Chapter 10: Specification of Robots
In 1990 the Hubble Space Telescope was carried into orbit, and its first pictures came back blurred. The fault was traced to the great mirror at the heart of the instrument, and what the investigators found is worth thinking about carefully.
The mirror had not been made carelessly. It had been ground and polished by machine to one of the smoothest surfaces ever produced, held to within a few nanometres of the shape the measuring instrument called correct. The trouble lay in the measuring instrument itself. A spacing inside it had been set about a millimetre out of position, and so it reported the wrong shape as the right one. The polishing machine then chased that wrong shape faithfully, over and over, until the outer edge of the mirror sat roughly two micrometres too flat — a few per cent of the width of a hair, and enough to ruin every image the telescope took until astronauts fitted corrective optics three years later.
So here was a machine that was extraordinarily consistent and extraordinarily fine in its movements, and still completely wrong. Fig. 10.0-1 shows the mirror during its manufacture. Consistency, fineness of movement, and correctness are three different properties, and a robot must be judged on all three separately.
Photograph
Fig. 10.0-1 – the Hubble Space Telescope’s primary mirror during manufacture or ground inspection: the full circular mirror face must be visible in a cleanroom setting, with technicians in protective suits alongside it to give a sense of its scale. Sourcing note: an archive photograph from the space agency or the mirror’s manufacturer showing the real mirror before launch; artist’s impressions, computer renderings, and in-orbit views of the assembled telescope are unusable.
This chapter will answer all five of these questions.
10.1 Specifying a Robot
A robot is almost always chosen before anyone has seen it doing the job it is being bought for. Someone must decide, from a printed sheet of numbers, whether a particular machine can place a component on a circuit board or lift a sack of grain onto a stack. That sheet of numbers is the robot’s specification.
A specification is not advertising. Every entry on it is a measurable claim, made under stated conditions, and it can be checked. If the sheet says that the arm returns to a taught point to within ±0.02 mm, that claim can be tested by sending the arm to the point a few hundred times and measuring where it actually stops each time.
10.1.1 Why a Robot Is Chosen by Its Numbers
Two robotic arms standing side by side may look almost the same and yet be built for entirely different work. Watching them move tells you very little, because both will move smoothly and both will stop where they were told to stop, as far as your eye can judge. The difference lives in quantities too small or too subtle to see.
Some of those quantities decide whether the robot can reach the work at all. Its reach fixes how far the end-effector can travel from the base. Its degrees of freedom fix how many independent ways the arm can move, and therefore whether it can approach a part from the required direction. Its payload — the mass the arm is rated to carry at its mounting flange, gripper included — fixes whether it can hold the part once it gets there.
The remaining quantities decide whether the work will be acceptable when the robot does reach it. A machine stacking sacks in a warehouse needs to lift 30 kg and does not care about a stray millimetre. A machine placing a resistor on a circuit board needs to be right to a fraction of a millimetre and will never be asked to lift more than a few grams. Neither could do the other’s job, and no single number captures that.
Why size and precision pull against each other
A robot rated for a heavy payload usually quotes a poorer positioning figure than a small assembly robot, and this is not carelessness on the manufacturer’s part. Heavier links deflect more under load, larger gearboxes carry more play between their teeth, and any small angular error at a joint is magnified by the length of the arm beyond it.
10.1.2 Resolution, Accuracy and Repeatability
In ordinary speech we call a careful machine “precise” and leave it at that. Engineering cannot afford a word that vague, because a robot’s positioning behaviour raises three separate questions, and a machine can answer one of them well while answering another badly.
The first question is how finely the robot can move at all. Every robot moves in steps, and below a certain step size it simply cannot go — the sensors cannot detect a smaller change, and the controller cannot command one. That smallest step is the robot’s resolution.
The second question is whether the robot goes where it was told. When the controller is commanded to place the end-effector at a particular point, the end-effector arrives somewhere, and the gap between the commanded point and the point actually reached is what accuracy describes.
The third question is whether the robot goes to the same place every time. Sent to the same point again and again under the same conditions, a robot does not stop at exactly the same spot each time; how tightly those stopping points cluster together is its repeatability.
Three questions, three answers
These three properties are independent, and it helps to picture them as shots at a target, as in Fig. 10.1-1. Shots may fall in a tight little cluster well away from the centre, which is good repeatability with poor accuracy. They may scatter widely but average out around the centre, which is poor repeatability with fair accuracy. Only when they fall in a tight cluster on the centre is the machine both accurate and repeatable.
Diagram
Fig. 10.1-1 – three circular target boards drawn side by side, each with a marked centre labelled “commanded point” and six small crosses marking where the end-effector actually stopped. Board (a): the six crosses in a tight cluster near the upper left edge, well off centre — captioned “repeatable but not accurate”. Board (b): the six crosses widely scattered but spread evenly around the centre — captioned “accurate on average but not repeatable”. Board (c): the six crosses in a tight cluster on the centre — captioned “accurate and repeatable”. A short note beneath the three boards reading “resolution is a different property: it fixes the smallest movement the robot can make between one shot and the next”.
Resolution sits apart from the other two, because it is not about where the shots land. It is about the fineness of the movements available to the machine in the first place, rather like the difference between a ruler marked only in centimetres and one marked in millimetres. A robot cannot correct an error smaller than the smallest step it is able to take.
10.1.3 Reading a Specification with Its Conditions
A performance number with no condition attached to it is not information. A repeatability figure measured with an empty gripper is not the figure you will get with a 5 kg part in the gripper; a figure measured on a machine that has been running for an hour is not the figure you will get in the first five minutes after switch-on. Table 10.1 shows an extract from the kind of sheet a supplier provides, with the condition written beside every value.
Read the table for what it does not say as much as for what it does. Repeatability is quoted; accuracy is not quoted at all, and that omission is normal across the industry. Repeatability can be measured on the factory floor with the robot’s own taught points, while accuracy has to be measured against an external standard, and it depends on how well each individual machine’s internal model matches its real geometry.
The rest of this chapter takes the three quantities in turn, examining what each one depends on, how far modern machines have pushed it, and what engineers do when the answer is not good enough.
10.2 Resolution
A kitchen scale that reads in whole grams cannot tell you that you have added half a gram of salt. It is not that the scale is wrong; it simply has no way of expressing an amount that small. Every measuring or moving system has such a limit, and in robotics that limit has a name.
10.2.1 The Smallest Incremental Movement
A robot does not move through a smooth, unbroken range of positions. Its sensors report position in discrete counts, its controller works in discrete numbers, and its drives are commanded in discrete increments. Between one available position and the next there is a gap, and the robot cannot stop anywhere inside that gap.
That gap is the robot’s resolution. If a joint can be commanded to 30.000°, 30.001° and 30.002° but to nothing between them, the resolution of that joint is 0.001°, and asking for 30.0005° will get you one of the neighbouring values.
Notice the two verbs in that definition, because they describe two different limits. Detecting a movement is the job of the sensors: if the position sensor on a joint cannot register a change, the robot has no knowledge that the joint moved at all. Achieving a movement is the job of the controller and the drive: even a change the sensor can see is useless if the controller cannot command a step that small. The robot’s working resolution is set by whichever of the two is coarser.
Fine steps are not the same as correct steps
It is worth separating resolution from the other two properties at once. A machine with very fine resolution can still be sent to the wrong place; it will simply get there in very small steps. Resolution says nothing about whether the robot’s idea of a position matches reality. It only says how finely that idea can be divided.
10.2.2 Resolution at the Joint and at the End-Effector
A rotary joint’s resolution is an angle, usually quoted in degrees. What matters to the work, though, is not the angle but the distance the tool moves, and those two are not the same number. A tiny rotation at the shoulder swings the gripper through a much larger distance than the same rotation at the wrist.
The link between them is the arc a rotating link sweeps out, shown in Fig. 10.2-1. For a small angle \(\theta\) measured in radians and a distance \(r\) from the joint axis to the point of interest, the arc length is
\[s = r\theta\]
where \(s\) is the distance travelled by that point in millimetres and \(r\) is measured in millimetres. Angles on a specification sheet are given in degrees, so they must first be converted, using \(1° = \pi/180\) rad.
Diagram
Fig. 10.2-1 – a single rigid link pivoting about a fixed revolute joint; the joint axis marked as a small circle at the left with the label “joint axis”, the link drawn as a straight bar of length r running to the right with r dimensioned along it, and the end of the link labelled “end-effector”; a second faint copy of the link drawn rotated by a small angle above the first, with the angle between them marked theta and the short arc between the two link tips marked s; a note beside the arc reading “s = r theta, with theta in radians”
The same joint on a longer arm therefore gives a coarser result: at 1.2 m the step becomes about 0.21 mm. This is also why a robot’s resolution at the end-effector is not a single fixed number. As the arm folds and extends, the distance from each joint axis to the tool changes, and the finest step the tool can take changes with it — best when the arm is drawn in, worst when it is stretched out at full reach.
10.2.3 How the Sensors Set the Resolution
A robot knows its own position only through its sensors. Whatever change the sensors cannot see, the controller cannot know about, and the actuators will never be told to correct. The sensing chain therefore places a hard ceiling on resolution that no amount of clever control can lift.
