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Chapter 16: Control Systems

In the 1780s, steam engines built by James Watt and Matthew Boulton were fitted with a device borrowed from windmill machinery: the centrifugal governor. Two heavy metal balls hung from arms on a spindle, and the engine itself turned that spindle. When the engine ran too fast, the balls swung outward, the arms lifted a collar, and a linkage partly closed the steam valve. The engine slowed down. When it ran too slow, the balls dropped, the valve opened wider, and the engine picked up speed again.

Nobody had to stand and watch the engine. The machine watched itself.

That small brass mechanism did what every self-correcting machine must do, including every modern robot. It measured what was actually happening, compared it with what should be happening, and used the difference between the two to change its own working.

It was not perfect. Some engines fitted with governors began to hunt: they sped up and slowed down in a steady swing that never quite settled. In 1868 James Clerk Maxwell examined the problem mathematically in a paper titled “On Governors”, and showed that the way a machine corrects itself decides whether it settles quietly at its target or keeps swinging past it.

A robot arm reaching for a marked point, a robot car staying on a line, and a two-wheeled robot balancing upright all face exactly that question.

Photograph

Fig. 16.0-1 – a centrifugal governor of the Watt type, mounted on a preserved steam engine; the two ball arms, the vertical spindle, the sliding collar and the linkage running to the steam valve must all be clearly visible in the frame; a side view is preferred over a distant view of the whole engine. Sourcing note: a museum or heritage-engine photograph is suitable; images in which the linkage to the valve is hidden, obscured by pipework, or cropped out are unusable.

NoteCuriosity Corner

Q1. When a robot arm is told to move to 45°, how does it find out whether it actually reached 45°? Q2. A microwave oven runs for exactly the time you set, even if the food is already hot. What kind of control is that, and what is it missing? Q3. If checking the result makes a machine more accurate, why are some robots still built with no sensors to check anything? Q4. Why can a machine that corrects its own mistakes still shoot past its target and swing back and forth? Q5. Which three parts must work together before a robot can measure a mistake and put it right?

By the end of this chapter you will be able to answer all five of these questions.

16.1 Why a Robot Needs a Control System

An overhead water tank in a house is usually fitted with a float valve. As the tank fills, the float rises; when it reaches a certain height, it presses on a lever and shuts the inlet. Nobody stands on the roof watching the water level, and yet the level is held roughly where it should be. The tank has been given a target, and the arrangement of float, lever and valve keeps it there.

That arrangement is a control system. In a house it holds a water level; in a robot it holds a joint angle, a wheel speed, a gripper force or a heading. The quantity being held is different every time, but the idea does not change: the machine is given a statement of what should happen, and its parts are arranged so that this actually happens.

NoteDefinition

A control system is an arrangement of parts that makes a device or machine behave in a specified, desired way, by acting on an input that states the required value of the quantity being controlled.

In everyday speech, “control” usually means the thing a person presses — a remote, a switch, a regulator knob. In engineering the word covers much more than the operator’s button. The control system is the whole arrangement that turns the desired value into real motion, including whatever measures the result and corrects it. The knob is only where the desired value is entered.

Doing as told is not the same as doing the job

A robot is expected to be precise, and precision is where the difficulty begins. Sending an instruction to a motor is easy. Making sure the instruction produced the intended result in the real world is a separate problem, and it is the problem this chapter is about.

16.1.1 Acting on an Instruction, and Checking the Result

Suppose a small robot car is told to run both its drive motors at full power for 3 seconds so that it travels straight ahead by about one metre. The instruction is carried out exactly. The motors do run, and they do stop after 3 s. The instruction has been obeyed perfectly.

Whether the robot has travelled one metre in a straight line is another matter. The battery may have discharged since the last run, so the same command now delivers less voltage and less speed. One wheel may slip on a dusty floor while the other grips. The two motors, even from the same batch, rarely produce exactly the same torque at the same voltage, so the robot curves slightly. Fig. 16.1-1 shows the gap between the path that was intended and the path that was taken.

Diagram

Fig. 16.1-1 – a small two-wheeled robot car at a start line; a dashed straight arrow labelled “intended path, 1 m” running forward to a marked target point; a solid curved arrow labelled “actual path” leaving the dashed line and ending to one side of the target; the sideways gap between the two end points marked and labelled “position error”; a note beside the robot reading “same command sent to both motors”

The robot has no idea any of this happened. Nothing in the command asked it to look. It has followed a time-based instruction, and time is the only thing it has kept track of.

Q. If both wheels are given exactly the same electrical signal, why does the robot still drift to one side?

A. Equal signals do not guarantee equal motion. Small differences in winding resistance, gearbox friction, tyre diameter and load on each side make one wheel turn a little faster than the other. Over a metre, a difference of even one per cent shows as a visible curve. The command cannot notice the drift because nothing in the machine is measuring it.

To close this gap, the robot needs a sensor: a device that converts a physical quantity such as distance, angle, tilt or speed into an electrical signal the controller can read. You have already met several of these. Their role here is new, though. A sensor is not being used to explore the surroundings; it is being used to report on the robot’s own performance, so that the machine can judge itself.

16.1.2 The Control System as the Robot’s Decision Maker

A robot works in a real environment, and real environments do not stay still. A load is heavier than expected. A floor slopes slightly. A gust of wind pushes a hovering drone sideways. A joint stiffens as its grease ages. Each of these is a disturbance: an outside influence that pushes the system away from the behaviour it was asked for.

Without a control system, a robot meets a disturbance by simply carrying on. The instruction was written before the disturbance existed, so the machine keeps executing it and the error keeps growing. With a control system, the robot compares what should be happening with what is happening, and changes what it is doing about it.

This changes what the machine can be trusted with. A robot arm asked to move to a coordinate can be commanded to that position and be relied on to arrive there with the correct orientation, at a chosen speed, without overshooting the point or reacting unpredictably when the load changes. That reliability is what makes a robot useful for repeated industrial work rather than for a single demonstration.

The correcting action is carried out by an actuator: the part that converts the controller’s electrical decision into physical action, such as a motor turning a joint or a servo swinging a gripper. Sensing gives the robot the information; the actuator is how it does something about it.

A control system is powerful, but it is not magic, and it is worth being honest about its limits from the start. It cannot make an arm reach a point that lies outside the span of its links. It cannot make a motor deliver more torque than it is built for. It can only make the best use of the mechanism it is given, and no control scheme rescues a machine that was mechanically wrong to begin with.

16.1.3 Setpoint, Measured Value and Error

Every control system starts from a statement of what should happen. That statement is the setpoint, also called the reference input: the desired value of the quantity being controlled, expressed in the same unit as the quantity itself. “Hold 24 °C”, “turn the shoulder joint to 45°”, “run the belt at 0.4 m/s” are all setpoints.

NoteDefinition

The setpoint is the desired value of the controlled quantity, supplied to the control system as its input, and stated in the same unit as that quantity.

Against the setpoint stands the measured value: what a sensor reports the quantity actually to be, at this instant. The difference between the two is the error, and the error is the single number a control system exists to reduce.

NoteDefinition

The error is the setpoint minus the measured value of the controlled quantity, taken at the same instant and in the same unit as that quantity. Its magnitude states how far the system is from its target, and its sign states in which direction.

Writing this as a relation,

\[e = \theta_{sp} - \theta_{meas}\]

where \(e\) is the error, \(\theta_{sp}\) is the setpoint and \(\theta_{meas}\) is the measured value. Here the symbols carry degrees because the controlled quantity is an angle; for a speed loop the same relation would be written in metres per second.

The order of subtraction matters. Taken this way round, a positive error means the joint has not yet reached the target and must move further in the commanded direction, while a negative error means it has gone past and must come back. Reverse the subtraction and every correction is applied backwards, which drives the joint away from its target instead of towards it. This is one of the commonest and most destructive mistakes in a control program, and it is a sign error, not an arithmetic one.

When the error reaches zero, the system is exactly where it was asked to be and no correction is called for. Fig. 16.1-2 shows the three quantities together on a single joint.

Diagram

Fig. 16.1-2 – a single revolute joint of a robot arm drawn at the origin of an angle scale; a dashed line at 45° labelled “setpoint 45°”; the solid link drawn at 41.5° labelled “measured value 41.5°”; the angle between the two arcs shaded and labelled “error = +3.5°”; a small arrow on the shaded arc showing the direction in which the joint must still move

error at a robot joint, and what it means at the end of the link

Given: setpoint of the shoulder joint, \(\theta_{sp} = 45.0°\); angle reported by the joint encoder, \(\theta_{meas} = 41.5°\); length of the link from the joint to its tip, \(r = 200 \text{ mm} = 0.200 \text{ m}\).

