Cisda2014 Minh

8/17/2019 Cisda2014 Minh

1/27

Control of a mobile agent using

only bearing measurements in triangular region

Minh Hoang Trinh 1, Kwang-Kyo Oh 2 and Hyo-Sung Ahn 1

1Distributed Control and Autonomous Systems Laboratory (DCASL),School of Mechatronics, Gwangju Institute of Science and Technology (GIST)

Gwangju, Republic of Korea2Automotive Components and Materials R&BD Group,

Korea Institute of Industrial Technology, Gwangju, Republic of Korea

IEEE CISDADecember 15, 2014

Control of a mobile agent using only bearing measurements in triangular region CISDA 2014 1 /27

http://find/

8/17/2019 Cisda2014 Minh

2/27

Outline

1 Introduction

2 Preliminaries and problem formulation

3 The proposed control law and stability analysis

4 Simulation and hardware experiment

5 Conclusion


http://find/

8/17/2019 Cisda2014 Minh

3/27

Outline

1 Introduction

MotivationLiterature review

2 Preliminaries and problem formulationPreliminariesProblem formulation

3 The proposed control law and stability analysisProposed control lawStability Analysis

4 Simulation and hardware experimentSimulationHardware experiment with quadrotors

5 ConclusionReferences


http://find/

8/17/2019 Cisda2014 Minh

4/27

Introduction

Robot navigation: the ability to determine its own position in itsframe of reference and then plan towards some goal location.

self-localization, path planning, map-building and map interpretation.

Landmark-based navigation: detect the landmark (mostly from optical sensors), compute relative location with landmarks (relative distances, bearing

angles), control law to reach desired location.


http://find/

8/17/2019 Cisda2014 Minh

5/27

Motivation

Bearing-only navigation algorithms:

Bio-inspiration: insects’ eyes can obtain good angle but poor rangeinformation1

Safety: a reserve solution when range sensors are malfunctioned

Economics: reduce sensors in the large systems

1R. Wehner, “Desert ant navigation: how miniature brains solve complex tasks”, In

Journal of Comparative Physiology A, 189(8), 2003, pp. 579–588.Control of a mobile agent using only bearing measurements in triangular region CISDA 2014 5 /27

http://find/

8/17/2019 Cisda2014 Minh

6/27

Literature review

Bearing-only navigation:

using directly bearings for navigation McLeman (2002): ”visual landmarks navigation” tactics in ants. Bekris et. al. (2004): “moving-toward-bisector” strategy (without

proof ). Loizou and Kumar (2007): a bearing-only control law with three

beacons (the mobile agent needs global frame information).

estimating distance from bearings to navigate M. Ye et. al. (2013): multi-agent self-localization

Deghat et. al. (2014): simultaneously estimate distance andcircumnavigation.


http://find/

8/17/2019 Cisda2014 Minh

7/27

Outline

1 Introduction

MotivationLiterature review




SimulationHardware experiment with quadrotors



http://find/http://goback/

8/17/2019 Cisda2014 Minh

8/27

Assumptions

Figure 1: The agent is inside the triangleA1,A2,A3.

Assumption 1

The agent’s initial position is inside the triangle formed by the

three stationary beacons and is not co-located with any beacon’s position.

Assumption 2

The agent measures the bearing angles β k , 0 ≤ β k

8/17/2019 Cisda2014 Minh

9/27

Bearing and bearing vector

Figure 2: The agent measures the bearings w.r.t. beacons A1,A2,A3.