Most position sensors on a robot end up as digital numbers, either because the sensor is digital to begin with or because an analogue-to-digital converter (ADC) turns a voltage into a count. The converter divides the whole measuring span into a fixed number of levels and reports which level the signal falls in. This dividing up of a continuous quantity into fixed levels is called quantisation, and the number of levels available is set by the bit depth — the number of binary digits in the converted value.
A converter of \(n\) bits provides \(2^n\) levels. An 8-bit converter offers 256 levels, a 10-bit converter 1024, and a 12-bit converter 4096. Reading a 0 V to 5 V signal with a 10-bit converter, one count is worth \(5\ \text{V} / 1024 = 4.88\) mV, so a change smaller than that leaves the reported number unchanged.
That result is worth pausing over. The mechanical parts of the two arms could be identical; the difference of nearly thirty times comes entirely from the choice of position sensor.
What else limits a sensor’s resolution
Bit depth is only the arithmetic limit. Three physical properties decide whether the last few bits mean anything at all.
The sensitivity of a sensor is how much its output changes for a given change in the quantity being measured. A sensor whose output barely moves as the joint turns will have its small movements lost among the levels of the converter, however many levels there are.
Electrical noise is the random fluctuation added to a signal by the sensor, the wiring and the power supply. If the reading wanders by a few counts on its own while nothing is moving, then those counts carry no information, and the sensor’s useful resolution is worse than its bit depth suggests.
Linearity describes how evenly the sensor’s output tracks the quantity it measures across the whole range. Poor linearity means that one count is worth a different amount of movement at one end of the travel than at the other, so a single resolution figure no longer describes the sensor’s behaviour everywhere.
10.2.4 How the Control System Sets the Resolution
Suppose the sensors are excellent and report a deviation of five micrometres. Nothing improves unless the control system can compute a correction that small and issue a command that small. The controller sets the second ceiling on resolution, and the finer of the two ceilings never rescues the coarser one.
Four properties of the control system matter here.
The command itself is quantised. Whatever the controller calculates must be delivered to the drive as a finite number — a count sent to a motor drive, a step to a stepper driver, or a duty cycle. The step size of that command fixes the smallest motion the drive can be asked for, in exactly the way bit depth fixes the smallest change a sensor can report.
The controller update rate is how many times per second the loop reads the sensors, computes the error and issues a new command. A loop that runs every 10 ms cannot respond to anything that happens and passes within 10 ms; between updates the robot is running on its last instruction.
The quality of the feedback path decides how much of the sensor’s information actually reaches the correction. A closed-loop scheme, of the kind studied earlier, compares measured position with commanded position continuously, and its ability to shave off the last few micrometres depends on how well its control algorithm is matched to the machine.
Finally, latency is the delay between a change happening and the system responding to it, and jitter is the variation in that delay from one cycle to the next. Steady, low latency lets a controller correct a deviation while it is still small. Jitter is more troublesome than plain delay, because a correction that arrives at an unpredictable moment can overshoot, and the motion becomes rough rather than fine.
The coarsest link governs
Put the whole chain together, as in Fig. 10.2-2, and a useful engineering rule appears. Movement passes through the sensor, the converter, the controller, the drive and the mechanism, and each stage has its own smallest step. The system’s resolution is set by the coarsest stage in that chain.
Diagram
Fig. 10.2-2 – flow: actual joint movement → position sensor (smallest change it can detect) → analogue-to-digital conversion (one count) → controller computation and update rate → command quantisation at the drive → actuator and gearbox (backlash and step size) → actual movement of the end-effector, with the last arrow returning to the position sensor to close the loop; beneath the row of blocks a single bracket spanning all of them labelled “system resolution is set by the coarsest stage”
This is why fitting a costlier sensor to a robot whose gearbox has half a degree of play changes nothing that matters. The money buys a finer measurement of a joint that still cannot be positioned finely.
10.2.5 Where High Resolution Is Needed
Resolution requirements differ by orders of magnitude from one kind of work to another, as Table 10.2 shows. The point of the table is not the individual figures, which vary widely between models, but the size of the gap between the top and the bottom of it.
The lower half of that table is the world of precision electronics and medical work, where a positional error of a few micrometres either scraps the product or endangers a patient. Chip manufacture is the extreme case, since the circuit features being aligned are themselves only a few tens of nanometres across.
High resolution is not free, and there are good reasons not to buy more of it than the task needs. Finer sensors cost more and demand cleaner power and better shielding to keep noise below one count. Faster control loops need more computing at every cycle. And, as noted above, none of it survives a mechanism with play in it, so the structure and the gearing must be improved alongside the electronics.
10.3 Current Trends in Resolution
Section 10.2 ended with a rule: the coarsest stage in the chain sets the resolution of the whole robot. Three developments in modern practice attack that rule from different directions — better position sensors at the joints, machines engineered to work below a micrometre, and cameras that watch the work itself rather than the machine.
10.3.1 Higher Resolution Encoders
10.3.1.1 What an Encoder Does
A motor turning a joint has no way of announcing how far it has gone. Something must watch the shaft and turn its rotation into a number the controller can read, and that device is an encoder.
The encoder is the reason a robot arm can work in the dark with no idea where its gripper is in the room. It does not see the gripper. It measures each joint angle, and the position of the gripper follows from those angles and the known lengths of the links.
Two consequences follow, and both matter for the rest of this chapter. First, the encoder’s resolution places a direct ceiling on the joint’s resolution. Second, an encoder reports only what the shaft it is attached to has done. Anything that happens after that shaft — play in the gearbox, flex in the link — is invisible to it.
10.3.1.2 The Optical Rotary Encoder
The commonest encoder on a robot joint works with light. A thin disk is fixed to the rotating shaft, and around its rim the disk carries alternating transparent and opaque segments, like the teeth of a very fine comb. A light source sits on one side of the disk and a photodetector on the other, as shown in Fig. 10.3-1.
Diagram
Fig. 10.3-1 – optical rotary encoder in exploded side view: the motor shaft at the left carrying a circular disk drawn edge-on and again face-on beside it, the face-on disk showing alternating clear and dark segments evenly spaced around its rim; a block labelled “light source (LED)” on one side of the disk and a block labelled “photodetector” directly opposite on the other side, with a dashed light ray drawn between them passing through one clear segment; to the right, the detector output drawn as a square wave labelled “output pulses”, with one complete high-low cycle marked “one segment”
As the shaft turns, each transparent segment lets the light through and each opaque segment blocks it. The photodetector therefore sees the light switch on and off, and its output is a train of square pulses. Counting the pulses gives the angle turned; counting them against time gives the speed.
A single detector cannot tell clockwise from anticlockwise, because the pulse train looks identical either way. A second detector is therefore placed so that its pulse train is a quarter of a cycle out of step with the first. This arrangement is called quadrature, and Fig. 10.3-2 shows why it works: whichever channel changes first tells the controller which way the shaft is going.
Diagram
Fig. 10.3-2 – quadrature output: two square waveforms drawn one above the other on a common time axis, labelled “channel A” and “channel B”, with B shifted a quarter of a cycle to the right of A; on the left half of the figure, above the pair, an arrow labelled “clockwise: A leads B”; on the right half, the same two waveforms with B shifted a quarter cycle to the left instead, labelled “anticlockwise: B leads A”; the rising edges that decide the direction marked with small dots
10.3.1.3 The Absolute Encoder and Its Angular Resolution
The encoder described so far counts changes. Switch such a device off and on again and it has a count, but no idea what the count means, because it never knew where it started. This kind is called an incremental encoder, and a robot fitted with one has to be driven to a known reference position after every power-up before it can be trusted.
An absolute encoder removes that ritual. Instead of one track of alternating segments, its disk carries several concentric tracks, each with its own pattern, read at the same instant by a row of detectors aligned across them, as in Fig. 10.3-3. Every angular position of the disk therefore produces its own unique combination of light and dark across the detectors — a distinct digital code for every position.
Diagram
Fig. 10.3-3 – absolute encoder disk drawn face-on with four concentric ring tracks; the innermost track divided into two halves, one clear and one dark; each successive track outward divided into twice as many alternating clear and dark sectors as the track inside it; a radial line drawn across all four tracks marked “detector array”, with one small detector symbol on each track and the pattern of light and dark at that radial line written beside it as a four-digit code; the shaft at the centre labelled “to joint shaft”
Each track contributes one binary digit, so an encoder with \(n\) tracks distinguishes \(2^n\) positions spread over one full turn. The smallest angular step it can distinguish is therefore
\[\text{Resolution} = \frac{360°}{2^n}\]
where \(n\) is the number of tracks, or equivalently the number of bits in the encoder’s output.
That example carries the whole trend in a single line. A ten-track disk on its own is far too coarse for a robot joint, and if it sat directly on the joint axis the tool would jump in steps of millimetres. Two things rescue it: the reduction gearing between motor and joint, and disks with far more tracks. Table 10.3 shows how quickly the step shrinks as tracks are added.
Each extra track doubles the number of codes and halves the step, which is why encoder specifications climb in bits rather than in degrees. Modern industrial servo motors commonly carry encoders of seventeen bits or more, and it is this steady increase in bit depth that the syllabus calls the trend towards higher resolution encoders.
10.3.1.4 Encoders in Electronics and Biomedical Manufacturing
Two industries have pushed encoder resolution harder than any other, and the syllabus names both.
