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

cs.ROMar 16, 2026

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

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

Organizations: Hong Kong Centre for Logistics Robotics and Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, China · State Key Laboratory of Intelligent Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, China · Thrust of Robotics and Autonomous Systems, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China

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.

Figures & tables

Fig. 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 $\boldsymbol{A}\boldsymbol{X}\boldsymbol{B}=\boldsymbol{Y}\boldsymbol{C}\boldsymbol{Z}$ .

Fig. 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.

Fig. 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.

Kinematic Error Level	Rotation Deviation (deg.)	Translation Deviation (mm)
Kinematic Error Level	mean $\pm$ std	mean $\pm$ std
low ( L )	0.054 $\pm$ 0.022	0.417 $\pm$ 0.161
medium-low ( ML )	0.103 $\pm$ 0.043	1.083 $\pm$ 0.412
medium ( M )	0.264 $\pm$ 0.089	1.818 $\pm$ 0.763
medium-high ( MH )	0.427 $\pm$ 0.143	2.861 $\pm$ 1.032
high ( H )	0.692 $\pm$ 0.278	5.567 $\pm$ 0.238

TABLE I: Distribution of robot end-effector pose error caused by different levels of kinematic errors in the simulation analysis (c.f., Section VII-A ).

Measurement Noise Level	Rotation Noise (deg.)	Translation Noise (mm)
Measurement Noise Level	mean $\pm$ std	mean $\pm$ std
low ( L )	0.032 $\pm$ 0.013	0.080 $\pm$ 0.034
medium-low ( ML )	0.080 $\pm$ 0.034	0.160 $\pm$ 0.067
medium ( M )	0.128 $\pm$ 0.054	0.479 $\pm$ 0.202
medium-high ( MH )	0.160 $\pm$ 0.067	0.798 $\pm$ 0.337
high ( H )	0.239 $\pm$ 0.101	1.277 $\pm$ 0.539

TABLE II: Distribution of measurement noise at different levels in the simulation analysis (c.f., Section VII-A ).

Fig. 4: Simulated evaluation results on different levels of kinematic errors under (a) medium and (b) high measurement noises (see Section VII-A 1. Each box plot summarizes the error distribution over 40 distinct test samples.

Fig. 5: Simulated evaluation results on different levels of measurement noises under (a) medium-low and (b) medium-high kinematic errors (see Section VII-A 2). Each box plot summarizes the error distribution over 40 distinct test samples.

Fig. 6: Simulated evaluation results on different amounts of calibration samples under (a) medium and (b) high kinematic errors with medium-high measurement noises (see Section VII-A 3). Each box plot summarizes the error distribution over 40 distinct test samples.

Fig. 7: Simulated ablation study regarding our initialization (i.e., SDP-Ini ) on different levels of kinematic errors under (a) medium and (b) high measurement noises (see Section VII-A 4). The curves report the mean deviations over 40 distinct test samples and the iterations to convergence of the unified optimizer with different initializations.

Fig. 8: Data acquisition in real-world experiments. (a) We synchronically adjust the attitudes of both robot arms, ensuring that distinct measurements are obtained. (b) Example images captured at different arm postures, where the calibration pattern is detected to estimate the tool pose in the sensor frame.

Method	Number of Calibration Samples
	20		40		60		80		100		120
	$e_{\boldsymbol{R}}$ (deg.)	$e_{\boldsymbol{t}}$ (mm)	$e_{\boldsymbol{R}}$ (deg.)	$e_{\boldsymbol{t}}$ (mm)	$e_{\boldsymbol{R}}$ (deg.)	$e_{\boldsymbol{t}}$ (mm)	$e_{\boldsymbol{R}}$ (deg.)	$e_{\boldsymbol{t}}$ (mm)	$e_{\boldsymbol{R}}$ (deg.)	$e_{\boldsymbol{t}}$ (mm)	$e_{\boldsymbol{R}}$ (deg.)	$e_{\boldsymbol{t}}$ (mm)
Wu [ 10 ]	1.3904	6.3091	1.1626	4.2898	1.0959	4.0361	1.0566	3.6754	0.9863	3.9177	0.9769	3.6729
Wang [ 13 ]	1.5661	6.2315	1.2790	4.2283	1.1347	4.0292	1.0737	3.6786	0.9798	4.3918	0.9568	4.1135
Jiang [ 12 ]	1.4773	6.2637	1.2465	4.3564	1.1920	4.1233	1.1049	3.8480	1.0134	4.3927	1.0101	4.0489
SDP-Ini (Ours)	1.1608	4.8550	0.8881	3.3454	0.7879	3.1204	0.7246	2.9885	0.7281	3.0956	0.7191	2.8456

TABLE III: Real-world evaluation results on different amount of calibration samples (see Section VII-B 1). Mean rotation/translation derivations of each method evaluated on 30 distinct test samples are reported. The best results are boldfaced .

Fig. 9: Real-world evaluation results on different amount of calibration samples (see Section VII-B 1). Here we use box plots to show the error distributions of each method over 30 distinct test samples.

Fig. 10: Real-world ablation study regarding our initialization (i.e., SDP-Ini ) on different amount of calibration samples (see Section VII-B 1). The curves report the mean deviations over 30 distinct test samples and the iterations to convergence of the unified optimizer with different initializations.

Fig. 11: The cooperative-measuring evaluation results (see Section VII-B 2). Below each method are the corresponding aligned point clouds obtained from multiple dual-arm postures, with each transformed point cloud visualized in a different color. The red-grid sphere denotes the minimum enclosing ball (MEB) computed from the centers of the spheres fitted to all transformed point clouds, while the blue points indicate the fitted sphere centers. The number reported below is the MEB radius $r_{\textnormal{MEB}}$ , where a smaller value indicates higher cooperative-measuring accuracy.

Fig. 12: The standard ball is rigidly attached to the tool-side end-effector and thus remains static in frame $\{E_{2}\}$ throughout the experiment. For each dual-arm posture, we can transform the ball point cloud measured in $\{L_{1}\}$ to the common frame $\{E_{2}\}$ via ( 54 ). As a result, the dual-arm system is now equivalent to a multi-view measuring setup.

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This article proposes a general optimization framework for solving hand-eye calibration problem. Unlike traditional methods, an iterative algorithm based on Lie algebra that achieves approximately global optimal solutions is developed. During the optimization process, the method strictly preserves the structural constraints of the calibration parameters and enables synchronized updates between calibration parameters. Recognizing that data used in real-word hand-eye calibration often contain uncertainty, especially in over-loading and large workspace industrial robot scenarios, which can significantly degrade accuracy, and accurately modeling such uncertainty is inherently difficult, this article avoids explicit uncertainty modeling. Instead, an uncertainty metric to evaluate the relative uncertainty between data sources is introduced and used to dynamically refine the iterative process. To further enhance convergence efficiency, an effective initial solution generation method that improves overall stability and accuracy is designed. Numerical simulations and real-world experiments validate the effectiveness of the proposed approach, and in synthetic datasets, the proposed approach improves the estimation accuracy by at least 67% under high-uncertainty conditions compared with the existing methods.

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