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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. X, NO. X, XXXXXX 2013 1

An Unknown Input HOSM Approach to EstimateLean and Steering Motorcycle Dynamics

Lamri Nehaoua, Dalil Ichalal, Hichem Arioui, Jorge Davila, Said Mammar, and Leonid Fridman

Abstract—This paper deals with state estimation of Pow-ered Single-Track vehicle and robust reconstruction of relatedunknown inputs. For this purpose, we consider an unknowninput high order sliding mode observer (UIHOSMO). First, amotorcycle dynamic model is derived using Jourdain’s principle.The strong observability of the obtained model is illustrated.Then, we consider both the observation of the PTW dynamicstates and the reconstruction of the lean dynamics and the rider’storque applied on the handlebar. Finally, several simulation casesare provided to illustrate the efficiency of the observer.

Index Terms—Motorcycle, HOSM, Strong Observability.

I. INTRODUCTION

REcently, the use of powered single-track vehicles isconstantly growing, upsetting driving practices and road

traffic. Unfortunately, this expansion is also inflected by animportant increase of motorcycle’s fatalities (20 times higherwhen driving a car). Recent statistics confirm this observationand consider riders as the most vulnerable road users. In2010, the French Agency of Road Safety made a findingof around 1000 deaths (25% of traffic fatalities), while thetraffic proportion of motorcycles does not exceed 1%. Severalresearch programs are launched to answer this issue and tofind solutions in term of preventive and / or active securitysystems, [1], [2].

The success, of the proposed security systems, dependsprimarily on the knowledge of the dynamics of motorcycleand, the evolution of its states (to be observed) stronglyinvolved by the rider’s action (to be reconstructed) and/or theinfrastructure features (slope, tilt, adherence, etc.). Regardingthe first point, several studies were carried out in order tounderstand the motorcycle dynamics (see, for example [3]–[6]), the stability analysis of powered two-wheeled (PTW),optimal and safe trajectories [7], [8] and the proposal of riskfunctions (see [9], [10], [31]) to detect borderline cases ofcontrol loss. These research are very few sustainable if theyare not propped by a system for estimating the dynamic statesof the bike.

The real measurement, by sensors, of all the bike statesis not conceivable. Thus, we propose to use observation

Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

L. Nehaoua, D. Ichalal, H. Arioui and S. Mammar arewith the IBISC Laboratory (EA-4526) of Evry Val d’EssonneUniversity. 40, Rue du Pelvoux, 91020 Evry, France.lamri.nehaoua,dalil.ichalal,hichem.arioui,[email protected]. Davila is with National Polytechnic Institute of Mexico, SEPI ESIME-UPT,[email protected]

L. Fridman is with National Autonomous University of Mexico (UNAM),Mexico. [email protected]

techniques to overcome measurement noise, expensive sensors,etc. Within this context, including all methodologies, very fewstudies exist [11]–[13].

The first work on the state estimation for motorcyclesdate back to 2008 [14]. Researches sustain this topic havecommonly concerned the estimation of the lean dynamicsand considerably less the steering one. Several approacheshave been experimented to estimate the roll angle: frequencyseparation filtering [16] or extended Kalman filters, [17].These techniques, performed under restrictive assumptions(dynamic steering is neglected, linear tire-road forces, etc.),suffer mainly from a relative robustness against the variationsof the forward velocity.

The issue of the steering angle estimation, and not the rider’storque reconstruction, is not well covered in literature as thelean angle estimation problem. Nonetheless, a recent resultshave been obtained in [18]–[21], where an LPV observerhas been used to design control strategy for a semi-activesteering. The approach is a simple gain scheduling for anLTI motorcycle model under three constant forward velocities.Unfortunately, no guarantees for stability or convergence of theproposed observer are given. In [22], [23], approaches basedon Takagi-Sugeno fuzzy models are used in order to covera large operating range of the system which allows to takeinto account some inherent nonlinearities in the motorcyclemodel. This aims to construct an observer able to estimatethe states in several situations where the nonlinear dynamicsare excited (cornering situation, saturated contact forces tire-road,...). These last works give a new direction for nonlinearobservation and constitute primary results in this topic.

To the best authors’ knowledge, the simultaneous estima-tion of the lean and the steering dynamics have never beenaddressed. The present work proposes a robust UIHOSMO[24], helping in states observation of motorcycle model and thereconstruction of rider’s action under parameter uncertainties.Sliding-mode observer (SMO) for system states estimationin the presence or absence of unknown inputs has been thesubject of several work. Nowadays, one can notice that obser-vation theory has matured and has succeed to deal with manytechnical issues [25], [26] where some restrictive conditionsrelated to the observability and the reconstruction of unknowninputs were released even suppressed.

To avoid filtering, the discontinuous output injection isreplaced by a continuous super-twisting algorithm (STA) [28].In this new version, the relative degree of the system’s outputswith respect to the unknown inputs must be equal to the systemorder. This restriction is resolved by the introduction of thehigh-order Sliding-mode observers (HOSMO) [24], based on

mailto:[email protected]

mailto:[email protected]




the high-order robust exact Sliding-mode differentiator [29],where the notion of strong observability and strong detectabil-ity were presented. It remains at least that the outputs relativedegree must exist which brings a novel restriction treatedby the development of the concept of weakly observablesubspaces detailed in [30].

In this paper, we suggest to use HOSMO higher order slid-ing mode observers. Theoretically, the HOSMO provide finitetime exact convergence for the system states and unknowninputs. Moreover, HOSMO ensure best possible accuracy withrespect to sampling steps and bounded disturbances withoutneed of a priori knowledge of the upper bound of disturbancesand the size of the sampling step [27].

