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Joint-Angle Invariance of Jacobian-Based Performance Indices for Serial Robotic Manipulators

Yaohua Zhou, Mingdao Lin, Jiayu Song, Xixin Zhao, Haidong Huang, Jiahu Qin, Xiaocong Li

arXiv:2610.29570v1cs.RO

TL;DR

Evaluating Jacobian-based performance indices across joint configurations can be costly, motivating structural simplifications. The paper identifies joint-angle invariance for specific indices and serial robots, then demonstrates reduced computational costs in task-layout optimization while maintaining comparable effectiveness.

  • Problem

    Jacobian-based indices can vary across the full joint space, making computation and visualization costly, while identifiable invariance properties are lacking.

    Source: B. Overview · p. 2

  • Method

    The paper formulates joint-angle invariance conditions for Jacobian-based indices and uses the resulting invariance to simplify performance evaluation and optimization.

    Source: B. Overview · p. 2, B. Properties and Proof of JAI · p. 4, U TABATU · p. 4

  • Results

    JAIBS reduces task-layout computational costs by approximately 45.7% to 73.7% versus CS, while the strategies' optimization effectiveness differs by no more than 0.1%.

    Source: B. Task layout optimization · p. 7

  • Takeaways & Limitations

    JAI-based performance maps and task-layout optimization can reduce computational effort while preserving effectiveness in the demonstrated applications.

    Source: §V · p. 7, B. Task layout optimization · p. 7

    JAI applies only when the selected task subspace is invariant under the transformations induced by first- and last-joint rotations.

    Source: U TABATU · p. 5

Abstract

from arXiv · show

This article investigates the joint-angle invariance (JAI) of Jacobian-based performance indices for serial robotic manipulators. We identify kinematic conditions under which representative performance indices, including manipulability, stiffness, and other Jacobian-derived indices, remain independent of the first and last joint angles. This property allows a systematic reduction in the effective configuration-space dimensionality, thereby simplifying the performance evaluation. The utility of JAI is demonstrated through two application scenarios: workspace performance map construction and task layout optimization. Comparative results show that the proposed JAI-based strategy reduces computational effort and storage requirements without compromising the effectiveness.

I. INTRODUCTION

The paper addresses the cost of evaluating Jacobian-based indices across joint configurations by introducing joint-angle invariance for specific serial robots. It reviews performance indices and prior applications, then outlines how invariance can simplify performance analysis and optimization.

I. INTRODUCTION

  • Jacobians map joint velocities to Cartesian twists and joint torques to wrenches, underpinning indices for motion capability and resistance to applied wrenches.Source: §I · p. 1

A. Related Work

  • Prior work covers workspace performance mapping,, robot design,, trajectory optimization –, and task-layout and manufacturing applications –.Source: A. Related Work · pp. 1–2

B. Overview

  • Joint-angle invariance makes specific indices independent of the first and last joint angles, addressing costly configuration-space evaluation and supporting performance-map construction and task-layout optimization.Source: B. Overview · p. 2

II. JACOBIAN-BASED PERFORMANCE INDICES FOR SERIAL ROBOTIC MANIPULATORS

  • The indices characterize directional, planar, volumetric, and condition-number-based kinematic or stiffness performance through geometric properties of manipulability and stiffness ellipsoids.Condition-number indices use 2-norm or Frobenius-norm measures; directional indices include MD and KD.Source: §II · pp. 2–3

A. Applicable Serial Robotic Manipulators

The described JAI configuration requires revolute first and last joints aligned with the base- and tool-plane normals; intermediate joints have no specific requirements. The performance indices assume a full-row-rank task Jacobian.

