Bicycling Dynamics in Bicycling Simulators

This page introduces a modular framework for bicycling dynamics subsystems, standardising definitions and interactions through five core models: multibody, stability, tyre, propulsion, and vertical vibration. Based on a review of 21 published solutions, it enhances transparency, reproducibility, and collaboration. Guiding questions address latency, manoeuvrability, and subsystem demands, balancing accuracy with computational efficiency to ensure validity and minimise simulator sickness.

A Framework for modular Bicycling Dynamics Subsystems for Human-in-the-Loop Bicycling Simulators

by Mathis Titgemeyer and Heather Kaths
Chair of Bicycle Traffic, University of Wuppertal

Background

Bicycling simulators are human-in-the-loop systems that offer a safe, fully controllable virtual environment for human factors research. These simulators integrate physical subsystems serving as human interfaces and virtual subsystems that generate the simulated environment, as outlined in the systematic framework for bicycling simulator technology.
At the core of this technology is the bicycling dynamics subsystem, which models the combined physical behaviour of the bicycle and the bicyclist that are connected through their contact points, referred to here as the bicycle-bicyclist system. This subsystem processes input actions from the human to calculate the system’s physical response, updating output variables that feed back into both the physical and environment subsystems. These updates generate corresponding sensory cues, ensuring the simulation’s interactivity.
Thus, the bicycling dynamics subsystem is central to simulator technology, directly influencing the realism and interactivity of the simulation which are key factors for advancing human factors research in bicycling.

The relevance of the bicycling dynamics subsystem for research quality

In bicycling simulators as in flight or driving simulators the dynamics subsystem fundamentally shapes the quality and reliability of research outcomes. Its primary role is to ensure the physical validity of the simulator by accurately replicating the dynamic behaviour of the bicycle-bicyclist system and generating corresponding sensory cues.
Physical validity is a prerequisite for behavioural validity in simulator-based experiments (Zöller 2015; Himmels et al. 2023). Inaccuracies in the dynamics subsystem can create sensory conflicts, where mismatched cues lead to simulator sickness, compromising both the validity and ethical integrity of the study (Matviienko et al. 2022; Kolff et al. 2025). Additionally, inconsistencies in model calibration and selection reduce the generalisability and reliability of findings across different studies (Schwarz et al. 2011).
The accuracy of the dynamic model must align with the specific research question, optimised for efficiency and minimised error rather than maximised for its own sake (Schwarz et al. 2011; Åström et al. 2005). Therefore, the dynamic model is an essential module in bicycling simulators, requiring careful integration and verification tailored to each use case. This ensures robust, reproducible, and ethically sound research outcomes.

Existing work on bicycling dynamics modelling and implementation in real-time applications

In the field of bicycling dynamics research, understanding and modelling bicycling dynamics serves diverse purposes. Modelling the bicycling dynamics can be applied to predict the self-stable conditions for specific bicycle geometries depending on the longitudinal speed: Åström et al. (2005) review different modelling approaches and individual effects on the dynamic stability of bicycles. Schwab and Meijaard (2013) present a comprehensive review on the control and dynamics of two-wheeled single track vehicles (bicycles and motorcycles). They outline the different approaches and introduce an extended version of the bicycling dynamics model by Whipple (1899) and a method for benchmarking linearised bicycling dynamics models Schwab and Meijaard (2013). Further applications of bicycling dynamics modelling include the development of steer-by-wire bicycles (Dialynas et al. 2018), automated bicycle robots (Tanaka and Murakami 2004), and using bicycling dynamics as an application of math and control theory for academic education (Åström et al. 2005).

The focus of this review is on the application of bicycling dynamics modelling to create operable bicycling dynamics subsystems for human-in-the-loop bicycling simulators. For this use case previous approaches can be grouped into three categories:

• Mechanical solutions: A mechanical apparatus designed to partially replicate real-world bicycling dynamics.
• Off-the-shelf solutions: Solutions provided with commercial hardware and software products.
• Full computational multi-DOF subsystems: Complex subsystems generated through manual and computer aided modelling methods.

Mechanical solutions

A bicycle is a mechanical system with unique dynamic characteristics due to its design as a single-track vehicle. Present simulators commonly rely on physically accelerating the human body to induce motion cues to the human tactile and vestibular senses. As every simulator requires a mechanical design for the vehicle mock-up it can be practical to add components for replicating parts of the dynamic behaviour to spare resources on complex control algorithms required for electro-mechanic actuators.
Powell et al. (2018) present a flywheel design as a simplistic mechanical solution for replicating the effect of the bicycle’s and bicyclist’s mass moment of inertia when accelerating in longitudinal direction.
The free roller design developed at the Swedish National Road and Transport Research Institute features rollers supporting the rear and front wheels over a width of approx. 2m (Ochel et al. 2025). When accelerating the rear wheel by pedalling, it drives the rear rollers via friction and a transmission system between rear and front rollers synchronises the speed and drives the front wheel in turn. Once a certain rotational velocity is exceeded, the bicycle remains stable in an upright position while allowing realistic lateral movement by small steering actions and balancing of the bicycle within the bounds of the rollers. While the physical validity of these small steering actions and lateral movements can be assumed to be high, the manoeuvres are limited to a straight corridor of the roller width: taking sharp turns and stops is not possible.
This shows that mechanical designs can provide efficient solutions for distinct aspects of simulator-based research, but can replicate only parts of the full scope of the dynamic behaviour of a bicycle and bicyclist. The development of complex motion base bicycling simulators is needed to allow scenarios with highly agile manoeuvres.

