# Design and Control of a Polycentric Knee Exoskeleton Using an Electro-Hydraulic Actuator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mechanism Design of the Polycentric Knee Joint

## 3. Mathematical Modeling of the Entire System, Including the EHA

## 4. Design of a Sliding Mode Control

## 5. Experimental Setting and Results

## 6. Discussion

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DoF | Degree of freedom |

EHA | Electro-hydraulic actuator |

SMC | Sliding mode controller |

ACL | Anterior cruciate ligament |

PCL | Posterior cruciate ligament |

ICR | Instantaneous center of rotation |

SEA | Series elastic actuator |

HA | Hydraulic actuator |

ROM | Range of motion |

MCU | Microcontroller unit |

SSE | Sum of square error |

RMSE | Root mean square error |

BLDC | Brushless DC |

I/O | Input/output |

CAN | Controller area network |

SPI | Serial peripheral interface |

FSR | Force-sensing resistor |

PID | Proportional-integral-differential |

PCHIP | Piecewise cubic Hermite interpolation |

Nomenclature | |

${\theta}_{p}$ | Relative angle between the ground link and the point $P$ within the coupler $(\xb0)$ |

$x$ | Position $\left(\mathrm{mm}\right)$ |

${p}_{n}$ | Coefficients of polynomial with respect to ${\theta}_{p}$ $(-)$ |

${q}_{n}$ | Coefficients of polynomial with respect to $x$ $(-)$ |

${D}_{m}$ | Ideal volumetric displacement of the motor $\left({\mathrm{mm}}^{3}/\mathrm{rev}\right)$ |

${\dot{\theta}}_{m}$ | Motor shaft speed $\left(\mathrm{rev}/\mathrm{s}\right)$ |

${Q}_{1}$ | Return flow from motor in pump, supplied flow in cylinder $\left({\mathrm{mm}}^{3}/\mathrm{s}\right)$ |

${Q}_{2}$ | Forward flow to motor in pump, return flow in cylinder $\left({\mathrm{mm}}^{3}/\mathrm{s}\right)$ |

${C}_{im}$ | Internal or cross-port leakage coefficient $\left({\mathrm{mm}}^{3}/\mathrm{s}/\mathrm{bar}\right)$ |

${C}_{em}$ | External leakage coefficient $\left({\mathrm{mm}}^{3}/\mathrm{s}/\mathrm{bar}\right)$ |

${P}_{1}$ | Pressure in the return chamber $\left(\mathrm{bar}\right)$ |

${P}_{2}$ | Pressure in the forward chamber $\left(\mathrm{bar}\right)$ |

${Q}_{L}$ | Load flow $\left({\mathrm{mm}}^{3}/\mathrm{s}\right)$ |

${P}_{L}$ | Pressure difference $\left(\mathrm{bar}\right)$ |

${V}_{1}^{0},{V}_{2}^{0}$ | Two chambers of initial condition $\left({\mathrm{mm}}^{3}\right)$ |

${\beta}_{e}$ | Bulk modulus $\left(\mathrm{bar}\right)$ |

${P}_{r}$ | Reference pressure $\left(\mathrm{bar}\right)$ |

${C}_{{im}^{\prime}}$ | Coefficient of the internal leakage $\left({\mathrm{mm}}^{3}/\mathrm{s}/\mathrm{bar}\right)$ |

${C}_{em{1}^{\prime}}$ | Coefficient of the external leakage from the return chamber $\left({\mathrm{mm}}^{3}/\mathrm{s}/\mathrm{bar}\right)$ |

${C}_{{em2}^{\prime}}$ | Coefficient of the external leakage from the forward chamber $\left({\mathrm{mm}}^{3}/\mathrm{s}/\mathrm{bar}\right)$ |

${A}_{1},{A}_{2}$ | Area of each chamber $\left({\mathrm{mm}}^{2}\right)$ |

${V}_{t}$ | Total hydraulic actuator volume $\left({\mathrm{mm}}^{3}\right)$ |

$\overline{A}$ | Average of cross-sectional area of chamber $\left({\mathrm{mm}}^{2}\right)$ |

$m$ | Mass of the load $\left(\mathrm{kg}\right)$ |

$B$ | Damping coefficient $\left(\mathrm{N}/\left(\mathrm{m}/\mathrm{s}\right)\right)$ |

$K$ | Spring coefficient $\left(\mathrm{N}/\mathrm{m}\right)$ |

${F}_{L}$ | Disturbance $\left(\mathrm{N}\right)$ |

$u$ | Motor shaft speed, control input $\left(\mathrm{rev}/\mathrm{min}\right)$ |

$\tilde{{s}_{r}}$ | Scaling factor $(-)$ |

$s$ | Sliding surface $(-)$ |

$\lambda $ | Strictly positive constant $(-)$ |

${x}_{d}$ | Desired position $\left(\mathrm{mm}\right)$ |

$e$ | Tracking error $\left(\mathrm{mm}\right)$ |

$\eta $ | Design parameter $(-)$ |

$\Phi $ | Boundary layer thickness $(-)$ |

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**Figure 1.**Overview of knee anatomy: (

