An electric kickscooter multibody model: equations of motion and linear stability analysis

Multibody System Dynamics https://doi.org/10.1007/s11044-024-09974-4 RESEARCH An electric kickscooter multibody model: equations of motion and linear stability analysis A.G. Agúndez1·D. García-Vallejo1·E. Freire2 Received: 18 October 2023 / Accepted: 12 February 2024 © The Author(s) 2024 Abstract In this work, a detailed multibody model of an electric kickscooter is presented. The model includes toroidal wheels as well as rear and front suspensions. The equations of motion are derived and linearized along the steady forward motion of the vehicle. Using an efficient linearization approach, suitable for complex multibody systems with holonomic and nonholonomic constraints, allows for obtaining the reduced linearized equations of motion as a function of the geometric, dynamic, wheels’, and suspensions’ parameters. The proposed electric kickscooter multibody model is validated with the stability results of a previously presented electric kickscooter benchmark. Since the resulting eigenvalues are parameterized regarding the design parameters, a detailed linear stability analysis of the system is performed. In particular, the influence on the stability of the toroidal geometry of the wheels, the elliptic cross-section of the toroidal wheels, the rider model, the steering axis inclination angle, the inertia tensor of the front frame, and the rear and front suspensions is analyzed. The model presented, together with the linearized equations of motion obtained in this work, enables a systematic analysis of the stability of these vehicles, which helps design new electric kickscooters with improved vehicle safety conditions and oriented to a wider range of potential users. Keywords Electric kickscooter ·Nonholonomic system ·Linearization ·Stability analysis 1Introduction Personal transportation in urban areas has been evolving in recent years, with more and more people increasingly opting for light and nonpolluting means of transport. Among the A.G. Agúndez [email protected] D. García-Vallejo dgv[email protected] E. Freire [email protected] 1Department of Mechanical Engineering and Manufacturing, Universidad de Sevilla, Sevilla, Spain 2Department of Applied Mathematics II, Universidad de Sevilla, Sevilla, Spain A.G. Agúndez et al. advantages that users find are economic savings, physical exercise, and respect for the environment [1]. In addition to these advantages, there is a growing problem in the centers of large and medium-sized cities, where the high number of motorized vehicles causes high pollution and permanent traffic jams. Many municipalities have already limited the use of combustion-engine vehicles in city centers, and this restriction is expected to become increasingly widespread and restrictive. Therefore, the use of light, nonpolluting, singleperson vehicles such as bicycles or scooters is emerging as a widely used alternative. Moreover, the use of these vehicles is being boosted introducing electric motors. Electric bicycles and scooters allow covering longer distances (or with greater slopes) while reducing the required physical effort of the rider, thus making them accessible to more people. Among the light unipersonal electric vehicles, one can find the bicycle, the scooter, the Segway, or the skateboard, with the electric bicycle being the most widespread among them. However, given its lightness, small size, and low physical effort required, the electric kickscooter (hereafter referred to as an e-scooter) is presented as a present and future alternative to the bicycle. In vehicle dynamics, bicycles and motorcycles have been research subjects for more than 120 years. The work by Whipple [2], in 1899, was the first paper devoted to the stability analysis of bicycles. Linear and nonlinear dynamics aspects of a simplified bicycle model, together with some experiments, were presented by Åström et al. [3], and Limebeer et al. [4] considered simple and complex models to perform a review of different bicycle and motorcycle aspects. Meijaard et al. [5] presented, in 2007, a remarkable work with a well-acknowledged benchmark bicycle model with a complete list of validated parameters. This benchmark has been extensively used in several experimental [6] and theoretical works [7–10] devoted to multibody dynamics. The linear stability of the steady forward and circular motions of the bicycle benchmark was meticulously analyzed by Basu-Mandal et al. [7] and Xiong et al. [9,11]. Furthermore, more advanced models of frames, tires, and riders were introduced in different extensions of the bicycle benchmark [12–17]. To describe the interaction of the electric drive components on the driving characteristics of electrically assisted bicycles, simulations of the driving behavior of bicycles with different positions of battery and motor in open- and closed-loop tests can be found by Bolk et al. [18]. With regard to motorcycles, Sharp [19] was one of the first authors to publish his stability studies by computing the eigenvalues of a motorcycle under different conditions, describing the three most characteristic linear modes of two-wheeled vehicles. Cooper [20] showed the importance of aerodynamic effects on the stability of motorcycles when traveling at high speeds, and Jennings [21] showed that the stability of a motorcycle is sensitive to suspension damping. Next, Sharp [22] studied the effect of