On Inverse Equations for Vehicle Dynamics

Abstract

Instead of writing equations which when solved yield the response of a vehicle to an input such as the front wheel steer angle, one can often invert the equations so that a response quantity is specified as an input and a new set of equations is solved yielding the steer angle required as an output. Using these equations one can discover the input steer angle a driver would need to impose in order to accomplish a specific maneuver for various vehicles.

It is shown that there are many possible inverse equation sets and that the eigenvalues of the inverse equations are hard to interpret since they may have little to do with the vehicle parameters. The linear single-input single-output case is studied first to fix ideas using a simple example. For the bicycle model vehicle, it is shown that any vehicle may have unstable inverse equations depending upon the response quantity used. Extensions to nonlinear and multiple-input multiple output systems are discussed.     

A Tractor Handling Study

Agricultural tractors are relatively slow moving vehicles but a good steer response under all working conditions is generally required. This paper evaluates the effect of ground slope, axle load, ground speed, tyres and ground surface on yaw rate response to steer input.     

Static Turning Analysis of Vehicles Subject to Externally Applied Forces - A Moment Arm Ratio Formulation

SUMMARY

A formulation for representing the static turning response of a two-axle vehicle due to applied external or control forces is expressed in terms of a simple ratio of two distances along the vehicle longitudinal axis. The two distancescoincide with points on the vehicle at which externally applied/ control forces and their reactive inertial forces act with respect to the vehicle neutral steer point. The resulting formulation is equivalent to the rotational equilibrium equation written with respect to the neutral steer point. The method allows a simple “visual analysis” of the steady turning process by showing how key forces and associated moment arms can change with respect to one another due to vehicle modifications or different operatingconditions, thereby affecting the static turning response of the vehicle.     

Impact of Dynamic Fluid Slosh Loads on the Directional Response of Tank Vehicles

SUMMARY

A comprehensive directional dynamics model of a tractor-tank trailer is developed by integrating a non-linear dynamic fluid slosh model to the three-dimensional vehicle dynamics model. The nonlinear fluid slosh equations are solved in an Eulerian mesh to determine dynamic fluid slosh loads caused by the dynamic motion of the vehicle. The dynamic fluid slosh forces and moments are coupled with the vehicle dynamics model to study the directional response characteristics of tank vehicles. The directional response characteristics of partially filled tank vehicles employing dynamic slosh model are compared to those employing quasi-dynamic vehicle model, for steady as well as transient directional maneuvers. Simulation results reveal that during a steady steer maneuver, the dynamic fluid slosh loads introduce oscillatory directional response about a steady-state value calculated from the quasi-dynamic vehicle model. The directional response characteristics obtained using the quasi-dynamic and dynamic fluid slosh models during transient steer inputs show good correlation. Based on this study, it can be concluded that the quasi-dynamic model can predict the directional response characteristics of tank vehicles quite close to that evaluated using the comprehensive fluid slosh model.     

Unified decision-making and control for highway collision avoidance using active front steer and individual wheel torque control

ABSTRACT

Collision avoidance is a crucial function for all ground vehicles, and using integrated chassis systems to support the driver presents a growing opportunity in active safety. With actuators such as in-wheel electric motors, active front steer and individual wheel brake control, there is an opportunity to develop integrated chassis systems that fully support the driver in safety critical situations. Here we consider the scenario of an impending frontal collision with a stationary or slower moving vehicle in the same driving lane. Traditionally, researchers have approached the required collision avoidance manoeuver as a hierarchical scheme, which separates the decision-making, path planning and path tracking. In this context, a key decision is whether to perform straight-line braking, or steer to change lanes, or indeed perform combined braking and steering. This paper approaches the collision avoidance directly from the perspective of constrained dynamic optimisation, using a single optimisation procedure to cover these aspects within a single online optimisation scheme of model predictive control (MPC). While the new approach is demonstrated in the context of a fully autonomous safety system, it is expected that the same approach can incorporate driver inputs as additional constraints, yielding a flexible and coherent driver assistance system.     

An Advanced Steering System with Active Kinesthetic Feedback for Handling Qualities Improvement

SUMMARY

Advanced Steering System with artificial steering wheel torque-active kinesthetic information feedback for improving handling qualities is discussed. Fundamentally the structure of the system may be considered to another form of model following control. In this system, a driver always remains in the control loop and receives steering control information which give him/her a direct hint to steer a steering wheel. This system works as a stability and control augmentation system of the vehicle to improve the vehicle handling qualities both in compensatory and pursuit control task, and is expected to reduce driver's workload. Effects of this system are analyzed in terms of man-machine system characteristics. Identification of driver dynamics was carried out to find why such improvement could be achieved. Availability of the proposed system is verified by analysis, simulator and proving ground tests.     

