Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4063046, 10 pages

http://dx.doi.org/10.1155/2016/4063046

## Structural Parameter Identification of Articulated Arm Coordinate Measuring Machines

Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming, China

Received 29 February 2016; Revised 14 June 2016; Accepted 4 September 2016

Academic Editor: Tomonari Furukawa

Copyright © 2016 Guanbin Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Precise structural parameter identification of a robotic articulated arm coordinate measuring machine (AACMM) is essential for improving its measuring accuracy, particularly in robotic applications. This paper presents a constructive parameter identification approach for robotic AACMMs. We first develop a mathematical kinematic model of the AACMM based on the Denavit-Hartenberg (DH) approach established for robotic systems. This model is further calibrated and verified via the practical test data. Based on the difference between the calculated coordinates of the AACMM probe via the kinematic model and the given reference coordinates, a parameter identification approach is proposed to estimate the structural parameters in terms of the test data set. The Jacobian matrix is further analyzed to determine the solvability of the identification model. It shows that there are two coupling parameters, which can be removed in the regressor. Finally, a parameter identification algorithm taking the least-square solution of the identification model as the structural parameters by using the obtained poses data is suggested. Practical experiments based on a robotic AACMM test rig are carried out, and the results reveal the effectiveness and robustness of the proposed identification approach.

#### 1. Introduction

The coordinate measuring machine (CMM) is a universal measuring instrument which can transform various geometric measurements into coordinate measurements [1]. This instrument has been widely used in the calibration and modeling of robotic systems. In particular, the articulated arm CMM (AACMM) is a new type of robot-like CMM with multiple degrees of freedom (DOF), which generally consists of a series of linkages connected by joints in series [2, 3]. The AACMM obtains the angles of joints by means of the angle encoder installed on the rotary joints. And the angles can be transformed into the three coordinates through the kinematic model. The AACMM possesses some specific and essential characteristics and advantages, for example, simple mechanical structure, small size, light weight, large measurement range, and flexible measurement in field [4].

However, the measuring accuracy of the AACMM is much lower than that of the orthogonal CMM [5, 6], which may greatly limit its applications. A potential strategy to improve the measuring accuracy of the AACMM is to choose high precision hardware components and to apply high requirements for manufacturing and assembly [3, 7]. However, the cost of the measuring machine increases dramatically by using more precise components. Moreover, in some specific applications, increasing the precision of individual components may not be able to ensure overall increased measuring precision and accuracy. In particular, small errors in some kinematic parameters may accumulate and thus influence the measuring accuracy of the AACMM greatly.

Another essential and economic way to eliminate errors of the structural parameters and to improve the measuring accuracy is to identify the robot’s structural parameters [8] by using appropriate parameter identification approaches. Generally, the structural parameter identification includes four steps [9]: (1) modeling: to establish a mathematical model describing the geometrical characteristics and kinematics of the robot; (2) measurement: to measure the coordinates of the end effector in the real world coordinate system; (3) identification: to identify the structural parameters of the obtained model by means of mathematical calculation of the data set; (4) compensation: to modify the parameters in the control system according to the identification results. These four steps are also applicable for the AACMM, and an improved identification method will be studied in this paper.

For the parameter identification of AACMM, Kovač and Frank [10] developed a new high precision device for the AACMM testing and calibration with the laser interferometer measurements along a line gauge beam. Santolaria et al. [11, 12] reported a method to calibrate an AACMM based on the Denavit-Hartenberg (DH) kinematic model parameters. These parameters are optimized by measuring a calibrated ball bar gauge located at different orientations and positions in the AACMM working space. Hamana et al. [13] presented a method, where the kinematic parameters of AACMM were calibrated using spherical center coordinates as the artifact. However, only part of the measuring space can be calibrated with the above methods. Thus, the AACMM cannot be calibrated by directly using these available results. In particular, the coupling relationships between the structural parameters and their effect on the measuring uncertainty of the AACMM were not considered in the above researches. Therefore, the robustness and efficiency of the identifications were affected due to those invalid calculations [14] of the redundant couplings.

In this paper, we propose an improved modeling and parameter identification method for AACMM robotic system. First, the kinematic model and structural identification matrix were established based on the DH method, and the coupling relationship between the structural parameters was obtained through further analysis of the structural identification matrix. Then the identification model of the AACMM was constructed, and a parameter estimation approach developed based on the LS method is proposed to identify the structural parameters. Practically collected data of the joint angles and coordinates of the probe are used to validate the model and identification approach. The redundancy embedded in the parameter matrix is further analyzed and eliminated to address the coupling effects and identifiability. Finally, practical experiments are conducted to verify the efficiency of the proposed identification method.

The advantages and the distinctive features of this proposed identification method in comparison to some other identification methods for AACMM (e.g., [12, 15, 16]) are as follows:(1)We do not need precise initial parameters, and even we do not need initial parameters (we can assign the initial parameters arbitrarily as long as they are not too exaggerated) in the identification. The identified values of the structural parameters can be solved through (18). However, in some available results, for example, [12, 15, 17], the initial identification parameters should be appropriately selected to achieve good identification results because an iteration calculation method is adopted.(2)In this paper, we conducted the coupling analysis such that those linearly dependant parameters are detected and removed from the parameters to be identified. Consequently, the calculation costs and the identification efficacy can be significantly improved. In fact, in our case study, only one time iteration calculation can provide fairly good results.(3)The time consumed by the proposed identification calculation is relatively shorter than the widely used iteration identification methods such as PSO, GA, and LS; that is, we can get the results just after one time iteration calculation.

The paper is organized as follows. Section 2 presents the kinematic modeling and validation; the parameter identification and the analysis are introduced in Section 3; and experimental results are given in Section 4. Section 5 provides conclusions.

#### 2. Kinematic Modeling and Verification

##### 2.1. Kinematic Modeling

As shown in Figure 1, the structure of the AACMM is similar to an articulated robot. Therefore, the AACMM model can be established by using available modeling methods developed for robotics. For the modeling of robotic kinematics, the most influential method is the Denavit-Hartenberg model (DH model) which has been widely used due to its clear physical meaning [18]. A homogeneous transform matrix is used to represent the spatial relations of adjacent joints coordinate systems [19]. Because all the adjacent joints of the AACMM are perpendicular [20], there is no nominally parallel problem, and we can use the DH method to establish the kinematic model.