Mechatronics: an introduction

CY3L2-08-8A

W.S. Harwin, January 2009

1. Actuators and transmissions (Power converters)

Measures: Power to weight ratios, efficiency, maximum stress, maximum strain, maximum speed, maximum torque. Repeatability, resolution, accuracy.

Actuators, are usually the package, but consider two variants, transforming one form of energy into another, and one changing the flow and effort relationship across ports.

Ultrasound motors then for example use piezo-ceramics to transduce electrical energy into mechanical energy, then use the elastomer ring to match the linear small displacement to a rotary large displacement.

Within the transducing element there may be different forms of energy storage eg electric motor takes electrical power and converts this to a B-field, and hence to a force.

Two primary variants of operationCyclic output: e.g. muscles/joints, cranes and robot limbs, power steering and tele-operation.

Set-points with perturbation. Conveyor belts, helicopter rotor-blades, internal combustion engines. The guralp seismometers.

Reversible field effected materials• Electrostrictive - Piezo ceramic

• ERFs MRFs (Electro-rheological fluids, Magneto-rheological fluids)

• Magnetostrictive

Reversible electronic and Ionic effected materials• EAPs PAMPs gels etc -possibly muscle

Faraday effect and electrostatic attraction• DC motors

• Micromachines

Reversible thermal changed materials• Liquid crystals, SMAs (Shape memory alloys)

fluid expansion• Pneumatic/hydraulic

Thermal expansion• IC engines, steam engines

Figure 1 energy flow

E_Field B-Field

Passive actuators (controlable dampers

ERF's MRF's

Reversable materials Piezoelectric Magnetostrictive

Maxwell actuators Electrostatic motors (micro machines

Solinoids/Moving coils/Electric motors

Definitions

Efficiency =PoweroutPowerin

volumetric power densityPoweroutVolume

, mass power densityPoweroutMass

passivity, passive if E t =∫0

tf vd E 0∀ t

Passive actuators don't move without external power.

Force/displacement curves

Agonist pairs or Actuator spring pairs

Frequency characteristicsFor mass dominated actuators we can consider a cyclic action that moves both the internal

(intrinsic) actuator mass and the external load mass. Assume that the actuator has a maximum force capacity of Fmax

M=M intrinsicM load if x=Ac sin t F=Ma=−M 2 Ac sin t

Huber and Ashby graphs

2. Cable and belt transmissions

Leather belt transmissions were common between 1830 and 1930 as a means of transmitting power from a drive shaft, that might be connected to a water mill or steam engine, to machines in a workshop. The use of belt transmissions declined with the introduction of electric motors in the 1930’s although belt transmissions still find a place for low power mechanisms.

Cable transmissions have several similarities to belt transmissions and are growing in popularity for robotic applications. These advantages include

Very low backlash - this is a major problem with reversing gears in a transmission mechanism

Small frictional losses in the system – an important consideration if the transmission mechanism is designed to be backdrivable.

Low inertia – an advantage when looking at systems with high accelerations

Large mechanical advantage is possible. In contrast to gears, a cable transmission can be designed to have a very small driver wheel and a very large driven wheel.

Consider the pulley arrangement shown in figure 1. If the torque balance on the input pulley is considered then, if the upper cable is at tension fl and the lower cable is at tension fh then the following formulae can be derived.

f hr in=T in f l r in

f l r out=T out f hr out

These two equations can be combined to derive the ‘gear’ or transmission ratio as

rout

rin

=−T out

T in

=n

Low tension cable fl

High tension cable fh

Power inTin in

Power outTout out

Region of slip

Region of slip

rout

rin

Figure 2 : Cable transmission

We now consider a unit mass of cable or belt going around the two pulleys. This unit mass of cable will have an unstretched length of l0=1/m, where m is the mass/unit length of the cable

While this constant mass is on the high tension side of the transmission it will be stretched by an amount

Δl=f h l0

EA

Where E is the a bulk modulus for a particular cable winding related to the Youngs modulus, and A is the area of the circle that just encompasses the cable.

Likewise on the low tension side of the transmission it will be stretched by an amount

Δl=f l l0

EA

For a boundary around either pulley the mass of cable travelling left to right on the top cable must, in any second, be the same as the mass of cable travelling right to left on the lower cable. Thus the speed of the cable at the higher tension must be

V h=l0

T1 f h/EA

where T is the time taken for unit mass to pass any point. Likewise for the cable at the lower tension

V l=l0

T1 f l /EA

As the cable gets laid onto the pulley it does so in its stretched condition. Thus the above formula can be used to derive a modified ‘gear’ or transmission ratio for the angular velocity of the assembly.

ωin

ωout

=nV h

V l

=n1 f h /EA 1 f l /EA

Thus to get any movement across the transmission a small amount of the mechanical advantage is lost. As the cable leaves a pulley it must accelerate or decelerate while on the pulley so as to extend or contract to the condition of the unsupported cable, that is it must stretch as it is taken

onto the high tension side, and shrink as it is taken onto the low tension side. The angle that this happens over is given by the basic equation for a cable slipping around a drum, that is

θ= 1μ loge f h−loge f l

where is the coefficient of friction between the cable and the pulley.

The efficiency of the transmission can now be estimated from the input and output power.

