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Chapter 17: Manipulators

In 1961 a machine went to work at a General Motors plant in New Jersey, doing a job that nobody wanted. Castings came out of the die-casting machine glowing hot, heavy and sharp at the edges, and someone had to lift each one clear and stack it. Burns and fumes were part of the shift. The machine that took the job over was called Unimate. It was a single jointed arm, driven by hydraulics, bolted to the floor, with a gripper at its tip. It did not look like a person, and it could not see anything at all. It simply moved.

What made it remarkable was not its strength but its memory. Its instructions were stored on a magnetic drum as a list of joint positions — turn this joint so far, lift that one so far, close the gripper, open it. The arm never knew where the casting was. It knew only the angles of its own joints. From those angles, the position of its own hand followed.

That is the idea this chapter is built on. An arm of the kind shown in Fig. 17.0-1 is a chain of rigid pieces joined end to end, driven by motors, ending in a tool. Tell it the angles and the tip arrives exactly where the geometry says it will, a thousand times a shift, without tiring. The engineer’s work is to know where “exactly” is, and to decide how the arm should travel there.

Photograph

Fig. 17.0-1 – an industrial articulated robotic arm at work on a factory line, photographed from the side so that the whole chain is visible: the fixed base bolted to the floor, the rigid links between the joints, at least three joints along the arm, and the tool at the tip (gripper or welding torch); the arm must fill the frame with the base clearly included, and no person should be inside the working area. Sourcing note: a stock or manufacturer-licensed photograph of a real installed industrial arm; images showing only a close-up of the gripper, or a rendered or computer-generated arm, are unusable.

NoteCuriosity Corner

Q1. A robot arm has no eyes to watch its own gripper. How, then, can it know exactly where the tip of the arm has reached? Q2. Why does adding one more joint change what an arm is able to do, and how do engineers count that ability in a single number? Q3. Why is a welding torch, a paint sprayer and a suction cup all treated as the same kind of part on a robot arm? Q4. When a robot polishes a surface, reaching the right position is not enough. What else must its controller keep watch on? Q5. Two arms are asked to press a delicate part into a tight hole. Why can an arm that gives a little sideways do this better than one that is stiff in every direction?

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

17.1 The Manipulator as a Machine

Reach across a table for a glass of water and watch what your own arm does. Your shoulder swings the whole arm towards the glass, your elbow sets how far out the hand goes, and your wrist turns the hand so that the fingers close the right way round. Three joints, a few rigid bones between them, and a hand at the end that does the actual work.

An industrial robot arm is built on exactly that skeleton, and for exactly that reason. It is the arrangement that lets a machine reach a great many different points from one fixed spot on the floor, and arrive at each of them holding a tool in a chosen direction.

17.1.1 What a Manipulator Is

A manipulator is the robotic arm itself — the mechanical structure that carries a tool to where the work is. The word is used in industry for the whole arm, from the part bolted to the floor up to the tool at the tip, together with the motors that drive it and the controller that commands it.

Two things separate a manipulator from an ordinary machine. The first is that it is programmable: the sequence of movements is stored as instructions and can be changed without rebuilding the machine. A lathe is built to spin a workpiece and will do nothing else; a manipulator can be taught to stack cartons this month and to tighten screws the next.

The second is that it returns. Give it the same instruction tomorrow and the tool comes back to the same place, within a fraction of a millimetre, shift after shift. That property is what makes an arm worth its cost, and you will meet it again as a formal specification later in your course.

NoteDefinition

A manipulator is a programmable mechanical arm, made of rigid links connected by joints, that extends from a fixed base to a tool at its free end, and whose motion is produced by actuators under the command of a controller.

The terms robotic arm and manipulator mean the same thing and are used interchangeably in this chapter and in industry.

The name is older than industrial robots. In the 1940s, laboratories handling radioactive material built mechanical arms so that an operator standing behind a shielded wall could work on the material on the other side. Those arms were called manipulators, and the word carried over to the programmed arms that followed.

Q. A crane also has a base, an arm and a hook, and it lifts far heavier loads. Is a crane a manipulator?

A. Not as the term is normally used. A crane has the mechanical structure, but its motion comes from an operator’s hands on the controls, so it neither stores a sequence nor returns to a point on its own. Fit a crane with a controller that executes a stored path and measures its own position, and the distinction begins to disappear — which tells you that programmability and repeatable return, not size or shape, are what the word is really pointing at.

17.1.2 The Arm as a Kinematic Chain

Look at the arm in Fig. 17.1-1 as a chain rather than as a single object. At the bottom is the base, the part fixed to the floor, a bench or a moving platform; it does not move relative to whatever it is mounted on, and every other motion is measured from it.

Diagram

Fig. 17.1-1 – a three-joint serial manipulator drawn in side view as a chain; the base hatched and labelled “base (fixed)”; the rigid segments between joints labelled “link 1”, “link 2”, “link 3”; each joint drawn as a small circle and labelled “joint 1”, “joint 2”, “joint 3” with a curved arrow at each showing the direction of rotation; the tool at the free tip labelled “end-effector”; a dashed line running from the base along the whole chain to the tip labelled “kinematic chain”

Above the base sit the links — the rigid segments of the arm. A link does not bend or stretch; it simply carries motion from the joint below it to the joint above it, and its length fixes how far the arm can extend.

Between two links sits a joint, the moving connection that allows one link to move relative to the other. Everything the arm does comes from its joints. The links only set the geometry; the joints supply the motion.

Joints come in two kinds, shown side by side in Fig. 17.1-2. A revolute joint allows one link to rotate about an axis relative to the next, like a door hinge or your own elbow, and its position is described by an angle. A prismatic joint allows one link to slide along a straight line relative to the next, like a drawer in its runners, and its position is described by a distance.

Diagram

Fig. 17.1-2 – two small joint diagrams side by side; on the left a revolute joint: two links joined by a pin, the axis of rotation marked with a dashed line, a curved arrow and the angle marked theta; on the right a prismatic joint: one link sliding inside a square guide on the other, a straight double-headed arrow along the sliding direction and the displacement marked d; each labelled “revolute joint (R)” and “prismatic joint (P)”

These two letters, R and P, are how arm configurations are named. An arm with two revolute joints one after the other is a 2R arm; an arm with a revolute joint followed by a prismatic joint is an RP arm. You have already worked with both of these in studying planar mechanisms, and you will use the 2R arm again in section 17.4.

Table 17.1 – Revolute joint vs prismatic joint | Feature | Revolute joint (R) | Prismatic joint (P) | |—|—|—| | Motion allowed | Rotation of one link about an axis | Sliding of one link along a straight line | | Joint variable | An angle, measured in ° or rad | A displacement, measured in mm or cm | | Everyday example | A door hinge, a bicycle pedal crank | A drawer runner, a table-saw fence | | Effect on the tip | Swings the tip along an arc | Moves the tip along a straight path | | Usual drive | Rotary motor, often through a gearbox | Lead screw, rack and pinion, or a cylinder | | Typical use in an arm | Shoulder, elbow and wrist motions | Vertical up-and-down motion, telescoping reach |

Joined end to end, base to link to joint to link, the whole assembly forms a kinematic chain. In an industrial arm the chain is open, meaning each link is connected to at most two others and the far end is free — free to hold a tool, which is the entire point of the arrangement.

NoteDefinition

A kinematic chain is an assembly of rigid links connected by joints so that the links can move relative to one another in a definite way. The chain is open when its links form a single unbranched series with one end fixed and the other end free.

At that free end is the end-effector — the tool the arm carries. It may be a gripper that closes on a component, a welding torch, a paint sprayer, a screwdriver spindle or a suction cup that lifts a sheet of glass. Whatever it is, it is the only part of the whole machine that touches the work.

NoteDefinition

An end-effector is the device attached to the free end of the last link of a manipulator, through which the arm acts on its surroundings.

Q3. Why is a welding torch, a paint sprayer and a suction cup all treated as the same kind of part on a robot arm?

A. Because they occupy the same place in the machine and play the same role in it. Each of them bolts to the free end of the last link, each is the single point through which the arm acts on the work, and each is powered and commanded separately from the joints that position it. The arm’s job is to bring that mounting point to a chosen position and direction; what is bolted there decides what task gets done. So an engineer designs the arm to deliver position, and treats every such tool as one interchangeable part called the end-effector.

One consequence of the chain arrangement is worth holding on to from the start. Motion accumulates along it: the tip’s position depends on every joint below it, not just on the last one. Turn the shoulder joint by half a degree and the gripper, a metre away, shifts by several millimetres. This is why the joints nearest the base are built heaviest and driven most precisely — their small errors arrive at the tool magnified.

17.1.3 Why Industry Uses Robotic Arms

A manipulator is not chosen because it is cleverer than a person. On any task calling for judgement, a trained worker is far ahead of it. It is chosen for four narrow strengths.

The first is consistency. An arm placing a component on a circuit board puts it in the same spot on the ten-thousandth board as on the first, because it is executing the same stored joint values each time. Human placement drifts with fatigue and attention; the arm’s does not.

The second is speed with that consistency held. Arms move quickly, and — more importantly — they do not need to slow down to stay accurate in the way a hurrying person does.

The third is endurance. An arm works three shifts, holds a heavy tool steady for hours, and does not tire on the last hour of the night shift.

The fourth is safety, and it is the reason many arms were installed in the first place. Welding fills the air with fumes and the workspace with intense light; spray painting fills it with solvent vapour; handling heavy or hot parts risks crush and burn injuries. Putting a machine at the point of danger removes a person from it.

Where an arm is the wrong answer

Set against these, a manipulator has real limits, and an engineer who cannot state them will specify the wrong machine.

Its base is fixed, so it can only work inside the region its links can reach. The work must be brought to the arm — on a conveyor, a fixture or a moving platform — which means designing the whole cell around it, not just buying the arm.

It also expects the world to be where it was told the world would be. A part sitting 20 mm from where the program assumes will be gripped 20 mm off centre, or missed altogether, unless sensors are added to find it first. Variety is expensive: every new part usually means new programming and often new fixtures.

That cost only pays back at volume. For a workshop making twelve different items a week in small numbers, the programming time alone can outweigh anything the arm saves.

And an arm repeats faults as faithfully as it repeats successes. A path programmed 2 mm too low will scrape every single workpiece, without hesitation and without noticing. Consistency is a strength only when what is being repeated is correct.

