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# Steering or leaning? New insights from optimal control of a bicycle

Jules Ronné, Jason K. Moore

► To cite this version:

Jules Ronné, Jason K. Moore. Steering or leaning? New insights from optimal control of a bicycle. Multidisciplinary Biomechanics Journal, 2025, 50th congress of the., 10.46298/mbj.16187. hal-05191739v2

HAL Id: hal-05191739

https://hal.science/hal-05191739v2

Submitted on 23 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Steering or leaning?
## New insights from optimal control of a bicycle

Jules Ronné*, Jason K. Moore

Department of BioMechanical Engineering, Delft University of Technology,
Delft, The Netherlands

* Corresponding author: jules.ronne@ik.me

Received date: 04/04/2025

Accepted date: 27/06/2022

Publication date: 27/10/2025

Keywords: bicycle, optimal control, strategy, steering

© 2025 The Authors
Licence CC-BY 4.0
Published by Société de Biomécanique

## 1. Introduction

Cyclists have two main control actions to maintain their balance and navigate: steering torque and lean torque. The literature commonly considers steering torque as the dominant control action. This assumption is supported by kinematic observations (Moore 2012) and system identification results (Dialynas et al. 2022). However, the respective use of these two torques has not yet been established. While steering torque is easily measurable, leaning torque is difficult to measure experimentally. Numerical simulation is suitable to address this question in a preliminary way (Sharp 2008). To further explore this issue, we used a numerical model allowing us to quantify the respective use of steering and leaning torques from optimal control.

## 2. Methods

### 2.1 Cyclist-bicycle model

A rigid multibody cyclist-bicycle model linearized around the vertical at a constant speed (u) was used (Figure 1. (a)). The bicycle itself has two degrees of freedom (DoF): roll angle (φ) and steer angle (δ). The wheels are assumed to not slip on the ground. The cyclist has lower limbs rigidly attached to the bicycle frame and its upper limbs have a DoF allowing rotation around the bicycle's longitudinal axis (θ). The generalized coordinates, defined as $q = [φ \ \delta \ \theta]^T$, verifies the following equation of motion (Schwab and Kooijman 2010):

$$M\ddot{q} + uC_1\dot{q} + [gK_0 + u^2K_2]q = T$$

Mass (M), damping (C₁) and stiffness matrices $K_0, K_2$) were computed using parameters of Moore (2012). They consider an urban bicycle (Batavus Browser model) and a cyclist of 1.80 meters and 86 kg. The vector of generalized forces, defined as $T = [0 \ T_δ \ T_θ]^T$, assumes two control actions: the steering torque (between the steering assembly and the frame), denoted $T_δ$, and the leaning torque (between the upper body and the frame), denoted $T_θ$.

### 2.2 Optimal control and Simulation

The use of control actions was simulated, assuming the cyclist minimized its control cost. The controllability Gramian was employed to calculate control actions that minimize the Minimum Transition Cost (MTC) for a maximal-performance state change.

The MTC was defined in Ronné (2024) as:

$$MTC = \int_{0}^{t_t} |T(t)| dt$$

where control actions are equally weighted.

The MTC is the minimum control cost that the system requires to complete a change of state for a given transition time $t_t$.

Two scenarios were simulated: Initiation Maneuver (IM) and Recovery Maneuver (RM). An IM scenario consists in reaching a given state from the vertical equilibrium. An RM scenario consists in reaching the vertical equilibrium from a given state.
Each scenario was simulated 200 times with random final (IM) and initial (RM) states within specified ranges (Table 1) and speeds (1-20 m/s). Three transition times (0.5, 1.25, and 2 s) were used, representing varying control intensities.

<table><caption>**Table 1.** Bounds of the ranges used to explore transitions in the space of state variables</caption><thead><tr><th>Input</th><th>φ [°]</th><th>δ [°]</th><th>θ [°]</th><th>Φ [°/s]</th><th>δ [°/s]</th><th>θ [°/s]</th></tr></thead><tbody><tr><td>Min</td><td>5</td><td>-35</td><td>-35</td><td>-100</td><td>-100</td><td>-100</td></tr><tr><td>Max</td><td>20</td><td>35</td><td>35</td><td>100</td><td>100</td><td>100</td></tr></tbody></table>

## 2.3 Inputs Control Cost Ratio (ICCR)

The control cost associated with control action i ($δ$ or $θ$) is defined as follows:

$$C_i = \int_{0}^{t_t} |T_i(t)| dt$$

where $T_i^*$ is the optimal torque applied to degree of freedom i. The Inputs Control Cost Ratio (ICCR), i.e. the metric used to quantify the respective use of the control actions, was defined as follows:

$$ICCR_{\delta} = \frac{C_{\delta}}{C_{\theta} + C_{\delta}}$$

The $ICCR_δ$ was calculated for both scenarios to describe the control strategy.

