Institute for Systems Research

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Author: search.filters.author.Andersson, Sean B.

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    The Hannay-Berry Phase of the Vibrating Ring Gyroscop
    (2004) Andersson, Sean B.; Krishnaprasad, Perinkulam S.; Krishnaprasad, Perinkulam S.; ISR; CDCSS
    In an analysis published in 1890 G.H. Bryan investigated the retrograde precession of the nodal points in a vibrating, rotating shell and wrote down a formula relating the rate of precession to the rate of rotation. This effect has been utilized in the design of various vibratory gyroscopes including modern MEMS-based devices. Existing analyses model these systems with a pair of harmonic oscillators coupled through the Coriolis force (the normal mode method). In this work we utilize the theory of moving systems developed by Marsden, Montgomery, and Ratiu to show that the nodal precession can be understood as a geometric phase with respect to the Cartan-Hannay-Berry connection. This approach allows us to explicitly characterize the simplifications of the linearizing assumptions common to previous analyses. Our results match those of Bryan for small amplitude vibrations of the ring. We use the inherently nonlinear nature of the moving systems approach to calculate a (small) correction to the rate of precession of the nodes.
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    Geometric Phases in Sensing and Control
    (2003) Andersson, Sean B.; Krishnaprasad, P.S.; ISR; CDCSS
    In many parameter-dependent systems, varying the parameters along a closed path generates a shift in the system depending only on the path itself and not on the manner in which that path is traversed. This effect is known as a geometric phase. In this thesis we focus on developing techniques to utilize geometric phases as engineering tools in both sensing and control. We begin by considering systems undergoing an imposed motion. If this motion is adiabatic then its effect on the system can be described by a geometric phase called the Hannay-Berry phase. Direct information about the imposed motion is obtained by measuring the corresponding phase shift. We illustrate this idea with an equal-sided, spring-jointed, four-bar mechanism and then apply the technique to a vibrating ring gyroscope. In physical systems the imposed motion cannot be truly adiabatic. Using Hamiltonian perturbation theory, we show that the Hannay-Berry phase is the first-order term in a perturbation expansion in the rate of imposed motion. Corrections accounting for the nonadiabatic nature of the imposed motion are then given by carrying the expansion to higher-order. The technique is applied to the vibrating ring gyroscope as an example. We also consider geometric phases in dissipative systems with symmetry. Given such a system with a parameter-dependent, exponentially asymptotically stable equilibrium point, we define a new connection, termed the Landsberg connection, which captures the effect of a cyclic, adiabatic variation of the parameters. Systems with stable, time-dependent solutions are handled by defining an appropriate dynamic phase. A simple example is developed to illustrate the technique. Finally we investigate the role of geometric phases in the control of nonholonomic systems with symmetry through an exploration of the H(3)-Racer, a two-node, one module G-snake on the three-dimensional Heisenberg group. We derive the governing equations for the internal shape of the system and the reconstruction equations relating changes in the shape to the overall motion. The controllability of the system is considered and the effect of various shape changes is explored through simulation.
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    Stochastic Language-based Motion Control
    (2003) Andersson, Sean B.; Hristu-Varsakelis, Dimitrios; ISR; CDCSS
    In this work we present an efficient environment representation based on the use of landmarks and language-based motion programs. The approach is targeted towards applications involving expansive, imprecisely known terrain without a single global map. To handle the uncertainty inherent in real-world applications a partially-observed controlled Markov chain structure is used in which the state space is the set of landmarks and the control space is a set of motion programs. Using dynamic programming, we derive an optimal controller to maximize the probability of arriving at a desired landmark after a finite number of steps. A simple simulation is presented to illustrate the approach.
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    Degenerate Gradient Flows: A Comparison Study of Convergence Rate Estimates
    (2002) Andersson, Sean B.; Krishnaprasad, Perinkulam S.; Krishnaprasad, Perinkulam S.; ISR
    Degenerate gradient flows arise in the context of adaptive control of linear systems when the usual gradient algorithm is used for the parameter update law. It is well known that in general parameter convergence is not guaranteed without further assumptions. The standard approach utilizes the notion of a persistently exciting input and different authors have derived different convergence rate estimates. In a recent paper Brockett re-examined this issue and developed a rate estimate using a property of symmetric matrices related to the condition number. In this paper we compare two well-known convergence rate estimates from the persistently exciting point of view with Brockett's estimate through a semianalytical numerical study. We establish a common footing by relating the assumptions of each theorem to the parameters specified under the persistently exciting condition. Our analysis shows that for all parameter values Anderson's result yields a tighter bound than the other two estimates. In each case the magnitude of the difference depends on the time it takes for the uniform observability condition to hold in the persistently exciting assumption. The shorter the time is, the larger the difference is.
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    The Berry-Hannay Phase of the Equal-Sided, Spring-Jointed, Four-Bar Mechanism: A Detailed Story
    (2002) Andersson, Sean B.; Krishnaprasad, Perinkulam S.; Krishnaprasad, Perinkulam S.; ISR
    In this work we apply the moving systems approach developed by Marsden, Montgomery, and Ratiu to a free-floating, equal-sided, spring-jointed, four-bar mechanism that is being slowly rotated about its central axis and derive a formula for the induced geometric phase. We investigate the phase for a few specific systems using both analytic analysis and simulation.
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    Directed Graphs and Motion Description Languages for Robot Navigation and Control
    (2001) Hristu-Varsakelis, Dimitrios; Andersson, Sean B.; ISR; CDCSS
    We propose a landmark-based representation of maps to be used for robotnavigation and exploration. Our approach is aimed towards mobilerobots that operate over expansive, imprecisely known terrain without asingle ``global'' map. Instead, a map is pieced together fromlocal terrain and navigation data stored in a directed graph. Each of thegraph's vertices contains information describing a landmark locally (e.g. adetailed map of that landmark's immediate surroundings). Thegeometric relationships between landmarks are unknown. Graph edgesstore language-based directions that enable a robot to steerbetween landmarks. These directions are written in the motiondescription language MDLe, reducing the complexity of the map and makingnavigation programs robot-independent. Furthermore, the proposedarchitecture iseconomical with respect to the amount of storage required to describefar-flung areas of interest. We present preliminary resultsdemonstrating our ideas using an indoor robot.