17.2 The Components of a Manipulator

An arm that only had links and joints would be a sculpture. To become a machine it needs something to drive the joints, something to carry that drive to them, something to sense what is happening, something to decide, and a supply of energy behind all of it.

Fig. 17.2-1 shows how those parts sit together. Read it as a loop rather than a list: the controller commands, the actuators move the joints, the sensors report what actually happened, and the controller commands again on the strength of that report.

Diagram

Fig. 17.2-1 – block diagram of a manipulator; blocks labelled “controller”, “actuators”, “transmission system”, “joints and links”, “end-effector”, “sensors” and “power supply”; a command arrow from controller to actuators, a mechanical-drive arrow from actuators through transmission system to joints and links, an arrow from joints and links to end-effector, an arrow from end-effector to a block outside the boundary labelled “workpiece”; a feedback arrow from sensors back to the controller, with sensor inputs tapped from the joints and from the end-effector; power arrows from the power supply to the controller and to the actuators, drawn in a distinct line style and labelled “power”

17.2.2 Actuators and the Transmission System

The joints do not move themselves. Each one is driven by an actuator, the component that converts supplied energy into mechanical motion. Actuators are the muscles of the arm: the controller’s decisions come to nothing unless an actuator carries them out.

NoteDefinition

An actuator is a device that converts electrical, hydraulic or pneumatic energy into mechanical motion, and so produces the movement of a joint.

Three kinds are used on manipulators. Electric motors, including DC and stepper motors, dominate modern arms because they are clean, easy to control precisely and easy to power. Hydraulic actuators, driven by pressurised oil, deliver very large forces for their size and were used on the early heavy arms; they need a pump and pipework, and a leak is both a mess and a hazard. Pneumatic actuators, driven by compressed air, are cheap and fast, but air is compressible, so a pneumatic cylinder is far better at going fully out or fully in than at stopping accurately halfway.

Table 17.2 – Electric, hydraulic and pneumatic actuators | Feature | Electric | Hydraulic | Pneumatic | |—|—|—|—| | Energy supplied as | Electric current | Pressurised oil | Compressed air | | Force or torque for its size | Moderate | Very high | Low to moderate | | Ease of precise positioning | Good — position is easily controlled and measured | Good, but needs valves and control gear | Poor at intermediate positions, because air compresses | | Supporting equipment needed | Drive electronics only | Pump, reservoir, pipework, filters | Compressor, air lines, dryer | | Typical use on an arm | Most modern industrial and educational arms | Heavy-payload arms, presses, older installations | Grippers, simple two-position motions | | Main drawback | Limited force without gearing | Leaks, noise, maintenance | Cannot hold an accurate mid-stroke position |

A motor by itself usually turns too fast and too weakly for a joint. A typical small motor spins in the thousands of revolutions per minute and produces a fraction of a newton metre, while an arm joint needs to turn slowly and hold a torque that may be tens of newton metres. Between them sits the transmission system — the gears, belts, chains or lead screws that carry motion from the actuator to the joint and change it on the way.

Gearing trades speed for torque. Reduce the speed by a factor and the torque available rises by very nearly the same factor, minus what friction takes.

Torque and speed at a joint driven through a gearbox

Given: A motor supplies a torque of 0.5 N·m at a speed of 3000 rpm. It drives an elbow joint through a gearbox of reduction ratio 50:1, whose efficiency is 85 %.

Find: The speed and the torque available at the joint.

Method: In a reduction gearbox the output turns slower than the input by the reduction ratio \(N\), so \(n_{out} = n_{in} / N\). If no energy were lost, the output torque would rise in the same proportion; the efficiency \(\eta\) accounts for what friction removes, so \(\tau_{out} = \tau_{in} \times N \times \eta\).

\(n_{out} = \dfrac{3000\ \text{rpm}}{50} = 60\ \text{rpm}\)

\(\tau_{out} = 0.5\ \text{N·m} \times 50 \times 0.85 = 21.25\ \text{N·m}\) Answer: The joint turns at 60 rpm with about 21 N·m of torque available. Check: Is 21 N·m sensible for an elbow? A torque of 21 N·m at the end of a 1 m link would hold a weight of \(21 / (9.81 \times 1) \approx 2.1\) kg against gravity. That is the right order for a small industrial arm, so the figure is believable — and it also shows why long, heavily loaded arms need far larger reductions.

Gearing brings its own cost, and it has a name. Backlash is the small free movement between gear teeth: when the motor reverses, it turns a little before the joint follows. At the tool that lost motion shows up as an error every time the arm changes direction, which is why precision arms use gear types designed to have very little of it.

On a small servo-driven arm in the laboratory, hold the base steady and gently rock the end-effector back and forth by hand while the arm is switched off. Estimate how far the tip moves before the joints take up the motion, and say which of that movement comes from the joints and which from bending of the links.

17.2.3 The End-Effector

The end-effector is chosen for the task, and it is usually the part that decides whether an installation works. An arm that can position beautifully is useless if the tool cannot hold the part.

Grippers are the most common family. A gripper holds a workpiece so that the arm can move it, and Fig. 17.2-2 shows the three arrangements you will meet most often. A parallel-jaw gripper closes two flat fingers on the sides of a part; a suction cup lifts a smooth flat surface by drawing air out from under it; a magnetic gripper picks up ferrous parts with an electromagnet that releases when the current is switched off.

Diagram

Fig. 17.2-2 – three grippers drawn side by side, each shown holding a workpiece; left, a parallel-jaw gripper with two flat fingers and arrows showing them closing inwards; centre, a suction cup on a flat plate with an arrow showing air being drawn out through the stem; right, a magnetic gripper on a steel block with the coil marked and the wire labelled “current on = holds”; each labelled with its name below

Process tools form the other family: welding torches, paint sprayers, screwdriver spindles, drills and cutters. These do not carry the work at all — the arm holds the tool against the work and moves it along a path.

Whatever the tool, one figure governs the choice of the arm behind it: the payload, the mass the arm can carry at its end-effector while still meeting its stated speed and accuracy. The payload figure always includes the end-effector itself. A 5 kg arm fitted with a 2 kg gripper can lift a 3 kg part, not a 5 kg one — a mistake that is made often enough to be worth remembering.

Photograph

Fig. 17.2-3 – close-up of a two-finger parallel-jaw industrial gripper mounted on the wrist of a robot arm, gripping a metal component; the mounting flange joining gripper to arm, the two jaws and the gripped part must all be clearly visible in the frame. Sourcing note: a stock or manufacturer photograph of a real gripper in use; illustrations, renders, and images where the gripper is empty or cropped away from the arm are unusable.

17.2.4 Sensors and Feedback

A manipulator that only sent commands outwards would be working blind. It would have no way of knowing whether a joint actually reached the angle it was told to reach, whether the gripper closed on a part or on nothing, or whether the tool is pressing too hard.

Sensors close that gap. They measure what is actually happening and report it back to the controller — an arrangement called feedback, and the single most important idea in making a machine behave as intended.

Table 17.3 – Sensors commonly fitted to a manipulator, what each measures and what the controller does with it | Sensor | What it measures | What the controller uses it for | |—|—|—| | Encoder at each joint | The angle of a revolute joint, or the displacement of a prismatic joint | Comparing the actual joint value with the commanded one, and correcting the difference | | Limit switch | Whether a joint has reached the end of its travel | Stopping motion before the mechanism is damaged; finding a known starting position | | Force or torque sensor at the wrist | The force the end-effector is exerting on the work | Controlling how hard the tool presses, and detecting a collision | | Proximity sensor | Whether an object is present within a short range | Confirming a part is in the fixture before the arm closes on it | | Vision camera | The position and orientation of a part in the scene | Correcting the programmed position when the part is not exactly where it was assumed to be |

The encoder is the one every joint has. It is a device fitted to the joint or to the motor driving it that reports the joint’s present angle or displacement as a number the controller can read. Without encoders an arm could be commanded but never corrected.

Q. If the controller already knows what angle it commanded, why does the joint need an encoder to tell it the angle?

A. Because commanding is not achieving. The load may be heavier than expected, friction may be higher on a cold morning, the belt may have stretched, or the arm may have struck something. The commanded value says what the controller wanted; the encoder says what the arm did. Control is the business of shrinking the difference between the two, and you cannot shrink a difference you cannot measure.

17.2.5 The Controller and the Power Supply

The controller is the computer of the arm. It holds the program, works out where each joint must be at each instant, sends the drive signals to the actuators, and reads the sensors to check the result. Every strategy in section 17.6 is something the controller does.

It also coordinates. A single joint moving alone is easy; six joints moving together so that the tool sweeps a smooth curve at constant speed is the controller’s real work, and it is why a manipulator is a computing problem as much as a mechanical one.

The power supply feeds both sides of the machine, and the two demands are quite different. The controller needs a small, clean, steady supply, because a dip in its voltage can reset it mid-motion. The actuators need a large one, and they take it in surges — the current drawn when a joint starts moving or holds a heavy load is many times the current drawn while idle. A supply sized for the idle case will collapse the first time the arm accelerates.

An arm holds its position by supplying power to its joints. When power is removed, an unbraked joint can swing down under the weight of the arm and whatever it is holding. On a laboratory servo arm, support the arm by hand and keep fingers clear of the joints before switching off or disconnecting anything, and never leave a load raised on an arm that has been powered down.

17.3 Degrees of Freedom and the Work Envelope

Two questions decide whether a particular arm can do a particular job. Can it get the tool into every position and direction the job asks for? And are those positions inside the region it can physically reach? The first question is answered by counting degrees of freedom, the second by looking at the work envelope.

17.3.1 Counting the Degrees of Freedom of an Arm

Hold your forearm still on a table and move only your wrist. The hand can turn, but it cannot go anywhere else — one joint has given you one independent way of moving. Free the elbow as well and the hand gains a second, quite different, freedom.

That is the whole idea. Each joint that can be driven independently gives the arm one more independent way to move, and the number of them is the arm’s degrees of freedom (DOF).

NoteDefinition

The degrees of freedom of a manipulator is the number of independent joint motions it has. For an open chain in which every joint is driven separately, it equals the number of driven joints.

Counting is easy; understanding what the count buys you is the part that matters. Think first about what is being positioned — not the arm, but the tool at its end.

