A Multi-objective Internal Preload Optimization Method of Redundantly Actuated Parallel Robots Based on Variable Impedance Control
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摘要: 由于冗余驱动的存在, 冗余驱动并联机器人系统逆动力学模型存在无限组可跟踪期望轨迹的控制力矩解, 这使得机器人在运行过程中具有完成附加任务的能力. 以实现骨科机器人的安全精准操控为目的, 提出了基于变阻抗控制的冗余驱动并联机器人多目标内力优化方法. 首先, 采用支链分解法对冗余驱动并联机器人的动力学进行建模. 其次, 为实现机器人的安全操作, 设计了冗余驱动并联机器人时变阻抗控制器, 利用李雅普诺夫理论分析了系统的稳定性; 在此基础上, 以消除冗余驱动并联机器人运动过程中的传动间隙为附加任务, 提出了一种以力矩传递性能、驱动功率和控制力为优化目标的多目标融合驱动力优化方法. 最后, 通过仿真实验与对比分析, 验证了所提方法的有效性, 实现了机器人系统传动间隙的消除.Abstract: Due to the existence of redundant actuation, the inverse dynamics model of the robot system has an infinite set of control torque solutions that can track the desired trajectory, which makes the redundantly actuated parallel manipulators capable of completing additional tasks during operation. In order to realize safe and precise operation of orthopedic robot, a multi-objective internal preload optimization method of redundantly actuated parallel robots based on variable impedance control is proposed in this paper. First, the dynamics of redundantly actuated parallel manipulator is modeled by using the branch chain decomposition method. Second, in order to realize safe operation, a time-varying impedance controller for redundantly actuated parallel robot is designed, and the stability of the robot system is analyzed by using the Lyapunov theory. On this basis, a multi-objective fusion internal preload optimization method is proposed, which takes torque transmission performance, driving power and control force as optimization objectives, to eliminate the backlash during the movement of redundantly actuated parallel robot. Finally, the effectiveness of the proposed method is verified through simulation experiments and comparative analysis, and the backlash of the robot system is eliminated.
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图 1 平面二自由度冗余驱动并联机器人结构简图
Fig. 1 Schematic of the planar 2-DOF redundantly actuated parallel robot
图 6
$s = (1,1,1)$ 时预载力矩优化参数曲线Fig. 6 Internal preload optimization parameter curve when
$s = (1,1,1)$ 
图 7
$s = (1,1,1)$ 时机器人主动关节力矩曲线Fig. 7 Torque curve of the robot in joint space when
$s = (1,1,1)$ 
图 8
$s = (-1,-1,-1)$ 时预载力矩优化参数曲线Fig. 8 Internal preload optimization parameter curve when
$s = (-1,-1,-1)$ 
图 9
$s = (-1,-1,-1)$ 时机器人主动关节力矩曲线Fig. 9 Torque curve of the robot in joint space when
$s = (-1,-1,-1)$ 
图 10 本文所提方法与固定阻抗控制方法的机器人末端位置误差曲线
Fig. 10 Position error curve of the robot of the proposed method and the fixed impedance control method
图 11 本文所提方法与固定阻抗控制方法的机器人末端交互力曲线
Fig. 11 Interaction force curve of the robot of the proposed method and the fixed impedance control method
图 12 时变阻抗控制方法的机器人主动关节力矩曲线
Fig. 12 Torque curve of the robot in joint space based on time-varying impedance control method
图 13 未考虑消除传动间隙的机器人主动关节力矩曲线
Fig. 13 Torque curve of the robot in joint space without consideration of elimination of backlash
图 14
$s = (1,1,1)$ 时多目标加权归一化内力优化方法下预载力矩优化参数曲线Fig. 14 Internal preload optimization parameter curve under the multi-objective weighted normalized optimization method when