accelerations and decelerations on stability, and Roe et al. [23] analyzed the effect of torsional stiffness on stability improvement. Splerings [24] studied the gyroscopic effect on stability, and Nishimi et al. [25] compared the frequencies and modes computed by eigenvalue calculation of a motorcycle model with those obtained with experimental modal analysis. A sensitivity analysis to assess the stability of the motorcycle, based on eigenvalue calculation and experimental work, was performed by Sharp [26]. The comparison of a modal analysis using eigenvalue computation and experimental results was also carried out by Cossalter et al. [27], and the stability analysis of a scooter was presented in [28]. The number of works devoted to the study of e-scooters has rapidly increased in recent years. The potential of e-scooters as a hybrid mode of transport, combined with public transport in large cities, was confirmed by Kostrzewska et al. [29]. Several studies related to safety and injuries exist. Unkuri et al. [30] focus on injuries in children and teenagers on e-scooters, and Griffin et al. [31] compare the severity of injuries in children (aged 2–12 An electric kickscooter multibody model: equations of motion and linear stability analysis... years) between traditional and motorized scooters. Mebert [32] carried out in Switzerland a study on injuries in adults aged 16-80 years, showing that the number of cases has increased in recent years. In the same way, a recent review performed by Kowalczewska et al. [33] concluded the rising number of accidents and injuries involving e-scooters. Moreover, the analysis of the stability and maneuverability of these vehicles to ensure the safety of the driver and other road users can be found in other works. García-Vallejo et al. [34]presented an e-scooter benchmark based on the SEAT eXS Kickscooter ES2, with its corresponding set of parameters. Using the linear equations of the bicycle benchmark by Meijaard et al. [5], in [34], it was shown that, for the numerical values of the presented e-scooter benchmark parameters, these vehicles are completely unstable in the velocity range 0–10 m/s. Next, Paudel et al. [35] presented the range of design parameters in the current designs of e-scooters and analyzed their performance in terms of self-stability, braking effect, and steady-state turning. Despite finding some self-stability velocity range, the results showed that the current e-scooter designs are not as stable as bicycles, and these vehicles were found unstable within the most common range of legislated riding velocity. Klinger et al. [36] carried out simulation studies and on-road tests for two scenarios of particular interest from a vehicle safety perspective: straight-line stability and emergency braking. Concerning the driver’s comfort, Asperti et al. [37] studied the vertical dynamic behavior by presenting a model that accounts for the mechanical impedance of the driver, and Cano-Moreno et al. [38] quantitatively assessed, based on the velocity and quality of the road, the effects of vibrations on the comfort and health of the e-scooter driver. Garman et al. [39], by designing a test course simulating an urban environment, performed a successful characterization of the e-scooter rider kinematics and vehicle dynamics. Brunner et al. [40] analyzed the impact of hand signals and rear blind spot checks on e-scooters stability, and Dozza et al. [41] introduced a framework for a data-driven evaluation of micro-mobility vehicles (bicycles and e-scooters) in field tests. Nevertheless, to the best of authors’ knowledge, a detailed e-scooter multibody model for methodologically analyzing the stability of these vehicles has not yet been presented. The objectives of this paper are to present a detailed multibody model of an e-scooter, obtain the linearized equations along the steady forward motion as a function of the main design parameters of the vehicle, and conduct a detailed linear stability analysis. The paper presents several contributions. First, an e-scooter multibody model, which considers the numerical values of the e-scooter benchmark by García-Vallejo et al. [34] and is based on the SEAT eXS Kickscooter ES2, is presented. First, compared to previous works existing in the literature, the model includes toroidal wheels instead of hoop-shaped wheels and rear and front suspensions. Second, another major contribution is the derivation of the reduced linearized equations of motion of the uncontrolled e-scooter multibody model along the steady forward motion. In the works by García-Vallejo et al. [34] and Paudel et al. [35], the linear stability analysis is conducted using the linear lean and steer equations of the bicycle benchmark by Meijaard et al. [5], which involve ad hoc linearization as opposed to linearization of fully nonlinear equations. The linearized equations by Meijaard et al. [5], based on the work by Papadopoulos [42], are derived using angular momentum balance about various axes and are not based on a systematic linearization of fully nonlinear equations of motion. In contrast, this paper develops a detailed nonlinear model of the e-scooter. An efficient linearization approach [43], previously used with the bicycle benchmark [43] and the waveboard [44] multibody models, is used in this work. With this linearization approach, the analytical expressions of the linearized equations are obtained not only as a function of the