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Chapter 7: Links and Joints

On the way to school, Nisha’s bus stops beside a road-widening site. A yellow excavator is working there. She watches its long arm swing to the left, drop down, scoop up a load of soil, lift it, swing back and empty it into a waiting truck. The whole machine never moves from its place. Only the arm moves.

Look carefully at that arm and you will see something useful. It is built from a few long, solid pieces of steel. Those pieces do not bend. All the bending happens at the points where two pieces meet. The steel gives the arm its strength. The meeting points give it its movement.

A robot arm in a car factory is built the same way. So is the arm you are reading with right now. Your upper arm and forearm are the solid pieces. Your shoulder, elbow and wrist are the meeting points.

Once you can see a machine as solid pieces plus meeting points, you can work out almost any movement it makes.

Diagram

Fig. 7.0-1 – a simple side view of an excavator arm beside a human arm, drawn in the same style. On the excavator, label the three straight steel pieces and mark the three bending points with small circles. On the human arm, label the upper arm and forearm and mark the shoulder, elbow and wrist with small circles, so the two look alike.

NoteCuriosity Corner

Q1. What is the difference between a link and a joint in a robot? Q2. Which joint lets a robot slide in a straight line, and which one lets it turn? Q3. Why does a robot arm bend only at certain points and not anywhere along its length? Q4. Why do some robots use soft, bendable parts instead of hard metal ones? Q5. What does it mean when someone says a robot has six degrees of freedom?

This chapter will answer all five of these questions.

7.1 The Moving Parts of a Robot

Think about picking up a pencil from your desk. Your arm has to reach out, tilt down, and close your fingers around it. Now imagine your arm was a single straight rod of steel from shoulder to fingertip. You could push it and pull it, but you could never reach around anything. The pencil would stay on the desk.

An arm is useful because it is not one piece. It is a few stiff pieces connected at points that can move. This is the basic idea behind the body of almost every robot that has to reach, lift or place something.

A stiff piece is called a link. A link does not bend or stretch during work. Its job is to hold shape, carry weight and keep the parts at the correct distance from each other. Links are usually made of aluminium, steel or strong plastic.

The movable point where two links meet is called a joint. A joint is a mechanical connection that lets one link move with respect to the link next to it. All the movement of the arm happens at the joints.

NoteDefinition

A link is a rigid or soft component of a robot that connects two joints and gives the robot its structure.

NoteDefinition

A joint is a mechanical connection between two links that allows them to move relative to each other.

NoteCuriosity Corner

Q1. What is the difference between a link and a joint in a robot?

A. A link is the solid piece that gives the robot its shape and carries the load. A joint is the movable connection between two links. Links hold position; joints create movement.

From base to end effector

A robot arm is built as a chain. Look at Fig. 7.1-1 as you read this. At the bottom is the base, the part fixed to the floor, a table, a wall or the ceiling. The base does not move. It takes the weight of everything above it and stops the arm from tipping over when it lifts a load.

Diagram

Fig. 7.1-1 – side view of a simple two-joint robot arm. From the bottom: a shaded floor, then a block labelled Base, then Link 0 rising upward, a circle labelled Joint 1, Link 1 going up and to the right, a circle labelled Joint 2, Link 2 going out to the right, and at the far end a small gripper labelled End effector.

Above the base, links and joints alternate: link, joint, link, joint, and so on. Each joint adds one more way for the arm to move, and each link carries that movement further out into the room.

At the far end of the chain is the end effector — the tool that actually does the job. It may be a gripper that holds objects, a welding tip, a suction cup, a paint spray head or a camera mount. Everything behind it exists only to bring the end effector to the right place, pointing the right way.

NoteExam Corner

Q. If all the movement happens at the joints, why do the links matter at all?

A. The links decide how far the arm can reach and how much weight it can hold. A joint by itself moves nothing useful; it is the link attached to it that swings out and carries the tool and the load.

Where the movement comes from

A joint does not move on its own. Something has to push or turn it. In most robots this is an electric motor; in heavy machines it is often a cylinder driven by oil or compressed air. Many joints also carry a small sensor that reports the joint’s present angle or position, so the controller knows where the arm actually is.

Because links stay stiff, the arm can only change shape at the joints. This is a good thing. It means the controller has a small, fixed set of things to control — one number for each joint — instead of a bendy arm whose shape could be anything.

NoteCuriosity Corner

Q3. Why does a robot arm bend only at certain points and not anywhere along its length?

A. Because the links are made stiff on purpose and only the joints are built to move. Fixing the bending points to a few known places keeps the arm strong and makes its position easy to calculate and control.

There is a cost to this design. Joints are the parts that wear out first, because they carry moving surfaces, motors and wiring. Adding more joints gives an arm more freedom to move, but also more weight, more cost and more parts that need servicing. A good robot design uses the smallest number of joints that can still do the job.

NoteSafety Note

A robot arm can start moving without warning while power is on, even if it has been still for a long time. Never put your hand between two links or near a joint to check a fitting. Switch the power off at the supply and wait for the arm to come to rest before touching any part of it.

7.2 Types of Motion in a Robot

Before you can choose the parts that make a robot move, you have to be clear about what kind of movement you actually want. “Move the box to the next table” and “turn the lid by a quarter” are both movement, but they are not the same kind of movement, and they need different parts.

Engineers sort robot movement into a few basic types. Once you can name the type, choosing the part becomes much easier. Three types matter most in robotics: linear motion, angular motion and circular motion. Fig. 7.2-1 shows all three together.

