sliding mode controller and flatness based set-point ...€¦ · keywords: sliding mode control,...

Sliding Mode Controller and FlatnessBased Set-Point Generator for a Three

Wheeled Narrow Vehicle

Nestor Roqueiro ∗ Marcelo G. de Faria ∗∗

Enric Fossas Colet ∗∗∗

∗ Universidade Federal de Santa Catarina, Florianopolis, BR (e-mail:[email protected]).

∗∗ Universidade Federal de Santa Catarina, Florianopolis, BR (e-mail:[email protected]).

∗∗∗ Universitat Politecnica de Catalunya, Barcelona, ES (e-mail:[email protected]).

Abstract: This paper presents a sliding mode controller for a narrow tilting three wheeledvehicle. A dynamic model of eighteenth order is presented for simulation purposes only. Designand analysis are based on a simpler third order model of a bicycle. The validity of thissimplification is verified through simulations. Because of the flatness of the bicycle, trajectoriesgiven by a presumed driver are redefined to take into account the limits of the vehicle stability.Finally, tilting and speed controllers are designed in the frame of sliding modes. Analyticaland simulation results performed in the simpler and complex model respectively validate thesedesigns.

Keywords: Sliding Mode Control, Flatness, Set-point generator, Tilting Vehicle, NonlinearModels

1. INTRODUCTION

Nowadays, automobile companies are involved in the de-sign of more efficient vehicles improving the energeticefficiency and making them smaller for the best use ofthe existing roads and streets. For this end and in orderto lighten the traffic in the cities, the design of smallervehicles with a better weight/load ratio are reported inAshmore [2006], Johannsen and et al. [2003], Gehre et al.[2001], Brink and Kroonen [2004], Gohl et al. [2006]. Thereis also a project in development at the “UniversidadeFederal de Santa Catarina” (Brazil). Specifically the aimof the project is the design of a three wheeled narrowvehicle for two passengers (Vieira et al. [2009]). However,because of their narrowness, this kind of vehicles havestability problems. A stable behavior can be achieved byallowing the vehicle to lean in curves, like a motorcycle.This strategy has been used in several concept vehicles, forinstance Mercedes Benz F 300 Life-Jet, General MotorsLean Machine, BMW Clever and Simple and one produc-tion vehicle, Carver by VanderBrink.

Recent publications address different topics related tocontrol of such vehicles. In Kidane et al. [2008], the authorsapplied two different types of control schemes known assteering tilt control (STC) and direct tilt control (DTC).The stability control algorithm for tilting vehicles needs tobe developed in such a way that no special operating skills

? Nestor Roqueiro acknowledges CAPES (Brazil) for a post-docgrant (Proc. 090609-3) in the Universitat Politecnica de Catalunya.E.Fossas acknowledges the support of Spanish government researchprojects DPI2010-15110 and DPI2008-01408.

are required by the driver. It means the driver will usethe steering input to follow a certain desired trajectorywithout regard to the vehicle’s tilt stability. That workuses a feedforward plus PID controllers for tilt stabilizationand a driver model that defines a look-ahead error of thetrajectory.

In Defoort and Murakami [2009] and Nenner et al.[2008], the authors deal with the robust stabilization andtrajectory-tracking problems of a riderless bicycle. In thelast article, a dynamic model which takes into accountgeometric-stabilization mechanisms due to bicycle trail ispresented. A posture controller combining second-ordersliding mode control and disturbance observer is derived.Then, a tracking controller based on the proposed pos-ture controller and the dynamic-inversion framework isdesigned. The dynamical model of a bicycle presentedwill be adopted in this work as a simplified model of thetricycle.

Several publications have presented the idea of a virtualdriver who has the ability to follow a path without falling.Frezza and Beghi [2003] take the roll angle as control inputinstead of steering angle avoiding to deal directly with leaninstability. Path tracking is defined as an optimizationproblem. In Saccon et al. [2008] a dynamic inversionof a simplified motorcycle model is used, allowing thecomputation of state and input trajectories correspondingto a desired output trajectory. A stabilizing feedback isobtained by standard Linear Quadratic Regulator (LQR).