The paper contributions are the following:• A motorcycle dynamic model is derived.• It is a first paper (for the authors best knowledge) when

the theory of strong observability/detectability are appliedto the problem of reconstruction for motorcycles steeringdynamics.

• To realize the reconstruction of the motorcycles states andunknown inputs, UIHOSM observers were chosen. Due tonon-minimum phase nature of the motorcycle dynamicsmodel, the direct estimation of the lean angle as a statevariable is not possible. To overcome this problem, themodel is rearranged in order to consider the lean angleas an unknown input to be reconstructed. The strongobservability property is then used.

• Various results for modeling and observation are dis-cussed and validated by using a high-end motorcyclesoftware simulator.

This paper is organized as follows: sections II and IIIare dedicated respectively to motorcycle modeling and thesynthesis of the UIHOSMO. Simulation results are given issection IV. The conclusion in section V wrap up the paper.

II. MOTORCYCLE DYNAMICS

This section highlights some outlines for the PTW mod-eling. Firstly, the model to be used for the validation step(section V) is described . Next, for observer synthesis, adedicated model is derived.

A. Modeling Assumptions

In figure 1, the considered geometrical configuration of themodeled vehicle is shown. The motorcycle is represented as aset of eight linked bodies: the main frame Gr which includesthe chassis and engine, the front body Gf which representsthe steering assembly, the lower body Gl which includes thelower suspension part and the front wheel hub, the swing-armbody Gs, the front and rear wheel R, the upper and the lowerparts of the rider body GB . The rider is considered as to besolidly attached to the main body Gr and, its posture stays inthe upright plan of the motorcycle symmetry plan.

The motion of the overall mechanical system is expressedat point v, origin of the vehicle reference frame ℜv . Theequation of motion, developed by using Jourdain’s principleof dynamics [33], allows to simulate 11 degrees of freedom(DOF): the velocity vector of point v (vx, vy , vz) the yaw,

pitch and roll of the main frame Gr (ψ, θ, ϕ), the handlebarsteer angle δ with respect to the rider torque input τ r appliedon the motorcycle’s handlebar, swing-arm motion µ and thetwo wheels spin.

Fig. 1. Geometrical representation of the motorcycle vehicle

The dynamic model of the motorcycle vehicle can beexpressed in a compact form by:

Mϑ = Q (1)

where the mass matrix M is symmetric and positive definiteand Q is the vector of the generalized efforts.

B. Linearized Model for Observer Synthesis

The study of the dynamics of motorcycle vehicles highlightstwo main modes of motion: in-plane mode representing themotorcycle movements in its plane of symmetry includingthe longitudinal motion and that of suspensions and the out-of-plane mode which describes the lateral dynamics whencornering [3], [32]. The last mode involves the roll inclination,the yaw rotation, the steering and the lateral motions of thebike. We consider here only the out-of-mode dynamics of thePTW. The coupling between the two modes is materialized,when necessary, by considering a variable longitudinal velocitythat appears in the lateral dynamics.

The motorcycle dynamic model (1) is linearized around thestraight-running trim trajectory and can be expressed by thefollowing state-space:

xv = Avxv +Bvτ r (2)

yv = Cvxv (3)

here, xv = [vy, ψ, ϕ, δ, ϕ, δ, Fyf , Fyr]T denotes the state

vector. Fyf and Fyr represent respectively the tires sideslipforces introduced in the state space representing the tirerelaxation. Av is a time-varying matrix related to the for-ward velocity vx while Bv is a time-invariant vector whereAv = Λ−1E and, Bv = Λ−1[0, 0, 0, 1, 0, 0, 0]T . Matrices Λand E are given appendix.

III. STATES AND UNKNOWN INPUTS ESTIMATION

In this section, we aim to estimate the motorcycle states andreconstruct both roll angle ϕ and rider’s torque τ r by using aUIHOSMO [24].




In our case, we consider that the measured variables arethe steering angle by means of a digital encoder sensor and,angular velocities by using inertial unit where the forwardvelocity can be also deduced. Sensor measurement noises andeffect vibration will be considered to be matched disturbance.

At first, we recall some important definitions about strongobservability and strong detectability of linear systems (forproofs see [34], [24]).

Consider the following SISO system, where x ∈ Rn, ζ ∈ R

is the unknown input and y ∈ R is the measured output:

x = Ax+Bu+Dζ (4)

y = Cx

Definition 3.1: ( [34]). s0 ∈ C is called an invariant zero ofthe triplet (A,D,C) if rank {R(s0)} < n+ rank {D}, whereR(s) is the Rosenbrock matrix of system (4):

R(s) =

[

sI −A −D

C 0

]

.

Let us recall ( [35]) that the relative degree of the output ywith respect to the unknown input ζ is the scalar r such that:

CAjD = 0 j = 1, · · · , r − 2,

CAr−1D 6= 0.

Definition 3.2: ( [34]). System (4) is called strongly observ-able if for any initial state x(0) and any unknown input ζ(t)),y(t) ≡ 0 for all t ≥ 0 implies that also x ≡ 0. Otherwise, (4)is called strongly detectable, if for any ζ(t) and x(0), y(t) = 0for all t ≥ 0 implies that x→ 0 as t→ ∞.

Proposition 3.1: ( [34], [24]). The following statements areequivalent:

1) The system (4) is strongly observable.2) The triplet (A,D,C) has no invariant zeros.3) The output y of system (4) has relative degree r = n

with respect to the unknown input ζ.

Otherwise, the following statements are also equivalent:

1) The system (4) is strongly detectable,2) The relative degree r of the system’s output y with

respect to the unknown input ζ exists, and the triplet(A,D,C) has a stable invariant zeros (system (4) isminimum phase).

