Force and Torque Amplifiers

Conceptually, an electric motor is a force or torque generator which forms a dynamic pair for the purpose of creating motion. Since force and velocity are closely related to each other, we need to examine their relationship.

Refer to Eq. 4.13. Force acting on a body times the linear velocity of the body describes power in the mechanical domain. In the electrical domain, power is the voltage times the current supplied to an electrical load (e.g.

a resistor). An electric motor is an electromechanical device which takes electrical power as input and produces mechanical power as output. If there is no dissipation of power, the output power must be equal to the input power.

Voltage I 1 Torque

• K Stator •

Electromagnet — ' \ ,

Current ^ ( c o i l s ) _ P o w e r _ ^ > ^ Velocity

Fig. 4.16 Power conversion in an electric motor.

Fig .4.16 illustrates how an electric motor converts power from the elec-trical domain to the mechanical domain. If the input power in the elecelec-trical domain applied to an electric motor is a constant p, torque r times angular

velocity u is a constant as well, that is,

T»UJ =p. (4.34) It is easy to see that the higher the velocity is, the lower the effective torque will be. If an electric motor does not have any inertial load, its ten-dency will be to output a very high angular velocity (thousands of rotations per minute). As a result, the torque which an electric motor can deliver is very low. This is an undesirable feature because the static frictional force between two bodies at rest is usually very large.

In order to trigger the relative motion between two bodies at rest, one needs to apply strong force or torque to overcome the static frictional force.

Therefore, it is necessary to alter the characteristic of an electric motor so that it can deliver the torque at a reasonable range of magnitude. Prom Eq. 4.34, we can see that the only way to increase torque is to reduce velocity if the power is to remain a constant. A device which allows us to reduce velocity without loss of power is commonly known as a speed reducer.

With reference to force or torque, a speed reducer can also be called a force or torque amplifier. That is an indispensable element inside an actuation system.

Fig. 4.17 Illustration of the input-output relationship in a speed reducer or torque amplifier for rotary motion.

Fig. 4.17 shows the relationship between the input and output in a speed reducer, or torque amplifier. To reduce speed or amplify torque related to a rotary motion, we connect this rotational motion to the input shaft of a speed reducer, or torque amplifier. The output motion from a speed reducer, or torque amplifier, will also be a rotational motion which is transmitted through the output shaft. If the velocity of the input motion is LUQ and the velocity of the output motion is w, the important parameter

Electromechanical System of Robots 141

of the speed reducer is the reduction ratio kr, that is,

kr = ^ . (4.35)

IV

If the torque of the input motion is T0, then the amplified torque of the output motion will be

T = kr»r0. (4.36)

In fact, a speed reducer or torque amplifier is a coupling device because it is composed of two independent bodies, as shown in Fig. 4.17. By coupling device, we mean a device which can be physically connected to another pair of bodies without losing any degree of freedom. For example, we can interface a dynamic pair (e.g. a motor) with a kinematic pair (e.g. a pair of links) through a coupling device such as a speed reducer or torque amplifier.

In this way, the degree of freedom of the kinematic pair is preserved.

In the following sections, we will discuss conceptually some examples of speed reducers or torque amplifiers.

4.3.2.1 Gear Mechanisms

A simple solution for speed reduction, or torque amplification, is to use a gear mechanism as shown in Fig. 4.18.

— Input shaft _ _ _ _ _ _ ^ _ ^

' ••;• ' •• ' ^ • E E L J I

Output shaft

(a) Gear mechanism (b) Real example

Fig. 4.18 A gear mechanism for speed reduction or torque amplification.

The input motion is supplied to the shaft of the gear having a smaller diameter, and the output motion is transmitted from the shaft of the gear having a larger diameter. If the numbers of teeth of the larger and smaller

gears are No and Ni respectively, the ratio of speed reduction will be

kr is also the ratio of amplification for the output torque.

However, there are three notable drawbacks with regard to the gear mechanism:

• First of all, the teeth of the two gears are in direct contact. After long-term use, there will be the problem of wear-and-tear. In addition to imprecision of machining and assembly, the undesirable problem of backslash (i.e. hysteresis) becomes inevitable.