A rigid body lying flat on a table can be placed by giving three numbers: how far along, how far across, and which way it is turned. That is three degrees of freedom in a plane. The same body free in space needs six: three to say where it is and three to say which way it is facing.

Degrees of freedom of a rigid body and of a serial arm Statement: A rigid body has 3 degrees of freedom in a plane and 6 in space. A serial manipulator has as many degrees of freedom as it has independently driven joints, and it can fully control the pose of its tool only if it has at least as many as the space requires. In words: Six numbers are needed to place a tool anywhere in space facing any direction, so an arm needs six independent joints to do that. Why it holds: Each independent joint variable can be chosen freely, so an arm with \(n\) joints can satisfy at most \(n\) independent conditions on its tool at once. Used for: Deciding how many joints a task needs before any arm is bought or built. Watch out: The count says how many independent motions exist, not how large they are. An arm can have six degrees of freedom and still fail a job because its links are too short or its joints cannot swing far enough.

Work down the ladder and the consequences become concrete. A single revolute joint sweeps the tool along an arc, and nothing else. Add a second and the tip can be brought to points spread over an area — but with both joint angles now used up in reaching that point, the direction the tool faces there is whatever the geometry leaves; you cannot choose it. Add a third joint in the same plane and the tool’s position and its facing can both be chosen.

The same argument in space gives six as the number for full freedom, which is why the classic factory arm has six joints. Fewer than six and some combinations of position and direction are simply unattainable — often perfectly acceptable, because many jobs never need them. An arm that only stacks boxes on a pallet keeps its gripper pointing down all day and has no use for the freedoms it lacks.

Q2. Why does adding one more joint change what an arm is able to do, and how do engineers count that ability in a single number?

A. Each independently driven joint adds one more freedom that can be chosen at will, and the tool’s pose needs a fixed number of freedoms to be fully specified — three in a plane, six in space. So a two-joint planar arm can reach a point but cannot also choose the angle the tool sits at, while a three-joint planar arm can do both. The single number engineers use is the degrees of freedom, which for an open chain is just the number of driven joints. Comparing that number with the number the task demands tells you at once whether an arm is capable of the job.

An arm may also carry more joints than the space needs — seven in place of six. The extra freedom does not let it reach anywhere new, but it lets it reach the same pose in more than one way, which is how an arm gets its tool past an obstacle or into a housing without folding itself into an awkward posture.

Q. A gripper opens and closes. Does that count as a degree of freedom of the arm?

A. No. The degrees of freedom of a manipulator count the joints that position and orient the tool. Opening and closing the gripper changes what the tool is doing, not where it is, so it is counted separately as a motion of the end-effector. The same reasoning excludes a welding torch’s trigger and a sprayer’s valve — useful motions, but not ones that move the tool anywhere.

17.3.2 Position and Orientation of the End-Effector

Reaching the right spot is only half of arriving. Picture an arm driving a screw into a panel. Its tip must sit exactly on the screw head, and the screwdriver must lie along the axis of the screw. Get the position right and the direction wrong, and the screw goes in crooked or not at all.

The first requirement is the position of the end-effector — where its tip is, given as coordinates measured from the base. The second is its orientation — which way the tool is facing there. Position and orientation taken together are called the pose of the end-effector, and a manipulator’s job, stated at its most compact, is to bring its end-effector to a commanded pose.

Orientation in space is described by three rotations, shown on the wrist in Fig. 17.3-1. Roll is rotation about the axis the tool points along, like a screwdriver turning. Pitch tips the tool up or down. Yaw swings it left or right. Those three, plus the three position coordinates, are the six numbers of a full pose.

Diagram

Fig. 17.3-1 – the wrist end of a robot arm with a tool attached; three mutually perpendicular axes drawn at the tool, the tool’s own pointing axis drawn as a dashed line; three curved arrows, one about each axis, labelled “roll (about the tool axis)”, “pitch (tips up and down)” and “yaw (swings left and right)”; the tool tip marked as a point and labelled “position (x, y, z)”

This is also where a limited arm shows its limits. A four-joint arm may be able to put the tool tip exactly on the screw head, yet be unable to tilt it to the angle the job needs, because it has no joint left with which to change that angle.

17.3.3 The Work Envelope

An arm bolted to the floor can only work where its links can take it. The set of all points its end-effector can be brought to is called the work envelope, or equivalently the workspace.

NoteDefinition

The work envelope of a manipulator is the region of space that its end-effector can reach, taking the lengths of the links and the travel limits of the joints into account.

The shape of that region follows directly from the arm’s geometry. Fig. 17.3-2 shows the simplest case, a two-link planar arm. With both joints free to turn, the tip can reach any point in a flat ring: the outer edge is where the arm is stretched straight, and the inner edge is where it is folded back on itself as tightly as the link lengths allow.

Diagram

Fig. 17.3-2 – plan view of a two-link planar arm; base marked at the centre with link lengths labelled L1 and L2; a shaded annulus (ring) drawn about the base; the outer circle labelled “outer radius = L1 + L2, arm straight” and the inner circle labelled “inner radius = |L1 − L2|, arm folded”; the unreachable disc inside the inner circle labelled “dead zone”; one arm posture drawn inside the ring reaching a point P

The reachable region of a two-link planar arm

Given: A planar arm has two revolute joints, with link lengths \(L_1 = 400\) mm and \(L_2 = 300\) mm. Both joints can turn without restriction.

Find: Whether the arm can reach points lying 750 mm, 500 mm and 60 mm from the base, measured in the plane of motion.

Method: With both joints free, the reachable region is a ring about the base. The farthest point is reached with the arm straight out, at a distance \(L_1 + L_2\) from the base; the nearest is reached with the arm folded back, at a distance \(|L_1 - L_2|\). A point is reachable if its distance \(r\) from the base satisfies \(|L_1 - L_2| \le r \le L_1 + L_2\).

\(r_{max} = L_1 + L_2 = 400\ \text{mm} + 300\ \text{mm} = 700\ \text{mm}\)

\(r_{min} = |L_1 - L_2| = |400\ \text{mm} - 300\ \text{mm}| = 100\ \text{mm}\)

The reachable band is therefore \(100\ \text{mm} \le r \le 700\ \text{mm}\).

At \(r = 750\) mm: outside, since \(750 > 700\). At \(r = 500\) mm: inside the band, so reachable. At \(r = 60\) mm: inside the dead zone, since \(60 < 100\). Answer: The arm can reach the point at 500 mm. It cannot reach the point at 750 mm, which lies beyond its full extension, nor the point at 60 mm, which lies inside the hole at the centre of its work envelope. Check: Both failures make physical sense. An arm cannot stretch past the sum of its links, and it cannot fold tighter than the difference between them. Your own arm has a dead zone too, but a very small one, because the upper arm and forearm are close to equal in length — which is why your hand can almost touch your shoulder.

Two practical points follow from the shape of an envelope.

The first is that joint limits cut pieces out of it. The ring in Fig. 17.3-2 assumed joints that turn freely; a real joint that swings only 270° leaves a wedge of the ring unreachable, and the true envelope of an industrial arm is usually an irregular solid rather than a tidy shape.

The second is that reach and payload trade against each other. The torque a joint must supply to hold a load is the load’s weight multiplied by its distance from that joint, so an arm fully extended is working its base and shoulder joints hardest. The same arm that lifts a part comfortably close in may not lift it at the edge of its envelope at all, and manufacturers state payload with that in mind.

The envelope also has to be respected as a physical space. Everything inside it can be struck by the arm at full speed, which is why an industrial installation fences the whole region off and interlocks the gate with the controller, so that opening it stops the machine.

17.4 The Mathematics of an Arm

Everything a manipulator’s controller knows about its own body comes from numbers on joints. The encoders report angles and displacements; nothing anywhere in the machine measures the tool tip directly. If the arm is to be commanded in terms of places in the workshop, someone has to turn joint readings into a position — and that is a geometry problem with an exact answer.

17.4.2 Forward Kinematics of a 2R Planar Arm

Take the simplest arm worth analysing: two links joined by two revolute joints, moving in a single plane. It is the 2R planar arm you met in studying mechanisms, and it is drawn with all its quantities marked in Fig. 17.4-1.

Diagram

Fig. 17.4-1 – a 2R planar arm; base frame at the shoulder with origin O and axes labelled x0 and y0; link 1 of length L1 from O to the elbow, making angle theta1 with the x0 axis, the angle marked with an arc from the x0 axis; link 2 of length L2 from the elbow to the tip, making angle theta2 with the extension of link 1, the angle marked with an arc from a dashed line continuing link 1; the tip labelled P(x, y); dashed lines dropped from P to both axes marking x and y; the elbow point labelled with coordinates x1, y1

The question forward kinematics answers is direct: given \(\theta_1\) and \(\theta_2\), where is the tip?

Work along the chain. Link 1 starts at the origin and runs a distance \(L_1\) in the direction \(\theta_1\), so the elbow sits at

\[x_1 = L_1 \cos\theta_1, \qquad y_1 = L_1 \sin\theta_1\]

Link 2 starts at the elbow. Its direction in the base frame is not \(\theta_2\): it has been carried round by link 1 as well, so it points at \(\theta_1 + \theta_2\). Adding its contribution to the elbow’s position gives the tip:

\[x = L_1 \cos\theta_1 + L_2 \cos(\theta_1 + \theta_2)\]

\[y = L_1 \sin\theta_1 + L_2 \sin(\theta_1 + \theta_2)\]

That pair of equations is the forward kinematic solution of the 2R planar arm. Feed in any two joint angles and it returns exactly one tip position.

NoteDefinition

Forward kinematics is the computation of the position and orientation of a manipulator’s end-effector from the known values of its joint variables and its link lengths.

Forward kinematics of the 2R planar arm Statement: For a two-link planar arm with revolute joints, \(x = L_1\cos\theta_1 + L_2\cos(\theta_1+\theta_2)\) and \(y = L_1\sin\theta_1 + L_2\sin(\theta_1+\theta_2)\), with the tool’s orientation in the plane given by \(\theta_1 + \theta_2\). In words: Each link contributes its own length in its own direction, and the directions add along the chain. Why it holds: A link is a fixed-length segment, so its contribution to the tip’s position is its length resolved along the base axes; because joint 2 is mounted on link 1, link 2’s direction in the base frame is the sum of the two joint angles. Used for: Turning encoder readings into a tool position, and predicting where a commanded pair of angles will put the tool before the arm is allowed to move. Watch out: The result is unique in one direction only. One pair of angles gives one position, but one position can usually be produced by two different pairs of angles.

