# Maximizing cycling efficiency: innovative bicycle drive mechanisms tailored to individual muscular capacities

Sabrina Otmani, Andrew Murray, Christine Azevedo Coste, François Bailly

► To cite this version:

Sabrina Otmani, Andrew Murray, Christine Azevedo Coste, François Bailly. Maximizing cycling efficiency: innovative bicycle drive mechanisms tailored to individual muscular capacities. Computer Methods in Biomechanics and Biomedical Engineering, 2025, pp.13. 10.1080/10255842.2025.2482894. hal-05006131

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Submitted on 5 Apr 2025

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<figure><img src="image_1.png" /></figure>
# Maximizing Cycling Efficiency: Innovative Bicycle Drive Mechanisms Tailored to Individual Muscular Capacities

Sabrina OTMANI *<fnref n="1" />, Andrew MURRAY ², Christine AZEVEDO COSTE<fnref n="1" />, François BAILLY<fnref n="1" />

April 5, 2025

## 1 Abstract

This study explores the customization and optimization of three distinct bicycle drive mechanisms, leveraging an individual's biomechanical data to maximize pedaling power throughput. Our approach utilizes torque/velocity/position relationships of the hip and the knee, so that the kinematics of the optimized designs allow the user to pedal with maximized joint torques and thus, enhance the power produced at the crank. The method is applied to the cases of two users with significantly distinct anthropometries, showing noticeable changes in the drive mechanisms and demonstrating its effectiveness for personalizing bicycle designs. The study highlights the importance of considering individual biomechanical factors, showing that even slight variations in design can lead to changes in the cycling kinematics, resulting in improved performance. Simulation results also show increased mean power throughput for more complex drive mechanisms compared to a classical one, regardless of the user profile. This suggests that such designs should be capable of accommodating a range of cyclists, from recreational users to high-performance athletes, as well as individuals and athletes with motor impairments. These findings underline the potential of biomechanically-informed personalized bicycle drive mechanisms to optimize pedaling efficiency and enhance performance across diverse user groups.

**Keywords:** Optimization, Performance, Anthropometry, Experimental data, Kinematics, Cycle models

Word count: 4304

## 1 Introduction

Cycling has steadily gained in importance due to its eco-friendly and cost-effective nature (Pucher and Buehler, 2017). Despite their massive use, bicycles<fnref n="1" /> have remained relatively unchanged over the years. In the area of performance cycling, cycling strategies and their impact on performance have been documented (Hansen and Smith, 2009; Bessot, Nicolas, 2018; Kamba et al., 2022) and mechanical improvements like adjusting pedal systems (Kuhn, 2012), chainring shapes (Lucia et al., 2004) or assistance mechanisms with the use of batteries (Sousa et al., 2007) have been explored, but the design of the bicycle itself was not significantly questioned.

In the area of rehabilitation cycling, and notably in the case of Functional Electrical Stimulation (FES) cycling -known to provide a variety of physical and mental health benefits for people with spinal cord injury (SCI) (Berkelmans, 2008; Chen et al., 2005), bicycles design has been questioned, mainly to avoid so-called dead centers, i.e., points on the pedaling cycle where torque production is made difficult by the circular pedal path of classical bicycles. In a number of studies, the modeling of the leg+crank kinematic chain is typically achieved using four-bar (Saravanan et al., 2017) or up to six-bar (Bazler et al., 2021; Agarwal et al., 2015) mechanisms. In this re-

<footnote>* Corresponding author: sabrinaotmani@gmail.com</footnote>

<footnote>†¹INRIA, Université de Montpellier, Montpellier, France</footnote>

<footnote>‡²Mechanical and Aerospace Engineering, University of Dayton, OH, USA</footnote>

<footnote n="1">¹The term *bicycle* is used herein to refer collectively to all cycles including bicycles, trikes and others.</footnote>
spect, a classical bicycle's crank mechanism and its rider correspond to a four-bar mechanism, further referred to as a traditional recumbent mechanism (TRM).

In order to personalize the mechanisms (essentially the lengths of their component parts) a set of rider parameters are required (seating position, physiological parameters: height, weight, segment sizes, joint torques, FES-related parameters). In Angeli et al. (2001), a six-bar coupler-driver mechanism (CDM) was introduced to circumvent the dead centers issue, which was then used in Gfohler and Lugner (2004) to study the influence of individual parameters on optimal muscle stimulation patterns and power output. This mechanism was also optimized in Rasmussen et al. (2004) using a simple biomechanical model of a paraplegic's lower limb (three muscle groups) providing simulated input data about the rider's muscular capacities. The optimization was performed using a feasible directions method with an overall finite-difference design-sensitivity analysis. In Lanese et al. (2022), in addition to the TRM and CDM, a third bicycle design was investigated, the six-bar crank-rocker mechanism (CRM). In this work, the three mechanisms were optimized to improve the power throughput of FES-stimulated riders with lower limbs paralysis. The optimization was performed using a random pursuit algorithm which is a questionable choice in a framework where the optimization problem derivatives can be computed. The riders joint torque data were taken from Szecsi et al. (2014), a study conducted on SCI volunteers pedaling at 60 rpm. In the absence of an accompanying dataset, authors from Lanese et al. (2022) inferred the torque data from the figures of Szecsi et al. (2014), a methodology that lacks precision.

The objective of this paper is to demonstrate that personalized innovative bicycle drive mechanisms can increase the power throughput compared to a classical bicycle. To perform the mechanism personalization, numerical optimization and algorithmic differentiation are used in combination with joint torque data obtained through parametrized analytical expressions derived in Anderson et al. (2007)

In the following, we first introduce our mechanical modeling framework. Then, the generic optimization algorithm is presented and used to customize three distinct bicycle models according to the rider's anthropometry, in order to maximize their power throughput. This approach is tested on two distinct user profiles (age, gender, stature), highlighting the versatility of the proposed approach and the importance of personalization in bicy-

cle design. This is achieved by incorporating their respective muscular torque capacities, in the form of joint position/velocity/torque relationships (Anderson et al., 2007), into the optimization process. Beyond the scope of this paper, the proposed methodology could be relevant to human-in-the-loop robotics, such as collaborative and rehabilitation robotics.

