6. No-hands Riding: Equilibrium and Steering

Introduction and synopsis

Already in the early days of bicycles people were fascinated by no-hands riding and attempted to understand the underlying physics. The fact that an unstable vehicle such as the bicycle can move on a stable trajectory, seemingly without any intervention of the rider, appeared to be miracle.

The first paper on no-hands riding based on a realistic model of bicycle geometry is probably due to Whipple (1) in the year 1899. In 1910 Klein and Sommerfeld (2) addressed no-hands riding in their classical treatise on the theory of the gyro. The conclusion to be drawn from this early work is that an autonomous bicycle without any active intervention of the rider has only a marginal velocity range in which it is stable. Unfortunately the seminal work of Klein and Sommerfeld did not succeed to define a state of the art for later work. Their results and conclusions were ignored by many later authors. In particular explanations of no-hands riding in physics textbooks and physics lectures are often outright wrong. To the knowledge of the author, there exists no consistent model of no-hands riding today. The purpose of this contribution is to close this gap.

First it has to be defined what needs to be explained.

• The bicycle has to be stable in the no-hands riding mode. This means that after a perturbation it returns to its original state.

• The rider has to have a possibility to steer the bicycle along a desired trajectory under no-hands riding conditions. This means that stable states different from σ = θ = 0 have to be possible.

The above statements already imply that we are dealing with a problem of control theory. The basic equation of motion of a bicycle is the equation of an inverted pendulum. This results in an instability which has to be counteracted by a control loop with negative feedback. Some of this negative feedback is provided by the steering system. If the frame is tilted, then a torque is induced on the handle bar, which turns the handle bar in the direction of the tilt. The resulting centrifugal force counteracts the capsizing torque and helps to stabilize the bicycle. The autonomous or automatic negative feedback, however, is in general not able to allow stable no-hands riding. At small speeds (less than about 14 km/h) the autonomous negative feedback becomes very large. This may lead to exponentially growing oscillations of the handle bar turn angle. Above 20 km/h the autonomous negative feedback is too small to stabilize the inverted pendulum. Active participation of the rider is thus necessary over most of the velocity range to stabilize no-hands riding.

Under no-hands conditions the system can suffer from three types of instability. The first is the self alignment instability. Below v_crit the symmetrical position of the handle bar is not stable. No-hands riding is thus impossible in this velocity range. In the expression below L is the distance between the contact points of front an back wheel respectively, Φ is the angle of the steering wheel with respect to horizontal and K is a correction factor. For Φ < 90° gyroscopic torques have a component in the steering axis. This is absorbed in the factor K. For typical bicycle parameters K ≈ 1.06 and v_crit is in the range 6 - 7 kph.

The second is the capsize instability. It occurs if the negative feedback is too small to stabilize the inverted pendulum (open loop gain smaller than 1). The third is the weaving instability characterized by exponentially growing rapid oscillations of the handlebar turn angle σ. The weaving instability is of little practical concern. Unless the negative feedback is very high, it will be suppressed by friction terms. Both instabilities will be discussed in detail later in the text.

All stability models based on gyroscopic effects miss a central point. A rolling wheel has a continuum of stable states. The autonomous bicycle has within its stability range only one stable state, the upright frame and zero turn angle. After a perturbation a rolling wheel changes from its original radius of curvature to a new radius. The autonomous bicycle returns to a straight trajectory.

No-hands riding thus has to be treated within the context of control theory. The feedback loop in the control circuit excludes angular momentum conservation and thus is in contradiction to gyroscopic models.

The output of the process to be controlled is the center of mass tilt angle θ_cm. Normally the desired output is θ_cm = 0. In this case the error signal is θ_cm. In general, the error signal is the difference between the target θ_cm and its actual value. The input parameter of the system is the frame tilt angle θ_f, which can be freely chosen by the rider. The rider acts as a controller which adjusts the input θ_f as a reaction to the error signal. The rider can tune the loop gain of the negative feedback loop by adjusting the amplitude of the reaction to the error signal, i.e. by adjusting the ratio θ_f/ θ_cm. As will be shown later, the ideal loop gain is slightly larger than one. If the loop gain is less than one the system will capsize, if it is too large, oscillatory instabilities may occur.

The criterion for stability against capsizing is G_loop(0) > 1, where G_loop(0) is the static loop gain in the feedback loop (including the rider). The static loop gain does not depend on dynamic parameters such as first derivative terms. This means that the famous gyroscopic term proportional to the frame tilt velocity does not enter the stability criterion at all. This term is is very often erroneously believed to be essential for stability of the bicycle. The fact, that it has nothing to do with stability against capsizing does not imply that it is unimportant. The dθ_f/dt gyroscopic term has an important influence on the form of the response to a transient as will be discussed later.