Measuring the second joint angle from the wrong reference What students do: They write \(x = L_1\cos\theta_1 + L_2\cos\theta_2\), treating \(\theta_2\) as an angle measured from the base axis. What happens: The computed tip is wrong for every posture except the one where \(\theta_1 = 0\), and the error grows as the shoulder swings. The mistake is invisible in a sketch, because both expressions look reasonable. The fix: Use \(\theta_1 + \theta_2\) in the second term. The joint measures the bend of link 2 relative to link 1, so link 1’s own rotation must be added to obtain link 2’s direction in the base frame.

Locating the tip of a 2R planar arm

Given: A 2R planar arm has link lengths \(L_1 = 250\) mm and \(L_2 = 200\) mm. The joints are set to \(\theta_1 = 30°\) and \(\theta_2 = 45°\).

Find: The coordinates of the tip in the base frame.

Method: Apply the forward kinematic equations of the 2R planar arm, taking care that the second link’s direction is \(\theta_1 + \theta_2\). Set the calculator to degree mode, or convert the angles to radians if the working is done in radians.

\(\theta_1 + \theta_2 = 30° + 45° = 75°\)

\(x = 250 \cos 30° + 200 \cos 75° = 250(0.8660) + 200(0.2588)\)

\(x = 216.5\ \text{mm} + 51.8\ \text{mm} = 268.3\ \text{mm}\)

\(y = 250 \sin 30° + 200 \sin 75° = 250(0.5000) + 200(0.9659)\)

\(y = 125.0\ \text{mm} + 193.2\ \text{mm} = 318.2\ \text{mm}\) Answer: The tip lies at approximately \(x = 268\) mm, \(y = 318\) mm in the base frame. Check: The tip’s distance from the base is \(r = \sqrt{268.3^2 + 318.2^2} \approx 416\) mm. The arm’s reachable band runs from \(|250 - 200| = 50\) mm to \(250 + 200 = 450\) mm, and 416 mm falls comfortably inside it. Both coordinates are positive, which matches an arm swung up and to the right as the angles describe — so the answer is consistent with the geometry.

Q1. A robot arm has no eyes to watch its own gripper. How, then, can it know exactly where the tip of the arm has reached?

A. It calculates the position instead of observing it. An encoder at each joint reports that joint’s angle, and the link lengths are fixed and known from the moment the arm was built. Forward kinematics turns those numbers into the tip’s coordinates: each link contributes its own length in its own direction, and the directions accumulate along the chain. So the arm knows where its hand is for the same reason you can say where your fingertip is with your eyes shut — the geometry of the limb, plus the angles of its joints, is enough.

Take the arm of the worked example and set \(\theta_2 = 0°\), leaving \(\theta_1 = 30°\). Predict the tip position before calculating, then compute it and confirm that the tip now lies exactly 450 mm from the base.

17.4.3 The Homogeneous Transformation Matrix

The geometric route works, but it was built by looking at a picture of one particular arm. Add joints, or move into three dimensions, and drawing triangles stops being practical. Engineers therefore use a second route to the same answer, one that can be applied mechanically to any chain.

The idea is to describe each link’s contribution as a single operation — a rotation followed by a shift along the rotated direction — and to write that operation as a matrix. Chaining links then becomes multiplying matrices, which a computer does without any picture at all.

A rotation alone can be written as a 2 × 2 matrix, but a shift cannot; a shift is an addition, not a multiplication. The trick that combines them is to write a planar point as three numbers, \((x, y, 1)\), and to use a 3 × 3 matrix. This is the homogeneous transformation matrix, and for a link of length \(L\) set at angle \(\theta\) it takes the form

\[T = \begin{bmatrix} \cos\theta & -\sin\theta & L\cos\theta \\ \sin\theta & \cos\theta & L\sin\theta \\ 0 & 0 & 1 \end{bmatrix}\]

The homogeneous transformation matrix in a plane Statement: A rotation by \(\theta\) combined with a translation of \(L\) along the rotated direction is written as the 3 × 3 matrix above, acting on points written as \((x, y, 1)\). In words: One matrix carries both the turning and the shifting that one link performs. Why it holds: The top-left 2 × 2 block is the ordinary rotation matrix, the top-right column is the translation resolved along the base axes, and the bottom row \((0, 0, 1)\) leaves the added third coordinate equal to 1 so that further matrices can be applied in the same way. Used for: Building the geometry of a whole chain by multiplying one matrix per link, and reading the tool’s position and orientation straight out of the result. Watch out: Matrix multiplication is not commutative. Multiplying the link matrices in the wrong order describes a different arm, and the mistake produces a plausible-looking answer.

17.4.5 What the Equations Tell the Engineer

Forward kinematics is used constantly inside a working arm. Every few milliseconds the controller reads the encoders and computes where the tool actually is, so that it can compare that position with where the tool was meant to be.

It is also used before anything moves. A commanded set of joint angles can be run through the equations to check that the resulting tool position lies where it should, and inside the work envelope, rather than discovering the answer by watching the arm swing into a fixture.

The reverse question is the harder one. Given a target point on the workpiece, what joint angles will put the tool there? That is inverse kinematics, and it is a different kind of problem: it may have no solution if the point lies outside the envelope, and it usually has more than one, since a 2R arm can normally reach a given point either elbow-up or elbow-down. Working it out is beyond this chapter’s treatment, but you should recognise the name and know why the controller needs it whenever a task is specified in workshop coordinates rather than in joint angles.

Two limitations of the mathematics are worth carrying away.

The first is that the equations describe an ideal arm. They assume the links are exactly \(L_1\) and \(L_2\) long, perfectly rigid, and that the encoders read the true joint angles. Real links are made to a tolerance and bend a little under load, and real gearing has backlash. The computed tip and the true tip therefore differ slightly, and closing that gap is the business of calibration.

The second is that not every part of the envelope behaves equally well. With the arm stretched almost straight, a large change in joint angles produces only a tiny movement of the tip in the outward direction; near that posture the arm loses fine control along one direction, and it is standard practice to plan paths that keep away from it. The geometry that tells you where the tool is also tells you where the arm is at its least useful.

17.5 Manipulator Designs Used in Industry

The same set of parts can be arranged in more than one way, and the arrangement decides what the arm is good at. Two designs cover most of what you will see on a factory floor: the articulated arm, which imitates a human arm, and the SCARA, which deliberately does not.

17.5.1 The Articulated Arm

An articulated arm is the classic factory robot: a series of revolute joints, one after another, arranged like a shoulder, an elbow and a wrist. Every joint rotates; there are no sliding joints anywhere in the chain.

Fig. 17.5-1 shows the usual six-joint layout. The first joint turns the whole arm about a vertical axis, swinging it left and right. The shoulder and elbow joints then set how far out and how high the wrist is carried. The last three joints sit close together at the wrist and supply roll, pitch and yaw, so that the tool can be pointed in any direction once it has arrived.

Diagram

Fig. 17.5-1 – a six-joint articulated arm drawn in side view with the base at the floor; each joint numbered 1 to 6 and drawn as a circle with a curved arrow showing its rotation; joint 1 labelled “waist – rotates about a vertical axis”, joint 2 “shoulder”, joint 3 “elbow”, and joints 4, 5, 6 grouped and labelled “wrist – roll, pitch, yaw”; the links between joints labelled; the end-effector at the tip labelled; a dashed outline around the reachable region labelled “work envelope – roughly spherical”

That division of labour is the design’s logic. The first three joints are the positioning joints, which decide where the wrist goes; the last three are the orienting joints, which decide which way the tool faces from there. Six joints give six degrees of freedom, and six is precisely what a full pose in space requires.

Because every joint rotates, the arm can fold, reach over an obstacle and come at a workpiece from underneath — much as your own arm can reach behind a stack of books without moving the stack. Its work envelope is a large, roughly spherical region, cut about by the travel limits of the individual joints.

The articulated arm Working principle: A chain of revolute joints, each driven by its own actuator, positions the wrist anywhere within a roughly spherical region; three further rotations at the wrist then set the direction the tool points. The controller computes every joint’s motion together so that the tool follows a single smooth path. Components: Fixed base; waist, shoulder and elbow joints with rigid links between them; a three-axis wrist assembly; the end-effector; encoders at every joint; controller and power supply. Advantages: Six degrees of freedom, so the tool can be brought to a chosen position in a chosen direction; a large and flexible work envelope; able to reach around obstacles and to follow curved paths in three dimensions. Limitations: A long chain of rotary joints is less stiff than a short one, so a heavily loaded arm deflects more than a compact design; the arm is slower than a SCARA on flat, repetitive work; it costs more and takes longer to program; and its wide envelope needs a correspondingly large fenced area. Where it is used: Welding car bodies, spray painting, machine loading, and palletising in warehouses.

17.5.2 The SCARA Arm

The SCARA — Selective Compliance Assembly Robot Arm — was designed for a narrower job and is much better at it. Its first two joints both rotate about vertical axes, so the arm swings in a horizontal plane like a folding table lamp seen from above. A third joint moves the tool straight up and down, and a fourth rotates the tool about its own vertical axis. That is four degrees of freedom: three rotary and one prismatic.

Diagram

Fig. 17.5-2 – a SCARA arm drawn in two views; on the left a plan view showing the base, the two horizontal links with their joints marked as circles rotating about vertical axes, and the reachable ring-shaped area shaded; on the right a side view showing the vertical prismatic joint carrying the tool up and down with a straight double-headed arrow, and the fourth rotary joint at the tool; joints labelled “rotary (vertical axis)”, “rotary (vertical axis)”, “prismatic (vertical)” and “rotary (tool axis)”; the swept volume in the side view outlined and labelled “work envelope”

The reachable area of the two horizontal joints is a ring about the base, for exactly the reason worked out in section 17.3: the arm cannot stretch past the sum of its links, nor fold closer than their difference. Sweeping that ring up and down with the vertical joint turns it into a hollow cylindrical volume, and joint limits usually trim it further into a kidney shape.

The name carries the design idea. Compliance is the willingness of a structure to give a little when a force is applied to it, and this arm’s compliance is selective: it can yield slightly in the horizontal directions, where its rotary joints allow a small deflection, while remaining stiff in the vertical direction, where the prismatic joint carries the load straight down its own axis.