In electronics manufacturing, the parts being handled have shrunk faster than anything else in engineering. A placement machine seating a component whose body is a millimetre long must put it down on pads a fraction of a millimetre wide, thousands of times an hour. The head that carries it is guided by encoders on the drive screws of each axis, and the machine can only place a part as precisely as those encoders can report where the head is. In wafer handling, where successive circuit layers must be laid over one another, the same argument applies at an even finer scale.
In biomedical manufacturing, the driver is not throughput but consequence. Devices such as stents and catheters are assembled from parts too small to adjust by hand, and a positioning error becomes a device that fails inside a patient. Encoders provide the position feedback that lets the machine hold a tolerance it cannot see, and they give something equally valuable: a record, cycle after cycle, that each device was made the same way as the last.
Both cases show the same reasoning. Higher resolution encoders are not bought because finer numbers are attractive; they are bought because the tolerance of the work fell below what the previous generation of sensors could resolve.
10.3.2 Sub-Micron Resolution
10.3.2.1 What a Micron Demands of a Machine
A micron, written properly as a micrometre, is one millionth of a metre: \(1\ \mu\text{m} = 10^{-6}\ \text{m} = 0.001\) mm. A human hair is roughly fifty to a hundred micrometres across, so a micrometre is a small fraction of the width of a hair.
Sub-micron resolution means the ability to detect or perform movements smaller than one micrometre. Reaching it changes what counts as a disturbance. At millimetre tolerances, a warm workshop is irrelevant; at sub-micron tolerances a change of one degree in temperature stretches a half-metre aluminium link by more than ten micrometres, which is ten times the tolerance. Vibration from a machine in the next bay, a draught across the frame, and the heat of the robot’s own motors all become sources of error.
The last clause is not decoration. A machine specified to sub-micron resolution achieves it only in the environment it was designed for, which is why such robots live in temperature-regulated, vibration-isolated rooms rather than on an open factory floor.
10.3.2.2 Semiconductor Manufacturing
The industry that needs this most is the manufacture of integrated circuits. Chips are built on a wafer, a thin circular slice of very pure crystalline silicon that serves as the base on which the circuits are formed. A wafer passes through the same machine many times, and each pass adds one layer of the circuit.
The critical requirement is alignment between those layers. Photolithography — the process of transferring a circuit pattern onto the wafer’s surface using light — lays down features far smaller than a micrometre. If one layer is placed even a fraction of a micrometre out of register with the layer below, the connections between them do not meet and the chips on that wafer are scrap.
Robotic handling appears at every stage. Robots load wafers into vacuum chambers and hold them still while material is etched away or deposited as a thin film; if the wafer shifts during the process, the layer is uneven. Robots present wafers to inspection instruments that examine them for defects. Robots carry finished dies into packaging, where connections a few tens of micrometres apart must be made without touching their neighbours. In none of these steps is a millimetre-class machine of any use.
10.3.2.3 Other Sub-Micron Applications
Sub-micron work is not confined to chips. Biomedical device manufacture uses it for the same reason as electronics — implantable devices, micro-needles and the laser machining of delicate components leave no room for a positioning error that a person could see.
Optical assembly is a third case, and an instructive one. Lenses, fibres and laser components must be aligned to a fraction of the wavelength of the light passing through them, and a misalignment of a micrometre is enough to distort the beam. Here the tolerance comes not from a manufacturing convention but from physics.
Precision alignment work in aerospace assembly, such as seating sensors and integrating satellite components, sits in the same class: the instrument’s later measurements are only as trustworthy as the alignment it was built with.
10.3.3 Integrated Vision Systems
10.3.3.1 What an Integrated Vision System Is
Every sensor discussed so far measures the robot. An encoder reports the angle of a shaft; from all the shaft angles the controller calculates where it believes the tool to be. That belief can be wrong for reasons no encoder can detect — a link that has flexed, a gear that has worn, a workpiece that is not quite where the fixture was supposed to hold it.
An integrated vision system attacks that gap by measuring the world instead of the machine.
The word “integrated” is doing real work in that definition. A camera that merely records the workcell for later viewing is not part of the control system. Here, images are captured, processed and turned into a correction inside the control loop, as Fig. 10.3-4 shows.
Diagram
Fig. 10.3-4 – flow: camera views the end-effector and the workpiece together → image processing extracts the actual positions of both → comparison with the desired relative position → correction computed by the controller → command to the joint drives → arm moves → the camera views the new positions, with the last arrow returning to the first block to close the loop; the joint encoder feedback drawn as a separate smaller loop returning from the arm to the controller, labelled “inner loop: joint angles only”; a note beside the camera block reading “measures tool relative to workpiece”
10.3.3.2 Resolution Enhancement through Visual Feedback
The gain from vision is easiest to see through an example. Suppose an arm has been commanded to place a component on a board, and every encoder reports exactly the angle it was asked for. The joints are where they should be, so the inner loop is satisfied and reports no error at all. But the link has warmed and lengthened by a few micrometres, and the tool is not where the controller believes.
A camera watching the tool and the board together sees that discrepancy directly, because it is measuring the one relationship that matters — where the tool is relative to the work. The controller then issues a small correction, and the placement is right despite an error its own joint sensors could never have found.
This is why vision is described as enhancing resolution rather than merely adding a sensor. The effective step size of the system is no longer set by the encoders alone; deviations below what the encoders can detect are picked up optically and corrected in real time, which is what the syllabus means by real-time adjustments based on visual feedback.
Vision brings its own limits, and they should be stated plainly. Processing an image takes time, so a vision correction arrives later than an encoder reading and cannot chase a fast disturbance. The result depends on lighting, on the surface finish of the part and on keeping the lens clean. And the camera must be able to see the tool and the work at the moment of interest, which a gripper or a spray of coolant may prevent. Vision extends what the joint sensors can do; it does not replace them.
10.4 Accuracy
Resolution tells you how finely a robot can move. It says nothing about whether the place it moves to is the place you asked for. That second question is the subject of this section, and it is the harder of the two to answer, because it cannot be settled from inside the machine.
10.4.1 Closeness to the Commanded Position
When a controller is told to put the end-effector at a particular point, it works out the joint angles that its internal model says will get the tool there, drives the joints to those angles, and reports the move complete. The tool is now somewhere. Whether that somewhere is the point you asked for is a separate matter entirely.
Two parts of that definition are easy to skip over and should not be. The positions must be expressed in the same frame, because a point means nothing until the frame it is measured in is named — a fact established when frames were introduced earlier in this course. And the conditions must be stated, because a robot carrying its rated load at full speed does not position its tool as well as the same robot moving slowly and empty.
In everyday speech, “accurate” and “precise” are used as though they were the same word. In engineering they are not. Accuracy is about being right; the tightness with which a machine repeats itself is a different property, dealt with in section 10.6.
10.4.2 Measuring the Accuracy Error
The gap between the commanded point and the achieved point is a distance in space, and it is measured as the straight-line distance between them. This quantity is the accuracy error, illustrated in Fig. 10.4-1.
Diagram
Fig. 10.4-1 – a set of three-dimensional axes labelled x, y and z with the origin marked “base frame”; a point marked C labelled “commanded position” and a second point marked A a short distance away labelled “achieved position”; a straight line joining C to A labelled E; dashed lines from C and A dropped to show the three separate differences along the axes, marked as the differences in x, in y and in z; a note reading “E is the straight-line distance between the two points”
If the commanded point has coordinates \((x_c, y_c, z_c)\) and the achieved point has coordinates \((x_a, y_a, z_a)\), all in millimetres and all in the same frame, then
\[E = \sqrt{(x_a - x_c)^2 + (y_a - y_c)^2 + (z_a - z_c)^2}\]
where \(E\) is the accuracy error in millimetres. Squaring each difference removes its sign, so an error to the left counts the same as an equal error to the right, and the square root returns a single distance rather than three separate components.
A single measurement of this kind describes one point only. A robot’s accuracy is not one number valid everywhere in its reach, so a proper statement of accuracy comes from measuring many points spread across the working region and quoting the largest error found, or a band that covers them all.
10.4.3 Positional, Angular and Path Accuracy
Reaching the right point is not always the whole of the requirement. Consider a welding torch. It must arrive at the seam, it must be tilted at the correct angle to the joint, and while welding it must travel along the seam and not wander off it. Those are three different demands, and accuracy is classified accordingly.
Positional accuracy concerns the location of the end-effector — how close the tool point comes to the target coordinates. This is the accuracy error just calculated.
Angular accuracy concerns the orientation of the end-effector — whether the tool is tilted and turned as instructed, quite apart from where its tip is. A drill can have its tip exactly on the marked spot and still be leaning by two degrees, and the hole it makes will be wrong.
Path accuracy concerns the whole journey rather than its endpoints — how closely the route actually travelled follows the route intended. Positional accuracy is judged at discrete points; path accuracy is judged continuously along the motion, so a robot may hit both ends of a stroke perfectly and still bow away from the intended line in the middle.
Fig. 10.4-2 shows the three side by side.
Diagram
Fig. 10.4-2 – three panels. Panel (a) “positional accuracy”: a target point marked with a cross and the tool tip drawn a short distance away, the gap dimensioned and labelled in mm. Panel (b) “angular accuracy”: a tool drawn at the correct point but tilted, with the intended tool axis drawn as a dashed line and the actual tool axis as a solid line, the angle between them marked in degrees. Panel (c) “path accuracy”: a dashed curve labelled “intended path” running between a start point and an end point, and a solid curve labelled “actual path” that meets it at both ends but bows away in the middle, with the widest gap dimensioned and labelled in mm
Path accuracy deserves a word of its own, because it is the one that worsens most sharply with speed. Following a curve requires several joints to move at once, each speeding up and slowing down on schedule, and any joint that lags leaves the tool off the line. Inertia, vibration and a changing load all pull the tool away from the path, and correcting for them in time needs a fast control loop.