$s = (1,1,1)$ 
图 15
$s = (1,1,1)$ 时多目标加权归一化内力优化方法下机器人主动关节力矩曲线Fig. 15 Torque curve of the robot in joint space under the multi-objective weighted normalized optimization method when
$s = (1,1,1)$ 
图 16
$s = (-1,-1,-1)$ 时多目标加权归一化内力优化方法下预载力矩优化参数曲线Fig. 16 Internal preload optimization parameter curve under the multi-objective weighted normalized optimization method when
$s = (-1,-1,-1)$ 
图 17
$s = (-1,-1,-1)$ 时多目标加权归一化内力优化方法下机器人主动关节力矩曲线Fig. 17 Torque curve of the robot in joint space under the multi-objective weighted normalized optimization method when
$s = (-1,-1,-1)$ 
图 19
$s = (-1,-1,-1)$ 时多目标加权归一化内力优化方法下机器人笛卡尔空间位置曲线Fig. 19 Position curve of the robot in Cartesian space under the multi-objective weighted normalized optimization method when
$s = (-1,-1,-1)$ 
图 20
$s = (-1,-1,-1)$ 时多目标加权归一化内力优化方法下机器人关节空间位置曲线Fig. 20 Position curve of the robot in joint space under the multi-objective weighted normalized optimization method when
$s = (-1,-1,-1)$ 
图 21
$s = (-1,-1,-1)$ 时本文所提方法与串联机器人时变阻抗控制方法的机器人末端位置曲线Fig. 21 Position curve of the robot of the method proposed in this paper and the time-varying impedance control method of the serial robot when s = (−1, −1, −1)
表 1 冗余驱动并联机器人物理参数
Table 1 Physical parameters of redundantly actuated parallel robots
$m_{i1}$ $m_{i2}$ $l_{i1}$ $l_{i2}$ $r_{i1}$ $r_{i2}$ ${I_{i1}} = {m_{i1}}r_{i1}^2$ ${I_{i2}} = {m_{i2}}r_{i2}^2$ 2.0 kg 2.0 kg 0.50 m 0.60 m 0.25 m 0.30 m $0.125\;{\rm{kg} } \cdot {\rm{m} }^2$ $0.180\;{\rm{kg} } \cdot {\rm{m} }^2$ 
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表 2 本文所提方法误差对比分析
Table 2 Comparison and analysis of the error of the proposed method
轨迹(m) ${\bar q_e}\;(\text{m})$ ${\dot {\bar q}_e}\;({\text{m/s}})$ ${F_e}\;({\text{N}})$ ${\bar d_x}\;(\text{m})$ ${\bar d_y}\; (\text{m})$ $\text{RMSE}\; (\text{m})$ $\text{JRMSE}\; (\text{rad})$ 式(34) ${[ {0.03} \;\;{ - 0.02} ]^{\rm{T}}}$ ${[{0.10}\;\;{ - 0.10}]^{\rm{T}}}$ 式(33) 0.0148 0.0090 0.0217 0.1141 式(37) ${[ {0.03}\;\;{ - 0.02} ]^{\rm{T}}}$ ${[ {0.10}\;\;{ - 0.10} ]^{\rm{T}}}$ 式(33) 0.0148 0.0090 0.0217 0.1113 式(34) ${[ {-0.02}\;\;{0.01} ]^{\rm{T}}}$ ${[{ - 0.05}\;\;{0.05}]^{\rm{T}}}$ 式(33) 0.0123 0.0069 0.0176 0.0954 式(37) ${[ {-0.02}\;\;{0.01} ]^{\rm{T}}}$ ${[ { - 0.05}\;\;{0.05} ]^{\rm{T}}}$ 式(33) 0.0123 0.0069 0.0176 0.0835 式(34) ${[ {0.05}\;\;{-0.03} ]^{\rm{T}}}$ ${[ { 0.20}\;\;{-0.15} ]^{\rm{T}}}$ 式(33) 0.0193 0.0110 0.0294 0.1508 式(37) ${[ {0.05}\;\;{-0.03} ]^{\rm{T}}}$ ${[ { 0.20}\;\;{-0.15} ]^{\rm{T}}}$ 式(33) 0.0193 0.0110 0.0294 0.1708 式(34) ${[ {0.03}\;\;{ - 0.02} ]^{\rm{T}}}$ ${[ {0.10}\;\;{ - 0.10} ]^{\rm{T}}}$ 式(38) 0.0107 0.0055 0.0166 0.0809 式(37) ${[ {0.03}\;\;{ - 0.02} ]^{\rm{T}}}$ ${[ {0.10}\;\;{ - 0.10} ]^{\rm{T}}}$ 式(38) 0.0107 0.0055 0.0166 0.0927 式(34) ${[ {0.03}\;\;{ - 0.02} ]^{\rm{T}}}$ ${[ {0.10}\;\;{ - 0.10} ]^{\rm{T}}}$ 式(39) 0.0092 0.0054 0.0161 0.0770 式(37) ${[ {0.03}\;\;{ - 0.02} ]^{\rm{T}}}$ ${[ {0.10}\;\;{ - 0.10} ]^{\rm{T}}}$ 式(39) 0.0092 0.0054 0.0161 0.0898
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