Gradient based adaptive trajectory tracking control … based adaptive trajectory tracking control for mobile robots Dinko Osmankovic´ University of Sarajevo Faculty of Electrical

2013 XXIV International Conference on Information, Communication and Automation Technologies (ICAT)October 30 – November 01, 2013, Sarajevo, Bosnia and Herzegovina

978-1-4799-0431-0/13/$31.00 ©2013 IEEE

Gradient based adaptive trajectory tracking control

for mobile robots

Dinko Osmankovic

University of Sarajevo

Faculty of Electrical Engineering

71000 Sarajevo, Bosnia&Herzegovina

email: [email protected]

Jasmin Velagic

University of Sarajevo

Faculty of Electrical Engineering

71000 Sarajevo, Bosnia&Herzegovina

email: [email protected]

Abstract—This paper presents a classical approach to modelreference adaptive control for trajectory tracking problem. Gra-dient based or MIT rule adaptation technique was applied to thewell known trajectory tracking controller and the mathematicalmodel of that adaptation is presented. The proposed solutionis compared to the controller without adaptation and to thefeedback linearisation based controller. In both simulation andexperimental environments the effectiveness of the proposedadaptive controller is shown and its use justified.

I. INTRODUCTION

Mobile robots are increasingly used in various fields ofindustry, service, surveillance etc. Several types of configu-rations (wheel number and type, actuation, single- or multi-body vehicle structure) are in use [1]. The most common forsingle-body robots are unicycles (e.g. differential drive mobilerobots), tricycles or Ackermann drive, and omnidirectionalrobots [2].

Trajectory tracking control is one of the most popularaspects of mobile robotics. There are many proposed con-trol structures for all kinds of mobile robots. Most of theseapproaches employ the kinematic model of the mobile robot,while some of them also include robot dynamics into controllerdesign. Although the problem of controlling certain classesof non-holonomic systems is virtually solved in theory, thebiggest problem that remains up to this day is transientperformance [3].

There are several adaptation schemes proposed. In [4]the velocities are generated using the kinematic model ofthe mobile platform. On the other hand, controller based onkinematic model and the backstepping method is proposed in[5]. In [6] the proposed algorithm is based on guidance schemebased on Lyapunov Guidance Control while [7] proposes MPCapproach. Feedback linearisation controller is presented in [3].Also, several fuzzy approaches to adaptive trajectory trackingproblem are proposed [8], [9], [10].

All of these methods try to solve the the problem oftransient performance. In [4] and [5] they include the dynamicmodel of WMR to obtain good transient performance, andthe stability proof is presented. The inclusion of the dynamicmodel increases complexity of controller design. Feedbacklinearisation method offer robust design of the trajectorytracking control, which gives very good performance, but alsohas increased design complexity. On the other hand, fuzzyapproach has no stability proof, but they produce very good

results in terms of transient performance and overall trajectorytracking.

In this paper we focus to more classical approach for thetrajectory tracking problem and propose the gradient basedmodel reference control approach to the trajectory trackingproblem which employs the MIT rule. We present the mathe-matical model of the adaptation and we try to justify its useby comparing it to the controller without adaptation and thefeedback linearisation based controller [3].

The paper is organised as follows. In section II we de-scribe the methodology behind our proposed solution and thenpresent the kinematic model, the trajectory tracking controllerand adaptation mechanism. Section III demonstrates the resultsobtained by our adaptive controller compared to its non-adaptive counterpart as well as the feedback linearisation basedcontroller. The paper concludes with some remarks about theproposed solution and some guidelines for further research aregiven in the last section.

II. METHODOLOGY

In this section we consider the methodology behind thegradient based adaptation mechanism for trajectory trackingcontrol.

First, we present the kinematic model of the mobile robot.The mobile robot used in this paper is the Pioneer P3-DX withdifferential drive. The same model is designed and created inthe ROS Stage simulator for the simulation purpose.

