# Data-driven biomechanical optimisation of bicycle and crank-pedal design for maximizing power output

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

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Sabrina Otmani, Andrew Murray, Christine Azevedo Coste, François Bailly. Data-driven biomechanical optimisation of bicycle and crank-pedal design for maximizing power output. Computer Methods in Biomechanics and Biomedical Engineering, 2025, 10.1080/10255842.2025.2586147. hal-05346904

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Submitted on 4 Nov 2025

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<figure><img src="image_1.png" /></figure>

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## Highlights

# Data-driven biomechanical optimisation of bicycle and crank-pedal design for maximizing power output

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

* A personalized crank-pedal mechanism that optimizes the pedal path kinematics to maximize power throughput.

* A data-driven approach that leverages joint position/velocity/torque relationships in the optimization of the crank-pedal mechanism to better exploit the user's torque-generating capabilities.

* A generic method to personalize bicycle design relying on a genetic algorithm, algorithmic differentiation and a gradient-based non-linear solver.
# Data-driven biomechanical optimisation of bicycle and crank-pedal design for maximizing power output

Sabrina Otmani<sup>a</sup>, Andrew Murray<sup>b</sup>, Christine Azevedo-Coste<sup>a</sup> and François Bailly<sup>a</sup>

## ARTICLE INFO

**Keywords:**

Cycling
Performance
Optimization
Personalization
Innovative Bike mechanism

## ABSTRACT

This paper presents a novel crank-pedal mechanism designed to optimize pedal-path kinematics. The goal of the design is to maximize power throughput by utilizing torque-generating capabilities produced by individual riders. The dimensions of the design are determined through an optimization algorithm that modifies the crank length, pedal shape, and frame geometry. The optimization uses the joint position, velocity, and torque relationships of a user. As such, the solver can take advantage of musculoskeletal motions that generate sustained large torques. The approach is tested in simulation with data from two user profiles, demonstrating similarities and variations that morphologies can produce in the design. In both cases, the optimized designs with the new crank-pedal mechanism improved the mean crank power during a crank revolution by approximately 15% compared to a traditional personalized bicycle. These results suggest that altering the dimensions of a bike using biomechanical data in the design process could have a significant impact on the pedaling performance of individuals, ranging from everyday users to athletes and individuals with motor impairments.

## 1. INTRODUCTION

General interest in cycling has grown steadily in recent decades due to its sustainability and affordability (Pucher and Buehler, 2017). Besides social or environmental considerations, cycling has been widely studied from a biomechanics perspective, demonstrating that the relationship between biomechanical assessments and cycling performance is both intricate and complex (Turpin and Watier (2020)), that small adjustments can have a substantial impact on lower limb cycling kinematics and kinetics (Menard et al. (2020); Bini (2021); Reynolds et al. (2025)) and that biomechanical numerical simulations can help us gaining insight in these questions (Sinclair et al. (2018); Clancy et al. (2023)).

Although not always performance-oriented, this increased attention has led to changes in the rotor pedaling system (Lucia et al., 2004; Santalla et al., 2002), the shape of the chainring (Rankin and Neptune, 2008), weight reduction (Groves, 2010), the introduction of electric assistance (Salmerón-Manzano and Manzano-Augliaro, 2018), and other ergonomic adjustments (Kennedy, 2014). Intriguingly, the underlying dimensions of the frame, crank, and pedal have remained relatively unchanged. Demonstrating the potential of bike design changes, Functional Electrical Stimulation (FES)-assisted cycling<fnref n="1" /> research is inspiring as it involves recumbent tricycles in which several dimensional changes are proposed. (Berkelmans, 2008; Chen et al., 2005; Lanese et al., 2022).

The kinematic chain defined by the frame, leg, and crank has been successfully modeled as a four-bar or six-bar mechanism (Bazler et al., 2021; Lanese et al., 2022; Otmani et al., 2025). To personalize these mechanisms, specifically the lengths of their components, various rider parameters are introduced, including seat position and physiological information (like height, weight, body segment sizes, and joint torques). Angeli et al. (2001) introduced a six-bar coupler-driver tricycle to address the dead center issue, the crank positions where torque production is most limited. This mechanism was later used in FES-assisted cycling by Gfohler and Lugner (2004) to examine how individual parameters affect optimal muscle stimulation patterns and power throughput. Rasmussen et al. (2004) optimized this mechanism using a simple biomechanical model of the lower limb of a person with paraplegia, incorporating data on muscular capacities of three muscle groups. Lanese et al. (2022) explored three tricycle designs: the traditional recumbent, the coupler-driver, and a six-bar crank-rocker with the aim of enhancing power throughput for FES-stimulated riders with lower limb paralysis. In their work, a random pursuit algorithm was used for optimization and incomplete joint torque data was used from Szecsi et al. (2014).

