Hand-Eye Calibration (2311.12655v2)

Abstract: Whenever a sensor is mounted on a robot hand it is important to know the relationship between the sensor and the hand. The problem of determining this relationship is referred to as hand-eye calibration, which is important in at least two types of tasks: (i) map sensor centered measurements into the robot workspace and (ii) allow the robot to precisely move the sensor. In the past some solutions were proposed in the particular case of a camera. With almost no exception, all existing solutions attempt to solve the homogeneous matrix equation AX=XB. First we show that there are two possible formulations of the hand-eye calibration problem. One formulation is the classical one that we just mentioned. A second formulation takes the form of the following homogeneous matrix equation: MY=M'YB. The advantage of the latter is that the extrinsic and intrinsic camera parameters need not be made explicit. Indeed, this formulation directly uses the 3 by 4 perspective matrices (M and M') associated with two positions of the camera. Moreover, this formulation together with the classical one cover a wider range of camera-based sensors to be calibrated with respect to the robot hand. Second, we develop a common mathematical framework to solve for the hand-eye calibration problem using either of the two formulations. We present two methods, (i) a rotation then translation and (ii) a non-linear solver for rotation and translation. Third, we perform a stability analysis both for our two methods and for the classical linear method of Tsai and Lenz (1989). In the light of this comparison, the non-linear optimization method, that solves for rotation and translation simultaneously, seems to be the most robust one with respect to noise and to measurement errors.

• The paper proposes a novel mathematical representation for hand-eye calibration using perspective matrices that bypasses explicit intrinsic and extrinsic parameter extraction.
• It introduces a unified framework employing unit quaternions to decouple rotation and translation through both closed-form and non-linear optimization methods.
• Experimental results show that the non-linear optimization approach outperforms the classical Tsai-Lenz method in noisy environments, enhancing calibration robustness.

An Academic Insight into the Hand-Eye Calibration Problem

The paper "Hand-Eye Calibration" by Radu Horaud and Fadi Dornaika explores the intricacies of determining the positional and orientational relationship between a sensor mounted on a robot hand and the hand itself, a problem commonly referred to as hand-eye calibration. This area of robotics is of significant importance for enabling precise manipulation and measurement tasks, such as object recognition and sensory alignment in robot workspaces.

Historically, the hand-eye calibration challenge has been primarily approached by solving a homogeneous matrix equation of the form $AX=XB$. The paper provides two critical contributions that extend beyond the classical solutions:

1. Alternative Formulation: The authors propose a novel mathematical representation of the hand-eye calibration problem characterized by the equation $MY=M'YB$. This formulation makes direct use of 3×4 perspective matrices ($M$ and $M'$) acquired from camera calibrations and provides an avenue that bypasses the need to explicitly extract intrinsic and extrinsic camera parameters. It allows for a broader application of different camera types, including single scan-line cameras, stereo vision systems, and range finders. This contrasts with the prior methods' focus on TV cameras and extends applicability.
2. Unified Mathematical Framework: The researchers introduce a mathematical framework that accommodates both the classical and the new formulations. This framework employs unit quaternions to model rotations, introducing two solutions: a closed-form solution that decouples the problem into rotation and translation, and a non-linear optimization method solving for rotation and translation simultaneously.
3. Stability Analysis and Experimental Verification: The authors conduct an in-depth stability analysis comparing the novel methods with the classical Tsai-Lenz method. The results highlight the robustness of the non-linear optimization approach, which exhibits superior performance in the presence of noise and errors in measurement. The closed-form solution, while efficient, demonstrated some susceptibility to calibration errors compared to the non-linear method.

The practical implications of this research are considerably significant. By incorporating methods that do not require delineation of intrinsic and extrinsic parameters, the proposed framework is versatile for real-world applications where sensor parameters cannot be easily isolated. Additionally, the non-linear optimization technique offers robustness crucial for scenarios involving significant noise and uncertainty, common in industrial and field robotics. The broader scope of applicational sensors offers new avenues for extended research in automated calibration and adaptive sensor alignment in dynamically changing environments.

In summation, this paper offers valuable insights and methodologies in hand-eye calibration, providing both theoretical underpinning and practical algorithms for enhanced accuracy and robustness in robotic sensor operations. The advancement proposed by Horaud and Dornaika sets the stage for future explorations into simultaneous hand-eye and robot calibration and refining sensor alignments across various robotic systems and configurations.