Autonomous Motorcycles for Agile Maneuvers, Part I: Dynamic Modeling

CONFIDENTIAL. Limited circulation. For review only. 1

Jingang Yi, Yizhai Zhang, and Dezhen Song

Abstract— Single-track vehicles, such as motorcycles, provide vG Velocity vector of the motorcycle frame (with an agile mobile platform. Modeling and control of motorcycles rear wheel set). for agile maneuvers, such as those by professional racing riders, γ ,γ Slip angles of the front and rear wheels, are challenging due to motorcycle’s unstable platform and f r complex tire/road interaction. As a first step attempting to respectively. understand how racing riders drive a motorcycle, in this two- λf ,λr Longitudinal slip values of the front and rear part paper we present a modeling and tracking control design of wheels, respectively. an autonomous motorcycle. In this first-part paper, we discuss ϕ, ψ Rear frame roll and yaw angles, respectively. a new dynamics model for the autonomous motorcycle. We consider the existence of lateral sliding velocity at each wheel ϕf The front steering wheel plane camber angle. contact point. Because of the importance of the tire/road inter- φ Motorcycle steering angle. action for vehicle stability and maneuverability, the dynamic φ Motorcycle kinematic steering angle (pro- modeling scheme also includes the motorcycle tire models. The g new nonlinear dynamic models are used for control systems jected steering angle on the ground plane). design in the companion paper with control input variables σ The front kinematic steering angle variable. are the front wheel steering angle and the angular velocities of m The total mass of the motorcycle rear frame front and rear wheels. and wheel. Js The mass moment of rotation of the steering NOMENCLATURE fork (with the front wheel set) about its rota- X, Y, Z A ground-fixed coordinate system. tion axis. x, y, z A wheel-base line moving coordinate system. l Motorcycle wheel base, i.e., distance between xw,yw,zw A front wheel plane coordinate system. C1 and C2. xB,yB,zB A rear frame body coordinate system. lt The front steering wheel trail. C1,C2 Front and rear wheel contact points on the h The height of the motorcycle center of mass. ground. r The front and rear wheel radius. Ffx,Ffy,Ffz The front wheel contact forces in the x, y, z- δ The rear frame rotation angle from its vertical axis directions. position. Frx,Fry,Frz The rear wheel contact forces in the x, y, z ξ The front steering axis caster angle. directions. R The radius of the trajectory of point C2 under vf , vr Velocity vectors of the front and rear wheel neutral steering turns. contact points, respectively. Cd The aerodynamics drag coefficient. vfx,vfy Front wheel contact point C1 velocities along kλ, kγ , kϕ Longitudinal, lateral, and camber stiffness co- the x-andy-axis directions, respectively. efficients of motorcycle tires, respectively. vrx,vry Rear wheel contact point C2 velocities along L(Lc) The (constrained) Lagrangian of the motorcy- the x-andy-axis directions, respectively. cle systems. vfxw ,vfyw Front wheel contact point C1 velocities along the xw-andyw-axis directions, respectively. I. INTRODUCTION vX ,vY Rear wheel contact point C2 velocities along the X-andY -axis directions, respectively. Single-track vehicles, such as motorcycles and bicycles, ωf ,ωr Wheel angular velocities of the front and rear have high maneuverability and strong off-road capabilities. wheels, respectively. In environments such as deserts, forests, and mountains, This work is supported in part by the National Science Foundation under mobility of single-track vehicles significantly outperforms grant CMMI-0856095. that of double-track vehicles. The recent demonstration of J. Yi is with the Department of Mechanical and Aerospace Engineering, the Blue Team’s autonomous motorcycle (Fig. 1(a)) in the Rutgers University, Piscataway, NJ 08854 USA. E-mail: [email protected]. Y. Zhang is with the Department of Information and Communication 2005 DARPA Grand Challenge autonomous ground vehicles Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi Province, competition has shown an example of the high-agility of the P.R. China. single-track platform [1]. D. Song is with the Department of Computer Science and Engineer- ing, Texas A&M University, College Station, TX 77843 USA. E-mail: Although the extensive study of the motorcycle dynamics [email protected]. have revealed some knowledge of motorcycle platform under Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 5, 2009. 2 CONFIDENTIAL. Limited circulation. For review only. steady motions, however, modeling and control of motorcy- them. There is a large body of work that studies motorcycle cles for agile maneuvers, such as those by professional racing stability and dynamics, and readers can refer to two recent re- riders, still remains a challenging task due to motorcycle’s view papers: one from a historical development viewpoint [2] intrinsic unstable platform and complex tire/road interaction. and the other from a control-oriented perspective [6]. Professional motorcycle riders can leverage the safety limits The modeling work can be considered as two groups [6]: a of the tire/road interaction, and maintain the vehicles at high simple inverted pendulum model and a multi-body dynamic performance while preserving safety. Understanding how model. For example, some simple second-order dynamic human drivers carry out these maneuvers not only advances models are presented in [7] to study the balance stability of our knowledge in vehicle dynamics and control, but also can a bicycle. Several researchers have studied the motorcycle be used for enhancing vehicle safety, such as designing new dynamics using multi-body dynamics [8]–[11]. The model driver assistance systems, for example, emergency obstacle developed in [9] is very comprehensive and contains various avoidance maneuvers. vehicle components. The model has been implemented in a simulation package called FastBike for the purposes of real-time simulations. Multi-body dynamics models are not suitable for the control system design due to their complexity while a inverted-pendulum model overly simplify the prob- lem and does not capture all of the dynamics and geometric characteristics. In [4], [12], mathematical models of a motorcycle are discussed using (constrained) Lagrange’s equations. In [13], (a) (b) experimental study of the motorcycle handling is compared with the mathematical dynamics model of a motorcycle with Fig. 1. (a) The Blue team autonomous motorcycle. (b) A Rutgers autonomous pocket bike. the rider. Stability and steering characteristics of a motorcycle are typically discussed using a linearization approach As a first step towards to to understand such high- with a consideration of a constant velocity [2], [6], [8], [14]– performance capabilities of the human drivers, and then de- [17]. A non-minimum phase property (unstable poles and sign human-inspired control algorithms for agile maneuvers, zeros in motorcycle dynamics) in these analyses explains the objective of this two-part paper is to develop a new the counter-steering phenomena and other steering stability modeling and control scheme for an autonomous motorcycle. observations. In [14], it is also demonstrated experimentally Comparing with existing study on the motorcycle dynamics the in-significance of the gyroscopic effect of the front wheel. and control, the main contribution of this study is the The concept of an autonomous bicycle without a rider new modeling and control system design with integrated has been proposed by several researchers [1], [4], [5], [18]– motorcycle dynamics with tire/road interaction. First, we do [21]. In this two-part paper, we extend the modeling and not enforce a zero lateral velocity nonholonomic constraint control design in [4], [5]. For the modeling part, we take a for the wheel contact points of the motorcycle system. Such constrained Lagrangian approach to capture the nonlinear dy- nonholonomic constraints are not realistic for high-fidelity namics of a motorcycle. Besides the consideration of control- vehicle modeling [2]. Second, we explicitly consider the oriented modeling approach that captures the fundamental tire/road interaction for designing control algorithms because properties of the motorcycle platform with a manageable of the importance of the tire/road interaction on motorcycle complexity, several new features have been adopted and dynamics. To our knowledge, there is no study that explicitly developed. First, we relax the zero lateral velocity of the considers such kinds of tire dynamics into the motorcycle wheel contact points and therefore allow wheel sliding in the control system design. Based on the new dynamics, in our models, which provides more realistic vehicle modeling [2]. companion paper [3], we extend the control system design Second, we explicitly consider the tire/road interaction for in [4], [5] for trajectory following maneuvers. designing control algorithms because of the importance of The remainder of the paper is organized as follows. We the tire/road interaction on motorcycle dynamics [22]. The review some related work in Section II. In Section III, study in [23] is probably the closest work to ours. The we discuss dynamic modeling of a riderless motorcycle. In authors in [23] employ a nonholonomic motorcycle dynamics Section IV, we present a motorcycle tire dynamics model and focus on the performance and maneuverability analysis and then integrate the tire dynamics with the motorcycle of motorcycles using the automotive tire/road interaction dynamics. Finally, we conclude the paper in Section V. characteristics.