The rider is not an ideal controller. Its response to an error signal has a time lag. If the time lag is large, then oscillations will occur. It is shown that the maximum allowable time lag is such, that an experienced rider can easily avoid oscillations.

No-hands riding treated as a linear control system.

The control loop is shown in Fig. 6. The input of the system is the frame tilt angle θ_f. Within a linearized model, the equations of motion consist in the simplest form of two second order linear differential equations. The first equation relates the frame tilt angle θ_f to the handlebar turn angle σ. The second equation relates the handlebar turn angle to the center of mass tilt angle θ_cm. The two linear differential equations can be transformed into two transfer functions.

The active control by the rider is represented by the transfer function T_rider(s). The rider will attempt to control the frame tilt angle such that the desired center of mass tilt angle and handle bar turn angle (and thus radius of curvature) results. In the oftenused model of an autonomous bicycle with a purely passive rider T_rider(s) = 1 holds. At this point it is not necessary to give a definition what a transfer function is and what the parameter s means. It is sufficient to know that in a control loop the transfer function of a chain is the product of the transfer functions in the chain. Thus

resolved for the perturbation P(s) this results in

G_loop(s) = T_f-σ(s)T_σ-cm(s)T_rider(s) is called the open loop gain of the system, whereas G(s) is the closed loop transfer function. If e.g., p(t) is a Dirac pulse, then its Laplace transform P(s) = 1. G(s) is thus the Laplace transform of the response function of the system in reaction to a Dirac pulse. Stability requires that the response transient decays to zero. From control theory it is known that a system is stable, i.e. perturbations are damped out, if the poles of G(s) (values of s at which G(s) becomes infinite) are located on the left hand side of the complex plane, in other words if no pole exists at which s has a positive real part. The poles are located at the solutions of T_f-σ(s)T_σ-cm(s)T_rider(s) = 1. The stability criterion thus is fulfilled, if all solutions of the equation G_loop(s) = 1 have a negative real part.

Stability of the autonomous bicycle.

In the autonomous bicycle the rider has no influence on the trajectory whatsoever. Thus T_rider(s) = 1 and the expression for the loop gain reduces to G_loop(s) = T_f-σ(s)T_σ-cm(s).

We first discuss the capsize instability. In the mathematical section it will be shown that no pole at a positive real value of s exists, if the static loop gain G_loop(0) = T_f-σ(0)T_σ-cm(0) is larger than one. The stability criterion for capsizing is thus G_loop(0) > 1. This is identical to the static equilibrium condition already discussed in chapters 3 and 5. Below we repeat the equilibrium equations derived in the previous sections.

The equilibrium position of the turn angle at a given frame lean angle is given by:

The transfer function T_f-σ(0) thus becomes

The factor K absorbs the effect of the component in the steering axis of the gyroscopic torque of the front wheel (see chapter 5). J_w is the moment of inertia of the front wheel.

Since the trail Δ is negative, K is slightly larger than one, approximately 1.06 for typical bicycle parameters.

In a similar fashion the transfer function T_σ-cm(0) is found.

The second term in the parenthesis reflects the gyroscopic torque of the two wheels and is negligible. It will be omitted in all further calculations. Combining the two expressions we obtain the static open loop gain as:

At high velocities the 1 - v_crit²/v² term is insignificant. Since K > 1 and sin(ϕ) < 1 the static loop gain at high velocity is smaller than one and the autonomous bicycle exhibits a capsize instability. At small velocity the v_crit²/v² term dominates and as long as v > v_crit the static loop gain is larger than one and no capsizing occurs. The limiting velocity for stability against capsizing is found to be

Opposed to the general belief, the front wheel gyroscopic torque (contained in the factor K) has a negative effect on stability against capsizing. Also, in opposition to widespread folklore, the stability criterion against capsizing is purely static and does not contain first order derivatives such as the famous dθ_f/dt gyroscopic term. This term is often erroneously believed to be the main cause of stability against capsizing.

From the above we cannot conclude that the autonomous bicycle is stable in the velocity range v_crit < v < v_capsize. Even if the capsizing instability is suppressed, a weaving instability may occur. The capsizing instability and the weaving instability are mutually exclusive. A necessary condition for the weaving instability is G_loop(0) > 1 which implies stability against capsizing.

