11Grippers and hands: holding on

Pick up a glass of water. You do not crush it, and you do not let it slip: your fingers squeeze about as hard as they need to and no harder, and you never think about it. Now wet your hands and try the same with a bar of soap. You squeeze, and the harder you squeeze, the more surely it shoots across the room.

Holding on comes down to two things: how hard the hand squeezes, and how much of that squeeze the surfaces give back as grip. This module puts a block in front of the arm from the last two modules, gives you a slider for each, and asks the arm to pick the block up and drop it in a bin.

The hand on the end

The module on a robot's parts called the piece at the end of the SO-101 its gripper, and gave its sixth servo one job: swinging one jaw against the other. The general word is end effector: the tool the arm carries to do the job, a gripper, a suction cup, a welding torch, a camera. The arm's job is to put the end effector somewhere; the end effector's job is everything after that.

A gripper is an end effector that holds by closing on things. The SO-101's is about as plain as they come: one jaw fixed to the wrist and one that swings on a hinge, driven by the same kind of servo as every other joint, with the same 2.94 newton meters standing still.

Squeeze

The jaw motor makes torque, a twist about the hinge. What the block feels is a push: the torque spread over the distance from the hinge to where the jaw meets the block, about eight centimeters here. Divide 2.94 newton meters by 0.08 meters and the jaws press the block with about 37 newtons, nearly four kilograms of weight. That push is the grip force: how hard the jaws squeeze what is between them.

The block is a 36 millimeter cube weighing 50 grams. A kilogram weighs about ten newtons, so the block weighs about half a newton, and the jaws squeeze it seventy times harder than gravity pulls it down. But the squeeze points sideways, into the faces of the block, and gravity pulls down. Something has to turn a sideways push into an upward hold, and that is friction.

Friction

Friction is the force that resists two touching surfaces sliding across each other. Press your palm flat on the table and try to slide it. It resists, and it resists more the harder you press, in proportion: twice the press, about twice the pull needed to slide, because the harder press flattens more of the two surfaces' tiny bumps against each other. A sheet of paper under your palm slides more easily at any pressure. So two things set the friction: how hard the surfaces are pressed together, and how grippy the pair is, which is a plain number. Rubber on most things is near 1, meaning each newton of press gives back about a newton of resistance to sliding. Wood on wood is nearer a third, hard plastic on hard plastic about a fifth, and wet soap in a wet hand a few hundredths.

Put the two ideas together. The swinging jaw presses one face of the block with the grip force, and the fixed jaw, which has no motor, presses the other face back exactly as hard, the way a wall presses back on your hand. Each face gives friction equal to that force times the grippiness; two faces, so twice that. That friction acts along the faces, which is to say up and down, and it is all that holds the block against its half-newton weight, plus the jolts of the carry.

The block and the jaws are both hard plastic, so as built the grippiness is about 0.2, and at full force the arithmetic is absurd: 37 newtons times 0.2, times two, is about 15 newtons of hold for half a newton of weight. Grip force at 0.30× as built means 0.88 newton meters and a squeeze of about 11 newtons; at 0.14×, 0.41 newton meters and about 5 newtons. Friction scales the block's grippiness: 1.00× is the plastic block as built, about 0.2, so 0.75× is 0.15 and 0.50× is 0.10.

The box

The arm starts folded, with the block on the table in front of it and the bin to one side. Press Lift and a fixed script runs: the arm reaches over the block with its jaws open, lowers them around it, closes, lifts, holds the block up for a moment, swings it over the bin, lets go, and comes home, and a fresh block is set on the table. You never need Reset between tries; it only puts the sliders back. The arm goes through the same motions every time, whatever happens to the block, and has no way of knowing whether it is holding anything.

The line under the controls says what the arm is doing and where the block is, and while the jaws close it also says how hard the jaw motor is pushing. Below it, the readouts give the jaw's angle, its motor's torque, and where the block is relative to the jaws. On the jaw angle, zero is the swinging jaw parallel to the fixed one, and positive numbers open it. The hinge sits to one side of the fixed jaw, so the swinging jaw can go a little past parallel, to minus 10 degrees, where its tip meets the fixed jaw's tip: the jaws shut on nothing. A jaw stalled about 13 degrees open has a block in it; a jaw at minus 10 has nothing.