Find: (a) the error at the joint; (b) how far the tip of the link is from where it should be.

Method: the error is \(e = \theta_{sp} - \theta_{meas}\). The distance travelled by a point at radius \(r\) when the link sweeps through an angle is \(s = r\theta\), which holds only when \(\theta\) is in radians, so the angle must be converted first using \(1° = \pi/180 \text{ rad}\).

\[e = 45.0° - 41.5° = +3.5°\]

\[e = 3.5° \times \frac{\pi}{180} = 0.0611 \text{ rad}\]

\[s = r\,e = 0.200 \text{ m} \times 0.0611 \text{ rad} = 0.0122 \text{ m}\]

the error is \(+3.5°\), and the tip of the link is about \(12.2 \text{ mm}\) short of its target position. The positive sign shows the joint must still rotate further in the commanded direction.

Check: the magnitude is sensible. A few degrees at the joint is a little over a centimetre at the tip of a 200 mm link, which is roughly the width of a thumb — small enough to look correct from across a room, and far too large for a gripper trying to enter a slot.

That last line is the practical point of the whole chapter. An error that is invisible at the joint is easily large enough to make the task fail at the tip, and it grows with the length of the arm. A machine that cannot measure this error cannot correct it, and the next three sections work through what a system that measures it looks like, and what a system that does not measure it costs.

16.2 The Parts of a Control System

However complicated a robot looks, its control arrangement can be broken into four elements: the input, the controller, the plant and the output. Every automated machine, from a room heater to an assembly line, is built from these same four, and learning to name them in an unfamiliar machine is a large part of learning to design one.

The four are not four boxes bolted to a chassis. They are four roles. A single microcontroller board may hold the input value in a variable and run the controller as a few lines of program, while the plant is the aluminum arm bolted in front of it. What matters is which role each part is playing.

Fig. 16.2-1 shows the four elements arranged in the order in which information travels through them.

Diagram

Fig. 16.2-1 – block diagram of a basic control system; three labelled rectangular blocks in a row, “Controller” and “Plant”, with an arrow entering the Controller from the left labelled “Input (setpoint)”, an arrow from Controller to Plant labelled “Control signal”, and an arrow leaving the Plant to the right labelled “Output”; no feedback path is to be drawn; all arrows point left to right

16.2.1 Input: the Setpoint or Reference Signal

The input to a control system is the setpoint, sometimes called the reference signal. It states what the system is required to achieve, and it enters the system before any physical action begins.

Where does the setpoint come from? Usually from one of two places. A human operator may supply it directly, by typing a temperature, turning a dial or pushing a joystick. Alternatively the robot’s own program supplies it, either as a fixed value written into the code or as a value computed by a higher-level planner that has decided where the arm should go next.

A setpoint need not always be a number. It may be a signal that changes over time, as when an arm is asked to follow a path rather than to reach a single point. It may even be a condition to be maintained rather than a value to be reached, as in the instruction given to a line-following robot: stay on the black line. In that case the underlying quantity being held at zero is the robot’s sideways offset from the line.

Q. A line-following robot is told to “stay on the black line”. Where is the number in that instruction?

A. The number is hidden, and finding it is the first step in designing the loop. The controlled quantity is how far the robot has strayed sideways from the centre of the line, and the setpoint for that quantity is zero. The sensors report the offset, and the controller works to drive it back to zero. An instruction that sounds verbal is still a numerical setpoint underneath.

16.2.2 Controller: the Decision-Making Unit

The controller is the decision-making part of the system. It reads the setpoint, works out what has to be done to achieve it, and issues an instruction to the parts that can act. In a school robot this is almost always a microcontroller such as an Arduino UNO, or a single-board computer such as a Raspberry Pi; in an industrial machine it may be a dedicated motion controller.

NoteDefinition

The controller is the element of a control system that receives the input, decides what corrective action is required, and produces the control signal that drives the rest of the system towards the desired behaviour.

Strictly speaking, the hardware is only where the controller lives. The controller itself is the method — the set of rules, written as a program, that turns a desired value into an instruction. Two robots with identical boards and identical motors can behave completely differently because the rules running inside them are different.

What the controller sends out is the control signal, also called the actuating signal: the electrical instruction that tells the driving parts how hard to act, and in which direction. For a direct current motor this is usually pulse width modulation (PWM), in which the supply is switched on and off rapidly and the fraction of time it stays on decides the average voltage the motor sees, and so its speed.

It is worth noticing what the controller is actually working with. It does not handle degrees or metres per second; it handles numbers read from pins and numbers written to pins. The engineer is the one who decides that a particular count corresponds to a particular angle. If that conversion is wrong, every decision built on it is wrong, however good the program looks.

16.2.3 Plant: the Physical System Being Controlled

The plant is the physical part of the system that is being controlled. It is where the real task actually happens: the movement, the rotation, the change of temperature.

NoteDefinition

The plant is the physical system being controlled, comprising the mechanical, electrical or other components whose behaviour is influenced by the control signal, and in which the controlled quantity is produced.

In a robotic arm, the plant is the joints, the motors and the moving links. In a drone, it is the propellers and the frame that respond to motor signals to produce lift and movement. In a conveyor belt system, it is the motorised belt that carries the objects along.

The actuator sits at the boundary between the controller and the plant. It receives the control signal, which is small and electrical, and produces the force or torque, which is large and physical. Many engineers simply count the actuator as part of the plant, since it is a physical component; when the two are named separately, the actuator is the part that applies the effort and the plant is what responds to it. Either way, no confusion arises so long as you can say which part decides and which part moves.

16.2.4 Output: the Real-World Action

The output is the actual effect produced in the real world: the value the controlled quantity has ended up at. It is what the setpoint was a statement about, and the two are directly comparable because they are quantities of the same kind, in the same unit.

A motor spins the wheels of a robot and the robot reaches a speed; that speed is the output. A gripper opens and the jaw separation reaches a certain width; that width is the output. A cooling fan runs and the enclosure settles at a certain temperature; that temperature is the output.

naming a component as the output What students do: write that “the output of the control system is the motor”, or “the output is the servo”. What happens: the answer cannot then be compared with the setpoint at all, because a setpoint of 45° cannot be compared with a motor. The whole logic of control, which rests on comparing two values of the same quantity, disappears. The fix: name the output as a quantity with a unit — the angle reached, the speed attained, the temperature held. The motor is part of the plant; the angle it produced is the output.

Table 16.1 sets the four elements side by side for three different machines. Reading across a row shows how differently the same role can be filled; reading down a column shows that a machine which looks nothing like a robot arm is built on exactly the same skeleton.

Table 16.1 – the four elements of a control system identified in three different machines, with the controlled quantity named in each case | Element | Robot arm joint | Line-following robot | Temperature-regulated enclosure | |—|—|—|—| | Input (setpoint) | Target joint angle, 45° | Sideways offset from the line, 0 mm | Air temperature, 25 °C | | Controller | Microcontroller running the joint program | Microcontroller reading the line sensors | Microcontroller reading the temperature sensor | | Control signal | PWM signal and direction pins to the motor driver | PWM signal to each of the two drive motors | Switching signal to the fan circuit | | Plant | Joint, gearbox, motor and link | Chassis, wheels and drive motors | Enclosure, its air and the fan | | Output (controlled quantity) | Angle actually reached, in degrees | Actual sideways offset, in millimetres | Actual air temperature, in degrees Celsius |

Notice that the control signal in every column is electrical and the output in every column is physical. That conversion, from a small signal to a real effect, is the whole reason a plant and an actuator exist.

16.2.5 Tracing the Flow of Information Through the System

It helps to follow a single command all the way through, because each element hands something different to the next. Fig. 16.2-2 traces one instruction to a robot arm joint.

Diagram

Fig. 16.2-2 – flow: setpoint of 45° entered in the program → controller decides how far and in which direction the joint must turn → control signal, a PWM waveform with a direction bit, sent to the motor driver → motor driver switches current to the joint motor → joint and link rotate → joint reaches an angle, which is the output; each arrow labelled with what it carries, marked in order as command, decision, signal, current, motion, result; no return arrow is to be drawn

Four different things travel along that chain. A desired value passes from the input to the controller. A decision passes from the controller to the driver as a low-current signal. Electrical power passes from the driver to the motor. Physical motion passes from the motor into the link, and ends as an angle.