II. RELATED WORK III. MOTORCYCLE DYNAMICS Mathematically modeling of a bicycle or a motorcycle Fig. 1(b) shows the Rutgers autonomous motorcycle pro- has been an active research area for many years. Although totype. The motorcycle is rear-wheel driving. Steering and some modeling differences have been discussed in [2], from velocity control are considered as control inputs for the control system design aspects, we consider bicycles and mo- riderless autonomous motorcycle. We do not consider the torcycles are similar, and hence do not explicitly distinguish weight shifting as one actuation mechanism as human drivers Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 5, 2009. CONFIDENTIAL. Limited circulation. For review only. 3 because the Blue Team motorcycle has previously demon- If we assume a small roll and steering angles, then from (3) strated an effective maneuverability only through vehicle we obtain an approximation steering and velocity control [1]. σ˙ cϕ = φ˙ cξ . (4) A. Geometry and kinematics relationships y xB The riderless motorcycle is considered as as a two-part ϕ B platform: a rear frame and a steering mechanism. Fig. 2(a) zB shows a schematic of the vehicle. We consider the following modeling assumptions: (1) the wheel/ground is a point con- G φ tact and thickness and geometry of the motorcycle tire are ξ neglected; (2) The motorcycle body frame is considered a ϕf point mass; and (3) the motorcycle moves on a flat plane and O2 Frx vertical motion is neglected, namely, no suspension motion. C2 h We denote C1 and C2 as the front and rear wheel Fry v O γ r 1 point points with the ground, respectively. As illustrated in r b Fig. 2(a), three coordinate systems are used: the navigation O X C1 Y N X, Y, Z F l Ffy frame ( -axis fixed on the ground), the wheel rz Z base moving frame (x, y, z-axis fixed along line C1C2), and φ B g the rear body frame (xB,yB,zB-axis fixed on the rear l ψ t Ffx vf y B α Ffz x frame). For the frame , we use (3-1-2) Euler angles and γf O C z represent the motion by the yaw angle ψ and roll angle ϕ.We r 3 y denote the unit vector sets for the three coordinate systems w x I J K i j k i j k w as ( , , ), ( , , ),and( B, B, B), respectively. It is straightforward to obtain that (a) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ iB i I 1 0 1 0 R(ψ) 0 γ x ⎣j ⎦ ⎣j⎦ ⎣ J ⎦ l f B = 0 R = 0 R k (ϕ) k (ϕ) 0 1 K v y B vr f ⎡ ⎤ ⎡ ⎤ δ φg γ c s 0 I r C ψ ψ 1 γ xw ⎣ ⎦ ⎣ ⎦ ψ f = − cϕ sψ cϕ cψ sϕ J , (1) y − K w sϕ sψ sϕ cψ cϕ C2 X where the rotation matrix O c s R(x)= x x − s c x x R α Y and cx := cos x, sx := sin x for angle x. We consider the trajectory of the rear wheel contact point C2, denoted by its coordinates (X, Y ) in N , as the motorcy- ψ O cle position. The orientation of the coordinate systems and α r the positive directions for angles and velocities follow the conversion of the SAE standard [17]. We consider the instantaneous rotation center of the motorcycle motion on the horizontal plane. Let Or denote the instantaneous rotation center and O denote the neutral r O instantaneous rotation center which is the intersection point r of the perpendicular lines of the front and rear wheel planes; (b) see Fig. 2. Under the neutral turning condition [10], the slip Fig. 2. A picture of the Blue team autonomous motorcycle [1]. angles of the front and rear wheels are the same, that is, The motion of the motorcycle on the XY plane can be λf = λr, and then the rotation center angles for Or and captured by the generalized coordinates (X, Y, ψ, ϕ, σ). Note σ Or are equal to the kinematic steering angle φg, namely, that the use of variable is to capture the steering impact α = α = φg.LetR denote the instantaneous radius of the on the motorcycle dynamics. The nonholonomic constraint trajectory of point C2 under neutral turning conditions. We of the rear wheel and the motion trajectory geometry imply define σ as the kinematic steering variable as the yaw kinematics equality l ˙ l ˙ σ := tan φ = . (2) vrx = Rψ = ψ. (5) g R σ From the geometry of the front wheel steering mecha- From a differential geometry viewpoint 1, we can partition nism [10], we find the following relationship, 1We here take a description of the base-fiber structure of nonholonomic tan φg cϕ = tan φ cξ . (3) dynamical systems with symmetry in [24]. Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 5, 2009. 4 CONFIDENTIAL. Limited circulation. For review only. the generalized velocities of the motorcycle as base velocities Incorporating the constraints (6), we obtain the constrained T 2 r˙ =[˙ϕ, vrx,vry, σ˙ ] and fiber velocities s˙ = ψ˙.Wethen Lagrangian Lc as write the constraints in (5) simply as 2 J 1 h b2 s˙ + A(r, s)r˙ =0, (6) L = s c2 σ˙ 2 + m 1 − σ s + σ2 v2 c 2 ϕ ϕ 2 rx 2 cξ 2 l l where A(r, s)= 0 − σ 00. l 2b 2bh Due to the steering mechanism and caster angle, the height 2 +vry + σvrxvry + cϕ σϕv˙ rx +2h cϕ ϕv˙ ry of the mass center of gravity of the motorcycle is changing l l under steering. As shown in Fig. 2(b), the height change 2 2 blt cξ +h ϕ˙ − mg h cϕ − σ sϕ . (12) ΔhG of the center of gravity G due to the steering action l can be calculated as [5] bl σ c The moment Ms on the rotating axis is obtained as Δh = δbs ≈ t ξ s , (7) G ϕ l ϕ lt − σ ≈ φ Ms = Ffy cϕf Ffz sϕf . (13) where we use a small angle approximation g from the 2 relationship (2). 1+(lt/r) Remark 1: In [4], [23], the steering axis is assumed to be vertical. This assumption simplifies the motorcycle dynamics The detailed calculation of (13) is given in Appendix I. and neglects a significant geometric stabilization mechanism, The equations of motion using the constrained Lagrangian 3 which is the “motorcycle trail” (denoted as lt in Fig. 2(a)) are obtained as [24] discussed in [7], [10], [14], [15]. The resulting model of d ∂L ∂L ∂L ∂L the motorcycle dynamics cannot capture the influence of the c − c +Ak c = − Cl r˙j +τ i,i,j=1,...,4, dt ∂r˙i ∂ri i ∂sk ∂s˙l ij steering angle φ on the roll dynamics when vrx =0. Namely, (14) one cannot use steering to stabilize the motorcycle. Such an i k where τ are the external forces/torques, Ai is the element observation is also pointed out in [6]. of connection A(r, s) at the kth row and ith column, and Given the roll angle ϕ and the steering angle φ, the camber l r s Cij denote the components of the curvature of A( , ) as angle of the front wheel can be approximated as [10] ∂Al ∂Al ∂Al ∂Al ϕf = ϕ + φ sξ . (8) Cl = i − j + Ak j − Ak i . (15) ij ∂rj ∂ri i ∂sk j ∂sk We consider the relationship between velocities of the rear wheel contact point C2 and the front wheel center O1.We From state variable σ, from (14), we obtain the steering r r ρ r write the position vector O1 = C2 + C2O1 , where C2 is dynamics as ρ i − k the position vector of point C2 and C2O1 = l B r B = li + r s j − r c k is the relative position vector of G.The d J 2 mgl b c ϕ ϕ s c σ˙ − t ξ s = τ + M . (16) ω = 2 ϕ ϕ s s angular velocity of the rear frame is represented as dt cξ l ϕ˙i + ψ˙k. Thus, we obtain v = r˙ + ω × ρ =(v − rψ˙ s )i + Considering a position feedback control of the steering angle O1 C2 C2O1 rx ϕ directly, we can reduce the dynamic equation (16) by a ˙ j k (vry + lψ + rϕ˙ cϕ) + rϕ˙ sϕ . (9) kinematic steering system as B. Motorcycle dynamics σ˙ = ωσ, (17) We use the constrained Lagrangian method in [24] to obtain the dynamics equation of the motion of the riderless where the input ωσ is considered as the virtual steering motorcycle. We consider the motorcycle as two parts: one velocity and given by dynamic extension rear frame with mass m and one steering mechanism with c2 mgl b c3 the mass moment of inertia Js. The Lagrangian L of the ξ − t ξ ω˙ σ = 2 (τs + Ms) 2 tan ϕϕ˙σ˙ + sϕ . motorcycle is calculated as Js cϕ lJs 1 1 L = J φ˙2 + mv · v − mg (h c −Δh ) (10) 2 s 2 G G ϕ G Similarly, we obtain the roll dynamics equation