Off-the-shelf solutions

In recent years, more mature hardware and software products for VR applications have been developed and simulator research has been diversified by interdisciplinary groups with more pragmatic approaches to technical development. In turn, a trend towards using off-the-shelf components for creating bicycling simulators can be observed.
Game engine software such as Unity (Unity Software Inc. 2024) and Unreal Engine (Epic Games Inc. 2025), commonly used to create the environment subsystem for simulators, provide physics engines capable of replicating vehicle dynamics which can be adapted to simulate single-track vehicles. Professional human-in-the-loop simulation software packages such as DYNA4 (Vector Informatik GmbH 2025), SILAB (WIVW 2024), and SimCreator (FAAC Inc. 2025) provide dedicated bicycling dynamics subsystems ready to be parameterised for individual applications. These packages are developed by experts of specialised companies with high experience in simulator technology, which ensures quality of the bicycling dynamics subsystem.Use of such available solutions can significantly reduce development effort. While game engine software can be a cost-efficient solution, professional software packages may require high financial investments. With both off-the-shelf solutions, the limited accessibility of the underlying equations and source code for reasons of protecting proprietary knowledge can limit research.
For many bicycling simulators, an off-the-shelf bicycle home trainer is converted to create a pedalling resistance subsystem. Designed for virtual bicycle races, the top models feature variable pedalling resistance torque automatically controlled by an integrated bicycling dynamics subsystem which can be configured with basic parameters such as the bicycle’s and bicyclist’s mass. Although reduced to pedalling resistance as the only output, these off-the-shelf solutions can take into account variables to replicate gradient effects, air resistance, rolling resistance, drive train efficiency and adjustable virtual gear ratio (Wahoo Fitness LLC 2023). These solutions require minimal technical knowhow and can be very efficient for bicycling simulations which focus on longitudinal dynamic effects. However, comprehensive documentation of the underlying models and parameters is again often unavailable due to protection of proprietary knowledge.
While the limitations of off-the-shelf solutions may be acceptable for simulator use cases with low physical validity requirements it conflicts with good research practices such as transparent and reproducible research methods. In conclusion, off-the-shelf solutions can be resource efficient, but this benefit comes at the cost of intransparency and lack of adaptability due to proprietary software and hardware.

Full computational multi-DOF subsystems

Computational models allow the highest freedom of technical design and controllability and thus are a promising solution for achieving a higher degree of physical validity. Using computational multi-DOF bicycling dynamics subsystems, high-tech bicycling simulators can simulate bicycle stability, generate active motion cues, and integrate lateral balance as a control action (Haasnoot et al. 2023).
Although many bicycling simulators have been developed since the late 1990’s, only few research groups have published detailed reports on the development of the bicycling dynamics subsystems of their simulators. In one of the earliest publications on the development of a bicycling dynamics subsystem, the research group at Shanghai Jiao Tong University presents a modelling approach for a hexapod motion platform simulator based on the theoretic works by Hand (1988) and divide it into a stability model and a vibration model which replicates vertical acceleration forces resulting from uneven road surfaces (He et al. 2005). Building up on this, Yin and Yin (2007a, 2007b) present equations and an approach to the control engineering of pedalling and steering torque force feedback and a simple implementation of brakes.
Shoman and Imine (2020a, 2021; Shoman; 2022) present their works on improving the bicycling simulator at the French IFSTTAR institute with an emphasis on tyre-road interaction forces and vertical vibration. The most comprehensive and continuous research on bicycling dynamics has been done at TU Delft leading to bicycling simulators of different designs and capabilities building up on the linearised equations by Meijaard et al. (2007). Schwab and Recuero (2013) developed and tested a model for a simulator with steering torque force feedback. Adding to this, Lee et al.’s (2017) publication of a bicycling dynamics model also includes pedalling torque force feedback and the full software code published open-source. Haasnoot et al. (2023; 2021) present a novel design for a bicycling simulator with lateral motion cueing enabling realistic roll and steer actions to navigate the virtual bicycle. Their detailed derivation of dynamic equations and development of control loops shows the complexity of developing motion base bicycling simulators with high physical validity.

Modular open-source dynamic models

While there is a wealth of literature on bicycling dynamics theory and even on bicycling dynamics modelling for bicycling simulators, only few models have been published as open-source software code. We could only identify three openly published bicycling dynamics models for simulators which are the “model based bicycle simulator” (Lee et al. 2017; Lee and Van Gelderen 2017), the Python library “SymBRiM” and the simulator software package “Sumonity” (Pechinger and Lindner 2024).

Despite extensive research into bicycling dynamics subsystems, critical theoretical gaps remain (particularly in bicycle stability, tyre-road interactions, and vertical vibration) while openly available models are scarce. Current implementations in simulator development vary widely in both the complexity of modelled effects and the transparency of reporting. This lack of standardisation hinders independent researchers from comparing, reproducing, or validating bicycling dynamics subsystems and the experiments conducted using them.
A modular framework for bicycling dynamics subsystems could address these challenges by enabling parallel development and validation of individual models. However, such an approach requires a structured framework to ensure compatibility and foster collaboration across research groups.
This page proposes a systematic framework for bicycling dynamics subsystems, building on the existing framework for bicycling simulator technology. By deriving functional models, input and output variables, and parameters through a systematic literature review, we establish a foundational architecture. This framework supports the systematic and parallel development of models, which can be combined into a modular subsystem to be tailored to the specific needs of individual bicycling simulators. The result is a flexible, reproducible, and collaborative approach to advancing bicycling dynamics research for simulators.

Method – A Literature Review

The preliminary exploratory literature review indicated a large number of publications on bicycling dynamics which do not consider integration in bicycling simulators. Hence, we conducted a search focusing on publications describing the development, implementation and verification of bicycling dynamics subsystems for bicycling simulators. Thereby, our search strategy was based on an existing systematic literature review on bicycling simulators which is described on the OpenBikeSim Design page.
All publication types were included, but works in languages other than English were excluded leading to a selection of 20 relevant publications. One doctoral dissertation reported two bicycling dynamics subsystems leading to 21 subsystems analysed.
To create a systematic framework for the bicycling dynamics subsystem we first extracted the input and output variables incorporated in all analysed subsystems and key parameters used. Through functional decomposition we then identified the key dynamic effects modelled. We clustered the effects into core models and defined how they are interacting to form the full bicycling dynamics subsystem. Comparing the different modelling approaches we derived overarching definitions for input and output variables and models. Taken together, this approach forms a systematic framework for bicycle dynamics subsystems based on previous published research.