Look carefully at Fig. 16.2-2 and one thing is missing. Nothing returns. The controller sends its instruction and hears nothing back, so it has no way of knowing what angle the joint actually reached. The system in Fig. 16.2-1 has exactly the same shape, and for the same reason.

That is not an oversight in the drawing. It is a real and widely used design, and the next section is about what such a system can and cannot do.

16.3 Open-Loop Control Systems

A system built exactly like the one traced at the end of the last section — a command going in, an action coming out, and nothing coming back — is called an open-loop control system. The name describes the shape of its block diagram: the path from input to output is a line with two loose ends, not a ring.

NoteDefinition

In an open-loop control system the output is determined by the input quantity alone; the output is not fed back to the input, and therefore has no influence on the control signal.

16.3.1 Control Without Feedback

In an open-loop system the controller acts on an assumption. It assumes that a particular command produces a particular result, because that is what happened when the machine was set up. Send full power for 3 s and the robot travels a metre; send 200 pulses to a stepper motor and the carriage moves 40 mm; run the pump for 20 s and the pot receives half a litre.

The assumption is not guessed. It comes from calibration: measuring, once and in advance, what a given command actually produces, and writing that relationship into the program. Calibration is real engineering work, and a well calibrated open-loop machine can be surprisingly good.

The weakness is that the calibration is a photograph of one moment. It was taken with a particular battery charge, a particular load, a particular floor surface and a particular amount of friction in the gearbox. Every one of those will change. The machine goes on obeying a relationship that has quietly stopped being true, and nothing in it is arranged to notice.

Fig. 16.3-1 shows this as a block diagram for a robot that dispenses water on a fixed timing.

Diagram

Fig. 16.3-1 – open-loop block diagram of a timed watering system; blocks in a row: “Input: run pump for 20 s” → “Controller: microcontroller timer” → “Actuator: pump and driver circuit” → “Plant: pot and soil” → arrow out labelled “Output: soil moisture actually reached”; a dashed box drawn round the empty space beneath the chain, labelled “no return path — the soil moisture is never measured”; all arrows point left to right

open-loop control Statement: the control signal is a function of the input alone, so the output has no effect on the control action. In words: the machine is told what to do, does it, and never checks the result. Why it holds: with no measurement of the output, there is no information available to change the command, so the command can only depend on what was decided before the action began. Used for: tasks where the relationship between command and result is stable and known, and where an occasional error does not matter. Watch out: accuracy depends entirely on the calibration staying true. Any disturbance, wear, load change or supply change passes straight through into the output, uncorrected.

16.3.2 Open-Loop Control in Robots and Everyday Machines

A microwave oven is the example most people already own. You set two minutes, and it heats for exactly two minutes. It never measures the temperature of the food, so it cannot know whether the food came out of the freezer or out of a warm kitchen. Whatever the state of the food, the timer runs out and the oven stops.

Q2. A microwave oven runs for exactly the time you set, even if the food is already hot. What kind of control is that, and what is it missing?

A. it is open-loop control. The oven acts on the input you gave it — a time — and produces heat for that duration. What it is missing is any measurement of the output. There is no temperature sensor reporting how hot the food has become, so there is no error to calculate and no reason for the oven to stop early or run longer. The setting is a command, not a target it checks.

A traffic signal running on a fixed cycle works the same way. Green for a set number of seconds, then red for a set number of seconds, whatever is actually on the road. If one approach is empty and another is choked, the cycle does not care; there is no measurement of traffic feeding back into the timing.

Robotics has its own well-known case: the stepper motor. A stepper turns through a fixed angle for each pulse it receives, and holds its position when it stops, so a driver can position a mechanism accurately by simply counting out pulses. Pen plotters, small milling tables and many 3D printing mechanisms are driven this way, with no sensor watching the axis at all.

That works beautifully until the carriage meets more resistance than the motor can overcome. The motor then fails to complete a step — it misses it — while the driver goes on counting as though nothing had happened. Every position after that is displaced by the missed step, and the fault is permanent for the rest of the job, as Fig. 16.3-2 shows.

Diagram

Fig. 16.3-2 – a pen plotter carriage on a horizontal axis; upper strip shows the intended pattern, a row of five evenly spaced marks; lower strip shows the actual pattern, with the first two marks correct and the last three all shifted to the left by the same small amount; the point of the shift labelled “one step missed here”; a note reading “the driver’s step count is unchanged; only the real position has moved”

Nothing in the machine is wrong afterwards, in the sense that the count and the command still agree perfectly. It is only reality that has moved.

16.3.3 Advantages of Open-Loop Control

Given all this, it is fair to ask why anybody builds an open-loop machine at all. There are four solid reasons.

Its design and operation are simple. With no measuring element and no comparison to arrange, there is less to design, less to build and much less to go wrong. Fewer parts also means fewer things that can fail in service.

It costs less. Sensors, the circuits that condition their signals and the processing capacity to handle them all cost money, and in a machine that will be made in thousands, leaving them out is a serious saving.

It is easy to implement, which is why almost every first robot in a school workshop is open-loop. A student can write a timed sequence and see a machine move on the same afternoon.

Its response is fast. No time is spent measuring the output and working out a correction, so the command takes effect at once. In a system that only has to switch something on, that directness is a genuine advantage.

Q3. If checking the result makes a machine more accurate, why are some robots still built with no sensors to check anything?

A. because checking costs something. Sensors, wiring and processing add money, weight, power draw and complexity, and every added part is another part that can fail. When the relationship between command and result is stable and well calibrated, and when a small error does not matter, an open-loop design does the job more cheaply, more simply and with a faster response. The engineering question is not “is feedback better?” but “is the error this task can tolerate larger than the error an open-loop design will produce?”

16.3.4 Limitations of Open-Loop Control

The limitations are the mirror image of the advantages, and they all follow from the same single fact: the output is never measured.

There is no error correction. The system cannot detect a mistake, so it certainly cannot put one right. If something goes wrong it carries on regardless, in exactly the manner it would have used had everything gone well.

Accuracy is low, and it drifts. A change in load, in supply voltage, in surface friction or in the mechanism’s own wear moves the output away from the intended value, and the machine has no means of pulling it back.

The system is not adaptive. A disturbance, an obstacle or a change in the environment leaves the original instruction untouched, because the instruction was decided before the disturbance existed.

There is no performance monitoring. At the end of the operation, nothing in the system can say whether the task succeeded. This matters more than it sounds. A machine that cannot report failure cannot be trusted to work unattended.

For these reasons open-loop control is unsuitable for the tasks that make robotics interesting. Balancing, navigation and object manipulation all require continuous adjustment while the task is under way, and continuous adjustment demands continuous measurement.

An open-loop drive keeps commanding full power into a jammed mechanism, because nothing tells it the motion has stopped. A stalled motor draws far more current than a turning one, and both the motor and the driver module heat rapidly. Disconnect the supply before freeing a jam, and never clear a stalled gearbox or wheel while the board is powered.

Program the robot to drive forward for a fixed 2 s at a fixed PWM value, and mark where the front edge stops. Repeat the same command five times from the same start line, measuring the distance each time in millimetres, and once more after the battery has run the robot for a few minutes. Record the spread between the shortest and longest run — that spread is the error an open-loop command hands you for free.

16.4 Closed-Loop Control Systems

The fault in every open-loop machine is the same, and so is the remedy: measure the output, and let the measurement come back and change the command. A system built that way is a closed-loop control system, also called a feedback control system, and the loop in the name is a real loop that can be traced with a finger on the block diagram.

NoteDefinition

A closed-loop control system is one in which the output is continuously measured and returned to be compared with the setpoint, and the controller acts on the difference between them so as to reduce it.

16.4.1 Feedback, the Summing Point and the Error Signal

Feedback is the returned measurement: a signal derived from the actual output and carried back to the input side of the system. It travels along the feedback path, which almost always begins at a sensor mounted on or near the plant.

Where the feedback arrives is called the summing point, drawn as a small circle with a cross in it. Two signals enter it — the reference input and the fed-back measurement — and one signal leaves it, the error. Fig. 16.4-1 shows the complete arrangement.

Diagram

Fig. 16.4-1 – block diagram of a closed-loop control system; from the left, an arrow labelled “Reference input (setpoint)” entering a small circle marked with a cross, the summing point; the incoming arrow marked with a plus sign; an arrow leaving the summing point labelled “Error signal” into a block labelled “Controller”; an arrow from Controller labelled “Control signal” into a block labelled “Plant”; an arrow leaving Plant to the right labelled “Output”; from that arrow a branch running downward and back to the left through a block labelled “Feedback element (sensor)” and returning into the summing point, with that incoming arrow marked with a minus sign; the whole return path labelled “Feedback path”

The minus sign on the returning arrow is the most important mark in the diagram. The measurement is subtracted from the setpoint, which is exactly the operation defined in section 16.1.3, and what leaves the summing point is therefore the error. Because the correction acts to oppose the departure from the setpoint, this arrangement is called negative feedback.

