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

Abstract

This document provides full details of second-order partial derivatives of rigid-body inverse dynamics. Several properties and identities using an extension of Spatial Vector Algebra for tensorial use are listed, along with their detailed derivations. Using those, the expressions for second-order derivatives are derived step-by-step in detail. The expressions build upon previous work by the authors on first-order partial derivatives of inverse dynamics. I. NOTATIONS, SYMBOLS AND ACRONYMSCartesian 3D vectors are denoted by lower-case letters with an overhead bar (v). Spatial vectors are denoted by lower-case bold letters (a). Matrices are denoted by upper-case bold letters (M ), while third-order tensors are denote by upper-case calligraphic letters (A). Acronyms:1) RHS : Right-hand-side 2) w.r.t : with respect to 3) FO : first-order 4) SO : second-order 5) SVA : Spatial Vector AlgebraSection II is an introduction on Spatial Vector Algebra (SVA) [1], also covered in Ref. [3, App A]. Section III provides an extension of SVA for spatial motion matrices, and spatial cross-product operators associated with them.Then, properties of spatial matrices (numbered K1-K16) are listed and derived in detail. Then, Sections IV and V provide the derivations of SO partial derivatives of ID w.r.t q and q respectively. Section VI provides the SO cross-derivatives w.r.t q and q, while Section VII gives the FO partial derivatives of M (q) w.r.t q. At the end, Section VIII provides the matrix form of expressions used in Algorithm developed. II. SPATIAL VECTOR ALGEBRA IDENTITIES AND PROPERTIES A. Spatial Vector AlgebraA body k with spatial velocity k v k ∈ M 6 in the body frame is decomposed in its angular and linear components as:where k ωk ∈ R 3 is the angular velocity of the body in a coordinate frame fixed to the body, while k vk ∈ R 3 is the linear velocity of the body-fixed point at the origin of the body frame. Spatial vectors can also be expressed in the ground frame. For example, the spatial velocity of the body k in the ground frame is denoted as 0 v k . In this case, the linear velocity is associated with the body-fixed point on body k that is coincident with the origin of

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

“…The resulting expressions and algorithm are very complicated, but are provided in the form of open-source algorithm with full derivation in Ref. [19] while keeping the current paper self-contained. The performance of this new SO algorithm is compared to AD using CasADi [20] in MATLAB.…”

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

“…III. Step by step derivation is provided in [45] for the interested reader, with unit tests available online [46]. For example, identity K1 is an extension of identity J1 in Ref.…”

“…C, and full step-by-step derivations in Ref. [45]. Given the extent of the full derivation, we again provide unit tests [47] to verify each of the results empirically.…”

“…Similarly, for Case C (j i ≺ k), taking partial derivative of (13b) results in a stricter case j ≺ i ≺ k. Appendix C lists the six expressions with full derivation in Ref. [45,Sec. VI].…”

“…[34] suggested the ABA-Zero-Algorithm (AZA), where the product of M −1 with any n-vector b can be computed by exploiting the form of the forward dyanmics: M −1 (q) (τ − C(q, q) q − g(q)) = ABA(q, q, τ , g) (50) By running ABA with zero q, zero generalized gravity g, and by inputting b instead of τ , the result M −1 b is obtained. With this approach, the usual O(N 3 ) Direct-Matrix-Multiply (DMM) for implementing the FO Forward-Dynamics derivatives (45) can replaced by the efficient n calls to the O(N ) AZA approach. This method results in computational savings when N > 50 [32].…”