Precise Optimization of Robotic Bipedal Walking Using Hamiltonian Dynamics

The two most common formulations of equation of motion for mechanical systems are derived by 1) vector analysis from Newton's Laws, and 2) energy analysis from Euler-Lagrange equations. A third alternative, derived from Hamilton's equations, is only well known in quantum mechanics and niche fields, specially if conservation of momentum is a central phenomenon. In Optimal Control, a core theorem is Pontryagin's Principle, whose main term - the control Hamiltonian - is inspired in the original Hamiltonian present in Hamilton's Equations. Even though there is a clear parallel between the two hamiltonians, dynamic formulations used in optimization are still largely based on Lagrangian mechanics and Classical mechanics. In this paper we present the competitiveness of Hamiltonian dynamics in the optimization of robotic bipedal walking - a complex nonlinear system - using its exact whole-body dynamics. With the use of Hamiltonian dynamics, we observed a significant increase in precision of optimization, with the trade-off of bigger computational time. Our results demonstrate the advantage of use of Hamiltonian dynamics for precise offline optimizations, and suggest the maintenance of use of Lagrangian dynamics for fast online optimizations.