Diagram

Fig. 7.2-1 – three small panels side by side, drawn in the same style. Panel 1 labelled Linear motion: a block on a horizontal rail with a straight double-headed arrow along the rail. Panel 2 labelled Angular motion: a straight bar pinned at one end by a small circle, with a curved arrow showing it swinging up and down through an angle marked with the symbol for an angle. Panel 3 labelled Circular motion: a wheel with a dot marked on its rim and a dashed circle showing the path the dot traces, with an arrow going all the way round.

7.2.1 Linear Motion

Push a drawer shut. Every part of the drawer travels the same distance in the same direction, and the drawer does not turn while it goes. That is linear motion — movement in a straight line, with no turning.

NoteDefinition

Linear motion is motion in which a body moves along a straight line, so that every point of the body moves the same distance in the same direction.

Linear motion is also called translational motion, because the whole body is simply shifted, or translated, from one place to another.

You see it all around a workshop. A box riding along a conveyor belt moves in a straight line. The print head of a 3D printer slides along its rail. A lift goes straight up and down. In each case the object arrives somewhere new but is still facing the same way as before.

Linear motion is the natural choice when a robot must place something at an exact spot along a line — feeding sheets into a cutting machine, or moving a nozzle across a fixed width.

7.2.2 Angular Motion

Now bend your elbow. Your forearm swings, but your elbow itself stays where it is. Your hand moves a long way, your wrist moves less, and the point right at the elbow hardly moves at all. Different points move different distances. This is not linear motion.

This is angular motion — motion in which a body turns about a fixed line called its axis. The axis is the imaginary line the body turns around; at your elbow, the axis runs across the joint from one side to the other.

NoteDefinition

Angular motion is motion in which a body turns about a fixed axis, so that it sweeps through an angle instead of travelling in a straight line.

Angular motion is measured as an angle, in degrees. A door on its hinges swings through an angle. The shoulder of a robot arm swings through an angle. A servo motor is often given a plain instruction such as “turn to 90 degrees”, and that number is an angle.

Angular motion is what lets a machine reach sideways without moving its base. The excavator in the chapter opening never left its spot, but its arm swung from the soil pile to the truck. That whole reach came from angular motion at three joints.

NoteExam Corner

Q. A robot arm swings its gripper from left to right across a table. Is that linear or angular motion?

A. Angular. The gripper travels along a curve, not a straight line, because the arm is turning about the joint at its shoulder. Points nearer the joint move a shorter distance than the gripper does.

7.2.3 Circular Motion

Watch the wheel of a moving robot. Take one point on the rim — a spot of paint, say. That spot goes round and round, always at the same distance from the centre, tracing a circle in the air. That is circular motion.

NoteDefinition

Circular motion is motion in which a point travels along a circular path, staying at the same distance from a fixed centre.

Wheels, fan blades, drill bits and pulleys all show circular motion. The turning goes on and on in the same direction, rather than swinging back and forth.

Angular motion and circular motion are closely related, and it helps to see how. When a wheel turns, the wheel as a whole is doing angular motion — it sweeps through an angle about its axle. At the same time, every single point on that wheel is doing circular motion around the axle. They are two views of the same event: the body turns through an angle, and each of its points rides around a circle.

NoteCompare Table

Table 7.1 – Angular motion vs circular motion | Feature | Angular motion | Circular motion | |—|—|—| | What is being described | The whole body turning about an axis | The path taken by one point on the body | | Measured as | An angle, in degrees | A path, a circle of a certain radius | | Usual extent in robots | A limited swing, often back and forth | Continuous turning, round and round | | Typical example | A robot elbow bending by 45 degrees | A robot wheel spinning as it drives |

NoteCommon Error

Calling every turning movement “circular motion” What goes wrong: A robot elbow that swings up by 30 degrees and comes back is often described as circular motion. But nothing completes a circle here. The fix: Use angular motion when you are talking about a body turning through an angle, especially a limited swing. Use circular motion when you are describing a point going all the way round a fixed centre, as on a wheel.

Circular motion at a wheel is how a mobile robot gets linear motion of its body. Each full turn of the wheel rolls the robot forward by the distance around the wheel’s rim.

NoteExample 7.1

How far a robot moves in one turn of its wheel

Given: A mobile robot has wheels of diameter \(d = 7\text{ cm}\). Take \(\pi = 3.14\).

Find: The distance the robot moves forward in one full turn of a wheel, and in 10 full turns.

Working: When a wheel rolls without slipping, one full turn carries the robot forward by the distance around its rim. That distance is

\[\text{distance in one turn} = \pi \times d\]

where \(d\) is the diameter of the wheel — the distance straight across it through the centre — and \(\pi\) is the constant that links a circle’s rim to its diameter.

\[\pi \times d = 3.14 \times 7\text{ cm} = 21.98\text{ cm}\]

For 10 turns: \(10 \times 21.98\text{ cm} = 219.8\text{ cm}\).

About 21.98 cm in one turn, and 219.8 cm (a little under 2.2 m) in 10 turns.

Notice the limit of that answer. It is correct only if the wheel does not slip. On loose sand or a wet floor the wheel can spin without carrying the robot the full distance, and a robot that counts wheel turns to work out where it is will then believe it has gone further than it really has.

NoteTry Yourself

Hold a ruler flat on your desk and slide it straight across without turning it — that is linear motion. Now press a pencil tip down on one end of the ruler and swing the other end — that is angular motion. Watch how far the swinging end travels compared with the middle of the ruler.

7.2.4 Motion in One Dimension and Two Dimensions

There is a second, simpler way to describe how a robot moves: by counting how many numbers you need to say exactly where it is.

