机械臂接触碰撞动力学分析

Dynamic Analysis of Manipulator with Contact Impact

针对机械臂在运行过程中的接触碰撞问题,基于碰撞过程中产生冲量和高斯最小约束原理,提出了一种确定接触碰撞后系统状态量的分析方法。首先,利用Lagrange方程,建立机械臂系统的动力学模型,并以此为基础根据经典碰撞理论与恢复系数方程,推导得到碰撞时系统的外部碰撞冲量求解模型;该模型中碰撞点的速度与碰撞冲量之间是解耦的,有利于计算求解。其次,根据高斯最小约束原理和求解得到的接触碰撞前的系统状态量,运用求多变量函数极值的方法建立机械臂系统接触碰撞后瞬时速度求解方程式。最后,以平面三连杆机械臂为实例进行碰撞动力学建模与仿真,研究分析系统发生接触碰撞时所受到的外部碰撞冲量大小以及碰撞前后机械臂角速度和各关节转角的变化规律,求解得到了机械臂发生碰撞后保持原有运动规律不变时各关节所需施加的驱动力矩。研究表明:碰撞发生瞬时各关节将产生刚性冲击;碰撞后瞬时各杆将产生抖动;碰撞产生的冲量由远端关节向近端关节以递减的方式变化,所得结论可为机械臂冲击后的运动轨迹控制提供一定的理论依据。

Aiming at the contact impact problem caused by the manipulator in the course of operating, a method of determining the state of the system after contact collision is proposed based on the impulse principle and Gauss＇ principle of least constraint. First of all, the dynamic model of the manipulator system is proposed based on Lagrange equation, besides the solution model of the external impulse of collision of the system under the collision is derived according to the classical collision theory and the equation involving coefficient of restitution ; the velocity of collision point is decoupled from the impulse of collision, which is beneficial to the calculation. Secondly, based on the Gauss＇ principle of least constraint and the state variables of the system before the contact collision, the instantaneous velocity solution equation of the manipulator system after collision is established according to the method of solving the extreme value of a muhivariable function; according to the speed after the impact, it can be determined whether the angular velocity of the manipulator is fit for the permited working range. Finally, taking a planar three link manipulator as an example for collision dynamics modeling and simulation result, the size of the external impulse of collision and the change regularity of the manipulators＇ angular velocities and each joint angle before and after the collision of the system are studied and analyzed, the driving moment required of each joint is given according to the change regularity of each joint angle before and after the collision. The research shows that the joint will have a rigid impact and each rod will shake at the collision instantaneous; the impulse of collision will be changed in the form of diminishing by the way of the distal joint to the proximal joint. The obtained conclusion in this paper may provide a theoretical basis for the motion control of the manipulator under the external impact.