# 2 Methods

## 2.1 Bicycle designs

In this paper, 3 bicycle designs are studied: the TRM, the CRM and the CDM, which development is based on the ones described in Lanese et al. (2022). While Lanese et al. (2022) focused on tricycles, we have generalized the models for cycles. They share the following features: angles ($\theta_i$) are defined positively from the vertical axis to the corresponding link $l_i$, counter clockwise, the ankle is locked at 90°, the lengths of thigh, shank and foot are measured on the typical profile, and the inclination of the back ($\theta_b$) is fixed at 57° (Diefenthaler et al., 2008). $\theta_1$, the angle between the vertical axis and $\vec{l}_1$, is set to 180°. $||\vec{l}_3||$ is a vector from the knee to the contact point between the foot and the pedal ($l_2$). Solving for the positions/velocities along one pedaling cycle requires the crank angle ($\theta_2$) as an input, the rest of the angles and parts/segments positions being deduced from the trigonometry. Computing the crank and ankle torques requires the hip and knee torques as inputs, the rest of the torques being deduced by quasi-static equations. Powers are then directly deduced from the velocities and torques. The specifics of each design are now presented:

**TRM** a classical four-bar mechanism (Figure 1, left: $l_1-l_2-l_3-l_{thigh}$). In this model, 3 parameters are optimized: $||\vec{l}_1||$, the distance between points O and A, $||\vec{l}_2||$, the crank length and $v_{crank}$, the crank velocity. Kinematics, ankle and crank torques are calculated using the free-body diagrams associated with the bodies in the following vector loop:

$$ \vec{l}_1 + \vec{l}_2 - \vec{l}_3 - \vec{l}_{thigh} = \vec{0} \qquad (1) $$

**CRM** a Watt II six-bar mechanism (Figure 1, middle: $\vec{l}_1, \vec{l}_7, \vec{l}_5, \vec{l}_3, \vec{l}_{thigh}$). In this model, 8 parameters are optimized: $||\vec{l}_1||$ the distance between points O and A, $||\vec{l}_2||$ the crank length, $||\vec{l}_4||$ the distance between points O and
$B, \theta_4$ the angle between the vertical axis and the $\vec{l}_4$ link $||\vec{l}_i||$, $i \in [5..7]$, the length of the corresponding links and $v_{crank}$, the crank velocity. Kinematics, ankle and crank torques are calculated using the free-body diagrams associated with bodies in following vector loops:

$$ \vec{l}_4 + \vec{l}_2 - \vec{l}_6 - \vec{l}_7 = \vec{0} \quad (2) $$

$$ \vec{l}_1 + \vec{l}_5 + \vec{l}_7 - \vec{l}_3 - \vec{l}_{thigh} = \vec{0} \quad (3) $$

**CDM** a Stephenson III six-bar mechanism (Figure 1, right: $\vec{l}_1, \vec{l}_4, \vec{l}_2, \vec{l}_8, \vec{l}_{3,4}, \vec{l}_{thigh}$). In this model, 9 parameters are optimized: $||\vec{l}_1||$ the distance between points O and A, $||\vec{l}_2||$ the crank length, $||\vec{l}_4||$ the distance between points O and B, $\theta_4$ the angle between the vertical axis and the $\vec{l}_4$ link, $||\vec{l}_i||$, $i \in [5..8]$, the length of the corresponding links, $\psi$ the angle between the $\vec{l}_5$ and $\vec{l}_6$ links and $v_{crank}$, the crank velocity. As for the CRM, the links $\vec{l}_1$ and $\vec{l}_4$ are considered fixed with the same definition. Links $\vec{l}_5, \vec{l}_6$ and $\vec{l}_8$ form a rigid triangular body which acts like the coupler of the system. $\vec{l}_5$ and $\vec{l}_7$ are no longer colinear. As for the previous designs, free-body diagrams and vector loops (Equations 4 and 5) were defined in order to calculate all the crank and ankle torques, powers and kinematics.

$$ \vec{l}_4 + \vec{l}_2 - \vec{l}_6 - \vec{l}_7 = \vec{0} \quad (4) $$

$$ \vec{l}_1 + \vec{l}_5 + \vec{l}_7 - \vec{l}_3 - \vec{l}_{thigh} = \vec{0} \quad (5) $$

## 2.2 Anthropometric parameters

For every bicycle design, hip and knee torques are required as inputs to solve for the ankle and crank torques. The computation of hip and knee torques for each typical profile was achieved by adapting the dataset from Anderson et al. (2007), which contains maximum torque values as a function of joint positions and velocities for several age, height, and gender categories. For that purpose and also to define the required lengths of the models, a set of anthropometric data (y) from two typical profiles, identified as M1 and F1, were collected. Limb lengths were precisely measured using standard tape measures, extending from proximal to distal endpoints. For the thigh, the measurement was realized between the greater trochanter and the bottom of the knee cap (head of the tibia). For the shank, the measurement was realized between the bottom of the knee cap and the lateral malleolus (de Leva, 1996). For the foot, the shoes size was used. The two profiles were chosen to be significantly different in terms of height, weight, age and limb lengths. This approach

<table><caption>Table 1: Anthropometric and personal data of typical profiles M1 and F1.</caption><thead><tr><th></th><th>M1</th><th>F1</th></tr></thead><tbody><tr><td>Gender</td><td>Male</td><td>Female</td></tr><tr><td>Age</td><td>34yrs</td><td>49yrs</td></tr><tr><td>Height (cm)</td><td>189</td><td>160</td></tr><tr><td>Weight (kg)</td><td>90</td><td>57</td></tr><tr><td>Thigh length (cm)</td><td>47.5</td><td>38</td></tr><tr><td>Shank length (cm)</td><td>44</td><td>39</td></tr><tr><td>Foot length (cm)</td><td>30</td><td>24</td></tr></tbody></table>

was employed to validate the methodology across varying anthropometries, demonstrating the adaptability of the approach in customizing bicycles and to study the impact of human body dimensions and proportions on the optimized designs. Detailed presentation of these parameters is provided in Table 1.