To calculate the mass center velocity, we take a similar c bhσ 2 − hσ hσ ϕ 2 approach as in (9) and obtain cϕ v˙rx + h cϕ v˙ry + h ϕ¨ + 1 sϕ vrx l l l v − ˙ i ˙ j k G =(vrx hψ sϕ) +(vry + bψ + hϕ˙ cϕ) + hϕ˙ sϕ . ltb cξ bh −g h sϕ + σ cϕ = − cϕ vrxωσ, (18) Plugging the above equations and (4)-(7) into (10), we l l obtain 2 Js 2 1 2 2 Readers can refer to [24] for the deﬁnition of the constrained Lagrangian L = σ˙ + m (v − hψ˙ s ) +(v + bψ˙ + hϕ˙ c ) + L 2 c2 2 rx ϕ ry ϕ c and also Chapter 5 of [24] for the Lagrange-d’Alembert principle for ξ nonholonomic constrained dynamical systems. 3 s 2 2 2 blt cξ Here the summation convention is used where, for example, if is of h ϕ˙ s − mg h c − σ s . (11) ∂Al ∂Al ϕ ϕ ϕ dimension m,thenAk j ≡ Σm Ak j . l i ∂sk k=1 i ∂sk Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 5, 2009. CONFIDENTIAL. Limited circulation. For review only. 5 longitudinal dynamics equation A. Tire kinematics relationships hσ 2 b2σ2 bσ bhσ Fig. 3 illustrates the kinematics of the tire/road contact. 1 − sϕ + v˙rx + v˙ry + cϕ ϕ¨ − 2 1 − v i j k v i j k l l2 l l Let c = vcx + vcy + vcz and o = vox + voy + voz