This research focuses on pedaling performance for able-bodied users. The approach herein combines the personalization of a traditional bicycle and a compatible novel pedal design to maximize power throughput. The traditional recumbent tricycle model defined in Lanese et al. (2022) is adapted to match a regular bicycle and a dedicated workflow is developed that integrates algorithmic differentiation and numerical optimization to implement a data-driven bicycle personalization. To fully account for a user's torque-generating capabilities, without an explicit muscle modeling, the proposed algorithm leverages exhaustive experimental muscular torque data capturing the complex position,

<figure><img src="image_2.png" /></figure>

<footnote><sup>a</sup>INRIA, Université de Montpellier, 860 Rue de St-Priest, Montpellier, 34090, France</footnote>

<footnote><sup>b</sup>Mechanical and Aerospace Engineering, University of Dayton, 300 College Park, Dayton, 45469, OH, USA</footnote>

<footnote>* This work was supported by Défi Clé Robotique centrée sur l'humain funded by Région Occitanie, France.</footnote>

<footnote>* Corresponding author</footnote>

<footnote>sabrinaotmani@gmail.com; sabrina.otmani@inria.fr (S. Otmani)</footnote>

<footnote>ORCID(s):</footnote>

<footnote n="1">¹FES-assisted cycling involves using electrical impulses to activate paralyzed or weak muscles, enabling rhythmic pedaling on a stationary or overground bike.</footnote>
velocity and torque relationships of the hip and knee joints. Unlike previous studies in which the torque data used for model personalization were inferred from a cycling task (Szecsi et al. (2014)), this study explores a wider range of mechanical solutions by incorporating full Range Of Motion (ROM) torque data (from an extensive isokinetic dynamometer dataset). To illustrate the capabilities of this approach on two individuals with contrasting morphologies, two cases were simulated and analyzed. First, a traditional bicycle with a personalized seat height and crank length is optimized to illustrate the personalization capabilities of the approach herein. Second, a customized bicycle incorporating the new pedal-crank mechanism illustrates how such a design modification can increase average power throughput by more than 15%.

This research emphasizes the potential of integrating biomechanical data into bicycle design. This adaptable, performance-oriented personalization methodology caters to an array of individuals, increasing the amount of power each can deliver through the pedal-crank mechanism. Furthermore, this study is a first step towards challenging conventional bicycle design by exploring the potential performance advantages of innovative mechanisms, such as our optimized crank-pedal system.

# 2. METHODS

## 2.1. Rider and bike modeling

The combination of the pedal-crank mechanism with the user's leg defines a four-bar mechanism (Figure 1) where the shank and the foot may be considered a single bar by locking the ankle at 90°. Compared to the typical model where the foot rests on the pedal at the end of the crank, an additional parameter is introduced: $\vec{l}_4$, a vector between the contact point of the foot and the joint at the end of the crank. By introducing this parameter, a novel pedal geometry is possible, while the traditional mechanism is achieved when $||\vec{l}_4|| = 0$ (Figure 1). As in a classical bike, the crank pedal mechanism has two revolute joints: one between the crank and the bike frame, and another one between the crank and the -possibly modified- pedal. Angles ($\theta_i$) are defined positively from the vertical axis to the corresponding link i clockwise.

The lengths of thigh, shank and foot represent the rider's anthropometry. The inclination of the back ($\theta_b$) is considered fixed at 57° which is consistent with Diefenthaler et al. (2008). The bicycle is composed of the following links: $||\vec{l}_1||$, the distance between points O and A, $||\vec{l}_2||$, the crank length, $||\vec{l}_3||$, the coupler length, $\delta$, the angle between the link from the knee to the metatarsal and $||\vec{l}_3||$ (Figure 1). Regardless of the norm of $\vec{l}_4$, this system remains a four-bar mechanism made of $\vec{l}_2, \vec{l}_1, \vec{l}_3$ and $\vec{l}_{thigh}$. Other angular values, such as the angle between $\vec{l}_2$ and $\vec{l}_4$ are deduced from the four-bar kinematics. $v_{crank}$ corresponds to the crank velocity. $\theta_1$, the angle between the vertical axis and $\vec{l}_1$, is set to 180° (this is arbitrary, as another value would not impact the results presented hereafter).

<table><caption>**Table 1**</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>Young</td><td>Old</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>31</td><td>24</td></tr></tbody></table>

$M_1$ and $F_1$ anthropometric and personal data. Age category defined as in Anderson et al. (2007): young (18 – 25) and old (> 65).

Determining the positions and velocities throughout one pedaling cycle requires the crank angle ($\theta_2$) as input, with the other angles and segment positions being deduced from trigonometry. Calculating the crank and ankle torques requires the hip and knee torques as inputs, with the remaining torques derived from quasi-static equations. Crank power is then directly calculated from these velocities and torques.

In this model, five parameters are optimized: $\mathbf{x} = [||\vec{l}_1||, ||\vec{l}_2||, ||\vec{l}_3||, \delta, v_{crank}]^T$. The coupler connects the knee joint to the contact point between the foot and the pedal, thereby directly influencing the systems kinematics. $||\vec{l}_4||$, which represents the length of the new pedal, is therefore derived from the optimized parameters to ensure the proper functioning of the system. In this study, the model is constrained to the sagittal plane. The mass of the bicycle part were not considered to simplify our model and the limbs masses of our subjects are only considered through the body weight in the computation of the joint torques. The upper body (including the pelvis) is considered as fixed and without any movements which could impact the joints orientation such as the hip.

The joint torques, powers, velocities and angles are computed for one crank cycle using the quasi-static equations developed from the free-body diagrams and vector loop:

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

## 2.2. Anthropometric parameters

Two profiles ($M_1$ and $F_1$) were considered in our study. A vector of anthropometric data, y, including thigh, shank and foot lengths, body mass, age and height was defined for each profile (Table 1), to parametrize the four-bar model. Personalized hip and knee torques were inferred from Anderson et al. (2007), which provides maximum torque surfaces at each joint given age, height and gender of the subject as well as angular position and velocity of the considered joint (see Section 2.4).
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 1:** Customized bicycle model featuring the novel pedal mechanism (left) and definitions of angles and links (right). Optimization variables are circled in red.</figcaption></figure>

