Lie Group Formulation of Recursive Dynamics Algorithms of Higher Order for Floating-Base Robots

The motions of mechanisms can be described in terms of screw coordinates by means of an exponential mapping. The product of exponentials (POE) describes the configuration of a chain of bodies connected by lower pair joints. The kinematics is thus given in terms of joint screws. The POE serves to express loop constraints for mechanisms as well as the forward kinematics of serial manipulators. Besides the compact formulations, the POE gives rise to purely algebraic relations for derivatives wrt. joint variables. It is known that the partial derivatives of the instantaneous joint screws (columns of the geometric Jacobian) are determined by Lie brackets the joint screws. Lesser-known is that derivative of arbitrary order can be compactly expressed by Lie brackets. This has significance for higher-order forward/inverse kinematics and dynamics of robots and multibody systems. Various relations were reported but are scattered in the literature and insufficiently recognized. This paper aims to provide a comprehensive overview of the relevant relations. Its original contributions are closed form and recursive relations for higher-order derivatives and Taylor expansions of various kinematic relations. Their application to kinematic control and dynamics of robotic manipulators and multibody systems is discussed.

Fast prototyping of new SE(3) estimation objectives remains awkward in practice. Modern Lie-group frameworks -- GTSAM, manif, Sophus, SymForce, Ceres -- target first-order workloads through different code-generation and automatic-differentiation strategies, each optimized for a particular seam between hand-derived geometry and generic differentiation. The remaining gap is a compact, AD-safe path from these first-order primitives to exact Hessians, observed-information matrices, and higher-order derivative tensors: the quantities needed for exact Newton steps, observed-information covariance estimates, and covariance correction. This paper presents a hybrid analytical/AD recipe for SE(3) negative log-likelihoods. The practitioner writes the NLL gradient once, generic over a scalar type, and places the analytical/AD seam at the point-action interface y = Tx. Closed-form Lie-group Jacobians are used up to this interface; AD is applied only beyond it. The same source is then instantiated with ordinary floating-point scalars for gradients, vector-seeded dual numbers for exact Hessians in a single forward-mode pass, and nested dual numbers for higher-order derivative tensors. On a representative 6-DoF, 5-landmark SE(3) NLL, the advocated seeded-Hessian path is approximately 5x faster than finite-differencing the AD gradient on this benchmark while matching a nested-AD oracle to machine precision. The implementation adds roughly 70 lines of analytical-Jacobian code over an AD-only baseline. We also identify and fix a removable singularity in the standard SO(3)/SE(3) scalar basis that would otherwise produce NaNs at the origin under seeded AD, and we audit which Lie-group derivative tensors require this stabilized basis. The result is a practical path from rapidly written SE(3) objectives to exact higher-order derivatives, with predictable runtime and no finite-difference tuning.

We propose an architecture for learning the dynamics of mechanical systems based on discrete forced Euler-Lagrange equations on Lie groups using only position data. By formulating the dynamics directly on manifold-valued configuration spaces, the method naturally respects the geometric structure of the systems and preserves geometric invariants and conservation laws. The reliance on position measurements alone makes the framework applicable in settings where velocity data are unavailable or noisy. The approach extends naturally to multibody systems, accommodates external control inputs, and demonstrates strong performance on both synthetic and real-world datasets.