## Abstract

We write nonlinear equations of motion for an idealized benchmark bicycle in two different ways and verify their validity. We then present a complete description of hands-free circular motions. Three distinct families exist. (i) A handlebar-forward family, starting from capsize bifurcation off straight-line motion and ending in unstable static equilibrium, with the frame perfectly upright and the front wheel almost perpendicular. (ii) A handlebar-reversed family, starting again from capsize bifurcation but ending with the front wheel again steered straight, the bicycle spinning infinitely fast in small circles while lying flat in the ground plane. (iii) Lastly, a family joining a similar flat spinning motion (with handlebar forward), to a handlebar-reversed limit, circling in dynamic balance at infinite speed, with the frame near upright and the front wheel almost perpendicular; the transition between handlebar forward and reversed is through moderate-speed circular pivoting, with the rear wheel not rotating and the bicycle virtually upright. Small sections of two families are stable.

## 1. Introduction

The dynamics of idealized bicycles is of academic interest due to the complexities involved in the behaviour of this seemingly simple machine, and is also useful as a starting point for studies of more complex systems, such as motorcycles with suspensions, flexibility and real tyres. In addition, reliable analyses of bicycles can provide benchmarks for checking general multibody dynamics simulation software. Meijaard *et al*. (2007) provide a detailed review of the scattered bicycle dynamics literature from 1869 to the present, with over 80 references. They then present multiply verified equations and analyses of near-straight hands-free motions of two benchmark bicycles. Here we present a detailed study of the *circular* hands-free motions of the first of these benchmark bicycles.

Of the many published analyses of a rigid-wheeled bicycle's near-straight motions, only a few are both general and correct (Meijaard *et al*. 2007). For circular motions, the analytical literature is even smaller and less verifiably correct. The fully nonlinear equations for bicycles are long, and their explicit form varies with the approach used in deriving them. For example, the two independent (and numerically cross-verified) sets of equations presented below differ greatly in length and defy full manual comparison. We advocate the view, therefore, that numerical agreement with many digits (say, 10 or more) between outputs (accelerations) from two sets of equations for several sets of *randomly* generated inputs may be taken as a reliable demonstration of equivalence, and we will use such checks below. In this light, we note that full nonlinear equations and/or simulations are presented by, among others, Collins (1963), Roland (1973), Psiaki (1979), Franke *et al*. (1990) and Lennartsson (1999), but we are unable to comment on their correctness here beyond a qualitative comparison between their results and ours. We also acknowledge the significant amount of work done on motorcycles, using more complex and therefore less easily verifiable models (e.g. Cossalter & Lot (2002) and Meijaard & Popov (2006), as well as references therein).

The topic of hands-free circular motions might be initially misunderstood because human riders can easily follow a wide range of circles at quite arbitrary speeds. But, in general, this is possible only by imposing handlebar torques or upper-body displacement from the symmetry plane. With zero handle torque (i.e. hands-free) and a centred rider, only a few discrete lean angles are possible at each speed.

We mention for motivation that straight-line motions of most standard bicycles, including the benchmark bicycle with its handlebar forward (HF) or reversed (HR), have a finite range of stable speeds. A bifurcation known as *capsize* occurs at the upper limit, where (by linear analysis) steady turns at all large radii can be sustained with zero steering torque. These HF and HR bifurcations are the origins of two distinct circular motion families.

Prior work on hands-free circular motions of bicycles and motorcycles is limited, incomplete and occasionally arguable in terms of the conclusions presented.

Kane's paper (Kane 1977) on steady turns of a motorcycle with front and rear point masses, linearized in the steer angle but not in the lean, contains plots of steer angle (to 12°) versus steer torque (which we take as zero), parameterized by lean, steer and turn radius. The zero-torque axis seems difficult to relate to our results below. Later, Man & Kane (1979) gave a fully nonlinear treatment, also incorporating distributed masses, tyre slip relations and the possibility of rider lean relative to the frame. The results in figs. 5 and 6 of that paper (and associated text) seem to imply that any desired turn radius can be achieved at a given speed, without altering the steer angle, and indeed that turn radius at a given speed is independent of the steer angle. This result, arguably lying outside normal bicycling experience, may be due to details of tyre slip modelling. No evidence of bifurcation from straight motion or other hands-free turning is apparent.

Psiaki (1979) found one solution family (off point B in figure 5), calculated eigenvalues and reported stability up to a lean of approximately 18°. The mechanical parameters of the bicycle were different, and speeds studied ranged from the capsize speed to approximately 2% below it. Our corresponding solutions below are unstable.

Franke *et al*. (1990) studied the nonlinear motions of a bicycle, modelled using some point mass simplifications, without providing full details of mechanical parameters, and allowing for lateral displacement of the rider. Their results (for zero rider displacement) show only one solution family (corresponding to our curve BA in figure 5), which like ours is unstable.

