Source-linked AI summary
Inverse kinematic solution for generic 3R positional robots using Conformal Geometric Algebra
Abhilash Nayak, Durgesh Haribhau Salunkhe
TL;DR
Generic 3R inverse kinematics has lacked a workspace-oriented geometric interpretation beyond joint-space conic intersections, motivating a more informative formulation. The paper uses Conformal Geometric Algebra to recast the model as the intersection of two circles, directly deriving a univariate polynomial in θ2. The method yields geometric insight into solution structure and, in the worked example, produces four complete inverse-kinematic solutions.
Problem
Previous geometric interpretations of generic 3R inverse kinematics were confined to joint-space conic intersections and did not clarify the robot’s kinematic model.
Method
The paper represents the generic 3R inverse kinematic model in CGA as the intersection of a fixed circle and a rotating circle, directly deriving a univariate polynomial in θ2.
Results
Four inverse-kinematic solutions are obtained for the worked problem: (0.0, 2.0, 1.0), (2.58, 0.326, 2.138), (−2.731, 1.56, −2.488), and (−1.341, −2.998, −1.753).
Takeaways & Limitations
The CGA formulation provides geometric insight into the number and distribution of solutions and unifies the inverse kinematic model without degeneracy conditions.
Abstract
from arXiv · showhide
The inverse kinematics of generic 3R robots has been investigated through multiple approaches, mainly algebraic methods involving the solution of certain equation sets. Previous geometric interpretations of the solution, characterized as the intersection of a pair of conics have been confined to the joint-space domain. In this article, we study the Inverse Kinematic Model (IKM) of 3R robots, using the advantages of Conformal Geometric Algebra (CGA) to provide further insights on its kinematic properties. Our approach directly yields a univariate polynomial in terms of theta_2 without the need to eliminate theta_1 and theta_3 by reframing the problem as the intersection of two circles, which are fundamental elements within this algebraic framework.
1 Introduction
Earlier inverse-kinematics methods derived univariate polynomials and geometric conic interpretations, but these interpretations remained tied to joint space and did not clarify the robot’s kinematic model. The paper proposes a CGA-based formulation for generic 3R robots that directly produces a polynomial in θ2 without degeneracy conditions.
- Prior approaches: Pieper’s method derives a univariate polynomial in t3 = tan θ3, then obtains θ1 and θ2 through backpropagation.It does not work when a1 = 0 and α1 = 0.
- Prior approaches: Selig reduced inverse kinematics to finding the intersection of a conic and a circle in the cos(θ1)-sin(θ1) plane.A related interpretation applied Pieper’s approach in the cos(θ3)-sin(θ3) plane to study solution counts and cuspidal properties.
- Research gap: Previous geometric interpretations described the eliminated polynomial but did not provide intuition about the robot’s kinematic model itself.
- Contribution: The proposed CGA method solves the generic 3R inverse kinematic model using geometric objects that provide insight into inverse-kinematic-solution number and distribution.
- Contribution: The approach directly yields a univariate polynomial in θ2, avoiding algebraic elimination of two of the three joint variables and removing degeneracy conditions.The authors present this as a unified inverse kinematic model for 3R robots.
2 CGA: Notations and basic operations
This section introduces the conformal geometric algebra representation used to model Euclidean primitives and robot transformations. It defines the algebraic products, conformal basis, direct and dual representations, and motor operations needed for kinematic analysis.
- CGA foundations: The 5-dimensional conformal geometric algebra G4,1 represents geometric objects and their duals using inner-product and outer-product null-space representations.Euclidean points are embedded as null vectors through the up() function.
- CGA operations: The geometric product combines the inner product, which yields a scalar, and the outer product, which yields a bivector.The inner product is commutative, whereas the outer product is antisymmetric.
- CGA operations: Orthogonal basis vectors satisfy ei · ej = 0 for i ≠ j, while their outer products form bivector basis elements denoted eij.
- CGA foundations: The basis change introduces null vectors e0 and e∞ representing the origin and infinity, respectively, within the conformal model.A multivector is a linear combination of the algebra’s 32 basis blades.
- Geometric primitives: Table 1 organizes direct and dual representations of 3D geometric primitives, with joins represented by ∧ and the dual regressive product by ∨.For planes, n is the normal vector and d is the distance from the origin; for spheres, pS is the center and r is the radius.
- Geometric primitives: CGA represents spheres directly as joins of four conformal points and dually through their conformal center and radius.The direct and dual forms let geometric elements be constructed in forms convenient for kinematic analysis.
- Kinematic transformations: CGA expresses robot rotations and frame transformations through rotors and motors parameterized by Denavit–Hartenberg quantities a, d, and α.The transformation motor is composed from translation and rotation versors.
3 Inverse kinematic model
The IKM of a generic positional 3R robot is reformulated in CGA as the intersection of a fixed circle and a circle rotating about the second joint axis. This produces θ2 candidates first, after which θ1 and θ3 are recovered geometrically from directed arcs and sign bivectors.
- Problem formulation: Inverse kinematics finds the joint angles θi for a target end-effector position P=(x,y,z), starting from the robot’s home configuration.The model uses points P0, P1, P2, and PH connected by revolute joints.
- Circle formulation: CGA recasts the prior conic-intersection interpretation as finding the intersection of a fixed circle CB and a rotating circle CAθ2.CB is represented as the intersection of a sphere and a plane, while CA is constructed similarly with a transformed joint-axis plane.
- Circle formulation: Rotating CA about the second joint axis with rotor R2 yields CAθ2, whose intersection x = CAθ2 ∨ CB depends on trigonometric functions of θ2.The rotor requires a bivector representing the plane of rotation perpendicular to the second joint axis.
- Solving for θ2: The condition x · x = 0 produces a univariate quartic polynomial in t2 = tan θ2, avoiding elimination of θ1 and θ3.This condition ensures that the circle intersection is a single real point.
- Recovering θ1 and θ3: For each selected θ2, θ1 is obtained from the directed arc on CB, while θ3 is obtained by rotating x back to CA and measuring the corresponding arc.The signs of both angles are adjusted using bivectors representing positive rotation about their respective joint axes.
- Example: In the numerical example, the quartic gives four θ2 values and the complete inverse-kinematic set contains four triples.The reported triples are (0.0, 2.0, 1.0), (2.58, 0.326, 2.138), (−2.731, 1.56, −2.488), and (−1.341, −2.998, −1.753).
4 Conclusions
The article presents the generic 3R inverse kinematic model as the intersection of two circles in Conformal Geometric Algebra, extending motion description toward orientations and generic 6R robots.
- The generic 3R inverse kinematic model is formulated as the intersection of two circles represented in Conformal Geometric Algebra.
- CGA extends the motion description to include orientations.
- The orientation capability will support studying the inverse kinematic model and kinematic properties of generic 6R robots.