Publications and Solutions reviewed

The literature review identified 21 descriptions of bicycling dynamics subsystems from 20 publications (including one doctoral dissertation presenting two distinct approaches) spanning the years 2005 to 2024. These subsystems were developed across eight international research labs, with TU Delft, IFSTTAR, and Shanghai Jiao Tong University contributing the majority of published research.
The development of advanced bicycling dynamics subsystems has been driven by both technical and scientific objectives, primarily focused on replicating the intricate interaction between bicycle and bicyclist. Technically, researchers aim to overcome the constraints of stationary setups by incorporating high-fidelity motion (such as roll and yaw acceleration) to enhance realism (Haasnoot 2021; Haasnoot et al. 2023; Bruzelius and Augusto 2018; Martinez Garcia 2021; Hernández-Melgarejo et al. 2020; Chen et al. 2007; Yin and Yin 2007a, 2007b; He et al. 2005). Engineering efforts also prioritise reducing system latencies to ensure precise and instantaneous sensory feedback, particularly for haptic steering torque and inertial resistance during critical manoeuvres like balancing or starting (Bruzelius and Augusto 2018; Dialynas et al. 2019; Lee et al. 2017; Powell 2017; Powell et al. 2018; Shoman and Imine 2021, 2020a).
From a theoretical perspective, there is a pressing need to quantify the human control loop in real-world bicycling. This is essential for refining models of active balance systems and developing steer-assist functionalities (Dialynas et al. 2018; Schwab and Recuero 2013; Hernández-Melgarejo et al. 2020). Collectively, these advancements reflect a broader shift: transforming human-in-the-loop bicycling simulators from basic exercise tools into high-fidelity research instruments capable of replicating the fundamental mechanics of bicycling.

Systematic Framework

Building on the systematic framework for bicycling simulator technology, we derived a detailed functional architecture for the bicycling dynamics subsystem. The review identified nine input variables from the physical subsystems and seven from the environment subsystem, with ten output variables exchanged between both physical and virtual subsystems.
The dynamic behaviour of the bicycle-bicyclist system was synthesised into four functional models (stability, tyre, propulsion, and vertical vibration) alongside one overarching multibody model defining the mechanical system. The resulting subsystem architecture, presented in figure ‘Architecture of the bicycling dynamics subsystem with inputs, outputs, parameters and models’, captures the functional principles of all reviewed bicycling dynamics solutions.
The following sections outline the input and output variables, each with a uniform definition synthesised from the literature. Where possible, we adopted the variable definitions from Meijaard et al. (2007) due to their foundational influence on bicycling dynamics research, harmonising them with other reported practices to improve clarity and comparability. The identified models replicating the dynamic behaviour of the bicycle-bicyclist system are structured into one multibody model and the four functional models. These are reviewed in subsequent sections, detailing the physical and mathematical approaches used in their derivation.

Input variables from physical subsystems

Steering angle $\delta$ and torque $T_\delta$

The handlebar steering subsystem provides two inputs to the bicycling dynamics subsystem which are the steering angle and the steering torque.
The steering angle is denoted by $\delta$ and given in radiant $[rad]$. We adopt the definition by Meijaard et al. (2007): “The steering angle $\delta$ is the rotation of the front handlebar frame with respect to the rear frame about the steering axis. A right turn of a forward-moving bicycle has $\delta>0$.” (Meijaard et al. 2007)
The first derivative of the steering angle $\delta$ with respect to time given in seconds $[s]$ is the steering angular velocity $\dot{\delta}$ $\left[\frac{rad}{s}\right]$ and the second is the steering angular acceleration $\ddot{\delta}$ $\left[\frac{rad}{s^2}\right]$ accordingly. Of the reviewed 21 solutions 20 report using the steering angle as an input variable, seven use the steering angular velocity and two use the steering angular acceleration.
The steering torque $T_\delta$ given in newton-metres $\left[Nm\right]$, is the externally applied torque acting on the handlebar assembly about the steer axis. It represents an action-reaction torque pair. A torque applied in the clockwise direction (when viewed from above) to the handlebar assembly, and an equal and opposite reaction torque applied to the rear frame. In simplified bicycling dynamics models, $T_\delta$ is typically assumed to be the torque directly applied by the bicyclist to initiate or control handlebar steering motion (Meijaard et al. 2007).
Four solutions implemented steering torque as an input variable.

Rear frame roll angle $\phi$ and torque $T_\phi$

The motion subsystem provides two input variables affecting the trajectory and balance of the bicycle which are the rear frame roll angle and the rear frame roll torque. In bicycling dynamics research this degree of freedom is also referred to as lean, but we use the term roll hence it is more commonly used in recent literature and more consistent across different vehicle simulators.
The roll angle, denoted by $\phi$ and given in radiant $[rad]$, describes the lean of the bicycle’s rear frame around the roll axis, defined as the line connecting the rear wheel ground contact point with the intersection point of the ground and the steering axis. A positive value of $\phi$ indicates a roll to the bicycle’s right side in forward direction. (Meijaard et al. 2007)
The first derivative of the roll angle $\phi$, with respect to time given in seconds $[s]$ is the roll angular velocity $\dot{\phi}$ $\left[\frac{rad}{s}\right]$ and the second derivative is the roll angular acceleration $\ddot{\phi}$ $\left[\frac{rad}{s^2}\right]$. The roll angle is reported as an input in four publications.
In compliance with the definition of the roll angle, the roll torque, denoted by $T_\phi$ and given in newton-metres $[Nm]$, is the net external torque acting on the bicycle-bicyclist system about the roll axis. It represents the component of the total torque that contributes to the leaning motion of the bicycle and is positive in the rightward direction. (Meijaard et al. 2007)
Although used in theoretical works on bicycling dynamics, none of the reviewed publications implemented roll torque as an input variable. Both publications by Yin and Yin (2007a, 2007b) are a notable exception as they used the lean angle, angular velocity, and angular acceleration of the bicyclist’s body relative to the rear frame as an input variable. This does not directly represent the defined roll angle, but was used as an indicator of the roll torque to calculate the roll angle of the bicycle as described above.

