含关节间隙的Delta机器人弹性动力学与振动特性分析

针对经济实用型并联机器人关节间隙对动平台位置精度与系统振动特性影响的问题,以Delta机器人为研究对象,利用数理统计原理对含关节间隙的Delta机器人支链进行了运动学分析,结合Lankarani-Nikravesh碰撞接触力模型与具有动态修正系数的Coulomb摩擦模型对关节间隙广义碰撞力进行了研究。利用空间有限元理论与拉格朗日方程,充分考虑主、从动臂的空间动力特性与运动协调关系,建立了Delta机器人弹性动力学模型,在定义杆件虚长度的基础上,将关节间隙产生的广义碰撞力结合到弹性动力学模型中,建立了含关节间隙的Delta机器人弹性动力学模型。借助FARO激光跟踪仪对间隙弹性动力学模型进行了验证分析,利用脉冲锤击法与Workbench软件仿真对Delta机器人的振动特性进行了研究分析。试验结果表明,考虑关节间隙时动平台中心点的运动轨迹较不考虑关节间隙时更靠近试验运行结果,验证了间隙弹性动力学模型的合理性与正确性,并且,系统前两阶非零固有频率的理论值与试验值的相对误差分别为3.544%和12.026%,两者相当接近。另外,由仿真结果可以发现3组从动臂是Delta机器人整机系统中最薄弱的环节。该研究可为经济实用型并联机器人的位置误差补偿与系统减振优化提供参考。

Elastic dynamics and analysis of vibration characteristics of Delta robot with joint clearance

Abstract: For the economical and practical parallel robot, there exists joint clearance in its mechanical structure, and the effect of joint clearance on the position precision of the moving platform and the vibration characteristics of the system can not be ignored. In order to analyze the problem, taking Delta robot as the study object, this paper puts forward a new thought that studies the elastic dynamic model with joint clearance and the vibration characteristics of the system. In the process of analysis, knowing that the joint clearance changes with the change of the robot motion state, which is random, the mathematical statistics principle is used innovatively to analyze the kinematics of the branched chain of Delta robot with joint clearance, and the mathematical expectation of joint clearance vector's probability density in the joint coordinate system is used to express joint clearance's numerical value quantitatively before the collision between shaft and shaft sleeve. Then, because the joint clearance is narrow, ignore the acceleration inertia force of the shaft in shaft sleeve, and only consider the collision force and the friction when shaft and sleeve collide. The probability that the shaft hits arbitrary point on the inner wall of the shaft sleeve is the same and obeys the normal distribution, and on the basis of this theory, combining with the Lankarani-Nikravesh collision contact force model and the Coulomb friction model with dynamic correction coefficient, the generalized collision force of joint clearance is studied. By using the finite element theory and the Lagrange equation, fully considering the active arm and the driven arm's spatial dynamic characteristics and motion coordination, the Delta robot system's elastic dynamic model is established. Based on the definition of bar virtual length, the generalized collision force produced by joint space is combined with the elastic dynamics model, the Delta robot elastic dynamic model with joint clearance is established. The system clearance elastic dynamic model is highly nonlinear time varying equation group, and for the characteristics of the equation group, the time discretization method is used to solve the problem, by using the Newmark algorithm with high stability, accuracy and computational efficiency. Then, with the aid of the FARO laser tracker, the clearance elastic dynamic model is verified, and the vibration characteristics of the Delta robot is analyzed by using the hammer impulse method and the simulation of the Workbench software. Experimental results show that the moving trajectory of the center point of the moving platform when considering the joint clearance is closer to the experimental results than that without the consideration of joint clearance. Then, the rationality and correctness of the clearance elastic dynamic model are verified, and the relative errors between theoretical and experimental values of the non zero natural frequency of the first 2 orders are 3.544% and 12.026%, respectively. The two results are very close, which also indirectly proves the correctness of the theoretical derivation of the clearance elastic dynamic model. In addition, from the simulation results it can be found that the 3 groups of subordinate moving arms are the weakest link in the whole Delta robot system. The study can provide a reference for economical and practical parallel robot's position error compensation and system vibration reduction optimization.