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Chapter 6: Planar and Spatial Mechanism

In the late 1970s, at Yamanashi University in Japan, a team led by Hiroshi Makino was studying a problem that sounds trivial and is not: getting a machine to push a pin into a hole. A human hand does it without thinking. It presses down firmly, and when the pin catches the edge of the hole, the hand lets it slide sideways by a fraction of a millimetre until it drops in.

The robot arms of that time could not manage this. They were built to move freely in every direction, and an arm that can be pushed sideways can also sag sideways under the weight it carries.

Makino’s answer was to take freedom away on purpose. He arranged the arm’s two main joints so that both turned about upright axes. The arm could sweep anywhere it liked across a horizontal plane, but it could not lift or tilt out of that plane at all. A separate axis moved the whole arm straight down. The result was a machine that was stiff exactly where it needed to press, and free exactly where the pin needed to wander.

That design became the SCARA arm — the Selective Compliance Assembly Robot Arm — and it is still assembling electronics today. Its lesson runs through this chapter: in a mechanism, what a machine is prevented from doing matters as much as what it can do.

Photograph

Fig. 6.0-1 – a SCARA robot arm at work on an assembly bench, viewed from the side so that both upright rotary joints of the horizontal arm and the separate vertical axis at the tool end are clearly visible; the workpiece or circuit board being assembled should be in frame. Sourcing note: a licensed industrial or stock photograph of a working SCARA arm; images where the arm is cropped, where the joints are hidden behind a cover, or where a manufacturer’s branding dominates the frame are unusable.

NoteCuriosity Corner

Q1. Why would a designer deliberately stop a robot arm from moving out of one plane, instead of giving it freedom in every direction? Q2. A door hinge, a bicycle pedal and a car windshield wiper look nothing alike. What do the three of them share? Q3. If a link moving in a plane can move in only three ways, how many ways can a link move in open space? Q4. A four-bar linkage and a slider-crank are built from the same number of links. Why does one swing back and forth while the other slides in a straight line? Q5. Some robot arms are a single chain of links, while others hold the tool with three or four arms at once. Which arrangement is steadier, and what does it cost?

Keep these five questions in mind as you read. By the end of this chapter you will be able to answer every one of them.

6.2 Planar Mechanisms

6.2.1 Motion Confined to a Single Plane

Open a door slowly and watch the handle. It travels along an arc. Now look at the whole door: every point on it — the top corner, the bottom corner, the hinge screws — also travels along an arc, and every one of those arcs lies flat. The door sweeps out a set of horizontal circles stacked above one another.

That is the essential picture. The motion of a door can be drawn completely on a single sheet of paper. You lose nothing by drawing only one of those stacked circles, because all the others are copies of it. A mechanism that behaves this way is a planar mechanism.

NoteDefinition

A planar mechanism is a mechanism in which all the links move in one plane or in planes parallel to it, so that the entire motion can be described in two dimensions.

The phrase “or in planes parallel to it” matters, and it is the part students most often drop. The links of a real machine are not all squeezed into one mathematical plane — they are stacked side by side so that they do not collide. The crank of a wiper sits a few centimetres behind the coupler. What makes the mechanism planar is that every one of those planes is parallel to the others, so no link ever moves towards or away from the plane it started in.

planar motion Statement: A rigid body in planar motion has three degrees of freedom — two translations within the plane and one rotation about an axis perpendicular to it. In words: To say where a body is in a plane, you need two numbers for its position and one angle for the way it faces. Why it holds: The plane removes the third translation and two of the three rotations available in space, leaving three of the original six independent motions. Used for: Describing and computing the position of every link in a two-dimensional mechanism, and the position of a planar robot arm’s tool. Watch out: Three degrees of freedom belong to a free body in the plane. A link inside a mechanism has fewer, because each joint it carries removes some of them.

What makes the axes line up

Behind the neat picture is a geometric condition on the joints. In a planar mechanism, the axis of every revolute joint is perpendicular to the plane of motion, and every one of those axes is therefore parallel to every other. The sliding direction of every prismatic joint lies in the plane.

Tilt one hinge axis by a few degrees and the mechanism stops being planar at once. The link attached to that hinge now swings out of the plane, drags the next link with it, and the whole assembly either binds or breaks. This is why a robot chassis built in the school workshop must have its motor shafts genuinely parallel: a mechanism that is planar on paper and skewed on the bench will fight itself.

Diagram

Fig. 6.2-1 – a four-link planar mechanism drawn in oblique view; the four revolute joint axes drawn as short dashed lines standing vertically, all parallel to one another and perpendicular to a shaded horizontal plane; each link drawn at a slightly different height above the plane so the stacking is visible; a caption arrow to the shaded plane reading “plane of motion”; a second arrow to one of the dashed axes reading “every revolute axis perpendicular to this plane”.

assuming flat parts make a mechanism planar What students do: Cut all the links from a single sheet of MDF or acrylic and conclude that the mechanism must be planar because everything in it is flat. What happens: If one joint axis is not perpendicular to the plane — a bolt tightened at an angle, a bracket bent out of true — the flat links still move out of plane, and the mechanism jams or the joints wear rapidly. The fix: Planarity is a property of the motion, not of the parts. Check that every revolute axis is parallel to every other and perpendicular to the plane of motion.

6.2.2 Features of a Planar Mechanism

Four properties follow from the definition, and together they explain why planar mechanisms are the ones you meet first, both in a textbook and on a workbench.

Motion described in two dimensions

The whole behaviour of the mechanism can be drawn on paper and worked out with plane geometry and trigonometry. Two coordinates and one angle fix any link completely. This is not a small convenience — it is the reason a planar arm can be analysed by hand while a spatial arm generally cannot.

Built from turning pairs and sliding pairs

Planar mechanisms use revolute joints and prismatic joints almost exclusively. Both are lower pairs, both are cheap to make accurately, and both are driven by actuators the school laboratory already has — a servo motor or a geared DC motor for a revolute joint, a lead screw driven by a stepper motor for a prismatic one.

6.2.3 Planar Motion in Everyday Machines

The syllabus asks you to recognise planar mechanisms in real life, and once you know what to look for they are difficult to avoid.

The bicycle pedal and crank

Press down on a pedal and the crank turns the chainwheel. The pedal spindle, the crank and the frame form a planar arrangement: the crank rotates in a vertical plane about the bottom bracket axis, and the pedal turns about its own axis parallel to that one. Nothing moves sideways, which is why the chain stays on the ring.

The windscreen wiper

A small motor turns a crank at constant speed. The crank pushes a coupler, and the coupler drives the wiper arm back and forth across the glass. Rotation goes in at one end and a controlled sweep comes out at the other, all of it in one plane a few millimetres above the windscreen. It is the neatest everyday demonstration of a linkage converting one kind of motion into another. Trace the parts in Fig. 6.2-2 from the motor outwards and you can see where the rotation stops and the sweeping begins.

Photograph

Fig. 6.2-2 – close view of a car windscreen wiper linkage with the cowl removed, showing the motor crank, the connecting coupler and the two wiper arm pivots; the parts must be distinguishable from one another and the pivots visible. Sourcing note: a licensed photograph or a clear workshop image of an exposed wiper linkage assembly; images where the linkage is buried under trim, heavily rusted, or shown only as a wiper blade on glass are unusable.

The door hinge

The simplest case of all: two links, the door and the frame, joined by a single revolute joint. One link is fixed, one degree of freedom remains, and the door can do nothing except swing in its plane. A hinge is a planar mechanism reduced to its bare minimum.

The hand pump lever

At a village or school hand pump, the handle is pushed down and a rod inside the casing rises. The handle turns about a pin near the head of the pump, and the far end of it lifts the pump rod almost vertically. Handle, link and pump body move in one plane; the water does the rest. It is a mechanism a student can watch working from a metre away, and every part of the motion is visible.

Q2. A door hinge, a bicycle pedal and a car windshield wiper look nothing alike. What do the three of them share?

A. All three are planar mechanisms. In each one, every link moves in a single plane or in planes parallel to it, and every revolute axis stands perpendicular to that plane — the hinge pin, the bottom bracket axle, the wiper pivot. Their sizes, materials and purposes are completely different, but their motion can be drawn on one sheet of paper, and the same geometry describes all three.

In robotics, planar mechanisms serve a further purpose. They are the models on which motion transmission, link geometry and joint coordination are first understood, because everything can be checked by drawing before any of it is trusted in three dimensions.

6.3 Classification of Planar Mechanisms by Motion

Two planar mechanisms may contain the same number of links and the same number of joints and still behave nothing like each other. What separates them is the kind of relative motion their links make: whether the links turn, whether they slide, or whether some do one and some the other.

This gives the first and most useful classification. Planar mechanisms are grouped by motion into purely rotational mechanisms, purely translational mechanisms, and combined rotational-translational mechanisms. Read the joints and you can place any mechanism in one of the three within a few seconds — all revolute joints, all prismatic joints, or a mixture.

6.3.1 Purely Rotational Mechanisms

In a purely rotational mechanism, every moving link turns about a pivot, and no link slides relative to another. The pivots lie in the plane of motion, and each axis of rotation stands perpendicular to that plane.

Because turning is the only motion available, such a mechanism is built entirely from revolute joints. A point on any moving link travels along a circular arc about its pivot, and its velocity at every instant points along the tangent to that arc.

What identifies a purely rotational mechanism
  • Every joint in the mechanism is a revolute joint.
  • The relative motion between two adjacent links is angular, never linear.
  • The velocity of a point on a link is tangential to the circular path that point follows.
  • The kinematic pairs are simple to manufacture and easy to maintain, since a pin in a bearing is about the cheapest accurate joint there is.
  • In a mechanism with a single degree of freedom, one angular input describes the position of every link in the system.

That last property is what makes these mechanisms attractive in robotics. Feed one number — the input angle — into the geometry and everything else follows. A servo motor or a stepper motor produces exactly that number, so the actuator and the mechanism fit together without any conversion in between.

Purely rotational planar mechanisms appear in robot arm joints where a controlled swing is needed, in servo-driven linkages that must repeat an angular movement accurately, and in gripper mechanisms where two claws open and close about fixed pivots.