## 2.3. Optimization

### 2.3.1. Objective function

Our formulation aims to find the combination of parameters, $\mathbf{x}^*$, that maximizes the average power throughput at the crank ($P_c$) during one crank cycle, for each typical profile studied and accounting for a set of constraints to be satisfied.

$$ 
\begin{array}{ll}
\mathbf{x}^* = \underset{\mathbf{x}}{\operatorname{argmax}} & J(\mathbf{x}, \mathbf{y}) \\
\text{s.t.} & \mathbf{x}_i^{\text{low}} \le \mathbf{x}_i \le \mathbf{x}_i^{\text{up}}, \quad (2) \\
& g(\mathbf{x}) \ge 0,
\end{array}
$$

with $J(\mathbf{x}, \mathbf{y})$ a specific cost function, ensuring the maximization of $P_c$ among other terms described in section 2.3.2, $\mathbf{x}_i^{\text{low}}$ and $\mathbf{x}_i^{\text{up}}$ the lower and upper bounds on $\mathbf{x}$ and $g(\mathbf{x})$ a structural constraint described in section 2.3.3.

### 2.3.2. Ankle torque penalization

The ankle joint is fixed at 90° in this study. This constraint is justified by previous research on lower limb joint motion during ergometer cycling, which observed that the ankle joint was centered around 90° while exhibiting a limited range of motion of 22° (Ericson et al. (1988)). Thus, immobilizing the ankle is considered a reasonable simplification for the present analysis. Although the ankle joint is locked, keeping the ankle torque within acceptable bounds is desirable because it ensures feasible ankle torques. For that purpose, a penalization term was added in the output of the objective function (3). To define these bounds, we consider the behavior of the foot during one crank revolution. There is

a first range when the foot pushes on the pedal and a second range when the foot pulls on the pedal. During the first range, the resistive torque is produced by the elastic component of the ankle, a similar mechanism is encountered during long jump movements Muraki et al. (2008). For the second range, the limits are based on the work of Moraux et al. (2013) defining ankle torques based on a person's height.

Following these two references, the plantar- and dorsi-flexion torque limits were respectively set at 200 Nm and 150 Nm for $M_1$, and at 175 Nm and 60 Nm for $F_1$. These prescribed values serve as the upper (dorsi-flexion) and lower (plantar-flexion) bounds for ankle torque in the mechanism design optimization. The ankle penalization function ($p_a$) is based on two sigmoid functions in which bounds are provided to smoothly comply with the optimization:

$$ p_a(\mathbf{x}, \mathbf{y}) = \frac{1}{1 + e^{(\tau_a^{\max} - \tau_a(\mathbf{x}, \mathbf{y}))/\tau_a^{\min}}} + \frac{1}{1 + e^{(\tau_a(\mathbf{x}, \mathbf{y}) + \tau_a^{\min})}}, \quad (3) $$

with $\tau_a$, the calculated ankle torque depending on the problem parameters $\mathbf{x}$ and the user's anthropometric data, $\mathbf{y}$, $\tau_a^{\max}$ and $\tau_a^{\min}$, the upper and lower bounds for the ankle torque. From this definition, the cost function introduced in Eq. 2 is defined as follows:

$$ J(\mathbf{x}, \mathbf{y}) = P_c(\mathbf{x}, \mathbf{y}) - w_a * p_a(\mathbf{x}, \mathbf{y}), \quad (4) $$

with $w_a$ a weight to modulate the importance of $p_a$ with regard to $P_c$.
<table><caption>**Table 2**</caption><thead><tr><th rowspan="2"></th><th colspan="2">Wide</th><th colspan="2">Reasonable</th></tr><tr><th>Min</th><th>Max</th><th>Min</th><th>Max</th></tr></thead><tbody><tr><td>l<sub>1</sub> (m)</td><td>0</td><td>2</td><td>0</td><td>1</td></tr><tr><td>l<sub>2</sub> (m)</td><td>0</td><td>2</td><td>0</td><td>1</td></tr><tr><td>l<sub>3</sub> (m)</td><td>0</td><td>2</td><td>0</td><td>0.7</td></tr><tr><td>v<sub>crank</sub> (rpm)</td><td>0</td><td>120</td><td>0</td><td>120</td></tr><tr><td>δ (°)</td><td>0</td><td>90</td><td>0</td><td>90</td></tr></tbody></table>

Limits for each link (lengths, velocity and angle) of the design for both ranges studied: wide and reasonable

### 2.3.3. Grashof Criterion

For a four-bar mechanism, the Grashof criterion states that the sum of the shortest *S* and longest link lengths *L* of a planar four-bar linkage must be less than (or equal) to the sum of the remaining two links (*P* & *Q*) for *S* to fully rotate with respect to a neighboring link (Grashof, 1890):

$$g(\mathbf{x}) = S + L - P - Q \le 0 \quad (5)$$

As in related works (Lanese et al., 2022; Bazler et al., 2021; Rodríguez-Molina et al., 2023), for a fully rotating crank, the Grashof criterion implies that link $\vec{l}_2$ is shorter than the three others included in the bicycle and rider system ($\vec{l}_1, \vec{l}_3, \vec{l}_{thigh}$).

### 2.3.4. Parameters bounds

Two sets of bounds on **x** were defined: wide (letting the solver exploit at best the torque-generating capabilities with few mechanical additions on the optimized bicycle) and reasonable limits (restricting the optimized bicycle to be similar to a standard one). For each situation, the limits of each parameter ($\mathbf{x}_i^{low}$ and $\mathbf{x}_i^{up}$ in (2)) are presented in Table 2. The bounds are the same for both typical profiles. As a maximal power throughput is wanted in our simulations for each design, 120rpm ($\approx$720 degrees/sec) is the maximal velocity allowed to reach to produce the best power throughput. This velocity corresponds to a common velocity of trained cyclist (Martin and Brown, 2009; McDaniel et al., 2014). Another scenario is studied, in which the crank length is enforced to be of 17cm, a typical dimension in most bicycles (Martin and Spirduso (2001)).

