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1. 北京航空航天大学 机械工程与自动化学院, 北京 100083;
2. 菏泽学院 机电工程学院, 菏泽 274015

Violation elimination of nominal models for manipulators constructed with Udwadia-Kalaba equation
LYU Guizhi1,2, LIU Rong1
1. School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. School of Electromechanical Engineering, Heze University, Heze 274015, China
Received: 2018-02-08; Accepted: 2018-04-27; Published online: 2018-05-15 12:09
Corresponding author. LIU Rong, E-mail:rliu@buaa.edu.cn
Abstract: In the nominal model of the manipulator trajectory tracking controller constructed by Udwadia-Kalaba equation, the initial conditions are difficult to satisfy the constraint equations, and the constraint violation is generated by the accumulation of errors in the process of numerical solution, which are all problems to be solved at present. Problems are solved by the method of eliminating violation errors directly. This method adds correction items to position and speed terms which are produced by the numerical solution process. The dynamic nominal model of the three-link manipulator is constructed, and the trajectory tracking simulation is carried out under the desired trajectory based on the Udwadia-Kalaba modeling idea. The simulation numerical results are corrected by using the traditional Baumgarte stability method and the proposed error direct elimination method. The results show that the direct elimination method can control the constraint violation in a smaller range more quickly, and is more suitable for the use of trajectory correction to manipulator dynamic nominal model.
Keywords: robotic manipulator     dynamic modeling     violation elimination     constrained manifold     Udwadia-Kalaba equation

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2 违约误差直接消除

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3 动力学建模 3.1 无约束操作臂的动力学建模

 图 1 三杆空间操作臂 Fig. 1 Three-link spatial manipulator

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3.2 受约束操作臂动力学名义模型

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4 数值仿真 4.1 无违约纠正的仿真

 图 2 操作臂各关节角度随时间变化曲线 Fig. 2 Curves of each manipulator joint's angle changing with time
 图 3 操作臂各关节角速度随时间变化曲线 Fig. 3 Curves of each manipulator joint's angular velocity changing with time

 图 4 操作臂末端在Cartesian空间运动轨迹 Fig. 4 Motion trajectories of end point of manipulator in Cartesian space
4.2 修正违约误差的仿真结果与讨论

 图 5 修正后工作空间操作臂末端轨迹 Fig. 5 Corrected trajectories of end point of spatial manipulator

 图 6 给定值和求解值之间的空间距离 Fig. 6 Spatial distance of position given values and solution values
5 结论

1) 相对于Baumgarte约束稳定法对操作臂末端点初始位置不满足约束方程时的渐近抑制修正，误差直接消除法可更为直接地将不满足约束方程的初始值修正为约束流形上的点，从而更便于机器人控制器中动力学名义模型的应用。

#### 文章信息

LYU Guizhi, LIU Rong
Violation elimination of nominal models for manipulators constructed with Udwadia-Kalaba equation

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(11): 2305-2311
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0076