Partial Derivatives of the Inverse Mass Matrix of Multibody Systems via Its Factorization

A closed expression for repeated partial derivatives of the inverse mass matrix is developed. It rests on the factorization of the inverse mass matrix in terms of a single configuration-dependent block-diagonal matrix. Thereupon, partial derivatives of the inverse mass matrix are given in terms of derivatives of the diagonal blocks. It turns out that the derivatives are nonvanishing only for a single block of this block-diagonal matrix. The result for open kinematic chains is extended to the mass matrix of mechanisms with kinematic loops