That combination is what makes it an assembly machine. When a pin is being pressed into a hole that is a fraction of a millimetre out of line, the chamfer on the hole pushes the pin sideways, the arm gives just enough to follow, and the pin slides home. A structure stiff in every direction would refuse to move sideways at all and would jam the pin or bend it.

Q5. Two arms are asked to press a delicate part into a tight hole. Why can an arm that gives a little sideways do this better than one that is stiff in every direction?

A. Because no two parts are ever aligned perfectly, and the small error has to go somewhere. When an arm can yield slightly in the horizontal plane, the sloping edge of the hole nudges the part into line and the arm follows that nudge — the alignment corrects itself as the part goes in. An arm that is rigid in every direction cannot follow; the misalignment turns instead into a sideways force on the part, which jams, scratches or breaks it. This is why the SCARA is built compliant horizontally and stiff vertically: it yields where yielding helps and holds firm where the insertion force must be delivered.

The SCARA arm Working principle: Two revolute joints turning about vertical axes move the tool anywhere within a ring-shaped horizontal area; a vertical prismatic joint then lowers or raises the tool, and a fourth rotary joint turns the tool about its own axis. The structure is deliberately stiff along the vertical axis and slightly compliant in the horizontal plane. Components: Fixed base; two horizontal links with vertical-axis rotary joints; a vertical prismatic joint, usually a lead screw or a ball spline; a tool rotation axis; the end-effector, most often a gripper or a suction cup; encoders, controller and power supply. Advantages: Very fast on flat, repetitive work; high stiffness in the vertical direction for accurate downward placement; horizontal compliance that helps parts seat during insertion; a compact footprint and simpler programming than a six-joint arm. Limitations: Only four degrees of freedom, so the tool cannot be tilted — it always points down; it cannot reach around or underneath anything; the work must be presented on a flat plane at a known height. Where it is used: Inserting components onto circuit boards, small-part assembly, screwdriving, and high-speed pick-and-place in electronics and packaging plants.

Photograph

Fig. 17.5-3 – a SCARA robot at work in an electronics assembly line, photographed so that the two horizontal links and the vertical tool axis are all visible, together with the flat work surface or conveyor beneath the tool. Sourcing note: a stock or manufacturer photograph of a real installed SCARA; renders, six-joint articulated arms mislabelled as SCARA, and images cropped to the tool alone are unusable.

17.5.3 Matching the Design to the Task

Neither design is better than the other; they answer different questions. Set them side by side and the trade-offs become clear.

Table 17.4 – The articulated arm and the SCARA compared | Feature | Articulated arm | SCARA | |—|—|—| | Degrees of freedom | Typically 6, all revolute | Typically 4 — three revolute and one prismatic | | Joint arrangement | Waist, shoulder, elbow and a three-axis wrist in one chain | Two vertical-axis rotary joints, a vertical prismatic joint, and a tool rotation | | Tool orientation | Can be set freely in any direction | Fixed pointing down; only rotation about the vertical is available | | Stiffness | Moderate, and it falls as the arm extends | High along the vertical axis, deliberately lower in the horizontal plane | | Work envelope | Large and roughly spherical, trimmed by joint limits | A hollow cylindrical volume, usually trimmed to a kidney shape | | Speed on flat repetitive work | Lower | Higher | | Suited to | Welding, painting, machine tending, palletising | Board assembly, insertion, screwdriving, fast pick-and-place on one plane |

The choice follows from the job. Ask first how the tool must be presented: if the task needs the tool tilted — a weld running around a curved seam, a spray gun following the contour of a car door — then four degrees of freedom will not do it, whatever else the arm can offer. Ask next about the shape of the workspace: a task that lives entirely on one flat surface at one height does not need a spherical envelope and should not pay for one.

Then weigh the ordinary engineering costs. A six-joint arm costs more, takes longer to program, needs a larger fenced area and is slower on flat repetitive work. A SCARA is cheaper, quicker and stiffer where it counts, and useless the moment the job asks it to tilt. Other arrangements exist in industry, such as gantry arms that slide along straight rails, but between them these two cover most installations.

Q. If an articulated arm has more degrees of freedom, why not simply use one for every job?

A. Because freedoms are not free. Every extra joint adds a motor, a gearbox, a length of link and one more source of backlash and deflection — so the arm becomes heavier, more expensive, slower to accelerate and less stiff. On a task that only ever needs the tool moved about one flat plane and set straight down, those two extra freedoms buy nothing and cost speed and accuracy. Good engineering supplies the freedoms the task needs, and no more.

17.6 Control Strategies of a Manipulator

Knowing where the tool is does not by itself move it anywhere. Something has to decide, moment by moment, what each actuator should be doing so that the arm behaves as intended. Those decision-making methods are the arm’s control strategies, and industry relies on three of them: position control, force control, and trajectory planning.

They are not alternatives to be chosen between once and for all. A single pick-and-place cycle may use a planned trajectory to travel near the part, position control to arrive, and force control for the last few millimetres of insertion.

17.6.1 Position Control

Position control is the strategy of commanding the arm to bring a joint, or the end-effector, to a specified position and then hold it there.

The method is a loop, drawn in Fig. 17.6-1. The controller is given a commanded position. It reads the encoders to find where the joint actually is. The difference between the two is the position error, and the controller drives the actuator in whatever direction shrinks that error. Then it measures again.

Diagram

Fig. 17.6-1 – flow: commanded position → comparator (drawn as a circle with a plus and a minus input) → controller → actuator → joint and link → actual position of the end-effector; a feedback arrow from an “encoder” block tapped at the joint back to the minus input of the comparator, labelled “measured position”; the arrow leaving the comparator labelled “position error”; the loop closed so that the last arrow returns to the comparator

Because the result of the action is measured and fed back into the next decision, this is closed-loop control. Its opposite, open-loop control, sends the command and simply assumes it was obeyed — cheaper, and blind. An arm that has been nudged off course, loaded more heavily than expected, or stiffened by cold grease is corrected by a closed loop and fails quietly with an open one.

NoteDefinition

Closed-loop control is control in which the actual output of a system is measured and compared with the commanded value, and the difference between the two is used to decide the next action.

Driving a single joint to a commanded angle Input: The commanded angle for the joint; the angle reported by the joint’s encoder; a tolerance within which the joint counts as having arrived. Output: A drive signal sent to the joint’s actuator, and a decision that the move is complete. Method:

1. Accept the commanded angle for the joint.
2. Repeat the following steps:
3.     Read the actual angle from the encoder.
4.     Compute error = commanded angle - actual angle.
5.     If the size of error is within the tolerance:
6.         Stop the actuator and report that the joint has arrived.
7.         Leave the loop.
8.     Otherwise, drive the actuator in the direction that reduces error,
       at a speed proportional to the size of error.
9. Until the joint has arrived, or the move has run out of time.

Why it works: The drive always acts in the direction that makes the error smaller, so the joint is pulled towards the commanded angle from either side. Making the speed proportional to the error slows the joint as it approaches, so it settles instead of shooting past. Watch out: If the joint is blocked, the error never falls and the actuator keeps driving against the obstruction until something overheats — which is why step 9 carries a time limit. Making the response too strong is the opposite failure: the joint overshoots, is corrected, overshoots the other way, and oscillates about the target.

There are two ways of stating what position is wanted, and the difference matters in practice.

In joint space control, the target is given as a set of joint values — this angle at the shoulder, that angle at the elbow. Each joint is simply driven to its own value. It is the simplest method and needs no geometry while the arm is running, but the tool takes whatever curved path the combination of joint motions happens to produce.

In Cartesian space control, the target is given as coordinates of the end-effector in the workshop: go to this point in space. That is how a person naturally specifies a task, but actuators do not understand points — they understand joint values. The controller must therefore convert the commanded position into joint angles, which is the inverse kinematics problem named in section 17.4.

Position control suits any task in which the tool moves freely without pressing on anything: transferring parts to a conveyor, tracing a path in the air, spraying at a fixed distance from a surface. Its weakness appears the moment the tool touches something. A position controller reacts to being blocked by pushing harder, because being blocked shows up as an error that refuses to shrink.

17.6.2 Force Control

Consider an arm asked to polish a curved panel. If it is told only where to be, a bump of half a millimetre on the panel leaves it either pressing far too hard or floating clear of the surface, because a stiff arm turns a tiny position mismatch into a large force.

Force control is the strategy of commanding and regulating the force the end-effector exerts, rather than only its position. A force or torque sensor at the wrist measures what the tool is actually pushing with, the controller compares that with the force the task calls for, and the actuator commands are adjusted to hold it.

The loop has the same shape as before — measure, compare, correct — but the quantity being regulated has changed. The arm is no longer asked to be in a place; it is asked to press with a given firmness, and it moves as much as it must in order to do so.

Q4. When a robot polishes a surface, reaching the right position is not enough. What else must its controller keep watch on?

A. The force the tool is applying to the surface. Polishing works only within a narrow band of pressure — too little and nothing is removed, too much and the surface is gouged or the tool stalls. A real panel is never exactly where the program assumes it is, and a stiff arm turns even a fraction of a millimetre of difference into a large change of force. So the controller reads a force sensor at the wrist and adjusts the arm’s motion to hold the contact force at the commanded value, letting the position float as the surface demands.

Table 17.5 – Position control and force control | Feature | Position control | Force control | |—|—|—| | What is commanded | A position, or a joint value | A contact force or torque | | What is measured back | Joint angle or displacement, from encoders | Force or torque, from a wrist sensor | | What the controller corrects | The difference between commanded and actual position | The difference between commanded and actual force | | Behaviour on meeting an obstruction | Pushes harder, since the error will not shrink | Yields, since the commanded force has been reached | | Suited to | Free motion — transferring, tracing, spraying | Contact tasks — polishing, insertion, delicate gripping | | Main risk | Damage to the part or the tool on unexpected contact | Instability if the response is made too strong; needs a fast, well-filtered sensor |

Force control is what lets an arm sand a casting, press a bearing into a housing, or close a gripper on a glass tube firmly enough to hold it and no firmer. It costs a sensor, a faster control loop and more careful tuning, and it is used where contact is part of the job rather than an accident.

17.6.3 Trajectory Planning

Suppose the tool is at one corner of a bench and must arrive at the other. Position control can hold it at the destination, but it says nothing about how the arm should travel there — and that journey is where an arm shakes itself, drops its load or strikes a fixture.