## 2.3 Optimization

### 2.3.1 Definition of the problem

The goal of the optimization is to determine the vector of parameters (x) for each design (TRM, CRM and CDM) and for each typical profile (M1 and F1) that maximizes an objective while satisfying a set of constraints (valid link lengths, range of motion, etc.):

$$ \begin{array}{ll} \max_{x} & J(\mathbf{x}, \mathbf{y}) \\ \text{s.t.} & x_i^{\text{low}} \le \mathbf{x}_i \le x_i^{\text{up}} \\ & g(\mathbf{x}) \ge 0 \end{array} \quad (6) $$

### 2.3.2 Objective function

The first term of the objective function was defined as the maximization of the mean power throughput produced over one full rotation of the bicycle crank:

$$ J_p = \bar{p}_{crank} \quad (7) $$

Although the ankle joint is locked (as in Lanese et al. (2022)), it remains necessary to account for the implicit torque required to maintain the ankle locked in position, to avoid non-feasible torque values.

The ankle torque bounds were defined for two distinct situations: when the foot pushes on the pedal and
<figure><img src="image_2.png" /><figcaption>Figure 1: The three models analyzed include the TRM (left), CRM (middle) and CDM (right). The optimized parameters are circled in red.</figcaption></figure>
when it pulls on it. As the ankle is naturally locked in dorsiflexion near 90°, the resistive torque observed when the foot pushes on the pedal is passively produced by the elastic components of the ankle and is thus similar to the ones observed in long jumps movements (Muraki et al., 2008). When the foot pulls on the pedal, Moraux et al. (2013) provides values of dorsiflexion ankle torque capacities based on height. Our ankle torques bounds were then defined based on these two papers. For typical profile M1, the plantar- and dorsi-flexion torque limits were respectively set at 200 Nm and 150 Nm. For typical profile F1, the plantar- and dorsi-flexion torque limits were respectively set at 175 Nm and 60 Nm. These prescribed values serve as the upper (dorsiflexion) and lower (plantarflexion) bounds for ankle torque in the mechanism design optimization for each profile. These ankle torque limits are considered in our optimization through a second term in the objective function ($J_a$, leveraging the sigmoid function, $sig$) which will be null unless the implicit ankle torque is out of these bounds. The ankle torque required by the optimized design ($\tau_a$) is deduced from the statics (sec. II.A) and is compared to its lower and upper bounds ($\tau_a^-$ and $\tau_a^+$)

$$J_a = sig(\tau_a - \tau_a^-) + sig(\tau_a^+ - \tau_a). \quad (8)$$

For the CDM, a third term in the objective function was introduced to account for angular limits on the hip joint, $J_h$ which would penalize the value of the objective function in case of limits violation. Finally, the global objective function is:

$$J = J_p - w_a * J_a - w_h * J_h, \quad (9)$$

with $w_a$ and $w_h$ weights to modulate the importance of $J_a$ and $J_h$ with regard to $J_p$.

## 2.4 Constraints

### 2.4.1 Grashof's constraint

Grashof's constraint ($g$ in (6)) is inherent to the presence of four-bar mechanisms in all studied designs (Grashof, 1890). This constraint guarantees that the shortest link of a four-bar mechanism can rotate freely, if the sum of the shortest and longest lengths (respectively $s$ and $l$) is less than or equal to the sum of the remaining two links (respectively $p$ and $q$).

$$g(\mathbf{x}) = s + l - p - q \le 0 \quad (10)$$

For the TRM, the Grashof constraint yields:

$$\| \vec{l}_2 \| + \| \vec{l}_1 \| \le \| \vec{l}_3 \| + \| \vec{l}_{thigh} \|. \quad (11)$$

For the CRM and CDM, the Grashof constraint yields:

$$\| \vec{l}_2 \| + \| \vec{l}_6 \| \le \| \vec{l}_4 \| + \| \vec{l}_7 \|. \quad (12)$$

### 2.4.2 Bounds of optimization variables

Bounds were defined for each parameter and for each design ($x_i^{low}$ and $x_i^{up}$ in (6)). All the bounds for each design are presented in Table 2 for both typical profiles. Also, the maximal velocity achievable was set to 120 rpm which correspond to 720 degrees/see. As a maximal power throughput is wanted in our simulations for each design, the velocity of 120 rpm was chosen as the maximal velocity allowed to reach. This velocity corresponds to a common velocity of trained cyclist for which the maximal power throughput is obtained (Martin and Brown, 2009; McDaniel et al., 2014). However, as 90 rpm is also a velocity commonly reported in bike studies, mean power throughput were computed for each subject and each architectures at 90 rpm.

