Dynamic modeling of parallel robots is one of the important methods for studying the motion and mechanical characteristics of robots. In the analysis of dynamic modeling for parallel robots (Part 1), we previously learned about the Newton Euler method and the Lagrange method. Today, we will continue to learn about the other two modeling methods: the virtual work principle method and the Kane equation method.

The principle of virtual work is a method for establishing the dynamic model of parallel mechanisms, based on the principle of virtual work in mechanics. Compared to other methods, its advantage is that there is no need to calculate joint internal forces, so the modeling process is relatively simple. The core idea of this method is to use virtual displacement to balance system energy and obtain the dynamic equation of the mechanism. By introducing virtual displacement, the virtual work of the system at the equilibrium position is zero, thus obtaining the motion equation of the mechanism at the equilibrium position. This method not only simplifies the mathematical process of modeling, but also provides a unified format for programming and solving the dynamic model of parallel mechanisms.

This method has been increasingly studied and applied in the dynamic modeling of parallel mechanisms. Its simplicity and applicability make the virtual work principle method a powerful tool for researchers to understand the dynamic characteristics of parallel mechanisms and design control algorithms. However, the principle of virtual work is an approximate method, so in some cases, it may not be possible to obtain a very accurate dynamic model, especially when considering system nonlinearity and complexity.

图1.Virtual work principle method

The Kane equation method is a method of establishing a rigid body dynamics model of a non free mass system using concepts such as partial velocity, angular velocity, generalized velocity, generalized active force, and generalized inertial force. Like the virtual work principle method, this method can eliminate the internal force term in the equation, avoid tedious differential operations, and the derivation process is more systematic. The core idea of this method is to use the Kane equation, which describes the generalized dynamic equation of a rigid body in a non free mass system, including generalized active forces and generalized inertial forces. Through this approach, the computational workload of dynamic modeling can be significantly reduced, making the dynamic characteristics of the system more clearly presented. However, due to the overly obscure principles, the Kane method has not been widely applied in the practical modeling of parallel mechanism dynamics.

图2. Kane's equation method

The dynamic equations of parallel mechanisms derived directly from the above methods usually present as complex coupled nonlinear functions of the platform pose or joint position. Such dynamic models pose challenges in solving and computing, with large computational complexity and long time consumption, making them relatively difficult to apply in real-time control. Among them, the nonlinear characteristics and complex coupling terms make directly solving dynamic equations complex and require high computational resources. Especially for large or high degree of freedom parallel mechanisms, this challenge is more apparent. Real time control requires fast and accurate calculations, and directly using these dynamic equations may result in excessive computational burden, limiting their widespread application in real-time applications.

To solve this problem, researchers usually use some optimization methods or approximation techniques, such as model simplification, numerical methods, neural networks, etc., to improve computational efficiency. These methods have to some extent reduced the computational burden, but still require a balance between accuracy and computational efficiency.