A Hierarchical Aggregation Modelling Method for Mobile Manipulators
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摘要: 移动机械臂因机械臂在动态作业过程中的耦合效应会影响移动平台的运动特性，增加了整个系统的复杂度和非线性，给系统建模带来了极大挑战. 为此提出了一种新的层级聚合建模方法. 该方法依据分析力学中Udwadia-Kalaba(U-K)理论的层级属性，首先将移动机械臂划分为3个子系统，并分别利用Lagrange方程建立各自的无约束动力学模型，然后基于移动机械臂机械结构上的约束利用Udwadia-Kalaba基本方程(UKE)建立整体系统模型. 此外，针对系统存在初始条件偏差的情况，利用基于Lyapunov稳定性理论来补偿初始条件偏差，以达到收敛理想轨迹的目的. 仿真结果验证了该文所提出的建模方法的可行性.
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关键词:
- Udwadia-Kalaba方法 /
- 层级聚合建模方法 /
- 移动机械臂 /
- 动力学建模
Abstract: The coupling effects of mobile manipulators on the motion characteristics of mobile platforms during the dynamic operation process, would increase the complexity and nonlinearity of the whole system and then bring great challenges to the system modelling. A new hierarchical aggregation modelling method was proposed to solve this issue. The method is based on the hierarchical properties of the Udwadia-Kalaba (UK) theory in the analytical mechanics. First, the mobile manipulator was divided into 3 subsystems, and the unconstrained dynamics of each one was modelled with the Lagrangian equations. Subsequently, the basic Udwadia-Kalaba equations (UKE) were employed to model the overall system, in view of the constraints within the mechanical structure of the mobile manipulator. In addition, the Lyapunov stability-based theory was used to compensate for the initial condition deviations to achieve convergence of the ideal trajectory. Simulation results validate the feasibility of the proposed modelling method. -
图 8 移动平台与机械臂各自轨迹误差
Figure 8. The respective trajectory errors of the mobile platform and the manipulator
图 9 层级聚合方法与拉氏方法计算效率对比
Figure 9. Comparison of computation efficiency between the hierarchical aggregation method and the Lagrange method
图 12 移动平台和末端执行器的轨迹误差
Figure 12. The mobile platform and end-effector trajectory errors
research institution robotic arm mass mr/kg platform mass mp/kg mass ratio δ University of Texas[2] 3.68 17.25 0.21 Ryerson University[3] 15.50 44.40 0.35 Hokkaido University[4] 4.40 20.56 0.21 Pukyong National University[5] 2.85 9.50 0.30 Iran University of Science and Technology[6] 0.72 6.00 0.12 Institute of Automation, CAS[7] 7.50 60.00 0.13 South China University of Technology[8] 2.00 10.00 0.20 
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表 2 机械臂几何参数
Table 2. Geometric parameters of the manipulator
joint number αi-1/(°) ai-1 di θi 1 0 0 0 θ1 2 90° 0 l1 θ2 3 0 l2 0 θ3 4 0 l3 0 0
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表 3 系统动力学参数表
Table 3. System dynamics parameters
object mass m/kg length l/m moment of inertia I/(kg·m2) mobile platform 50 2 joint 1 2 0.5 0.625 joint 2 3 0.7 0.122 5 joint 3 2 0.5 0.042 end-effector 0.5 0.1 0.000 4
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表 4 约束条件参数
Table 4. Constraint parameters
constraint initial condition $\begin{gathered}F_1: x(t)=\frac{\pi}{2} t; F_2: y(t)=\sin \left(\frac{\pi}{2} t\right); \\ F_3: \varphi(t)=\cos \left(\frac{\pi}{2} t\right); \\ F_4: x_{\mathrm{mp}}(t)=0.75+0.25 \cos (\pi t); \\ F_5: y_{\mathrm{mp}}(t)=\frac{0.25 \sqrt{2}}{2} \sin (\pi t); \\ F_6: z_{\mathrm{mp}}(t)=0.25-\frac{0.25 \sqrt{2}}{2} \sin (\pi t)\end{gathered}$ $\begin{gathered}x_0=1; y_0=1; \varphi_0=0; \\ \theta_1=-\frac{\pi}{2}; \theta_2=\frac{\pi}{2}; \theta_3=\frac{\pi}{6}; \\ \dot{x}_0=1; \dot{y}_0=1; \dot{\varphi}_0=0; \\ \dot{\theta}_1=-\frac{\pi}{2}; \dot{\theta}_2=\frac{\pi}{2}; \dot{\theta}_3=\frac{\pi}{6}\end{gathered}$
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