# Configuration Design and Kinematic Performance Analysis of a Novel Spatial 8R Hip Joint Rehabilitation Mechanism

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Configuration Design of the Spatial 8R Mechanism Configuration

_{11}and R

_{18}are parallel to the Y axis of the static coordinate system, the axes of R

_{12}and R

_{16}are parallel to the X axis of the static coordinate system, and the axes positions of R

_{13}, R

_{14}and R

_{15}, which are not described in detail, are clear. The mechanism is symmetrical from left to right, and the arrangement of the revolute pairs on one side is described as follows. The static coordinate system O

_{b}-xyz and the moving coordinate system O

_{m}-uvw are established on the static and moving platforms of the mechanism, as shown in the figure. The origin O

_{b}of the static coordinate system is at the midpoint of revolute pairs R

_{11}and R

_{21}. The x-axis direction is the same as the O

_{b}R

_{21}direction, while the y-axis direction is perpendicular to R

_{11}R

_{21}inward, and the z-axis direction is perpendicular upward. The origin om of the moving coordinate system is at the rotation center of the revolute pair R

_{27}. The axis of the vertical revolute pair R

_{27}faces up in the direction of the u axis, the axis of the vertical revolute pair R

_{27}is inward in the direction of the v axis, and the direction of the w axis is left along the axis of the revolute pair R

_{27}. The branch chain coordinate system R

_{1i}-x

_{1i}y

_{1i}z

_{1i}(i = 1, 8) is established at the rotation center of the revolute pair R

_{1i}. x-axis direction of the branch chain coordinate system. R

_{11}-x

_{11}y

_{11}z

_{11}is oriented along the R

_{11}axis, the y

_{11}axis is oriented to the left along the connecting direction of R

_{11}R

_{12}, and the z

_{11}axis faces vertically downward.

_{21}R

_{22}= 47 mm; R

_{22}R

_{23}= 45 mm; R

_{23}R

_{24}= 160 mm; R

_{24}R

_{25}= 120 mm; R

_{25}R

_{26}= 33 mm;

^{1}R

_{27}R

_{28}= 172 mm;

^{2}R

_{27}R

_{28}= 127 mm.

_{27}R

_{28}.

_{11}, 5R branch chain, is

_{11}can control the motion of the mechanism around the x axis. R

_{11}of the 6R branch chain is thus used as the drive, and the drive is installed at R

_{17}of the 2R branch chain, so the mechanism will carry out decoupled motion around the x axis and y axis.

## 3. Kinematic Performance Analysis of the Spatial 8R Mechanism

#### 3.1. Forward/Inverse Kinematics

_{2i}-x

_{2i}z

_{2i}(i = 1~8) was established at the rotation center of the revolute pair R

_{2i}. The direction of each branch chain coordinate system is shown in Figure 2, where the coordinate system at R

_{22}is the same as that at R

_{26}, and the coordinate system at R

_{25}and R

_{21}, and z axis of R

_{23}and R

_{24}, is the same as that at R

_{21}. The coordinate system R

_{27}-x

_{27}z

_{27}coincides with the moving coordinate system O

_{m}-uvw. The D-H parameters of the 6R and 2R branches are shown in Table 1 and Table 2.

_{i−}

_{1}is the rotation degree from z

_{i−}

_{1}to z

_{i}around the x

_{i−}

_{1}axis. a

_{i−}

_{1}is the distance from z

_{i−}

_{1}to z

_{i}along the x

_{i}axis. θ

_{i}is the rotation degree from x

_{i−}

_{1}to x

_{i}around the z

_{i}axis. d

_{i}is the distance from x

_{i−}

_{1}to x

_{i}along the z

_{i}axis. g, m, n, k, l

_{1}, l

_{2}and l

_{3}are constants, and θ

_{1}, θ

_{2}, θ

_{3}, θ

_{4}, θ

_{5}, θ

_{6}, θ

_{7}and θ

_{8}are variables.

_{i}stands for sin(θ

_{i}), c

_{i}stands for cos(θ

_{i}),${\mathrm{c}}_{ij}$ stands for $\mathrm{cos}\left({\theta}_{i}+{\theta}_{j}\right)$, and ${\mathrm{s}}_{ij}$ stands for $\mathrm{sin}\left({\theta}_{i}+{\theta}_{j}\right)$, the same as below.

_{26}-x

_{26}z

_{26}relative to the static coordinate system O

_{b}-xyz, which is the basic equation for kinematic analysis of 6R branch chains.

_{1}= θ

_{2}= θ

_{3}= 0°, θ

_{4}= 90°, θ

_{5}= 90° and θ

_{6}= 0°, and the calculation result is

_{7}= θ

_{8}= 0° into ${}_{2}{}^{0}\mathit{T}{}_{2}$, we can verify that it is correct.

