Chapter 5: Degrees of Freedom and Mechanical System Design in TinkerCAD
In the late 1970s a team of engineers in Japan, led by Hiroshi Makino at the University of Yamanashi, set out to solve a very ordinary factory problem. Electronic parts had to be pushed straight down into holes on a circuit board, thousands of times a day, quickly, and without bending a single pin.
Arms that could move and turn in every possible direction already existed. They were not the obvious answer. Every extra movement an arm can make is one more thing to control, one more motor to pay for, and one more direction in which the tool can wobble at the exact moment a pin meets a hole.
So the team did something that still shapes robot design today: they took movements away on purpose. Their arm swung horizontally at the shoulder and at the elbow, turned the tool about a vertical axis, and slid straight down. Four movements, and no more. Vertically it was stiff, so it could press a part firmly home. Horizontally it could give a little, so a pin arriving a fraction of a millimetre off centre would settle into its hole instead of snapping.
That machine, the SCARA arm, went on to become one of the most widely used assembly robots in the world. Not because it could do everything, but because someone had counted carefully what it actually needed to do.
Photograph
Fig. 5.0-1 – a SCARA assembly robot at work on a bench or production line, seen from the side so that the two horizontal swinging joints, the vertical sliding shaft and the tool at its lower end are all clearly visible in one frame; the flat, roughly circular area it sweeps over the worktable should be apparent. Sourcing note: a real photograph of an industrial SCARA arm in an assembly setting is required, not a rendering or a stock illustration; images cropped so tightly that the vertical shaft or the base is cut off are unusable, as is any image in which the arm is hidden behind machine guarding.
Keep these five questions in mind as you read. This chapter will answer every one of them.
5.1 Links: The Rigid Bodies of a Robot
Look at your own arm for a moment. Between your shoulder and your elbow there is a single bone that does not stretch, does not bend and does not shorten. All the movement happens at the joints; the bone simply carries that movement outward and holds up whatever your hand is holding.
A robot arm is built on exactly the same idea. It is made of stiff pieces that do not change shape, connected by places where motion is deliberately allowed. Those stiff pieces are called links, and everything else in this chapter rests on them.
5.1.1 What a Link Is
A link is one rigid piece of a robot’s mechanical structure. It may be a machined aluminum bar in an industrial arm, a laser-cut acrylic plate in a school robot, or the moulded shell of a mobile robot’s chassis. What makes it a link is not what it is made of, but the fact that we treat its size and shape as fixed while the robot works.
Before that idea can be stated properly, one term from mechanics is needed. A rigid body is a body whose particles stay at fixed distances from one another, so the body does not bend, stretch or compress no matter what forces act on it. It is an idealisation — no real material behaves this way perfectly — but it is an extremely useful one.
A link is not merely a spacer between two joints. It houses the joints at its two ends, carries the actuators — the motors or cylinders that produce the robot’s motion — and holds the sensors, wiring and any load the robot is handling. In an assembly-line arm, one link may contain a motor, a gearbox, an encoder and a cable channel inside a single hollow casting.
The load a robot is designed to carry at its outer end has its own name. The payload is the maximum mass, including the gripper and the object held, that the robot can handle at a stated reach and speed without loss of accuracy. Every link between the base and that load must be stiff enough and strong enough to carry it.
The arrangement is easiest to see in a picture. In Fig. 5.1-1 the links are the long straight members, the joints are the circled connections between them, and the tool at the far end is the part that actually does the work.
Diagram
Fig. 5.1-1 – a four-link articulated robot arm drawn from the side; the fixed base hatched and labelled “base”; each rigid member labelled “link 1”, “link 2”, “link 3” in order outward; each connection between two members marked with a small circle and labelled “joint”; the gripper at the outer end labelled “end-effector”; a small arrow at each joint showing the direction of permitted rotation
That tool has a name too. The end-effector is the device attached to the outermost link that interacts with the world — a gripper, a welding torch, a suction cup, a paint nozzle or a camera mount.
Notice how the naming runs. The base is fixed to the floor or the bench and never moves. Links run outward from it, one after another, and the end-effector sits at the far end. Everything the robot achieves is the accumulated effect of small motions at the joints, carried outward by rigid links.
5.1.2 Characteristics of a Link
Three properties of a link decide how the finished robot behaves: how stiff it is, how heavy it is and how that weight is spread out, and what shape it has. These are worth taking one at a time, because each one has a direct and visible consequence at the end-effector.
5.1.2.1 Rigidity
In robotics each link is normally assumed to behave as a perfectly rigid body. That is, the link is taken not to bend, stretch, compress or twist under the forces acting on it during operation.
This assumption is made deliberately, and for three good reasons. It simplifies the mathematics enormously, because the position of the end-effector then depends only on the joint values and the fixed link lengths. It improves kinematic precision, because a link whose length is known and constant gives a predictable end-effector position. And it improves structural stability, because a design built around stiff members holds its shape under load instead of flexing unpredictably.
Absolute rigidity, of course, is impossible. Every real material deflects a little under load, and a long arm at full stretch does droop by a small amount — often a fraction of a millimetre, which is a great deal when the robot is placing a component on a circuit board. This is exactly why a manufacturer’s rated payload usually falls as the arm extends further from its base, and why designers choose stiff materials and deep, hollow cross-sections rather than thin solid ones.
5.1.2.2 Mass and Inertia
The mass of a link matters twice over. It adds to the gravitational load the actuators must hold up, and it adds to the load that must be accelerated and stopped every time the robot moves. A heavy link therefore needs a larger motor, a stronger gearbox and more energy, and it slows the whole arm down.
But mass alone does not tell the whole story. What resists a rotation is not simply how much mass there is, but how far that mass sits from the axis of rotation. This is described by the moment of inertia — a measure of how a body’s mass is distributed about the axis it turns around.
Two links of identical mass can behave completely differently. A long, slender link has a much higher moment of inertia about its shoulder than a short, compact link of the same mass, because most of its material sits far from the axis. This is the same effect you feel when you swing a long broom by its handle rather than a short one of equal weight.
\[I_A = \frac{1}{3} \times 0.60 \ \text{kg} \times (0.30 \ \text{m})^2 = \frac{1}{3} \times 0.60 \times 0.09 = 0.018 \ \text{kg}\cdot\text{m}^2\]
\[I_B = \frac{1}{3} \times 0.60 \ \text{kg} \times (0.15 \ \text{m})^2 = \frac{1}{3} \times 0.60 \times 0.0225 = 0.0045 \ \text{kg}\cdot\text{m}^2\]
\[\frac{I_A}{I_B} = \frac{0.018}{0.0045} = 4\]
Answer: \(I_A = 0.018\) kg·m² and \(I_B = 0.0045\) kg·m². Link A resists rotation four times as strongly as link B. Check: The two links have exactly the same mass, yet one is four times harder to swing. That is expected, because \(I\) depends on \(L^2\), so doubling the length must multiply the moment of inertia by four. The magnitudes are also sensible: a light half-metre link should have a moment of inertia of the order of hundredths of a kg·m², not tens.
The design lesson is direct. If a link must be long, keep its material close to the axis and its far end light — which is why industrial arms taper outward, and why heavy motors are mounted near the base rather than out at the wrist.
5.1.2.3 Geometry
The geometry of a link is its shape, its dimensions and its overall structural form. Geometry is not decoration; it is chosen from the motion the link must produce, the space the robot has to work in, and the loads the link must carry.
Straight links of uniform cross-section are used wherever motion is meant to be along a line. Prismatic mechanisms, Cartesian robots, slider blocks and linear stages all use them, because a straight member slides cleanly along a straight guide and is easy to make accurately.
Curved or non-uniform links are chosen for three reasons. They allow a compact motion pathway, letting the arm fold into a smaller space. They let the workspace be enlarged without adding an extra joint, since a bent link can reach around an obstruction. And in spatial mechanisms they provide the specific changes of orientation that a straight member cannot.
Diagram
Fig. 5.1-2 – two link shapes side by side, both labelled with their joint positions at each end; on the left a straight uniform bar labelled “straight link – used in linear stages, slider blocks and Cartesian robots”; on the right a curved link of the same end-to-end distance labelled “curved link – reaches around an obstruction without an extra joint”, with a small hatched obstacle drawn inside the curve and a dashed line showing the straight path that would have collided with it
5.1.3 Choosing a Material for a Link
No single material is best for every link, so the choice is always a set of trade-offs. Four questions decide it: how stiff must the link be, how light can it be made, how easily can it be cut and drilled in the workshop available, and what will it cost.
Metals are chosen where stiffness and strength matter most. Aluminum alloys give a high strength for their weight, machine easily and resist corrosion, which suits arms that must move quickly and precisely. Steel is chosen where higher strength and rigidity are essential, as in heavy-duty robots, but its greater weight forces larger actuators.
Lighter engineered materials are chosen where speed and energy matter. Composite materials such as carbon fibre reinforced polymer are exceptionally stiff for their weight, which improves how quickly an arm can accelerate and stop.
In school and low-cost robots, acrylic sheet and ABS plastic are common, because they can be laser-cut or 3D printed quickly and cheaply. They are perfectly adequate for learning, and quite unsuitable for high loads or high precision. Beyond these four questions sit the working conditions themselves: a link for a robot in a wet field, a paint booth or a food plant must also survive moisture, chemicals and cleaning.
The point to carry forward is that a link is chosen, not merely supplied. Its material, its shape and its mass are engineering decisions, and each one shows up later as speed, accuracy, cost or reach at the end-effector.
