[Paper Review] Polynomial Invariants and SAGBI Bases for Multi-screws

Deborah Crook, Peter Donelan|arXiv (Cornell University)|Jan 15, 2020

Polynomial and algebraic computation19 references4 citations

TL;DR

This paper develops polynomial invariants for multiple screws using SAGBI basis theory, providing a computational framework to classify serial robot manipulators. It derives explicit generating sets for screw pairs and conjectures a generating set for screw triples, linking these invariants to Denavit–Hartenberg parameters and offering a mathematically rigorous alternative to traditional kinematic parameterization.

ABSTRACT

Polynomial invariants for robot manipulators and their joints arise from the adjoint action of the Euclidean group on its Lie algebra, the space of infinitesimal twists or screws. The aim of this paper is to determine basic sets of generating polynomials for multiple screws. Techniques from the theory of SAGBI bases are introduced. As a result, a complete description is provided of the polynomial invariants for screw pairs and some results for screw triples are obtained. The invariants are shown to be related to Denavit-Hartenberg parameters.

Motivation & Objective

• To develop a systematic computational method for identifying polynomial invariants of multiple screws under the adjoint action of the Euclidean group.
• To provide a mathematically rigorous alternative to Denavit–Hartenberg parameters by expressing kinematic invariants algebraically.
• To extend invariant theory to non-reductive groups like the Euclidean group by leveraging SAGBI bases for subalgebras of polynomial rings.
• To establish connections between screw invariants and classical geometric invariants such as pitch, rotation angles, and common perpendiculars.
• To conjecture a complete set of generators for the ring of invariants of three screws, based on computational evidence and symmetry analysis.

Proposed method

• Applies SAGBI (Subalgebra Gröbner) basis theory to compute generating sets for the ring of polynomial invariants under the adjoint action of the Euclidean group on its Lie algebra.
• Uses Plücker coordinates $(oldsymbol{ ho}, oldsymbol{v})$ to represent screws and defines invariants as homogeneous polynomials in these coordinates.
• Employs the Klein form $oldsymbol{ ho}_i oldsymbol{ ho}_j$ and the Killing form $oldsymbol{ ho}_i oldsymbol{ ho}_i$ as fundamental invariants for screw pairs.
• Derives invariants such as $oldsymbol{ ho}_i oldsymbol{v}_i$, $oldsymbol{ ho}_i oldsymbol{v}_j + oldsymbol{ ho}_j oldsymbol{v}_i$, and the scalar triple product $[oldsymbol{ ho}_1, oldsymbol{ ho}_2, oldsymbol{ ho}_3]$ for screw triples.
• Compares derived invariants with Denavit–Hartenberg parameters, showing that $ ext{cos} heta$ and $d ext{sin} heta$ are rational functions of the polynomial invariants.
• Uses computational enumeration of SAGBI bases to identify candidate generators and conjecture a minimal generating set for three-screw invariants.

Experimental results

Research questions

• RQ1 What is a complete and minimal generating set for the ring of polynomial invariants of two screws under the Euclidean group action?
• RQ2 Can a generating set for the invariants of three screws be systematically identified using computational algebraic geometry?
• RQ3 How are the classical Denavit–Hartenberg parameters related to polynomial invariants of screw systems?
• RQ4 To what extent can SAGBI bases be used to compute and verify invariants in non-reductive group actions?
• RQ5 Are the conjectured invariants for three screws algebraically independent, and do they generate the full invariant ring?

Key findings

• A complete generating set of polynomial invariants for screw pairs is explicitly determined using SAGBI basis techniques, with the invariants being $oldsymbol{ ho}_i oldsymbol{ ho}_j$, $oldsymbol{ ho}_i oldsymbol{v}_i$, and $oldsymbol{ ho}_i oldsymbol{v}_j + oldsymbol{ ho}_j oldsymbol{v}_i$ for $i < j$.
• For screw triples, the paper conjectures that the ring of invariants is generated by the pairwise dot products $oldsymbol{ ho}_i oldsymbol{ ho}_j$, the self-pitches $oldsymbol{ ho}_i oldsymbol{v}_i$, the symmetric mixed products $oldsymbol{ ho}_i oldsymbol{v}_j + oldsymbol{ ho}_j oldsymbol{v}_i$, the scalar triple product $[oldsymbol{ ho}_1, oldsymbol{ ho}_2, oldsymbol{ ho}_3]$, and the sum $[{f v}_1,oldsymbol{ ho}_2,oldsymbol{ ho}_3] + [oldsymbol{ ho}_1,{f v}_2,oldsymbol{ ho}_3] + [oldsymbol{ ho}_1,oldsymbol{ ho}_2,{f v}_3]$.
• The invariant $z_{123} + z_{231} + z_{312}$ is identified as the sum of rotation invariants, corresponding to the trace of the commutator-like expression in screw triple systems.
• The twist angle $ heta$ and displacement $d$ in Denavit–Hartenberg parameters are shown to be rational functions of the fundamental polynomial invariants: $ ext{cos} heta = rac{oldsymbol{ ho}_1 oldsymbol{ ho}_2}{ orm{oldsymbol{ ho}_1} orm{oldsymbol{ ho}_2}}$ and $d ext{sin} heta = rac{oldsymbol{ ho}_1 oldsymbol{v}_2 + oldsymbol{ ho}_2 oldsymbol{v}_1}{ orm{oldsymbol{ ho}_1} orm{oldsymbol{ ho}_2}}$.
• The offset parameter $b$ for a triple of screws is shown to be an $SE(3)$ invariant, though it has not yet been expressed in terms of the conjectured generators.
• The paper demonstrates that SAGBI bases are a viable computational tool for exploring invariants in non-reductive group actions, despite the lack of general finite generation theorems for such groups.