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933386&type= 4.74133301 3.80999994 的构建进行优化。将模型构建中的不确定参量放入集合 <math id="M104"><mfenced open="{" close="" separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>O</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>O</mi></mrow></msub><mo>∈</mo></mrow></mfenced></mrow></mfenced><mfenced open="" close="}" separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>O</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933391&type= 23.28333473 5.67266655 ， <math id="M105"><msub><mrow><mi>W</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933439&type= 5.50333309 3.72533321 集合为存在常数上界的有界集合，因此可以将集合 <math id="M106"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933400&type= 4.74133301 3.80999994 优化为集合 <math id="M107"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933594&type= 11.59933376 3.80999994 ： <math display="block" id="M108"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub><mo>=</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933416&type= 33.86666870 4.40266657 . <math id="M109"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933594&type= 11.59933376 3.80999994 区域如 图4 图4 所示（彩图见期刊电子版），在紫色区域内。将不确定度集合 <math id="M110"><msub><mrow><mi>W</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933439&type= 5.50333309 3.72533321 以绿色外接圆来表示，在外接圆上做两条平行 <math id="M111"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933594&type= 11.59933376 3.80999994 集合边界的切线，此时 <math id="M112"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933594&type= 11.59933376 3.80999994 集合扩展了一个常数上界的距离，由于存在 <math id="M113"><msub><mrow><mi>W</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933439&type= 5.50333309 3.72533321 的约束给USV提供了避障缓冲区域。 3.3　水面无人艇动力学特性约束 2 由于USV自身存在一定的动力学特性，因此USV在一定时间内所做出的运动状态改变是有限的。许多不在 <math id="M114"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933594&type= 11.59933376 3.80999994 集合内的速度矢量对USV本身而言是不可能达到的，因此通过增加动态窗口的尺寸缩小可行速度的选择范围，进而获取USV的最优航行速度。 设置某时刻的时间窗口为 <math id="M115"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934047&type= 3.55599999 2.79399991 ，计算USV在 <math id="M116"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934047&type= 3.55599999 2.79399991 时间内的速度窗口 <math id="M117"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933559&type= 3.04800010 3.72533321 和角速度窗口 <math id="M118"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933470&type= 3.64066648 3.72533321 ： <math display="block" id="M119"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>v</mi><mfenced open="|" close="" separators="|"><mrow><mi>v</mi></mrow></mfenced><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>-</mo><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>+</mo><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933476&type= 57.23466873 8.04333401 ， <math display="block" id="M120"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>ω</mi><mfenced open="|" close="" separators="|"><mrow><mi>ω</mi></mrow></mfenced><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>-</mo><msub><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>,</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>+</mo><msub><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933481&type= 61.72200012 8.04333401 ， 式中： <math id="M121"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933911&type= 2.87866688 3.72533321 和 <math id="M122"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933488&type= 3.55599999 3.72533321 分别表示USV当前的速度和角速度， <math id="M123"><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933493&type= 3.64066648 5.67266655 和 <math id="M124"><msub><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933498&type= 4.31799984 5.67266655 分别为USV保持加速状态下的最大加速度和角加速度， <math id="M125"><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933502&type= 3.72533321 5.67266655 和 <math id="M126"><msub><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933507&type= 4.40266657 5.67266655 分别为USV保持减速状态下的最大加速度和角加速度。 