Definition: The bearing vector

û k := p Ak − p

p Ak − p =

p k

p k = 1∠β k . (1)

where k ∈ 1, 2, 3 and 1 is x -axis unit vector in the agent’s local frame.


http://find/

8/17/2019 Cisda2014 Minh

10/27

The subtended bearing

(a) Case 1: α3 = ϑ3 (b) Case 2: α3 = 2π − ϑ3

Let ϑk = |β k −1 − β k +1|, 0 ≤ ϑk

8/17/2019 Cisda2014 Minh

11/27

Assumptions

Assumption 3

The desired location p ∗ is inside the triangular A1,A2,A3. The agent knows the subtended bearing angles α∗1, α

∗

2, α∗

3 at desired location, whichsatisfy

3

k =1

α∗

k = 2π, (3a)

Ak < α∗k ≤ π. (3b)The single-integrator dynamics is used to model the agent:

ṗ = u ,

where p , u ∈ R2 are the position of the agent and the control inputrespectively.


http://find/

8/17/2019 Cisda2014 Minh

12/27

Problem formulation and proposed control law

Figure 4: p ∗ is the desired position where three subtended bearing angles are α∗1 , α∗

2 , α∗

3 .

Problem 1

Under Assumptions 1-3, design a control law for the agent to reach to its desired location asymptotically.


http://goforward/http://find/http://goback/

8/17/2019 Cisda2014 Minh

13/27

Outline

1 IntroductionMotivationLiterature review








8/17/2019 Cisda2014 Minh

14/27

Proposed Control Law

Proposed control law using bearing-only measurements

ṗ = u = u 1 + u 2 + u 3, (4)

where

u 1 = k u (α1 − α∗

1)û 1 = k u e 1û 1

u 2 = k u (α2 − α∗

2)û 2 = k u e 2û 2

u 3 = k u (α3 − α∗

3)û 3 = k u e 3û 3,

and e k = αk − α∗

k , k ∈ {1, 2, 3}: the subtended bearing error.



8/17/2019 Cisda2014 Minh

15/27

Stability Analysis

Lemma 2Under the control law ( 4 ), the agent will never escape from the triangle A1A2A3 if it is initially positioned inside that region.

Figure 5: Illustration of Lemma 2 ’s proof.

Proof.

Consider a case when the agent ison the side A2A3. Since û 2 = −û 3and e 1 = π − α1 > 0, u 1 drives the

agent into the triangle.Other cases can be treatedsimilarly.


http://find/

8/17/2019 Cisda2014 Minh

16/27

Stability Analysis

Figure 6: The unique equilibrium pointinside the triangle.

Lemma 3

There is a unique point inside the triangle A1,A2,A3 satisfying all three

subtended angles α∗1, α∗

2, α∗

3 in the Assumption 3 .

Lemma 4

There is a unique equilibrium point of system ( 4 ) inside the triangle A1A2A3.


Th b d d b i ’ d i

http://find/

8/17/2019 Cisda2014 Minh

17/27

The subtended bearings’ dynamics

α̇1 = (

−1

r 3 sin α2 +

−1

r 2 sin α3)e 1 +

1

r 3 sin α1e 2 +

1

r 2 sin α1e 3

= −g 11e 1 + f 12e 2 + f 13e 3

α̇2 = 1

r 3sin α2e 1 − (

1

r 3sin α1 +

1

r 1sin α3)e 2 +

1

r 1sin α2e 3

= f 21e 1 − g 22e 2 + f 23e 3

α̇3 = 1

r 2sin α3e 1 +

1

r 1sin α3e 2 − (

1

r 1sin α2 +

1

r 2sin α1)e 3

= f 31e 1 + f 32e 2 − g 33e 3

where r k = p − p Ak , k ∈ {1, 2, 3}. Note that

−g kk + f (k +1)k + f (k −1)k = 0,


S bili l i

http://find/

8/17/2019 Cisda2014 Minh

18/27

Stability analysis

Let α = [ α1 α2 α3 ]T , e =

e 1 e 2 e 3

T ⇒ α̇ = ė , and

ė = M (e )e (5)

where

M (e ) =

−g 11 f 12 f 13f 21 −g 22 f 23

f 31 f 32 −g 33

.