    This paper appeared in the 2002 IEEE Int'l Conference on Robotics andAutomation
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    The Berry-Hannay Phase of the Equal-Sided Spring-Jointed Four-Bar Mechanism
    (2001) Andersson, Sean B.; Krishnaprasad, Perinkulam S.; Krishnaprasad, Perinkulam S.; ISR
    In this work we apply the moving systems approach developed by Marsden, Montgomery, and Ratiu to a free-floating, equal-sided, spring-jointed, four-bar mechanism that is being slowly rotated about its central axis and derive a formula for the induced geometric phase.
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    The MDLe Engine -- A Software Tool for Hybrid Motion Control
    (2000) Hristu, Dimitrios; Krishnaprasad, Perinkulam S.; Andersson, Sean B.; Zhang, F.; Sodre, P.; D'Anna, L.; ISR; CDCSS
    One of the important but often overlooked practical challenges in motion control for robotics and other autonomous machines has to do with the implementation of theoretical tools into software that will allow the system to interact effectively with the physical world. More often than not motion control programs are machine-specific and not reusable, even when the underlying algorithm does not require any changes.

    The work on Motion Description Languages (MDL) has been an effort to formalize a general-purpose robot programming language that allows one to incorporate both switching logic and differential equations. Extended MDL (MDLe) is a device-independent programming language for hybrid motion control, accommodating hybrid controllers, multi-robot interactions and robot-to-robot communications.

    The purpose of this paper is to describe the "MDLe engine," a software tool that implements the MDLe language.

    We have designed a basic compiler/software foundation for writing MDLe code. We provide a brief description of the MDLe syntax, implementation architecture, and functionality. Sample programs are presented together with the results of their execution on a set of physical and simulated mobile robots.