The Control Basis: An Action Architecture for Computational - - PDF document

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

The Control Basis: An Action Architecture for Computational Development

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Laboratory for Perceptual Robotics – College of Information and Computer Sciences

The Control Basis

In place of a relatively small set of special purpose developmental reflexes, an exhaustive array of closed-loop control relations is proposed that tile a high dimensional state space with multiple lower-dimensional attractors. the landscape of attractors is modeled as a discrete-event dynamical system within which the robot designer can overlay a time-varying system of logical constraints on the learner to support exploration- based developmental learning algorithms.

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proximo/distal cephalon/caudal quasi-static/dynamic

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Potential Functions

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The value of a scalar potential at the location of a particle in a field represents the energy that will be liberated if the particle is released from this configuration. e.g. the gravitational potential of a particle of mass m near the Earth is the work required to move particle from the surface of the Earth to altitude h. The gradient of the potential field defines a force acting on the particle that returns the system to its equilibrium state.

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Potential Functions – Spring-Mass-Damper

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For the SMD, the potential function is the energy stored in the spring which is released when the spring is allowed to assume its original shape Hooke’s law

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Equilibrium Point Theory - Differential Geometry

Critical points – places where the gradient vanishes

stable fixed points unstable critical points

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minima saddle point maxima

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Potential Functions and Local Minima

Curvature in the Neighborhood of a Critical Point a critical point is said to be degenerate if it also has zero curvature excluding degenerate critical points, gradient descent will converge to type 0 critical points exclusively

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Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Potential Functions and Local Minima

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convex – if the Hessian of f is positive semi-definite over domain q, it has ≤ 1 stable +ixed points on the interior of 3 harmonic – if the trace of the Hessian then f has no local minima

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Harmonic Functions

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soap films, laminar fluid flow, steady state temperature in thermally conductive media, voltage distribution in electrically conductive media,

• exclude local minima (and maxima)
• only type 1 critical points (saddle points) (sets of measure zero)
• gradient flow produces non-intersecting streamlines
• hitting probability of a random walk --- use in path planning

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Navigation Functions

analyticity - infinitely differentiable (C∞ continuous) such that its Taylor series about q0 converges to f (q) for q in the neighborhood of q0. polar - gradients (streamlines) terminate at a unique minimum. functions that contain type 1 minima exclusively are polar Morse – functions whose isolevel curves are single points, closed curves, or closed curves that join at critical points … Morse functions cannot include degenerate critical points admissibility - Potential fields for robot control require bounded torque at

• bstacle boundaries (and everywhere else in the interior subset of

configuration space as well).

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Action Representation

• actions re-code state space using

classifiers in phase portrait

• “funneling” - fixed point sensory

geometry relative to features afforded by the environment

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Parametric Landscape of Attractors

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Funnel-ing the World Using TRACK/SEARCH Actions

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action: closed-loop feature (s) tracker where sensor viewpoint is controlled with kinematic chain t

Control Basis: TRACK primitive

T: TRACK

a =

visual foveation – contact force tracking any feature of any signal

• bjectives sensors effectors

X X

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

State – Takens’ Embedding Theorem

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a history of 2n + 1 time-delayed observations of the same quantity, e.g. [φ(t), φ(t − τ ), φ(t − 2τ ), . . . , φ(t − 2nτ )], is sufficient to represent an n-dimensional dynamical system. related to similar arguments from information theory where the information in a signal is related to the time derivatives at time t that can be approximated using finite-temporal differences [y(t), (y(t) − y(t − τ ))/τ, (y(t) − 2y(t − τ ) + y(t − 2τ ))/ τ2 , … ] The time history of feedback in the phase portrait (φ, ̇ #), likewise, reveals the dynamics of systems—in particular, we are interested in controlled dynamical systems that reveal the influence of the interactions between a controller and stimuli (driving functions) embedded in the environment.

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Control Basis – TRACK State Representation

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state for the two-finger grasp on the irregular planar triangle is the probability distribution over membership in four independent phase portrait models {m0, m1, m2, m3}

Laboratory for Perceptual Robotics – College of Information and Computer Sciences 16

State and Closed-Loop Control Dynamics

state is membership in models m0, m1, m2, m3, m4

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Quasistatic-Dynamic Shaping

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state: g(a) = 0 undefined = 1 transient = 2 converged state: g(a) = [ m0 m1 m2 m3 ] but the infant robot can bootstrap skills and accumulate models at the same time by using a quasi-static subset of models

Laboratory for Perceptual Robotics – College of Information and Computer Sciences 18

The Control Jacobian

NO_REFERENCE TRANSIENT CONVERGED

(or ∇" = 0) (or ∇" > 0)

Laboratory for Perceptual Robotics – College of Information and Computer Sciences 19

Summary: Control Basis TRACK Actions

f

s t ENV

J = ∂φ(σ ) ∂uτ ∈ R1xn

vector of changes in setpoints scalar m < n fewer rows than columns redundant (underconstrained)

Laboratory for Perceptual Robotics – College of Information and Computer Sciences 20

Multi-Objective Control

where, the annihilator of J Appendix A.9

Laboratory for Perceptual Robotics – College of Information and Computer Sciences 21

Multi-Objective Control

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

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Control Basis: POSTURAL primitive

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Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

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Combining SEARCH and TRACK

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tendon routing in the human finger

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

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Control Basis: SEARCH primitive

• the orient counterpart of

TRACK actions

Book 10 m 20 m Clamp

saturation motion

tilt pan tilt pan

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S:SEARCH

(TRACK)

Laboratory for Perceptual Robotics – College of Information and Computer Sciences

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Laboratory for Perceptual Robotics – Department of Computer Science 25

EXAMPLE: Coordinated Human-Robot Search

s

t

• n average, HR team performed 40% better than human alone