geometric and inertial parameters of the benchmark by Meijaard et al. [5] but also as a function of the toroidal wheels’ and suspensions’ parameters of the e-scooter. A.G. Agúndez et al. Lastly, since the eigenvalues are parameterized in terms of these design parameters, a detailed linear stability analysis is performed by considering a wide range of scenarios. In this way, studying the influence of the different design parameters on the stability of these vehicles will help develop new e-scooters with improved vehicle safety conditions and oriented to a wider range of potential users, including elderly and physically impaired individuals. The following structure is considered. After the Introduction part, Sect. 2describes the e-scooter multibody model in detail. Particular attention is paid to describing the model of toroidal wheels and the front and rear suspensions, and the nonlinear equations of motion are derived. Next, Sect. 3presents the linearized equations of motion along the steady forward motion. In Sect. 4, the linear stability results are shown and discussed, and finally, Sect. 5 summarizes the main conclusions obtained from this work. 2 Description of the e-scooter multibody model In this section, the e-scooter multibody model is presented. The model considers the geometric and dynamic parameters of the e-scooter benchmark proposed by García-Vallejo et al. [34], corresponding to the SEAT eXS Kickscooter ES2. The modeling of the toroidal wheels and the rear and front suspensions are described in detail, and the nonlinear equations of motion are presented. 2.1 Description of the multibody model The multibody model presents seven rigid bodies: the rear and front wheels are bodies Rand F, respectively; the rear body and frame assembly, which includes a rigid rider, is body B; the front handlebar is designated as body H; the rear and front suspensions are represented as bodies SRand SF, respectively; and lastly, the global reference frame is denoted as body 1. The origin of the global frame is O1, and the origins of the body reference frames are located at the respective centres of mass Gj, with j={R,F,B,H,SR,SF}. To describe the system, a set of n=15 coordinates is used, with the n×1 vector of coordinates xgiven by: x=xbybzbψbφbθbδsθ SRθRθFξRξFηRηFT.(1) The coordinates xb,yb,andzbare Cartesian coordinates that locate the centre of mass GB; ψbis the yaw angle; φbis the lean angle; and θbis the pitch angle. The triplet {ψb,φb,θb} allows orientating body Bin space. The steering angle δcorresponds to the rotation of the handlebar with respect to body B. The coordinate srepresents the distance between GH and GFand considers the spring elongation of the front suspension, and θSRrepresents the rotation of body SRwith respect to body B, due to the rear suspension. The rotations of the rear and front wheels are given by θRand θF, respectively. Lastly, ξR,ξF,ηRand ηFare angular coordinates used to describe the toroidal geometry of the wheels. Figure 1 shows the numbering of the bodies, the set of generalized coordinates of the system and the body reference frames, and a lateral view of the multibody model with the main geometric parameters is depicted in Fig. 2. The list of parameters of the e-scooter, classified into geometric, dynamic, wheels’ and suspensions’ parameters, is shown in Table 1of Appendix A. The numerical values of these parameters correspond to those of the SEAT eXS Kickscooter ES2. To obtain the moment An electric kickscooter multibody model: equations of motion and linear stability analysis... Fig. 1 Multibody model of the SEAT eXS Kickscooter ES2: numbering of the bodies and body reference frames. The generalized coordinates of the multibody system, presented in Eq. (1), are shown, except for the coordinate θSR, shown in Fig. 4(b); the coordinate s, depicted in Fig. 5(b); and the angular coordinates ξR, ξF,ηRand ηF, which are shown in detail in Fig. 6. The rigid rider is represented in a simplified manner to facilitate the visualization of the multibody model of inertia tensors of the bodies, a CAD model of the e-scooter, shown in Fig. 3, is created. Figure 3(a) shows a view of the e-scooter without including the rider, and Fig. 3(b) depicts the vehicle with the rider model. A rigid rider with mass mh=75 kg is considered in this work. As detailed in Table 1of Appendix A, the mass of body Bincludes the masses of the deck, md=2.5 kg, and the rider, and is given by mB=md+mh=77.5 kg. Note that, in Figs. 1and 2, the shape of the rider has been simplified in order to ease the visualization of the multibody model. The orientation matrices of the body reference frames, expressed as a function of the elemental rotation matrices, are given by: RB(x)=RψbRφbRθb,RSR(x)=RBRθSR,RR(x)=RSRRθR,(2) RH(x)=RBRνRδ,RSF(x)=RH,RF(x)=RSFRθF,(3) where Rνconsiders the steer axis tilt angle denoted by νin Fig. 1. A detailed view of the rear suspension is shown in Fig. 4. In particular, Fig. 4(a) shows a view of the rear suspension of a SEAT eXS Kickscooter ES2, and Fig. 4(b) illustrates a scheme with body SRand the coordinate θSR.Notethat,asshowninFig.4(b), the rear suspension is modeled by means of a torsion spring, with stiffness constant kr, and a damper with damping coefficient dr. Similarly, Fig. 5(a) shows the front suspension of the SEAT A.G. Agúndez et al. Fig. 2 Lateral view of the e-scooter multibody model with the main geometric parameters. A detailed list of the parameters with