6.3.1.1 Four-bar Linkage

The four-bar linkage is the mechanism to learn first, because a surprising number of machines are a four-bar linkage wearing a disguise.

It has four links joined in a loop by four revolute joints. One link is fixed and acts as the frame. The link driven by the motor is the crank. The link at the far side of the loop, which does the useful work, is the rocker. The link joining crank to rocker, touching the frame at neither end, is the coupler.

Diagram

Fig. 6.3-1 – four-bar planar linkage; the ground link drawn horizontally at the bottom, hatched to show it is fixed; the crank drawn short and rising from the left ground pivot; the coupler drawn spanning across to the right; the rocker drawn rising from the right ground pivot; each of the four revolute joints marked with a small open circle; the four links labelled frame, crank, coupler and rocker; a curved arrow at the crank pivot showing continuous rotation, and a double-headed curved arrow at the rocker showing its back-and-forth swing.

Follow the motion in Fig. 6.3-1. The motor turns the crank steadily. The crank pushes and pulls the coupler, which is free at both ends and therefore moves in a complicated way — it swings and shifts at the same time. The coupler in turn drives the rocker, which can only pivot about its own fixed point, so it swings out and back through a limited angle. Continuous rotation has gone in and a controlled oscillation has come out, with no gears, no belts and no electronics.

four-bar linkage Working principle: Four rigid links form a closed loop through four revolute joints, with one link fixed as the frame. Rotation of the crank is transmitted through the coupler to the rocker. Because the loop is closed, fixing the crank angle fixes the position of every other link, so the output motion is fully determined by the input. Components: Frame (fixed link), crank (input link), coupler (connecting link), rocker (output link), and four revolute joints. Advantages: Only one actuator is needed for the whole mechanism; the joints are simple pins that are cheap and hard-wearing; the output motion repeats exactly on every cycle; and the character of the motion can be changed by changing link lengths alone. Limitations: The output swing is confined to a fixed range that cannot be varied once the linkage is built; the coupler’s motion is awkward to describe mathematically; and the loop can jam at positions where the drive direction becomes ambiguous. Where it is used: Windscreen wiper drives, hand-operated pump levers, folding table and chair frames, bicycle suspension links, and robot gripper jaws.

6.3.1.2 Crank-rocker Mechanism

A four-bar linkage does not always behave the way just described. Whether the crank can turn all the way round, or can only rock back and forth like the rocker, depends on the four link lengths. When the proportions are right and the crank does rotate continuously while the output link oscillates, the arrangement is called a crank-rocker mechanism.

The condition that decides this is Grashof’s condition, named after Franz Grashof, who set it out in the nineteenth century.

Grashof’s condition Statement: For a four-bar linkage, the sum of the lengths of the shortest and longest links must be less than or equal to the sum of the lengths of the remaining two links if continuous relative rotation is to occur between at least one pair of links. In words: Add the shortest and the longest. If that total does not exceed the total of the other two, some link in the loop can go all the way round. Why it holds: The linkage can only pass through the positions where the crank folds back on itself and stretches out fully if the remaining two links are long enough to span the gap that these positions demand. Fail the condition and the loop simply cannot close in those postures, so the crank stalls partway and reverses. Used for: Deciding, before anything is cut or drilled, whether a set of link lengths will give continuous rotation or only a rocking motion. Watch out: The condition tells you that some link can rotate fully; it does not by itself tell you which one. That depends on which link is fixed. When the shortest link is one of the two attached to the frame, it becomes the crank and the linkage is a crank-rocker.

checking whether a set of link lengths gives a crank-rocker

Given: A four-bar linkage is to be built with link lengths \(l_1 = 30\) mm (intended crank), \(l_2 = 80\) mm (coupler), \(l_3 = 70\) mm (rocker) and \(l_4 = 90\) mm (frame). The 30 mm link is attached to the frame and is to be driven by a servo motor.

Find: Whether the crank can rotate continuously, and what kind of four-bar this is.

Method: Apply Grashof’s condition. Identify the shortest and longest links, add their lengths, and compare that sum with the sum of the other two. Then check where the shortest link sits in the loop.

Shortest link \(s = 30\) mm; longest link \(l = 90\) mm.

The remaining two links are \(p = 80\) mm and \(q = 70\) mm.

\[s + l = 30\ \text{mm} + 90\ \text{mm} = 120\ \text{mm}\]

\[p + q = 80\ \text{mm} + 70\ \text{mm} = 150\ \text{mm}\]

\[s + l = 120\ \text{mm} \le p + q = 150\ \text{mm}\]

The condition is satisfied. The shortest link, 30 mm, is attached to the fixed 90 mm frame, so the shortest link is the crank.

The condition holds with 30 mm to spare, so the 30 mm link rotates continuously while the 70 mm link oscillates. The linkage is a crank-rocker.

Check: Is the margin sensible? The crank must be able to lie flat against the frame and also stretch straight out from it. Folded, the crank and frame span \(90 - 30 = 60\) mm, which the coupler and rocker can bridge since they can fold to \(80 - 70 = 10\) mm and stretch to 150 mm. Extended, the span is \(90 + 30 = 120\) mm, again within 150 mm. Both extreme postures close, so continuous rotation is physically possible.

Diagram

Fig. 6.3-2 – the same four-bar linkage drawn twice, side by side, in its two extreme rocker positions; in each drawing the crank is shown at a different angle, once folded back along the frame and once stretched out in line with it; the two extreme rocker positions joined by a shaded arc labelled “swing angle of the rocker”; the frame hatched in both drawings.

Fig. 6.3-2 shows what the rocker actually does. It reaches a furthest position, stops, and returns — not because anything tells it to stop, but because the geometry of the closed loop gives it nowhere further to go. That reversal happens at exactly the same place on every cycle, which is why wiper blades sweep the same arc for years.

assuming every four-bar linkage has a rotating crank What students do: Choose four link lengths that look reasonable, fit a motor to the shortest link, and expect it to spin. What happens: The motor turns partway, the linkage locks solid, and the motor stalls and draws heavy current until something gives — usually the joint, sometimes the motor. The fix: Test Grashof’s condition on paper before cutting the links. If the shortest and longest together exceed the other two, no link will rotate fully, and the mechanism must be driven as a rocker instead.

A crank-rocker running on a bench has a closing gap between the coupler and the frame at each end of the stroke. Keep fingers, sleeves and hair clear of that gap while the motor is powered, clamp the frame to the bench so the mechanism cannot walk off the edge, and disconnect the supply before adjusting any link length.

Cut four cardboard strips of 30 mm, 80 mm, 70 mm and 90 mm between hole centres and join them into a loop with split pins, pinning the 90 mm strip to a board. Turn the 30 mm strip by hand through a full circle and note that it goes round. Now replace the 80 mm strip with a 40 mm one, test Grashof’s condition again on paper, and predict what will happen before you turn it.

6.3.2 Purely Translational Mechanisms

In a purely translational mechanism, every point of every moving link travels along a straight line, and no link turns relative to another. Each link keeps the same orientation from the beginning of the motion to the end.

Picture a lift moving between floors. The floor of the lift stays horizontal throughout, the walls stay vertical, and every point in the cabin — the ceiling light, the handrail, a coin lying on the floor — moves the same distance in the same direction at the same speed. That is translation.

What identifies a purely translational mechanism
  • Motion takes place along parallel straight lines.
  • The mechanism uses prismatic joints, which permit sliding and nothing else.
  • The orientation of every moving link stays unchanged relative to the reference frame.
  • At any instant, the velocity and acceleration of every point on a link are equal in both magnitude and direction, since the whole link is moving as one.

That last point has a practical consequence worth holding on to. In a rotating link, the tip moves faster than the middle, so a designer must think about where on the link a force is applied. In a translating link there is no such distinction, which makes these mechanisms easy to reason about and easy to control.

6.3.2.1 Sliding Table or Cross-head Mechanism

A sliding table, also called a cross-head, is a block that runs back and forth inside a straight guide or slot. The block carries the tool or the workpiece, and the guide forces it to keep its orientation for the whole of its travel.

Diagram

Fig. 6.3-3 – a sliding table mechanism in plan view; a straight guide drawn as two parallel rails, hatched on their outer edges to show they are fixed; a rectangular block seated between the rails and labelled “sliding table”; a double-headed straight arrow along the rails labelled with the joint variable d; a connecting rod entering from the left and pinned to the block at a small open circle labelled “revolute joint to the driving link”.

The block in Fig. 6.3-3 is driven by an external force or by a link connected to a crank. As the driving link moves, it pushes or pulls the block, which slides smoothly along the guide. The block itself never rotates; only the straight-line motion parallel to the guide takes place.

This principle is at work in piston and cylinder assemblies, in the tool slides of shaping machines, and in linear robotic actuators wherever a precise straight-line movement is needed within a plane.

6.3.2.2 Cartesian Manipulator

A Cartesian manipulator is a robot built entirely from prismatic joints arranged along mutually perpendicular axes, usually named X, Y and Z. Each axis is driven by its own linear actuator — a lead screw turned by a stepper motor, a rack and pinion, or a pneumatic cylinder.

To reach a point, the controller simply moves each axis by the required amount. There is no geometry to solve: the coordinates of the target are the commands. If the tool must go 40 mm further right and 25 mm further back, the X axis moves 40 mm and the Y axis moves 25 mm, and that is the whole calculation.

The tool keeps the same orientation everywhere it goes, and the region it can reach — its workspace — is a plain rectangular box. Nothing is out of reach inside that box and nothing at all is reachable outside it.

Photograph

Fig. 6.3-4 – a desktop 3D printer with its cover removed, viewed at an angle so that at least two of its perpendicular linear axes are visible, with the guide rails, the lead screw or belt drive, and the carriage carrying the tool head all identifiable. Sourcing note: a licensed photograph of a Cartesian-style printer or a small gantry machine; images that show only the finished print, or in which the rails and carriage are concealed behind panels, are unusable.