在此基础上USV需要保证它在与会遇船舶碰撞之前可以及时制动，那么速度窗口和角速度窗口就需要再次被约束： <math display="block" id="M127"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>≤</mo><mroot><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow><mrow/></mroot><mo>,</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>≤</mo><mroot><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><msub><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933514&type= 48.25999832 6.94266701 . 在求出速度窗口和角速度窗口后，还能够求出USV在 <math id="M128"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934047&type= 3.55599999 2.79399991 时间内实现状态改变的艏向窗口 <math id="M129"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933523&type= 2.96333337 3.72533321 ： <math display="block" id="M130"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>θ</mi><mfenced open="|" close="" separators="|"><mrow><mi>θ</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msub><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub><mi mathvariant="normal">Δ</mi><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msub><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub><mi mathvariant="normal">Δ</mi><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933529&type= 91.77866364 10.24466610 ， 式中 <math id="M131"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933919&type= 2.79399991 3.72533321 表示USV此时的艏向。 将速度窗口 <math id="M132"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933559&type= 3.04800010 3.72533321 与艏向窗口 <math id="M133"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933575&type= 2.96333337 3.72533321 相结合构建出USV的可行速度集合 <math id="M134"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">V</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933553&type= 4.40266657 3.72533321 ，其中 <math id="M135"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933559&type= 3.04800010 3.72533321 为可行速度大小的集合， <math id="M136"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933575&type= 2.96333337 3.72533321 为可行速度方向的集合。将 <math id="M137"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">V</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933583&type= 4.40266657 3.72533321 集合与前文所述的 <math id="M138"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933594&type= 11.59933376 3.80999994 集合相结合，求取此时USV的可行无碰速度（Reachable Avoidance Velocity， <math id="M139"><mi mathvariant="normal">R</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933600&type= 8.29733276 2.87866688 ）集合： <math display="block" id="M140"><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>V</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>V</mi><mfenced open="|" close="" separators="|"><mrow><mi>V</mi><mo>∈</mo><mi>R</mi><mi>V</mi><mo>,</mo><mi>V</mi><mo>∉</mo><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933608&type= 52.23933029 5.67266655 . 在构建速度障碍区域时，将其他对象（包括障碍船）的速度转换到USV的绝对速度空间，因此本节中动力学特性的约束是针对USV与障碍艇两者一同建立的。 3.4　国际海上避碰规则约束 2 在实现动态避障的过程中，考虑使用速度障碍法的优势在于速度障碍集合 <math id="M141"><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933612&type= 4.74133301 3.80999994 中已经提供了USV与会遇船舶（障碍物）的相对位置。如 图5 图5 所示，USV的绝对速度空间可以分为4个区域，分别为 <math id="M142"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933618&type= 11.59933376 3.80999994 ， <math id="M143"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933625&type= 4.14866638 3.80999994 ， <math id="M144"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933633&type= 4.14866638 3.80999994 和 <math id="M145"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933641&type= 4.14866638 3.80999994 ，图示分割线与 <math id="M146"><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933649&type= 11.59933376 3.80999994 的底端相切且与该区域的角平分线垂直。 