Cossalter *et al*. (1999) evaluated steady turns of a motorcycle with toroidal wheels and various tyre parameters. In the upper left plot of their fig. 5 (rigid tyres), a line of zero steer torque is plotted on axes of curvature and speed. It seems equivalent to most of our curve BA, with a bifurcation of around 5 m s^{−1}, and with turn radius decreasing to approximately 5 m at a speed of 4 m s^{−1}. No other circular motion families are apparent.

Lennartsson (1999) presented a thoroughgoing analysis of circular motions of a bicycle, with different parameters, where he finds all but one of the solution families, which we will present later (our curve CDE in figure 5—even though he separately determines the HR bifurcation that gives rise to it). For yet other parameter values, Aström *et al*. (2005) again overlooked the same family.

In this context, precise numerical values of coordinates and velocities corresponding to several circular motions of the benchmark bicycle will serve a benchmarking purpose of their own. With this motivation, as also to present for the first time a complete picture of the circular motions of at least one idealized bicycle, we take up for study the primary benchmark bicycle of Meijaard *et al*. (2007).

We find, nominally, four different one-parameter families of circular motions of the benchmark bicycle, most easily understood through the eight limiting points they connect pairwise. Physically, two of these families merge into one, leaving three in all. We also examine the stability of these families, and find only two small intervals of stability (for the benchmark bicycle parameters).

Beyond the confirmed nonlinear equations, and some precise results for hands-free turns, the main contribution of this paper is to describe previously unknown non-trivial limiting cases and the circular motion families that connect them.

## 2. Mechanical model, coordinates and notation

Our bicycle model is mechanically identical to that of Meijaard *et al*. (2007), though our chosen coordinates and symbols differ on the surface, as enumerated in the electronic supplementary material.

The bicycle model has four rigid bodies: a rear frame with rider rigidly attached; a front frame (fork and handlebar assembly); and two wheels. These are connected by frictionless hinges at the steering head and two wheel hubs. The axisymmetric wheels make knife-edged dissipationless no-slip contact with the horizontal ground. There is no propulsion. The parameter values used here also have lateral symmetry, though our equations allow asymmetry.

We begin with nine generalized coordinates, not all independent. Three coordinates *x*, *y* and *z* specify the position of the centre of the rear wheel in a global reference frame XYZ (figure 1). The same point is attached to the rear frame as well. Three angles *θ*, *ψ* and *ϕ* (in (3, 1, 3) Euler angle sequence as described below) specify the rear frame configuration. The rear wheel rotation relative to the rear frame is *β*_{r}. The front fork rotation (steering angle) relative to the rear frame is *ψ*_{f}. Finally, the front wheel rotation relative to the front fork is *β*_{f}.

There are six constraints on the bicycle. Of these, two are holonomic, requiring the two wheels to touch the ground. Four are non-holonomic, expressing no-slip at each wheel (two equations per wheel).

Note that we have introduced at least one unnecessary coordinate, namely *z*. It is eliminated in Lagrange's equations below using the holonomic constraint of normal contact between the rear wheel and the ground, but retained in the Newton–Euler equations. Another holonomic constraint, involving normal contact between the front wheel and the ground, could in principle be used to eliminate one more generalized coordinate (such as the third Euler angle *ϕ*), but the contact condition is analytically complicated and this constraint is retained in Lagrange's equations below along with the four non-holonomic constraints of rolling without slip.

We now describe how an instantaneous configuration of the bicycle is obtained, using the generalized coordinate values, starting from the reference configuration. In the non-standard reference configuration chosen here (figure 1), the bicycle is flat on the ground. Its lateral symmetry plane coincides with the *XY* plane.

The rear frame is first rotated by *θ* about *e*_{3} (the *Z*-direction). This determines the eventual heading direction through a yawing motion. (In discussing rear frame rotations, it may help to imagine a point like P being held fixed, though the rotation matrix does not depend on it.) Next, the rear frame is rotated by *ψ* about the body-fixed *e*_{1} axis (what was *X* before the first rotation). This determines the lean of the bicycle through a rolling motion. Finally, the rear frame is rotated by *ϕ* about the body-fixed *e*_{3} axis (what was *Z* at the start). This rotation (pitching motion) is not arbitrary: it will eventually be determined by front wheel-to-ground contact. Holding the rear frame fixed, the rear wheel is rotated by *β*_{r}, the front fork by *ψ*_{f} and the front wheel relative to the front fork by *β*_{f}. After these rotations, the bicycle is translated so that the rear wheel centre is at (*x*, *y*, *z*). Now the bicycle is in the instantaneous configuration. A straight-running bicycle will have *ψ*=*π*/2 (initially putting the bicycle below ground) and *ϕ*=*π* (bringing the bicycle back up to point along the negative *x*-axis).

