Robust observer-based H controller design for motorcycle lateral dynamics

Transcription

1 Robust observer-based H controller design for motorcycle lateral dynamics A. Ferjani b, I. Zaidi b, M. Chaabane b a PB 1173, 3038, STA laboratory, Sfax university, Tunisia Abstract The present work deals with the design problem of a robust observer based controller for a motorcycle system using LPV approach. The designed model is specifically uncertain and disturbed one, whose uncertainties are related to variations of both the cornering stiffness and the longitudinal velocity. The nonlinear motorcycle model is firstly transformed on an uncertain LPV model with two vertices, then an observer based H robust controller is designed. Both the controller and observer gain matrices are computed by solving an unified convex optimization problem under LMI constraints using YALMIP solver. Numerical simulation results are given to illustrate the effectiveness of the designed method. Keywords: Non linear motorcycle model, LPV model, observer-based controller, Linear Matrix Inequalities (LMIs). Nomenclature Corresponding author address: amelelfirjani@gmail.com (A. Ferjani) Preprint submitted to Journal of LATEX Templates May 8, 2018

2 Symbol Table 1: Illustration of parameters of motorcycle Signification M, M r, M f total weight of the motorcycle, mass of the rear and front portions. l f distance from the gravity center of the rear frame to the front wheel contact. l r distance from the gravity center of the rear frame to the rear wheel contact. C f1, C f2, C r1, C r2 σ r, σ f cornering front and rear stiffness. the rear and front tire relaxation lengths. ɛ, ρ, r steering head angle, curvature of the road, road curvature radius. η pneumatic trail. µ, h coefficient of friction of the road, height of the bike s center of gravity. ζ f ζ r F zf, g e distance from the center of gravity of the front frame to the ground. distance between the gravity centers of the rear and front frames. front wheel load, the gravity force. distance from the center of gravity of the front frame to the fork. ψ, δ, φ yaw angle, steering angle and roll angle v x, v y longitudinal and lateral velocities. i fy, i ry I rz inertia s polar moments of the front and rear wheel. polar moment of inertia of the camber inertia of the rear wheel. 1. Introduction In recent years, more and more people are moving towards fast, light, less expensive means of transport and small jigs. Among these means of transport, motorized two-wheeled vehicles are increasingly popular because of the great freedom of driving and the possibility of avoiding traffic congestion and parking difficulties. At the present time, scooters and motorcycles have become necessary means of transport in our society and the last statics of sales of these vehicles reinforce this statement. The increase in the number of motorcycles and scooters is, un- 2

3 fortunately, followed by an increase in accidents on the road. In these circumstances, safety in two-way transport has become a central concern for public bodies and car manufacturers. Several preventive and repressive measures have been put in place to warn drivers of the risks associated with certain behavior on the steering wheel and the handlebars. On the other hand, several research projects have been initiated in order to provide solutions in the field of prevention and driver assistance systems. In the last years, many motorcycle equipment manufacturers focus on active and semi-active safety systems. The first work to stabilize a motorcycle has been developed in [1], by considering a simple model linear for describe the motorcycle behavior. The control of motorcycles has been investigated in [2] and [3] and the problem of observation and estimation of unmeasurable state variables have been studied in [4],[5], [6], [7]. Overall, motocycle safety system design is still an open problem and little results exist in the literature. To the best of our knowledge, there is no studies that deal with both the motorcycle stability and estimation in presence of disturbance, except the work [7] which has studied the two problems separately, without considering external disturbance. Overall, safety system design for motorcycle is still an open problem and little results exist in the literature. In the modelling aspect, we develop an uncertain polytope which can cover the time-varying longitudinal velocity range and has few vertices. Moreover, the nonlinear tyre model is reformulated into a Linear Parameter-Varying (LPV) model with norm-bounded uncertainties. LPV models have been studied in many works like [8]. Motivated by these observations, a robust H observer based controller will be investigated. The observer will be used to estimate the unmeasured variables of the motorcycle like the yaw rate, the roll rate and the steer rate and the controller ensures the robust stability although the variation of the road conditions and external disturbances. The less conservative LMI based design conditions of both the observer and the controller are proposed and which can be solved in only step. The rest of the paper is structured as follows: the analysis of the motorcycle 3