当USV的绝对速度落在 <math id="M147"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933675&type= 4.14866638 3.80999994 区域时，USV向远离会遇船舶处航行， <math id="M148"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933675&type= 4.14866638 3.80999994 区域的数学表示如下： <math display="block" id="M149"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>v</mi><mfenced open="|" close="" separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow></mfenced><mo>⋅</mo><mfenced separators="|"><mrow><mi>v</mi><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mo>∉</mo><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933683&type= 71.12000275 6.34999990 . 当USV的绝对速度落在 <math id="M150"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933693&type= 4.14866638 3.80999994 区域时，USV从左侧驶过会遇船舶，其数学表示如下： <math display="block" id="M151"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>v</mi><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow></mfenced><mo>×</mo><mfenced separators="|"><mrow><mi>v</mi><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>z</mi></mrow></msub><mo><</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mo>∉</mo><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub><mo>,</mo><mi>v</mi><mo>∉</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933701&type= 88.39199829 9.14400005 ， <math display="block" id="M152"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>v</mi><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow></mfenced><mo>×</mo><mfenced separators="|"><mrow><mi>v</mi><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>z</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>v</mi><mo>∉</mo><mi>W</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>O</mi><mi>U</mi><mi>O</mi></mrow></msub><mo>,</mo><mi>v</mi><mo>∉</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933710&type= 88.30733490 9.14400005 ， 式中： <math id="M153"><msub><mrow><mfenced open="[" close="]" separators="|"><mrow/></mfenced></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933722&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933718&type= 5.41866684 5.07999992 为提取向量的 <math id="M154"><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933753&type= 1.69333339 2.79399991 轴分量的数学符号，即获取该向量 z 轴方向的数值。以USV的几何中心为坐标原点，航行方向为 <math id="M155"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933736&type= 2.03200006 2.79399991 轴正向，与 <math id="M156"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933736&type= 2.03200006 2.79399991 轴顺时针夹角90°方向为 <math id="M157"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933745&type= 1.86266661 3.72533321 轴正向， <math id="M158"><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933753&type= 1.69333339 2.79399991 轴正向垂直指向纸面向内。 当USV的绝对速度落在 <math id="M159"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933761&type= 4.14866638 3.80999994 区域时，USV从右侧驶过会遇船舶，结合COLREGS可以得出满足国际海上避碰规则的可行无碰速度集合（COLREGS based Reachable Avoidance Velocity， <math id="M160"><mi mathvariant="normal">C</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934007&type= 10.83733368 2.87866688 ）。 当USV与会遇船舶处于追越或者左交叉局面时： <math display="block" id="M161"><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>A</mi><mi>V</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>v</mi><mfenced open="|" close="" separators="|"><mrow><mi>v</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>V</mi></mrow></msub><mo>,</mo><mi>v</mi><mo>∈</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>⋃</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933778&type= 50.54600143 5.67266655 . 