The machine in Fig. 6.3-4 is the version of this design most students have seen. Its tool head is carried by one axis, that axis is carried by a second, and the bed or the frame provides the third. Cartesian arrangements are used in coordinate robots doing pick-and-place work in factories, in computer numerical control machines and 3D printers where the tool head must follow guided rails, and in automatic sliding systems such as door actuators and linear sorting mechanisms.

The limitation is bulk. A prismatic joint needs a guide running the entire length of its travel, so a Cartesian machine that reaches one metre must be more than one metre wide. A rotating arm of the same reach folds into a fraction of that space.

6.3.3 Combined Rotational-Translational Mechanisms

Most real machines do not choose. Some of their links rotate while others slide, and this mixture — also called a general planar mechanism — is by far the most common type in robotics, because it is the only one that can convert between rotary and linear motion.

What identifies a combined mechanism
  • The mechanism contains both revolute (R) and prismatic (P) joints.
  • Some links undergo angular displacement while others undergo linear displacement.
  • Such systems are often closed-chain arrangements, able to transmit high forces or to generate a specific path.
  • The velocity and acceleration behaviour is more complex than in either pure type, because a constant input speed does not produce a constant output speed.

Table 6.2 – the three classes of planar mechanism by motion | Feature | Purely rotational | Purely translational | Combined rotational-translational | |—|—|—|—| | Joints used | Revolute only | Prismatic only | Revolute and prismatic together | | Path of a point on a moving link | Circular arc | Straight line | Curved or straight, depending on the link | | Orientation of a moving link | Changes continuously | Unchanged throughout | Changes for some links, not for others | | Relation between input and output speed | Fixed by the geometry, varies through the cycle | Direct and uniform | Varies through the cycle, often strongly | | Space occupied for a given reach | Compact, since links fold | Large, since guides span the full travel | Intermediate | | Typical actuator | Servo or stepper motor | Lead screw, rack and pinion, pneumatic cylinder | One of each | | Everyday example | Windscreen wiper linkage | 3D printer gantry | Engine piston and crank |

6.3.3.1 Slider-crank Mechanism

The slider-crank mechanism converts rotary motion into straight-line motion, or straight-line motion into rotary motion, entirely within one plane. It has four parts: a crank, a connecting rod acting as the coupler, a slider, and a fixed frame.

Diagram

Fig. 6.3-5 – slider-crank mechanism in side view; the frame hatched along the bottom; a crank pivoted to the frame at the left, drawn at about 45 degrees with a curved arrow showing continuous rotation; a connecting rod pinned to the crank end at one revolute joint and to the slider at another, both marked with small open circles; the slider drawn as a block inside a horizontal guide at the right, with a double-headed straight arrow along the guide; the two extreme slider positions marked with faint vertical dashed lines labelled “end of stroke”.

Follow Fig. 6.3-5 through one turn of the crank. As the crank swings round its fixed pivot, it drives the connecting rod, which pushes and pulls the slider along its straight guide. One complete revolution of the crank produces one complete stroke of the slider — out and back once. Two joints in this mechanism are revolute and one is prismatic, which is what places it in the combined class.

slider-crank mechanism Working principle: A rotating crank drives a connecting rod whose far end is pinned to a slider constrained to a straight guide. The crank’s circular motion is resolved by the connecting rod into reciprocating motion of the slider, and the same linkage run backwards turns a push on the slider into rotation of the crank. Components: Frame, crank, connecting rod (coupler), slider, two revolute joints and one prismatic joint. Advantages: Converts between rotary and linear motion with a single moving loop and no gearing; carries high forces along the line of the slider; and repeats its stroke identically every cycle. Limitations: The slider’s speed is not uniform even when the crank turns at a constant rate; the mechanism passes through two positions each cycle where the connecting rod is in line with the crank and can transmit no turning effort; and side thrust from the connecting rod wears the guide. Where it is used: Internal combustion engines, piston pumps and compressors, mechanical presses, and robotic linear actuators driven from a rotary motor.

Q. If the motor turns the crank at a steady speed, does the slider move at a steady speed too?

A. No. The slider comes to a complete stop at each end of its stroke, reverses, and reaches its greatest speed somewhere near the middle. The crank pin is moving at a constant speed around its circle, but only the component of that motion along the guide direction reaches the slider, and that component varies through the turn. This is why a piston engine vibrates and why a slider-crank is a poor choice when a load must be moved at a constant rate.

Q4. A four-bar linkage and a slider-crank are built from the same number of links. Why does one swing back and forth while the other slides in a straight line?

A. Because of the joints, not the links. Both mechanisms have four links, but the four-bar linkage joins them with four revolute joints, so every moving link can only turn about a pivot and the output link swings through an arc. In the slider-crank, one of those revolute joints has been replaced by a prismatic joint, and a prismatic joint permits nothing except sliding along a straight guide. Swap that single joint and rotation at the input comes out as straight-line motion at the output.

6.3.3.2 Planar Robotic Arm with Mixed Joints

A planar robotic arm that combines a revolute joint with a prismatic joint is known by the order of those joints along the chain: RP if the revolute joint comes first, PR if the prismatic joint does. Such an arm reaches points that neither a purely rotating arm nor a purely sliding arm could reach with two joints.

In an RP arm, the first joint is revolute, so the whole arm swings about a fixed pivot at the base. The second joint is prismatic, so the outer link extends and retracts along its own axis. The revolute joint chooses the direction in which the arm points; the prismatic joint chooses how far along that direction the tool sits. Between them, the two joint variables — an angle and a length — place the end-effector anywhere within a circular region around the base.

Diagram

Fig. 6.3-6 – an RP planar arm; a hatched base at the origin of a set of X and Y axes; a revolute joint at the base marked with an open circle and a curved arrow labelled theta; a link rising from it at an angle, drawn as an outer tube with an inner rod sliding within it; a straight double-headed arrow along the rod labelled d; the tool point at the end of the rod marked P; dashed lines dropping from P to the X and Y axes.

Fig. 6.3-6 shows why this arrangement suits certain tasks so well. To reach a target, the arm turns until it points at the target and then extends until it arrives — direction first, distance second, exactly the way a person points at something and then walks towards it.

A PR arm reverses the order. The first joint slides the arm to a chosen position along a straight line, and the second joint then turns the tool at the end. This suits a task where the tool must be carried along a row and then oriented differently at each stop, such as placing components along a line of holes and turning each one to face the right way.

Mixed-joint planar arms are widely used in industrial automation, in pick-and-place systems, and in inspection machines, wherever a combination of radial reach and linear extension is needed. Their appeal is practical: the hybrid motion gives a more flexible workspace and more path options than a two-joint arm of either pure type, while the kinematics stay simple enough to compute quickly.

6.4 Classification of Planar Mechanisms by Structural Arrangement

The first classification asked what kind of motion the links make. The second asks a different question altogether: how are the links wired together? Do they form a line, with a beginning and an end, or do they form a loop that closes back on itself?

This is not a matter of tidiness. The answer decides how motion travels through the mechanism, how the control system must be written, and how rigid and accurate the finished machine can be. The two categories are open-chain mechanisms and closed-chain mechanisms.

6.4.1 Open-chain Mechanisms

Hold your arm out. The shoulder is anchored to your body, the elbow follows the shoulder, the wrist follows the elbow, and your hand at the far end is attached to nothing. That is an open-chain mechanism: links joined one after another in a series, one end fixed and the other end free.

NoteDefinition

An open-chain mechanism is a mechanism in which the links are connected in series without forming a loop, so that one end is fixed to the base and the other end is free to move.

Each link is joined to the next by a revolute or a prismatic joint, and motion passes along the chain from one joint to the next until it reaches the free end. Because there is no loop to constrain anything, every joint contributes its own degree of freedom to the system. Three revolute joints in series give three degrees of freedom; add a fourth joint and you have four.

Diagram

Fig. 6.4-1 – an open-chain planar mechanism; a hatched base at the left carrying the first revolute joint; three links drawn end to end, each joined to the next by a revolute joint marked with a small open circle; the free end of the last link carrying a gripper labelled “end-effector”; a curved arrow at each joint; a bracket beneath the whole chain labelled “no loop: the far end connects to nothing”.

The operation of the chain in Fig. 6.4-1 is straightforward to picture. An actuator drives the first joint; the link attached to it moves, and in doing so it carries the whole of the rest of the chain with it. The second actuator then moves its own link relative to that new position, and so on until the end-effector arrives where it is wanted.

Serial manipulators in robotics

Open chains are what almost every robot arm is. A robot built this way is called a serial manipulator, and the family runs from the smallest hobby arm to the large industrial machines on a car assembly line.

The simplest planar member of the family is the 2R planar manipulator: two revolute joints in sequence, with two links. Turn both joints and the end-effector moves anywhere within a two-dimensional workspace. Add a third revolute joint and you have a 3R planar manipulator, with more reach and the ability to arrive at a point with the tool facing a chosen direction rather than whatever direction the geometry happens to give.

The price of openness

An open chain is flexible, easy to build and easy to control, but it pays for that in two ways.

It is less rigid than a closed loop, because the only thing holding the end-effector in place is the stiffness of every link and every joint between it and the base. Load the tool and the whole chain deflects a little at every joint.

It also accumulates error. A small angular error at the shoulder is multiplied by the entire length of the arm beyond it, so a fraction of a degree at the base can become several millimetres at the tool.

Q. Why does a small error at the first joint matter more than the same error at the last joint?

A. Because of the length it acts through. An angular error at a joint swings everything beyond that joint. At the base, everything beyond it is the whole arm, so a 1° error on a 400 mm arm shifts the tool by about 7 mm. The same 1° error at the last joint swings only the short final link, perhaps 60 mm long, and moves the tool by about 1 mm. Errors from all the joints then add together along the chain, which is why long serial arms are calibrated joint by joint.

Even so, the simplicity of construction and control makes open chains the natural choice for pick-and-place operations, welding, painting and assembly-line automation, where reach and flexibility matter more than absolute stiffness.

6.4.2 Closed-chain Mechanisms

Now join the free end of the last link back to the first link. The chain becomes a loop, and the result is a closed-chain mechanism, which behaves quite differently.

