The 1st Joint International Conference on Multibody System DynamicsMay 25–27, 2010, Lappeenranta, FinlandLateral dynamics of a bicycle with passive rider modelA. L. Schwabℵ and J. D. G. KooijmanℵℵLaboratory for Engineering MechanicsDelft University of TechnologyMekelweg 2, NL 2628 CD Delft, The NetherlandsPhone: 31-15-2782701, Fax: 31-15-2782150email: [email protected]: [email protected] paper addresses the influence of a passive rider on the lateral dynamics of a Whipple-like bicyclemodel. In the original Whipple  model the rider is assumed to be rigidly connected to the rear frameof the bicycle and there are no hands on the handlebar. Contrary, in normal bicycling the arms of a riderare connected to the handlebar and the rider can use both steering and upper body rotations for control.From observations [2, 6] two distinct rider postures can be identified. The first posture is where the upperbody leans forward with the arms stretched to the handlebar, and the upper body twists while steering. Thesecond rider posture is an upright one where the upper body stays fixed with respect to the rear frame andwhere the arms, hinging at the shoulders and the elbows, exert the control force on the steering. Modelscan be made where neither posture adds any degrees of freedom to the original Whipple-like bicycle model.For both posture cases the open loop, or uncontrolled, dynamics of the bicycle-rider system is investigatedand compared to the rigid rider model by examining the eigenvalues and eigenmotions in the 0 to 10 m/sforward speed range. It is shown that such a passive rider can dramatically change the eigenvalues and itsstructure with respect to those of the rigid rider model.Keywords: Bicycle dynamics, human control, nonholonomic systems, multibody dynamics.1 INTRODUCTIONThe bicycle is an intriguing machine as it is laterally unstable at low speed and stable, or easy to stabilize,at high speed. During the last decade a revival in the research on dynamics and control of bicycles hastaken place . Most studies use the so-called Whipple model  of a bicycle. In this model a hands-freerigid rider is fixed to the rear frame. However, from experience it is known that some form of control isrequired to stabilize the bicycle and/or carry out tracking operations. This control is either done by steeringor by performing some set of upper body motions. The precise control used by the rider is currently understudy [2, 6]. Here we focus on steering and the contribution of passive body motions on the uncontrolleddynamics of a bicycle. In a previous study  it has been shown that passive lateral upper-body motionshave little effect on the uncontrolled dynamics of a bicycle.From observations [2, 6] two distinct rider postures can be identified. The first posture is where the upperbody leans forward with the arms stretched to the handlebar, and the upper body twists while steering, ascan be seen in Figure 1a. The second rider posture is an upright one where the upper body stays fixed withrespect to the rear frame and where the arms, hinging at the shoulders and the elbows, exert the controlforce on the steering, shown in Figure 1b. Models can be made where neither posture adds any degrees offreedom to the original Whipple-like bicycle model. For both posture cases the open loop, or uncontrolled,dynamics of the bicycle-rider system is investigated and compared to the rigid rider model by examining theeigenvalues and eigenmotions in the 0 to 10 m/s forward speed range. The paper is organized as follows.First the original bicycle model is presented. Next the extension of this model with a twisting upper body orflexed arms is presented and the stability of the lateral motions are compared to those of a rigid rider model.The paper ends with some conclusions.
(a) Rider A on the Stratos bicycle(b) Rider A on the Browser bicycleFigure 1: Bicycling on a treadmill, two distinct postures: a) Rider A on the Stratos bicycle with forwardleaned body and stretched arms. b) Rider A on the Browser bicycle with an upright body and flexed arms.2 BICYCLE MODELThe basic bicycle model used is the so-called Whipple  model which recently has been benchmarked .The model, see Figure 2, consists of four rigid bodies connected by revolute joints. The contact between theknife-edge wheels and the flat level surface is modelled by holonomic constraints in the normal directionand by non-holonomic constraints in the longitudinal and lateral direction. In this original model it isassumed that the rider is rigidly attached to the rear frame and has no hands on the handlebar. The resultingnon-holonomic mechanical model has three velocity degrees of freedom: forward speed v, lean rate ϕ̇ andsteering rate δ̇.For the stability analysis of the lateral motions we consider the linearized equations of motion for smallperturbations about the upright steady forward motion. These linearized equations of motion are fullydescribed in Meijaard 2007 . They are expressed in terms of small changes in the lateral degrees offront frame (fork andHandlebar), Hrear frame includingrider Body, BRear wheel, RFront wheel, FλPzwQcxsteer axisFigure 2: The bicycle model: four rigid bodies (rear wheel R, rear frame B, front handlebar assembly H,front wheel F) connected by three revolute joints (rear hub, steering axis, front hub), together with thecoordinate system, and the degrees of freedom.
