Timothy M. Caldwell

Switching mode generation and optimal estimation with application to skid-steering

Skid-steered vehicles, by design, must skid in order to maneuver. The skidding causes the vehicle to behave discontinuously during a maneuver as well as introduces complications to the observation of the vehicles state, both of which affect a controllers performance. This paper addresses estimation of contact state by applying switched system optimization to estimate skidding properties of the skid-steered vehicle.In order to treat the skid-steered vehicle as a switched system, the vehicles ground interaction is modeled using Coulomb friction, thereby partitioning the system dynamics into four distinct modes, one for each combination of the forward and back wheel pairs sticking or skidding. Thus, as the vehicle maneuvers, the system propagates over some mode sequence, transitioning between modes over some set of switching times. This paper presents second-order optimization algorithms for estimating these switching times. We emphasize the importance of the second-order algorithm because it exhibits quadratic convergence and because even for relatively simple examples, first-order methods fail to converge on time scales compatible with real-time operation. Furthermore, the paper presents a technique for estimating the mode sequence by optimizing a relaxation of the switched system.

An adjoint method for second-order switching time optimization

Switched systems evolve over a sequence of continuous modes of operation, transitioning between modes in a discrete manner. Assuming a mode sequence is known, the evolution of a switched system is dictated by the set of times for which the modes transition. This paper presents second-order optimization of these switching times and compares its convergence with first-order switching time optimization. We emphasize the importance of the second-order method because it exhibits quadratic convergence and because even for relatively simple examples, first-order methods fail to converge on time scales compatible with real-time operation.

Projection-based optimal mode scheduling

This paper develops an iterative optimization technique that can be applied to mode scheduling. The algorithm provides both a mode schedule and timing of that mode schedule with convergence guarantees. Moreover, the algorithm takes advantage of a line search, and the number of iterations in the line search is bounded. There are two key ingredients in the algorithm. First, a projection operation is used that takes arbitrary curves and maps them to feasible switching controls. Second, a descent direction that incorporates the projection is calculated using the mode insertion gradient. Similar to derivative-based finite dimensional optimization, the convergence guarantees and sufficient decrease criteria follow from a local approximation of the cost in the direction of the search direction, but this local approximation is not the standard quadratic approximation. An example demonstrates the steps to implement the optimization algorithm and illustrates convergence.

Projection-based switched system optimization: Absolute continuity of the line search

The line search is considered for the problem of numerical switched system optimization using projection-based techniques. Switched system optimization may be formulated as an infinite dimensional optimal control problem where the switching control design variables are constrained to the integers. Projection-based techniques handle the integer constraint by considering an equivalent problem with unconstrained design variables but where the cost is dependent on the projection of the design variables to the constrained set of feasible switched system trajectories. This paper is concerned with the line search step of the projection-based optimization procedure. The main result provides sufficient conditions on the descent direction so that the update rule is absolutely continuous with respect to the step size.

Optimal parameter identification for discrete mechanical systems with application to flexible object manipulation

We present a method for system identification of flexible objects by measuring forces and displacement during interaction with a manipulating arm. We model the objects structure and flexibility by a chain of rigid bodies connected by torsional springs. Unlike previous work, the proposed optimal control approach using variational integrators allows identification of closed loops, which include the robot arm itself. This allows using the resulting models for planning in configuration space of the robot. In order to solve the resulting problem efficiently, we develop a novel method for fast discrete-time adjoint-based gradient calculation. The feasibility of the approach is demonstrated using full physics simulation in trep and using data recorded from a 7-DOF series elastic robot arm.

Second-order optimal estimation of slip state for a simple slip-steered vehicle

We present a method to optimally estimate when a slip state transition occurs for a slip-steered vehicle. A slip-steered vehicles contact state with the ground must slip sideways in order for the vehicle to turn. This slipping generates uncertainties for an autonomous controller. These uncertainties can be be reduced if an estimate of when the vehicle switches between slipping and sticking is provided to the controller. We present an estimator that optimally determines when a switch between slipping and sticking occurs by comparing simulations of the slip-steered vehicle with its measured configurations. We demonstrate that steepest descent-based optimization has slow convergence and show how this issue can be rectified by using Newtons Method. This is a primary stress of our paper. The paper concludes with the introduction of an algorithm that uses second-order optimization in a manner that is appropriate for online implementation.

Projection-based switched system optimization

The Pontryagin Maximum Principle is applied to the switched system optimization problem resulting in a generalization of a well known necessary condition for switching time optimality. The switched system optimization may be formulated as an infinite dimensional problem where the switching control design variables, at any given time, are constrained to the integers. This paper analyzes projection-based techniques for handling the integer constraint. The necessary condition derived in this paper uses the cost composed with the projection of the design variables onto the feasible set. A specific form of projection is considered and two candidate projections are proposed-one projects the immediate value of the switching control and neglects the state while the second is variable on the projected state error.

Single integration optimization of linear time-varying switched systems

This technical note considers the switching time optimization of time-varying linear switched systems subject to quadratic cost-also potentially time-varying. The problem is formulated so that only a single set of differential equations need to be solved prior to optimization. Once these differential equations have been solved, the cost may be minimized over an arbitrary number of modes and mode sequences without requiring additional simulation. The differential equations that need to be simulated are as smooth as the systems vector fields, despite the fact that the optimization itself is nonsmooth. The technique is illustrated in examples from the literature.

Projection-based iterative mode scheduling for switched systems

Abstract This paper describes a method for scheduling the events of a switched system to achieve an optimal performance. The approach has guarantees on convergence and computational complexity that parallel derivative-based iterative optimization but in the infinite dimensional, integer constrained setting of mode scheduling. In comparison to methods relying on mixed integer programming, the presented approach does not require a priori discretizations of time or state. Furthermore, in comparison to embedding and relaxation methods, every iteration of the algorithm returns a dynamically feasible solution. A large class of problems call for optimal mode scheduling. This paper considers a vehicle tracking problem and a high dimensional multimachine power network synchronization problem. For the power network example, both single horizon and receding horizon approaches prevent instability of the network, and the receding horizon approach does so at near real-time speeds on a single processor.

Power Network Regulation Benchmark for Switched-Mode Optimal Control

Abstract Power network regulation is presented as a benchmark problem for assessing and developing switched-mode optimal control approaches like mode scheduling, sliding window scheduling and modal design. Power network evolution modeled by the swing equations and coupled with controllable switching components is a nonlinear, high-dimensional problem. The proposed benchmark problem is the 54 generator IEEE 118 Bus Test Case composed of 106 states. Open questions include scalability in state and number of modes of operation, as well as real-time implementation, reliability, hysteresis, and timing constraints. Can the entire North American power network be regulated? Can every transmission line have independent switching control authority?