A dimension is one independent direction in which something can move. If a robot slides along a straight rail, you only need one number — how far along the rail it is. Movement like this is called motion in one dimension. A lift is another example: one number, its height, tells you everything about its position.

Now put a small robot on a flat table top where it can go forwards, backwards, left and right. One number is no longer enough. You need two — how far across, and how far along. This is motion in two dimensions, and the two numbers are usually written as \(x\) and \(y\), where \(x\) is the distance measured across the table and \(y\) is the distance measured along it. Fig. 7.2-2 shows both cases.

Diagram

Fig. 7.2-2 – two panels. Panel 1 labelled One dimension: a small robot sitting on a straight horizontal rail, with a single arrow along the rail and one distance marked from the left end to the robot. Panel 2 labelled Two dimensions: a top view of a rectangular table with an x axis along the bottom edge and a y axis up the left edge, a small robot somewhere on the table, and dashed lines dropping from the robot to each axis to show its two distances.

The pen of a plotter drawing a map moves in two dimensions across the paper. A floor-cleaning robot moves in two dimensions across a room, because it stays on the floor and only its position on that floor changes. Adding a third number for height would give motion in three dimensions, which is what a full robot arm works in.

NoteExam Corner

Q. A 3D printer head moves left and right along a rail, and the rail itself moves forward and back. Is the head moving in one dimension or two?

A. Two. Its position on the flat printing area needs both numbers — how far left or right, and how far forward or back. Either movement on its own would be one dimension; together they cover a plane.

7.3 Joints in a Robot

You now know what kinds of motion a robot may need. The next question is a practical one: what part actually produces each kind? A machine cannot slide “in general”. Something has to be built into it that can only slide, or can only turn.

That part is the joint. The type of joint you fit decides the type of motion you get, and nothing else about the arm can change that. A joint that slides will never turn.

Every joint needs something to drive it. The part that supplies the movement — an electric motor, or a cylinder pushed by oil or compressed air — is called an actuator. The joint decides the kind of motion; the actuator provides the force or the turning effort that causes it.

Robots use three main types of joints: prismatic joints, revolute joints and spherical joints.

7.3.1 The Prismatic Joint

Think of a drawer in a desk, or the sliding window in a classroom. The moving part rides along a track, and it can go only one way — forwards and backwards along that track. It cannot turn, and it cannot leave the track.

A prismatic joint works exactly like this. One part slides over another along a shared straight axis, so the joint produces linear motion. Because of the sliding action it is also called a slider. Fig. 7.3-1 shows the idea.

Diagram

Fig. 7.3-1 – a horizontal rail labelled fixed member with a block sitting on it labelled sliding member. A straight double-headed arrow above the block shows movement along the rail. A dashed line runs through the rail and the block labelled axis of the joint. Mark the sliding distance with a small letter x.

NoteHow It Works

Prismatic joint What it is: A joint made of two parts — a fixed member and a sliding member that moves along it in a straight line. How it works: An actuator extends or pulls back the sliding member along the shared axis. The drive is usually a motor turning a lead screw, or a cylinder driven by compressed air or oil. Used for: The vertical column of a pick-and-place machine on an assembly line, which lifts straight up and lowers straight down over a conveyor. Watch out: It can only reach points along one straight line. The open rail also collects dust and grit, and a dirty rail makes the joint stiff or jams it altogether.

Prismatic joints have some useful properties. They are stiff, so they hold their position well when a load pushes against them. The load is spread along the rail rather than concentrated at one point, which suits heavy work. They usually run more quietly than a geared rotating joint. And in machines built in large numbers, a simple slide is cheap to make.

7.3.2 The Revolute Joint

Now think of a door. It is fixed to the wall by hinges, and it swings open through an angle. It never slides sideways; it only turns.

A revolute joint works the same way. It allows angular motion about a single fixed axis, which is why it is also called a hinge joint or rotary joint. It has a fixed member and a rotating member, and the rotating member swings with respect to the fixed one. See Fig. 7.3-2.

Diagram

Fig. 7.3-2 – a fixed base block with a pin through it shown as a small circle labelled axis of rotation. A bar is attached to the pin and drawn in two positions, one solid and one dashed, with a curved arrow between them and the angle between the two positions marked. Label the base fixed member and the bar rotating member.

NoteHow It Works

Revolute joint What it is: A joint that lets one link turn about a single fixed axis with respect to the link next to it. How it works: The controller sends the actuator an angle and a speed. A motor, usually through a gearbox, turns the rotating member to that angle, and a sensor at the joint reports the angle reached. Used for: The shoulder and elbow of an industrial robot arm, which together swing the tool across the working area. Watch out: A small error in angle at the joint becomes a large error at the far end of a long arm, so the joint’s angle sensor and gears have to be accurate.

That last point is worth working out with real numbers, because it explains why robot joints are built so carefully.

NoteExample 7.2

How much a one-degree error moves the tool

Given: A robot arm is 60 cm long from the joint to the tip of its gripper. The joint turns to the wrong angle by 1 degree. Take \(\pi = 3.14\).

Find: How far the gripper ends up from where it should be.

Working: As the arm turns, the gripper travels along a circle of radius 60 cm. A full turn of 360 degrees would carry it once around that circle, a distance of

\[2 \times \pi \times r = 2 \times 3.14 \times 60\text{ cm} = 376.8\text{ cm}\]

where \(r\) is the length of the arm from the joint to the gripper. One degree is one part in 360 of that:

\[\frac{376.8\text{ cm}}{360} = 1.0467\text{ cm}\]

About 1.05 cm — a little over one centimetre.

