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Scale-Invariant Manipulability Shape Tracking Across Heterogeneous Manipulators

Geunwoo Kwon, Dong-gyu Lee, Kai Li, Soonwoong Hwang, Wansoo Kim

arXiv:2609.36784v1cs.RO

TL;DR

Full-matrix manipulability tracking penalizes scale differences even when transferring orientation and axis ratios across differently sized or structured systems. The paper proposes unit-determinant, scale-invariant shape tracking integrated with concurrent tasks; simulations report shape transfer across robot and human-reach settings, with lower concurrent-task errors than Full tracking in the human-reach evaluation.

  • Problem

    Full-matrix tracking includes absolute-scale differences, although heterogeneous systems may share manipulability orientation and semi-axis ratios while differing in scale.

    Source: §I · pp. 1–2

  • Method

    The Shape method tracks unit-determinant manipulability representatives under AIRM and integrates its objective with position and end-effector direction tasks in a constrained joint-velocity QP.

    Source: §I · p. 2, §V · p. 7

  • Results

    In simulations, Shape tracking reproduced a common target shape on three robot followers and the human target's forward-elongated dual force shape on four robots, with smaller concurrent-task errors than Full tracking at the final hold and endpoint.

    Source: §V · pp. 7–8

  • Takeaways & Limitations

    The results support transferring manipulability geometry alongside concurrent tasks without selecting an absolute target scale in the evaluated robot and human-reach settings.

    Source: §V · pp. 7–8

    The evaluation is limited to kinematic simulations and a single human reaching motion; physical task execution with dynamics and contact, and broader human motions, remain future work.

    Source: §V · p. 8

Abstract

from arXiv · show

When transferring manipulability across systems with different sizes and kinematic structures, matching absolute ellipsoid scale may be unnecessary when the goal is to reproduce orientation and semi-axis length ratios. Full-matrix tracking, however, penalizes both shape and absolute-scale differences, even when only shape matching is required. We therefore propose a scale-invariant manipulability shape-tracking method that treats matrices differing only by a positive scalar factor as equivalent and uses their unit-determinant representatives. We derive the differential of the unit-determinant shape representative and an orthonormal coordinate representation of the tangent tracking residual under the affine-invariant Riemannian metric (AIRM). The resulting scale-invariant objective is integrated with position and end-effector direction tasks in a constrained joint-velocity quadratic program. Simulations with four heterogeneous robots evaluate robot-to-robot and human-to-robot transfer. On three followers, the proposed method achieves endpoint shape distances of 9.30 x 10^-5 without scale tuning. With robot-specific target scales tuned during motion, the Full method retains endpoint axis-ratio errors of 0.19-0.31 on KR500 and UR20. For human reaching with concurrent tasks, the proposed method yields dual force shapes elongated along X like the human target on all four robots, with endpoint position errors of 2.4-5.6% of reference arm length versus up to 75% for the Full method tracking the original human ellipsoid.

I. INTRODUCTION

Manipulability ellipsoids describe motion and force transmission through their principal directions and semi-axis lengths, whose ratios capture directional structure. Because heterogeneous systems can share orientation and axis ratios while differing in scale, the paper proposes tracking unit-determinant shapes rather than requiring absolute-scale matching.

A. Manipulability Ellipsoids

  • A manipulability ellipsoid represents motion and force transmission capabilities under prescribed Euclidean joint-velocity or joint-torque norm constraints.Its principal directions and semi-axis lengths describe capabilities, while semi-axis ratios capture relative directional structure.Source: §I · p. 1, A. Manipulability Ellipsoids · pp. 2–3