10.4.4 The End-Effector’s Ability to Reach the Desired Location
Whatever happens inside the machine, a robot is judged at one place only: the working end. The end-effector may be a gripper, a welding torch, a cutting tool, a camera or a surgical instrument, and the practical measure of accuracy is how precisely that tool can be placed where it is wanted.
The point on the tool that the controller actually tracks is called the tool centre point (TCP) — the tip of a welding wire, the centre of a gripper’s closed jaws, the focus of a laser head. Every commanded position refers to this point, and the desired location for it is stated in a chosen frame: the robot’s base frame for a position fixed relative to the machine, or a workpiece frame for a position fixed relative to the part being worked on.
The consequences of missing that location are set by the task, not by the size of the error. A tenth of a millimetre is nothing to a machine stacking cartons and enough to leave a component bridging two pads on a circuit board. In surgical work, a deviation well under a millimetre can carry a tool into tissue that was meant to be untouched. In drilling an aircraft panel, a hole slightly out of place weakens the joint it was meant to strengthen. The same number is trivial or unacceptable depending entirely on what is being made.
10.4.5 What Limits the Accuracy of a Robot
Six things stand between a commanded position and the position actually reached, and it is worth seeing how different they are from one another.
The first is the kinematic model — the set of link lengths, joint axis directions and offsets that the controller uses to calculate where the tool is from the joint angles. Every real machine differs a little from its drawing, and wherever the model disagrees with the metal, the calculation is wrong even though the joints are exactly where they were commanded.
The second is the zero position of each joint. The angle an encoder calls zero may not be the joint’s true mechanical zero, and such an offset shifts every subsequent movement of that joint by a fixed amount, which then accumulates down the chain of links.
The third is sensor resolution, discussed in section 10.2: an error smaller than the smallest detectable step cannot be found, let alone corrected.
The fourth is mechanical stiffness. Links and joints deflect under load, so an arm holding a heavy part droops slightly, and the droop is larger when the arm is extended than when it is folded in.
The fifth is the control system, whose speed and stability decide whether a deviation is corrected while it is still small.
The sixth is the environment — temperature changes that alter the length of links, vibration reaching the frame from nearby machinery, and forces applied by the process itself.
Why most of this can be corrected
Look again at that list and one feature stands out. A wrong link length, a joint zero offset, a droop under a known load — none of these varies randomly from run to run. Given the same command in the same conditions, they produce the same error every time. An error of this kind is called a systematic error, and a systematic error, once measured, can be modelled and subtracted.
That single observation is what the whole of section 10.5 is built on. It also explains something that puzzles people meeting these ideas for the first time: a robot can be consistent to a hundredth of a millimetre and still be a millimetre away from where it was told to go, cycle after faithful cycle.
10.5 Current Trends in Accuracy
Section 10.4 closed on a useful observation: most of what spoils a robot’s accuracy is systematic. The same command in the same conditions produces the same error, which means the error can be measured, described and cancelled. Three lines of work follow from that idea — measuring the machine’s true geometry and correcting its model, correcting for the changes that happen while the machine runs, and measuring the tool’s real position from outside the robot altogether.
10.5.1 Enhanced Calibration Techniques
10.5.1.1 Sources of Error in Joints and Links
In the controller’s model of the arm, every link has exactly the length on the drawing, every joint axis points exactly where the design says, and every encoder reads zero at the joint’s true zero. No real machine is built that way. Parts are made to tolerances, they are assembled by hand, they are bolted to a floor that is not perfectly flat, and they wear.
Four differences between the model and the metal account for most of the resulting error, and Fig. 10.5-1 shows the first three of them on a simple arm.
Diagram
Fig. 10.5-1 – three small panels, each showing the same two-link arm with a fixed base. Panel (a) “joint zero offset”: the arm drawn where the controller believes it is, in dashed outline at 0°, and drawn again in solid outline rotated by a small angle, with the gap between the two marked and labelled “encoder zero not at true mechanical zero”. Panel (b) “link length deviation”: the first link drawn in dashed outline at its design length, marked l, and the same link in solid outline slightly longer, with the small extra length dimensioned and labelled “actual length differs from design”. Panel (c) “axis misalignment”: the second joint’s axis drawn as a dashed line perpendicular to the link, and the actual axis drawn as a solid line tilted by a small angle from it, with the angle marked and labelled “assembly tilt”
A joint zero offset means the position the encoder calls zero is not the joint’s true mechanical zero. Every movement of that joint is then shifted by a constant angle, and because each link hangs from the one before it, the shift is carried down the whole chain to the tool.
A link length deviation means a link is not the length the model assumes. A difference of one or two millimetres in a long link sounds trivial and is not, because it displaces every point the tool is sent to.
An axis misalignment means a joint axis does not point where the design intended, either twisted about the link or tilted away from the perpendicular. This distorts the shape of the whole motion, so the error changes as the arm moves rather than staying constant.
Backlash is the small amount of free play between meshing gear teeth, or in bearings and couplings. When a joint reverses direction, the drive must take up this play before the joint begins to move, so a small commanded movement produces no movement at all, and the joint arrives at a slightly different place depending on which way it approached.
Two further sources are not fixed properties of the assembly at all. Links and joints flex under load, so the arm droops by an amount that depends on what it is carrying and how far it is stretched out. And metal expands as it warms, so the effective link lengths change through a working shift. These belong to the next subsection rather than to calibration, for a reason that will become clear.
10.5.1.2 The Calibration Process
Calibration is the work of finding out what the machine’s geometry really is and correcting the controller’s model to match.
Notice what is being changed and what is not. No part of the robot is dismantled, straightened or replaced. Only numbers inside the controller change. The arm was always capable of reaching the right point; it simply did not know which joint angles would take it there.
Step 7 is done by a mathematical fitting technique, most often least squares, which chooses the parameter values that minimise the total squared error across all the poses. Step 10 is the step most often skipped and the one that matters most, since a calibration that has not been checked against fresh poses is only a claim.
Diagram
Fig. 10.5-2 – flow: drive the arm to a known pose → record the joint encoder readings → measure the true end-effector position with an external instrument → compare with the position predicted by the kinematic model → estimate corrections to the model parameters → load the corrected model into the controller → measure a fresh set of validation poses → decision point labelled “errors within tolerance?”; the “yes” arrow leaving to a final block labelled “calibration accepted”, the “no” arrow returning to the first block
Enhanced calibration, the trend the syllabus names, is this procedure taken further than it once was: more measured poses, instruments that measure to micrometres, models that include axis tilts and twists rather than link lengths alone, and corrections for the deflection the arm suffers under a known load. A six-axis industrial arm whose accuracy is around ±1.0 mm as delivered can typically be brought to a few tenths of a millimetre by a full kinematic calibration of this kind.
10.5.1.3 What Calibration Can and Cannot Correct
Calibration corrects systematic geometric error, and it is remarkably good at it. It cannot touch anything that varies from one cycle to the next.
Backlash is the clearest example of the boundary. Its size can be measured, but which way it acts depends on the direction from which the joint approached, so a single corrected link length cannot absorb it. Deflection under load is a second example: it is systematic for a stated payload, and a calibration performed with an empty gripper does not describe the arm carrying a five-kilogram part. Temperature is a third, since it changes while the machine works.
There is also something calibration cannot do at all, and it is worth stating plainly: calibration improves accuracy but does not improve repeatability. It changes the numbers the controller computes with, not the mechanical play, flexing or noise that make one cycle differ from the next. A robot that returns to a taught point within ±0.02 mm before calibration will still return to within ±0.02 mm afterwards — but the point it returns to will now be much closer to the point that was asked for.
10.5.2 Dynamic Error Compensation
10.5.2.1 Sources of Dynamic Error
A calibrated robot is correct for the conditions in which it was calibrated. Then it runs for eight hours, and the conditions change. Holding accuracy through those hours is the work of dynamic error compensation.
Five families of dynamic error occur in practice.
Thermal effects come first, because they are the largest and the most predictable. Motors, gearboxes and control electronics all produce heat, and the surrounding air changes temperature through the day, so the metal of the arm expands and contracts.
Mechanical wear is slower but relentless. Gear teeth lose material, bearings develop clearance, and sliding surfaces wear unevenly. Backlash therefore grows, stiffness falls, and friction changes, so the machine that was calibrated a year ago is not quite the machine running today.
Load variation comes third. Many robots handle parts of different masses, and a heavier part deflects the links more and demands more torque to accelerate, so both the position reached and the path taken to reach it change with the load.
Environmental conditions come fourth: humidity that promotes corrosion and stiffens joints, dust that enters gearboxes and settles on optical sensors, and extremes of temperature that thicken lubricants or degrade seals.
Fifth are electrical and control effects. Motor winding resistance rises as the motor heats, altering the torque a given current produces. Sensors drift slowly with temperature and age, so their readings develop a bias. In fast motion, small delays in signals become significant compared with the time available to correct.
