Chapter 7: Robot Kinematics
In 1969, at Stanford University’s Artificial Intelligence Laboratory, an engineer named Victor Scheinman built a robotic arm that looked nothing like the heavy hydraulic machines then appearing in car factories. It was all-electric, it had six joints, and one of those joints did not rotate at all — it slid, extending the arm outwards like a telescope. It came to be known simply as the Stanford Arm.
The unusual design was not decoration. The arm was meant to be driven by a computer, and the computers of that time were slow. Every time the machine moved, the computer had to answer one question: given the angle of each joint right now, where in the room is the tip of the arm? Scheinman chose the arrangement of joints partly so that this question had an answer the computer could work out quickly, using ordinary trigonometry rather than heavy numerical searching.
That question has not changed. A modern arm welding a car body, a rover folding its instrument arm down onto Martian soil, and a two-jointed arm you might build on a school bench all ask it. The joints report their angles; the geometry of the links turns those angles into a position.
The questions below are the ones this chapter will answer.
Photograph
Fig. 7.0-1 – an early computer-controlled robotic arm in a laboratory setting, of the Stanford Arm type; the full arm must be visible from its fixed base to its gripper, with the individual joints and the straight rigid links between them clearly distinguishable, and the sliding (telescoping) section visible; a plain laboratory background is acceptable but the arm must not be cropped at the base or the gripper. Sourcing note: an archival or museum photograph under a licence permitting educational print reproduction; a modern rendering or an artist’s impression is not usable, and neither is an image in which the arm is partly hidden behind equipment or people.
7.1 What Robot Kinematics Studies
A robot arm is a machine built to put something in the right place. The something may be a gripper closing on a bolt, a welding tip touching a seam, or a camera looking at a plant leaf from a fixed distance. Whatever the tool, the machine has done its job only when that tool arrives at the intended position.
The arm itself cannot move the tool directly. It can only turn or slide its joints. Every position the tool reaches is therefore a consequence of the joint settings and of the fixed lengths of the parts between them. Working out that consequence is what this chapter is about.
7.1.1 Motion Described Without Force
Think of an ordinary door on its hinges. If you know the width of the door and the angle it has swung through, you can say exactly where the handle is. It does not matter whether the door was pushed gently by a child or flung open by a gust of wind. The handle’s position depends on the geometry, not on the force.
The study of motion in a robot works the same way. Robot kinematics looks only at how the parts of a robot move and where they end up, and deliberately leaves out the forces and torques that cause the motion. It answers where and in which direction, never how hard.
This is why kinematics is described as descriptive rather than causal. It describes the geometry of a movement that has happened or is being planned; it does not explain what made it happen. The branch of study that does bring forces and torques into the picture is called dynamics, and it is what an engineer turns to when asking how powerful a motor must be or how the arm will behave when it accelerates a heavy load. Kinematics comes first, because you cannot ask how hard to push until you know where you are going.
Kinematics rests on one important assumption: the parts between the joints are treated as rigid. A rigid part does not stretch, bend or twist, so its length stays the same no matter what it is carrying. Real metal does flex slightly, but for the arms and mechanisms in this course the assumption holds well enough to be useful.
Because kinematics needs nothing except geometry, it can be used long before any hardware exists. A designer can test on paper whether a proposed arm will reach every point on a workbench, and a controller can convert joint readings into a tool position thousands of times a second while the robot is running.
7.1.2 Links, Joints and the End-Effector
Every arm, however complicated, is built from just two kinds of part, and it is worth fixing their names before any equation appears. Fig. 7.1-1 shows them on a simple two-jointed arm of the sort you might build in the school workshop.
Diagram
Fig. 7.1-1 – anatomy of a planar robotic arm: a fixed base plate hatched to show it is bolted to the table and labelled “ground link (base)”; a first revolute joint at the base drawn as a small circle and labelled “joint 1”; a straight rigid bar from it labelled “link 1”; a second revolute joint drawn as a small circle at the far end of link 1 and labelled “joint 2”; a second straight rigid bar labelled “link 2”; a two-finger gripper at the free tip labelled “end-effector”; a curved arrow at each joint showing the direction of permitted rotation; a dashed outline of the whole arm in a second position to show that only the joints change
A link is a rigid part of the robot whose shape and length do not change. In a school-built arm a link is usually a strip of MDF, acrylic or aluminum; in a factory arm it is a cast metal housing. A joint is the connection between two links, and it is the only place where relative motion is allowed. Links carry the geometry; joints carry the movement.
One link is special. The ground link, also called the base, is fixed to the table, the floor or the vehicle, and it does not move. Everything else is measured with respect to it. Without a fixed part to measure from, saying that the gripper is at 25 cm would mean nothing at all.
When links and joints are connected one after another, the assembly is called a kinematic chain.
At the free end of the chain sits the part that actually does the work.
The end-effector is the point of interest in almost every kinematic calculation. The joints matter only because they decide where the end-effector goes. This is exactly like your own arm: your shoulder and elbow are not the point of the exercise, but the position of your hand depends entirely on them.
7.1.3 Joint Variables of Revolute and Prismatic Joints
Two kinds of joint appear throughout this chapter, and the whole of robot kinematics at this level is built from them. A revolute joint allows one link to rotate relative to the next about a fixed axis, like a door hinge or your elbow. A prismatic joint allows one link to slide along a straight line relative to the next, like the sliding section of a telescopic ladder or a drawer in its runners.
Each of these joints permits exactly one independent movement, so a single number is enough to say where the joint has been set. That number is called the joint variable, and Fig. 7.1-2 shows what it measures in each case.
Diagram
Fig. 7.1-2 – two joints drawn side by side. Left: a revolute joint, shown as a circle with a dot at its centre joining a fixed hatched link to a movable straight link; the angle between the two links marked θ with a curved arrow showing the direction of increasing θ; caption text “joint variable: angle θ, measured in degrees”. Right: a prismatic joint, shown as a rectangular sleeve fixed to a hatched base with a straight rod sliding through it; the length of rod extended beyond the sleeve marked d with a straight double-headed arrow along the sliding direction; caption text “joint variable: displacement d, measured in millimetres”
The joint variable of a revolute joint is written as \(\theta\) and stated in degrees or radians; the joint variable of a prismatic joint is written as \(d\) and stated in a unit of length such as millimetres or centimetres. Keeping these units straight matters more than it first appears, because a trigonometric function expects an angle and will silently accept a wrong one.
A second point is easy to miss. A joint variable is always measured relative to the link before it in the chain, not relative to the ground. The elbow of an arm is bent by 40° with respect to the upper link, whatever the shoulder happens to be doing at the time. Section 7.4 shows how these relative angles are added together to give a position in the fixed base frame.
The letters in the last row are used constantly. A joint that rotates is an R joint and one that slides is a P joint, and an arm is named by reading its joints in order from the base outwards — hence the 2R and RP configurations this chapter works through. Section 7.3.3 returns to this naming in detail.
7.1.4 The Two Directions of the Kinematic Question
The relationship between joint variables and end-effector position can be used in either direction, and Fig. 7.1-3 sets the two directions side by side.
Diagram
Fig. 7.1-3 – two horizontal arrows drawn one above the other between two labelled boxes. Upper arrow points left to right, from a box reading “joint variables (θ₁, θ₂ or θ, d) known” to a box reading “position and orientation of end-effector”; the arrow labelled “forward kinematics”. Lower arrow points right to left, from a box reading “desired position of end-effector” to a box reading “joint variables required”; the arrow labelled “inverse kinematics”. A note beside the upper arrow reads “always one answer”; a note beside the lower arrow reads “may have several answers, or none”
Going one way, the joint variables are known and the position of the end-effector is to be found. A controller reading its joint encoders is in exactly this situation: the sensors report the angles, and the machine must work out where its tool currently is.
Going the other way, the destination is known and the joint settings needed to get there are to be found. This is called inverse kinematics, and it is the question a robot faces when a person points at a bolt and says “pick that up”.
The two directions are not equally easy. Forward kinematics has exactly one answer, because a given set of joint settings puts the arm in one shape and nowhere else, and that answer follows from ordinary trigonometry. Inverse kinematics can have several answers — an arm can often reach the same point with its elbow up or its elbow down — and for a point outside the arm’s reach it has none at all. Its solution needs methods beyond this chapter, and you will meet it in practical work rather than here.
