The bicycle that steers itself upright

The weight a frequency can find worked at the bottom of a well, where a parabola gives a clean frequency and nothing grows or decays. Every minimum is a parabola explained why that local calculation has such a long reach. The same idea can be applied to a state that has no well at all. Take a state of motion, disturb it slightly, and keep only what is linear in the disturbance. The result is a matrix, and its eigenvalues contain everything: oscillation in their imaginary parts, growth and decay in their real parts. That procedure, applied to a bicycle running upright in a straight line, gives a result that took 130 years to be calculated reliably and is still widely misreported.

The push that comes out sideways mentions the question of why a bicycle stays up, and it quotes the conclusion that gyroscopes do not fully explain it. People are generally surprised to find that a bicycle can balance with nobody on it at all. A bicycle running down a slope without a rider, at more than a walking pace, goes straight and corrects its own wobble and only falls over when it slows down. The stability is a property of the machine’s geometry and mass distribution, not a skill of the rider. This has a clean mathematical description that does not have a simple physical explanation.

Four roots #

Francis Whipple published the right equations in 1899. His model has four rigid parts: rear frame, front fork, and two knife-edge wheels, connected by hinges and rolling without slipping. That rolling constraint, whose odd consequences the ball a turntable will not throw off shows in a simpler machine, is what makes this system unusual: the wheel cannot go sideways, but a wheel can reach any position by a suitable path, so the number of ways the bicycle can move at an instant is smaller than the number of coordinates needed to describe its position. Such systems are called non-holonomic, and they lack the tidy conservative form on which the action that knows where every path ends relies. For that reason a rolling machine with no friction at all can still have eigenvalues with negative real parts, and its motion can appear to die away.

Linearised around upright rolling at speed $v$, with lean angle $\phi$ and steering angle $\delta$, the equations take a form that is both compact and remarkable:

$$M\ddot{q} + vC_1\dot{q} + \left(gK_0 + v^2K_2\right)q = 0, \qquad q = (\phi, \delta).$$

The two-by-two matrices $M$, $C_1$, $K_0$, $K_2$ depend only on the geometry and mass distribution. Twenty-five parameters are needed to define a bicycle, and many published calculations over the decades had errors in them, so that Arend Schwab, Jim Papadopoulos, Andy Ruina and Jaap Meijaard set out in 2007 to verify a benchmark. They checked by several independent methods, and they published the matrices to fourteen digits, with their eigenvalues, so that any future calculation could be tested.

The characteristic equation is a quartic in the growth rate, so there are four roots. All four are computed here from the benchmark’s parameters. Before anything was drawn, the matrices were built from those parameters and compared with the published numbers, and the roots at standstill were checked against their published values, $\pm 3.13$ and $\pm 5.53$ per second. These are the same bicycle as an upside-down double pendulum, falling over. The edges of the stable band came out at 4.292 and 6.024 m/s, matching the published values.

Three things a bicycle can do #

The roots fall into three types, and each has a physical meaning.

Capsize is a real root that sits close to zero. At high speed, it is positive but small: a slow fall into a tightening spiral, taking many seconds to develop. Its existence surprises people who think that going faster is always more stable. It is the reason a fast bicycle without a rider eventually falls over, and it is what a rider corrects continually without being aware of it, since its time constant is far longer than any reaction time. Most riders, in fact, think of a fast bicycle as very stable. It is in the sense that matters to them.

Weave is a pair of complex roots, an oscillation in which the bicycle sways and steers together, with the two coordinates passing motion back and forth rather as the two pendulums that will not stop swapping do. At low speed, its growth rate is positive: a wobble grows until the bicycle falls. It is what any learner fights. Its frequency grows steadily with speed, as the dashed line in the figure shows. At the weave speed, the real part of its growth rate crosses through zero, and above it the wobble dies.

Castering is a real root with a large negative value. It describes the front fork quickly aligning with the direction of travel, as a shopping trolley’s front wheel does when pushed. It is a fast decay and is not seen as motion, but it contributes much of the machine’s stable response at speed.

Between the weave speed and capsize speed, in the benchmark’s case between 4.3 and 6.0 m/s, about 15 to 22 km/h, there is no growth anywhere. That is a very ordinary cycling speed. The machine really does ride itself there. Its geometry was never designed for this, although real bicycles have evolved by trial and error to have stable bands of about this kind.

One shove at three speeds #

The eigenvalues are complete, but a time history shows what they mean in a form that is more easily believed.