A. Applicable Serial Robotic Manipulators

  • The applicable robot has revolute first and last joints whose axes are perpendicular to the base and tool planes, respectively; intermediate joints have no specific requirements.Source: A. Applicable Serial Robotic… · p. 4, Figure 2 · p. 4
  • The base plane is where the robot is mounted, and the tool plane is perpendicular to the last link; frames are typically aligned with the first and last joint axes.Source: A. Applicable Serial Robotic… · p. 4
  • The evaluated task Jacobian must have full row rank, so m ≤ n; this includes redundant and nonredundant robots, with lower-dimensional indices evaluated in the selected attainable subspace.Source: A. Applicable Serial Robotic… · p. 4

B. Properties and Proof of JAI

The section frames JAI as invariance of Jacobian-derived quadratic forms under changes to the first and last joint angles. Its applicability to lower-dimensional tasks depends on whether the selected task subspace is invariant under the induced transformation.

B. Properties and Proof of JAI

  • JAI allows the first and last joint angles to be set to zero, increasing matrix sparsity and computational efficiency.Source: B. Properties and Proof of JAI · p. 4
  • For lower-dimensional evaluation, JAI applies only when the selected task subspace is invariant under the transformation induced by rotating the first and last joints.The derivation is carried out in 6-dimensional Cartesian space, while task evaluation may use an l-dimensional subspace.Source: B. Properties and Proof of JAI · p. 4
  • Under a pure z-axis rotation represented by adjoint operator A, the body velocity and body Jacobian transform, and the same transformation applies to the manipulability and stiffness quadratic-form matrices.The matrices (JJ)^−1 and K^−2 are denoted uniformly by B.Source: B. Properties and Proof of JAI · p. 4
  • MD, MP, MV, KD, KP, and KV share the form 1/p det(U^TBU), providing a common expression for their JAI analysis.The six indices are grouped using U, whose columns are orthonormal.Source: B. Properties and Proof of JAI · p. 4

U TABATU

Theorem 1 states that the performance condition holds for all symmetric positive definite B exactly when U and A^TU have identical column spaces. The proof addresses sufficiency and necessity separately.

U TABATU

  • For every symmetric positive definite B, the condition holds if and only if Col(U) = Col(A^TU).Theorem 1 assumes A is orthogonal and U has orthonormal columns.Source: U TABATU · p. 4
  • Under column-space equality, A^TU = UC for an orthogonal matrix C, completing the sufficiency argument.The proof establishes C^TC = I and (det(C))^2 = 1.Source: U TABATU · p. 4, U TBU · p. 4

U TBU

  • Necessity is approached by contradiction, using B = I + αbb^T for α > 0 and arbitrary b, together with the matrix determinant lemma.Source: U TBU · pp. 4–5

U TABATU

JAI for the six directional, planar, and volumetric indices requires an AT-invariant task subspace; four additional indices are invariant through an orthogonal similarity transformation. In 3D, the first and last joint rotations geometrically rotate the ellipsoid about the z-axis while preserving the supported indices.

U TABATU

  • Equality of the quadratic-form matrices is equivalent to equality of the orthogonal projection matrices, preserving the ellipsoid semi-axis lengths and thus MC,2, MC,F, KC,2, and KC,F.The proof also establishes necessity by contradiction when the column spaces differ.Source: U TABATU · p. 5
  • The six directional, planar, and volumetric indices exhibit JAI only when the task subspace is AT-invariant; Table II lists representative admissible bases.The criterion is Col(U) = Col(ATU), and the listed bases yield indices unaffected by q1Δ and qnΔ.Source: U TABATU · p. 5, Table II · p. 5
  • An arbitrary lower-dimensional task space does not automatically satisfy JAI; the selected subspace must meet the stated column-space condition.The projected twist and Jacobian are evaluated within that selected subspace.Source: U TABATU · p. 5
  • The four indices MC,2, MC,F, KC,2, and KC,F are independent of q1Δ and qnΔ under the stated orthogonal similarity transformation.The transformation preserves eigenvalues, and J's singular values are the square roots of the eigenvalues of JJT.Source: U TABATU · p. 5
  • In 3D, rotating the first and last joints rotates the manipulability/stiffness ellipsoid about the z-axis while maintaining directional, planar, and volumetric JAI under the stated condition.The passage specifies the simplest case ψ = 0.Source: U TABATU · p. 5

IV. APPLICATION SCENARIOS

The section presents performance map construction and task layout optimization as application scenarios.

