# System for Evaluation and Compensation of Leg Length Discrepancy for Human Body Balancing

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

#### 2.1. Anthropometric Human Body Model

#### 2.2. Human Body Balancing

#### 2.3. System Description

- a mechanical set of two 3-RPS parallel manipulators with mobile force plates;
- an electronic system for control, measuring and communication with the PC;
- a vision system with two cameras; and
- a PC application used for control and data collection.

## 3. Mathematical Model of the System

#### 3.1. The Geometry of the 3-RPS Parallel Manipulator

#### 3.2. Inverse Kinematic of the 3-RPS Parallel Manipulator

- ${}^{A}{\widehat{\mathit{s}}}_{i}$ is the unit vector of the prismatic joint;
- ${}^{A}\mathbf{p}$ is the position vector of the moving platform center in the frame {A};
- ${}^{A}{\mathbf{R}}_{B}$ is the rotation matrix of the moving frame {B} with respect to the frame {A};
- ${}^{B}{\mathbf{b}}_{i}$ is the position vectors of spherical joints in the frame {B}; and
- ${}^{A}{\mathbf{a}}_{i}$ is the position vectors of revolute joints in the frame {A}.

#### 3.3. Forward Kinematics of the 3-RPS Parallel Manipulator

#### 3.4. Force Plates

- a is the distance between the sensors along the x-axis;
- b is the distance between the sensors along the y-axis; and
- l is the distance between the origins of the left and right local force plate frame.

## 4. Human Body Balancing Algorithm

#### 4.1. Simulation Results

^{®}, as shown in Figure 9 (the model is available for download at [31]). The model of the 3-RPS parallel manipulators and the anthropometric model of the human body standing on force plates were both made with the use of SimMechanics ToolBox. The ${\mathrm{IKP}}_{L}$ and ${\mathrm{IKP}}_{R}$ blocks calculate the reference lengths of the 3-RPS parallel manipulators’ linear actuators with Equation (7). The virtual models of the 3-RPS parallel manipulators with moving force plates calculate the FKP and CoP for the left and the right leg. The HBBA was realized with a state machine and according to the flowchart shown in Figure 8b. The positions of the markers used in the simulation are shown in Figure 3, and they were measured on specifically defined points of the human body virtual model.

#### 4.1.1. Scenario 1: Human Body with LLD

#### 4.1.2. Scenario 2: Human Body with LLD and Scoliosis

## 5. Mechatronic System Design

#### 5.1. Mechanical Design

#### 5.2. Electronic Design

#### 5.2.1. Master Device

#### 5.2.2. Slave Device

#### 5.3. Software Design

#### 5.3.1. Firmware for Microcontrollers

#### 5.3.2. PC Application for Control and Data Collection

#### 5.3.3. Vision System

#### 5.4. Experimental Results

#### 5.4.1. Experiment 1: Healthy Population—Left and Right Leg Load Distribution

#### 5.4.2. Experiment 2: Shift in the CoM Caused by a Force Plate Height Difference Shift

#### 5.4.3. Experiment 3: A Healthy Volunteer with a Simulated LLD

## 6. Conclusions

## Supplementary Files

Supplementary File 1## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LLD | Leg Length Discrepancy |

CT | Computerized Tomography |

MRI | Magnetic Resonance Imaging |

ASIS | Anterior Inferior Iliac Spine |

3-RPS | 3-Revolute-Prismatic-Spherical |

CoM | Center of Mass |

CoP | Center of Pressure |

3-DOF | Three Degrees of Freedom |

IKP | Inverse Kinematic Problem |

FKP | Forward Kinematic Problem |

IR | Infra-red |

BMI | Body Mass Index |

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**Figure 2.**12-segment anthropometric human body model with variable leg length and scoliosis (RF, Right Foot; LF, Left Foot; RLL, Right Lower Leg; LLL, Left Lower Leg; RUL, Right Upper Leg; LUL, Left Upper Leg; PEL, Pelvis; LT, Lower Trunk; UT, Upper Trunk; H, Head; RTA, Right Total Arm; LTA, Left Total Arm). Left leg length is given as ${l}_{L}$, right leg length is ${l}_{R}$, and the angle of scoliosis is $\varphi $: (

