Comparison and analysis of multibody dynamics formalisms for solving optimal control problem

Optimal Control methods are increasingly used for the control of multibody systems (MBS). This work analyzes the different dynamic formulations and compare their performances in solving Optimal Control Problem. The focus is on minimal coordinates and the derivation of the dynamics via the recursive methods for treelike MBS (i.e., the so-called Newton-Euler and Order-N recursive algorithms). The different formulations are introduced and their derivations are discussed. A benchmark case study (i.e., a 3D series manipulator balancing an inverted pendulum) is modeled and a series of manipulation tasks (movement of the end effector in the 3D space) are performed. The OCP is formulated and solved with the help of the CasADi software while the dynamic formulations are generated by the Robotran software. Results show that the implicit and semi-explicit formulations derived via the Newton-Euler recursive algorithm lead to faster computation of the OCP than the explicit formulations. This is explained by a more compact expression for the implicit dynamics. However, a lower number of high local minima is observed with the explicit formulations for the most extreme robot manipulations.