freedom (the rear frame roll angle, ϕ, and the steering angle, δ) from the upright straight ahead configuration(ϕ, δ) (0, 0), at a forward speed v, and have the formMq̈ vC1 q̇ [gK0 v 2 K2 ]q f ,T(1)Twhere the time-varying variables are q [ϕ, δ] and the lean and steering torques f [Tϕ , Tδ ] . Thecoefficients in this equation are: a constant symmetric mass matrix, M, a damping-like (there is no realdamping) matrix, vC1 , which is linear in the forward speed v, and a stiffness matrix which is the sum ofa constant symmetric part, gK0 , and a part, v 2 K2 , which is quadratic in the forward speed. The forces onthe right-hand side, f , are the applied forces which are energetically dual to the degrees of freedom q.The entries in the constant coefficient matrices M, C1 , K0 , and K2 can be calculated from a non-minimalset of 25 bicycle parameters as described in Meijaard 2007 . A procedure for measuring these parametersfor a real bicycle is described in , where measured values for the bicycles used in this study can be foundin Table 2. Then, with the coefficient matrices the characteristic equation,()det Mλ2 vC1 λ gK0 v 2 K2 0,(2)can be formed and the eigenvalues, λ, can be calculated. These eigenvalues, in the forward speed rangeof 0 v 10 m/s, are presented for example for the Stratos bicycle with a rigid rider in Figure 4a. Inprinciple there are up to four eigenmodes, where oscillatory eigenmodes come in pairs. Two are significantand are traditionally called the capsize mode and weave mode. The capsize mode corresponds to a realeigenvalue with eigenvector dominated by lean: when unstable, the bicycle just falls over like a capsizingship. The weave mode is an oscillatory motion in which the bicycle sways about the headed direction. Thethird remaining eigenmode is the overall stable castering mode, like in a caster wheel, which correspondsto a large negative real eigenvalue with eigenvector dominated by steering.At near-zero speeds, typically 0 v 0.5 m/s, there are two pairs of real eigenvalues. Each pair consistsof a positive and a negative eigenvalue and corresponds to an inverted-pendulum-like falling of the bicycle.The positive root in each pair corresponds to falling, whereas the negative root corresponds to the timereversal of this falling. When speed is increased two real eigenvalues coalesce and then split to form acomplex conjugate pair; this is where the oscillatory weave motion emerges. At first this motion is unstablebut at vw 4.7 m/s, the weave speed, these eigenvalues cross the imaginary axis in a Hopf bifurcationand this mode becomes stable. At a higher speed the capsize eigenvalue crosses the origin in a pitchforkbifurcation at vc 7.9 m/s, the capsize speed, and the bicycle becomes mildly unstable. The speed range forwhich the uncontrolled bicycle shows asymptotically stable behaviour, with all eigenvalues having negativereal parts, is vw v vc .3 PASSIVE RIDER MODELSThe original Whipple model can be extended with a passive rider. From observations where riding on a largetreadmill (3 5 m) [2, 6], two distinct postures emerged which will be moddeled. In the first posture modelthe upper body is leaned forward and the arms are stretched and connected to the handlebar whereas theupper body is allowed to twist, see Figure 3a. The second posture model has a rigid upper body connected tothe rear frame and hinged arms at the shoulder and elbow connected to the handlebar, see Figure 3b. Neithermodel adds any extra degree of freedom to the original Whipple model. This means that the number andstructure of the linearized equations of motion (1) stays the same, only the entries in the matrices change.For the modelling of the geometry and mass properties of the rider, the method as described by Moore et al.2009  is used. Here the human rider is divided into a number of simple geometric objects like cylinders,blocks and a sphere of constant density see Figure 6a. Then with the proper dimensions and the estimatesof the individual body part masses the mechanical models can be made. For rider A used in this study thisdata can be found in Table 3, whereas the calculation of the necessary skeleton points is given in Table 4.The geometry and mass properties of the two bicycles used in this study where measured by the procedureas described in  and the results are presented in Table 2.The complete model of the bicycle with passive rider was analyzed with the multibody dynamics softwarepackage SPACAR . SPACAR handles systems of rigid and flexible bodies connected by various joints