For the MIMO system case, the strong observability and de-tectability can be reformulated by reconsidering the definitionof the relative degree. For this, let be the system (4), wherex ∈ R

n, y ∈ Rm is the output vector and ζ ∈ R

m is theunknown input vector.

The output y is said to have the vector relative degree r =[r1, · · · , rm] with respect to the unknown input ζ( [34]), if

CiAsDj = 0 i, j = 1, · · · ,m, s = 1, · · · , ri − 2

CiAri−1Dj 6= 0

and

det

C1Ar1−1D1 · · · C1A

r1−1Dm

......

...

CmArm−1D1 · · · CmArm−1Dm

6= 0

Lemma 3.1: ( [24]). Let the output y of (4) have the vectorrelative degree r = [r1, · · · , rm] with respect to the unknowninput ζ. Then the vectors C1, ...,C1A

r1−1, ..., Cm, ...,CmArm−1 are linearly independent.

A. Case 1: estimation of the roll angle

The motorcycle dynamics system (2) has n = 8 states, oneoutput which corresponds to the steering angle δ, measured bya digital encoder sensor, and one unknown input correspondingto the rider’s torque τ r applied on the motorcycle handlebar.From definition (3.1), the triplet (Av,Dv,Cv) has one un-stable invariant zero which makes the motorcycle dynamicsto be a non-minimum phase. In addition, the system’s outputhas relative degree r = 2 < n with respect to the unknowninput ζ. It follows from proposition (3.1) that the motorcycledynamics system (2 ) is neither observable nor detectable, andthus for all vx in the allowable velocities range.

In order to make the system observable, the motorcyclemodel (2 ) is rewritten in such way that the roll angle ϕ appearsas an unknown input rather than a state variable. In fact, theunstable invariant zero is a direct consequence of the counter-steering phenomena generated by the motorcycle roll. In thiscase, the new system equation are:

xp = Apxp +Bpτ r +Dpϕ, (5)

yp = δ = Cpxp,

where xp = [vy, ψ, ϕ, δ, δ, Fyf , Fyr]T denotes the state vector.

The output yp of system (5) has a relative degree r = 2with respect to the unknown input ϕ. In addition, the triplet(Ap,Dp,Cp) has 5 stable invariant zeros for all vx in theallowable velocities range. It results from proposition (3.1),that (5) is strongly detectable. This definition implies thatonly r system’s states can be estimated exactly while theobservation of the remaining states are asymptotically exact.

To estimate the system’s state vector xp and the unknowninput ϕ, it is necessary to separate the clean states from thosecontaminated by the unknown input. To achieve this, system(5) is transformed to a new coordinates system ξp = Txp suchthat the closed-loop system dynamics (Ap −CpLp,Dp,Cp)is expressed by:

ξ11 = ξ12 + b11τ r (6)

ξ12 = a1ξ11 + a2ξ12 + a3ξ21 + · · ·+ a7ξ25 + b12τ r + dϕϕ

ξ2 = A21ξ1 +A22ξ2 +B2τ r

yp,new = ξ11,

where ξTp = [ξT1 , ξT2 ] ∈ R

n and ξT1 = [ξ11, ξ12] ∈ Rr and

ξT2 = [ξ21, · · · , ξ25] ∈ Rn−r. Since the system is strongly

detectable, the observer gain Lp can be chosen such that (Ap−




LpCp) is Hurwitz. It follows that the state observer have theform:

zp = Apz +Bpτ r +Lp(yp −Cpz) (7)

ϑ2 = A21ϑ1 +A22ϑ2

xp = z + T−1ϑ,

in which xp is the vector of the estimated states, z ∈ Rn and

ϑ ∈ Rn is given by (recall that n = 7 and r = 2):

ϑ =

[

ϑ1

ϑ2

]

l Rr

l Rn−r

=

v1

v2

ϑ2

in this equation, vector ϑ components are composed from theobserver’s nonlinear injection v ∈ R

r+k+1. As reported in[24], since the unknown input ϕ is bounded with an upperbound |ϕ| ≤ ϕmax and the k successive derivatives of ϕare bounded by the same constant ϕ′

max, then by selectinga sufficiently large λ, v can be obtained by using a (r+ k)-thorder differentiator as following (for k = 1):

v1 = −3λ1

4 |v1 − yp +Cpz|3

4 sign(v1 − yp +Cpz) + v2

v2 = −2λ1

3 |v2 − v1|2

3 sign(v2 − v1) + v3 (8)

v3 = −1.5λ1

2 |v3 − v2|1

2 sign(v3 − v2) + v4

v4 = −1.1λsign(v4 − v3)

In addition, the reconstruction of the unknown input ispossible by using:

ϕ =1

dϕ(v4 − a1v1 − a2v2 − a3ϑ21 − · · · − a7ϑ27 − b12τ r)

(9)

B. Estimation of the Roll Angle and Rider’s Torque

In this case study, the system equations are given by:

xp = Apxp +Dppζ, (10)

ypp =

[

δ

ψ

]

= Cppxp,

Dpp =[

Dp Bp

]

, ζT =[

ϕ τ r]

.

From definition (3.1), the output ypp in (10) has a relativedegree vector r = [2, 1] with respect to the unknown inputvector ζ. In addition, the triplet (Ap,Dpp,Cpp) has 4 stableinvariant zeros for all vx in the allowable velocities range. Itresults from Lemma (3.1), that (10) is also strongly detectable.This definition implies that only rT = r1 + r2 system’sstates can be estimated exactly while the observation of theremaining states are asymptotically exact.

