Multi-body systems such as robot mechanisms are usually modelled numerically by a Lagrangian approach, using a generalised coordinate vector q to describe system position. In the much used Independent Coordinates method, q has as many components as there are degrees of freedom. The equations of motion (EoM) reduce to an ODE - straightforward to solve but messy, with no clear connection to the physics. At an opposite extreme are Natural Coordinates (NCs), where q holds world-frame cartesian coordinates of suitable points on the moving parts. Then the EoM are an index 3 differential algebraic equation (DAE) system - the numerics are harder - typically using index reduction methods and a solver of the DASSL family - but the Lagrangian formulation is simple, sparse and human-readable.

DAETS is a DAE code to solve high-index DAEs by Taylor series expansion, developed by Ned Nedialkov and me over the last 15 years. It is ideally suited to (smooth) mechanism problems, which it solves directly with no index-reduction stage. Using DAETS, algorithmic differentiation (AD) and our take on NCs, we create and solve the EoM directly from the Lagrangian function in a seamless run time process, with no computer algebra system used.

Though the architecture of a classical NCs algorithm and of ours seem utterly different, they are surprisingly alike "under the bonnet". For instance a key Jacobian matrix turns out to be identical in both methods. The talk will explore similarities and differences, and issues of computational efficiency and user convenience.