PID Controls of Motors Howie Choset thanks to

PID Controls of Motors Howie Choset (thanks to Gary Fedder) http: //www. library. cmu. edu/ctms/examples/motor. htm

Controls • • • Review of Motor Model Open Loop (controller-free) Response Proportional Control – Stable – Faster Response = Bigger Overshoot – Steady State Error • PI Control – Maintain Stability – Decrease Steady State Error = Bigger Overshoot • PID Control – Derivative term reduces overshoot, settling time • Feed Forward – Overcome damping

Mass-Spring-Damper Model (Analogy) Model of mass spring damper system z(t) position, z(t) velocity t 0 initial time, z(t 0) initial position & velocity

Review of Motor Model • electric inductance (L) = [H] (VL = L di/dt) * input (V): Source Voltage * output (theta): position of shaft * The rotor and shaft are assumed to be rigid • moment of inertia of the rotor (J) [kg. m^2/s^2] * damping ratio of the mechanical system (b) [Nms] * electromotive force constant • Ke is volt (electromotive force) per radians per second (V/ rad/sec) • torque constance • Kt is torque amp (Nm/Amp) * electric resistance (R) = [ohm]

Review of Motor Model Torque is proportional to current (Lenz’s Law) Back emf is proportional to motor speed (Faraday’s Law) Mechanical Equation of Motion Assume (K=Ke=Kt) Electrical Equation of Motion Solve for sq/V

Transfer Function of Motor (with Approximations). = Open Loop Transfer Function = Can rewrite function in terms of an electrical and mechanical behavior . Electrical time constant on motor is much smaller example motor with equivalent time constants . = For small motors, the mechanical behavior dominates (electrical transients die faster).

Open Loop Response (to a Step) • Apply constant voltage • Slow response time (lag) • Weird Apples-to-Orange relationship between input and output – If you want to set speed, what voltage do you input? – Weird type of steady state error • No reaction to perturbations Input Voltage Plant Output Speed

Closed Loop Controller Give it a velocity command get a velocity output Ref + error - Controller voltage Controller Evaluation Steady State Error Rise Time (to get to ~90%) Overshoot Settling Time (Ring) (time to steady state) Stability Plant

Close the loop analogy

Stability Asymptotic Stability:

Closed Loop Response (Proportional Feedback) Proportional Control Easy to implement Input/Output units agree Improved rise time Steady State Error (true) P: Rise Time vs. Overshoot P: Rise Time vs. Settling time R + error - Voltage = Kp error Controller voltage Plant

Closed Loop Response (PI Feedback) Proportional/Integral Control No Steady State Error Bigger Overshoot and Settling Saturate counters/op-amps P: Rise Time vs. Overshoot P: Rise Time vs. Settling time I: Steady State Error vs. Overshoot Ref + error - Voltage = (Kp+1/s Ki) error voltage Plant

Closed Loop Response (PID Feedback) Proportional/Integral/Differential Quick response Reduced Overshoot Sensitive to high frequency noise Hard to tune P: Rise Time vs. Overshoot P: Rise Time vs. Settling time I: Steady State Error vs. Overshoot D: Overshoot vs. Steady State Error R + error - Voltage = (Kp+1/s Ki + s. Kd) error voltage Plant

Feed Forward Volt Decouples Damping from PID To compute Try different open loop inputs and measure output velocities For each trial i, Tweak from there. . R + error - + volt + Controller Plant

Follow a straight line with differential drive Error can be difference in wheel velocities or accrued distances Make both wheels spin the same speed asynchronous – false start wheels can have slight differences (radius, etc) Make sure both wheels spin the same amount and speed false start More complicated control laws – track orientation m 1 vref = vref + K 1 * thetaerror + K 2 * offset error modeling kinematics of robot dead-reckoning

Encoders

Encoders – Incremental Photodetector Encoder disk LED Photoemitter

Encoders - Incremental

Encoders - Incremental • Quadrature (resolution enhancing)

Where are we? • If we know our encoder values after the motion, do we know where we are?

Where are we? • If we know our encoder values after the motion, do we know where we are? • What about error?

Encoders - Absolute § More expensive § Resolution = 360° / 2 N where N is number of tracks 4 Bit Example