As before, to estimate the system’s state xp and the un-known input vector ζ, it is necessary to separate the cleanstates from those contaminated by the unknown inputs. Toachieve this, system (10) is transformed to a new coordinates

system ξp = Txp such that the closed-loop system dynamics(Ap −CppLpp,Dpp,Cpp) is expressed by:

ξ11 = ξ12, (11)

ξ12 = a11,11ξ11 + a11,12ξ12 + a11,13ξ13 + a12,11ξ21+

· · ·+ a12,14ξ24 + d11ϕ+ d12τ r,

ξ13 = a11,21ξ11 + a11,22ξ12 + a11,23ξ13 + a12,21ξ21+

· · ·+ a12,24ξ24 + d21ϕ+ d22τ r,

ξ2 = A21ξ1 +A22ξ2,

yTpp,new =

[

ξ11 ξ13]

,

where ξTp = [ξT1 , ξT2 ] ∈ R

n and ξT1 = [ξ11, ξ12, ξ12] ∈ RrT

and ξT2 = [ξ21, · · · , ξ24] ∈ Rn−rT . Next, the state observer is

designed as:

z = Apz +Lpp(ypp −Cppz) (12)

ϑ2 = A21ϑ1 +A22ϑ2

xp = z + T−1ϑ

in which xp is the vector of estimated states, z ∈ Rn and

ϑ ∈ Rn is given by (recall that n = 7, r = [2, 1] and rT = 3):

ϑ =

[

ϑ1

ϑ2

]

l RrT

l Rn−rT

=

v1,1

v1,2

v2,1

ϑ2

Once again, vi ∈ RrM+k+1 is the nonlinear part of the

observer where i = 1, · · · ,m and rM = max(ri). Eachunknown input ζi is bounded with |ζi| ≤ ζi,max and the(rM − ri+ k) successive derivatives of ζi are bounded by thesame constant ζ ′i,max, consequently the auxiliary variable vi

is a solution of the discontinuous vector differential equationby considering k = 1 as:

v1,1 = −3λ1

4 |v1,1 − yp1+Cp1

z|3

4 sign(v1,1 − yp1+Cp1

z)

+ v1,2

v1,2 = −2λ1

3 |v1,2 − v1,1|2

3 sign(v1,2 − v1,1) + v1,3 (13)

v1,3 = −1.5λ1

2 |v1,3 − v1,2|1

2 sign(v1,3 − v1,2) + v1,4

v1,4 = −1.1λsign(v1,4 − v1,3)

v2,1 = −2λ1

3 |v2,1 − yp2+Cp2

z|2

3 sign(v2,1 − yp2+Cp2

z)

+ v2,2

v2,2 = −1.5λ1

2 |v2,2 − v2,1|1

2 sign(v2,2 − v2,1) + v2,3 (14)

v2,3 = −1.1λsign(v2,3 − v2,2)

Finally, the reconstruction of the unknown input vector ispossible by using:[

ϕ

τ r

]

= D−1

([

v1,3

v2,2

]

−

[

a11,11 · · · a12,14

a11,21 · · · a12,24

]

ϑ

)

D =

[

d11 d12

d21 d22

]

(15)




Remark 3.1: In the full relative degree case, all system’sstates can be exactly estimated and the observability matrix

P =

C1

...

C1An−1

...

Cm

...

CmAn−1

(16)

can be chosen for the coordinates transformation (T = P ).However, in the strongly detectable case, only n−nu system’sstates are exactly estimated where nu is the number ofinvariant zeros (see [30] for a deeper study of the problem). Apossible choice of the transformation matrix T can be foundin [36].

IV. VALIDATION RESULTS

We recall here that the aim of our work is to addressthe states and unknown inputs estimation in order to deriveefficient warning systems for riders. The context concernsmainly the urban situations (the linear and nonlinear dynamicsare almost identical) and less sport motorcycles which arecharacterized by a strong nonlinear dynamics. This issue willbe addressed in future.

In order to proof the efficiency of the proposed observer formotorcycle application, we shall respect the following steps:

• A comparison of our nonlinear model with one of themodels using in BikeSim simulator. The comparison isachieved under two different scenarios. A discussionis given to highlights the differences between the twomodels.

• The second step concerns the linearization of our pro-posed nonlinear model around trim trajectories.

• Based on the linear model, a high order sliding modeobserver is synthesized. To overcome the neglected non-linearities, a robustness study is carried-out with respectto the forward speed time variations and parameter un-certainties.

• Finally, the HOSM observer is applied to the nonlinearmodel to estimates the roll (ϕ) and the steering dynamics(τ r).

A. Assessment with BikeSim Simulator

Despite some differences, both models shows a close be-havior (by considering the same motorcycle parameters anddegrees of freedom). Differences can be explained by theconsidered modeling assumptions summarized in the followingpoint:

• The lateral velocities of both models are not identical.In our model, the lateral speed is expressed at point v(figure 1), whereas, in BikeSim simulator, the same speedis calculated at the main frame center of gravity.

• In our modeling approach, the aerodynamic effort andflexibilities of the steering mechanics are neglected. Inaddition, the tire/road contact point is considered to bestatic and do not move throughout the tire circumference.

Figure 2 shows the result of a first scenario consisting ofdouble lane change maneuver with longitudinal velocity of 100km/h. Besides lateral velocities, all the other states are veryclose. The small differences are due to the precedent hypoth-esis assumed before. For example, the chattering phenomenawe can observe on the steering dynamics is due to the lack ofthe flexibilities in our model. The same scenario is performedfor 50km/h (figure 3) for which the previous remarks alsohold.

A second scenario consists of a cornering maneuver witha constant radius maneuver with time varying forward speedin the interval [50,100] km/h (figure 4). The both behaviorsare similar. Nevertheless, a small difference of 0.2◦ appearsbetween the two steering angles. This is due, mainly, tothe fact that the BikeSim simulator aims to compensates theaerodynamic effort, which is not the case of our model.

