Local POE model for robot kinematic calibration

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Abstract

A robot kinematic calibration method based on the local frame representation of the product-of-exponentials (Local POE) formula is introduced. In this method, the twist coordinates of the joint axes are expressed in their respective local (body) frames. The advantages of this new approach are threefolds: (1) revolute and prismatic joints can be uniformly expressed in the twist coordinates based on the line geometry; (2) the twist coordinates of the joint axes can be set up with simple values because the local frames can be arbitrarily defined on the links; (3) the kinematic parameters described by the twist coordinates vary smoothly that makes the method robust and singularity-free. By assuming that the kinematic errors exist only in the relative initial poses of the consecutive link frames, the kinematic calibration models can be formulated in a simple and elegant way. The calibration process then becomes to re-define a set of new local link frames that are able to reflect the actual kinematics of the robot. This method can be applied to robot manipulators with generic open chain structures (serial or tree-typed). The simulation and experiment results on a 4-DOF SCARA type robot and a 5-DOF tree-typed modular robot have shown that the average positioning accuracy of the end-effector increases significantly after calibration.

Introduction

Most robots have their kinematic model implemented, assuming nominal arrangement and description of their coordinate systems. Due to the inherent kinematic errors such as machining and assembly errors, the actual kinematic parameters of a robot will differ from their nominal values as implemented in the robot controller, lowering the positioning accuracy of the robot. Kinematic calibration serves as a solution to improve the absolute positioning accuracy of a robot. It is addressed as an integrated process of modeling, measuring, and numerically identifying the actual characteristics of a robot.

In order to formulate the calibration model, several kinematic modeling methods have been employed such as the Denavit–Hartenberg (D–H) parameterization approach [1], [2], [3], [4], [5], [6], [7], the continuous and parametrically complete (CPC) modeling approach [8], [9], [10], the zero reference position modeling approach [11], [12], [13], [14], and the product-of-exponentials (POE) formulation approach [15], [16], [17], [18], [19], [20].

The D–H kinematic formulation method [21] uses a minimum set of parameters to describe the relationship between adjacent joint axes for the formulation of the kinematics. However, being adopted for calibration modeling, this method is not amenable to direct identification as all the parameters in the D–H model are stringently defined and are unique to the particular robot configuration concerned [5]. In addition, it has the singularity problem when neighboring joint axes are nearly parallel. A number of researchers use modified forms of D–H formulation or other modeling techniques to overcome the singularity problem. Hayati and co-workers [1], [2], [3], [4] introduce an angular alignment parameter βi in place of the di parameter in the D–H model to represent a small misalignment between two consecutive parallel axes, and an additional linear parameter for handling prismatic joints. Stone [5] uses a six-parameter representation S-model in which two additional parameters are added to the D–H model to allow for arbitrary placement of link frames. Zhuang and co-workers [8], [9], [10] propose a six-parameter CPC model. In this model, a singularity-free line representation consisting of four line parameters are adopted to ensure parametric continuous, while two additional parameters are used to allow arbitrary placement of the link coordinate frames to make the model complete. Kazerounian and co-workers [11], [12] and Mooring [13], [14] develop their calibration model based on a zero reference position analysis method as described in [22], [23]. This method describes the manipulator kinematics in terms of the axes directions and locations in the zero reference position. The error parameters considered in this model are redundant, and the elimination of redundant parameters requires assumptions that rotation and translation errors in certain directions to be negligible [14].

The POE representation method can also be considered as a zero reference position method. It describes the joint axes based on line geometry. Hence, it is uniform in modeling manipulators with both revolute and prismatic joints. Significantly, the kinematic parameters in the POE model vary smoothly with changes in joint axes so that the model can handle certain kinematic singularity problems encountered in other kinematic parametrization methods. These features make the POE formula very suitable for robot kinematic calibration. Park and Okamura [16], [17] propose a novel calibration model based on the base (global) frame representation of the POE formula for open chain manipulators. This model assumes that the kinematic errors exist in each of the joint axes and the initial pose of the tool frame. Since all the kinematic parameters are expressed in the base frame, the attachment of the local frames to each of the joints (or links) is unnecessary.