I. INTRODUCTION

  • Human screwing movements show phase-dependent velocity-ellipsoid geometry changes not fully captured by determinant and condition number.This motivates describing and transferring posture-dependent directional capabilities through orientation and semi-axis ratios.Source: §I · p. 1
  • Full-matrix tracking includes orientation, semi-axis ratios, and absolute-scale differences, whereas shape tracking excludes uniform-scale differences.Earlier transfer approaches retained reference scale and adapted it to the robot before tracking.Source: §I · pp. 1–2
  • The Shape method transfers ellipsoid orientation and semi-axis length ratios across different kinematic systems without matching absolute scale.It treats positive scalar multiples as equivalent and uses unit-determinant representatives.Source: §I · p. 2
  • The Shape method integrates with position and end-effector direction objectives in a constrained joint-velocity QP and transfers robot-to-robot and human-to-robot shape without target-scale selection.The experiments also examine what constant target-scale factors can and cannot achieve for the Full method.Source: §I · p. 2

B. Riemannian Geometry of SPD Matrices

The paper represents symmetric positive-definite matrices with affine-invariant Riemannian geometry, using a tangent-space inner product and associated norm. Exponential and logarithm maps connect points and tangent vectors, and the induced geodesic distance is the norm of the logarithmic displacement.

B. Riemannian Geometry of SPD Matrices

  • At an SPD matrix P, tangent vectors are identified with real symmetric matrices and measured with the AIRM inner product.The inner product is ⟨U,V⟩_P = tr(P^-1 U P^-1 V).Source: B. Riemannian Geometry of SPD Matrices · p. 3
  • The AIRM norm is induced by the tangent-space inner product and uses the Frobenius norm of the whitened tangent vector.The passage identifies P^(1/2) as the SPD square root.Source: B. Riemannian Geometry of SPD Matrices · p. 3
  • Riemannian exponential and logarithm maps are mutually inverse, and the geodesic distance equals the norm of the logarithm map from P to Q.The matrix exponential and principal matrix logarithm define the corresponding maps.Source: B. Riemannian Geometry of SPD Matrices · p. 3

III. SCALE-INVARIANT MANIPULABILITY SHAPE TRACKING

The Shape method tracks unit-determinant manipulability representatives, preserving orientation and semi-axis length ratios without requiring absolute-scale matching. The paper also formulates full-matrix tracking as an AIRM tangent-rate residual with scalar-gain logarithmic feedback.

III. SCALE-INVARIANT MANIPULABILITY SHAPE TRACKING

  • The Full method tracks complete reference manipulability ellipsoids, including scale and shape, while the Shape method tracks orientation and all semi-axis length ratios without absolute-scale matching.The Shape method is the proposed scale-invariant approach; the Full method is the full-matrix baseline.Source: §III · p. 3

A. Manipulability Differential and Full-Matrix Tracking

  • The manipulability Jacobian is a third-order derivative tensor whose slices give the matrix derivative with respect to each joint coordinate.The mode-3 contraction is identified with a D × D matrix, and derivatives are evaluated at the current configuration.Source: A. Manipulability Differential and… · p. 3
  • The Full method compares the robot-induced manipulability rate with a logarithmic feedback command using the AIRM tangent norm.It uses the logarithmic feedback of with a scalar gain, rather than the Euclidean norm of the original vectorized least-squares formulation.Source: A. Manipulability Differential and… · p. 3

B. Unit-Determinant Shape Representation

The unit-determinant representative separates manipulability shape from absolute scale, preserving principal directions and semi-axis ratios. The method tracks shape through a differential and an orthonormal tangent-space residual, although exact tracking can be limited by robot kinematics, constraints, and competing tasks.

B. Unit-Determinant Shape Representation

  • Normalizing by the determinant-derived scale produces a unit-determinant representative that preserves principal directions and all semi-axis ratios.For any positive scalar rescaling of the original matrix, the normalized representative is unchanged.Source: B. Unit-Determinant Shape Representation · p. 3
  • The Shape method compares normalized current and reference matrices without matching their original scales; the achieved scale may still vary with configuration.The normalization divides each semi-axis length by the square root of the determinant-derived scale.Source: B. Unit-Determinant Shape Representation · p. 3

C. Scale-Invariant Shape Tracking

  • The derivative tensor of the normalized shape matrix defines its rate, while an orthonormal tangent coordinate map preserves the AIRM norm of shape residuals.The residual has m−1 coordinates: two for D = 2 and five for D = 3.Source: C. Scale-Invariant Shape Tracking · pp. 3–4
  • Exact shape tracking is not guaranteed by the reduced tangent representation and may be limited by kinematics, joint constraints, or competing task objectives.The representation does not guarantee that the shape Jacobian has full row rank or that the reference is reachable.Source: C. Scale-Invariant Shape Tracking · p. 4

D. Geometric Properties

The geometric decomposition separates shape and scale components of the tracking residual. Shape tracking is invariant to positive rescaling of the reference, while the Shape method omits the scalar scale-rate term and preserves the stated velocity–force shape duality.