Since in the velocity range of capsize stability the feedback gain is larger than one, a phase shift of 180⁰ or more will lead to exponentially increasing oscillations. The equations of motion of a bicycle are such that only first order derivative terms can prevent the immediate onset of oscillations below v_capsize and thus prevent the direct transition from capsizing to weaving instabilities. Klein and Sommerfeld (2) have analyzed equations of motion without any dissipative terms and have shown that the dθ_f/dt gyroscopic term reduces the phase shift and thus has a stabilizing effect against the weaving instability. In historic bicycles with a heavy front wheel, it provided a stability island between 16 km/h and 20 km/h. Much more efficient in preventing oscillations are dissipation terms. This will be discussed later in more detail.

In conclusion it can be stated that the autonomous bicycle exhibits a capsize instability above about 20 km/h and tends to oscillatory instabilities below 16 km/h if dissipative terms are neglected in the equations of motion. Including friction terms shifts the onset of the weaving instability to lower velocities. Since no-hands riding is easily possible above 20 km/h, the autonomous bicycle is not a suitable model for no-hands riding.

Mathematical description of no-hands riding

This section is mathematically somewhat more complex than the rest of this site. The use of differential equations and of the Laplace transform formalism is unavoidable. To keep the formalism simple, the simplest model for the equations of motion is used. This is a model in which all mass is concentrated in the center of mass. More sophisticated model can be found in the paper by Meijnaard et al. (3).

The equation for the center of mass lean angle θ_cm as a function of the handle bar turn angle σ then becomes:

J is the moment of inertia of the system bicycle-rider with respect to the x (tilt)-axis, H is the z-coordinate of the center of mass, M the total mass, L the distance between the ground contact points of the two wheels, ϕ the angle of the steering axis and A the horizontal distance between rear hub and center of mass (see section 2). The first term in the equation represents the change in angular momentum with respect to the lean axis, the second the torques of the centrifugal force and kink force respectively and the last term the lean torque due to gravity.

By applying a Laplace transform, the differential equation in time space is transformed into an ordinary equation in terms of the Laplace variable s. Time derivatives transform into multiplications by the variable s. The equation can then be easily solved. The inverse operation, transforming the solution back into time space, is in general not possible in analytic form. To discuss the stability of a control system, a reverse transformation is not necessary. The stability can be investigated directly in Laplace space.

The Laplace transform of the above equation becomes:

Resolving this equation for θ_cm(s) results in the transfer function

The equation for the handle bar turn angle is

where θ_f is the frame lean angle, J_s the moment of inertia of the steering system, J_w the moment of inertia of the front wheel and λ a phenomenological friction term. λ results predominantly from the friction between front tire and ground when turning the handlebar. The second term is the famous gyroscopic term describing the torque resulting from tilting the frame. Please remember: The sign of the trail Δ is negative.

The first term in the above equations represents the change in angular momentum of the steering system, the second the gyroscopic torque proportional to the tilt velocity, the third is a phenomenological friction term, the forth the torque induced by gravity and the last the torque induced by the centrifugal force.

The Laplace transform becomes

resolving for for σ(s) leads to the transfer function

The open loop gain is given by

For positive values of s the slope is positive in the interval of interest. Thus a positive solution of G_loop(s) -1 exists only if G_loop(0) < 1. The stability criterion is a purely static criterion with no first derivative terms entering. In particular, the dθ_f/dt gyroscopic term is not contained in the criterion.

A loop gain higher than one excludes a capsize instability but not growing oscillations. Growing oscillations are a potential problem in the velocity range below 20 km/h. They correspond to complex solutions of G_loop(s) -1 with a positive real part. A plausible value of the friction term λ in G_loop(s) is sufficient to suppress the weaving instability above about 10 km/h.

For a more elaborate discussion of stability, including the weaving instability, we use the method of the Nyquist plot.

This method allows to determine the location of the poles of the function G(s) = G_loop(s)/(1-G_loop(s)) without computing the poles. A pole in G(s) corresponds to a zero in the function F(s) =(1-G_loop(s)). The Nyquist plot is based on plotting the frequency dependence of F(s), i.e., F(iω) with ω running from - infinity to + infinity with the imaginary part of F(iω) in the vertical axis and the real part in the horizontal axis. Because the transfer functions are Laplace transforms, G_loop(s) approaches zero for infinite values of s. The Nyquist curve starts at 1 and ends at 1 and thus forms a closed loop.