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

Too little squeeze

The slow close is the module on motors, seen from the other end. The jaw's hinge has drag, as every joint does, and in this simulation getting the jaw swinging costs about a quarter of a newton meter. At full force that is nothing; at 0.41 it is over half of what the motor has, so little is left to move the jaw and it swings slowly. Once the jaw is stalled against the block nothing is moving, the drag costs nothing, and the whole torque goes into the squeeze. Five newtons at a grippiness of 0.2 on two faces is 2 newtons of hold for half a newton of weight: plenty.

At 0.10× the motor has 0.29 newton meters, barely more than the jaw costs to move. What is left swings the jaw so slowly that the arm, which waits a few seconds for it and then gets on with the script, has lifted before the jaw reaches the block. A gripper cannot hold with force it does not have.

Too little friction

By the arithmetic above, this should not happen. 37 newtons of squeeze at a grippiness of 0.08 is still 6 newtons of hold, twelve times the block's weight. The bar of soap has arrived.

The trouble is the shape of the grip. The fixed jaw lies flat against one face of the block. The swinging jaw stopped 13 degrees short of parallel, because that is where the block stopped it, so the block sits in a shallow V, narrow toward the wrist and open toward the tips. A surface can only push at right angles to itself; whatever it does along itself is friction, already counted. So the fixed jaw pushes straight into the block, but the swinging jaw's face is tilted, and its push goes mostly into the block and a little along it, toward the open end of the V, which is out of the jaws. The steeper the tilt, the bigger the slice of the squeeze that leaks that way. At 13 degrees it is a small slice, but never zero.

Only friction holds the block against that leak, and friction is also a slice of the squeeze, twice the grippiness. Squeeze harder and both grow together, so the squeeze cancels out of the contest, and grippiness alone decides: above some line friction wins at any grip force, below it, it loses at any grip force. In this simulation the line is at a grippiness of about 0.10, which is 0.50× on the slider. At 0.40× the jaws lift the block and the leak walks it out of them within a second, however hard they squeeze; at 0.50× it is a toss-up from one go to the next; from 0.75× up it always holds. That is why soap escapes, and why you hold a wet glass by wrapping your fingers underneath rather than pinching harder.

Both at once

Set both sliders far down, Friction to 0.75× and Grip force to 0.14×, and the block still goes into the bin: what matters is what the two multiply to, and 5 newtons at 0.15 on two faces is 1.5 newtons of hold, three times the weight. For this light block the weight itself is never the limit: at the weakest grip that closes, 5 newtons, it would need a grippiness of only 0.05, below the slider's floor. A block four times heavier would need 0.2, and the same 5 newtons would drop it at 0.75×.

Real grippers spend on both sides: rubber pads buy grippiness, flat parallel faces keep the pushes in line, and when neither is enough they stop relying on friction: fingers that curl underneath, a suction cup, or a cage that closes around the object so its shape cannot escape. Your hand does all of these without being told.

The arm cannot feel

Run the friction try once more and watch the readouts as the block slips out: the jaw snaps from 13 degrees open to minus 10, shut on nothing, and the motor's torque falls to 0.00. The arm carries on to the bin, opens its empty jaws over it, and goes home as if nothing had happened.

The jaw told the whole story. Stalled at 13 degrees with the motor straining, it had the block; snapped to minus 10 with the motor relaxed, because a jaw hanging straight down where it was sent needs nothing to stay there, it had nothing. The jaw servo measures that angle hundreds of times a second, as the module on a robot's parts said, and the readout shows it. The script never asks, and so it lifts, carries and releases an empty hand with complete confidence. That stall is the makings of a sense of touch, and a real robot that picks things up reads it, or a proper touch sensor, to know whether the grasp worked. That is for a later part of this course, on how robots sense.

What the simulation leaves out

The block is a hard cube that never deforms, and the jaw tips are hard too. Real jaw pads are rubber, which flattens against an object and grips several times better than plastic, and real objects are round, soft or wet. And a real jaw motor squeezing a fragile object at 37 newtons breaks it, which is why an egg is a harder problem than a block. That, too, is for the part on sensing.