5.2 Joints: Where Motion Is Allowed
A pile of rigid links is not a machine. It becomes one only where two links are connected in a way that lets them move with respect to each other, and that connection is the joint.
Think of a door. The door leaf and the frame are both rigid, and neither of them is interesting on its own. The hinge between them is what turns two flat boards into something that opens. It also decides exactly how they may move: the door can swing, and it can do nothing else.
5.2.1 What a Joint Is
A joint is a mechanical connection between two links that permits a specific type of relative motion between them, and prevents every other type.
The second half of that sentence matters as much as the first. A hinge does not merely allow rotation; it also refuses to let the door slide sideways, lift off the frame or twist about its own width. A joint is a permission and a restriction at the same time.
This is also where a common misunderstanding begins, so it is worth naming at once.
5.2.2 Characteristics of a Joint
Four things describe a joint completely, and a designer must settle all four before the joint can be built.
Type of motion allowed
Every joint is defined first by the motion it permits. That motion may be rotational, translational, or a combination of the two, depending on the mechanical design and on what the task demands. A pick-and-place machine that must drop a part straight down needs a joint that slides; a shoulder that must sweep an arm across a bench needs one that turns.
Range of motion
No joint moves without limit. A revolute joint may be restricted to 300° of rotation because a cable inside would otherwise be wound tight; a sliding joint stops at the end of its rail. These limits come from physical construction, from deliberate mechanical stops fitted for the purpose, and from safety requirements that keep the machine inside a controlled boundary.
A joint’s range is as much a part of its specification as the motion it allows. A wrist that can rotate only 90° is a different machine from one that can rotate 270°, even though both are revolute.
Actuation method
The joint must be driven by something. Electrical motors are the usual choice in the robots you will build and in most modern industrial arms. Hydraulic systems, using pressurised oil, deliver very high force and are used in large, heavy machines. Pneumatic cylinders, using compressed air, are fast, simple and well suited to two-position movements such as opening and closing a gripper. Some joints are not driven at all; a passive joint simply follows the motion imposed on it by the rest of the mechanism.
The actuation method decides how much force or torque the joint can deliver, how smoothly it can be controlled, and how the robot must be supplied — with electricity, oil pressure or compressed air.
Sensing at the joint
A joint that moves but cannot report where it is can only be commanded, never corrected. Sensing elements are therefore built into most joints: encoders, which count increments of shaft rotation; potentiometers, whose resistance changes with angle; and force sensors, which report the load the joint is carrying. Between them they supply position, velocity and applied load, and it is this feedback that makes accurate, coordinated motion possible.
5.2.3 Types of Joints
Four joint types cover almost everything you will meet in this course. They differ in one respect only: what kind of relative motion each one permits.
5.2.3.1 Prismatic Joint
A prismatic joint allows one link to slide linearly along a fixed axis relative to another link, producing pure translational motion with no rotation at all. A drawer in a cupboard is a prismatic joint, and so is the vertical shaft that carried the SCARA arm’s tool straight down onto a circuit board.
Its motion is described by a single number: how far the sliding link has moved along its axis. Prismatic joints are used in Cartesian robots, in vertical lifting mechanisms and in sliding actuators, wherever precise straight-line positioning is needed for pick-and-place work, material handling or automated tooling.
Diagram
Fig. 5.2-1 – prismatic joint; an outer rectangular block shown as the fixed link, hatched, with an inner rectangular block sliding inside it as the moving link; the axis of the joint drawn as a dashed line through both blocks with a double-headed straight arrow along it labelled “translation along the joint axis”; a note stating “no rotation permitted”
5.2.3.2 Revolute Joint
A revolute joint allows one link to rotate about a fixed axis relative to another, producing pure rotation with no sliding. The door hinge is the everyday example, and its motion is described by a single angle.
The revolute joint is by far the most widely used joint in articulated industrial robots, and there are three plain reasons for that. It is mechanically simple, and therefore reliable and cheap to make well. It is easy to drive directly with a motor, which naturally produces rotation. And by combining a few of them a designer obtains smooth, curved paths through space without any long sliding surface to wear or jam. You will find revolute joints at the shoulder, elbow and wrist of almost every manipulator.
Diagram
Fig. 5.2-2 – revolute joint; two rectangular links overlapping at one end and joined by a pin drawn as a filled circle; the lower link hatched and labelled “fixed link”, the upper labelled “moving link”; the axis of rotation drawn as a dashed line through the pin, perpendicular to the plane of the links; a curved arrow about the pin labelled “rotation about the joint axis”, with the joint angle marked between the two links
5.2.3.3 Cylindrical Joint
A cylindrical joint allows a link both to rotate about an axis and to slide along that same axis. Two motions are permitted, and they are independent of each other: the link can turn without sliding, slide without turning, or do both together.
A key being pushed into a lock while also being turned is close to the idea. Because the rotation and the translation share one axis, the connected link can be brought to any point on a cylindrical surface around that axis. Cylindrical joints appear in cylindrical-coordinate robotic arms, where a tool must be lifted, lowered and swung about a central column for material handling and positioning.
Diagram
Fig. 5.2-3 – cylindrical joint; a solid cylindrical shaft passing through a collar; the collar hatched and labelled “fixed link”, the shaft labelled “moving link”; the common axis drawn as a dashed line through the centre of the shaft; a straight double-headed arrow along the axis labelled “translation” and a curved arrow around the axis labelled “rotation”, with a note stating “both motions share the same axis”
5.2.3.4 Spherical Joint
A spherical joint, also called a ball-and-socket joint, allows a link to rotate freely about three perpendicular axes that intersect at a single point. It permits three independent rotations and no sliding at all. Your own shoulder and hip work this way, which is why you can raise, swing and twist your arm without the joint itself going anywhere.
Its strength is also its difficulty. Because three rotations happen at one point, controlling them all at once demands sophisticated actuators and sensors, so spherical joints are rarely used in heavy industrial manipulators. They are common in humanoid robots, robotic wrists and bio-inspired mechanisms, where natural, multi-directional movement and fine orientation control matter more than raw load capacity.
Diagram
Fig. 5.2-4 – spherical joint; a ball at the end of a rod seated inside a partially cut-away socket; the socket hatched and labelled “fixed link”, the rod labelled “moving link”; three mutually perpendicular dashed axes drawn through the centre of the ball, each carrying a curved arrow labelled “rotation 1”, “rotation 2” and “rotation 3”; a note stating “all three axes meet at the centre of the ball; no sliding permitted”
Comparing the four
Set side by side, the pattern behind the four types becomes plain. They are different answers to one question: how many independent motions should this connection allow, and of what kind?
5.2.4 Links and Joints Together: The Kinematic Chain
Links and joints are never studied separately for long, because a robot is what you get when you alternate them: base, joint, link, joint, link, and so on out to the end-effector. That assembly has a name.
A kinematic chain is a set of links connected by joints in such a way that motion at any joint produces a definite, predictable motion of the whole assembly.
Chains come in two arrangements, and the difference decides how the mechanism behaves. In an open kinematic chain the links form a single unbranched line: one end is fixed to the base and the other is free and carries the end-effector. Almost every robotic arm is an open chain. In a closed kinematic chain the links form a loop, so that the last link connects back to the first. The wiper mechanism of a vehicle and the pedal-crank arrangement of a bicycle are closed chains.
Diagram
Fig. 5.2-5 – two chains side by side, both drawn as stick mechanisms with links as straight bars and revolute joints as small circles; on the left, an open chain labelled “open kinematic chain” – a hatched fixed base, then joint, link 1, joint, link 2, ending at a free end labelled “end-effector”, with a curved arrow at each joint; on the right, a closed chain labelled “closed kinematic chain” – four bars forming a quadrilateral with a circle at each of the four corners, the lower bar hatched and labelled “fixed link”, and a curved arrow on the left-hand bar labelled “input”
Whichever arrangement is used, each joint carries one number that says how far it has moved from its zero position. That number is the joint variable: an angle, usually written \(\theta\), for a revolute joint, and a linear displacement, usually written \(d\), for a prismatic joint. Give every joint variable of a chain a value and the position of every link — and therefore of the end-effector — is completely fixed.
This is the idea the rest of the chapter is built on. The links fix the lengths, the joints fix what may move, and the joint variables say how much. What remains is to count how many of those variables a mechanism truly needs, and that count is the subject of the next section.
5.3 Degrees of Freedom
A chess rook and a chess bishop both move, but they do not have the same freedom. The rook is confined to ranks and files, the bishop to diagonals. If you want to describe where a piece can go, counting its possible squares is far less useful than knowing what kinds of movement it is allowed at all.
Machines are described the same way, and the count that does it is the degrees of freedom.
5.3.1 Translation and Rotation
Every motion a rigid body can perform, however complicated it looks, is built from just two kinds of movement.
Translation is linear movement, in which every point of the body shifts by the same amount in the same direction. A lift rising in its shaft translates; so does a drawer sliding open.
Rotation is angular movement about an axis, in which the body turns while the axis itself stays put. A ceiling fan rotates; so does a door on its hinge.
Take a book lying on your desk. Slide it towards you without turning it — that is pure translation. Now spin it where it lies without moving its centre — that is pure rotation. Slide and spin it at the same time and you have a general motion, but it is still nothing more than those two ingredients combined.