Note that *z* and *ϕ* must ensure contact between the wheels and the ground (*ψ* determines *z* and *ψ* and *ψ*_{f} together determine *ϕ* as discussed later). The bicycle thus has seven independent configuration variables, though we use nine for convenience.

Finally, we describe here our notation for rotation matrices, in the form of products of matrices corresponding to rotations about known axes. Given vectors ** a** and

**, the cross product**

*b***×**

*a***is written using matrix components as**

*b**S*(

*a*)

*b*, whereLet a body rotate about unit vector through angle

*θ*. A vector

**fixed in the body then gets rotated to**

*r***′=**

*r**R*(

*n*,

*θ*)

**, where the rotation matrixIn particular,**

*r***may be the position vector from any point fixed in the body to any other point fixed in the body, with neither necessarily lying on the axis of rotation.**

*r*## 3. Lagrange's equations of motion

Finding Lagrange's equations for the bicycle is routine, if tedious and error-prone. Details of the calculations outlined below may be found in Basu-Mandal (in preparation) and in the electronic supplementary material for this paper.

First, the kinetic and potential energies of the system at an arbitrary configuration are found. This is straightforward and not presented here to save space.

For the contact constraint equations, we equate the vector velocities of the ground contact points to zero, giving six scalar equations, including four no-slip conditions and two vertical direction equations, which are differentiated holonomic constraints. Of the latter two, the one for the rear wheel is easily integrated and is used in our Lagrangian formulation to eliminate *z* in terms of lean *ψ*; the one for the front wheel is retained in differentiated form due to analytical difficulties.

Thus, for Lagrange's equations, we retain five velocity constraints.

We now have eight degrees of freedom (*z* being eliminated). Very long equations of motion are found in the usual way (the Matlab m-file is 3.5 MB in size; for the Maple file, see electronic supplementary material).

## 4. Numerical solution of the nonlinear equations

We have eight second-order ordinary differential equations (ODEs; Lagrange's equations) and five velocity constraints which we can differentiate to get second-order ODEs also.1 The original eight ODEs also have five Lagrange multipliers: there are 13 unknowns and 13 equations.

Of the 16 *initial conditions* needed to specify the initial-value problem, five velocities are obtained from the five wheel constraint equations. Of the remaining 11 initial conditions (eight coordinates and three velocities), we can arbitrarily specify only 10 because *ψ*(0) and *ψ*_{f}(0) determine *ϕ*(0). We choose to solve for and from the velocity constraint equations in terms of the remaining, arbitrarily specified, initial conditions.

We now consider calculation and uniqueness of *ϕ*(0). Figure 2 shows a view in which the front wheel and the ground plane both reduce to lines. We have(4.1)where P, S and Q are as shown in figure 1; *R*_{rf} and *R*_{ff} are the rear frame and front frame rotation matrices; and (for notation) is the position vector from P to S when the bicycle is in the reference position.

Also, , where *r*_{2} is the radius of the front wheel and . The *z* component of , equated to zero, yields an implicit relation in *ϕ*, *ψ* and *ψ*_{f} only, free of velocities. Thus, *ϕ* is determined by *ψ* and *ψ*_{f}.

There are eight possible configurations in which the wheels could touch the ground, as sketched in figure 3, with only one of them being of actual interest. For a vertical, straight-running bicycle, the appropriate initial condition is *ϕ*(0)=*π* and *ψ*=*π*/2 (figure 3, leftmost). This is because, in the reference configuration (figure 1), both wheels touch the *x*-axis and the bicycle lies in the negative *y* half-plane. Choosing *ϕ*(0)=0 here corresponds to figure 3, second from left. Only these two are recognized by our formulation (*lowermost* point of the wheel touches the ground).

In general (non-vertical, non-straight) initial choices of coordinates, there again seems to be two possible choices of *ϕ*(0) (the first two cases of figure 3). Choosing the one closer to *π* usually gives the solution of interest, which we verify *post facto* from animation (electronic supplementary material). We ignore the cases where no solutions for *ϕ*(0) exist.

With the above determination of *ϕ*(0), we can now numerically integrate the equations of motion. A nonlinear simulation of the benchmark bicycle is provided by Meijaard *et al*. (2007). On simulating the bicycle using identical initial conditions, we obtain a visual match with their results (see our electronic supplementary material and fig. 4 of their paper). We will make more accurate comparisons in §5.

## 5. Newton–Euler equations

We now obtain the equations of motion using the Newton–Euler approach, which is easily programmed into, say, Matlab for a fully numerical evaluation of the second derivatives needed for numerical integration. Though straightforward, the implementation is less routine than Lagrange's equations and is presented here fully.

### (a) Momentum balance

We begin with free body diagrams of the rear wheel, the rear wheel and the rear frame, the front wheel and the entire bicycle (figure 4*a–d*).

