Almost all papers in IEEE or robotic textbooks introduced methods of how to calculate the analytical Jaocobian. Nevertheless, none of them mentioned how to deal with the its derivatives. What is so important for knowing the derivatives of Jacobian? When using second-order control, the Derivatives of Jacoiban becomes a major issue; When applying constraints on a humanoid robot stanced leg, it could definitely not be negligible.
Inaccurate approximate approach
Some people try to use the approximate approach; however, the result is terribly inaccurate when sampling time is large due to the derivative of joint's position.
Solutions
I did the deduction based on the Kinematics and proposed an algorithm. The program written in MATLAB could calculate both the numerical and symbolic types just as you wish. it's worth noting that the numerical result is REAL SOLUTION instead of an approximate one.
Results
The animation below shows the second order control of a robotic arm based on the knowledge of the Jacobian derivative. The dynamic performance of the robotic arm largely enhanced by taking into concern of the velocity and acceleration.
Position
The blue line is the position of the end-effector, and the red line is the desired trajectory.
Orientation
The blue line is the orientation of the end-effector, and the red line is the desired trajectory.