Proceedings, Bicycle and Motorcycle Dynamics 2013Symposium on the Dynamics and Control of Single Track Vehicles,

11–13 November , 2013, Narashino, Japan

Dynamics and Control of a Steer-by-Wire Bicycle

A. L. Schwabℵ, Nick Appleman∗

ℵLaboratory for Engineering MechanicsDelft University of Technology

Mekelweg 2, NL-2628 CD Delft, The Netherlandse-mail: a.l.schwab@tudelft.nl

∗NovatronixBalthasar van der Polweg 212

2628 AX Delft, The Netherlandsemail: nick.appelman@novatronix.nl

ABSTRACT

This work is focused on the modeling and ex-perimental validation of a steer-by-wire systemfor bicycles. The purpose for this system is tomodify and enhance the lateral stability of a bi-cycle at low forward speeds. Case studies showadditional capabilities of a steer-by-wire systemon bicycles to influence its dynamic behavior,by providing a dynamic response comparable toa bicycle with a virtually different geometry oreven the ability to stabilize an inherently unsta-ble bicycle. A steer-by-wire bicycle prototypewas designed and build by replacement of themechanical connection between handlebar- andsteering-assembly by electronic actuators and acustom digital controller. The steer-by-wire bi-cycle prototype equipped with sensors, measur-ing the forward speed and roll-rate was subse-quently used to experimentally evaluate the pro-posed control algorithms. Preliminary rider testsshowed a perceived near-to-identical behaviourof the steer-by-wire system to a mechanical con-nection. Adding lateral stability enhancement atlow speed by active steer-torque control was per-ceived as beneficial by the rider.

Keywords: bicycle, steer-by-wire, stability, con-trol, rider control.

1 INTRODUCTIONAlready for some time, electronic enhance-

ments regarding vehicle behavior has made itsway into the aviation and automotive industry bythe term ”by-wire” technology. Electronic sen-sors and actuators are used to replace traditionalmechanical systems in which software is used tooperate the actuators in a way that is not possiblewith traditional mechanical systems.

The use of steer-by-wire technology can alsooffer great opportunities on single-track vehicleslike motorcycles, scooters and bicycles. Single-track vehicles can be laterally highly unstable,especially at low forward speeds and they requirea relative high amount of rider control [1, 2]. Byreplacing the mechanical connection between thehandlebar- and steering-assembly with electronicactuators and adding sensors to measure the stateof the system, a controller can be used to controlthe dynamic behavior of the bicycle.

In open literature there is no research avail-able which experimentally evaluates a steer-by-wire system on single-track vehicles. Only a fewtheoretical publications proposing enhancementsin motorcycle handling [3, 4] are available. It re-mains questionable if the removal of the countersteering behavior as proposed by Marumo andNagai [3] will be beneficial. The possibility ofa lane keeping assistance system on motorcyclesby Katagiri et al. [4] on the other hand can greatly

1

improve safety. This is also demonstrated bySeiniger et al. [5] by actively assisting the mo-torcycle rider’s steer input to hold its driving pathduring extensive in-corner braking manoeuvres.

steering actuator

handlebar actuator

on-off switch

battery packmicro controller

controller

FIGURE 1. Prototype of the steer-by-wire bicyclewith steering and handlebar actuators, sensors, actua-tor controllers, battery pack and microcontroller.

The work presented here, is focused on themodeling and experimental validation of a steer-by-wire control strategy to modify and enhancethe lateral stability of a bicycle at low forwardspeeds. Case studies show additional capabilitiesof a steer-by-wire system on bicycles to influenceits dynamic behavior, by providing a dynamic re-sponse comparable to a bicycle with a virtuallydifferent geometry or even the ability to stabi-lize an inherently unstable bicycle. A steer-by-wire bicycle prototype was designed and buildby replacement of the mechanical connection be-tween handlebar- and steering-assembly by elec-tronic actuators and a custom digital controller,see Figure 1. The steer-by-wire bicycle proto-type equipped with sensors, measuring the for-ward speed and roll-rate is subsequently used toexperimentally evaluate the proposed control al-gorithms.

