引用本文: 李高峰, 许杉, 孙雷, 刘景泰. 面向非完全姿态约束问题的旋转矩阵群中的垂线理论[J]. 机器人, 2018, 40(2): 195-205.
LI Gaofeng, XU Shan, SUN Lei, LIU Jingtai. The Theory of Perpendicular Curve in Rotation Group Aiming at Incomplete Orientation Constraint Problems[J]. ROBOT, 2018, 40(2): 195-205.
 Citation: LI Gaofeng, XU Shan, SUN Lei, LIU Jingtai. The Theory of Perpendicular Curve in Rotation Group Aiming at Incomplete Orientation Constraint Problems[J]. ROBOT, 2018, 40(2): 195-205.

## The Theory of Perpendicular Curve in Rotation Group Aiming at Incomplete Orientation Constraint Problems

• 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．

Abstract: In many robot applications, the complete orientation constraint in 3D 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 or 4-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.

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