<table><thead><tr><th rowspan="2">M1</th><th colspan="2">TRM</th><th colspan="2">CRM</th><th colspan="2">CDM</th></tr><tr><th>Min</th><th>Max</th><th>Min</th><th>Max</th><th>Min</th><th>Max</th></tr></thead><tbody><tr><td>||<span style="text-decoration: overline;">l</span><sub>1</sub>|| (m)</td><td>0</td><td>1</td><td>0.8</td><td>1</td><td>0.8</td><td>1</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>2</sub>|| (m)</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0.1</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>4</sub>|| (m)</td><td>/</td><td>/</td><td>0.2</td><td>1</td><td>0.1</td><td>0.4</td></tr><tr><td>θ<sub>4</sub> (°)</td><td>/</td><td>/</td><td>30</td><td>90</td><td>40</td><td>90</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>5</sub>|| (m)</td><td>/</td><td>/</td><td>0</td><td>0.4</td><td>0.1</td><td>0.4</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>6</sub>|| (m)</td><td>/</td><td>/</td><td>0.15</td><td>1</td><td>0</td><td>0.2</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>7</sub>|| (m)</td><td>/</td><td>/</td><td>0.1</td><td>1</td><td>0</td><td>0.2</td></tr><tr><td>ψ (°)</td><td>/</td><td>/</td><td>/</td><td>/</td><td>0</td><td>30</td></tr><tr><td>crank<sub>vel</sub>(°/s)</td><td>0</td><td>720</td><td>0</td><td>720</td><td>0</td><td>720</td></tr></tbody></table>

<table><caption>Table 2: Bounds of the parameters for the TRM, CRM and CDM designs for M1 (top) and F1 (bottom).</caption><thead><tr><th rowspan="2">F1</th><th colspan="2">TRM</th><th colspan="2">CRM</th><th colspan="2">CDM</th></tr><tr><th>Min</th><th>Max</th><th>Min</th><th>Max</th><th>Min</th><th>Max</th></tr></thead><tbody><tr><td>||<span style="text-decoration: overline;">l</span><sub>1</sub>|| (m)</td><td>0</td><td>1</td><td>0.6</td><td>1</td><td>0.6</td><td>0.8</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>2</sub>|| (m)</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0.06</td><td>0.2</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>4</sub>|| (m)</td><td>/</td><td>/</td><td>0.2</td><td>1</td><td>0</td><td>0.4</td></tr><tr><td>θ<sub>4</sub> (°)</td><td>/</td><td>/</td><td>30</td><td>90</td><td>0</td><td>90</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>5</sub>|| (m)</td><td>/</td><td>/</td><td>0</td><td>0.3</td><td>0.1</td><td>0.4</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>6</sub>|| (m)</td><td>/</td><td>/</td><td>0.15</td><td>1</td><td>0.1</td><td>0.5</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>7</sub>|| (m)</td><td>/</td><td>/</td><td>0.1</td><td>1</td><td>0.1</td><td>0.5</td></tr><tr><td>ψ (°)</td><td>/</td><td>/</td><td>/</td><td>/</td><td>0</td><td>90</td></tr><tr><td>crank<sub>vel</sub>(°/s)</td><td>0</td><td>720</td><td>0</td><td>720</td><td>0</td><td>720</td></tr></tbody></table>
## 2.5 Hip/Knee torque data and interpolation

Position/velocity/torque relationships used in this study are derived from Anderson et al. (2007), which proposed analytical expressions for maximum voluntary joint torque based on joint angle and angular velocity. These joints-specific relationships include passive and active torque components and they are parameterized by gender, age, height, weight and the direction of movement (extension or flexion). However, in their original form, these

<figure><img src="image_3.png" /><figcaption>Figure 2: Position/velocity/torque relationship for knee flexion (F1) from Anderson et al. (2007), approximated by radial basis functions and extended outside the original definition domain. A. Out of the original domain and of the margin B. Out of the original domain and in the margin C. In the original domain</figcaption></figure>

equations are difficult to use in a gradient-based optimization framework because of the many if/else statements and the non-differentiable functions. Radial basis functions (RBF) were used to approximate these functions, an interpolation method suitable for high-dimensional scattered data interpolation. To let the solver search outside the original domain of angular joint positions and velocities acquired in Anderson et al. (2007), the input data are smoothed to zero to cover a larger set of positions and velocities by means of a transition margin (Fig. 2 and Tab. 3). This allowed us to interpolate torque surfaces

for each joint, each movement direction and each type of users, providing analytical and differentiable expressions compatible with gradient-based solvers. As these data are inputs to the algorithm solving for the crank torque, this step was crucial for the solver to interact with extensive biomechanical data.

<table><caption>Table 3: Bounds defined in Anderson et al. (2007) and acceptable margins considered for both angular position and velocity for each joint.</caption><thead><tr><th></th><th>Hip</th><th>Knee</th></tr></thead><tbody><tr><td>Min. angular position (°)</td><td>-28.6</td><td>0</td></tr><tr><td>Max. angular position (°)</td><td>74.4</td><td>103.1</td></tr><tr><td>Min. angular velocity (°/s)</td><td>-200</td><td>-200</td></tr><tr><td>Max. angular velocity (°/s)</td><td>250</td><td>400</td></tr><tr><td>Acceptable margin</td><td>10</td><td>10</td></tr></tbody></table>

## 2.6 Solvers: a two-stage approach

For each bicycle design and profile, the optimization was conducted in two-stages: an initial solution was found using an early-stopped genetic algorithm, which was then refined with a gradient-based solver.