### 2.3.5. Two-stage optimization framework

An initial parameters search is performed using PYGAD, a genetic algorithm optimization Python library (Gad, 2021). This search warm-starts a refinement step carried out with Interior Point OPTimer (IPOPT) (Wächter and Biegler, 2006), a Non-linear problem (NLP) solver interfaced with an algorithmic differentiation software, CasADi (Andersson et al., 2018). This combination integrates symbolic computation with automatic differentiation to accurately and efficiently compute gradients, Jacobians and Hessians in the non-linear solver. Algorithmic differentiation is preferred

over finite differences in gradient-based solvers due to its superior accuracy, efficiency and stability.

## 2.4. Joint position/velocity/torque relationship approximation

### 2.4.1. Original torque values

The torque-generating capacities of the riders are derived from Anderson et al. (2007), which provides values for joint torques obtained during maximum voluntary contraction as a function of joint angle and angular velocity. In this work, isometric, concentric and eccentric maximum voluntary contractions were collected during hip extension, hip flexion, knee extension and knee flexion. Then, parametrized equations of maximal joint torque surfaces were developed, depending on age category, gender, height, weight, studied joint (hip or knee) and studied movement (extension or flexion) of the subject.

### 2.4.2. Optimization-compliant approximation

In the process of integrating joint torque data into bicycle design optimization, a significant challenge arises in the need to adapt the original equations to the requirements of numerical optimization. In Anderson et al. (2007), the joint position/velocity/torque relationships are described as not-always-differentiable conditional equations which are incompatible with a gradient-based optimization framework. A reformulation of these relationships is proposed herein using Radial Basis Functions (RBF) interpolation, a method particularly useful to interpolate high dimensional scattered data (Hardy, 1971; Wright, 2003), with differentiable properties to comply with gradient-based optimization. In particular, multiquadric RBF are efficient for spatial approximations (Kansa, 1990). This symbolic approximation of the parametric relationships is hard-coded, using Scipy Virtanen et al. (2020) and CasADi, to be leveraged efficiently by the nonlinear solver.

### 2.4.3. Domain expansion

The original torque surfaces are restricted to joint position and velocity bounds specified in Anderson et al. (2007). To explore a wide range of mechanical solutions, the behavior of the interpolation outside of this initial domain needs to be specified. To accomplish this, a transition margin is added which linearly goes to a low torque value (< 15 Nm) when the model goes outside the bounds as seen in Table 3 and Figure 2. Depending on the angular position and velocity combination, three different cases can occur: i) If the combination strictly lies inside the bounds of Anderson et al. (2007), the original torque is fitted; ii) If the combination lies in the transition margin, a linearly decaying torque is fitted; iii) If the combination lies out of the bounds and of the transition margin, a low torque value is fitted. The expanded domain is illustrated in Figure 2 for the hip flexion of $M_1$. For the benefit of the solver, this domain expansion eliminates the limitations of the original dataset,
<table><caption>**Table 3**</caption><thead><tr><th></th><th>Hip</th><th>Knee</th></tr></thead><tbody><tr><td>Min. angular position (r̄)</td><td>-28.6</td><td>0</td></tr><tr><td>Max. angular position (r̄)</td><td>74.4</td><td>103.1</td></tr><tr><td>Min. angular velocity (r̄/s)</td><td>-200</td><td>-200</td></tr><tr><td>Max. angular velocity (r̄/s)</td><td>250</td><td>400</td></tr><tr><td>Transition margin</td><td>10</td><td>10</td></tr></tbody></table>

Bounds defined in Anderson et al. (2007) and transition margin considered for both angular position and velocity.

while implicitly imposing a penalty if it moves outside the set as low torque will ultimately reduce the cost function.

## 2.5. Optimization Scenari

In order to showcase the versatility of our approach, several scenari are optimized, ranging from extreme alterations of the pedaling kinematics to small, performance-improving adjustments:

* Fixing the crank length at a standard dimension (17 cm), for both the traditional pedal design (|| $\vec{l}_4$|| = 0) and the novel pedal design (|| $\vec{l}_4$|| > 0)

* Constraining the optimization (Eq.2) to reasonable bounds (Table2) for both the traditional pedal design (|| $\vec{l}_4$|| = 0) and the novel pedal design (|| $\vec{l}_4$|| > 0)

* Constraining the optimization (Eq.2) to wide bounds (Table2) only for the novel pedal design (|| $\vec{l}_4$|| > 0)

# 3. RESULTS

The results of the optimization are now presented focusing on several aspects: crank power throughput for one leg (Table 4, all optimization times are under 5s), optimized bicycle design (Figure 3, 4), trajectory of the foot around the metatarsal (Table 5), matching of the design to standard bicycle dimensions in terms of crank length, the length between the seat and crank axis of rotation for instance (Table 6). The joint angular positions during one crank revolution and a graphical representation of how several mechanisms exploit the riders' torque-generating capacities are also presented.

In Table 4, the results of the optimization are categorized through a comparison of the personalized bicycle with the novel pedal design (|| $\vec{l}_4$|| > 0) and the personalized traditional bicycle design (|| $\vec{l}_4$|| = 0).