5.3.2 The Six Degrees of Freedom of a Free Rigid Body
Now imagine a body floating freely in space, with nothing attached to it — a small satellite, or a drone hovering in still air. How many independent movements can it make?
It can translate along each of the three axes: along X, along Y and along Z. It can also rotate about each of the same three axes. There is no seventh independent movement, because any other motion you can devise is some combination of these six.
The three rotations have their own names, borrowed from aircraft and used throughout robotics. Rotation about the X axis is roll, rotation about the Y axis is pitch, and rotation about the Z axis is yaw.
Diagram
Fig. 5.3-1 – a rectangular body drawn in three dimensions at the origin of a right-handed set of axes labelled X, Y and Z; a straight double-headed arrow along each axis labelled “translation along X”, “translation along Y” and “translation along Z”; a curved arrow about each axis labelled “rotation about X (roll)”, “rotation about Y (pitch)” and “rotation about Z (yaw)”; a caption note stating “a free rigid body in space: six independent movements”
This is the largest number of independent movements any single rigid body can have, and it gives us the count we have been building towards.
The degrees of freedom (DOF) of a mechanical system is the number of independent movements it can perform. Equivalently, it is the number of independent variables needed to describe the position and orientation of the system completely.
Those two statements are the same fact seen from two sides. A free rigid body has six degrees of freedom, and it takes exactly six numbers — three coordinates and three angles — to say precisely where it is and how it is oriented. Give five of them and the body’s pose is still undecided.
A drone in open air is the clearest everyday example. It can fly forward and back, left and right, up and down, and it can roll, pitch and yaw, so it possesses all six degrees of freedom. That is precisely why flying one takes practice: six independent things can be happening at once.
5.3.3 How a Joint Constrains Motion
Attach that free body to something and the freedom starts to disappear.
Fit a door leaf into a hinge and it can no longer translate at all, nor can it roll or pitch. Of the six movements a free body had, five have been removed by the hinge, and one is left: rotation about the hinge axis. The door has one degree of freedom.
This is the essential role of a joint. It does not create motion; it permits a chosen part of the motion and removes the rest. Every joint, every rail and every bearing you add to a design is a constraint, and constraints only ever subtract.
The count that survives all the constraints in a mechanism is what we call its degrees of freedom, and engineers also refer to it as the mechanism’s mobility. It is the number of independent inputs a designer must supply — and, in a powered machine, very close to the number of motors that must be paid for.
5.3.4 Degrees of Freedom Contributed by Each Joint Type
We can now put a number against each of the joint types from section 5.2, because the “independent motions permitted” column of Table 5.1 was the degrees of freedom count all along.
A revolute joint contributes one degree of freedom, and so does a prismatic joint — one rotation and one translation respectively. A cylindrical joint contributes two, because its rotation and its slide are independent of each other. A spherical joint contributes three, all of them rotations.
For an open kinematic chain — an ordinary robot arm, running from a fixed base out to a free end-effector — these numbers simply add up. Every joint’s freedom is available at the end of the chain, because there is no loop to interfere.
\[\text{DOF} = 1_{\text{(revolute)}} + 1_{\text{(revolute)}} + 1_{\text{(revolute)}} + 2_{\text{(cylindrical)}} = 5\]
Answer: The arm has 5 degrees of freedom. Check: The count is sensible: it is more than the three needed to reach any point in space, so the arm has some control of orientation, but it is fewer than the six a completely free body has. This arm can therefore reach a target position, and can orient its tool in some ways but not in every way — which is exactly what a five-axis machine is expected to do.
5.3.5 Counting Degrees of Freedom in Real Mechanisms
The counting is best learnt on real machines, so here are several, arranged from the simplest upward.
5.3.5.1 A Single Revolute Joint and the Four-Bar Linkage
A shaft turning in a bearing, a door on its hinge, a robot wheel on its axle: each is one link rotating relative to another about a fixed axis. The body can turn freely about that axis and cannot translate in any direction, so each has one degree of freedom.
Now take a mechanism that looks far more complicated. A four-bar linkage is formed by connecting four rigid links in a loop with four revolute joints, one of the links being fixed as the frame. The wiper mechanism of a vehicle is exactly this: a motor turns one link, and the wiper blade sweeps back and forth across the windscreen.
Four links and four joints — and the whole assembly still has just one degree of freedom. Turn the input link, called the crank, through some angle, and the positions of the coupler and the follower are then completely decided. You cannot choose them separately, however hard you push.
Diagram
Fig. 5.3-2 – four-bar planar linkage drawn as four straight bars forming a quadrilateral; the lowest bar hatched and labelled “fixed link (frame)”; the other three labelled “crank”, “coupler” and “rocker”; each of the four corners marked with a small open circle and labelled “revolute joint”; a curved arrow at the crank labelled “input rotation”; a dashed outline showing the position of the coupler and rocker after the crank has turned, with a curved arrow at the rocker labelled “resulting motion”
The reason is the loop. In an open chain each joint is free to move on its own, but here the fourth link closes the circuit back to the frame, and the links must move in a fixed relationship or the loop would tear apart. Closing a loop costs freedom, and that is what makes the four-bar so useful: a single motor produces a completely repeatable, coordinated motion of every part.
5.3.5.2 The Two-Link RR Arm
A two-link planar robotic arm, usually called the RR mechanism because it has two revolute joints in succession, is the simplest arm worth studying. The first link turns at the shoulder, the second turns at the elbow, and the end-effector is carried at the far end.
Each joint contributes one rotation, and the chain is open, so the arm has two degrees of freedom. Two numbers — the shoulder angle and the elbow angle — fix the position of every part of the arm.
Diagram
Fig. 5.3-3 – a two-link RR planar arm; a hatched fixed base at the left with a small circle marking the shoulder joint; link 1 running from the shoulder to a second small circle marking the elbow joint; link 2 running from the elbow to a point marked “end-effector”; the shoulder angle labelled theta1 measured from the horizontal and the elbow angle labelled theta2 measured from the line of link 1; a curved arrow at each joint showing the direction of rotation
Two is enough to place the end-effector anywhere within the flat region the arm can reach. It is not enough to place it there at any angle you please. Describing a tool completely on a flat surface takes three numbers — two for position and one for its orientation — and this arm supplies only two of them. Where the end-effector goes, its orientation follows from the joint angles rather than being chosen.
5.3.5.3 The Differential Drive Mobile Robot
A differential drive robot is the classroom workhorse: a chassis with two independently driven wheels, usually with a castor to keep it upright. Drive both wheels forward at the same speed and it moves straight ahead. Drive them at equal speeds in opposite directions and it spins about the midpoint of the axle without going anywhere.
So the robot has two independent motions available to it: translation forward or backward, and rotation about its own vertical axis. Two motors, two independent commands.
Diagram
Fig. 5.3-4 – top view of a differential drive robot; a rectangular chassis with a driven wheel drawn on the left edge and another on the right edge, joined by a dashed axle line through the chassis centre; a castor drawn at the front; a straight arrow along the chassis centre line labelled “forward and backward translation”; a curved arrow about the midpoint of the axle labelled “rotation about the vertical axis”; a crossed-out sideways arrow at the chassis side labelled “sideways motion not available”
Notice carefully what it cannot do. It cannot slide sideways, however its wheels are driven. To move one metre to its left it must turn, drive, and turn back — which is exactly why parking a car takes several movements rather than one.
5.3.5.4 The Cartesian Robot
A Cartesian robot, also called a three-axis robot, moves along three perpendicular directions — X, Y and Z — using three prismatic joints driven by linear actuators. Each contributes one translation, so the machine has three translational degrees of freedom.
The consequence is very clean. It can bring its tool to any point inside its rectangular working volume, and it cannot change the tool’s orientation at all: whatever direction the tool points at the start, it points there always. A 3D printer and a CNC milling machine both work on this principle, and for those tasks the fixed orientation is not a defect but a convenience.
Diagram
Fig. 5.3-5 – a Cartesian robot drawn as a gantry; a hatched fixed frame; a horizontal beam sliding along the frame labelled “X axis, prismatic joint”; a carriage sliding along that beam labelled “Y axis, prismatic joint”; a vertical shaft sliding through the carriage labelled “Z axis, prismatic joint”; a straight double-headed arrow along each of the three; a tool at the lower end of the vertical shaft, with a note stating “the tool stays parallel to itself at every position”
The four counts side by side
5.3.6 Why the Degrees of Freedom Must Match the Task
The count is not an academic exercise. It is the first design decision made about any robot, because it settles how many actuators must be bought, controlled, powered and maintained.
The rule is to start from the task, not from the machine. Ask how many numbers are needed to describe what the tool must achieve. Placing a tool anywhere on a flat surface, at any angle, takes three. Placing a tool anywhere in space, in any orientation, takes six. A machine with fewer degrees of freedom than its task needs will simply be unable to do part of the job, and no amount of clever programming will rescue it.
Fewer degrees of freedom, where the task genuinely permits it, is a positive advantage. Every freedom removed is a motor not bought, a control loop not tuned, a cable not routed and a direction in which the tool cannot wobble. A machine that can only move where it is supposed to move is cheaper, faster, stiffer and more reliable than one that could move anywhere.
More degrees of freedom than the minimum are worth having only for a reason. An arm with seven joints doing a six-number task can reach the same point in more than one posture, which lets it work around an obstacle or avoid folding into itself. That extra ability is paid for in cost, weight and control complexity, so it is added deliberately and not by habit.