Influence of Liquid Load Shift on the Dynamic Response of Articulated Tank Vehicles

SUMMARY

The influence of the lateral load shift on the dynamic response characteristics of an articulated tank vehicle is investigated assuming inviscid fluid flow conditions. A quasi-dynamic roll plane model of a partially filled cleanbore tank of circular cross-section is developed and integrated to a three-dimensional model of the articulated vehicle, assuming constant forward speed. The destabilizing effects of liquid load shift are studied by comparing the directional dynamics of the partially filled tank vehicle to that of an equivalent rigid cargo vehicle subject to steady steer input. Dynamic response characteristics demonstrate that the stability of a partially filled tank vehicle is adversely affected by the Liquid load shift The distribution of cornering forces caused by the liquid load shift yield considerable deviation of the path followed by the liquid tank vehicle. The influence of the vehicle speed on the dynamics of the liquid tank vehicle is also investigated for variations in the fill levels and fluid density.     

Development and Evaluation of a High Voltage Supply Unit for Electrorheological Fluid Dampers

SUMMARY

Controllable dampers using electrorheological (ER) fluids have attracted considerable interest in recent years. They are proposed for use in semi-active suspensions for ground vehicles. The main advantages of ER fluid dampers are their fast response, ease of control, simple construction and low power requirements. This paper describes the development and testing of a high-voltage supply unit for modulating the damping force of an ER fluid damper. Experimental results on the vibration isolation characteristics of an ER fluid damper with different control strategies, obtained using a quarter-car model test rig, are also presented.     

Comparisons of vehicle stability controls based on 4WS,Brake, Brake-FAS and IMC techniques

This paper proposes three different kinds of vehicle stability control systems all based on internal model control (IMC) strategy which are 4WS (4 wheel steer: front- and rear-wheel active steer) IMC, Brake-FAS (brake and front-wheel active steer) IMC and Brake IMC, respectively. Inverse system method is introduced to solve the nonlinearity coupled with brake involved vehicle stability control systems. Based on an 11-DOF (degrees of freedom) Matlab/Simulink^® vehicle model testified by CarSim7^®, simulations combined with different driving manoeuvres and road surfaces are performed, and detailed comparisons and analyses are given based on simulation results.     

Real-time characterisation of driver steering behaviour

In recent years the application of driver steering models has extended from the off-line simulation environment to autonomous vehicles research and the support of driver assistance systems. For these new environments there is a need for the model to be adaptive in real time, so the supporting vehicle systems can react to changes in the driver, their driving style, mood and skill. This paper provides a novel means to meet these needs by combining a simple driver model with a single-track vehicle handling model in a parameter estimating filter – in this case, an unscented Kalman filter. Although the steering model is simple, a motion simulator study shows it is capable of characterising a range of driving styles and may also indicate the level of skill of the driver. The resulting filter is also efficient – comfortably operating faster than real time – and it requires only steer and speed measurements from the vehicle in addition to the reference path. Adaptation of the steer model parameters is demonstrated along with robustness of the filter to errors in initial conditions, using data from five test drivers in vehicle tests carried out on the open road.

Abbreviations: ADAS: advanced driver assistance systems; CG: centre of gravity; CAN: controller area network; EKF: extended Kalman filter; GPS: global positioning system; UKF: unscented Kalman filter     

Optimal control of brakes and steering for autonomous collision avoidance using modified Hamiltonian algorithm

ABSTRACT

This paper considers the problem of collision avoidance for road vehicles, operating at the limits of friction. A two-level modelling and control methodology is proposed, with the upper level using a friction-limited particle model for motion planning, and the lower level using a nonlinear 3DOF model for optimal control allocation. Motion planning adopts a two-phase approach: the first phase is to avoid the obstacle, the second is to recover lane keeping with minimal additional lateral deviation. This methodology differs from the more standard approach of path-planning/path-following, as there is no explicit path reference used; the control reference is a target acceleration vector which simultaneously induces changes in direction and speed. The lower level control distributes vehicle targets to the brake and steer actuators via a new and efficient method, the Modified Hamiltonian Algorithm (MHA). MHA balances CG acceleration targets with yaw moment tracking to preserve lateral stability. A nonlinear 7DOF two-track vehicle model confirms the overall validity of this novel methodology for collision avoidance.     

Advanced Ground Vehicle Suspension Systems - A Classified Bibliography

SUMMARY

This bibliographical list classifies the extensive literature related to the dynamic analysis and control design aspects of advanced ground vehicle suspension systems. It is restricted to the design of advanced suspensions for ride quality and safety. Brief generalized notices and definitions are presented.     

Design of Optimal Four-Wheel Steering System

SUMMARY

Optimal design of the four wheel steering (4WS) system of the ground vehicle is studied. 4WS vehicles with the optimal control scheme are considered first. General formulation of the optimal control law is developed based on the linear quadratic regulator theory. The vehicle speed function (VSF) based 4WS vehicle with a simple feedback controller is considered as a special case of the optimal system. Two new designs of the VSF 4WS system are proposed and their performances are compared with the optimal 4WS systems and the existing VSF 4WS system. The first system is designed for the maximum stability while the second system is designed to emulate the response of the optimal 4WS vehicle. Advantages of the new VSF designs are discussed.     

