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		a
	A
S	σ	U

repeatability = ±r = 3s accuracy = ( S – U) + e

39.1.2.2 - Control Resolution

•Spatial resolution is the smallest increment of movement into which the robot can divide its work volume. Spatial resolution depends on two factors: the systems control resolution and the robots mechanical inaccuracies. It is easiest to conceptualize these factors in terms of a robot with 1 degree of freedom.

•Control resolution - is determined by the robot’s position control system and its feedback measurement system. It is the controllers ability to divide the total range of movement for the particular joint into individual increments that can be addressed in the controller. The increments are sometimes referred to as “addressable parts”. The ability to divide the joint range into increments depends on the bit storage capacity in the control memory. The number of separate, identifiable increments (addressable points) for a particular axis is,

# of increments = 2n

where n is the number of control bits

•example - A robot with 8 bit control resolution can divide a motion range into 256 discrete positions. The control resolution is about (range of motion)/256. The increments are almost always uniform and equal.

•If mechanical inaccuracies are negligible, Accuracy = Control Resolution/2

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39.1.2.3 - Payload

•The payload is always specified as a maximum value, this can be before failure, or more commonly, before serious performance loss.

•Static considerations,

-gravity effects cause downward deflection of the arm and support systems

-drive gears and belts often have noticeable amounts of slack (backlash) that cause positioning errors

-joint play (windup) - when long rotary members are used in a drive system and twist under load

-thermal effects - temperature changes lead to dimensional changes in the manipulator

•Dynamic considerations,

-acceleration effects - inertial forces can lead to deflection in structural members. These are normally only problems when a robot is moving very fast, or when a continuous path following is essential. (But, of course, during the design of a robot these factors must be carefully examined)

•e.g.

Consider a steel cantilever beam of length L, width B and height H, fixed at one end and with a force P, applied at the free end due to the gravitational force on the load.

P

L

B

H

τ

			δ	=	deflection of beamtip caused by point load
δ		PL3	E =		Youngs modulus = 30×10	6	(psi)
	=	---------
		3EI			BH3
			I	=
					---------- for rectangular beam

12

**Note: this deflection does not include the mass of the beam, as might be important in many cases.

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1a. Gravity Effects (payload)

Say, P = 100(lbs)

L = 60(in)

B = 4 (in)

H = 6 (in)

δ payload = 0.0033 (in)

If accuracy = 0.01 then the gravity effects are less

If accuracy = 0.001 then the gravity effects are too large

Aside: Note that the length has a length cubed effect on the tip deflection,

so if a second similar link was added to the robot, the deflection would increase 8 times, a third link would increase deflection by 81 times.

1b. Gravity effects (robot link mass)

δ	=	ω L4	ω	=	weight	= 0.91	lb
		---------8EI			---------------length		----
							in
	δ link mass =		0.00066 (in)

δ total = 0.0033 + 0.00066 = 0.00396

Aside: If the deflection were too large, then we could use lighter link materials, or larger annular (round tubular) members. Annular members allow actuators, and instrumentation inside.

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2. Drive Gear and Belt Drive Play

assume we are using gears, or timing belts, that do not mesh perfectly

The gears do not mesh perfectly, and the resulting space is ‘D’

The input drive has to move a distance ‘D’ before the output engages, and motion begins (this is often after a direction change). This error is multiplied by the

gear ratio between input gears and the final position of the robot arm. Similar errors occur for chains, belts, and other types of errors.

Aside: Some errors can be taken out of the system by using very precise gearing, or anti-backlash gearing that uses springs to hold the input gear against the drive gear. It is also possible to compensate for this in software.

With good gearing, Backlash can be held to less than 0.010 (in), but special design is required when accuracies of 0.001 (in) are desired.

3. Joint Flexibility - ( the angular twist of the joints, rotary drives, shafts, under the load)

θ	32LT	θ =	twist of the cantilevered link in radians
	= --------------
	π D4G	L =	distance of the applied moment from the fixed end

T = the applied moment

G = the polar moment of inertia

D = the effective diameter of application of the moment

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4. Thermal effects
δ thermal	= α∆ TL		α = coefficient of linear thermal expansion
If for the previous values we consider,
	–6	in
α =	6.5×10	--------	( for steel)
		inF

∆ T = T1(working temp.) – T0(calib. temp.) = 80F – 60F = 20F δ thermal = 0.0078 (in)

Major errors in accuracy can result from thermal expansion/contraction

5. Acceleration Effects

The robot arm, and payload are exposed to forces generated by acceleration.This applies mainly to the payload mass, but also to the link mass. These forces cause bending moments that must be added to the masses considered before.

Fpayload = Mpayloadrpayloadω '

Flink = Mlinkrcentroidω '(approximate)

The robot arm also experiences radial forces due to centripetal forces. These lead to elongation of the arm, but are often negligible.

Fpayload = Mpayloadrpayloadω 2

And, if the centre of rotation moves, we must also consider coriolis forces, these could potentially result in a ‘whip’ effect. This does occur in multilink robots.

δFpayloadL3

=-----------------------

3EI

6.Combine cartesian components of deflection into one vector

δ accuracy =

( ∑ δ xi) 2

+ ( ∑ δ yi) 2

+ ( ∑ δ zi) 2

*** Remember to compare to control resolution