their numerical values can be found in Table 1of Appendix A Fig. 3 CAD model of the e-scooter SEAT eXS Kickscooter ES2, with a rigid rider of mass mh=75 kg eXS Kickscooter ES2, and Fig. 5(b) presents a scheme with body SFand the coordinate s. The front suspension is modeled as a linear spring with stiffness constant kfand a damper with damping coefficient df. An electric kickscooter multibody model: equations of motion and linear stability analysis... Fig. 4 Rear suspension of the e-scooter multibody model Fig. 5 Front suspension of the e-scooter multibody model The position vectors of the centres of mass Gj, expressed in the global reference frame, are computed as follows: rGB=xbybzbT,rA=rGB+RB¯ rGBA, rGSR=rA+RSR¯ rAGSR,rGR=rA+RSR¯ rAGR,(4) rI=rGB+RB¯ rGBI,rGH=rI+RH¯ rIGH, rGF=rGH+RH¯ rGHGF,rGSF=rGF+RH¯ rGFGSF, A.G. Agúndez et al. where rAand rIare the position vectors of points Aand I, shown in Fig. 2.InEqs.(4), ¯ rGBA,¯ rAGSR,¯ rAGR,¯ rGBI,¯ rIGH,¯ rGHGFand ¯ rGFGSFare vectors, expressed in the body reference frames, given by: ¯ rGBA=d−xB0RR−zBT,¯ rAGSR=−d/200 T, ¯ rAGR=−d00 T,¯ rGBI=xI00 T,(5) ¯ rIGH=00−bT,¯ rGHGF=00−sT, ¯ rGFGSF=00eT, where xBand zBare the horizontal and vertical distances, respectively, between the centre of mass GBand the rear contact point P;xIis the horizontal distance between GBand the auxiliar point I;bis the distance between Iand the centre of mass GH;dis the distance between the auxiliar point Aand the centre of mass GR;eis the distance between GFand GSF;andRRis the radius of the rear wheel. These parameters are shown in Fig. 2,andtheir numerical values are presented in Appendix A. 2.2 Description of the toroidal wheels The wheels of the e-scooter are modeled as tori with elliptic cross-section. To parameterize the surfaces of the toroidal wheels, the angular coordinates ξiand ηi,presentedinEq.(1), with i={R,F}, are used. The toroidal wheels present a major radius ρiand a minor radius ai, verifying the following relation: ρi+ai=Ri,(6) where Riis the radius of the hoop-shaped wheel of equivalent radius. In the elliptic cross-section, the minor axis of the ellipse is bi, and the major axis corresponds to the previously mentioned minor radius of the torus ai. The elliptic profile ri,in polar form relative to its centre, is given by: ri(ηi)=aibi (bicos(ηi))2+(aisin(ηi))2.(7) Note that the elliptic profile is parameterized in terms of the coordinate ηi. Figure 6(a) presents a three-dimensional view of the toroidal wheel, with the coordinates ξiand ηi,and Fig. 6(b) represents a cross-section of the wheel, where the geometric parameters ρi,ai, and bican be seen. The geometry of the wheel is completely described by defining the nondimensional parameters μiand σi, with i={R,F}: μi=ai ρi ,σ i=bi ai ,(8) where μiis the torus aspect ratio, given by the ratio of the minor to the major radius of the wheel, and σiis the aspect ratio of the elliptical cross-section, defined as the ratio of the mi- nor to the major radius of the ellipse. Note that the hoop-shaped wheel case of the e-scooter benchmark in Ref. [34] is also captured with this model and is obtained by degenerating the tori for μi=σi=0, leading to ρi=Ri. The inertia tensors of the rear and front wheels used An electric kickscooter multibody model: equations of motion and linear stability analysis... Fig. 6 Model of the toroidal wheel. A three-dimensional view of the wheel is shown in Fig. 6(a), which depicts the angular coordinates ξiand ηi, with i={R,F}, that describe the toroidal geometry. The local reference frames, the centre of mass Gi, the centre of the torus tube Ci, and the contact points with the ground, denoted as Pand Qfor the rear and front wheels, respectively, are also shown in Fig. 6(a). A crosssection of the wheel is depicted in Fig. 6(b), illustrating the major radius of the torus, ρi, and the minor and major axes of the elliptic cross-section, biand ai, respectively in the e-scooter benchmark [34] are substituted by the inertia tensors of an elliptic torus, parameterized in terms of the cross-section parameters ρi,ai,andbi, whose expressions can be found by Diaz et al. [45]: Iixx =Iizz =1 8mi4ρ2 i+3a2 i+2b2 i,(9) Iiyy =1 4mi4ρ2 i+3a2 i,(10) with i={R,F}. The contact of the e-scooter wheels with the ground leads to the following set of holonomic constraints: C(x)=rPZrQZn·tLRn·tTRn·tLFn·tTFT=0,(11) where rPZand rQZare the Z-components of the position vectors of the contact points, rP and rQ;tLiand tTiare the longitudinal and transversal tangent vectors to the contact points; and nis the normal vector to the ground surface. These vectors are given by: rP=rGR+RR¯ rGRP,rQ=rGF+RF¯ rGFQ, tLR=RR ∂¯ rGRP ∂ξR ,tTR=RR ∂¯ rGRP ∂ηR ,(12) tLF=RF ∂¯ rGFQ ∂ξF ,tTF=RF ∂¯ rGFQ ∂ηF . A.G. Agúndez et al. The size of the Jacobian matrix Jin Eq. (54)is(2n−2m−l) ×(2n−2m−l) =14 ×14. Therefore, the linearized equations of motion of the e-scooter are given by the following fourteen equations: ˙ ˜xb=˜vxb,(55) ˙ ˜ φb=˜ωφb,(56) ˙ ˜ δ=˜ωδ,(57) ˙ ˜s=˜vs,(58) ˙ ˜ θSR=˜ωθSR,(59) ˙ ˜vxb=1˜s+2˜ θSR+3˜vs+4˜ωθSR,(60) ˙ ˜ωφb=5˜ φb+6+7v2˜ δ+8v˜ωφb+9v˜ωδ,(61) ˙ ˜ωδ=10 ˜ φb+11 +12v2˜ δ+13v˜ωφb+14v˜ωδ,(62) ˙ ˜vs=15 ˜s+16 ˜ θSR+17 ˜vs+18 ˜ωθSR,(63) ˙ ˜ωθSR=19 ˜s+20 ˜ θSR+21 ˜vs+22 ˜ωθSR,(64) ˙ ˜yb=γ1v˜ δ+γ2˜ωφb+γ3˜ωδ+γ4v˜ ψb,(65) ˙ ˜ ψb=γ5v˜ δ+γ6˜ωδ,(66) ˙ ˜ θR=γ7˜vxb+γ8˜vs+γ9˜ωθSR,(67) ˙ ˜ θF=γ10 ˜vxb+γ11 ˜vs+γ12 ˜ωθSR.