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Review Articles

On the Stability and Control of the Bicycle

[+] Author and Article Information
Robin S. Sharp

School of Engineering, University of Surrey, Guildford GU2 7XH, UK

Appl. Mech. Rev 61(6), 060803 (Oct 08, 2008) (24 pages) doi:10.1115/1.2983014 History: Received January 31, 2008; Revised March 25, 2008; Published October 08, 2008

After some brief history, a mathematical model of a bicycle that has become a benchmark is described. The symbolic equations of motion of the bicycle are given in two forms and the equations are interpreted, with special reference to stability. The mechanics of autostabilization are discussed in detail. The relationship between design and behavior is shown to be heavily speed-dependent and complex. Using optimal linear preview control theory, rider control of the bicycle is studied. It is shown that steering control by an optimal rider, especially at low speeds, is powerful in comparison with a bicycle’s self-steering. This observation leads to the expectation that riders will be insensitive to variations in design, as has been observed in practice. Optimal preview speed control is also demonstrated. Extensions to the basic treatment of bicycle dynamics in the benchmark case are considered so that the modeling includes more realistic representations of tires, frames, and riders. The implications for stability predictions are discussed and it is shown that the moderate-speed behavior is altered little by the elaborations. Rider control theory is applied to the most realistic of the models considered and the results indicate a strong similarity between the benchmark case and the complex one, where they are directly comparable. In the complex case, steering control by rider-lean-torque is feasible and the results indicate that, when this is combined with steer-torque control, it is completely secondary. When only rider-lean-torque control is possible, extended preview is necessary, high-gain control is required, and the controls are relatively complex. Much that is known about the stability and control of bicycles is collected and explained, together with new material relating to modeling accuracy, bicycle design, and rider control.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 5

Preview gain sequences for tight (top), medium (middle), and loose (bottom) controls at 0.01 s intervals

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Figure 6

Diagrammatic representation of a bicycle tracking a roadway path, with the whole system referenced to ground. Such a description in discrete time implies that the road sample values pass through a shift-register operation at each time step. The dynamics of the shift-register are easy to specify mathematically.

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Figure 7

Illustration of the time-invariant optimal controls, corresponding to an infinite optimization horizon and a white-noise disturbance. The control involves state feedback with the system state including both vehicle and roadway contributions.

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Figure 8

Feedback gains K13–K16 as functions of speed for high-authority control with 0.01 s sampling interval

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Figure 9

Feedback gains K13–K16 as functions of speed for medium-authority control with 0.01 s sampling interval

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Figure 10

Feedback gains K13–K16 as functions of speed for low-authority control with 0.01 s sampling interval

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Figure 13

Preview gains K20–K2(n−1) as functions of speed for low-authority control with 0.01 s sampling interval

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Figure 14

Full preview distance needs of the benchmark bicycle as a function of speed, for different levels of control authority. The preview necessary is roughly proportional to speed. Looser control implies the need for more preview, amounting to perhaps 12.5 s in a loose case but only about 2.2 s in a tight one.

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Figure 15

Frequency response of the lateral displacement of the front tire contact point of the controlled system at 6 m/s with 334 preview points and a q1 weighting of 100. The tracking is virtually perfect for excitation frequencies less than 1 rad/s, with the limit set by gain attenuation.

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Figure 16

Double lane change simulation results for the rider-controlled bicycle at 6 m/s with 334 preview points and a q1 weighting of 100. The upper plot shows the path demand and the track followed by the front wheel contact point, the center plot shows the steer torque needed, and the lower plot shows the roll and steer angles.

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Figure 19

Eigenvalues of the modified bicycle over 0.2–10 m/s speed range, with inclusion of a front tire crown radius of 0.01 m. This figure should be compared with the benchmark case of Fig. 3. The influence of the crowning of the front tire is to reduce the range of autostable operation. The speeds at which the weave stabilizes and the capsize destabilizes are both reduced, the latter the more so.

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Figure 20

Eigenvalues of the modified bicycle over 0.2–10 m/s speed range, with inclusion of a front tire crown radius of 0.04 m. The autostable region has disappeared, with common zero-crossing point at about 3.3 m/s. The capsize mode instability for higher speeds is significantly worsened by the front tire crowning.

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Figure 21

Eigenvalues of the modified bicycle over 0.2–10 m/s speed range, with inclusion of a rear tire crown radius of 0.08 m. The influence of the crowning of the rear tire is weaker than that for the front and opposite in sense. In this quite extreme case the autostable speed range extends to much higher speeds.