**a**) anatomical structure of the human knee; (

**b**) changes in the shape and tension of the anterior cruciate ligament (ACL) and posterior cruciate ligament (PCL).

**Figure 2.**Range of motion of the designed polycentric knee exoskeleton: (

**a**) design modeling of the polycentric knee exoskeleton; (

**b**) normal angle and workspace of the robot during the gait cycle.

**Figure 3.**Design of the knee structure and determination of the number of degrees of freedom (DoF): (

**a**) calculation point of the degrees of freedom; (

**b**) configuration of a polycentric knee with coupler point $P$. ICR is instantaneous center of rotation.

**Figure 6.**Schematic diagram of an axial-piston pump: cam plate; shaft; cylinder block and piston shoes; cylinder valve.

**Figure 9.**Simulation results of sliding mode control: (

**a**) results of position performance in mm; (

**b**) tracking error in mm.

**Figure 11.**The experimental setup: (

**a**) hardware configuration including encoder, loadcell, and force-sensing resistor (FSR); (

**b**) schematic of experimental configuration; (

**c**) EHA unit with motor and pump; (

**d**) microcontroller unit and motor driver for motor control.

**Figure 12.**Experimental results of sliding mode control and proportional-integral-differential (PID) control: (

**a**,

**c**,

**e**) results of angular position performance; (

**b**,

**d**,

**f**) angle tracking error.

**Figure 13.**Driving capacity as frequency changes: (

**a**) magnitude variation in angle tracking; (

**b**) phase variation in angle tracking.

$\mathit{P}$-Value | $\mathit{Q}$-Value | ||
---|---|---|---|

${p}_{1}$ | 1.571 × 10^{−8} | ${q}_{1}$ | –1.807 × 10^{−9} |

${p}_{2}$ | –4.164 × 10^{−6} | ${q}_{2}$ | 5.788 × 10^{−7} |

${p}_{3}$ | 0.0004789 | ${q}_{3}$ | –7.285 × 10^{−5} |

${p}_{4}$ | –0.03192 | ${q}_{4}$ | 0.004161 |

${p}_{5}$ | –0.6195 | ${q}_{5}$ | –0.6389 |

${p}_{6}$ | 148 | ${q}_{6}$ | 91.73 |

Parameter | Specification |
---|---|

$x$ | $\le 150\mathrm{mm}$ |

$u$ | $\le 5000\text{}\mathrm{rpm}$ |

${V}_{t}$ | $41,233{\mathrm{mm}}^{3}$ |

$m$ | $1.6\mathrm{kg}\left(1.4\mathrm{kg}\le \mathrm{m}\le 1.8\mathrm{kg}\right)$ $\widehat{m}=\sqrt{2.52}\cong 1.587\mathrm{kg}$ |

${\beta}_{e}$ | $17,200\mathrm{bar}$ |

$\overline{A}$ | $274.8894{\mathrm{mm}}^{2}$ |

$B$ | $0.05\mathrm{N}/\left(\mathrm{m}/\mathrm{s}\right)$ |

${D}_{m}$ | $0.8\mathrm{cc}/\mathrm{rev}$ |

Component | Parameter | Specification | Component | Parameter | Specification |
---|---|---|---|---|---|

Hydraulic cylinder | Bore size | 20 mm | Motor | Input voltage | 24 V |

Rod size | 10 mm | Watts | 200 W | ||

Maximum allowable pressure | 3.5 MPa | Speed limit | 5000 rpm | ||

Stroke length | 150 mm | MCU | System clock | 200 MHz | |

Hydraulic pump | Displacement | 0.8 cc | Interrupt time | 0.002 s | |

Hydraulic oil | Model | ISO VG 46 | Mass | Weight | 1.6 kg (shank) |

Bulk modulus | 17,200 bar | Encoder | Degree | 0–360° |

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## Share and Cite

**MDPI and ACS Style**

Lee, T.; Lee, D.; Song, B.; Baek, Y.S.
Design and Control of a Polycentric Knee Exoskeleton Using an Electro-Hydraulic Actuator. *Sensors* **2020**, *20*, 211.
https://doi.org/10.3390/s20010211

**AMA Style**

Lee T, Lee D, Song B, Baek YS.
Design and Control of a Polycentric Knee Exoskeleton Using an Electro-Hydraulic Actuator. *Sensors*. 2020; 20(1):211.
https://doi.org/10.3390/s20010211

**Chicago/Turabian Style**

Lee, Taesik, Dongyoung Lee, Buchun Song, and Yoon Su Baek.
2020. "Design and Control of a Polycentric Knee Exoskeleton Using an Electro-Hydraulic Actuator" *Sensors* 20, no. 1: 211.
https://doi.org/10.3390/s20010211