(a) Forward leaned rider, stretched arms.(b) Upright rider, flexed arms.Figure 3: Two distinct bicycle models which include a passive rider: a) Rider with forward leaned body andstretched arms. b) Rider with upright body and flexed arms.in both open and closed kinematic loops, and where parts may have rolling contact. SPACAR generatesnumerically, and solves, full non-linear dynamics equations using minimal coordinates (constraints areeliminated). SPACAR can also find the numeric coefficients for the linearized equations of motion based ona semi-analytic linearization of the non-linear equations. This technique has been used here to generate theconstant coefficient matrices M, C1 , K0 , and K2 from the linearized equations of motion (1) which serveas a basis for generating the eigenvalues of the lateral motions in the desired forward speed range.3.1 FORWARD LEANED PASSIVE RIDERIn this posture model the upper body is leaned forward and the arms are stretched and connected to thehandlebar, see Figure 3a. The leaned upper body is allowed to twist about its longitudinal axis when steered.The upper body also needs a pitching degree of freedom but in a first order approximation the pitchingmotions is zero. This also follows directly from symmetry arguments. The linearized equations of motionare derived as described above together with the eigenvalues of the lateral motions. These eigenvalues areshown in Figure 4b. For comparison the eigenvalues for a rigid rider, that is the rider is rigidly attached tothe rear frame and there are no hands on the handlebar, are shown in Figure 4a.Compared to the rigid rider solutions there are some small changes in the eigenvalues but the overall structure is still the same. Most noticeable is that the stable speed range goes up and the the frequency of theweave motion goes down. This can be explained as follows. Adding this passive rider model makes twomajor changes to a fully rigid rider model. The first is that the attached passive mechanism of arms andtwisting upper body adds a mass moment of inertia to the steering assembly. Looking at the entries in themass matrix this increases the diagonal mass term M (2, 2) for the steering degree of freedom δ, from 0.25kgm2 to 0.69 kgm2 . The off-diagonal terms increase slightly. The effect on the eigenvalues is that theadded mass increases the weave speed and decreases the weave frequencies overall. The second changeis the added stiffness to the steering assembly due to the compression forces exerted by the hands on thehandlebar when leaning forward. This effects more entries in the matrices of the linearized equations ofwhich the most noticeable are the changes in the constant symmetric stiffness matrix K0 . The diagonalterm for the steering stiffness, K0 (2, 2), increases from 6.8 Nm/rad to 3.2 Nm/rad and the off-diagonalterms decrease by 50 %. The effect on the eigenvalues of this increased stiffness is an increased capsizespeed and an overall increase of weave frequencies. However the two effects together, result in little changecompared to the rigid rider model as described above. It should also be noted that the more the direction of