NoteDefinition

A closed-chain mechanism is a mechanism in which the links form one or more closed loops, so that motion is transmitted through more than one path and the loop geometry constrains the relative motion of the links.

In a closed chain, the links are not merely carrying one another — they are holding one another in place. Motion travels from the input through several interconnected paths at once, and the geometry of the loop settles what every link is allowed to do. Give the mechanism one input motion and the position of every other link is decided for you. There is nothing left to choose.

The four-bar linkage of section 6.3.1.1 is the classic example: four links and four revolute joints forming a single loop, in which driving the crank determines the coupler and the rocker completely. The slider-crank is another, differing only in that one revolute joint has been replaced by a prismatic one.

Parallel manipulators

In robotics, the closed-chain idea produces the parallel manipulator, in which several kinematic chains run from a common fixed base to the same moving platform or tool. All the chains act on the tool at the same time, so its position is the result of what all of them do together.

A planar example is the five-bar mechanism used in some drawing machines and light pick-and-place heads. Two motors sit side by side on the fixed base, each driving its own two-link chain, and the two chains meet at a single pin that carries the pen or the gripper. Neither motor controls the pen by itself; both do, jointly.

Diagram

Fig. 6.4-2 – a planar five-bar parallel mechanism; a hatched horizontal base at the bottom carrying two revolute joints spaced apart, each with a curved arrow and labelled “driven joint”; from each, a link rising inwards to a second revolute joint; from each of those, a further link rising to meet at a single common revolute joint at the top carrying a small tool symbol labelled “tool point”; the closed loop formed by base, four links and the shared joint shaded lightly to make the loop visible.

Because each chain in Fig. 6.4-2 shares the load, the structure is stiff and the tool position is accurate. The motors also stay on the base rather than riding on the arm, so the moving parts are light and can be accelerated quickly.

The working principle behind every closed chain is constraint and cooperation: each link’s movement is restricted by the others, so all the parts must move in harmony. That brings higher structural rigidity, better accuracy and less accumulated error. It also brings harder motion control, since no joint can be commanded in isolation, and a smaller workspace, since the loop cannot be stretched beyond the point where it would come apart.

6.4.3 Why the Structure Decides the Behaviour

The clearest way to see the difference is to count degrees of freedom in two mechanisms that appear to be equally complicated.

Take an open chain with four revolute joints: the fixed base, and four moving links running outward from it to a free end. That mechanism has four degrees of freedom, one for every joint, and four motors are needed to command it.

Now take four revolute joints arranged in a closed loop — the four-bar linkage, whose four links include the fixed frame. The same number of joints, made in the same way, and the mechanism has one degree of freedom. Turn the crank and everything else is settled. One motor commands the whole thing.

The joints did not change. Closing the loop tied them together, and each tie removed a freedom.

adding up joints to find the degrees of freedom of a loop What students do: Count four revolute joints in a four-bar linkage and conclude that it has four degrees of freedom. What happens: The design is fitted with four motors that immediately fight one another, because the geometry only permits one independent motion. The joints bind, the current rises, and the linkage or the motors are damaged. The fix: Count freedoms only in an open chain by adding joints. In a closed chain, every loop imposes constraints that subtract from the total — a single-loop four-bar with four revolute joints has one degree of freedom, not four.

Table 6.3 – open-chain versus closed-chain mechanisms | Feature | Open-chain mechanism | Closed-chain mechanism | |—|—|—| | Arrangement of links | Connected in series, one free end | Connected in one or more loops, no free end | | Degrees of freedom | Equal to the total contributed by the joints | Fewer than the joint total, because the loop constrains the links | | Number of actuators needed | One per joint | One per degree of freedom, usually far fewer than the joint count | | Rigidity | Lower, since the load passes through a single path | Higher, since several paths share the load | | Error behaviour | Errors accumulate along the chain | Errors are partly cancelled by the loop | | Workspace | Large, roughly the reach of the extended chain | Smaller, bounded by where the loop can still close | | Control | Simple: each joint can be commanded on its own | Complex: joint motions are interdependent | | Typical use | Robot arms for welding, painting, pick-and-place | Linkages, presses, and high-speed parallel manipulators |

There is no better structure, only a better match to the task. If the job is to reach into a machine, around an obstacle and out again, the reach of an open chain is worth its softness. If the job is to place a component on a board a thousand times an hour without drifting, the stiffness of a closed chain is worth its restricted workspace.

This trade-off — reach against rigidity, freedom against precision — returns later in the chapter, when the same two structures are met again in three dimensions.

6.5 Classification of Planar Mechanisms by Function

The first two classifications describe what a mechanism is. The third describes what it is for.

Every planar mechanism, however it is built, performs one or more of three jobs: it generates a required motion, it guides a point along a required path, or it transmits force from one link to another. In robotics these three functions correspond to the three things a robot must do — move its joints, follow a trajectory, and press on the world hard enough to matter.

6.5.1 Motion Generation Mechanisms

A motion generation mechanism is designed to produce a specific movement of one link relative to another within the plane. What it controls is the relationship between the input and output links — how far the output turns for a given turn of the input — and not the exact route traced by any particular point.

The crank-rocker of section 6.3.1.2 is the standard case. The crank rotates continuously and the rocker swings through a fixed angle, out and back, once per revolution. If the design brief says the output link must sweep 70° and return, the designer chooses link lengths until it does. Where the middle of the coupler happens to travel while all this is going on is nobody’s concern.

In robotics, motion generation mechanisms are used in robot arms, in servo joints, and in automated linkages that must produce accurate and repeatable rotational or oscillatory movement — the swing of a welding head between two positions, the stroke of a painting arm across a panel, the opening and closing of a gripper.

The requirement, in short, is a relationship between two angles. Anything that delivers it correctly is a valid solution.

6.5.2 Path Generation Mechanisms

A path generation mechanism is designed so that one chosen point on one of its links traces a required path or trajectory in the plane. Here the route is the whole requirement, and the orientation of the link carrying the point does not matter.

The point that does the tracing is called the coupler point, because it usually sits on the coupler — the link that is pinned to two moving links and to the frame at neither end. Being attached to nothing fixed, the coupler swings and shifts at once, and a point on it can trace curves of remarkable variety.

Diagram

Fig. 6.5-1 – a four-bar linkage drawn with its frame hatched, crank on the left and rocker on the right; a coupler point marked with a solid dot on the coupler, set clearly to one side of the line joining the coupler’s two pins, and labelled “coupler point”; a dashed closed curve drawn through that dot showing the complete path it traces during one revolution of the crank, labelled “coupler curve”; three faint outlines of the linkage at different crank angles to show how the point moves along the curve.

Fig. 6.5-1 shows the essential trick. Move the coupler point to a different place on the coupler, or change one link length, and the dashed curve changes shape entirely. Straight portions, figures of eight and long flat arcs are all available from the same four links, which is why path generation linkages appear in machines that must guide something along an awkward route without a computer to tell it where to go.

In robotics, path-generating mechanisms guide an end-effector along a defined path in manipulator arms and pick-and-place robots, so that the tool moves smoothly and arrives without stopping at every corner.

treating motion generation and path generation as the same requirement What students do: Describe a four-bar linkage as a path generation mechanism because its links obviously move along paths. What happens: The wrong quantity gets designed for. A linkage tuned to give a 70° output swing may send its coupler point along a curve that fouls the workpiece, and a linkage tuned to trace a curve may give an output swing nothing like the one required. The fix: Ask what the specification actually fixes. If it fixes the relationship between input and output angles, the function is motion generation. If it fixes the route a particular point must follow, the function is path generation.

6.5.3 Force or Torque Transmission Mechanisms

A force or torque transmission mechanism is designed to transfer or amplify force from one link to another within the plane. Its geometry provides mechanical advantage, so that a small input force produces a larger output force.

NoteDefinition

Mechanical advantage is the ratio of the output force produced by a mechanism to the input force applied to it, for a given position of the mechanism.

The phrase “for a given position” is not decoration. In a linkage the mechanical advantage changes as the links move, and in some mechanisms it changes enormously. A toggle mechanism — two links pushed towards a straight line — has a modest advantage while the links are at an angle and an extremely large one as they approach alignment. Presses, riveting tools and crimping pliers all exploit this.

Diagram

Fig. 6.5-2 – a toggle mechanism; a hatched fixed pivot on the left, a hatched vertical guide with a sliding block on the right; two links of equal length joined at a central revolute joint, the left link running from the fixed pivot to that joint and the right link running from the joint to the block; a straight arrow pushing down on the central joint labelled “input force”; a larger straight arrow at the block pushing right labelled “output force”; the two links drawn once at a wide angle and once, faintly, almost in line, with a note reading “output force grows as the links approach alignment”.

The slider-crank of section 6.3.3.1 does the same job in reverse. Rotary motion at the crank becomes a powerful linear thrust at the slider, which is exactly what a mechanical press or a piston pump needs. In robotics, force transmission mechanisms appear in grippers that must hold a part firmly, in actuators, and in pressing tools that need strong, precisely controlled movement.

finding the gripping force produced by a lever-type gripper jaw

Given: One jaw of a robot gripper is a lever pivoted at a fixed pin. A servo pushes on the lever with a force of \(F_{in} = 12\) N at a perpendicular distance of \(d_{in} = 60\) mm from the pivot. The gripping face of the jaw lies at a perpendicular distance of \(d_{out} = 20\) mm from the same pivot.

Find: The force the jaw applies to the object it is holding, and the mechanical advantage of the arrangement.

Method: The jaw is a single rigid link in equilibrium about its pivot, so the moment produced by the input force about the pivot equals the moment resisted at the gripping face. Using \(\tau = F \times d\) with all distances measured perpendicular to the force, \(F_{in} \times d_{in} = F_{out} \times d_{out}\).