The initial solution was determined using PyGAD, a Python library dedicated to genetic algorithm optimization (Gad, 2023). As a heuristic stochastic optimization method, PyGAD does not provide an optimal solution but a good solution satisfying the constraints. To refine the solutions, the optimization problem was formulated in a symbolic manner, using CasADi, an algorithmic differentiation software (Andersson et al., 2018). Then, the constrained non-linear optimization problem was solved using Interior Point OPTimizer (IPOPT) (Wächter and Biegler, 2006), an interior point library in which accuracy and speed are improved by the exact computation of gradients, Jacobians and Hessians performed by CasADi. The initial solution found with PyGAD was used as a warm start for the second optimization.
<figure><img src="image_4.png" /><figcaption>Figure 3: TRM (left), CRM (middle) and CDM (right) designs for M1 (top) and F1 (bottom). The dotted blue points represent the trajectory of the contact point (metatarses) during one crank cycle for a crank velocity of 120 rpm.</figcaption></figure>
<table><caption>Table 4: Optimal parameters found for the TRM, CRM and CDM designs for both typical profiles for a crank velocity of 120 rpm</caption><thead><tr><th rowspan="2"></th><th colspan="2">TRM</th><th colspan="2">CRM</th><th colspan="2">CDM</th></tr><tr><th>M1</th><th>F1</th><th>M1</th><th>F1</th><th>M1</th><th>F1</th></tr></thead><tbody><tr><td>||<span style="text-decoration: overline;">l</span><sub>1</sub>|| (m)</td><td>0.43</td><td>0.41</td><td>0.8</td><td>0.65</td><td>0.8</td><td>0.8</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>2</sub>|| (m)</td><td>0.09</td><td>0.08</td><td>0.09</td><td>0.08</td><td>0.08</td><td>0.07</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>4</sub>|| (m)</td><td>/</td><td>/</td><td>0.21</td><td>0.21</td><td>0.14</td><td>0.22</td></tr><tr><td>θ<sub>4</sub> (°)</td><td>/</td><td>/</td><td>89.95</td><td>89.95</td><td>39.53</td><td>11.46</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>5</sub>|| (m)</td><td>/</td><td>/</td><td>0.4</td><td>0.3</td><td>0.34</td><td>0.29</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>6</sub>|| (m)</td><td>/</td><td>/</td><td>0.15</td><td>0.15</td><td>0.08</td><td>0.1</td></tr><tr><td>||<span style="text-decoration: overline;">l</span><sub>7</sub>|| (m)</td><td>/</td><td>/</td><td>0.16</td><td>0.13</td><td>0.12</td><td>0.24</td></tr><tr><td>ψ (°)</td><td>/</td><td>/</td><td>/</td><td>/</td><td>29.79</td><td>38.38</td></tr><tr><td>crank<sub>vel</sub>(°/s)</td><td>719.63</td><td>719.63</td><td>719.63</td><td>719.63</td><td>719.63</td><td>719.63</td></tr></tbody></table>

# 3 Results

## 3.1 Optimal bicycle designs and foot trajectory

The optimal parameters found are presented in Table 4 for each case study (M1 and F1) and each design. For every bicycle design and for both typical profiles, the maximal possible crank velocity was reached. The crank length $R_2$ (optimized around 8 cm depending on the case study) tends to be shorter than classical crank lengths observed in the cycling market (17cm). For each optimized bicycle design, the effect of considering anthropometric data for M1 and F1 is clear. For instance, with TRM and CRM, F1 ends up with a smaller $||<span style="text-decoration: overline;">l</span><sub>1</sub>||$ (meaning a shorter distance between the seat and the crank shaft). For the CDM, $<span style="text-decoration: overline;">l</span><sub>7</sub>$ is smaller and more inclined for M1 than for F1. The differences are small except for CDM where large differences are observed for $||<span style="text-decoration: overline;">l</span><sub>4</sub>||,||<span style="text-decoration: overline;">l</span><sub>5</sub>||,||<span style="text-decoration: overline;">l</span><sub>7</sub>||$ and $θ_4$ (mean variation between F1 and M1 of: 2% for TRM, 3% for CRM and 32% for CDM, M1 considered as the reference). In Fig. 3, the images were rendered at the same crank angle for M1 and F1 for each design. Different cycling positions can be observed between the two typical profiles across all designs and, hence, different joint kinematics.

Another way to understand the impact of the different optimized designs is to look at the trajectory of the foot during one crank cycle indicated by the dotted blue lines

in Figure 3. As presented in the previous sections, the different designs lead to different behaviors and, thus, to different foot trajectories. For the TRM model, we observe a classical circular trajectory of the contact point which corresponds to the pedaling kinematics of a classical bicycle (with a shorter crank than usual). For the CRM model, we observe a "press behavior". The movement is curvilinear for both typical profiles with a larger magnitude for M1 than F1. Finally for the CDM model, we observe a reniform trajectory, of different shape for the two typical profiles, with a similar quick flexed movement at the end of the trajectory (when the foot goes up). These results show the necessity of considering anthropometric data in the optimization of bicycles design. The mechanism parts are tailored to the muscular capacities of a user, leading to a customized foot trajectory and consequently, an optimized power throughput.

## 3.2 Mean power throughput

<table><caption>Table 5: Comparison of mean power throughput ($J_p$) produced and its standard deviation for F1 and M1 and for each optimal design for one leg at 90 rpm (top) and 120 rpm (bottom)</caption><thead><tr><th rowspan="2">90 rpm</th><th colspan="2">TRM</th><th colspan="2">CRM</th><th colspan="2">CDM</th></tr><tr><th>M1</th><th>F1</th><th>M1</th><th>F1</th><th>M1</th><th>F1</th></tr></thead><tbody><tr><td>J<sub>p</sub> (W)</td><td>377.49</td><td>173.0</td><td>1187.8</td><td>493.1</td><td>1254.18</td><td>693.29</td></tr><tr><td>std J<sub>p</sub> (W)</td><td>25.88</td><td>13.60</td><td>194.57</td><td>58.73</td><td>145.75</td><td>87.68</td></tr></tbody><thead><tr><th rowspan="2">120 rpm</th><th colspan="2">TRM</th><th colspan="2">CRM</th><th colspan="2">CDM</th></tr><tr><th>M1</th><th>F1</th><th>M1</th><th>F1</th><th>M1</th><th>F1</th></tr></thead><tbody><tr><td>J<sub>p</sub> (W)</td><td>403.05</td><td>184.6</td><td>1305.51</td><td>568.1</td><td>1523.76</td><td>686.14</td></tr><tr><td>std J<sub>p</sub> (W)</td><td>19.14</td><td>10.43</td><td>94.93</td><td>50.54</td><td>130.9</td><td>65.76</td></tr></tbody></table>