A trajectory is a time-based plan of the motion: where the tool should be at every instant, how fast it should be going, and how sharply it should be speeding up or slowing down. Trajectory planning is the business of producing that plan before the motion begins.

The planner works within limits. No joint may be driven beyond its travel or its speed; the path must clear fixtures and stay inside the fenced area; and the motion must be smooth, because an abrupt start jolts the structure and can shift whatever the gripper is holding. Fig. 17.6-2 shows the usual answer: the speed is ramped up, held, and ramped down again, so that no instant demands an impossible acceleration.

Diagram

Fig. 17.6-2 – a graph of speed against time for one move; horizontal axis labelled “time (s)”, vertical axis labelled “speed of the tool (m/s)”; the curve rises in a straight ramp, runs flat at a level marked “maximum permitted speed”, then falls in a straight ramp to zero; the three parts labelled “accelerate”, “constant speed” and “decelerate”; the area under the curve shaded and labelled “area = distance travelled”

The finished plan is handed to the position controller, which executes it step by step. The planner decides the route and the timetable; the controller keeps the arm on them.

Three kinds of trajectory cover most work, and Fig. 17.6-3 sets them side by side. A point-to-point (PTP) move carries the tool from one place to another with no control over the shape of the path between them; it is the quickest option and is used when nothing lies in the way. A linear trajectory holds the tool on a straight line between the two points, which matters when the tool must pass through a gap or lay a straight bead of weld. Circular and spline trajectories follow a specified curve, and are used for work such as tracing a round seam or painting along a contour.

Diagram

Fig. 17.6-3 – three paths drawn between the same two marked points A and B on one diagram; a curved swooping path labelled “point-to-point: shape of the path is not controlled”; a straight path labelled “linear trajectory”; an arc labelled “circular trajectory”; a rectangular obstacle drawn between A and B which the point-to-point path passes through and the other two avoid

Checking a planned move against the tool’s speed limit

Given: A palletising arm must carry a carton 800 mm along a straight path. The programmer has asked for the move to be completed in 1.2 s. The manufacturer states a maximum tool speed of 1.0 m/s. Take the simplest smooth plan, in which the tool accelerates uniformly for the first half of the move and decelerates uniformly for the second half.

Find: Whether the planned move stays within the speed limit, and if not, the shortest time in which it can be made.

Method: For a plan that ramps the speed up and straight back down, the speed–time graph is a triangle, and the distance covered is the area under it, \(d = \tfrac{1}{2} v_{peak} t\). Rearranging gives the peak speed the plan demands, which is the figure that must be compared with the limit.

\(d = 800\ \text{mm} = 0.800\ \text{m}\)

\(v_{peak} = \dfrac{2d}{t} = \dfrac{2 \times 0.800\ \text{m}}{1.2\ \text{s}} = 1.33\ \text{m/s}\)

Since \(1.33\ \text{m/s} > 1.0\ \text{m/s}\), the planned move exceeds the limit.

Shortest permitted time: \(t_{min} = \dfrac{2d}{v_{max}} = \dfrac{2 \times 0.800\ \text{m}}{1.0\ \text{m/s}} = 1.6\ \text{s}\) Answer: The move as planned would need a peak speed of about 1.33 m/s and is not permitted. Completing it within the 1.0 m/s limit takes at least 1.6 s. Check: The average speed of the original plan was only \(0.800 / 1.2 = 0.67\) m/s, comfortably inside the limit — which is exactly the trap. A tool that starts from rest and stops again must exceed its average speed somewhere in the middle, so a plan is always checked against its peak, never its average.

Q. Why plan a trajectory at all? Why not send the arm straight to the goal and let the position controller sort it out?

A. Because a controller handed a distant target sees an enormous error and answers it with everything it has. The arm lunges away at full drive, the load shifts in the gripper, the structure flexes and rings, and the tool arrives too fast to stop cleanly. The path taken is also nobody’s decision, so nothing guarantees it misses the fixture on the way. Planning replaces one huge error with a stream of small ones, each easy for the controller to follow, and keeps the route under the engineer’s control.

17.7 Manipulators at Work

Everything in this chapter so far has been about what an arm is and how it is made to behave. This section looks at the jobs those capabilities were actually bought for, and at what each job demands of the machine.

Read each one as an engineering problem first. What has to be moved or done, how accurately, how often, and what makes the task hard for a person? The choice of arm, tool and control strategy follows from those answers.

17.7.1 Pick-and-Place Operations

The most widespread use of manipulators is also the simplest to state: take an item from where it arrives and put it where it is wanted. A biscuit packaging plant needs biscuits lifted off a moving belt and set into trays; an electronics plant needs components taken from a feeder and placed on a board; a pharmaceutical line needs strips laid into cartons. The work is repetitive, fast and unrelenting, which is exactly what people are worst at and arms are best at.

Fig. 17.7-1 shows the cycle the arm repeats. Notice that it is a loop with a sensing step in it, not a fixed dance — the arm is told where the next item is before it reaches for it.

Diagram

Fig. 17.7-1 – flow: item detected on conveyor → position of item reported to controller → arm moves above the item → tool lowered and gripper closed → arm lifts and transfers to the tray → tool lowered and gripper opened → arm returns to the waiting position, with the last arrow returning to the first block to close the cycle

Where every item lies flat and is placed flat, a SCARA does the job faster than anything else, and the tool simply points down throughout. When the item must be turned over or set at an angle, the extra freedoms of an articulated arm are needed.

The end-effector is chosen from the item rather than from the arm: suction cups for smooth flat surfaces such as sachets or glass, soft jaws for food, magnetic heads for steel parts. Between picks the arm travels point-to-point through clear space for speed, then switches to a straight vertical approach for the last few centimetres, so that it comes down onto the item and not across it.

The measure of success here is cycle time — the seconds taken per item — and it is set as much by how quickly the arm can accelerate and settle as by how fast it can travel.

17.7.2 Welding

Welding was one of the first jobs given to industrial arms, and for good reason: it is skilled, exhausting and dangerous. The arc is blindingly bright, the fumes are harmful, and the sparks and heat make protective clothing essential.

Two kinds of welding are automated, and they ask different things of the machine. In spot welding, used to join sheet metal at chosen points — the panels of a vehicle body, for instance — the arm carries a heavy electrode gun to each point in turn, squeezes, passes a current, and moves on. What matters is reaching a great many awkwardly placed points quickly and holding the gun square to the metal.

Photograph

Fig. 17.7-2 – several articulated robot arms carrying spot welding guns working on a vehicle body shell on an automotive assembly line, photographed from outside the safety fencing so that at least two arms, the body shell and the fenced boundary are visible. Sourcing note: a stock or manufacturer photograph of a real production line; renders, close-ups of a single weld, and images with people inside the working area are unusable.

In arc welding the demands change entirely. Here the torch must run along a seam holding a steady angle and, crucially, a steady speed, because the amount of metal deposited at each point depends on how long the torch lingers there. Run too fast and the weld is thin; slow down at a corner and it piles up. This is a trajectory planning problem before it is anything else, and it needs the six freedoms of an articulated arm to keep the torch angle correct as the seam curves.

The gain is consistency. Every weld on every unit receives the same speed, the same angle and the same current, which is difficult for even a skilled welder to sustain through a full shift.

17.7.3 Assembly

Assembly means fitting parts together: pressing a bearing into a housing, seating a connector, driving screws, inserting components into a board. The difficulty is that the clearances are often smaller than the positional error of the machinery around them.

This is where selective compliance earns its keep. A SCARA presses the part down rigidly while giving slightly in the horizontal plane, so a chamfer or a lead-in guides the part home instead of jamming it. Where the fit is tighter still, a force sensor at the wrist lets the controller push until a set force is felt and stop, rather than pushing to a commanded depth against an unknown obstruction.

Vision is common in assembly cells for the same reason. A component that has arrived a few millimetres out of place in its tray will be picked crookedly by an arm working from a stored position, so a camera measures where it actually is and the controller shifts the programmed position to match.

The honest limit is worth stating: the arm has no idea what it is holding. If a component is fed upside down and nothing senses it, the arm will fit it upside down, perfectly, on every unit until someone notices.

On a small servo arm in the laboratory, program a move between two marked points on the bench and run it ten times, marking on paper where the tip lands each time. Measure the spread of the marks, then repeat with the arm approaching the point from the opposite direction and compare the two spreads.

17.7.4 Painting and Coating

Spray painting is a coating problem disguised as a motion problem. The thickness of paint laid down at a point depends on how far the gun is from the surface, what angle it is held at, and how long it dwells there. Hold all three steady and the film is even; let any of them wander and the finish shows it as runs, thin patches or a visible band.

An articulated arm suits this because the surfaces are rarely flat. Following the curve of a vehicle door or the moulding of a cabinet means continuously tilting the gun to keep it perpendicular while sweeping it across at constant speed — position, orientation and velocity all being controlled at once.

There are two further gains. A person spraying stands in a cloud of solvent vapour and fine paint mist for hours; a machine does not need to be there at all, and the booth can be sealed and extracted. And because the arm’s passes overlap by the same amount every time, less paint is wasted than with hand spraying. Booths of this kind are electrically hazardous areas, since solvent vapour is flammable, and arms built for them are constructed accordingly.

17.7.5 Material Handling and Palletising

Material handling covers the movement of raw material, parts and finished goods around a plant — loading a machine tool, unloading an oven, transferring castings between stations. Palletising is the particular case of stacking finished goods on a pallet for storage or dispatch.

Stacking is more demanding than it sounds. Cartons or sacks must be laid in an interlocking pattern, as in Fig. 17.7-3, so that the stack holds together when it is moved; each layer is often rotated relative to the one below it, and the arm must place items accurately enough that the stack stays square as it grows.

Diagram

Fig. 17.7-3 – plan view of two successive layers on a pallet; the lower layer showing five cartons in one interlocking arrangement, the upper layer showing the same five cartons rotated by 90° so the joins do not line up; the pallet outline drawn beneath both; an arm with a vacuum head shown at the side placing a carton, with the placing order numbered 1 to 5 on the lower layer

The demands on the arm here are payload and reach rather than freedom of orientation. The tool — usually a vacuum head or a clamp — stays pointing down throughout, so few degrees of freedom are needed, but a full sack held at the far edge of a pallet puts a large torque on the base and shoulder joints, for the reason set out in section 17.3.3. This is why palletising arms are heavily built and why their stated payload is quoted with the reach it applies at.