Get that sign the wrong way round and the system does not merely work badly; it works energetically in the wrong direction, driving the output further from the target with every pass round the loop.

closed-loop (feedback) control Statement: the control signal is a function of the error, where the error is the setpoint minus the measured output. In words: the machine keeps looking at what it has actually done, and acts on the gap between that and what it was asked to do. Why it holds: subtracting the measurement from the setpoint produces a signal that shrinks as the output approaches the target, so the correction naturally fades to nothing at the moment the target is reached. Used for: any task where the result must be right in spite of load changes, wear, friction or outside disturbances — joint positioning, speed regulation, balancing, heading control. Watch out: the loop trusts its sensor completely. A wrong measurement is obeyed as faithfully as a right one, and a correction that is too strong or applied too late can set the system oscillating instead of settling.

A loop is not run once. It runs over and over — read, compare, correct, read again — many times each second, and each pass is a fresh attempt on a slightly smaller error. The interval between passes matters as much as the arithmetic inside them. A loop that repeats a hundred times a second can catch a drone tipping; a loop delayed by a long wait in the program may only notice the fall after it has become unrecoverable.

16.4.2 Closed-Loop Control in Robots

A robotic arm in industry carries an encoder on each joint: a sensor that reports how far the shaft has actually turned. The controller compares that reported angle with the commanded angle and applies torque in whichever direction reduces the difference, continuing until the difference is negligible. The same arm running its motors open loop would be far less precise, because friction and payload would decide where it really stopped.

Q1. When a robot arm is told to move to 45°, how does it find out whether it actually reached 45°?

A. it measures the joint with a sensor, usually an encoder on the joint shaft, and feeds that measurement back to be subtracted from the setpoint. The result is the error. While the error is positive the controller keeps driving the joint further; when the error reaches zero the arm is where it was asked to be and the correction stops. Without that returned measurement the arm knows only what it was told, never what it did.

A self-balancing robot on two wheels is the same idea applied to something that will fall over if the idea fails. Gyroscopes and accelerometers report the tilt angle, the setpoint is upright, and the controller drives the wheels forward or backward to bring the base under the falling body, as Fig. 16.4-2 shows.

Diagram

Fig. 16.4-2 – a two-wheeled self-balancing robot seen from the side, leaning slightly forward; a vertical dashed line through the wheel axle labelled “setpoint: upright”; the body’s centre line labelled “measured tilt angle”; the angle between them shaded and labelled “error”; a horizontal arrow at the wheels pointing in the direction of the lean, labelled “wheels driven forward to bring the base under the body”; a small block labelled “gyroscope and accelerometer” marked on the body

An autonomous drone holds its position in the same way against a very unhelpful environment. Wind pushes it off course, its sensors detect the drift, and the controller changes the thrust of individual motors to bring it back. Stable hovering is not the absence of disturbance; it is a loop correcting disturbance continuously.

Conveyor systems in a packing hall use photoelectric sensors or encoders to watch belt speed and the position of items on it, so the controller can hold the spacing between packets or stop the belt exactly where a packet must be picked up. The line-following robot of section 16.2.1 belongs in the same family: its sensors measure the sideways offset, and its setpoint for that offset is zero.

16.4.3 Advantages of Closed-Loop Control

Accuracy is higher, and it stays higher. Because the output is measured against the setpoint continuously, a departure is detected while it is small and corrected before it grows.

Error correction is automatic. Whether the disturbance is a heavier load, a stiffer joint, a slope in the floor or a drop in battery voltage, the system does not need to be told which one has occurred. All of them show up as the same thing — an error — and all of them are answered by the same corrective action.

The system adapts to disturbances in real time, which is what allows a robot to work in an environment it cannot fully predict. This is the property that separates a machine that works in a laboratory from one that works in a field.

Behaviour stays consistent. Two runs of the same task give closely similar results even though the conditions were not identical, and that consistency is what makes automated production possible.

Safety improves. Because the system is watching its own output, it can recognise a condition that should not be occurring — a speed above the permitted maximum, motion in the wrong direction, a joint that is not moving although full effort is being applied — and act on it before damage is done.

16.4.4 Limitations of Closed-Loop Control

Complexity increases. Sensors, their mounting, their wiring, the conditioning of their signals and the program that interprets them are all additional work, and each is an additional way for the machine to fail.

Cost increases with it. An encoder on every joint, and a controller with the capacity to service them all, is a substantial part of the price of an industrial arm.

The initial response is slightly slower. The loop must acquire a reading and compute a correction before it acts, and that processing takes time — a small delay in a fast microcontroller, a noticeable one in a system with slow sensors or a heavily loaded processor.

Power consumption is higher. Sensors draw current, the processor works harder, and the actuators are continually making small corrections rather than sitting still. On a battery-powered robot this is not a footnote; it is flight time or run time lost.

Two further limitations deserve more attention than they usually receive. The first is that the loop believes its sensor absolutely. A sensor that has come loose, or is reading through dust, or is returning a value in the wrong unit, produces an error signal that is confidently wrong, and the controller acts on it with full authority. The second is that feedback can make a system worse behaved rather than better: a correction that is too aggressive, or that arrives too late, can carry the output past the target and set up a swing that never settles. Section 16.6 returns to this.

16.4.5 Choosing Between Open-Loop and Closed-Loop Control

The distinction now needs to be drawn sharply, because in design work it is the first decision made and the hardest to reverse later. Table 16.2 sets the two side by side.

Table 16.2 – open-loop control system compared with closed-loop control system | Feature | Open-loop control system | Closed-loop control system | |—|—|—| | Use of feedback | Output is not measured or returned | Output is measured and returned to the summing point | | Basis of control action | The input command alone | The error between setpoint and measured output | | Error correction | None; a deviation goes uncorrected | Automatic and continuous while the loop runs | | Accuracy under changing conditions | Falls as load, supply or wear change | Held, because the change appears as an error and is corrected | | Response to a disturbance | No response; the original command continues | The disturbance is detected and opposed | | Performance monitoring | The system cannot tell whether the task succeeded | The output is known at every instant | | Complexity and cost | Simple and cheaper; fewer components | More complex and costlier; sensors and processing added | | Power consumption | Generally lower | Higher, from continuous sensing and correction | | Speed of initial response | Faster, as no measurement precedes the action | Slightly slower, by the time taken to measure and compute | | Risk of oscillation | None; the system cannot chase its own output | Present if the correction is badly proportioned or delayed | | Suitability in robotics | Simple, non-critical tasks with a stable command-to-result relationship | Precision tasks: positioning, balancing, navigation, manipulation |

Reading the table as a designer rather than as a list, three questions decide the choice. Does the relationship between command and result stay stable in the conditions this machine will actually meet? How much error can the task tolerate before it fails? And is there a sensor that can measure the controlled quantity at all, at a price and a rate the machine can afford?

If the relationship is stable and the tolerance is generous, an open-loop design is the better engineering answer, not the lazy one. If either condition fails, no amount of careful calibration will save an open-loop machine, because the thing that defeats it will happen after the calibration was done.

Most real robots are not purely one or the other. A single machine may position its arm under closed-loop control while switching a cooling fan open loop, because the arm must be right to a millimetre and the fan only has to be on. The question is asked separately for each quantity being controlled.

16.5 Sensors, Controllers and Actuators in the Control Loop

A modern robot is a mechatronic machine: electronics and mechanics designed together, neither of them making sense on its own. When the block diagram of section 16.4 is replaced by real hardware, the loop is carried by three kinds of part — a sensor, a controller and an actuator — working round the plant.

Q5. Which three parts must work together before a robot can measure a mistake and put it right?

A. a sensor, a controller and an actuator. The sensor measures what is actually happening and reports it as a signal. The controller compares that report with the setpoint, finds the error and decides on a correction. The actuator turns that decision into physical action. Remove any one of the three and the loop breaks: with no sensor there is nothing to compare, with no controller nothing to decide, and with no actuator nothing changes in the world.

Each of the three has to be adequate for the job, and “adequate” means something specific in each case.

16.5.1 The Sensor as the Measuring Element

In the control loop the sensor is the measuring element. It acts as the system’s sense organ, converting a physical quantity — position, speed, distance, tilt, temperature or pressure — into a signal the controller can read.

