The Backward floating base Jacoiban method is a general method for inverse kinematic solution. This method can be used to solve IK and FK for every kind of robotic arm systems, including both serial and parallel arm. All types of joints such as spherical, prismatic, universal joints are included in this method.
Motivation from Humanoid Robot
This idea was inspired by whole body jacobian method applied on humanoid robot. Traditionally, the Jacoiban method is used only on the serial arm. They used Jacobian to solve the end effector position in terms of joint’s position. The base for this serial robotic arm is fixed. We called it fixed Jacobian method. In parallel arm, people often solve the kinematic trains from the separate fixed base(ground) to the same moving base(floating base), making extra constraints at the effector(moving base), where all the branches are merged. However, we can think this problem from a different view. We treat the top disk as the origin of the kinematic trains, and the parallel leg and hybrid arm as the links grown from this floating base. So the system can be simplified as tracking the end-effector of every branch.
And in the hybrid arm case, the Target position and orientation for the parallel arm are fixed, meaning that the velocity for the position and orientation are zero. All we have to do is to track the targets of the moving arm and the floating base. This backward notion really helps us to solve the complicated parallel and serial arm system easily and succinctly.