Pedalling angle $\omega$ and torque $T_\omega$ and motor power $P_{motor}$

The propulsive power generated by the human and converted to propulsion by the mechanics of the bicycle result from the input variables of the pedalling angular velocity, also called cadence, and the pedalling torque which are both provided by the pedalling resistance subsystem. Optionally an additional motor power can be simulated replicating a pedelec support based on input from a human-machine interface (HMI) subsystem.
The pedalling angle, denoted by $\omega$ and expressed in radiant $[rad]$, represents the instantaneous angular position of the right crank lever. The angle increases positively in the direction of forward pedalling, defined as clockwise when viewed from the right-hand side of the bicycle. The reference position $\omega=0$ corresponds to the configuration in which the right crank lever points directly forward and is parallel to the bicycle’s roll axis, under the condition that the steering angle $\delta=0$. The pedalling angle correlates with the rotation angle of the rear wheel $\theta_R$ according to the gear ratio when a fixed gear is used. In contrast, when a freewheel is present in the drivetrain, this correlation exists only when the pedalling torque $T_\omega\gg0$.
The first derivative of the pedalling angle $\omega$, with respect to time given in seconds $[s]$ is the pedalling angular velocity $\dot{\omega}$ $\left[\frac{rad}{s}\right]$, also called cadence, and the second derivative is the roll angular acceleration $\ddot{\omega}$ $\left[\frac{rad}{s^2}\right]$.
The pedalling torque $T_\omega$ given in newton-metres $[Nm]$, is the torque in direction of the pedalling angular velocity $\dot{\omega}$ as defined above. The product of the pedalling torque and the pedalling angular velocity represent the pedalling power input $P_\omega$ by the bicyclist to propel the bicycle.
Remarkably, the pedalling angle is not implemented in any published solution, but 17 publications report using the rear wheel angular velocity and one also the rear wheel acceleration as an input variable. Accordingly, three publications use the rear wheel torque as an input variable instead of the pedalling torque which is used in none of the reported solutions.
The motor power denoted by $P_{motor}$ and given in watts $[W]$ is an optional input from an additional human-machine interface (HMI) subsystem. For the simulation of a pedelec drive, the motor power represents the additional propulsive power of an auxiliary motor in the drive train of the bicycle. None of the reviewed publications report an option to simulate pedelec assistance by a motor power input variable.

Braking torque $T_{brake}$

Braking torques applied at the front and rear wheels are denoted by $T_{brake-F}$ and $T_{brake-R}$ respectively, and are measured in newton-metres $[Nm]$. These torques act opposite to the direction of wheel rotation and produce a decelerating effect when the corresponding wheel has non-zero angular velocity. The braking torque inputs are provided by the braking subsystem which replicates the physical characteristics of the braking device to be simulated. For the modelling purpose it is important to emphasise the obvious: Braking torque serves to oppose the net propulsive power of the bicycle, but cannot actively generate reverse propulsion. Its effect is limited to reducing forward motion, and it contributes no power when the wheel is stationary. Of the reviewed publications two implemented braking torque as an input and six used a force-based approach implementing braking force to oppose the propulsive forces.

Input variables from the environment subsystem

Naturally, the bicycle-bicyclist system physically interacts with the environment it is moving through. The corresponding variables depend on the current location of the system in space and time, namely the spatial position within the virtual environment and the temporal situation within the simulation scenario. The predominant interactions occur at the points of contact between the bicycle-bicyclist system and the environment which are the tyre-road contact points and the windage area. In the following the input variables provided by the environment subsystem are presented.

Gradient $\alpha$

The gradient also called slope or incline, represents the inclination of the road surface. It is denoted by $\alpha$ and given in radiant $[rad]$. At a non-zero gradient the gravity vector is not perpendicular to the road surface which results in horizontal gradient force acting on the bicycle-bicyclist system. To describe the effects of this gradient force the vector can be divided into longitudinal and lateral components at the tyre-road surface contact points of front and rear wheel. The longitudinal component of the gradient force acting in direction of the intersection between wheel plane and road surface plane adds up to the force accelerating or decelerating the bicycle. The lateral component of the gradient force perpendicular to the longitudinal force on the road surface plane adds up to the tyre forces. Since both wheels can be positioned at an angle to each other (steering angle $\delta>0$), the gradient force must be considered separately for each tyre-road surface contact point depending on the local gradient.
While the function for modelling the gradient forces is described in propulsion model section below, the required input variables described in this section are the local longitudinal and lateral gradients at front and rear tyre-road surface contact points. They are denoted by $\alpha_{x-F}$, $\alpha_{y-F}$, $\alpha_{x-R}$, $\alpha_{y-R}$ respectively and given in radiant $[rad]$.
Of the reviewed solutions twelve implemented gradient input into the equations for the longitudinal effects and three explicitly neglect it. Seven publications report implementation of lateral gradient input for modelling tyre forces.

Tyre-road surface adhesion $\mu$

The friction between the bicycle tyres and the road surface is determined by the unitless tyre-road surface adhesion coefficient $\mu$ which depends on characteristics of both components’ materials (Gressmann 2017). The tyre’s contribution to the adhesion coefficient depends on the tyre profile, compound material, pressure and temperature and may be modelled as part of the tyre model. The road surface’s influence on the adhesion coefficient depends on the surface profile (smooth or uneven), material (asphalt or gravel), condition (icy or wet) and temperature. It is a local variable defined in the environment subsystem.
Since its locality the tyre-road surface adhesion coefficient needs to be provided for each tyre-road surface contact point at front and rear which are denoted by $\mu_F$ and $\mu_R$ respectively.
In the reviewed solutions, only the developments by Shoman and Imine (2020a, 2020b, 2021; Shoman 2022) implement tyre-road surface adhesion as an input variable. The majority of publications assume sufficient friction between the tyre and road surface and neglect potential slip.

