Derivation and analysis of the Jacobian matrices (Jacobian matrix)

When it comes to inverse kinematics (IK), the most important part is the use of the Jacobian matrix to represent the relationship between the target State and the variable group. Refer to the "Introduction to inverse kinematics with Jacobian transpose, pseudoinverse and damped Least" squares "for specific references. Today we mainly introduce the derivation and application of the Jacques ratio matrix in the positive kinematics. May use some of my previous writing about the coordinate transformation of the relevant knowledge, interested can be read.
Now let's assume that we already know some prior knowledge, such as that the robotic arm is connected by some joints (Joint), which are connected from the base (base) to the end controller (end-effector), and the joints generally have only one degree of freedom (either rotating or panning) and the end controller has six degrees of freedom. As shown in 1, we have described the structure of the robotic arm very clearly. We are now going to solve how to find out the six degrees of freedom of the end controller through the degrees of freedom of each joint . Here in the introduction of a concept of instantaneous dynamics, that is, any joint motion, the end controller will follow the motion, if the time is small enough, we can be seen as the joint and the end of the controller are instantaneous changes. Now our problem is transformed into the relationship between finding ΔQ(joint changes) and ΔX(the change of the end controller).

Figure 1

Let's start by identifying some variables, and we're generally talking about angular changes for rotational joints (revolute), and for translational joints (prismatic), the distance changes are generally discussed. In order to unify the representation, we collectively refer to the angle and distance as the channel volume, recorded as Q. At the same time, we recorded the target State (that is, six degrees of freedom of the end controller) as a vector group X={x1,x2,x3,x4,X5, X6}. Well, the preparation is almost there, now let's deduce the formula. We assume that each of the items in X is obtained by Q (in fact, as well) and is recorded as x=f (q). So we can get a set of equations:

Figure 2

Based on the above equation, we can extend it. We make a full differential on each of the items in X , and the right side of the equation biases all q . The specific expansion is as follows, and we can convert the left-hand expansion of Figure 3 into the matrix form on the right:

Figure 3

Some conclusions can be found, such as that the line of a matrix is equal to the number of items in X , that is, the number of degrees of freedom of the end controller, and the matrix column equals the number of items in Q , that is, the number of joints. We now naturally associate the ΔQ and ΔX mentioned above. The matrix we get here is generally called the Jacobian matrix (Jacobian matrix), recorded as J. The above equation can be simply written as:

Figure 4

Here, the deduction of the Jacobian matrix is complete. The main link between ΔQ and ΔX, is an important part of instantaneous kinematics. In robotics, the Jacobian matrix is mainly used to calculate the angular velocity and line velocity of the end controller, andΔQ and ΔX can be used to express speed in the case of a small time slice. In inverse kinematics, we usually use the degree of freedom of the end controller to inverse the degrees of freedom of each node, which is how we can get Q changes by multiplying the ΔX by the inverse of the Jacobian matrix .

Write this today and go on to further analysis tomorrow.

Derivation and analysis of the Jacobian matrices (Jacobian matrix)