**a**) human body without LLD and scoliosis (${l}_{R}={l}_{L},\varphi =0$); (

**b**) human body with LLD (${l}_{R}<{l}_{L},\varphi =0$); (

**c**) human body with LLD (${l}_{R}>{l}_{L},\varphi =0$); (

**d**) human body with scoliosis (${l}_{R}={l}_{L},\varphi >0$); and (

**e**) human body with LLD and scoliosis (${l}_{R}>{l}_{L},\varphi >0$).

**Figure 3.**Human Body Balancing: (

**a**) human body with LLD ($Co{M}_{x}<0$); and (

**b**) balanced human body ($Co{M}_{x}=0$).

**Figure 5.**Display of the mechanical set with two 3-RPS parallel manipulators with moving force plates: (

**a**) with the patient on the force plates; and (

**b**) with the application used to diagnose the load on each of the patient’s legs.

**Figure 8.**Flowchart: (

**a**) Turning the system on and moving the 3-RPS parallel manipulators to their initial height; and (

**b**) Human Body Balancing Algorithm.

**Figure 10.**Scenario 1: (

**a**) Initial state of the human body with LLD (algorithm iteration k = 0); (

**b**) algorithm iteration k = 1; (

**c**) algorithm iteration k = 2; (

**d**) algorithm iteration k = 3; (

**e**) algorithm iteration k = 4; and (

**f**) the human body is balanced (algorithm iteration k = 5).

**Figure 11.**Scenario 1: (

**a**) $Co{M}_{x}$, the position of the human body’s CoM; (

**b**) reference height of the left (${z}_{Lref}$) and right (${z}_{Rref}$) force plate, current height of the left (${z}_{L}$) and right (${z}_{R}$) force plate; (

**c**) the load on the left (${m}_{L}$) and right (${m}_{R}$) leg; (

**d**) the balancing trigger signal (Q) marking the start of a new HBBA iteration; (

**e**) position of the LASIS and RASIS markers, which are located on the pelvis; and (

**f**) left and right force plate height difference ($\Delta z$), which corresponds with the LLD evaluation ($\Delta l$) in the HBBA’s final step.

**Figure 12.**Scenario 2: (

**a**) Initial state of the human body with LLD and scoliosis (algorithm iteration k = 0); (

**b**) algorithm iteration k = 1; (

**c**) algorithm iteration k = 2; (

**d**) algorithm iteration k = 3; and (

**e**) the human body with LLD and scoliosis is balanced (algorithm iteration k = 4).

**Figure 13.**Scenario 2: (

**a**) $Co{M}_{x}$, the position of the human body’s CoM; (

**b**) reference height of the left (${z}_{Lref}$) and right (${z}_{Rref}$) force plate, current height of the left (${z}_{L}$) and right (${z}_{R}$) force plate; (

**c**) the load on the left (${m}_{L}$) and right (${m}_{R}$) leg; (

**d**) the balancing trigger signal (Q) marking the start of a new HBBA iteration; (

**e**) position of the LASIS and RASIS markers, which are located on the pelvis; and (

**f**) left and right force plate height difference ($\Delta z$), which corresponds with the LLD evaluation ($\Delta l$) in the HBBA’s final step.

**Figure 14.**Parts of the mechanical set with two 3-RPS parallel manipulators with moving force plates: (

**a**) handrail and steps of the mechanical set; and (

**b**) left and right 3-RPS parallel manipulators with moving force plates.

**Figure 18.**Measurements of loads on the left ${m}_{L}$ and right ${m}_{R}$ legs and the CoM along the x-axis ($Co{M}_{x}$) for three chosen volunteers: (

**a**) Volunteer 1, the loads on the left ${m}_{L}$ and the right ${m}_{R}$ legs; (

**b**) Volunteer 1, CoM along the x-axis ($Co{M}_{x}$); (

**c**) Volunteer 2, the loads on the left ${m}_{L}$ and the right ${m}_{R}$ legs; (

**d**) Volunteer 2, CoM along the x-axis ($Co{M}_{x}$); (

**e**) Volunteer 3, the loads on the left ${m}_{L}$ and the right ${m}_{R}$ legs; and (

**f**) Volunteer 3, CoM along the x-axis ($Co{M}_{x}$).

**Figure 19.**Experiment 2: Shift of the CoM along the x-axis ($Co{M}_{x}$) depending on a force plate height difference $\Delta z$.