Convert the distances to metres: \(d_{in} = 60\ \text{mm} = 0.060\ \text{m}\) and \(d_{out} = 20\ \text{mm} = 0.020\ \text{m}\).

\[\tau = F_{in} \times d_{in} = 12\ \text{N} \times 0.060\ \text{m} = 0.72\ \text{N·m}\]

\[F_{out} = \frac{\tau}{d_{out}} = \frac{0.72\ \text{N·m}}{0.020\ \text{m}} = 36\ \text{N}\]

\[\text{Mechanical advantage} = \frac{F_{out}}{F_{in}} = \frac{36\ \text{N}}{12\ \text{N}} = 3\]

The jaw grips with a force of 36 N, three times the force applied by the servo.

Check: The mechanical advantage should equal the ratio of the two distances, and \(60\ \text{mm} \div 20\ \text{mm} = 3\), which matches. A gripping force of 36 N is roughly the weight of a 3.7 kg mass, which is a believable clamping force for a small gripper and well within what a hobby servo can produce through a 3:1 lever.

Q. If a linkage multiplies force three times, does it also move three times as far?

A. The opposite. The gripping face sits closer to the pivot than the point the servo pushes, so it moves through a shorter distance — one third as far — while producing three times the force. Work is force multiplied by distance, and a linkage cannot create work out of nothing. Whatever a mechanism gains in force it gives up in movement, and a designer who needs both a firm grip and a wide jaw opening has to add a second stage rather than wish the trade away.

One mechanism, three functions

These three functions are not three families of hardware. The same four-bar linkage can be described under all three headings depending on what its designer cared about — the swing of the rocker, the curve traced by a coupler point, or the force delivered at the output. The classification describes the design requirement, not the parts in the box.

Table 6.4 – the three functional classes of planar mechanism | Feature | Motion generation | Path generation | Force or torque transmission | |—|—|—|—| | What the design fixes | The relationship between input and output link positions | The route traced by one chosen point | The force or torque delivered at the output | | What is allowed to vary freely | The path of any individual point | The orientation of the link carrying the point | The exact path, provided the force is delivered | | Quantity the designer works with | Angles of the input and output links | Coordinates of the coupler point through the cycle | Mechanical advantage at the working position | | Typical example | Crank-rocker driving an oscillating output | Four-bar with a coupler point tracing a curve | Toggle mechanism, slider-crank press | | Use in robotics | Servo joints, welding and painting arms, gripper actuation | Guiding an end-effector along a defined trajectory | Gripper jaws, pressing and clamping tools |

6.6 Spatial Mechanisms

6.6.1 Motion in Three-Dimensional Space

Everything so far has been confined to a sheet of paper. Real work is not. A welding torch has to reach behind a bracket and arrive tilted at 30° to the seam. A hand lifting a cup from a shelf rises, moves forward, and turns the cup upright, all at once. No planar mechanism can do any of this.

A spatial mechanism is one whose links and joints move in three-dimensional space rather than in a single plane. Each link can move along all three directions and rotate about three perpendicular axes, so a free link has the full six degrees of freedom established in section 6.1.2.

NoteDefinition

A spatial mechanism is a mechanism in which the motion of the links and joints takes place in three-dimensional space, so that a link may both translate along and rotate about three mutually perpendicular axes.

Diagram

Fig. 6.6-1 – a single rigid link drawn in three-dimensional view at the centre of a set of X, Y and Z axes; three straight double-headed arrows along the three axes labelled “translation along X”, “translation along Y” and “translation along Z”; three curved arrows encircling the three axes labelled “rotation about X”, “rotation about Y” and “rotation about Z”; a caption note reading “six independent motions, and no seventh”.

Fig. 6.6-1 is worth studying for a moment, because it contains the whole difference between the two halves of this chapter. Three of those six arrows survive in a plane. The other three are what a spatial mechanism buys, and what it charges for.

The links of such a mechanism can move in any direction and rotate about several axes at once, producing combinations of motion that are difficult to picture and harder to draw. A robotic manipulator can carry its end-effector forward and backwards, up and down, left and right, and rotate it about its own axes at the same time. This is precisely why spatial machines are described qualitatively at this stage and analysed with matrix methods later.

6.6.2 Joints Used in Spatial Mechanisms

A spatial mechanism is built from rigid links connected by kinematic joints that permit motion in three dimensions. Four kinds do most of the work.

  • Revolute joint (R): allows relative rotation about a single axis.
  • Prismatic joint (P): allows linear sliding along one axis.
  • Spherical joint (S): allows rotation about all three perpendicular axes.
  • Cylindrical joint (C): allows both rotation and translation along the same axis.

The first two are the joints of section 6.1.2, unchanged. What is new is that their axes no longer have to be parallel to one another. A revolute joint whose axis points sideways instead of upwards turns the same link out of the plane, and that single change is what lifts a mechanism into three dimensions.

The other two joints have no planar equivalent at all. A spherical joint is the ball-and-socket arrangement of a hip or a shoulder, or of the tie rod ends under a car: the ball can turn any way it likes inside its socket but cannot be pulled out of it, giving three rotations and no translation. A cylindrical joint is a shaft that is free both to spin inside its housing and to slide along it, like a telescopic aerial that also twists — one rotation and one translation, sharing a single axis.

Diagram

Fig. 6.6-2 – four small three-dimensional sketches in a row, each captioned with its name and letter; revolute joint drawn as a pin through two brackets with a curved arrow about the pin axis; prismatic joint drawn as a square rod inside a square sleeve with a straight double-headed arrow along the rod; spherical joint drawn as a ball seated in a socket with three curved arrows about three perpendicular axes through the ball centre; cylindrical joint drawn as a round shaft inside a sleeve with both a curved arrow about the shaft axis and a straight double-headed arrow along it; beneath each sketch a line giving the degrees of freedom it contributes.

Table 6.5 – joints used in spatial mechanisms | Joint | Letter | Motion allowed | Degrees of freedom | Everyday example | |—|—|—|—|—| | Revolute | R | Rotation about one axis | 1 | Door hinge, motor shaft in a bearing | | Prismatic | P | Sliding along one axis | 1 | Drawer runner, telescopic slide | | Spherical | S | Rotation about all three perpendicular axes | 3 | Hip joint, tie rod end, camera ball head | | Cylindrical | C | Rotation about and sliding along the same axis | 2 | Telescopic aerial that also twists |

Read the table by the fourth column. A designer chooses joints the way a shopper chooses coins: the total must come to the number of degrees of freedom the task needs, and there are several ways to make it up. Six revolute joints in series give six degrees of freedom, and so, in principle, do two spherical joints — but the six revolute joints can each be driven by a motor, while a spherical joint is difficult to actuate directly. That practical difference is why almost every industrial arm is built from revolute joints and why spherical joints appear mostly where the motion is driven from somewhere else.

Spatial mechanisms met in practice include 6R serial robotic arms, Stewart platforms, SCARA robots and the joints of humanoid robots. The SCARA arm on the opening page belongs on that list for a reason worth noticing: its two main joints turn about parallel upright axes, which is a planar arrangement, but the separate vertical axis carries the whole arm up and down, so the tool moves in three-dimensional space.

Hold a marker pen in your hand and demonstrate each of the six independent motions in turn, naming them as you go. Now lay the pen flat on the desk and slide it about without lifting it, and list which of the six you have lost and which remain.

6.6.3 Characteristics of a Spatial Mechanism

Five properties describe what these mechanisms offer and what they demand.

Three-dimensional motion

Each link moves in all three directions, giving broad reach and considerable flexibility. The mechanism can approach a target from above, from the side, or from behind, which a planar mechanism cannot do at all.

Higher degrees of freedom

Spatial mechanisms typically have three to six degrees of freedom, which allows both the position and the orientation of the tool to be controlled. Six is the significant number: with six degrees of freedom, a tool can in principle be placed at any point within its reach and at any angle, since six independent motions match the six a free body has.

Complex geometry and control

Working out where the tool is, given the joint variables, requires three-dimensional transformations and matrix operations rather than the plane trigonometry that served in the earlier sections. Every joint also needs its own actuator, its own driver and its own feedback, so the electronics grow along with the mathematics.

High precision and flexibility

These mechanisms suit operations that demand exact positioning and complex trajectories — following a weld seam around a curved joint, or carrying an instrument along a path chosen to avoid everything else in the way.

Open-chain or closed-chain design

The structural choice of section 6.4 reappears here unchanged. A spatial mechanism may be built as an open chain or as a closed one, and the choice is made on the application’s requirements for speed, accuracy and load-bearing capacity.

A caution about words is needed here, because two quite different ideas share the same phrase. In the study of mechanisms, a closed loop means that the links themselves form a physical loop — the closed chain of section 6.4.2. In control systems, a closed loop means something else entirely: that the controller measures what actually happened and corrects itself. A machine can be an open chain under closed-loop control, and often is. When you meet the phrase, check whether the sentence is about linkages or about feedback.

What it costs

The syllabus asks for the advantages, and they are real, but a designer weighs them against a list that is just as real. More degrees of freedom means more motors, more mass to carry, more cost, more to calibrate and more to go wrong. Every joint adds play, and in a serial spatial arm that play accumulates exactly as it did in the planar case. There are also postures in which a spatial arm loses the ability to move in a particular direction no matter how its joints turn, and these must be found and avoided during design rather than discovered on the factory floor.

The engineering judgement is therefore the same one made at the start of this chapter. Take the third dimension when the task genuinely requires it, and stay in the plane when it does not.

6.7 Types of Spatial Manipulator

Spatial manipulators fall into two main categories, separated by their kinematic structure, their motion characteristics and the way their joints are arranged. These two arrangements determine how the robot moves and what kind of task it performs well in three-dimensional space.

The division will look familiar. It is the open-chain and closed-chain distinction of section 6.4, met again outside the plane, where the consequences are larger.

6.7.1 Serial Manipulators

A serial manipulator consists of a series of links connected end to end by joints, forming an open kinematic chain. The base is fixed, and the end-effector is at the far end with nothing beyond it.

Each joint contributes one or more degrees of freedom, usually rotational or translational, and the combined movement of all the joints produces the motion of the end-effector. Nothing moves the tool by itself; the tool goes where the accumulated effect of every joint puts it.

