Vehicular Optimal Control

Chapter 9 deals with the solution of minimum-time and minimum-fuel vehicular optimal control problems. These problems are posed as fuel usage optimization problems under a time-of-arrival constraint, or minimum-time problems under a fuel usage constraint. The first example considers three variants of a simple fuel usage minimization problem under a time-of-arrival constraint. The first variant is worked out theoretically, and serves to highlight several of the structural features of these problems; the other two more complicated variants are solved numerically.The second example is also a multi-stage fuel usage minimization problem under a timeof- arrival constraint.More complicated track and vehicle models are then employed; the problem is solved numerically. The third problem is a lap time minimization problem taken from Formula One and features a thermoelectric hybrid powertrain. The fourth and final problem is a minimum-time closed-circuit racing problem featuring a racing motorcycle and rider.

Advanced Vehicle Modelling

The aim of this chapter is to discuss some of themodelling techniques used to replicate the important features of bicycle, car, motorcycle, and driver dynamics. These more ‘advanced’ vehicle models will include a number of features that were neglected in models described in Chapters 4 and 5, including suspensions with anti-dive and/or anti-squat geometries, environmental influences such as aerodynamic effects, threedimensional road geometries, and tyre models of the type described in Chapter 3.

Tyres

Chapter 3 focuses on modern tyre modelling. While classical two-dimensional nonholonomic- constraint models work reasonably well at very low speeds, these models are not acceptable in realistic applications. This chapter aims to explain the mechanisms and modelling issues related to the generation of tyre forces and moments. Physical models such as the brush and string models are used to clarify the basic concepts. Building upon these findings, the empirical models widespread in vehicle dynamics analyses are discussed. An overview of some of the advanced models currently used in the industry is also given.

A History of Road Vehicles

Chapter 1 is almost entirely discursive and covers the early history of road vehicles, outlining some of the important technological achievements that underpin the development of modern road vehicular transport. The focus is on bicycles, motorcycles, and cars; the history of steering mechanisms for four-wheeled vehicles is considered early on. Several early engine, suspension, and tyre developments are also discussed.

Optimal Control

Chapter 8 focuses on nonlinear optimal control and its applications. The chapter begins by introducing the fundamentals of optimal control and prototypical problem formulations. This is followed by the treatment of first-order necessary conditions including the Pontryagin minimum principle, dynamic programming, and the Hamilton–Jacobi–Bellman equation. Singular arcs and bang–bang controls are relevant in the solution of many minimum-time and minimum-fuel problems and so these issues are discussed with the help of examples that have been worked out in detail.This chapter then turns towards direct and indirect numericalmethods suitable for solving large-scale optimal control problems numerically.The chapter concludes with an example relating to the calculation of an optimal track curvature estimate from global positioning system (GPS) data.

Precursory Vehicle Modelling

Chapter 4 considers a number of precursory car and bicycle models, as well as an important oscillatory phenomenon commonly referred to as shimmy. The chapter begins by considering a classical single-track car model that will be used to analyse such things as yaw stability, the effects of acceleration and braking, and some of the influences of cornering. After that, the Timoshenko–Young bicycle model that has steering and rolling freedoms is considered. Bicycle stability is then considered with the help of a point-mass bicycle model. Shimmy, including gyroscopic shimmy, is discussed in the concluding part of the chapter. The reader will also be introduced to the fundamental notions of oversteer and understeer.

Ride Dynamics

Chapter 6 dealswith road surfacemodelling and vehicle suspension systems, and their ride dynamics. A wide variety of car and motorcycle suspension configurations are now available. While most of these systems appear ‘very different’ from each other, many of their important properties can be analysed within a common ride-dynamics framework.The chapter begins with an analysis of the simple two-degree-of-freedom single-wheel-station (quarter-car) model.The validity of this model is discussed, and several of its properties studied, including its mode shapes, its invariant equation and the associated interpolation constraints, its frequency response characteristics, its design compromises, and its suspension component-value optimization.The singletrack suspensionmodel that can be used formotorcycles, and single-track carmodels, is then discussed, while in the last part of the chapter a full-vehicle suspension model is considered. The chapter finishes with some introductory remarks relating to suspension synthesis from a passive circuit-theoretic perspective.

The Whipple Bicycle

Chapter 5 provides a comprehensive review of the bicycle model presented inWhipple’s seminal 1899 paper.The set of nonlinear differential equations that describe the general motion of a bicycle and rider are derived and then linearized to study the smallmotions about a straight-running trim condition at a given constant speed.The stability of the Whipple model is discussed and some of the handling characteristics of bicycles are analysed using control-theoretic ideas. Extensions relating to accelerating and braking vehicles, tyred vehicles, and vehicles with flexible frames are also considered.

Topics in Mechanics

Chapter 2 provides a comprehensive review of the classical mechanics required when building vehicle models. Both vector-based methods and the variational approach to classical mechanics are reviewed, and efforts to highlight the links between the two approaches have been made. A wide range of illustrative examples with a particular focus on non-holonomic systems are studied, including Chaplygin’s sleigh, rolling balls, and rolling discs. Equilibria and stability, and the connection between timereversal symmetry and dissipation, are also briefly discussed.