The central issue in S. Fuchshumer and Rittenschober[2005] is the discussion of the differential flatness of the

Preprints of the 18th IFAC World CongressMilano (Italy) August 28 - September 2, 2011

Copyright by theInternational Federation of Automatic Control (IFAC)

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planar holonomic bicycle model of a car. The bicycle modelis obtained from the four wheeled vehicle by mergingtogether the front and the rear wheels to a single (massless)front and rear wheel, respectively, located at the longitudi-nal axis of the car. The vehicle dynamics control design isaccomplished following the flatness based control theory.

Our goal is to design a controller able to tilt the vehiclethrough steering angle, taking into account the expectedlimits to avoid losses of tire grip. Furthermore, it shouldtrack the trajectory given by the driver through the steer-ing wheel as close as possible. Because it is a light vehicle,its mass changes significantly with full or partial load,shifting the center of gravity. Thus, a robust controllershould be used. The controller should be transparent tothe driver and reach the optimum tilt angle, improving thevehicle safety and its performance. The aim of this paper isto show how these objectives can be reached using slidingmode control for low level controllers (longitudinal speedand tilt angle).

First, we briefly present a model of a tricycle with ninedegrees of freedom that will represent the real system.Then, we propose a simple model of a bike that leans as anapproximation of three-wheeled vehicle for the project andanalysis of the controllers. Simulations allow us to ensurethat both models are similar enough to design controllersusing the simpler one. A flatness based approach fortrajectory tracking is presented and used to define theset points for low level controllers. Next section showsbriefly the project of sliding mode low level controllers.We present some simulations of closed loop system usingthe nine degrees of freedom model with and withoutperturbations. Finally, conclusions and some topics onfuture developments close the paper.

2. THE THREE WHEELED TILTABLE VEHICLE

This work starts from the tilting tricycle model of sixdegrees of freedom reported in Vieira et al. [2009]. Threeextra degrees of freedom were added to allow modelling in-dependent vertical movements of each wheel. In addition,lateral wind and road roughness were modelled as externaldisturbances.

The vehicle consists of a central mass (body 2) and threewheels, two located in the front (bodies 3 and 4) andone in the rear (body 1). The origin of the coordinatesystem is the contact point of the rear wheel with the roadand the clockwise convention is adopted (Figure 1). Thethree wheels are assumed to be always in contact with theground.

In this frame, the velocity of each of the masses is defined.It results in a model of nine degrees of freedom withpositions and velocities described as (see Figure 2):

• Longitudinal vehicle movement (x), x = u.• Transverse vehicle movement (y), y = v.• Rear wheel vertical movement (z1), z1 = w1.

• Main body vertical movement (z2), z2 = w2 , w.• Right side front wheel vertical movement (z3), z3 =w3.• Left side front wheel vertical movement (z4), z4 = w4.• Rotation with respect to Z axis (ψ).• Rotation with respect to X axis (φ).

Fig. 1. Inertial coordinate system

• Rotation of the body 2 with respect to an axis parallelto Y axis (θ). This is due to a movement of body 2(not punctual) with respect to its own rotation centerwhich will be considered to be placed in its masscenter. The rotation around its transverse axis causesa movement that interferes with the behaviour of thefront and rear suspensions.

In this way the vertical movements of all of the bodiesare taken into account independently, allowing to simulatesuspensions. Vehicle rotation with respect to Y axis (α)will be modelled by a variable parameter in velocitiesformulation. The variable α represents the inclination ofthe road.

Fig. 2. Velocities definition for mass 1

To create the dynamical model, a multi-body analysisapproach was chosen, following two steps to achieve thefinal model. They are:

• Velocity equations: They represent the way that ve-locities works on bodies in 9 degree of freedom. Theseequations are based on vehicle velocity and representthe different components for each body.

