Application of Jacobian matrix transpose (Jacobian transpose) in force and torque

In addition, the Jacobian matrix can reflect the relationship between the force exerted on the end effector and the torque exerted on the joint, as we have analyzed in detail the use of the Jacobian matrix to solve the velocity of the end effector by the given joint velocity. This is the kind of relationship that is mainly introduced today.

Before that, let's start by saying the following two points for the knowledge to be used:

(1) Principle of virtual work: for an Object , you only need to consider the active force , without any internal force, because of internal forces is generated by the structure , this force does not cause the object to move. And for a static equilibrium system, the effect of all external forces, after the virtual displacement, the virtual work done , the sum equals zero.

(2) Cross-multiplication matrix: 1, the fork by the front of the vector can be transformed into a matrix, we call this matrix is a cross-multiplication matrix, we can see that this is an anti-called Matrix. Its most important property is a=-aT.

Figure 1

Then get to the point. 2, we can be similar to the relationship between line velocity and angular velocity, by means of the cross-multiplication of the relationship between the output and torque.

Figure 2

According to the formula of Figure 2 and the properties of the cross-multiplication matrix, we can do further derivation, we can get the most basic formula of two robots, as follows:

Figure 3

We now offer another method of derivation, according to the principle of virtual work, we make virtual displacements to the end effector, since the sum of the virtual work is 0, we can deduce as follows:

Figure 4

This allows us to link forces and torques through the transpose of the Jacobian matrix.

Application of Jacobian matrix transpose (Jacobian transpose) in force and torque