10Gravity and momentum: why arms sag and overshoot

Hold a full shopping bag out in front of you with a straight arm and keep the arm level. Look at your hand: it is a little lower than you meant, and the heavier the bag, the lower. Now swing your empty arm quickly across your body and stop it dead at a point you have picked. It does not stop there. It carries on a little, and you haul it back.

Robot arms do both, for the same two reasons you do. The first is gravity, which pulls on every link and every load, all the time. The second is that anything with mass, once moving, wants to keep moving. The arm in the box is the SO-101 from the last module, with the same servo at its shoulder lift, and the box can do one thing no workshop can: turn gravity off.

The box

The arm holds its elbow and wrist straight, as before, so that from the shoulder out it is one long lever with the hand on the end. On this joint, bigger numbers tip the arm down and smaller ones raise it: plus 35 degrees is level, and about minus 52 is straight up. The servo has been told to hold minus 10, so the arm points forward and up at about 45 degrees. Swing up tells it to go to minus 70, a little past straight up, as fast as it can, and Swing down brings it back the same way. Payload puts a steel ball in the hand. Gravity is the new knob: 1.0× Earth is the world as it is, 2.0× is a planet with twice the pull, and 0.0× is orbit.

The chart shows the target and the angle, the speed, and the motor's torque. The line under the chart says how far the arm is from its target, what the motor is giving, and, after each swing, the fastest the arm went and how far past the target it traveled.

Why arms sag

Recall from the module on a robot's parts what a servo does: it reads where its shaft is, compares that with where it was told to be, and nudges its motor until the two agree. Nearly agree, in fact, and this module is about the nearly. The servo has one piece of information, the gap between where it is and where it was told to be. It does not know what is hanging off it, or which way is down. So the gap is the only thing its push can be made from, and the rule almost every servo uses is the simplest one there is: a fixed push for every degree of gap. A gap of two degrees gets twice the push of a gap of one, and no gap gets no push at all.

Now hang an arm on it. Gravity twists the shoulder downward, and the twist never stops. In this pose it is about half a newton meter, less than the last module's 0.7, because that arm was near level and this one is raised 45 degrees, and the higher a stick is raised the shorter the weight's lever. To hold the arm up, the servo must twist back with the same half a newton meter, and by its rule it can only do that if there is a gap. So the arm settles a little below where it was told, exactly far enough below that the push the gap earns equals the pull. The gap is how the servo finds out how hard to push. For the servo in the box, it comes to about four degrees for every newton meter.

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

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

Every reading fits the rule. Twice the gravity is twice the pull, so twice the gap. Half a kilogram in the hand adds about 1.5 newton meters of pull, a little under the last module's 3.3 per kilogram for the same reason, the shorter lever, and the gap grows by five or six degrees to match. With no gravity there is nothing to hold up, so there is no gap.

That last one deserves a closer look, because it is where two ideas that everyday speech runs together come apart. Mass is how much stuff a thing is made of. It is measured in kilograms, and it is the same everywhere. Weight is the pull of gravity on that mass: on Earth, about ten newtons for every kilogram, half that on a planet with half the gravity, and nothing at all in orbit. The Payload knob adds mass. The Gravity knob changes what that mass weighs. Sag comes from weight, which is why the Gravity knob can remove it.

With 500 g still in the hand, drag Gravity to 2.0×: the arm sinks all the way to the table, the motor giving all it has; back at 1.0× it lifts itself off again. Half a kilogram where gravity is twice Earth's weighs what a full kilogram weighs here, and the last module showed what a kilogram does to this arm: it asks the motor for more than its 2.94 newton meters, and the arm ends up resting on the table. Same mass, twice the weight, same collapse.

Real arms sag for two more reasons the simulation leaves out: there is a little slack in any stack of gears, and a long link bends a little under a load. Neither is the main cause. The servo's gap is there whenever there is a load to hold, and no stiffer link removes it. Controllers have a trick for shrinking that gap, and it is one of the subjects of Part 3; the reason for the gap is what you have just measured.