• Equation of motion: Building equations to kinetic andpotential energy based on the previously definition,we can define a Lagrangian Equation of Motion, thatdescribes a differential equation set.

The basic geometry of body 2 is depicted in Figure 3 andgeometric definitions of bodies 3 and 4 are depicted inFigure 4.

The full model of the vehicle was presented in Roqueiroand Fossas [2010]. As in Leal et al. [2008], using theLagrangian formulation, a second order vector dynamicmodel can be written as:

M(q)q(t) + Cq(t) + K(q)q(t) = F(q, q, t) (1)

where M(q), C and K(q) are respectively the inertia,damping and stiffness matrices, q is the freedom de-grees vector defined by q = {x, y, z1, z2, z3, z4, ψ, φ, θ} and


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Fig. 3. Geometric definitions of mass 2

Fig. 4. Geometric definitions of masses 3 and 4

F(q, q, t) the vector of external forces and momenta actingon the vehicle.

3. TILTABLE BICYCLE

The bicycle models presented in Getz [1995] and Frezzaand Beghi [2003] begin with a description of a simpli-fied kinematic model of a car considering the wheels inits medium plane. Second, it incorporates the dynamiccharacteristics of the inverted pendulum. Thirdly, the tra-jectory problem is solved with kinematic model. Then,a viable solution to avoid the bicycle falling is obtainedthrough an optimization process. In this article, dynamicsmodel is defined instead of the kinematics one prior tosolve the reference’s tracking problem.

When riding a non-autonomous tilting tricycle, which isour bicycle-like vehicle case study, the desired trajectorygiven by the driver should be followed through the steeringwheel and the accelerator pedal. The stable tilting dy-namic behavior of the vehicle have priority over trajectoryand speed. This means that if some reference should berelaxed, first it must be the speed in order to meet the tra-jectory and angle of inclination. Failure to meet these twospecifications means that the reference trajectory shouldbe relaxed in order to prevent vehicle falling. Through thesteering wheel the driver sets the desired values of frontwheels angle δd(t) and with the accelerator pedal the driversets the vehicle velocity ud(t). So, we model the tiltingbehavior and then we will project a controller in order todrive as close as possible to the desired trajectory. We usea simple model for a bike that uses Newton’s second lawto model the tilting dynamics as:

φ(t) = g sin(φ(t)) +cos(φ(t))

a4u2(t)δ(t) (2)

It is just a force balance between gravitational momentumand fictitious inertial momentum, with g the gravitationalacceleration, u the longitudinal speed of the vehicle a4the vehicle’s lenght and δ the steering angle. The angle

φ is restricted by the road grip to a maximum value|φ (t)| < φmax.

For x − axis longitudinal movement, the model can bewritten as:

mu(t) =2nTm(t) ηT

d− 1

2Cxu2(t)Aρ (3)

It is a force balance which considers the thrust and thedrag force of wind, where n is the transmission reduction,Tm the traction torque, d the wheel diameter, ηT the trans-mission efficiency, Cx the aerodynamic drag coefficient forlongitudinal flow, A the vehicle’s frontal area and ρ the airdensity.

A comparison between the two models used in this work ispresented below. The controller project, including analysisof closed loop stability, is facilitated using the simplifiedmodel. Obviously, some dynamic behaviours can not bedescribed. In this case we are interested in controlling twovariables directly, the leaning angle φ and the speed u.Then, a simple model considering the dynamics of this twovariables could be enough. Equations (2) and (3) are takenas a simplified model. The other objective is to follow atrajectory given by the driver. The trajectory is definedby the velocity ud and the steering angle δd as a firstapproximation. A comparison between those two modelsgives a qualitative and perhaps quantitative informationon the behaviour of the models adopted. The followingfigures show some comparative results.

In the simulation shown in the Figure 5 we used the samedriver inputs for δd(0.001rad) and velocity ud(26m/s).The speed control is done with a Sliding Mode controller(Roqueiro et al. [2010]), and tilt control is done with PIDwith gain schedule. We note that the dynamic behaviour issimilar with the same steady states values. The oscillationsseen in the dynamic response of the bicycle and that donot appears in tricycle’s response are due to interactionsbetween the front wheels and the central body.