4 described by an LPV model is introduced in Section 3. Hereafter, in Section4, the analysis and the design problems of the robust observer based controller are studied. To show the performance of the proposed observer in different cases, some simulations are given in Section 5. Finally, conclusions are given in Section Notations and Preliminaries The following lemmas are used to prove our results: Lemma 2.1. [15] Given a positive scalar ɛ and two matrices G and H, the following inequality holds: G T H + H T G ɛg T G + 1 ɛ HT H (1) Lemma 2.2. [16] Considering a negative definite matrix Ξ < 0, a given matrix Z, and a scalar µ > 0, the following inequality holds: Z T ΞZ < µ(z + Z T ) µ 2 Ξ 1 (2) 3. Motorcycle Model Analysis 3.1. Nonlinear model of the motorcycle The model used in this work describes the motorcycle lateral and roll dynamics, which is obtained by considering the Sharp s motorcycle model [9] (see Figure 1). The dynamics of a motorcycle can be represented by a model with four equations [10], [11] describing the lateral motion mainly caused by lateral forces (F yf, F yr ) and the yaw and roll motions under rider s steering actions. The movements that correspond to lateral, roll, yaw and steering motions respectively, are expressed by the following equations: M( v y + v x ψ) + M f ζ r ψ + d1 φ + Mf e δ = F yf + F yr d 1 v y + b 2 φ + a2 ψ + b1 δ + b5 v x ψ + d 3 v x δ = Mx M f ζ r( v y + v x ψ) + a 2 φ + a3 ψ + a1 δ a4 v x φ d2 v x δ = Mz M f e v y + b 1 φ + a1 ψ + c1 δ d3 v x φ + c3 v x ψ + K δ = M s (3) 4

5 Figure 1: Illustration of motorcycle s motion Using the moment principle, the following expressions can be obtained: Mx = b 4 sin φ b 3 sin δ Mz = l f F yf l r F yr Ms = b 3 sin φ c 2 sin δ ηf yf + τ (4) where coefficients a i, b i, c i and d i are given in AppendixA.: Lateral and roll forces have the following expressions: F yf = C f1 α f + C f2 γ f, F yr = C r1 α r + C r2 γ r (5) where α f = ( vy+l f α r = ( ψ η δ v x vy lr ψ v x ) γ f = φ + δ sin(ε) γ r = φ ) δ cos(ε) 5

6 Substituting equations (4) and (5) in model (3), we obtain the following form: M v y + M f ζ r ψ + d1 φ + Mf e δ = a 11 (v x)v y + a 12 (v x) ψ + a 14 φ + a 15 (v x) δ + b 16 δ d 1 v y + b 2 φ + a2 ψ + b1 δ = a22 (v x) ψ + a 24 φ + a 25 (v x) δ + b 26 δ M f ζ r v y + a 2 φ + a3 ψ + a1 δ = a31 (v x)v y + a 32 (v x) ψ + a 33 (v x) φ + a 34 φ + a 35 (v x) δ + b 36 δ M f e v y + b 1 φ + a1 ψ + c1 δ = a51 (v x)v y + a 52 (v x) ψ + a 53 (v x) φ + a 54 φ + a 55 (v x) δ + b 56 δ + τ (6) Where motorcycle parameters are defined in Table 1 and parameters a ij and b ij are given in AppendixA. [ Let s consider state vector x(t) = v y ψ φ φ δ ] T, system (6) can be written as follows: Eẋ (t) = A 11 (v x )x (t) + B 11 δ(t) + B 21 τ(t) (7) where: E = M M f ζ r d 1 0 M f e M f ζ r a 3 a 2 0 a 1 d 1 a 2 b 2 0 b M f e a 1 b 1 0 c 1 ] a 11 (v x) a 12 (v x) 0 a 14 (v x) a 15 (v x) 0 a 22 (v x) 0 a 24 a 25 (v x) A 11 (v x) = a 31 (v x) a 32 (v x) a 33 (v x) a 34 a 35 (v x) a 51 (v x) a 52 (v x) a 53 (v x) a 54 a 55 (v x) [ B 11 = b 16 b 26 b 36 0 b 56 T [ ] T B 21 = Matrix E is a nonsingular matrix, so, system (7) can be expressed as follows: ẋ (t) = A(v x )x (t) + B 1 δ(t) + B 2 u(t) (8) 6