This chapter therefore works in the forward direction throughout: joint variables in, end-effector position out, for planar chains in the 2R and RP configurations.
7.2 Planar Open Chain Mechanisms
Before the position of an end-effector can be calculated, the kind of mechanism carrying it has to be pinned down. The syllabus is specific about which kind this chapter works with: the planar open chain. Both halves of that name are restrictions, and each one makes the mathematics that follows possible.
7.2.1 Motion Confined to a Single Plane
Lay a robot arm flat on a table and switch it on. If every part of it stays in contact with the tabletop as it moves, sweeping about but never lifting, then every point of the mechanism travels in one flat surface. A mechanism whose parts all move in a single plane, or in planes parallel to it, is a planar mechanism.
For this to happen, the joints must be arranged correctly. In a planar arrangement the axis of every revolute joint is perpendicular to the plane of motion, so each rotation sweeps the following link around within that plane rather than tilting it out of it, and every prismatic joint slides along a line lying in the plane. Fig. 7.2-1 shows the arrangement.
Diagram
Fig. 7.2-1 – a two-link planar arm drawn in the XY-plane, viewed at a slight angle so that the plane is visible as a shaded flat sheet; the base fixed at the origin with the X-axis and Y-axis labelled in the plane; the axis of each of the two revolute joints drawn as a short dashed line standing perpendicular to the shaded plane and labelled “joint axis perpendicular to plane of motion”, the two axes clearly parallel to one another; curved arrows at both joints showing that each rotation keeps the following link inside the shaded plane; the end-effector marked as a point on the plane
The advantage of this restriction is simplicity. A point in a plane needs only two numbers to locate it, \(x\) and \(y\), and the direction in which a body points there is a single angle measured from the \(x\)-axis. The arm’s whole position and orientation can therefore be described with three quantities and worked out with ordinary trigonometry.
By contrast, a spatial mechanism moves in three-dimensional space, so its links can tilt out of any one plane. Locating its end-effector needs three position coordinates and three angles of orientation, and the calculation grows accordingly. Many working robots are spatial, but their mathematics is built on exactly the geometric reasoning you are about to learn on the planar case.
Plenty of useful machines are genuinely planar. A pick-and-place arm that lifts components off a flat conveyor and sets them on a flat tray, a pen plotter drawing on a sheet, and a windscreen wiper all do their work in one plane. Where the task itself happens on a surface, an arm restricted to that surface is lighter, cheaper and easier to control than one that can move anywhere.
7.2.2 The Ground Link and the Chain of Links
An open chain is put together in a strict order, and that order is what allows the position of one link to be worked out from the one before it.
Three conditions define the arrangement:
- One link is fixed. This is the ground link, and it does not move. It carries the first joint and provides the reference from which all positions are measured.
- Every other link is joined to exactly one link before it and at most one link after it, so the parts run in a single sequence outward from the base.
- The last link in the sequence is free. It is connected only to the link before it, and it carries the end-effector.
Fig. 7.2-2 shows this sequence for a three-link arm, and the numbering matters: joint 1 connects the ground link to link 1, joint 2 connects link 1 to link 2, and so on outwards.
Diagram
Fig. 7.2-2 – a planar open chain of three links; a hatched fixed base labelled “ground link” at the left; revolute joint 1 drawn as a circle on the base, then link 1, then revolute joint 2, then link 2, then revolute joint 3, then link 3 ending in a gripper labelled “end-effector”; each joint numbered outward from the base and each link labelled with its length l1, l2, l3; a curved arrow at each joint showing the direction of permitted rotation; an arrow running along the whole chain labelled “sequence runs base to end-effector, and never returns”
This one-way sequence is the reason forward kinematics can be done step by step. The position of joint 2 is fixed once joint 1 has been set, because link 1 has a known length and now a known direction. The position of joint 3 follows once joint 2 has been set. Each stage depends only on what has already been worked out, so the chain can be walked from the base outwards until the end-effector is reached. Section 7.4 does precisely this for two links.
7.2.3 Why an Open Chain Is Not a Loop
Trace a finger along the arm in Fig. 7.2-2, starting at the base and moving outwards. You arrive at the gripper and stop. There is no path back to the base through the mechanism itself, and that is what makes the chain open.
Now think of a different machine. The linkage that swings a bus door, or the crank and connecting rod driven by a bicycle pedal, forms a closed chain: the links join up so that following them round brings you back to where you started. Fig. 7.2-3 places the two kinds side by side.
Diagram
Fig. 7.2-3 – two mechanisms drawn side by side for comparison. Left, labelled “open chain”: a hatched fixed base, revolute joint 1, link 1, revolute joint 2, link 2, ending in a free gripper; a dotted arrow traced from the base to the gripper with the note “path ends at the free tip”. Right, labelled “closed chain (four-bar linkage)”: a hatched ground link at the bottom with a revolute joint at each of its ends; a crank rising from the left joint, a coupler across the top and a rocker down to the right joint, all four revolute joints drawn as circles and each link named; a dotted arrow traced right round the loop with the note “path returns to its starting link”; a curved arrow on the crank showing the direction of the driving rotation
The difference is not cosmetic; it changes how freely the mechanism moves. In an open chain each joint can be set independently of the others, because nothing beyond it holds it back. In a closed chain the links must satisfy the loop at every instant, so turning one joint forces particular movements at the rest. That is exactly why a four-bar linkage is useful — it turns one input rotation into a fixed, repeatable output motion — and exactly why it is unsuitable as a general-purpose arm.
7.2.4 Degrees of Freedom of a Planar Open Chain
Each joint in an open chain contributes one independent movement, because each revolute joint can be set to any angle and each prismatic joint to any extension without regard to the others. The count of these independent movements is the mechanism’s degrees of freedom.
For a planar open chain the count is simple: one degree of freedom for every joint. A single revolute joint gives one, a 2R arm gives two, and an arm with three joints gives three. The ground link contributes nothing, since it never moves, and the end-effector contributes nothing either, since it is rigidly attached to the last link and goes wherever that link goes.
Now compare that count with what a task in a plane demands. To place a tool completely in a plane you must specify three things: the coordinate \(x\), the coordinate \(y\), and the angle at which the tool points. A 2R arm supplies only two independent settings, so it can be commanded to put its tip at a chosen point or to hold a chosen orientation, but not both at once. Once the two joint angles have been chosen to reach the point, the direction of the gripper is already decided, and nothing is left over to adjust it with. Fig. 7.2-4 shows the difference a third joint makes.
Diagram
Fig. 7.2-4 – two planar arms drawn side by side, both reaching the same marked target point P. Left, labelled “2R arm, 2 DOF”: hatched base, revolute joints 1 and 2, links l1 and l2, gripper at P with its pointing direction shown as a short arrow; a note reading “point reached, but the gripper’s direction is whatever the geometry gives”. Right, labelled “3R arm, 3 DOF”: hatched base, revolute joints 1, 2 and 3, links l1, l2 and l3, gripper at the same point P, the third link drawn twice in faint outline at two clearly different pointing angles; a note reading “same point reached, direction now chosen freely”
This is a piece of engineering judgement, not merely arithmetic. A robot that stacks identical boxes on a shelf needs its gripper to arrive both at the right place and square to the shelf, so two joints in the plane will not do. A robot that lowers a probe onto a marked point, where the direction the probe points hardly matters, is well served by two.
The rule to carry forward is that a planar open chain can be fully commanded in the plane only when it has three degrees of freedom, and that this chapter’s 2R and RP arms, having two joints each, control position alone.
7.3 Kinematic Diagrams
Before an arm can be analysed it has to be drawn, and the drawing an engineer uses for this purpose looks almost nothing like the machine. It is stripped down to lines, circles and boxes. That stripping down is not laziness; it is what makes the analysis possible.
7.3.1 Why a Schematic Replaces the Drawing
A photograph of an industrial arm shows a cast housing, a cable loom, cooling fins, mounting bolts and a manufacturer’s paintwork. None of that has the slightest effect on where the gripper ends up. The position of the gripper depends on exactly two things: the length of each link measured from one joint to the next, and the setting of each joint. Everything else in the picture is noise.
A kinematic diagram keeps the two things that matter and throws the rest away. Fig. 7.3-1 shows the same arm drawn both ways.
