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Dynamics and Control of Drifting in Automobiles

March, 2013

Abstract

In rally racing, it is quite common for drivers to deliberately corner with the rear tires of their racecar operating at their friction limits. Under such circumstances, the rear tires are said to be saturated. This practice is intriguing because it directly opposes the principles on which current vehicle safety systems are based; these systems react to rear tire saturation by restoring the vehicle to an operating regime in which the tires are not saturated. By deliberately saturating the rear tires when cornering, some of the world’s best drivers are clearly utilizing control possibilities that have not yet been harnessed in automotive control design. Through a more thorough understanding of these drivers’ cornering techniques, it may be possible to take advantage of these control possibilities in future safety systems, especially in the design of collision mitigation and avoidance systems for agile autonomous vehicles.

This dissertation focuses upon drifting, a cornering technique that involves steadystate operation with rear tire saturation. Drifts are typically characterized by countersteer, large sideslip angles, and significant rear wheelspin as a result of large rear drive torques. A drift corresponds to operation at an open-loop unstable equilibrium condition of a vehicle at which the rear tires are saturated. This dissertation examines the characteristics of “drift equilibria” using simple vehicle models and establishes three key characteristics of these equilibria.

The first characteristic is that yaw rate variations have a significant influence upon the sideslip dynamics around drift equilibria; in fact, the direct effect of a steering input upon the sideslip dynamics around a drift equilibrium is ultimately outweighed by the effect of that input acting through the yaw dynamics. The second characteristic is near-saturation of the front lateral force at drift equilibria, meaning that lateral v control authority through front steering is nearly unidirectional in character. The third and final characteristic is that the rear drive force input has significant lateral control authority around drift equilibria; this is as a result of the coupling between the rear tire lateral and longitudinal forces that arises from rear tire saturation.

These characteristics shape the design of a control algorithm that drifts a vehicle by stabilizing a desired drift equilibrium. The drift controller developed in this dissertation utilizes a successive loop structure in which yaw rate variations are used to control the sideslip dynamics in an outer loop and tire forces are used to control the yaw dynamics in an inner loop. The controller coordinates front lateral force (via steering) and rear drive force such that the front lateral force is used for lateral control whenever additional cornering force is available at the front tire, but the rear drive force is used for lateral control whenever the front tire is friction limited.

When implemented on a steer- and drive-by wire test vehicle, the drift controller achieves sustained, robust drifts while operating on a surface where friction varies considerably. Furthermore, stability analyses such as phase portraits and a numericallyvalidated Lyapunov function demonstrate the controller’s ability to stabilize a desired drift equilibrium and create a sizable region of convergence around that equilibrium.

Nevertheless, the dual role of rear drive force as a longitudinal and lateral control input when drifting presents challenges from a stability analysis standpoint. A fundamental incompatibility between lateral and longitudinal control objectives arises when using the rear drive force for lateral control that makes it difficult to provide analytical stability guarantees for the drift controller. This same incompatibility also makes it difficult to develop a drift controller with a sufficiently large region of feasibility that explicitly prescribes stable lateral and longitudinal dynamics.

One way of addressing this issue is through the use of a third input for lateral control. Towards this end, this dissertation presents a drift controller design that incorporates differential control of drive and brake torques at the rear wheels as an additional means to generate a yaw moment. This drift controller has a fairly large region of feasibility while also enabling a straightforward, analytical demonstration of stability.