Embedded Systems Control Laboratory for Perceptual Robotics Department

Embedded Systems Control Laboratory for Perceptual Robotics – Department of Computer Science

Basic Tools of Control Theory • Open- and Closed-Loop Control • Laplace Transform (just briefly) • The Canonical Spring-Mass-Damper • Equilibrium Setpoint Control • Qualitative Second-Order Response • Closed-Loop Transfer Function • Time- and Frequency-Domain Response Laboratory for Perceptual Robotics – Department of Computer Science 2

Motor Circuits α-motor neurons initiate overt motion--they’re fast • on average, each will innervate approximately 200 individual muscle fibers. • relatively slow γ-motor neuron regulates muscle tone by setting the reference length of the spindle receptor. • Golgi tendon organ measures the tension in the tendon and inhibits the αmotor neuron if it exceeds safe levels • Laboratory for Perceptual Robotics – Department of Computer Science 3

Negative Feedback • If the measured spindle length is greater than the reference, the α-motor neuron cause a contraction of the muscle tissue • if the measured spindle length is less than the reference, the α-motor neuron is inhibited, allowing the muscle to extend Negative Feedback. . . the α-motor neuron changes its output so as to cancel some of its input. . . Laboratory for Perceptual Robotics – Department of Computer Science 4

Negative Feedback • first submitted for a patent in 1928 by Harold S. Black • Black’s patent application was met with great skepticism, reportedly associated with a perpetual motion machine • subsequently, it was the basis for Watt’s governor • spawned the field of cybernetics • now heralded as a fundamental principle of stability in compensated dynamical systems Laboratory for Perceptual Robotics – Department of Computer Science 5

The Muscle Stretch Reflex • Laboratory for Perceptual Robotics – Department of Computer Science 6

Open-Loop Control an initiating stimulus causes a response without further stimulation SR reflexes - e. g. withdrawl reflex • Laboratory for Perceptual Robotics – Department of Computer Science 7

Closed-Loop Control a (time-varying) setpoint is achieved by constantly measuring and correcting in order to actively reject disturbances • autopilot • Norbert Weiner - cybernetics (helmsman), homeostasis, endocrine system living things as feedback regulators • Laboratory for Perceptual Robotics – Department of Computer Science 8

Laplace Transform assume: x(t) ~ Cest • L( d/dt (x(t)) ) = s L( x(t) ) • L( d 2/dt 2 (x(t)) ) = s 2 L( x(t) ) . . . a linear differential equation with constant coefficients and a finite number of terms is Laplace-transformable. . . they transform into polynomials in s… Laboratory for Perceptual Robotics – Department of Computer Science 9

Laplace Transform assume: x(t) ~ Cest • L( d/dt (x(t)) ) = s L( x(t) ) • L( d 2/dt 2 (x(t)) ) = s 2 L( x(t) ) an Nth order differential equation an. Dn + an− 1 Dn− 1+ …+ a 0 = 0 into an Nth order polynomial ansn + an− 1 sn− 1+ …+ a 0 = 0 Laboratory for Perceptual Robotics – Department of Computer Science 10