A nonlinear mathematical model for a bicycle (original) (raw)
The Lagrangian Method for a Basic Bicycle
Journal of Applied Mathematics and Physics, 2014
The ground plan in order to disentangle the hard problem of modelling the motion of a bicycle is to start from a very simple model and to outline the proper mathematical scheme: for this reason the first step we perform lies in a planar rigid body (simulating the bicylcle frame) pivoting on a horizontal segment whose extremities, subjected to nonslip conditions, oversimplify the wheels. Even in this former case, which is the topic of lots of papers in literature, we find it worthwhile to pay close attention to the formulation of the mathematical model and to focus on writing the proper equations of motion and on the possible existence of conserved quantities. In addition to the first case, being essentially an inverted pendulum on a skate, we discuss a second model, where rude handlebars are added and two rigid bodies are joined. The geometrical method of Appell is used to formulate the dynamics and to deal with the nonholonomic constraints in a correct way. At the same time the equations are explained in the context of the cardinal equations, whose use is habitual for this kind of problems. The paper aims to a threefold purpose: to formulate the mathematical scheme in the most suitable way (by means of the pseudovelocities), to achieve results about stability, to examine the legitimacy of certain assumptions and the compatibility of some conserved quantities claimed in part of the literature.
An advanced model of bicycle dynamics
European Journal of Physics, 1990
A theoretical model of a moving bicycle is presented for arbitrary bicycle geometries at finite angles. The non-linear equations of motion are derived and solved with the help of a computer. The solutions are tested for energy conservation, and examined with respect to inherent stability. For common bicycles, velocity and lean angle ranges of self-stable motion are predicted.
Reduced Dynamics of the Non-holonomic Whipple Bicycle
Journal of Nonlinear Science
Though the bicycle is a familiar object of everyday life, modelling its full nonlinear three-dimensional dynamics in a closed symbolic form is a difficult issue for classical mechanics. In this article, we address this issue without resorting to the usual simplifications on the bicycle kinematics nor its dynamics. To derive this model, we use a general reduction based approach in the principal fiber bundle of configurations of the three-dimensional bicycle. This includes a geometrically exact model of the contacts between the wheels and the ground, the explicit calculation of the kernel of constraints, along with the dynamics of the system free of any external forces, and its projection onto the kernel of admissible velocities. The approach takes benefits of the intrinsic formulation of geometric mechanics. Along the path toward the final equations, we show that the exact model of the bicycle dynamics requires to cope with a set of non-symmetric constraints with respect to the structural group of its configuration fiber bundle. The final reduced dynamics are simulated on several examples representative of the bicycle. As expected the constraints imposed by the ground contacts, as well as the energy conservation are satisfied, while the dynamics can be numerically integrated in real time.
Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007
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 by rates of frame lean, steer angle and rear wheel rotation. We construct the terms in the governing equations methodically for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equations by hand in two ways and also checked them against two nonlinear dynamics simulations. In the century-old literature, several sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternativ...
Modelling, Control System Design and Simulation of an Autonomous Bicycle
Modelling, Identification and Control, 2014
This paper presents the mathematical modelling and control system design of an autonomous bicycle. The nonlinear equations of motion have been derived and the proposed control system has been used in the simulation of the dynamical behaviour of the bicycle. With the assumption of rolling without slipping condition for the wheels-ground interaction, the system is constrained by nonholonomic equations, and the equations of motion are highly nonlinear. Unlike many other approaches present in related literature, the dynamical model is preserved in simulations in its original nonlinear form without any simplifying assumptions and linearization. Numerical results of the simulations show that the proposed closedloop control system is achievable. Design of the experimental system has been based on a commercially available bicycle. The mechanical modifications and control system hardware have been designed according to the simulation results.
Control and Trajectory Planning of an Autonomous Bicycle Robot
Computation
This paper addresses the modeling and the control of an autonomous bicycle robot where the reference point is the center of gravity. The controls are based on the wheel heading’s angular velocity and the steering’s angular velocity. They have been developed to drive the autonomous bicycle robot from a given initial state to a final state, so that the total running cost is minimized. To solve the problem, the following approach was used: after having computed the control system Hamiltonian, Pontryagin’s Minimum Principle was applied to derive the feasible controls and the costate system of ordinary differential equations. The feasible controls, derived as functions of the state and costate variables, were substituted into the combined nonlinear state–costate system of ordinary differential equations and yielded a control-free, state–costate system of ordinary differential equations. Such a system was judiciously vectorized to easily enable the application of any computer program writ...
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 level ground. The mass distribution and geometry are otherwise arbitrary. This conservative non-holonomic system has a 7-dimensional accessible configuration space and three velocity degrees of freedom parameterized by rates of frame lean, steer angle and rear-wheel rotation. We construct the terms in the governing equations methodically for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equations by hand in two ways and also checked them against two non-linear dynamics simulations. In the century-old literature several sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternative formulations of the equations of motion or alternative numerical solutions. Further, the results here can also serve as a check for general-purpose dynamics programs. For the benchmark bicycles we accurately calculate the eigenvalues (the roots of the characteristic equation) and the speeds at which bicycle lean and steer are self-stable, confirming the century-old result that this conservative system can have asymptotic stability.
2006
People have been successfully building and riding bicycles since the 1800s, and many attempts have been made to describe the motion of these machines mathematically. However, common acceptance of the correct linearized equations of motion for a bicycle has remained elusive. In his 1988 master's thesis at Cornell University, Scott Hand derived the equations again and performed the first known extensive survey of the literature, finding and documenting the mistakes made in previous attempts. The question remained however of what mistakes, if any, Mr. Hand and his advisors made. The subsequent advent of cheap and plentiful computing power and the development of numerical methods to take advantage of it provide an opportunity to confirm, once and for all, the correct linearized equations of motion for an idealized bicycle. That is exactly what A. L. Schwab, J. P. Meijaard, and J. M. Papadopoulos have done in their recent paper.
Development of Efficient Nonlinear Benchmark Bicycle Dynamics for Control Applications
IEEE Transactions on Intelligent Transportation Systems, 2015
We present a symbolic method for modeling a nonlinear multibody bicycle with holonomic and nonholonomic constraints. The method, developed for robotic multibody dynamics, is applied to a benchmark bicycle, in which all six ground contact constraint equations are eliminated, leaving analytic coupled ordinary differential equations corresponding to the bicycle rear body roll, steer angle, and rear wheel rotation degrees of freedom without any approximation. We have shown that the nonlinear dynamics of the bicycle satisfies an underactuated manipulator equation and demonstrated an analytic method to solve the vehicle pitch angle from a quartic equation. This reduced analytic model offers insights in understanding complex nonlinear bicycle dynamic behaviors and enables the development of an efficient model suitable for real-time control outside of the linear regime.
The Dynamics of a Bicycle on a Pump Track -- First Results on Modeling and Optimal Control
arXiv (Cornell University), 2023
We investigate the dynamics of a bicycle on an uneven mountain bike track split into straight sections with small jumps (kickers) and banked corners. A basic bike-rider model is proposed and used to derive equations of motion, which capture the possibilities to accelerate the bicycle without pedaling. Since this is a first approach to the problem, only corners connected by straight lines are considered to compute optimal riding strategies. The simulation is validated with experimental data obtained on a real pump track. It is demonstrated that the model effectively captures the longitudinal bike acceleration resulting from the relative vertical motion between the rider's upper body and the bicycle. Our numerical results are in good analogy with real rider's actions on similar tracks.