Basics of Robotics (2) - Differential Motion and Velocity - Calculation of Jacobian Matrix, Inversion of Jacobian Matrix, Calculation of Joint Motion Velocity

Basics of Robotics (2) - Differential Motion and Velocity - Calculation of Jacobian Matrix, Inversion of Jacobian Matrix, Calculation of Joint Motion Velocity

---

Knowledge points in this article:
differential motion of coordinate system, differential change between coordinate systems, differential motion of robot and robot hand coordinate system, calculation of Jacobian matrix, inversion of Jacobian matrix, between Jacobian matrix and differential operator associate

---

---

1. Jacobian matrix

The Jacobian matrix represents the geometric relationship of the mechanism components over time. It can convert the differential motion or velocity of a single joint into the differential motion or velocity of a point of interest (such as an end effector), and can also compare the motion of a single joint with the entire mechanism. movement linked. Since the straightness of the joint angle changes with time, the size of each element of the Jacobian matrix also changes with time, so the Jacobian matrix is ​​time-dependent.
In simple terms, the Jacobian matrix is ​​able to link the differential motion or velocity of two points in the robot, so the information stored in the Jacobian matrix can be understood as the characteristics of the robot structure between the two points, but because the two points linked by the matrix The point is differential, so it is also called the Jacobian matrix.

2. Differential motion of the coordinate system

1. Differential translation

2. Differential rotation

1. Differential rotation around a reference axis

2. Differential rotation around the general axis q

3. Differential transformation (translation + rotation)

1. Differential transformation of the coordinate system

2. Differential transformation between coordinate systems

Directly use the formula to calculate the differential operator relative to its own coordinate system

3. Calculation of the Jacobian matrix

Remember the Jacobian formula here! ! ! !
direct examples set of formulas

4. The relationship between the Jacobian matrix and the differential operator

The meaning of this passage can be understood as:
the differential value and Jacobian matrix of known robot joint movement

Using this formula, the matrix D can be obtained, that is, the differential motion of the robot hand dx, dy, dz, $\delta$ x，$\delta$y，$\delta$z

The matrix D has been obtained, and then substituted into the above formula to obtain the differential operator $D$

Then the differential operator Δ \DeltaSubstituting$\Delta \; into\; the\; above\; formula,\; we\; can\; get\; [\; dT\; ]$

Using the above formula again, the new pose of the robot hand can be obtained

example

Five, Jacobian matrix inversion

It is known that the speed requirement of the robot hand is the matrix D. In order to make the robot hand meet the speed requirement, it is necessary to calculate the speed of each joint of the robot, which is the matrix $D_{}$, so the inverse of the Jacobian matrix also needs to be calculated.

In this paper, the inverse motion equation is used to calculate the velocity of the joint, and the detailed explanation is as follows

Example 1: Using the known Jacobian to inversely calculate the joint velocity

Example 2: Use the inverse motion equation to directly find the joint velocity

This problem is to use the inverse motion equation to directly calculate the joint velocity. Because the Jacobian matrix is ​​unknown, the
calculation amount of the Jacobian matrix is ​​very large, and it is very troublesome to use the obtained Jacobian matrix to use the formula to calculate the joint velocity.
So a better method is to use the inverse motion equation to directly calculate the joint velocity

Use the Jacobian matrix to find out whether there is a degenerate point in the robot's workspace.
Global degeneration: The robot will lose one or more degrees of freedom in some special postures.
Local degeneration: The robot will appear in a certain joint under certain conditions. There is no solution, which can be solved by adjusting the equation of motion.
You can use methods such as finding the determinant of the Jacobian matrix
Reference:
https://max.book118.com/html/2017/0525/109105829.shtm

---

Summarize

By studying the contents of this chapter and using the formula,
the velocity of the robot’s joints is known, and the Jacobian matrix can be used to obtain the velocity of the robot hand; the
velocity of each joint of the robot can be obtained by calculating the inverse of the Jacobian matrix when the velocity of the robot’s hand is known.
At the same time, I also learned the method of finding the joint speed without using the Jacobian matrix. Using the inverse differential motion equation of the robot, it is possible to determine the speed of each joint to produce the desired robot hand speed.
Know the robot inverse motion equation and inverse motion differential equation, that is, know the position and speed of the robot in space.

The forward motion equation and inverse motion equation of kinematics in the first chapter are to obtain the position of the robot; the differential motion in this chapter is to obtain the motion speed of the robot and the speed of each joint on the basis of the known position; the next chapter is dynamic The scientific analysis is based on the known motion speed and joint speed of the robot, how to make each joint of the robot rotate and how much force the driver drives to meet the desired motion speed of the robot.

This article mainly refers to: Introduction to Robotics Analysis, Control and Application 2nd Edition (Saeed B. Niku)