10.5.2.2 Adjusting for Temperature Change
Metal changes length with temperature according to
\[\Delta L = \alpha L_0 \Delta T\]
where \(\Delta L\) is the change in length in millimetres, \(L_0\) is the original length in millimetres, \(\Delta T\) is the temperature change in degrees Celsius, and \(\alpha\) is the coefficient of linear thermal expansion of the material, in reciprocal degrees Celsius. Table 10.7 gives typical values for materials used in robot structures.
That last row explains a design choice students often find puzzling — why a costly composite is used for the links of high-precision machines. It is not chosen for lightness alone. It barely moves when the temperature does.
Compensating for that growth takes four steps, and they run in sequence.
The temperature must first be measured where it matters. Thermocouples, resistance temperature detectors or infrared sensors are fitted to joints, motors and structural links, because heating is not uniform — the joint nearest a hard-working motor runs far hotter than the outer link.
The expansion is then predicted. The controller holds a thermal model of each link, using the material’s coefficient, the link’s length and the measured temperature, and computes how much that link has grown at this moment.
The kinematic model is then updated with the new effective link lengths, and the joint angles for the next commanded point are recalculated from the updated model. The tool therefore arrives at the correct place using slightly different joint angles from the ones it used when the machine was cold.
Finally, whatever the model fails to predict is caught by feedback. Vision or laser measurement of the actual tool position picks up the residue, and in tasks involving contact, force feedback flags an unexpected resistance that may indicate a dimensional shift.
10.5.2.3 Adjusting for Wear and Environmental Factors
Wear is handled in much the same spirit: measure the change, then command around it. As gear teeth wear, the backlash at a joint grows, and its size can be estimated from the small movement observed when the joint reverses. The controller then adds an equal correction in the opposite direction whenever that joint changes direction, so the lost motion is taken up before the commanded move begins.
Increased friction is handled through the torque the controller asks for. Comparing the motion the encoders report with the motion that was commanded shows a joint moving sluggishly, and the control loop raises the current to that motor until the actual motion matches the intended one. The joint keeps its behaviour even as the mechanism ages.
Vibration and shock arriving from neighbouring machinery are measured rather than resisted. Accelerometers and gyroscopes mounted on the arm sense the disturbance as it happens, and the controller issues an opposing correction to hold the tool on its path.
Humidity and dust are dealt with differently again, because they attack the sensors rather than the mechanism. Optical sensors clouded by dust and connections affected by moisture produce noisy or inconsistent readings, so systems designed for such places carry more than one sensor for a critical measurement and filter the signals, ignoring a reading that has begun to disagree with the others.
10.5.3 Feedback Mechanisms
10.5.3.1 Laser-Based Tracking
Every correction described so far rests on knowing where the tool actually is. The robot’s own encoders cannot supply that, since they measure joints rather than the tool. An external instrument is needed, and the most precise one in common use measures with light.
A laser tracking system directs a beam at a small retroreflective target — an optical device that returns light back along the direction it arrived from — mounted on the end-effector. The returned beam is analysed to find the target’s distance and direction, and the head steers itself to keep the beam on the target as the arm moves, as in Fig. 10.5-3.
Diagram
Fig. 10.5-3 – a robot arm on the right with a small spherical retroreflective target mounted beside its end-effector; a laser tracker head on a tripod on the left with a beam drawn from the head to the target and a second arrow drawn along the same line back to the head, labelled “outgoing beam” and “returned beam”; the tracker head connected by a line labelled “measured position” to a box marked “controller”, and the controller connected by a line labelled “corrected command” to the robot’s base; a note beside the target reading “measures where the tool actually is, not where the joints say it is”
Two measuring techniques are used. Laser interferometry determines a change in distance by combining the returned beam with a reference beam and reading the interference pattern that results; it detects changes down to the nanometre level but tracks a change in distance rather than an absolute one. Laser triangulation finds distance from the position at which the reflected spot lands on a detector; it is less precise than interferometry but faster, which suits a moving target.
10.5.3.2 Real-Time Correction
Measurement becomes correction when the instrument is wired into the control loop. The tracker reports where the tool is; the controller compares that with where the tool should be at this instant on the commanded path; the difference is turned into a correction and sent to the joint drives. The cycle repeats continuously while the motion is in progress.
The robot has therefore stopped executing its programme blindly. It is executing it while being watched, and adjusting as it goes. If thermal growth pushes the tool 0.2 mm off the path, the correction begins before the move has finished, rather than being discovered when the part is inspected.
Four benefits follow from measuring the tool externally rather than trusting the joints.
The measurement is far finer than the joint encoders can offer, so deviations well below the robot’s own resolution at the tool become visible. Errors are caught as they arise rather than accumulating quietly across a shift. The machine adapts continuously instead of waiting for the next scheduled recalibration. And because the instrument measures the real position, it catches every cause of error at once — including causes nobody modelled, such as a vibration from a passing forklift or an unexpected force from the process itself.
That last point is the strongest argument for external feedback, and also the reason it is not universal. A model-based correction only cancels the errors it was built to expect; a measurement cancels whatever is actually there. Against this stand the cost of the instrument, the need for a clear line of sight, and the fact that a vision or laser measurement takes time to produce, so the correction always arrives a little after the event it is correcting.
10.6 Repeatability
Ask someone to sign their name twenty times and the signatures will not be identical, but they will be recognisably the same. Ask a robot to go to the same point twenty times and something similar happens: it stops in almost, but not exactly, the same place. How tightly those twenty stopping points cluster is the property this section is about, and it is the one figure almost every manufacturer is willing to print.
10.6.1 Returning to the Same Position Under Identical Conditions
An industrial robot is usually programmed by being shown what to do. An operator drives the arm to a position, saves it, and the controller stores the joint angles at that moment. Such a stored position is called a taught point, and the programme is a list of them. What the robot must then do, thousands of times a shift, is return to those points.
Read the phrase “identical conditions” carefully, because it is what makes the figure meaningful and also what makes it easy to abuse. A repeatability figure measured with an empty gripper at half speed on a fully warmed machine describes those circumstances and no others.
Notice also what the spread is measured against. It is taken about the mean of the positions the robot actually reached, not about the position it was told to reach. Repeatability asks whether the robot agrees with itself. It never asks whether the robot is right.
Diagram
Fig. 10.6-1 – a close-up view of one taught point: a small cross labelled “commanded point” and, a short distance away from it, six small dots clustered together labelled “positions actually reached over six cycles”; a dot at the centre of the cluster marked with a small square and labelled “mean of the achieved positions”; a dashed circle drawn around the cluster and centred on that mean, its radius dimensioned and labelled “repeatability band, ±”; a separate straight arrow drawn from the cross to the mean and labelled “accuracy error”
That example shows both properties living side by side in one set of measurements. This robot wanders by twelve micrometres and sits three micrometres off target, so along this axis it is both repeatable and accurate. Alter one link length in its controller by half a millimetre and the second number would jump to about 0.5 mm while the first stayed at 0.012 mm.
10.6.2 Repeatability Compared with Accuracy
The two words are treated as synonyms so often that it is worth setting them against each other item by item.
Two consequences of that table matter more than the rest.
The first is that repeatability is almost always the better figure on the same machine, often by a factor of ten or more. That is no coincidence. Repeatability excludes every systematic error by construction, because a wrong link length in the model sends the arm to the same wrong place on every cycle — which ruins accuracy and leaves repeatability untouched.
The second is that repeatability sets a floor beneath accuracy. Calibration can shift the cluster of achieved positions onto the target, but it cannot shrink the cluster. A robot that scatters over ±0.1 mm can never be made accurate to much better than that, however carefully its model is corrected.
Which of the two a task actually needs
A robot programmed by teaching needs repeatability far more than accuracy. The operator drives the arm to the real component and saves that position, so whatever the model has wrong is built into the taught point and cancels out. All that matters afterwards is that the arm can get back there.
A robot programmed from a drawing needs accuracy as well. Points taken from a computer model of the part are expressed in the geometry of the design, and the arm must convert them into joint angles through its own model. Every mismatch between that model and the real machine now shows up in the work, which is precisely the case that calibration was developed for.
10.6.3 Factors Affecting Repeatability
Everything that varies from one cycle to the next widens the cluster. Seven influences account for most of what is seen in practice.
Mechanical wear and backlash come first. Play between gear teeth lets a joint settle at slightly different places depending on how the last movement loaded it, and that play grows as the teeth wear.
Joint and link stiffness comes second. Links that flex under load deflect by an amount that depends on how the load is swinging at the instant the arm stops, so a springy arm halts at slightly different places on different cycles even with the same part in the gripper.
Thermal effects come third, and they are the reason a repeatability figure is always quoted for a warmed-up machine. Through the first half hour of a shift the links are still growing, so the stopping point creeps steadily in one direction rather than scattering about a mean.
Control system resolution comes fourth. If the sensing and command chain cannot resolve a deviation of five micrometres, the controller cannot remove it, and that deviation appears as scatter.
Payload variation comes fifth. A robot set up for a light part behaves differently with a heavy one, because the extra mass increases both the deflection of the links and the effort the joints must make to bring the motion to a stop.
Environmental disturbance comes sixth: vibration from neighbouring machines, shocks through the floor, draughts across the frame, and dust settling on optical sensors.