Diagram
Fig. 7.3-1 – the same two-jointed arm drawn twice, side by side. Left panel titled “as the machine looks”: a pictorial arm with a solid moulded base, a shoulder motor housing, a thick shaped upper arm, an elbow motor, a tapering forearm, cables running along the links and a two-finger gripper. Right panel titled “as the kinematic diagram shows it”: a hatched ground symbol, a small circle for the shoulder revolute joint, a plain straight line labelled l₁ for the upper arm, a small circle for the elbow revolute joint, a plain straight line labelled l₂ for the forearm, and a short open fork at the tip marked “end-effector”; joint variables θ₁ and θ₂ marked with curved arrows. A caption line between the panels reads “same geometry, no shapes, no sizes, no housings”
Three things follow from working with such a diagram. Positions, and later velocities, can be calculated from a few labelled lengths and angles. The joint variables become visible at a glance, so it is obvious what the machine can and cannot be commanded to do. And constraints show up plainly — a joint that cannot help reach the target, or two joints that duplicate each other’s effect, are far easier to spot on a schematic than on a solid model.
The diagram is also what lets one piece of analysis serve many machines. A steel factory arm and a cardboard arm built in the school workshop reduce to the same diagram if their joints are arranged alike, so the equations derived once apply to both.
7.3.2 Standard Symbols for Links and Joints
Kinematic diagrams use a small, settled set of symbols, so that a diagram drawn in one workshop can be read in another. Fig. 7.3-2 shows the ones needed for the planar chains in this chapter.
Diagram
Fig. 7.3-2 – symbol set for planar kinematic diagrams, drawn as five labelled items in a row. (1) Ground: a short horizontal line with diagonal hatching beneath it, labelled “fixed ground link”. (2) Link: a plain straight line with a small dot at each end, labelled “rigid link, length marked beside it”. (3) Revolute joint: a small circle with a dot at its centre joining two lines, with a curved arrow around it, labelled “revolute joint (R), variable θ”. (4) Prismatic joint: a small rectangular sleeve with a straight rod passing through it and a double-headed straight arrow along the rod, labelled “prismatic joint (P), variable d”. (5) End-effector: a short two-pronged fork at the end of a line, labelled “end-effector”
Notice what the symbols deliberately do not record. There is no indication of the thickness of a link, the material it is cut from, the size of the motor at a joint or the load the arm can carry. Those are real engineering questions, but they belong to the design of the structure and the drive, not to the geometry of the motion.
7.3.3 Naming a Configuration by Its Joint Sequence
An arrangement of joints is named by listing the joints in order from the ground link outwards, writing R for each revolute joint and P for each prismatic joint. The list is the configuration of the arm.
An arm whose base joint rotates and whose elbow joint also rotates is an RR configuration. Because both letters are the same, this is usually written 2R, and the two names mean exactly the same thing. An arm whose base joint rotates and whose second joint slides is an RP configuration.
The order is part of the name and cannot be shuffled. RP describes an arm that first turns and then extends along the direction it is pointing. PR describes an arm that first slides along a fixed track and then turns at the end of that slide. These two machines reach quite different sets of points, so writing one name when you mean the other is a real error, not a slip of notation.
Because each letter stands for one joint, and each joint of these types supplies one degree of freedom, counting the letters gives the degrees of freedom of the chain directly. RP has two letters and two degrees of freedom; RRP has three.
7.3.4 Reading Diagrams of Common Planar Configurations
Fig. 7.3-3 shows five planar configurations as kinematic diagrams. Reading them is a skill worth practising, because a diagram tells you the shape of the region an arm can reach before a single number is calculated.
Diagram
Fig. 7.3-3 – five planar kinematic diagrams drawn as five labelled panels, each with x and y axes and a hatched ground symbol at the origin. Panel 1 titled “RR (2R)”: revolute joint at the origin, link l₁ rising at angle θ₁ from the x-axis, second revolute joint at its far end, link l₂ at angle θ₂ measured from the extension of link 1, free tip marked. Panel 2 titled “RP”: revolute joint at the origin, link of fixed length L at angle θ from the x-axis, a prismatic sleeve at its far end with a rod extending outward along the same line by a distance d, free tip marked. Panel 3 titled “RRP”: revolute joint at the origin, link l₁ at angle θ₁, revolute joint, link l₂ at angle θ₂, prismatic extension a along the direction of link 2, free tip marked. Panel 4 titled “RPR”: revolute joint at the origin, prismatic extension a along the link at angle θ₁, then a revolute joint carrying a short link l₁ at angle θ₂, free tip marked. Panel 5 titled “PRR”: a horizontal prismatic slider on the x-axis with displacement a, carrying a revolute joint with link l₁ at angle θ₁, a second revolute joint with link l₂ at angle θ₂, free tip marked. In every panel each joint variable is marked and each fixed link length is labelled
Read each diagram in the same order. Find the hatched ground symbol, because that is where measurement begins. Move outwards along the chain, noting for each joint whether it is a circle or a sleeve. Note which lengths are fixed labels and which are variables that change as the arm moves. Finally, look at the free tip, because that is the point the equations will locate.
Doing this for the RR arm in panel 1 tells you that both variables are angles and both link lengths are fixed, so the tip must always lie at a distance from the base somewhere between the folded length and the fully stretched length. Doing it for the RP arm in panel 2 tells you that one variable is an angle and the other is a length along the arm’s own direction, so the tip can be brought in towards the base and pushed out again along any chosen line. These two arms are the ones the next two sections work through in full.
The remaining panels are read the same way, and they show how quickly the possibilities multiply. The PRR arm in panel 5 rides on a straight track, so it can carry its whole two-jointed reach along a production line — an arrangement common where a single arm has to serve several workstations.
7.4 Forward Kinematics of the 2R Configuration
The 2R arm is the simplest machine that can still reach a chosen point on a surface, and almost every larger arm contains one inside it. Two rigid links, two revolute joints, and a tool at the tip: from that, a complete set of position equations can be built with nothing more than the sine and cosine of an angle.
7.4.1 Setting Up the Base Frame and the Joint Angles
Nothing can be calculated until it is settled where measurement starts and in which direction the angles are counted. That settling is the whole content of Fig. 7.4-1, and it is worth doing slowly, because most errors in kinematics are made here rather than in the trigonometry.
Diagram
Fig. 7.4-1 – 2R planar arm with its base frame. Hatched ground at the lower left with a revolute joint circle at the origin O; horizontal axis to the right labelled x₀ and vertical axis upwards labelled y₀, the pair labelled “base frame, fixed to the bench”; link 1 drawn as a straight line from O of length labelled L₁, rising to a second revolute joint circle at point A; the angle between the x₀ axis and link 1 marked θ₁ with a curved arrow showing the counterclockwise direction; a light dashed line drawn through A parallel to the x₀ axis; link 2 drawn from A of length labelled L₂ to the end-effector at point P; a light dashed line extending link 1 beyond A, with the angle between that extension and link 2 marked θ₂ and a curved arrow; the total angle between the dashed horizontal at A and link 2 marked (θ₁ + θ₂); point P marked with a small gripper symbol and dashed lines dropped from P to both axes marking x and y
The base frame is the fixed reference attached to the ground link. Its origin \(O\) sits at the centre of the first joint, its \(x_0\) axis runs horizontally to the right along the bench, and its \(y_0\) axis runs perpendicular to that in the plane of motion. Every position calculated in this section is a position in this frame, and quoting a position without naming the frame it belongs to is meaningless.
Angles are counted counterclockwise as positive, which is the ordinary mathematical convention and the one that makes \(\cos\) and \(\sin\) behave as expected. The two joint variables are:
- \(\theta_1\), the angle of link 1 measured from the \(x_0\) axis;
- \(\theta_2\), the angle of link 2 measured from the direction of link 1, not from the \(x_0\) axis.
The second measurement is the one students most often get wrong, and there is a physical reason for the convention. The encoder at the elbow is bolted between link 1 and link 2. It can only report how much link 2 has been turned relative to the link it is mounted on. It has no idea where the horizontal is, and no way of finding out. Every joint sensor in every arm reports a relative quantity, so the equations are built from relative quantities and converted to the fixed frame afterwards.
Two lengths complete the description. \(L_1\) is the distance from joint 1 to joint 2, and \(L_2\) is the distance from joint 2 to the tip of the end-effector. Both are fixed by the construction of the arm and never change while it moves.
