Nth Order Analytical Time Derivatives of Inverse Dynamics in Recursive and Closed Forms

Shivesh Kumar, Andreas Müller

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

Apart from one key point, the candidate solution matches the paper’s closed-form nth-derivative identities and uses the same Leibniz/binomial machinery. However, it replaces the paper’s Ȧ = A a − A a A with Ȧ = A b − A a A and propagates b(·) into the higher-order formula for A(n). This change is inconsistent with the paper’s central identity J̇ = −A a J (derived via aX ≡ 0) and with the stated kinematic higher-order formula A(n) that only involves a(·). Therefore, the paper’s derivations are correct as written, while the model’s modification of Ȧ and A(n) is incorrect.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript presents clear, general closed-form and recursive formulas for nth-order time derivatives of the EOM for serial chains using Lie/spatial notation. The statements are correct and consistent, and the provided identities (especially Eqs. (34), (39)–(47)) are immediately implementable. A small notational ambiguity in the statement of Ȧ (Eq. (27)) could be clarified to avoid misinterpretation. The work is useful for practitioners needing higher-order derivatives for motion planning and control, and complements recent efforts in analytical and automatic differentiation.