## 3.1. Mean Crank Power

Overall, higher mean crank power values are obtained with the novel crank-pedal mechanism, regardless of the condition ($M_1/F_1$, reasonable limits with || $\vec{l}_2$|| = 17cm, reasonable limits without restrictions on $\vec{l}_2$) compared to the mechanism without the new crank-pedal mechanism.

Increases in mean crank power of 9% and 13%, respectively, are observed for $M_1$ and $F_1$, when comparing mechanisms with || $\vec{l}_4$|| > 0 and || $\vec{l}_4$|| = 0 and without restrictions on $\vec{l}_2$.

Comparing the designs with || $\vec{l}_4$|| > 0, for both $M_1$ and $F_1$, the mean crank power is higher for the optimization with wide limits than the others. For $M_1$, the design with wide limits results in a mean crank power 10% higher than the design with reasonable limits and 27% higher than the design with reasonable limits and || $\vec{l}_2$|| = 17cm. For $F_1$, the design with wide limits results in a mean crank power 3% higher than the design with reasonable limits and 27% higher than the design with reasonable limits and || $\vec{l}_2$|| = 17cm.

Table 4, viewed from the perspective of crank length, shows that designs with || $\vec{l}_2$|| = 17cm have lower mean crank power compared to the two others designs for both typical profiles and regardless of the presence of the novel crank-pedal mechanism.

Figure 5 shows the temporal evolution of power production during one crank revolution. All design conditions exhibit a similar power production profile except for the one with wide limits. At the start of the cycle, the power reaches a local minimum while the other conditions reach local maxima. At around 75% of the cycle duration, the largest peak observed for the wide limits condition is sustained longer than in the others (both for $M_1$ and $F_1$). A singular behavior is observed for $M_1$ when || $\vec{l}_2$|| = 17cm and || $\vec{l}_4$|| > 0 with a drastic decrease in crank power at the end of the cycle. A negative and undesirable power throughput is observed for $F_1$ with || $\vec{l}_2$|| = 17cm and || $\vec{l}_4$|| = 0 and for $M_1$ for both designs with || $\vec{l}_2$|| = 17cm when the crank angle ($\theta_2$) is around 300°. Overall, Figure 5 indicates that the higher power throughput of designs featuring the novel crank-pedal mechanism stems from an earlier and more sustained peak torque production near the end of the crank revolution.

## 3.2. Foot trajectory

Depending on the optimization condition, foot (metatarsals) trajectories have similar shapes for $M_1$ and $F_1$ (Table 5). Foot trajectories for the personalized traditional bicycle ($\vec{l}_4$ = 0) are circular as the foot directly rests in contact with the tip of the original crank. Enforcing || $\vec{l}_2$|| = 17cm results in a larger foot trajectory than without (observing that the optimization tends to decrease the crank length). For the conditions featuring the novel crank-pedal mechanism || $\vec{l}_4$|| > 0, foot trajectories are no longer restricted to be circular and exhibit more general shapes. For the condition with wide limits, the trajectory is more stretched out than the others for both $M_1$ and $F_1$. The trajectory shapes obtained are larger for $M_1$ than for $F_1$ for all the designs, as might be expected due to the overall larger dimensions of the rider. The novel crank-pedal mechanism offers a variety of possible foot trajectories translating into changes in lower limb kinematics that better exploit joint torques.

## 3.3. Personalized bicycle geometry

Regarding the designs obtained with the wide limits, the optimized design has longer links, especially for $\vec{l}_1$, $\vec{l}_4$ and
<table><caption>**Table 4**</caption><thead><tr><th rowspan="3"></th><th colspan="4">||<span style="text-decoration: overline;">l</span><sub>4</sub>|| > 0</th><th colspan="4">||<span style="text-decoration: overline;">l</span><sub>4</sub>|| = 0</th></tr><tr><th colspan="2">Wide limits</th><th colspan="2">Reasonable limits</th><th colspan="2">Reasonable limits<br/>- l<sub>2</sub>=17cm</th><th colspan="2">Reasonable limits</th><th colspan="2">Reasonable limits<br/>- l<sub>2</sub>=17cm</th></tr><tr><th>M1</th><th>F1</th><th>M1</th><th>F1</th><th>M1</th><th>F1</th><th>M1</th><th>F1</th><th>M1</th><th>F1</th></tr></thead><tbody><tr><th>Mean Crank Power (W)</th><td>499</td><td>231</td><td>453</td><td>223</td><td>393</td><td>182</td><td>415</td><td>198</td><td>307</td><td>132</td></tr><tr><th>Std Crank power (W)</th><td>24</td><td>11</td><td>19</td><td>11</td><td>24</td><td>11</td><td>19</td><td>25</td><td>41</td><td>21</td></tr></tbody></table>

Comparison of mean crank power produced, its standard deviation (std) rounded to the nearest unit for $F_1$ and $M_1$ between several optimization conditions for $||\vec{l}_4|| = 0$ and $||\vec{l}_4|| > 0$

<table><caption>**Table 5**</caption><thead><tr><th rowspan="2"></th><th colspan="4">||<span style="text-decoration: overline;">l</span><sub>4</sub>|| > 0</th><th colspan="4">||<span style="text-decoration: overline;">l</span><sub>4</sub>|| = 0</th></tr><tr><th>Wide Limits</th><th>Reasonable Limits</th><th>Reasonable Limits<br/>l<sub>2</sub> = 17 cm</th><th>Reasonable Limits</th><th>Reasonable Limits<br/>l<sub>2</sub> = 17 cm</th></tr></thead><tbody><tr><th>M<sub>1</sub></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>F<sub>1</sub></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

Foot trajectory at the metatarsals for both typical profiles and for each optimized design studied.

