Video

-Calibration Experiments

Abstract

Multi-robot co-manipulation shows great potential to address the limitations of using single robot in complicated tasks such as robotic surgeries. However, the dynamic setup poses great uncertainties in the circumstances of robotic mobility and unstructured environment. Therefore, the relationships among all the base frames (robot-robot calibration) and the relationships between the end-effectors and the other devices such as cameras (hand-eye calibration) and tools (tool-flange calibration) have to be determined constantly in order to enable robotic cooperation in the constantly changing environment. We formulated the problem of hand-eye, tool-flange and robot-robot calibration to a matrix equation AXB=YCZ. A series of generic geometric properties and lemmas were presented, leading to the derivation of the final simultaneous algorithm. In addition to the accurate iterative solution, a closed-form solution was also introduced based on quaternions to give an initial value. To show the feasibility and superiority of the simultaneous method, two non-simultaneous methods were also proposed for comparison. Furthermore, thorough simulations under different noise levels and various robot movements were carried out for both simultaneous and non-simultaneous methods. Experiments on real robots were also performed to evaluate the proposed simultaneous method. The comparison results from both simulations and experiments demonstrated the superior accuracy and efficiency of the simultaneous method.

Problem Formulation

Measurement Data:

Homogeneous transformations from the robot bases to end-effector (A and C), and from tracker to marker (B).

Unknowns:

Homogeneous transformations from one robot base frame to another (Y), and from eye/tool to robot hand/flange (X and Z).

The measurable data A, B and C, and the unknowns X, Y and Z form a transformation loop which can be formulated as, AXB=YCZ.

Approaches

Non-simultaneous Methods

3-Step Method

In the non-simultaneous 3-Step method, the X and Z in (1) are separately calculated as two hand-eye/tool-flange calibrations which can be represented as an AX = XB problem in the first and second steps. This results in two data acquisition procedures, in which the two manipulators carry out at least two rotations whose rotational axes are not parallel or anti-parallel by turns while the other one being kept immobile. The last unknown robot-robot relationship Y could be solved directly using the previously retrieved data by the method of least squares.

2-Step Method

The non-simultaneous 2-Step method formulates the original calibration problem in successive processes which solve AX = XB firstly, and then the AX = YB. The data acquisition procedures and obtained data are the same with the 3-Step method. In contrast to solving robot-robot relationship independently, the 2-Step method solves tool-flange/hand-eye and robot-robot transforms in an AX = YB manner in the second step. This is possible because equation AXB = YCZ can be expressed as (AXB)inv(Z) = YC, which is in an AX = YB form with the solution of X known.

Simultaneous Method

Non-simultaneous methods face a problem of error accumulation, since in these methods the latter steps use the previous solutions as input. As a result, the inaccuracy produced in the former steps will accumulate to the subsequent steps. In addition to accuracy, it is preferred that the two robots participating the calibration procedure simultaneously, which will significantly save the total time required.

In regards to this, a simultaneous method is proposed to improve the accuracy and efficiency of the calibration by solving the original AXB = YCZ problem directly. During the data acquisition procedure, the manipulators simultaneously move to different configurations and the corresponding data set A, B and C are recorded. Then the unknown X, Y and Z are solved simultaneously.

Experiments Results

Ample real experiments have been conceived and carried out under different configurations to evaluate the proposed methods. As shown in Fig. 1, the experiments involved a Staubli TX60 robot (6 DOFs, averaged repeatability 0.02mm), a Barrett WAM robot (4 DOFs, averaged repeatability 0.05mm) and a NDI Polaris optical tracker (RMS repeatability 0.10mm). The optical tracker was mounted to the last link of the Staubli robot, referred to as sensor robot. The corresponding reflective marker was mounted to the last link of the WAM robot, referred to as marker robot.

To demonstrate the superiority of the simultaneous method in the real experimental scenarios, a 5-fold cross-validation approach is implemented for 200 times for all the calibration methods under all system configurations. For simultaneous method, after data alignment and RANSAC processing, 80% of the remaining data are randomly selected to calculate unknown X, Y, and Z, and 20% are used as test data to evaluate the performance. For 2-Step and 3Step methods, after calculating the unknowns by each method, same test data from the simultaneous method are used to evaluate their performances. Results show that our simultaneous method is superior that the other methods.

Institute & People Involved

National University of Singapore (NUS): Liao Wu; Hongliang Ren, Member, IEEE

The Chinese University of Hong Kong (CUHK): Jiaole Wang, Student Member, IEEE; Max Q.-H. Meng, Fellow, IEEE

Publications

[1]. Liao Wu†, Jiaole Wang†, Lin Qi, Keyu Wu, Max Q.-H. Meng, Hongliang Ren. Simultaneous hand-eye, tool-flange and robot-robot calibration for comanipulation by solving the AXB = YCZ problem. IEEE Transactions on Robotics. 2016, 32(2): 413-428. († equally contributing authors) [pdf][code][video][pub]

[2]. Jiaole Wang†, Liao Wu†, Max Q.-H. Meng, Hongliang Ren. Towards simultaneous coordinate calibrations for cooperative multiple robots. In: Proceedings of 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2014). Chicago, Illinois, USA, 2014: 410-415. († equally contributing authors) [pdf][pub]