B. Model linearization

In order to synthesize the HOSM observer, the proposednonlinear model is linearized with respect to the trim tra-jectories (roll angle close to zero). The dynamic behaviorand tire road dynamics are linearized separately. The previousscenarios are simulated and presented below.

The double lane change scenario, under two different ve-locities (50 and 100 KhM), is presented on figures 5 and 6.We can notice that the behavior of the two cases are similar.This is mainly due to the constant velocity characteristics.

This is not obvious with the second scenario (constant radiuswith acceleration), where the differences are clear. Of course,this scenario is far from our addressed context and urbansituation. The both models, under constant acceleration, donot fit at all (figure 7), because, among other reasons, of thehuge camber angle and tire forces (point contact hypothesis).

C. Observer design

In this section, the UIHOSMO is constructed for the pre-sented motorcycle model. Some results and discussions areprovided to illustrate the effectiveness and the ability of theUIHOSMO in estimating simultaneously the dynamic statesand both roll angle and the applied torque by the rider onthe handlebar. The observer is designed in such a way toestimate all the dynamic states and unknown inputs from onlythe knowledge of steering angle δ(t) and the yaw rate ψ(t).The parameters λi, i = 1, 2 of the differentiator, in equations13 and 14, are chosen as following: λ1 = λ2 = 5000.The Luenberger gain Lpp in equation 12 is computed bypole placement at the eigenvalues: −15, −30, −45, −60,−150, −165, −180. With these parameters, the UIHOSMO isimplemented with initial conditions x(0) = [0.1745 0.1745 −0.3491 − 0.0873 0.3491 50 50]. Finally, validations arecarried-out by using the nonlinear model.




0 2 4 6 8 10 12−40

−30

−20

−10

0

10

20

30

40

time (s)

τ r (N

.m)

0 2 4 6 8 10 12−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time (s)

δ (

deg)

0 2 4 6 8 10 12−20

−15

−10

−5

0

5

10

15

20

25

30

time (s)

φ (

deg)

0 2 4 6 8 10 12−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

vy (

m/s

)

0 2 4 6 8 10 12−15

−10

−5

0

5

10

time (s)

dot

ψ (

deg/s

)

0 2 4 6 8 10 12−50

−40

−30

−20

−10

0

10

20

30

40

50

time (s)

dotφ

(deg/s

)

Fig. 2. Double lane change at 100 km/h. In blue, simulation of the nonlinear model and in red, simulation of Bikesim

0 5 10 15 20 25−10

−8

−6

−4

−2

0

2

4

6

8

10

time (s)

τ r (N

.m)

0 5 10 15 20 25−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time (s)

δ (

deg)

0 5 10 15 20 25−15

−10

−5

0

5

10

15

20

time (s)

φ (

deg)

0 5 10 15 20 25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

vy (

m/s

)

time (s)0 5 10 15 20 25

−20

−15

−10

−5

0

5

10

15

time (s)

dot

ψ (

deg/s

)

0 5 10 15 20 25−30

−20

−10

0

10

20

30

time (s)

dot

φ (

deg/s

)

Fig. 3. Double lane change at 50 km/h. In blue, simulation of the nonlinear model and in red, simulation of Bikesim

0 2 4 6 8 10 12−12

−10

−8

−6

−4

−2

0

time (s)

τ r (N

.m)

0 2 4 6 8 10 12−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time (sec)

δ (

deg)

0 2 4 6 8 10 12−35

−30

−25

−20

−15

−10

−5

0

time (s)

φ (

deg)

0 1 2 3 4 5 6 7 8 9 10−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

time (s)

vy (

m/s

)

0 2 4 6 8 10 12−2

0

2

4

6

8

10

time (s)

dot

ψ (

deg/s

)

0 2 4 6 8 10 12−10

−8

−6

−4

−2

0

2

4

time (s)

dot

φ (

deg/s

)

Fig. 4. Constant radius with acceleration from 50 km/h to 100 km/h. In blue, simulation of the nonlinear model and in red, simulation of Bikesim




0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

δ (

deg)

0 1 2 3 4 5 6 7 8 9 10−20

−10

0

10

20

30

40

time (s)

φ (

deg)

0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

vy (

m/s

)

0 1 2 3 4 5 6 7 8 9 10−15

−10

−5

0

5

10

time (s)

dot

ψ (

deg/s

)

0 1 2 3 4 5 6 7 8 9 10−800

−600

−400

−200

0

200

400

600

time (s)

Fyf (N

)

0 1 2 3 4 5 6 7 8 9 10−1000

−500

0

500

time (s)

Fyr

(N)

Fig. 5. Double lane change at 100 km/h. In blue, simulation of the linear model and in red, simulation of nonlinear model

0 2 4 6 8 10 12 14 16 18 20−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time (s)

δ (

deg)

0 2 4 6 8 10 12 14 16 18 20−20

−15

−10

−5

0

5

10

15

20

time (s)

φ (

deg)

0 2 4 6 8 10 12 14 16 18 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

time (s)

vy (

m/s

)

0 2 4 6 8 10 12 14 16 18 20−20

−15

−10

−5

0

5

10

15

time (s)

dot

ψ (

deg/s

)

0 2 4 6 8 10 12 14 16 18 20−500

−400

−300

−200

−100

0

100

200

300

400

time (s)

Fyf (N

)

0 2 4 6 8 10 12 14 16 18 20−500

−400

−300

−200

−100

0

100

200

300

400

time (s)

Fyr

(N)

Fig. 6. Double lane change at 50 km/h. In blue, simulation of the linear model and in red, simulation of nonlinear model

0 1 2 3 4 5 6 7 8 9 10−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

time (s)

δ (

deg)

