The Control Basis: An Action Architecture for Computational Development Laboratory for Perceptual Robotics – College of Information and Computer Sciences 2 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. proximo/distal cephalon/caudal quasi-static/dynamic 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. Laboratory for Perceptual Robotics – College of Information and Computer Sciences 3
Potential Functions 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 4 Potential Functions – Spring-Mass-Damper 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 5
Equilibrium Point Theory - Differential Geometry saddle minima maxima point stable fixed points unstable critical points Critical points – places where the gradient vanishes Laboratory for Perceptual Robotics – College of Information and Computer Sciences 6 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 Laboratory for Perceptual Robotics – College of Information and Computer Sciences 7
Potential Functions and Local Minima 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 8 Harmonic Functions 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 9
Navigation Functions analyticity - infinitely differentiable (C ∞ continuous) such that its Taylor series about q 0 converges to f (q) for q in the neighborhood of q 0 . 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 obstacle boundaries (and everywhere else in the interior subset of configuration space as well). Laboratory for Perceptual Robotics – College of Information and Computer Sciences 10 Action Representation Parametric Landscape of Attractors • actions re-code state space using classifiers in phase portrait • “funneling” - fixed point sensory geometry relative to features afforded by the environment Laboratory for Perceptual Robotics – College of Information and Computer Sciences 11
Funnel-ing the World Using T RACK /S EARCH Actions Laboratory for Perceptual Robotics – College of Information and Computer Sciences 12 Control Basis: T RACK primitive objectives sensors effectors X X T: T RACK action: closed-loop feature ( s ) tracker where sensor viewpoint is controlled with kinematic a = chain t visual foveation – contact force tracking any feature of any signal Laboratory for Perceptual Robotics – College of Information and Computer Sciences 13
State – Takens’ Embedding Theorem a history of 2n + 1 time-delayed observations of the same quantity, e.g. [ φ (t), φ (t − τ ), φ (t − 2 τ ), . . . , φ (t − 2n τ )], 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 − 2 τ ))/ τ 2 , … ] 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 14 Control Basis – T RACK State Representation state for the two-finger grasp on the irregular planar triangle is the probability distribution over membership in four independent phase portrait models { m 0 , m 1 , m 2 , m 3 } Laboratory for Perceptual Robotics – College of Information and Computer Sciences 15
State and Closed-Loop Control Dynamics state is membership in models m 0 , m 1 , m 2 , m 3 , m 4 Laboratory for Perceptual Robotics – College of Information and Computer Sciences 16 Quasistatic-Dynamic Shaping state: g ( a ) = [ m 0 m 1 m 2 m 3 ] but the infant robot can bootstrap skills and accumulate models at the same time by using a quasi-static subset of models state: g ( a ) = 0 undefined = 1 transient = 2 converged Laboratory for Perceptual Robotics – College of Information and Computer Sciences 17
The Control Jacobian NO_REFERENCE (or ∇" > 0) TRANSIENT (or ∇" = 0) CONVERGED Laboratory for Perceptual Robotics – College of Information and Computer Sciences 18 Summary: Control Basis T RACK Actions s f ENV t scalar J = ∂ φ ( σ ) ∈ R 1 xn m < n ∂ u τ fewer rows than columns redundant (underconstrained) vector of changes in setpoints Laboratory for Perceptual Robotics – College of Information and Computer Sciences 19
Multi-Objective Control where, the annihilator of J Appendix A.9 Laboratory for Perceptual Robotics – College of Information and Computer Sciences 20 Multi-Objective Control Laboratory for Perceptual Robotics – College of Information and Computer Sciences 21
Control Basis: P OSTURAL primitive Laboratory for Perceptual Robotics – College of Information and Computer Sciences 15 22 Laboratory for Perceptual Robotics – College of Information and Computer Sciences Combining S EARCH and T RACK tendon routing in the human finger Laboratory for Perceptual Robotics – College of Information and Computer Sciences 15 23 Laboratory for Perceptual Robotics – College of Information and Computer Sciences
Control Basis: S EARCH primitive motion S:S EARCH tilt pan saturation tilt pan ( T RACK ) Book Clamp • the orient counterpart of 20 m T RACK actions 10 m Laboratory for Perceptual Robotics – College of Information and Computer Sciences 15 24 Laboratory for Perceptual Robotics – College of Information and Computer Sciences EXAMPLE: Coordinated Human-Robot Search s t on average, HR team performed 40% better than human alone Laboratory for Perceptual Robotics – Department of Computer Science 25 25
Recommend
More recommend
