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A Unified Calibration Framework for Coordinate and Kinematic Parameters in Dual-Arm Robots

Tianyu Huang, Bohan Yang, Bin Li, Wenpan Li, Haoang Li, Wenlong Li, Yun-Hui Liu

Abstract

Precise collaboration in vision-based dual-arm robot systems requires accurate system calibration. Recent dual-robot calibration methods have achieved strong performance by simultaneously solving multiple coordinate transformations. However, these methods either treat kinematic errors as implicit noise or handle them through separated error modeling, resulting in non-negligible accumulated errors. In this paper, we present a novel framework for unified calibration of the coordinate transformations and kinematic parameters in both robot arms. Our key idea is to unify all the tightly coupled parameters within a single Lie-algebraic formulation. To this end, we construct a consolidated error model grounded in the product-of-exponentials formula, which naturally integrates the coordinate and kinematic parameters in twist forms. Our model introduces no artificial error separation and thus greatly mitigates the error propagation. In addition, we derive a closed-form analytical Jacobian from this model using Lie derivatives. By exploring the Jacobian rank property, we analyze the identifiability of all calibration parameters and show that our joint optimization is well-posed under mild conditions. This enables off-the-shelf iterative solvers to stably optimize these parameters on the manifold space. Besides, to ensure robust convergence of our joint optimization, we develop a certifiably correct algorithm for initializing the unknown coordinates. Relying on semidefinite relaxation, our algorithm can yield a reliable estimate whose near-global optimality can be verified a posteriori. Extensive experiments validate the superior accuracy of our approach over previous baselines under identical visual measurements. Meanwhile, our certifiable initialization consistently outperforms several coordinate-only baselines, proving its reliability as a starting point for joint optimization.

A Unified Calibration Framework for Coordinate and Kinematic Parameters in Dual-Arm Robots

Abstract

Precise collaboration in vision-based dual-arm robot systems requires accurate system calibration. Recent dual-robot calibration methods have achieved strong performance by simultaneously solving multiple coordinate transformations. However, these methods either treat kinematic errors as implicit noise or handle them through separated error modeling, resulting in non-negligible accumulated errors. In this paper, we present a novel framework for unified calibration of the coordinate transformations and kinematic parameters in both robot arms. Our key idea is to unify all the tightly coupled parameters within a single Lie-algebraic formulation. To this end, we construct a consolidated error model grounded in the product-of-exponentials formula, which naturally integrates the coordinate and kinematic parameters in twist forms. Our model introduces no artificial error separation and thus greatly mitigates the error propagation. In addition, we derive a closed-form analytical Jacobian from this model using Lie derivatives. By exploring the Jacobian rank property, we analyze the identifiability of all calibration parameters and show that our joint optimization is well-posed under mild conditions. This enables off-the-shelf iterative solvers to stably optimize these parameters on the manifold space. Besides, to ensure robust convergence of our joint optimization, we develop a certifiably correct algorithm for initializing the unknown coordinates. Relying on semidefinite relaxation, our algorithm can yield a reliable estimate whose near-global optimality can be verified a posteriori. Extensive experiments validate the superior accuracy of our approach over previous baselines under identical visual measurements. Meanwhile, our certifiable initialization consistently outperforms several coordinate-only baselines, proving its reliability as a starting point for joint optimization.
Paper Structure (20 sections, 5 theorems, 91 equations, 12 figures, 3 tables)

This paper contains 20 sections, 5 theorems, 91 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Notations and Preliminaries
  4. Algorithm Overview
  5. Unified Calibration of Coordinate and Kinematic Parameters
  6. Error Model via Product-of-Exponentials Formula
  7. Analytical Jacobian via Lie Derivatives
  8. Identifiability Analysis
  9. SDP Relaxation for Certifiable Coordinate Initialization
  10. Quadratically Constrained Quadratic Programming
  11. Semidefinite Relaxation
  12. Certifiably Correct Solution
  13. Experiment Results
  14. Simulation Analysis
  15. Real-world Experiments
  16. ...and 5 more sections

Key Result

Proposition 1

Define a homogeneous state vector as Then $\boldsymbol{f}_i$ and $\boldsymbol{g}_i$ are linear in $\boldsymbol{w}$, i.e., where $\boldsymbol{\Omega}_{f_i},\ \boldsymbol{\Omega}_{g_i}$ are introduced as auxiliary matrices and constructed from the measurements.

Figures (12)

  • Figure 1: Illustration of a canonical dual-arm robot system. The sensor-side robot is equipped with a camera. The tool-side robot holds a calibration tool (e.g., a chessboard). The closed-loop pose chain in the dual-arm system can be defined by $\boldsymbol{A} \boldsymbol{X} \boldsymbol{B} = \boldsymbol{Y} \boldsymbol{C} \boldsymbol{Z}$.
  • Figure 2: Pipeline of the proposed unified calibration algorithm. Given a set of measurements under multiple dual-arm postures, we construct a consolidated Lie-algebraic error model and iteratively refine the coordinate and kinematic parameters on the $\mathrm{SE}(3)$ manifold. Before our iterative optimization, the coordinate transforms are initialized by a certifiably correct SDP relaxation solver, while the nominal kinematic parameters directly serve as the initial guess.
  • Figure 3: Real-world experiments are conducted on a dual-arm robot system that contains an AUBO-i10 arm equipped with an industrial 3D camera and an AUBO-i10 arm equipped with a calibration tool. The calibration tool consists of a high-precision ChArUco chessboard and a standard ceramic ball.
  • Figure 4: Simulated evaluation results on different levels of kinematic errors under (a) medium and (b) high measurement noises (see Section \ref{['subsec:sim_exper']}1. Each box plot summarizes the error distribution over 40 distinct test samples.
  • Figure 5: Simulated evaluation results on different levels of measurement noises under (a) medium-low and (b) medium-high kinematic errors (see Section \ref{['subsec:sim_exper']}2). Each box plot summarizes the error distribution over 40 distinct test samples.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1: Tightness Guarantee of Relaxation
  • Lemma 1
  • proof