105weaveEigenvalues, Re , Im [1/s]Eigenvalues, Re , Im [1/s]10weave0capsize 5caster 1004 vw6v [m/s]2vc 85weave0capsize 5caster 1010weave026 vw48vc 10v [m/s](a) Eigenvalues for the Stratos bicycle with fully rigid rider,hands-free.(b) Eigenvalues for the Stratos bicycle with a rider with stretchedarms, hands on the handle bars and a yawing upper body.Figure 4: Eigenvalues for the lateral motions of a bicycle-rider combination with a) a fully rigid rider andhands-free; b) with a rider with stretched arms, hands on the handle bars and a yawing upper body accordingto the model from Figure 3a.105weaveEigenvalues, Re , Im [1/s]Eigenvalues, Re , Im [1/s]10weave0capsize 5caster 10024vw 6vc 810v [m/s](a) Eigenvalues for the Browser bicycle with fully rigid rider andhands-free.5weavecapsize0weave 5caster 100246810v [m/s](b) Eigenvalues for the Browser bicycle with a rider with rigidupper body and flexed arms and hands on the handle bars.Figure 5: Eigenvalues for the lateral motions of a bicycle-rider combination with a) a fully rigid rider andhands-free; b) with a rider with rigid upper body and flexed arms and hands on the handle bars according tothe model from Figure 3b.the stretched arms is parallel to the steer axis, the less is the change in the dynamics compared to the rigidrider model.3.2 UPRIGHT PASSIVE RIDERIn the upright posture the rigid upper body is connected to the rear frame and the arms are hinged at theshoulder and elbow and connected via the hands to the handlebar, see Figure 3b. The linearized equationsof motion are derived as described above together with the eigenvalues of the lateral motions. These eigenvalues are shown in Figure 5b. For comparison the eigenvalues for a rigid rider, that is the rider is rigidlyattached to the rear frame and there are no hands on the handlebar, are shown in Figure 5a.Compared to the rigid rider solutions there are dramatic changes in the eigenvalue structure. The stableforward speed range has disappeared completely because the weave speed has decreased to zero and thecapsize motion is always unstable. Note that the weave motion is now always stable but gets washed outby the unstable capsize. This dramatic change can be explained as follows. By adding the hinged armsto the handlebar a stable pendulum-type of oscillator has been added to the steer assembly. Although this
oscillator stabilizes the initial unstable weave motion it kills the stable eigen-dynamics of the bicycle. Thesteer assembly is not able to stabilize the lateral motion by the steer-into-the fall mechanism. The addedmass is most noticeable in the diagonal steering related term M (2, 2) which increases from 0.25 kgm2 to0.46 kgm2 . More dramatic is the change in the constant symmetric stiffness matrix K0 , here the steeringrelated stiffness K0 (2, 2) increase from an unstable 6.6 Nm/rad to a stable 2.3 Nm/rad, which partlyexplains the dramatic change in the eigenvalue structure.4 CONCLUSIONSAdding a passive upper body to the three degree of freedom Whipple model of an uncontrolled bicycle,without adding any extra degrees of freedom, can change the open-loop dynamics of the system. In the caseof a forward leaned rider with stretched arms and hands on the handle bars there is little change. However,an upright rider position with flexed arms and hands on the handle bars changes the open-loop dynamicsdrastically and ruins the self stability of the system.Future work is direct towards the comparison of the control effort of the human rider in both postures.ACKNOWLEDGEMENTThanks to Jason Moore for measuring the bicycles and riders during his Fulbright granted year (2008/2009)at TUDelft and thanks to Batavus for supplying the bicycles.REFERENCES J ONKER , J. B., AND M EIJAARD , J. P. SPACAR - Computer program for dynamic analysis of flexiblespatial mechanisms and manipulators. In Multibody systems handbook (1990), W. Schiehlen, Ed.,Berlin, Germany: Springer, pp. 123–143. KOOIJMAN , J. D. G., AND S CHWAB , A. L. Some observations on human control of a bicycle. In 11thmini Conference on Vehicle System Dynamics, Identification and Anomalies (VSDIA2008), Budapest,Hungary (Nov. 2008), I. Zobory, Ed., Budapest University of Technology and Economics, p. 8. KOOIJMAN , J. D. G., S CHWAB , A. L., AND M EIJAARD , J. P. Experimental validation of a model ofan uncontrolled