并联机器人逆动力学建模的几何代数方法

Geometric Algebra-based Method for Inverse Dynamic Modeling of Parallel Robots

逆动力学建模是并联机器人动力学性能评估和实际控制的基础。以几何代数为数学工具,提出并联机器人的解析逆动力学建模通用方法。根据几何代数的外积性质,可直接确定并联机器人中各分支的关节速度和加速度幅值,建立机器人驱动速度和被动速度之间的映射关系,并推导出并联机器人分支中各杆件的速度和加速度。结合虚功原理建立并联机器人的逆动力学模型,并确定驱动力/力矩的解析表达式。关节速度和加速度计算不需要判断几何关系、进行求导或求偏微分,并且整个计算过程中仅涉及加法和乘法,无需求解符号线性方程组,提高计算效率。以六自由度并联机器人6-UPS和三自由度并联机器人2PUR-PRU为例(U:虎克铰,P:移动副,S:球铰,R:转动副),得到在给定轨迹下两个机器人驱动力的理论结果,通过与ADAMS软件仿真结果对比验证所提方法的正确性。同时,通过与基于牛顿-欧拉方程的自然正交补法进行计算效率对比,证明该方法具有计算高效的优点,可替代ADAMS仿真和原先方法用于并联机器人逆动力学建模及分析。

Inverse dynamic modeling is the basis of dynamic performance evaluation and actual control of parallel robots. Using geometric algebra as a mathematical tool, a general method for the analytical inverse dynamic modeling of parallel robots is proposed.Based on the property of outer product, the intensities of velocities and accelerations of joints in the limbs can be directly determined,and then the mapping relationship between the actuated and passive velocities of robots is developed, followed by the derivation of velocities and accelerations of centers of mass of links. Combined with the principle of virtual work, the inverse dynamic modeling of parallel robots can be established, and then analytical expressions of actuated forces/torques can be obtained. No judgment of geometric relations, and no derivations or partial derivations are involved in the calculation of joint velocities and accelerations. In addition, only the addition and multiplication are involved in the whole calculation process, and there is no need to solve the symbolic linear equations, which significantly improves the calculation efficiency. Taken two parallel robots, 6-UPS with six degrees of freedom(DOFs) and 2PUR-PRU with three DOFs(where U denotes a universal joint, P denotes a prismatic joint, S denotes a spherical joint, and R denotes a revolute joint), as examples, the actuated forces of two parallel robots in the given trajectories can be obtained, and the correctness of the proposed theoretical method is verified by the simulations of ADAMS software. Meantime, by comparing the calculation efficiency with the natural orthogonal complement method based on the Newton-Euler equation, it is proved that the proposed method has the advantage of high efficiency, and can be chosen as an alternative to ADAMS simulation and original theoretical methods for the inverse dynamic modeling and analysis of parallel robots.