[Control] Introduction to dynamic modeling --> Newton-Euler method and Lagrange method

1 Robot dynamics modeling method

Multi-body system dynamics has formed a variety of modeling and analysis methods. Early dynamics research mainly included Newton-Euler vector mechanics methods and analytical mechanics methods based on Lagrange equations. This method is easier to solve for simple rigid body systems with fewer degrees of freedom because it has fewer equations and a smaller amount of calculation. However, for complex rigid body systems, as the degree of freedom increases, the number of equations will increase sharply and the amount of calculation will increase.

With the development of the times, computer technology has made rapid progress. Although computer programming can be used to solve the dynamic equations, solving the joint angular velocity at the next moment requires a suitable numerical integration method and writing a program. Although this method can solve the equation, however, because this programming method is not universal, programming needs to be solved for each specific problem, which is relatively inefficient. Therefore, if the computational issues can be considered while modeling dynamics, and the versatility of modeling and solution can be considered during the modeling process, this problem can be solved better. Among them, the more famous methods include Kane method, variational method, Roberson-Wittenburg method, and spinor method and other multi-body dynamics research methods.

1.1 Newton-Euler method

The Newton-Euler method is the earliest dynamic modeling and analysis method used. Newton's equation describes the relationship between the external force, mass and center of mass acceleration of a translational rigid body, while the Euler equation describes the external moment of a rotating rigid body. , the relationship between angular acceleration, angular velocity and inertia tensor, so the Newton-Euler equation can be used to describe the relationship between the force, inertia and acceleration of a rigid body and establish the dynamic equation of the rigid body.

Newton's equation (rigid body translation): external force, mass, center of mass acceleration
Euler's equation (rigid body rotation): moment, angular acceleration, angular velocity, inertia tensor

This method analyzes the stress of each rigid body in the system, so the physical meaning is clear and expresses the complete force relationship of the system. When the number of rigid bodies is small, the amount of calculation is small. However, as the number of rigid bodies increases, the number of equations increases, resulting in a larger amount of calculation and lower calculation efficiency.

A summary of the Newton-Euler method is as follows:

1. Among the Newton-Euler equations, Newton's equation is mainly used to solve the translation problem of rigid bodies, and the Euler equation is mainly used to solve the rotation problem of rigid bodies;
2. Any motion of any rigid body can be synthesized by translation and rotation. The translation of force will generate torque, and the translation of torque can be performed directly;
3. The force analysis of rigid bodies can be concentrated on one point;
4. The Newton-Euler equation modeling of multi-body systems is only one of the modeling algorithms for dynamics;
5. The Newton-Euler equation currently established is only the dynamic equation of a multi-rigid body system in free motion space, and external forces and moments can be introduced during static analysis; however, the contact situation of multi-rigid bodies needs to be carried out separately, because the multi-rigid body Contact is a very complex situation involving many circumstances;
6. Compared with single rigid body dynamics, multi-rigid body dynamics analysis needs to introduce multi-rigid body kinematic analysis. Kinematic analysis needs to solve the linear velocity and angular velocity of the rigid body, and then solve the linear acceleration and angular acceleration of the rigid body.

1.2 Lagrangian method

The Lagrange equation is another classic dynamic modeling method. The Newton-Euler equation can be considered as a force balance method for solving dynamic problems, while the Lagrange equation uses another idea. , which establishes a dynamic model based on the energy of the system.

Different from the Newton-Euler method in the modeling process, it can avoid the forces between internal rigid bodies and simplify the modeling process. The disadvantage is that its physical meaning is unclear, and for complex systems, the differential operation of the Lagrangian function will become very cumbersome.

2 Classification of robot dynamics modeling methods

Classification of Robot Dynamics Modeling Methods

The relationship between mechanical quantities and motion quantities

Ref.

1. Robot Dynamics Modeling: Robotic Arm Dynamics
2. [Dynamics] Robotic arm dynamics modeling (Newton-Euler method)
3. Robot Dynamics: Forward Dynamics and Inverse Dynamics of Robotic Arm
4. LQR lateral control based on vehicle kinematics model (1)