7 where A(v x ) = E 1 A 11 (v x ), B 1 = E 1 B 11, B 2 = E 1 B 21 and u(t) = τ(t) To simplify the development, v x is written as v. In the following, we mention that the yaw control is only tripped at a non zero Table 2: Main parameter values for the motorcycle Symbol Value M 248.1Kg C f1, C f N.rad 1, 9385N.rad 1 C r1,c r N.rad 1, N.rad 1 l f 0.829m l r 0.585m of the longitudinal velocity. For this, we propose to accommodate the variation of this speed Motorcycle uncertain LPV model using polytopic approach and reducing vertices Let consider that the longitudinal speed v is time varying in interval [v m, v M ]. Then, variable is time varying in [, ]. For pair (v, 1 ), we can used a v v M v m v rectangular polytope [12] to describe it. Since the choices for the parameter set (v, 1 ) only occur on the solid line and v most of the area inside the rectangle is not achievable. So, the description using a rectangular polytope could be conservative. Moreover, if the number of the polytope vertices increases, the computational load and the complexity also increase. In the following, we propose to use the idea proposed by [13] to reduce the number of the polytope vertices and also to simplify the modeling complexity. Based on [13] and considering matrix A(v x ) defined in Section 3.1, for the set of the variables (v, 1 ), we consider a rectangular polytope given in Figure 2. v As it is shown in Figure 2, the rectangular polytope obtained by the four vertices Ω 1 Ω 2 Ω 3 Ω 4 illustrates the variation of the set of variables (v, 1 v ). A 7

8 Figure 2: Illustration of transforming the rectangular polytope to the uncertain straight line segment. straight line D 1 crosses Ω 1 and Ω 3, then shifting the straight line until the tangent to the hyperbola y = 1 v, we obtain another straight line D 2. D 2 cross the straight line Ω 1 Ω 2 in the point p 2 and in the other hand, the Ω 3 Ω 4 in the point p 4. Then, the line crossing the middle of the lines Ω 1 p 2 and Ω 3 p 4 via p 1 and p 3. The sweep above and below of D 3 along of the two segments [Ω 1 p 2 ] and [Ω 3 p 4 ] can achieve all possibilities of most of the area inside the polytope. The obtained two new vertices are defined by the following coordinates: p 1 = (v v M v m 1 m, + + ( vm + v M 2v mv M vmvm 2v mv M 1 vmv M )N(t)) p 3 = (v M, vm v M ( vm + v M 1 )N(t)) 2v mv M vmvm 2v mv M vmvm Since N(t) is less than one, the uncertain straight line segment crossing points p 1 and p 3 can cover the whole parallelogram Ω 1 p 2 p 4 Ω 3. Then, based on the 8