denote the velocities of the contact point and the wheel center hσ hσ bhσ 2 hσ in the frame B, respectively. We deﬁne the longitudinal slip sϕ cϕ ϕv˙ rx − sϕ ϕ˙ = − −2 1 − sϕ l l l l ratio λs and lateral side slip ratio λγ , respectively, as 2 − h 2b σ b bh 1 vcx rωw − vcy s v + v + v + c ϕ˙ ω + F − λs := ,λγ := tan γ = , (22) l ϕ rx l2 rx l ry l ϕ σ m rx vcx vcx

1 1 2 where ωw is the wheel angular velocity. √ (Ffx + σFfy) − Cdv , (19) m 1+σ2 m rx and lateral dynamics equation bσ bv 1 v˙ +˙v + h c ϕ¨ − h s ϕ˙ 2 = − rx ω − F l rx ry ϕ ϕ l σ m ry 1 ϕ + √ (Ffy − σFfx) . (20) m 1+σ2 O In (19), Cd is the aerodynamic drag coefﬁcient. T Let q˙ := [ϕv ˙ rx vry] denote the generalized velocity Fx of the motorcycle and we rewrite the above dynamics equa- C tions (18)-(20) in a compact matrix form as F vc ⎡ ⎤ y y ωσ F v vcy ⎢ ⎥ z cx γ ⎢Ffx⎥ x Mq K B ⎢ ⎥ ¨ = m + m ⎢Ffy⎥ , (21) z ψ ⎣ ⎦ Frx Fig. 3. Schematic of the tire kinematics. Fry where matrices For the front wheel, the camber angle is different (8), and M11 M12 M = the velocity relationship between C1 and the wheel center M21 M22 B ⎡ ⎤ O1 in is then 2 bhσ ˙ − ⎢ h cϕ h cϕ ⎥ vfx = vfox + rψ sϕ,vfcy = vfoy rϕ˙ f cϕ, ⎢ l ⎥ ⎢ 2 ⎥ vfz = vfoz − rϕ˙ f sϕ . (23) ⎢ bhσ hσ b2σ2 bσ ⎥ = ⎢ − ⎥ , ⎢ cϕ 1 sϕ + 2 ⎥ ⎢ l l l l ⎥ Using the relationship (9) and (8), we simplify the above ⎣ ⎦ bσ velocity calculation and obtain h c 1 ϕ l ⎡ ⎤ vfx = vrx,vfy = vry − rφ˙ sξ cϕ +lψ.˙ (24) c c − 1 − hσ s hσ ϕ v2 + g h s + ltb ξ σ c ⎢ l ϕ l rx ϕ l ϕ ⎥ 1 From the side slip ratio (22) of the rear wheel, we have Km = ⎣ − hσ hσ bhσ 2 − 2 ⎦ , 2 1 sϕ cϕ ϕv˙ rx + sϕ ϕ˙ Cdvrx l l 2 l m ˙ − ˙ h sϕ ϕ˙ vry vfy rφ sξ cϕ lψ λrγ = tan γr = − = − − and vrx vfx vrx ⎡ ⎤ 2 − bh c r tan ξ cϕ l ϕ vrx 0000 = tan γ − ω + σ, (25) ⎢ 1 σ 1 ⎥ f σ B B − √ − √ 0 vrx m = ⎣ ω m 1+σ2 m 1+σ2 m ⎦ . 1 1 − bvrx − √σ √ − vfy 1+ 2 1+ 2 0 where γ := φg −γf and tan γ = − ; see Fig. 2. We also l m σ m σ m f f vfx use relationships (4) and (5) in the last step above. Moreover, In the above matrix Bm, from (2) and the geometry and kinematics of the front wheel hσ h b2σ b bh B =2 1 − s s − v − v − c ϕ.˙ (Fig. 2), we have ω l ϕ l ϕ l2 rx l ry l ϕ ≈ It is clear that the control inputs in (17) and (21) are the σ = tan φg = tan(γf + γf ) tan γf + tan γf r tan ξ c2 virtual steering velocity ωσ and the wheel traction/braking ϕ − F F = λrγ + ωσ σ + λfγ. forces f and r. vrx