## 3. Results and discussion

In IM scenarios (Figure 1. (b)), $ICCR_δ$ indicates a majority of steering torque control (around 70 %) from 5 m/s for the two longest transition times. This ratio shows a local minimum of 40 % around 5 m/s. For $t_t = 0.5s$, $T_δ$ dominated below 5 m/s, then $ICCR_δ$ showed a minimum of 40 % at 10 m/s. In RM scenarios (Figure 1. (c)), for $t_t = 2s$, $T_δ$ dominated the control, averaging over 60 %. For $t_t = 0.5s$, control is dominated by steering torque below 5 m/s, then the ratio reversed above this speed and converges to 40 %. Surprisingly $ICCR_δ$ appears very insensitive to the out-of-balance state used. These results provide a nuance to the observations of Sharp (2008), showing steering torque to be predominant in a path-tracking task.

(a) Dynamical Model

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(b) Initiation Maneuver

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(c) Recovery Maneuver

<figure><img src="image_5.png" /><figcaption>**Figure 1.** (a) Dynamical model, (b) and (c) show the $ICCR_δ$ values obtained for the RM and IM scenarios respectively. Solid line: $t_t = 0.5s$, dashed: $t_t = 1.25s$ and dotted: $t_t = 2s$. The corridor represents the 95 % confidence interval around the mean value over the 200 simulations.</figcaption></figure>

Credit: Jules Ronné.

These results are most likely specific to the physical properties used in the model. Indeed, vehicle dynamics and cyclist-bicycle inertia ratio influence the control strategy. Also, the model used remains simple and could be improved by limiting cyclist's torque capacities. The Gramian approach allows control actions to be calculated with a low-complexity simulation, but more advanced approaches such as Model Predictive Control seem more flexible for improving the model.

## 4. Conclusions

Our analysis shows that steering torque is the primary control action used during optimal finite-horizon transitions, but its dominance can be affected by factors such
as speed, transition time, and scenario. Notably, leaning
torque also plays a significant role and its optimal usage
varies strongly with speed, as suggested by leaning angle
observations in Moore (2010).

These findings lay the groundwork for future exper-
imental studies to validate our results and explore how
cyclists' control strategies change with experience and
age, providing insight into the complex relationships
between cyclist behavior and bicycle dynamics.

Conflict of Interest Statement

None

Contributor Roles

JR: Conceptualization, Methodology, Software, Resources, Writing original draft, Writing- review & editing; JKM: Software, Resources.

Funding

No specific funding for this study.

Data, software, code availability

Cyclist-bicycle model: https://bicycleparameters.readthedocs.io/stable/gallery/examples/plot_mooreriderlean2012.html

Analysis Code: https://github.com/jrPhD/Steering_or_leaning

References

Moore, J. K. (2012). *Human Control of a Bicycle* (Doctoral dissertation, University of California, Davis). Retrieved from https://moorepants.github.io/dissertation/

Dialynas, G., Christoforidis, C., Happee, R., & Schwab, A. L. (2022). Rider control identification in cycling taking into account steering torque feedback and sensory delays. *Vehicle System Dynamics*, 1–25. https://doi.org/10.1080/00423114.2022.2048865

Sharp, R. S. (2008). On the stability and control of the bicycle. *Applied Mechanics Reviews*, 61(6), 060803. https://doi.org/10.1115/1.2983014

Schwab, A. L., & Kooijman, J. D. G. (2010). Controllability of a bicycle. *5th Asian Conference on Multibody Dynamics 2010*, 6. Kyoto, Japan. https://api.semanticscholar.org/CorpusID:59142837

Ronné, J. (2024). Quantification de la maniabilité des vélos: Évaluations et contributions aux approches expérimentales et théoriques (Thèse de doctorat, Université de Lyon). Retrieved from https://theses.hal.science/tel-04882867v1

Moore, J. K., Hubbard M., Schwab, A.L., Kooijman, J.D.G. & Peterson, Dale L. (2010), Statistics of bicycle rider motion, *Procedia Engineering*, 2(2), 2937–2942, https://doi.org/10.1016/j.proeng.2010.04.091