_{i}stands for sin(θ

_{i}), c

_{i}stands for cos(θ

_{i}),

**T**is the pose matrix, (x, y, z) is the position coordinates of the origin O

_{m}of the moving coordinate system in the static coordinate system O

_{b}-XYZ, and α, β and γ are the rotation angles of the moving platform around the OZ axis, OY axis and OX axis in the static coordinate system, respectively.

_{21}can control the rotation of the mechanism around the x axis, and a drive is installed at R

_{21}, so the rotation angle of R

_{21}around the x axis is equal to that of the moving platform. Moreover, the moving platform is concentrically matched with the revolute pair R

_{27}, and its rotation angle around the y axis is equal to that of R

_{27}.

_{21}and θ

_{27}of R

_{21}and R

_{27}(corresponding to the angles θ

_{1}and θ

_{7}in the D-H parameter in Table 2).

#### 3.2. Velocity Jacobian Matrix and Dexterity Analysis of the Spatial 8R Mechanism

#### 3.3. Manipulability Analysis of the Spatial 8R Mechanism

_{i}) approaches infinity; thus, these positions are the singular configuration of the mechanism, where β

_{i}represents the coordinate position on the β axis.

## 4. Analysis of Hip Joint Motion Laws

_{11}R

_{21}, the y axis faces vertically up, and the z axis is perpendicular to the xy plane. The drive is added to R

_{11}, R

_{21}, R

_{18}and R

_{28}. The forward flexion, posterior extension, abduction and adduction movements of the hip joint were simulated in ADAMS. The variations in the velocities and angular velocities of the moving platform for a single movement are shown in Figure 6.

_{x}represents the velocity curve and ω

_{x}represents the angular velocity curve. In forward flexion of the hip joint, the leg is lifted forward; thus, the velocity curve along the y axis changes considerably, and the velocity curve along the x axis changes as a joint linkage to ensure smoother forward flexion movement. There is no velocity in the z axis direction, so the velocity is 0. Since the forward flexion movement rotates around the x axis, the change in the angular velocity curve in the x axis direction is more obvious. The change in the angular velocity curve around the y axis is smaller than that around the x axis, and there is no rotation around the z axis; thus, the corresponding angular velocity curve is 0. The analysis of the velocity and angular velocity curves of other movements was carried out in the same manner. As seen in Figure 6, the movement of this mechanism can conform to the regular motion of the human hip joint, and therefore can be used to help patients recover.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 6.**Variations in the velocity and angular velocity of the moving platform y for a single movement. (

**a**) Variations in the velocity and angular velocity of flexion motion. (

**b**) Variations in the velocity and angular velocity of the extension motion. (

**c**) Variations in the velocity and angular velocity of abduction motion. (

**d**) Variations in the velocity and angular velocity of adduction motion.

i | α_{i−1}/(°) | a_{i−1}/mm | θ_{i}/(°) | d_{i}/mm |
---|---|---|---|---|

1 | 90 | g | θ_{1} | 0 |

2 | 90 | 0 | θ_{2} | l_{1} |

3 | 90 | 0 | θ_{3} | 0 |

4 | 0 | h | θ_{4} | 0 |

5 | 0 | m | θ_{5} | 0 |

6 | 90 | 0 | θ_{6} | l_{2} |

i | α_{i−1}/(°) | a_{i−1}/mm | θ_{i}/(°) | d_{i}/mm |
---|---|---|---|---|

1 | 90 | k | θ_{7} | 0 |

2 | 90 | 0 | θ_{8} | l_{3} |

Motion Type | Range of Motion/(°) |
---|---|

Flexion | 0~125° |

Extension | 0~30° |

Adduction | 0~60° |

Abduction | 0~40° |

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**MDPI and ACS Style**

Jia, Z.; Li, R.; Wang, Y.; Liu, J.; Liang, H.
Configuration Design and Kinematic Performance Analysis of a Novel Spatial 8R Hip Joint Rehabilitation Mechanism. *Appl. Sci.* **2022**, *12*, 12488.
https://doi.org/10.3390/app122312488

**AMA Style**

Jia Z, Li R, Wang Y, Liu J, Liang H.
Configuration Design and Kinematic Performance Analysis of a Novel Spatial 8R Hip Joint Rehabilitation Mechanism. *Applied Sciences*. 2022; 12(23):12488.
https://doi.org/10.3390/app122312488

**Chicago/Turabian Style**

Jia, Zengyu, Ruiqin Li, Yuan Wang, Juan Liu, and Hailong Liang.
2022. "Configuration Design and Kinematic Performance Analysis of a Novel Spatial 8R Hip Joint Rehabilitation Mechanism" *Applied Sciences* 12, no. 23: 12488.
https://doi.org/10.3390/app122312488