Constraint violation suppression for dynamics modeling of lower limb rehabilitation robot
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摘要:
U-K理论为获得约束多体系统的解析动力学方程提供了新的理念，但由于数值近似和截断误差等因素的影响，动力学方程在位置和速度层面上存在约束违约。Baumgarte约束违约稳定法（BSM）通过约束修正得到稳定的动力学方程。然而，Baumgarte参数的选择通常涉及一个试错过程，可能会出现失效的仿真结果。为此，利用经典的四阶Runge-Kutta法研究了Baumgarte参数选取问题，创建了基于BSM修正后的U-K理论的机器人系统解析动力学方程。以下肢康复机器人为研究对象仿真分析，结果表明：利用所提方法可以有效抑制约束违约，关节角度误差控制在-5×10-3（°）~5×10-3（°）范围内；关节角速度误差控制在-2×10-4~2×10-4 rad/s范围内；机器人末端执行器运行轨迹能够很好地贴近系统预定的目标。
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关键词:
- U-K理论 /
- 约束违约 /
- Baumgarte约束违约稳定法(BSM) /
- Baumgarte参数 /
- 下肢康复机器人
Abstract:The U-K theory provides a new concept for obtaining the explicit dynamic equation of constraint multibody system. However, one consequence of the numerical approximation and truncation error is the constraint violation of the dynamic equation at the position and velocity level. Baumgarte's constraint violation stability methods (BSM) provide a stable dynamic equation by constraint modification. Nevertheless, the selection of Baumgarte parameters usually involve a trial-and-error process, which may result in the failure of simulation results. Consequently, the Baumgarte parameters selection problem is studied by using the classical fourth-order Runge-Kutta method, and the explicit dynamic equation of robot system based on the modified U-K theory by BSM is established. Furthermore, the lower limb rehabilitation robot is taken as the research object for simulation analysis. The results show that the constraint violation can be effectively suppressed. The joint angle errors are controlled within the range of -5×10-3(°)-5×10-3(°), the joint angular velocity errors are controlled within the range of -2×10-4 rad/s-2×10-4 rad/s, and the operation trajectory of the robot end-effector can be well close to the predetermined target of the system.
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图 2 下肢康复机器人系统示意图
Figure 2. Schematic diagram of lower limb rehabilitation robot system
图 3 修正后的关节角度误差对比(不同步长和Baumgarte参数)
Figure 3. Comparison of corrected joint angle errors (different step lengths and Baumgarte parameters)
图 4 修正后的关节角速度误差对比(不同步长和Baumgarte参数)
Figure 4. Comparison of corrected joint angular velocity errors (different step lengths and Baumgarte parameters)
图 5 修正前后数值解与理论解的对比(h=0.01, α=10, β=300)
Figure 5. Comparison of numerical solutions and theoretical solutions before and after correction (h=0.01, α=10, β=300)
图 6 修正前后误差的对比(h=0.01, α=10, β=300)
Figure 6. Comparison of errors before and after correction (h=0.01, α=10, β=300)
表 1 不同Baumgarte参数取值
Table 1. Different Baumgarte parameter values
步长 Baumgarte参数 h=0.01 
h=0.01 
h=0.001 
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