One degree sounds tiny, but it has put the gripper more than a centimetre off target. On an arm twice as long the same error would be twice as large.

Revolute joints are the most common joints in robotics. They are mechanically simple, easy to drive with an ordinary motor, and they lose little energy in transmitting movement. They are used in robot arms, legs and wheel drives, and they can be fitted with damping so that a sudden knock does not pass straight through the machine.

NoteCuriosity Corner

Q2. Which joint lets a robot slide in a straight line, and which one lets it turn?

A. The prismatic joint slides in a straight line and produces linear motion. The revolute joint turns about a fixed axis and produces angular motion.

NoteExam Corner

Q. A robot must lift a tray straight up without tilting it. Which joint suits the job better?

A. A prismatic joint. It moves the tray along a straight vertical line and keeps it facing the same way. A revolute joint would swing the tray along a curve and tilt it as it went.

7.3.3 The Spherical Joint

Hold your arm out and move your hand around. You can raise it, lower it, take it forward and back, and roll it — all from the same shoulder. One connection is giving you movement in several directions at once. A hinge could not do this.

A spherical joint is built for exactly this. It has two parts: a ball element, which is a ball-shaped piece free to turn inside the second part, the socket, which is the hollow cup holding it. Because the ball can turn inside the socket in any direction, the joint gives turning about all three directions in space — the X, Y and Z co-ordinates. It is also called a ball-and-socket joint. Fig. 7.3-3 shows a cut-through view.

Diagram

Fig. 7.3-3 – a cut-through side view of a ball-and-socket joint. Show a cup-shaped part labelled socket mounted on a base, with a ball inside it labelled ball element and a rod coming out of the ball. Draw three curved arrows around the rod showing that it can tilt in two directions and roll about its own length.

The human shoulder is the everyday example of a ball-and-socket joint, and artificial replacement joints copy the same shape. In robots, spherical joints appear in wrists, grippers and end effectors, where the tool must be pointed in many different directions without moving the whole arm. In vehicles they are used in driveshafts and steering linkages.

Their strength is range. A spherical joint gives a wide, smooth movement in three directions from a single compact connection, and it can soak up a sudden shock instead of passing it on to the rest of the machine.

Their weakness is control. Because three movements share one connection, it is hard to fit a separate motor and angle sensor for each one. So in many machines the spherical joint is not driven directly. It is left free to follow, while other joints do the driving — which is why a robot wrist is often built from two or three revolute joints arranged to meet at a point, rather than from one true ball-and-socket joint.

7.3.4 Comparing the Three Joints

The choice of joint follows from the job. Ask what motion is needed, how much room there is, how heavy the load will be, and how exactly the position must be held. Table 7.2 sets the three types side by side.

NoteCompare Table

Table 7.2 – Prismatic, revolute and spherical joints | Feature | Prismatic joint | Revolute joint | Spherical joint | |—|—|—|—| | Also called | Slider | Hinge or rotary joint | Ball-and-socket joint | | Motion produced | Linear motion | Angular motion | Turning in three directions | | Directions of movement | One, along a straight axis | One, about a fixed axis | Three, about a single point | | Main parts | Fixed member and sliding member | Fixed member and rotating member | Ball element and socket | | Typical use in a robot | Vertical lift of a pick-and-place machine | Shoulder and elbow of a robot arm | Wrist and gripper mountings | | Main limitation | Reaches only along one line; rail must stay clean | Small angle errors grow along a long arm | Hard to drive and measure each direction separately |

NoteCommon Error

Thinking a joint can be made to move differently by programming it What goes wrong: Students sometimes expect that a prismatic joint can be told to turn, or a revolute joint told to move in a straight line, by writing the right instruction. The fix: The kind of motion is fixed by the mechanical build of the joint, not by the program. A program can only choose how far and how fast that one motion goes. To get a different kind of motion, you must combine joints of different types — which is what the next section is about.

7.4 Links in a Robot

A joint on its own does nothing useful. Bolt a motor to a table and let it turn: something spins, but nothing is carried anywhere. The joint only becomes useful when a piece of the robot is attached to it and swings out into the room.

That piece is the link, and it decides three important things. It decides how far the robot can reach, because reach depends on link length. It decides how much the robot can carry, because the link has to take the weight without breaking or sagging. And it decides where the next joint sits, which shapes the whole arm.

Links have some other jobs too. The last link of an arm usually carries the end effector, so the tool is really part of the final link. Links spread the forces from the load down the arm towards the base. And because a link is a hollow or channelled piece of metal, it is a convenient place to house the motors, gears and wiring that drive the joints.

The weight a robot is able to carry is called its payload. Payload is limited mainly by the links and the joint actuators — a long, thin link will bend under a load that a short, thick one carries easily.

Links are of two broad types: rigid links and soft links.

7.4.1 Rigid Links

Press down on the middle of a steel scale held at both ends and it barely moves. That stiffness is exactly what most robot arms are built from.

A rigid link is a link that does not bend or change shape while the robot works. Rigid links are made of materials that resist being deformed — aluminium, steel, or composite materials for lightness with strength. They hold their length and their shape whether the robot is empty-handed or carrying a full load.

This matters for accuracy. If a link keeps its exact shape, then knowing the angle of every joint is enough to know where the tool is. See Fig. 7.4-1, where the same link is shown carrying a load and staying straight.

Diagram

Fig. 7.4-1 – two panels. Panel 1 labelled Rigid link: a straight bar fixed at one end to a joint, with a weight hanging from the far end, and the bar drawn perfectly straight. Panel 2 labelled Soft link: the same arrangement, but the bar is drawn curving downward under the weight, with a dashed straight line showing where a rigid link would have been.