Tyre-road resistance

When rolling, the bicycle’s tyres generate resistance that counteracts the power propelling the bicycle forward. According to Gressmann (2017), this resistance consists of the three components which are the flexing resistance, rolling resistance and road resistance. The flexing resistance (original German: “Walkwiderstand”) describes the power loss that occurs due to the not fully elastic deformation of the tyres at the contact area. This effect depends on the pressure and structure of the tyres. It is captured by a unitless flexing resistance parameter for front and rear wheel and denoted by $i_{flex-F}$ and $i_{flex-R}$ respectively.
The rolling resistance (original German: “Abrollwiderstand”) describes the power loss resulting from the displacement of the actual rolling point in front of the theoretical rolling point (at right angles to the road surface under the tyre axis). Again, this effect depends on the tyre pressure and tyre structure and is captured by a unitless rolling resistance parameter for front and rear wheel and denoted by $k_{roll-F}$ and $k_{roll-R}$ respectively.
The road resistance (original German: “Fahrbahnwiderstand”) describes the power loss caused by small irregularities in the road surface, such as fine gravel. Consequently, it is an input variable provided by the environment subsystem for each tyre-road contact point individually. The local road resistance is a unitless factor and is denoted by $i_{road-F}$ and $i_{road-R}$ respectively.
The majority of the reviewed solutions use simplified tyre models with rigid knife edged tyres with point contact to the road surface neglecting power loss due to rolling resistance. Bruzelius and Augusto (2018) implement a constant rolling resistance force which is down scaled for lower velocities. Ten solutions implement variable rolling resistance equations depending on the longitudinal velocity. However, all of them combine the parameters for the three components of rolling resistance outlined above in one constant parameter. Only the solutions by Shoman and Imine implement a variable road adhesion coefficient for modelling rolling resistance.

Road surface profile $z_{road}$

The road surface profile represents vertical variations of the ground level due to uneven road surfaces such as gravel roads or potholes in asphalt pavement. In comparison to gradient its effect is only considered in the longitudinal direction of travel, and the variations are of much higher frequency and lower amplitude than the variations in gradient. The effect of the road surface profile is predominantly the cause of vertical displacement of the front and rear wheel resulting in vertical vibration of the bicycle-bicyclist system. (Shoman 2022).
Theoretically, it also influences tyre forces and propulsion resistance power (Gressmann 2017). However, these effects are assumed to be captured in the tyre-road surface adhesion and tyre rolling resistance variables.
The road surface profile is a variable input provided by the environment for the front and rear tyre-road surface contact point locally. It is denoted by $z_{road-F}$ and $z_{road-R}$ respectively and given in millimetres $[mm]$.
The solutions by He et al. (2005) and Shoman and Imine (2020a, 2021; 2022) use a road surface profile variable to simulate vertical vibrations.

Aerodynamic effects on bicycling

Above approximately 15 km/h, air drag becomes the dominant resistance force opposing the longitudinal movement of the bicycle-bicyclist system (Gressmann 2017). This drag arises as the system moves through air, with its vectorial strength and direction determined by the system’s forward motion and ambient wind.
Gressmann (2017) categorises total wind drag into three components: pressure resistance, air friction resistance, and induced resistance. Induced resistance generates an upward force perpendicular to the airflow, but its effect is negligible due to the irregular shape of the system and the motion of the bicyclist’s legs (Gressmann 2017). Pressure and air friction resistance act along the airflow vector, opposing propulsion in headwind or no-wind conditions, while tailwind reduces drag. Sidewind may decrease drag by reducing the exposed windage area, and advanced aerodynamic designs, such as optimised wheels, can even harness sidewind for propulsion. However, such technologies remain limited to competitive bicycling and are not relevant for mobility-focused research. Similarly, slipstreaming, despite its significant drag reduction, is uncommon in everyday bicycling and is disregarded here.
Lateral sidewind components cause drift, necessitating counter-steering which results in additional resistance (Schwab et al. 2018). Compared to longitudinal aerodynamic forces this effect is minor and is neglected in the present context. Air friction resistance is also small and negligible (Gressmann 2017), leaving pressure resistance as the primary consideration for everyday bicycling dynamics.
Pressure resistance depends on environmental variables: the wind vector $\vec{v_{wind}}$ given in metres per second $\left[\frac{m}{s}\right]$, and the ambient air density, $\rho_{air}$ given in kilogram per cubic metre $\left[\frac{kg}{m^3}\right]$. The air density varies depending on the air temperature and pressure. However, in this context of bicycling mobility it can be assumed to be constant.
Bicycle-bicyclist system-specific attributes further influence air drag, including the unitless air drag coefficient $c_w$ which reflects the shape and surface properties as well as the windage area, $A$ given in square metres $[m^2]$, perpendicular to the airflow. While rotating wheel spokes contribute additional drag, this effect is less influential for everyday bicycling mobility and is omitted here.
Of the reviewed solutions twelve implement air density as a constant parameter and none as a variable input. No publication does consider ambient wind effects and reduce resistance power by air drag to the component created by the bicycle-bicyclist system’s movement.
Air drag coefficient and windage area are implemented as constant parameters in 13 solutions. No reviewed solution does implement air drag effects on the lateral movement and stability of the system.

Output variables

The following subsections describe the output variables provided by the bicycling dynamics subsystem to the other subsystems.

Velocity vector of the bicycle-bicyclist system $\vec{v_R}$

While the bicycling dynamics subsystem may compute velocities for different points of the bicycle-bicyclist system internally the output variable relevant to the other subsystems is the velocity of the rear tyre-road contact point R. The translational velocity vector is denoted by $\vec{v_R}$ and given in metres per second $[m/s]$. Its vector components are defined by the local coordinate system, also called frame, with origin in point R with $\dot{x}_R$ being the longitudinal, $\dot{y}_R$ the lateral and $\dot{z}_R$ the vertical component.
The transformation from the local frame R to the global fixed frame in point O is achieved through a rotation matrix $M_{rot}$, which accounts for the relative orientation between the two frames. The orientation is defined by three Euler angles which are the roll ($\theta$), pitch ($\phi$), and yaw ($\psi$), corresponding to rotations about the x, y, and z axes, respectively. These rotation angles are also output variables of the bicycling dynamics subsystem. The rotation matrix $M_{rot}$ is derived by composing the elementary rotation matrices $R$ in the ZYX sequence (yaw, pitch, roll):

$$M_{rot} = R_z(\psi) \cdot R_y(\phi) \cdot R_x(\theta),$$

The velocity vector $\vec{v_R}$ in the local frame is transformed to the global frame $v_O$ via:

$$v_O = M_{rot} \cdot \vec{v_R}$$

The translational and rotational velocities were integrated as variables for updating the virtual environment in all reviewed solutions. However, for motion cueing only a part of the reviewed studies implemented translational and rotational accelerations (longitudinal n=4, lateral n=6, vertical n=4, roll n=8, pitch n=6, yaw n=6).