0 1 2 3 4 5 6 7 8 9 10−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

time (s)

φ (

deg)

0 1 2 3 4 5 6 7 8 9 10−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

time (s)

vy (

m/s

)

0 1 2 3 4 5 6 7 8 9 10−2

0

2

4

6

8

10

12

14

16

time (s)

dot

ψ (

deg/s

)

0 1 2 3 4 5 6 7 8 9 10−200

0

200

400

600

800

1000

time (s)

Fyf (N

)

0 1 2 3 4 5 6 7 8 9 10−200

0

200

400

600

800

1000

1200

time (s)

Fyr

(N)

Fig. 7. Constant radius with acceleration from 50 km/h to 100 km/h. In blue, simulation of the linear model and in red, simulation of nonlinear model




0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

vy (

m/s

) 0 0.2 0.4−0.2

0

0.2

0.4

0 1 2 3 4 5 6 7 8 9 10−15

−10

−5

0

5

10

time (s)

dot

ψ (

deg/s

)

0 0.1 0.2 0.3−2

0

2

4

6

0 1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

2

3

4

5

time (s)

dot

δ (

deg/s

)

0 0.5 1−5

0

5

0 1 2 3 4 5 6 7 8 9 10−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

δ (

deg) 0 0.2 0.4

−2

0

2

0 1 2 3 4 5 6 7 8 9 10−30

−20

−10

0

10

20

30

40

time (s)

φ (

deg) 0 0.2 0.4

−20

0

20

40

0 1 2 3 4 5 6 7 8 9 10−40

−30

−20

−10

0

10

20

30

40

50

60

time (s)

τ r (N

.m) 0 0.2 0.4

−50

0

50

100

Fig. 8. Double lane change at 100 km/h. In blue, model results and in red, estimation result

0 1 2 3 4 5 6 7 8 9 10

−10

−5

0

5

10

15

20

25

time (s)

err

ors

vy (m/s)

φ (deg)

τr (N.m)

0 1 2 3 4 5 6 7 8 9 10−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

time (s)

err

ors

dot δ (deg/s)

dotψ (deg/s)

δ (deg)

Fig. 9. Double lane change at 100 km/h. Estimation error

In the first scenario, we consider again the double lanechange maneuver with forward speed at 100 km/h. In fig-ure 8, some estimated states and the two unknown inputsare depicted. One can conclude that the observer providessatisfactory results. Since the total relative degree is equalto 3, the yaw rate (ψ), the steering rate (δ) and the steeringangle (δ) are exactly estimated with finite-time convergence asshown by small zoom onside each sub-figure. The remainingstates, namely the lateral speed (vy), the roll rate (ϕ), thetire lateral forces (Fyf , Fyr) and the two unknown inputsare estimated in finite-time with a bounded estimation error(figure 9). Moreover, it is possible to deal with the problemof transient phase by tuning the observer’s parameters as

discussed in [37].

0 1 2 3 4 5 6 7 8 9 10−30

−20

−10

0

10

20

30

40

time (s)

φ (

de

g)

0 1 2 3 4 5 6 7 8 9 10−40

−30

−20

−10

0

10

20

30

40

50

60

time (s)

τ r (N

.m)

Fig. 10. Double lane change at 100 km/h. Robustness with respect toparameter variations (+20% variation on the nominal mass and inertia values).In blue, model results, in red estimation result by using nominal parameterand in black, estimation result by introducing uncertainty

In order to illustrate the performances of the proposedobserver in the presence of modeling uncertainties, the ob-server is designed by using the linear motorcycle nominalmodel. In the nonlinear model, all bodies’ mass and inertiaare considered with a 20% variation with respect to theirnominal values. As shown in figure 10, the observer ensuresan acceptable estimations for almost the state variables and theunknown inputs. Once again, Three of eight states are exactlyestimated despite uncertainty whereas, the remaining variable




0 1 2 3 4 5 6 7 8 9 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time (s)

vy (

m/s

) 0 0.2 0.4−0.2

0

0.2

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (s)

δ (

deg) 0 0.1 0.2

−2

0

2

4

0 1 2 3 4 5 6 7 8 9 10−30

−20

−10

0

10

20

30

40

50

time (s)

φ (

deg)

0 0.5 1−50

0

50

0 1 2 3 4 5 6 7 8 9 10−40

−20

0

20

40

60

80

100

time (s)

τ r (N

.m) 0 0.2 0.4

0

50

100

Fig. 11. Double lane change at 90 km/h. Robustness with respect to the forward speed vx. In blue, model results and in red, estimation result

0 1 2 3 4 5 6 7 8 9 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time (s)

vy (

m/s

)

0 0.2 0.4

−0.2

0

0.2

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (s)

δ (

deg) 0 0.1 0.2

−2

0

2

4

0 1 2 3 4 5 6 7 8 9 10−30

−20

−10

0

10

20

30

40

50

time (s)

φ (

deg) 0 0.5 1

−50

0

50

0 1 2 3 4 5 6 7 8 9 10−40

−20

0

20

40

60

80

100

time (s)

τ r (N

.m) 0 0.2 0.4

0

50

100

Fig. 12. Double lane change at 110 km/h. Robustness with respect to the forward speed vx. In blue, model results and in red, estimation result




are estimated in finite-time with an acceptable bounded esti-mation error.

As explained previously, the longitudinal velocity vx isconsidered constant, but in real situations, forward speed isalso subject to variations. The observer is constructed bychoosing a nominal value vx = 100 km/h but in the simulatedsystem, a 10% variation with respect to the nominal value isassumed. Estimation results are depicted in figures 11 and 12.For exact estimation, the steering angle δ estimation is givenas an example where for bounded error estimation, the lateralspeed estimation vy is shown. However, the estimations ofthe unknown inputs are visibly affected by the time varyinglongitudinal speed, in particular, at the maximum values.