Regardless of the crank velocity studied and regarding the mean power throughput, M1 has a higher $J_p$ than F1 for all designs (Tab. 5), which is expected given the respective muscular capacities of each typical profile (Tab. 1). Indeed, for 120 rpm, the mean power throughput of M1 is 54% higher in TRM, 56% higher in CRM and 55% higher in CDM than F1. The gap between M1 and F1 is then almost constant for all the designs. For 90 rpm, the mean power throughput of M1 is 54% higher in TRM, 58% higher in CRM and 44% higher in CDM than F1. The mean power throughput obtained is also higher
with a crank velocity of 120 rpm than 90 rpm except for $F_1$ with the CDM. The increase of power throughput follows the design's complexity (TRM followed by CRM and then CDM) which is expected since more complex mechanisms can lead to more arbitrary foot trajectories (position and velocity) likely to be closer to a zone of optimal torque generation. At 120 rpm, the mean power throughput gap between CRM and CDM is smaller than the one between TRM and CRM for $M_1$ (14% for $M_1$ between CRM/CDM and 69% between CRM/TRM) and $F_1$ (17% for $M_1$ between CRM/CDM and 64% between CRM/TRM). Here again, the gap is almost constant between each pair of designs for both typical profiles.

## 3.3 Hip and knee trajectories

<figure><img src="image_5.png" /><figcaption>Figure 4: Hip (top) and Knee (bottom) angular positions for $M_1$ and $F_1$ found for each design for a crank velocity of 120 rpm. The horizontal axis represents the crank angle $\theta_2$ in degrees where 0° corresponds to the vertical upper position. The vertical axis corresponds to the joint (hip and knee) angular position in degrees.</figcaption></figure>

**Hip joint** For the hip angular position, a higher range of movement (RoM) is observed using the CDM and CRM designs compared to the use of the TRM for both typical profiles. $F_1$ has almost the same RoM with CRM and CDM as well as the same maximal extension and flexion, as seen in Figure 4 and Table 6. A difference is observed in the timing where the maximal extension is achieved for these two designs. For $M_1$, there is a steady increase of

the RoM between the three designs (26° for TRM, 43° for CRM and 64° for CDM). Thus, the RoM of CRM is 37% higher than the RoM of TRM and 32% lower than the RoM of CDM.

**Knee joint** For the knee angular position, the difference between CRM and CDM for $M_1$ is different than for the hip joint: indeed, the RoM of CDM is $\approx 13\%$ higher than the RoM of CRM and the RoM of CRM is 72% higher than the one of TRM. For $M_1$, the maximal flexion is the same for CRM and CDM but the maximal extension is higher for CDM than CRM especially for the knee (30% of gap for the maximal extension for $M_1$ between CRM and CDM for the knee vs 19% for the hip). For $F_1$, the difference between the CRM and the CDM lies in the timing where the maximal extension is achieved, its value (120° in CRM and 97° in CDM) and the RoM obtained for $F_1$ is higher for CRM than CDM (Table 6).

Globally, using CRM and CDM allows having more extension and flexion for the hip and knee joints. Depending on what behavior is desired (more flexed, more extended, in a rehabilitation context for instance), one design can produce a better output compared to the others.

<table><caption>Table 6: Range of motion (RoM), maximal extension and flexion for the hip and knee of $M_1$ and $F_1$ for each design expressed in degrees (°) for a crank velocity of 120 rpm.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="2">TRM</th><th colspan="2">CRM</th><th colspan="2">CDM</th></tr><tr><th>Hip</th><th>Knee</th><th>Hip</th><th>Knee</th><th>Hip</th><th>Knee</th></tr></thead><tbody><tr><td rowspan="3">M1</td><td>RoM(°)</td><td>26</td><td>22</td><td>43</td><td>95</td><td>64</td><td>109</td></tr><tr><td>Max. Extension (°)</td><td>71</td><td>40</td><td>59</td><td>105</td><td>85</td><td>130</td></tr><tr><td>Max. Flexion (°)</td><td>45</td><td>18</td><td>16</td><td>10</td><td>21</td><td>21</td></tr><tr><td rowspan="3">F1</td><td>RoM(°)</td><td>26</td><td>28</td><td>56</td><td>110</td><td>53</td><td>82</td></tr><tr><td>Max. Extension (°)</td><td>71</td><td>50</td><td>69</td><td>120</td><td>67</td><td>97</td></tr><tr><td>Max. Flexion (°)</td><td>45</td><td>22</td><td>13</td><td>10</td><td>14</td><td>15</td></tr></tbody></table>

# 4 Discussion

This paper presents three bicycle designs optimized using torque data from the literature and anthropometric data from two distinct typical profiles ($M_1$ and $F_1$) to produce a maximal power throughput. The objective of this paper is to demonstrate the increase in power throughput provided by innovative bicycle drive mechanisms compared to classical ones. To ensure a fair comparison, all the
mechanisms presented were personalized using riders anthropometries, to best exploit their muscular capabilities. The main contributions of this work are as follows:

* **Torque Data:** Incorporation of exhaustive hip and knee torque data into the optimization process for three distinct bicycle mechanisms, considering two typical user profiles. These torque values are derived from parameterized analytical expressions that link joint torque, angular position, and angular velocity.

* **Optimization Framework:** Development of a two-stage numerical optimization combined with algorithmic differentiation: a genetic algorithm for warm-starting and a gradient-based solver to refine the solution.

* **Application and Validation:** Application of our method to two distinct user profiles (differing in age, gender, and stature) to demonstrate the versatility of our approach and highlight the importance of customizing bicycle designs based on individual user data, particularly regarding power throughput.

