Alessandro Rucco

Computing minimum lap-time trajectories for a single-track car with load transfer

In this paper we address the minimum lap-time problem for a single-track rigid car which includes tire models and load transfer. Given a planar track including lane boundaries, our goal is to find a trajectory of the car minimizing the lap time subject to tire and steering limits. By using a new set of coordinates, the time-dependent system is transformed into a “space-dependent” (and space-variant) system. The choice of a suitable set of coordinates and the partition of the dynamics into a “longitudinal” one and a “transverse” one, allows us to convert the minimum time problem into a fixed horizon constrained optimal control problem. Based on a projection operator nonlinear optimal control technique, we propose a minimum lap-time strategy to push the rigid car to the limit of its handling capabilities. Finally, we provide numerical computations that: (i) show the effectiveness of the proposed strategy, and (ii) allow us to highlight important features of minimum lap-time trajectories.

Trajectory optimization for constrained UAVs: A Virtual Target Vehicle approach

In this paper we propose a novel approach for trajectory optimization for constrained Unmanned Aerial Vehicles (UAVs). With regard to the classical trajectory optimization problem, we take a Virtual Target Vehicle (VTV) perspective by introducing a virtual target that plays the role of an additional control input. Based on a nonlinear projection operator optimal control technique and extending the concepts of the maneuver regulation framework, we propose a trajectory optimization based strategy to compute, for any given desired path with a specified desired speed profile, the (local) optimal feasible trajectory that best approximates the desired one. The optimization procedure takes explicitly into account the extra flexibility of the VTV by changing (during the transient period) the velocity of the virtual target with the benefit of improving the convergence of the solver to obtain the optimal feasible path, and also avoid the singularities that occur in some maneuver regulation techniques described in the literature. We provide numerical computations for three testing scenarios that illustrates the effectiveness of the proposed strategy.

An Efficient Minimum-Time Trajectory Generation Strategy for Two-Track Car Vehicles

In this paper, we propose a novel approach to compute minimum-time trajectories for a two-track car model, including tires and (quasi-static) longitudinal and lateral load transfer. Given the car model and a planar track, including lane boundaries, our goal is to find a trajectory of the car minimizing the traveling time subject to steering and tire limits. Moreover, we enforce normal force constraints to avoid wheel liftoff. Based on a projection operator nonlinear optimal control technique, we propose a minimum-time trajectory generation strategy to compute the fastest car trajectory. Numerical computations are presented on two testing scenarios, a 90° turn and a real testing track. The computations allow us to both demonstrate the efficiency and accuracy of the proposed approach and highlight important features of the minimum-time trajectories. Finally, we integrate our strategy into a commercial vehicle dynamics software, thus computing minimum-time trajectories for a complex multibody vehicle model. The matching between the predicted trajectory and the one of the commercial toolbox further highlights the effectiveness of the proposed methodology.

Optimal Control Based Dynamics Exploration of a Rigid Car With Longitudinal Load Transfer

In this brief, we provide optimal control-based strategies to explore the dynamic capabilities of a single-track car model that includes tire models and longitudinal load transfer. First, we propose numerical tools to analyze the equilibrium manifold of the vehicle. That is, we design a continuation and predictor-corrector numerical strategy to compute the cornering equilibria on the entire range of operation of the tires. Second, as a main contribution of this brief, we explore the system dynamics by the use of nonlinear optimal control techniques. Specifically, we propose a combined optimal control and continuation strategy to compute aggressive car trajectories. To show the effectiveness of the proposed strategy, we compute aggressive maneuvers of the vehicle inspired to testing maneuvers from virtual and real prototyping.

Dynamics exploration of a single-track rigid car model with load transfer

In this paper we explore the dynamics of a single-track car model. We develop a model of a rigid car inspired to the well known bicycle model. The bicycle model is a planar rigid model that approximates the vehicle as a rigid body with two wheels. However, the bicycle model does not allow to describe the effect of load transfer, since it does not model the suspensions. Using an explicit formulation of the holonomic constraints imposed on the rigid model, we are able to model the load transfer of the car. The resulting model can be seen as a limit condition of a model with suspensions whose stiffness goes to infinity. The load transfer allows to have a more accurate model for the tires. We use a standard model known as Pacejka model that provides empirical curves describing the forces generated by the tires. With this model in hand, we perform an analysis of the equilibrium manifold of the vehicle and, as main contribution of the paper, we explore the trajectories of the system by use of novel nonlinear optimal control techniques. These techniques allow us to compute aggressive trajectories of the car vehicle and study how the vehicle behaves depending on its parameters. We compute trajectories for the vehicle on a real car testing track.