0 2 4 6 8 10 12 14 16 18 20−3

−2

−1

0

1

2

3

4Tilting Angle Comparison − Closed Loop

Time (s)

Tilti

ng

(D

eg

rees)

Tricycle

Bicycle

Fig. 5. Tilting angle comparison between tricycle andbicycle models at constant speed.

A simulation with system in open loop for speed and PIDGain Schedule controller for tilt angle is shown in Figure6. Simultaneous changes were implemented in the speedand steering angle in order to test the effects of coupling.

The conclusion is that this model can be used successfullyfor project control and analysis of the dynamic propertiesof the closed loop system.

Any controller based on the bicycle model ought to dealwith strong oscillations in transient. So, if it succeed, it


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0 2 4 6 8 10 12 14 16 18 20−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4Tilting Angle Comparison − Open Loop

Time (s)

Tilti

ng

(D

eg

rees)

Tricycle

Bicycle

Fig. 6. Tilting angle comparison between tricycle andbicycle models, changing speed.

will show better performance in the three-wheeled vehicle.A similar approach to the design of controllers through theuse of simplified models of two wheeled vehicles is used inKidane et al. [2007], Kidane et al. [2008].

4. THE TRAJECTORY PLANNING PROBLEM

From current and targeted steering angle and linear veloc-ity values provided by the driver, a continuous referencetrajectory is planned. The system given by equations (2)and (3) is flat and the tilting angle and the linear velocityare flat outputs. Thus, from the driver’s goal values, atilting target is computed and reference trajectories forthe tilting angle and the linear velocity are planned.

We say that a tilting target φk+1 is feasible if |φk+1| ≤φmax. Hence, using equation (2), we define the tiltingtarget as:

φk+1 = sign(δk+1) max

{∣∣∣∣arctan

(u2k+1 δk+1

a4 g

)∣∣∣∣ , φmax

}If φk+1 = φmax, uk+1 is calculated again from equation (2)so that δk+1 can be reached. Thus, from φk, φk+1, uk anduk+1, the reference trajectories are defined by:

φk+1k (t) = φk + (φk+1 − φk)

(t− tk)

(tk+1 − tk)(4)

uk+1k (t) = uk + (uk+1 − uk)

(t− tk)

(tk+1 − tk)(5)

Remark that these trajectories yield uniform rectilinearmovements (second derivatives are zero). Furthermore,feedfoward controller can be obtained from equations (2)and (5) together with the actuator dynamics.

There are some pre-established conditions that agree tothe solution of the trajectory problem. Namely, if thetarget steering angle and linear velocity yield a non feasibletarget tilting angle, the linear velocity must be reduced sothat the target steering angle is kept.

5. STATEMENT OF THE CONTROL PROBLEM

In four wheeled vehicles, the rotation with respect to the Xaxis (angle φ, also called rolling) is not desired. However,two wheeled vehicles (i.e. bikes and motorcycles) takebenefit of rolling angle to compensate fictitious inertialforce effects when cornering, improving the stability and

the performance. The bearing angle φ can be dynamicallyadjusted (controlled) through a tilting mechanism. Oneof the aims of this work is to carry out a controller suchthat, acting on the steering wheels, it corrects the tiltingangle φ in a way that forces to the wheel’s plane arecancelled. The desired tilting angle φd is given in equation(4). Furthermore, there is a second control objective: totrack the forward velocity.

5.1 Sliding mode control

In Roqueiro et al. [2010] and Moreira et al. [2009], sin-gle input controllers were designed for the tricycle. Theselected system input was the front wheels steering angle,and the selected output was the bodywork tilting angle. Inthis paper, a two-input two-output sliding mode controllerbased in the model described in section 3 is reported. Theinput variables are the front wheels steering angle δ andthe motor torque Tm. The output variables are: φ, thevehicle tilting angle and u, the vehicle speed.