Based on the argument method of function theory it can be shown that the number of clockwise encirclements of the origin equals the number of zeros of F(s) in he right-hand plane plus the number of poles in the right-hand plane.

In our case the transfer function T_σ-cm(s) is unstable and has a pole on the right-hand side at s = sqrt(HMg/J). The other transfer function T_f-σ(s) is stable for v > v_crit. It has no pole on the right-hand side. Thus the system is stable, if the number of clockwise encirclements of the origin equals one.

Fig. 11 demonstrates that stability against oscillations can be reached with modest and realistic damping rates. The main effect of the damping is to shift the intersection point on the x-axis to the right. If, on the contrary, the damping is decreased, the intersection point moves to higher negative x-values and becomes singular for zero damping. This illustrates the artificial character of a system without damping.

In the absence of the dθ_f/dt gyroscopic term, the red eigenvalue would remain oscillatory (complex conjugated) up to 20 km/h instead of 17 km/h, but still with a negative real part and thus stable. The transition to a capsize instability is unaffected and remains at 20 km/h.

Active no-hands riding

As shown in the previous section, stability against capsizing is impossible above about 20 km/h for the autonomous bicycle. This contradicts experience. In reality no-hands riding becomes easier with increasing velocity even above 20 km/h. This is due to the active interaction of the rider with the bicycle. The rider has basically one degree of freedom, the frame lean angle ϴ_f which can be chosen independently of the center of mass lean angle. In our simple model all body motions of the rider are absorbed into the parameter ϴ_f.

The rider must solve the following stability problems:

• Preventing capsizing. This means ensuring that G_loop(0) is equal or larger than one.

• Enabling steering in the no-hands riding mode.

Preventing the weaving instability is not an issue. The weaving instability is automatically suppressed mainly by the friction term and to a lesser degree by the dϴ/dt gyroscopic term.

We first address stability against capsizing. G_loop(0) > 1 can be achieved at all speeds above the critical velocity by

The simplest approach is to assume a constant T_rider(s) = T_rider(0). T_rider(0) is simply the ratio between the center of mass lean angle and the frame lean angle chosen by the rider.

G_loop(0) can thus be adjusted to any desired value at any speed by adjusting the frame lean angle.

Realistic model of active no-hands riding

As stated above, the autonomous bicycle is not stable with respect to capsizing for velocities above 20 kph. In this velocity range the rider has to actively stabilize the system. This can be done by properly adjusting the frame lean angle with respect to the center of mass lean angle. However, the assumption of a frequency independent correction factor T_rider = θ_f/θ_cm used in the previous section is not realistic. In the following we assume that the system is subject to a time dependent disturbance induced for instance by a wind gust, by surface irregularities, by pedaling etc. This induces deviations of the lean angle ϴ_cm from zero. The target of the rider is to keep ϴ_cm(t) as small as possible. To model this we make the following assumptions: i) T is the target value for θ_f/θ_cm ii) the rider reacts to the perturbation with a finite reaction time iii) in the absence of actions of the rider the system is stiff (θ_f = θ_cm).

The time lag is modelled by a memory function. It takes into account that after a perturbation pulse the system is initially stiff and will the approach the target value of T within a characteristic time τ_rider.

For M(t) the simplest form is used.

The consequence of the memory function is, that θ_f(t) lags behind the value targeted by the rider by a characteristic time τ_rider. After a Laplace transform we obtain the rider transfer function as

T_rider is no longer simply a multiplication factor, but depends on s. For long times t >> τ_rider (corresponding to s = 0) it reduces to the value without time lag T_rider(0) = T, for short times t << τ_rider (corresponding to s >> 1/ τ_rider) it is equal to one. The open question is, whether stability can be achieved with a reasonable reaction time of the rider.

The green curve shows the autonomous bicycle (T = 1) which exhibits a capsize instability (curve does not encircle the origin). The red curve shows active no-hands riding (T = 1.3) with a reaction time of the rider of 0.5 sec. The curve encircles the origin clockwise, the system is stable. In the case of the blue curve, the reaction time of 4 sec is too long to attain stability. The curves cross at the x-axis at the right side of the origin and thus circles the origin counterclockwise.

Fig. 15 confirms the result of the Nyquist diagram of Fig. 14. The system is stable up to a reaction time of more than 1 sec. For longer reaction times the system behaves similar to an intoxicated rider. The control loop is too sluggish to balance out perturbations.

In conclusion no-hands riding above 20 km/h is unproblematic. Approporiate adjustments of the frame lean angle by the rider provide a robust method to regain and maintain stability and to steer the bicycle into turns.

