In order to better perform rehabilitation training on the ankle joint complex in the direction of dorsiflexion/plantarflexion and inversion/eversion, especially when performing the isokinetic muscle strength exercise, we need to calibrate the kinematic model to improve its control precision. The ankle rehabilitation robot we develop is a parallel mechanism, with its movements in the two directions driven by two linear motors. Inverse solution of positions is deduced and the output lengths of the two UPS kinematic branches are calibrated in the directions of dorsiflexion, plantarflexion, inversion, and eversion, respectively. Motion of each branch in different directions is fitted in high-order form according to experimental data. Variances, standard deviation, and goodness of fit are taken into consideration when choosing the best fitting curve, which ensures that each calibration can match the most appropriate fitting curve. Experiments are conducted to verify the effectiveness of the kinematic calibration after finishing the calibration, and the errors before and after calibration of the two kinematic chains in different directions are compared, respectively, which shows that the accuracy after calibration has been significantly improved.

Rehabilitation training is an effective way to help patients restore their ankle joint complex’s (AJC) motor abilities for patients with ankle injuries. The movement of AJC has three Degrees of Freedom (DOFs), Dorsiflexion/Plantarflexion (DO/PL), Inversion/Eversion (IN/EV), and Adduction/Abduction (AD/AB) [

Structure of the ankle complex with its movement.

To augment conventional physical therapy, many robotic ankle rehabilitation devices have been developed to provide repetitive, task-specific, interactive treatment of the impaired limb and monitor its motor recovery [

Besides that, isometric and isotonic exercises have also been developed for muscle strength exercises of AJC [

To realize the isokinetic muscle strength exercise in DO/PL and IN/EV direction using our developed 2-UPS/RRR parallel ankle rehabilitation robot (PARR), we need to have the precise kinematic model of the two UPS kinematic branches. U, P, S, and R stand for universal, prismatic, spherical, and revolute joint, respectively, and the underlined letter represents the actuated joint. Considering machining and assembly errors, the designed robot needs to be calibrated firstly to make the positional inverse solution obtained by theoretical analysis more accurately.

The conventional approach toward kinematic calibration generally begins with formulating the problem in terms of constraint equations that are derived from the kinematic model of the robot. In the data acquisition phase, the pose of the moving platform and the corresponding actuated joint coordinates are obtained. Finally, a suitable optimization method utilizes the obtained data to determine the actual geometry [

In this work, we focus on the kinematic calibration of our 2-UPS/RRR PARR according to field experiments and high-order data fitting, with variances, standard deviation, and goodness of fit as the criterion of fitting evaluation. The rest of the paper is organized as follows. Section

The mechanical structure of the developed 2-UPS/RRR PARR is shown in Figure

Mechanical structure of the developed 2-UPS/RRR PARR.

ROM of the AJC [

Motion direction | ROM ( | MAW ( |
---|---|---|

Dorsiflexion | 20.3 | 30.0 |

Plantarflexion | 37.6 | 45.0 |

Inversion | 14.5 | 22.0 |

Eversion | 10.0 | 22.0 |

Abduction | 15.4 | 36.0 |

Adduction | 22.0 | 36.0 |

Details of mechanical design of the developed PARR are shown in our previous work [

The kinematic coordinate system of the parallel 2-UPS/RRR rehabilitation robot is shown as in Figure

Coordinate system of the parallel mechanism and its UPS branch. (a) Schematic diagram of the mechanism. (b) UPS branch.

The coordinate system of the UPS branch is shown in Figure

The inverse solution of positions is to deduce the inputs of three branches based on the output angles of the moving platform around the fixed coordinate system, that is, the above-mentioned

By combining (

According to the parameters of the developed 2-UPS/RRR PARR, the coordinate values of points

For our PARR, the angle of AD/AB

From (

During the movement of DO/PL

The length changes of

During the movement of IN/EV

The length changes of

We can see that

The error model of the kinematic calibration is as in (

During the calibration, firstly, we control the output lengths of

The result of calibration in DO/PL is as in Figure

Error of

The second-order fitting, third-order fitting, fourth-order fitting, fifth-order fitting, and sixth-order fitting are, respectively, performed on the errors between actual and theoretical values of

Error fitting of

Fitting results of

2nd-order | 3rd-order | 4th-order | 5th-order | 6th-order | |
---|---|---|---|---|---|

Var | 0.0317 | 0.0310 | 0.0305 | 0.0190 | 0.0104 |

Std | 0.1779 | 0.1776 | 0.1746 | 0.1379 | 0.1019 |

0.4062 | 0.4186 | 0.4280 | 0.6435 | 0.8051 |

After comprehensive consideration of variances, standard deviations, goodness of fit, and computation complexity according to the fitting result, we define the criterion of choosing the optimal fitting curve as in the following equation:

According to Figure

The second-order fitting, third-order fitting, fourth-order fitting, fifth-order fitting, and sixth-order fitting are, respectively, performed on the errors between actual values and theoretical values of

Error fitting of

Fitting results of

2nd-order | 3rd-order | 4th-order | 5th-order | 6th-order | |
---|---|---|---|---|---|

Var | 0.037a | 0.0351 | 0.035 | 0.0329 | 0.0326 |

Std | 0.1923 | 0.1874 | 0.1872 | 0.1813 | 0.1805 |

0.6600 | 0.6772 | 0.6778 | 0.6979 | 0.7005 |

According to Figure

The result of

Error of

The second-order fitting, third-order fitting, fourth-order fitting, fifth-order fitting, and sixth-order fitting are, respectively, performed on the errors between actual values and theoretical values of

