# Six-Bar Linkage Models of a Recumbent Tricycle Mechanism to Increase Power Throughput in FES Cycling

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## Abstract

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## 1. Introduction

## 2. Design Approach

- Model the traditional recumbent tricycle (TRT). Kinematic and quasi-static models are constructed to simulate the experimental work published by Szecsi et al. The TRT includes a continuously rotating crank, which serves as the dependent variable for the published hip-joint and knee-joint moment data as shown in Figure 1. Along with the joint moment vs. crank angle data, the TRT model also accepts the tricycle dimensions and the lower limb dimensions of the cyclist.
- Power output of the P1P2 and P1P3 riders. The performance of the TRT riders is analyzed, using the respective joint moment data, over a full revolution of the crank angle ${\theta}_{2}$. The moment at the crank center ${M}_{A}$ produced by one one leg is determined throughout 0 $\le {\theta}_{2}<$ 360 ${}^{\circ}$. The average power produced by one leg of the cyclist is$$\overline{P}=\frac{{\dot{\theta}}_{2}}{2\pi}{\int}_{0}^{2\pi}{M}_{A}d{\theta}_{2},$$
- Transform the muscle data. The kinematics of alternative tricycle designs will differ from the TRT. The crank angle is an inappropriate dependent variable for the exploration presented in this paper. In order to apply the joint moment data to alternative designs, the data dependency is shifted from crank angle to the appropriate joint angle.
- Model the crank rocker tricycle (CRT) and the coupler driver tricycle (CDT). Kinematic and quasi-static models are created for the CRT and CDT, both utilizing tricycle dimensions, rider dimensions and transformed muscle data.
- Optimize the CRT and CDT dimensions. Optimizations are performed on the CRT and CDT models with the objective of improving upon the cycle-averaged power produced by each group (P1P2 and P1P3) of riders.

## 3. Traditional Recumbent Tricycle Model

## 4. Joint Moment Transformation

## 5. Alternate Tricycle Models

#### 5.1. The CDT Model

#### 5.2. The CRT Model

## 6. Optimization

## 7. Results and Discussion

#### 7.1. Manufacturability

#### 7.2. Joint Torque Curve Continuity

#### 7.3. Power Distribution

#### 7.4. Design Recommendation

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Knee and hip moment data reported in Szecsi P1P2 and P1P3 groups, adapted from Ref. [21], 2014 Clarivate Analytics Web of Science, shown as the black curve. The numerical approximations are shown in blue. The curves are plotted from an absolute crank angle of 22${}^{\circ}$ to correspond with Szecsi’s data, which is measured from top dead center.

**Figure 2.**Kinematic model for the TRT mechanism, adapted from Ref. [23], 2014 Elsevier Ltd.

**Figure 4.**The color curves represent the power calculated by the TRT model at the crank center attributed to the knee (cyan), hip (magenta) and total (blue). The calculated data overlays the black curves that represent the power reported by Szecsi.

**Figure 6.**Kinematic model for the CDT mechanism, adapted from Ref. [23], 2014 Elsevier Ltd.

**Figure 8.**Kinematic model for the CRT mechanism adapted from Ref. [23], 2014 Elsevier Ltd.

**Figure 10.**The design space shown in green bounds the region of usable crank center and rocker arm pivot locations for both the CRT and CDT designs, in units of meters.

**Figure 11.**Joint torque curves for CRT design R2 and CDT design D1. Note that the axes limits are kept the same as Figure 5 for comparative purposes.

**Figure 12.**CRT instantaneous power curve at crank center for design R2 and CDT instantaneous power curve at crank center for design D1. Plot (

**a**) shows the CRT design R2 instantaneous power curve for the P1P2 group. Plot (

**b**) shows the CRT design R2 instantaneous power curve for the P1P3 group. Plot (

**c**) shows the CDT design D1 instantaneous power curve for the P1P2 group. Plot (

**d**) shows the CDT design D1 instantaneous power curve for the P1P3 group.

**Table 1.**Designs R1 and R2 were selected from the CRT optimization results and designs D1 and D2 were selected from the CDT optimization results for further analysis to identify the best design for spinal cord injured FES cyclists. The powers listed are in watts (W). For reference, the TRT is shown in Figure 2, the CDT in Figure 6 and the CRT in Figure 8.

