When dealing with overactuated or underactuated serial or parallel manipulators the jacobian is not square and therefore not invertable. The reason is that either the desired vector (for example end-effector velocity) cannot always be obtained (in the underactuated case) or there are multiple ways to obtain the desired vector (in the overactuated case). In the underactuated case, usually what you want to do is calculate the "best effort" while in the overactuated case, a choice needs to be made for a certain combination based on some metric.
A generalised inverse can be used to calculate this and the moore-penrose pseudoinverse is a commonly used form. However, it can be shown that this is merely a mathematical fomularion that has no meaning dynamically speaking, while the described choices are strongly related to the dynamics of the system!
Several years ago we started formulating a relation between inverting the causality of a dynamic model in bond-graph language. The idea was that this could provide a physics-based formulation of a weighted generalised inverse. However, due to other priorities this work was never finished. For the parallel manipulator case we have a good basis, but for the serial manipulator there are many open points.
This assignment would consist of:
- Diving into the current literature of the topic to see which other relevant works exist
- Pick up and finalise the existing work, complete the theory by investigating model inversions of manipulators in bond-graph language and the effect of various dynamics of such a system.