多体系统轨迹跟踪的瞬时最优控制保辛方法

随着近年来机器人在各行业领域的广泛应用，对机器人的动力学与控制性能不断提出新的要求，特别是对设计越来越复杂、操作越来越灵巧的智能机器人，要求其能够对目标轨迹实现高精度跟踪以满足实际工作需求. 因此，针对机器人多体系统对目标轨迹跟踪的任务需求，基于微分代数方程提出瞬时最优控制保辛方法. 首先，采用多体动力学绝对坐标建模方法建立机器人系统的普适动力学方程，即微分代数方程；然后，采用保辛方法将连续时间域内的微分代数方程进行离散化，进而得到以当前位置、速度和拉式乘子为未知量的非线性代数方程组；其次，通过引入对目标轨迹跟踪以及对控制加权的瞬时最优性能指标，根据瞬时最优控制理论获得当前最优控制输入；最后，通过离散时间步的更新完成对目标轨迹的跟踪任务. 为了验证本文方法的有效性，以双摆轨迹跟踪控制为例进行了数值仿真，结果表明:针对机器人轨迹跟踪任务所提出的瞬时最优控制保辛方法能够实现对目标轨迹的高精度跟踪，且瞬时最优控制由受控微分代数方程推导获得，更具一般性，能够适应其他复杂多体系统的轨迹跟踪控制问题. 

SYMPLECTIC METHOD FOR INSTANTANEOUS OPTIMAL CONTROL OF MULTIBODY SYSTEM TRAJECTORY TRACKING

With the wide application of robot in various fields, the new requirement on dynamics and control perfor-mance for robot has been continually proposed. Especially for the intelligent robot with much more complex system and flexibility of operation, the high accuracy of trajectory tracking should be satisfied for practical mission requirement. Therefore, the aim of this paper is to satisfy the requirement of trajectory tracking mission of robot multibody system, and then the symplectic method based on differential-algebraic equations for instantaneous optimal control is proposed. First, the general dynamic equation of robot should be established by absolute coordinates of multibody system, i.e., differential-algebraic equations; then, the differential-algebraic equations are discretized in the domain of continuous-time by symplectic method, and then the present position/velocity/Lagrange multiplier are taken as unknown variables of nonlinear equations; afterward, the combination of objective tracking trajectory and weighted control input are in-troduced as the performance of instantaneous optimal control. The optimal control input is obtained by the theory of instantaneous optimal control;finally, the tracking mission for the objective trajectory can be continuously implemented with the updated time step. In order to test the effectiveness of the proposed method, the trajectory tacking problem of double pendulum is taken as an example, and numerical simulations show that the proposed symplectic method for in-stantaneous optimal control can obtain high accuracy tracking results, meanwhile, the proposed symplectic method based on differential-algebraic equations can be applied for other trajectory tracking mission of complex multibody system.