9 work of [14], each point inside the parallelogram Ω 1 p 2 p 4 Ω 3 can be represented by a linear combination of the new uncertain vertices p 1 and p 3 as follows: where: µ 1 = p n = µ 1 p 1 + µ 2 p 3 (9) v v m and µ 2 = v M v and µ 1 + µ 2 = 1 v M v m v M v m The description of the two parameters v and 1, in expression (9), has only two v vertices and it can overlay all the possibilities for the set of v and 1 v. To take into account the road condition variations, the cornering stiffness are given by the following equations: C f1 = C f1 + C f1, C f2 = C f2 + C f2 Cr1 = C r1 + C r1, C r2 = C r2 + C r2 (10) Considering the representation in equations (8) and (9), the obtained model is: 2 ẋ(t) = µ i ((A i + A i )x(t) + (B 1 + B 1 )δ(t) + B 2 u(t)) (11) i=1 We define the state matrices by the first term containing nominal parameters. So, the state matrices of the multiple model are written as: f 1 c 11 f 2 c 11 Mv m 0 C f2 + C r2 f 3 ηc 11 f 2 c 11 f 4 c 11 M f ζ r v m a 4 v m C f2 l f C r2 l r d 2 v m + l f f 3 ηc 11 A 1 = 0 b 5 v m 0 b 4 d 3 v m f 3 ηc 11 l f f 3 ηc 11 c 3 v m d 3 v m b 3 ηc f2 K η 2 f 3 c 11 f 1 c 12 f 2 c 12 Mv M 0 C f2 + C r2 f 3 ηc 12 f 2 c 12 f 4 c 12 M f ζ r v M a 4 v M C f2 l f C r2 l r d 2 v M + l f f 3 ηc 12 A 2 = 0 b 5 v M 0 b 4 d 3 v M f 3 ηc 12 l f f 3 ηc 12 c 3 v M d 3 v M b 3 ηc f2 K η 2 f 3 c 12 The uncertain ones are given by: A 1 = G 11 N(t)H11, A 2 = G 12 N(t)H12 9

10 where G 11 = G 12 = f 11 f 21 0 C f2 + C r2 ηf 31 f 21 f 41 0 C f2 l f C r2 l r ηl f f ηf 31 ηl f f 31 0 η C f2 η 2 f 31 f 12 f 22 0 C f2 + C r2 ηf 32 f 22 f 42 0 C f2 l f C r2 l r ηl f f ηf 32 ηl f f 32 0 η C f2 η 2 f 32 N N N = N 0 0 N 0 N H 11 = I(5), H 12 = I(5) C f1 cos(ε) + C f2 sin(ε) l f (C f1 cos(ε) + C f2 sin(ε)) B 1 = b 3 0 (C f1 cos(ε) + C f2 sin(ε) where B 1 = G 2 N(t)H2 G 2 = C f1 cos(ε) + C f2 sin(ε) l f ( C f1 cos(ε) + C f2 sin(ε)) 0 0 ( C f1 cos(ε) + C f2 sin(ε) H 2 = I(5, 1) with: c 11 = v M v m 1 +, d 11 = v m + v M 2v m v M vm v M 2v m v M c 12 = v m v M 1 +, d 12 = v m + v M 2v m v M vm v M 2v m v M 1 vm v M 1 vm v M 10

11 f 1 = C f1 C r1, f 2 = (C f1 l f C r1 l r ), f 3 = C f1 and f 4 = (l 2 f C f1 + l 2 rc r1 ). For more details, see [14]. 4. Analysis and Synthesis of the robust H observer based controller The objective is to design a robust observer based H controller such that the closed-loop system is globally stable and the H performances are guaranteed. Based on LPV model (11), the observer and the controller are defined as ˆx(t) = 2 µ i (A iˆx(t) + L i (y(t) ŷ(t)) + B 2 u(t)) + B 1 δ(t) i=1 ŷ(t) = C ˆx(t) u(t) = 2 µ i K iˆx(t) i=1 (12) where ˆx(t) is the estimated state and K i (i = 1 : 2) are the controller gains to be determined, L i are the observer gains to be determined, as the yaw rate, the roll rate and the steering rate can be generally measured, C is an adequate matrix which is given by: C = Defining the estimation error as: The derivative of the error estimation is given by: e 0 (t) = x(t) ˆx(t) (13) 2 ė 0 (t) = A i x(t) + (A i L i C)e 0 (t) + B 1 δ(t) (14) i=1 By using (11), (12) and (14), we can express the augmented system in the following form: 2 Ẋ(t) = µ i (ĀiX(t) + Bδ(t)) (15) i=1 11