The mechanisms in Table 5.2 illustrate the point together. A wiper needs one coordinated sweep and gets one degree of freedom. A printer needs to reach every point in a box and gets three translations. An assembly arm needs to place parts and turn them and gets four. Each design begins with a count, and the mechanism is built to deliver that count and nothing more.
5.4 3D Modelling in Robotics
Everything so far has been about deciding what a robot should be able to do: which links, which joints, how many degrees of freedom. At some point those decisions have to become an actual object with actual dimensions, and that is where mistakes become expensive.
A link cut 10 mm too short cannot be lengthened. A bracket whose holes do not line up must be scrapped. So before anything is cut, drilled or printed, the machine is built once in a computer, where a mistake costs nothing but a click.
5.4.1 What 3D Modelling Is
3D modelling in robotics is the process of creating a virtual three-dimensional representation of a robot, or of any of its parts — the frame, an arm, a joint, a wheel, a gripper — before the part is physically made.
The model is not a picture. A picture shows one view; a model holds the true three-dimensional geometry, so it can be turned, sectioned, measured and assembled with other models. Software of this kind is called computer-aided design (CAD) software.
In robotics the model is used above all to study spatial relationships — how the parts sit with respect to one another, whether a link will strike the base when the arm folds, whether the gripper’s fingers meet where they should. Through such a model a designer can watch how a robot will move, reach and interact, long before any material is spent on it.
Diagram
Fig. 5.4-1 – on the left, four basic solid shapes drawn separately and labelled “cube”, “cylinder”, “cone” and “sphere”; on the right, the same shapes arranged into a simple two-link robot arm, with a cube labelled “base”, a sphere labelled “shoulder joint”, two stretched cylinders labelled “link 1” and “link 2”, and a pair of thin cuboids labelled “gripper”; a broad arrow between the two halves labelled “assembled by moving, scaling and rotating”
5.4.2 Why a Robot Is Modelled Before It Is Built
Five purposes explain why this stage is never skipped in serious design work.
Visualisation and conceptualisation
The model lets designers view the robot from every angle, try out competing design ideas side by side, and judge both appearance and mechanical sense before manufacturing begins. Rejecting a bad idea at this stage costs an afternoon; rejecting it after fabrication costs the material and the workshop time as well.
Simulation of movements
Once the parts exist as models, they can be animated to study how the robot moves within its working region. Rotations, translations and the motion of each joint can be played out, and interferences — a link striking the frame, a cable pulled taut — show up on screen rather than on the bench.
Measurement and accuracy
Every dimension in a model is exact and can be verified. Hole spacings, shaft diameters and link lengths can be checked against each other before anything is cut, so that parts fit together on the first attempt rather than the third.
Material selection and weight estimation
CAD software allows a material to be assigned to each component, after which the mass, the weight distribution and the centre of gravity of the whole assembly can be estimated. This matters more than it sounds: it is how a designer discovers, before building, that an arm is too heavy for its intended motor.
\[V = 20.0 \ \text{cm} \times 3.0 \ \text{cm} \times 0.6 \ \text{cm} = 36 \ \text{cm}^3\]
\[m_{\text{acrylic}} = 1.19 \ \text{g/cm}^3 \times 36 \ \text{cm}^3 = 42.8 \ \text{g}\]
\[m_{\text{aluminum}} = 2.70 \ \text{g/cm}^3 \times 36 \ \text{cm}^3 = 97.2 \ \text{g}\]
Answer: About 43 g in acrylic and about 97 g in aluminum. Check: The magnitudes are sensible for a bar roughly the size of a school ruler — tens of grams, not kilograms. The aluminum version is heavier by the ratio of the densities, \(2.70 / 1.19 \approx 2.3\), which matches the two masses obtained. Densities vary a little between grades, so both figures are estimates, not guarantees.
Integration with manufacturing
When the design is settled, the model itself becomes the manufacturing instruction. It can be exported directly to a 3D printer or a CNC machine, so the part that is made is the part that was designed, with no redrawing in between.
5.4.3 Designing Structures by Manipulating Shapes in a Virtual Space
Designing in a virtual space means creating, arranging and refining digital models in a three-dimensional environment. The method is deliberately simple: start from basic geometric shapes and change them until they become the part you want.
Cubes, cylinders, cones and spheres are the building blocks. Each can be resized, rotated, stretched and combined with others to form the components of a robot — arms, joints, bases and grippers. Working this way, a designer can adjust the dimensions of every component precisely, look at the design from any angle, try different materials and finishes, and test how parts move against one another without any risk of damage.
5.4.3.1 Moving, Scaling and Rotating Along the Three Axes
Three operations do most of the work, and all three act along or about the same three axes you met in section 5.3.
Translation moves a shape along the X, Y or Z axis to place it where it belongs. It changes position and nothing else.
Rotation spins a shape about one or more axes to orient it correctly. It changes orientation and leaves position and size untouched.
Scaling increases or decreases the size of a shape. Scaling the same amount along all three axes keeps the shape’s proportions; scaling along one axis only stretches it, which is how a stock cylinder becomes a long, slim link.
Notice that this is the same vocabulary as the mechanics: translation and rotation about three axes. The difference is that here you are moving a shape in a design, not a link in a machine.
Diagram
Fig. 5.4-2 – a single cuboid shown three times against a set of X, Y and Z axes; in the first view a straight arrow along the X axis labelled “translation – changes position”; in the second a curved arrow about the Z axis labelled “rotation – changes orientation”; in the third the cuboid drawn stretched along the Z axis with a double-headed arrow labelled “scaling – changes size”; each view carrying a note of what is left unchanged by that operation
5.4.3.2 Extrusion, Union and Subtraction
Two further operations turn simple solids into real parts.
Extrusion extends a two-dimensional outline into a three-dimensional object — a circle drawn on a plane becomes a cylinder, a rectangle becomes a bar. It is the usual way of turning a sketched profile into a solid.
Union and subtraction combine shapes. Union merges several shapes into a single object, which is how a base plate and its raised boss become one part. Subtraction removes the volume of one shape from another, which is how holes, slots and hollows are made: place a cylinder where the hole should be and subtract it, and a bolt hole appears.
Subtraction is worth dwelling on, because almost every mounting hole in every robot you build is made this way. A pivot hole is not drawn; it is a cylinder that has been taken away.
Diagram
Fig. 5.4-3 – two rows, each showing a before-and-after pair with an arrow between them; the upper row labelled “union”: a flat plate and a small cylinder overlapping, becoming a single solid part with the boss standing on the plate; the lower row labelled “subtraction”: the same plate with a narrow cylinder positioned through it, becoming the plate with a clean round hole where the cylinder stood; a note under the lower row stating “the subtracted shape defines the hole”
For example, in designing a robotic arm, cylinders may be stretched to form the links, spheres placed as joint markers, and cuboids combined to build the base. Used together, these operations give both flexibility and precision, which is what allows realistic mechanical designs to be produced entirely on screen.
5.4.4 Applications of 3D Modelling in Robotics
The same technique serves at every stage of robotics work.
- Designing industrial manipulators, and testing different link and joint configurations before committing to one.
- Simulating autonomous mobile robots moving through the environments they will have to navigate.
- Creating humanoid robots, in order to analyse human-like motion.
- Developing educational and prototype robots before any physical implementation.
- 3D printing robotic parts and enclosures directly from the model.
Behind all five lies the same argument. Every question you can answer in the model is a question you do not have to answer with cut material, and in robotics the questions are numerous. What remains is to learn a tool that lets you do it, and that is where the next section begins.
5.5 TinkerCAD as a 3D Modelling Tool
Professional CAD packages are powerful and unforgiving. They must be installed, they demand a capable machine, and they expect you to know a great deal before you can draw anything at all. For a first robot that is the wrong place to begin.
TinkerCAD was built for exactly this situation: a tool simple enough to be learnt in an afternoon, yet real enough that what you design can be printed and bolted onto a robot.
5.5.1 What TinkerCAD Is
TinkerCAD is a beginner-friendly, browser-based tool for creating and visualising 3D models. It is developed by Autodesk and runs entirely inside a web browser, so nothing has to be installed and no advanced technical knowledge is needed to start.
Because it runs online, a design can be opened from any computer with an internet connection. That is a genuine advantage in a school laboratory, where students move between machines and work is continued from home. The trade-off is equally plain: without a connection, the tool is unavailable.
The interface works by drag-and-drop. Basic geometric shapes — cubes, cylinders, spheres and cones — are dragged from a panel onto the working area and then resized, rotated and combined into mechanical and structural designs. The same shapes that were described in section 5.4 as the building blocks of a model are, here, literally items in a list you pick from.
Photograph
Fig. 5.5-1 – screen capture of the TinkerCAD 3D design editor with a simple robot part under construction; the shapes panel on the right, the workplane grid in the centre with one selected shape showing its dimension boxes, and the view controls and Group button visible. Sourcing note: a current capture of the actual editor is required, taken at a resolution where the dimension figures on the selected shape are legible; captures cropped so that the shapes panel or the workplane grid is cut off are unusable, and any capture showing a student’s account name or e-mail address must be excluded or the identifying region removed.
Its real strength is the bridge it builds. A beginner can see, in a single session, how separate shapes combine into a mechanical structure; a more advanced user can take the same model on to 3D printing or into professional CAD for refinement.
5.5.2 The Workplane and the Three Axes
The flat grid you design on is called the workplane. It is the reference surface of the virtual space: shapes placed on it sit at height zero, and every position is measured from it.