7.4.2 Locating the Elbow
Take the arm one link at a time. Link 1 starts at the origin, has length \(L_1\) and points at \(\theta_1\) above the \(x_0\) axis. Its far end is the elbow, marked \(A\) in Fig. 7.4-1, and finding it is a single right-angled triangle.
Drop a perpendicular from \(A\) to the \(x_0\) axis, as Fig. 7.4-2 shows. The link itself is the hypotenuse, the horizontal side is the \(x\) coordinate of \(A\), and the vertical side is its \(y\) coordinate.
Diagram
Fig. 7.4-2 – resolving link 1 into components: origin O with x₀ and y₀ axes; a straight line of length L₁ from O at angle θ₁ to the x₀ axis, ending at point A; a dashed vertical line dropped from A to the x₀ axis meeting it at right angles, with the right angle marked; the horizontal side labelled “L₁ cos θ₁” and the vertical side labelled “L₁ sin θ₁”; the angle at O marked θ₁
From the definitions of cosine and sine in a right-angled triangle,
\[x_A = L_1 \cos \theta_1\]
\[y_A = L_1 \sin \theta_1\]
These two expressions are the complete answer for a one-joint arm. An arm with a single revolute joint sweeps its tip around a circle of radius \(L_1\), and the pair of equations above is the parametric description of that circle.
7.4.3 Locating the End-Effector
Now add the second link. It begins at the elbow \(A\), has length \(L_2\), and points at \(\theta_2\) away from the direction of link 1.
The first task is to find the angle link 2 makes with the fixed \(x_0\) axis, since only that angle can be used with the base frame. Draw a line through \(A\) parallel to the \(x_0\) axis, as in Fig. 7.4-3. The direction of link 1, extended past the elbow, makes the same angle \(\theta_1\) with this parallel line as it does with the axis itself, because the two lines are parallel. Link 2 is turned a further \(\theta_2\) from that extension. The angle of link 2 from the horizontal is therefore the sum:
\[\phi = \theta_1 + \theta_2\]
Diagram
Fig. 7.4-3 – adding the second link: point A shown with a dashed horizontal line through it parallel to the x₀ axis and a dashed line extending link 1 beyond A; the angle between the dashed horizontal and the extension of link 1 marked θ₁; the angle between that extension and link 2 marked θ₂; link 2 drawn from A of length L₂ to point P; a dashed vertical line dropped from P to the dashed horizontal through A, with the right angle marked, the horizontal side labelled “L₂ cos (θ₁ + θ₂)” and the vertical side labelled “L₂ sin (θ₁ + θ₂)”
With that angle known, link 2 is resolved exactly as link 1 was. Measured from the elbow, it moves the tip a horizontal distance \(L_2 \cos(\theta_1 + \theta_2)\) and a vertical distance \(L_2 \sin(\theta_1 + \theta_2)\).
The end-effector at \(P\) is reached by starting at the origin, travelling to the elbow, and then travelling along link 2. Its coordinates are therefore those of the elbow plus the components just found:
\[x = x_A + L_2 \cos(\theta_1 + \theta_2)\]
\[y = y_A + L_2 \sin(\theta_1 + \theta_2)\]
7.4.4 The Forward Kinematic Equations
Substituting the elbow coordinates gives the result in its final form, in terms of the two link lengths and the two joint variables alone.
\[x = L_1 \cos \theta_1 + L_2 \cos(\theta_1 + \theta_2)\]
\[y = L_1 \sin \theta_1 + L_2 \sin(\theta_1 + \theta_2)\]
Here \(x\) and \(y\) are the coordinates of the end-effector in the base frame, in the same length unit as \(L_1\) and \(L_2\); \(\theta_1\) and \(\theta_2\) are the joint variables in degrees or radians.
Table 7.4 shows what the equations produce for one particular arm, with \(L_1 = 12\) cm and \(L_2 = 8\) cm, at five different joint settings.
Read the last column first. The first and third rows have the arm fully stretched, and the tip sits at exactly \(L_1 + L_2 = 20\) cm from the base, though in quite different directions. The last row has the elbow folded right back on itself, and the tip has come in to \(L_1 - L_2 = 4\) cm. Every other setting lies between those two extremes. Notice also the second and third rows: two completely different pairs of joint angles give the same tool orientation of 90°, but put the tool in entirely different places.
7.4.5 Orientation of the End-Effector
A tool usually has to arrive pointing in a particular direction as well as at a particular place. A gripper approaching a component from the side is not the same as a gripper approaching it from above, even if the jaws close on the same point.
For the 2R arm the end-effector is carried on link 2, so it points wherever link 2 points. Its orientation in the base frame is the angle already derived:
\[\phi = \theta_1 + \theta_2\]
measured counterclockwise from the \(x_0\) axis. This is the third quantity needed to place a body in a plane, alongside \(x\) and \(y\).
It is worth stopping on what that means for a two-jointed arm. There are only two joint variables available, and all three of \(x\), \(y\) and \(\phi\) are built from the same two numbers. Choose \(x\) and \(y\), and \(\phi\) is decided for you; insist on a particular \(\phi\), and you lose the freedom to choose the point. This is the practical face of the degree-of-freedom count from section 7.2.4, and it is why an arm that must both reach a point and approach it at a set angle is built with a third joint.
7.4.6 The Reachable Workspace of a 2R Arm
Before asking what joint angles put the tool at a point, it is worth asking whether any joint angles can. The set of points the tool can be brought to is called the reachable workspace.
For a planar 2R arm the answer follows from the geometry of the two links, and it depends only on the distance of the point from the base:
\[r = \sqrt{x^2 + y^2}\]
Stretch the arm straight out, with \(\theta_2 = 0°\), and the tip is as far from the base as it can get: \(r = L_1 + L_2\). Fold the elbow right back, with \(\theta_2 = 180°\), and link 2 lies along link 1 pointing back towards the base, leaving the tip at \(r = |L_1 - L_2|\). Every other elbow setting places the tip somewhere between these two, and by turning \(\theta_1\) the whole pattern can be swung to any direction. The reachable workspace is therefore the ring-shaped region between two circles, shown in Fig. 7.4-4.
Diagram
Fig. 7.4-4 – reachable workspace of a 2R planar arm: two concentric circles centred on the base joint at the origin; the outer circle labelled “radius L₁ + L₂, arm fully stretched” and the inner circle labelled “radius |L₁ − L₂|, elbow fully folded”; the ring between them shaded and labelled “reachable workspace”; the disc inside the inner circle left blank and labelled “cannot be reached”; a small inset arm drawn stretched straight out touching the outer circle, and a second inset arm drawn folded back with its tip touching the inner circle
A point is reachable when its distance from the base satisfies
\[|L_1 - L_2| \le r \le L_1 + L_2\]
The absolute value matters, because the inner radius is a distance and cannot be negative whichever link is longer.
The blank disc at the centre surprises people. An arm cannot reach a point that is too close to its own base, for the same reason you cannot touch your own shoulder with your fingertips: the forearm has to fold somewhere, and past a certain point it runs into the upper arm. If the two links are made equal in length, the inner radius becomes zero and the hole closes up, so the workspace is a full disc — which is one reason equal-length links are a common design choice.
That last point generalises. The ring in Fig. 7.4-4 is the workspace of an ideal arm whose joints turn freely through any angle. A real arm has joint limits, a mounting bracket in the way and cables that will not wrap indefinitely, so its true workspace is a portion of the ring, and it must be checked against the actual machine before a layout is committed to.