Consider the rear wheel (figure 4*a*). There are three forces acting on it: its weight through P; an axle force also at P; and a force at the ground contact R. There is also an unknown bearing moment with no component along the bearing axis . Angular momentum balance about P giveswhich in matrix notation is(5.1)where *S* is a skew symmetric matrix and ** α** is the angular acceleration (a vector). We eliminate by taking the dot product with , obtaining a scalar equation (rearranged so that unknowns are on the left-hand side)(5.2)

In equation (5.2), we could write in place of , but keeping the unknown *F*_{rw} on the right-hand side eases assembly of the matrix equations.

We now consider angular momentum balance for the rear wheel and the rear frame, together, about the point S on the fork axis (figure 4*b*). Eliminating *M*_{1} by taking the dot product with , we have the scalar equation(5.3)where ** a** is the acceleration (a vector). For clarity, we mention the roles of various terms in equation (5.3). The first term on the left-hand side and the first two on the right-hand side involve the form

**r**×

**F**(moment of a force). Each rigid body contributes the terms , of which the first appears on the left-hand side and the second on the right-hand side. The last two terms on the left-hand side are

**r**×

*m*

**a**terms, one each for the rear frame and wheel.

Angular momentum balance for the front wheel about Q (figure 4*c*) gives(5.4)

Now consider the entire bicycle (figure 4*d*). Linear momentum balance gives(5.5)where *m*_{tot} is the total mass. Angular momentum balance about G gives(5.6)The two momentum balance equations for the entire bicycle have three scalar components each. Thus, so far we have nine scalar equations.

### (b) Constraint equations

The constraints are the same as before, but now we will not eliminate *z*. That is, we will retain six constraint equations. Similar to differentiation of five velocity constraint equations above, we now directly differentiate two vector equations.

To begin, we write (in figure 2, read ‘rear’ in place of ‘front’, P in place of Q and R in place of F)(5.7)whereThe velocity of P can be found in two wayswhere is zero (no-slip contact). Differentiating and rearranging, we get(5.8)in matrix notation, where and is(5.9)Similarly, the front wheel constraint equation in matrix form is(5.10)with similar expressions for , and . We do not reproduce them here. We thus have six more scalar equations (equations (5.8) and (5.10)).

So far, we have the following unknown vectors: , , , , , , , , and . Thus, we have 30 scalar unknowns and 15 equations so far. More equations come from kinematic relations among the unknowns.

### (c) Further kinematic relations

We begin withDifferentiating, we get (in matrix notation)(5.11)

Similarly, we can write relations between *a*_{H} and *a*_{G} and between *a*_{Q} and *a*_{H} as(5.12)(5.13)

With nine more scalar equations (equations (5.11)–(5.13)), we still need six.

The angular accelerations of the four rigid bodies are interrelated. We haveDifferentiating with respect to time, we have (in matrix notation)(5.14)

Similarly, we can write(5.15)(5.16)Thus, we have introduced three more unknowns, , and , giving a total of 33 unknowns. But we have added equations (5.14)–(5.16), all vector equations. So we now have 33 independent simultaneous linear algebraic equations as well.

A key step remains. We have , the angular acceleration of the rear frame. But we still need , and . For the rear frame, it can be shown thatof the form . Then The above can be used to solve for , once *α*_{rf} is known from Newton–Euler equations.

We now have a numerical procedure for obtaining the second derivatives of the system coordinates. The choice of initial conditions is subject to the same constraints as for the Lagrange's equations case, except that initial conditions for *z* and must also be given and must satisfy the system constraints.

### (d) Verification

For several arbitrary (random) choices of coordinates and their first derivatives, the second derivatives obtained from the Lagrange and Newton–Euler sets of equations matched to machine precision (table 1). We conclude that our two sets of equations are equivalent. We will verify in §6 that our Maple equations, linearized about straight motion and with numerical benchmark parameters inserted, match the numbers of Meijaard *et al*. (2007).

## 6. Stability of straight motion

To study the stability of straight motion, we use our Lagrangian equations. Since the equations are very long, we substituted the benchmark system parameters before the linearization.

Let *ϵ* be infinitesimal, *v* be the nominal forward velocity of the bicycle and *t* be time. We make the following substitutions into the equations of motion:Setting *ϵ*=0 gives the solution corresponding to exactly straight-line motion.

The constraint forces (or *λ*s) still need to be found, from a set of linear equations. Barring *λ*_{5}=309.30353…, all are zero. Accordingly, we substituteWe now expand the equations of motion about *ϵ*=0 and drop terms of (*ϵ*^{2}). The (*ϵ*) terms give the linearized equations of motion (omitted for brevity).