(68) In the system of equations (55)–(68), k, with k=1...22, are the coefficients of the linearized dynamic equations, and γp, with p=1...12, are the coefficients associated with the linearized nonholonomic constraints. The analytical expressions of these coefficients have been obtained in terms of the geometric, inertial, wheels’ and suspensions’ parameters. Their expressions, which are overly long to be shown here, are available to the reader on request. Some comments about the linear system of equations (55)–(68) are included below. Among the linearized equations of motion (55)–(68), Eqs. (61)and(62) are the lean and steer equations, respectively. In the works by García-Vallejo et al. [34] and Paudel et al. [35], the linearized lean and steer equations of the e-scooter were obtained by particularizing the linear equations of the bicycle benchmark by Meijaard et al. [5] for the parameters of different e-scooters designs. The lean and steer equations derived by Meijaard et al. [5]are given by: M1¨ q+vC1˙ q+gK0+v2K2q=f,(69) where q=φδ Tis the generalized coordinates vector, including the lean and steering angles; M1is the inertia matrix; C1is the velocity dependent ‘damping’ matrix; K0is the gravitational stiffness matrix; and K2includes the gyroscopic and centrifugal effects. Lastly, fis the generalized torque vector, including the lean disturbance torque Tφand the steering An electric kickscooter multibody model: equations of motion and linear stability analysis... torque Tδ, with f=TφTδT.Forf=0,Eq.(69) can alternatively be written as: ¨ ˜ φ=β1˜ φ+β2+β3v2˜ δ+β4v˙ ˜ φ+β5v˙ ˜ δ, (70) ¨ ˜ δ=β6˜ φ+β7+β8v2˜ δ+β9v˙ ˜ φ+β10 v˙ ˜ δ. (71) The structure of the lean and steer equations (61)and(62) obtained in the present work is the same as that of the benchmark equations (70)and(71) by Meijaard et al. [5]. In particular, the coefficients k, particularized for the hoop-shaped wheels case (μi=σi=0), the rigid case (kr→∞,kf→∞) and the undamped scenario (dr=df=0), are the same as the coefficients βs, with s=1...10, of Eqs. (70)and(71). Nevertheless, the linearized lean and steer equations (69) of the bicycle benchmark by Meijaard et al. [5] are computed in a different way from the linearized equations (61)and(62). The linearized equations by Meijaard et al. [5] involve ad hoc linearization instead of systematic linearization of the nonlinear equations of motion and are derived based on Papadopoulos [42] using angular momentum balance about various axes. Furthermore, only the linearized equations associated with the lateral dynamics of the bicycle are obtained by Meijaard et al. [5]. In contrast, the linearized equations (55)–(68) derived in this work are obtained from the linearization of the nonlinear equations of motion (27)–(29). Using the linearization approach of Ref. [43], the analytical expressions of the linearized equations are obtained not only as a function of the geometric and dynamic parameters of the benchmark by Meijaard et al. [5] but also as a function of the toroidal wheels’ and suspensions’ parameters of the e-scooter. In the work by García-Vallejo et al. [34], the equation associated with the forward motion is ˙vxb=0, being decoupled from the remaining equations. Nevertheless, in the present work, the addition of the rear and front suspensions (and subsequently the coordinates θSRand s) leads to Eq. (60), where it can be seen that the forward motion is coupled with the suspensions coordinates and their velocities. This coupling found in the model with suspensions is associated with the squat, representing the effect of the inertia on the suspensions. 4 Results and discussion In this section, a detailed linear stability analysis is performed by resorting to the linearized equations of motion (55)–(68). The evolution of the eigenvalues with the forward speed is studied for different scenarios of interest. The chart of Fig. 8summarizes the scenarios considered in the linear stability analysis of the e-scooter. Validation of the e-scooter multibody model presented in this work First, the model of the present work is validated with the results presented by García-Vallejo et al. [34]. As previously mentioned, Eqs. (61)and(62), derived by linearizing the fully nonlinear equations of motion, yield the linearized equations by Meijaard et al. [5], used in Ref. [34], when particularized for the hoop-shaped wheels case (μi=σi=0), the rigid case (kr→∞,kf→∞) and the undamped scenario (dr=df=0). Obtaining the same results allows for validating the multibody model presented in this work and the linearization performed, given that the linear equations by Meijaard et al. [5] are computed using an ad hoc approach instead of linearizing the nonlinear equations of motion. The evolution of the eigenvalues with the forward speed is shown in Fig. 9(b) and corresponds to the same linear stability results of the e-scooter benchmark presented by García-Vallejo et al. [34] A.G. Agúndez et al. Fig. 8 Linear stability analysis of the e-scooter: eigenvalues sensitivity analysis Fig. 9 Eigenvalues from the linearized speed analysis for the bicycle and e-scooter benchmarks. In contrast to the bicycle, no self-stability velocity range exists for riding the e-scooter with hands-off An electric kickscooter multibody model: equations of motion and linear stability analysis... Fig. 10 Influence of the moments of inertia of the front frame (body H): evolution of the real and imaginary parts of the eigenvalues from the linearized speed analysis, varying the inertia tensor of the front handlebar. The scenarios