7.4.7 Worked Problems on the 2R Arm
The three problems below cover the questions this section makes it possible to answer: where the tool is, how it is oriented, and whether a target can be served at all. In each case, make sure the calculator is in degree mode before any trigonometric function is used, since a value entered in degrees and evaluated in radians produces a plausible number and a badly wrong position.
\[\theta_1 + \theta_2 = 40° + 25° = 65°\]
\[x = L_1 \cos \theta_1 + L_2 \cos(\theta_1 + \theta_2) = 12 \cos 40° + 8 \cos 65°\]
\[x = 12(0.7660) + 8(0.4226) = 9.19 + 3.38 = 12.57 \text{ cm}\]
\[y = L_1 \sin \theta_1 + L_2 \sin(\theta_1 + \theta_2) = 12 \sin 40° + 8 \sin 65°\]
\[y = 12(0.6428) + 8(0.9063) = 7.71 + 7.25 = 14.96 \text{ cm}\]
\[\phi = \theta_1 + \theta_2 = 65°\]
Answer: The end-effector is at \((12.57 \text{ cm},\ 14.96 \text{ cm})\) in the base frame, with its tool pointing at 65° from the \(x_0\) axis. Check: The distance from the base is \(r = \sqrt{12.57^2 + 14.96^2} = \sqrt{158.0 + 223.8} = 19.54\) cm. That lies between the folded reach of 4 cm and the stretched reach of 20 cm, and sits near the outer edge, which is what a nearly straight elbow of 25° should give. Both coordinates are positive, matching an arm raised into the first quadrant.
\[L_1 + L_2 = 15 + 10 = 25 \text{ cm}, \qquad |L_1 - L_2| = |15 - 10| = 5 \text{ cm}\]
For point \(M\):
\[r_M = \sqrt{9^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97} = 9.85 \text{ cm}\]
\[5 \text{ cm} \le 9.85 \text{ cm} \le 25 \text{ cm}\]
For point \(N\):
\[r_N = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} = 3.61 \text{ cm}\]
\[3.61 \text{ cm} < 5 \text{ cm}\]
Answer: Point \(M\) is reachable. Point \(N\) is not: it lies inside the inner circle of the workspace, in the dead region around the base. Check: The two results are consistent with the shape of the workspace. \(M\) is comfortably inside the ring, while \(N\) is only 3.61 cm from the base — closer than the 5 cm the arm is left with when its elbow is folded fully back. Moving the arm’s base 6 cm further from \(N\) would bring that target into range, which is the practical fix on a bench layout.
\[\theta_1 + \theta_2 = 30° + 90° = 120°\]
\[x = 10 \cos 30° + 10 \cos 120° = 10(0.8660) + 10(-0.5000) = 8.66 - 5.00 = 3.66 \text{ cm}\]
\[y = 10 \sin 30° + 10 \sin 120° = 10(0.5000) + 10(0.8660) = 5.00 + 8.66 = 13.66 \text{ cm}\]
\[\phi = 120°\]
Answer: The end-effector is at \((3.66 \text{ cm},\ 13.66 \text{ cm})\), with the tool pointing at 120° from the \(x_0\) axis. Check: With the elbow at a right angle, the two links form the perpendicular sides of a right-angled triangle, so the tip must be at \(r = \sqrt{10^2 + 10^2} = 14.14\) cm from the base. The computed point gives \(\sqrt{3.66^2 + 13.66^2} = \sqrt{13.4 + 186.6} = 14.14\) cm, which agrees. The negative cosine has correctly pulled the tip back towards the \(y_0\) axis, as bending the elbow to the left should.
7.5 Forward Kinematics of the RP Configuration
The second configuration named in this course replaces the elbow with a slide. The arm turns to face a direction, then extends along that direction like a telescope. It is a common arrangement wherever a machine has to reach out along a chosen line — a probe pushed towards a plate, a fork sliding a load into a rack, a nozzle advanced towards a surface — and its equations turn out to be even simpler than those of the 2R arm.
7.5.1 The Rotating Link and the Sliding Extension
Reading the joints from the base outwards gives R then P, so this is the RP configuration. Fig. 7.5-1 sets it up in the same base frame used throughout the chapter.
Diagram
Fig. 7.5-1 – RP planar arm with its base frame. Hatched ground at the origin O carrying a revolute joint circle; horizontal axis to the right labelled x₀ and vertical axis upwards labelled y₀, the pair labelled “base frame, fixed to the bench”; a straight link of fixed length labelled L drawn from O at an angle θ above the x₀ axis, ending at point A drawn as a small rectangular sleeve; a rod continuing outward from the sleeve along exactly the same straight line, its extended portion labelled d with a straight double-headed arrow showing the sliding direction; the tip of the rod marked P with a small tool symbol; the angle at O marked θ with a curved arrow showing the counterclockwise direction; a dashed line along the whole arm from O through A to P labelled “single straight line: L and d share one direction”; dashed lines dropped from P to both axes marking x and y
There are two joint variables. The first is \(\theta\), the angle of the arm measured counterclockwise from the \(x_0\) axis, supplied by the revolute joint at the base. The second is \(d\), the distance the rod has been pushed out beyond the fixed link, supplied by the prismatic joint. The fixed link has length \(L\), which does not change.
One feature of this arrangement does all the work in the derivation. The prismatic joint slides along the same straight line that the fixed link lies on, so the extension adds to the reach without changing the direction at all. Turning the base swings the whole assembly; extending the rod pushes the tool further along the line it is already pointing down. The two motions never interfere.
7.5.2 Locating the End-Effector
Work outwards from the origin, exactly as before. The fixed link runs from \(O\) at angle \(\theta\) and has length \(L\), so its far end \(A\) — the mouth of the sleeve — sits at
\[x_A = L \cos \theta\]
\[y_A = L \sin \theta\]
Now extend the rod. It travels a distance \(d\) in the direction the arm already points, which is still the direction \(\theta\). Resolving that displacement into components gives a horizontal part \(d \cos \theta\) and a vertical part \(d \sin \theta\). Both use the same angle \(\theta\), because the slide has not turned the arm at all.
Adding this displacement to point \(A\) gives the position of the end-effector at \(P\):
\[x = L \cos \theta + d \cos \theta\]
\[y = L \sin \theta + d \sin \theta\]
7.5.3 The Forward Kinematic Equations
Both terms in each equation share a common factor, so the result tidies up neatly:
\[x = (L + d) \cos \theta\]
\[y = (L + d) \sin \theta\]
The orientation of the tool is the direction of the single straight arm it is mounted on:
\[\phi = \theta\]
Here \(L\) and \(d\) are in the same unit of length, \(\theta\) is in degrees or radians, and \(x\) and \(y\) come out in the unit used for \(L\) and \(d\).
These equations should look familiar from another part of the course. Writing \(r = L + d\) turns them into \(x = r \cos \theta\) and \(y = r \sin \theta\), which is exactly the conversion from polar coordinates to Cartesian coordinates. That is not a coincidence: an RP arm is a machine built to work in polar coordinates. Its two joints set the two polar quantities directly, one supplying the direction and the other the radius.
Table 7.5 shows the equations at work on an arm with \(L = 18\) cm.
Compare the first two rows, and then the fourth and fifth. In each pair the angle is unchanged and only the rod has been pushed out by 6 cm, and in each pair the tool has moved exactly 6 cm directly outwards from the base. Extending the prismatic joint always moves the tool along the straight line joining it to the base, never sideways. Fig. 7.5-2 shows that radial path.
Diagram
Fig. 7.5-2 – radial motion of an RP arm: origin O with x₀ and y₀ axes; a straight dashed ray drawn from O at an angle θ, extending well beyond the arm; the fixed link L drawn in solid line along this ray from O to the sleeve at A; three tool positions marked along the same ray beyond A, at extensions labelled d = 0, d = 5 cm and d = 10 cm; a straight arrow along the ray from the first tool position to the last labelled “the tool travels along a straight radial line”; a second, lighter copy of the whole arm drawn at a different angle to show that the ray itself swings with θ
7.5.4 The Reachable Workspace of an RP Arm
The prismatic joint of a real arm cannot slide indefinitely. It has a shortest setting \(d_\text{min}\), often zero when the rod is fully drawn in, and a longest setting \(d_\text{max}\) fixed by the length of the rod and the slide. These two limits fix the reach.
At the shortest setting the tool sits at a distance \(L + d_\text{min}\) from the base; at the longest, at \(L + d_\text{max}\). Every intermediate extension puts it somewhere between. Since \(\theta\) can swing the arm to any direction, the reachable workspace is again a ring, and a point at distance \(r = \sqrt{x^2 + y^2}\) from the base is reachable when
\[L + d_\text{min} \le r \le L + d_\text{max}\]
Diagram
Fig. 7.5-3 – reachable workspace of an RP arm: two concentric circles centred on the base joint at the origin, the inner labelled “radius L + d_min, rod fully retracted” and the outer labelled “radius L + d_max, rod fully extended”; the ring between them shaded and labelled “reachable workspace”; the disc inside the inner circle blank and labelled “cannot be reached — the fixed link is too long”; a second panel beside it showing the same two circles but with only a wedge of the ring shaded between two radial lines marked θ_min and θ_max, labelled “with the base joint limited to a range of angles, the workspace becomes an annular sector”
The unreachable hole at the centre has a different cause from the one in the 2R arm. There, the tool could not come closer than \(|L_1 - L_2|\) because the elbow ran out of fold. Here it cannot come closer than \(L + d_\text{min}\) because the fixed link simply holds the tool out at arm’s length and there is no joint that can shorten it. Choosing a short fixed link and a long travel therefore gives a wide ring; a long fixed link with a short travel gives a narrow one far from the base.
