8Reach: where the hand can and cannot go

Sit at a table, keep your back against the chair, and think about where your fingertip can go. Not the far corner of the table. Not the point an inch in front of your own shoulder, because your arm cannot fold that small. Not the wall behind you, because your shoulder does not turn that far. Everywhere else, yes. All of those yeses together make a shape in the air around you, and you know that shape so well you never think about it.

A robot arm has one too. Last module asked, for one place at a time, whether the robot could find joint angles to put its hand there. This module asks that question for every place at once. The workspace is every place a robot can put its hand, taken all together. Robot people also call it the work envelope. It is a solid shape in the air around the robot, and this module is about what that shape looks like and where it comes from.

Where the edges come from

The robot in the box is the SO-101, the small arm whose parts you labeled in an earlier module. Its joints, from the bottom: the shoulder turn, which swings the whole arm left and right on the base; the shoulder lift, the hinge where the upper arm starts, which tips the arm forward and back; the elbow; the wrist bend; the wrist twist; and the jaw. When this module says "the shoulder" on its own, it means the shoulder lift, because that hinge is where the arm's length begins.

The shape has an outside edge, an inside edge, a missing slice, and a floor. Each comes from one plain fact about the arm.

The outside edge comes from length. Stretch the arm dead straight and it is as long as it will ever be. Every place at exactly that distance from the shoulder makes the surface of a ball with the shoulder at its center, and nothing outside that ball can be touched. This arm has about 41 centimeters of arm past its shoulder, so that is the ball's radius. The shoulder turn's axis sits 3 centimeters behind the shoulder, so the center swings a little as the arm turns, which is one reason every number here says "about".

The inside edge comes from joint limits. Last module's flat arm had an inside edge because its two pieces differed in length by 10 centimeters. This arm's comes from its hinges stopping. To touch a place close to its own shoulder, an arm has to fold nearly flat, forearm back along the upper arm. This elbow bends to about 97 degrees, a right angle and a little more, and stops; a fold that stops at a right angle leaves the hand out at the corner of an L, never back beside the shoulder. The wrist bend, stopping at about 95, pulls the fingertips in a little further and no more, and the shoulder motors and the base sit where the hand would need to go. So there is a hollow around the base that the hand never enters, wrapping up and over behind the shoulder: close to the shoulder means folded, and this arm cannot fold far.

The missing slice comes from the shoulder turn. On this arm that turn stops about 110 degrees each way from straight ahead, so seen from above the arm can face a fan of about 220 degrees and no more. Behind the base is a wedge of about 140 degrees that the arm cannot turn to face, so nothing in it can be reached by turning. The one way in is over the top: the shoulder lift can lean the whole arm backward over the base, until it too reaches its limit, about 100 degrees from the middle of its travel. You will find what that buys in the box.

The floor is the table. The arm cannot put its hand below the surface it stands on, so whatever the ball would have been under the table is gone.

Put those together: a thick shell of a ball, hollow near the base, with a wedge cut out of the back and its bottom sliced off flat. It is not a circle and not a box. It is the arm's link lengths and joint limits made visible, and no two robots have the same one.

One more thing follows from last module. At the outside edge the arm is stretched straight, so one shape reaches a place there, with the jaws pointing straight out along the arm. Deeper inside there is slack: elbow high or elbow low both reach, as you saw last module, so the jaws can come at the same place from more than one side. The hand can touch everything in the workspace; near the edge, only one way.

Try it

A gold star floats in front of the arm, and the arm's fingertips follow it the way the hand did last module, the robot finding the angles itself. Four joints do the work: the shoulder turn, the shoulder lift, the elbow and the wrist bend. The wrist twist and the jaw hold still, because neither moves the fingertips much. A blue dot marks the point the arm is placing, between the tips of the two jaws.