V. CONCLUSION

The present paper deals with the problem of observer designfor simultaneously estimating states and unknown inputs for acomplex system such that a motorcycle vehicle. For that pur-pose, an Unknown Inputs High Order Sliding Mode observeris proposed and validated.

Three main contributions are detailed. The first one con-cerns the validation of the motorcycle nonlinear model withprofessional simulator data and thus, by considering two mainscenarios: a double lane change at different forward speedand a cornering along a constant radius with forward speedvariation. The second contribution concerns a full comparisonbetween the nonlinear and the linear models for the sameprevious scenarios. Conclusion about the effect of the non-linearities on the motorcycle driving behavior is discussed.Finally, the effectiveness of the observer is demonstrated on anonlinear model and with a different initial conditions.

It is shown that, by introducing some rearrangement, theobservability of a given model can be achieved, and the re-sulting model representation fulfill the detectability condition.Validation of the observer is proven despite the presence ofparameter uncertainties and/or forward speed variation. It isshown that all measurable signals and their successive deriva-tives are exactly estimated with a finite-time convergence.This is guaranteed if the plant model can be transformedto the normal form by using a suitable transformation. Theremaining states are estimated with a finite-time convergenceand a bounded error estimation. Nevertheless, a wide variationof the forward speed can be seriously affect the observerperformance. This issue constitute our future work.

VI. ACKNOWLEDGEMENT

J. Davila gratefully acknowledges the CONACyT(Mexico)grant 151855, SIP-IPN grant 20120218 and CDA-IPN. L. Frid-man gratefully acknowledges the CONACyT(Mexico) grant132125, PAPIIT, UNAM, grant IN113613.

motorcyclevx, vy : longitudinal and lateral velocityϕ, ψ, δ: roll, yaw and steer rotationsτr : rider torqueFyf , Fyr : lateral forceM : motorcycle mass matrixQ: motorcycle generalized effort vec-

tornotationsx, x: derivatives of a variable x w.r.t

timex: estimate of a variable xxT : transpose of vector or matrix xxf , xr denotes front and rearmotorcyclemGr

, mGf, mGl

, mGs165.13, 9.99, 7.25, 8 [kg]

mGBu, mGBl

, mRf,

mRr

43.52, 25.84, 0, 14.7 [kg]

xvGr, xvGf

, xvGl,

xvGs

39, 65.07, 80.13, -25 [cm]

xvGBu, xvGBl

, xvRf,

xvRr

-23.04, 13.4, 95, -42 [cm]

zvGr, zvGf

, zvGl, zvGs

53.2, 104.34, 60.27, 34 [cm]zvGBu

, zvGBl, zvRf

,zvRr

80.01, 50.4, 29, 29 [cm]

xGf, xGl

, xRf10.2, 6.04, 6.9 [cm]

zGf, zGl

, zRf35.51, -10.88, -45.49 [cm]

Ixf , Ixl, Ixub, Ixlb 1.341, 0, 0.9, 0.5 [Kg/m2]Izf , Izl, Izr , Izub, Izlb 0.4125, 0, 14.982, 0.9, 0.5 [Kg/m2]Cxzr , Cxzub 3.691, 0.433 [Kg/m2]iyrf , iyrr wheels spin inertia 0.484, 0.638

[kg.m2]ρf , ρr wheels radius 29, 29 [cm]η tire trail -4.89 [cm]Fzf front tire normal force 1543.5 [N]σf , σr tire’s relaxation [m]ǫ castor angle -0.4189 [rad]Cδ handlebar damping 12.6738

[N.m−1.s]g gravity force 9.81 [N]Cf1, Cr1 front and rear tire’s sideslip stiff-

ness -22408, -17657 [N/rad]Cf2, Cr2 front and rear tire’s camber stiff-

ness -1056.3, -518.49 [N/rad]

VII. APPENDIX

Λ =

a11 a12 a13 a14 0 0 0 0

a22 a23 a24 0 0 0 0

a33 a34 0 0 0 0

a44 0 0 0 0

⋆ 1 0 0 0

1 0 0

0 σf 0

σr

E =

0 b12 0 0 0 0 1 1

0 b22 b23 b24 0 0 xvRfxvRr

0 b32 0 b34 b35 b36 0 0

0 b42 b43 Cδ b45 b46 η 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

b71 b72 0 b74 b75 b76 −vx 0

b81 b82 0 0 b85 0 0 −vx




parameters aij

a11 =∑

mi, a12 =∑

i mixvi, a13 = −

∑

mizvi, a14 =∑

j mjxj , a22 =∑

i mix2

vi+sin ǫ2(Ixf+Ixl)+cos ǫ2(Izf+Izl)+2Izr+Izub+Izlb, a23 = −

∑

i mixvizvi−(Ixf+Ixl−Izf − Izl) cos ǫ sin ǫ+2Cxzr +Cxzub, a24 =

∑

j mjxvjxj +

(Izf + Izl) cos ǫ, a33 =∑

i miz2

vi + 2Ixr + Ixlb + Ixub +sin ǫ2(Izf +Izl)+cos ǫ2(Ixf +Ixl), a34 = −

∑

j mjzvjxj+

sin ǫ(Izf + Izl), a44 =∑

j mjx2

j + Izf + Izlparameters bijb12 = −

∑

mivx, b22 = −vx∑

i mixvi, b23 =−(iyrf/ρf + iyrr/ρr)vx, b24 = −iyrf/ρf sin ǫvx, b32 =(∑

i mizvi + iyrf/ρf + iyrr/ρr)vx, b34 = iyrf/ρf cos ǫvx,b35 =

∑

i mizvig, b36 = −g∑

j mjxj + Fzfη, b42 =−(

∑

j mjxj − iyrf/ρf sin ǫ)vx, b43 = −iyrf/ρf cos ǫvx,b45 = Fzη− g

∑

j mjxj , b46 = Fz sin ǫη− g sin ǫ∑

j mjxj ,b71 = Cf1, b72 = Cf1xvRf

, b74 = Cf1η, b75 = Cf2vx,b76 = vx(−Cf1 cos ǫ + Cf2 sin ǫ), b81 = Cr1, b82 =Cr1xvR1