Rigid links also make a robot repeatable. Send the arm to the same set of joint angles a thousand times and the tool arrives at the same spot each time, because nothing in between has stretched or flexed. On an assembly line, that is the whole point.

NoteExample 7.3

Working out the reach of an arm

Given: A robot arm has two rigid links. The link from the shoulder to the elbow is 40 cm long, and the link from the elbow to the gripper is 35 cm long. Treat the joints themselves as taking up no space.

Find: The furthest and the nearest distance from the shoulder at which the gripper can be placed.

Working: The gripper is furthest away when both links point the same way, so that the two lengths add:

\[40\text{ cm} + 35\text{ cm} = 75\text{ cm}\]

It is nearest when the elbow is folded right back, so that the second link points back along the first and the two lengths subtract:

\[40\text{ cm} - 35\text{ cm} = 5\text{ cm}\]

The gripper can be placed between 5 cm and 75 cm from the shoulder.

That result is only true because the links are rigid. If they stretched or bent, the reach would change with every load, and no fixed answer would exist.

NoteExam Corner

Q. The arm above needs to reach 90 cm. Can this be done by turning the joints further?

A. No. Turning changes direction, not distance. The 75 cm limit comes from the link lengths, so the arm would need longer links — or the whole robot would have to be moved closer.

Rigid links have an honest drawback. They are heavy, so the motors have to be stronger and use more power. And because they do not give way, a collision is a hard one: if a rigid arm strikes an obstacle, or a person, the full force is delivered.

7.4.2 Soft Links

Now think about picking up a ripe tomato. Your fingers close gently and give way where the fruit pushes back. A stiff metal gripper would either miss it or crush it.

A soft link is a link that is meant to bend or deform in response to an outside force. Soft links are also called flexible links. They are made from soft polymers, elastomers such as rubber, or inflatable sections, so they change shape and then return to it.

Softness solves problems that stiffness cannot. A soft link can push its way past an obstacle instead of knocking it over, which helps a robot work in a cluttered or awkward space. A soft gripper moulds itself around whatever it holds, so one gripper can pick up objects of many shapes and sizes without a separate tool for each. And when a soft arm bumps into something, it gives way rather than delivering the whole force. See Fig. 7.4-2.

Diagram

Fig. 7.4-2 – photo of a soft robotic gripper made of curved rubber-like fingers, closed gently around a single piece of fruit resting on a table.

NoteCuriosity Corner

Q4. Why do some robots use soft, bendable parts instead of hard metal ones?

A. Because bending is useful. A soft link changes shape to fit around an object it is holding, squeezes past obstacles in a crowded space, and absorbs the force of a knock instead of passing it on. A rigid link can do none of these.

The cost of softness is knowing where things are. With a rigid arm, the joint angles tell you where the tool is. With a soft link, the shape depends on the load as well, so the same joint angles can put the tip in different places on different days. Soft links also carry far less payload than metal ones of the same size.

NoteCompare Table

Table 7.3 – Rigid links vs soft links | Feature | Rigid link | Soft link | |—|—|—| | Behaviour under load | Keeps its shape | Bends or deforms, then returns | | Usual materials | Aluminium, steel, composites | Soft polymers, elastomers, inflatable sections | | Payload | High | Low | | Position accuracy | High and repeatable | Changes with the load | | Best suited to | Assembly lines and repeated exact movements | Handling delicate objects and working in cluttered spaces |

NoteSafety Note

A soft link is not automatically a safe link. The motor, pump or air supply driving it can still be strong enough to trap a finger. Treat a soft robot like any other machine: switch off the power supply and let any stored air pressure out before you handle it.

7.5 Making a Robot Move the Way You Want

Take a real task. A bottle comes along a conveyor belt. The robot must pick it up, carry it across to a box standing on a table, lower it in, and let go.

No single joint can do that. A prismatic joint could lower the gripper, but it could never carry the bottle across to the box. A revolute joint could swing across to the box, but it could not come straight down onto the bottle without tilting it. The task needs both.

This is the real work of robot design: not choosing one joint, but arranging several joints and links so that their separate motions add up to the movement the job needs.

7.5.1 Joining Links and Joints into a Chain

Robot arms are built as a chain. Start at the base. Add a joint. Attach a link to it. Add another joint at the far end of that link. Attach the next link. Carry on until you reach the end effector.

Each joint in the chain contributes its own one motion, and each link carries everything beyond it further out. What makes the chain powerful is that the joints move at the same time. Two joints, each giving a simple swing, combine into a path that neither could produce alone.

Here is the important consequence. A robot arm made only of revolute joints can still move its gripper in a straight line — but only if the joints turn together in a carefully matched way. As the shoulder swings the arm out, the elbow must fold in by just the right amount at every instant to cancel the curve. Fig. 7.5-1 shows the same arm at three moments during such a move.

Diagram

Fig. 7.5-1 – a two-link arm drawn three times over the same base, in light, medium and dark line. In each drawing the shoulder joint has swung further and the elbow has folded by a different amount, but the gripper sits on the same horizontal dashed line, at three points along it. Label the dashed line “straight path of the gripper” and label the shoulder and elbow joints.

So the type of motion at a joint and the type of motion of the tool are not the same thing. Turning joints can produce a straight-line path for the tool, if the controller works out the angles. Choosing joints that match the task simply makes that job easier: a task that is mostly up-and-down is far simpler with a prismatic joint doing the lifting.

