Design Realization lecture 20 John Canny 10/30/03

Last time  Real-time programming

This time  Mechanics – Physics and Motors

Review of physics  Newton’s law for translation: F = m a F in Newtons, m in kg, a in m/s 2.  Acceleration a = dv / dt  Kinetic energy E = ½ m v 2 E in Joules, m in kg, v in m/s.

Physics of translation  Momentum p = m v and so F = dp / dt  In the absence of force, momentum is conserved.  Momentum conservation implies energy conservation.

Physics of rotation  Rotation is more complex; Euler’s equation: T = I  +  x I  T (torque) in N-m,  in radians/sec,  in radians/sec 2, I in kg-m 2,  = d  / dt  I is a 3x3 matrix, not necessarily diagonal.  If T = 0, then I  = -  x I  which is usually non-zero. So  is non-zero,  changes with time, and the object wobbles.

Physics of rotation  Angular momentum is q = I   The rotation equation simplifies to T = dq / dt because dq/dt = I d  /dt + dI/dt  = I  +  x I   So even though an object wobbles when there is no external force, the angular momentum is conserved: q = I 

Physics of rotation  Kinetic energy of rotation is ½  T I   In the absence of external torque, kinetic energy of rotation is conserved.  But angular momentum conservation does not imply energy conservation.

Work  Work done by a force = F x (Joules) where x is the distance (m) through which the force acts.  Work done by a torque = T  (Joules)

Power  Power is rate of doing work.  Power of a force = F v (Watts).  Power of a torque = T  (Watts).  Power often expressed in horsepower = 746 Watts

Motors  Motors come in several flavors:  DC motors  Stepper motors  (AC) induction motors  (AC) Single-phase motors  (AC) Synchronous motors  The first two are highly controllable, and usually what you would use in an application. But we quickly review the others.

3-phase AC  Three or four wires that carry the same voltage at 3 equally-spaced phases:  Single phase AC requires two wires (only 1/3 the current or power of 3-phase).

AC induction Motors  Induction motors – simple, cheap, high-power, high torque, simplest are 3-phase.  Speed up to 7200 rpm: speed ~ 7200 / # “poles” of the motor.  Induction motors are brushless (no contacts between moving and fixed parts). Hi reliability.  Efficiency high: %

Single-phase AC Motors  Single-phase (induction) motors – operate from normal AC current (one phase). Household appliances.  Single-phase motors use a variety of tricks to start, then transition to induction motor behavior.  Efficiency lower: 25-60%  Often very low starting torque.

Synchronous AC Motors  Designed to turn in synchronization with the AC frequency. E.g. turntable motors.  Low to very high power.  Efficiency ??

DC Motors  DC motor types:  DC Brush motor  “DC” Brushless motor  Stepper motor

DC Brush Motors  A “commutator” brings current to the moving element (the rotor).  As the rotor moves, the polarity changes, which keeps the magnets pulling the right way. DEMODEMO  Highly controllable, most common DC motor.

DC Brush Motors  At fixed load, speed of rotation is proportional to applied voltage.  Changing polarity reverses rotation.  To first order, torque is proportional to current.  Load curve:  Motors which approximate this ideal well are called DC servo motors.

DC Brushless Motors  Really an AC motor with electronic commutation.  Permanent magnet rotor, stator coils are controlled by electronic switching. DEMODEMO  Speed can be controlled accurately by the electronics.  Torque is often constant over the speed range.

Stepper Motors  Sequence of (3 or more) poles is activated in turn, moving the stator in small “steps”.  Very low speed / high angular precision is possible without reduction gearing by using many rotor teeth.  Can also “micro- step” by activating both coils at once.

Driving Stepper Motors  Note: signals to the stepper motor are binary, on-off values (not PWM).  In principle easy: activate poles as A B C D A… or A D C B A…Steps are fixed size, so no need to sense the angle! (open loop control).

Driving Stepper Motors  But in practice, acceleration and possibly jerk must be bounded, otherwise motor will not keep up and will start missing steps (causing position errors).  i.e. driver electronics must simulate inertia of the motor.

Stepper Motor example  From Sherline CNC milling machine:  Step angle: 1.8°  Voltage: 3.2 V  Holding torque: 0.97 N-m  Rotor inertia: 250 g-cm 2  Weight: 1.32 lb (0.6 Kg.)  Length: 2.13" (54 mm)  Power output = 3W  Precision stepper motor: 0.02° /step, 1 rpm, 3W

DC Motor example  V = 12 volts  Max Current = 4 A  Max Power Out = 25 W  Max efficiency = 74%  Max speed = 3500 rpm  Max torque = 1.4 N-m  Weight = 1.4 lbs  Forward or reverse (brushed)  Many DC motors of all sizes available new and surplus for < $10

DC Motors – micro sizes  From Micromo:  Conventional (brush) DC motor: 6mm x 15mm  13,000 rpm  0.11 m Nm  Power 0.15 W  V from 1.5 to 4.5 V

Brushless DC Motors  From Micromo:  Brushless DC motor: 16mm x 28mm  65,000 rpm  50 m Nm  Power 11 W  V = 12 V

DC Motors – gearing  Gearing allows you to trade off speed vs. torque.  An n:1 reduction gearing decreases speed by n, but increases torque by n.  Ratios from 10:1 to many 1000s :1 are available in compact “gearheads” that attach to motors.

DC Motors – gearing  But gears cost efficiency (20% - 50%)  Gears decrease precision (due to backlash).  Reduction gear train is normally not backdriveable (can’t use for “force control”).

DC torque motors  Some high-end motors are available for direct drive servo or force applications (no gears).  They have low speed (a few rpm), high precision (with servo-ing), and moderate torque.  Typically have large diameter vs. length, and use rare-earth magnetic material.  Cost $100’s (but maybe less as surplus).

Sensors  Shaft encoders can be fitted to almost any DC motor. They provide position sensing.  Many motor families offer integrated encoders.  Strain gauges can be used to sense force directly. Or DC brush motor current can be used to estimate force.

Linear movement  There are several ways to produce linear movement from rotation:  Rotary to linear gearing:

Linear movement  Ball screws: low linear speed, good precision  Motor drives shaft, stages move (must be attached to linear bearing to stop from rotating).

Linear movement  Belt drive: attach moving stage to a toothed belt:  Used in inkjet printers and some large XY robots.

True Linear movement  There are some true linear magnetic drives.  BEI-Kimco voice coils:  Up to 1” travel  100 lbf  > 10 g acceleration  6 lbs weight  500 Hz corner frequency.  Used for precision vibration control.

Summary  AC motors are good for inexpensive high-power applications where fine control isnt needed.  DC motors provide a range of performance:  DC brush: versatile, “servo” motor, high speed, torque  DC brushless: speed/toque depend on electronics  Stepper: simple control signals, variable speed/accuracy without gearing, lower power  Direct-drive (torque) motors, expensive, lower torque  Linear actuation via drives, or voice coils.