Steering in the no-hands mode

Up to now, only stable states with ϴ_f = ϴ_cm =0 were considered. Stability against capsizing requires G_loop(0) larger or equal to one. If it is larger than one, then the only stable state is ϴ_f = ϴ_cm =0, which is a straight trajectory. In order to steer or maneuver, the rider has to sustain G_loop(0) = 1 to keep a given radius of curvature stable and to deviate slightly from G_loop(0) = 1 to steer the bicycle. This is not an easy task and restricted to expert no-hands riders. A stable predefined radius of curvature can be achieved by a simple proportionality control of the form:

σ_target is the targeted handle bar turn angle and c an amplification factor. In order to steer the bicycle into a stable turn, the rider first has to introduce a perturbation, i.e. to lean the frame into the desired direction and then to tune the lean angle to reach the loop gain of the above expression.

Fig. 17 shows that a simple control function is sufficient to stabilize the bicycle at a given handle bar turn angle and thus at a given radius of curvature.

The role of the dθ_f/dt gyroscopic term

There has been a considerable amount of discussion and controversy about the role of this term. In some textbooks it is considered to be the dominant term providing bicycle stability. It is argued that a lean of the bicycle induces a torque on the handlebar in the direction of the lean. This torque is supposed to turn the handlebar such, that the resulting centrifugal force straightens the bicycle. It is trivial to see the error in this argument. If the equation of motion is integrated, then a lean angle does not induce a turn angle, but a turn velocity. This also implies that if the lean angle returns to its original value after an excursion, the turn angle does not. The torque induced by the return only stops the turn velocity.

In this section it was shown, that the term has no effect on the capsize instability. The stability against capsizing is given by purely static parameters. First derivative terms do not enter.

Klein and Sommerfeld have shown that in the absence of the dθ_f/dt term the autonomous bicycle without dissipative terms would be instable at all speeds. The dθ_f/dt term can provide a small stability island for 16 km/h < v < 20 km/h by suppressing the weaving instability in this range. In this section it was shown that the essential term to obtain stability against weaving is a dissipative term. It results mostly from friction between the front tire and ground. The conclusion thus is that the dθ_f/dt term is irrelevant for stability of the bicycle.

The role of the dθ_f/dt term becomes clear by looking at the response function of the steering system to a Dirac pulse. The response function is obtained from an inverse Laplace transform applied to the transfer function of the steering system. The result is somewhat complex, but can be represented in the form:

In the above equation the second term is due to the dθ_f/dt term. It introduces a phase shift into the otherwise purely sinusoidal expression. The cosine part is an instant reaction of the system to the pulse. Without the damping term, however, the system would be purely oscillatory. The dθ_f/dt term has a stabilizing effect by reducing the time lag of the handlebar angle to a perturbation. Since it is not dissipative, it cannot efficiently dampen oscillations. During the negative slope of the curve in Fig. 16, the gyroscopic torque changes sign and slows down the return to zero.

Fig. 18 refers to the velocity of 15 kph, a velocity at which the autonomous bicycle is stable with respect to capsizing. The effect of perecession is most pronounced in the phase of negative sign of dθ_s/dt. In this phase the precession term has to opposite sign compared to the centrifugal term. It efficiently dampens overshoot and fastens return to equilibrium.

In the velocity range in which the autonomous bicycle is stable, the positive effect of the precession to the pulse response function has the character of "nice to have". It makes no hands riding easier, but is not mandatory.

Above 20 kph the autonomous bicycle is instable. There exists a positive real solution. It turns out, that precession is crucial for no hands riding at v > 20 kph.

Opposite to the "nice to have" effect in the velocity range in which the autonomous bicycle is stable, precession is crucial in the instable range in which the rider has to actively provide stability.

To my knowledge nobody has realized before the importance of precession at v > 20 kph. The most important contribution of precession to bicycle riding has gone unnoticed.

References

(1) F. J. W. Whipple. The stability of the motion of a bicycle. Quarterly Journal of Pure and Applied Math., 30, pp. 312–-348 (1899)

(2) F. Klein und A. Sommerfeld. Über die Theorie des Kreisels, Heft IV.Teubner, Leipzig, 1910. pp. 863–-884

Download Paper of Klein und Sommerfeld (warning: 17 MB!!!)

(3) J. P. Meijnaard, Jim M. Papadopoulos, Andy Ruina and A. L. Schwab, Proc. R. Soc. A 463, 2007 (2084): 1955–1982

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