Its first requirement is easy to state and easy to get wrong: it must measure the quantity that is actually being controlled. If the setpoint is a joint angle, a sensor reporting motor current is not feedback for that loop, however useful it may be for something else.

Its second requirement is resolution — the smallest change in the quantity that the device can distinguish. Resolution has to be finer than the tolerance the task demands. A sensor that can only tell one degree from the next cannot help a loop that has to hold half a degree; the error will simply be invisible to it.

Fig. 16.5-1 shows the arrangement of a single-joint test rig of the kind a school workshop can build, with the encoder mounted where it can see the shaft.

Photograph

Fig. 16.5-1 – a geared direct current motor with a rotary encoder fitted at the rear of the motor shaft, photographed close up against a plain background; the motor body, the gearbox, the output shaft and the encoder disc or module with its ribbon of wires must all be identifiable in the frame. Sourcing note: a component photograph or workshop bench photograph is suitable; images in which the encoder is hidden behind the mounting bracket, or in which the motor is buried inside an assembled robot, are unusable.

is this encoder good enough for the tolerance demanded?

Given: an encoder producing 20 pulses per revolution of the motor shaft; a gearbox of ratio 48:1 between the motor shaft and the joint; a link of length \(r = 150 \text{ mm} = 0.150 \text{ m}\) from the joint to its tip; the task requires the joint angle to be held within \(0.5°\).

Find: the smallest change in joint angle the encoder can detect, whether it meets the tolerance, and what that smallest step amounts to at the tip of the link.

Method: the gearbox turns the motor shaft 48 times for one turn of the joint, so the counts per revolution of the joint are \(20 \times 48\). The angle per count is \(360°\) divided by that figure. The tip movement follows \(s = r\theta\), with \(\theta\) converted to radians using \(1° = \pi/180 \text{ rad}\).

\[\text{counts per joint revolution} = 20 \times 48 = 960\]

\[\text{angle per count} = \frac{360°}{960} = 0.375°\]

\[0.375° \times \frac{\pi}{180} = 0.006545 \text{ rad}\]

\[s = 0.150 \text{ m} \times 0.006545 \text{ rad} = 0.00098 \text{ m} = 0.98 \text{ mm}\]

the encoder resolves \(0.375°\) at the joint, which is finer than the \(0.5°\) demanded, so it is adequate. One count corresponds to about \(1.0 \text{ mm}\) of movement at the tip of the link.

Check: the magnitude is sensible. A step of about a millimetre at the tip of a 150 mm link is roughly the thickness of a coin, and the tolerance of \(0.5°\) works out to about \(1.3 \text{ mm}\) at the same radius — comfortably larger than one count, which is exactly the relationship a usable measuring element must have.

Two further properties decide whether a sensor can serve in a loop. It must respond faster than the loop repeats, or the controller will keep acting on stale information. And its reading must be steady enough that random fluctuation is not mistaken for a real error, since the controller cannot tell noise from news.

16.5.2 The Controller as the Deciding Element

The controller is the deciding element. Each time round the loop it reads the feedback, subtracts it from the setpoint, and converts the error into a control signal of the right size and direction.

Everything it works with is a number. The encoder gives it counts, not degrees; the analogue input gives it a value between 0 and 1023, not millimetres. The engineer supplies the conversion between the two, and a loop built on a wrong conversion behaves confidently and wrongly, because the arithmetic inside it is faultless.

The controller also decides how often the loop runs, and that is a design choice, not an accident. Any time the program spends elsewhere is time the loop is blind.

a long wait inside a control loop What students do: put delay(500) inside the loop that reads the sensor and drives the motor, usually to slow down the printed output so it can be read. What happens: for half a second the controller neither measures nor corrects anything. A joint that is drifting or a robot that is tipping goes uncorrected through that whole interval, and the correction that finally arrives is based on a reading that is already old. The machine looks sluggish and unstable, though the control arithmetic is perfectly correct. The fix: keep the loop free of blocking waits. Time the loop with millis() and let it repeat as fast as the sensor allows, printing only occasionally if printing is needed at all.

16.5.3 The Actuator as the Correcting Element

The actuator is the correcting element, the part that carries the controller’s decision into the physical world. In robots this is usually a motor or a servo; in industrial plant it may be a hydraulic cylinder or a pneumatic device. Whatever the type, it converts a small electrical signal into force, torque or motion.

For an actuator to close a loop it must have authority: enough torque or force to move the plant against friction, gravity and load. A loop with an underpowered actuator is a loop that measures its error accurately, computes the correct answer, and then cannot do anything about it.

It must also be able to act in both directions, because a correction is as often a reduction as an increase. And it needs a driver between it and the controller. A microcontroller pin can supply only a few tens of milliamperes, while even a small geared motor draws hundreds, so the pin carries the decision and a driver module such as the L298N carries the current.

There is one honest limitation worth knowing before you meet it on the bench. Below a certain size, a control signal produces no movement at all, because it cannot overcome static friction in the gearbox. The loop then settles with a small error it can measure but cannot remove, and the joint sits stubbornly a fraction of a degree away from its target.

A closed-loop robot moves on its own initiative. Nudge a joint under active position control and the motor will drive back at once to correct the error, with no command from anyone. Clamp the base to the bench before switching on, keep fingers clear of joints and pinch points while a loop is running, and disconnect the supply before adjusting the mechanism.

16.5.4 Closing the Loop in a Simple Robot

Putting the three elements together gives the method that every position loop follows, whatever the hardware. The method is written out before any program, because the program is only a translation of it.

closed-loop position control of a robot joint Input: the setpoint angle for the joint, in degrees; the angle reported by the joint encoder each time round the loop. Output: a control signal to the motor driver, consisting of a direction and a magnitude, repeated until the joint is within tolerance of the setpoint. Method:

1. Read the setpoint angle.
2. Read the encoder count and convert it to the measured angle in degrees.
3. Compute error = setpoint angle - measured angle.
4. If the magnitude of error is less than the tolerance, send zero drive and go to step 8.
5. If error is positive, set the direction to move the joint forward; otherwise set it to reverse.
6. Set the drive magnitude in proportion to the magnitude of error, limited to the maximum
   the motor and driver may be given.
7. Send the direction and magnitude to the motor driver.
8. Wait until the next loop interval, then repeat from step 2.

Why it works: the drive is proportional to the error, so a joint far from its target is driven hard and a joint nearly there is driven gently. As the joint approaches the setpoint the error shrinks, the drive fades with it, and the motion comes to rest at the target instead of arriving at full speed. Watch out: three boundary cases decide whether this works on real hardware. If the tolerance in step 4 is smaller than the encoder’s resolution, the condition can never be satisfied and the joint hunts round the target forever. If the drive computed in step 6 falls below the level that overcomes static friction, the joint stops short and stays there. And if the loop interval in step 8 is long, every correction is based on an old reading.

Notice what step 6 does not say. It does not say “drive at full power until the target is reached”, which is the instinctive first attempt and the reason so many first attempts overshoot. Making the correction proportional to the error is the first real idea in control design, and the next section takes it further.

16.6 Proportional-Integral-Derivative Control

Closing the loop tells a robot that it is wrong. It does not by itself tell the robot how hard to push back. That second question has its own answer, and the most widely used answer in engineering is the PID control system, named for the three corrective actions it combines: proportional, integral and derivative.

16.6.1 Why a Simple On-Off Correction Is Not Enough

The obvious way to use an error is to ignore its size. Drive at full power while the joint is short of the target, and switch off the moment it arrives. This is on-off control, and it is the first thing most people try.

It fails for a reason that has nothing to do with electronics. A joint arriving at full speed carries momentum, and a motor switched off does not stop instantly. The joint sails past the target, which is called overshoot. The error is now negative, so the controller drives at full power the other way, and the joint sails past again. The output swings back and forth about the setpoint instead of settling on it.

Q4. Why can a machine that corrects its own mistakes still shoot past its target and swing back and forth?

A. because the correction takes time to have its full effect, and because a moving mass does not stop the instant the driving effort is removed. If the controller keeps applying a strong correction right up to the target, the momentum already built up carries the output past it. The error then reverses sign, the correction reverses too, and the same thing happens in the other direction. The cure is not a bigger correction but a better-shaped one: ease off as the target approaches, and take account of how fast the output is closing on it.

Making the drive proportional to the error, as the algorithm in section 16.5.4 did, is a large improvement. The correction now fades as the target is approached, so the joint slows on its way in rather than arriving at full speed.

