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.

The coordinate measuring machine (CMM) is a universal measuring instrument which can transform various geometric measurements into coordinate measurements [

However, the measuring accuracy of the AACMM is much lower than that of the orthogonal CMM [

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 [

For the parameter identification of AACMM, Kovač and Frank [

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., [

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 (

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.

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

As shown in Figure

The structure of the AACMM.

In this paper, a 6-DOF AACMM is studied. The schematic structure of this system is shown in Figure

According to the DH method, there are four groups of structural parameters in the AACMM, for example, linkage length

The coordinates of the AACMM were established followed by the DH method, as shown in Figure

The estimated structural parameters of the AACMM.

Linkage number | | | | |
---|---|---|---|---|

1 | 0 | 376 | 0 | −90 |

2 | 62 | 0 | 0 | −90 |

3 | 0 | 751 | 0 | −90 |

4 | 62 | 0 | 0 | −90 |

5 | 0 | 500 | 0 | −90 |

6 | 0 | 15 | 0 | 90 |

Coordinate systems of the AACMM.

According to the principle of homogeneous transformation, the transform from the coordinate system

The probe coordinates in the reference coordinate system

There is one group of parameters which are the variables in (

To validate the proposed kinematic model (

The interface of the acquisition software of the AACMM.

By comparing the coordinates of the probe calculated by the software and the reference coordinates (which can be treated as the true values of the coordinates of the probe), we found that the calculated coordinates of the probe are very close to the reference coordinates. The differences between the two group coordinates are always less than 1.5 mm, which means that the movement uncertainty of the AACMM is about 1.5 mm. In this sense, the kinematic model (

It is known that the accuracy of the AACMM is affected greatly by the manufacture, assembly, and component selection [

From (

To show the effect of uncertainties, we calculate the differential of (

Thus (

It is shown that (

We denote

Equation (

From (

It is clear that the uncertain structural parameters

In (

We denote

Since

According to the analysis presented in last section, we know that there are two structural parameters which are related to others and do not need to be identified in terms of (

The overall identification algorithm has been carried out to solve (

The flowchart of the identification procedure.

In the literature, there are many different identification methods, for example, PSO and GA. However, these algorithms need much more computational costs, which may limit their applicability for AACMM. The time consumed by the proposed identification calculation is relatively shorter than the widely used iteration identification methods such as PSO and GA; that is, we can get the results just after one time iteration calculation.

The step-by-step implementation procedure can be given as follows:

initialize the structural parameters of

calculate

calculate

calculate the least squares solution of

set

To show the effectiveness of the suggested modeling and identification algorithms, an experimental study was carried out based on an AACMM test rig, which is shown in Figure

The articulated arm coordinate measuring machine.

In the data acquisition procedure of the AACMM, many poses are used to obtain enough information about the AACMM. The information is very important for the following identification. And acquiring poses is commonly used in the calibration of AACMM and robots [

The movement uncertainties of AACMM before compensation.

Directions | Maximum (max) (mm) | Standard deviation (SD) (mm) | Average (ave) (mm) | Absolute average (AA) (mm) |
---|---|---|---|---|

| 1.485 | 0.639 | −0.071 | 0.524 |

| 1.366 | 0.478 | 0.439 | 0.545 |

| −0.978 | 0.358 | −0.308 | 0.421 |

The movement uncertainties in the directions of

The significant uncertainties shown in Table

From the experimental studies, it is found that the movement uncertainty becomes relatively stable when the number of identification poses is greater than 30, and it is almost invariant when the number of identification poses is larger than 50. Therefore, the joint angles and reference coordinates of 50 poses were acquired to identify the structural parameters of the AACMM, and the identification algorithm presented in Section

The identified structural parameters of AACMM.

Linkage number | | | | |
---|---|---|---|---|

1 | −0.121 | 376.500 | 0.087 | −89.986 |

2 | 61.942 | 0.016 | 0.066 | −90.026 |

3 | 0.0324 | 750.658 | 0.001 | −89.997 |

4 | 62.225 | −0.795 | −0.021 | −89.931 |

5 | −0.036 | 500.287 | 0.071 | −89.956 |

6 | 0.057 | 15 | 0 | 90 |

Compared to Table

The movement uncertainty of AACMM after identification.

Directions | Maximum (max) (mm) | Standard deviation (SD) (mm) | Average (ave) (mm) | Absolute average (AA) (mm) |
---|---|---|---|---|

| −0.112 | 0.044 | −0.003 | 0.036 |

| 0.116 | 0.046 | 0.013 | 0.038 |

| 0.117 | 0.053 | 0.007 | 0.044 |

The movement uncertainties in the directions of

The comparison of the movement uncertainties in the directions of

In the direction of

In the direction of

In the direction of

A constructive parameter identification approach for articulated arm coordinate measuring machines has been presented in this paper. A structural kinematic model is established based on the DH method and verified through experiments. Based on the difference between the coordinates of the probe calculated by the kinematic model and the reference coordinates, a mathematical parameter identification model is further developed to decrease the uncertainties in the DH model. The analysis of the Jacobian matrix in the identification model shows that there are two structural parameters which are related to others in our case studies. Therefore, these structural parameters are removed from the identification model. Then the structural parameter identification with the aim to get the least-square solution of the identification model can be carried out by using the obtained poses and the reference coordinates of the AACMM. To facilitate practical implementations, experimental studies have been conducted. These experimental results have revealed the effectiveness of the proposed structural parameter identification for AACMM.

The proposed modeling and identification method can be extended to the calibration of serial robots, where the identification method is required due to their dynamic operation environments. This will be further studied in our future work.

The authors declare that they have no competing interests.

This work was supported by the National Natural Science Foundation of China (Grant no. 51465027).