IV. TIRE DYNAMICS MODELS Therefore, we obtain the relationship between the front and rear wheel side slip ratios as follows. In this section, we discuss how to capture the motorcycle 2 tire/road interaction. We particularly like to present a friction r tan ξ cϕ λfγ =2σ − ωσ − λrγ. (26) forces modeling scheme for motorcycle dynamics (21). vrx Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 5, 2009. 6 CONFIDENTIAL. Limited circulation. For review only.

Similarly, we can obtain the slip ratio calculation of the front The values of the longitudinal, corning, and cambering wheel as follows. First, we obtain the longitudinal velocity coefficients, kλ, kγ , kϕ, depend on the normal load Fz. of the contact point C1 as Due to the acceleration and deceleration, the normal load F is changing during motion. For front and rear wheels, ≈ z vfxw = vfx cφg +vfy sφg vrx cφg +(vry + σvrx) sφg the normal loads Ffz and Frz are obtained respectively as 1 2 = √ 1+σ vrx + σvry . b h l − b h 1+σ2 F = mg − mv˙ ,F = mg + mv˙ , (31) fz l l Gx rz l l Gx Therefore, by the definition (22), we obtain the front wheel where v˙Gx is the longitudinal acceleration of the motorcycle longitudinal slip ratio G v˙ √ at the mass center . The relationship between Gx and the rω r 1+σ2 acceleration of point C2 is obtained as λ =1− f =1− ω . (27) fs 2 f − ˙ − ¨ − ˙ 2 − ˙ vfxw (1 + σ )vrx + σvry v˙Gx =˙vrx vryψ hψ sϕ bψ 2hψϕ˙ cϕ . The calculation of the above relationship is given in Ap- B. Modeling of frictional forces pendix II. In this paper, we use the tire models in [25] to Modeling of the frictional forces between the tire and the calculate the dependence of the stiffness coefficients on the road surface is complex. Here we focus on modeling of the normal load. longitudinal force F and lateral force F because of their x y F (x) importance in motorcycle dynamics and control. Stable The tire/road frictional forces depend on many factors, Unstable such as slip and slip angles, vehicle velocity, normal load, Fmax and tire and road conditions, etc. It is widely accepted that the pseudo-static relationships, namely, the mathematical models of the longitudinal force Fx and slip λ,andthe lateral force Fy and slip angle γ, are the most useful F = α F characteristics to capture the tire/road interaction. To capture k s x max tire/road friction characteristics, we propose to approximate the friction forces by a piecewise linear relationship shown in Fig. 4. Let F (x) denote the frictional force as a function x of independent variable x. The piecewise linear function O xm xmax F (x) captures the property of the tire/road forces: when Fig. 4. Linear approximation of the tire/road frictional force F (x). 0 ≤ x ≤ xm, F (x)=kx, and when xm

