# Mechanism Design and Performance Analysis of a Sitting/Lying Lower Limb Rehabilitation Robot

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Range of Motion Analysis of Lower Limbs in Humans’ Sitting/Lying Position

_{1}and the length of the calf be l

_{2}. The fixed coordinate system is denoted as o

_{0}-x

_{0}y

_{0}z

_{0}, the coordinate system of the hip joint is denoted as o

_{1}-x

_{1}y

_{1}z

_{1}, the coordinate system of the knee joint is denoted as o

_{2}-x

_{2}y

_{2}z

_{2}, and the coordinate system of the ankle joint is denoted as o

_{3}-x

_{3}y

_{3}z

_{3}. Additionally, θ

_{i}(i = 0, 1, 2) is the rotation angle of each joint relative to axis x

_{i}(i = 0, 1, 2), and θ

_{3}is the rotation angle of the lower limb about axis z

_{0}, where counterclockwise rotation is defined as the positive direction. According to Table 1, considering that the human body is in the sitting position, the actual flexion and extension angle θ

_{0}of the hip joint will be limited. In this paper, 0 ≤ θ

_{1}≤ 45°, 0° ≤ θ

_{2}≤ 45°, and −150 ≤ θ

_{3}≤ 0° are selected. The D-H method is used to calculate the space coordinates of the end of the ankle joint (x

_{c}, y

_{c}, z

_{c}), which can be expressed as Equation (1).

_{1}= 550 mm, l

_{2}= 430 mm and l

_{1}= 402 mm, l

_{2}= 313 mm, are selected for spatial motion analysis and a Monte Carlo method was used to draw the three-dimensional motion range of lower limbs in the sitting position. In Figure 2a, the yellow–green areas indicate the range of motion of l

_{1}= 550 mm and l

_{2}= 430 mm, and the blue areas indicate the range of motion of l

_{1}= 402 mm and l

_{2}= 313 mm. In this paper, the ankle joint is required to carry out plane rehabilitation exercise at $\mathrm{z}=-200$ mm, and the range of motion bounded by $\mathrm{z}\le -200$ mm is selected, as shown in Figure 2a, and the x–y projection of this bounded range of motion is obtained, as shown in Figure 2b.

## 3. Mechanical Structure Design

#### 3.1. Mechanical Structure Description

_{1}A

_{2}A

_{3}A

_{4}, as shown in Figure 5. The patient can select the suitable closed ring by adjusting the seat height with different chain lengths A

_{1}A

_{4}.

#### 3.2. Selection of Motor Drive

#### 3.2.1. Precision Linear Module Selection

_{e}= 610 mm > S.

#### 3.2.2. Selection of Drive Motor and Reducer

_{1}represents torque generated by acceleration; T

_{2}represents load torque (torque at constant speed); T

_{3}represents preloading torque; T

_{4}represents other torques.

_{L}represents the moment of inertia, ${F}_{a}$ represents the axial load, I represents the lead, σ represents the efficiency, ${F}_{n}$ represents the preload load, $\beta $ represents the lead angle of the lead screw. It is assumed that ${F}_{a}=500$ N and σ = 80%.

## 4. Kinematics Analysis of Mechanism

#### 4.1. Forward and Inverse Kinematics

_{1}and C

_{2}represent sliders, and P represents the end point. When the two sliders move in a straight line on the guide rails, the end point P realizes two degree of freedom movement.

_{1}and L

_{2}.

_{1}are (x

_{1}, 0), and the coordinates of point C

_{2}are (x

_{2}, 0). It is easy to obtain the inverse solutions for x

_{1}and x

_{2}according to the geometric relationship:

_{1}is the included angle between AP and the x direction, and α

_{2}is the included angle between BP and the x direction. The forward kinematics solution can be obtained as follows:

#### 4.2. Jacobian Matrices and Singularity Analysis

#### 4.2.1. Jacobian Matrix

#### 4.2.2. Singularity Analysis

#### 4.3. Condition Number

## 5. Trajectory Planning

_{i}(i = 0, 1, …, 5) represents the coefficients to be solved.

