Robot arm: move the joints, how it works step by step

What the model shows

Watch a car body go down a welding line and you will see orange arms dip in, touch a seam and pull back, hundreds of times an hour. Each of those arms has to put a tool at an exact point in space, pointing in an exact direction. The motors cannot be told "go there". They can only be told an angle. This page is about the arithmetic in between.

The model is a medium-sized industrial arm with six revolute joints, bolted to the floor between two stands. The shoulder axis is 520 mm above the floor and 160 mm in front of the base axis, the upper arm is 620 mm long, the forearm 600 mm, and the tool centre point (the spot between the fingertips) is 240 mm beyond the wrist. These are model values chosen to look and behave like a typical arm of this class, not the figures of a particular product.

You can drag the blue ball and the arm will follow, or use the six joint sliders and watch the ball follow the arm. The readout in the corner shows all six angles, the position of the tip in millimetres, how fast the tip is moving and which joint is turning fastest.

Why an arm needs six joints

A rigid object in space has six independent ways to move. It can slide in three directions and turn about three axes. A gripper that has to pick a part from any position, at any angle, needs control over all six, so a general-purpose arm has six joints.

The first three joints do most of the travelling. J1 turns the whole arm about the vertical axis, J2 leans the upper arm at the shoulder and J3 tilts the forearm at the elbow. Between them they decide where the wrist goes. The last three joints sit in the wrist. J4 rolls the forearm about its own length, J5 bends the wrist and J6 rolls the flange that carries the tool. They decide which way the tool points.

In this model, as in most industrial arms, the axes of J4, J5 and J6 all pass through one point called the wrist centre. That design choice turns out to matter a great deal, as the next sections show.

From joint angles to the tip position

Going from angles to a position is called forward kinematics, and it is the easy direction. Each joint rotates everything beyond it, so you can chain the rotations together, starting at the base.

In the vertical plane of the arm, measure distances out from the J1 axis (call it r) and up from the floor (h). With J2 measured from upright and J3 measured from a level forearm, the wrist centre sits at

• r = 160 + 620 sin J2 + 600 cos(J2 + J3)
• h = 520 + 620 cos J2 − 600 sin(J2 + J3)

all in millimetres. At the starting pose J2 is 15° and J3 is 0°, which gives r = 160 + 160.5 + 579.6 = 900.0 mm and h = 520 + 598.9 − 155.3 = 963.6 mm. J5 is 75°, which adds to the 15° downward lean of the forearm to point the gripper straight down, so the tip is 240 mm lower, at 723.6 mm. J1 then swings that whole plane around the vertical axis.

The wrist joints add three more rotations on top. The model does this with ordinary 3 × 3 rotation matrices, one per joint, multiplied in order. Move any slider and the page simply runs this chain again.

Working backwards from the tip

Inverse kinematics is the other direction: given where the tip must be and which way the gripper must point, find the six angles. That is what a robot controller needs every time it plans a straight-line move, and it is much harder. The equations are non-linear, there may be several answers, and there may be none.

There are two broad ways to solve it. A numerical solver starts from the current angles, works out how a small change in each joint would move the tip (the Jacobian matrix), takes a step that reduces the error and repeats until the error is tiny. That works for almost any arm. For arms whose last three axes meet in one point, there is also a closed-form solution that needs no iteration, and the model uses that.

Splitting the problem at the wrist centre

Because the three wrist axes cross at one point, turning J4, J5 or J6 never moves that point. So the wrist centre can be found before any joint angle is known: start at the target tip and step 240 mm backwards along the direction the gripper should point.

Now the problem falls into two smaller ones.

1. Position. J1 must face the wrist centre, so J1 is simply its compass bearing seen from above. In the arm's plane, the shoulder, the elbow and the wrist centre form a triangle with two known sides (620 and 600 mm) and a third side D, the measured distance from shoulder to wrist centre. The law of cosines gives the elbow angle: sin J3 = (620² + 600² − D²) ÷ (2 × 620 × 600). J2 is then the direction to the wrist centre minus the angle the elbow bend adds.
2. Orientation. With J1 to J3 known, the rotation they produce is known. Whatever is left over between that and the required gripper rotation has to come from the wrist, and three wrist angles can be read straight out of that leftover rotation matrix.