, b74 = Cf1η, b75 = Cr2vxi = Gr, Gf , Gl, Gs, GRf

, GRr , GBl, GBu

j = Gf , Gl, GRf

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Lamri NEHAOUA received the Dipl.-Ing. degreeon control systems from the University of Sétif,Algeria, in 1999, the M.S. degree in computer visionfor robotics applications from Clermont-Ferrand IIUniversity, France, in 2002, and the Ph.D. degreein robotics from University of Evry Val d’Essonne,France, in 2008. Currently, he is an Associate Pro-fessor with the Université d’Évry. His main interestare driving simulators, driver assistance, modeling,control and observation of complex system.

Dalil ICHALAL received his Master’s degree fromUniversité Paul Cézanne Aix Marseille 3 (France)in 2006 and PhD degree from National PolytechnicInstitute of Lorraine (France) in 2009. In 2010, hejoined the department of electrical engineering ofUniversité of Evry-Val d’Essonne (France) and theLaboratory of Informatique, Biologie Integrative etsystèmes complexes (IBISC). His research interestsinclude state estimation and observer design, faultdiagnosis and fault tolerant control of nonlinear sys-tems with application on vehicles and motorcycles.

Hichem ARIOUI received the Dipl.-Ing. degreeon control systems from Badji Mokhtar University,Annaba, Algeria, in 1998 and the M.S. and Ph.D.degrees in robotics and automation from Universityof Evry, France, in 1999 and 2002, respectively. Hewas with the IBISC Lab. University of Evry, wherehe worked on the control of haptic display undervarying time delay in the frame of his Ph.D. thesis.Since 2003, he has been an Associate Professor ofelectrical engineering with the University of Evry.His research interest includes modeling and control

of haptic interaction and, more recently, design and control of drivingsimulator applications.

Jorge DAVILA was born in Mexico City, Mexico,in 1977. He received the BSc and MSc degreesfrom the National Autonomous University of Mex-ico (UNAM) in 2000 and 2004, respectively. In2008, he received the Ph.D. degree and was awardedwith the Alfonso Caso medal to the best Graduatestudent from the UNAM. He currently holds a fullProfessor position at the Section of Graduate Studiesand Research of the National Polytechnic Institute ofMexico. His professional activities are concentratedin the fields of observation of linear systems with un-

known inputs, observers for switching systems, nonlinear observation theory,robust control design, high-order sliding-mode control and their applicationto the aeronautic and aerospace technologies.

Saïd Mammar (M’02 - SM’11) received the Dipl.-Ing. degree from the École Supérieure d’Électricité,Gif-sur-Yvette, France, in 1989, the Ph.D. degreein automatic control from the Université Paris XI-Supélec, Orsay, France, in 1992, and the HDRdegree from the Université d’Évry val d’Essonne,France, in 2001. From 1992 to 1994, he held aresearch position with the French National Instituteon Transportation Research and Safety, Versailles,France, where he was involved in research on trafficnetwork control. From 1994 to 2002, he was an

Assistant Professor with the Université d’Évry val d’Essonne, where hehas been a Professor since 2002. From 2006 to September 2009, he wasa Scientific and University Attaché with the French Embassy, Hague, TheNetherlands. He is the Head of the IBISC Lab. since January 2010. Hisactual research interests include robust control, vehicle longitudinal and lateralcontrol for driving assistance, and intelligent transportation systems.

Leonid M. FRIDMAN received the M.S. degree inmathematics from Kuibyshev (Samara) State Uni-versity, Samara, Russia, in 1976, the Ph.D. degreein applied mathematics from the Institute of ControlScience, Moscow, Russia, in 1988, and the Dr.Sc. degree in control science from Moscow StateUniversity of Mathematics and Electronics, Moscow,Russia, in 1998. From 1976 to 1999, he was withthe Department of Mathematics, Samara State Ar-chitecture and Civil Engineering University. From2000 to 2002, he was with the Department of Post-

graduate Study and Investigations at the Chihuahua Institute of Technology,Chihuahua, Mexico. In 2002, he joined the Department of Control Engineeringand Robotics, Division of Electrical Engineering of Engineering Faculty atNational Autonomous University of Mexico (UNAM), México. His researchinterests are variable structure systems. Prof. Fridman is an Associate Editorof the Journal of Franklin Institute, Nonlinear Analysis: Hybrid Systems, theConference Editorial Board of IEEE Control Systems Society, Vice-Chair ofTC on Variable Structure Systems and Sliding mode control of IEEE ControlSystems Society and Member of TC on Discrete Events and Hybrid Systemsof IFAC. He is an Author and Editor of 6 books and 12 special issues and morethan 350 technical papers on sliding mode control. He is a winner of Scopusprize for the best cited Mexican Scientists in Mathematics and Engineering2010. He was working as an invited professor in 18 universities and researchcenters of Australia, China, France, Germany, Italy, Israel, and Spain.

Documents

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. X, NO. …verona.fi-p.unam.mx/~lfridman/publicaciones/papers/ieeetvt.pdf · 0018-9545 (c) 2013 IEEE. Personal use is permitted, but