当USV与会遇船舶处于对遇或者右交叉局面时： <math display="block" id="M162"><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>A</mi><mi>V</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>v</mi><mfenced open="|" close="" separators="|"><mrow><mi>v</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>V</mi></mrow></msub><mo>,</mo><mi>v</mi><mo>∈</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>⋃</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933787&type= 50.54600143 5.67266655 . 3.5　结合最优避障速度的航迹重规划 2 由于 <math id="M163"><mi mathvariant="normal">C</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934007&type= 10.83733368 2.87866688 集合中的速度 <math id="M164"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933824&type= 1.86266661 2.79399991 和艏向 <math id="M165"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933841&type= 1.77800000 2.79399991 都是连续的，因此可行速度矢量仍然不是有限的选择，最优避障速度的确定仍然十分困难。针对该问题，本文以离散化网格的方式处理 <math id="M166"><mi mathvariant="normal">C</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934007&type= 10.83733368 2.87866688 集合中的速度与艏向，等间距网格离散速度 <math id="M167"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933824&type= 1.86266661 2.79399991 为 <math id="M168"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933833&type= 3.38666677 2.87866688 个速度，同理，等间距网格离散艏向 <math id="M169"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933841&type= 1.77800000 2.79399991 为个 <math id="M170"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933853&type= 2.87866688 2.87866688 艏向，再将其任意组合配对成速度矢量，最终得到离散为 <math id="M171"><mi>M</mi><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933961&type= 6.26533365 2.87866688 个速度矢量的 <math id="M172"><mi mathvariant="normal">C</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934007&type= 10.83733368 2.87866688 集合。 上述获取的 <math id="M173"><mi>M</mi><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933961&type= 6.26533365 2.87866688 个速度矢量需要建立该体系的评价标准来判断该速度矢量的优劣，最终选择最优航行速度。对于这 <math id="M174"><mi>M</mi><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933961&type= 6.26533365 2.87866688 个速度矢量而言，其代价可以按如下公式进行计算： <math display="block" id="M175"><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>-</mo><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933889&type= 89.32333374 11.76866722 ， <math display="block" id="M176"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><msub><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></mrow></mfenced></mrow></mfrac></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933897&type= 29.46400070 10.66800022 . 式中： <math id="M177"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934000&type= 3.04800010 3.72533321 是USV到下一个局部子目标点的目标速度， <math id="M178"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933911&type= 2.87866688 3.72533321 是USV此时的速度大小， <math id="M179"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933919&type= 2.79399991 3.72533321 是USV此时的艏向， <math id="M180"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933926&type= 3.64066648 3.72533321 是下一个局部子目标点的位置， <math id="M181"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933933&type= 3.64066648 3.72533321 是USV此刻的位置， <math id="M182"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933940&type= 3.64066648 3.72533321 和 <math id="M183"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933949&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933946&type= 3.64066648 3.72533321 是两个比例系数。代价评价函数 <math id="M184"><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933969&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933965&type= 3.89466691 3.72533321 以速度改变量和目标朝向结合考虑，远离目标点和速度改变量大的速度集合实现需要加大的代价。