Take the grasp position on the left-hand stand: tip at 620 mm out, 285 mm up and 600 mm to the side, gripper pointing down. The wrist centre is 240 mm above it, at 525 mm. Seen from above it lies 862.8 mm from the J1 axis on a bearing of 44.06°, so J1 = −44.06°. In the arm's plane r = 702.8 mm and h = 5 mm, so D = 702.8 mm. Then sin J3 = 0.3366 and J3 = 19.67°. The direction to the wrist centre is 89.59° from upright and the elbow bend accounts for 53.50°, so J2 = 36.09°. To point the gripper down, J5 must make up the rest of 90°: 90 − 36.09 − 19.67 = 34.24°.

Checking the solver both ways

Numbers like these are easy to get subtly wrong, so the page's solver was tested with the same method the browser uses. The test picks 20,000 random sets of joint angles inside the joint limits, computes the tip position and gripper rotation with forward kinematics, feeds those back into the inverse solver and runs forward kinematics again on the answer. Sets within about 1° of J5 = 0 are left out, for a reason explained below. For the remaining 19,836 the worst tip error was 7 × 10⁻¹² mm, and every set came back to its original angles to within 10⁻¹³ radian.

The browser makes the same check on every frame as you drag: the readout's tip position comes from running forward kinematics on the angles the solver chose, so if the ball and the readout ever disagree, something is wrong.

How far the arm can reach

The wrist centre can never be more than 620 + 600 = 1,220 mm from the shoulder axis, when the arm is fully stretched. With the gripper held level that puts the tip at most 1,620 mm from the J1 axis. Drag the ball beyond that and the arm stretches as far as it can toward it. The ball turns red and the readout says how many millimetres short the tip is. Ask for a point 2 m away at shoulder height and the tip stops at 1,617 mm, 386 mm short.

The reach is not a perfect shell, because the joints have limits. In the model J2 can lean 60° back and 100° forward, and J3 can go 85° up and 80° down. Turn on Show reach to see the region in the arm's plane where the fingertips can go with the gripper at its current angle. Change the gripper angle and the whole shape shifts, because the tip sits 240 mm from the wrist centre in a different direction. The ring on the floor marks the 340° that J1 can sweep.

What goes wrong when the wrist lines up

Look again at the orientation step. The wrist angles are read from a rotation that the three wrist joints have to produce. When J5 is exactly zero, the J4 and J6 axes lie on one line. Turning J4 forward and J6 back by the same amount then leaves the gripper exactly where it was, so there are endless pairs of angles that do the same job, and the solver cannot tell them apart. This is called a wrist singularity. That is also why the round-trip test leaves out angle sets very close to J5 = 0: the solver returns a different, equally valid pair, so the angles cannot be compared.

The real trouble comes near the singularity rather than at it. Step 6 of the guide holds the gripper level, pointing straight out, and slides the tip 440 mm sideways, 1,155 mm above the floor, reaching 250 mm/s at the middle of the run. That path passes 20 mm from the line where the forearm points exactly along the gripper. Far from the crossing the wrist barely moves. Close to it, J4 has to roll the J5 bend from one side to the other, and J6 has to undo the roll so the gripper does not twist. Over the whole run J4 goes from −81.2° to +81.2°, and 39.2° of that happens while the tip covers the middle 20 mm. At its fastest J4 has to turn 2.04° for every millimetre of tip travel, which at 250 mm/s means 510° per second. At 1,000 mm/s it would be about 2,040° per second, and a path 10 mm closer roughly doubles both figures.