bicycle. Multibody System Dynamics 19 (2008), 115–132. M EIJAARD , J. P., PAPADOPOULOS , J. M., RUINA , A., AND S CHWAB , A. L. Linearized dynamicsequations for the balance and steer of a bicycle: a benchmark and review. Proceedings of the RoyalSociety A 463 (2007), 1955–1982. M OORE , J. K., H UBBARD , M., KOOIJMAN , J. D. G., AND S CHWAB , A. L. A method for estimatingphysical properties of a combined bicycle and rider. In Proceedings of the ASME 2009 InternationalDesign Engineering Technical Conferences & Computers and Information in Engineering Conference(DETC2009, Aug 30 – Sep 2, 2009, San Diego, CA, 2009). M OORE , J. K., KOOIJMAN , J. D. G., AND S CHWAB , A. L. Rider motion identification during normalbicycling by means of principal component analysis. In MULTIBODY DYNAMICS 2009, ECCOMASThematic Conference (29 June–2 July 2009, Warsaw, Poland, 2009), M. W. K. Arczewski, J. Fraczek,Ed. S CHWAB , A. L., KOOIJMAN , J. D. G., AND M EIJAARD , J. P. Some recent developments in bicycledynamics and control. In Fourth European Conference on Structural Control (4ECSC) (2008), A. K.Belyaev and D. A. Indeitsev, Eds., Institute of Problems in Mechanical Engineering, Russian Academyof Sciences, pp. 695–702. W HIPPLE , F. J. W. The stability of the motion of a bicycle. Quarterly Journal of Pure and AppliedMathematics 30 (1899), 312–348.
A Measured Bicycle and Rider dataThis appendix summarizes the measured geometry and mass data of the bicycles and rider used. The firstbicycle is the Stratos which can be characterized as a hybrid bicycle. The second bicycle is the Browserwhich is a standard Dutch city bicycle.252221152324301317 19 161812112820311014 293 226 587 942716(a) Bicycle rider model with skeleton points.(b) Bicycle geometryFigure 6: a) Bicycle rider model with skeleton points and b) bicycle geometry.ParameterBottom bracket heightChain stay lengthFork lengthFront hub widthHandlebar lengthRear hub widthSeat post lengthSeat tube angleSeat tube lengthStem lengthWheel baseTrailHead tube angleRear wheel radiusFront wheel radiusSymbolhbblcslfwf hlhbwrhlspλstlstlswcλht 90 λrRrFValue for Stratos Value for Browser0.290 m0.295 m0.445 m0.460 m0.455 m0.455 m0.100 m0.100 m-0.090 m0.190 m0.130 m0.130 m0.195 m0.240 m75.0 68.5 0.480 m0.530 m0.190 m0.250 msee Table 1see Table 1see Table 1see Table 1see Table 1Table 1: Bicycle geometry dimensions for the Stratos and the Browser bicycle according to figure 6b.
ParameterWheel baseTrailSteer axis tiltGravityForward speedSymbolwcλgvValue for Stratos1.037 m0.0563 m16.9 9.81 N/kgvarious m/sValue for Browser1.121 m0.0686 m22.9 9.81 N/kgvarious m/sRear wheel RRadiusMassInertiarRmR(IRxx , IRyy )0.338 m3.96 kg(0.0916, 0.1545) kgm20.341 m3.11 kg(0.0884, 0.1525) kgm2(0.3267, 0.4825) m7.22 kg 0.372870 0.03835 00.716870 0.0383500.45473kgm2(0.2799, 0.5348) m9.86 kg 0.527140 0.11442 01.317610 0.1144200.75920kgm2(0.9089, 0.7296) m3.04 kg0.176840 0.02734 00.144430 0.0273400.04464kgm2(0.8632, 0.7467) m3.22 kg0.253380 0.07205 00.245370 0.0720500.09558kgm20.340 m3.334 kg(0.09387, 0.15686) kgm20.344 m2.02 kg(0.0904, 0.1494) kgm2Rear Body and frame assembly BCentre of mass (xB , zB )Massm B IBxx 0 IBxz 0 IByy 0 InertiaIBxz 0 IBzzFront Handlebar and fork assembly HCentre of mass (xH , zH )Massm H IHxx 0 IHxz 0 IHyy 0 InertiaIHxz 0 IHzzFront wheel FRadiusMassInertiarFmF(IRxx , IRyy )Table 2: Parameters for the Stratos and the Browser bicycle for the bicycle model from figure 2.ParameterChest circumferenceForward lean angleSymbolcchλf lHead circumferenceHip joint to hip jointLower arm circumferenceLower arm lengthLower leg circumferenceLower leg lengthShoulder to shoulderTorso lengthUpper arm circumferenceUpper arm lengthUpper leg circumferenceUpper leg 4 m63.9 (on Stratos)82.9 (on Browser)0.58 m0.26 m0.23 m0.33 m0.38 m0.46 m0.44 m0.48 m0.30 m0.28 m0.50 m0.46 mRider massmBr72.0 kgHead massLower arm massLower leg massTorso massUpper arm massUpper leg mBr0.028mBr0.100mBrTable 3: Anthropomorphic data for rider A according to figure 6a.