Calibration and maintenance come seventh, and they need to be separated carefully. Servicing a robot — replacing worn bearings, adjusting gear preload, re-lubricating joints — genuinely improves repeatability, because it removes the play and friction that made cycles differ. Recalibrating the controller’s model does not, since that changes only the numbers the controller computes with. Maintenance restores the mechanism; calibration restores the model. A machine needs both, for different reasons.
10.7 Current Trends in Repeatability
Section 10.6 listed everything that makes one cycle differ from the next: play, flex, heat, noise, changing load. Improving repeatability means attacking that list, and modern practice does so in three ways — by engineering the mechanism so that less varies, by letting the controller adjust itself as conditions change, and by holding a band well under a millimetre in work where consistency is the whole point.
10.7.1 Sub-Millimeter Repeatability
10.7.1.1 What Sub-Millimeter Repeatability Means
Sub-millimeter repeatability means that a robot returns to a taught position with a spread of less than one millimetre across repeated cycles. In practice the phrase is applied to machines quoting bands such as ±0.2 mm, ±0.05 mm or ±0.02 mm.
The phrase describes consistency, not correctness. A machine may hold ±0.02 mm about a point that lies two millimetres from where the design intended, and it is still, by this definition, sub-millimeter repeatable. That is not a quibble; it is why the two figures are always quoted separately.
What has to be right for the band to hold
Reaching such a band is the result of four decisions taken together, each of which has appeared earlier in this chapter.
The sensing must resolve well below the band. Encoders of seventeen bits or more report joint position in steps far finer than the tolerance, so a deviation is detected while it is still a small fraction of the allowance.
The mechanism must not move on its own. Backlash and flex are what make one cycle differ from the next, so the structure is made stiff and the gearing tight, which is the subject of section 10.7.2.
Slow drifts must be compensated. Thermal growth and wear do not scatter the stopping points; they walk them steadily in one direction, and the compensation methods of section 10.5.2 are what stop that walk from consuming the whole band.
Feedback must confirm the result. Laser tracking or vision measures where the tool actually went, so a machine drifting outside its band is found by measurement rather than by a batch of rejected parts.
10.7.1.2 Spot Welding in Automotive Production
Nowhere is consistency valued more plainly than in the welding of a vehicle body. A body shell is held together by several thousand spot welds, each made by clamping two or more steel panels between a pair of electrodes and passing a heavy current through the joint for a fraction of a second. The heat melts a small nugget of metal where the panels meet, and they are fused when it solidifies.
Photograph
Fig. 10.7-1 – several robotic arms fitted with spot welding guns working on a car body shell on a production line, photographed from outside the cell so that at least two complete arms, the welding guns with their pairs of electrodes, and the body shell held in its fixture are all visible in one frame. Sourcing note: a photograph of a real automotive body shop; renderings, computer-generated images, and close-ups showing only sparks or a single gun are unusable.
Every one of those welds must land on the flange it was meant for. If the electrodes come down a few millimetres away, the weld may catch only one panel or sit too close to an edge, and the joint is weaker than the design assumed. Put a weld where an earlier one already sits and the current runs through the existing nugget instead of forming a new one. Repeated across a shell, small inconsistencies produce bodies whose strength differs from one another, which is unacceptable in a structure meant to behave predictably in a collision.
Notice which of the two properties this job actually needs. The weld positions are taught on a real body shell held in a real fixture, so the robot’s own idea of where those points lie in space never enters the question. What matters is that it returns to each of them, thousands of times a shift, for years. A robot that places every weld 0.3 mm away from the position in the design file, but places it there every single time, builds sound bodies. A robot that averages perfectly and scatters over two millimetres does not.
That first limitation is worth dwelling on. Repeatability is measured against the robot’s own returns, but the work is done on a part held by something else. If the fixture is loose, no amount of repeatability in the arm rescues the result, because the target itself has moved.
10.7.2 Improved Mechanical Design
Compensation corrects errors after they arise. Mechanical design prevents them, and it is the more durable of the two approaches because it needs no sensor, no model and no computing.
Backlash is the first target. Fig. 10.7-2 shows why it exists: between meshing gear teeth there must be a small clearance, or the teeth would jam, and that clearance becomes lost motion whenever the joint reverses.
Diagram
Fig. 10.7-2 – two meshing gear teeth drawn greatly enlarged, the driving tooth on the left in contact with the right-hand face of the driven tooth, and the small gap on the opposite face dimensioned and labelled “backlash”; below this pair, the same two teeth drawn again after the drive has reversed, now touching on the opposite faces with the gap moved to the other side; a curved arrow between the two drawings labelled “the drive turns through this gap before the joint moves at all”
Three arrangements are used at robot joints, and they trade cost against play, as Table 10.9 sets out.
A harmonic drive achieves its very small backlash by a different principle from ordinary gearing. Instead of a few teeth meeting at a point, a flexible toothed component is deformed into engagement with a rigid one over a large arc, so many teeth share the load at once and the clearance at any single pair matters far less. A direct drive removes the question altogether by removing the gearbox, at the price of a much larger motor for the same torque at the joint.
Stiffness is the second target. A link that deflects under load stops at a slightly different place depending on how that load is swinging, so designers choose materials and cross-sections that resist bending — steel alloys where the extra mass is acceptable, carbon fibre composites where it is not — and keep the reach no longer than the job needs, since deflection grows sharply with length.
Bearings are the third. Preloading a bearing, which means assembling it with a deliberate internal force, removes the small clearance that would otherwise let a shaft shift sideways as the load direction changes.
None of this is free. A stiffer arm is a heavier arm, and a heavier arm needs larger motors, accelerates more slowly and costs more to run. Harmonic drives and direct drives are expensive. Improved mechanical design buys repeatability with mass and money, which is why it is applied to the joints that need it rather than to every joint equally.
10.7.3 Adaptive Control Systems
10.7.3.1 Why Fixed Control Parameters Are Not Enough
A conventional controller is tuned once — on a particular machine, carrying a particular load, at a particular speed. Its response to a given error is then fixed. That serves while the machine and its work stay as they were on the day of tuning, and real production rarely obliges.
Four changes defeat fixed parameters. The payload varies, so a correction that settles a light part neatly overshoots with a heavy one, because both the moving mass and the torque needed to stop it have changed. The mechanism wears, so friction rises and play grows, and a correction sized for a new gearbox is too small for a worn one. The surroundings shift, so links lengthen with heat and vibration arrives from neighbouring machines. And small errors accumulate: a slight overshoot repeated every cycle walks the result out of tolerance over a long run.
An adaptive control system answers all four in the same way — by changing its own settings while it runs.
Fig. 10.7-3 shows the structure. The ordinary feedback loop is still there, comparing measured position with commanded position. Above it sits a second, slower loop that watches how well the first is performing and adjusts its settings.
Diagram
Fig. 10.7-3 – block diagram: a block labelled “commanded position” feeding a summing point; the summing point feeding a block labelled “controller”; the controller feeding a block labelled “joint drives and arm”; the arm’s measured position returning to the summing point along a feedback line labelled “position feedback from encoders”; above this loop a block labelled “adaptation mechanism” taking input arrows from the measured response and from the error signal, and sending an output arrow down into the controller block labelled “updates the controller’s parameters”; the inner loop annotated “fast: corrects the error” and the outer path annotated “slower: corrects the controller”
10.7.3.2 Pick-and-Place Operations in Logistics and Packaging
The clearest use of adaptive control is the work the syllabus names: the repetitive pick-and-place motion of logistics and packaging. A robot lifts an item from a moving conveyor and sets it into a carton or a tray, then does it again, several times a minute, through a whole shift.
Everything about that task fights consistency. The items differ in mass and in how that mass is distributed, so no single tuning suits them all. The motion is fast, so inertia matters and the arm is still settling as it releases. The cycle repeats so often that wear becomes measurable within months. And the tolerance is real: an item set down a few millimetres out of place will not seat in its compartment, and a jammed carton stops the line downstream.
An adaptive controller treats each cycle as information. It compares the motion obtained with the motion commanded, adjusts the torque and the timing it will use next time, and so keeps successive placements inside the same band whether the item weighs 200 g or 2 kg, and whether the machine is new or a year old.
The gains are consistency across long runs and the efficiency that follows from it — fewer misplaced items, fewer jams, less rework, and less downtime spent retuning. The cost is complexity: an adaptive system needs good sensing, careful design to keep the outer loop stable, and, as the snapshot above warns, the discipline of reporting whatever it has quietly compensated for.
10.8 Summary
Specifying a robot
- The specification of a robot is the set of measurable performance quantities that say what it can do — payload, reach, degrees of freedom, resolution, accuracy and repeatability — each stated with the conditions under which it was measured.
- A performance figure without its condition is not information. Payload, speed, temperature and the part of the reach being used all change the result.
- Resolution, accuracy and repeatability answer three different questions: how finely can the robot move, does it go where it was told, and does it go to the same place every time. A machine can answer one well and another badly.
- Data sheets normally quote repeatability and omit accuracy, because repeatability can be measured on site against the robot’s own taught points, while accuracy needs an external reference.
Resolution
- Resolution is the smallest incremental movement a robot can detect or achieve, stated at a named joint or at the end-effector.
- Detecting is the sensors’ limit and achieving is the controller and drive’s limit; the coarser of the two governs.
- A rotary joint’s angular resolution becomes a distance at the tool through the arc relation \(s = r\theta\), with \(\theta\) in radians. The same joint therefore gives a coarser step on a longer arm, and a robot’s resolution at the tool changes as the arm folds and extends.
