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

cs.ROOct 21, 2026

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

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

Organizations: College of Information Science and Technology, Eastern Institute of Technology, Ningbo 315200, China · Department of Automation, University of Science and Technology of China, Hefei 230027, China · Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China · Department of Interdisciplinary Information Studies, The University of Tokyo, Japan ·

Abstract

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.

Figures & tables

Category	Performance index	Mathematical Form	Description
Kinematic performance	Directional manipulability index	$M_{D}=\dfrac{1}{\sqrt{\bm{\xi}^{\rm T}(\bm{JJ}^{\rm T})^{-1}\bm{\xi}}}$	Manipulability along a unit velocity direction $\bm{\xi}$
	Planar manipulability index	$M_{P}=\dfrac{1}{\sqrt{\det(\bm{P}^{\rm T}(\bm{JJ}^{\rm T})^{-1}\bm{P})}}$	Manipulability within a plane defined by $\bm{P}\in\mathbb{R}^{m\times 2}$
	Volumetric manipulability index	$M_{V}=\dfrac{1}{\sqrt{\det(\bm{Q}^{\rm T}(\bm{JJ}^{\rm T})^{-1}\bm{Q})}}$	Manipulability over a subspace defined by $\bm{Q}\in\mathbb{R}^{m\times l}(3\leqslant l\leqslant m)$
	Jacobian condition number index (2-norm)	$M_{C,2}=\dfrac{\sigma_{\min}(\bm{J})}{\sigma_{\max}(\bm{J})}$	Kinematic isotropy: ratio of extreme singular values
	Jacobian condition number index (Frobenius-norm)	$M_{C,F}=\dfrac{m}{\sqrt{{\rm trace}\left(\bm{JJ}^{\rm{T}}\right){\rm trace}\left((\bm{JJ}^{\rm{T}})^{-1}\right)}}$	Normalized isotropy considering all singular values
Stiffness performance	Directional stiffness index	$K_{D}=\dfrac{1}{\sqrt{\bm{\zeta}^{\rm T}\bm{K}^{-2}\bm{\zeta}}}$	Stiffness along a unit force direction $\bm{\zeta}$

TABLE I: Jacobian-Based Performance Indices for Serial Manipulators.

Fig. 1: Cartesian manipulability/stiffness ellipsoid of the robot when $m=3$ . Those Jacobian-based performance indices presented in Table I possess distinct geometric interpretations within this ellipsoid.

Fig. 2: Applicable serial manipulator configuration for JAI of Jacobian-based performance indices.

	Dimension	Mathematical Form
$\bm{U}_{xy}$	(i) $l=2$	$[\cos\psi\,\bm{e}_{x1}+\sin\psi\,\bm{e}_{y1},-\sin\psi\,\bm{e}_{x1}+\cos\psi\,\bm{e}_{y1}]$ , $\psi\in[0,2\pi)$
	(ii) $l=2$	$[\cos\psi\,\bm{e}_{x2}+\sin\psi\,\bm{e}_{y2},-\sin\psi\,\bm{e}_{x2}+\cos\psi\,\bm{e}_{y2}]$ , $\psi\in[0,2\pi)$
	(iii) $l=2$	$\dfrac{1}{\sqrt{1+\psi^{2}}}\,[\bm{e}_{x1}+\psi\bm{e}_{x2},\bm{e}_{y1}+\psi\bm{e}_{y2}]$ , $\psi\neq 0$
	(iv) $l=4$	$\begin{gathered}\left[\cos\psi\,\bm{e}_{x1}+\sin\psi\,\bm{e}_{y1},-\sin\psi\,\bm{e}_{x1}+\cos\psi\,\bm{e}_{y1},\right.\\[-3.41418pt] \left.\cos\varphi\,\bm{e}_{x2}+\sin\varphi\,\bm{e}_{y2},-\sin\varphi\,\bm{e}_{x2}+\cos\varphi\,\bm{e}_{y2}\right],\\[-3.41418pt] \psi,\varphi\in[0,2\pi)\\ \end{gathered}$
$\bm{U}_{z}$	(i) $l=1$	$\cos\psi\,\bm{e}_{z1}+\sin\psi\,\bm{e}_{z2},\ \psi\in[0,2\pi)$
	(ii) $l=2$	$[\bm{e}_{z1},\bm{e}_{z2}]$

TABLE II: Representative

\bm{A}^{\rm T}

-invariant subspaces spanned by orthonormal basis vectors.