Fig. 1. Kinematic model of differential drive mobile robot

If we assume that robot R has velocity vR, generalizedcoordinates can be written as q = [xR yR θR]

T and thefollowing constraint can be applied:

xR · sin(θ)− yR · cos(θ) = 0 (1)

The full kinematic model of WMR becomes [11]:

xy

θ

=

[

cos(θ) −a sin(θ)sin(θ) a cos(θ)

0 1

]

[

vRωR

]

(2)

where

J =

[

cos(θ) −a sin(θ)sin(θ) a cos(θ)

0 1

]

(3)

is a Jacobian matrix and a is a displacement betweencentral point between wheels P0 and the actual robot’s centreof rotation Pr. In the case of Pioneer P3-DX its value isa = 10cm = 0.1m. If h = [x y]T we define the errorsas:

h = hd − h

θ = θd − θ. (4)

In controller design we require that the kinematic controllerminimizes the position and orientation errors, i.e. lim

t→+∞

ρ(t) =

0 and limt→+∞

θ(t) = 0 where ρ(t) is Euclidean distance between

robot and desired point at time instance t. This is satisfied if

limt→+∞

h = 0 and limt→+∞

θ(t) = 0.

The main problem in the kinematic controller design is tofind manoeuvrability vector vc to achieve the desired opera-tional motion. The proposed kinematic controller is describedwith the following equation [4]:

vc = J−1(

vd + Ltanh(

L−1Kh))

(5)

where vd is desired velocity vector, vc is command velocity,and L and K are positive-definite diagonal control weightmatrices. In the case of differential drive WMR this matrixequation becomes:

[

vcωc

]

=

[

cos(θ) sin(θ)

−1

asin(θ)

1

acos(θ)

]

xd + Ix tanh

(

kx

Ixx

)

yd + Iy tanh

(

ky

Iyy

)

(6)

where x = xd − xR and y = yd − yR, kx and ky arecontroller gains and Ix and Iy are saturation constants, while(xd, yd) is a desired position. This controller is asymptoticallyLyapunov stable [4].

A. Gradient based adaptive controller

We propose the MIT rule based adaptation for the kine-matic controller for the trajectory tracking. Let the criterionbe:

J =1

2e2. (7)

We choose to adapt parameters kx and ky of the kinematiccontroller in order to minimize the position error of the mobilerobot. Therefore:

kx = −γxex∂ex

∂kx

ky = −γyey∂ey

∂ky. (8)

where γx and γy are adaptation rates for x and y robot po-sitions. The desired behaviour of the system with the controllercan be described by the following equations:

xm = −a1xm + b1xd

ym = −a2ym + b2yd. (9)

Expressing the errors ex and ey in terms of x and y andsubstituting (9) into it we get the following equations:

∂ex

∂kx= −

Ix

k2xtanh−1(lx)

∂ey

∂ky= −

Iy

k2ytanh−1(ly) (10)

where lx =x− xm − 2xd

Ixand ly =

y − ym − 2ydIy

. Since

these values are arguments of atanh function, they need to beproperly scaled so the value of the values in previous equationsdo not quickly grow (to infinity). This can be achieved byincreasing the values of Ix and Iy . This, for now, has to bedone empirically.

Since we are using ROS, we need to discretise (8) whichdescribes controller adaptation. For that, we propose finitedifference method. The values for controller gains for t + 1are:

kx(t+ 1) = −Tγxex∂ex

∂kx+ kx(t)

ky(t+ 1) = −Tγyey∂ey

∂ky+ ky(t) (11)

where T is sample time of the controller (standard valuefor T is 0.1 s).

The equations (11) describe the adaptation of controllergains kx and ky of the controller described by equation 6.

In the next section we present the simulation results ob-tained with ROS Stage simulator. We also compare this tosome other types of adaptive and robust methods for trajectorytracking.

