# Optimal Design of Lower Limb Rehabilitation System Based on Parallel and Serial Mechanisms

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## Abstract

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## 1. Introduction

- The location of the drive guides of the active manipulator, because of the intersection of their axes at one point in the upper right side of a patient user;
- Not a convenient shape for the moving platform, as in the form of an equilateral triangle;
- Limited location of fastenings of mechanisms to the platform center;
- The used optimization algorithms for the generation of random parameters and search by climbing to the top;
- Consideration of intersections of only active manipulator links;
- The use of the average size of a human limb with no customizing capability.

- A novel hybrid modular structure of a robotic system for the rehabilitation of the lower limbs is designed as based on two modules identical in structure, including an active 3-PRRR manipulator for moving the patient’s foot and a passive orthosis based on the RRR mechanism for supporting the lower limbs;
- A method for parametric synthesis of a hybrid robotic system with a modular structure is formulated, taking into account the generated levels of parametric constraints depending on the ergonomics and manufacturability of the proposed design.

## 2. Motion Requirements from Lower Limb Biomechanics

- The limbs are not abducted (leg abduction angle ${\gamma}_{i}=0$) while performing one cycle ($t\in \left[0;360\right]$) of gait simulation;
- Abduction of the left leg to ${\gamma}_{1}=-30\xb0$;
- Performing one cycle of gait simulation;
- Simultaneous adduction of the left leg to the angle ${\gamma}_{1}=0\xb0$ and abduction of the right leg to ${\gamma}_{2}=30\xb0$;
- Performing one cycle of gait simulation;
- Abduction of the right leg ${\gamma}_{2}=0\xb0$.

## 3. Mathematical Model

## 4. Optimization Design Problem

- Selection of optimization parameters, which includes both continuous and discrete.
- 1.1.
- We use the link lengths as continuous optimization parameters ${L}_{BCij}$, ${L}_{CDij}$, guide positions ${x}_{Bi1}$,${z}_{Bi1}$,${y}_{Bi1}$,${z}_{Bi1}$,${x}_{Bi3}$,${y}_{Bi3}$ (Figure 5), and horizontal dimensions of the platforms.
- 1.2.

- Selection of optimization criterion. Due to the fact that as a result of optimization it is necessary to determine the geometric parameters at which the compactness of the structure is ensured, we write the criterion function in the following form:$$F={\sum}_{i=1}^{2}{\sum}_{j=1}^{3}\left({L}_{BCij}+{L}_{CDij}\right)\to min$$
- The optimization constraint is the reachability of all points of the trajectory described earlier in Section 2 and the absence of intersections for each of these points, that is as follows:$${N}^{-}=0$$

## 5. Numerical Simulation

- Continuous:
- Link sizes: ${L}_{BCij}\in \left[200;900\right]$, ${L}_{CDij}\in \left[200;900\right]$;
- Coordinates of the guides:$${x}_{B11}\in \left[-2000;-50\right],{z}_{B11}\in \left[-1500;1500\right],{y}_{B12}\in \left[0;2000\right],$$$${z}_{B12}\in \left[-1500;1500\right],{x}_{B13}\in \left[-50;2000\right],{y}_{B13}\in \left[0;2000\right],$$$${x}_{B21}\in \left[50;2000\right],{z}_{B21}\in \left[-1500;1500\right],{y}_{B22}\in \left[0;2000\right],$$$${z}_{B22}\in \left[-1500;1500\right],{x}_{B23}\in \left[50;2000\right],{y}_{B23}\in \left[0;2000\right],$$
- Platform sizes: ${d}_{x}\in \left[100;300\right]$, ${d}_{y}\in \left[100;300\right]$.

- Discrete:
- Options for attaching kinematic chains to moving platforms ${p}_{ij}\in \mathrm{1,2}$;
- Variants of configurations of kinematic chains ${l}_{ij}\in 1,2.$

#### 5.1. Constraint Level 1

#### 5.2. Constraint Level 2

#### 5.3. Constraint Level 3

#### 5.4. Constraint Level 4

Level | Target Function | Platform Connection | Chain Configuration | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Left (I) Module Chains | Right (II) Module Chains | Left (I) Module Chains | Right (II) Module Chains | ||||||||||