Steering resistance torque $T_{\delta-res}$

The steering resistance torque $T_{\delta-res}$ $[Nm]$ is the reaction torque against the steering torque $T_\delta$ as described above. It was integrated in 15 solutions as an output variable to control haptic handlebar steering resistance feedback.

Pedalling resistance power $P_{\omega-res}$ and pedalling resistance torque $T_{\omega-res}$

The pedalling resistance torque $T_{\omega-res}$ is the reaction torque against the pedalling torque $T_\omega$ $[Nm]$ as described above. Consequently, the pedalling resistance power $P_{\omega-res}$ $[W]$ is the power acting against the human pedalling power input $P_\omega$ and the pedelec motor power $P_{motor}$.
Haptic pedalling resistance was implemented in 20 of the reviewed solutions.

Multibody model

A multibody model is a prerequisite for analysing the dynamics of interconnected bodies undergoing large translational and rotational displacements, with motion governed by kinematics and force equilibrium (Shabana 2020). In the bicycling dynamics application, the bicycle-bicyclist system is divided into bodies with defined properties, namely geometry, mass, flexibility, and damping. Free-body diagrams identify interconnections between these bodies and the environment, using theoretical joints to constrain relative motion to specific degrees of freedom (DOF). Strategic placement of coordinate systems, also called frames, aligns with key points and axes to simplify the derivation of equations of motion. Thus, the multibody model is essential for deriving equations that describe the bicycle-bicyclist system’s kinematic and dynamic behaviour.

Most reviewed solutions use a single multibody model, but some employ multiple models to address distinct phenomena, such as bicycling stability and vertical vibration. For stability modelling, seven publications adopt Whipple’s (1899) four-rigid-body model: front and rear wheels, rear frame with the bicyclist’s body, and front frame (Whipple 1899). Others vary in defining the bicyclist’s body: three studies split it between front and rear frames, while another three separate the upper body from the rear frame to incorporate leaning actions. All stability models use rigid bodies connected by frictionless revolute joints.

For vertical vibration analysis, four studies create an additional multibody model to simplify the system to two bodies with defined mass, stiffness, and damping. Shoman and Imine (2021) divide the system into a front body (front wheel, front frame, and a fraction of the bicyclist’s body) and a rear body (rear wheel, rear frame, and the majority of the bicyclist’s body), with tyre-road surface contact points modelled as spring-damper elements (Shoman and Imine 2021). He et al. (2005) separate the bicyclist’s body from the bicycle, connecting them via spring-damper elements at the saddle, handlebar, and tyre-ground contact points (He et al. 2005). As the differences in the multibody models can also lead to differences in the equations of motion, transparent documentation is an important prerequisite for comparable bicycling dynamics subsystems.

Propulsion model

The propulsion model replicates the longitudinal $\left(x\right)$ real-world dynamics of a bicycle-bicyclist system, where forward movement is influenced by mass moment of inertia, elevation gain, and friction losses. These factors collectively determine the power required for propulsion to bicycle at a certain forward speed $\dot{x}_R$. In the following, all powers acting on the propulsion of the bicycle-bicyclist system are presented and their implementation in previous solutions reviewed.
The resistance power output $P_{\omega-res}$ $[W]$ is determined by the bicycling dynamics subsystem computing the equilibrium of all powers acting on the propulsion of the bicycle-bicyclist system which are the human pedalling power input $P_\omega$ $[W]$, motor power input $P_{motor}$ $[W]$, brake power input $P_{brake}$ $[W]$, air drag power $P_{air}$ $[W]$, gradient power $P_{grad}$ $[W]$, inertia power $P_{inertia}$ $[W]$, tyre-road resistance power $P_{tire}$ $[W]$, mechanical friction resistance power $P_{mech}$ $[W]$, cornering power $P_{corner}$ $[W]$, and vibration damping power $P_{vib}$ $[W]$.

$$\sum\limits P=0=P_{\omega-res}+P_\omega+P_{motor}+P_{brake}+P_{air}+P_{grad}+P_{inertia}+P_{tire}+P_{mech}+P_{vib}+P_{corner}$$

The review of published propulsion model solutions shows significant gaps in the reporting.
Six published solutions did not include any details about the propulsion power modelling at all and the other publications reported input variables for power modelling such as brake force, but did not provide detail on how this input was integrated in the propulsion model. The reported modelling approaches are reviewed for each effect acting on the propulsion as follows:

• Pedalling power input was reported for three solutions explicitly. The other solutions reporting on the propulsion modelling mentioned pedalling input variables such as pedalling angular velocity, but did not describe how the human propulsion input was modelled.
• Motor power input from a virtual pedelec drive was implemented in none of the reviewed solutions.
• Brake power was only mentioned in one publication, but detailed reports on the modelling were missing in all solutions. This is in contradiction to the reporting of brake input variables which were reported for twelve solutions as described above.
• Air drag power was modelled in 13 solutions consistently as a function of air density, air resistance coefficient, and frontal area, as constant parameters, and forward speed cubed as the sole variable. However, no solution accounted for ambient wind effects.
• Gradient power based on elevation changes of the road was implemented in eleven solutions, as a function of the system’s total mass, gradient angle in longitudinal direction, and forward speed.
• Inertia power, due to changes in the longitudinal speed was modelled in eleven solutions, using constant translational and rotational mass moments of inertia and variable forward velocity squared.
• Tyre-road resistance power, implemented in eleven publications, replicates the energy loss from not fully elastic tyre deformation against the road surface. In previous solutions, this effect was modelled using constant parameters for tyre and ground rolling resistance, system mass, gravitational acceleration and variable forward velocity. It remains unclear whether publications incorporating vertical vibration models included vertical acceleration forces at the front and rear wheels in tyre-road resistance equations.
• Mechanical friction resistance power models replicating friction losses in the drive train and wheel bearings were reported for three solutions.
• Vibration damping power losses due to vertical oscillation energy absorbed by damping bodies in the bicycle-bicyclist system were modelled in five solutions.
• Cornering power affecting the bicycle-bicyclist system’s propulsion during steering manoeuvres are described in fundamental bicycling mechanics literature (Gressmann 2017; Wilson and Schmidt 2020), but were not modelled in previous solutions.