5.2 Sliding surfaces

The outputs to be regulated to zero are the errors in theoutput variables. Namely, zero tilting error and zero speederror in the X axis with relative degrees two and one,respectively. Then, the following sliding functions

S1 = τd

dt(φd − φ) + (φd − φ) (6)

S2 = ud − u (7)

are selected. The closed loop system is input-output de-coupled. Input δ manages S1 while the input Tm managesS2. Indeed, the Jacobian of the switching function is a2×2-matrix with zeroes in the diagonal. Then, the controlaction defined by

δ =

{−k1 if S1 > 0,k1 if S1 < 0,

(8)

Tm =

{−k2 if S2 > 0,k2 if S2 < 0,

(9)

yields (S1, S2) = (0, 0), provided that |Tm,eq| < k2 and

|δeq| < k1. Hence, u = ud and τ ddt (φd − φ) = (φd − φ)in the subset of the intersection {x | S1 = 0 ∧ S2 = 0}defined by the preceding inequalities.

Naturally, nobody expects a bang-bang action for inputsTm and δ that are continuous functions. However, discon-tinuous control actions can be designed for the correspond-ing actuators (two DC-motors with torque and position asoutputs). In the Laplace domain, they can be modelled as

GTm(s) =b

s+ 1(10)

Gδ(s) =a

s(a s+ 1)(11)

Taking the actuators into account, two new switchingfunctions S1 and S2 have to be considered.

S1 = p1d

dt(φd − φ) (12)

S2 = p2d

dt(ud − u) (13)


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where pi(ddt

)for i = 1, 2 are polynomials in the variable

ddt of degrees 2 and 3 respectively. Furthermore, the couple

(S1, S2) has a well defined vector relative degree (4, 2), but

the new input-output system (v1, v2; S1, S2) is no longerdecoupled. Hence, the discontinuous control is designedfor the new input variables(

v1v2

)= ∇SG

(w1

w2

)(14)

where G is the input matrix. See details in the appendix.

Thus, the controllers

vi =Vi,m + Vi,M

2+V1,m − V1,M

2sign(Si), for i = 1, 2

locally force S1 = 0 and S2 = 0 in finite time. These equal-ities, in turn, yield z1 = z1,d and z5 = z5,d asymptotically.

In the original input variables the controllers result in:

w1 =l(V1,m+V1,M

2 +(V1,m−V1,M )sign(S1)

2

)cos(z1) · z25

−

2 · z3(V2,m+V2,M

2 −(V2,m−V2,m

2

)sign(S2)

)z5

w2 =

V2,m+V2,M

2 +(V2,m−V2,M

2

)sign(S2)

α · b

(15)

As for the discontinuous gains V1,m, V1,M , V2,m, V2,M ,they will define the sliding domain which must includethe target dynamics. There is a constrain on these valuesgiven by the power of the traction motor. However, thegain corresponding to the steering angle can be adjustedby appropriate gears.

5.3 Tracking a reference

First the model is simulated in extreme conditions withoutperturbations in order to observe the ideal sliding dynam-ics. The selected velocities are 4m/s, 8.3m/s and 30m/sto reflect maximum and minimum velocities and an inter-mediate cruising speed in town. The simulation consistsin turning the steering wheel to the driver’s desired angle(±0.3rad, ±0.07rad and ±0.003rad) for the three selectedvelocities and then going back to the rest position.

Table 1 shows the Root Mean Square (RMS) error and themax absolute error achieved in this simulation.

Table 1. Simulation results.

30 m/s 8.3 m/s 4 m/s

RMS Error (rad) 0.0024 0.0047 0.0054

Max. Abs. Error (rad) 0.0243 0.0475 0.0531

5.4 Tracking a reference in the perturbed model

In the second simulation, wind perturbation and roadroughness will be considered. Specifically, a lateral windof 15m/s in both directions for high speed (at simulationtime 30 and 50, ceasing at 70 seconds), 10m/s for cruisingspeed (at simulation time 130, 150 and ceasing at 70seconds) and 2.5m/s (at simulation time 230, 250 andceasing at 270 seconds) for low speed. The speed changes

occurs at time 90 seconds (from 30 to 8.3 m/s) and time190 (from 8.3 to 4 m/s). The road roughness perturbationis modelled by means of a random signal of frequency0.2Hz and width 0.5cm, generating a vertical force in eachwheel.

