Vibration and impact of launching, inner and outer pressure difference, and thermal deformation of the space capsule will change the transformation between the pose measurement system and the space robot base. It will be complicated, even hard, to measure and calculate this transformation accurately. Therefore, an error modeling method considering both the distance error and the rotation error of the end-effector is proposed for self-calibration of the space robot on orbit, in order to avoid the drawback of frame transformation. Moreover, according to linear correlation of the columns of the identification matrix, unrecognizable parameters in the distance and rotation error model are removed to eliminate singularity in robot kinematic calibration. Finally simulation tests on a 7-DOF space robot are conducted to verify the effectiveness of the proposed method.

Space robots can assist astronauts to reach and expand their maintenance work areas, improving operational efficiency and safety [

The actual pose of the space robot end-effector can hardly be measured by an external measuring device due to the extreme orbital environment, so the internal sensing system mounted on its end-effector is adopted for measurement during self-calibration. Many researchers have devoted efforts to kinematic self-calibration of robot manipulators. Angulo and Torras [

Works on self-calibration above adopted the absolute pose/position error model for kinematic calibration. They have to describe the end-effector pose errors under the robot base frame, making it inevitable to identify the transformation matrix between the measurement system frame and the robot base frame before calibration. However, this transformation matrix is very complicated to measure and calculate accurately, even hardly possible to obtain in unmanned environments such as on orbit [

In this paper, we propose an error modeling method considering both the distance and the rotation error of the end-effector of the space robot. The remainder of this paper is organized as follows. In Section

Different from kinematic calibration of robot manipulators on the ground, the space robot has large structural size and extreme working environment, making it impossible to measure the end pose of the space robot using external measuring equipment, so its own hand-eye vision system [

Kinematic self-calibration system. The space robot and the checkerboard calibration plate are mounted on the outside of the space capsule, and a hand-eye camera is adopted to measure the pose of the checkerboard.

As illustrated in Figure

Skeleton of the coordinate frames for self-calibration. The coordinate frames of the robot base, the checkboard, and all the joints are defined, while the related lengths are marked.

However, vibration and impact of launching, inner and outer pressure difference, and thermal deformation of the space capsule will change the pose of the checkerboard with respect to the robot base frame, making it complicated to measure and calculate the transformation matrix

As the basis of error modeling, kinematic modeling is aimed at describing the relation between any two adjoining link coordinate frames with as few parameters as possible. But inappropriate kinematic parameters might not meet three fundamental principles, namely, completeness, model continuity, and minimality [

Without loss of generality, we assume that the number of kinematic parameters used to describe the relation between any two adjoining link coordinate frames is

Kinematic model: the kinematic model of the space robot relates the configuration

The robot configurations are not exactly known due to the existence of sensor noise. Similarly, defects in manufacturing and assembly result in kinematic errors. The difference between the nominal value

Absolute pose error model [

In Figure

Distance error model [

Schematic of the relative pose error model. The robot moves from the configuration

General rotation transformation: suppose that

Equivalent angle and axis of rotation [

It should be noted that the equivalent angle and axis of rotation have the following three important properties.

For a certain rotation matrix

When

Differential rotation by equivalent angle and axis of rotation: any differential rotation transformation

Differential rotation by 3-dimensional differential rotation angles [

If the equivalent angle and axis of rotation of a certain differential rotation matrix

It can be obtained by Equations (

Considering that

Thus, Equation (

Differential rotation by the rotation matrix [

It should be noted that

The two kinds of differential rotation matrix satisfy

We can avoid identifying the transformation matrix between the measurement system frame and the robot base frame by using the distance error. This is true for equivalent angle of rotation to describe the variation of the end-effector orientation.

As shown in Figure

Suppose that

According to Proposition

Then, the differential rotation matrix between

Equation (

Suppose that the equivalent angle and axis of rotation corresponding to

According to Equation (

Rotation error model: as the robot configuration changes from

Substituting Equation (

Similar to the derivation of distance error, we make the rotational axis

Derivation of the rotation error model. The rotational axis

According to Proposition

According to the third property of the equivalent angle and axis of rotation,

Finally, the rotation error model in Equation (

In summary, the distance and rotation error model is obtained by Equations (

So we can obtain the identification matrix of the distance and rotation error model as

When there are redundant parameters in the error model, the identification matrix is rank defect, and measurement noises will affect the accuracy and robustness of parameter identification seriously, inferring the necessity of removal of redundant parameters. Redundant parameters in the error model will be discussed in the next section.

A modified DH method, termed as MDH method [

The coordinate frame is established by the MDH method as shown in Figure

Coordinate frame established by the MDH method. The angle between the

Nominal length of the rods in the kinematic self-calibration system of the space robot.

