David J. N. Limebeer

Mechanical Steering Compensators for High- Performance Motorcycles

This paper introduces the idea of using mechanical steering compensators to improve the dynamic behavior of high-performance motorcycles. These compensators are seen as possible replacements for a conventional steering damper and comprise networks of springs, dampers, and a less familiar component called the inerter. The inerter was recently introduced to allow the synthesis of arbitrary passive mechanical impedances, and finds a potential application in the present work. The design and synthesis of these compensation systems make use of the analogy between passive electrical and mechanical networks. This analogy is reviewed alongside the links between passivity, positive reality, and network synthesis. Compensator design methods that are based on classical Bode-Nyquist frequency-response ideas are presented. Initial designs are subsequently optimized using a sequential quadratic programing algorithm. This optimization process ensures improved performance over the machine’s entire operating regime. The investigation is developed from an analysis of specific mechanical networks to the class of all biquadratic positive real functions. This aspect of the research is directed to answering the question: “What is the best possible system performance achievable using any simple passive mechanical network compensator?” The study makes use of computer simulations, which exploit a state-of-the-art motorcycle model whose parameter set is based on a Suzuki GSX-R1000 sports machine. The results show that, compared to a conventional steering damper, it is possible to obtain significant improvements in the dynamic properties of the primary oscillatory modes, known as “wobble” and “weave.” DOI: 10.1115/1.2198547

Optimal control for a Formula One car with variable parameters

The minimum-lap-time optimal control problem for a Formula One race car is solved using direct transcription and nonlinear programming. Features of this work include significantly reduced full-lap solution times and the simultaneous optimisation of the driven line, the driver controls and multiple car set-up parameters. It is shown that significant reductions in the driven lap time can be obtained from track-specific set-up parameter optimisation. Reduced computing times are achieved using a combination of a track description based on curvilinear coordinates, analytical derivatives and model non-dimensionalisation. The curvature of the track centre line is found by solving an auxiliary optimal control problem that negates the difficulties associated with integration drift and trajectory closure.

Optimal control of Formula One car energy recovery systems

The utility of orthogonal collocation methods in the solution of optimal control problems relating to Formula One racing is demonstrated. These methods can be used to optimise driver controls such as the steering, braking and throttle usage, and to optimise vehicle parameters such as the aerodynamic down force and mass distributions. Of particular interest is the optimal usage of energy recovery systems (ERSs). Contemporary kinetic energy recovery systems are studied and compared with future hybrid kinetic and thermal/heat ERSs known as ERS-K and ERS-H, respectively. It is demonstrated that these systems, when properly controlled, can produce contemporary lap time using approximately two-thirds of the fuel required by earlier generation (2013 and prior) vehicles.

Relaminarisation of Reτ = 100 channel flow with globally stabilising linear feedback control

The problems of nonlinearity and high dimension have so far prevented a complete solution of the control of turbulent flow. Addressing the problem of nonlinearity, we propose a flow control strategy which ensures that the energy of any perturbation to the target profile decays monotonically. The controller’s estimate of the flow state is similarly guaranteed to converge to the true value. We present a one-time off-line synthesis procedure, which generalises to accommodate more restrictive actuation and sensing arrangements, with conditions for existence for the controller given in this case. The control is tested in turbulent channel flow (Re_τ = 100) using full-domain sensing and actuation on the wall-normal velocity. Concentrated at the point of maximum inflection in the mean profile, the control directly counters the supply of turbulence energy arising from the interaction of the wall-normal perturbations with the flow shear. It is found that the control is only required for the larger-scale motions, specifically those above the scale of the mean streak spacing. Minimal control effort is required once laminar flow is achieved. The response of the near-wall flow is examined in detail, with particular emphasis on the pressure and wall-normal velocity fields, in the context of Landahl’s theory of sheared turbulence.

Influence of Road Camber on Motorcycle Stability

This paper studies the influence of road camber on the stability of single-track road vehicles. Road camber changes the magnitude and direction of the tire force and moment vectors relative to the wheels, as well as the combined-force limit one might obtain from the road tires. Camber-induced changes in the tire force and moment systems have knock-on consequences for the vehicle’s stability. The study makes use of computer simulations that exploit a high-fidelity motorcycle model whose parameter set is based on a Suzuki GSX-R1000 sports machine. In order to study camber-induced stability trends for a range of machine speeds and roll angles, we study the machine dynamics as the vehicle travels over the surface of a right circular cone. Conical road surfaces allow the machine to operate at a constant steady-state speed, a constant roll angle, and a constant road camber angle. The local road-tire contact behavior is analyzed by approximating the cone surface by moving tangent planes located under the road wheels. There is novelty in the way in which adaptive controllers are used to center the vehicle’s trajectory on a cone, which has its apex at the origin of the inertial reference frame. The results show that at low speed both the weave- and wobble-mode stabilities are at a maximum when the machine is perpendicular to the road surface. This trend is reversed at high speed, since the weave- and wobble-mode dampings are minimized by running conditions in which the wheels are orthogonal to the road. As a result, positive camber, which is often introduced by road builders to aid drainage and enhance the friction limit of four-wheeled vehicle tires, might be detrimental to the stability of two-wheeled machines.

