Optimal linear time-invariant preview steering control for motorcycles

Abstract

Prior research into the application of optimal linear preview control theory to the steering of a car by a driver is extended into the domain of the motorcycle. The simple car model employed previously is replaced by a much more elaborate motorcycle model, and the control mode is changed from a fixed control for the car to a free control for the motorcycle. Handlebar torque is always the main control input but, in some cases, the rider's upper body lean torque is also included as a control. The machine speed is considered to be constant. The objective of the optimal control is to minimize a weighted sum of path-tracking errors and control power. The time-invariant control corresponding to an infinite optimization horizon and a white noise disturbance is found for each of several cases, involving variations in machine speed and performance priorities.Results demonstrate the relatively minor role of body lean torque control and the strong relationship between the optimal tracking steering control and the motorcycle oscillatory modes of motion. The extent of rider preview necessary for full control is found for cases involving two controls and only one control.The optimal controls are installed on the machine for which they were derived, and simulation results from tracking tasks are shown. The simulations demand the transformation of the problem from a global description, in which the optimal control is found, to a local description corresponding to the rider's view. It is concluded that a motorcycle rider model representing a useful combination of steering control capability and computational economy has been established. The model yields new insights into the rider and motorcycle behaviour.

Closed-Loop Planar Motion Control of a Steerable Probe With a “Programmable Bevel” Inspired by Nature

“…In order to construct a feedback control strategy for the probe, we adopt an algorithm developed for car-like robots [29,[31][32][33][34][35][36] which has received significant attention over the years and is generally expressed in the chained form representation [29,[31][32]36]. In this method, the bicycle model, which is similar to the one developed here for the flexible probe, can be converted into the single chained form, which has two new control inputs (u 1 , u 2 ) and four new states (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) as follows:…”

On the stability and control of unicycles

“…Solutions for the control of systems that include time delays tend to be easier in discrete time, and that is the approach preferred here. Appropriate theory for optimal-linear-preview control exists (Tomizuka & Whitney 1975;Tomizuka 1976;Tomizuka & Rosenthal 1979;Louam et al 1988Louam et al , 1992Prokop & Sharp 1995), which has been applied to steering and speed control of various road vehicles (Sharp & Valtetsiotis 2001;Sharp 2005Sharp , 2006Sharp , 2007aSharp ,c,d,e, 2008Thommyppillai et al 2009a,b). The detailed theory required is included in Thommyppillai et al (2009b) and it is replicated here as the electronic supplementary material, appendix D. It is now reviewed briefly and then applied to longitudinal and lateral control of the unicycle.…”

“…The trim state and the initial conditions define the course of the unicycle if only the trim controls are utilized, so that differences between the (x, y) positions implied by the trim and those demanded by the path are used, together with the optimal gains, to derive the control perturbations necessary to track the path. The optimal controls are obtained in a fixed frame of reference, owing to the simplicity of the road model in such a reference system, but general path-tracking is feasible only if the controls are applied in a local, rider's view, reference frame (Sharp & Valtetsiotis 2001;Sharp 2005Sharp , 2006Sharp , 2007a. At each time step, T s , therefore, the position and orientation of the unicycle are used to transform the road data belonging to the current time up to the preview horizon, nT s ahead, into the local frame of the rider.…”

“…Over the last several years, linear optimal preview control theory has been applied to car driving and to motorcycle and bicycle riding (Sharp & Valtetsiotis 2001;Sharp 2005Sharp , 2006Sharp , 2007aSharp -e, 2008Thommyppillai et al 2009a,b) and this article is about the application of the theory to the longitudinal and lateral unicycle stabilization and path-following-control problem. The strategy for dealing with varying speed and higher amplitude nonlinear motion is to design linear preview controllers that are optimal for the locality, in a state-space sense, of a trim state and to schedule, using suitable indicators of the appropriate neighbouring trim states, over the gain sets and trim states, such that the current control is always well matched to the present operating conditions.…”

Advances in the development of a virtual car driver

“…The part of the programme considered here involves the extension of previous separate works on longitudinal control [27,29] and lateral control [25,26,28,30,34] to combined longitudinal and lateral controls. Recent work on the computation of optimal linear H 2 preview controls, including the writing of a MAT-LAB toolbox [11,12] see http://code.google.com/p/preview-control-toolbox/, is also newly exploited.…”