Photograph

Fig. 6.7-1 – a six-axis industrial robot arm on a factory floor, photographed from the side with the arm extended, so that the base rotation joint, the two large arm joints, and the three compact wrist joints near the tool are all separately visible; a gripper or welding tool should be fitted at the end. Sourcing note: a licensed industrial or stock photograph; images in which the arm is folded so the wrist joints cannot be counted, or in which cabling and enclosures hide the joint housings, are unusable.

Count the joints in Fig. 6.7-1 from the base outward and you find six, each a revolute joint. Three near the base position the tool in space; three at the wrist orient it. This 6R arrangement is the workhorse of industrial robotics precisely because six degrees of freedom is the smallest number that lets a tool be both placed anywhere in reach and turned to any angle.

Serial manipulators are flexible and can reach many points within a large workspace, but they suffer from lower rigidity and from the error accumulation described in section 6.4.1 — now over a longer chain, and with gravity pulling on it from a direction the plane never had to consider.

serial manipulator Working principle: Links are joined end to end in an open chain from a fixed base to a free end. Each joint is driven by its own actuator, and the position and orientation of the end-effector is the combined result of all the joint variables together. Components: Fixed base, a series of rigid links, one revolute or prismatic joint between each pair of links, an actuator and position sensor at every joint, and an end-effector at the free end. Advantages: A large workspace for the space the machine occupies; a tool that can be approached to a target from many directions, including around obstacles; independent control of each joint; and a structure that is simple to build and to extend. Limitations: Lower rigidity, since a single chain of joints carries the whole load; errors that accumulate from base to tool; and each actuator having to carry the weight of every link, joint and motor beyond it, which limits both payload and speed. Where it is used: Welding, painting and assembly on production lines, pick-and-place work, laboratory handling, humanoid robot arms and SCARA assembly machines. Six-revolute-joint industrial arms of this type are supplied by several manufacturers, including ABB, KUKA and FANUC.

6.7.2 Parallel Manipulators

A parallel manipulator consists of several kinematic chains, called limbs, that connect the fixed base to the end-effector platform. Unlike a serial manipulator, where one chain does everything in turn, all the limbs work simultaneously to control the platform’s position and orientation.

The structure offers high stiffness, high accuracy and considerable load-bearing capacity, because the load at the platform is divided among the limbs and travels back to the base along several short paths instead of one long one. In exchange, the workspace is smaller and the control is more complex, since no limb can be commanded without regard to what the others are doing.

Photograph

Fig. 6.7-2 – a delta robot mounted above a moving conveyor in a packing line, viewed from the side or slightly below so that all three limbs are visible running from the fixed overhead frame down to the small moving platform carrying the gripper; products on the conveyor should be in frame. Sourcing note: a licensed photograph of a delta-configuration pick-and-place robot in operation; images that show only the enclosure, or in which fewer than three limbs are visible, are unusable.

Two configurations account for most parallel manipulators in service. A Stewart platform uses six extendable legs to control a top platform in six degrees of freedom, and is used in flight simulators and motion control systems. A delta robot, shown in Fig. 6.7-2, uses three or more parallel arms connected to a common moving platform, and is used for high-speed pick-and-place operations.

Diagram

Fig. 6.7-3 – a Stewart platform; a hatched hexagonal fixed base at the bottom and a smaller hexagonal moving platform above it; six legs drawn between them in three crossed pairs, each leg shown as a two-part extendable strut with a straight double-headed arrow along it; the joints at both ends of each leg marked as small balls and labelled “spherical joint”; a note reading “six legs, six degrees of freedom of the platform”.

Fig. 6.7-3 shows why the arrangement is so stiff. Push down on the platform and every one of the six legs resists, each along its own length, where a strut is at its strongest. There is no long cantilever anywhere in the machine and no joint asked to resist a bending moment on its own.

parallel manipulator Working principle: Several kinematic chains run from a common fixed base to a single moving platform. The actuators, usually mounted on the base, drive the limbs together, and the platform arrives at the position and orientation that satisfies all the limbs at once. Components: Fixed base, two or more limbs each made of links and joints, a moving platform carrying the end-effector, and one actuator per degree of freedom, generally mounted on the base. Advantages: High structural rigidity and positional accuracy; large loads relative to the machine’s own weight; very high speed and acceleration, since the limbs are light and the motors do not move; and errors that do not accumulate, because no chain is long. Limitations: A workspace that is small compared with the space the machine occupies; limbs that can collide with each other or with the workpiece near the edges of that workspace; motion control that must solve for all limbs together; and an inability to reach around obstacles. Where it is used: Flight simulators and motion platforms, high-speed packing and sorting lines, precision positioning stages, and machine tools requiring great stiffness.

Q. If parallel manipulators are stiffer and faster, why is the serial arm still the common one?

A. Because most industrial tasks need reach and access more than they need stiffness. A parallel manipulator’s limbs surround its own workspace, so it cannot put a tool inside a car body or behind a bracket — the limbs would collide with the work. A serial arm reaches into awkward places by folding and unfolding, which is exactly what welding, painting and machine tending demand. The parallel machine wins where the work comes to the robot on a conveyor and the same short movement is repeated all day.

Q5. Some robot arms are a single chain of links, while others hold the tool with three or four arms at once. Which arrangement is steadier, and what does it cost?

A. The multi-limbed one — the parallel manipulator — is steadier. Its load is shared among several short chains instead of travelling down one long one, so it deflects less, its errors do not accumulate, and its motors can sit on the fixed base, leaving the moving limbs light enough to accelerate very hard. The cost is reach. Its workspace is small for the size of the machine, its limbs get in each other’s way near the edges of that workspace, it cannot reach around an obstacle, and no limb can be controlled on its own, so the control problem is considerably harder.

Comparing the two spatial arrangements

Table 6.3 compared open and closed chains as structures. The comparison below takes the same distinction to the level of a working machine, where the questions are about where the motors sit and what the robot can be asked to do.

Table 6.6 – serial versus parallel spatial manipulators | Feature | Serial manipulator | Parallel manipulator | |—|—|—| | Kinematic structure | One open chain from base to tool | Several limbs from base to a shared platform | | Where the actuators sit | On the moving links, one at each joint | Usually on the fixed base | | Mass that has to be accelerated | The links, joints and motors beyond each joint | The limbs only, which can be made light | | Achievable speed and acceleration | Moderate, limited by the mass being carried | High, which is why packing lines use them | | Workspace relative to machine size | Large | Small | | Access to awkward positions | Good; the arm can fold around obstacles | Poor; the limbs surround the working region | | Payload relative to own weight | Lower | Higher | | Typical application | Welding, painting, assembly, machine tending | High-speed pick-and-place, motion simulators, precision stages |

6.8 Choosing Between Planar and Spatial Designs

6.8.1 Comparing Planar and Spatial Mechanisms

A designer rarely asks whether a mechanism should be planar or spatial in the abstract. The question arrives attached to a task, and the task answers it: count the independent motions the job genuinely requires, and build for that number and no more.

Consider a machine that places components onto circuit boards lying flat on a bench. The board never tilts. The components are round or square and can be set down facing any way. Such a task needs two motions across the board and one down onto it — three, and none of the three is a tilt. A spatial arm with six degrees of freedom would do the job, and would spend most of its cost and most of its mass on freedoms the task never uses.

Now change one detail. Suppose some components must be inserted at an angle into a sloping bracket. The task has just acquired a rotation out of the plane, and no planar mechanism can supply it. Everything else about the machine may stay the same; that single requirement has settled the argument.

Diagram

Fig. 6.8-1 – the same pick-and-place task drawn twice, side by side, with a circuit board on a bench in both halves; on the left, a planar arrangement with two revolute joints turning about parallel upright axes plus a separate vertical slide, the tool shown always pointing straight down, captioned “three degrees of freedom, tool orientation fixed”; on the right, a six-revolute-joint spatial arm reaching the same board with its tool tilted at an angle to the surface, captioned “six degrees of freedom, tool orientation free”; both halves labelled with the joint types.

Fig. 6.8-1 puts the trade in one picture. The machine on the left is cheaper, faster, stiffer and easier to program, and it can only ever set a component down flat. The machine on the right can do everything the left-hand one can, and more, at a cost in money, mass, calibration and complexity that is not small.

Table 6.7 – planar versus spatial mechanisms | Feature | Planar mechanism | Spatial mechanism | |—|—|—| | Region of motion | One plane, or planes parallel to it | Three-dimensional space | | Degrees of freedom of a free link | 3 — two translations and one rotation | 6 — three translations and three rotations | | Joints commonly used | Revolute and prismatic, with all revolute axes parallel | Revolute, prismatic, spherical and cylindrical, with axes in any direction | | Method of analysis | Plane geometry and trigonometry, workable by hand | Three-dimensional transformations and matrix operations | | Number of actuators for a typical machine | Few | Many, typically three to six | | Rigidity for a given cost | High, since flat plates resist out-of-plane loads well | Lower, since loads arrive from every direction | | Control of tool orientation | Not available without an added axis | Available | | Typical use | Wipers, gantries, presses, grippers, planar arms | Industrial arms, humanoid joints, motion platforms | | Main limitation | Cannot reach out of the plane or tilt the tool | Cost, mass, calibration effort and control complexity |

Q1. Why would a designer deliberately stop a robot arm from moving out of one plane, instead of giving it freedom in every direction?

A. Because every freedom that is removed is a freedom that no longer has to be paid for, held up or controlled. Confining an arm to a plane means its structure only has to resist loads in that plane, so it can be made stiff cheaply; it needs fewer motors, so it is lighter and faster; and its geometry can be worked out with plane trigonometry instead of three-dimensional transformations. The refusal to move is itself useful: an arm that cannot be pushed out of its plane will not sag out of it either, which is exactly what a machine pressing a part into a hole needs.

This is also why planar mechanisms are studied first. They serve as simplified models for understanding motion transmission, link geometry and joint coordination, and every one of those ideas carries directly into three dimensions, where the geometry is harder but the reasoning is unchanged.

6.8.2 Applications in Robotics

Both kinds of mechanism are in service, usually in the same building and often in the same machine.