At 3 m/s the wobble grows with each swing. The curve has been cut off once the lean reaches half a radian, past which any linear model is meaningless and a real machine is on its way to the ground. At 5 m/s the shove dies away in a couple of seconds, and what is left is an upright bicycle moving in a slightly different direction. At 8 m/s, the sway disappears even sooner, and then what remains is a tiny, slowly growing lean: the capsize.

Karl Åström, Richard Klein and Anders Lennartsson used a bicycle at Lund to demonstrate this for teaching control, and J. P. Meijaard and others have reproduced it. A particularly direct test used a bicycle running on a treadmill; a riderless bicycle at the right speed does in fact show this weaving and recovery exactly as the calculation says. That is a good reason for regarding the linear equations as reliable within their range, since they contain neither approximation about the rider nor any other free parameter.

The front steers into the fall #

The figure shows the mechanism in its most visible form. A lean to the right produces a rightward steer. This puts the wheels’ contact line under the mass, in the same way that a person balances a broom on a hand by moving the hand underneath. Steer is ahead of lean, which is what a control engineer calls phase lead, and it is exactly what a balancing mechanism requires. It overshoots, leans the other way, and the two oscillate in a decaying fashion. A rider feels this as the bar turning by itself when the hands come off.

It is easy to find components of the steer. At the fork, a leaning frame with trail has a torque that turns the wheel towards the fall, since the ground’s reaction on the contact patch, behind the steering axis, acts with a lever arm. The front assembly’s own mass is ahead of the steering axis in most bicycles and tends to fall forward and to the side, another steering torque. A spinning front wheel, if it leans, has a gyroscopic steer towards the same side. All of those turn the wheel in the right direction, and they appear as entries in the matrices. Each one adds to the off-diagonal coupling and is not a separate explanation for stability.

Notice something odd about how energy behaves here. The model has no friction, and its total energy is conserved. Yet the wobble decays. Where does the energy go? It goes into the forward speed. A wobble is a small transverse motion, and when it dies, its energy returns to the bicycle’s forward motion, at a level of second order in the amplitude. The stable motion is not an attractor in the usual dissipative sense, even though it looks that way. Time-reversed, the same equations would give a growing wobble at the expense of speed. The egg that keeps turning after it is stopped has a related oddity in a similar rolling constraint, and the spool that rolls towards the hand pulling it another in simpler form. A purely conservative system can have asymptotically stable motion only because rolling constraints are not of the type that Hamiltonian mechanics covers.

What the stability depends on #

Popular explanations invoke gyroscopes or trail, so a calculation can test them directly. Change one parameter of the benchmark and find the stable band again.

The calculation seems at first to vindicate both. Without front-wheel spin, this machine has no stable speed in the range up to 14 m/s. Without trail it retains only a tiny band, from 2.9 to 3.1 m/s. Yet this is what can be said of this one design, and it gives a misleading picture of what matters. The Science paper by Jodi Kooijman, Meijaard, Papadopoulos, Ruina and Schwab in 2011 took a direct route to this question. They built a two-mass-skate bicycle, with its wheels counter-rotating so that there is no angular momentum to speak of, and the trail slightly negative. Conventional wisdom would give it no chance. By placing its front and rear masses suitably, they found a stable band, and they demonstrated it by shoving the machine and watching it recover.

So neither gyroscopes nor trail are necessary. They are also not sufficient: they showed machines with both that are unstable at all speeds. The explicit equations show that the machine’s stability depends on the combination of all twenty-five parameters. Several necessary conditions are known: a steer-into-the-fall coupling, and steer leading lean, as the figure shows. No general and simple necessary and sufficient condition has been found, even for this simple model. That is a useful warning about physical explanation. A correct calculation in a low-dimensional system may fail to produce any one-sentence reason, and the one-sentence reasons are nearly all wrong.

The chemist David Jones became curious about this in the 1970s. He tried to build an unrideable bicycle, and his attempt to remove the gyroscopic effect by adding a counter-rotating wheel at the front gave a bicycle that could still be ridden, but with difficulty when released. Removing trail made it nearly unmanageable. Jones’s conclusion favoured trail, and that was widely taken as the answer for about three decades, though the numbers show that trail by itself is not a complete explanation. For a real bicycle, his results stand: the combination of the two is a good one for humans.