**Figure 20.**Experiment 3: (

**a**) Volunteer 3 placing the left leg on the force plate; (

**b**) a researcher helps Volunteer 3 set the left and right legs on their respective marked locations; (

**c**) Volunteer 3 is balanced (

**d**) LLD simulation by shortening the right (lengthening the left) leg by 15 mm (initial algorithm state k = 0); (

**e**) algorithm iteration k = 1; and (

**f**) Volunteer 3 is balanced (algorithm iteration k = 2).

**Figure 21.**Experiment 3: (

**a**) $Co{M}_{x}$, the position of the human body’s CoM; (

**b**) reference height of the left (${z}_{Lref}$) and right (${z}_{Rref}$) force plate, current height of the left (${z}_{L}$) and right (${z}_{R}$) force plate; (

**c**) the load on the left (${m}_{L}$) and right (${m}_{R}$) leg; (

**d**) the balancing trigger signal (Q) marking the start of a new HBBA iteration; and (

**e**) left and right force plate height difference ($\Delta z$), which corresponds with the LLD evaluation ($\Delta l$) in the HBBA’s final step.

**Table 1.**Scenario 1: Figure 11 response values at the end each iteration of the HBBA.

Iteration | ${\mathit{CoM}}_{\mathit{x}}$ [mm] | ${\mathit{z}}_{\mathit{L}}$ [mm] | ${\mathit{z}}_{\mathit{R}}$ [mm] | $\Delta \mathit{z}$ [mm] | ${\mathit{m}}_{\mathit{L}}$ [kg] | ${\mathit{m}}_{\mathit{R}}$ [kg] | LASIS [mm] | RASIS [mm] |
---|---|---|---|---|---|---|---|---|

k = 0 | −19.35 | 424.40 | 424.40 | 0 | 43.22 | 31.78 | 869.8 | 899.8 |

k = 1 | −7.06 | 443.75 | 424.40 | 19.35 | 39.53 | 35.47 | 889.1 | 899.8 |

k = 2 | −2.42 | 450.81 | 424.40 | 26.41 | 38.19 | 36.81 | 896.1 | 899.8 |

k = 3 | −0.81 | 453.23 | 424.40 | 28.83 | 37.73 | 37.27 | 898.5 | 899.8 |

k = 4 | −0.27 | 454.04 | 424.40 | 29.64 | 37.58 | 37.42 | 899.4 | 899.8 |

k = 5 | −0.09 | 454.31 | 424.40 | 29.91 | 37.53 | 37.47 | 899.7 | 899.8 |

**Table 2.**Scenario 2: Figure 13 response values at the end of each iteration of the HBBA.

Iteration | ${\mathit{CoM}}_{\mathit{x}}$ [mm] | ${\mathit{z}}_{\mathit{L}}$ [mm] | ${\mathit{z}}_{\mathit{R}}$ [mm] | $\Delta \mathit{z}$ [mm] | ${\mathit{m}}_{\mathit{L}}$ [kg] | ${\mathit{m}}_{\mathit{R}}$ [kg] | LASIS [mm] | RASIS [mm] |
---|---|---|---|---|---|---|---|---|

k = 0 | 14.88 | 424.40 | 424.40 | 0 | 33.39 | 41.61 | 894.8 | 869.8 |

k = 1 | 4.60 | 424.40 | 439.28 | −14.88 | 36.22 | 38.78 | 894.8 | 884.6 |

k = 2 | 1.48 | 424.40 | 443.88 | −19.48 | 37.09 | 37.91 | 894.8 | 889.2 |

k = 3 | 0.47 | 424.40 | 445.36 | −20.96 | 37.37 | 37.63 | 894.8 | 890.7 |

k = 4 | 0.17 | 424.40 | 445.83 | −21.43 | 37.46 | 37.54 | 894.8 | 891.2 |

**Table 3.**Experiment 1: Measurement results of the CoM along the x-axis ($Co{M}_{x}$), mass, loads on the left (${m}_{L}$) and right (${m}_{R}$) legs, the positions of the LASIS and RASIS anatomical points, and the BMI. The loads on the left and right legs are shown as percentages [%] of the human body mass.