$\vec{l}_3$, $\vec{l}_1$ reaches 1.99 meter for $M_1$ and $F_1$. $\vec{l}_4$ is approximately 1.52 meters for $M_1$ and 1.72 meters for $F_1$. $\vec{l}_3$ is close to 1.93 meters for $M_1$ and 1.89 meters for $F_1$ (Figure 3, 4). Obviously, this design is problematic compared to the ones obtained with the other conditions due to the fact that the rider is higher (2 meters above) and in consequence, the cycle frame would require more material and would not be easy to manipulate. Versus bicycles available on the market, all optimized designs are deemed to be acceptable except for the design achieved by allowing wide dimensions.

Examining the designs with reasonable limits, the design for $M_1$ is observed to be more obtainable as compared to $F_1$ (Table 6). A longer $\vec{l}_4$ link is observed for $F_1$ which can be limiting in the manufacturability of the associated bicycle design and in the usability of the bicycle. Having a customized pedal ($\vec{l}_4$) for the optimized bicycle design helps the user to develop higher mean crank power compared to the one without $\vec{l}_4$ and produces a more realistic design with reasonable limits, with and without $\vec{l}_2 = 17cm$.

<table><caption>**Table 6**</caption><thead><tr><th></th><th><i>M</i><sub>1</sub></th><th><i>F</i><sub>1</sub></th></tr></thead><tbody><tr><td><i>l</i><sub>1</sub></td><td>0.61</td><td>0.75</td></tr><tr><td><i>l</i><sub>2</sub></td><td>0.11</td><td>0.09</td></tr><tr><td><i>l</i><sub>3</sub></td><td>0.59</td><td>0.69</td></tr><tr><td><i>l</i><sub>4</sub></td><td>0.19</td><td>0.48</td></tr><tr><td><i>v</i><sub>crank</sub> (rpm)</td><td>119.93</td><td>119.93</td></tr><tr><td><i>delta</i>(r̄)</td><td>11.45</td><td>41.25</td></tr></tbody></table>

Reasonable parameters found using the optimization for $M_1$ and $F_1$ for the designs with $l_4 > 0$ without restrictions on $l_2$.

Regarding the optimized length of $\vec{l}_2$, we observed shorter lengths (11cm for $M_1$ and 9cm for $F_1$) compared to the traditional length seen in the market (17cm) which shows the impact of the subject morphology in the optimization and the adjustment. In terms of manufacturability, it is not limiting to have shorter crank.
For both examined profiles, the maximum crank velocity is almost reached for the bicycle designs with reasonable limits and optimized $\vec{l}_2$ (Table 6). The primary distinction between the two typical profiles, regardless of their anthropometry, lies in the parameter $\delta$, which represents the angular gap between the link connecting the knee joint to the metatarsal and the link $\vec{l}_3$. This value dictates the position of the limb during the cycle trial. This can be explained by the anthropometric difference of the two typical profiles as the shank length of the two typical profiles is notably different. This difference also induced the difference in $\vec{l}_4$ between the two typical profiles for the design with reasonable limits which links the foot (around the metatarsal) of the typical profile to the crank (Figure 3, 4).

For $F_1$, a highly flexed knee is observed for the designs with reasonable limits (3) which is not representative of cycling. For the designs with $\vec{l}_2 = 17cm$, the crank velocity is lower than the ones of the others designs which reach the maximal value possible. Indeed, the crank velocity found is below 100 rpm for $M_1$ and below 56 rpm for $F_1$.

### 3.4. Joint torque exploration

An interesting way to analyze these results is to overlay the maximal joint position/velocity/torque surfaces, along with the phase portraits of the corresponding joints, while pedaling on various personalized mechanisms (Figure 8 for M1 and F1). This representation allows to quickly visualize how the modified pedaling kinematics leverages at best the maximal torque that the user's body can produce. The more the optimization is relaxed, the more the phase portraits are shifted towards higher torque areas resulting in higher mean crank torque over a crank revolution (Table 4). For M1 (Figure 8, top row), this is particularly remarkable both for the hip and knee joints, where the wide limits solutions are substantially dilated (larger range of motion) and shifted towards areas of high torque. The same can be noticed for the knee joint of F1 (Figure 8, bottom row), where the presence of the new crank pedal mechanism ($||\vec{l}_4|| > 0$) allows for more angular velocity and larger ranges of motions, resulting in higher power at the crank.

### 3.5. Joint angular position

Finally, Figures 6 and 7 present the impact of the several optimized bicycle designs on the kinematics of the hip and knee joints.

**Hip Joint:** The design with $||\vec{l}_2|| = 17cm$ and $||\vec{l}_4|| = 0$ (green) results in the largest hip flexion, for $M_1$ and $F_1$ ($\ge 85\ddot{r}$).

For $M_1$, the hip ROM are the smallest for the bicycle designs with $||\vec{l}_4|| > 0$ and $||\vec{l}_2|| = 0$ resulting from reasonable parameters and with $\vec{l}_2$ optimized (around 30$\ddot{r}$). The maximal ROM are obtained for the designs with $||\vec{l}_2|| = 17cm$ with $||\vec{l}_4|| > 0$ and $||\vec{l}_4|| = 0$ (around 50$\ddot{r}$).

Same observations are made for $F_1$ for the maximal ROM (around 55$\ddot{r}$) and the smaller ROM is obtained for the designs with $||\vec{l}_4|| = 0$ and $\vec{l}_2$ optimized (around 42$\ddot{r}$).

