Basics of Robotics (3) - Dynamics Analysis and Force-Lagrangian Mechanics, Establishment of Robot Dynamics Equations, Establishment of Dynamic Equations for Multi-DOF Robots

Basics of Robotics (3) - Dynamics Analysis and Force-Lagrangian Mechanics, Establishment of Robot Dynamics Equations, Establishment of Dynamic Equations for Multi-DOF Robots

This chapter mainly includes Lagrangian mechanics, Lagrangian functions and establishment of solutions, dynamic equations of multi-degree-of-freedom robots, static analysis of robots, transformation of forces and moments between coordinate systems, and is mainly combined with examples for mastering and understanding

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1. Lagrangian Mechanics

Dynamics analysis is to study how hard the robot should be driven. Although the position and speed of the robot can be obtained according to the kinematic equation + differential motion, when the robot actually moves is determined by the force of the motor or driver, so carry out The dynamic analysis is to obtain the force generated by each joint motor under the desired speed and position, that is, the dynamic equation of the robot needs to be established.
There are many ways to establish dynamic equations, such as Newtonian mechanics method, Lagrangian mechanics method, Newton-Euler method and so on.
The Newton-Euler formula can be considered as a force balance method to solve dynamic problems; the
Lagrange formula is an energy-based dynamic method.

Because the author of the Newton-Eulerian method has not yet learned it, it will be supplemented after further in-depth study. This article mainly explains the Lagrangian mechanics method.

1. Example 1: Dynamic equations with linear motion and rotation

2. Example 2: Dynamic equation with centripetal acceleration and Coriolis acceleration

3. Example 3: Dynamic equation with moment of inertia

This question involves finding the formula for the moment of inertia.

2. Dynamic equation of multi-degree-of-freedom robot

The dynamic equation of a multi-degree-of-freedom robot will be much more complicated, but it is also possible to define the Lagrangian function by first calculating the kinetic energy and potential energy of the connecting rod and joint, and then by deriving the Lagrangian function to the joint variable get the kinetic equation.

The above is the process of calculating the total kinetic energy of a multi-degree-of-freedom robot, which can be understood as:
1. The coordinates of a point on the robot may not be based on the base coordinate system, so coordinate transformation will be involved;
2. It is necessary to calculate the position matrix and joint variables. The guide involves rotating joints and sliding joints. Q and U are introduced for the convenience of subsequent representations.
3. It is necessary to obtain the kinetic energy of the connecting rod speed, which is convenient for processing and representation, and the trace of the matrix is ​​introduced.
4. To obtain the inertia, a pseudo-inertia matrix is ​​introduced. Easy to express

Example: Multi-DOF Robot Dynamic Equations

The idea of ​​solving the dynamic equation example of a multi-degree-of-freedom robot:
directly substitute into the formula of the dynamic equation of a multi-degree-of-freedom robot, and calculate D, U, J according to the formula, and finally obtain a complete dynamic equation

3. Static analysis of the robot

example

Knowing the Jacobian matrix of the robot and the expected force and moment of the robot hand, the above formula can be used to obtain the torque of each joint in the corresponding situation, so that the robot
can be controlled through the joint torque.
Because the Jacobian matrix is ​​related to the configuration of the robot, if the configuration of the robot changes, the Jacobian matrix will also change. Therefore, in
order to obtain the joint torque in real time, it is necessary to recalculate the Jacobian matrix in real time.

4. Transformation of forces and moments between coordinate systems

example

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The transformation of force and moment between coordinate systems can be understood as:
the known robot hand is based on the matrix under the A coordinate system [ $F_A$], namely force and moment,
want to obtain the force and moment of the robot hand based on the B coordinate system [ $F_B$], which can be obtained by substituting the above formula, force and moment [ $F_B$];
That is, the force and moment of the robot hand in the A coordinate system [ $F_A$] and the force and moment of the robot hand in the B coordinate system [ $F_B$] have the same practical effect.

Summarize

This chapter establishes kinetic equations and derives kinetic equations. When the dynamic equation of the robot is known, it can be used to estimate the power required for each joint when driving the robot at a certain speed and acceleration, and can also be used to select the appropriate driver for the robot.
In fact, in most cases, the simplified form under assumptions is used, that is, some unimportant items are ignored to reduce the amount of calculation; at the same time, it can be observed which item is more important to the robot according to the obtained dynamic equation. Important, by comparing terms, it is possible to determine the importance of certain terms in the equations of motion and their effect on the total required forces and moments.
For example, in the case of weightlessness, the gravity term can be ignored, and the inertia term is dominant; in the case of a slow robot movement, the centripetal acceleration term and the Coriolis acceleration term can be ignored .

Through the study of the three chapters, knowing the robot's position, speed, and the force and torque required to generate the position and speed, the robot hand at the end can be controlled more accurately, and the robot hand can be controlled to achieve the desired position and speed.
The next chapter describes how to smoothly generate line motion and how to plan trajectory on the basis that the robot can reach the desired point