D. Geometric Properties

  • Replacing the reference matrix with any positive scalar multiple leaves its normalized shape, shape error, and Shape-method cost unchanged at a fixed configuration.The shape and scale residual components are Frobenius-orthogonal, separating their contributions to the full tracking cost.Source: D. Geometric Properties · p. 4
  • The Shape method omits precisely the scalar log-scale rate-tracking term from the Full method’s manipulability cost.The full residual decomposes into a traceless shape residual and a scalar scale residual.Source: D. Geometric Properties · p. 4
  • Under the stated norm conventions, normalized velocity and force shapes share principal axes and have reciprocal corresponding semi-axis lengths.The force manipulability matrix is the inverse of the velocity matrix under those conventions.Source: D. Geometric Properties · p. 4

E. Integration with Task-Space Control

The control formulation combines task-space and manipulability residuals as weighted soft objectives in a constrained joint-velocity quadratic program. The supplied table passages identify phase-mean tracking errors and Full-method target-scale sensitivity, but do not provide their numerical comparisons.

E. Integration with Task-Space Control

  • The constrained joint-velocity command minimizes combined task-space and manipulability objectives, with the Shape or Full cost selected for manipulability tracking.The formulation includes joint-velocity limits and one-step joint-position bounds; task and manipulability residuals are minimized jointly as weighted soft objectives.Source: E. Integration with Task-Space Control · p. 4
  • Table I reports phase means for tracking errors, while Table II addresses target-scale sensitivity of the Full method.The supplied table passages contain titles only, so they do not establish specific values or a method comparison.Source: Table I · p. 5, Table II · p. 5

A. Experimental Setup

The experiments compare full manipulability-matrix tracking with unit-determinant shape tracking across heterogeneous robots. Shape tracking reaches very small endpoint shape distances, while total AIRM distance remains higher because it also includes scale differences.

A. Experimental Setup

  • The kinematic simulations use four robots: seven-joint FR3 and Gen3, and six-joint KR500 and UR20, with differing arm lengths and link proportions.The joint-velocity QP runs at 500 Hz with joint speeds bounded by 0.6 rad/s.Source: A. Experimental Setup · p. 5

B. Robot-to-Robot Manipulability Transfer

  • The comparison transfers a time-varying FR3 manipulability reference to Gen3, KR500, and UR20 using three active joints for XZ motion and no position or direction objectives.The source motion is held, moved forward and downward, then held again; the manipulability reference can continue changing during the final hold.Source: B. Robot-to-Robot Manipulability… · p. 5
  • At 11 s, Shape reaches ds(11) ≈ 9.30 × 10^-5 on all three followers, while Full remains at 1.0696 on KR500 and 0.7576 on UR20.Source: B. Robot-to-Robot Manipulability… · p. 6
  • Shape has larger mean total AIRM distances for every follower and phase because total distance includes scale differences that Shape does not penalize.The results distinguish matching orientation and semi-axis ratios from matching absolute scale.Source: B. Robot-to-Robot Manipulability… · p. 6

C. Target-Scale Sensitivity

The target-scale sensitivity experiment tests whether constant scaling can substitute for omitting scale from the tracking objective. Robot-specific tuning brings Full close to Shape during motion, but residual endpoint shape errors remain on KR500 and UR20.