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Autom., Leuven, Belgium, 1998, pp. 2670– developed motorcycle dynamics. The control inputs for the 2675. [19] S. Lee and W. Ham, “Self-stabilzing strategy in tracking control of proposed motorcycle dynamics are the front wheel steering unmanned electric bicycle with mass balance,” in Proc. IEEE/RSJ Int. angle and the angular velocities for the front and rear wheels. Conf. Intell. Robot. Syst., Lausanne, Switzerland, 2002, pp. 2200– The trajectory tracking and balance control systems design 2205. [20] Y. Tanaka and T. Murakami, “Self sustaining bicycle robot with is based on the new dynamic model and presented in the steering controller,” in Proc. 2004 IEEE Adv. Motion Contr. Conf., companion paper [3]. Kawasaki, Japan, 2004, pp. 193–197. Currently, we plan to extend the motorcycle dynamics [21] ——, “A study on straight-line tracking and posture control in electric bicycle,” IEEE Trans. Ind. Electron., vol. 56, no. 1, pp. 159–168, 2009. models in two directions. First, we will relax the neutral [22] R. Lot, “A motorcycle tires model for dynamic simulations : Theo- driving approximation and present a general yaw dynamics retical and experimental aspects,” Meccanica, vol. 39, pp. 207–220, model. Second, a coupled longitudinal and lateral motorcycle 2004. [23] J. Hauser and A. Saccon, “Motorcycle modeling for high-performance tire dynamics will be developed and the LuGre dynamic maneuvering,” IEEE Control Syst. Mag., vol. 26, no. 5, pp. 89–105, friction model is currently used to capture the coupled 2006. tire/road friction characteristics. [24] A. Bloch, Nonholonomic Mechanics and Control.NewYork,NY: Springer, 2003. [25] R. S. Sharp, S. Evangelou, and D. J. N. Limbeer, “Advances in the ACKNOWLEDGMENTS modelling of motorcycle dynamics,” Multibody Syst. Dyn., vol. 12, pp. 251–283, 2004. The ﬁrst author thanks Dr. N. Getz at Inversion Inc. for his helpful suggestions and support. The authors are grateful APPENDIX I to Prof. S. Jayasuriya at Texas A&M University, Dr. E.H. CALCULATION OF Ms Tseng and Dr. J. Lu at Ford Research and Innovation Center We consider the front wheel center O1 and the projected for their helpful discussions and suggestions. steering axis point C3 on the ground surface. Since the Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 5, 2009. 8 CONFIDENTIAL. Limited circulation. For review only. frictional moment is independent of the coordinate system. We can setup a local coordinate system xf yf zf by rotating the coordinate system xyz around the z-axis with an angle φg (origin at contact point C1). Let (if , if , if ) denote the unit vectors along the xf ,yf ,zf -axis directions, respectively. In the new coordinate system, we obtain the coordinates of − O1 and C3 as (0,rsϕf , r cϕf ) and (lt, 0, 0), respectively. We write the front wheel friction force vector F f as F − i − j − k f = Ffx f Ffy f Ffz f r − i n and the vector C3C1 = lt f . The directional vector O1C3 of the steering axis O1,C3 is then

ltif − r sϕ j + r cϕ kf n = f f f . O1C3 2 2 lt + r

Therefore, the friction moment Ms about the steering axis is calculated as r × F · n Ms =( C3C1 f ) O1C3 lt = Ffy cϕ −Ffz sϕ . 2 f f 1+(lt/r)

APPENDIX II CALCULATION OF ACCELERATION v˙ G

Taking the time derivative of the mass center velocity vG and considering the moving frame xyz’s angular velocity ω =˙ϕi + ψ˙k, we obtain δv v˙ = G + ω × v =(˙v − hψ¨ s −hψ˙ϕ˙ c )i + G δt G rx ϕ ϕ 2 2 (˙vry + bψ¨ + hϕ¨ cϕ −hϕ˙ sϕ)j +(hϕ¨ sϕ +hϕ˙ cϕ)k

+(ϕ ˙i + ψ˙k) × vG 2 =(˙vrx − vryψ˙ − hψ¨ sϕ −bψ˙ − 2hψ˙ϕ˙ cϕ)i +(˙vry + 2 2 vrxψ˙ + bψ¨ + hϕ¨ cϕ −hψ˙ sϕ −2hϕ˙ sϕ)j +(vryϕ˙ + 2 hϕ¨ sϕ +bψ˙ϕ˙ +2hϕ˙ cϕ)k, v δ G v where δt denotes the derivative of G by treating the xyz- coordinate as a ﬁxed frame.

Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 5, 2009.