Even so, two problems survive. If the proportional response is made strong enough to move a stiff joint quickly, it is often strong enough to overshoot. And a proportional response has a subtler defect: a joint held against gravity or against friction needs a permanent, non-zero drive just to stay where it is. Since a proportional controller produces zero drive at zero error, the joint quietly settles at a small error where the drive it produces exactly balances the load. That leftover is called the steady-state error, and no amount of patience removes it.

Fig. 16.6-1 shows the three behaviours together.

Diagram

Fig. 16.6-1 – graph of joint angle against time; horizontal dashed line across the plot labelled “setpoint”; three curves starting from the same point at the left; curve A labelled “on-off control” swinging above and below the setpoint in large oscillations that do not settle; curve B labelled “proportional only” rising smoothly and levelling out just below the setpoint, with the small vertical gap marked and labelled “steady-state error”; curve C labelled “proportional, integral and derivative” rising smoothly, with only a slight overshoot, and settling on the setpoint line; axes labelled “time” and “joint angle”

16.6.2 The Three Corrective Actions

A PID controller answers the two surviving problems by adding two more terms to the proportional one. Each of the three looks at the error in a different way — at what it is now, at what it has been, and at where it is going — and the control signal sent to the actuator is the sum of the three.

The proportional action responds to the present size of the error. The bigger the departure from the setpoint, the stronger the correction. It does the main work of driving the output towards the target, and on its own it leaves the steady-state error described above.

The integral action responds to accumulated past error. A small error that persists is added up over time, and the correction grows steadily until it is large enough to remove the error completely. This is precisely what eliminates a steady-state offset. Its cost is sluggishness: an accumulated correction takes time to build, and takes time to unwind again once the target is reached, which can itself cause overshoot.

The derivative action responds to the rate at which the error is changing. If the output is closing on the setpoint quickly, the derivative term pushes back against the approach before the target is reached, damping the motion. It is the term that anticipates, and it is what reduces overshoot and sudden swings. Its cost is sensitivity to noise: a jumpy sensor reading looks, to a derivative term, like a violently changing error.

Table 16.3 – the three corrective actions of a PID controller | Feature | Proportional (P) | Integral (I) | Derivative (D) | |—|—|—|—| | What it responds to | The size of the error now | The error accumulated over past time | The rate at which the error is changing | | Question it answers | How far off am I? | How long have I been off? | How fast am I closing in? | | Main effect | Provides the bulk of the corrective drive | Removes the leftover steady-state error | Damps the approach and reduces overshoot | | Behaviour when it is too strong | Overshoot and oscillation | Slow, drifting response that overshoots late | Jerky drive; sensor noise is amplified | | Behaviour when it is absent | The response is weak and slow | A small permanent error remains | The output overshoots and swings before settling |

PID control Statement: the control signal is formed by combining three corrective actions derived from the same error — one proportional to its present value, one to its accumulated past value, and one to its rate of change. In words: react to how far off you are, to how long you have been off, and to how quickly things are changing. Why it holds: the three terms address three different failures. Proportional action alone cannot supply a drive at zero error; integral action supplies it. Neither can see motion coming; derivative action does. Used for: motor speed regulation, robotic arm position control and balance in self-balancing robots — tasks where both precision and a smooth approach matter. Watch out: the strength of each term has to be chosen for the particular machine, usually by experiment. Values that suit one robot will make another oscillate, sit sluggishly short of its target, or shudder on a noisy sensor reading.

16.6.3 PID Control in a Self-Balancing Robot

A two-wheeled robot that stands upright, in the manner of a small self-balancing scooter, is the clearest demonstration of all three actions at once, because it fails visibly the moment any of them is wrong.

The gyroscope and accelerometer report the tilt angle; the setpoint is upright; the error is the lean. Each pass round the loop, the proportional action reacts to how far the robot is leaning, the integral action to a small lean it has been holding for some time — the kind produced by a battery mounted slightly off centre — and the derivative action to how fast the lean is growing. The sum of the three sets how hard, and in which direction, the wheels are driven so as to bring the base back under the body.

Diagram

Fig. 16.6-2 – three small panels side by side, each showing the same two-wheeled robot leaning forward; panel 1 labelled “Proportional: reacts to how far it has leaned”, with the lean angle marked; panel 2 labelled “Integral: reacts to a small lean held over time”, with a row of three faint outlines of the same slight lean and a clock symbol; panel 3 labelled “Derivative: reacts to how fast the lean is growing”, with two outlines a moment apart and an arrow showing the increasing angle; a caption strip beneath all three reading “the three corrections are added together and sent to the wheels”

This process repeats many times each second, which is why a well-tuned balancing robot appears simply to stand still. What looks like stillness is a rapid sequence of small corrections, each one made before the previous departure had time to become visible.

Push the robot gently and the same three actions answer the push: the proportional term reacts to the displacement, the derivative term to the speed of the shove, and the integral term to any residual lean left afterwards. It recovers and stands upright again, which no open-loop machine could do, because no open-loop machine would know it had been pushed.

With the robot clamped so that its wheels are clear of the bench, set a joint or wheel to drive at full effort until it reaches a marked target, and watch how far past the mark it runs. Then change the drive so that it falls in proportion to the remaining distance, and compare where it stops. Note in your file which version reached the mark faster, and which one stayed there.

16.7 Control for Accurate, Efficient and Safe Robot Performance

A control system is not added to a robot for its own sake. It is added because three demands are made of every working machine: that it does the job correctly, that it does not waste what it is given, and that it does not hurt anyone or destroy itself. Each of the three is met, or missed, in the design of the loop.

16.7.1 Accuracy and Stability

Feedback improves accuracy by making the machine’s own result part of its decision. A joint that has fallen short of its target produces an error, and the error produces the drive that removes it. The improvement is real, and it is the reason precision robots are built this way.

But feedback improves accuracy only up to the honesty of the measurement. The loop drives the measured value to the setpoint, not the true value. If the encoder was zeroed at a position that was not truly zero, or if the conversion from counts to degrees is out by a small factor, the loop will work faithfully and settle the joint in the wrong place, every time, without any sign of distress.

Q. A robot arm returns to the same wrong point again and again, with hardly any scatter. Is that a control fault?

A. No. Returning to the same point repeatedly is excellent repeatability, which means the loop is doing its job: it is holding the measured value at the setpoint reliably. Landing on the wrong point is poor accuracy, and that comes from the relationship between the measurement and the real world — a zeroing error, a wrong conversion factor, or a shifted sensor mounting. The cure is calibration, not a change to the controller.

The second requirement is stability: the property of settling at the setpoint and staying there rather than oscillating about it or drifting away. Stability is not guaranteed by closing the loop; as section 16.6 showed, a correction that is too strong or arrives too late can make behaviour worse than doing nothing.

An unstable loop is easy to recognise on the bench. The joint buzzes about its target, or the balancing robot rocks in a swing that grows rather than dies away. What that behaviour indicates is not a broken component but a badly proportioned correction, and the fix is in the design of the control action rather than in the hardware.

16.7.2 Efficiency and the Cost of Correction

Correction is not free. Every pass round the loop, the sensor draws current, the processor works, and the actuator is asked to do something. A loop that fusses continuously — nudging the motor first one way and then the other because sensor noise keeps producing tiny errors — burns energy and wears the mechanism without improving the result at all.

The usual answer is a deadband: a small band on either side of the setpoint within which the error is treated as zero and no correction is issued. The band is set a little wider than the sensor’s noise and a little narrower than the tolerance the task demands. Fig. 16.7-1 shows the difference this makes.

Diagram

Fig. 16.7-1 – graph of joint angle against time, drawn twice one above the other; upper plot labelled “no deadband”: the trace jitters continuously in small rapid corrections about the setpoint line, with a row of arrows beneath labelled “motor driven on every pass”; lower plot labelled “with a deadband”: a shaded horizontal band centred on the setpoint line and labelled “deadband, wider than the sensor noise”, the trace entering the band and then running flat with no corrections, and only one arrow beneath, labelled “motor driven only when the error leaves the band”; both plots share the axes labels “time” and “joint angle”

Efficiency also depends on how the approach is shaped. A drive that fades as the target is neared, and stops without overshoot, does the job in one movement. A drive that overshoots and comes back does the same job two or three times over, using two or three times the energy and adding that much wear to the gears.

On a battery-powered robot this arithmetic shows up directly as running time.

what continuous correction costs in running time

Given: a battery of capacity \(C = 2200 \text{ mAh}\); the robot’s steady draw with its loop settled and the drive idle, \(I_1 = 350 \text{ mA}\); the draw when the loop is jittering and the motors are being nudged constantly, \(I_2 = 470 \text{ mA}\).

