Todd D. Murphey

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.

Scalable Variational Integrators for Constrained Mechanical Systems in Generalized Coordinates

We present a technique to implement scalable variational integrators for generic mechanical systems in generalized coordinates. Systems are represented by a tree-based structure that provides efficient means to algorithmically calculate values (position, velocities, and derivatives) needed for variational integration without the need to resort to explicit equations of motion. The variational integrator handles closed kinematic chains, holonomic constraints, dissipation, and external forcing without modification. To avoid the full equations of motion, this method uses recursive equations, and caches calculated values, to scale to large systems by the use of generalized coordinates. An example of a closed-kinematic-chain system is included along with a comparison with the open-dynamics engine (ODE) to illustrate the scalability and desirable energetic properties of the technique. A second example demonstrates an application to an actuated mechanical system.

Convergence-Preserving Switching for Topology-Dependent Decentralized Systems

Stability analysis of decentralized control mechanisms for networked coordinating systems has generally focused on specific controller implementations, such as nearest-neighbor and other types of proximity graph control laws. This approach often misses the need for the addition of other control structures to improve global characteristics of the network. An example of such a situation is the use of a Gabriel graph, which is essentially a nearest-neighbor rule modified to ensure global connectivity of the network if the agents are pairwise connected through their sensor inputs. We present a method of ensuring provable stability of decentralized switching systems by employing a hysteresis rule that uses a zero-sum consensus algorithm. We demonstrate the application of this result to several special cases, including nearest-neighbor control laws, Gabriel graph rules, diffuse target tracking, and hierarchical heterogeneous systems.

Feedback control methods for distributed manipulation systems that involve mechanical contacts

In this paper we introduce feedback control methods for distributed manipulation systems thatmove objects via rolling and slipping point contacts. Because of the intermittent nature of these mechanical contacts, the governing mechanics of these systems are inherently nonsmooth. We first present a methodology to model these non-smooth mechanical effects in a manner that is tractable for non-smooth control analysis. Using these models, we show that when considerations of these non-smooth effects are taken into account, a class of traditional open-loop distributed manipulation control methods cannot stabilize objects near an equilibrium. However, stability can be achieved through the use of feedback, and we present non-smooth feedback laws with guaranteed stability properties. We then describe an experimental modular distributed manipulation test-bed upon which one can implement a variety of control schemes. Experiments with this test-bed confirm the validity of our control algorithms. Multimedia extensions include videos of these experiments.

Modeling Forces and Moments at the Base of a Rat Vibrissa during Noncontact Whisking and Whisking against an Object

During exploratory behavior, rats brush and tap their whiskers against objects, and the mechanical signals so generated constitute the primary sensory variables upon which these animals base their vibrissotactile perception of the world. To date, however, we lack a general dynamic model of the vibrissa that includes the effects of inertia, damping, and collisions. We simulated vibrissal dynamics to compute the time-varying forces and bending moment at the vibrissa base during both noncontact (free-air) whisking and whisking against an object (collision). Results show the following: (1) during noncontact whisking, mechanical signals contain components at both the whisking frequency and also twice the whisking frequency (the latter could code whisking speed); (2) when rats whisk rhythmically against an object, the intrinsic dynamics of the vibrissa can be as large as many of the mechanical effects of the collision, however, the axial force could still generate responses that reliably indicate collision based on thresholding; and (3) whisking velocity will have only a small effect on the transient response generated during a whisker–object collision. Instead, the transient response will depend in large part on how the rat chooses to decelerate its vibrissae after the collision. The model allows experimentalists to estimate error bounds on quasi-static descriptions of vibrissal shape, and its predictions can be used to bound realistic expectations from neurons that code vibrissal sensing. We discuss the implications of these results under the assumption that primary sensory neurons of the trigeminal ganglion are sensitive to various combinations of mechanical signals.

Second-Order Switching Time Optimization for Nonlinear Time-Varying Dynamic Systems

This technical note gives a method for calculating the first and second derivatives of a cost function with respect to switching times for systems with piecewise second-differentiable dynamics. Differential equations governing the linear and bilinear operators required for calculating the derivatives are presented. Example optimizations of linear and nonlinear systems are presented as evidence for the value of second-order optimization methods. One example converges in 33 iterations using second-order methods whereas the first-order algorithm requires over 30 000 iterations.

A method of cooperative control using occasional non-local interactions

Current approaches to distributed control involving many robots generally restrict interactions to pairs of robots within a threshold distance. While this allows for provable stability, there are performance costs associated with the lack of long-distance information. We introduce the acute angle switching algorithm, which allows a small number of long-range interactions in addition to interactions with nearby neighbors, without sacrificing provable stability. We prove several formal properties of the acute angle switching algorithm, including system-wide connectivity. Further, we show simulation results demonstrating the efficacy and robustness of multi-robot systems based on the acute angle switching algorithm

Dynamic Modeling and Motion Planning for Marionettes: Rigid Bodies Articulated by Massless Strings

We consider the problem of modeling a robotic marionette. Marionettes are highly under-actuated systems that can only be controlled remotely by moving strings. We present a mixed dynamic-kinematic modeling technique that removes the controller dynamics from the marionette, resulting in a clean abstraction that represents the dynamics of the marionette in a natural way. As an example, a model is derived for a single arm moving in a plane. A model for a three-dimensional marionettes is also shown. Finally, an expansive-space tree (EST) motion planner is used to find a path from an input configuration to a goal for a puppet arm with seven degrees of freedom

Switching Rules for Decentralized Control with Simple Control Laws

We introduce a novel method for enforcing stability on a decentralized control system. In contrast to previous work, our approach allows for the use of a wide variety of simple control laws, while still providing for a formal proof of stability. Our motivating example uses a simple geometric switching function coupled with PD control that has an intuitive interpretation as a virtual spring mesh. Building on this example, we show a general proof technique that applies to a large class of decentralized control systems. Furthermore, we describe additional cases that illustrate how our technique can be applied to useful systems that are straightforward to implement.

Trajectory generation for underactuated control of a suspended mass

The underactuated system under consideration is a magnetically-suspended, differential drive robot utilizing a winch system to articulate a suspended mass. A dynamic model of the system is first constructed, and then a nonlinear, infinite-dimensional optimization algorithm is presented. The system model uses the principles of kinematic reduction to produce a mixed kinematic-dynamic model that isolates the modeling of the system actuators from the modeling of the rest of the system. In this framework, the inputs become generalized velocities instead of generalized forces facilitating real-world implementation with an embedded system. The optimization algorithm automatically deals with the complexities introduced by the nonlinear dynamics and underactuation to synthesize dynamically feasible system trajectories for a wide array of trajectory generation problems. Applying this algorithm to the mixed kinematic-dynamic model, several example problems are solved and the results are tested experimentally. The experimental results agree quite well with the theoretical showing promise in extending the capabilities of the system to utilize more advanced feedback techniques and to handle more complex, three-dimensional problems.