Finding optimal bicycle designs for both profiles shows the versatility of our method and its potential for different typical profiles including people with different capabilities or different anthropometries. For each design, we observed similar structures for both typical profiles, with some differences in link lengths and foot trajectories (e.g., wider trajectories). These differences remained consistent across all designs studied (TRM, CRM, and CDM). Regarding the optimized mean power throughput, the values obtained for *M1* are consistent with the literature for the TRM. Indeed, in Martin and Brown (2009), thirteen trained cyclists performed a thirty second maximal cycling trial on an isokinetic bicycle ergometer at 120 rpm. The average power produced at the pedal for a complete revolution during the three seconds of maximal cycling is 540 W (standard deviation -std-: 34 W) for a complete revolution which is in the same magnitude as *M1* for TRM. The lower power throughput produced by both typical profiles in the TRM compared to the literature can be due to the fact that these typical profiles are not trained cyclists: indeed, the joint torques considered in our study are obtained from active adults and the typical profiles studied have a different anthropometry compared to a common trained cyclist. For the CRM and CDM, the average power throughput observed is higher than for the TRM for both profiles. For *M1*, the power throughput with CRM and CDM is of the same order of magnitude as those in Padulo et al. (2017), where lower-limb power was compared between squat and leg press movements.

The maximal power output obtained was 835 W (std: 164 W) for the leg press and 1366 W (std: 384 W) for the squat. As the movement for CRM looks more like a leg press than traditional cycling, these results are interesting to put in perspective. Our different results for CRM and CDM compared to TRM can be explained by the difference in these designs with the classical bike which leads to more arbitrary foot trajectories likely to produce higher joint torques. However, these results need to be validated in an experimental setup with the CRM and CDM designs. Although both six-bar mechanisms are deemed suitable and will yield acceptable mechanical design, their increased weight might impair the simulation results.

Regarding the joint angular position, having different designs allows different kinematics, with notably larger RoMs especially when using CRM and CDM for both profiles. Differences appear in the timing where the maximal extension/flexion appears and the value of the maximal extension between these two designs. In both typical profiles, the TRM presents a more flexed behavior of the knee compared to the other designs.

In our study, data from Anderson et al. (2007) was used for the hip and knee joint as inputs. These values are obtained during maximum voluntary contraction which can also explain the high mean power throughput obtained for the several designs. In that dataset, the authors studied the hip with the knee extended which does not match the kinematics of pedaling, and which has a major impact on the hip torque value because of bi-articular muscles such as the *rectus femoris*. Other torque values should be obtained in the future to address this issue.

During the optimization, several penalties were considered for each joint regarding the torque or joints angular positions. In this study, PyGAD and IPOpt were chosen to solve the Non-linear problem (NLP). Other methods should be tested to optimize our results as the behavior of IPOpt can vary regarding the initial conditions and the parameters bounds (especially for CRM and CDM). The technique presented combines kinematics, statics, the data from joint torque studies, and optimization to define bike designs that maximize power throughput. As the dynamics were ignored in this initial work, the results should be qualified accordingly. Long-term considerations should incorporate dynamic factors, and these results, as well as future findings, should be tested under experimental conditions. Also, the ankle is considered locked at 90° as in Lanese et al. (2022) and as a first hypothesis to simply our models but this can differ from an actual cycling situation. It presents a limitation and
can possibly impact results obtained for the three models. The experimental validation of the results has not yet been carried out. For future works, as experimental validation of the results has not yet been carried out, a prototype of the optimized TRM design will soon be manufactured to validate the simulation. Also, the maximal velocity allowed was set to 120 rpm which allows having the maximal power throughput compared to other velocity (McDaniel et al., 2014) however it does not represent the common established velocity for traditional bikes (90 rpm). Indeed, due to the lack of complete power data at 90 rpm in the literature (Martin and Brown, 2009), the presentation and discussion were focused on the results obtained at 120 rpm. Further comparisons of results obtained at different crank velocity (at 90 rpm for example) should be performed. Experimental acquisitions will be conducted to adjust the input of joint torques (hip and knee) to obtain joint torques in configurations that more closely resemble cycling.

In summary, an approach to optimize bicycle design dimensions driven by experimentally collected torque and anthropometric data from two distinct rider profiles is proposed for three distinct mechanisms. These customized designs enhanced the mean power throughput produced while leveraging experimental joint torque data and satisfying a set of constraints for both typical profiles. The simulation results show that higher mean power throughout are observed for both typical profiles with more complex designs (CRM and CDM) compared to the classical bicycle. The differences in the lengths of the model links between the two typical profiles across the considered bicycle designs, as well as the mean power throughput achieved in simulation for each design, can be explained by the differences in their anthropometries. These results highlights the need to integrate some biomechanical knowledge in bicycle development to reach higher level of performance and possibly disrupt existing historical technological solutions. More generally, the methodology presented in this study applies to any human-in-the-loop mechanism design, such as collaborative or rehabilitation robotics.

## Acknowledgement

This work was supported by "Défi Clé Robotique centrée sur l'humain" funded by Région Occitanie, France.

## Conflict of interest statement

The authors confirm that there are no conflicts of interest regarding the work described in the current manuscript.

## Appendices

A technical report which presents the equations for all the designs of the position, velocity and static analysis is available at: https://hal.science/hal-04647805

## References

Agarwal, S., Badduri, J., and Bandyopadhyay, S. (2015). Optimal synthesis of six-bar function generators. In *Proc. 14th IFToMM World Congress, Taipei, Taiwan, October*, pages 25–30.

Anderson, D. E., Madigan, M. L., and Nussbaum, M. A. (2007). Maximum voluntary joint torque as a function of joint angle and angular velocity: Model development and application to the lower limb. *Journal of Biomechanics*, 40(14):3105–3113.