The workplane defines the three axes directly. Two of them, X and Y, lie in the plane of the grid, and the third, Z, rises vertically out of it. A shape is positioned by how far it sits along X and Y and how high it is raised along Z, which is exactly the three-number description of a point that you have already met.
Dimensions are entered in millimetres, and a snap grid fixes the step by which a shape moves when dragged. A fine snap gives precise placement, a coarse one gives quick approximate work, and any dimension can in any case be typed in exactly rather than dragged.
When a shape is selected, small handles appear on it. Corner handles scale it in the plane, a top handle changes its height, an arrow moves it across the workplane, a separate handle raises it above the workplane, and curved arrows rotate it about each axis.
Diagram
Fig. 5.5-2 – a single cuboid selected on a gridded workplane; the workplane’s X and Y directions labelled along the grid and the Z direction labelled by a vertical dashed line rising from the shape; the corner scaling handles marked as small squares and labelled “scale in X and Y”, the top handle labelled “scale in Z”, a flat arrow at the base labelled “move on the workplane”, a raised arrow above the shape labelled “lift along Z”, and three curved arrows labelled “rotate about X”, “rotate about Y” and “rotate about Z”; the shape’s length, width and height dimension boxes shown with the note “any dimension may be typed exactly”
5.5.3 Solid Shapes, Holes and Grouping
Every shape in TinkerCAD carries one of two properties. A Solid shape adds material. A Hole shape removes it — but only once it has been combined with the solid it sits inside.
That combining is done with the Group command, and it is the single most important idea in the tool. Position a narrow cylinder through a plate, set the cylinder to Hole, select both, and group them: the cylinder disappears and a clean round hole is left in the plate. Group two solids instead and they merge into one object.
This is the union and subtraction of section 5.4, made concrete. Union is grouping two solids; subtraction is grouping a solid with a hole.
Grouping also serves a second purpose, one that matters when a robot has many parts. A finished sub-assembly — a joint and the link attached to it, say — can be grouped so that it moves as a single item, keeping its internal alignment while you position it against the rest of the machine. Ungroup reverses this whenever a part inside must be edited again.
5.5.4 Aligning, Dimensioning and Viewing a Design
Getting shapes into roughly the right place is easy; getting them exactly concentric, flush or symmetrical is what separates a model that assembles from one that does not.
The Align tool does this work. It takes several selected shapes and lines them up on a chosen face or on their common centre, in any of the three directions. Aligning a pin to the centre of a hole by eye is guesswork; aligning them with the tool is exact, and every pivot in the mechanisms of the next section depends on it.
Diagram
Fig. 5.5-3 – two views of the same three shapes side by side; on the left, labelled “before aligning”, a plate, a tube and a small cylinder scattered at different positions with their centre lines drawn as dashed lines that do not coincide; on the right, labelled “after aligning on the common centre”, the same three shapes stacked concentrically with a single dashed centre line passing through all three; a note stating “aligned in X and Y, then stacked in Z”
Three further tools complete the working set. The Ruler places a measuring reference on the workplane, so that a shape’s position and size can be read and set as exact figures from a chosen origin. Duplicate copies a shape with all its dimensions intact, which is how a matched pair of links or a row of identical holes is made. Mirror produces a reflected copy, which is how the second finger of a gripper is made from the first.
Finally, the view itself is a tool. Rotating the workspace to look at the design from the front, the side and from above will reveal misalignments that are invisible from any single angle — a link that appears to touch its joint from the front but stands 2 mm clear when seen from the side.
With the workplane, the shapes, the Solid and Hole properties, and the grouping and aligning tools, you now have everything needed to build a real mechanism. The next section does exactly that, three times over.
5.6 Designing Different Structures Using TinkerCAD
Everything in this chapter now comes together. You know what links and joints are, you can count the degrees of freedom a mechanism has, and you have the tools to build shapes on a workplane. What follows is three mechanisms built from those shapes — one open chain and two closed ones — each chosen because it teaches something the earlier sections only described.
Work through them at a computer rather than reading them. The value is not in the finished picture but in discovering, while aligning a pivot, why alignment mattered in the first place.
5.6.1 The RR (Revolute–Revolute) Mechanism
The RR mechanism — the two-link planar arm of section 5.3.5.2 — is the simplest arm worth building. Two rigid links are connected by two revolute joints, usually called the shoulder and the elbow, in imitation of a human arm. The first link is attached to a fixed base; the second carries the end-effector, which does the actual picking, placing or pointing.
Modelling it teaches one thing above all: that a joint is a place where two parts must be concentric, and that everything else follows from getting that right.
Building the arm
- Drag a tube onto the workplane and resize it into a short, hollow circular base. Increase the wall thickness a little so the joint looks solid and stable. This tube represents the shoulder joint, the first revolute joint, about which link 1 will turn.
- Drag a cylinder onto the workplane for link 1. Set its length to the reach you want and reduce its diameter so that it looks proportionate against the base tube.
- Use Align to bring the cylinder concentric with the tube, so that it appears to emerge from the hollow centre of the joint. Group the two shapes into a single assembly: the shoulder joint and its link.
- Drag a second tube for the elbow joint. Set its outer diameter to match the end of link 1 so the two meet cleanly, and position it at the free end of that link.
- Insert a smaller cylinder through the second tube to act as the pivot pin. It marks the axis about which the second rotation occurs, and it is what makes the joint readable as a joint rather than a step in the arm.
- Duplicate the first cylinder to create link 2, and shorten it slightly so that the arm tapers outward as a real arm does. Align it concentrically with the pivot hole of the elbow.
- Make the end-effector from two thin boxes. Resize them into gripping fingers, rotate them slightly in opposite directions so that they form an open claw, and place them at the end of link 2.
- Group in two stages, not one: the shoulder joint with link 1 as one assembly, and the elbow joint, link 2 and the claw as the other. The arm then holds together while the two parts remain distinct.
That last step deserves attention, because it is where most first attempts go wrong.
Turn the second assembly by hand about the elbow and watch the claw sweep. Turn the first about the shoulder and watch the whole arm swing. Those two independent rotations are the arm’s two degrees of freedom, seen directly.
Photograph
Fig. 5.6-1 – screen capture of a completed RR arm model in the TinkerCAD editor, viewed from a three-quarter angle so that both joints and both links are clearly separated; the base tube, first cylinder link, elbow tube with its pivot pin, second link and the two claw fingers must all be distinguishable in the frame. Sourcing note: a capture of an actual completed model is required, not an illustration; images in which the two links overlap so far that the elbow joint is hidden are unusable, and any account name shown in the interface must be removed.
5.6.2 The Four-Bar Linkage
The four-bar linkage consists of four rigid bars joined by four revolute pivots, with one bar held fixed. You met it in section 5.3.5.1 as the mechanism that has four joints and only one degree of freedom. Building it is the quickest way to feel why.
Here the modelling problem is different from the arm’s. All four pivot holes must lie in one plane and be spaced exactly, because a loop that does not close cannot move at all.
Building the linkage
- Drag a box onto the workplane and resize it into a long, thin, flattened bar. This is the fixed link, the frame of the mechanism.
- Place a narrow cylinder near each end of the bar, set both cylinders to Hole, align them accurately, and group them with the bar. Two clean pivot holes appear, and this link will now stay stationary.
- Duplicate the bar twice and shorten each copy to make the crank and the follower. Give each of them a pivot hole at both ends by the same method, and group each one so that it becomes a separate component.
- Place the crank at the left-hand pivot of the fixed link and the follower at the right-hand pivot, checking with Align that the holes sit truly over one another. This alignment is what will allow the mechanism to turn smoothly.
- Make the coupler from another bar, sized so that its length matches the distance between the free ends of the crank and the follower. Give it a pivot hole at each end and place it between those two ends, closing the loop.
- Insert small cylindrical pins through the four pivot points to represent the joints, and use Align once more to check that every hole lies on the same plane.
Now turn the crank by hand with the rotation handles, a little at a time, and watch. The coupler shifts, the follower swings, and neither of them can be moved on its own — turning the one input link decides where every other link must be. That is one degree of freedom, produced by four bars and four joints.
Diagram
Fig. 5.6-2 – a four-bar linkage as modelled from solid shapes, drawn in three-quarter view; the flattened bar at the bottom hatched and labelled “fixed link, two holes subtracted”; the shorter bar at the left labelled “crank”, the bar at the right labelled “follower”, and the bar joining their free ends labelled “coupler”; a small cylindrical pin drawn through each of the four pivot holes and labelled “pin”; a curved arrow at the crank labelled “input turned by hand” and a curved arrow at the follower labelled “resulting swing”; a note stating “all four holes lie on one plane”
The mechanism is worth viewing from several angles once it is built. Rotating the workspace shows immediately whether one of the bars is standing a few millimetres proud of the others, which is the usual reason a first attempt looks wrong when it is turned.
5.6.3 The Cam and Follower Mechanism
The cam and follower does something the first two mechanisms cannot: it changes one kind of motion into another. It has two main parts — the cam, which rotates, and the follower, which moves up and down in response to the cam’s shape.
Its whole behaviour comes from a single design choice, namely that the cam’s surface is not a constant distance from its axis of rotation.
Building the mechanism
- Drag a cylinder onto the workplane and size it into a disc for the cam. Lay it flat on the workplane so that it represents a horizontal rotating disc.