III. SIMULATION RESULTS

In this section we present the simulation results of theproposed adaptive kinematic controller for the trajectory track-ing mobile robot. We also compare the proposed adaptivecontroller with the controller without adaptation, and thefeedback linearisation controller presented in [3] described bythe following equation:

v =vd cos(θd − θ) + k1((xd − x) cos θ + (yd − y) sin θ)

ω =ωd + k2sign(vd) · ((yd − y) cos θ − (xd − x) sin θ)+

+ k3(θd − θ) (12)

with recommended values for gains:

k1 = k3 = 2ζ√

ω2d(t) + bv2d(t), k2 = b|vd(t)|,

ζ ∈ (0, 1) ∧ b > 0. (13)

Three typical tests are run. We have chosen standard curvesused in testing the trajectory tracking control:

1) circular trajectory (low complexity)2) lemniscate trajectory (average complexity)3) lissajous trajectory (high complexity).

For all tests we used the following values for control andadaptation parameters explained in section II:

• a1 = a2 = 10,

• b1 = b2 = 1,

• Ix = Iy = 25,

• γ = 0.98,

• T = 0.1s,

• kx0 = ky0 = 0.1.

Also, for the safety reasons, the linear and angular veloci-

ties are limited to |vR| ≤ 1m

sand |ωR| ≤ 0.6

rad

s.

The size of the grid box on all figures presenting trajecto-ries from ROS Rviz tool is 0.1× 0.1m.

A. Circular trajectory

In this test, the robot is trying to follow the circulartrajectory. In the Fig. 2 the resulting trajectories generated bypreviously mentioned three controller types are shown.

It can be seen that the proposed adaptation is justified.Although all controllers have good tracking ability, adaptivecontroller and feedback linearisation controller produce bettercontrol when error is large (i.e. at start when robot is distantfrom the trajectory).

0 5 10 15 20 25

0

0.5

1

1.5

2

t(s)

err

or

for

positio

n x

(m

)

non adaptive

adaptive

feedback linearisation

0 5 10 15 20 25

0

0.5

1

t(s)

err

or

for

positio

n y

(m

)

non adaptive

adaptive


Fig. 3. Positional errors for the considered controllers for circular trajectorytest

0 5 10 15 20 25

−0.5

0

0.5

1

1.5

2

t(s)

linear

velo

city (

m/s

)

non adaptive

adaptive


0 5 10 15 20 25

−0.5

0

0.5

1

1.5

2

2.5

t(s)

angula

r velo

city (

rad/s

)

non adaptive

adaptive


Fig. 4. Linear and angular velocities for the considered controllers for circulartrajectory test

In Figs. 3 - 5 the position errors, velocity signals, andgains of the considered controllers are illustrated. From thesefigures we can also see the justification of using gradient basedadaptation. But, the pay-off is the velocity signal which hasworsen compared to the controller without adaptation. But it’sstill superior to the feedback linearisation based controller.

B. Lemniscate trajectory

In this test we generate a more complex trajectory. In Fig.6 the resulting trajectories generated by previously mentionedthree controller types are presented. In this case the adaptivecontroller also produces superior results compared to thecontroller without adaptation [4] and feedback linearisationcontroller with parameters proposed in [3].

In Figs. 7 - 9 the position errors, velocity signals, and gainsof the considered controllers are shown.

Similar conclusion can be drawn as in the previous test. Theadaptive controller produces better tracking but at the expenseof somewhat worse velocity signals.

(a) Trajectory generated by the controllerwithout adaptation

(b) Trajectory generated by the controllerwith adaptation

(c) Trajectory generated by the feedbacklinearisation controller

Fig. 2. Trajectory tracking controller comparison for circular trajectory




Fig. 6. Trajectory tracking controller comparison for lemniscate trajectory

0 5 10 15 20 250

1

2

3

4

5

t(s)

kX

non adaptive

adaptive

0 5 10 15 20 250

1

2

3

4

5

t(s)

kY

non adaptive

adaptive

Fig. 5. Controller gains for circular trajectory test

C. Lissajous trajectory

In this test we generate a very complex Lissajous trajectory.In Fig. 10 the resulting trajectories generated by previouslymentioned three controller types are presented. In this casethe adaptive controller also produces superior results comparedto the controller without adaptation and feedback linearisationcontroller.

In Figs. 11 - 13 the position errors, velocity signals, andgains of the considered controllers are shown.