How much trail it takes #

For one fixed design with its masses kept, the effect of trail can be mapped by scanning. The stable band opens at about zero trail, on this machine at nearly $-1$ cm, then widens quickly with increasing trail. Its upper edge, the capsize speed, rises faster than the weave speed does. By 20 cm of trail it extends well beyond typical riding speeds. Bicycle frames in fact use values from about 5 to 10 cm, and motorcycles and racing machines have other values, with tradeoffs in handling. More trail makes a bicycle more stable and harder to turn, and a rider wants some of each. The modern frame geometry settled on these values by the 1890s without any of this calculation. Within a few years of J. K. Starley’s Rover safety bicycle of 1885, makers had settled on a raked fork, which riders judged preferable, and it is striking that their preferences happened to fall in a broad part of the stable region.

Other questions in this space have been answered in the same way. A steering damper reduces weave at the expense of capsize, or vice versa. Motorcycles have a high-speed weave that is a real safety problem, and a low-speed wobble, in which frame flexibility also has a part. Both were studied by R. S. Sharp in the 1970s by the same eigenvalue method with more parameters and with tyre models, showing this kind of analysis is not only an academic exercise.

Why a matrix can settle what intuition cannot #

It is worth asking why this question stayed unsettled so long, when the matrices are small. One reason is that every proposed explanation is partly right, since each mechanism contributes a real term to the off-diagonal coupling. A gyroscopic steering torque, a trail torque and a mass-offset torque all turn the wheel towards the fall, and so each could be tested and found to act in the expected direction. What none could tell is whether that term, added to the others, puts all four roots on the left of the axis. The sum may have a sign that no term has alone. James Clerk Maxwell met the same difficulty in a steam engine’s regulator, as the governor that needed friction to work tells, and the root conditions he began were completed by Routh and Hurwitz. Stability in a system with four roots is a property of the whole, and it can be destroyed by strengthening a term that seems to act helpfully, by pushing another root across.

The other reason is one of simple error. Papers on the subject since Whipple often got the equations wrong, and the 2007 authors, in their survey of about thirty derivations, found few correct and several contradicting one another. Their publishing the result to fourteen digits, together with a recipe to reproduce it, ended that. It is a rare case in classical mechanics where a benchmark was needed in order to end a long disagreement over a system with only three degrees of freedom.

Riders and other departures #

A human is not a rigid mass rigidly attached to the frame. A rider moves relative to the frame, steers actively and looks ahead. Most of what riders do is control, and a bicycle that is not self-stable can still be ridden with little difficulty: skilled riders have managed bicycles that have been deliberately designed with no stable band. The eigenvalues, however, help to explain how easy a machine is to ride, since a stable or near-stable bicycle needs less control. Riding experts find that the feel of a bicycle tracks these values to some degree. A heavy rider’s mass also greatly changes the matrices and moves the band, usually downward in speed.

The model also ignores a good deal. Its wheels have knife-edge contact and do not slip. Real tyres have contact patches and flex, and these become important at high speed, as motorcyclists know. The frame is rigid, and a real frame has compliance that creates shimmy at certain speeds. Those effects enlarge a model considerably and require experimental data. Yet the Whipple model’s predictions of weave and capsize match the experiments quite closely, in the low-speed range that bicycles occupy.

No part of this applies at large lean angles: the linear equations describe only small disturbances, and they say nothing of what happens when a machine is heeled over in a turn. The band computed here says what happens in straight, steady running. It also does not cover how to turn. A rider must first steer slightly the opposite way to lean into a curve, by what is called countersteering, and that non-minimum-phase behaviour is also contained in the matrices. It is one of the features that make steering a bicycle harder to explain than to do.

Still open: why simple rules fail for such a small system #

The governing equations have been known since 1899 and verified since 2007. The eigenvalues can be computed by hand. Yet the question “what makes this stable?” has not reduced to a statement simpler than the list of matrix entries. Explicit Hurwitz conditions exist for a quartic, and they translate into inequalities in the twenty-five parameters, without obvious physical meaning. Efforts continue to find design rules that would help someone creating a new machine, and these have become more relevant as stability is designed in for folding bicycles, cargo bikes and electric scooters.

Meanwhile an experimental question remains about riders’ preferences, since whether a bike can be ridden hands-free is a different matter from whether it feels safe to steer, and engineers have no theoretical measure of handling quality that matches how riders rate a machine. It is a curious state of affairs for a machine with a long history: one of the most efficient forms of transport, whose dynamics are fully known, can be ridden by a child and still lacks a good answer to why it stays upright.

Linearised about upright rolling, a bicycle has four roots. The benchmark’s band between 4.292 and 6.024 m/s is where all four decay, so a shove dies and the machine rides itself, with the front steering under its own fall. Gyroscopes and trail each move that band, and a machine with neither has been built with a band of its own. There are four numbers and twenty-five parameters, and the one-sentence reason everyone wants does not exist.