Design | ${\overline{\mathit{P}}}_{\mathit{P}1\mathit{P}2}$ (W) | ${\overline{\mathit{P}}}_{\mathit{P}1\mathit{P}3}$ (W) | ${\overline{\mathit{P}}}_{\mathbf{av}}$ (W) | ${\mathit{\varphi}}_{\mathit{k}}$ | ${\mathit{\varphi}}_{\mathit{h}}$ | |
---|---|---|---|---|---|---|

TRT | 14.70 | 16.20 | 15.45 | 69–122${}^{\circ}$ | 85–120${}^{\circ}$ | |

CRT | R1 | 23.98 | 22.79 | 23.39 | 76–110${}^{\circ}$ | 111–118${}^{\circ}$ |

R2 | 21.36 | 29.00 | 25.18 | 73–112${}^{\circ}$ | 94–102${}^{\circ}$ | |

CDT | D1 | 31.29 | 29.34 | 30.32 | 70–121${}^{\circ}$ | 106–118${}^{\circ}$ |

D2 | 26.19 | 35.83 | 31.01 | 69–121${}^{\circ}$ | 92–109${}^{\circ}$ |

**Table 2.**Designs R1, R2, D1, D2 Optimized Design Parameter Results for further analysis to identify the best design for spinal cord injured FES cyclists. Link dimensions (and vectors) are in cm and $\lambda $ is a dimensionless ratio.

Design | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{6}$ | ${\mathit{R}}_{7}$ | ${\mathit{R}}_{8}$ | ${\mathit{R}}_{9}$ | ${\mathit{R}}_{10}$ | $\mathit{\lambda}$ | |
---|---|---|---|---|---|---|---|---|---|

TRT | [−74, 30]${}^{\mathsf{T}}$ | 15 | 7 | - | - | - | - | 1.31 | |

CRT | R1 | [−70, 11]${}^{\mathsf{T}}$ | 3 | 7 | [−50, 14]${}^{\mathsf{T}}$ | 17 | 51 | 5 | 2.03 |

R2 | [−62, 32]${}^{\mathsf{T}}$ | 4 | 8 | [−49, 1]${}^{\mathsf{T}}$ | 15 | 44 | 9 | 1.81 | |

CDT | D1 | [−62, 25]${}^{\mathsf{T}}$ | 16 | 10 | [−38, 9]${}^{\mathsf{T}}$ | 14 | 34 | 22 | 2.10 |

D2 | [−64, 28]${}^{\mathsf{T}}$ | 15 | 5 | [−29, 13]${}^{\mathsf{T}}$ | 7 | 32 | 19 | 2.04 |

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**MDPI and ACS Style**

Lanese, N.A.; Myszka, D.H.; Bazler, A.L.; Murray, A.P.
Six-Bar Linkage Models of a Recumbent Tricycle Mechanism to Increase Power Throughput in FES Cycling. *Robotics* **2022**, *11*, 26.
https://doi.org/10.3390/robotics11010026

**AMA Style**

Lanese NA, Myszka DH, Bazler AL, Murray AP.
Six-Bar Linkage Models of a Recumbent Tricycle Mechanism to Increase Power Throughput in FES Cycling. *Robotics*. 2022; 11(1):26.
https://doi.org/10.3390/robotics11010026

**Chicago/Turabian Style**

Lanese, Nicholas A., David H. Myszka, Anthony L. Bazler, and Andrew P. Murray.
2022. "Six-Bar Linkage Models of a Recumbent Tricycle Mechanism to Increase Power Throughput in FES Cycling" *Robotics* 11, no. 1: 26.
https://doi.org/10.3390/robotics11010026