The paper is organized as follows. After thisbrief introduction the model for the system de-sign is described and some simulation results areshown. Next the experimental setup is describedand some preliminary test results are shown. Thepaper ends with some conclusions.

2 SYSTEM DESIGN AND SIMULATIONThe model for the steer-by-wire system de-

sign is based on the three degree of freedom

Whipple/Carvallo [1] bicycle model. This modelis extended by separating the handlebar assem-bly from the front steering assembly, which in-troduces an additional rotational degree of free-dom, see Figure 2. The lateral degrees of free-dom of this extended model are: the rear frameroll angle φ , the front assembly steering angleδ , and the handlebar steering angle θ . Since weare only interested here in the lateral dynamics,the forward speed v, which is a degree of free-dom of the Whippel/Carvallo model, is treatedas a parameter. Combining the lateral degrees

φ

δ

zw

FR

BS

H

t

Ox

α

θ

y δ

θTφ

Tδ

Tθ

(a)

φ

δ

zw

FR

BS

H

t

Ox

α

θ

y δ

θTφ

Tδ

Tθ

(b)

FIGURE 2. Steer-by-wire bicycle model, side view(a) and top view (b), together with the lateral degreesof freedom, rear frame roll angel φ , front frame steer-ing angle δ , and handlebar steering angle θ , and somegeometry variables. This model, based on the Whip-ple/Carvallo bicycle model [1], shows the addition ofa separate handlebar body H and the possibility tohave unequal front wheel steering δ and handlebar θ

relations.

of freedom in a generalized coordinate vectorq = [θ ,φ ,δ ]T , the linearized equations of motionfor the extended bicycle model can be expressedby

M̄q̈+ C̄q̇+ K̄q = f̄, (1)

2

with the mass matrix M̄, damping matrix C̄ andstiffness matrix K̄ given by,

M̄ =

[Iθ 00 M

], C̄ =

[0 00 vC1

],

K̄ =

[0 00 gK0+ v2K2

], (2)

and the right-hand side forcing term f̄ =[Tθ ,Tφ ,Tδ ]

T , which contains the handlebartorque Tθ , the rear frame roll torque Tφ (usu-ally zero), and the front steering assembly torqueTθ . The matrices M,C1,K0, and K2, are thetwo-by-two matrices from the linearized equa-tions of motion of the original Whipple/Carvallomodel [1], Iθ is the mass moment of inertia of thehandlebar assembly, v is forward speed and g isthe gravitational acceleration.

2.1 Handlebar tracking controlTo minimize the difference between the han-

dlebar angle θ and the steering assembly angle δ ,tracking control needs to be implemented. In thisway the steer-by-wire system should behave likean ordinary, mechanically steered bicycle, whenthe rider applies a steer torque at the handle-bar. A proportional-differential (PD) controller isimplemented which provides an action-reactiontorque TPD to the steering assembly and the han-dlebar, of the form,

TPD = Kp(θ −δ )+Kd(θ̇ − δ̇ ), (3)

with proportional gain Kp, and differential gainKd . The forcing term in 1 then becomes,

f̄ =

Tθ

Tφ

Tδ

=

Th−TPD

0TPD

, (4)

with the rider applied steer torque Th at the han-dlebar, and zero applied roll angle torque. Thesteer-by-wire bicycle model can be visualized ina block diagram as shown in Figure 3. The han-dlebar block represents the rotational inertia ofthe handlebar while the controller block repre-sents the tracking controller layout. For the PD-controller, the proportional and differential gainsin 3 are chosen such that a critically damped sys-tem response is obtained to ensure a fast and ac-curate response without overshoot.