Development and numerical validation of a reduced-order two-track car model

In this paper we propose a novel reduced-order car model and show how it can be used for parameter fitting and dynamic analysis of complex car vehicles. The proposed two-track model consists of a rigid body interacting with the ground at four contact points and including tire models and load transfer. The model does not include suspension models, thus keeping a reasonable level of complexity. Load transfer (both longitudinal and lateral) is taken into account by explicitly imposing the holonomic constraints (for contact) and computing the reaction forces of the ground at the four contact points. Since the vehicle interacts with the ground at four contact points, it results in a hyper-static structure so that the reaction forces are not uniquely determined. Using the Principle of Least Work, we obtain a compatibility equation, providing for a unique resolution of the forces. The compatibility equation is parametrized by four strain parameters, one for each contact point. These parameters are related to those of the suspension, and provide a means for accounting for some dynamic aspects of the suspension without introducing additional states (and parameters) into the dynamics. We provide numerical computations validating the proposed model with respect to a multi-body virtual prototype on aggressive maneuvers.

A Model Predictive Control-Based Architecture for Cooperative Path-Following of Multiple Unmanned Aerial Vehicles

This chapter proposes a sampled-data model predictive control (MPC) architecture to solve the decentralized cooperative path-following (CPF) problem of multiple unmanned aerial vehicles (UAVs). In the cooperative path-following proposed scenario, which builds on previous work on CPF, multiple vehicles are required to follow pre-specified paths at nominal speed profiles (that may be path dependent) while keeping a desired, possibly time-varying, geometric formation pattern. In the proposed framework, we exploit the potential of optimization-based control strategies with significant advantages on explicitly addressing input and state constraints and on the ability to allow the minimization of meaningful cost functions. An example consisting of three fixed wing UAVs that are required to follow a given desired maneuver illustrates the proposed framework. We highlight and discuss some features of the UAVs trajectories.

A Predictive Path-Following Approach for Fixed-Wing Unmanned Aerial Vehicles in Presence of Wind Disturbances

In this paper we address the path-following problem for fixed-wing Unmanned Aerial Vehicles (UAVs) in presence of wind disturbances. Given a desired path with a specified airspeed profile assigned on it, the goal is to follow the desired maneuver while minimizing the control effort. We propose a predictive path following control scheme based on trajectory optimization techniques, to compute feasible UAV trajectories, combined with a sample-data Model Predictive Control (MPC) approach, to handle the wind field. By explicitly addressing the wind field, the UAV exploits the surrounding environment thus extending its capabilities in executing the desired maneuver. We provide numerical computations based on straight line and circular paths under various wind conditions. The computations allow us to highlight the benefits of the proposed control scheme.

Optimal UAV Rendezvous on a UGV

An important restriction that arises when fixed-wing Unmanned Aerial Vehicles (UAVs) need to perform several types of persistent missions is the fact that they have limited fuel capacity, which consequently it requires periodic refueling to accomplish the mission. Typically, refueling stations are immobile and the UAV may have to waste fuel and operation time in traveling to the station. Instead, we propose the use of Unmmaned Ground Vehicle (UGV) as a refueling unit where the UAV will rendezvous onto a moving UGV. To accomplish this task successfully with a good performance, the UAV needs to determine an optimal trajectory in 3D taking UAV and UGV dynamics and kinematics into account. Additionally, it is desirable to control the trajectory aggressiveness generated by the UAV through a human operator. In this paper, we propose an optimal control formulation to generate a tunable UAV trajectory for rendezvous on a moving UGV. By a suitable choice of a weighting term in our problem setting, we are able to control some performance features of the UAV trajectory. Numerical results are presented to validate our approach using a nonlinear optimal control solver.

A Sampled-Data Model Predictive Framework for Cooperative Path Following of Multiple Robotic Vehicles

We present a sampled-data model predictive control (MPC) framework for cooperative path following (CPF) of multiple, possibly heterogeneous, autonomous robotic vehicles. Under this framework, input and output constraints as well as meaningful optimization-based performance trade-offs can be conveniently addressed. Conditions under which the MPC-CPF problem can be solved with convergence guarantees are provided. An example illustrates the proposed approach.