C. Target-Scale Sensitivity

  • The multipliers are selected per robot by sweeping constant c values to minimize motion-phase ds RMS on that follower’s realized trajectory, giving Full favorable robot-specific tuning.The selected values are 1.189 for Gen3, 11.31 for KR500, and 5.657 for UR20; a demonstration-only pipeline lacks the follower-trajectory information used here.Source: C. Target-Scale Sensitivity · p. 6
  • With each robot’s own selected multiplier, Full has motion-phase mean shape distances ds = 0.2210–0.2221, close to approximately 0.2210 with Shape.Using the Gen3-selected multiplier instead gives ds = 1.6470 for KR500 and 0.7410 for UR20.Source: C. Target-Scale Sensitivity · p. 6
  • At the endpoint, Shape achieves ds(11) ≈ 9.30 × 10^-5 for all three followers, while Full retains ds(11) = 0.0684 for KR500 and 0.1113 for UR20 even with their selected multipliers.KR500 and UR20 retain axis-ratio errors despite small major-axis errors.Source: C. Target-Scale Sensitivity · p. 7
  • For six combinations using a multiplier selected for a different robot, scale differences account for 66.4–97.9% of final-hold squared total AIRM distance.Source: C. Target-Scale Sensitivity · p. 7
  • Constant target scaling depends on both robot and evaluation phase, whereas Shape tracks orientation and semi-axis length ratios without selecting a target-scale multiplier.Source: C. Target-Scale Sensitivity · p. 7

D. Human-to-Robot Transfer with Concurrent Tasks

The human-to-robot experiment transfers a recorded reach’s directional geometry while tracking end-effector position and direction. Across four robots, Shape reproduces the human target’s forward force-shape elongation, with lower endpoint errors than uncalibrated Full tracking; the evaluation remains limited to kinematic simulations.

D. Human-to-Robot Transfer with Concurrent Tasks

  • At the endpoint, all four robots using Shape reproduce dual force shapes elongated along X, like the human target.The recorded forward reach combines wrist displacement with increasing elongation of the human dual force ME along world +X.Source: D. Human-to-Robot Transfer with… · p. 7
  • At the endpoint, Shape yields lower position, direction, and shape errors than Full for all four robots.For KR500, endpoint position error is 75.104% of Lr with Full and 5.603% with Shape; for UR20, it is 67.018% and 3.360%, respectively.Source: D. Human-to-Robot Transfer with… · p. 7
  • Neither method received a robot-specific target-scale selection in this experiment, so Full’s errors characterize uncalibrated transfer rather than full-matrix tracking generally.The experiment combines the scale mismatch with a competing position task.Source: D. Human-to-Robot Transfer with… · p. 7

V. CONCLUSION

  • In kinematic simulations, Shape reproduced a common target shape on three robot followers with small endpoint errors and no robot-specific target-scale adjustment.Robot-specific scaling brought Full’s shape tracking close to Shape during motion, but effectiveness depended on the robot and fitting phase.Source: §V · p. 7
  • For the recorded human reach, Shape reproduced the target dual force shape’s forward elongation while tracking end-effector position and direction.Without robot-specific scale calibration, all four robots had lower position, direction, and shape errors with Shape than with Full at the endpoint and during the final hold.Source: §V · p. 8
  • The evaluation is limited to kinematic simulations of manipulability transfer through robot motion.Future work will examine physical task execution with dynamics and contact, and a broader range of human motions.Source: §V · p. 8

APPENDIX A

Appendix A constructs an orthonormal basis for the space of symmetric traceless matrices and uses it to represent the AIRM tangent tracking residual.

APPENDIX A

  • Dividing the generalized Gell–Mann matrices by √2 gives a Frobenius-orthonormal basis of Sym0(D).The appendix specifies basis ordering for D = 2 and D = 3.Source: Appendix A · p. 8
  • The shape residual U lies in the tangent space at P, and its normalized form X is symmetric and traceless.The appendix establishes this using the unit-determinant current shape P and the relative matrix R, which is SPD with determinant one.Source: Appendix A · p. 8
  • At fixed P, the coordinate representation is linear in joint velocity, and orthonormality establishes equivalence of the two tracking formulations.The appendix states CP(U) = JM q̇ − kM e_s and uses the basis orthonormality to prove equivalence of (15) and (16).Source: Appendix A · p. 8
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