12 where X(t) = x(t) e 0 (t) Ā i = A i + B 2 K i + A i A i B 2 K i A i + L i C (16) and B = B 1 + B 1 B 1 (17) To improve motorcycle lateral dynamics stability, the objective in H controller design is to possess robustness against external disturbance (steering angle) and uncertainties (cornering stiffness and the longitudinal velocity) that guarantees a given attenuation level of disturbance rejection attenuation. The objective of this work is to design an observer based H controller such that the following requirements are satisfied: Ensure the stability for closed loop system (15) with the controller when δ(t) = 0. Ensure disturbance rejection for system (15) such that: X(t) γ 2 δ(t) (18) The stability conditions of augmented system (15) are given in terms of LMIs in the following theorem and can be solved in one step. Theorem 4.1. For given scalars µ and ɛ 2, system (15) is asymptotically stable via the robust observer based H controller (12), if there exist positive definite matrices Z and P 2 and matrices M i, J i and S 1, linear variables ɛ 1, ɛ 3 and ɛ 4 and a positive scalar γ which guarantees the H performance such that the 12

13 following LMIs are verified: θ i1 ZH1 T ZH1 T B 2 M i B 1 0 ε 1 I ε 2 I µZ 0 µi µI µi 0 θ i2 ε 2 P 2 G 1i P 2 G 2 0 µi ε 2 I ε 4 I 0 0 θ 3 H2 T ε 3 I where < 0 (19) θ i1 = ZA T i + A i Z + M T i B T 2 + B 2 M i + S 1 + ε 1 G 1i G T 1i + ε 3 G 2 G T 2 θ i2 = A T i P 2 + P 2 A i J T i P 2 J i C + I θ i3 = γ 2 I + ε 4 H T 2 H 2 And the gains of the controller and the observer are respectively: K i = M i Z 1 (20) Proof: The Lyapunov function is chosen as: L i = P 1 2 J i (21) V (X(t)) = X T (t) P X(t) where the Lyapunov matrix P is chosen as: P = P P 2 13

14 The derivation of the Lyapunov function gives: 2 V (X(t)) = µ i ((ĀiX(t) + Bδ(t)) T P X(t) + X T (t) P (ĀiX(t) + Bδ(t)(t))) (22) i=1 Then, objective (18) is guaranteed by ensuring the following inequality: V (X) + X T X γ 2 δ T δ 0 (23) Therefore, we have: 2 i=1 µ i X δ T ĀT i P + P Āi + I P B X B T P γ 2 I δ Then, inequality (23) holds if the following condition is satisfied: ĀT i P + P Āi + I P B 0 (24) B T P γ 2 I By substituting the expressions of Āi and B defined in (16) and (17), inequality (24) is equivalent to: ϖ i P 1 B 2 K i + A T i P 2 P 1 (B 1 + B 1 ) ϑ i P 2 B 1 0 (25) γ 2 I where ϖ i = A T i P 1 + P 1 A i + Ki T BT 2 P 1 + P 1 B 2 K i + A T i P 1 + P 1 A i + I ϑ i = A T i P 2 + P 2 A i C T L T i P 2 P 2 L i C + I In the following step, we expand the matrix (25) in three matrices that is: Θ i = T i + Λ i + Λ T i with: T i = ψ i P 1 B 2 K i P 1 B 1 φ i 0 γ 2 I Λ i = P 1 A i A T i P 2 P 1 B P 2 B