- Make the disc eccentric. Either use an oval shape in place of the circle, or combine a smaller cylinder offset from the centre of a larger one. Either way, the distance from the axis to the surface must vary as the disc turns, because that variation is what will lift the follower.
- Drag a long, thin box onto the workplane for the follower rod, and stand it vertically above the cam. Extend its upper end so that its movement will be easy to see.
- Add a small cylinder at the lower end of the rod as the follower head, and position it so that it rests on the cam surface. The contact between this head and the cam surface is what determines the follower’s vertical motion.
- Add a second box around the rod, with a vertical slot, to act as the guide. The slot allows the follower to move only up and down, imitating the constrained motion of a real mechanism.
- Check the assembly from the side. The follower head should sit cleanly on the cam surface, neither floating above it nor sunk into it, and the guide should keep the rod aligned with the cam through a full turn.
Turn the cam by hand and watch the follower. As the cam rotates, the varying height of its surface pushes the follower head upward and lets it fall again, so a steady rotation at the cam becomes a rise-and-fall at the rod. This is the conversion of rotary motion into reciprocating motion, and it is the reason cams appear in engines, in printing machines and in automatic feeders on production lines.
Diagram
Fig. 5.6-3 – cam and follower assembly seen from the side; an eccentric disc lying flat and labelled “cam”, with its axis of rotation marked by a dashed vertical line offset from the centre of the disc profile and a curved arrow labelled “rotary input”; a vertical rod above it labelled “follower”, ending in a small cylinder labelled “follower head” resting on the cam surface; a slotted block around the rod, hatched and labelled “guide – permits vertical motion only”; a straight double-headed arrow beside the rod labelled “reciprocating output”; a dashed outline showing the follower at its highest position
The three mechanisms compared
Read the third row across and the chapter’s central idea appears in one line. Two joints in an open chain gave two degrees of freedom; four joints in a closed loop gave one; and a mechanism with a single input can still produce a completely different kind of output motion. The count is decided by the arrangement, never by the number of parts.
5.7 Workspace of a Robotic Manipulator
Stand still, keep your feet planted, and try to touch every point in the room. You cannot. There is a shell of space around you that your hand can sweep — no further out than your arm reaches, and no closer in than your folded elbow allows — and everything outside it is simply unavailable without moving your feet.
A robot arm lives under the same restriction, and far more strictly, because it cannot walk. The region it can reach is a fixed property of the machine, decided the day its links were cut.
5.7.1 What the Workspace Is
A manipulator is a robot arm: an open kinematic chain of links and joints, mounted on a base, carrying an end-effector that acts on the world. Every manipulator has a definite, measurable region in which it can work.
The workspace, also called the work envelope, is the region of space within which the manipulator’s end-effector can reach and operate. It represents all the positions the arm can attain, given the arrangement of its joints, the lengths of its links and the limits imposed by its mechanical design.
The reason a workspace exists at all is that both quantities in a manipulator are bounded. Its links are of fixed length, and its joints can move only within specific angular or linear limits. An arm therefore cannot reach every point in its environment; it operates only within a particular geometric region, and outside that region it is powerless no matter how it is programmed.
This has a consequence that is easy to state and easy to forget. A robot is selected for a task by first checking that the task lies inside the machine’s workspace. No amount of clever control will move an end-effector to a point that the links cannot physically bring it to.
5.7.2 Reachable Workspace and Dexterous Workspace
Not every point in the workspace is equally useful, and the difference has a name.
The reachable workspace includes every point the end-effector can touch in at least one orientation. It forms the outer boundary of the robot’s movement capability, and it is the figure a manufacturer quotes as the machine’s reach.
The dexterous workspace includes only those points that the end-effector can reach in every required orientation. Since reaching a point in one particular way is easier than reaching it in all required ways, the dexterous workspace is always the smaller of the two, and it lies inside the reachable workspace.
Your own arm shows the difference. You can touch the wall directly behind your shoulder with your fingertips, at one awkward angle only — that point is in your reachable workspace. Try to press your palm flat against it, at any orientation you choose, and you cannot; the point is outside your dexterous workspace.
Diagram
Fig. 5.7-1 – plan view of a manipulator’s two workspaces; a hatched base at the centre with the arm drawn schematically; a large outer region shaded lightly and labelled “reachable workspace – attainable in at least one orientation”, and a smaller region inside it shaded more darkly and labelled “dexterous workspace – attainable in every required orientation”; the outer boundary labelled “arm fully extended” and the inner boundary labelled “arm fully folded”; a note stating “the dexterous workspace always lies within the reachable workspace”
This distinction becomes essential wherever the orientation of the tool matters as much as its position. A welding torch must meet the seam at a particular angle; a screwdriver must arrive along the axis of the screw. For such tasks the dexterous workspace, not the quoted reach, is the figure that decides whether the robot can do the job.
5.7.3 Factors Affecting the Workspace
Five things between them decide the size and shape of the region a manipulator can work in.
The types of joints used
Joint type sets the basic geometry. Revolute joints sweep their links along circular or spherical paths, so they produce rounded workspaces. Prismatic joints slide along straight axes, so they produce linear or box-shaped regions. An arm built entirely of revolute joints and one built entirely of prismatic joints will have workspaces of completely different character even if they occupy the same volume.
The lengths of the links
Link lengths set the scale. The further a link extends, the further the end-effector can be carried from the base, so the outer boundary of the workspace grows directly with the sum of the link lengths.
The limits of each joint
A joint that can rotate through 180° sweeps half the arc of one that can rotate through 360°. Joint limits, whether they come from mechanical stops, from internal cables or from safety requirements, cut portions out of the region that the link lengths alone would suggest.
The placement of the base
The same arm mounted on the floor, on a wall or hanging from the ceiling serves three different regions of a room. Base placement does not change the shape of the workspace, but it decides which part of the actual working area that shape covers, and it is often the cheapest way to fix a reach problem.
Physical obstructions
Finally, the robot must share its space with the world, and sometimes with itself. Fixtures, conveyors, guarding and the robot’s own base and body may all prevent the arm from reaching positions that are geometrically available, which reduces the effective workspace below the theoretical one.
5.7.4 Workspace Shapes of Common Configurations
Because joint types set the geometry, each standard configuration has a characteristic workspace shape, and recognising them is a quick way to judge whether a machine suits a task.
A revolute–revolute (RR) arm has a workspace resembling an annular region — a flat ring. The outer boundary is traced when the arm is fully stretched, and the inner boundary when the two links fold towards each other as far as they can.
\[R = l_1 + l_2 = 250 \ \text{mm} + 200 \ \text{mm} = 450 \ \text{mm}\]
\[r = |l_1 - l_2| = |250 \ \text{mm} - 200 \ \text{mm}| = 50 \ \text{mm}\]
\[A = \pi \left[ (450 \ \text{mm})^2 - (50 \ \text{mm})^2 \right] = \pi \times 200000 \ \text{mm}^2 = 6.28 \times 10^{5} \ \text{mm}^2\]
Answer: The reachable region is a ring of outer radius 450 mm and inner radius 50 mm, with an area of about \(6.28 \times 10^{5}\) mm², which is 0.63 m². Check: The magnitude is sensible for an arm whose reach is a little under half a metre — a working area of roughly two-thirds of a square metre, comparable to a small desk top. Note also that if the two links were made equal in length, the inner radius would fall to zero and the ring would close into a full disc.
Diagram
Fig. 5.7-2 – plan view of a planar RR arm and its annular workspace; the shoulder at the centre marked as a small circle on a hatched base; link 1 of length l1 and link 2 of length l2 drawn in a mid-range posture with the end-effector marked; an outer circle of radius R = l1 + l2 drawn as a solid line and labelled “arm fully extended”; an inner circle of radius r = l1 − l2 drawn as a solid line and labelled “arm fully folded”; the ring between them shaded and labelled “reachable region”; the inner disc left unshaded and labelled “cannot be reached”
The other standard configurations follow the same logic.
An RP (revolute–prismatic) mechanism produces a sector-shaped workspace — a wedge or slice — because the prismatic joint slides in and out along a line while the revolute joint swings that line about the base.
A SCARA robot, the arm of this chapter’s opening story, typically has a doughnut-shaped, or toroidal, workspace: a ring in the horizontal plane, given thickness by the vertical travel of its sliding joint.
A six-degree-of-freedom industrial arm has a roughly spherical workspace, with certain regions near the base that cannot be reached because the arm is unable to fold that far into itself.
A Cartesian robot, with three prismatic joints, has a rectangular box workspace, and a cylindrical-coordinate arm sweeps out a hollow cylinder.
Diagram
Fig. 5.7-3 – four robot configurations in a row, each drawn schematically above the shape of its workspace; from left to right, “articulated (revolute joints)” above a sphere with a small unreachable region shaded near its base, “cylindrical” above a hollow cylinder, “Cartesian” above a rectangular box, and “SCARA” above a flat ring given vertical thickness; each workspace shape drawn semi-transparent so the machine inside remains visible, and each labelled with its name
Read that table alongside Table 5.2 and the chapter closes on itself. The joints decide the degrees of freedom, the degrees of freedom and the link lengths decide the workspace, and the workspace decides whether a given robot can do a given job at all. A machine is chosen, in the end, by matching a shape in space to a task in space.
5.8 Summary
Links and joints
- A link is a rigid body forming part of a robot’s mechanical structure, whose dimensions and shape are assumed to stay constant during operation. Links house the joints, carry the actuators and sensors, and support the payload.