This time the star moves in three dimensions, and the line under the box gives its place as three numbers: x is centimeters forward of the base, y centimeters to the arm's left, z centimeters above the table, all measured from the point on the table under the middle of the base. The shoulder turn's axis is at x 4; the shoulder, the center of the ball, is at about x 7, y 0, z 12. The same line says whether the fingertips are on the star or how far short they are, and below it, as last module, "The arm's answer" lists the four joint angles.

Drag the star with the mouse or a finger and it slides in a plane facing you. Drag the empty background instead and the view turns around the arm; roll the wheel to zoom. The box opens looking over the arm's left shoulder from behind and above, and the shape you are about to paint is bigger than that first view, so zoom out early. The buttons under the box nudge the star a centimeter at a time in fixed directions, whichever way you are looking. Once you have clicked one, the keyboard works too: arrow keys for left, right, up and down, Page Up and Page Down for forward and back, and Shift for five-centimeter steps.

The star is gold when the robot's search puts the fingertips on it and gray when it cannot; as last module, gray is the robot's verdict. New this time: wherever the star has been, it leaves a dot behind, green if the fingertips could reach it there and gray if not. Sweep the star through the air and the dots paint the workspace. Clear dots wipes them; Reset wipes them and puts the star back where it starts, at x 30, y 0, z 15.

Loading the simulation. The robot runs in your browser, so the first one takes a few seconds to arrive.

ResetPauseSlowLabels

This is easier on a larger screen, but it works here too.

What you saw

Your loops painted a slice of the ball: an arc of green on the outside and, inside it, hugging the base, a hollow with no green at all. Where the outer arc lies is the stretched length. Nudged straight forward from the start, at 15 centimeters up, the star turns gray at about x 50 and comes back to gold at 48; take the shoulder's x of 7 away from that and you have about 41 centimeters. Nudged straight up from the start, it grays at about z 46 and comes back to gold at about 45, a place 23 centimeters in front of the shoulder and 33 above it, which along the slant from the shoulder is about 40 again. Two points on the same ball. Directly over the shoulder, where the slant is straight up, the top is a little over 50. The hollow is the inside edge: the elbow and the wrist stopping, and the arm's own body in the way.

Seen from above, the dots made a fan. Ahead of the arm and out to both sides, green. Behind it, gray, through a wedge of roughly 140 degrees, and while the star was back there the shoulder turn read 110, its limit, with the arm stretched toward a place it could not face. That wedge is one joint limit, drawn on the floor.

The last try visited the hollow and the wedge. At x 10, z 5, the wrist bent to its limit of 95 degrees and the fingertips still hung 4 centimeters short; nudged forward, the star turned gold at x 12, the hollow's edge on the table. At x −20, z 40, behind the base and high up, the star was gold: the arm got there by leaning backward over its own base, the shoulder lift at −90. Straight down behind the base at x −30, z 15, the star was gray, the fingertips well over 10 centimeters short, and the shoulder lift read −100, its limit. Leaning back, the arm can dangle its hand into the top of the wedge and no lower; to reach the table behind itself it would have to lean further than its shoulder allows, or turn round, which it cannot. So the shape is not the same front and back: a pocket high up behind the arm, and nothing low down.

One honesty about the dots. Gray means the robot's search, the guess-and-correct from last module, could not find angles for that place. Right at the edge, a place that is only just reachable may paint gray because the search gave up a few millimeters short. The shape you painted is the shape the robot can actually use.

Why a robot cares

Nobody buys a robot arm for its own sake. They buy it to reach things: parts on a conveyor, a weld along a seam, a shelf. Every one of those has to sit inside the workspace, and the check is done on paper before the robot is bolted down, because a robot that cannot reach its part is a very expensive ornament. It happens anyway, and the fixes are all clumsy: move the part, move the robot, buy a bigger one, or put the whole robot on a rail so its workspace slides along with it.

And inside is not enough. A part right at the outer edge can be touched one way only, with the arm stretched straight and no bend to spare. Robot people keep their work well inside the shape, where the arm has room to bend and the hand has a choice of approaches. The next module is about what happens when the arm gets there: how hard it can push, how much it can lift, and why the answer changes with how far it is reaching.