In practice the base joint is often restricted too — by a cable loom, a mounting bracket or the layout of the workcell. When \(\theta\) is confined between a minimum and a maximum, the ring becomes a wedge-shaped portion of a ring, as the second panel of Fig. 7.5-3 shows. This annular sector is the workspace of a great many real machines that only ever need to serve one side of themselves.
7.5.5 Worked Problems on the RP Arm
\[L + d = 18 + 6 = 24 \text{ cm}\]
\[x = (L + d) \cos \theta = 24 \cos 50° = 24(0.6428) = 15.43 \text{ cm}\]
\[y = (L + d) \sin \theta = 24 \sin 50° = 24(0.7660) = 18.39 \text{ cm}\]
\[\phi = \theta = 50°\]
Answer: The end-effector is at \((15.43 \text{ cm},\ 18.39 \text{ cm})\), with the tool pointing at 50° from the \(x_0\) axis. Check: The distance from the base must be exactly \(L + d\), since the whole arm is one straight line. Testing this, \(\sqrt{15.43^2 + 18.39^2} = \sqrt{238.1 + 338.2} = 24.01\) cm, which is 24 cm to within rounding. The angle is above 45°, so \(y\) should exceed \(x\), and it does.
\[L + d_\text{min} = 20 + 0 = 20 \text{ cm}, \qquad L + d_\text{max} = 20 + 12 = 32 \text{ cm}\]
For point \(S\):
\[r_S = \sqrt{14^2 + (-18)^2} = \sqrt{196 + 324} = \sqrt{520} = 22.80 \text{ cm}\]
\[20 \text{ cm} \le 22.80 \text{ cm} \le 32 \text{ cm}\]
For point \(T\):
\[r_T = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164} = 12.81 \text{ cm}\]
\[12.81 \text{ cm} < 20 \text{ cm}\]
Answer: Point \(S\) is reachable. Point \(T\) is not, because it lies closer to the base than the fixed link is long. Check: The results are consistent with the geometry. \(S\) lies below the \(x_0\) axis, which is no obstacle to an arm whose base can turn right round, and its distance falls comfortably inside the ring. \(T\) is only 12.81 cm out, and even with the rod fully retracted the tool is held 20 cm from the base, so no setting of the joints can bring it in that far.
At \(d = 0\), \(L + d = 15\) cm:
\[x_1 = 15 \cos 120° = 15(-0.5000) = -7.50 \text{ cm}\]
\[y_1 = 15 \sin 120° = 15(0.8660) = 12.99 \text{ cm}\]
At \(d = 10\) cm, \(L + d = 25\) cm:
\[x_2 = 25 \cos 120° = 25(-0.5000) = -12.50 \text{ cm}\]
\[y_2 = 25 \sin 120° = 25(0.8660) = 21.65 \text{ cm}\]
Displacement of the tool:
\[\Delta x = -12.50 - (-7.50) = -5.00 \text{ cm}, \qquad \Delta y = 21.65 - 12.99 = 8.66 \text{ cm}\]
\[\text{distance} = \sqrt{(-5.00)^2 + 8.66^2} = \sqrt{25.0 + 75.0} = 10.00 \text{ cm}\]
Answer: The tool moves from \((-7.50 \text{ cm},\ 12.99 \text{ cm})\) to \((-12.50 \text{ cm},\ 21.65 \text{ cm})\), a straight-line distance of 10.00 cm directly away from the base along the 120° direction. Check: The distance travelled equals the extension of 10 cm, exactly as it must when the rod slides along the direction the arm already points. Both points also satisfy \(y/x = -1.732 = \tan 120°\), confirming that they lie on the same ray from the origin, so the tool has not drifted sideways at all.
7.6 Choosing a Configuration for a Task
Two arms have now been analysed, and both can be built with the same motors, the same controller and the same amount of aluminum. Deciding which one a job needs is an engineering judgement, and the kinematics worked out in the last two sections is what that judgement rests on.
7.6.1 Comparing the 2R and RP Arms
The two configurations differ in almost every respect except the one that matters most to a casual glance: both can put a tool somewhere on a flat surface.
The single most useful line in that table is the one about the path of the tool. Bending the elbow of a 2R arm swings the tool along a curve; extending the rod of an RP arm drives it along a straight radial line. Where a task is naturally radial — pushing a tray into a slot, advancing a sensor towards a plate — the RP arm does it with one joint moving at a constant rate, while the 2R arm must coordinate both joints continuously to hold a straight line.
The second most useful line is the one about obstructions, illustrated in Fig. 7.6-1. A 2R arm can fold its elbow to one side and bring its tool round a low barrier standing between the base and the target. An RP arm is a straight rod: if anything stands on the line from the base to the target, the arm cannot get there at all.
Diagram
Fig. 7.6-1 – reaching past an obstruction, drawn as two panels sharing the same layout. Both panels show a hatched base at the origin, a small solid block labelled “obstruction” standing part way along the direct line to a target point marked X. Left panel titled “2R arm”: link 1 rising steeply from the base, the elbow bent so that link 2 comes down onto the target from above, the whole arm clearing the block; the elbow angle marked θ₂. Right panel titled “RP arm”: the straight arm drawn as a single line from the base towards the target, shown striking the obstruction, with the blocked portion drawn as a dashed line and marked with a cross
7.6.2 Matching Workspace Shape to the Job
Selecting a configuration begins with the region the tool actually has to serve, not with the arm. Draw the working area on the bench, mark where the arm can be mounted, and then ask which workspace covers it.
Fig. 7.6-2 shows the two rings drawn to the same outer radius, and the contrast at the centre is stark.
Diagram
Fig. 7.6-2 – workspaces compared at equal outer reach, drawn as two panels of the same size and scale, each with a base at the centre and an outer circle of the same radius labelled “30 cm”. Left panel titled “2R arm, L₁ = 16 cm, L₂ = 14 cm”: inner circle of radius 2 cm, the wide shaded ring between them labelled “reachable”; the small blank inner disc labelled “2 cm”. Right panel titled “RP arm, L = 20 cm, d from 0 to 10 cm”: inner circle of radius 20 cm, the narrow shaded ring between them labelled “reachable”; the large blank inner disc labelled “20 cm”
An arm cannot be chosen on outer reach alone. Both arms in the figure reach 30 cm, but the 2R arm serves nearly the whole disc while the RP arm serves only a 10 cm band at its outer edge. If the components to be picked are scattered over the bench, the first arm is the only candidate. If they arrive on a rack that always sits between 20 cm and 30 cm from the mounting point, the second arm covers the job with a simpler mechanism and a shorter, stiffer structure.
Arm A: \(\ \lvert L_1 - L_2 \rvert = \lvert 16 - 14 \rvert = 2 \text{ cm}\), and \(L_1 + L_2 = 16 + 14 = 30 \text{ cm}\).
\[\text{workspace of A: } 2 \text{ cm} \le r \le 30 \text{ cm}\]
Arm B: \(\ L + d_\text{min} = 20 + 0 = 20 \text{ cm}\), and \(L + d_\text{max} = 20 + 12 = 32 \text{ cm}\).
\[\text{workspace of B: } 20 \text{ cm} \le r \le 32 \text{ cm}\]
The required band \(22 \text{ cm} \le r \le 30 \text{ cm}\) lies inside both.
Answer: Both arms cover the required distances, but Arm A is the sound choice for this layout. The guard rail stands on the straight line from the base to part of the rack, and only the 2R arm can bend its elbow to bring the tool over it. Arm B would be the better choice on an unobstructed bench, since its straight radial motion matches the rack geometry with a simpler mechanism. Check: The arithmetic is consistent with the two arms’ extremes: Arm A just reaches 30 cm when fully stretched, so pick points at exactly 30 cm leave it straight out with no margin, and a slightly longer link 2 would be prudent. Arm B has 2 cm of unused travel beyond 30 cm, which is a sensible margin. Neither result depends on orientation, which is correct, since the reachability test ignores the angle the tool arrives at.