考虑到局部避障过程中目标导向的重要性，两个比例系数的权重应满足 <math id="M185"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>></mo><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933957&type= 12.36133385 3.72533321 。 对 <math id="M186"><mi>M</mi><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933961&type= 6.26533365 2.87866688 个组合分别计算 <math id="M187"><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933969&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933965&type= 3.89466691 3.72533321 ，选取代价最小的速度 <math id="M188"><msubsup><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933992&type= 4.57200003 3.97933316 作为最优航行速度。考虑到实际避障过程中，AIS系统可以给远程提供大量信息，因此可以在全局规划的同时实现远程的航迹重规划进而实现动态避障。采用最优速度 <math id="M189"><msubsup><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933992&type= 4.57200003 3.97933316 进行航行虽然可以保证USV避免碰撞，但 <math id="M190"><msubsup><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933992&type= 4.57200003 3.97933316 并不能保证USV能够准确向下一个目标点行驶。那么在考虑避障的同时，还需要实时更新航线规划，通过模拟USV沿速度 <math id="M191"><msubsup><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20933992&type= 4.57200003 3.97933316 的航行过程，判断下一个目标点的目标速度矢量 <math id="M192"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934000&type= 3.04800010 3.72533321 是否在 <math id="M193"><mi mathvariant="normal">C</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934007&type= 10.83733368 2.87866688 集合内，若不在，那么需要不断循环模拟直到满足 <math id="M194"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>A</mi><mi>V</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934012&type= 14.30866718 4.06400013 。根据USV每次模拟时的位置矢量 <math id="M195"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934019&type= 3.55599999 3.72533321 和模拟花费的时间即可计算出新规划目标点的位置矢量 <math id="M196"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934027&type= 3.21733332 3.72533321 ： <math display="block" id="M197"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msubsup><mi>n</mi><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934039&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934038&type= 26.92399979 4.06400013 ， 式中： <math id="M198"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934042&type= 1.86266661 2.79399991 表示模拟循环次数， <math id="M199"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934047&type= 3.55599999 2.79399991 表示模拟的时间周期。 4 实验结果与分析 1 4.1　无人艇动态避障算法流程 2 获取动态障碍物感知信息后，首先将相关感知信息转换为USV与障碍船之间的相对运动参数。然后，根据碰撞危险度计算模型可以判断当前时刻是否存在碰撞危险，如果存在碰撞危险，则划分会遇局面和构建速度障碍区域，针对当前障碍区域和约束条件通过代价函数计算出最佳避障速度，并依据该理论速度改变无人船的航向或速度来避开动态障碍物。这段由于动态避障形成的偏离原先规划好的全局路径的局部路径则为本文所述的动态避障重规划航迹。最后，再通过航迹重规划方法来判断何时开始返航，并重新规划到全局路径中的局部目标点的返航路径。算法流程如 图6 图6 所示。 4.2　仿真实验与结果分析 2 国际海上避碰规则将船舶之间的会遇局面划分为追越、正面相遇和交叉相遇3种情况 ［ 12 12 ］ ，其中交叉相遇还可划分为左相遇和右相遇两种情况。在仿真实验中，首先设计了以追越场景为例的动态避障实验，用于对比本文提出的改进速度障碍法与传统速度障碍法的避障效果。仿真实验参数设置如下： （1）USV：初始航速为10 m/s，初始航向为正北方向，实验中表示USV的外接圆半径为7.3 m； （2）追越艇：初始航速为6 m/s，初始航向为正北方向，实验中表示追越艇的外接圆半径为7.3 m； （3）目标点：设置在USV正北方向的250 m处； （4）发生碰撞：当USV与追越艇的外接圆出现交集时，即USV与追越艇的几何中心距离小于14.6 m，两船发生碰撞行为，此时动态避碰实验失败。 传统速度障碍法与改进速度障碍法的仿真实验结果分别如 图7 图7 和 图8 图8 所示。 图7 图7 和 图8 图8 分别为仿真实验中采用两种动态避碰方法得到的船舶航行轨迹和船舶相对距离。在采用传统的速度障碍法时，速度障碍区域构造不够完善且未进行实时的航迹重规划，在实现动态避障的过程中USV存在与追越艇碰撞的可能性，如 图7 图7 （a）中22 s时，两艇的相对距离小于14.6 m，发生碰撞。而在使用改进的速度障碍法时，首先对速度障碍区域的构造进行了改进，并且在航行过程中不断进行航迹重规划，可以观察到在 图8 图8 （b）中20 s左右的时间存在一个波峰，通过航迹重规划增加USV与追越艇的相对距离延缓返航时间，避免了碰撞的发生。根据以上的仿真结果可以说明改进的速度障碍法构造的速度障碍区域更加合理，在实时规划航迹的状态下提高了动态避障的安全性与可靠性。 在上述仿真实验的基础上，为了进一步验证改进速度障碍法的有效性，仿真实验在单船会遇避障中针对正面相遇、追越、左交叉相遇和右交叉相遇4种常见会遇局面设置了对应碰撞冲突场景，并在V-REP软件中进行仿真实验。初始实验参数设置如 表1 表1 所示。 类　别 航速/（m·s -1 ） 艏向角 初始坐标 外接圆直径/m 水面无人艇 10 180 （110，0） 14.6 对遇船 5 0 （-130，0） 14.6 追越船 4 180 （40，0） 14.6 左交叉船 10 90 （0，-110） 14.6 右交叉船 10 315 （-64.7，64.7） 14.6 图9 图9 为USV在COLREGS中4种常见会遇场景时采取的避障措施，其中USV航迹表示为绿色，会遇船舶航迹表示为红色（彩图见期刊电子版）。 图10 图10 是单船对遇实验中水面USV航向角和两船距离的变化情况， 表2 表2 是单船会遇避障实验结果。