Stability model

The stability model replicates the characteristic properties of a single-track bicycle as best defined in Meijaard et al.’s (2007) fundamental work: “A controlling bicyclist can balance a forward-moving bicycle by turning the front wheel in the direction of an undesired lean. This moves the ground contact points under the bicyclist, just like an inverted pendulum can be balanced by accelerating the support point in the direction of lean. Some uncontrolled bicycles can balance themselves. The torques for the self-correcting steer motions can come from various geometric, inertial and gyroscopic features of the bicycle.” (Meijaard et al. 2007)
Consequently, the interacting variables to be computed by the stability model include the roll and yaw angle and the lateral movement which can be represented in a simulator via visual and motion cues and the steering torque. Equations to model this behaviour were reported in 15 publications. Since there is a large body of research on modelling bicycle stability as described in the introduction, all existing bicycling dynamics subsystems were developed on the base of existing stability models. The fundamental works referenced in the analysed publications include Åström et al. (2005) (n=2), the works by Caro et al. (Caro et al. 2013; Caro et al. 2015) (n=4), Hand (1988) (n=3), and most popularly Meijaard et al. (2007) (n=6). Despite the variety in the base equations used, the presented stability models have in common that they are linear and inspired by the original Whipple (1899) model.
For computing the steering torque ten publications reported equations based on the stability model which varied largely in level of detail modelled and the modelling approach used. All ten publications reported the steering angle as an input variable to the model. The implementation of other input variables differed largely among the reported equations and included the angle, speed or acceleration of roll (n=5) or yaw (n=1) and the longitudinal speed (n=5) in the bicycle’s direction of travel.
The effects causing the steering torque which are replicated in the models include the front tyre force, the rotational mass moment of inertia of the front frame assembly and steering parts of the human body such as the lower arms, the frictional losses in the bearings of the bicycle’s headset, the gyroscopic effect of the rotating front wheel, and the gravitational force when the roll angle is larger than zero. Similar to the stability equations, the developers referenced earlier works as the base of their models which include the equations developed by Åström et al. (2005) (n=2), which in one case were combined with a damping term developed by (Sharp 2008), the equations by Meijaard et al. (2007) (n=4), and by Chen (2004) (n=1). In summary, the stability model and its derived steering torque equations are grounded in foundational research, primarily linear and inspired by the Whipple model, with variations in input variables and modelled effects.

Tyre model

The tyres of a real-world bicycle serve multiple critical functions: they provide adhesion for controlled steering and power transmission, and they dampen vertical vibrations. However, the tyre models discussed here focus solely on their impact on steering manoeuvres and longitudinal acceleration.
Most reviewed bicycling dynamics subsystems model tyres as rigid, knife-edged bodies with single-point road contact, using simplified non-holonomic constraints. These models assume infinite adhesion, ensuring all forces are transmitted without slip.
Seven publications employ a linear tyre model to calculate tyre forces and slip angles, assessing their influence on the bicycle’s longitudinal speed and yaw rate. While all assume constant tyre parameters such as cornering stiffness and camber Shoman and Imine (2021) introduce tyre-road adhesion as a variable input from the environment subsystem.
Unlike the stability model, none of the publications reference original works for their tyre models, except for a video tutorial cited by Martinez Garcia (2021). This omission reduces the transparency of the tyre models presented.

Vertical vibration model

The perceived comfort of riding a bicycle is strongly affected through vertical vibrations and the bicycle’s capability of damping these vibrations before they reach the bicyclist (Gressmann 2017; Wilson and Schmidt 2020). This effect can be modelled by a multibody system with spring-damper elements between the individual bodies as described above. For analysis and modelling, the multibody system is described by differential equations which allow the computation of vertical accelerations and forces at certain points of interest such as the contact points of the bicycle and the bicyclist’s body.
While vertical vibrations can also be induced through movements of the bicyclist such as the pedalling motion, the four publications which presented a vertical vibration model focused on forces induced through an uneven road profile. The road profile was implemented as vertical displacements at the front and rear tyre-ground contact points depending on the bicycle’s forward speed. All four publications used real-world data to replicate the road profile. Thus, the modelling of vertical vibrations in bicycles primarily focuses on road-induced forces, using real-world data to simulate road profiles and their impact on bicyclist comfort.