Unlike Park and Okamura's model [16], [17] in which the base frame representation of POE formula is employed, we propose a calibration model based on the local frame representation of the POE formula, termed as the Local POE formula. In the local POE formula, all the joint axes are expressed in their respective local frames. The major advantage of this formula is that the local coordinate frames can be arbitrarily assigned onto their corresponding links. Therefore, in the presence of kinematic errors in the robot structure, one can always assume that the kinematic errors only exist in the initial poses of the consecutive local frames. The calibration algorithm is, according to the measurement data, to find a set of new local coordinate frames in the neighborhood of their original ones that precisely describe the actual kinematics of the robot. Since the calibration algorithm also allows the twist coordinates of the joint axes and the joint displacements to retain their nominal values when expressed in the new local frames, the calibration model can be greatly simplified and the kinematic constraints such as zero-pitch screws need not to be included into the algorithm.

Because of the local POE representation, the proposed calibration algorithm can uniformly deal with general open chain robots regardless of the types of joints, the number of branches, and the number of degrees of freedom. The kinematic parameters in this calibration model vary smoothly with changes in joint axes which makes the model singularity-free. Special descriptions are unnecessary when adjacent joint axes are close to parallel. Since the joint axes are expressed in their local frames, this calibration model is not only applicable to conventional industrial robots, but is especially useful for modular reconfigurable robotic systems [24]. The model can be easily modified by adding or removing local frames when modules are added in or removed from the robot system during reconfiguration.

The remaining sections of this paper are organized as follows. Section 2 briefly introduces the local POE formula for robot kinematics. The formulation of the calibration models for a general serial type robot is addressed in Section 3. A computer simulation example for calibrating a 4-DOF SCARA type robot is then presented in Section 4. For validation of the proposed calibration algorithm, the calibration experiment conducted on a 5-DOF tree-type modular robot is described in Section 5. This paper is summarized in Section 6. Finally, the geometric background related to the POE formula is provided in Appendix A.

Section snippets

The local POE formula for robot kinematics

Brocket [25] shows that forward kinematic equation of an open chain robot containing either revolute or prismatic joints can be uniformly expressed as a product of matrix exponentials. Because of its compactness, the POE formula has been shown to be a useful modeling tool in robotics [26], [27]. Depending on the coordinate frames used for describing the joint axes, the POE formula may have different expressions. For the local frame representation of the POE formula, the twists of joints are

Basic considerations

The nominal forward kinematics T0,n+1(q1,q2,…,qn) defined by Eq. (8), denoted as T in brief, is a function of the local frame initial poses T(0), joint twists s, and joint displacements q, where T(0)=[T0,1(0),T1,2(0),…,Tn−1,n(0),Tn,n+1]T, s=[s1,s2,…,sn]T, and q=[q1,q2,…,qn]T. Mathematically,T=f(T(0),s,q).The calibration model can be obtained by linearizing the forward kinematic equation such thatδTT−1=fT(0)δT(0)+fsδs+fqδqT−1,where δTT−1 denotes the pose error at the tool frame expressed

Computer simulation

In this section, a simulation example for calibrating a 4-DOF SCARA type robot (Fig. 5) is given to demonstrate the effectiveness of the calibration algorithm. This example is presented because

  • 1.

    it allows the verification of the assumptions made during modeling;

  • 2.

    it demonstrates the capability of the calibration algorithm in handling a robot with different joint types (revolute and prismatic) and the robustness in dealing with the singularity problems inherent in the conventional D–H representation;

Calibration experiment

To further validate the effectiveness and generality of the calibration algorithm for industrial applications, a calibration experiment is also conducted. The experimental setup consists of an articulated type coordinate measuring machine, SpinArm, by Mitutoyo of Japan, and a tree-type modular robot (to be calibrated) from Amtec GmbH of Germany, shown in Fig. 9.

The Mitutoyo SpinArm is a 6-DOF articulated coordinate measuring machine. The accuracy for positioning measurement is ±0.00008 m. The

Summary

A robot kinematic calibration method based on local representation of the POE formula is proposed in this article. By taking advantage of the local POE formula where the local coordinate frames can be arbitrarily assigned, the kinematic calibration becomes a process of redefining a set of new local coordinate frames to reflect the robot actual geometrical characteristics. Identification of the kinematic parameters is achieved by an iterative least squares algorithm. The advantages of this new

Acknowledgements

This project is funded by the Nanyang Technological University, Singapore, under Applied Research Grants RG64/96 and RG29/99, and the Gintic Institute of Manufacturing Technology, Singapore, under University Collaborative Project Grant, U97-A-006.

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