No motor can follow that, so a real controller has to slow the tip down, stop with an error, or let the path bend away from the straight line. Robotics textbooks give two revolute joint axes lining up as a basic case of a singularity: in that pose the arm loses the ability to move the tool in some direction. The cure is planning: tilt the gripper a few degrees or move the path so the wrist never comes close.

Moving a block from one stand to the other

The pick-and-place cycle in the model has eight moves. A joint move takes the arm above the block. Then three straight-line moves follow: down to the grasp height, back up after the fingers close, and 1.2 m across to the other stand. After that it lowers again, opens the fingers and lifts clear.

Every straight-line move is planned by asking the inverse solver for the angles at each frame. The gripper's rotation stays fixed in space throughout, so as J1 swings from −44.06° to +44.06°, J6 turns by the same 88° to keep the block from spinning in the fingers.

Each move speeds up and slows down smoothly. With that profile the top speed is 1.5 times the average, so a move of length L planned for a top speed v takes 1.5 L ÷ v. At 1 m/s the 1.2 m transfer takes 1.8 seconds.

Sharing the floor with people

A conventional industrial arm works inside a fence, and opening the gate stops it. Collaborative robots, or cobots, are meant to work next to people, and the safety rules for them set numbers you can check.

ISO 10218-1, the safety standard for industrial robots, says that under reduced speed control the speed of the tool centre point shall not exceed 250 mm/s. Switch on Collaborative mode and the model caps the tip at that figure; the 1.2 m transfer now takes 7.2 seconds instead of 1.8.

ISO/TS 15066 describes four ways people and robots can share space: a safety-rated monitored stop, hand guiding, speed and separation monitoring, and power and force limiting. For power and force limiting it lists how hard a robot may press on each body region. For the hands and fingers the limit is 140 N when the hand is clamped against something (quasi-static contact), and twice that, 280 N, for a brief knock where the hand can move away (transient contact). The table lists 65 N for the face, but the same document says contact with the face, skull and forehead is not permissible at all.

The same document turns force limits into speed limits. Model the hand as a spring with a stiffness of 75 N/mm and a mass of 0.6 kg, and the robot as an effective mass equal to half its moving parts plus the payload. The highest safe speed is then v = F ÷ √(μk), where μ is the reduced mass of the two. For an arm with 30 kg of moving parts carrying 2 kg, the effective mass is 17 kg, μ is 0.58 kg and the transient limit for the hand works out at 1.34 m/s. A heavier arm or payload brings that down, which is one reason cobots are light. These are example figures for this model; the standard requires a risk assessment with the real robot's values.

Where the model is simpler than a real arm

• The joints move exactly where they are told. Real joints have reduction gearboxes, often strain wave gears, with a little backlash and elasticity, and real links bend under load.
• There is no dynamics. A real controller also has to limit acceleration, torque and motor current, and slows down near singularities and joint limits.
• The solver always picks the answer nearest the current angles. The geometry allows up to eight answers for one tip pose: J1 facing the point or turned away from it, the elbow up or down, and the wrist flipped or not. The model only uses the elbow-up family.
• Collisions between the arm, the stands and the floor are not checked, apart from keeping the target above the floor.
• The cobot mode only caps speed. It does not measure contact force.

A short history

George Devol filed his patent for a programmable transfer arm in 1954, and it was granted in 1961. The machine built from it, Unimate, went to work on a General Motors line in Ewing Township, New Jersey, in 1961. In 1969 Victor Scheinman at Stanford University designed the Stanford arm, an all-electric arm with six degrees of freedom whose links were laid out so the arm solutions were quick to compute. In 1973 KUKA built its first robot, FAMULUS, and ASEA introduced the IRB 6, the first fully electric, microprocessor-controlled industrial robot; the first one went to Magnussons in Genarp, Sweden, to wax and polish steel tubes bent at 90°. Cobots came later: J. Edward Colgate and Michael Peshkin of Northwestern University invented them in 1996.

