面向非完全姿态约束问题的旋转矩阵群中的垂线理论
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摘要
在很多机器人领域,有很多给定任务要求并不需要3维空间的完整姿态约束,而是只限定了目标姿态某一轴的方向.这类问题称为非完全姿态约束问题.针对该问题,定义了旋转矩阵群中测地线的垂足和垂线,并给出了其解析表达式.基于垂线理论,提出了沿垂线规划的点到点轨迹生成算法.最后通过仿真实验求解操作臂的轨迹规划问题,仿真以6自由度的PUMA 560为平台,并依次固定第6关节、第4关节得到5自由度和4自由度的操作臂.提出的算法不仅适用于具有功能冗余的6自由度操作臂,也可应用于不具有冗余特性的5自由度和4自由度操作臂.实验结果验证了提出的理论和算法在解决非完全姿态约束问题上的通用性.同时,该算法还能避免参数表示方法中的奇异性问题.
In many robot applications, the complete orientation constraint in 3 D Euclidean space is not necessary to fulfill given tasks. In these tasks, only one direction of the target orientation is constrained. These problems are called incomplete orientation constraint problems. Aiming at such problems, the definition and the analytical expressions are given for the foot of perpendicular and the perpendicular curve of geodesic in rotation group. Based on the perpendicular curve theory, a point-to-point trajectory generating algorithm by following the perpendicular curve is proposed. Finally, trajectory planning of manipulators is solved by simulation experiments. And the PUMA 560 with 6 DOFs(degrees of freedom) is taken as the platform, of which the 6-th and the 4-th joints are fixed successively to get 5-DOF and 4-DOF manipulators.The proposed algorithm can not only be applied to 6-DOF manipulators with function redundancy, but also the 5-DOF or4-DOF manipulators without function redundancy. The experimental results demonstrate that the proposed method is highly universal in solving incomplete orientation constraint problems. Moreover, the algorithm can avoid the representational singularity problem in parametrization methods.
In many robot applications, the complete orientation constraint in 3 D Euclidean space is not necessary to fulfill given tasks. In these tasks, only one direction of the target orientation is constrained. These problems are called incomplete orientation constraint problems. Aiming at such problems, the definition and the analytical expressions are given for the foot of perpendicular and the perpendicular curve of geodesic in rotation group. Based on the perpendicular curve theory, a point-to-point trajectory generating algorithm by following the perpendicular curve is proposed. Finally, trajectory planning of manipulators is solved by simulation experiments. And the PUMA 560 with 6 DOFs(degrees of freedom) is taken as the platform, of which the 6-th and the 4-th joints are fixed successively to get 5-DOF and 4-DOF manipulators.The proposed algorithm can not only be applied to 6-DOF manipulators with function redundancy, but also the 5-DOF or4-DOF manipulators without function redundancy. The experimental results demonstrate that the proposed method is highly universal in solving incomplete orientation constraint problems. Moreover, the algorithm can avoid the representational singularity problem in parametrization methods.
引文
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[2]Schuetz C,Baur J,Pfaff J,et al.Evaluation of a direct optimization method for trajectory planning of a 9-DOF redundant fruit-picking manipulator[C]//IEEE International Conference on Robotics and Automation.Piscataway,USA:IEEE,2015:2660-2666.
[3]徐朋,赵东标,程锦翔,等.冗余机器人逆运动学解流形的多目标优化[J].机器人,2016,38(6):704-710.Xu P,Zhao D B,Cheng J X,et al.Multi-objective optimization for inverse kinematics solution manifolds of redundant robots[J].Robot,2016,38(6):704-710.
[4]申浩宇,吴洪涛,陈柏,等.基于主从任务转化的冗余度机器人避障算法[J].机器人,2014,36(4):425-429.Shen H Y,Wu H T,Chen B,et al.Obstacle avoidance algorithm for redundant robots based on transition between the primary and secondary tasks[J].Robot,2014,36(4):425-429.
[5]阳方平,李洪谊,王越超,等.一种求解冗余机械臂逆运动学的优化方法[J].机器人,2012,34(1):17-21,31.Yang F P,Li H Y,Wang Y C,et al.An optimization method for solving the inverse kinematics of redundant manipulator[J].Robot,2012,34(1):17-21,31.
[6]Beeson P,Ames B.TRAC-IK:An open-source library for improved solving of generic inverse kinematics[C]//IEEE-RAS International Conference on Humanoid Robots.Piscataway,USA:IEEE,2015:928-935.
[7]Chan T F,Dubey R V.A weighted least-norm solution based scheme for avoiding joint limits for redundant joint manipulators[J].IEEE Transactions on Robotics and Automation,1995,11(2):286-292.
[8]Starke S,Hendrich N,Magg S,et al.An efficient hybridization of genetic algorithms and particle swarm optimization for inverse kinematics[C]//IEEE International Conference on Robotics and Biomimetics.Piscataway,USA:IEEE,20161782-1789.
[9]陈鹏,刘璐,余飞,等.一种仿人机械臂的运动学逆解的几何求解方法[J].机器人,2012,34(2):212-216.Chen P,Liu L,Yu F,et al.A geometrical method for inverse kinematics of a kind of humanoid manipulator[J].Robot,201234(2):212-216.
[10]Shimizu M,Kakuya H,Yoon W K,et al.Analytical inverse kinematic computation for 7-DOF redundant manipulators with joint limits and its application to redundancy resolution[J]IEEE Transactions on Robotics,2008,24(5):1131-1142.
[11]Wang Y S,Sun L,Yan W B,et al.An analytic and optima inverse kinematic solution for a 7-DOF space manipulator[J]Robot,2014,36(5):592-599.
[12]Sciavicco L,Siciliano B.Modelling and control of robot manipulators[M].Berlin,Gernamy:Springer-Verlag,2000.
[13]Huo L G,Baron L.The joint-limits and singularity avoidance in robotic welding[J].Industrial Robot,2008,35(5):456-464.
[14]王英石,孙雷,刘景泰.基于空间几何方法的乒乓球机器人最优姿态求取[J].机器人,2014,36(2):203-208.Wang Y S,Sun L,Liu J T.Optimal pose solution based on space geometry method for pingpong robot[J].Robot,2014,36(2)203-208.
[15]Murray R M,Li Z X,Sastry S S.A mathematical introduction to robotic manipulation[M].New York,USA:CRC Press,1994.
[16]戴建生.旋量代数与李群、李代数[M].北京:高等教育出版社,2014:119-149.Dai J S.Screw algebra and Lie groups and Lie algebras[M].Beijing:Higher Education Press,2014:119-149.
[17]魏武,叶春台,袁银龙.基于群论的六足机器人运动空间研究[J].机器人,2016,38(5):522-530.Wei W,Ye C T,Yuan Y L.Workspace analysis of a six-legged robot based on group theory[J].Robot,2016,38(5):522-530.
[18]Yu Y,Yang S,Wang M,et al.High performance full attitude control of a quadrotor on SO(3)[C]//IEEE International Conference on Robotics and Automation.Piscataway,USA:IEEE,2015:1698-1703.
[19]Hartley R,Trumpf J,Dai Y,et al.Rotation averaging[J].International Journal of Computer Vision,2013,103(3):267-305.