Where planar mechanisms are found

Planar arrangements do the work that lives on a surface. Gantry machines — computer numerical control routers and 3D printers — move a tool over a bed using perpendicular prismatic axes. Grippers close their jaws through planar linkages. Conveyor diverters, sliding doors and linear sorting mechanisms are planar. So are the wiper linkages, hand pump levers and press toggles met earlier in this chapter. Wherever a task is flat, a planar mechanism is the economical answer.

Industrial automation

Spatial manipulators do the assembly, welding, painting and material handling on production lines, because parts arrive at angles and tools must be presented to them squarely. A single arm may be asked to weld a seam that curves in three dimensions, which requires both position and orientation to change continuously along the path.

Medical robotics

In surgical systems such as the da Vinci system, precise three-dimensional motion allows instruments to be moved through small openings and worked inside the body, where the surgeon’s hand movements at a console are converted into much smaller movements of the instrument tips.

Photograph

Fig. 6.8-2 – the instrument arms of a surgical robot positioned over an operating table in a theatre, viewed so that the separate arms and the slender instrument shafts are clearly distinguishable; the surgeon’s console may appear in the background. Sourcing note: a licensed clinical or manufacturer photograph; images showing surgical wounds, blood or identifiable patients or staff are unusable.

Aerospace and remote handling

Spatial manipulators are used for satellite servicing, for exploration tasks where a machine works in a place a person cannot go, and for remote handling of hazardous objects, including bomb disposal. What all of these share is distance: a person operates the machine from somewhere safe, and the manipulator supplies the reach and the dexterity that the person’s own arms cannot.

Simulation systems

Flight simulators and motion platforms use parallel spatial mechanisms to reproduce movement realistically. Six extendable legs beneath a cabin can tilt, lift and swing it in every direction, which is why a simulator can convince a trainee pilot that an aircraft is banking while the whole machine stays inside a room.

3D printing and machining

Accurate tool positioning and orientation in manufacturing rests on mechanisms of both kinds. A simple printer is a planar gantry with a third axis; a five-axis machining centre is a spatial mechanism, able to tilt the tool so that a curved surface can be cut in one setup instead of several.

Humanoid and service robots

Where a machine must move in a lifelike way among people, three-dimensional motion is unavoidable. Shoulders, wrists and necks are spatial joint arrangements, and a humanoid arm is a serial spatial manipulator with the same strengths and the same accumulated errors as the industrial arm on a factory floor.

Look closely at any of these machines and the same short list of ideas keeps appearing: links held rigid, joints chosen for the motion they permit, chains left open for reach or closed for stiffness. The rest of this course builds on that list rather than replacing it.

6.9 Summary

Planar mechanisms
  1. A planar mechanism is one in which all links move in a single plane or in planes parallel to it, so the whole motion can be described in two dimensions. Every revolute axis is perpendicular to the plane of motion; every prismatic sliding direction lies in it.
  2. Its four defining features: motion described in two dimensions; no link having more than three degrees of freedom; joints that are turning pairs and sliding pairs; and behaviour governed by the relative lengths of the links.
  3. Planar motion is what a door hinge, a bicycle pedal and crank, a windscreen wiper linkage and a hand pump lever all have in common, however different they look.
  4. Planarity is a property of the motion, not of the parts. Flat links with one skewed axis do not make a planar mechanism.
Classification by motion
  1. Purely rotational mechanisms use revolute joints only. Every point on a moving link travels a circular arc, adjacent links move angularly relative to one another, and in a single-degree-of-freedom system one angular input fixes the whole configuration. Examples: the four-bar linkage and the crank-rocker mechanism.
  2. The four-bar linkage has four links — frame, crank, coupler and rocker — joined by four revolute joints in a closed loop. Continuous rotation of the crank becomes a controlled oscillation of the rocker.
  3. Grashof’s condition decides whether continuous rotation is possible: with \(s\) the shortest link, \(l\) the longest and \(p\) and \(q\) the other two, at least one link can rotate fully when \(s + l \le p + q\). If the shortest link is attached to the frame, it is the crank and the linkage is a crank-rocker.
  4. Purely translational mechanisms use prismatic joints only. Every link keeps its orientation, motion is along parallel straight lines, and every point on a link has the same velocity and acceleration. Examples: the sliding table or cross-head mechanism, and the Cartesian manipulator, whose workspace is a plain rectangle and whose commands are simply the target coordinates.
  5. Combined rotational-translational mechanisms contain both revolute and prismatic joints, so some links turn while others slide. Examples: the slider-crank mechanism, which converts between rotary and reciprocating motion, and the planar robotic arm with mixed joints, in RP or PR form, whose two joint variables are an angle and a length.
  6. A slider-crank does not deliver uniform speed. The slider stops at each end of its stroke and is fastest near mid-stroke, even when the crank turns steadily.
Classification by structural arrangement
  1. An open-chain mechanism has its links in series with one end fixed and the other free. Every joint adds a degree of freedom, motion is transmitted outward from the base, and errors accumulate along the chain onto the end-effector. A robot arm built this way is a serial manipulator; a 2R planar manipulator is the simplest useful case, and a 3R adds reach and dexterity.
  2. A closed-chain mechanism has its links forming one or more loops. Motion travels along several interconnected paths, the loop constrains the links to one another, and a single input determines the position of every link. The four-bar linkage and the slider-crank are closed chains.
  3. Closing the loop takes freedom away rather than adding it: four revolute joints in an open chain give four degrees of freedom, while the same four joints in a four-bar loop give one.
  4. The trade is reach against rigidity. Open chains give a large workspace, good access and simple control, with lower stiffness and accumulated error. Closed chains give stiffness, accuracy and load capacity, with a smaller workspace and interdependent, harder control.
Classification by function
  1. A motion generation mechanism fixes the relationship between the input and output link positions, and does not care what path any individual point follows.
  2. A path generation mechanism fixes the route traced by one chosen point, the coupler point, and does not care about the orientation of the link carrying it.
  3. A force or torque transmission mechanism delivers mechanical advantage — the ratio of output force to input force at a given position — so that a small input force produces a larger output. Toggle mechanisms and slider-crank presses do this.
  4. Mechanical advantage is bought with movement, never for nothing: a jaw that grips three times as hard moves one third as far.
  5. The same four-bar linkage can be described under all three headings. The classification names the design requirement, not the hardware.
Spatial mechanisms
  1. A spatial mechanism is one whose links and joints move in three-dimensional space, so that a link may translate along and rotate about three mutually perpendicular axes.
  2. Four joints do most of the work: the revolute joint (R), allowing rotation about a single axis; the prismatic joint (P), allowing sliding along one axis; the spherical joint (S), allowing rotation about all three perpendicular axes; and the cylindrical joint (C), allowing both rotation and translation along the same axis. They contribute one, one, three and two degrees of freedom respectively.
  3. Its characteristics: three-dimensional motion giving broad reach; higher degrees of freedom, typically three to six, allowing control of both position and orientation; complex geometry and control requiring three-dimensional transformations and matrix operations; high precision and flexibility for exact positioning and complex trajectories; and a structure that may be open-chain or closed-chain according to the demands for speed, accuracy and load capacity.
  4. Examples include 6R serial robotic arms, Stewart platforms, SCARA robots and humanoid robot joints.
  5. “Closed loop” means two different things. In mechanisms it means the links form a physical loop; in control it means the controller measures the result and corrects itself.
Spatial manipulators
  1. A serial manipulator is an open chain of links from a fixed base to a free end, with each joint contributing its own degrees of freedom and the end-effector’s motion being the combined result. It offers a large workspace and good access, at the cost of rigidity, accumulated error, and every motor having to carry everything beyond it.
  2. A parallel manipulator connects a fixed base to a moving platform through several limbs working simultaneously. Its actuators usually sit on the base, so the moving parts are light and fast, and the shared load gives high stiffness, accuracy and payload — but the workspace is small, the limbs can collide, and no limb can be controlled alone.
  3. Stewart platforms control a top platform in six degrees of freedom using six extendable legs, and are used in flight simulators and motion control systems. Delta robots use three or more parallel arms on a common moving platform for high-speed pick-and-place work.
Choosing between them
  1. Count the independent motions the task genuinely needs. A task that lives on a surface and never tilts its tool needs a planar mechanism; the moment the tool’s orientation must change out of plane, a spatial mechanism is unavoidable.
  2. Every freedom removed is one that need not be built, carried, calibrated or controlled — and an arm that cannot move out of its plane cannot sag out of it either.
  3. Planar mechanisms are studied first because they are the simplified models on which motion transmission, link geometry and joint coordination are understood, and that reasoning carries unchanged into three dimensions.
  4. Both kinds are in service together: planar gantries, grippers and sorting mechanisms alongside spatial arms in industrial automation, medical robotics, aerospace and remote handling, simulation systems, machining and 3D printing, and humanoid and service robots.