The human case for automating this work is straightforward. Lifting weights repeatedly at awkward angles, all shift, is a well-known cause of back injury.

Table 17.7 – What each application asks of the arm | Application | Arm usually chosen | Typical end-effector | Control strategy that dominates | |—|—|—|—| | Pick-and-place | SCARA for flat work, articulated when orientation changes | Suction cup, parallel-jaw or soft gripper | Point-to-point position control, with a straight approach | | Spot welding | Articulated, high payload | Welding gun | Position control to many taught points | | Arc welding | Articulated | Welding torch | Trajectory planning at constant tool speed | | Assembly | SCARA, or articulated for larger products | Gripper or driver spindle | Position control with force control on insertion | | Painting and coating | Articulated | Spray gun | Trajectory planning holding distance, angle and speed | | Material handling and palletising | Articulated, high payload and long reach | Vacuum head or clamp | Point-to-point position control |

Reading down the table, the pattern is plain. The arm itself contributes only two things to every one of these jobs: it puts a tool exactly where it is told, and it does so again and again without drifting. Everything else — which tool, where the work sits, what path to take, what to measure — is supplied by the engineer who designs the cell around it.

17.8 Summary

What a manipulator is
  1. A manipulator, or robotic arm, is a programmable mechanical arm of rigid links connected by joints, running from a fixed base to a tool at its free end, driven by actuators under a controller’s command.
  2. Two properties separate it from an ordinary machine: its sequence of motions is programmed and can be changed, and it returns to the same position repeatedly.
  3. The links are rigid and fix the geometry; all motion comes from the joints. Joined base to tip they form an open kinematic chain, with one end fixed and the other free to carry a tool.
  4. A revolute joint (R) rotates one link about an axis and its joint variable is an angle; a prismatic joint (P) slides one link along a line and its joint variable is a displacement. These letters name configurations such as 2R and RP.
  5. The end-effector is the device at the free end through which the arm acts on its surroundings — a gripper, a welding torch, a sprayer, a driver spindle or a suction cup. It is the only part that touches the work.
  6. Motion accumulates along the chain, so a small angular error at a joint near the base appears at the tool magnified by the distance out to it.
The parts and what each must be good at
  1. The base carries every force the arm generates and must be stiff and firmly anchored; links are wanted stiff and light at once, which pull against each other; joints have travel limits the controller must respect.
  2. An actuator converts electrical, hydraulic or pneumatic energy into mechanical motion. Electric drives suit most modern arms, hydraulic drives give large force for their size, and pneumatic drives are fast and cheap but poor at holding an intermediate position because air compresses.
  3. The transmission system — gears, belts, chains or lead screws — carries motion from actuator to joint and trades speed for torque, with \(\tau_{out} = \tau_{in} \times N \times \eta\) and \(n_{out} = n_{in}/N\) for a reduction ratio \(N\) and efficiency \(\eta\).
  4. Backlash is the free play between gear teeth. It appears at the tool as an error each time the direction of motion reverses.
  5. Payload is the mass the arm can carry at its end-effector while still meeting its stated speed and accuracy, and it includes the mass of the end-effector itself.
  6. Sensors supply feedback — the measurement of what actually happened. An encoder at each joint reports the joint’s angle or displacement; limit switches, force or torque sensors, proximity sensors and vision cameras add the rest.
  7. The controller holds the program, computes what each joint must do and coordinates all of them together. The power supply must be clean and steady for the controller and large enough for the surges the actuators draw.
Freedoms, pose and reach
  1. The degrees of freedom of an open-chain manipulator is the number of independently driven joints. Opening and closing a gripper is not counted, since it does not move the tool anywhere.
  2. A rigid body needs 3 degrees of freedom to be placed in a plane and 6 in space, which is why a full-freedom industrial arm has six joints. Fewer freedoms leave some combinations of position and direction unattainable; a seventh freedom reaches nothing new but offers more than one posture for the same pose.
  3. Position says where the end-effector’s tip is; orientation says which way the tool faces, described by roll, pitch and yaw. Together they are the pose, and delivering a commanded pose is the arm’s task stated at its shortest.
  4. The work envelope is the region the end-effector can reach, set by the link lengths and the joint travel limits. For a two-link planar arm with free joints it is a ring, reaching out to \(L_1 + L_2\) and no closer in than \(|L_1 - L_2|\).
  5. Reach and payload trade against each other, because the torque a joint must hold equals the load’s weight multiplied by its distance from that joint. Everything inside the envelope can be struck by the arm, which is why an industrial installation fences it and interlocks the gate.
The mathematics of the arm
  1. A position is meaningless until the frame it is measured in is named; for a manipulator that is a fixed base frame. Link lengths are fixed by construction; joint variables change as the arm moves.
  2. The first joint’s angle is measured from a base axis, but the second joint’s angle is measured from the direction of link 1, because that joint moves relative to the link carrying it.
  3. Forward kinematics computes the position and orientation of the end-effector from the joint variables and link lengths. For the 2R planar arm, \(x = L_1\cos\theta_1 + L_2\cos(\theta_1+\theta_2)\) and \(y = L_1\sin\theta_1 + L_2\sin(\theta_1+\theta_2)\), with the tool’s orientation equal to \(\theta_1 + \theta_2\).
  4. One pair of joint angles gives exactly one tip position, but one tip position can usually be produced by two different pairs of angles.
  5. The same geometry can be written as homogeneous transformation matrices, in which one 3 × 3 matrix carries both the rotation and the translation of a link, acting on points written as \((x, y, 1)\).
  6. Chaining links becomes matrix multiplication: \(T_{02} = T_{01} \cdot T_{12}\). The product’s top-right column holds the tool’s position and its top-left block holds the tool’s orientation, so a single matrix delivers the full pose. The order of multiplication cannot be changed.
  7. Because the 2R arm’s orientation is forced to be \(\theta_1 + \theta_2\) — the same two angles already spent on reaching the point — it cannot choose position and orientation independently.
  8. Inverse kinematics is the reverse problem: finding the joint angles that put the tool at a stated point. It may have no solution outside the envelope and usually has more than one inside it.
  9. The equations describe an ideal arm. Link tolerances, deflection under load and backlash make the true tip differ from the computed one, which is what calibration addresses; and near full extension the arm loses fine control in the outward direction, so paths are planned away from that posture.
Designs used in industry
  1. The articulated arm is a chain of revolute joints — waist, shoulder and elbow to position the wrist, and a three-axis wrist to orient the tool. Six freedoms and a roughly spherical envelope let it reach around obstacles and follow curved paths, at the cost of stiffness, speed on flat work, price and floor space.
  2. The SCARA has two rotary joints about vertical axes, a vertical prismatic joint and a tool rotation: four freedoms, a hollow cylindrical envelope, and a tool that always points down.
  3. Compliance is a structure’s willingness to give under force. The SCARA is selectively compliant — able to yield slightly in the horizontal plane while staying stiff vertically — so a part being pressed into a hole is guided into line instead of jamming.
  4. Choosing between them starts with whether the tool must be tilted and what shape the workspace is, then weighs cost, speed, stiffness and space. Freedoms an application does not need are freedoms that cost speed, stiffness and money.
Control strategies
  1. Position control drives each joint until the position error — the difference between the commanded and the measured joint value — falls to zero. Measuring the result and feeding it back makes the system closed loop; an open-loop system commands and never checks.
  2. In joint space control the target is given directly as joint values; in Cartesian space control it is given as the tool’s position in workshop coordinates, which the controller must convert into joint values by inverse kinematics.
  3. A joint-space move does not carry the tool in a straight line, because each joint turns at its own rate. Where the path matters, it must be planned in Cartesian space.
  4. Force control regulates the force the tool applies instead of the position it occupies, using a force or torque sensor at the wrist. It is essential on contact, where a small position error against a rigid surface becomes a very large force. In practice the two are combined — force into a surface, position across it.
  5. Trajectory planning produces a trajectory: a time-based plan giving position, velocity and acceleration at every instant, worked out within joint limits, speed limits, obstacles and a requirement of smoothness. It decides the route and the timetable; the position controller keeps the arm on them.
  6. Point-to-point trajectories control only the endpoints, linear trajectories hold the tool on a straight line, and circular or spline trajectories hold it on a defined curve. Smooth acceleration keeps forces within what the actuators can supply and stops the structure from vibrating.
Manipulators at work
  1. Pick-and-place moves items from where they arrive to where they are wanted, with the end-effector chosen from the item and cycle time as the measure of success.
  2. Spot welding takes a heavy gun to many taught points; arc welding runs a torch along a seam at a steady angle and, above all, a steady speed, since the metal deposited depends on how long the torch dwells at each point.
  3. Assembly turns on clearances smaller than the positional error around them, which is why selective compliance, force control on insertion and vision correction all belong to it. The arm cannot tell that a component is upside down unless something senses it.
  4. Painting and coating need constant distance, constant angle and constant speed together, since film thickness depends on all three; automation also removes a person from a booth full of solvent vapour and reduces paint wasted.
  5. Material handling and palletising need payload and reach rather than freedom of orientation, with the tool pointing down and interlocking layers stacked so the load holds together — and with the joint torques worst when the load is at full reach.
  6. Across every application the arm contributes only two things: it puts a tool exactly where it is told, and it repeats that without drifting. The tool, the fixture, the path and the sensing are the engineer’s design.

