Analytical differentiation of the articulated-body algorithm: a geometric multilinear approach

This paper presents a new geometric algorithm to efficiently compute the analytical differentiation of the Articulated-body Algorithm (ABA) with respect to the state variables. Despite the fact that ABA solves the forward dynamics problem in linear-time for branched-multibody systems, its explicit differentiation is not straightforward, and computationally demanding tensor products and contractions appear. The proposed algorithm makes use of Lie algebraic and multilinear operations to recursively exploit the underlying sparsity of the linearization problem. As a result, the arithmetic complexity is dramatically reduced without affecting the analytical solution. Since the linealization of forward dynamics is of great importance in robot trajectory optimization, a differential dynamic programming solver is employed to demonstrate the performance of our algorithm with humanoid robot models such as N{\footnotesize AO} and HRP-2. In addition, we provide the computational cost with different robots using an optimized C++ implementation that can be found at https://github.com/garechav/geombd_crtp.git}{https://github.com/garechav/geombd\_crtp.