15 with ψ i = A T i P 1 + P 1 A i + K T i B T 2 P 1 + P 1 B 2 K i + I φ i = A T i P 2 + P 2 A i C T L T i P 2 P 2 L i C + I According to expressions of A i and B 1 and using Lemma 2.1, inequality (25) holds if there exist real scalars ɛ 1, ɛ 2, ɛ 3 and ɛ 4 satisfying: with Θ i = Π i Υ i Ξ i 0 (26) Π i = A T i P 1 + P 1 A i + K T i B T 2 P 1 + P 1 B 2 K i + I + ε 1 1 HT 1 H 1 + ε 1 P 1 G 1i G T 1iP 1 + ε 1 2 HT 1 H 1 + ε 3 P 1 G 2 G T 2 P 1 Ξ i = [ ] Υ i = P 1 B 2 K i P 1 B 1 χ i 0 0 γ 2 I + ε 1 3 HT 2 H 2 + ε 4 H T 2 H 2 χ i = A T i P 2 + P 2 A i C T L T i P 2 P 2 L i C + I + ε 2 P 2 G 1i G T 1iP 2 + ε 1 4 P 2G 2 G T 2 P 2 Pre-post multiplying (26) by the matrix diag(z, Y ) with Z = P 1 1 and Y = diag(z, I), we obtain: with: Π i = ZΠ i Z diag(z, Y )Θ i diag(z, Y ) = Π i = ZA T i + A i Z + M T i B T 2 + B 2 M i + S 1 + ε 1 1 ZHT 1 H 1 Z + ε 1 G 1i G T 1i + ε 1 2 ZHT 1 H 1 Z + ε 3 G 2 G T 2 Ῡ i Ξi Ῡ i = ZΥ i Y [ ] = B 2 M i B 1 15

16 Ξ i = Y Ξ i Y = Zχ iz 0 0 γ 2 I + ε 1 3 HT 2 H 2 + ε 4 H T 2 H 2 Matrices M i, J i and S 1 are given by: M i = K i Z, J i = P 2 L i, S 1 = ZZ Then, χ i is rewritten as follows: χ i = A T i P 2 + P 2 A i C T J T i J i C + I + ε 2 P 2 G 1i G T 1iP 2 + ε 1 4 P 2G 2 G T 2 P 2 Using Lemma 2.2, there exists a positif scalar µ such that matrices Ξ i can be rewritten : Using Schur s complement, (27) can be written as: 2µY Ξ i = µi µi Ξ i Y Ξ i Y 2µY µ 2 Ξ 1 i (27) Which is equivalent to: 2µZ 0 µi 0 2µI 0 µi Ξ i = χ i 0 γ 2 I + ε 1 3 HT 2 H 2 + ε 4 H2 T H 2 Using Schur s complement, sufficient conditions in Theorem 4.1 are established. 5. Simulation results To show the performance and the effectiveness of the proposed observer based controller law, we have considered both the variation of the longitudinal velocity as given in Figure 3 and the cornering stiffness. The steering angle is given by Figure 4. 16

17 vx(m.s 1 ) T(sec) Figure 3: Longitudinal velocity 0.4 Steering angle Figure 4: Steering angle 17

18 Longitudinal velocity is varying between v m = 11m.s 1 and v M = 18m.s 1. The reduced model using two vertices approach is firstly parameterized. Taking into account the uncertainties of the cornering stiffness which are 12% of the nominal values given in Table 2, the observer based robust state feedback controller is then designed. Figures 5-9 show the performance of the designed observer based controller by considering fixed parameters µ = 1 and [ ] T ε 2 = 0.8 and the following initial conditions x(0) = [ ] T and ˆx(0) = K i and L i controller gains are given by: [ K 1 = 1.0e ] [ K 2 = 1.0e ] L 1 = 1.0e + 03 L 2 = 1.0e + 03 Matrix Z is given by:

19 Z = Parameters ɛ 1, ɛ 3 and ɛ 4 are obtained as: ɛ 1 = ɛ 3 = , ɛ 4 = Figures 5-9 show the results of a driving test with varying longitudinal velocity, Figures 5-8 present the comparison between one of the states of the model and its estimate from the robust H observer based controller. These figures show the good estimation, stabilization and the robustness with respect to uncertainties. It can be seen there is a close approximation of the estimated and measured lateral velocity, yaw rate, roll rate and roll angle of the motorcycle. The yaw rate however shows a deviation which is related to steering angle of the motorcycle. We can also see that the designed robust H observer based controller improves the stability and safety under some unmeasured states and cornering manoeuver. 6. Conclusion In this paper, the problem of the design of the observer based robust control of the lateral dynamic of the motorcycle is studied. Using LPV approach and some geometrical transformations, the nonlinear motorcycle model is rewritten in uncertain LPV model form. Then, a robust observer based controller has been designed using LMI approach. To prove the performance of the controller, simulations using Matlab-SIMULINK toolbox have been presented. The comparison of the estimated states and its measured, then the stabilization and 19