17.9 Key Terms

  1. Actuator — a device that converts electrical, hydraulic or pneumatic energy into mechanical motion, and so drives a joint.
  2. Articulated arm — a manipulator whose joints are all revolute, arranged like a shoulder, an elbow and a wrist, usually with six degrees of freedom.
  3. Backlash — the small free movement between gear teeth, which appears at the tool as an error whenever the direction of motion reverses.
  4. Base — the fixed part of a manipulator, anchored to the floor or a frame, from which all motion is measured.
  5. Cartesian space control — position control in which the target is stated as the tool’s position in workshop coordinates, which the controller converts into joint values.
  6. Closed-loop system — a system in which the actual output is measured and compared with the desired value, and the difference is used to correct the drive.
  7. Compliance — the willingness of a structure to give slightly when a force is applied to it.
  8. Controller — the computer of the arm, which holds the program, computes each joint’s motion, drives the actuators and reads the sensors.
  9. Degrees of freedom (DOF) — the number of independent joint motions a manipulator has; for an open chain, the number of independently driven joints.
  10. DOF — degrees of freedom.
  11. Encoder — a device fitted to a joint or its motor that reports the joint’s present angle or displacement to the controller.
  12. End-effector — the tool attached to the free end of the last link, through which the arm acts on its surroundings.
  13. Feedback — the return of a measurement of what actually happened to the controller, so that it can correct what it commands next.
  14. Force control — a strategy in which the controller regulates the force the tool applies rather than the position it occupies.
  15. Forward kinematics — the computation of the end-effector’s position and orientation from the known joint variables and link lengths.
  16. Gripper — an end-effector that holds a workpiece so that the arm can move it.
  17. Homogeneous transformation matrix — a single matrix that carries both the rotation and the translation contributed by one link, so that a chain can be built by matrix multiplication.
  18. Inverse kinematics — the computation of the joint variables that would place the end-effector at a stated position and orientation.
  19. Joint — the moving connection between two links, which allows one to move relative to the other.
  20. Joint limits — the ends of a joint’s permitted travel, beyond which the mechanism would be damaged.
  21. Joint space control — position control in which the target is stated directly as a set of joint values, each joint being driven to its own number.
  22. Joint variable — the quantity that specifies one joint’s position: an angle for a revolute joint, a displacement for a prismatic joint.
  23. Kinematic chain — an assembly of rigid links connected by joints so that they can move relative to one another in a definite way.
  24. Link — a rigid segment of the arm, which carries motion from one joint to the next and does not bend or stretch.
  25. Link length — the fixed distance between the two joints a link connects, which sets how far the arm can extend.
  26. Manipulator — a programmable mechanical arm of rigid links and joints, running from a fixed base to a tool at its free end, driven by actuators under a controller’s command.
  27. Open-loop system — a system that sends its command without measuring the result or correcting for it.
  28. Orientation — the direction the end-effector is facing at a given place, described in space by roll, pitch and yaw.
  29. Payload — the mass the arm can carry at its end-effector, including the end-effector itself, while still meeting its stated speed and accuracy.
  30. Pitch — rotation that tips the tool up or down.
  31. Point-to-point (PTP) — a trajectory that controls only the start and goal positions, leaving the path between them to whatever the joint motions produce.
  32. Pose — the position and orientation of the end-effector taken together.
  33. Position — where the tip of the end-effector is, given as coordinates in the base frame.
  34. Position control — a strategy in which the controller drives each joint until its measured value matches the commanded one.
  35. Position error — the difference between the commanded joint value and the value actually measured.
  36. Prismatic joint (P) — a joint that allows one link to slide along a straight line relative to the next.
  37. Programmable — able to have its stored sequence of motions changed without the machine being rebuilt.
  38. PTP — point-to-point.
  39. Revolute joint (R) — a joint that allows one link to rotate about an axis relative to the next.
  40. Robotic arm — another name for a manipulator, used interchangeably with it.
  41. Roll — rotation of the tool about the axis it points along.
  42. SCARA — Selective Compliance Assembly Robot Arm; a four-degree-of-freedom manipulator with two vertical-axis rotary joints, a vertical prismatic joint and a tool rotation.
  43. Trajectory — a time-based plan of motion giving the position, velocity and acceleration of the manipulator at every instant between start and goal.
  44. Trajectory planning — the strategy that produces such a plan within joint limits, speed limits, obstacles and a requirement of smoothness.
  45. Transmission system — the gears, belts, chains or lead screws that carry motion from an actuator to a joint and trade speed for torque.
  46. Work envelope — the region of space the end-effector can reach, set by the link lengths and the joint travel limits.
  47. Yaw — rotation that swings the tool left or right.

17.10 Objective Questions

  1. The number of independent quantities needed to specify the pose of a rigid body in space is:
  1. 3
  2. 4
  3. 6
  4. 9
  1. The joint variable of a prismatic joint is:
  1. an angle, measured in degrees or radians
  2. a displacement, measured in millimetres
  3. a torque, measured in newton metres
  4. an angular speed, measured in revolutions per minute
  1. Choose the correct option for the assertion and reason given below. Assertion (A): A 2R planar arm cannot choose the position of its tool and the direction the tool faces independently of each other. Reason (R): In a 2R planar arm the tool’s orientation equals the sum of the two joint angles, and both angles have already been used in reaching the position.
  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. Match each item in Column A with the statement in Column B that describes what it does.
Column A Column B
a. Encoder i. Anchors the arm and carries the forces it generates
b. Transmission system ii. Reports a joint’s present angle to the controller
c. End-effector iii. Trades the actuator’s speed for torque at the joint
d. Base iv. Acts directly on the workpiece
  1. Fill in the blanks.
  1. The small free movement between gear teeth is called ______, and it shows itself at the tool whenever the direction of motion ______.
  2. For a two-link planar arm with freely turning joints, the outer boundary of the work envelope lies at a distance ______ from the base and the inner boundary at a distance ______.
  1. State whether each of the following is true or false.
  1. The stated payload of an arm is the mass it can carry after the mass of the end-effector has been subtracted.
  2. In force control the controller regulates the force the tool applies rather than the position it occupies.
  3. A move commanded in joint space carries the tool along a straight line between the two points.
  4. Opening and closing a gripper adds one degree of freedom to a manipulator.
  1. Write the full form of each of the following abbreviations: DOF, PTP, SCARA.

17.11 Check Your Concepts

  1. Define a manipulator, and state the two properties that separate it from an ordinary production machine such as a lathe.

  2. Name the two kinds of joint used in manipulators. For each, state the motion it permits, its joint variable and the unit in which that variable is measured.

  3. Explain why every joint of an industrial arm carries an encoder, even though the controller already knows what value it commanded that joint to take.

  4. Differentiate between position control and force control on three points of difference, and give one task for which each is the appropriate strategy.

  5. A 2R planar arm has link lengths \(L_1 = 300\) mm and \(L_2 = 200\) mm. The joints are set to \(\theta_1 = 40°\) and \(\theta_2 = 25°\). Compute the coordinates of the tip in the base frame, then show by calculation whether this point lies inside the arm’s work envelope.

  6. A packaging line must lift flat foil sachets from a moving belt and lay them flat into cartons standing on a fixed table at the same height. Justify, with reasons, your choice of arm design, of end-effector, and of the trajectory type used for the transfer between belt and carton. State one limitation of the arrangement you have chosen.

17.12 Diagram, Circuit & Output Reading

  1. The block diagram of the position control of one joint contains the following blocks and paths: a block giving the desired joint angle, feeding a comparator; an arrow from the comparator to a block marked “controller”; an arrow from the controller to a block marked “actuator”; an arrow from the actuator to a block marked “joint and link”; and a feedback path returning from the joint to the comparator through a block whose label has been left blank.
  1. Name the blank block and state what it measures.
  2. Name the quantity travelling along the arrow from the comparator to the controller, and state how it is obtained.
  3. Predict what the arm would do if the feedback path were cut while the arm was carrying a load heavier than the one it was programmed for.
  1. A cell must place a 3 kg component, held by a gripper of mass 1.5 kg, at a point 900 mm from the base, with the tool tilted 30° from the vertical. The three arms available are described below.

Table 17.8 – Typical data for the three arms available, listing degrees of freedom, maximum reach and rated payload | Arm | Degrees of freedom | Maximum reach | Rated payload | |—|—|—|—| | P | 4 (three rotary, one prismatic, tool always vertical) | 800 mm | 6 kg | | Q | 6 (all rotary) | 1200 mm | 5 kg | | R | 6 (all rotary) | 700 mm | 12 kg |

  1. State which arm can do the job.
  2. For each of the other two, name the single specification that rules it out.
  3. State what the load on the chosen arm is, and explain why that figure is not simply the mass of the component.
  1. A student building a two-link planar arm predicts the tip position using \(x = L_1\cos\theta_1 + L_2\cos\theta_2\) and \(y = L_1\sin\theta_1 + L_2\sin\theta_2\). On testing, the predicted and measured positions agree whenever \(\theta_1 = 0°\), but the disagreement grows steadily as the shoulder joint is swung further round.
  1. Identify the single fault in the student’s expressions.
  2. Write the corrected expressions.
  3. Explain why the fault stays hidden when \(\theta_1 = 0°\).
  1. An arm is built with four driven joints: a rotary joint at the base turning about a vertical axis, a second rotary joint turning about a vertical axis at the elbow, a prismatic joint that raises and lowers the tool vertically, and a rotary joint that turns the tool about its own vertical axis.
  1. State the number of degrees of freedom of this arm.
  2. State whether this arm can present its tool pointing horizontally, and give the reason.
  3. Describe the shape of its work envelope, seen first in plan view and then as a solid.

17.13 Case-Based Questions

  1. A school workshop team is building a small two-link planar arm to move components about a bench. The link lengths are \(L_1 = 200\) mm and \(L_2 = 150\) mm, each joint is driven by its own servo, and both joints can turn freely. The team marks two target points on the bench, one 120 mm from the base and one 380 mm from the base, both measured in the plane in which the arm moves.
  1. Calculate the inner and outer boundaries of the arm’s work envelope, and state for each target point whether the arm can reach it, with the reason.
  2. With the joints set to \(\theta_1 = 60°\) and \(\theta_2 = 30°\), calculate the coordinates of the tip in the base frame.
  3. The team finds that the tip stops a few millimetres short of the marked point every time the arm approaches it from one particular direction, but lands correctly when it approaches from the other. Name the most likely mechanical cause, and suggest one way of reducing its effect.
  1. A warehouse is installing an arm to stack cartons onto pallets. Each carton has a mass of 8 kg. The vacuum head used to lift them has a mass of 4 kg. The arm is rated to carry 15 kg at a reach of 1.2 m. The far corner of the pallet lies 1.4 m from the base of the arm, and the finished stack is 1.6 m high.
  1. Show by calculation whether the arm can lift a carton within its rating at a reach of 1.2 m.
  2. Explain what changes when the same carton has to be placed at the far corner of the pallet, 1.4 m from the base, and why the manufacturer quotes payload together with a reach.
  3. To save floor space, one member of the team proposes leaving out the safety fencing around the cell and relying on operators to stay clear while the arm is running. Evaluate this proposal, and state what the cell should have instead.