Note that we do not differentiate the five velocity constraint equations. Instead, we solve them for , , , and , and substitute into Lagrange's eight equations of motion, differentiating as needed (e.g. differentiating where is needed). We solve the resulting eight linear equations for the five , , and . Of these, we find . There remain two equations giving and . Dropping tildes, they are (we retained more decimal places than shown here)(6.1)

(6.2)

These equations match with those of Meijaard *et al*. (2007) completely and the eigenvalues obtained from the two systems match to 14 decimal places. In particular, straight motion of the HF bicycle is stable at speeds between 4.2924 and 6.0243 m s^{−1}. Further straight-motion stability results are therefore not presented here.

## 7. Circular motions

### (a) Finding hands-free circular motions

In circular motions, *x* and *y* vary sinusoidally and *z* is constant. The rear wheel centre traverses a circle of radius (say) *R*. The first Euler angle *θ* (heading) grows linearly with time. The second Euler angle *ψ* (roll), the third Euler angle *ϕ* (pitch), the steering rotation angle *ψ*_{f} and the wheel spin rates and are all constant.

We seek a triple subject to some conditions, dependent on a free parameter *R*, as follows.

(i) Given *ψ* and *ψ*_{f}, *ϕ* is found as discussed earlier. (ii) *ψ* determines *z* (used in our Newton–Euler equations) and . (iii) The initial values of *θ*, *x*, *y*, *β*_{r} and *β*_{f} are arbitrarily taken as zero. (iv) Setting , and with given (or chosen), the velocity constraint equations give , , , and . (We find that ). (v) Having all initial conditions required for the Newton–Euler equations, we find the second derivatives of the coordinates. (vi) Finally, we define a vector function with *R* as a parameter(vii) We numerically find an *R*-parameterized family of zeros of the above map, where steer and lean acceleration vanish.

As a preamble to what follows, note that the existence of one circular hands-free motion implies the existence of three others, by symmetry. First, one may create a mirror image of the configuration, for example, leaning and steering rightwards instead of leftwards. Second, the rotational velocities of both wheels may be reversed, without affecting inertial forces and moments.

### (b) Plotting hands-free circular motions

The number of families of circular motion solutions, not counting symmetries, is three or four depending on how we count. Consider, initially, the schematic in figure 5*a*,*b*, depicting lean and steer, respectively, as a function of front wheel speed.

In figure 5*a*, lean is defined as *ψ*−*π*/2, positive when the bicycle leans left, and shown here from 0 (upright) to *π*/2 (flat on the ground). In figure 5*b*, steer angle is plotted from 0 (straight) to *π*/2 (perpendicular leftwards). The front wheel speed (really, speed of the front contact point) is , and is plotted from 0 to ∞ in a non-uniform scaling ( is obtained from using the velocity constraint equations).

For the two thick curves (one solid and another dashed), and is plotted instead. For these thick curves, corresponds to the bicycle moving forward with a reversed handlebar (i.e. turned beyond *π*/2). In figure 5*b*, for the thick curves corresponding to reversed handlebars, *π* has been added to the steer *ψ*_{f}.

For visualization, separation between nearly coincident portions of light and thick curves is exaggerated. Points E and F coincide in reality, as do G and H.

Handlebar asymmetry plays a role in the solutions obtained. Turning the handle by *π* (i.e. reversing the handlebar) effectively gives a slightly different bicycle. If the front wheel centre was exactly on the fork axis, the front fork plus handlebar centre of mass was exactly on the fork axis, and an eigenvector of the front fork assembly's central moment of inertia matrix coincided with the front fork axis, then handlebar reversal would give exactly the same bicycle. The reader might wish to consider the thick lines in figure 5 not as alternatively plotted curves at all, but regularly plotted curves for a different bicycle whose front assembly is a reversed version of the benchmark's. Here, we avoid this expedient in favour of consistency.

### (c) Limiting motions

We now discuss various special motions and limiting configurations which help to understand the hands-free circular motions of the benchmark bicycle.

#### (i) HF and HR bifurcations to large-radius turns

By equations (6.1) and (6.2), four eigenvalues govern stability of straight motion. For hands-free circular motions with very large radii, all sufficiently small leans must give steady solutions, implying a zero eigenvalue. There is only one such point with handlebar forward (HF): the ‘capsize’ bifurcation at the upper limit of the stable speed range for straight motion (noted by Whipple 1899). With the handlebar reversed (HR), there is a similar stable speed range with its own capsize point. These points are labelled B and C in figure 5. Near these points, lean and steer (off 0 or −*π* as appropriate) are both small and the radius is large.

#### (ii) HF and HR flat spinning

There are limiting circular motions where the lean approaches *π*/2 (i.e. lying flat on the ground), steer approaches zero, turn radius approaches a small finite limit and velocity approaches infinity. (The contact points are substantially displaced around each wheel.) Such solutions exist for both HF and HR configurations, i.e. with steer approaching 0 and −*π*. These points are labelled F and E, respectively, in figure 5. These limits require infinite friction.

