Udwadia-Kalaba方法在四足机器人单腿轨迹跟踪控制违约问题的应用

The Application of the Udwadia-Kalaba Method in Addressing Constraint Violations in Single-Leg Trajectory Tracking Control of Quadruped Robots

针对四足机器人由于初始条件与约束方程不相容和数值积分对低阶约束造成的违约的问题,提出了一种基于Udwadia-Kalaba方法以解决以上四足机器人单腿动力学违约问题的轨迹跟踪控制方法,该方法利用拉格朗日方程建立无约束的动力学方程,将系统的预定轨迹视为系统的约束关系,联系正运动学得到的末端位置建立约束方程.首先,对于初值违约问题,设置了各关节不为零的初角速度,基于Udwadia-Kalaba方法所进行的末端轨迹跟踪误差较大,利用Baumgarte稳定性方法以减小系统误差,并与PID控制下得到的末端轨迹进行对比,验证了对于修正初值违约问题,Baumgarte稳定性方法的优越性;接着,考虑数值积分对低阶约束造成的违约的问题,将其中一个约束轨迹设置为和时间有关的线性方程,利用Udwadia-Kalaba方法进行轨迹跟踪控制,发现其轨迹偏差较大,而后利用Baumgarte稳定性方法对末端轨迹进行修正,达到了使末端轨迹与轨迹规划贴合的目的.

In response to the issue of violation in trajectory tracking of single leg dynamics for quadruped robots due to the incompatibility between initial conditions and constraint equations,as well as the violation caused by numerical integration on low-order constraints,a trajectory tracking control method based on the Udwadia-Kalaba method is proposed.This method utilizes the Lagrangian equation to establish unconstrained dynamic equations,considers the predetermined trajectory of the system as a constraint relationship,and establishes constraint equations based on the end position obtained from direct kinematics.Firstly,to address the problem of initial value violation,non-zero initial angular velocities of each joint are set.It is found that the trajectory tracking error based on the Udwadia-Kalaba method is relatively large.To mitigate this issue,the Baumgarte stability method is employed to reduce system errors,and a comparison with the end trajectory obtained under PID control is conducted,validating the superiority of the Baumgarte stability method in correcting initial value violations.Subsequently,considering the violation caused by numerical integration on low-order constraints,one of the constraint trajectories is set as a time-dependent linear equation.Trajectory tracking control is performed using the Udwadia-Kalaba method,which results in significant trajectory deviation.Then,the Baumgarte stability method is utilized to correct the end trajectory,achieving the goal of aligning the end trajectory with the trajectory planning.