IV. APPLICATION SCENARIOS

  • The section considers performance map construction as an application scenario.Source: §IV · p. 5
  • The section also considers task layout optimization as an application scenario.Source: §IV · p. 5
  • Together, the two scenarios demonstrate applications of the proposed approach.Source: §IV · p. 5

A. Performance map construction

The JAI-based strategy constructs performance maps without sampling the first and last joints, reducing computation and storage while producing nearly identical distributions to full-space sampling.

A. Performance map construction

  • The study constructs performance maps for the 7-DOF Franka Research 3, whose mechanism configuration satisfies the proposed JAI.Joint configurations are sampled within their limits and performance values are obtained through forward kinematics.Source: A. Performance map construction · p. 5
  • Computational load falls to 1/100 of the conventional strategy, with CPU time decreasing to 1.723 s.The JAI-based strategy avoids sampling q1 and q7 and sets both to zero when calculating the index.Source: A. Performance map construction · p. 6
  • Storage data size falls to 1/10 of its original, and the .mat file storage requirement decreases by 48.6%.Source: A. Performance map construction · p. 6
  • Relative differences remain at 10^-10 for M and 10^-11 for K, indicating almost identical performance distributions between strategies.The passage attributes observed discrepancies mainly to numerical approximation rather than violation of JAI.Source: A. Performance map construction · p. 6

B. Task layout optimization

JAI reduces the task-layout optimization from three variables to two while preserving effectiveness, and JAIBS improves computational efficiency and stability compared with CS. The section demonstrates the approach for planar plotting and curved-surface polishing.

B. Task layout optimization

  • Table III compares computational efficiency and storage space between CS and JAIBS.The supplied table caption identifies the comparison but does not provide its numerical outcomes.Source: Table III · p. 6
  • The optimization is constrained to a horizontal plane, with translations along x and y and rotation about z as its three variables.The task layout pose is denoted pw = [xw, yw, φw].Source: B. Task layout optimization · p. 6
  • Rotating the task layout about the base z-axis makes its y-coordinate zero, allowing the original three-variable pose to be represented by [xw, 0, φw] without changing trajectory performance.The corresponding first-joint adjustment is q1∆ = φw∆, where φw∆ = −atan2(yw, xw).Source: B. Task layout optimization · pp. 6–7
  • The method is demonstrated on planar spiral plotting with a Franka Research 3 robot and curved-surface polishing with an IRB 4400 robot.Plotting optimizes planar manipulability; polishing optimizes directional stiffness along the tool-frame z-axis.Source: B. Task layout optimization · p. 7
  • Both strategies perform comparably, with optimization-effectiveness differences within 0.1%, but JAIBS yields more consistent task layouts across initial conditions and computational procedures.Task B, using closed-form inverse kinematics, has smaller deviations and stronger computational stability than Task A, which uses numerical solutions.Source: B. Task layout optimization · p. 7
  • JAIBS reduces computational costs by approximately 45.7% to 73.7% compared with CS while maintaining performance and improving stability.The 3-dimensional task layout effectively has only 2 DOFs because of JAI, primarily concerning the first joint.Source: B. Task layout optimization · p. 7

V. CONCLUSIONS

The study introduces JAI for Jacobian-based performance indices of serial robots and demonstrates its use in performance-map construction and task-layout optimization. It reports computational-efficiency gains while preserving effectiveness.

V. CONCLUSIONS

  • JAI applies to specific Jacobian-based performance indices for serial robots, which remain independent of the first and last joint angles.The study presents applications in performance-map construction and task-layout optimization.Source: §V · p. 7
  • Across the demonstrated applications, JAIBS achieves substantial computational-efficiency gains while preserving effectiveness.The conclusion identifies potential relevance to robot design, optimization, and planning where computational efficiency is needed.Source: §V · p. 7
  • Table IV presents the task-layout optimization results for CS and JAIBS.The caption identifies the methods compared but supplies no numerical values.Source: Table IV · p. 8
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