Rod | |||||
---|---|---|---|---|---|

Length (m) | 2.3 | 0.5 | 5 | 6.5 | 2.55 |

According to the modeling rules of the MDH method, we obtain the nominal kinematic parameters of the 7-DOF space robot as shown in Table

Nominal kinematic parameters of the 7-DOF space robot.

Joint | |||||
---|---|---|---|---|---|

1 | 0 | 0 | 2.3 | 0 | 0 |

2 | 0 | −90 | 0.5 | 0 | 0 |

3 | 0 | −90 | 0.5 | 0 | 0 |

4 | 5 | 0 | 0.5 | 0 | 0 |

5 | 5 | 0 | 0.5 | 0 | 0 |

6 | 0 | 90 | 0.5 | 0 | 0 |

7 | 0 | 90 | 0.5 | 0 | 0 |

Since the relation between the camera frame and the end-effector has been calibrated on the ground and it is assumed to be unchanged on orbit, the coordinate frame of the end-effector coincides with that of the last joint. Therefore, the end-effector pose of a robot manipulator with n DOFs can be calculated as

It should be noted that nonsingularity of an error model indicates its identification matrix is column full rank. That is to say, any column of the identification matrix is linearly independent. The kinematic parameters of the robot can be sorted into three groups by its corresponding column in the identification matrix.

Independent parameters, whose corresponding column is linearly independent with each other

Relative parameters, whose corresponding column is linearly dependent with another one

Ineffective parameters, whose corresponding column is the zero vector, indicating that they have no effect on the pose error of the robot end-effector

We differentiate Equation (

Local transfer matrix of the MDH method [

Global transfer matrix

Schematic of error transferring. Kinematic errors are transferred to the end-effector pose by the local transfer matrix

Parameter independence in the error model can be determined just by analysing the redundant parameters of adjacent joints [

The full column rank of

According to Equations (

Since

Suppose

Considering that

Next, three typical singularities are discussed to analyze the parameter independence between

The two adjacent joints are parallel but not collinear, indicating that

According to Equations (

The two adjacent joints are parallel and collinear, indicating that

According to Equations (

The two adjacent joints are orthogonal with

According to Equations (

Based on the analysis above, the identifiability of kinematic parameters in the absolute pose error model of the 7-DOF space robot can be obtained and shown in Table

Identifiability of kinematic parameters in the absolute pose error model of the 7-DOF space robot.

Identifiability | Kinematic parameters |
---|---|

Independent parameters | |

Relative parameters | |

Ineffective parameters | None |

Only the relative parameters

Equation (

(1) The distance-related identification matrix: we can obtain the distance-related identification matrix with respect to

The position of the end-effector can be written as

For the first joint,

Substitute Equation (

We assume that the space robot moves from the configuration

Obviously, the kinematic parameters

For the last joint,

For the

The rotation-related identification matrix

We can obtain the rotation-related identification matrix by Equations (

For the first joint,

By Equations (

For the last joint,

By Equations (

By Equation (

For the

(2) The distance and rotation error model: in summary, the identifiability of kinematic parameters in the distance and rotation error model of the 7-DOF space robot can be obtained and shown in Table

Identifiability of kinematic parameters in the distance and rotation error model of the 7-DOF space robot.

Identifiability | Kinematic parameters |
---|---|

Independent parameters | |

Relative parameters | |

Ineffective parameters |

Since

The process of calibration simulation is shown in Figure

Flowchart of calibration simulation. Encoder noise and pose measurement noise are added to achieve more reliable simulation results.

Whether it is the absolute pose error model or the proposed distance and rotation error model, the least squares method is a powerful tool to identify the kinematic errors against sensor noises. The specific application of least squares method is as

For the distance and rotation error model,

The end-effector poses of the space robot are measured by a hand-eye camera, so the camera has to point to the target checkerboard. Besides, the end-effector of the space robot under the selected measurement configurations has to be close to the checkerboard in order to ensure measurement accuracy.

For the sake of convenience and economy, we select one configuration for comparison as

The corresponding end-effector frames of the measurement configurations. A total of 52 configurations are selected as measurement configurations, one of which is defined as

In practical applications, the actual end-effector pose of a robot can be obtained by the hand-eye camera, while the actual robot configurations can be measured by the encoders. However, in simulation applications, measuring noises and encoder noises are added according to their respective distributions in order to imitate the influence of sensor errors. These measuring noises meet normal distribution as shown in Table

Normal distribution of measuring noises.

Measuring noise | End-effector position (m) | End-effector orientation (°) | Robot configuration (°) |
---|---|---|---|

Mean | 0 | 0 | 0 |

Standard deviation | 0.001 | 0.01 | 0.005 |

According to Section

Kinematic parameter errors of the 7-DOF space robot.