7.6.3 What Forward Kinematics Cannot Decide
Forward kinematics is a narrow tool used constantly, and knowing its boundary is part of using it well.
It answers one question only: given the joint variables and the link lengths, where is the tool and how is it pointing? It does not answer the reverse question of what joint values will bring the tool to a chosen point, which is the business of inverse kinematics and is taken up in practical work.
It says nothing at all about force. Whether the motor at the shoulder can hold the arm up with a 200 g load at full reach, how the arm behaves when it accelerates, and whether the structure will vibrate after a fast move are all questions of dynamics.
It also assumes an ideal machine. Real link lengths differ from their drawings by a fraction of a millimetre; real joints have a little free play before they take up motion; real links bend under load. An arm whose equations are perfect can still miss its target, which is why a manufactured arm is calibrated — its actual link lengths and joint offsets are measured and fed back into the controller’s equations.
Finally, the workspace it predicts is a geometric ideal. It takes no account of joint limits, of cables that will not wrap, of the arm colliding with its own base or with the bench, or of an obstruction like the guard rail in the last example. Every workspace calculated on paper must be checked against the real machine before a bench layout is fixed.
None of this weakens the method. It places it: forward kinematics is the first calculation an arm designer does and the calculation an arm’s controller repeats thousands of times a second, and every later question — reach, speed, strength, accuracy — is asked against the geometry it establishes.
7.7 Summary
What kinematics studies
- Robot kinematics is the study of the motion and position of a robot’s joints, links and end-effector, without considering the forces or torques that cause that motion. It is descriptive, not causal.
- Questions about force, torque, motor size and behaviour during acceleration belong to dynamics, which is a separate study built on top of the geometry.
- Kinematics assumes rigid links: each link keeps its length and shape whatever it carries. A link that sags under load places the tool slightly away from the calculated point.
The parts of an arm
- A link is a rigid part whose length does not change; a joint is the connection between two links and the only place relative motion is allowed.
- The ground link is fixed, and every position is measured with respect to it. A position quoted without naming its reference frame means nothing.
- A kinematic chain is links joined by joints so that motion passes from one to the next; the end-effector is the tool at the free end, and its position and orientation are what the calculation aims at.
- A revolute joint rotates and its joint variable is an angle \(\theta\); a prismatic joint slides and its joint variable is a displacement \(d\). Each supplies one independent movement.
- A joint variable is always measured relative to the previous link, because a joint sensor sits between two links and can detect nothing else.
Planar open chains
- A planar mechanism moves in a single flat plane: all revolute axes are perpendicular to that plane and all prismatic directions lie in it. A point then needs only \(x\) and \(y\), and orientation is a single angle.
- A planar open chain mechanism runs from a fixed ground link through links connected in sequence to a free last link, forming no closed loop. Tracing it from the base ends at the free end and never returns.
- Joints are numbered from the base outwards, and motion accumulates in that direction: a joint near the base carries everything beyond it, so an error there displaces the tool more than the same error near the tool.
- An open chain lets every joint be set independently and reaches far; a closed chain constrains its joints but is stiffer and shares the load over several paths.
- A rigid body in a plane has three degrees of freedom (DOF) — \(x\), \(y\) and orientation. In an open chain the DOF equals the number of joints, so a two-jointed planar arm can be sent to a chosen point but cannot also be given a chosen orientation.
Kinematic diagrams
- A kinematic diagram shows only links, joints and the type of motion each joint allows, leaving out shapes, sizes and appearance, because the tool position depends only on link lengths and joint settings.
- Standard symbols are used: hatching for the ground link, a plain line for a link, a circle for a revolute joint, a sleeve and rod for a prismatic joint, and a fork for the end-effector.
- A configuration is named by reading the joints from the ground link outwards, writing R for revolute and P for prismatic. RR is also written 2R. The order is part of the name, so RP and PR are different machines.
- Two arms with identical kinematic diagrams place their tools alike, but the diagram says nothing about material, stiffness, joint limits, speed or load capacity.
Forward kinematics of the 2R arm
- The base frame has its origin at the first joint, with \(x_0\) horizontal and \(y_0\) perpendicular to it in the plane; angles are counted counterclockwise as positive.
- The angle of a link in the fixed frame is the sum of all the joint angles from the base up to it, so link 2 lies at \(\theta_1 + \theta_2\) from the \(x_0\) axis, not at \(\theta_2\).
- The elbow sits at \(x_A = L_1\cos\theta_1\) and \(y_A = L_1\sin\theta_1\), and the end-effector is found by adding the components of link 2 to it.
- The forward kinematic equations of the 2R arm are
\[x = L_1 \cos \theta_1 + L_2 \cos(\theta_1 + \theta_2)\]
\[y = L_1 \sin \theta_1 + L_2 \sin(\theta_1 + \theta_2)\]
- The orientation of the tool is \(\phi = \theta_1 + \theta_2\). With only two joints, \(x\), \(y\) and \(\phi\) are all built from the same two numbers, so choosing the point fixes the orientation.
- The reachable workspace is the set of points the tool can be placed at, ignoring orientation. For the 2R arm it is the ring \(\lvert L_1 - L_2 \rvert \le r \le L_1 + L_2\), where \(r = \sqrt{x^2 + y^2}\).
- The outer radius comes from the arm stretched straight and the inner radius from the elbow folded fully back. Equal links close the central hole in theory, though joint limits leave a small dead region in a real arm.
Forward kinematics of the RP arm
- In the RP configuration a revolute joint at the base sets the direction \(\theta\) and a prismatic joint extends the tool by \(d\) along that same direction, so the whole arm remains one straight line.
- The extension must be resolved like any displacement, giving \(d\cos\theta\) and \(d\sin\theta\); it is not added to \(x\) alone.
- The forward kinematic equations of the RP arm are
\[x = (L + d) \cos \theta\]
\[y = (L + d) \sin \theta\]
with the tool orientation \(\phi = \theta\). 4. Writing \(r = L + d\) shows that these are the polar-to-Cartesian conversion: an RP arm sets the two polar quantities directly, one joint per quantity. 5. Extending the prismatic joint moves the tool along a straight radial line from the base, never sideways. 6. The reachable workspace is the ring \(L + d_\text{min} \le r \le L + d_\text{max}\). Its central hole exists because the fixed link holds the tool out and no joint can shorten it. If the base joint is limited in angle, the ring becomes an annular sector.
Choosing between the configurations
- The two arms differ in the path the tool follows, in the size of the unreachable hole, and in whether the arm can bend round an obstruction; equal outer reach does not mean equal coverage.
- The 2R arm suits scattered targets on an open bench and can fold its elbow past a barrier. The RP arm suits radial tasks such as loading along a line, using a simpler mechanism, but a straight arm cannot avoid anything standing in its way.
- Forward kinematics does not give the joint values needed to reach a chosen point, which is inverse kinematics; it says nothing about force, speed or stiffness; and its ideal workspace ignores joint limits, cabling and collisions, so it must be checked against the real machine.
7.8 Key Terms
- 2R configuration — a planar arm whose two joints are both revolute, so its joint variables are the two angles \(\theta_1\) and \(\theta_2\); also written RR.
- Base frame — the fixed reference attached to the ground link, with its origin at the first joint, in which every calculated position of the end-effector is expressed.
- Closed chain — a chain in which the last link is connected back to the ground link, so tracing the links from the base returns to the starting point.
- Degrees of freedom (DOF) — the number of independent variables that must be specified to fix the position of every part of a mechanism; in a planar open chain it equals the number of joints.
- DOF — degrees of freedom.
- Dynamics — the study of motion together with the forces and torques that cause it, as distinct from kinematics.
- End-effector — the device at the free end of a kinematic chain, such as a gripper, welding tip or sensor, whose position and orientation the robot is controlled to achieve.
- Forward kinematics — the computation of the position and orientation of the end-effector from known joint variables and fixed link lengths.
- Ground link — the fixed link of a mechanism, also called the base, with respect to which all motion and all positions are measured.
- Inverse kinematics — the computation of the joint variables needed to place the end-effector at a desired position and orientation.