20 8 6 4 v y v ye v y (m.s 1 ) T(sec) Figure 5: Lateral velocity ψ(rad.s 1 ) ψ ψ e T(sec) Figure 6: Yaw rate 20

21 6 4 φ φ e φ(rad.s 1 ) T(sec) Figure 7: Roll rate φ φ e φ(rad) T(sec) Figure 8: Roll angle 21

22 τ(n.m) x T(sec) Figure 9: Control input the robustness with respect to uncertainties for the LPV representation of lateral dynamic motorcycle model under some unmeasured states and cornering manoeuver are only studied in this manuscript. AppendixA. Coefficients and Parameters a 1 = M f eζ r + I k fzcos(ε), a 2 = M f ζ f ζ r C rx + (I fz I fx )sin(ε)cos(ε) a 3 = M f ζ 2 r + I fz + I fx sin(ε) 2 + I fz cos(ε) 2, a 4 = i fy R f + iry R r b 1 = M f eζ f + I fz sin(ε), b 2 = M f ζ 2 f + Mrh2 + I rx + I fx cos(ε) 2 + I fz sin(ε) 2 b 3 = ηf zf M f eg, b 4 = ( M f ζ f + M rh ) g, b 5 = M f ζ f + M rh + i fy R f c 1 = I fz + M f e 2, c 2 = (ηf zf M f eg)sin(ε), c 3 = M f e + i fy R f sin(ε) d 1 = M f ζ f + M rh, d 2 = i fy R f sin(ε), d 3 = i fy R f cos(ε) + iry R r a 11 (v x) = C f1+c r1, a v 12 (v x x) = ( C f1l f C r1 l r + Mv v x x) a 14 = (C f2 + C r2 ), a 15 (v x) = ηc f1, b v 16 = (C x f1 cos(ε) + C f2 sin(ε)) 22

23 sin φ a 22 (v x) = b 5 v x, a 24 = b 4 φ, a 25(v x) = d 3 v x, b 26 = b 3 a 31 (v x) = l f C f1 l rc r1, a v 32 (v x x) = (M f ζ rv x + C 2 2 f1l f +Cr1 l r ) v x a 33 (v x) = a 4 v x, a 34 = (C f2 l f C r2 l r) a 35 (v x) = ( ηc f1l f v x + d 2 v x), b 36 = l f (C f1 cos(ε) + C f2 sin(ε)) a 51 (v x) = ηc f1, a v 52 (v x x) = ( ηc f1l f c v 3 v x x) a 53 (v x) = d 3 v x, a 54 = (ηc f2 + b 3 sin φ φ ) a 55 (v x) = ( η2 C f1 v x + K), b 56 = (η(c f1 cos(ε) + C f2 sin(ε)) + c 2 ) References [1] N. Getz, Dynamic inversion of nonlinear maps with applications to nonlinear control and robotics, Master s thesis, Department of Electrical Engineering and Computer Sciences, University of California (1995). [2] V. Dankan, D. V. Kishore, Shivashankar, A. C. Ramachandra, C. Pandurangappa, Optimization of motorcycle pitch with non linear control, IEEE International Conference On Recent Trends In Electronics Information Communication Technology, India, 2016, pp [3] S. A. Evangelou, Control of motorcycles by variable geometry rear suspension, IEEE Multi-Conference on Systems and Control, Yokohama. Japan, 2010, pp [4] U. Nenner, R. Linker, P.-O. Gutman, Robust feedback stabilization of an unmanned motorcycle, Control Engineering Practice, 18 (8) (2010), pp [5] H. Dabladji, D. Ichalal, H. Arioui, S. Mammar, Estimation of lateral dynamics and road curvature for two-wheeled vehicles: A HOSM observer approach,19th IFAC World Congress on International Federation of Automatic Control (IFAC 2014), Cape Town, South Africa, 2014, pp

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