Joint | |||||
---|---|---|---|---|---|

1 | 0 | 0 | 0 | 0 | 0 |

2 | 0.01 | 0.05 | 0.01 | 0.05 | 0.05 |

3 | 0.01 | 0.05 | 0.01 | 0.05 | 0.05 |

4 | 0.01 | 0.05 | 0.01 | 0.05 | 0.05 |

5 | 0.01 | 0.05 | 0.01 | 0.05 | 0.05 |

6 | 0.01 | 0.05 | 0.01 | 0.05 | 0.05 |

7 | 0.01 | 0.05 | 0.01 | 0.05 | 0.05 |

The purpose of robot calibration is to obtain the accurate kinematic parameters representing the robot structure and the exact estimation of the end-effector pose. The least squares method is adopted to obtain the calibrated kinematic parameters

The calibration residuals by the distance error model, by the distance and rotation error model, and by the absolute pose error model are shown in Tables

Calibration residuals by the distance error model.

Joint | |||||
---|---|---|---|---|---|

1 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

2 | 0.00054 | 0.00445 | 0.00094 | 0.00038 | −0.04111 |

3 | 0.00063 | 0.01097 | 0.00015 | 0.03997 | 0.11269 |

4 | 0.00355 | 0.00632 | −0.00861 | 0.01106 | 0.15176 |

5 | 0.00442 | −0.07475 | 0.02887 | 0.05620 | −0.04298 |

6 | 0.00012 | −0.20839 | −0.00284 | −0.01352 | −0.02700 |

7 | −0.00194 | −0.29798 | 0.00152 | −0.00042 | −0.03749 |

Calibration residuals by the distance and rotation error model.

Joint | |||||
---|---|---|---|---|---|

1 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

2 | 0.00056 | 0.00050 | 0.00051 | 0.00176 | −0.04111 |

3 | 0.00041 | −0.00313 | 0.00015 | 0.03701 | 0.00586 |

4 | 0.00081 | 0.00485 | −0.00861 | 0.01164 | 0.01368 |

5 | 0.00053 | −0.00637 | 0.01005 | 0.00310 | −0.04298 |

6 | 0.00048 | −0.01963 | 0.00047 | −0.05903 | −0.02700 |

7 | −0.00159 | −0.01215 | 0.00045 | −0.00042 | −0.03749 |

Calibration residuals by the absolute pose error model.

Joint | |||||
---|---|---|---|---|---|

1 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

2 | 0.00017 | 0.00140 | −0.00024 | 0.00247 | −0.04111 |

3 | 0.00016 | 0.00262 | 0.00015 | 0.04160 | 0.00187 |

4 | −0.00005 | 0.00014 | −0.00861 | −0.00283 | 0.00287 |

5 | −0.00009 | 0.00569 | 0.00890 | 0.00251 | −0.04298 |

6 | 0.00005 | −0.00458 | −0.00003 | −0.04436 | −0.02700 |

7 | −0.00006 | 0.00457 | −0.00003 | −0.02722 | 0.00050 |

Some of the calibration residuals are relatively big in Tables

For the distance and rotation error model,

For the absolute pose error model,

Equations (

Moreover, we come to a conclusion that the distance and rotation error model does better than the distance error model in identification accuracy of the kinematic parameters, but a little weaker than the absolute pose error model.

Finally, 500 robot configurations are selected randomly to serve as the validation group. The end-effector position estimate errors of validation configurations are calculated, respectively, with different error models, and maximum and average of these estimate errors are analysed to compare calibration performance.

Figure

The end-effector position errors of 500 confirm configurations.

The histograms of the end-effector position errors of 500 confirm configurations.

The statistical characteristics of the end-effector position errors of 500 confirm configurations.

Location | Max (mm) | Mean (mm) |
---|---|---|

Before calibration | 38 | 19 |

By distance error | 21 | 11 |

By distance and rotation error | 4.8 | 2.5 |

By absolute error | 1.1 | 0.5 |

Whether it is the identification accuracy of the kinematic parameters shown in Tables

Proposed in this paper is an error model involving both the distance and the rotation error of the space robot end-effector. The error model can avoid identifying the transformation matrix between the measurement system frame and the robot base frame, suitable for self-calibration of the space robot. Besides, identifiable parameters in the distance and rotation error model are confirmed to eliminate singularity in robot kinematic calibration. Finally, we conduct the calibration simulation and compare differences in calibration performance between these models. Statistical results indicate that the proposed error model does better in the accuracy of the robot end-effector position estimate and of the kinematic parameter identification than the only distance error model. However, it still matters that observability of the distance and rotation error model be studied as an indicator of measurement configuration optimization, which will significantly reduce the number of configurations required for calibration. Besides, information fusion provides a powerful tool to deal with uncertainty and external disturbance of pose measurement and application of filtering algorithms in robot calibration is worthy of attention. From the operational point of view, the light conditions to carry on calibration processes should be also taken into consideration. In summary, there are still more future works and challenges to adopt the proposed method in practical application.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (61573066 and 61327806).