A bicycle, the benchmark one from the literature, seen from above, and you are on it. The arrow keys push on the handlebar; signs on the road say which lane to be in, thirty metres ahead of where it starts to count. Press left to go left and you will lean right and go right, because a bicycle turns by leaning and leans by steering the other way. To go left you steer right first, briefly, then follow the lean. Everyone who rides does this and almost nobody knows it; the keys make it impossible not to notice.
The bicycle is the Whipple model in the benchmark form of Meijaard, Papadopoulos, Ruina and Schwab: two degrees of freedom, the lean and the steer, linearised about riding straight at a constant speed, with the published mass, geometry and gyroscopic terms of their benchmark machine; the path follows from the steer angle and the trail. Left to itself it balances between 4.3 and 6.0 m/s, wobbles over at 3 and slowly capsizes at 8. The road is drawn much wider than it is long, so the lane change you see is a metre and a half over thirty; side gusts kick the lean now and then. A key is 1.5 N·m on the handlebar, ramped in over sixty milliseconds; a lean past thirty degrees is a fall, and so is the ditch, each costing a second on the ground and two points. Cost is the mean distance from the lane in force, in metres over the ride, plus two per fall. The riders on the ladder ride your speed on your road: no hands at all; a rider who steers toward the lane and keeps the front wheel straight, with no lean in the loop; and the textbook, a linear quadratic regulator on lean, steer, their rates, heading and offset, which countersteers because the optimum does.
Steering the front wheel to the right moves the wheels right, out from under you, and the bicycle falls left; the only way to stay up is to steer left into the fall, and now you are turning left, leaning left, with the wheel pointed left. Every left turn is a right turn first. That is not a quirk of bicycles but the signature of a right-half-plane zero: the stabilised bicycle, lean loop closed, answers a request to lean with a path that goes the wrong way for a fraction of a second before it goes the right way, and the zero sits exactly where the open-loop lean pole was, at about √(g/h), three per second. A zero there sets a speed limit on the whole loop: ask the path to answer faster than the zero and the wrong-way part grows instead of shrinking, which is what the rider who steers toward the lane discovers, on the ground, every second or so. The textbook rider does not know any of this and countersteers anyway, because the optimum through the model does. Your hands learn it the same way, and the trail behind you shows the little hook every lane change starts with.
A zero in the right half-plane cannot be cancelled by the controller, because cancelling it would put the pole it hides in the right half-plane, unstable and unobservable. So its effect stays in the loop: an inverse response, and a limit on bandwidth of roughly half the zero's position, past which the sensitivity function must exceed one somewhere by more than the design can afford. The waterbed of the Bode integral is pinned there. Rear-wheel-steered bicycles have such a zero in the open loop, at v ⁄ a with a the distance to the rear contact, and every attempt to build a ridable one has failed; Åström, Klein and Lennartsson tell the story, with the transfer functions.
The front-steered bicycle's zero is subtler: it is not in the plant from torque to path, but in the closed loop from a requested lean to the path once the lean is stabilised, and it lands on the open-loop pole at √(g/h). That is the countersteer. Any rider, a cascade of lean feedback inside path feedback, inherits it, and the path loop can be no faster than about a second and a half per lane change without overshooting or falling.
Two facts and a consequence. The lean is an inverted pendulum, unstable at about three per second, and the steer is the only input to it: steering moves the contact line under the mass, which is what accelerates the lean, and it accelerates it away from the steer. And a steady turn needs a steady lean with the steer pointing into it, in the ratio above. Put them together: to arrive in a left turn, leaning left and steering left, you must first accelerate the lean leftward, which means steering right; then catch the lean by steering left, and hold. Fajans measures it on real motorcycles and bicycles; at speed, riders press forward on the left grip to go left.
The rider is a cascade. The inner loop steers into the lean to keep the pendulum up, fast; the outer loop asks that inner loop for a lean to reach a lane, slow. The linear quadratic regulator on the ladder is both loops in one gain, and its first move on every sign is the countersteer, a torque the wrong way for a third of a second; you can see it in the hook at the start of its trail. Press only in the direction you want and you supply the wrong first move without the second.
The eigenvalues of that four-state system move with speed. At rest there are two unstable real ones, the bicycle falling over. As speed rises they pair into a weave, an oscillation of lean and steer together, which crosses into the left half-plane at 4.29 m/s; a slow real capsize mode crosses back out at 6.02. Between those speeds the bicycle rides itself hands-free, steering into every fall because the front assembly is arranged to do so: the wheel's contact is behind the steer axis, and the spinning wheel precesses into the lean. That was the folklore, gyroscopes and trail; Kooijman and colleagues built a bicycle with neither, counter-rotating wheels and negative trail, and it rode itself too. What is needed is that the front assembly falls into a lean faster than the rear, and there are many ways to arrange that.
The speed knob here crosses both boundaries. At three metres a second the weave grows and a gust will have you on the ground unless you damp it by hand; at eight the capsize is unstable, slowly, and the bicycle wants to spiral into whichever lean it has. Riders manage both without thinking, which says something about the loop you close without knowing it.