When the designs with $||\vec{l}_2|| = 17cm$ are compared, $\vec{l}_4$

is observed to produce a different behavior of the joint with real flexed movement of the hip joint. The hip joint evolution for each design is observed to be the same for both typical profiles with small differences between the design with reasonable limits with and without $||\vec{l}_2|| = 17cm$.

**Knee Joint:** For the knee joint, different behavior is seen between $M_1$ and $F_1$. For $M_1$, the maximum extension is reached for the design with wide parameters followed by the designs with reasonable limits without $||\vec{l}_4|| = 0$ and with $||\vec{l}_2|| = 17cm$ (green line).

For $F_1$, the maximum extension is reached for the designs with reasonable limits without the new pedal and with $||\vec{l}_2|| = 17cm$ (green line) followed by the design with the new pedal and $||\vec{l}_2|| = 17cm$ (blue line).

The maximal angular position is higher for $F_1$ (74$\ddot{r}$) than $M_1$ (71$\ddot{r}$). The pattern of evolution for each typical profile is the same regarding all the designs. The amplitude and the starting points vary.

A smaller ROM is observed for the designs without the new pedal and with $\vec{l}_2$ optimized ($M_1$: 23$\ddot{r}$ and $F_1$: 26$\ddot{r}$) compared to the other designs. For $F_1$, the knee behavior is similar for the designs with wide and reasonable parameters with $\vec{l}_2$ optimized. All of these differences highlight the need to study each joint for each typical profile and for the different designs, justifying a customized approach to the design.

## 4. DISCUSSION

This paper presents a framework to maximize individuals' power throughput on bicycles personalized to their torque-generating capacities. It also introduces a novel crank-pedal mechanism that increases the amount of power that one is able to produce at the crank. The presented work relies on a two-stage optimization algorithm altering the bike geometry given biomechanical data of the user to induce a pedaling kinematics that generates the most torque. Two distinct typical profiles (gender, age, height, weight and limb lengths) are studied to showcase the versatility of the proposed approach. The addition of the new crank-pedal mechanism in the design of the bicycle allows both profiles to produce higher power throughput than without (Table 4). Our results explain this improvement at several levels. First, the feet kinematics resulting from the incorporation of the novel crank-pedal are more variable than when the feet directly rests on the crank tip (Table 5). This more versatile workspace of the mechanical system means that the same pedaling task can be performed with a larger variety of kinematic trajectories of the knee and hip joints. As the maximal producible joint torque depends on the joints position and velocity, when optimized, these types of mechanisms offer more power possibilities than regular ones. These adjustments are thus more complex and tailored to the user's torque-generating capacities than simply increasing the crank's moment arm.
Regarding average power values presented in our study, they are consistent with Martin and Brown (2009), in which thirteen trained male cyclists performed a thirty second maximal cycling trial on an isokinetic bicycle ergometer at 120rpm (average power of 549W per leg for a complete crank revolution) and with Elmer et al. (2011), in which thirteen male cyclists performed a thirty second maximal cycling trial on an isokinetic bicycle ergometer at 120 rpm (average power of 563W per leg for a complete crank revolution). With a traditional bicycle design, we obtain for $M_1$ an average power of 368W, this lower value being explained by the fact that the torque-generating capacities derived from Anderson et al. (2007) were not acquired on trained athletes, but on regular people. Having a closer look at the joints phase portraits induced by the several bicycle designs, joint torques produced are higher during the extension than during the flexion (both for the hip and knee joints). These results are consistent with Elmer et al. (2011). The unique combination of a gradient-based solver, algorithmic differentiation and optimization-compliant approximations of torque data constitutes the methodological core of our approach. Although it does not guarantee convergence towards global optima, to the best of our knowledge, this is the first instance of such a global numerical approach being applied to bicycle design personalization with a new crank-pedal mechanism. Other optimisation-based frameworks exist and study the impact of the sitting positions (Zongxing et al., 2021), the seat tube angle (Bisi et al., 2012) or the influence of the saddle seat back (Menard et al., 2020) on joints during cycling but none of them questions the crank pedal design for performance. The use of algorithmic differentiation was key to the success of our approach, as finite-difference-based gradient descent would not converge most of the time (due to inexact gradients and Hessians). In future work, a multistart strategy could be explored to mitigate the local convergence issue and improve the likelihood of reaching global optima.

The inclination of the cyclist's back was considered fixed and corresponds to a mean inclination during cycling Diefenthaler et al. (2008). In future work, this inclination could be included as an optimization variable to study its influence on the maximum crank power (it could then be indirectly enforced through the saddle height and handlebar).

Also, to simplify our model, the ankle is being locked at 90° by model design, as in Lanese et al. (2022). This simplification might overlook the interactions of bi-articular muscles and underestimate the importance of the ankle in trained cyclists Crossley et al. (2024). To mitigate this approximation, we accounted for the torques needed to lock the ankle in this position in the objective function, and ensured they would lie within physiological bounds.

For the designs with $||\vec{l}_2|| = 17$ cm, negative power throughputs were observed for both subjects (Figure 5), which are not physically possible. This behavior arises due to the constraints imposed on $||\vec{l}_2||$, leading to a singularity in the problem-solving process. Future work will focus on enhancing the robustness of the solver to prevent such

singularities and ensure more reliable solutions.