Figure 7 shows the system’s performance under tiltingcontrol. It is worth to remark that the velocity controlsystem is not significantly affected by this perturbation.

Figure 8 shows the steering (control action) performed bythe vehicle to achieve the tracking of the reference tiltingangle.

0 50 100 150 200 250 300−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time (s)

Angle

(ra

d)

Tilting

Reference

Fig. 7. Reference and tilting angle under wind and roaddisturbance.

0 50 100 150 200 250 300−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Time (s)

Angle

(ra

d)

Steering

Fig. 8. Steering angle (control action).

The road roughness has small influence over the system’sresponse. However, strong lateral wind could cause majortilting peaks, which are quickly rejected by the controller.Table 2 shows the error results for this simulation.

Table 2. Simulation with disturbances.

30 m/s 8.3 m/s 4 m/s

RMS Error (rad) 0.0388 0.0066 0.0054

Max. Abs. Error (rad) 0.3692 0.0475 0.0533

6. CONCLUSSIONS AND FURTHER RESEARCH

Simulation results show that a three dimension flat modelof a bicycle is sufficient to design sliding mode controllersfor a three wheeled narrow vehicle. The controllers yielda good performance of the eighteenth order system even


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when it is submitted to lateral wind perturbations. Whenlinear actuators are included into the model, the systemresults in a two-input two-output coupled system withwell-defined vector relative degree. It has been decoupledfrom a change of variables in the inputs for control designpurposes. The improvement of the model including tire-road effects is left as a further research.

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APPENDIX

The full model: bicycle plus actuators is a system of ODEin R6. Let z = (z1, z2, z3, z4, z5, z6) and w = (w1, w2) bethe state and the input variables respectively. z1 = φ thetilting angle, z3 = δ the steering angle, z4 = u the forwardvelocity and z6 = Tm the traction torque. w1 and w2 arethe actuator inputs. The vector field that describes thedynamics is

dz

dt= f(z) +G ·wT (16)

where

f(z) =

z2

g · sin(z1)− cos(z1)

l· z3 · z25

z4

−1

a· z4, α · z6 − β · z25

−z6

G =

0 00 00 01 00 00 b

It is straightforward to check that the vector relativedegree of (z1 − z1d, z5 − z5d) is well defined and it is equalto (4, 2). Thus, the switching functions can be defined

by S1 = p1

(d

dt

)(z1 − z1d), S2 = p2

(d

dt

)(z5 − z5d),

where p1(ddt

)and p2

(ddt

)are Hurwitz polynomials in the

derivative with respect to time of degrees three and onerespectively. Since these polynomials are designed by theuser, the corresponding poles can be placed appropriatelyso that higher derivatives could not be beard in mind.

Remember that the system is input-output decoupledwhen δ and Tm are the inputs. However, when the actu-ators dynamics are considered this is no longer true. Onegets,

∂S

∂z·G =

(cos(z1) · z25

l

2 cos(z1) · z3 · z5 · α · bl

0 α · b

)Thus, one proceed as usual in order to simplify the controldesign and a new set of inputs v = (v1, v2) is considered;

namely, vT = ∂S∂z · G · w

T . Let us assume discontinuousgains Vi,m < Vi,M so that Vi,m ≤ vi,eq ≤ Vi,M , then

vi =Vi,m + Vi,M

2+V1,m − V1,M

2sign(Si), for i = 1, 2

locally force S1 = 0 and S2 = 0 in finite time, which, inturn yield z1 = z1,d and z5 = z5,d asymptotically.


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sliding mode controller and flatness based set-point ...€¦ · keywords: sliding mode control,...

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