NoteExam Corner

Q. If revolute joints can produce a straight path anyway, why fit a prismatic joint at all?

A. Because it needs no coordination. One prismatic joint moving on its own gives a perfectly straight line, while revolute joints must be matched against each other continuously. The prismatic joint is also stiffer for lifting a load straight up.

Now put the bottle task into steps, using an arm that has a turning base, a shoulder joint, an elbow joint, a prismatic joint that raises and lowers the gripper, and a gripper as its end effector.

NoteAlgorithm

Pick a bottle from a conveyor and place it in a box Input: The position of the bottle on the conveyor and the position of the box. Output: The bottle standing inside the box. Steps:

1. Turn the base joint until the arm faces the conveyor.
2. Swing the shoulder and elbow joints so that the gripper is directly above the bottle.
3. Extend the prismatic joint to lower the gripper around the bottle.
4. Close the gripper.
5. Pull back the prismatic joint to lift the bottle clear of the belt.
6. Turn the base joint until the arm faces the box.
7. Swing the shoulder and elbow joints so that the gripper is above the box.
8. Extend the prismatic joint to lower the bottle into the box.
9. Open the gripper.
10. Pull back the prismatic joint and return all joints to their starting positions.

Why it works: Each step asks one kind of joint for the one motion it is built to give — turning for reaching across, sliding for going straight down — so the separate motions add up to the whole task.

Read the steps again and notice something. Steps 3, 5, 8 and 10 all use the prismatic joint. Vertical movement happens four times in a single cycle. That is why a machine built for this job usually gets a prismatic joint for lifting even though revolute joints could manage it.

7.5.2 Identifying Links and Joints in a Given System

Being able to look at a machine and name its links and joints is a skill worth practising. It is also how you begin to understand any machine you have never seen before. Work through it in a fixed order.

Reading a machine, step by step

Start at the base — the part fastened to the floor, the wall or the frame, which does not move. Then follow the machine outwards from there. At every place where two parts move against each other, stop and ask one question: does this connection slide, or does it turn? A sliding connection is a prismatic joint. A connection that turns about one axis is a revolute joint. A connection that can turn in several directions from one point is a spherical joint. Between any two joints is a link. The last item in the chain, the part that touches the work, is the end effector.

Try this on an industrial robot arm of the kind used for welding and pick-and-place work, shown in Fig. 7.5-2.

Diagram

Fig. 7.5-2 – photo of a six-axis industrial robot arm bolted to a factory floor, seen from the side, with the gripper at the end clearly visible.

Following the chain from the floor upwards gives the list in Table 7.4.

NoteTable

Table 7.4 – the parts of an industrial robot arm, with the type of each part and the motion it gives | Part of the arm | Link or joint | Type | Motion it gives | |—|—|—|—| | Bolted foot on the floor | Base | Fixed | None | | Waist | Joint | Revolute | Turns the whole arm left and right | | Lower arm | Link | Rigid | Carries the arm upward and outward | | Shoulder | Joint | Revolute | Raises and lowers the lower arm | | Upper arm | Link | Rigid | Carries the reach further out | | Elbow | Joint | Revolute | Folds the upper arm | | Wrist | Joint | Revolute | Tilts and rolls the tool | | Gripper | End effector | — | Holds and releases the object |

The pattern is easy to see once it is written down: joint, link, joint, link, all the way out to the tool. Every joint in this particular arm is a revolute joint, which is common in industrial arms because they are compact, easy to drive and easy to seal against dust.

NoteExam Corner

Q. An arm has four links between its base and its gripper. How many joints does it have?

A. Four. The chain alternates joint, link, joint, link, starting with a joint at the base, so there is one joint in front of each link.

The same reading works on everyday machines. On a folding table lamp, the heavy weight on the desk is the base, each straight arm is a rigid link, each place where the arms pivot is a revolute joint, and the shade holding the bulb is the end effector. On an excavator, the cab sits on a revolute joint that turns it, the boom and the dipper arm are rigid links, and the bucket is the end effector.

NoteTry Yourself

Pick up a pair of scissors and a stapler. For each one, decide which part is the base, which parts are links, and what type of joint connects them. Write down the type of motion each joint gives.

7.6 Degrees of Freedom

Two robot arms stand side by side in a workshop. One can only slide a tool up and down a post. The other can reach into a machine, tilt its tool sideways and pull a part out through a gap. You need a short way of saying how much movement a robot has, without describing every joint.

That measure is its degrees of freedom.

Count the independent ways the robot can move — independent meaning each one is a movement the others cannot produce. A slider on a post has one. The arm reaching into the machine has several. This count is written in short as degrees of freedom (DoF).

NoteDefinition

The degrees of freedom of a robot is the number of independent ways it can move in its workspace.

The workspace, sometimes called the work envelope, is simply the whole region of space the robot’s tool is able to reach.

Counting the degrees of freedom

The counting rule follows straight from the joints. Every axis that a joint can turn about, or extend along, is one degree of freedom.

A revolute joint turns about one axis, so it gives one degree of freedom. A prismatic joint extends along one axis, so it also gives one. A spherical joint turns about three axes from a single point, so it gives three. Add up the contributions of all the joints and you have the degrees of freedom of the robot.

NoteExample 7.4

Counting the degrees of freedom of a robot arm

Given: An arm built from a revolute joint at the base, a revolute joint at the shoulder, a revolute joint at the elbow, one prismatic joint that raises and lowers the tool, and a spherical joint at the wrist.

Find: The total degrees of freedom of the arm.

Working: Take the joints one at a time and add what each contributes.

