Jacob Philippus Meijaard

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

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 alternative formulations of the equations of motion or alternative numerical solutions. Further, the results here can also serve as a check for general purpose dynamic 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.

A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects

A new bicycle design points to the importance of mass distribution for stability. A riderless bicycle can automatically steer itself so as to recover from falls. The common view is that this self-steering is caused by gyroscopic precession of the front wheel, or by the wheel contact trailing like a caster behind the steer axis. We show that neither effect is necessary for self-stability. Using linearized stability calculations as a guide, we built a bicycle with extra counter-rotating wheels (canceling the wheel spin angular momentum) and with its front-wheel ground-contact forward of the steer axis (making the trailing distance negative). When laterally disturbed from rolling straight, this bicycle automatically recovers to upright travel. Our results show that various design variables, like the front mass location and the steer axis tilt, contribute to stability in complex interacting ways.

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

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 consists of four rigid bodies, viz a rear frame, a front frame being the front fork and handlebar assembly, a rear wheel and a fiont wheel, which are connected by revolute joints The contact between the knife-edge wheels and the flat level surface is modelled by holonomic constiaints in the normal direction and by non-holonomic constraints in the longitudinal and lateral direction The rider is rigidly attached to the rear frame with hands free from the handlebar This system has three degrees of freedom, the roll, the steer, and the forward speed For the benchmark we consider the linearized equations for small perturbations of the upright steady forward motion The entries of the matrices of these equations form the basis for comparison Three diffrent kinds of methods to obtain the results are compared pencil-and-paper, the numeric multibody dynamics program SPACAR, and the symbolic software system AutoSim Because the results of the three methods are the same within the machine round-off error, we assume that the results are correct and can be used as a bicycle dynamics benchmark

COMPARISON OF THREE-DIMENSIONAL FLEXIBLE BEAM ELEMENTS FOR DYNAMIC ANALYSIS: FINITE ELEMENT METHOD AND ABSOLUTE NODAL COORDINATE FORMULATION

Three formulations for a flexible spatial beam element for dynamic analysis are compared: a finite element method (FEM) formulation, an absolute nodal coordinate (ANC) formulation with a continuum mechanics approach and an ANC formulation with an elastic line concept where the shear locking of the asymmetric bending mode is suppressed by the application of the Hellinger–Reissner principle. The comparison is made by means of an eigenfrequency analysis on two stylized problems. It is shown that the ANC continuum approach yields too large torsional and flexural rigidity and that shear locking suppresses the asymmetric bending mode. The presented ANC formulation with the elastic line concept resolves most of these problems.Copyright

Comparison of Three-Dimensional Flexible Beam Elements for Dynamic Analysis: Classical Finite Element Formulation and Absolute Nodal Coordinate Formulation

Three formulations for a flexible spatial beam element for dynamic analysis are compared: a Timoshenko beam with large displacements and rotations, a fully parametrized element according to the absolute nodal coordinate formulation (ANCF), and an ANCF element based on an elastic line approach. In the last formulation, the shear locking of the antisymmetric bending mode is avoided by the application of either the two-field Hellinger‐Reissner or the three-field Hu‐Washizu variational principle. The comparison is made by means of linear static deflection and eigenfrequency analyses on stylized problems. It is shown that the ANCF fully parametrized element yields too large torsional and flexural rigidities, and shear locking effectively suppresses the antisymmetric bending mode. The presented ANCF formulation with the elastic line approach resolves most of these problems. DOI: 10.1115/1.4000320

HOW TO DRAW EULER ANGLES AND UTILIZE EULER PARAMETERS

This article presents a way to draw Euler angles such that the proper operation and application becomes immediately clear. Furthermore, Euler parameters, which allow a singularity-free description of rotational motion, are discussed within the framework of quaternion algebra and are applied to the kinematics and dynamics of a rigid body.

A review on bicycle dynamics and rider control

This paper is a review study in dynamics and rider control of bicycles. The first part gives a brief overview of the modelling of the dynamics of bicycles and the experimental validation. The second part focuses on a review of modelling and measuring human rider control, together with the concepts of handling and manoeuvrability and their experimental validation. The paper concludes with the open ends and promising directions for future work in the field of handling and control of bicycles.

Lateral dynamics of a bicycle with a passive rider model: stability and controllability

This paper addresses the influence of a passive rider on the lateral dynamics of a bicycle model and the controllability of the bicycle by steer or upper body sideway lean control. In the uncontrolled model proposed by Whipple in 1899, the rider is assumed to be rigidly connected to the rear frame of the bicycle and there are no hands on the handlebar. Contrarily, in normal bicycling the arms of a rider are connected to the handlebar and both steering and upper body rotations can be used for control. From observations, two distinct rider postures can be identified. In the first posture, the upper body leans forward with the arms stretched to the handlebar and the upper body twists while steering. In the second rider posture, the upper body is upright and stays fixed with respect to the rear frame and the arms, hinged at the shoulders and the elbows, exert the control force on the handlebar. Models can be made where neither posture adds any degrees of freedom to the original bicycle model. For both postures, the open loop, or uncontrolled, dynamics of the bicycle–rider system is investigated and compared with the dynamics of the rigid-rider model by examining the eigenvalues and eigenmotions in the forward speed range 0–10 m/s. The addition of the passive rider can dramatically change the eigenvalues and their structure. The controllability of the bicycles with passive rider models is investigated with either steer torque or upper body lean torque as a control input. Although some forward speeds exist for which the bicycle is uncontrollable, these are either considered stable modes or are at very low speeds. From a practical point of view, the bicycle is fully controllable either by steer torque or by upper body lean, where steer torque control seems much easier than upper body lean.

Linearized equations for an extended bicycle model

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 models, the pure-rolling conditions of the knife-edge wheels are introduced as non-holonomic constraints and the rider is assumed to be rigidly attached to the rear frame. There are two degrees of freedom for the lateral motion, the lean angle of the rear frame and the steering angle. In the present paper, the model is extended in several ways, while the simplicity of having only two degrees of freedom is retained. The extensions of the model comprise the shape of the tires, which are allowed to have a finite transverse radius of curvature, the effect of a pneumatic trail and a damping term due to normal spin at the tire contact patch, the gradient of the road, the inclusion of driving and braking torques at the wheels and the aerodynamic drag at the rear frame.

A review on motorcycle and rider modelling for steering control

The paper is a review of the state of knowledge and understanding of steering control in motorcycles and of the existing rider models. Motorcycles are well known to have specific instability characteristics, which can detrimentally affect the riders control, and as such a suitable review of these characteristics is covered in the first instance. Next, early models which mostly treat riding as a regulatory task are considered. A rider applies control based on sensory information available to him/her, predominantly from visual perception of a target path. The review therefore extends to cover also the knowledge and research findings into aspects of road preview control. Here, some more emphasis is placed on recent applications of optimal control and model predictive control to the riding task and the motorcycle–rider interaction. The review concludes with some open questions which naturally present a scope for further study.