As an example we use the parameters ofthe benchmark bicycle [1], with a handlebarinertia of I0 = 0.001 kgm2, proportional gainKp = 90 Nm/rad, and differential gain Kd = 0.6Nms/rad. The performance of this tracking con-trol is shown in Figure 4 by comparing the steerstiffness transfer functions of the benchmark bi-cycle without and with the steer-by-wire system.The steer stiffness transfer function is defined asHBB(s) = Tδ (s)/δ (s) for the rigid connection andas HSBW (s) = Tθ (s)/θ(s) for the steer-by-wirebicycle. The proposed system shows, in a for-ward speed range of 0 to 10 m/s, good trackingperformance in a frequency range of 0 to 3 Hz.Above 3 Hz the steer stiffness drops of due to thefinite bandwidth of the tracking system.

TδPD

1s2J

controllerhandlebar

+−

Te

Tθ+

−Th+

−

θ δ

δbicycle

φTφ

+s Kd

Kp+

ε

ε

PD

FIGURE 3. Block diagram of the steer-by-wire bi-cycle model, which includes the handlebar trackingcontroller (PD) and steer torque feedback to the han-dlebar, with the rear frame roll angle φ , front assem-bly steering angle δ , handlebar steering angle θ , bythe rider applied handlebar steering torque Th, frontassembly steering torque Tδ , and the handlebar track-ing controller and handlebar feedback steer-torqueTPD.

2.2 Lateral stability enhancementThe unstable lateral motions at low speed

can be stabilized by adding a steer-torque con-trol system to the handlebar tracking control sys-tem. For the design of the low speed stabilitycontroller we follow the work of Ruijs and Pace-jka [6] who proposed a steer-into-the fall con-troller [2] for a motorcycle robot at sub-weavespeed, which uses the roll rate of the rear frameas input and steer-torque as system output, withlinear regressive gain scheduling as a function offorward speed. This controller has successfully

3

10−1

100

101

0

5

1010

−1

100

101

102

103

frequency [Hz]

Steer stiffness T/ − Benchmark Bicycle

forward speed v [m/s]

mag

nitu

de [N

m/r

ad]

(a)

10−1

100

101

0

5

1010

−1

100

101

102

103

frequency [Hz]

Steer stiffness T/ − Steer−by−Wire Bicycle

forward speed v [m/s]

mag

nitu

de [N

m/r

ad]

(b)

FIGURE 4. Magnitude of the steer stiffness trans-fer function, as perceived by the rider, as a functionof forward speed v and the frequency s, (a) on thebenchmark bicycle HBB(s), and (b) the steer-by-wirebicycle HSBW (s). The steer stiffness at the handle-bars changes significantly as a function of input fre-quency as well as the forward speed of the bicycle. Athigher frequencies the steer stiffness is primarily de-fined by the mass- and inertia properties of the bicy-cle. A significant drop in the steer stiffness magnituderelation occurs at the weave speed (vw = 4.29 m/s)and the corresponding weave frequency (0.55 Hz) ofthe bicycle shown by the downward resonance-likepeak. The steer stiffness of the steer-by-wire bicy-cle at higher frequencies are primarily defined by thestiffness and damping properties of the PD-controller,whereas the anti-resonance-like upward peak at lowforward speeds is caused by the PD-controller coeffi-cients and the mass- and inertia properties defined inthe system matrix.

been implemented [7] on the benchmark bicyclemodel and on scaled down prototype [8]. Theproposed stabilizing control takes on the form ofan additional steer torque TSE applied at the steer-

ing assembly,

TSE = Ks(vavg− v)φ̇ ∀ v < vavg (5)

which is proportional to the rear frame roll rateφ̇ , the forward speed v of the bicycle, and a con-stant Ks. The magnitude of this speed depen-dency is linearly decreased in magnitude up tovavg, which is around the weave speed, as no sta-bilizing control is required inside the auto-stablespeed region. At higher forward speeds no sta-bilizing control torque is applied, as the unstablecapsize mode is relatively slow and easy to con-trol. The forcing term in the linearized equationsof motion (1) is now f̄= [Th−TPD,0,TPD+TSE ]

T .This leads to an extended block diagram for thecomplete system, as shown in Figure 5.