Find: the running time in each case, and the time lost to the jitter.

Method: for a roughly constant current, running time is capacity divided by current, \(t = C / I\). With capacity in milliampere-hours and current in milliamperes, the result comes out directly in hours.

\[t_1 = \frac{2200 \text{ mAh}}{350 \text{ mA}} = 6.3 \text{ h}\]

\[t_2 = \frac{2200 \text{ mAh}}{470 \text{ mA}} = 4.7 \text{ h}\]

\[\Delta t = 6.3 \text{ h} - 4.7 \text{ h} = 1.6 \text{ h}\]

about \(6.3 \text{ h}\) of running when the loop settles quietly, against about \(4.7 \text{ h}\) when it does not — roughly \(1.6 \text{ h}\), or a quarter of the working time, lost to corrections that achieve nothing.

Check: the magnitude is sensible, and if anything optimistic. A cell delivers less than its rated capacity at a heavier draw, so the second figure in practice would be lower still, which only widens the gap.

16.7.3 Safety and the Detection of Unsafe Conditions

The same measurement that makes a robot accurate also lets it recognise that something has gone wrong. This is the safety argument for closed-loop design, and it is separate from the accuracy argument.

A loop that knows the commanded speed and the actual speed can detect an overspeed. A loop that knows the commanded direction and the measured direction can detect a reversal, which usually means a motor has been wired backwards. A loop that sees full drive being applied while the measured position does not change has detected a stall, and can cut the drive before the motor overheats. An open-loop machine can detect none of these, because it measures nothing.

Software limits work the same way: the controller refuses a command that would take a joint beyond the angle its mechanism allows, because it can compare the requested angle with a known limit. This prevents a great many collisions between an arm and its own frame.

What software limits cannot do is protect against the controller itself failing. If the program crashes or a driver latches on, the very part that was supposed to enforce the limit is the part that has failed. This is why serious machines carry protection that does not depend on the controller at all — a limit switch wired to interrupt the drive directly, and an emergency stop that removes power rather than politely asking the software to stop.

Never treat a software limit as the only thing standing between a moving joint and a person. Before working on or near the mechanism, isolate the supply at the switch or disconnect the battery, and confirm the drive is dead by trying to move the joint by hand. A program that has stopped responding may still be holding an output high.

Taken together, these three demands explain why so much design effort goes into a part of the robot that never touches the work. Accuracy decides whether the task is done correctly, efficiency decides how long the machine can keep doing it, and safety decides whether it may be allowed near people at all. All three are settled by how the loop is built.

16.8 Summary

  1. A control system is an arrangement of parts that makes a device or machine behave in a specified, desired way, by acting on an input that states the required value of the quantity being controlled.
  2. Sending an instruction and achieving a result are two different things. Battery voltage, load, friction, wheel slip and small differences between nominally identical motors all mean that an obeyed command can still produce a wrong outcome.
  3. A disturbance is an outside influence that pushes a system away from the behaviour it was asked for. Real environments supply them constantly, and a machine that cannot answer them cannot be trusted to work unattended.
  4. The setpoint is the desired value of the controlled quantity, stated in the same unit as that quantity. The measured value is what a sensor reports the quantity actually to be.
  5. The error is the setpoint minus the measured value, taken at the same instant and in the same unit: \(e = \theta_{sp} - \theta_{meas}\). Its magnitude says how far off the system is and its sign says in which direction. Reversing the subtraction drives every correction the wrong way.
  6. A small angular error at a joint becomes a much larger displacement at the tip of a link, since \(s = r\theta\) with the angle in radians. An error invisible at the joint can be large enough to fail the task at the gripper.
  7. Every control system is built from four elements: the input, the controller, the plant and the output. These are roles rather than separate boxes, and one microcontroller may carry more than one of them.
  8. The input is the setpoint or reference signal, supplied by an operator or by the robot’s own program. An instruction that sounds verbal, such as “stay on the line”, still has a numerical setpoint underneath it — in that case, zero sideways offset.
  9. The controller receives the input, decides what action is needed, and issues the control signal, also called the actuating signal. The hardware only hosts the controller; the controller itself is the set of rules the program applies.
  10. The plant is the physical system being controlled, in which the controlled quantity is produced. The actuator sits between controller and plant, converting a small electrical signal into force, torque or motion.
  11. The output is the actual value reached by the controlled quantity, named with its unit — an angle, a speed, a temperature. It is not a component, and naming a component as the output destroys the comparison on which control depends.
  12. In an open-loop control system the output is determined by the input alone; it is not measured and does not influence the control signal. Such a system acts on a relationship established once by calibration.
  13. Open-loop control is simple, cheap, easy to implement and fast in response, because nothing is measured before acting. A microwave timer, a fixed-cycle traffic signal and a stepper-driven plotter axis all work this way.
  14. Open-loop control offers no error correction, low and drifting accuracy, no adaptation to disturbance and no performance monitoring. It is therefore unsuited to balancing, navigation and object manipulation, which need continuous adjustment.
  15. In a closed-loop control system the output is measured continuously, returned along the feedback path, and compared with the setpoint at the summing point; the controller then acts on the difference.
  16. Because the measurement is subtracted from the setpoint, the arrangement is called negative feedback, and the correction opposes the departure from the target. A sign reversed here drives the output away from the setpoint rather than towards it.
  17. Closed-loop control gives higher and better-sustained accuracy, automatic error correction whatever the cause, adaptation to disturbance, consistent behaviour and the ability to detect unsafe conditions.
  18. Its costs are greater complexity, higher price, a slightly slower initial response, higher power consumption, complete dependence on the honesty of the sensor, and the possibility of oscillation if the correction is badly proportioned or delayed.
  19. The choice between the two designs turns on three questions: whether the command-to-result relationship stays stable in service, how much error the task can tolerate, and whether the controlled quantity can be measured at an affordable price and rate. Many robots use both, one quantity at a time.
  20. A robotic control loop is carried by three kinds of part: sensors, controllers and actuators. The sensor measures, the controller decides, the actuator corrects; remove any one and the loop breaks.
  21. A sensor must measure the quantity actually being controlled, resolve changes finer than the tolerance demanded, respond faster than the loop repeats, and read steadily enough that noise is not mistaken for error.
  22. A controller works only in numbers read from and written to pins, so the conversion between counts and physical units is the engineer’s responsibility. Time spent waiting inside the loop is time the system is blind and uncorrected.
  23. An actuator must have the authority to move the plant against friction, gravity and load, must act in both directions, and needs a driver between it and the controller because a logic pin cannot supply motor current. Below a certain drive it cannot overcome static friction, leaving a small error it can measure but not remove.
  24. On-off correction overshoots, because a moving mass does not stop when the drive is removed; the error then reverses and the output swings about the setpoint. Making the drive proportional to the error fades the correction as the target is approached.
  25. A proportional response alone leaves a steady-state error wherever a permanent drive is needed to hold position against gravity or friction, since zero error produces zero drive.
  26. A PID control system combines three corrective actions on the same error: proportional to its present size, integral to its accumulated past value, and derivative to its rate of change. The proportional action does the main work, the integral action removes the steady-state error, and the derivative action damps the approach and reduces overshoot.
  27. Each PID action has a cost when it is too strong: proportional gives oscillation, integral gives a slow drifting response, derivative amplifies sensor noise. The strengths must be chosen by experiment for the particular machine.
  28. Feedback drives the measured value to the setpoint, not the true value. Returning repeatedly to the same wrong point is good repeatability with poor accuracy, and the cure is calibration rather than a change of controller.
  29. Stability is the property of settling at the setpoint and staying there. Closing the loop does not guarantee it; a correction that is too strong or too late produces buzzing, rocking or a growing swing.
  30. Correction costs energy and wear. A deadband set wider than the sensor noise and narrower than the task tolerance stops needless nudging, and a movement that reaches the target once, without overshoot, does the work a jittering loop repeats.
  31. For a roughly constant draw, running time is \(t = C / I\), so continuous unnecessary correction is paid for directly in hours of battery life lost.
  32. Measurement is what lets a robot recognise overspeed, a reversed direction, a stall or a joint limit being exceeded, and act before damage is done. Software limits cannot protect against failure of the controller itself, which is why serious machines carry limit switches and an emergency stop that remove power independently of the program.