#### (iii) HF upright static equilibrium

With the handlebar turned almost *π*/2, there is a static equilibrium with a lean angle of *exactly* zero, labelled A in figure 5. With a symmetric handlebar, it is obvious that an upright equilibrium exists with exactly *π*/2 steer. But with the left–right asymmetry due to finite steer of an asymmetric handlebar, it seems surprising that the lean remains exactly zero, because equilibrium implies two conditions (handlebar torque and net bicycle-tipping torque both zero) on the one remaining variable (steer). To understand this, imagine locking the handlebar at a variety of near-*π*/2 steer angles, at each of which the equilibrium lean angle is determined. Select the locked steer angle that gives tipping equilibrium with zero lean. Equilibrium ensures zero net moment on the bicycle about the line (say *L*) of intersection between the rear frame symmetry plane and the ground. Of all forces on the bicycle, the two not in the symmetry plane are the front fork assembly weight and the front wheel ground contact force: they are therefore in moment equilibrium about *L*. But these two forces are then in moment equilibrium about the handle axis as well, and locking is not needed, explaining the zero lean at A.

#### (iv) Pivoting about a fixed rear contact point

It seems possible, for each steer angle, to find an angle of lean such that the normal to the front wheel rolling direction passes through the rear contact; in such a configuration, the bicycle rotates about the rear contact, which remains stationary. One imagines that by properly choosing both the steer angle and the front wheel speed, we might simultaneously achieve roll and steer balance with the rear contact at rest. Such a motion does exist: the steer is close to *π*/2 (handle turned left); the rear frame is nearly upright; the front wheel follows a circle at a definite speed; and the bicycle pivots about a vertical axis through the rear wheel contact. Such motions were found for both HF and HR configurations (steer: −*π*/2). Nearby points, defined for plotting convenience at exactly zero lean, are labelled G and H, respectively, in figure 5. Noting that all motions of the bicycle are time-reversible, a pivoting motion can be reversed to give another motion where the front wheel speed and the handlebar are both reversed. Thus, the HF and HR pivoting solutions actually coincide, as do G and H, although they are sketched distinct for visualization.

#### (v) High-speed dynamic equilibrium

Envisioning that terminal points occur in pairs, an expected eighth is found as K in figure 5. This configuration involves small lean, near-*π*/2 steer and speed approaching infinity. It may be viewed as a perfectly dynamic counterpart to the static solution at A. It seems that the normal ground force at one wheel must become negative beyond some high speed for such a motion, but our analysis assumes sustained contact and ignores this question.

### (d) Description of the circular motion families

We can now connect appropriate pairs of endpoints to describe four circular motion families found for the benchmark bicycle.

One HF family connects points B and A. The bicycle first bifurcates from HF straight motion, with steer and lean increasing while speed decreases, until a maximum lean angle is reached. Thereafter, steer continues to increase and velocity continues to decrease, while lean decreases towards upright. The final perfectly upright state is approached via extremely slow motion, superficially like the pivoting points G=H but with the rear contact not quite fixed.

An HR family starts at C, bifurcating from HR straight motion. First, the previous pattern is followed (attaining a maximum lean with continuously decreasing speed), but then at a near-cusp point labelled D, a qualitatively different curve is followed. The steer then decreases towards HR straightness, while lean and speed increase, as the bicycle approaches the flat and fast limit point E.

A third circular motion family, for continuity in the discussion, may be thought of as starting from the HF flat and fast limit F (the path radii of the rear wheel centre differ at F and E). Velocity and lean angle decrease, while steer increasingly deviates until the rear frame is upright at pivoting motion G.

A fourth circular motion family starts with the identical pivoting motion at H (now considered HR), with lean increasing/steer decreasing up to a near-cusp at point J, and then reversing that trend to achieve a near upright lean and a nearly perpendicular steer, as the speed goes to infinity at K. But since G is essentially the same as H (except for an inconsequential speed reversal), the third and fourth families are actually one (G=H could be removed from the list of terminal points). This combined family—FGHJK—joins HF and HR configurations.

By this count, we have three circular motion families in all.

With this background, we consider qualitatively why thick curves ED and JH lie so close to FG (figure 5). In figure 5*a*,*b*, we actually see the broken curve FGHJ+DE *folded* at GH. In essence, this says that starting in an upright condition with the steer essentially *π*/2, an added leftward or rightward amount of steering leads to a bicycle with HR and HF configuration, respectively; these two configurations may be viewed as almost identical bicycles (due to ‘small’ handlebar asymmetry), and hence dynamic equilibria obtained are almost identical as well. Without handlebar asymmetry, the coincidence would be perfect.

In figure 5*b*, we have exaggerated the closeness of points G and H. In reality, due to the handlebar asymmetry, they have a small vertical separation, with G lying slightly above *π*/2 and H slightly below it. Actual numerical and graphical results presented below have no such misrepresentations.