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | ||

1 | 7040.7 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 |

2 | 7265.1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

3 | 7739.7 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |

4 | 7784.1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 |

Level | Guide Positions (mm) | Platforms | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Left (I) Module | Right (II) Module | d_x (mm) | d_y (mm) | |||||||||||

x_B11 | z_B11 | y_B12 | z_B12 | x_B13 | y_B13 | x_B21 | z_B21 | y_B22 | z_B22 | x_B23 | y_B23 | |||

1 | −1028.0 | −629 | 2000 | −957 | −1114 | 1227 | 419 | −1162 | 1844 | −355 | 682 | 2000 | ||

2 | −1042.6 | −1031.0 | 1588.2 | −1092.6 | −1042.6 | 1588.2 | 1165.1 | −861.4 | 1588.2 | −1096.6 | 1165.1 | 1588.2 | 140.8 | 272.7 |

3 | −1066.5 | −1239.2 | 1567.9 | −1208.0 | −1066.5 | 1567.9 | 1043.0 | −1280.4 | 1567.9 | −1283.0 | 1043.0 | 1567.9 | 152.5 | 300.0 |

4 | −884.6 | 247.1 | 1677.1 | −1300.6 | −884.6 | 1677.1 | 891.6 | 148.7 | 1677.1 | −25.0 | 891.6 | 1677.1 | 148.9 | 281.7 |

Level | Link Sizes (mm) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Left (I) Module | Right (II) Module | |||||||||||

L_ BC11 | L_ CD11 | L_ BC12 | L_ CD12 | L_ BC13 | L_ CD13 | L_ BC21 | L_ CD21 | L_ BC22 | L_ CD22 | L_ BC23 | L_ CD23 | |

1 | 519.3 | 573.8 | 900.0 | 443.4 | 408.5 | 569.0 | 656.7 | 631.8 | 685.9 | 544.6 | 285.2 | 822.5 |

2 | 603.1 | 777.0 | 467.4 | 726.6 | 674.2 | 397.4 | 675.7 | 661.9 | 450.7 | 704.3 | 382.7 | 744.1 |

3 | 780.6 | 725.0 | 568.8 | 783.5 | 559.8 | 465.5 | 725.0 | 814.9 | 628.0 | 682.1 | 494.2 | 512.3 |

4 | 790.9 | 799.0 | 599.1 | 757.0 | 393.8 | 552.2 | 790.9 | 799.0 | 599.1 | 757.0 | 393.8 | 552.2 |

## 6. Experimental Investigations

- Operation of the mechanical safety device to ensure patient safety.
- Practicing the movement of limbs in the sagittal plane.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Axes of rotations of the hip joint [25].

**Figure 2.**The planned sequence of movements as reference joint trajectories: (

**a**) Hip joint flexion ${\alpha}_{i}$; (

**b**) Knee joint flexion ${\beta}_{i}$; (

**c**) Hip joint abduction ${\gamma}_{i}$.

**Figure 3.**Reference movement of the center of the ankle joint: (

**a**) three-dimensional view; (

**b**) in projection on the XY plane; (

**c**) in projection on the XZ plane; (

**d**) in projection on the YZ plane.

**Figure 5.**Kinematic scheme of rehabilitation system: (

**a**) the kinematic design (I is the left module for the left limb; II is the right module for the right limb), (

**b**) an alternative design for the kinematic design.

**Figure 7.**Checking the links intersection: (

**a**) model for links; (

**b**) first case type; (

**c**) second type case.

**Figure 9.**Distribution of continuous parameters for constraint level 1: (

**a**) guide positions; (

**b)**link sizes.

**Figure 10.**An example of the relative position of guides at different levels of constraints: (

**a**) Level 1; (

**b**) Level 2.

**Figure 11.**Distribution of continuous parameters for constraint level 2: (

**a**) guide positions; (

**b**) link sizes.

**Figure 13.**Changing the coordinates of the center ${P}_{i}$ of the platforms during the trajectory development process.

**Figure 14.**Distribution of continuous parameters for constraint level 3: (

**a**) guide positions; (

**b**) link sizes.

**Figure 15.**Distribution of continuous parameters for constraint level 4: (

**a**) guide positions; (

**b**) link sizes.

**Figure 18.**Control system block diagram for the prototype in Figure 17.

**Figure 19.**Test position of the safety device: (

**a**) without applying load, (

**b**) compression of the upper pair of elastic elements and movement of the orthosis when the load directed along the Y–axis is exceeded.