η= PoutPin

=1 f h /EA 1 f l /EA

≈1−f h− f l

EA

since, in general, EA will be large when compared to the tension in the cable this efficiency is still close to one. It can be seen that to get any transmission of power, the cable must slip as it leaves the pulley. Clearly this does not happen in the similar case of a toothed belt leaving a toothed pulley. Does this mean that toothed belts are a better solution? The answer is no. A toothed belt must go through the same process of stretch and shrink as it goes around the toothed pulley. This will either happen as an instantaneous process as the belt tooth ‘snaps’ away from the pulley, or by the toothed belt riding up on the teeth of the pulley in order to gain enough extension to reach the high tension condition. A similar argument applies as the toothed belt leaves the pulley on the low tension side.

Although other transmission systems are possible in robotics, there are clear advantages to a well designed cable transmission, as evidenced by the WAM (whole arm manipulator) robot manufactured by Barrett Technologies and the PHANToM haptic interface manufactured by Sensable Devices.

Reference:

W.T. Townsend and J.K. Salisbury “The efficiency limit of belt and cable drives” Journal of mechanisms, transmissions and automation in design, 110 pp 303-307

3. Ultrasound MotorsBased on piezoelectric effectBenefits

● High torque at low speeds (no need for gearing)● Silent, lightweight, compact,

However● Poor start torque characteristics

One example is in the autofocus lens motors of cannon cameras

Piezoelectric effect

EitherApply a force to a piezoelectric material to generate an electric field.

orApply an electric field to a piezoelectric material to generate a force

In fig 6.3: Stress in dielectric (the piezoelectric material) is

d=FA

where A is the area of the plates/material.

With no E field this causes a strain

=hh

. Thus Youngs Modulus Myoungs=d

Piezoelectric stress constant gives relationship between material stress and Electric field

E≈Vh

and piezoelectric effect gives

g=Ed

Thus simplified relationship is

d=M youngs −Eg

The piezoactuator also has an associated capacitance and it is possible to resonate the electrical system so as to maximise the E-Field and thus the piezoelectic effect. Note material is anisotropic in both g and Myoungs., so complete figures would relate stress to tensile and shear strains in each axis as well as E-field in each axis.

Ultrasound motor construction and operationThree layers

1. Piezoelectric stack2. Elastic body3. Coupling ring

E.M Field actuatorsElectric Motors

Force on a charge in an Electrical and Magnetic field is as follows

If we assume a wire of cross sectional area So, length l, in a uniform field, B then the total number of charges in the wire is

Also if the mean charge velocity is v then the current in the wire is

Assuming that the wire is only in a magnetic field, then the force due to a current i is

Set up the motor as follows (Two pole motor, 1 winding)

The force on one strand of wire in the armature is thus

Hence the torque generated on one loop is

Typically current passes into the armature via brushes to a number of commuting segments. Thus the torque must be averaged over the angle it is applied

Can calculate the mean torque by integration, let m= number of windings x number of poles/2

Hence motor torque equation is

Faradays induction law

Where a wire is moving in a magnetic field, a voltage is created. This is implicit in Faradays law that states

For our armature We are looking at the area of the loop as projected onto the plane normal to the magnetic field.If the loop Thus the total flux coupled with the coil is

The constant is identical to that for the induced torque, although they are often quoted as different numbers in motor specifications, the torque constant in units of torque/current, and the generator constant as Volts / revolution speed.

The armature also will have a resistance and an inductance when entirely separated from the magnetic field. Hence one approximation for the action of the motor is

Armature shearConsider an ideal motor where the winding causes an average shear over the armature of Overall torque is then T=2 r2 l where r is the armature radius and l is the field washed length. This formula can be rewritten as T=2V where V is the volume of the armature.

motor type Torque/unit rotor volume Stress

KNm /m3 KN /m2

Industrial servo 20-45 10-22

Aerospace machines 45-75 22-38

Liquid cooled generator 130-220 65-110

Types of motor

Permanent Magnet motorsPermanent magnet motors use permanent magnets to provide a constant uniform B field, thus all the terms for the force and velocity equations can be grouped in a single constant.

Pancake motor

Brushless DC motors Brushless motors use a permanent magnet as the rotor and use electronic commutation to determine which of 3n windings to energise.

Shunt wound and series wound motorsSome motors use separate rotor and stator windings connected in either series or shunt (parallel) with the armature to provide the magnetic field. For a series motor the B field is roughly proportional to current (all other factors fixed), so

Stepper and switched reluctance motors

Table 1: Characteristics of DC motors

Motor Type Start

torqueRotor Inertia Cost

Reliability

Electronic commutation

?Comments

Permanent Magnet High Med Low Good No Small size, linear

Brushless DC motor

High Low High very good

Yes Linear, common in small motor applications

Series wound DC

Very high Med Low Good No

Widely used with both AC and DC current supplies

Shunt wound DC

Med Med Low Good No Constant torque for a wide range of velocities

Printed Circuit Motor

High Low High Good NoLow rotor inertia makes it suitable for applications requiring high acceleration

Permanent magnet stepper motor

Med Med Med Good YesGood positioning possible providing torque limit not exceeded

Variable reluctance stepper motor

Low Med Low Good YesGood positioning possible providing torque limit not exceeded

Piezo electric motor

Low Low High Very good

Yes Operates at low speeds without need for gearing