%% Matlab code for Skeleton Grid Points see Figure 1b%% Adapted Table 10 from MooreHubbardKooijmanSchwab2009r1 [0 0 0];r2 [0 0 -rR];r3 r2 [0 wrh/2 0];r4 r2 [0 -wrh/2 0];r5 [sqrt(lcs 2-(rR-hbb) 2) 0 -hbb];r6 [w 0 0];r7 r6 [0 0 -rF];r8 r7 [0 wfh/2 0];r9 r7 [0 -wfh/2 0];r10 r5 [-lst*cos(last) 0 -lst*sin(last)];% calculate f0f0 rF*cos(laht)-c*sin(laht);r11 r7 [-f0*sin(laht)-sqrt(lf 2-f0 2)*cos(laht) 0 f0*cos(laht)-sqrt(lf 2-f0 2)*sin(laht)];r12 [r11(1)-(r11(3)-r10(3))/tan(laht) 0 r10(3)];r13 r10 [-lsp*cos(last) 0 -lsp*sin(last)];% determine mid knee angle and mid knee positiona1 atan2((r5(1)-r13(1)),(r5(3)-r13(3)));l1 sqrt((r5(1)-r13(1)) 2 (r5(3)-r13(3)) 2);a2 acos((l1 2 lul 2-lll 2)/(2*l1*lul));%r14 r13 [lul*sin(a1 a2) 0 lul*cos(a1 a2)];r15 r13 [lto*cos(lafl) 0 -lto*sin(lafl)];r16 r12 [-ls*cos(laht) 0 -ls*sin(laht)];r17 r16 [0 lss/2 0];r18 r16 [0 -lss/2 0];r19 r17 [-lhb 0 0];r20 r18 [-lhb 0 0];r21 r15 [0 lss/2 0];r22 r15 [0 -lss/2 0];% determine left elbow positiona1 atan2((r19(1)-r21(1)),(r19(3)-r21(3)));l1 sqrt((r19(1)-r21(1)) 2 (r19(3)-r21(3)) 2);a2 acos((l1 2 lua 2-lla 2)/(2*l1*lua));%r23 r21 [lua*sin(a1-a2) 0 lua*cos(a1-a2)];r24 r23 [0 -lss 0];r25 r15 [ch/(2*pi)*cos(lafl) 0 -ch/(2*pi)*sin(lafl)];r26 r5 [0 lhh/2 0];r27 r5 [0 -lhh/2 0];r28 r14 [0 lhh/2 0];r29 r14 [0 -lhh/2 0];r30 r13 [0 lhh/2 0];r31 r13 [0 -lhh/2 0];Table 4: Skeleton points code according to Figure 6a and 6b.
Figure 1: Bicycling on a treadmill, two distinct postures: a) Rider A on the Stratos bicycle with forward leaned body and stretched arms. b) Rider A on the Browser bicycle with an upright body and ﬂexed arms. 2 BICYCLE MODEL The basic bicycle model used is the so-called W