In this study, three types of constraints were considered: Grashof's constraints, parameter bounds, and ankle torque limits. Additional constraints, such as those on knee or hip angular position, could be beneficial to prevent non-feasible movements during cycling as it's observed for $F_1$. The studied designs are built on the mathematical representation of the traditional bicycle presented in Lanese et al. (2022) because we wanted them to be as close as possible to traditional bikes, to deliver a first demonstration of our approach. In future works, it might be relevant to study more exotic mechanisms such as the coupler-driver or the crank-rocker. Our approach being generic, it would easily scale to other mechanisms. It should however be noted that, while the standardized dimensions of bicycles may not be optimal for every individual, they represent an excellent compromise that balances functionality, cost, tradition, and regulatory requirements. Therefore, any novel solutions seeking a wide adoption (apart from niche applications) should fit within these compromises, offering improvements without disrupting the benefits of standardization and mass production.

Similarly to Lanese et al. (2022) but at odds with most biomechanical simulation on cycling Clancy et al. (2023), we chose to rely on an implicit modeling of muscle capacities (opposite to the explicit modeling required in, e.g., Opensim). Although torque-driven models constrained by dynamometer data do reflect some aspects of muscular capacity (such as maximum voluntary torque output), they do not fully capture the complexity of muscle physiology (e.g., fatigue, co-contraction, individual muscle forces, or neuromuscular control). In future works, our approach could be translated to a more conventional biomechanical simulation framework de Sousa and Font-Llagunes (2024) in which it would be instructive to fully account for muscular capacities and for inertial and Coriolis effects. To complement this work, future works could also include a sensitivity analysis of the optimization parameters on the power output, to compare their respective influence.

Lower limb joint torque values were derived from Anderson et al. (2007), in which hip joint torque data were acquired with the knee extended. This is not representative of the bike riding kinematics and might be problematic as bi-articular muscles, such as the hamstrings, spanning the hip and knee joints and creating a kinematic dependency relationship on the torque capacities of the involved joints. This dependency was not captured in our reference dataset and, to the best of our knowledge, in any other dataset publicly available. In the future, applying this framework with a dataset capturing the impact of the knee kinematics on the output produced by the hip joint and vice versa should lead to more accurate simulation. We expect it to produce higher power at the crank, as an extended knee lowers the hip torque capacity. In the same vein, our framework would allow to study more varied populations such as athletes or people with muscle impairments resulting in dedicated bicycle designs. In this research, we centered the focus on mean crank power while
also aiming to maintain the torque produced around the
ankle. However, it's worth noting that our study serves as
an initial step toward better meeting the expectations and
capabilities of users. Our results show also that there is
a strong dependence to the typical profile anthropometry.
Thus, design consideration that are based on individual users
will lead to bicycles of significantly different design.

5. CONCLUSION

In this paper, a numerical approach to personalize bi-
cycle and pedal-crank mechanism design driven by com-
puted torque from the literature and anthropometric data is
proposed. Our approach led to simulated increases in the
mean crank power (15.25%) produced while enforcing a set
of constraints such as joint torque limitations and a certain
realism in the bicycle design. The simulation results show
that the addition of the new crank-pedal mechanism allows
having a higher mean crank power compared to classical
designs. Defining reasonable bounds to obtain a realistic de-
sign prevents the solver from exploring infeasible solutions
that could lead to singularities, from which it may struggle
to escape. This approach ensures convergence toward an
optimal solution that balances realism, mean crank power,
and satisfaction. These results are a first step towards a more
complete personalization of bicycle designs for the needs
of specific users such as athletes seeking peak performance
or individuals on the path to rehabilitation. Future work
will focus on improvements of the methodology and addi-
tional investigations of the new pedal ||ṫ₄|| > 0. Also,
the optimized designs should be tested experimentally in
order to validate our results and help transition simulation
insights into real-world application. Other parameters which
are crucial to the performance such as aerodynamics and
bicycle weight should also be studied and included in future
works.

Conflict of interest statement

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

6. Appendices

A patent application has been filed to the INPI, the french
National Institute of Industrial Property. The registration
number is: FR2409775.

Acronyms

FES Functional Electrical Stimulation.

IPOPT Interior Point OPTimizer.

NLP Non-linear problem.

RBF Radial Basis Functions.

ROM Range Of Motion.

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<figure><img src="image_5.png" /><figcaption>**Figure 2:** Optimization-compliant approximation of $M_1$'s hip flexion torque as a function of angular position and velocity, on an expanded domain based on the original one defined in Anderson et al. (2007): **A.** Original domain, **B.** Transition margin, **C.** Low-value padding.</figcaption></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 3:** Reasonable bicycle design for $M_1$ (top) and $F_1$ (bottom) found using the optimization with $\vec{l}_4$. Black links represent the limbs of the typical profile studied. Black points represent the center of rotation which can make the skeleton appear disproportionate.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 4:** Non reasonable design with $\vec{l}_4$ for $M_1$ obtained with wide limits in the optimization. Black links represent the limbs of the typical profile studied. Black points represent the center of rotation which can make the skeleton appear disproportionate.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 5:** Crank power production during one pedaling cycle for $M_1$ (top) and $F_1$ (bottom) with one leg for each optimization scenario.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 6:** Hip joint angular positions by $M_1$ (top) and $F_1$ (bottom) for all the designs optimized studied.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 7:** Knee joint angular positions by $M_1$ (top) and $F_1$ (bottom) for all the designs optimized studied.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 8: Superimposition of maximal joint position/velocity/torque surfaces (background maps) and phase portrait (u-shaped colored curves) of the corresponding joint while pedaling on different personalized mechanisms. Each subplot corresponds to a joint and a motion direction for a specific subject. The angular position and velocity for each joint and for each design when $\theta_2 = 0$- is represented by a black cross (X).</figcaption></figure>