Andersson, J. A. E., Gillis, J., Horn, G., Rawlings, J. B., and Diehl, M. (2018). CasADi – A software framework for nonlinear optimization and optimal control. *Mathematical Programming Computation*.

Angeli, T., Gföhler, M., Eberharter, T., Lugner, P., Rinder, L., and Kern, H. (2001). Optimization of the pedal path for cycling powered by lower extremity muscles activated by functional electrical stimulation. *Computer Methods in Biomechanics and Biomedical Engineering-3. Gordon and Breach*.

Bazler, A. L., Myszka, D. H., and Murray, A. P. (2021). The Redesign of a Recumbent Tricycle Using a Crank Rocker Mechanism to Increase Power Throughput in FES Cycling. *International Design Engineering Technical Conferences and Computers and Information in Engineering Conference*, 8A.

Berkelmans, R. (2008). Fes cycling. *Journal of Automatic Control*, 18(2):73–76.

Bessot, Nicolas (2018). Effects of pedal rate and power output on cycling kinematics. *Mov Sport Sci/Sci Mot*, (101):23–31.
613 Chen, S.-C., Lai, C.-H., Chan, W. P., Huang, M.-H., Tsai, H.-W., and Chen, J.-J. J. (2005). Increases in bone mineral density after functional electrical stimulation cycling exercises in spinal cord injured patients. *Disability and Rehabilitation, 27*(22):1337-1341.

618 de Leva, P. (1996). Joint center longitudinal positions computed from a selected subset of chandler's data. *Journal of Biomechanics, 29*(9):1231-1233.

621 Diefenthaler, F., Carpes, F., Bini, R., Mota, C., Carlos, A., and Guimarães, S. (2008). Methodological proposal to evaluate sagittal trunk and spine angle in cyclists: Preliminary study. *Brazilian Journal of Biomotricity*.

625 Gad, A. F. (2023). Pygad: An intuitive genetic algorithm python library. *Multimedia Tools and Applications*, pages 1-14.

628 Gfohler, M. and Lugner, P. (2004). Dynamic simulation of fes-cycling: influence of individual parameters. *IEEE Trans. on Neural Systems and Rehabilitation Engineering*, 12(4):398-405.

632 Grashof, F. (1890). Theoretische maschinenlehre. *Voss*, 3.

634 Hansen, E. A. and Smith, G. (2009). Factors affecting cadence choice during submaximal cycling and cadence influence on performance. *International Journal of Sports Physiology and Performance*, 4(1):3-17.

638 Kamba, M., Naito, H., Ozaki, H., Machida, S., and Kata-moto, S. (2022). Effect of gear ratio and cadence on gross efficiency and pedal force effectiveness during multistage graded cycling test using a road racing bicycle. *Sports*, 11:5.

643 Kuhn, B. (2012). A comparison of three bicycle pedal types and power output. *Master's Theses*.

645 Lanese, N. A., Myszka, D. H., Bazler, A. L., and Murray, A. P. (2022). Six-bar linkage models of a recumbent tricycle mechanism to increase power throughput in fes cycling. *Robotics*, 11(1).

649 Lucia, A., Balmer, J., Davison, R., Perez, M., Santalla, A., and Smith, P. (2004). Effects of the rotor pedalling system on the performance of trained cyclists during incremental and constant-load cycle ergometer tests. *Internat. Journal of Sports Medicine*, 25(7):479-485.

654 Martin, J. C. and Brown, N. A. (2009). Joint-specific power production and fatigue during maximal cycling. *Journal of Biomechanics*, 42(4):474-479.

657 McDaniel, J., Behjani, N. S., Elmer, S. J., Brown, N. A., and Martin, J. C. (2014). Joint-specific power-pedaling rate relationships during maximal cycling. *Journal of Applied Biomechanics*, 30(3):423-430.

661 Moraux, A., Canal, A., Ollivier, G., Ledoux, I., Doppler, V., Payan, C., and Hogrel, J.-Y. (2013). Ankle dorsi- and plantar-flexion torques measured by dynamometry in healthy subjects from 5 to 80 years. *BMC musculoskeletal disorders*, 14:104.

666 Muraki, Y., Ae, M., Koyama, H., and Yokozawa, T. (2008). Joint torque and power of the takeoff leg in the long jump. *International Journal of Sport and Health Science*, 6:21-32.

670 Padulo, J., Migliaccio, G. M., Ardigò, L., Leban, B., Cosso, M., and Samozino, P. (2017). Lower limb force, velocity, power capabilities during leg press and squat movements. *International Journal of Sports Medicine*, 38:1-7.

675 Pucher, J. and Buehler, R. (2017). Cycling towards a more sustainable transport future. *Transport reviews*, 37(6):689-694.

678 Rasmussen, J., Christensen, S. T., Gföhler, M., Damsgaard, M., and Angeli, T. (2004). *Structural and Multidisciplinary Optimization*, 26:132-138.

681 Saravanan, S., Rathish, R., Balakrishnan, D. N., Venkata-tramanan, C., Vignesh, M., Vijay, M., and Samuel, R. M. (2017). Design and fabrication of bicycle using four bar mechanism. *Internat. Journal of Scientific Research in Science, Engineering and Technology*, 3(2).

686 Sousa, D. M., Costa Branco, P. J., and Dente, J. A. (2007). Electric bicycle using batteries and supercapacitors. In *2007 European Conference on Power Electronics and Applications*, pages 1-10.

690 Szecsi, J., Straube, A., and Fornusek, C. (2014). A biomechanical cause of low power production during fes cycling of subjects with sci. *Journal of neuroengineering and rehabilitation*, 11:1-12.

694 Wächter, A. and Biegler, L. (2006). On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. *Mathematical programming*, 106:25-57.