- The rigid-body assumption is adopted deliberately, because it simplifies the mathematics, improves kinematic precision and improves structural stability. It is an approximation: real links deflect under load, which is why a manufacturer’s rated payload falls as the arm extends.
- A link’s mass adds to the gravitational load and to the load that must be accelerated. Its moment of inertia describes how that mass is distributed about the axis of rotation, so a long slender link resists rotation far more strongly than a compact link of the same mass.
- Link geometry follows function. Straight, uniform links suit prismatic mechanisms, Cartesian robots, slider blocks and linear stages; curved or non-uniform links are used for compact motion pathways, to enlarge the workspace without adding a joint, and where spatial mechanisms need specific orientation changes.
- Link material is chosen from stiffness, weight, ease of fabrication, cost and working conditions. Aluminum gives strength for its weight, steel gives rigidity at the cost of mass, composites give stiffness at low weight, and acrylic and ABS plastic suit low-cost educational robots but not high loads or high precision.
- A joint is a mechanical connection between two links that permits one specific type of relative motion and constrains all others. Permitting one motion and blocking the rest is the joint’s real function.
- A joint is described by four characteristics: the type of motion it allows, its range of motion, its actuation method — electrical motors, hydraulic systems, pneumatic cylinders or passive mechanisms — and the sensing elements fitted to it, such as encoders, potentiometers and force sensors that measure position, velocity and applied load.
- The four joint types studied are the prismatic joint (one translation along a fixed axis), the revolute joint (one rotation about a fixed axis), the cylindrical joint (one rotation about, and one independent translation along, the same axis) and the spherical or ball-and-socket joint (three rotations about axes meeting at one point).
- Revolute and prismatic joints dominate real robots because each permits exactly one motion, giving one value to command and one value to read.
- A kinematic chain is a set of links connected by joints such that motion at the joints produces a determinate motion of the assembly. In an open chain the links run from a fixed base to a free end carrying the end-effector, and each joint can be commanded separately; in a closed chain the links form a loop, and moving one fixes the motion of the rest.
Degrees of freedom
- Every motion of a rigid body is built from translation, which is linear movement along an axis, and rotation, which is angular movement about an axis.
- The degrees of freedom of a mechanical system is the number of independent movements it can perform, equal to the number of independent values needed to specify its position and orientation completely.
- A free rigid body in three-dimensional space has six degrees of freedom: three translational, along X, Y and Z, and three rotational — roll about X, pitch about Y and yaw about Z. A rigid body confined to a plane has three: two translational and one rotational.
- Joints do not give freedom, they remove it. Each joint permits its own motions and blocks the rest, and what survives is the mechanism’s freedom.
- Degrees of freedom contributed by each joint type: revolute 1, prismatic 1, screw 1, cylindrical 2, planar 3, spherical 3. A screw joint contributes one, not two, because the thread ties its rotation and translation together, and only independent motions are counted.
- Counted mechanisms: a single revolute joint or rotating shaft has 1; a four-bar linkage has 1, despite its four links and four joints, because the closed loop and the fixed link lengths leave only one free choice; a two-link RR arm has 2; a differential drive robot has 2 independently controlled motions, against the 3 values needed to describe its pose in the plane, which is why it cannot move sideways directly; a Cartesian robot has 3 translational degrees of freedom and cannot change its tool’s orientation.
- Fixing both position and orientation anywhere in three-dimensional space requires six degrees of freedom. Each extra degree of freedom costs an actuator, a sensor, control effort, mass and one more direction of unwanted movement, so a machine is given the freedom its task needs and no more.
3D modelling and TinkerCAD
- 3D modelling is the creation of a virtual three-dimensional representation of a robot or its components, with real dimensions, before anything is built. The software used is called computer-aided design software.
- Its purposes are visualisation and conceptualisation, simulation of movements, measurement and accuracy, material selection and weight estimation, and integration with manufacturing, where the model is exported directly to a 3D printer or CNC machine.
- Assigning a material to a model turns a shape into an engineering estimate, since mass follows from the model’s volume and the material’s density, \(m = \rho V\). These masses decide motor sizing, acceleration and stability.
- Designs are built from simple solids — cubes, cylinders, cones and spheres — using six operations: translation, rotation, scaling, extrusion, union and subtraction. Scaling may be uniform, preserving proportions, or along one axis only.
- On screen every shape has all six degrees of freedom; modelling is the stage at which a designer decides which of them the finished machine will keep.
- TinkerCAD is a beginner-friendly, browser-based tool from Autodesk for creating and visualising 3D models, requiring no installation and working by drag-and-drop. Designs are stored online, shared by link and exported for 3D printing.
- Shapes rest on the workplane, whose X and Y axes lie in the surface and whose Z axis points upward. Position and size can be typed in exactly rather than dragged approximately.
- Every shape is either a solid or a hole, and nothing is cut until shapes are grouped: grouping fuses the solids and removes the material overlapped by any hole, giving both union and subtraction from one operation.
- Parts that must be concentric or level are aligned with the align tool, dimensions that must match real hardware are set with the ruler, and the model is turned and viewed from several directions to reveal collisions invisible in one view.
- TinkerCAD holds shapes in place but does not simulate mechanical connections, so a modelled joint does not rotate on its own; motion is studied by rotating a part by hand and reasoning about what the neighbouring parts must do.
- Three mechanisms were modelled. The RR arm, built from bars with bossed ends, shows that the distance between pivot hole centres, not the length of the bar, is the effective link length. The four-bar linkage shows one input angle fixing the position of every other link. The cam and follower converts rotary motion into reciprocating motion of the follower, guided by a slot that acts as a prismatic joint, so the whole mechanism has one degree of freedom.
- For an eccentric circular cam of radius \(R\) with its shaft offset by \(e\), the follower rises between \(R - e\) and \(R + e\), giving a total lift of \(2e\).
Workspace
- The workspace, or work envelope, of a manipulator is the region containing every position its end-effector can reach, fixed by the joint types, the link lengths and the joint limits. It is definite and calculable, not approximate.
- For a planar RR arm the workspace is an annular ring, with outer radius \(r_{\text{out}} = l_1 + l_2\) when the arm is fully stretched and inner radius \(r_{\text{in}} = |l_1 - l_2|\) when it is fully folded. Equal link lengths remove the dead zone at the centre.
- The reachable workspace contains every point the end-effector can touch in at least one orientation; the dexterous workspace contains only those points it can reach in every required orientation, and is therefore always smaller. Points on the outer boundary are reachable but not dexterous, since a fully stretched arm has only one possible configuration.
- Five factors shape the workspace: the types of joints used, since revolute joints sweep arcs and prismatic joints sweep straight lines; the lengths of the links; the limits on each joint’s travel; the placement of the base on floor, wall or ceiling; and physical obstructions, including the robot’s own structure.
- Characteristic shapes follow from the configuration: annular for an RR arm, sector-shaped for an RP arm, a rectangular box for a Cartesian robot, a hollow cylinder for a cylindrical robot, part of a hollow sphere for a spherical robot, and an approximately spherical shell for an articulated arm, with unreachable regions close to its base.
- The chain of reasoning runs one way through the whole chapter: links and joints decide the degrees of freedom, the degrees of freedom and the link lengths decide the workspace, and the workspace decides whether the machine can do the job it was built for.
5.9 Key Terms
- 3D modelling — The creation of a virtual three-dimensional representation of an object or assembly, with defined dimensions and shape, before it is physically built.
- ABS (acrylonitrile butadiene styrene) — A common moulded and 3D-printed plastic, easy to fabricate and suitable for low-cost robots but not for high loads or high precision.
- Actuator — The device, such as a motor or a cylinder, that supplies the motion at a joint.
- CAD (computer-aided design) — Software used to create, dimension, view and analyse models of parts and assemblies before manufacture.
- Cam — A rotating element with a shaped profile that imparts a prescribed motion to a follower held against it.
- Cartesian robot — A robot using three prismatic joints along mutually perpendicular axes, giving three translational degrees of freedom and a box-shaped workspace.
- Closed chain — A kinematic chain whose links form a loop, so that moving one link fixes the motion of all the others.
- CNC (computer numerical control) — Machine control in which cutting or shaping tools follow paths supplied as digital instructions, often exported from a CAD model.
- Cylindrical joint — A joint permitting one rotation about an axis and one independent translation along the same axis, contributing two degrees of freedom.
- Dexterous workspace — The set of points the end-effector can reach in every required orientation; always smaller than the reachable workspace.
- Differential drive robot — A mobile robot with two independently driven wheels, able to drive forward and backward and to turn about its own centre, but not to move sideways directly.
- DOF (degrees of freedom) — The number of independent movements a mechanical system can perform, equal to the number of independent values needed to fix its position and orientation.
- End-effector — The device fitted to the free end of the last link, such as a gripper or a tool, through which the robot acts on its environment.
- Extrusion — The modelling operation that extends a two-dimensional outline into a three-dimensional solid.
- Follower — The part held in contact with a cam, which is displaced as the cam rotates.
- Four-bar linkage — A closed chain of four rigid links joined by four revolute joints, having one degree of freedom.
- Grouping — In TinkerCAD, the operation that merges selected shapes into one object, fusing the solids and cutting away the material overlapped by any hole.
- Hole — In TinkerCAD, a shape marked as empty space, which removes material from the solid it overlaps when the two are grouped.
- Joint — A mechanical connection between two links that permits one specific type of relative motion and constrains all others.
