# Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review

@article{Meijaard2007LinearizedDE, title={Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review}, author={Jacob P. Meijaard and Jim Papadopoulos and Andy Ruina and Arend L. Schwab}, journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, year={2007}, volume={463}, pages={1955 - 1982} }

We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally symmetric ideally hinged parts: two wheels, a frame and a front assembly. The wheels are also axisymmetric and make ideal knife-edge rolling point contact with the ground level. The mass distribution and geometry are otherwise arbitrary. This conservative non-holonomic system has a seven-dimensional accessible configuration space and three velocity degrees of freedom parametrized…

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## References

SHOWING 1-10 OF 252 REFERENCES

### Benchmark results on the linearized equations of motion of an uncontrolled bicycle

- Engineering
- 2005

In this paper we present the linearized equations of motion for a bicycle as, a benchmark The results obtained by pencil-and-paper and two programs are compaied The bicycle model we consider here…

### Linearized equations for an extended bicycle model

- Engineering
- 2006

The linearized equations of motion for a bicycle of the usual construction travelling straight ahead on a level surface have been the subject of several previous studies [1], [2]. In the simplest…

### Hands-free circular motions of a benchmark bicycle

- PhysicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2007

We write nonlinear equations of motion for an idealized benchmark bicycle in two different ways and verify their validity. We then present a complete description of hands-free circular motions. Three…

### Experimental validation of a model of an uncontrolled bicycle

- Engineering
- 2008

Abstract
In this paper, an experimental validation of some modelling aspects of an uncontrolled bicycle is presented. In numerical models, many physical aspects of the real bicycle are considered…

### Derivation of the Linearized Equations for an Uncontrolled Bicycle

- Engineering
- 2004

Abstract: The linearized equations of motion for an uncontrolled bicycle are derived. The bicycle has its usual construction and the contact between the wheels and the road is modelled by holonomic…

### Dynamics of Flexible Multibody Systems with Non-Holonomic Constraints: A Finite Element Approach

- Mathematics
- 2003

In this article it is shown how non-holonomic constraints can beincluded in the formulation of the dynamic equations of flexiblemultibody systems. The equations are given in state space formwith the…

### Observations on the controllability of motion of two-wheelers

- Mathematics
- 2001

Abstract This paper presents an investigation into the motion of two-wheelers: everyday motorcycles and bicycles. The emphasis is on controllability of the system for small lateral deviations from…

### Direct determination of periodic solutions of the dynamical equations of flexible mechanisms and manipulators

- Engineering
- 1991

A method based on finite elements, characterized by generalized strains as functions of the nodal co-ordinates, is the basis for the computer program SPACAR for the kinematic and dynamic analysis of…

### Control for an autonomous bicycle

- MathematicsProceedings of 1995 IEEE International Conference on Robotics and Automation
- 1995

A controller is derived which, using steering and rear-wheel torque, causes a model of a riderless bicycle to recover its balance from a near fall as well as converge to a time parameterized path in the ground plane.

### Symbolic vector/dyadic multibody formalism for tree-topology systems

- Physics
- 1991

A multibody formalism is presented that can be applied to automatically generate efficient equations of motion for a system of rigid bodies in a tree topology using a computer algebra language that supports vector/dyadic algebra, small variable simplification options, and the automated introduction of new symbols.