6.10 Key Terms

  1. Cartesian manipulator — A robot built entirely from prismatic joints along mutually perpendicular axes, so that the target coordinates are the commands and the tool keeps a fixed orientation.
  2. Closed-chain mechanism — A mechanism whose links form one or more closed loops, so that motion travels along several paths and one input fixes the position of every link.
  3. Coupler — The link of a four-bar linkage that joins the crank to the rocker and is attached to the frame at neither end.
  4. Coupler point — A chosen point on the coupler whose path through the cycle is the object of a path generation design.
  5. Crank — The input link of a linkage, driven by the actuator; in a crank-rocker it rotates continuously.
  6. Crank-rocker mechanism — A four-bar linkage proportioned so that the crank rotates fully while the output link swings back and forth through a limited angle.
  7. Cylindrical joint (C) — A joint permitting both rotation about and sliding along the same axis, contributing two degrees of freedom.
  8. Degrees of freedom (DOF) — The number of independent motions a body or system can make; three for a body in a plane and six for a body in space.
  9. End-effector — The part at the free end of a robot chain that acts on the world, such as a gripper, suction cup or welding torch.
  10. Four-bar linkage — A closed chain of four links — frame, crank, coupler and rocker — joined by four revolute joints.
  11. Grashof’s condition — The rule that continuous relative rotation is possible in a four-bar linkage when the sum of the shortest and longest link lengths does not exceed the sum of the other two.
  12. Joint — A kinematic pair as built into a machine, permitting certain relative motions between two links and preventing the rest.
  13. Joint variable — The single quantity that describes how far a joint has moved: an angle for a revolute joint, a displacement for a prismatic joint.
  14. Kinematic chain — A set of links connected by joints so that their relative motions are constrained.
  15. Kinematic pair — Two links connected so that they can move relative to each other in a definite way.
  16. Link — A body treated as rigid, whose points keep fixed distances from one another while it carries motion and force between joints.
  17. Mechanical advantage — The ratio of the output force a mechanism produces to the input force applied to it, at a given position of the mechanism.
  18. Mechanism — A kinematic chain with one link fixed, so that a given input produces a definite and repeatable output motion.
  19. Open-chain mechanism — A mechanism whose links are joined in series without a loop, with one end fixed to a base and the other end free.
  20. Parallel manipulator — A robot in which several limbs connect a fixed base to a single moving platform, all working at once to control the tool.
  21. Planar mechanism — A mechanism in which all links move in one plane or in planes parallel to it, so the motion can be described in two dimensions.
  22. Prismatic joint (P) — A joint permitting sliding along one straight axis and nothing else, contributing one degree of freedom.
  23. Revolute joint (R) — A joint permitting rotation about one fixed axis and nothing else, contributing one degree of freedom.
  24. Rocker — The output link of a four-bar linkage, which swings back and forth about its own fixed pivot.
  25. SCARA — Selective Compliance Assembly Robot Arm; an assembly robot whose main joints turn about parallel upright axes, with a separate vertical axis for the tool.
  26. Serial manipulator — A robot arm built as an open chain, in which the end-effector’s motion is the combined result of all the joints in series.
  27. Slider-crank mechanism — A planar mechanism of crank, connecting rod, slider and frame that converts between rotary and reciprocating straight-line motion.
  28. Spatial mechanism — A mechanism whose links and joints move in three-dimensional space, translating along and rotating about three perpendicular axes.
  29. Spherical joint (S) — A joint permitting rotation about all three perpendicular axes and no translation, contributing three degrees of freedom.
  30. Workspace — The region of space a manipulator’s end-effector can reach.

6.11 Objective Questions

  1. Which of these joints permits rotation about all three mutually perpendicular axes and no translation at all?
  1. Revolute joint
  2. Prismatic joint
  3. Spherical joint
  4. Cylindrical joint
  1. A four-bar linkage is built with link lengths 25 mm, 60 mm, 55 mm and 70 mm, with the 70 mm link fixed as the frame and the 25 mm link attached to it and driven by a motor. What will the mechanism do?
  1. The shortest link will rotate continuously while the opposite link swings back and forth
  2. No link will rotate fully, and every moving link will only rock
  3. Only the coupler will rotate fully, while the driven link stays still
  4. All four links will rotate continuously about their own pivots
  1. Assertion (A): A four-bar linkage needs only one motor, even though it contains four joints. Reason (R): Closing a kinematic chain into a loop constrains the links to one another, so the mechanism has fewer degrees of freedom than its joint count suggests.
  1. Both A and R are true, and R is the correct explanation of A
  2. Both A and R are true, but R is not the correct explanation of A
  3. A is true, but R is false
  4. A is false, but R is true
  1. Assertion (A): A delta robot can accelerate its tool much harder than a serial arm of comparable size. Reason (R): A delta robot is usually mounted on a frame above the conveyor rather than beside it.
  1. Both A and R are true, and R is the correct explanation of A
  2. Both A and R are true, but R is not the correct explanation of A
  3. A is true, but R is false
  4. A is false, but R is true
  1. Match each item in Column A with the description in Column B that fits it.
Column A Column B
a. Cylindrical joint i. Six extendable legs controlling a platform in six degrees of freedom
b. Cartesian manipulator ii. The point whose traced curve is the object of the design
c. Coupler point iii. Rotation about and sliding along the same axis
d. Stewart platform iv. A rectangular workspace, with the target coordinates as the commands
  1. Fill in the blanks.
  1. In a crank-rocker mechanism, the link driven by the motor is the ______, and the link that swings through a limited angle is the ______.
  2. A rigid body has ______ degrees of freedom when moving in a plane and ______ degrees of freedom when moving in space.
  3. A mechanism whose links form one or more loops is called a ______ mechanism.
  1. Give the full form of DOF and of SCARA, and name the joint represented by each of the letters R, P, S and C.

6.12 Check Your Concepts

  1. Define a planar mechanism. Explain why the words “or in planes parallel to it” belong in the definition, using a machine of your choice as an example.

  2. Differentiate between a revolute joint and a prismatic joint on at least three points, giving one everyday example of each.

  3. Explain the working of a slider-crank mechanism with the help of a labelled sketch. Name its four parts, state which of its joints are revolute and which is prismatic, and give one limitation of the mechanism.

  4. Give reasons: the positioning error of a serial manipulator is far larger at the end-effector than at the base, even when every joint is made to the same standard of accuracy.

  5. One jaw of a robot gripper is a lever pivoted at a fixed pin. A servo pushes on the lever with a force of 15 N at a perpendicular distance of 75 mm from the pivot, and the gripping face lies 25 mm from the same pivot. Calculate the gripping force and the mechanical advantage, and state what the arrangement gives up in return for the extra force.

  6. Compare open-chain and closed-chain arrangements under the headings rigidity, workspace, error accumulation and control complexity. Then justify which of the two you would choose for a machine that must repeat the same short movement several thousand times an hour inside a fixed area, and say what your choice makes difficult.

6.13 Diagram, Circuit & Output Reading

  1. A workshop team proposes the four-bar linkage described in the table below. The 120 mm link is bolted to the bench and the 40 mm link is driven by a motor.

Table 6.8 – proposed link lengths for a four-bar linkage | Link | Length | Intended role | |—|—|—| | Link 1 | 120 mm | Fixed to the bench | | Link 2 | 40 mm | Driven by the motor | | Link 3 | 100 mm | Joins link 2 to link 4 | | Link 4 | 90 mm | Carries the tool |

  1. Apply Grashof’s condition to these lengths, showing your working.
  2. State whether link 2 will rotate continuously, and name the type of four-bar this is.
  3. The team now replaces link 3 with a 55 mm link and keeps everything else the same. Test the condition again and predict what the motor will do when it is switched on.
  1. A machine is built as follows. Three revolute joints are connected in series, and the axes of all three are vertical and parallel to one another. A prismatic joint is then added at the end of the chain, sliding in the vertical direction and carrying the tool.
  1. State how many degrees of freedom the machine has, and give your reasoning.
  2. Are the three revolute joints alone a planar or a spatial arrangement? Justify your answer by referring to the joint axes.
  3. State what the prismatic joint adds that the three revolute joints could not supply, and name one thing this machine still cannot do.
  1. A student’s drawing of a planar four-bar linkage shows the frame hatched at the bottom, a crank and a rocker rising from its two ends, and a coupler joining them. Three of the four joints are drawn as pin joints. The fourth, where the coupler meets the rocker, is drawn as a ball seated in a socket.
  1. Identify the single fault in the drawing.
  2. Explain what would happen to the motion of the linkage if it were actually built this way.
  3. State the correction needed, and give the reason in one line.
  1. A packing company is choosing between two machines for a fixed workstation. Their published figures are given below.

Table 6.9 – comparison of two candidate machines, typical manufacturer figures | Specification | Machine A | Machine B | |—|—|—| | Structure | Open chain, six revolute joints | Three limbs from a fixed overhead base to a moving platform | | Reach | 900 mm | 800 mm diameter working circle | | Payload | 6 kg | 1 kg | | Repeatability | ±0.05 mm | ±0.10 mm | | Maximum picks per minute | 60 | 150 | | Tool orientation | Free in all directions | Limited to rotation about the vertical |

  1. Name the structural class of each machine, giving the feature in the table that identifies it.
  2. The task is to lift 400 g cartons off a conveyor at 120 picks per minute and set them upright in a tray. State which machine fits, quoting the two figures that decide it.
  3. Explain why the machine you chose in part (b) is restricted to a light payload, referring to how its limbs must be built if it is to reach that many picks per minute.

6.14 Case-Based Questions

  1. A school robotics club is building a machine to press a school crest onto flat card. A servo motor turns a crank through complete revolutions. A connecting rod runs from the crank pin to a slider that carries the stamp, and the slider runs in a vertical guide above the bench. The card lies flat on the bench, and the stamp must always come down square to the card.
  1. Name the mechanism the club has built, classify it under the motion-based classification of planar mechanisms, and state which of its joints are revolute and which is prismatic.
  2. The servo turns at a steady rate, but a member of the club notices that the stamp does not descend at a steady speed. Explain why this happens and state where in the stroke the stamp is moving fastest.
  3. The club wants roughly three times the stamping force without changing the servo. Suggest one change to the mechanism that would achieve this, and state clearly what the machine gives up in exchange. Add one precaution the club should follow while the machine is running on the bench.
  1. A district warehouse is automating one aisle. Small identical boxes arrive flat on a conveyor and must be lifted and set down flat in cartons alongside, at 100 boxes per minute. The boxes are never tilted, and the tool never needs to face in any direction other than straight down. The floor area available above the conveyor is limited, and the aisle floor must stay clear for staff to walk through.
  1. List the independent motions this task actually requires, and use your list to decide whether a planar or a spatial arrangement is the smaller sufficient choice.
  2. The warehouse is offered a six-revolute-joint serial arm mounted on the floor beside the conveyor, and a three-limbed parallel robot mounted on a frame above it. Recommend one, giving three reasons drawn from the situation described, and state one disadvantage your choice brings with it.
  3. Six months later the warehouse begins handling a second product that must be inserted at an angle into a sloping tray. Explain what this new requirement changes about the recommendation in part (b), and give the reason in terms of degrees of freedom.