• Secondly, the distance between the input shaft and the output shaft depends on the diameters of the two gears. More gears must be added if one wants to alter this distance.

• Thirdly, the ratio of reduction is not very high (normally less than 100).

4.3.2.2 Pulley-and-Timing Belt Assemblies

In order to overcome the first two drawbacks of a gear mechanism, a com-mon solution is to use the pulley-and-timing belt assembly as shown in Fig. 4.19.

Timing belt

Y Input shaft ^^^^^m^^mi

^ ^ i** •* r >HBH

Output shaft M B S H E . -'If

(a) Pulley and timing-belt (b) A real example

Fig. 4.19 Pulley-and-timing belt assembly for speed reduction or torque amplification.

In a pulley-and-timing belt assembly, the timing belt is a flex spline with teeth. The whole timing belt is normally made of soft material (e.g.

rubber). The direct contact of the two pulleys is therefore replaced by the soft contact between the timing belt and the two pulleys. As a result, the effect of wear and backslash is not as serious as that of a gear mechanism.

Electromechanical System of Robots 143

Since the length of the timing belt is selectable, it is easy to alter the distance between the input shaft and the output shaft. Just as with a gear mechanism, the speed reduction is equal to the ratio between the numbers of teeth of the two pulleys. If the number of teeth on the larger pulley is No and the number of teeth on the smaller pulley is Nt, the ratio of speed reduction (or the ratio of torque amplification) will be

k - ^

Again, this ratio is not very high (typically less than 100).

4.3.2.3 Harmonic-Drive Devices

The best device for speed reduction, or torque amplification, is the har-monic drive, as shown in Fig. 4.20.

Wave generator * ^.pl-I" H r V ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ H t - 'f* Input motion \ ^~"~"^.. ^ ^ £ j ^ B B S ^ ^ ^ ^ ^ ^ ^ ^ ^ B I B f r ' . ' a f a :

-earner \ ™ " j ^ ^ H S a P ^ ^ K M ^ ^ ^ ^ ^ r ^ f l B i ^ k ^i 1

Output motion • • . R i g i ( J c i r c u l a r' s p,i n e

carrier

(a) Cross - section view (b) Real example

Fig. 4.20 Harmonic drive for speed reduction or torque amplification.

Refer to Fig. 4.20. A harmonic drive is composed of: a) an elliptical wave generator, b) a flex spline with teeth, and c) a rigid circular spline with teeth, and d) the supporting body. The elliptical wave generator is the input motion carrier. Its axis of rotation is the input shaft. And, the rigid circular spline with teeth is the output motion carrier, if the flex spline is fixed onto the supporting body. Alternatively, the flex spline is the output motion carrier, if the circular spline is fixed onto the supporting body. The axis of rotational motion is the output shaft.

Assume that the output motion carrier is the flex spline. The role of the wave generator is to keep the flex spline in contact with the circular spline at the two ends along the major axis of the elliptical wave generator.

It is easy to see that there will be zero displacement between the flex spline

and the circular spline if they have the same numbers of teeth. Let Nf be the number of teeth on the flex spline and Nc the number of teeth on the circular spline. If Nf = Nc + 1, it is easy to see that the circular spline will shift one tooth after one full rotation of the wave generator. Similarly, if Nf = Nc + 2, the circular spline will shift two teeth after one full rotation of the wave generator. In order to make the flex spline shift Nf teeth, the wave generator has to make Nf/2 rounds of full rotation. If Nf = Nc + 2, the ratio of speed reduction or torque amplification will be

kr = Nf/2.

Interestingly, kr is linearly proportional to Nf. This explains why the ratio of speed reduction of a harmonic drive can be very high (easily over several hundred). Since the contact forces between the flex spline and the circular spline are along the major axis of the wave generator (perpendicular to the tangential direction of each pair of teeth in contact), there is no backslash effect, as long as there is no slippage between a pair of teeth in contact.

The only problem with a harmonic drive is that the input and output shafts are coaxial. If this is not desirable, a good solution is to combine the use of the harmonic drive (for achieving better ratio and no backslash) with the pulley-and-timing belt assembly (for achieving the offset between input and output shafts).