Why arms overshoot

When you pressed Swing up, the servo saw a gap of sixty degrees and, by its rule, asked its motor for far more than the 2.94 newton meters it has, so the motor gave everything, and the arm took off.

The trouble begins near the target. The gap shrinks, so the push shrinks, but the arm is now moving at over 200 degrees per second, and a moving arm does not stop because you have stopped pushing it. Momentum is what a moving thing carries by moving: its mass times its speed. Taking momentum away needs a force applied for a time, and the more momentum, the more force or the more time. The instant the arm crosses its target, the gap is on the other side, so the servo pushes the other way, harder the further past the arm goes: in the bottom panel the torque crosses zero and climbs on the other side. That is the brake. But a brake that starts from nothing and can never exceed 2.94 newton meters takes time to remove that much momentum, and the arm keeps going while it does, and stops some way past. Overshoot is that distance: how far past its target a moving thing travels before it is stopped.

Now the arm is on the wrong side of the target, so the servo hauls it back. It picks up speed again, crosses again, overshoots again, by less. What you are watching is a weight bobbing on a spring, because that is what a servo with this rule is: a spring that pulls toward the target, harder the further you stretch it. The motor in the box still has only its two numbers; the joint it turns has a little friction, in the box as in a real gearbox, and every pass bleeds some of the motion off through it, so the bobbing dies away and the arm is still.

Weightless, not massless

This is the experiment the box exists for. With gravity off, the sag is gone: no weight, no pull, no gap. The overshoot is not gone, because it never came from weight. It came from mass and speed, and the Gravity knob changes neither. Add a kilogram to the hand and there is no weight to hold, but there is a kilogram more to get moving and to stop, and it sits at the far end of the lever, where it counts most. With the same motor doing the stopping, the arm goes further past.

The line under the chart adds a detail: the heavier arm never went quite as fast, because the same motor had more to get moving. Momentum is mass times speed, so it carried more momentum even so, a lot more mass at a little less speed, and more momentum takes the brake longer to remove. The bobbing lasts longer for the same reason: friction bleeds off about the same on each pass, and there is more to bleed.

This is why the robot arms on the International Space Station move so slowly. Nothing they carry weighs anything up there, but a spacecraft that weighs tons on Earth has the same mass in orbit, and once it is moving, stopping it takes the same force for the same time as it would on the ground. An arm that got it moving briskly could not stop it in time.

Gravity and the brake

Back on Earth, with Gravity at 1.0× and 500 g in the hand, press Swing up: about 16 degrees past, close to the weightless 15 with the same ball. Press Swing down and the arm drops about 48 degrees past, until the fingertips tap the table, before the motor hauls it back up.

Gravity takes part in the stopping too, on whichever side the arm's position gives it. At the top of the swing the arm is close to upright, and a stick held straight up twists nothing: the weight has almost no lever, so gravity hardly joins in, and the overshoot comes out close to the weightless one. It is a little larger, not smaller, because minus 70 is a little past upright, and past upright the weight tips the arm over the top rather than holding it back.

At the bottom of the swing the arm is nearer level, the lever is long, and swinging down into it the weight pulls with the motion. The motor's 2.94 newton meters must first cancel that pull before what is left can take any momentum away, and with half a kilogram in the hand there is not much left. So the arm falls a long way past before it is caught. You saw the same drop in the last module when a loaded arm was lowered; now you have both halves of the reason: momentum to take away, and a brake that gravity has weakened.

Why real robots move the way they do

Everything in the box is why a real robot arm does not go flat out and stop dead. Its controller slows the arm before it arrives, so there is less momentum to remove at the target. It moves a heavy load more slowly than a light one. It lowers a load more gently than it lifts one. And it is why the arm's own weight matters so much to whoever designs it: every gram of link is mass to be started and stopped as well as weight to be held, which is why robot arms are built as light as they can be made. Part 3 is about how a controller manages all this. The physics it is managing is what you have just driven.