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Figure 22

Eigenvalues of the modified bicycle over 0.2–10 m/s speed range, with inclusion of overturning moments due to tire crowning and side-forces and aligning moments in response to lateral slip, turn slip, and wheel camber. Tire crown radii are 0.01 m for the rear and 0.008 m for the front, at which level their influences are small.

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Figure 25

Eigenvalues of modified bicycles over 0.2–20.2 m/s speed range in 0.5 m/s steps, showing the influence of frame torsional damping coefficient on weave and wobble modes. The stiffness assumed is 7000 Nm/rad and the twist joint is located at coordinates (xT=1.01 m; zT=−0.9 m). The wobble mode damping is strongly influenced by the variations.

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Figure 26

Eigenvalues of modified bicycles over 0.2–20.2 m/s speed range in 0.5 m/s steps, showing the influence of rider-upper-body-lean freedom on the properties of the weave mode. The low-speed behavior is again largely unaffected by the compliance.

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Figure 27

Main feedback gains relating to steer torque for the fully developed two-control bicycle model with medium-authority control (q1=100), as functions of speed

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Figure 1

Diagrammatic bicycle with front-frame mass, mH, and rear-frame mass, mB, adapted from Refs. 1,8-10,15. Contrasting with the references, front and rear wheel masses and inertias are incorporated into the respective frames, as appropriate for axially symmetric bodies. Only the spin inertia of each wheel needs to be specified individually. The number of parameters needed to define the bicycle is thereby reduced without any loss of generality.

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Figure 2

Variation of effective steering system stiffness through speed for the benchmark bicycle, showing a change of sign at 1.723 m/s

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Figure 3

Eigenvalues of the benchmark bicycle over 0.2–10 m/s speed range. The uncontrolled machine will capsize quite quickly at very low speeds and very slowly for speeds above about 6 m/s. The oscillatory weave mode forms at 0.6843 m/s from the coalescence of two positive real roots and stabilizes at 4.29 m/s. The capsize mode passes from stable to unstable at 6.02 m/s.

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Figure 4

Responses in steer and roll angles of the benchmark bicycle at 4.6 m/s to a steer-torque input of 0.1 Nm from zero initial conditions. The nonminimum phase steering response is clear and the bicycle turns in a stable state when the transient response has decayed.

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Figure 17

Eigenvalues of the nominal bicycle over 0.2–10 m/s speed range with 2 m∕s2 acceleration. This figure should be compared with the benchmark case of Fig. 3. The differences caused by the acceleration are surprisingly large.

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Figure 18

Eigenvalues of the nominal bicycle over 0.2–10 m/s speed range with 2 m∕s2 deceleration. The deceleration causes the stabilization of the weave mode to occur at significantly higher speed and the capsize mode remains stable through the speed range studied.

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Figure 23

Eigenvalues of the benchmark and modified bicycles over 0.2–20.2 m/s speed range in 0.5 m/s steps. Low-speed and high-speed behaviors are predicted differently by the four models, but in the midrange, they give similar results.

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Figure 24

Eigenvalues of modified bicycles over 0.2–20.2 m/s speed range in 0.5 m/s steps, showing the influence of frame torsional compliance on the properties of the weave mode. The twist joint is located at coordinates (xT=1.01 m; zT=−0.9 m). The low-speed behavior is largely unaffected by the compliance.

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Figure 28

Main feedback gains relating to rider-lean torque for the fully developed two-control bicycle model with medium-authority control (q1=100), as functions of speed

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Figure 29

Main feedback gains for the fully developed steer-torque-controlled bicycle model for q1=100, as functions of speed

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Figure 30

Main feedback gains for the fully developed rider-lean-torque-controlled bicycle model for q1=100, as functions of speed

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Figure 31

Preview gains K20–K2(n−1) for the fully developed bicycle model with both steer-torque and rider-lean-torque controls and medium-authority (q1=100), as functions of speed. Note the relative smallness of the rider-lean-torque values.

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Figure 32

Preview gains K20–K2(n−1) for the fully developed bicycle model with steer-torque control and q1=100, as functions of speed. These gains are almost the same as for the two-control case, see Fig. 3.

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Figure 33

Preview gains K20–K2(n−1) for the fully developed bicycle model with rider-lean-torque control and q1=100. These gains are much smaller than those relating to steer torque, see Figs.  3132. Also, about three times the preview distance is now required for full benefit and the sequence is more complex than for steer-torque control.

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Figure 11

Preview gains K20–K2(n−1) as functions of speed for high-authority control with 0.01 s sampling interval

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Figure 12

Preview gains K20–K2(n−1) as functions of speed for medium-authority control with 0.01 s sampling interval

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