_{II}offers a shorter path than l

_{I}, and l

_{III}offers an even shorter path:

## 6. Experiment and Evaluation

#### 6.1. Experimental Setup

_{1}, the training track of volunteer Ⅱ was l

_{2}, and the training track of volunteer Ⅲ was l

_{3}.

_{a}(t) of the slider. Then, with the position information x

_{d}(t) of the next discrete point expected in the TXT document, the position deviation e(t) is obtained by using Equation (22). According to the position deviation e(t), the instantaneous velocity v(t) is calculated and sent to the motor through the serial port to realize real-time tracking.

_{p}and K

_{d}represent the proportional regulation coefficient and differential regulation coefficient, respectively.

#### 6.2. Experimental Result and Evaluation

_{1}and slider C

_{2}, the x value and y value of the end-effector are obtained, as shown in Figure 14.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Range of motion analysis of ankle joint axis. (

**a**) Lower limb range of motion bounded by z ≤ −200 mm. (

**b**) The x-y projection of the bounded range of motion in (

**a**).

**Figure 13.**The value of x

_{1}and x

_{2}over time. (

**a**) The terminal x

_{1}value varies with time. (

**b**) The terminal x

_{2}value varies with time.

**Figure 14.**Terminal trajectories and y-value variation with time. (

**a**) The end of the track. (

**b**) The terminal x value varies with time. (

**c**) The terminal y value varies with time.

**Figure 15.**The deviation of x and y values over time. (

**a**) The deviation of the terminal x value. (

**b**) The deviation of the terminal y value.

**Figure 16.**Track tracking error of sliders. (

**a**) Track tracking error of slider C

_{1}. (

**b**) Track tracking error of slider C

_{2}.

**Figure 17.**The participants’ joint motion curves. (

**a**) The terminal θ

_{1}value varies with time. (

**b**) The terminal θ

_{2}value varies with time. (

**c**) The terminal θ

_{3}value varies with time.

Joint | Datum Plane | Movement | Angle Range (°) |
---|---|---|---|

Hip | Sagittal plane | Flexion (lying pos.) | 0~125 |

Flexion (sitting pos.) | 0~45 | ||

Coronal plane | Abduction (lying pos.) | 0~45 | |

Adduction (sitting pos.) | 0~45 | ||

Knee | Sagittal plane | Flexion | −150~0 |

Ankle | Sagittal plane | Dorsiflexion | 0~20 |

Flexion | 0~45 |

Control Component | Model | Basic Parameters | Number |
---|---|---|---|

Upper computer | HP 15-bc011TX | i5-6300HQ CPU @ 2.30 GHz | 1 |

Motor | SDGA-02C12BD | 0.2 KW, 36 V, 0.64 N.m | 4 |

Linear module | NDC86-1510-740-1-P-F0-S2 | 610 mm | 4 |

Speed reducer | 60ZDF5-400T1 | 5:1 | 4 |

Actuators | TSDA-C11A | RS-232 | 4 |

Relay board | WF-16i-16o | RS-485 | 1 |

Encoder | / | 2500 p/r | 4 |

Software | QT 5.9.7 | / | 1 |

Volunteer | Gender | Thigh Length | Calf Length |
---|---|---|---|

Ⅰ | Male | 560 mm | 450 mm |

Ⅱ | Male | 500 mm | 435 mm |

Ⅲ | Male | 480 mm | 390 mm |

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**MDPI and ACS Style**

Dong, F.; Li, H.; Feng, Y.
Mechanism Design and Performance Analysis of a Sitting/Lying Lower Limb Rehabilitation Robot. *Machines* **2022**, *10*, 674.
https://doi.org/10.3390/machines10080674

**AMA Style**

Dong F, Li H, Feng Y.
Mechanism Design and Performance Analysis of a Sitting/Lying Lower Limb Rehabilitation Robot. *Machines*. 2022; 10(8):674.
https://doi.org/10.3390/machines10080674

**Chicago/Turabian Style**

Dong, Fangyan, Haoyu Li, and Yongfei Feng.
2022. "Mechanism Design and Performance Analysis of a Sitting/Lying Lower Limb Rehabilitation Robot" *Machines* 10, no. 8: 674.
https://doi.org/10.3390/machines10080674