On Second-Order Derivatives of Rigid-Body Dynamics: Theory & Implementation

Understand

Model-based control for robots has increasingly been dependent on optimization-based methods like Differential Dynamic Programming and iterative LQR (iLQR).

• These methods can form the basis of Model-Predictive Control (MPC), which is commonly used for controlling legged robots.
• Computing the partial derivatives of the dynamics is often the most expensive part of these algorithms, regardless of whether analytical methods, Finite Difference, Automatic Differentiation (AD), or Chain-Rule accumulation is used.
• Since the second-order derivatives of dynamics result in tensor computations, they are often ignored, leading to the use of iLQR, instead of the full second-order DDP method.