Error fitting of

Fitting results

2nd-order | 3rd-order | 4th-order | 5th-order | 6th-order | |
---|---|---|---|---|---|

Var | 0.0484 | 0.036 | 0.035 | 0.0349 | 0.0347 |

Std | 0.2201 | 0.1898 | 0.1870 | 0.1869 | 0.1862 |

0.8707 | 0.9038 | 0.9067 | 0.9067 | 0.9075 |

According to Figure

The second-order fitting, third-order fitting, fourth-order fitting, fifth-order fitting, and sixth-order fitting are, respectively, performed on the errors between actual values and theoretical values of

Error fitting of

Fitting results of

2nd-order | 3rd-order | 4th-order | 5th-order | 6th-order | |
---|---|---|---|---|---|

Var | 0.0351 | 0.0349 | 0.0336 | 0.0336 | 0.0317 |

Std | 0.1875 | 0.1869 | 0.1834 | 0.1834 | 0.1781 |

0.6881 | 0.6901 | 0.7015 | 0.7016 | 0.7186 |

According to Figure

The result of

The error of

The second-order fitting, third-order fitting, fourth-order fitting, fifth-order fitting, and sixth-order fitting are, respectively, performed on the errors between actual values and theoretical values of

Error fitting of

Fitting results of

2nd-order | 3rd-order | 4th-order | 5th-order | 6th-order | |
---|---|---|---|---|---|

Var | 0.0483 | 0.036 | 0.036 | 0.035 | 0.0347 |

Std | 0.2197 | 0.1898 | 0.1870 | 0.187 | 0.1862 |

0.5239 | 0.6446 | 0.6552 | 0.6552 | 0.6581 |

According to Figure

The second-order fitting, third-order fitting, fourth-order fitting, fifth-order fitting, and sixth-order fitting are, respectively, performed on the errors between actual values and theoretical values of

Error fitting of

Fitting results of

2nd-order | 3rd-order | 4th-order | 5th-order | 6th-order | |
---|---|---|---|---|---|

Var | 0.0352 | 0.0349 | 0.0336 | 0.0336 | 0.0317 |

Std | 0.1875 | 0.1869 | 0.1834 | 0.1834 | 0.1781 |

0.9098 | 0.9104 | 0.9137 | 0.9138 | 0.9187 |

According to Figure

Through the above analysis, we can obtain the calibration results of the controlled lengths of

In order to better verify the calibration effect of the PARR, we conducted relevant experiments in the directions of DO/PL and IN/EV, respectively. The method is to control the input of

Firstly, we control the lengths of

Prototype of the 2-UPS/RRR PARR.

The experiment results of

Experiment results of

1.875 | 0.748 | 1.423 | 1.974 | 0.452 | −0.099 |

3.75 | 1.43 | 2.719 | 3.582 | 1.031 | 0.168 |

5.625 | 2.453 | 4.662 | 5.66 | 0.963 | −0.035 |

7.5 | 3.427 | 6.51 | 7.471 | 0.99 | 0.029 |

9.375 | 4.498 | 8.54 | 9.44 | 0.835 | −0.065 |

11.25 | 5.521 | 10.474 | 11.378 | 0.776 | −0.128 |

13.125 | 6.495 | 12.312 | 13.293 | 0.813 | −0.168 |

15 | 7.176 | 13.596 | 14.663 | 1.404 | 0.337 |

16.875 | 8.248 | 15.61 | 16.826 | 1.265 | 0.049 |

18.75 | 9.271 | 17.526 | 18.855 | 1.224 | −0.105 |

20.625 | 10.293 | 19.436 | 20.803 | 1.189 | −0.178 |

22.5 | 11.365 | 21.43 | 22.742 | 1.07 | −0.242 |

24.375 | 12.387 | 23.326 | 24.532 | 1.049 | −0.157 |

26.25 | 13.41 | 25.214 | 26.371 | 1.036 | −0.121 |

28.125 | 14.481 | 27.183 | 28.56 | 0.942 | −0.435 |

In order to show the error changes before and after calibration more intuitively during the DO movement, we compare the errors before and after calibration in the same figure, as in Figure

Error comparison of

Similarly, we can obtain the experiment results of

Error comparison of

Error comparison of

Error comparison of

Based on the experiment results, the average errors before and after calibration are shown in Figure

Average error comparison of

From (

Kinematic calibration of the two UPS kinematic branches of the developed 2-UPS/RRR PARR is conducted and described in this paper in detail in order to improve the control precision for the rehabilitation training like isokinetic muscle strength exercise. Motion of each branch in different directions is fitted in high-order form according to experimental data. Variance, standard deviation, and goodness of fit are taken into consideration when choosing the best fitting curve. Experiments have been conducted, which show that the accuracy after calibration has been significantly improved and verify the effectiveness of the kinematic calibration.

In the future, we will study the repeatability performance of the kinematic calibration, as the repeatability of the kinematic calibration is an important aspect for the validation of the procedure. In addition, we will also focus on the kinematic calibration in the direction of AD/AB, compliant and interactive control strategies, as well as multimode rehabilitation training method.

All the data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to acknowledge the support of the National Natural Science Foundation of China (nos. 61903011 and 51675008), the Natural Science Foundation of Beijing Education Committee (no. KM202010005021), Beijing Natural Science Foundation (no. 3204036), and the Beijing Postdoctoral Research Foundation (no. Q6001002201901).