Fig. 3: Geometric representation of JAI in 3-dimensional Cartesian space ( $m=3$ ). When the first and last joints rotate, due to $(\bm{J}_{\Delta}\bm{J}_{\Delta}^{\rm T})^{-1}=\bm{A}(\bm{J}\bm{J}^{\rm T})^{-1}\bm{A}^{\rm T}$ and $\bm{K}_{\Delta}^{-2}=\bm{A}\bm{K}^{-2}\bm{A}^{\rm T}$ , the manipulability/stiffness ellipsoid exhibits a rotation about the $z$ -axis. The invariants comprise four quantities: (i) the intercept of the ellipsoid on the $z$ -axis (deep green line segment); (ii) the cross-sectional area of the ellipsoid within the $xy$ -plane (dusty pink plane); (iii) the volume of the ellipsoid (light purple ellipsoid); (iv) the lengths of all semi-axes of the ellipsoid (three dashed line segments in red, green, and blue).

Strategy	Computational load	CPU time (s)	Data size	Storage space (KB)
CS	$10^{7}\times\text{Sampling}$	163.887	$10^{7}\times\text{Sampling}$	35682
JAIBS	$10^{5}\times\text{Sampling}+\text{Rotation}+\text{Replication}$	1.723	$10^{6}\times\text{Sampling}$	18329

TABLE III: Comparative results of computational efficiency and storage space between CS and JAIBS.

Panel of Figure 7 — Fig. 4: Pointwise relative performance difference maps between CS and JAIBS. (a) $\mathcal{M}$ . (b) $\mathcal{K}$ .

Fig. 5: Different task layouts on a horizontal plane.

Panel of Figure 9 — Fig. 6: Application scenarios for task layout optimization. (a) Task A: plotting using Franka Research 3. (b) Task B: polishing using IRB 4400.

Task	Strategy	No.	Average performance	Task layout (m,m,rad)	Equivalent task layout (m,m,rad)	CPU Time (s)	Average CPU time (s)
Task A	CS	$1$	$0.0967$	$[0.4988,0.0503,1.8742]$	$[0.5014,0,1.7737]$	$14.794$	$17.604$
		$2$	$0.0967$	$[0.4949,0.0813,1.9370]$	$[0.5015,0,1.7742]$	$23.367$
		$3$	$0.0967$	$[0.5012,-0.0249,1.7236]$	$[0.5018,0,1.7732]$	$14.650$
	JAIBS	$1$	$0.0967$	$[0.5015,0,1.7830]$	$[0.5015,0,1.7830]$	$6.181$	$6.231$
		$2$	$0.0967$	$[0.5015,0,1.7828]$	$[0.5015,0,1.7828]$	$6.364$
		$3$	$0.0967$	$[0.5015,0,1.7828]$	$[0.5015,0,1.7828]$	$6.150$

TABLE IV: Task layout optimization results of CS and JAIBS.

Robotics

Robotics studies the design and control of machines that perceive, move, and act in physical environments.

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The Twist Decomposition of Serial Robots Under Lower-Mobility Tasks

Luc Baron, Damien Chablat

IICiMed · Polytechnique Montréal, box 6079, sta. CV, QC, H3C 3A7 Canada · box 6079, sta. CV, QC, H3C 3A7 Canada+2

Robot Kinematics Robotics Inverse Kinematics

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This paper introduces a twist decomposition framework for serial manipulators performing lower mobility tasks. Rather than relying on Jacobian null-space projections, the method separates the end-effector twist into task and redundant components using geometrically defined twist projectors. This formulation provides a direct and intuitive distinction between task-relevant and task-irrelevant motions in operational space, enabling a compact inverse kinematics scheme that naturally handles both manipulator and task redundancy.

Abstract

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