TδPD + SE1

s2J

controllerhandlebar

+−

Te

Tθ+

−Th+

−

θ δ

δ

bicycle

φTφ

+s Kd

Kp +

ε

ε

PD + SE

+

Ks

φ

vavg-v

+

sφ φ

FIGURE 5. Block diagram of the steer-by-wire bi-cycle model with enhanced lateral stability at lowspeed, which includes the combined handlebar track-ing controller (PD) and low speed stabilization con-troller (SE).

By applying this low speed stabilization con-troller with proportionality constant Ks = 10Ns2/rad and vavg = 5 m/s, the weave speed is de-creased from 4.3 m/s to 1.0 m/s and by such mak-ing the bicycle self-stable from 1 m/s until highforward speed, since the unstable capsize mode isrelatively slow and easy to control. This changein the eigenvalues and the selfstable speed rangeis illustrated in Figure 6.

3 EXPERIMENTAL SETUPA conventional Dutch city bicycle (Batavus

Browser) is converted to a steer-by-wire bicy-cle by replacement of the mechanical steeringconnection with electronic actuators and sensors,

4

0 2 4 6 8 10−15

−10

−5

0

5

10

vw

= 4.3 m/s

vc= 6.02 m/s

vc

vw

v [m/s]

λ [r

ad/s

]

(a)

0 2 4 6 8 10−15

−10

−5

0

5

10

vw

= 1.02 m/s

vc= 6.02 m/s

vc

vw

v [m/s]

λ [r

ad/s

]

(b)

FIGURE 6. Eigenvalues λ from the linearized sta-bility analysis for the original steer-by-wire bench-mark bicycle model (a) in comparison to the eigen-values for the same model with added lateral stabilitycontrol (b), in the forward speed range of 0 < v < 10m/s. The lateral stability control enlarged the stableforward speed range by pushing the weave speed vw

back from 4.3 to 1.0 m/s, while retaining the capsizespeed vc.

see Figure 1. Two identical Maxon EC90 brush-less DC motors and EPOS2 power-amplifiers ac-tuate the steering- and handlebar-assembly, asshown in Figure 7. To increase the maximumoutput torque, the actuators are connected us-ing a timing-belt construction with an amplifi-cation ratio of 132/20 which allows for a maxi-mum instantaneous steer torque of about 15 Nm.Separate Altheris FCP22AC position- and Sili-con Sensing CRS03 angular rate sensors are usedon the handlebar- and steering assembly to pro-vide the controller the required state information.The roll sensor uses an InvenSense IDG-500 an-gular rate gyro to measure the rear frame roll rate.A small Maxon DC motor, spring-loaded against

position sensor

position sensor

angular rate sensor

angular rate sensor

motor amplifier (2x)

(a)

powerrelays (2x)

fuse (2x)

forwardspeedsensor

roll sensor

(b)

FIGURE 7. Steer-by-wire bicycle component lay-out, showing the physical placement of the steering-and handlebar assembly, battery pack and controller,with (a) front assembly with sensors, motors, and mo-tor amplifiers, and (b) rear rack with battery pack, mi-crocontroller, and forward speed sensor.

the rear wheel is used to measure the forwardspeed of the bicycle. A Microchip dsPIC33Fequipped Explorer16 micro-controller is used tocontrol the system. A 24 V Super-B lithium-ion battery pack powers the actuators and micro-controller.

The proposed PD-controller values of Kp =90 Nm/rad and Kd = 0.6 Nms/rad of the bench-mark bicycle simulation would in practice resultin unrealistic high actuator torques. Also, dur-ing the experiments it became apparent that pre-sumable un-modeled actuator- and controller dy-namics cause the handlebar assembly to becomemildly unstable.

The steer-by-wire prototype subsequentlyutilizes a double PD-controller configurationwith slightly lower and unequal gain coefficients.The double PD-controller configuration, one inthe steer torque path (Kp = 15 Nm/rad, Kd = 1.5Nms/rad) and one in the handlebar torque path(Kp = 8 Nm/rad, Kd = 0.6 Nms/rad), was used tomaximize the feedback torque and tracking per-

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formance without forcing the handlebar assem-bly into an unstable mode.