Discussion

Our review revealed significant variety and complexity in bicycling dynamics subsystems, which is expected given that the self-stability of single-track bicycles remains incompletely understood despite over a century of research.
To address these findings, we highlight the most critical aspects of bicycling dynamics research for bicycling simulators and propose guiding questions for each. The diversity of bicycling dynamics subsystems is substantial but difficult to assess due to the low level of detail reported in publications and the scarcity of open-source models. This review included only English-language literature and did not search Git repositories, which may have resulted in missed open solutions for bicycling dynamics subsystems. Nevertheless, this is the first comprehensive review of bicycling dynamics subsystems specifically for bicycling simulators. The diversity presented through our review underscores the need for reporting guidelines and encourages authors to publish their models openly. Until such guidelines exist, researchers should consider: What is the minimal reported information required for an independent researcher to compare and reproduce the bicycling dynamics subsystem?
The effects modelled or omitted vary widely, indicating that the development of bicycling dynamics subsystems is heavily influenced by the intended use case of the bicycling simulator. A modular approach is therefore recommended to allow easy interchange of models. As there is no straightforward answer to which simulator technology best suits a given application (Wynne et al. 2019), we advise researchers to reflect on the bicycling task by asking: How agile are the manoeuvres to be simulated, and which dynamic effects are most important for controlling the bicycle-bicyclist system? Additionally, developers should consider the tolerances for input and output variables that can be afforded without compromising the meaningfulness of experimental results.
None of the reviewed bicycling dynamics subsystems provided a full assessment of physical validity. While some publications reference the benchmark test proposed by (Meijaard et al. 2007) for stability models, no approaches were identified for validating tyre, vertical vibration, or propulsion models, nor complete subsystems. This highlights the need for best practices in validation, similar to those proposed for driving simulators (Schwarz et al. 2011). Until such practices are established, it is essential to evaluate for each subsystem and application: Which aspects of the bicycling dynamics simulation are most relevant for validation, and how can they be assessed against comparative data? Open resources, such as benchmarking tools and datasets, could significantly enhance the quality and transparency of future models and their validation.
Publications on high-fidelity motion platform bicycling simulators often stress the importance of accurate bicycling dynamics subsystems to elicit realistic bicyclist behaviour and reduce simulator sickness (Haasnoot 2021; Shoman 2022). However, evaluations of motion base simulators have shown varying degrees of simulator sickness, partly attributed to sensory conflict and suspected high latency in the control loop (Bruzelius and Augusto 2018; Haasnoot 2021; Haasnoot et al. 2023; Shoman and Imine 2020b). This necessitates the development of high-performance control loops, requiring bicycling dynamics subsystems with minimal computational demands. Balancing accuracy with computational efficiency is crucial, prompting the question: Which latency in the control loop can be afforded without causing severe sensory conflict or simulator sickness, and how much computing time does each component require?
Our review presented technical solutions for bicycling dynamics subsystems in three categories: mechanical, off-the-shelf, and computational. Simple mechanical or off-the-shelf solutions can offer advantages over complex control loops with computational models. However, simple solutions cannot reproduce all physical effects playing into the dynamics of highly agile manoeuvres. Developers should critically assess their engineering ambitions and ask: Which mechanical, off-the-shelf, or computational technologies are the most suitable technical solution for the dynamic effects to be simulated?
Lastly, the bicycling dynamics subsystem is the core subsystem of a bicycling simulator, strongly influencing overall system performance and validity. Therefore, it should not be developed in isolation but with consideration of all other system parts by asking: Which requirements do the physical and environment subsystems place on the bicycling dynamics subsystem in terms of input and output variables, performance, and accuracy?
Until guidelines for bicycling dynamics subsystems are established, we recommend that developers use the proposed guiding questions to inform the design of new models. We further encourage the open publication of subsystems and models to foster transparency and reproducibility in the field. Future research should prioritise the development of efficient validation methods tailored to specific use cases, as current approaches lack comprehensive assessment. Addressing these gaps will not only improve the reliability of bicycling simulators but also advance the broader understanding of bicycle-bicyclist system dynamics. Such progress is essential for ensuring realistic behaviour and minimising simulator sickness in future studies.

Guiding questions for bicycling dynamics subsystems specific to the simulator use case:
1. Which latency in the control loop can be afforded without causing severe sensory conflict and simulator sickness?
2. How agile are the manoeuvres to be simulated, and which dynamic effects are most important for controlling the bicycle-bicyclist system?
3. What demands for input and output variables do the physical subsystems and the environment subsystem place?
4. Which inaccuracies in input and output variables can be afforded without endangering the meaningfulness of the results?
5. Which aspects of the bicycling dynamics simulation are most relevant for validation, and how can they be assessed against comparative data?
6. Which mechanical or computational solutions are most suitable for the dynamic effects to be simulated?

Conclusion

The presented work addressed the critical research gap in the standardisation and transparency of bicycling dynamics subsystems for human-in-the-loop bicycling simulators. Despite extensive research, the field lacks a unified framework to guide the development, validation, and comparison of these subsystems, which are central to simulator realism and research validity. The scarcity of openly available models and inconsistent reporting practices further hinder reproducibility and collaboration across research groups.
To fill this gap, we conducted a systematic literature review of 21 bicycling dynamics subsystems, extracting and synthesising their input and output variables, functional models, and architectural principles. By decomposing these subsystems into five core models (multibody, stability, tyre, propulsion, and vertical vibration) we derived a modular framework that standardises definitions and interactions. This method ensured compatibility with existing research while providing a foundation for future development.
The primary scientific contribution of this work is the proposal of a systematic framework that enables parallel development and validation of individual models, tailored to specific simulator use cases. This framework enhances transparency, reproducibility, and collaboration in the field. Future research should prioritise the development of open-source models, comprehensive validation methods, and reporting guidelines to address the current lack of standardisation and ensure the reliability of bicycling simulators for advancing human factors research.

Acknowledgements

We thank Prof. Dr.-Ing. Malte Rothhämel for his valuable feedback on the framework and modular bicycling dynamics subsystems.

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Appendix A1 – Table of reviewed Publications

Reference	Publication type
Bruzelius and Augusto (2018)	project report
Martinez Garcia (2021)	master thesis
Shoman and Imine (2021)	journal article
Shoman and Imine (2020a)	conference paper
Shoman and Imine (2020b)	conference paper
(Shoman 2022) (two bicycling dynamics subsystems presented)	doctoral dissertation
Hernández-Melgarejo et al. (2020)	journal article
Chen et al. (2007)	journal article
He et al. (2005)	journal article
Yin and Yin (2007b)	journal article
Yin and Yin (2007a)	journal article
Dialynas (2020)	doctoral dissertation
Dialynas et al. (2019)	journal article
Schwab and Recuero (2013)	conference paper
Lee et al. (2017)	conference paper
(Haasnoot 2021)	master thesis
(Haasnoot et al. 2023)	journal article
Powell (2017)	master thesis
Powell et al. (2018)	journal article
Stroh (2016)	bachelor thesis