Base, revolute: 1

Shoulder, revolute: 1

Elbow, revolute: 1

Lift, prismatic: 1

Wrist, spherical: 3

\[1 + 1 + 1 + 1 + 3 = 7\]

The arm has 7 degrees of freedom.

Notice what was left out of that sum. The gripper opening and closing is not counted. Opening a gripper does not move the robot to a new position or point it in a new direction; it is the tool doing its own job, and it is normally listed separately from the arm’s degrees of freedom.

NoteCommon Error

Counting links, or counting the gripper, as degrees of freedom What goes wrong: An arm with five links is described as having five degrees of freedom, or a gripper’s open-and-close is added into the total. The fix: Count axes of movement at the joints, not parts. A link on its own adds no freedom, and a gripper’s grip is the tool’s action, not the arm’s movement.

Why six is a common number

Now think about placing an object exactly in space — not just getting it to the right spot, but getting it to face the right way as well.

To say where the object is, you need three numbers: how far across, how far along, and how high. To say which way it is facing, you need three more, one for the turn about each of those three directions. These three turns have names borrowed from aircraft: roll, pitch and yaw. Fig. 7.6-1 shows all six together.

Diagram

Fig. 7.6-1 – a small block at the centre of three axes drawn at right angles to each other. Along each axis, a straight double-headed arrow labelled with the movement it gives: up and down, left and right, forward and back. Around each axis, a curved arrow labelled roll, pitch and yaw. Give the figure the overall label “six ways a body can move in space”.

So six independent movements are needed to put an object anywhere within reach and point it in any direction. That is why so many industrial robot arms are built with exactly six degrees of freedom, usually from six revolute joints. Fewer than six, and there will be positions the tool can reach but cannot face correctly.

NoteCuriosity Corner

Q5. What does it mean when someone says a robot has six degrees of freedom?

A. It means the robot has six independent ways to move — enough to place its tool at any reachable point and also point it in any direction. Three of these fix the position of the tool and three fix the direction it faces.

NoteExam Corner

Q. A robot’s links are replaced with longer ones. Do its degrees of freedom increase?

A. No. Longer links give a bigger workspace, so the robot reaches further, but the number of independent ways it can move is unchanged. Degrees of freedom depend on the joints, not on link length.

More freedom is not always better

An arm with more degrees of freedom than the strict minimum can reach the same point in more than one way. Fig. 7.6-2 shows why that is useful: with an extra joint, the arm can bend its elbow upward and come at a target over an obstacle instead of straight through it.

Diagram

Fig. 7.6-2 – two panels showing the same base, the same target point and the same box-shaped obstacle in between. Panel 1 labelled Two joints: the arm reaches straight towards the target and its middle link passes through the obstacle, marked with a cross. Panel 2 labelled Three joints: the arm is folded so that its elbow rises above the obstacle and the gripper still arrives at the same target, marked with a tick.

That flexibility has a price. Each extra joint adds a motor, a sensor, wiring, weight and cost, and it gives the controller one more thing to work out. It also adds one more part that can fail. Robots are therefore built with the smallest number of degrees of freedom the job genuinely needs. A machine that only stacks boxes on a flat floor may need four; a machine that welds a curved seam inside a car body needs six.

7.7 Summary

1. A robot arm is built as a chain of stiff pieces called links, connected at movable points called joints. The base is the fixed part it all stands on, and the end effector is the tool at the far end.
2. Links give the robot its reach, its strength and its shape. Joints give it all of its movement. An actuator, usually a motor or a cylinder, supplies the force that drives each joint.
3. Linear motion is movement in a straight line, with no turning. It is also called translational motion.
4. Angular motion is a body turning about a fixed axis, measured as an angle in degrees.
5. Circular motion is a point travelling round a fixed centre along a circular path. When a body turns, the body shows angular motion while each of its points shows circular motion.
6. Motion in one dimension needs one number to describe a position; motion in two dimensions needs two, usually written as x and y.
7. A prismatic joint slides one part along another on a shared axis and gives linear motion. It is stiff and quiet, but reaches only along one straight line.
8. A revolute joint turns about a single fixed axis and gives angular motion. It is simple, accurate and the most common joint in robotics, but a small angle error grows into a large error at the end of a long arm.
9. A spherical joint has a ball turning inside a socket and gives movement about three axes from one point. It has a wide range but is difficult to drive and measure.
10. The type of motion a joint gives is fixed by how it is built, not by the program.
11. Rigid links keep their shape under load, which makes a robot accurate and repeatable. They are heavy, and they deliver the full force in a collision.
12. Soft links bend and return, so they can hold delicate objects, squeeze past obstacles and absorb a knock. Their position is harder to predict and they carry less payload.
13. Payload is the weight a robot can carry, and it is limited mainly by the links and the joint actuators.
14. Real tasks need several joints working together. Joints of different types are combined so that their separate motions add up to the movement the task needs.
15. To read a machine, start at the base and follow it outwards, asking at each moving connection whether it slides or turns, and treating the part between two joints as a link.
16. The degrees of freedom of a robot is the number of independent ways it can move in its workspace.
17. A revolute joint and a prismatic joint each contribute one degree of freedom; a spherical joint contributes three. Adding these up gives the robot’s total.
18. Six degrees of freedom are needed to place a tool anywhere in space and point it in any direction — three for position and three for the turns called roll, pitch and yaw.
19. More degrees of freedom give more ways to reach a point, but add weight, cost, control work and more parts that can fail.