A two-joint example with two answers

The six-axis solver is easier to follow if you first do the same job with two joints. Take a flat arm with a 620 mm upper link and a 600 mm forearm, the lengths of the model, and put its shoulder at the origin. The angles here are the ordinary planar ones, so they are not the same numbers as the J2 and J3 sliders: the shoulder angle is measured from the horizontal, and the elbow angle is how far the forearm bends away from the straight line of the upper link.

Say the wrist must reach a point 800 mm out and 300 mm up. Its distance from the shoulder is D = √(800² + 300²) = 854.4 mm. The law of cosines gives the elbow bend:

• cos(elbow) = (D² − 620² − 600²) ÷ (2 × 620 × 600) = (730,000 − 384,400 − 360,000) ÷ 744,000 = −0.01935

so the elbow bends by 91.11°. A cosine does not say whether the angle is positive or negative, and that is the whole reason for two answers. With a bend of −91.11° (elbow up) the shoulder angle is 65.15° and the elbow sits at (260, 563) mm, above the line from shoulder to wrist. With +91.11° (elbow down) the shoulder angle is −24.04° and the elbow sits at (566, −253) mm, below it. Put either set of angles back into the forward equations and you get exactly (800, 300). Both arms are real, both are valid, and a controller has to pick one. The six-axis arm adds two more sources of doubling, J1 facing away and the wrist flipped, which is how one tool pose can have up to eight solutions.

Which one to pick is a practical matter. The elbow-down answer here would put the elbow 253 mm below the shoulder, which on a floor-mounted arm may mean hitting the base or the table. Controllers usually choose the solution closest to the current angles, so the arm does not suddenly swing its elbow through the other side in the middle of a move. The model does the same, and that is why you never see it flip.

How much torque each joint needs

The joints do not all work equally hard. Gravity pulls on every link, and the pull on a joint is the weight times the horizontal distance from that joint to the weight. The shoulder carries everything beyond it, at the longest distances.

Here is a rough estimate with assumed masses, since the model has none. Say the upper arm is 20 kg with its centre 310 mm from the shoulder, the forearm with its wrist motors is 12 kg with its centre 300 mm beyond the elbow, and the gripper holds a 6 kg block at the stretched-out reach of 1.22 m. With the whole arm held horizontal:

• Upper arm: 20 × 9.81 × 0.31 = 61 N·m
• Forearm: 12 × 9.81 × 0.92 = 108 N·m
• Load: 6 × 9.81 × 1.22 = 72 N·m
• Shoulder total: about 241 N·m

The elbow has to support only what lies beyond it: 12 × 9.81 × 0.30 + 6 × 9.81 × 0.60 = 71 N·m. The wrist carries far less, which is why wrist motors are small and sit back on the forearm. This also shows why the shoulder is the biggest motor on the arm, and why the same arm lifts much less when stretched out than when the load is held close to the base. At a quarter of the distance, the same block would cost the shoulder a quarter of the torque.

A servo motor cannot produce 241 N·m directly. Divide by the gearbox ratio and its efficiency. With a ratio of 120 and an assumed efficiency of 80 percent, the motor needs 241 ÷ (120 × 0.8) = 2.5 N·m, about what a motor of a few hundred watts can deliver. The cost is speed: a motor turning at 3,000 rpm moves the joint at 25 rpm, or 150° per second.

Why the motors need a reducer

That ratio is why almost every joint on a real arm has a reduction gearbox next to the motor. A motor is light and fast but weak. A gearbox trades speed for torque, and in a robot it also has to be stiff, have almost no play and fit in a joint. Two designs dominate.

A strain wave gear, sold under the name Harmonic Drive, has three parts: an elliptical wave generator on the motor shaft, a thin flexible steel cup with outside teeth (the flexspline), and a rigid ring with inside teeth (the circular spline). The circular spline has two more teeth than the flexspline. As the ellipse turns, the flexspline's teeth engage the ring at two opposite regions, and every half turn of the input moves the flexspline back by one tooth. A flexspline with 200 teeth therefore gives a ratio of 200 ÷ 2 = 100 from one stage (this tooth count is an example). The maker says the gear has zero backlash because the teeth stay engaged, and that its size and weight do not depend on the ratio.