Optimal Control of a Formula One Car on a Three-Dimensional Track—Part 2: Optimal Control

The optimal control of a Formula One car on a three-dimensional (3D) track is studied. The track is described by its geodesic and normal curvatures, and its relative torsion. These curvature parameters are obtained from noisy measurement data using the optimal estimation technique described in Part 1. The optimal control calculations presented are based on the aforementioned track model and a vehicle model that is responsive to the geometric features of a 3D track. For vehicle modelling purposes the track is treated as a plane tangent to a nearby point on the tracks spine. This tangent plane moves under the car and is orthogonal to the principal normal vector m at the nearby spine point. Results are presented that compare 2D and 3D minimum-lap-time results, with the two compared. The Barcelona Formula One track studied in Part 1 is used again as an illustrative example.

Aeroelastic Control of Long-Span Suspension Bridges

The modeling, control, and dynamic stabilization of long-span suspension bridges are considered. By employing leading- and trailing-edge flaps in combination, we show that the critical wind speeds for flutter and torsional divergence can be increased si g nificantly. The relatively less well known aerodynamic properties of leading-edge flaps will be studied in detail prior to their utilization in aeroelastic stability and control system design studies. The optimal approximation of the classical Theodorsen circulation function will be studied as part of the bridge section model building exercise. While a wide variety of control systems is possible, we focus on compensation schemes that can be implemented using passive mechanical components such as springs, dampers, gearboxes, and levers. A single-loop control system that controls the leading- and trailing-edge flaps by sensing the main deck pitch angle is investigated. The key finding is that the critical wind speeds for flutter and torsional divergence of the sectional model of the bridge can be greatly increased, with good robustness characteristics, through passive feedback control. Static winglets are shown to be relatively ineffective.

Faster, Higher, and Greener: Vehicular Optimal Control

Vehicular optimal control problems have been studied extensively since the early part of the 20th century. Progress in solving these problems has been driven primarily by applications in space and atmospheric flight, including launch vehicles, Earth-based and interplanetary space orbital transfer, and high-performance supersonic aircraft. In all of these applications, the ability to solve increasingly complex optimal control problems has been made possible by advances in high-speed computing. The mathematical and computing techniques being developed are now so diverse, and the range of applications of optimal control so broad, that a comprehensive review of the entire scope of vehicular optimal control is an impossible task. Instead, we will attempt only to provide a flavor of the scope and variety of these problems.

Suppression of Burst Oscillations in Racing Motorcycles

Burst oscillations occurring at high speed, and under firm acceleration, can be suppressed with a mechanical steering compensator. Burst instabilities in the subject racing motorcycle are the result of interactions between the wobble and weave modes under firm-acceleration at high speed. Under accelerating conditions, the wobble-mode frequency (of the subject motorcycle) decreases, while the weave mode frequency increases so that destabilizing interactions can occur. The design analysis is based on a time-separation principle, which assumes that bursting occurs on time scales over which speed variations can be neglected. Even under braking and acceleration conditions linear time-invariant models corresponding to constant-speed operation can be utilized in the design process. The influences of braking and acceleration are modeled using d’Alembert-type inertial forces that are applied at the mass centers of each of the model’s constituent bodies. The resulting steering compensator is a simple mechanical network that comprises a conventional steering damper in series with a linear spring. In control theoretic terms, this network is a mechanical lag compensator. A robust control framework was used to optimize the compensator design because it is necessary to address the inevitable uncertainties in the motorcycle model, as well as the nonlinearities that influence the machine’s local behavior as the vehicle ranges over its operating envelope.

Frequency response identification of the dynamics of a Tokamak plasma

We describe the application of an H/sup /spl infin// system identification procedure to a Tokamak. The work is motivated by the need to create linear models which are suitable for controller design and which may be used to validate different models derived from physics principles. The paper develops an H/sup /spl infin// system identification algorithm and demonstrates its successful application to the Tokamak configuration variable (TCV). Each of the required steps is detailed, from the design of identification experiments to the creation of low-order models from a combination of Hankel model reduction and Chebychev approximation. The method described is a worst-case identification technique, in that it aims to minimize the H/sup /spl infin// error between the identified model and the true plant. Such a model is particularly well suited for robust controller design. The identified model of TCV is compared with various physics-based models.