7.8 Key Terms

1. Actuator — the part that supplies the movement at a joint, such as an electric motor or a cylinder driven by oil or compressed air.
2. Angular motion — motion in which a body turns about a fixed axis, sweeping through an angle instead of travelling in a straight line.
3. Base — the fixed part of a robot, fastened to the floor, a wall, a table or the ceiling, which carries the weight of everything above it.
4. Circular motion — motion in which a point travels along a circular path, staying at the same distance from a fixed centre.
5. Degrees of freedom (DoF) — the number of independent ways a robot can move in its workspace.
6. Dimension — one independent direction in which something can move; one dimension needs one number to fix a position, two dimensions need two.
7. End effector — the tool at the far end of a robot arm that does the actual job, such as a gripper, a welding tip or a suction cup.
8. Joint — a mechanical connection between two links that allows them to move relative to each other.
9. Linear motion — motion in which a body moves along a straight line, every point moving the same distance in the same direction; also called translational motion.
10. Link — a rigid or soft component of a robot that connects two joints and gives the robot its structure.
11. Payload — the weight a robot is able to carry.
12. Pitch — the turn of a body about the axis running across it from side to side, tipping its front end up or down.
13. Prismatic joint — a joint in which one part slides over another along a shared straight axis, giving linear motion; also called a slider.
14. Revolute joint — a joint that allows turning about a single fixed axis, giving angular motion; also called a hinge joint or rotary joint.
15. Rigid link — a link that does not bend or change shape while the robot works.
16. Roll — the turn of a body about its own long axis.
17. Soft link — a link made to bend or deform in response to an outside force and then return to shape; also called a flexible link.
18. Spherical joint — a joint with a ball element turning inside a socket, giving movement about three axes from a single point; also called a ball-and-socket joint.
19. Yaw — the turn of a body about its upright axis, swinging its front end left or right.

7.9 Objective Questions

1. (The joint that allows one part to slide over another along a shared straight axis is the)

   1. revolute joint
   2. prismatic joint
   3. spherical joint
   4. fixed joint

2. (Fill in the blanks.)

   1. The tool at the far end of a robot arm that does the actual job is called the ________.
   2. A link that does not bend or change shape while the robot works is called a ________ link.
   3. The weight a robot is able to carry is called its ________.

3. (State whether each statement is true or false.)

   1. All the movement of a robot arm happens at its links.
   2. Angular motion is measured as an angle, in degrees.
   3. A spherical joint is also known as a ball-and-socket joint.

4. (The number of degrees of freedom given by a single spherical joint is)

   1. one
   2. two
   3. three
   4. six

5. (Match each item in Column A with the correct description in Column B.)

   Column A	Column B
   a. Revolute joint	i. Supplies the force or turning effort at a joint
   b. Soft link	ii. The fixed part that carries the weight of the whole arm
   c. Base	iii. Turns about a single fixed axis
   d. Actuator	iv. Bends under an outside force and then returns to shape

6. (A robot arm is built from four revolute joints and one prismatic joint, and carries a gripper at its end. State the number of degrees of freedom of the arm.)

7. (A mobile robot has wheels of diameter 10 cm. Taking \(\pi = 3.14\), write the distance the robot moves forward in one full turn of a wheel.)

7.10 Concept Questions

1. (Define a link and a joint. State one clear difference between them.)

2. (A wheel of a mobile robot is spinning as the robot drives forward. Explain how angular motion and circular motion are both present in this one wheel.)

3. (A packing machine has to lower a bottle straight down into a carton, without tilting it. Give a reason why a prismatic joint is a better choice for this movement than a revolute joint, even though revolute joints can also produce a straight path.)

4. (Describe one situation in which a rigid link is the right choice and one in which a soft link is the right choice. Give a reason for each.)

5. (A robot arm measures 45 cm from a revolute joint to the tip of its gripper. Because of a fault in its angle sensor, the joint stops 2 degrees away from the angle it was told to reach. Taking \(\pi = 3.14\), work out how far the gripper ends up from its correct position. Show your working.)

6. (A workshop replaces the rigid links of a robot arm with links made from a material that bends slightly whenever the arm carries a load. Explain what would happen to the accuracy of the arm and why.)

7.11 Case-Based Questions

1. (A school science laboratory has a small robot arm bolted to a bench. It is used to lift sample trays from the bench and place them inside a cupboard. Its parts are listed below, in order from the bench outwards.)

NoteTable

Table 7.5 – the parts of the laboratory robot arm, in order from the bench outwards | Part | Description of how it moves | |—|—| | Foot | Bolted to the bench; does not move | | Waist | Turns the whole arm left and right about an upright axis | | Lower arm | A stiff aluminium tube | | Shoulder | Raises and lowers the lower arm about a single axis | | Upper arm | A stiff aluminium tube | | Lift | Slides the gripper straight up and down | | Gripper | Opens and closes to hold a tray |

1. Name the type of each joint in this arm.
2. Work out the number of degrees of freedom of the arm, and state clearly what you have counted and what you have left out.
3. The trays must be lifted straight up without tilting. Name the part of the arm that does this, and give a reason why it suits the job.

2. (A fruit farm sets up a packing station. Tomatoes arrive on a conveyor belt that runs in a straight line. A robot arm above the belt picks up each tomato and places it in a tray. The farm first fits a metal gripper, but too many tomatoes are damaged. The gripper is then replaced with one made of soft, curved rubber fingers.)

   1. Name the type of motion the conveyor belt gives to the tomatoes, and give a reason for your answer.
   2. Explain why the soft gripper damages fewer tomatoes than the metal one.
   3. State one disadvantage the farm should expect from using soft links in the gripper.