A cycloidal gear, sold by Nabtesco as the RV reducer, uses two stages. A spur gear stage first slows the motor down, then crankshafts move two lobed discs around a ring of pins. The discs have one lobe fewer than there are pins, so each turn of the crankshaft advances the output by one pin. Ratios run from about 30:1 to over 300:1, backlash is under 1 arc minute, and the housing contains large bearings that carry the load of the arm. Nabtesco says its cycloidal gearbox can absorb up to five times its rated torque in an emergency stop. The RV design reached the market in 1985.

Strain wave gears are light and compact, and cycloidal gears carry large loads and shocks, so designers tend to weigh those traits joint by joint. Neither type is free of elasticity, and a joint with a 100:1 gear winds up slightly under load, which is one reason a real arm's tip is a little off from the computed one.

Real robots and real numbers

The first industrial robot arm worked in a die-casting plant. In 1961 a Unimate designed from George Devol's patent was installed at a General Motors plant in the Trenton, New Jersey area, where it unloaded castings from a die-casting press, a hot and dangerous job. The Henry Ford museum keeps that first machine. IEEE Spectrum reports that the prototype weighed about 1,360 kg and had to invent its own control and memory system.

More than 60 years later, arms are common. The International Federation of Robotics counted 542,000 industrial robots installed worldwide in 2024, and 4,664,000 in operation. China installed 295,000 of them, 54 percent of the world total. Japan installed 44,500 and the Republic of Korea 30,600, the fourth largest market by annual installations. Behind these numbers sit arms that look like this model: six joints, a spherical wrist and a reduction gear in each axis.

Safety rules developed alongside. ISO 10218-1 sets the 250 mm/s limit under reduced speed control used by the Collaborative mode, and ISO/TS 15066 describes four collaborative methods with body-region force limits. Its content has been folded into the 2025 edition of ISO 10218-2, according to a German robot integrator. The 280 N figure for a brief knock to the hand and the 140 N figure for a clamped hand are limits for contact. They are not the force a robot can exert in general, which depends on its payload and speed.

Medical robots use the same kinematics for a different job. The US Food and Drug Administration describes robotically-assisted surgical devices as a surgeon's console linked to a cart with three or four hinged arms that hold the camera and instruments, and notes that the device is not actually a robot because it cannot operate without direct human control. The agency has cleared such devices for procedures such as gall bladder removal and hysterectomy, and has not authorized any of them specifically for treating or preventing cancer.

Writing an arm down as a DH table

Forward kinematics needs a description of the arm: where each joint axis sits relative to the one before it. Robot makers use a compact convention named after Jacques Denavit and Richard Hartenberg. Each joint gets one row of four numbers. d is the distance along the previous joint axis, a is the length of the common perpendicular to the next axis, α is the twist between the two axes, and θ is the joint angle itself. Six rows describe a six-axis arm, and multiplying the six small transforms gives the pose of the tool.

The arm on this page can be written that way. Its first row is d = 520 mm and a = 160 mm with a twist of 90°, which places the shoulder axis above and in front of J1. The second row is a = 620 mm, the upper arm, with θ = J2 minus 90° because the model counts J2 from upright. The fourth row is d = 600 mm, the forearm to the wrist centre, and the last row is d = 240 mm, the wrist centre to the point between the fingertips. The table in the Controls tab lists all six rows. A check run with 5,000 random poses found that the DH product and the direct formula on this page agree to better than a millionth of a millimetre.

Universal Robots publishes the same kind of table for its arms, together with the mass and centre of mass of every link. For the UR5e, which shares its table with the UR7e, the rows give d1 = 162.5 mm, a2 = −425 mm, a3 = −392.2 mm, d4 = 133.3 mm, d5 = 99.7 mm and d6 = 99.6 mm. With every joint at zero the flange therefore sits at x = −425 − 392.2 = −817.2 mm, y = −(133.3 + 99.6) = −232.9 mm and z = 162.5 − 99.7 = 62.8 mm, which the check reproduces exactly. The offsets d4 and d5 are why a UR arm has its characteristic stepped wrist: its three wrist axes do not meet in one point.

