Details of Second-Order Partial Derivatives of Rigid-Body Inverse Dynamics

TL;DR

This paper provides detailed derivations and properties of second-order partial derivatives of rigid-body inverse and forward dynamics, extending previous work to include external forces and impact dynamics, using Spatial Vector Algebra.

Contribution

It introduces comprehensive formulas and identities for second-order derivatives in rigid-body dynamics, including external forces and impact scenarios, building on prior first-order derivative work.

Findings

Derived second-order derivatives for inverse and forward dynamics

Extended derivatives to systems with external forces

Included derivatives for impact and KKT forward dynamics

Abstract

The details of second-order partial derivatives of rigid-body Inverse/Forward dynamics are provided. Several properties and identities using Spatial Vector Algebra are listed, along with their detailed derivations. The expressions build upon previous work by the author on first-order partial derivatives of inverse dynamics. The first/second-order derivatives are also extended for systems with external forces. Finally, the KKT Forward dynamics and Impact dynamics derivatives are derived.