The animation shows the second order control of a hybrid arm based on the knowledge of the Jacobian derivative. The dynamic performance of the robotic arm is largely enhanced by taking into concern of the velocity and acceleration.
The Jacobian derivative for a hybrid arm could not be found based on the past research due to its high complexity. However, by introducing the floating base backward Jacobian mehtod, the parallel arm can be separated into coupling serial arm. In this way, finding the augmented Jacobian would not be a problem.
Jacobian and dJacobian Matrix representation
The picture below shows the matrix representation for the Jacobian and its derivative. The code was written in MATLAB, which is much easier and fast to construct the simulation with Linear Algebra embedded function.