Random Vehicle Vibrations as Effected by Dry Friction in Wheel Suspensions

SUMMARY

The changes of vibration comfort and stability of vehicles as effected by the value of frictional force generated in laminated springs are discussed and the probability of departure of the tyres from the ground is described.

A partly automated analogue computer simulation being equivalent to a great number of highway experiments was applied in the study.

The computer model was excited by several stochastic roadprofile - analogue signals generated by filters from the wide-band random signal of a digital noise generator.     

Stationar- und Ubergangsverhalten von Sattel- und Lasttugen bei der Kreisfahrt: Lineare Berechnungen

ABSTRACT

Steady and Transient Turning of Tractor-Semitrailer and Truck-Trailer Combinations: A Linear Analysis

A simplified analysis is made of the yaw stability and control of the two types of the commercial vehicle combinations (tractor-semitrailer, truck-trailer) at a constant forward velocity during steady and transient turning. The combined vehicle is treated as a linear dynamic system (Fig. 2). The steer angle at the front wheels of the tractor (or truck) and the steady-state responses if the road verhicle train (yaw rate, articulation angles and sideslip angle) are calculated (Equations 18 to 25). Exploratory calculations are performed to determine the influence of the cornering stiffness of the tires for the two types of the vehicle combinations upon the steady-state responses (Figs. 7 to 10). For a linear simplified model of articulated vehicle the steady-state turning behaviour is stable also under conditions of rather high driving speed (70 km/h). A simplified analysis of the transient turning behaviour of the two types of road trains has shown the tractor-semitrailer to preserve stability even under driving speeds exceeding 70 km/h (Fig. 13), whereas the truck-trailer combinations appear to become oscillatory unstable if the driving speed rises above the 60 km/h margin (Fig. 14).     

On Inverse Equations for Vehicle Dynamics

Instead of writing equations which when solved yield the response of a vehicle to an input such as the front wheel steer angle, one can often invert the equations so that a response quantity is specified as an input and a new set of equations is solved yielding the steer angle required as an output. Using these equations one can discover the input steer angle a driver would need to impose in order to accomplish a specific maneuver for various vehicles.

It is shown that there are many possible inverse equation sets and that the eigenvalues of the inverse equations are hard to interpret since they may have little to do with the vehicle parameters. The linear single-input single-output case is studied first to fix ideas using a simple example. For the bicycle model vehicle, it is shown that any vehicle may have unstable inverse equations depending upon the response quantity used. Extensions to nonlinear and multiple-input multiple output systems are discussed.     

Static Turning Analysis of Vehicles Subject to Externally Applied Forces - A Moment Arm Ratio Formulation

A formulation for representing the static turning response of a two-axle vehicle due to applied external or control forces is expressed in terms of a simple ratio of two distances along the vehicle longitudinal axis. The two distancescoincide with points on the vehicle at which externally applied/ control forces and their reactive inertial forces act with respect to the vehicle neutral steer point. The resulting formulation is equivalent to the rotational equilibrium equation written with respect to the neutral steer point. The method allows a simple “visual analysis” of the steady turning process by showing how key forces and associated moment arms can change with respect to one another due to vehicle modifications or different operatingconditions, thereby affecting the static turning response of the vehicle.     

Aerodynamics of Road- and Rail Vehicles

SUMMARY

The technical state-of-the-art of aerodynamics of ground transportation vehicles is reviewed. Currently available theoretical calculation methods and experimental simulation techniques as well as typical results illustrating the impact of aerodynamics on vehicle performance and running characteristics are summarized and the interactions between vehicle system dynamics and aerodynamics are adressed. Correlation of theoretical and experimental data show the present potential of vehicle aerodynamics and point to fields in which further research work is necessary.     

Power Requirements for Traversing Uneven Roadways

SUMMARY

Vehicles which travel on uneven roadways or rough surfaces require power beyond that associated with air drag, rolling resistance or other sources of friction even though kinetic and potential energy may be conserved on the average. This is true because damped relative motions within the vehicle dissipate energy, and, even for nearly rigid vehicles, energy is lost at impact with the ground whenever the vehicle loses contact with the ground surface due to the finite downward acceleration of gravity.

Using elementary vehicle models, the nature and magnitude of the component of propulsive force associated with these energy loss mechanisms is estimated. In certain speed ranges, this force is found to vary dramatically with speed for several types of periodic roadway profiles studied. While the force due to unevennesss may be small compared to other forces for high-speed vehicles operating on smooth surfaces, it can be the major source of required power for off-road vehicles operating on very rough terrain.