That one detail changes how the arm is solved. Without a spherical wrist, the position and orientation problems no longer separate at a wrist centre. The see-through arm on the stage is solved numerically instead: start from the last answer, compute how the flange moves for a tiny change of each joint (the Jacobian), and step the joints to shrink the error, with a little damping so the steps stay small near a singularity. Run 60 times on random poses for each of the eight tables, the solver lands within 0.04 mm of the target.

Three ways to lose a direction

A singularity is a pose where the Jacobian loses rank: some direction of tool motion would need infinite joint speed. For an arm like this one, with a spherical wrist, the determinant of the Jacobian factors neatly. It equals L2 × L3 × |cos J3| × r × |sin J5|, where L2 and L3 are the 620 mm and 600 mm links and r is the horizontal distance of the wrist centre from the J1 axis. The check compares this formula with the numerical determinant on 2,000 random poses and finds them equal.

Each factor is one kind of singularity. When sin J5 is zero, J4 and J6 line up: the wrist singularity this exhibit started with. When cos J3 is zero, the upper arm and forearm lie in one line, either stretched out or folded back: the elbow singularity, at the edge of the reach area. When r is zero, the wrist centre is directly above J1: the shoulder singularity, where turning J1 spins the wrist in place instead of moving it sideways.

The Manipulability bar in the readout shows this determinant as a share of its largest value inside the joint limits. Singularity map colours the reach area by the position factors, |cos J3| × r, so the pale bands are the places to avoid when planning a fast straight move. Universal Robots arms have the same three kinds of singularity, although their wrist offsets move them slightly.

Joint moves and straight moves

A robot program can ask for a move in two ways. A joint move turns every joint from its start angle to its end angle together, each at its own even pace. A straight move keeps the tool on a straight line and solves the inverse kinematics all the way. Both start and stop at the same poses, so the difference is the path between them.

Between the two stands, a straight move covers 1.200 m in 1.80 seconds at 1,000 mm/s. A joint move between the same poses only turns J1 and J6, so the tip swings along an arc of radius 863 mm around the J1 axis. That arc is 1.327 m long and takes 1.99 seconds, but the arm never comes near a singularity. The traces on the stage show both paths.

Past the wrist singularity the difference is dramatic. The straight move slides the tip 440 mm at up to 250 mm/s while J4 peaks at 510° per second. The joint move between the same two poses takes a path only 5 mm longer and J4 never exceeds 10° per second. Real controllers offer both moves for this reason. Joint moves are faster and safer when the path in between does not matter, and straight moves are used where the tool has to follow a line, such as a weld seam or a gluing bead, after the programmer has checked the path for singularities.

Payload and joint torque on a real cobot

The torque panel uses the same principle as the estimate above: the torque about a joint is the rate at which potential energy changes when that joint turns. For the model arm it uses the assumed masses from the estimate, upper arm 20 kg and forearm 12 kg, and reproduces the 241 N·m at the shoulder and 71 N·m at the elbow for the arm held horizontal with 6 kg in the gripper.

For the Universal Robots arms it uses the link masses and centres of mass from the maker's table. Held straight out (every joint at zero except the wrist joint J4 at −90°), with its rated 7.5 kg at the flange, a UR7e needs about 129 N·m at the shoulder and 56 N·m at the elbow. A UR20, with 20 kg and a 1,750 mm reach, needs about 657 N·m at the shoulder. Those numbers show why rated payload and reach trade against each other: the UR16e carries 16 kg at only 900 mm, and the UR30 carries 30 kg at 1,300 mm. The figures here are static. A real controller adds the torque needed to accelerate the links, so the motors need more than this at full speed.

