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Optimization-based robot control strategies often rely on first-orderdynamics approximation methods, as in iLQR. Using second-order approximationsof the dynamics is expensive due to the costly second-order partial derivativesof the dynamics with respect to the state and control. Current approaches forcalculating these derivatives typically use automatic differentiation (AD) andchain-rule accumulation or finite-difference. In this paper, for the firsttime, we present analytical expressions for the second-order partialderivatives of inverse dynamics for open-chain rigid-body systems with floatingbase and multi-DoF joints. A new extension of spatial vector algebra isproposed that enables the analysis. A recursive algorithm with complexity of$\mathcal{O}(Nd^2)$ is also provided where $N$ is the number of bodies and $d$is the depth of the kinematic tree. A comparison with AD in CasADi showsspeedups of 1.5-3$\times$ for serial kinematic trees with $N> 5$, and a C++implementation shows runtimes of $\approx$51$\mu s$ for a quadruped.

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