- Joint — the connection between two links, and the only place where relative motion between them is allowed.
- Joint variable — the single quantity that fixes the setting of a one-degree-of-freedom joint: an angle for a revolute joint, a displacement for a prismatic joint.
- Kinematic chain — an assembly of rigid links connected by joints, arranged so that the motion of one link is passed on to the next.
- Kinematic diagram — a simplified schematic drawing showing only the links, the joints and the motion each joint allows, without the shapes, sizes or appearance of the parts.
- Link — a rigid part of a mechanism whose shape and length do not change as the mechanism moves.
- Prismatic joint — a joint that allows one link to slide along a straight line relative to the next; its joint variable is a length, written \(d\).
- Reachable workspace — the set of all points at which the end-effector can be placed by some combination of joint settings, ignoring the orientation it arrives with.
- Revolute joint — a joint that allows one link to rotate about a fixed axis relative to the next; its joint variable is an angle, written \(\theta\).
- Robot kinematics — the study of the motion and position of a robot’s joints, links and end-effector, without considering the forces or torques that cause that motion.
- RP configuration — a planar arm whose base joint is revolute and whose second joint is prismatic, so the arm turns to a direction and then extends along it.
7.9 Objective Questions
- Which of the following is not required in order to calculate the position of the end-effector of a planar arm by forward kinematics?
- The length of each link
- The value of each joint variable
- The mass of each link
- The origin and axes of the base frame
- A 2R planar arm has \(L_1 = 18\) cm and \(L_2 = 11\) cm. The radius of the circle inside which its end-effector cannot be placed is
- 7 cm
- 11 cm
- 18 cm
- 29 cm
- Assertion (A): For a 2R planar arm, link 2 makes an angle of \(\theta_1 + \theta_2\) with the \(x_0\) axis of the base frame. Reason (R): A joint sensor is mounted between two links and can measure only the turn of one link relative to 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): An RP planar arm cannot place its tool at a point closer to the base than the length of its fixed link. Reason (R): In an RP arm the prismatic joint slides along a direction perpendicular to the fixed link.
- 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 entry in Column B.
| Column A | Column B |
|---|---|
| a. Revolute joint | i. Drawn with hatching; all measurement starts here |
| b. Prismatic joint | ii. Drawn as a circle; its variable is an angle |
| c. Ground link | iii. Drawn as a fork at the free end; the point the equations locate |
| d. End-effector | iv. Drawn as a sleeve with a rod; its variable is a length |
- Fill in the blanks.
- In a planar open chain, the number of degrees of freedom is equal to the number of ____.
- The orientation of the end-effector of an RP arm, measured from the \(x_0\) axis, is equal to ____.
- Tracing an open chain from the ground link outwards ends at the ____ link and never returns to the start.
- A mechanism whose parts all move in a single flat plane is called a ____ mechanism.
- Give the full form or meaning of each of the following as used in this chapter: DOF; the letter R in a configuration name; the letter P in a configuration name; the name 2R.
7.10 Check Your Concepts
Define robot kinematics. State one question about a robot arm that kinematics cannot answer, and name the branch of study that does answer it.
Differentiate between a revolute joint and a prismatic joint on the basis of the motion allowed, the joint variable and its unit, and give one everyday example of each.
An arm’s elbow sensor reports 30°. Explain why this reading is not the angle of the forearm measured from the horizontal, and state how the angle in the base frame is obtained.
A planar arm has two revolute joints. Explain why it can be commanded to a chosen point on the bench but cannot also be made to arrive with a chosen tool orientation. State what change to the design would remove this restriction.
A 2R planar arm has \(L_1 = 14\) cm and \(L_2 = 9\) cm. Its joints are set to \(\theta_1 = 25°\) and \(\theta_2 = 50°\).
- Calculate the coordinates of the end-effector in the base frame.
- State the orientation of the tool.
- Verify that the calculated point lies inside the reachable workspace of the arm.
- A workcell requires a tool to be placed at points lying between 12 cm and 26 cm from the mounting point, and some of those points lie behind a fixture 6 cm high standing on the bench. Two designs are proposed: a 2R arm with \(L_1 = 15\) cm and \(L_2 = 11\) cm, and an RP arm with a fixed link \(L = 12\) cm and an extension from 0 cm to 14 cm. Compare the two designs on reach, on the shape of the path the tool follows, and on their ability to work past the fixture, and recommend one with reasons.
7.11 Diagram, Circuit & Output Reading
- A kinematic diagram shows the following, read from left to right: a short horizontal line with hatching beneath it; a small circle joined to it, with a curved arrow drawn around the circle; a plain straight line of labelled length 16 cm leading from the circle; a small rectangular sleeve at the far end of that line with a rod passing through it and a straight double-headed arrow along the rod; and a two-pronged fork at the tip of the rod.
- Name the configuration of this arm and state its degrees of freedom.
- Name each joint variable and state the unit in which it is expressed.
- State the shape of the path traced by the fork when only the second joint is moved.
- The table below describes four planar arms available in a workshop.
| Arm | Configuration | Dimensions |
|---|---|---|
| P | 2R | \(L_1 = 10\) cm, \(L_2 = 10\) cm |
| Q | 2R | \(L_1 = 16\) cm, \(L_2 = 6\) cm |
| R | RP | \(L = 8\) cm, extension 0 cm to 10 cm |
| S | RP | \(L = 14\) cm, extension 2 cm to 9 cm |
- State the inner and outer radius of the reachable workspace of each arm.
- Name the arms that can place a tool at a point 9 cm from the base.
- Name the arms that cannot reach a point 21 cm from the base, and give the reason in each case.
- A student calculates the end-effector position of a 2R arm with \(L_1 = 12\) cm and \(L_2 = 8\) cm, set to \(\theta_1 = 30°\) and \(\theta_2 = 60°\), and writes the following working.
\[x = 12 \cos 30° + 8 \cos 60° = 10.39 + 4.00 = 14.39 \text{ cm}\]
\[y = 12 \sin 30° + 8 \sin 60° = 6.00 + 6.93 = 12.93 \text{ cm}\]
- Identify the single mistake in the working.
- Write the corrected working and give the correct coordinates.
- State how far the student’s answer is from the true position of the tool along each axis.
- A figure shows two circles drawn concentric about the base joint of a planar arm, of radius 5 cm and 25 cm, with the region between them shaded.
- State what the shaded region represents and what the blank disc at the centre represents.
- Give the two link lengths of a 2R arm that would produce exactly this shaded region.
- Give the fixed link length and the range of extension of an RP arm that would produce exactly the same shaded region.
- Give the coordinates of one point in the plane that neither of these arms can reach, and explain why.
7.12 Case-Based Questions
- A school workshop has built a 2R planar arm and bolted it to a bench. Its links measure \(L_1 = 20\) cm and \(L_2 = 12\) cm, and its base joint sits at the origin of the base frame, with the \(x_0\) axis running along the front edge of the bench. Three marked positions on the bench are to be served by the arm:
| Position | Coordinates |
|---|---|
| A | (28 cm, 6 cm) |
| B | (5 cm, 4 cm) |
| C | (0 cm, 30 cm) |
- Calculate the inner and outer limits of the arm’s reachable workspace, and determine which of the three positions the arm can serve.
- With the joints set to \(\theta_1 = 35°\) and \(\theta_2 = 20°\), calculate the position and orientation of the end-effector, and state whether the tool is then closer to A or to C.
- For the position the arm cannot serve, suggest two different changes — one to the arm and one to the bench layout — that would bring it into range, and justify each.
- A storage rack is to be loaded by an RP planar arm mounted on the floor beside it. The arm has a fixed link of \(L = 30\) cm, a prismatic joint whose extension runs from 5 cm to 25 cm, and a base joint restricted to angles between 0° and 90°. Two rack slots are to be reached, at (40 cm, 20 cm) and at (10 cm, 25 cm) in the arm’s base frame. A support pillar stands on the floor on the straight line between the arm’s base and the first slot.
- State the shape of this arm’s reachable workspace and calculate its inner and outer radii.
- Determine whether each of the two slots lies within that workspace, showing your working.
- A 2R arm with the same outer reach is offered as an alternative. Discuss whether it would serve this installation better, taking the pillar and the shape of the loading motion into account, and state which arm you would fit.