**Figure 20.**Test compression of elastic elements with simultaneous excess load along: (

**a**) X–axis, simultaneously along the X and Y, (

**b**) when a movement causes a simultaneous excess of load on both axes, compression of all pairs of elastic elements occurs.

**Figure 21.**Acquired trajectory of test movement of the end-effector of the active manipulator during the rehabilitation process.

**Figure 23.**Discrepancy between experimental and simulated values of angles when practicing rehabilitation movements.

**Table 1.**Anthropometric measurements of the lower limbs, [26].

Measurement | Country with Max Value for Women | Value, mm | Country with Max Value for Men | Value, mm |
---|---|---|---|---|

Thigh circumference | Kenya | 720 | Thailand | 660 |

Calf muscle circumference | Kenya | 416 | Japan | 422 |

Length buttock-knee | The Netherlands | 664 | The Netherlands | 703 |

Foot length | Kenya | 270 | The Netherlands | 296 |

Foot width | The Netherlands | 107 | The Netherlands | 116 |

Calf length | The Netherlands | 483 | The Netherlands | 538 |

Thigh width in sitting position | USA | 501 | The Netherlands | 438 |

Without Orthosis | Taking into Account the Orthosis | |
---|---|---|

Thigh circumference | 720 | 815 |

Calf muscle circumference | 422 | 610 |

Length buttock-knee | 703 | 703 |

Foot length | 296 | 326 |

Foot width | 116 | 176 |

Calf length | 538 | 738 |

Thigh width in sitting position | 501 | 531 |

№ Level | Constraints Level | Number of Optimization Parameters |
---|---|---|

1 | No assumptions | 38 |

2 | Constraint on equality of $Y$ coordinates of all guides, equality of $X$ coordinates of guides for the left and right legs | 33 |

3 | Level 2 + constraint on the coordinate ranges of guides based on the ranges of their variable coordinates | 33 |

4 | Level 3 + constraint on equal lengths of module links ${L}_{BC1j}={L}_{BC2j}$, ${L}_{CD1j}={L}_{CD2j}$ | 27 |

${\mathit{\theta}}_{\mathit{i}}\xb0$ | ${\mathit{L}}_{\mathit{O}\mathit{E}},$ $\mathbf{m}\mathbf{m}$ | ${\mathit{L}}_{\mathit{E}\mathit{F}},$ $\mathbf{m}\mathbf{m}$ | ${\mathit{L}}_{\mathit{F}\mathit{G}},$ $\mathbf{m}\mathbf{m}$ | ${\mathit{L}}_{\mathit{G}\mathit{H}},$ $\mathbf{m}\mathbf{m}$ | ${\mathit{d}}_{\mathit{E}\mathit{F}},$ $\mathbf{m}\mathbf{m}$ | ${\mathit{d}}_{\mathit{F}\mathit{G}},$ $\mathbf{m}\mathbf{m}$ | ${\mathit{d}}_{\mathit{G}\mathit{H}},$ $\mathbf{m}\mathbf{m}$ | ${\mathit{d}}_{\mathit{Z}},$ $\mathbf{m}\mathbf{m}$ | ${\mathit{d}}_{\mathit{l}\mathit{i}\mathit{n}\mathit{k}},$ $\mathbf{m}\mathbf{m}$ |
---|---|---|---|---|---|---|---|---|---|

90 | 135.5 | 703 | 738 | 326 | 259 | 194 | 176 | 170 | 80 |

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**MDPI and ACS Style**

Malyshev, D.; Perevuznik, V.; Ceccarelli, M.
Optimal Design of Lower Limb Rehabilitation System Based on Parallel and Serial Mechanisms. *Machines* **2024**, *12*, 104.
https://doi.org/10.3390/machines12020104

**AMA Style**

Malyshev D, Perevuznik V, Ceccarelli M.
Optimal Design of Lower Limb Rehabilitation System Based on Parallel and Serial Mechanisms. *Machines*. 2024; 12(2):104.
https://doi.org/10.3390/machines12020104

**Chicago/Turabian Style**

Malyshev, Dmitry, Victoria Perevuznik, and Marco Ceccarelli.
2024. "Optimal Design of Lower Limb Rehabilitation System Based on Parallel and Serial Mechanisms" *Machines* 12, no. 2: 104.
https://doi.org/10.3390/machines12020104