16.9 Key Terms

  1. Actuator — the part of a system that converts the controller’s electrical decision into physical action, such as a motor turning a joint or a servo moving a gripper.
  2. Calibration — measuring in advance what a given command actually produces, and writing that relationship into the program so the machine can act on it later.
  3. Closed-loop control system — a control system in which the output is continuously measured and returned for comparison with the setpoint, the controller acting on the difference so as to reduce it.
  4. Control signal — the electrical instruction issued by the controller, stating how hard the driving parts must act and in which direction; also called the actuating signal.
  5. Control system — an arrangement of parts that makes a device or machine behave in a specified, desired way, by acting on an input that states the required value of the quantity being controlled.
  6. Controller — the element that receives the input, decides what corrective action is required, and produces the control signal.
  7. Deadband — a small band on either side of the setpoint within which the error is treated as zero and no correction is issued, used to stop needless correction caused by sensor noise.
  8. Disturbance — an outside influence, such as a load change, a slope or a gust of wind, that pushes a system away from the behaviour it was asked for.
  9. Encoder — a sensor fitted to a shaft that reports how far it has actually turned, used as the feedback element in joint position loops.
  10. Error — the setpoint minus the measured value of the controlled quantity, taken at the same instant and in the same unit; its magnitude gives the distance from the target and its sign the direction.
  11. Feedback — a signal derived from the actual output and carried back to the input side of the system for comparison with the setpoint.
  12. Negative feedback — an arrangement in which the measured output is subtracted from the setpoint, so that the resulting correction opposes the departure from the target.
  13. Open-loop control system — a control system in which the output is determined by the input quantity alone; the output is not fed back and has no influence on the control signal.
  14. Output — the actual value reached by the controlled quantity, named with its unit, such as the angle attained, the speed held or the temperature reached.
  15. Overshoot — the amount by which the output passes beyond its setpoint before the control action brings it back.
  16. PID (proportional-integral-derivative) control system — a closed-loop control method that forms its control signal from three corrective actions on the same error: proportional to its present size, integral to its accumulated past value, and derivative to its rate of change.
  17. Plant — the physical system being controlled, comprising the components whose behaviour is influenced by the control signal and in which the controlled quantity is produced.
  18. PWM (pulse width modulation) — a way of setting the average voltage delivered to a device by switching the supply on and off rapidly and varying the fraction of time it stays on.
  19. Resolution — the smallest change in a quantity that a measuring device can distinguish, which must be finer than the tolerance the task demands.
  20. Sensor — a device that converts a physical quantity such as position, angle, distance, tilt or temperature into an electrical signal the controller can read.
  21. Setpoint — the desired value of the controlled quantity, supplied to the control system as its input and stated in the same unit as that quantity; also called the reference input.
  22. Stability — the property of a control system of settling at its setpoint and remaining there, rather than oscillating about it or drifting away.
  23. Steady-state error — the small permanent error left when a proportional response alone must hold a load, since zero error produces zero drive; also called steady-state offset.
  24. Summing point — the place in a closed-loop system, drawn as a small circle with a cross, where the fed-back measurement is subtracted from the setpoint to produce the error.

16.10 Objective Questions

  1. In a control system, the quantity that states what the system is required to achieve is called the
  1. output
  2. setpoint
  3. plant
  4. control signal
  1. A machine that carries out its instruction and never measures the result is best described as
  1. a closed-loop control system
  2. a feedback control system
  3. an open-loop control system
  4. a PID control system

  1. Fill in the blank: the point in a closed-loop block diagram where the fed-back measurement is subtracted from the reference input is called the ____________.

  2. State whether the following is true or false, giving the correct statement where it is false: in a closed-loop system, the feedback signal is added to the setpoint so that the correction reinforces the motion already taking place.

  3. Match each item in Column A with the description in Column B that fits it.

Column A Column B
a. Sensor i. Converts the controller’s decision into force or motion
b. Controller ii. The physical system in which the controlled quantity is produced
c. Actuator iii. Reports the actual value of the controlled quantity
d. Plant iv. Computes the error and issues the control signal
  1. Assertion (A): a proportional-only controller usually leaves a small permanent error when a joint must be held against gravity. Reason (R): a proportional controller produces zero drive when the error is zero.
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  1. Assertion (A): closing the loop around a robot joint guarantees that the joint will settle at the correct physical position. Reason (R): a closed-loop controller drives the measured value of the controlled quantity towards the setpoint.
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.

16.11 Check Your Concepts

  1. Define a control system, and give the full form of PID.

  2. Name the four elements of a control system. For a fan-cooled enclosure held at 25 °C, state what fills each of the four roles, and name the output with its unit.

  3. Differentiate between an open-loop control system and a closed-loop control system on any four points, giving one example of each from robotics.

  4. The shoulder joint of a robot arm is commanded to 30.0°, and its encoder reports 27.2°. The link from the joint to its tip is 250 mm long. Calculate the error at the joint and the distance by which the tip falls short of its intended position, and state what the sign of the error tells the controller to do.

  5. A robot arm under closed-loop control returns again and again to almost exactly the same point, but that point is 4 mm away from the hole it is supposed to enter. Give reasons for this behaviour, and state where the fault must be looked for.

  6. A school team is building a robot that must (i) position a gripper on a marked target to within one millimetre and (ii) switch on a cooling fan whenever the robot is running. Justify, with reasons, the type of control you would choose for each of the two tasks, and state one cost that your choice for the first task imposes on the robot.

16.12 Diagram, Circuit & Output Reading

  1. A block diagram is drawn with the following blocks and arrows. An arrow labelled “40 °C” enters a block labelled “microcontroller”; an arrow labelled “switching signal” runs from that block to a block labelled “heater and driver”; an arrow runs from there to a block labelled “drying chamber”; an arrow labelled “chamber temperature” leaves the chamber to the right and ends there. Identify the type of control system shown, name the element that each block is acting as, and state what would have to be added, and where, to convert it to the other type.

  2. A student wires a joint drive as follows: the geared direct current motor’s two terminals are connected directly to two digital output pins of the microcontroller board; the encoder is powered from the 5V and GND pins of the same board and its signal goes to pin 2; the board is powered from a 9 V battery. Identify the single fault in this arrangement, state what would happen when the program switches the motor on, and describe the correct arrangement.

  3. A joint carries an encoder producing 12 pulses per revolution of the motor shaft, and the gearbox between the motor shaft and the joint has a ratio of 30:1. The task requires the joint angle to be held within 0.2°. Calculate the counts per revolution of the joint and the smallest change in joint angle the encoder can detect, and state with a reason whether this sensor is suitable for the task.

  4. The following method is written for a joint position loop. Read it and answer the questions below it.

1. setpoint = 60.0
2. read the measured joint angle
3. error = measured angle - setpoint
4. if error is positive, drive the joint forward
5. otherwise, drive the joint in reverse
6. repeat from step 2

The joint starts at a measured angle of 52.0°, and driving forward increases the measured angle. State the value of the error on the first pass, the direction the joint is driven, and what happens to the measured angle over the passes that follow. Name the single change that would make the method correct.

16.13 Case-Based Questions

  1. A warehouse uses a small wheeled trolley robot to carry cartons along a straight aisle to a picking station 3.00 m from its start point. In the first version, the robot is simply told to drive both motors at a fixed power for 4 s and then stop. The team measures where the front edge of the robot actually stops on five consecutive runs, and records the following.
Run Distance travelled (m)
1 3.04
2 2.97
3 2.92
4 2.86
5 2.79

The picking station will accept the carton only if the robot stops within 100 mm of the 3.00 m mark.

  1. Identify the type of control system used in this first version, and justify your answer with reference to the block diagram it would have.
  2. Using the recorded data, state whether the robot meets the requirement, and give two physical reasons for the trend you can see across the five runs.
  3. The team decides to close the loop. Name the quantity that must be controlled, state its setpoint, name a sensor that could measure it in this aisle, and state one disadvantage the change will bring.
  1. For a school science exhibition, a team builds a two-wheeled robot that must stand upright on its own. Tilt is measured by a gyroscope and accelerometer, and the wheels are driven by geared direct current motors through a driver module. In testing, the team reports two problems. First, the robot rocks steadily backwards and forwards about the upright position and never settles. Second, when it does briefly settle, it holds a small permanent lean to one side, which the team traces to a battery mounted slightly off the centre line.
  1. Explain why this task cannot be attempted with open-loop control, whatever calibration is carried out beforehand.
  2. Name the corrective action of a PID controller that addresses each of the two reported problems, and explain in each case how that action produces the improvement.
  3. The team must test the robot on a workshop bench with the class watching. State two precautions they should take before switching on, and give the reason for each.