with α=1, α=0.5, and α=0.1 are considered As shown in Fig. 9(a), the uncontrolled bicycle presents a self-stability velocity range, given by vw<v<v c. The lower bound vwis the weave speed, corresponding to the stabilization of the weave mode, and the upper bound is the capsize speed vcfor which the uncontrolled bicycle becomes unstable. In contrast, Fig. 9(b) shows that for the numerical values of the SEAT eXS Kickscooter ES2 (see Table 1of Appendix A), the e-scooter is completely unstable when ridden with hands-off. Influence of the inertia tensor of the front frame First, the influence of the inertia tensor of the front frame (body H) on the stability is studied. To this end, the evolution of the eigenvalues with the forward velocity vis computed for different moments of inertia of body H. Taking as reference the inertia tensor ¯ IHshown in Table 1of Appendix A,which corresponds to the SEAT eXS Kickscooter ES2, the modified inertia tensor of body H, denoted by ¯ IH, is given by: ¯ IH=α¯ IH,(72) where αis a constant that allows for modifying ¯ IH. In this case, the mass of body His assumed to be redistributed in such a way that the position of the centre of mass GHis not modified. The variation of the moments of inertia of the front frame greatly impacts on the stability of the e-scooter. These variations could arise, for instance, from modifying the location of the e-scooter battery, removing it from the handlebar. Figures 10 (a) and (b) show the evolution of the real and imaginary parts of the eigenvalues with the forward velocity, for the scenarios of α=1, α=0.5andα=0.1. Despite for α=1 the e-scooter is completely unstable, Fig. 10 (a) shows that a decrease in αleads to the appearance of a self-stability velocity range. In particular, for α=0.5, the self-stability velocity range is given by 6.22 < v<9.18 m/s, and for α=0.1, this range broadens to 4.56 <v<9.18 m/s. Influence ofthe rider model (inertia tensor ofbody B)As described in Sect. 2.1, the body B of the multibody model is comprised of the e-scooter deck and the rider. Due to the reduced mass of the e-scooter compared to the rider’s mass, its modeling is of essential importance. A.G. Agúndez et al. Fig. 11 Influence of the moments of inertia of body B: evolution of the real parts of the eigenvalues with the forward speed, considering the rider model in Table 1(Bm) and an alternative inertia tensor of body B presented by Paudel et al. [35] (Alt) A single e-scooter design presents varying performance depending on the rider’s anthropometric characteristics or their riding style. To illustrate the rider’s influence on stability, the evolution of the eigenvalues with the forward speed is computed for the rider model considered in the e-scooter benchmark by García-Vallejo et al. [34] and in this study, as depicted in the CAD model shown in Fig. 3(b) and with the inertia tensor ¯ IBpresented in Table 1of Appendix A. This model is referred to as ‘Bm’ in Figs. 11 (a) and (b). Additionally, the eigenvalues are computed for an alternative rider model with standing posture, provided by Paudel et al. [35], which is designated as ‘Alt’ in Figs. 11 (a) and (b). The inertia tensor of the body Bconsidered by Paudel et al. [35], denoted by ¯ IB, is given by: ¯ IB=⎛ ⎝ 19.21 0 −0.037 0¯ I Byy 0 −0.037 0 1.78 ⎞ ⎠kg m2.(73) Note that the moment of inertia ¯ I Byy in Eq. (73) is not provided in Ref. [35], as it has no influence on the lateral balancing of the e-scooter. Figure 11 shows the evolution of the real part of the eigenvalues with the forward velocity, considering both rider models. Two scenarios are studied: Fig. 11 (a) considers the inertia tensor of the front frame shown in Table 1(α=1), and Fig. 11 (b) depicts the case of α=0.1. In Fig. 11 (a), it can be seen that the e-scooter is completely unstable for both riders’ models. In contrast, Fig. 11 (b) shows a self-stability velocity range for both inertia tensors of body B. For the rider model considered in this study, the e-scooter is found to be stable for 4.56 <v<9.18 m/s, and for the inertia tensor by Paudel et al. [35], 4.53 <v<9.18 m/s. It is important to note that the evolution of the eigenvalues in Fig. 11 (b), for the scenario ‘Alt’, is qualitatively the same as that shown by Paudel et al. [35]. Influence of the toroidal wheels The influence of the wheels’ parameters on the stability of the e-scooter is analyzed. The toroidal geometry of the wheels is considered by means of nonzero values of the tori aspect ratios μi. Figures 12 (a) and (b) show the evolution of the real and imaginary parts of the eigenvalues with the forward speed in the hoop-shaped and An electric kickscooter multibody model: equations of motion and linear stability analysis... Fig. 12 Influence of the tori aspect ratios μi: comparison of the evolution of the eigenvalues with forward velocity, in the hoop-shaped and torus-shaped scenarios Fig. 13 Influence of the tori aspect ratios μi: comparison of the evolution of the eigenvalues with forward velocity, in the hoop-shaped and torus-shaped scenarios, considering the inertia tensor of body Bpresented by Paudel et al. [35] torus-shaped scenarios, respectively. The results in Fig. 12 are obtained for the numerical values of the e-scooter parameters in Table 1. In particular, the linearized speed analysis is performed for μR=μF=0.3, which constitute realistic values of e-scooter wheels. As shown in Fig. 12 (a), the toroidal geometry greatly impacts on the