Similar conclusion can be drawn as in the previous test.The adaptive controller produces better tracking, but at theexpense of somewhat worse velocity signals.

0 5 10 15 20 25

0

0.5

1

1.5

2

t(s)

err

or

for

positio

n x

(m

)

non adaptive

adaptive


0 5 10 15 20 25

0

0.5

1

t(s)

err

or

for

positio

n y

(m

)

non adaptive

adaptive


Fig. 7. Positional errors for the considered controllers for lemniscatetrajectory test

D. Experimental validation

Since the code is written for ROS, the same code isrun for the experimental setup with Adept Pioneer P3-DXmobile robot shown in Fig. 14. In this case, the velocitycommands are sent directly to the robot driver instead of ROSStage simulator. Since the interface has the same architecture(geometry msgs/Twist) there was no need to change the codein order to run it on the mobile platform.

Results obtained during the experimental validation drawto the same conclusion as in the simulation environment(although the simulation was configured to mimic the realenvironment as much as possible, e.g. precise robot model,slippage etc.). Since the running code is the same and the




Fig. 10. Trajectory tracking controller comparison for Lissajous trajectory

0 5 10 15 20 25

−0.5

0

0.5

1

1.5

2

t(s)

linear

velo

city (

m/s

)

non adaptive

adaptive


0 5 10 15 20 25

−0.5

0

0.5

1

1.5

2

2.5

t(s)

angula

r velo

city (

rad/s

)

non adaptive

adaptive


Fig. 8. Linear and angular velocities for the considered controllers forlemniscate trajectory test

0 5 10 15 20 250

1

2

3

4

5

t(s)

kX

non adaptive

adaptive

0 5 10 15 20 250

1

2

3

4

5

t(s)

kY

non adaptive

adaptive

Fig. 9. Controller gains for lemniscate trajectory test

simulation environments mimics the real one, the differencesare quite small as presented in Fig. 15.

IV. CONCLUSION

In this paper we proposed a gradient based technique(MIT rule based adaptation) for controller adaptation. Forthe kinematic model of the differential drive mobile robot,we developed a adaptation mechanism for trajectory tracking

0 5 10 15 20 25

0

0.5

1

1.5

2

t(s)

err

or

for

positio

n x

(m

)

non adaptive

adaptive


0 5 10 15 20 25

0

0.5

1

t(s)

err

or

for

positio

n y

(m

)

non adaptive

adaptive


Fig. 11. Positional errors for the considered controllers for lissajous trajectorytest

0 5 10 15 20 25

−0.5

0

0.5

1

1.5

2

t(s)

linear

velo

city (

m/s

)

non adaptive

adaptive


0 5 10 15 20 25

−0.5

0

0.5

1

1.5

2

2.5

t(s)

angula

r velo

city (

rad/s

)

non adaptive

adaptive


Fig. 12. Linear and angular velocities for the considered controllers forlissajous trajectory test

that is shown to be stable and robust even for very complextrajectory curves.

The proposed solution is tested on three different scenarios:circular, lemniscate and Lissajous tracking trajectory. In allthree scenarios it can be concluded that the adaptation mech-anism for the considered controller is justified. It producesbetter results than the non-adaptive controller and feedback

0 5 10 15 20 250

1

2

3

4

5

t(s)

kX

non adaptive

adaptive

0 5 10 15 20 250

1

2

3

4

5

t(s)

kY

non adaptive

adaptive

Fig. 13. Controller gains for lissajous trajectory test

Fig. 14. Mobile platform Pioneer P3-DX

(a) Trajectory generated in simulationenvironment

(b) Trajectory generated in experimen-tal environment

Fig. 15. Trajectory tracking controller comparison for simulated andexperimental environments

linearisation based controller.

The biggest drawback of using this approach is that con-troller produces somewhat worse velocity commands so thefuture work will deal with this problem.

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Documents

Gradient based adaptive trajectory tracking control … based adaptive trajectory tracking control for mobile robots Dinko Osmankovic´ University of Sarajevo Faculty of Electrical