The rigid inverse dynamics modeling of parallel robots is based on the known size and inertia parameters of the mechanism, considering the gravity and external load effects of the parts, and determining the joint driving force/torque based on the end motion trajectory (position, velocity, and acceleration). This lays the foundation for evaluating the dynamic performance of the mechanism, designing the structure, estimating motor parameters, and controlling the actual prototype.

The commonly used rigid body inverse dynamics modeling methods for parallel mechanisms are mainly divided into three types: Newton Euler method, Lagrangian method, and virtual work principle method, while others include Kane method. Today, we will mainly learn about the Newton Euler method and the Lagrange method.

The Newton Euler method is a commonly used method in robot dynamics, based on Newton's kinematic laws and Euler equations, providing a theoretical framework for studying and controlling the motion of robots. This method has a simple principle and a clear derivation process, making it widely used in the dynamic analysis of parallel mechanisms.

The core idea of the Newton Euler method is to use Newton's laws of kinematics to express the relationship between the forces and torques between various parts of a robot. Meanwhile, the relationship between the angular momentum and angular kinetic energy of the robot system is described through the Euler equation. This method has intuitive and physical significance, and can more clearly reflect the dynamic behavior of the robot system. However, there are also some challenges in the practical application of the Newton Euler method. Firstly, this method requires a large amount of algebraic and matrix calculations, resulting in a relatively complex calculation process and requiring high computational resources. Secondly, when the robot system is relatively large or complex, the derivation and solution of equations may become cumbersome and prone to introducing errors.

Nevertheless, due to its solid theoretical foundation, clear thinking, and validated in many industrial applications and research fields, the Newton Euler method remains a powerful tool in dynamic analysis. When choosing to use it, it is necessary to balance the computational complexity and the applicability of the method in order to better meet the needs of specific robot systems.

Figure 1. Newton Euler method

The Lagrangian method is a commonly used method for modeling robot dynamics, which is characterized by focusing on the kinetic and potential energy of the system without explicitly considering the internal forces of the mechanism during the modeling process. In the Lagrangian method, the dynamic equations of the system can be obtained by taking the derivative of the Lagrangian function. This function includes the difference between the kinetic energy and potential energy of the system. By applying the Euler Lagrange equation to the partial derivative of the Lagrange function with respect to the generalized coordinates, the motion equation of the system can be derived.

The advantage of this method is that it can succinctly and intuitively express the dynamic characteristics of the robot system without considering detailed information about the internal forces of the mechanism. However, compared to methods such as the Newton Euler method, the Lagrangian method introduces concepts such as generalized forces and requires partial differential calculations, making the process relatively cumbersome.

Figure 2. Lagrangian method

The Newton Euler method is suitable for relatively simple parallel robot systems, but in complex systems, there may be issues of large computational complexity and accumulated errors.

The Lagrangian method has better mathematical properties and a simpler form compared to the Newton Euler method. It is suitable for complex parallel robot systems and can facilitate controller design and optimization analysis.

In summary, the Newton Euler method and the Lagrangian method are two commonly used dynamic modeling methods for parallel robots, and the selection of an appropriate method depends on the specific system characteristics and research needs.