Lecture 6:Control Problems and SolutionsCS 344R: RoboticsBenjamin KuipersBut First, Assignment 1:Follower s• A follower is a control law where the robotmoves forward while keeping some errorterm small.– Open-space follower– Wall follower– Coastal navigator– Color follower• Due October 4.Control Laws Have Conditions• Each control law includes:– A trigger: Is this law applicable?– The law itself: u = Hi(y)– A termination condition: Should the law stop?Open-Space Follower• Move in the direction of large amounts ofopen space.• Wiggle as needed to avoid specular reflections.• Turn away from obstacles.• Turn or back out of blind alleys.• Try to be elegant and robust.Wall Follower• Detect and follow right or left wall.• Implement the PD control law taught inclass.• Respond to step-changes in environment orset-point.• Tune to avoid large oscillations.• Terminate on obstacle or wall vanishing.Coastal Navigator• Join wall-followers to follow a complex“coastline”• When a wall-follower terminates, make theappropriate turn, detect a new wall, andcontinue.• Inside and outside corners, 90 and 180 deg.• Orbit a box, a simple room, or the desks!Color Follower• Move to keep a desired color centered inthe camera image.• Train a color region from a given image.• Follow an orange ball on a string, or abrightly-colored T-shirt.• How quickly can the robot respond?Problems and Solutions• Time delay• Static friction• Pulse-width modulation• Integrator wind-up• Chattering• Saturation, dead-zones, backlash• Parameter driftUnmodeled Effects• Every controller depends on its simplifiedmodel of the world.– Every model omits almost everything.• If unmodeled effects become significant,the controller’s model is wrong,– so its actions could be seriously wrong.• Most controllers need special case checks.– Sometimes it needs a more sophisticated model.Time Delay• At time t,– Sensor data tells us about the world at t1 < t.– Motor commands take effect at time t2 > t.– The lag is dt = t2 − t1.• To compensate for lag time,– Predict future sensor value at t2.– Specify motor command for time t2.t1t2tnowPredicting Future Sensor Values• Later, observers will help us make betterpredictions.• Now, use a simple prediction method:– If sensor s is changing at rate ds/dt,– At time t, we get s(t1), where t1 < t,– Estimate s(t2) = s(t1) + ds/dt * (t2 - t1).• Use s(t2) to determine motor signal u(t) thatwill take effect at t2.– "Smith predictor"Static Friction (“Stiction”)• Friction forces oppose the direction of motion.• We’ve seen damping friction: Fd = − f(v)• Coulomb (“sliding”) friction is a constant Fcdepending on force against the surface.– When there is motion, Fc = η– When there is no motion, Fc = η + ε• Extra force is needed to unstick an object andget motion started.Why is Stiction Bad?• Non-zero steady-state error.– (runaway pendulum story)• Stalled motors draw high current.– Running motor converts current to motion.– Stalled motor converts more current to heat.• Whining from pulse-width modulation.– Mechanical parts bending at pulse frequency.Pulse-Width Modulation• A digital system works at 0 and 5 volts.– Analog systems want to output control signalsover a continuous range.– How can we do it?• Switch very fast between 0 and 5 volts.– Control the average voltage over time.• Pulse-width ratio = ton/tperiod. (30-50 µsec)tontperiodPulse-Code Modulated Signal• Some devices are controlled by the lengthof a pulse-code signal.– Position servo-motors, for example.0.7ms20ms1.7ms20msBack EMF Motor Control• Motor torque is proportional to current.• Generator voltage is proportional to velocity.• The same physical device can be either amotor or a generator.• Switch back and forth quickly, as in PWM.Drive as a motor Sense as a generator20msBack EMF Motor ControlIntegrator Wind-Up• Suppose we have a PI controller• Motion might be blocked, but the integralis winding up more and more control action.• Reset the integrator on significant events.! u(t) = "kPe(t) " kIe dt0t#+ ub! u(t) = "kPe(t) + ub˙ u b(t) = "kIe(t)Chattering• Changing modes rapidly and continually.– Bang-Bang controller with thresholds set tooclose to each other.– Integrator wind-up due to stiction near thesetpoint, causing jerk, overshoot, and repeat.Dead Zone• A region where controller output does notaffect the state of the system.– A system caught by static friction.– Cart-pole system when the pendulum ishorizontal.– Cruise control when the car is stopped.• Integral control and dead zones can combineto cause integrator wind-up problems.Saturation• Control actions cannot grow indefinitely.– There is a maximum possible output.– Physical systems are necessarily nonlinear.• It might be nice to have bounded error byhaving infinite response.– But it doesn’t happen in the real world.Backlash• Real gears are not perfect connections.– There is space between the teeth.• On reversing direction, there is a short timewhen the input gear is turning, but theoutput gear is not.Parameter Drift• Hidden parameters can change the behaviorof the robot, for no obvious reason.– Performance depends on battery voltage.– Repeated discharge/charge cycles age the battery.• A controller may compensate for smallparameter drift until it passes a threshold.– Then a problem suddenly appears.– Controlled systems make problems harder to findUnmodeled Effects• Every controller depends on its simplifiedmodel of the world.– Every model omits almost everything.• If unmodeled effects become significant,the controller’s model is wrong,– so its actions could be seriously wrong.• Most controllers need special case checks.– Sometimes it needs a more sophisticated
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