Research on Constraint Following Control of Flexible Joint Manipulators Based on Singular Perturbation
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摘要: 针对二连杆柔性关节机械臂，提出了一种基于奇异摄动理论和Udwadia-Kalaba(U-K)方法的控制方法. 设计步骤主要分为两步：第一，基于奇异摄动法对系统进行降阶，把系统拆分为快、慢系统，不仅降低了求解系统的阶次，而且克服了系统柔性；第二，基于U-K方法设计了快、慢系统的状态反馈约束跟随控制律，能使快、慢系统约束跟随误差收敛到零，即使系统初始不满足约束条件，该方法不需要借助Lagrange乘子和伪广义速度等辅助变量，可以同时处理完整约束和非完整约束. 将以上方法运用在二连杆柔性关节机械臂系统中，解决了二连杆柔性关节机械臂的柔性振荡和约束跟随的问题. 使用MATLAB进行仿真，并且与传统PID控制进行了对比，验证了所提出的方法的有效性与优越性.Abstract: For the control of 2-link flexible joint manipulators, a control method based on the singular perturbation theory and the Udwadia-Kalaba (U-K) method was proposed. The control design was implemented by 2 steps. First, the system order was reduced based on the singular perturbation method and the system was divided into fast and slow sub-systems, to simplify the solution process and overcome the system flexibility. Second, the state feedback constraint following control law for the fast and slow sub-systems was designed with the U-K method, to make the constraint following errors of the fast and slow sub-systems converge to zero, even if the system can't initially satisfy the constraints. The proposed method can deal with holonomic and nonholonomic servo constraints at the same time without the auxiliary variables of the Lagrange multiplier and the pseudo generalized velocity. The method was applied to 2-link flexible joint manipulator systems to solve the flexible oscillation and constraint following problems. Through simulations on MATLAB, and was compared with the traditional PID control, to verify the effectiveness and superiority.
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图 5 转杆1的角速度$ \dot{q}^{(2)}$
Figure 5. The angular velocity $ \dot{q}^{(2)}$ of rotary rod 1
图 6 转杆2的角速度$ \dot{q}^{(4)}$
Figure 6. The angular velocity $ \dot{q}^{(4)}$ of rotary rod 2
表 1 二连杆柔性机械臂参数
Table 1. Parameters of the 2-link flexible manipulator
parameter symbol value moment of inertia of joint 1 J1/(kg·m2) 0.135 elastic coefficient of joint 1 K1/(N·m/rad) 10 000 moment of inertia of joint 2 J2/(kg·m2) 0.150 elastic coefficient of joint 2 K2/(N·m/rad) 10 000 mass of link 1 m1/kg 8 length of link 1 l1/m 0.6 center position of link 1 lc1/m 0.3 mass of link 2 m2/kg 8 length of link 2 l2/m 0.6 center position of link 2 lc2/m 0.3 
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