evolution of the eigenvalues, and the results vary considerably with respect to the hoop-shaped scenario. Nevertheless, despite the toroidal wheels, it can be seen that the e-scooter remains completely unstable in all the velocity range. Figure 13 (a) shows a comparison between the hoop and toroidal wheel scenarios, when the inertia tensor of body Bis considered by Paudel et al. [35]. Similar results as those of Fig. 12 (a) are obtained, with the e-scooter being completely unstable. Lastly, Fig. 12 (b) shows that the self-stability velocity range found in Fig. 11 (b), when a value of α=0.1was considered, vanishes due to the destabilizing effect of the toroidal wheels. A.G. Agúndez et al. Fig. 14 Influence of the ellipses’ aspect ratios σi: comparison of the evolution of the eigenvalues with forward velocity, for σi=1 (circular cross-section), σi=0.9andσi=1.1 Influence of the elliptic cross-section in the toroidal wheel The results presented so far correspond to the case of a toroidal wheel with circular cross-section (σi=1). An elliptic profile of the wheel cross-section can be considered with σi=1. Figures 14 (a) and (b) show the evolution of the real and imaginary parts, respectively, of the eigenvalues with the forward speed. The scenarios of σi=0.9andσi=1.1 are compared with the circular-cross section case (σi=1). It can be seen that the inclusion of the elliptic profile leads to variations in the evolution of the eigenvalues, with a lower value of σresulting in a stabilizing effect. Note that, in any case, the e-scooter multibody model is unstable, existing eigenvalues with positive real parts. Influenceofthesteeraxistiltangle The influence of the steer axis tilt angle νon the stability is studied. As shown in Fig. 15 and described in Appendix A,thetrailc, the position of the centre of mass GHand the magnitudes xI,band lare a function of the angle ν. Figure 15 shows, for an arbitrary steering axis angle ν, the variation of these magnitudes with respect to the reference values shown in Table 1(corresponding to ν∗=18◦and represented with the superscript ∗). Figures 16 (a) and (b) show, for the hoop-shaped and toroidal wheel scenarios, respectively, the evolution of the real part of the eigenvalues with the forward speed for three different steering axis inclination angles: ν∗=18◦(reference value), ν=16◦and ν=20◦. Note that an increase in νresults in a reduction of the real parts of the eigenvalues. Nevertheless, it can be seen that for the numerical values of the e-scooter parameters presented in Table 1, the variation of the steering axis angle νdoes not lead to any self-stability velocity range; eigenvalues with positive real parts exist in all cases. Influence of the rear andfront suspensions So far, the results shown correspond to the rigid case. The results of the multibody model considering rear and front suspensions are now compared with the rigid scenario. In this case, the numerical values of the rear and front stiffness constants, obtained with the quasi-static tests and provided in Eqs. (25)and(26), are used instead of kr→∞and kf→∞. Similarly, the null values of the rear and front damping coefficients in the undamped case, dr=df=0, are substituted by the numerical values obtained with the dynamic tests and presented in Eqs. (25)and(26). In the rigid case, the masses of the rear and front suspensions (mSRand mSF)aresettobenull,andthe An electric kickscooter multibody model: equations of motion and linear stability analysis... Fig. 15 Variation of the parameters c,xI,b,land position of GHwith the steer axis tilt angle ν Fig. 16 Influence of the steering axis tilt angle νon the stability numerical values of mBand mHare those found in Table 1(mB=77.5kgandmH=8kg). In contrast, in the scenario considering the suspensions, mSR=mSF=0.5 kg and the masses of bodies Band Hare reduced, with mB=77 kg and mH=7.5 kg. Therefore, the total mass of the e-scooter is the same in both scenarios. Furthermore, given the low influence of the inertia tensors of bodies SRand SFon the stability, ¯ ISRand ¯ ISFcan be assumed to be negligible (see Table 1). Figures 17 (a) and (b) show the evolution of the real part of the eigenvalues of the escooter with hoop-shaped and toroidal wheels, respectively, considering the suspensions A.G. Agúndez et al. Fig. 17 Linearized speed analysis of the e-scooter multibody model with rear and front suspensions: comparison with the rigid case and the rigid case. The most important aspect to highlight due to the introduction of the suspensions is the appearance of two complex conjugate pairs of eigenvalues. The numerical values of the real parts of these pairs, which can be seen in Figs. 17 (a) and (b) in the suspensions’ scenario (labeled as ‘Susp’), are independent of the forward speed and are strongly dependent on the damping coefficients drand df. Moreover, the imaginary parts of these complex conjugate pairs strongly depend on the stiffness constants of the rear and front suspensions. Concerning the remaining eigenvalues, it can be seen that the rear and front suspensions, for this particular set of numerical values of kr,kf,drand dfdo not lead to major differences with respect to the rigid scenario (labeled as ‘Rigid’ in Figs. 17 (a) and (b)). To illustrate the result of the linearization, the Jacobian matrix that leads to the eigenvalues shown in Fig. 17 (b) is provided in Appendix C. 5Conclusions In this work, an e-scooter multibody model with toroidal