Variable | ${\mathit{CoM}}_{\mathit{x}}$ | Mass [kg] | ${\mathit{m}}_{\mathit{L}}$ [%] | ${\mathit{m}}_{\mathit{R}}$ [%] | Height [cm] | LASIS [cm] | RASIS [cm] | BMI [kg m${}^{-2}$] |
---|---|---|---|---|---|---|---|---|

Mean | −2.99 | 75.26 | 51.12 | 48.88 | 174.9 | 101.2 | 101.2 | 24.45 |

SD | 8.45 | 14.15 | 2.95 | 2.95 | 10.6 | 7.1 | 7.1 | 2.89 |

Min | −27.08 | 53.68 | 44.90 | 41.94 | 155.0 | 88.0 | 88.0 | 18.72 |

Max | 15.38 | 108.92 | 58.06 | 55.10 | 193.0 | 115.0 | 115.0 | 29.76 |

**Table 4.**Measurement results of the CoM along the x-axis ($Co{M}_{x}$), mass, loads on the left (${m}_{L}$) and right (${m}_{R}$) legs, the positions of the LASIS and RASIS anatomical points for three chosen volunteers.

Subject | ${\mathit{CoM}}_{\mathit{x}}$ | Mass [kg] | ${\mathit{m}}_{\mathit{L}}$ [kg] | ${\mathit{m}}_{\mathit{R}}$ [kg] | Height [cm] | LASIS [cm] | RASIS [cm] | |
---|---|---|---|---|---|---|---|---|

Mean [mm] | SD [mm] | |||||||

Volunteer 1 | 13.25 | 0.798 | 91.15 | 40.29 | 50.86 | 178 | 104 | 104 |

Volunteer 2 | −9.80 | 0.490 | 55.34 | 29.44 | 25.90 | 157 | 96 | 96 |

Volunteer 3 | −0.21 | 0.861 | 81.14 | 40.65 | 40.49 | 179 | 106 | 106 |

**Table 5.**Experiment 3: Figure 21 response values from for each iteration of the HBBA.

Iteration | ${\mathit{CoM}}_{\mathit{x}}$ [mm] | ${\mathit{z}}_{\mathit{L}}$ [mm] | ${\mathsf{z}}_{\mathit{R}}$ [mm] | $\mathbf{\Delta}\mathit{z}$ [mm] | ${\mathit{m}}_{\mathit{L}}$ [kg] | ${\mathit{m}}_{\mathit{R}}$ [kg] |
---|---|---|---|---|---|---|

k = 0 | 26.4 | 425.0 | 410.0 | 15.0 | 32.16 | 48.98 |

k = 1 | 4.4 | 425.0 | 423.2 | 1.8 | 39.21 | 41.94 |

k = 2 | −0.6 | 425.0 | 425.4 | −0.4 | 40.73 | 40.46 |

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## Share and Cite

**MDPI and ACS Style**

Vrhovski, Z.; Obrovac, K.; Nižetić, J.; Mutka, A.; Klobučar, H.; Bogdan, S. System for Evaluation and Compensation of Leg Length Discrepancy for Human Body Balancing. *Appl. Sci.* **2019**, *9*, 2504.
https://doi.org/10.3390/app9122504

**AMA Style**

Vrhovski Z, Obrovac K, Nižetić J, Mutka A, Klobučar H, Bogdan S. System for Evaluation and Compensation of Leg Length Discrepancy for Human Body Balancing. *Applied Sciences*. 2019; 9(12):2504.
https://doi.org/10.3390/app9122504

**Chicago/Turabian Style**

Vrhovski, Zoran, Karlo Obrovac, Josip Nižetić, Alan Mutka, Hrvoje Klobučar, and Stjepan Bogdan. 2019. "System for Evaluation and Compensation of Leg Length Discrepancy for Human Body Balancing" *Applied Sciences* 9, no. 12: 2504.
https://doi.org/10.3390/app9122504