可以看出，USV初始航向角设置为180°，当会遇船舶进入无人船的感知范围时，开始计算无人船与会遇船之间的相对参数，然后根据第2节的 式（21） 式（21） 计算USV与会遇船之间的碰撞危险度。当碰撞危险度超过设定阈值时开始避障流程，即搭建速度障碍区域、结合约束条件和代价函数确定最佳避障速度。从 图10 图10 中可看出，艏向角偏离设定值时开始避障行为，之后USV根据障碍物情况重新规划航行目标点，在USV避开障碍物后航向角开始变化直至朝向目标方向，USV结束避障行为，并根据调整后的方向航行至目标点。在开始避障后，USV和会遇船舶的相对距离变化为先减小后增加，即在避障过程中两船的相对位置会存在一个极值。在避障仿真实验中，两船的最小距离处于29~31 m之间，约为2~3个船长的距离，说明USV能够在保障航行安全的情况下实现对障碍物的规避并最终航行至目标点。 1,2; 会遇 类型 避障起始时间/s 避障结束时间/s 航迹重规划点 两船最小距离/m 对遇 11 20 （-45.1， 30.9） 29.8 追越 0 15 （-9.7，-32.2） 29.2 左交叉 5.7 13 （24.1， -29.6） 30.6 右交叉 6 13 （18.9， 21.5） 29.4 4.3　实船实验与结果分析 2 根据COLREGS对避碰场景的划分，本文设置了对遇、追越和左交叉和右交叉4种避碰场景下的避碰实验来验证基于航迹重规划的USV动态避障方法的有效性。实船实验在广州市南沙区龙穴岛海域进行，实验场地如 图11 图11 所示。 在USV启动之前需要在上位机预先设置水面USV需要达到的目标点，并在水面相应位置启动障碍船。水面USV与障碍船的初始实验参数设置如 表3 表3 ~ 表6 表6 所示。由于实验场地海况一般为2~3级，为尽可能减小风浪对USV航行的影响，综合考虑选择12节的航速进行实验。 类别 航速（节） 艏向角/（°） 初始经纬度/（°） 目标点经纬度/（°） 无人艇 12 -31.3 22.66617，113.675 29 22.667 71，113.674 28 障碍艇 12 -117.9 22.66754，113.675 74 - 类别 航速（节） 艏向角/（°） 初始经纬度/（°） 目标点经纬度/（°） 无人艇 12 -31.2 22.667 35，113.674 64 22.668 90，113.673 63 障碍艇 12 60.4 22.667 78，113.673 23 - 类别 航速（节） 艏向角/（°） 初始经纬度/（°） 目标点经纬度/（°） 无人艇 12 -29.6 22.668 53，113.674 07 22.670 10，113.673 11 障碍艇 12 154.1 22.669 98，113.673 32 - 类别 航速（节） 艏向角/（°） 初始经纬度/（°） 目标点经纬度/（°） 无人艇 12 -28.1 22.670 32，113.673 06 22.672 32，113.671 92 障碍艇 6 -29.2 22.670 93，113.672 75 - 避碰实验使用的USV尺寸为7.5 m×2.0 m，障碍艇尺寸为10 m×2.4 m。考虑到传感器误差和海面波浪的影响，本文将叠加USV尺寸后的障碍艇的模型尺寸设为24 m×5 m，当USV位于障碍物膨化模型内时意味着即将发生碰撞。 图12 图12 为USV在COLREGS中4种常见会遇场景时采取的避障措施，其中USV航迹表示为蓝色轨迹点，会遇船舶航迹表示为红色轨迹点。与仿真结果基本一致，在刚开始航行时USV和会遇船舶均按照给定速度航行，由于波浪和风等外界条件的影响，此时USV和会遇船的航向均受到环境因素的影响。当CRI较高时，USV会根据和会遇船舶之间的相对位置和方向进行判断并结合COLREGS重新规划航迹，之后USV按照重规划后的航线航行，最后根据COLREGS成功避开会遇船舶，到达预设的目标点。当会遇船在USV碰撞危险距离内时，开始统计两船之间的距离变化和USV的航向角变化，如 图13 图13 和 图14 图14 所示。由于船舶惯性和船舶侧推动力的影响，可以看到USV在会遇船进入避障范围内时避障过程存在着一定的延时，反映到距离图则是两船之间距离的减少，反映到航向角图则表现为USV航向角一段时间内的稳定。度过短暂的延时期后，可以看见USV的航向角出现了明显的变化，同时两船之间的距离开始拉远，最后成功地避开了会遇船。 表7 表7 是单船会遇避障实验结果，可以看到，在单船会遇避障实验中，USV距离障碍物最近时的位置均在障碍物模型之外，这表示水面USV能够在保证足够避障安全裕量的前提下驶离会遇船舶并最终抵达目标点，证明了基于航迹重规划的水面USV动态避障方法的有效性。 会遇类型 避碰起始时间/s 返航时间/s 返航点经纬度/（°） 两船最小距离/m 右交叉 8 17 22.671 26 ， 113.672 20 24.1 左交叉 9 18 22.667 94， 113.673 93 30.2 对遇 7 16 22.669 34， 113.673 84 17.3 追越 8 22 22.671 26， 113.672 20 19.6 5 结　论 1 本文通过矩形包围框对速度障碍区域的几何模型进行描述，同时速度不确定度对速度障碍区域的运动模型进行描述，从而实现对可行驶速度区域的模型建立，同时将COLREGS与USV自身的动力学特征相结合，将USV可行驶速度区域集合进一步缩小，最终通过代价函数的计算选择最优航行速度 <math id="M200"><msubsup><mrow><mi>v</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20934196&type= 4.57200003 3.97933316 ，并在碰撞危险度达到设定的阈值时进行航迹重规划。由仿真和实物实验结果分析可知，基于航迹重规划的动态避障方法在面对包括对遇、追越、左交叉和右交叉会遇冲突时能够根据COLREGS实现避让行为并继续航行至目的地。在航速12节的情况下，避碰过程中的路径重规划时间在15 s以下，而且能保证两个船长位置的避障安全裕量，这表明该方法能有效应对实际海上复杂交通环境下各种可能的碰撞情况，实现水面USV实时动态的局部路径规划。下一步将结合基于异构信息融合的感知方法和水面USV实际应用场景进行复杂交通环境下多船动态避障方法的研究。 参考文献 孙兆伟 ， 刘雪奎 ， 吴限德 ， 等 . 用于通信保障航天器的遗传蚁群融合路径规划 ［J］. 光学 精密工程 ， 2013 ， 21 （ 12 ）： 3308 - 3316 . doi: 10.3788/ope.20132112.3308 http://dx.doi.org/10.3788/ope.20132112.3308 用于通信保障航天器的遗传蚁群融合路径规划 SUN ZH W ， LIU X K ， WU X D ， et al . Path planning based on ant colony and genetic fusion algorithm for communication supporting spacecraft ［J］. Opt. Precision Eng. ， 2013 ， 21 （ 12 ）： 3308 - 3316 . （in Chinese） . doi: 10.3788/ope.20132112.3308 http://dx.doi.org/10.3788/ope.20132112.3308 Path planning based on ant colony and genetic fusion algorithm for communication supporting spacecraft 颜雪松 ， 胡成玉 ， 姚宏 ， 等 . 精英粒子群优化算法及其在机器人路径规划中的应用 ［J］. 光学 精密工程 ， 2013 ， 21 （ 12 ）： 3160 - 3168 . doi: 10.3788/ope.20132112.3160 http://dx.doi.org/10.3788/ope.20132112.3160 精英粒子群优化算法及其在机器人路径规划中的应用 YAN X S ， HU CH Y ， YAO H ， et al . Elite particle swarm optimization algorithm and its application in robot path planning ［J］. Opt. 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