wheels and suspensions has been proposed. The model is based on the SEAT eXS Kickscooter ES2. With respect to previous works in the literature, which compute the linearized lean and steer equations by using ad hoc linearization [34,35], the nonlinear equations of motion were linearized in this paper. To this end, an efficient linearization approach, devoted to multibody systems with holonomic and nonholonomic constraints, allowed the reduced linearized system of equations along the steady forward motion of the e-scooter to be obtained. The linearized equations were computed analytically as a function of the geometric, dynamic, toroidal wheels’ and suspensions’ parameters of the e-scooter. The multibody model presented in this work was validated by comparing the linear stability results along the steady forward motion with the e-scooter benchmark results by García-Vallejo et al. [34]. The same eigenvalues from the linearized speed analysis were obtained. In contrast to bicycles, for the numerical values of the e-scooter parameters considered in this work, the multibody model was shown to be completely unstable. The linear system of equations presented in Sect. 3.2 allowed performing a detailed linear stability analysis. First, An electric kickscooter multibody model: equations of motion and linear stability analysis... it was shown that the variation of the moments of inertia of the front frame (body H)ledto significant changes in the stability of these vehicles. In particular, a reduction of the inertia tensor of the front frame (associated, for instance, with removal of the battery from the handlebar) gave rise to the appearance of a self-stability velocity range. Next, the influence of the rider model, considered by modifying the inertia tensor of body B, was studied. For both the rider model of the e-scooter benchmark [34] and Paudel et al. [35], no self-stability velocity range was found. The combined effect of the inertia tensor of body Bfound by Paudel et al. [35], together with the reduction of the inertia tensor of the front frame, led to an evolution of the eigenvalues with the forward speed qualitatively similar to that of Ref. [35]. Furthermore, the inclusion of toroidal wheels in the multibody model allowed studying their effect on the stability of the vehicle. In all the cases, the toroidal geometry resulted in no self-stability velocity range. Regarding the geometry of the cross-section of the toroidal wheel, a lower value of the ellipse aspect ratio results in a stabilizing effect. Similarly, the unstable behavior remained in spite of the modification of the steer axis tilt angle. Lastly, the introduction of the rear and front suspensions led to the emergence of two complex conjugate pairs of eigenvalues. The real parts of these eigenvalues, which were shown to be independent of the forward speed, are highly dependent on the damping coefficients of the suspensions, while the imaginary parts are highly dependent on the stiffness constants of the suspensions’ springs. For the particular set of numerical values of the stiffness and damping constants, obtained from the quasi-static and dynamic tests, no remarkable changes with respect to the rigid scenario were found in the remaining eigenvalues. The electric kickscooter multibody model presented, together with the linearized equations of motion obtained in this work, enables a systematic analysis of the stability of these vehicles, which helps in designing new e-scooters with improved vehicle safety conditions and oriented to a wider range of potential users. In future work, the stability analysis could be extended to more complex trajectories, such as periodic ones. Moreover, a key aspect that could be improved is the rider model, considering different postures and riding styles, the relative motion of the upper body with respect to the board, and the effect of the body support via the arms and hands on the handlebar. Lastly, the influence of different tire models on stability could also be studied. Appendix A: List of the e-scooter multibody model parameters The list of parameters of the e-scooter multibody model is shown in Table 1. These parameters are classified into geometric, dynamic, wheels’ and suspensions’ parameters. Table 1 List of the e-scooter multibody model parameters and numerical values Symbol Numerical value Geometric parameters Wheel base w0.925 m Trail (ref. value) c0.034 m Steer axis tilt (ref. value) νπ/10 rad (18◦) Distance A,GRd0.15 m Distance GSF,GFe0.1 m A.G. Agúndez et al. 42. Papadopoulos, J.M.: Bicycle Steering Dynamics and Self-Stability: A Summary Report on Work in Progress. Cornell Bicycle Research Project. Cornell University, Ithaca (1987) 43. García-Agúndez, A., García-Vallejo, D., Freire, E.: Linearization approaches for general multibody systems validated through stability analysis of a benchmark bicycle model. Nonlinear Dyn. 103(1), 557–580 (2021) 44. Agúndez, A.G., García-Vallejo, D., Freire, E., Mikkola, A.: A reduced and linearized high fidelity waveboard multibody model for stability analysis. J. Comput. Nonlinear Dyn. 17(5), 051010 (2022) 45. Diaz, R.A., Herrera, W.J., Martinez, R.: Using symmetries and generating functions to calculate and minimize moments of inertia (2004). Preprint. arXiv:physics/0404005 46. Schiehlen, W.: Multibody system dynamics: roots and perspectives. Multibody Syst. Dyn. 1(2), 149–188 (1997) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.