Choosing a gripper

The tool on the flange matters as much as the arm. A two finger gripper squeezes a part from its sides, so it suits parts with parallel faces and can hold them firmly against sideways forces. Robotiq lists its 2F-85 with an 85 mm stroke and a 5 kg payload, and the wider 2F-140 with a 140 mm stroke and 2.5 kg, each weighing about 1 kg. A suction cup holds a part by its top face through a vacuum, which suits boxes, sheet metal and glass but needs a flat, sealed surface. An electromagnet holds steel parts by their top face and lets go when the current stops. In the model the cup and the magnet hold the 70 mm block by its top, so the tool centre point sits 35 mm higher than with the fingers, and the pick and place cycle adjusts its heights for that.

The numbers in the robot list

Each entry in the robot list comes from the maker's product page or a space agency page, checked on 27 October 2026, and the source link sits under each entry. Payload is the rated payload, and reach is the maker's figure, which for most arms is the distance from the J1 axis to the wrist or flange at full stretch. Yaskawa gives separate horizontal and vertical reach, and the list uses the horizontal one. Space arms are listed with what their agencies give: Canadarm2 is 17 m long and can handle loads of up to 116,000 kg in orbit, the European Robotic Arm is 11 m long with seven joints and moves payloads of 8 t, and the arm on the Perseverance rover is 2.1 m long with five degrees of freedom. Values a source does not state are left out rather than estimated.

Common misunderstandings

"The arm sees where its hand is." In this model and in most industrial arms, the controller does not look at the tool. It reads each joint's encoder, runs forward kinematics and assumes the links are the length it was told. If a link bends or a gear winds up, the real tip differs from the computed one. That is why cameras and calibration are added when accuracy matters.

"Moving from A to B means moving in a straight line." Only if the controller plans it that way. A joint move turns every joint evenly from start to end, and the tip sweeps along a curve. A straight-line move needs the inverse solver at every frame, which is what the model does for the three straight-line moves in the pick-and-place cycle, and it is the kind of move that can run into a singularity.

"More joints always means more skill." Six are enough to place and aim a tool. A seventh joint adds redundancy, so the same tool pose can be reached with different elbow positions, and that helps avoid obstacles. It also means the solver has infinitely many answers instead of a handful, so it must be told how to choose.

"A cobot is safe by itself." The standards judge a whole application. A gentle arm holding a sharp knife is not safe, and speed limits that are fine for a hand may not be fine for the head. ISO/TS 15066 requires a risk assessment with the real robot, tool and payload; the figures in this exhibit are examples.

"250 mm/s is the cobot speed limit." It is the limit for reduced speed control in ISO 10218-1. In power and force limiting, the allowed speed comes from the force limit and the effective mass, as in the 1.34 m/s example above, and can be much higher.

Things to try on the screen

1. Tap Part names, then drag the blue ball slowly in a circle around the arm. Watch the J1 reading follow the ball's bearing from above, and see which joint readouts change when the ball moves closer to the base.
2. Turn on Show reach and move the Gripper angle slider from 0 to 90. The blue region shifts, because the tip hangs 240 mm from the wrist centre in a new direction. Find the angle where the arm can reach highest.
3. Drag the ball farther than the arm can stretch. It turns red, and the readout says how far short the tip is. About 1.6 m at the level gripper is the limit.
4. Set J5 to 0. The J4 and J6 axes now lie along one line, so turn J4 one way and J6 back by the same amount: the gripper stays where it was. That is the wrist singularity from the section above, seen directly.
5. Start Play, then switch on Collaborative mode, and time one transfer. It takes 7.2 seconds instead of 1.8.
6. Try the six joint sliders without touching the ball, and write down the J2 and J3 pair that puts the tip on the floor between the stands. Then check it by dragging the ball there.
7. Open the DH bench in the Controls tab, pick UR3e, and drag the ball out to a stand. The see-through arm stops short and the panel says by how many millimetres, because its reach is 500 mm.
8. Turn on Singularity map, then drag the ball across a pale band and watch the Manipulability bar fall toward zero.