Note that the hands-free-motion plots can also provide qualitative information about the sign of steer torque away from the plotted curves. For example, consider the light (HF) curves in the steer plot of figure 5. Recalling that the horizontal axis is also a line of zero steer torque, one can imagine increasing the steer angle at a speed just below B (such as 5 m s^{−1} in figure 6). The torque will become non-zero (negative, as it happens), attain a peak negative value, reduce to zero as the BA curve is crossed, then increase to a peak positive value and then drop again as the FG curve is approached. Thus, steer torque may vary significantly in both sign and slope (i.e. ‘stiffness’) as one alters turn radius, posing something of a control problem for the rider attempting to corner quickly at lean angles up to *π*/4 and steer angles up to *π*/12.

### (e) Accurate plots, with four-way symmetry

As mentioned above, for each circular motion, another is obtained if all speeds are reversed, and every left-leaning solution also implies a right-leaning one, where (*ψ*, *ψ*_{f}) are replaced by (*π* −*ψ*, −*ψ*_{f}). The resulting four-way symmetry in the solutions is represented (actual numerics) in figure 6, where the infinite horizontal scale of is mapped to a finite range using the arctangent of (4 is an arbitrary scaling parameter chosen for better visualization).

All the curves represent numerically obtained solutions, while the labelled thick dots indicate the correspondence with figure 5. Here, roll (*ψ*) has been plotted from 0 to *π*, instead of lean from −*π*/2 to *π*/2. Steer (*ψ*_{f}) has been plotted from −*π* to *π*. As mentioned above, in figure 5, the thick lines actually show *π*+*ψ*_{f} against ; in figure 6, we plot *ψ*_{f} against , obtaining a curve in the third quadrant.

The steer curves provide another vantage on the earlier described near-symmetry about *π*/2. The lean curves are harder to untangle, unless one reflects points HJK through the origin and CDE through the vertical axis. Then a reflected curve (say C′D′E′) is visible in the first quadrant and (similarly primed) K′J′H′GF appears in the first and fourth.

### (f) Some precise (benchmark) numerical values

The graphical results discussed above were presented, after some trial and error, in terms of variables allowing simple *post facto* interpretation. Here, in terms of our original variables, we report some precise numerical results for benchmarking.

Table 2 lists some initial conditions for steady circular hands-free motions. The radius *R* traversed by the rear wheel centre is also provided. These were independently verified through simulations by Arend Schwab (using Spacar).

In addition, we now list some special numerical values.

We have separately sought and found a static equilibrium of the bicycle at *ψ*_{f}=1.3397399115 and *ψ*=*π*/2 (corresponding to point A in figures 5 and 6). The corresponding rear wheel centre radius *R* is 0.2771720012 m.

Point B in figures 5 and 6 corresponds to a straight-ahead capsize speed of 6.0243 m s^{−1}. Point C corresponds to a straight-ahead (HR) capsize speed of 7.9008 m s^{−1}. In linearized analysis of straight riding, capsize occurs at that speed where the handlebar ‘torsional stiffness’ vanishes, permitting any arbitrary turn to be maintained with zero handlebar torque. This forward speed is the unique solution of a linear equation in *V*^{2}.

Point E corresponds to an HR flat motion with *R*≈3.3049 m (from numerical extrapolation). Point F corresponds to another flat motion (handlebar forward) with *R*≈3.0087 m (numerical extrapolation). These configurations are hard to evaluate precisely due to geometrical (contact) and mathematical (Euler angles) singularities.

Point G (=H) represents the solution *ψ*=*π*/2, *ψ*_{f}=1.6416430491, and *R*=−0.0415586589 (here *R*<0 because *ψ*_{f}>*π*/2 and the bicycle moves in a circle that curves right instead of left). Since this point is not a limiting motion, its definition is somewhat arbitrary: rather than an upright frame, we might instead specify minimum front wheel velocity or some other condition.

Point K can be found precisely by setting gravity to zero, choosing any non-zero speed, and seeking a unique circular motion. (This is asymptotically equivalent to finite gravity and infinite speed.) We have found *ψ*=1.6679684551, *ψ*_{f}=−1.6922153670 and *R*=0.0666827859 m.

## 8. Stability of circular motions

For stability analysis of circular motions, the generalized coordinates *x* and *y* are replaced by new ones defined by *x*=−*R* sin *Χ* and *y*=*R* cos *Χ*. *R* and remain constant during origin-centred circular motions. Also, two new configuration-dependent unit vectors are introduced. These arewhich are in radial and circumferential directions in the plane of the ground. Finally, the in-ground-plane vector constraint equations of no-slip are *not* retained in terms of *x* and *y* components, but instead retained in terms of components along and . This makes the constraint forces (hence Lagrange multipliers) constant during the circular motions of interest. Lagrange's equations are then obtained in the usual way, for the new set of generalized coordinates.