- Kinematic chain — A set of rigid links connected by joints, in which motion at the joints produces a determinate motion of the whole assembly.
- Link — A rigid body forming part of a robot’s mechanical structure, whose dimensions and shape are assumed constant during operation.
- Moment of inertia — A measure of how a body’s mass is distributed about an axis of rotation, determining how strongly the body resists being turned about it.
- Open chain — A kinematic chain whose links run in a single line from a fixed base to a free end carrying the end-effector, each joint being commanded independently.
- Payload — The maximum mass a robot can carry at its end-effector, at a stated reach and speed, without loss of accuracy.
- Pitch — Rotation about the Y-axis.
- Prismatic joint — A joint permitting one translation along a fixed axis and no rotation, contributing one degree of freedom.
- Reachable workspace — The set of points the end-effector can touch in at least one orientation.
- Reciprocating motion — Repeated back-and-forth movement along a straight line, such as that of a cam follower.
- Revolute joint — A joint permitting one rotation about a fixed axis and no translation, contributing one degree of freedom.
- Rigid body — A body whose particles remain at fixed distances from one another, so that it does not bend, stretch or compress under applied forces.
- Roll — Rotation about the X-axis.
- Rotation — Angular movement about an axis, in which the axis stays fixed and other points travel in arcs around it.
- RP (revolute–prismatic) — A two-joint configuration in which a revolute joint swings the assembly while a prismatic joint extends the reach, producing a sector-shaped workspace.
- RR (Revolute–Revolute) mechanism — A two-link planar arm with two revolute joints, usually called the shoulder and the elbow, having two degrees of freedom.
- Scaling — The modelling operation that increases or decreases the size of a shape, either uniformly or along one axis only.
- Spherical joint — A ball-and-socket joint permitting three rotations about axes meeting at a single point, contributing three degrees of freedom.
- Subtraction — The modelling operation that removes one shape from another, used to create holes, slots and hollows.
- TinkerCAD — A browser-based 3D modelling tool from Autodesk in which designs are built by dragging, resizing and combining basic geometric shapes.
- Translation — Linear movement along an axis, in which every point of the body moves the same distance in the same direction.
- Union — The modelling operation that combines several shapes into a single solid object.
- Workplane — The flat reference surface in a 3D modelling tool on which shapes rest and from which their heights are measured.
- Workspace — The region of space containing every position a robot’s end-effector can reach, fixed by the joint types, the link lengths and the joint limits; also called the work envelope.
- Yaw — Rotation about the Z-axis.
5.10 Objective Questions
- A ball-and-socket joint connecting two links contributes how many degrees of freedom to the mechanism?
- One
- Two
- Three
- Six
- A planar RR arm has link lengths of 140 mm and 90 mm, and both joints can turn freely. The radius of the circular region near the base that the end-effector cannot reach is:
- 45 mm
- 50 mm
- 90 mm
- 230 mm
- Assertion (A): A screw joint contributes only one degree of freedom, even though the nut both turns and advances. Reason (R): In a screw joint the rotation and the translation are tied together by the thread, so fixing one fixes the other.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Assertion (A): The dexterous workspace of a manipulator can never be larger than its reachable workspace. Reason (R): A point in the dexterous workspace must be attainable in every required orientation, which is a stricter condition than being attainable in at least one.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Match each item in Column A with the correct item in Column B.
| Column A | Column B |
|---|---|
| a. Prismatic joint | i. Three rotations about axes through one point |
| b. Cylindrical joint | ii. One translation along a fixed axis |
| c. Spherical joint | iii. Rotation about the Z-axis |
| d. Yaw | iv. One rotation about, and one translation along, the same axis |
- Fill in each blank with the correct word, phrase or expression.
- In TinkerCAD, a shape marked as a ________ removes material from the solid it overlaps when the two are grouped.
- A rigid body confined to a plane has ________ degrees of freedom.
- The modelling operation that extends a two-dimensional outline into a three-dimensional solid is called ________.
- For an eccentric circular cam of radius \(R\) with its shaft offset by \(e\), the total lift of the follower is ________.
- State whether each statement is true or false.
- A four-bar linkage has four degrees of freedom because it is built from four links.
- Opening and closing a gripper is counted among the degrees of freedom of the arm carrying it.
- Increasing the length of a link increases the reach of an arm and also increases its droop under load.
- A differential drive robot can move directly sideways without first turning.
- Write the full form of each of the following abbreviations used in this chapter: DOF, CAD, CNC, ABS.
5.11 Check Your Concepts
Define a link. State the three reasons for which a link is assumed to be perfectly rigid, and give one honest limitation of that assumption.
Differentiate between an open chain and a closed chain, giving one example of each and stating how the two differ in the way their joints can be commanded.
A cylindrical joint contributes two degrees of freedom and a screw joint contributes one, although both permit a rotation and a translation. Explain the reason for the difference.
Explain the working of a cam and follower mechanism, and state how the guide that surrounds the follower rod restricts its motion. Name the joint type that the guide forms.
A planar RR arm is to be built with link lengths \(l_1 = 120\) mm and \(l_2 = 90\) mm, with both joints free to turn through complete circles.
- Calculate the outer and inner radii of its workspace.
- Calculate the area of the region its end-effector can reach.
- State what happens to the inner radius if \(l_2\) is increased to 120 mm, and explain why a designer might want that change.
- A school laboratory must choose one machine for a task in which small cylindrical jars are lifted from a table and placed on shelves arranged around a central column, always upright. Compare the Cartesian, cylindrical and articulated configurations against the shape of the workspace each produces, the ability of each to change the orientation of the tool, and the control effort each demands. Recommend one configuration for this task, and justify your choice.
5.12 Diagram, Circuit & Output Reading
- An arm is built with the following joints, in order from the fixed base outward: a revolute joint rotating about a vertical axis, a prismatic joint sliding vertically, a second revolute joint rotating about a vertical axis, and a two-finger gripper at the end.
- State the number of degrees of freedom of the arm, showing how you arrived at the figure.
- State one motion of the gripper that this arm cannot produce, and name the kind of joint that would have to be added to produce it.
- A planar mechanism is drawn as follows. A horizontal bar is hatched and fixed to the bench, with a pivot at each of its two ends, 90 mm apart. A 30 mm bar is pivoted at the left-hand pivot, and a 60 mm bar is pivoted at the right-hand pivot. An 85 mm bar joins the free ends of these two bars through a pivot at each of its own ends. A motor turns the 30 mm bar through complete revolutions.
- Name the ground link, the crank, the coupler and the rocker in this mechanism.
- State the number of degrees of freedom of the mechanism, with a reason.
- Describe the motion of the 60 mm bar while the motor turns steadily, and name one everyday machine that uses this arrangement.
- A student models a link in TinkerCAD in the following steps. A box is created 80 mm long, 16 mm wide and 6 mm thick, and placed on the workplane. Two cylinders of 4 mm diameter and 20 mm height are created and positioned with their centres 10 mm in from each end of the box and centred across its width. Both cylinders are left set as solids. All three shapes are then selected and grouped. The student expects to see a bar with a pivot hole near each end.
- State what the student will actually see after grouping, and identify the single mistake in the procedure.
- State the correction needed, in one line.
- Assuming the mistake is corrected, state the effective length of this link, and explain why that figure differs from 80 mm.
- The table below gives the link lengths and joint travels of three planar RR arms being considered for a workshop bench.
| Arm | \(l_1\) (mm) | \(l_2\) (mm) | Shoulder travel | Elbow travel |
|---|---|---|---|---|
| P | 120 | 100 | 360° | 360° |
| Q | 200 | 60 | 180° | 300° |
| R | 130 | 130 | 360° | 270° |
- State which of the three arms can reach a point 250 mm from the shoulder axis, showing the figure you used for each.
- State which arm has the largest unreachable region near its base, and give the radius of that region.
- Identify the arm whose workspace has no unreachable region at the centre, and explain what feature of its design produces this.
5.13 Case-Based Questions
- A school robotics club is building a two-link planar arm from acrylic sheet to move small components between two marked positions on a workbench. Link 1 is cut as a bar 140 mm long and link 2 as a bar 80 mm long. Both bars are 16 mm wide and 4 mm thick, and each carries a 4 mm pivot hole centred 10 mm in from each end. The shoulder joint is fixed to the bench, and both joints can turn freely through complete circles. The two marked positions lie 45 mm and 150 mm from the shoulder axis, measured along the bench.
- Determine the effective length of each link, and hence the outer and inner radii of the arm’s workspace.
- State, with reasons, whether the arm as built can reach each of the two marked positions.
- Propose one change to the link lengths that would let the arm serve both positions without reducing its maximum reach, and justify your proposal.
- A workshop group is building a plotting rig in which a marker pen must be lifted clear of the paper and lowered again exactly once for every revolution of a slowly turning motor. The pen must rise by 20 mm. The group decides to model an eccentric cam and follower in TinkerCAD before printing the parts. The disc available in their design is 60 mm in diameter and 10 mm thick, and the motor shaft is 5 mm in diameter. The follower rod is to be held in a guide block with a vertical slot cut through it, with the pen fixed to the top of the rod.
- Determine the offset at which the shaft hole must be placed in the disc, and state the highest and lowest positions of the follower head above the shaft axis.
- Name the joint formed by the guide block and the follower rod, name the joint at the cam shaft, and state the number of degrees of freedom of the complete mechanism.
- State one check the group should carry out on the model before exporting it for 3D printing, and give the reason for that check.