- A digital sensor divides its span into \(2^n\) levels, where \(n\) is the bit depth, so one count of a 10-bit converter on a 0 V to 5 V span is worth 4.88 mV.
- Bit depth is only the arithmetic limit. Sensitivity, noise and linearity decide whether the last few counts carry any information at all; a reading that wanders by three counts on its own has an effective resolution three counts wide.
- On the control side, resolution is limited by the step size of the commands issued, the controller update rate, the quality of the feedback, and latency together with its variation, jitter.
- The system’s resolution is set by the coarsest stage in the chain of sensor, converter, controller, drive and mechanism. A finer sensor on a joint with half a degree of play changes nothing.
- Resolution should be matched to the tolerance of the work rather than maximised, since finer sensing costs money, demands cleaner power and faster computing, and is wasted on a mechanism that cannot use it.
Current trends in resolution
- An encoder converts the mechanical motion of a shaft, joint or actuator into an electrical signal from which the controller determines position, and from successive positions, speed and direction.
- An optical rotary encoder interrupts a beam of light with a patterned disk on the shaft, producing a train of pulses. A second detector a quarter cycle out of phase, called quadrature, reveals the direction of rotation.
- An incremental encoder counts steps and loses its position at power-off, so the robot must be driven to a reference position before use. An absolute encoder carries one track per bit and produces a unique code for every position, which is known the instant power returns.
- The angular resolution of an absolute encoder at its own shaft is \(360°/2^n\), where \(n\) is the number of tracks. Gear reduction between motor and joint divides that step further by the gear ratio.
- Encoder resolution rises in bits, each extra track doubling the codes and halving the step. Electronics and biomedical manufacturing drive this trend, because their tolerances fell below what earlier sensors could resolve.
- Sub-micron resolution is the ability to detect or execute movements smaller than one micrometre, \(1\ \mu\text{m} = 10^{-6}\ \text{m}\), and it holds only in controlled conditions of temperature, vibration and cleanliness.
- Semiconductor manufacture is its principal home, because successive circuit layers laid on a wafer must register with one another to a fraction of a micrometre. Biomedical device manufacture, optical assembly and precision aerospace alignment make the same demand.
- An integrated vision system links cameras and image processing directly into the control unit, so that observations of the work adjust the motion while it is in progress.
- Vision measures the tool relative to the workpiece, which is the one relationship the joint encoders cannot see, so it corrects deviations below the encoders’ detection limit. It is slower than encoder feedback, depends on lighting and a clear view, and cannot replace fine joint sensing, because the correction it calls for must still be commanded.
Accuracy
- Accuracy is the closeness of the position actually reached by the end-effector to the position commanded, both in the same stated frame, under stated conditions.
- The accuracy error is the straight-line distance between the commanded point and the achieved point, \(E = \sqrt{(x_a - x_c)^2 + (y_a - y_c)^2 + (z_a - z_c)^2}\), and it varies across the working region, so a proper statement covers many points.
- Accuracy is classified as positional accuracy, angular accuracy and path accuracy: the location of the tool, its orientation, and how closely the route travelled follows the route intended. Path accuracy is judged continuously and worsens most sharply with speed.
- The point the controller tracks is the tool centre point, and every commanded position refers to it, expressed in the base frame or in a workpiece frame.
- Accuracy is limited by the kinematic model’s mismatch with the real machine, joint zero offsets, sensor resolution, mechanical stiffness, control system precision, and environmental effects.
- Most of these produce the same error on every cycle. Such a systematic error can be measured, modelled and subtracted, which is why a robot can be consistent to a hundredth of a millimetre and still a millimetre from its target.
Current trends in accuracy
- The geometric sources of error are joint zero offset, link length deviation, axis misalignment, backlash and mechanical play, elastic deflection under load, and thermal expansion.
- Calibration measures the positions actually reached at a set of known poses, compares them with the positions the model predicts, and corrects the model’s joint offsets, link lengths and axis orientations. Nothing mechanical is altered; only numbers in the controller change.
- The process runs as data collection, error detection, parameter estimation, model update and validation, and the validation step is the one that must not be skipped.
- Calibration improves accuracy but does not improve repeatability. It moves the cluster of achieved positions onto the target; it cannot shrink the cluster.
- Dynamic error compensation adjusts control parameters while the robot runs, to counter errors that change with time. Its sources are thermal effects, mechanical wear, load variation, environmental conditions, and electrical and control-related factors such as sensor drift and signal delay.
- Thermal growth follows \(\Delta L = \alpha L_0 \Delta T\). An aluminum link of 450 mm warming by 18 °C grows by about 0.186 mm, which alone exceeds a 0.1 mm tolerance.
- Temperature is compensated in four steps: measure the temperature at the links, predict the expansion from a thermal model, recalculate the joint angles from the updated link lengths, and catch the residue with vision or force feedback.
- Wear is compensated by commanding around the growing backlash and by raising motor torque as friction rises; vibration is measured by accelerometers and gyroscopes and countered; dust and humidity are met with redundant sensing and filtering.
- Laser-based tracking aims a beam at a retroreflective target on the end-effector and measures the tool itself rather than the joints. Laser interferometry reads the interference pattern of the returned beam and resolves changes to the nanometre level; laser triangulation reads the position of the reflected spot on a detector and is less precise but faster.
- Wired into the control loop, this measurement gives real-time correction: fine measurement, immediate detection, continuous adaptation, and robustness to error sources nobody modelled. It needs line of sight, costs a great deal, and always reports a little after the event.
Repeatability
- Repeatability is the closeness of agreement between the positions reached when the same command is executed repeatedly, approached along the same path, under identical conditions of payload, speed and temperature.
- It is measured as a ± band about the mean of the robot’s own achieved positions, not about the commanded point, so it asks whether the robot agrees with itself and never whether it is right.
- Repeatability is usually the better figure on the same machine, because it excludes systematic errors by construction, and it sets a floor beneath accuracy: calibration cannot make a robot accurate to better than it is repeatable.
- A robot programmed by teaching needs repeatability above all, since the taught point absorbs whatever the model has wrong. A robot programmed from a design file needs accuracy as well.
- Repeatability is affected by mechanical wear and backlash, joint and link stiffness, thermal effects, control system resolution, payload variation, environmental disturbance, and calibration and maintenance.
- Servicing the mechanism improves repeatability; recalibrating the model does not. Maintenance restores the mechanism, calibration restores the model, and a machine needs both.
Current trends in repeatability
- Sub-millimeter repeatability is a return band of less than one millimetre about the mean of the returns, held across many cycles under identical conditions. It rests on sensing finer than the band, a mechanism that does not move on its own, compensation for slow drifts, and feedback that confirms the result.
- Spot welding in automotive production is the standard case. Weld positions are taught on a real shell, so consistency, not absolute accuracy, decides whether the body is sound — and the fixture holding the shell matters as much as the arm.
- Improved mechanical design prevents error instead of correcting it: low-backlash gearing such as a harmonic drive, or direct drive with no gearing at all; stiff materials and sections; and preloaded bearings. The price is mass, cost and slower motion.
- An adaptive control system adjusts its own parameters and strategy in response to changes in the robot’s dynamics or environment, so that its corrections stay correct as payload, wear and conditions change.
- Repetitive pick-and-place work in logistics and packaging is where it earns its place, holding placements inside the same band across varying items and an ageing machine. Its danger is that adaptation can hide a developing fault, so an adaptive system should report what it has compensated for.
10.9 Key Terms
- Absolute encoder — a rotary encoder whose disk carries one patterned track per bit, so that every angular position produces its own unique code and the position is known the instant power is applied.
- Accuracy — the closeness of the position actually reached by the end-effector to the position it was commanded to reach, both expressed in the same reference frame.
- Accuracy error — the straight-line distance between the commanded position and the position actually reached.
- Adaptive control system — a control system that adjusts its own control parameters while running, in response to changes in the robot’s dynamics or environment.
- ADC — analogue-to-digital converter.
- Analogue-to-digital converter (ADC) — a device that turns a continuously varying voltage into a whole number, by dividing the measuring span into a fixed set of levels.
- Angular accuracy — how closely the orientation of the end-effector matches the orientation commanded, measured as an angle.
- Backlash — the free play between meshing gear teeth or in bearings and couplings, which must be taken up before a reversing joint begins to move.
- Bit depth — the number of binary digits in a converted reading, which fixes the number of levels available as \(2^n\).
- Calibration — the process of measuring where the end-effector actually goes at a set of known poses and correcting the controller’s kinematic parameters so that its model matches the real machine.
- Direct drive — a joint driven straight from the motor shaft with no gearbox, and therefore with no gear backlash, at the cost of a much larger motor.
- Dynamic error compensation — continuous adjustment of a robot’s control parameters while it runs, to counteract errors that change with time such as thermal growth, wear and changing payload.
- Encoder — a device that converts the mechanical motion of a shaft, joint or actuator into an electrical signal from which position, speed and direction can be determined.
- Harmonic drive — a compact, high-ratio reduction drive in which a flexible toothed element engages a rigid one over a large arc, giving effectively no backlash.
- Incremental encoder — an encoder that counts steps from wherever counting began, and which therefore loses its position when power is removed.
- Integrated vision system — cameras and image-processing software linked to the robot’s control unit so that what the camera observes adjusts the motion while the task is in progress.