In these new equations, we seek circular motions by noting that *R*, *ψ*, *ψ*_{f} and *ϕ* are constants; is a constant; as well (i.e. not an extra unknown); and are constants; and all the five Lagrange multipliers are constants as well. Thus, there are 12 constants to be determined. Meanwhile, we have eight equations of motion, five velocity constraint equations (including one that is actually a differentiated holonomic constraint) and a holonomic constraint equation to enforce front wheel contact with the ground. That is, we have 14 equations and 12 unknowns. The following lines of thought help to clarify the situation.

The Lagrange multiplier (say *λ*_{2}) corresponding to the no-slip constraint at the rear wheel, in the direction, turns out to be zero; this is expected because there is no propulsive thrust, and one of the equations of motion reduces to exactly *λ*_{2}=0. We drop this equation, but retain *λ*_{2} as an unknown and expect our subsequent calculation to rediscover that *λ*_{2}=0 (an automatic consistency check). So we now have 13 equations and 12 unknowns.

We retain the holonomic (front wheel contact) constraint equation in our calculations to ensure that a correct value for *ϕ* is obtained. But this automatically ensures that the velocity constraint equation in the normal direction at the front wheel contact is identically satisfied, and so we drop that equation. We then have 12 equations and 12 unknowns.

As may be anticipated, it turns out that the -direction no-slip equation is identically satisfied at the rear wheel, leaving 11 equations and 12 unknowns. This suggests, in line with prior calculations, that there are one or more one-parameter solution families. As before, we choose *R*, and solve 11 equations and 11 unknowns (see electronic supplementary material for further discussion).

All quantities of interest (including the Lagrange multipliers) are now treated as *ϵ*-order time-varying perturbations of the nominal solutions corresponding to circular motion; the equations of motion (including velocity constraint equations) are linearized in terms of *ϵ*. The (*ϵ*) equations obtained from the velocity constraint equations are differentiated to get a full second-order system. These are solved for the (perturbations in) Lagrange multipliers and second derivatives of generalized coordinates; and a constant coefficient system is obtained in terms of the eight degrees of freedom used in our formulation. We then obtain a non-minimal set of 16 eigenvalues. Of these, 10 are zero (see electronic supplementary material for discussion). Of the remaining six non-zero eigenvalues, two are found to be exactly ±i*ω* (where *ω* is already known for the circular motion). These two eigenvalues merely represent the same circular motion shifted to a nearby circle. There remain four non-trivial eigenvalues, which are tabulated for the motions reported in table 3. These accurate eigenvalues can serve a benchmarking purpose. They are consistent to three or four decimal places with correspondingly (in)accurate eigenvalues found numerically using finite differences from the Newton–Euler equations (electronic supplementary material). The latter, being quicker, were used to check the stability of the circular motions obtained above.

All of the circular motions of the benchmark bicycle with straight (forward) handlebar turn out to be unstable. Of the reversed-handlebar motions, relatively few are stable. These are shown in figure 7 (recognizable as the second-quadrant representation of points D and J from figure 6), by means of individual thick dots corresponding to our discrete sampling of the underlying continuous curves. We avoid here the sign change on speed used in figure 5; so the two curves are reflected versions of CDE and HJK of figure 5*a*.

## 9. Conclusions

In this paper we have, first, obtained two independent sets of fully nonlinear equations of motion for a bicycle. Of these two, the first (Lagrange/Maple) allows analytical linearization and is used to numerically cross-check with Meijaard *et al*. (2007). The second set (Newton–Euler/Matlab) is good for rapid simulation.

We have studied circular motions of a benchmark bicycle, obtaining mathematically four (physically, three) different one-parameter families of such solutions. Barring Lennartsson (1999) and Aström *et al*. (2005), each missing one solution family, no other study of circular motions has reliably reported these multiple solution families. We have described the solution families obtained in terms of their endpoints in the plotting plane. These endpoints have been intuitively interpreted and described. Precise numerical values for some motions have been provided for benchmark purposes. A stability analysis has also been carried out of the circular motions, and precise eigenvalues reported for some chosen points. Most of the circular motions obtained turn out to be unstable for the benchmark bicycle, though this may not remain the case for other reasonable designs.

## Acknowledgments

Arend Schwab and Andy Ruina read drafts of the paper and provided useful technical and editorial comments. Arend Schwab also verified several of our numerical results and helped locate some errors.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.1849 or via http://www.journals.royalsoc.ac.uk.

↵More robust numerical simulation would use coordinate partitioning on velocities, where we use a partial state vector, and a subset of velocities is used, in each time step, to find the other velocities from the velocity constraint equations, ensuring sustained satisfaction thereof (electronic supplementary material).

- Received November 29, 2006.
- Accepted March 30, 2007.

- © 2007 The Royal Society