Advanced Vehicle Dynamics

The dynamic performance of a vehicle is mainly determined by the interaction of its tires and road. A vehicle can only move and maneuver by the force systems generated under the tires. In this chapter, we introduce the required coordinate frames to determine the location and orientation of tires in the vehicle body coordinate frame; the mathematical equation to calculate longitudinal and lateral forces; and individual equations needed to develop dynamic equations of vehicles in the following chapters. The resultant force system that a tire receives from the ground is at the center of the tireprint and can be decomposed along x _t, y _t, and z _t axes of the tire coordinate frame T. The interaction of a tire with road generates a three-dimensional (3D) force system including three forces and three moments. The force system at the tireprint of a loaded, rolling, steered, cambered tire includes: forward force F _x, lateral force F _y, vertical force F _z, aligning moment M _z, roll moment M _x, and pitch moment M _y. The forward force F _x and lateral force F _y are the most significant forces in vehicle maneuvering. To accelerate or brake a vehicle, a longitudinal force must be developed between the tire and the ground. When a torque T is applied to the spin axis of a tire, longitudinal slip ratio s occurs and a longitudinal force F _x is generated at the tireprint proportional to s. The tire lateral force F _y is a function of two angles of the tire: sideslip angle α and camber angle γ. The F _x and F _y take the tire load F _z, sideslip α, longitudinal slip s, and the camber angle γ as input. We adopt the proportional-saturation model for longitudinal and lateral slips of tire. When α = 0, a small longitudinal slip s < s _s generates the longitudinal force F _x∕F _z = C _s s, and when s = 0, a small sideslip angle α < α _s generates a lateral force of F _y∕F _z = −C _α α. When there exists a longitudinal slip s < s _s and then we also introduce a sideslip α < α _s, the longitudinal force will reduce. Similarly, when there exists a longitudinal slip s < s _s, the lateral force will drop. The elliptic mathematical model introduces the analytical expression of the interaction F _x∕F _z and F _y∕F _z. Because the longitudinal and lateral forces are affected by the vertical force F _z on the tire, there must be a model to calculate the weight transfer during forward and lateral acceleration. Such equations are calculated in this chapter.

In this chapter we study the planar model of vehicles to examine maneuvering by steering as well as the wheel torque control. The wheel torque and steer angle are the inputs and the longitudinal velocity, lateral velocity, and yaw rate are the main output variables of the planar vehicle dynamics model. The planar vehicle dynamic model is the simplest applied modeling in which we assume the vehicle remains parallel to the ground and has no roll, no pitch, and no bounce motions. The planar motion of vehicles has three degrees of freedom: translation in the x and y directions, and a rotation about the z-axis. The longitudinal velocity v _x along the x-axis, the lateral velocity v _y along the y-axis, and the yaw rate \(r=\dot {\psi }\) about the z-axis are the outputs of the dynamic equations of motion.

By ignoring the roll motion as well as the lateral load transfer between left and right wheels, we define a simplified two-wheel model for the vehicle.

The four-wheel planar vehicle model is an extension to the two-wheel planar vehicle model to include the lateral weight transfer. The four-wheel planar model provides us with better simulation of drifting vehicles. This model is capable to simulate drift of vehicles as well as simulation of different tire-wheel interaction for all four tires of a vehicle.

The roll vehicle dynamic model is well expressed by four kinematic variables: the forward motion x, the lateral motion y, the roll angle φ, and the yaw angle ψ, plus four equations for the dynamics of each wheel. In the roll model of vehicle dynamics, we do not consider vertical movement z and pitch motion θ. The model of a rollable rigid vehicle is more exact and more realistic compared to the vehicle planar model. Using roll dynamic model, we are able to analyze the roll behavior of a vehicle in maneuvers. Angular orientation of the vehicle is expressed by three angles: roll φ, pitch θ , and yaw ψ, and the vehicle angular velocities are expressed by their rates: roll rate p, pitch rate q, and yaw rate r.

A rolled vehicle introduces new reactions in the tires of the vehicle that must be considered in the development of the dynamic equations of motion. The most important reactions are:

An extra steer angle appears because of the wheel roll. The roll steer is a result of suspension mechanisms that provide some steer angle when the vehicle rolls and the mechanism deflects. The wheel roll steering is assumed to be proportional to the vehicle roll angle φ. Therefore, the actual steer angle δ _a of such a tire will be δ _a = δ + δ _φ.

In this chapter we introduce bicycle as well as four-wheel roll models with independent in-wheel motors. The four-wheel roll vehicle model is the best practical vehicle mathematical model. This model provides us with in-wheel torques T _i, tire slips α _i, s _i, β _i, tire and vehicle forces F _x, F _y, \(F_{x_{i}}\), \(F_{y_{i}}\), \(F_{z_{i}}\), velocity components of the vehicle v _x, v _y, ω _i, as well as yaw and roll angular variables φ, ψ, p, r. This model is an extension to the two-wheel roll vehicle model to include the lateral weight transfer as well as roll effects on vehicle dynamics. The four-wheel roll vehicle model is an excellent model to simulate drifting of vehicles.

Passenger cars are developed to move on smooth paved pre-designed roads. To keep vehicles on road, we need a steering mechanism to provide steer angle as an input to the vehicle dynamic system. Ideally, all wheels of a vehicle should be able to steer independently such that the vehicle follows the desired path at the given speed. In this chapter we review steer and road dynamics.

Roads are made by continuously connecting straight and circular paths by proper transition turning sections. Having a continuous and well-behaved curvature is a necessary criterion in road design. The clothoid spiral is the best smooth transition connecting curve in road design which is expressed by parametric equations called Fresnel Integrals. The curvature of the clothoid curve varies linearly with arc length and this linearity makes clothoid the smoothest driving transition curve. Having a road with linearly increasing curvature is equivalent to entering the path with a steering wheel at the neutral position and turning the steering wheel with a constant angular velocity. This is a desirable and natural driving action.

Ideally, perpendicular lines to all wheels of a vehicle intersect at a single point called the kinematic center of rotation. When a vehicle is moving very slowly, we may assume the velocity vector of each wheel is in their tire plane. Therefore, the perpendicular lines to the tire planes intersect at the kinematic center of rotation of the vehicle, somewhere on the rear axis. However, when the vehicle moves faster, the actual center of rotation will move away from the kinematic center of rotation. Steering mechanism relates the last and right steerable wheels and provide a mathematical relationship to calculate all steer angles based on the angle of the steering wheel or the steer angle one of the wheels.