A Kinematic Calibration Method of a 3T1R 4-Degree-of-Freedom Symmetrical Parallel Manipulator
Abstract
:1. Introduction
2. Kinematics of a Symmetrical 3T1R Parallel Manipulator
2.1. Structure of the 3T1R Parallel Manipulator
2.2. kinematics Based on local POE formula
2.3. Branched Chain Kinematics Based on the Local POE Formula
2.4. Kinematics of the Parallel Manipulator Branched Chain i Based on the Local POE Formula
3. Establishing the Kinematic Error Model for this 3T1R Parallel Manipulator
3.1. Establishment of the Kinematic Error Model from a Single Branched Chain
3.2. Establishment of the Overall Kinematic Error Model
4. Method to Reduce the Number of Sensors Used in Passive Joints
5. A Recursive Least Squares Method to Identify the Parameters in the Kinematic Error Model
6. Simulation Results
- 1.
- Use the numerical forward kinematics algorithm to obtain the joint displacements and joint angles of 20 different parallel manipulator poses;
- 2.
- Assign errors to kinematic parameters, such as , , and , as shown in Table 1.
- 3.
- Simulate the actual initial pose using ;
- 4.
- The actual joint twist is computed as ;
- 5.
- The actual joint variable is computed as ;
- 6.
- The recursive calibration algorithm is used to identify the kinematic errors of the parallel manipulator.
7. Pre-Processing Compensation
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
3T1R parallel manipulator | The parallel manipulator can achieve three degrees of freedom of translation along the X, Y, and Z axes and one degree of freedom of rotation around the Z axis |
POE formula | The product of exponentials formula |
D-H convention | The Denavit-Hartenberg convention |
Nomenclature
n | The number of passive joints |
i | Represents the parallel manipulator , branched chains |
Represents the parallel manipulator , branched chains | |
j | The number of joints on each branched chain |
Forward kinematics of these parallel manipulator branched chains i | |
The initial pose of the coordinate system relative to the coordinate system | |
After calibration, the initial pose of the coordinate system relative to the coordinate system | |
The actual initial pose of the coordinate system relative to the coordinate system | |
Standard representation of joint variable; it represent the joint angle or joint displacement | |
Standard representation of the nominal joint variable | |
Standard representation of the joint variable after calibration | |
Base coordinate system of the parallel manipulator | |
, | Base coordinate system of these parallel manipulator branched chains i and |
, | The translational motion coordinate system of these parallel manipulator branched chains i and |
, | The rotational motion coordinate system where it is connected to the modules of these parallel manipulator branched chains i and |
The coordinate system of link rotates around the D point of these parallel manipulator branched chains i and | |
The parallel manipulator moving platform rotational motion coordinate system around the point | |
Midpoint coordinate system of the end effector | |
Forward kinematics after calibration of the parallel manipulator | |
The nominal pose of the end effector | |
The actual pose of the end effector | |
The command pose of the end effector during error compensation | |
The end pose after compensation | |
The adjoint transformation of , also written as , | |
The representation of the end error on these parallel manipulator branched chains i in the base coordinate system | |
The error of t in coordinate system | |
Error Jacobian matrix of these parallel manipulator branched chains i | |
J | Error Jacobian matrix of the entire parallel manipulator |
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B-11 | 0.02 | ||
11-12 | 0.02 | ||
12-13 | 0.02 | ||
13-14 | 0.02 | ||
14-P | \ | \ | |
B-21 | 0.02 | ||
21-22 | 0.02 | ||
22-23 | 0.02 | ||
23-24 | 0.02 | ||
24-P | \ | \ |
Kinematic Errors | ||
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Zhang, F.; Chen, S.; He, Y.; Ye, G.; Zhang, C.; Yang, G. A Kinematic Calibration Method of a 3T1R 4-Degree-of-Freedom Symmetrical Parallel Manipulator. Symmetry 2020, 12, 357. https://doi.org/10.3390/sym12030357
Zhang F, Chen S, He Y, Ye G, Zhang C, Yang G. A Kinematic Calibration Method of a 3T1R 4-Degree-of-Freedom Symmetrical Parallel Manipulator. Symmetry. 2020; 12(3):357. https://doi.org/10.3390/sym12030357
Chicago/Turabian StyleZhang, Fengxuan, Silu Chen, Yongyi He, Guoyun Ye, Chi Zhang, and Guilin Yang. 2020. "A Kinematic Calibration Method of a 3T1R 4-Degree-of-Freedom Symmetrical Parallel Manipulator" Symmetry 12, no. 3: 357. https://doi.org/10.3390/sym12030357
APA StyleZhang, F., Chen, S., He, Y., Ye, G., Zhang, C., & Yang, G. (2020). A Kinematic Calibration Method of a 3T1R 4-Degree-of-Freedom Symmetrical Parallel Manipulator. Symmetry, 12(3), 357. https://doi.org/10.3390/sym12030357