Alessandro Saccon

Optimal Control on Lie Groups: The Projection Operator Approach

Many nonlinear systems of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. Examples range from aircraft and underwater vehicles to quantum mechanical systems. In this paper, we develop an algorithm for solving continuous-time optimal control problems for systems evolving on (noncompact) Lie groups. This algorithm generalizes the projection operator approach for trajectory optimization originally developed for systems on vector spaces. Notions for generalizing system theoretic tools such as Riccati equations and linear and quadratic system approximations are developed. In this development, the covariant derivative of a map between two manifolds plays a key role in providing a chain rule for the required Lie group computations. An example optimal control problem on SO(3) is provided to highlight implementation details and to demonstrate the effectiveness of the method.

Achievable motorcycle trajectories

The authors show that a (simple, nonholonomic) motorcycle can exactly track a large class of smooth trajectories in the plane. Instability and nontrivial dynamic coupling make the exploration of aggressive motorcycle trajectories a rather challenging task. Previously (Hauser et al., 2004), we developed optimization techniques for constructing a suitable roll trajectory that (approximately) implements the desired plane trajectory. In that work, we found that the tracking error is usually quite small leading to the natural question: Given a smooth trajectory in the plane, does there exist a bounded roll trajectory that allows a simple motorcycle model to exactly track the plane trajectory? In this paper, we develop a technique for proving that such exact tracking is possible and apply it to a number of example cases. Our technique is based on the nonlinear system inversion work of Devasia and Paden (1998). Indeed, our algorithm is in the class that they propose. Unfortunately, we have been unable to directly use their results as the motorcycle system does not appear to satisfy the specific conditions required.

A virtual rider for motorcycles: An approach based on optimal control and maneuver regulation

Recently developed optimization and nonlinear control strategies are applied to drive a multi-body motorcycle model along a specified path with an associated velocity profile. The resulting control scheme is based on three main pillars: a dynamic inversion procedure to compute the input-state trajectories corresponding to a desired maneuvering task, an inverse optimal control heuristic for designing closed loop dynamics, and a maneuver regulation controller to overcome limitations of standard trajectory tracking scheme. The controller (virtual rider) is implemented on a commercial simulation software. Simulation results are presented.

Trajectory Exploration of a Rigid Motorcycle Model

This paper introduces a rigid motorcycle model that captures many important aspects of real motorcycle dynamics including sliding and load transfer. The model is used to demonstrate a dynamic inversion procedure which maps a desired flatland trajectory into a corresponding (state-input) trajectory for the rigid motorcycle model. This inverse trajectory is the solution of an optimal control problem that is computed using the projection operator approach for the optimization of trajectory functionals, a recently developed optimization technique. The effectiveness of the proposed strategy is illustrated using a trajectory computation for a realistic path that is traversed with a demanding speed profile. The rigid motorcycle model detailed in this paper is also of interest as a nontrivial example of a mechanical system with nonideal holonomic constraints.

A Virtual Rider for Motorcycles: Maneuver Regulation of a Multi-Body Vehicle Model

This work develops a virtual rider that can be used to make a multi-body two-wheeled vehicle follow a specified ground path with a prescribed velocity profile. The virtual rider system is based on a simplified motorcycle model, the sliding plane motorcycle, which is composed of a single rigid body with two ground contact points. This reduced order nonlinear system was presented in an earlier work, together with a dynamic inversion procedure for computing a state-control trajectory corresponding to the desired task. This dynamic inversion procedure is combined in this work with a maneuver regulation controller to yield a nonlinear feedback control strategy. A transverse coordinate system that is consistent with the mechanical symmetries of ground vehicles is constructed and used in the development of the maneuver regulation controller. An inverse optimal control strategy, which also exploits the mechanical symmetries, is developed to shape the dynamic response of the closed loop system. Numerical results with the virtual rider driving a multi-body vehicle through a demanding maneuver with lateral accelerations reaching 1 g are presented.

Model predictive for path following with motorcycles: application to the development of the pilot model for virtual prototyping

Controlling a riderless motorcycle is a challenging problem because the dynamics are nonlinear and non-minimum phase. In this paper, an innovative control strategy is proposed for driving a motorcycle along a given path, tracking a speed profile given as a function of the arc-length of the path. The solution is based on model predictive control. Exploiting the possibility given by MPC to work on trajectories, we invert the cause-effect structure of the problem and act as if the roll angle was an input. We then determine, among a polynomial set of roll angle trajectories, the optimal one in terms of the error at preview distance from the target path. By inverting the dynamics, we compute the steering and longitudinal controls needed to track the computed roll trajectory.

Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking

This paper addresses the sensitivity analysis for hybrid systems with discontinuous (jumping) state trajectories. We consider state-triggered discontinuities in the state evolution, potentially accompanied by mode switching in the control vector field. For a given trajectory with state jumps, we show how to construct an approximation of the nearby perturbed trajectory corresponding to a given variation of the initial condition and input signal. A major complication in the construction of such an approximation is that, in general, the jump times corresponding to a nearby perturbed trajectory are not equal to those of the nominal one. The main contribution of this work is the development of a notion of error to clarify in which sense the approximate trajectory is, at each instant of time, a first-order approximation of the perturbed trajectory. This notion of error naturally finds application in the (local) tracking problem of a time-varying reference trajectory of a hybrid system. To illustrate the possible use of this new error definition in the context of trajectory tracking, we outline how the standard linear trajectory tracking control for nonlinear systems could be generalized for hybrid systems.

Energy-Optimal Motion Planning for Multiple Robotic Vehicles With Collision Avoidance

We propose a numerical algorithm for multiple-vehicle motion planning that explicitly takes into account the vehicle dynamics, temporal and spatial specifications, and energy-related requirements. As a motivating example, we consider the case where a group of vehicles is tasked to reach a number of target points at the same time (simultaneous arrival problem) without colliding among themselves and with obstacles, subject to the requirement that the overall energy required for vehicle motion be minimized. With the theoretical setup adopted, the vehicle dynamics are explicitly taken into account at the planning level. This paper formulates the problem of multiple-vehicle motion planning in a rigorous mathematical setting, describes the optimization algorithm used to solve it, and discusses the key implementation details. The efficacy of the method is illustrated through numerical examples for the simultaneous arrival problem. The initial guess to start the optimization procedure is obtained from simple geometrical considerations, e.g., by joining the desired initial and final positions of the vehicles via straight lines. Even though the initial trajectories thus obtained may result in intervehicle and vehicle/obstacle collisions, we show that the optimization procedure that we employ in this paper will generate collision-free trajectories that also minimize the overall energy spent by each vehicle and meet the required temporal and spatial constraints. The method developed applies to a very general class of vehicles; however, for clarity of exposition, we adopt as an illustrative example the case of wheeled robots.

Guaranteeing stable tracking of hybrid position-force trajectories for a robot manipulator interacting with a stiff environment

This work considers the control of a manipulator with the aim of executing desired time-varying motion-force trajectories in the presence of a stiff environment. In several situations, the interaction with the environment constrains just one degree of freedom of the manipulator end-effector. Focusing on this contact degree of freedom, a switching position-force controller is considered to perform the hybrid motion-force tracking task. To guarantee input-to-state stability of the switching closed-loop system, a novel stability result and sufficient conditions are presented. The switching occurs when the manipulator makes or breaks contact with the environment. The analysis shows that to guarantee closed-loop stability while tracking arbitrary time-varying motion-force profiles with a rigid manipulator, the controller should implement a considerable (and often unrealistic) amount of damping, resulting in inferior tracking performance. Therefore, we use the stability analysis technique developed in this paper to analyze a manipulator equipped with a compliant wrist. Guidelines are provided for the design of the wrist compliancy while employing the switching control strategy, such that stable tracking of a motion-force reference trajectory can be achieved and bouncing of the manipulator against the stiff environment can be avoided. Numerical simulations are presented to illustrate the effectiveness of the approach.

Second-order-optimal filters on Lie groups

We provide an explicit formula for the second-order-optimal nonlinear filter for state estimation of systems on general Lie groups with disturbed measurements of inputs and outputs. Optimality is with respect to a deterministic cost measuring the cumulative energy in the unknown system disturbances (minimum-energy filtering). We show that the resulting filter will depend on the choice of affine connection, thus encoding the nonlinear geometry of the state space. For the case of attitude estimation, where we are given a second order (dynamic) mechanical system on the tangent bundle of the special orthogonal group SO(3), and where we choose the symmetric Cartan-Schouten (0)-connection, the resulting filter has the familiar form of a gradient observer combined with a matrix Riccati differential equation that updates the filter gain.