4 PRELIMINARY TEST RESULTSPreliminary rider tests with the steer-by-wire

system showed very satisfactory behaviour of thesystem. After gaining some confidence, the ridercould not distinguish the handling of the steer-by-wire bicycle from a bicycle with rigid steer-ing connection. Only when looking at the frontsteering assembly while applying a rider steeringtorque, the rider was able to see a small phase lagbetween the handlebar and the steering assembly.This small phase lag had no influence on he per-ceived handling during normal operation.

More qualitative results were obtained withthe stability enhanced steer-by-wire bicycle. Inriding a straight track at low forward speed at 5km/h (1.4 m/s), two cases were compared; withand without lateral stability enhancement. Dur-ing these tests the steer rate of the handlebar wasrecorded as a measure for rider control input.Figure 8 shows the power spectral density (PSD)of the steer rate. As a measure of steer effort,the area under the PSD curve, indicated by Pe,is used. Clearly, steer effort without haptic feed-back (Pe = 2890) is much larger than with hap-tic feedback (Pe = 980). Moreover, the powerspectral density of the stability enhanced bicycleat low forward speed (5 km/h) was shown to becomparable to the PSD at normal forward speedof 20 km/h (not shown here). This clearly indi-cates the benefit of such a stability enhancementsystem at low forward speed. Rider steer effortat low speed is reduced which makes the bicyclemore easy to ride and could lead to future im-plementation on bicycles tailored for elderly orthose physically impaired.

5 CONCLUSIONSA steer-by-wire bicycle has been designed

and build. Preliminary rider tests showed a per-ceived near-to-identical behaviour of the steer-by-wire system to a mechanical connection.Adding lateral stability enhancement for lowspeed stability by active steer-torque control re-duced the steer-effort of the rider considerablyand was perceived as beneficial by the rider.

In future research the steer-by-wire bicyclewill serve as a versatile experimental platform for

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

PS

D [(

deg/

s)2 /H

z]

frequency [Hz]

Pe=980

(a)

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

PS

D [(

deg/

s)2 /H

z]

frequency [Hz]

Pe=2890

(b)

FIGURE 8. Power spectral density (PSD) of thehandlebar steer rate for a bicycle with (a) regularsteer torque feedback and (b) no steer torque feed-back on the handlebar, riding a straight track at aforward speed of 5 km/h. As a measure of steer ef-fort, the area under the PSD curve, indicated by Pe,is used. Clearly, steer effort without haptic feedback(Pe = 2890) is much larger than with haptic feedback(Pe = 980).

identifying human rider control in bicycling.

ACKNOWLEDGEMENTThanks to the Accell Group for providing the

Batavus Browser bicycle, and to Jodi Kooijmanfor his guidance and support in this project.

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[2] J.D.G. Kooijman, J.P. Meijaard, J.M. Pa-

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[3] Y. Marumo and M. Nagai. Steering con-trol of motorcycles using steer-by-wire sys-tem. Vehicle System Dynamics, 45(5):445–458, 2007.

[4] N. Katagiri, Y. Marumo, and H. Tsunashima.Controller design and evaluation of lane-keeping-assistance system for motorcycles.Journal of Mechanical Systems for Trans-portation and Logistics, 2(1):43–54, 2009.

[5] P. Seiniger, K. Schroter, and J. Gail. Perspec-tives for motorcycle stability control sys-tems. Accident Analysis and Prevention,2011.

[6] P.A.J. Ruijs and H.B. Pacejka. Recent re-search in lateral dynamics of motorcycles.In Procedings of 9th IAVSD Symposium onThe Dynamics Of Vehicles on roads andon tracks, Sweden June 24-28 1985, vol-ume supplement to Vehicle System Dynam-ics, Volume 15, pages 467– 480, 1986.

[7] A.L. Schwab, J.D.G. Kooijman, and J.P.Meijaard. Some recent developments in bi-cycle dynamics and control. In Fourth Eu-ropean Conference on Structural Control,page 8, 2008.

[8] Joep Mutsaerts. Lego NXTbike-GS bicycle with active stability.http://bicycle.tudelft.nl/schwab/Bicycle/.

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