- ISO 9283 — the international standard that lays down how the pose accuracy and pose repeatability of an industrial robot are to be tested, so that different makers’ figures can be compared.
- Jitter — the variation, from one cycle to the next, in the delay between an event and the system’s response to it.
- Kinematic model — the set of link lengths, joint axis directions and offsets that the controller uses to work out where the tool is from the joint angles.
- Laser interferometry — a measuring technique that finds a change in distance by combining a returned laser beam with a reference beam and reading the interference pattern produced.
- Laser triangulation — a measuring technique that finds distance from the position at which a reflected laser spot lands on a detector.
- Latency — the delay between a change occurring and the control system responding to it.
- LED — light-emitting diode.
- Linearity — how evenly a sensor’s output tracks the quantity it measures across the whole of its range.
- Path accuracy — how closely the route actually travelled by the end-effector follows the route intended, judged continuously throughout the motion.
- Payload — the mass a robot is rated to carry at its mounting flange, including the mass of the gripper or tool.
- Positional accuracy — how close the tool point comes to the target coordinates, measured as a distance.
- Quadrature — the arrangement of two encoder output channels a quarter of a cycle apart in phase, from which the direction of rotation is determined.
- Quantisation — the division of a continuously varying quantity into a fixed set of levels for measurement or command.
- Repeatability — the closeness with which a robot returns to the same taught point on repeated attempts, under identical conditions of payload, speed, path of approach and temperature.
- Resolution — the smallest incremental movement a robot can detect or achieve, stated at a named joint or at the end-effector.
- Sensitivity — how much a sensor’s output changes for a given change in the quantity it is measuring.
- Specification — the set of measurable performance quantities describing what a robot can do, each stated with the conditions under which it was measured.
- Spot weld — a joint made by clamping overlapping metal sheets between two electrodes and passing a heavy current for a fraction of a second, fusing a small nugget of metal where the sheets meet.
- Sub-micron resolution — the ability to detect or execute movements smaller than one micrometre, sustained under controlled conditions of temperature, vibration and cleanliness.
- Sub-millimeter repeatability — a return band of less than one millimetre about the mean of a robot’s stopping points, held across repeated cycles under identical conditions.
- Systematic error — an error that takes the same value every cycle under the same conditions, and which can therefore be measured, modelled and subtracted.
- Taught point — a position stored by driving the arm there and recording the joint angles it was actually in, rather than by calculating coordinates.
- TCP — tool centre point.
- Tool centre point (TCP) — the point on the end-effector that the controller tracks, such as the tip of a welding wire or the centre of a gripper’s closed jaws, and to which every commanded position refers.
10.10 Objective Questions
- The angular resolution of a 12-bit absolute encoder, measured at its own shaft, is closest to:
- 0.0275°
- 0.0879°
- 0.3516°
- 1.4063°
- Assertion (A): A robot may have an excellent repeatability figure and a poor accuracy figure at the same time. Reason (R): Repeatability compares the positions the robot reaches with one another, while accuracy compares them with the position that was commanded.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Assertion (A): A full kinematic calibration improves the repeatability of a robot arm. Reason (R): Calibration corrects the joint zero offsets, link lengths and axis orientations held in the controller’s model.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Match each item in Column A with its description in Column B.
| Column A | Column B |
|---|---|
| a. Laser interferometry | i. reduction gearing with effectively no backlash |
| b. Harmonic drive | ii. mounted on the end-effector so that a beam returns along its own path |
| c. Retroreflective target | iii. senses vibration reaching the arm from nearby machinery |
| d. Accelerometer | iv. finds a change in distance from an interference pattern |
Fill in the blanks: A robot that stops at the same wrong place on every cycle is displaying a ______ error, while scatter of the stopping points about their own mean is caused by ______ error.
State whether each of the following is true or false.
- The resolution of a rotary joint, measured at the end-effector, is the same whether the arm is folded in or stretched out.
- An incremental encoder reports the correct joint position as soon as power is restored after a shutdown.
- A repeatability figure on a data sheet holds only for the payload and speed at which it was measured.
Give the full form of each of the following abbreviations used in this chapter: ADC, TCP, LED.
A sensor producing 0 V to 5 V is read by a 10-bit analogue-to-digital converter. State, in millivolts, the change in the sensor’s output that corresponds to one count.
10.11 Check Your Concepts
Define resolution as it applies to a robot, and state one condition that must accompany a resolution figure if the figure is to mean anything.
Name the three types of accuracy, and state in one line for each what it compares.
Differentiate between an incremental encoder and an absolute encoder, and describe one situation in which each would be the better choice.
Give reasons why the accuracy of a robot arm is usually worse near the outer limit of its reach than close to its base.
A rotary joint has an angular resolution of 0.005°, and the end-effector is 800 mm from the joint axis. Calculate the smallest distance through which that joint can move the end-effector. State what this distance becomes when the arm is folded so that the tool is 300 mm from the same axis, and explain what the two results together tell you about quoting a single resolution figure for a robot.
A workcell is to be programmed offline from a computer model of the part, rather than by driving the arm to each position and teaching it. Explain how this changes the relative importance of accuracy and repeatability for that cell, describe the steps of the kinematic calibration that should be carried out before the cell is commissioned, and state two kinds of error that this calibration will not remove.
10.12 Diagram, Circuit & Output Reading
- The extract below is taken from the data sheets of two robots offered for a task that requires a 3 kg part to be placed within ±0.1 mm of a position 900 mm from the base of the robot.
| Quantity | Robot A | Robot B |
|---|---|---|
| Payload | 5 kg at the mounting flange, including gripper | 12 kg at the mounting flange, including gripper |
| Reach | 850 mm | 1300 mm |
| Repeatability | ±0.03 mm at rated payload and speed, after warm-up | ±0.12 mm at rated payload and speed, after warm-up |
| Accuracy | not quoted | not quoted |
- State, with a reason, which of the two robots can be considered for the task at all.
- Explain why the repeatability figures alone do not settle whether the task can be met.
- Name one further piece of information you would ask the supplier for, and say what you would do with it.
- An optical rotary encoder is fitted to a joint shaft. Its disk carries 500 segments around its rim, and it produces two output channels, A and B. In the first part of a recorded run, channel A rises a quarter of a cycle before channel B, and 125 complete pulses are counted on channel A. The shaft then pauses, after which channel B begins to rise before channel A.
- Through what angle did the shaft turn during the first part of the run?
- What does the change in the order of the rising edges tell you about the second part of the run?
- State what this encoder cannot tell the controller about the position of the tool at the far end of the mechanism.
- A trimmer potentiometer is used to measure the angle of a model joint. Its wiper is connected to
A0on an Arduino UNO and one end of its track is connected to5V, but the other end of the track has been left unconnected. On the Serial Monitor, the reading sits close to 1023 and hardly changes as the shaft is turned.
- Identify the fault in the connections and explain why it produces this reading.
- State the correction needed.
- Explain what the resolution of this measurement would be, in degrees per count, once the fault is corrected, if the potentiometer turns through 270° from one end of its track to the other.
- A laser tracker is used to record where a robot’s tool centre point comes to rest at one taught position, at intervals through a single production shift. The table gives the mean of thirty returns and the spread about that mean at each interval.
| Time into the shift | Mean x-coordinate of the returns | Spread about the mean |
|---|---|---|
| 0 h | 400.000 mm | ±0.02 mm |
| 2 h | 400.055 mm | ±0.02 mm |
| 4 h | 400.098 mm | ±0.02 mm |
| 6 h | 400.120 mm | ±0.02 mm |
- State which of the robot’s two positioning properties has changed through the shift and which has not, giving your reason from the data.
- Name the most likely physical cause of the change and explain the mechanism by which it produces this pattern.
- Explain why running a kinematic calibration at the start of the shift would not have prevented this.
10.13 Case-Based Questions
- A packaging line handles pouches of three sizes, of mass 0.4 kg, 1.2 kg and 2.5 kg, which arrive mixed on a conveyor. A robot is to lift each pouch and place it into a carton, and the carton compartments allow a placement error of ±0.5 mm. The robot chosen quotes a repeatability of ±0.08 mm at its rated payload of 3 kg, and the line will run three shifts a day.
- Explain why the cell may not achieve the ±0.08 mm quoted on the data sheet, and identify the property of the work that is responsible.
- Describe two measures, one mechanical and one in the control system, that would help the cell hold its placements inside ±0.5 mm across the mixed run.
- The supplier offers an adaptive control option. State what it would adjust and why, and state one thing the maintenance team should monitor if it is fitted.
- In a vehicle body shop, welds are placed on flanges that allow a positional error of ±0.5 mm. The cell is checked at the start of a shift and is well within tolerance. By the fourth hour, measurements show that the welds along one side of the shell are being placed about 0.3 mm further out than they were at the start, in a consistent direction. The arm concerned carries steel links, the longest of which is 600 mm, and the coefficient of linear thermal expansion of steel is about \(12 \times 10^{-6}\) /°C.
- Identify the most likely cause of the drift and explain how it produces a shift that is consistent in direction rather than random.
- Calculate the change in the length of the 600 mm link for a temperature rise of 15 °C, and comment on your answer in relation to the ±0.5 mm allowed.
- Describe two ways the cell could be corrected — one that predicts the error from a model and one that measures the tool’s actual position — and state one limitation of each.