The system (5) is defined inMe = ( A1 − α∗1, π − α∗1] × ( A2 − α∗2, π − α∗2] × ( A3 − α∗3, π − α∗3].In Me : g kk ≥ 0, f jk ≥ 0 for j , k ∈ {1, 2, 3}

The column sums of M are zero.

Theorem 5

Under Assumptions 1– 3 the origin of the system ( 5 ) is asymptotically stable.


S bili A l i

http://find/

8/17/2019 Cisda2014 Minh

19/27

Stability Analysis

Proof.

Consider the Lyapunov function: V (e ) =3

k =1λk =

3k =1

|e k |:

V is positive definite in Me . λk = |e k |: convex, positive, Lipschitz continuous in Me − {0}. ⇒ λk

is differentiable everywhere except at e k = 0. The upper-right derivative of λk at e k = 0 is D

+λk (e k ) = 1.

V̇ is negative definite in Me . The result is followed by considering three cases: e k = 0, e k > 0 and

e k

8/17/2019 Cisda2014 Minh

20/27

Outline








Si l ti

http://find/

8/17/2019 Cisda2014 Minh

21/27

Simulation

Three beacons: A1(−1;0),A2(4; 0) and A3(0; 5); Desired position:

α∗

1 = α∗

2 = α∗

3 = 2π/3.

(a) Trajectories under control law (4). (b) Angle errors corresponding to the trajectory

from the initial position (2.5; 1.5).

Figure 7: Simulation Results


Quadrotor platform

http://find/

8/17/2019 Cisda2014 Minh

22/27

Quadrotor platform

Quadrotor platform

Figure 8: A quadrotor used in experiments

Quadrotor’s Modules

Controller: Atmega 2560

Sensors: Accelerometer, Gyro

sensor, Magnetometer, Sonarsensor, Barometer, GPS.

Actuators: 4 brushless DCmotors

Communication: Zigbee


Hardware experiment

http://find/

8/17/2019 Cisda2014 Minh

23/27

Hardware experiment

Experiment’s setup & goal

three quadrotors acts asstationary beacons

a quadrotor flies to desiredlocation satisfying:α∗1 = α

∗

2 = α∗

3 = 120o .

Figure 9: Trajectory

Experiment Record


Outline

http://cisda2014.mp4/http://find/

8/17/2019 Cisda2014 Minh

24/27

Outline








Conclusion


8/17/2019 Cisda2014 Minh

25/27

Conclusion

Summary A navigation control law using only bearing measurements with three

stationary beacons. Analysis using Lyapunov stability theory: the agent asymptotically

reaches desired location. Simulation and hardware experiment.

Further research directions Extend the navigation control law to entire plane. Analyze performance of the navigation control law under noise.


Q & A


8/17/2019 Cisda2014 Minh

26/27

Q. & A.

Thank you!


References


8/17/2019 Cisda2014 Minh

27/27

References

[1] M. A. McLeman et. al., “Navigation using visual landmarks by the ant leptothoraxalbipennis”, Insectes Sociaux , 2002.

[2] R. Wehner, “Desert ant navigation: how miniature brains solve complex tasks”, Journal of Comparative Physiology A, 2003

[3] K. Bekris et. al., “Angle-Based Methods for Mobile Robot Navigation: Reaching the EntirePlane”, ICRA, LA, 2004.

[4] S. Loizou et. al., “Biologically inspired bearing-only navigation and tracking,” CDC , 2007.

[5] M. Basiri et. al., “Distributed control of triangular formations with angle-only constraints,”Systems and Control Letters , 2010.

[6] A. Bishop, “Distributed bearing-only formation control with four agents and a weak controllaw,” Proc. of the 9th IEEE Int. CCA , 2011.

[7] A. N. Bishop et. al., “Control of triangle formations with a mix of angle and distanceconstraints”, in Conference on Control Applications , 2012.

[8] M. Ye et. al., “Multiagent Self-Localization Using Bearing Only Measurements”, 52nd IEEEConference on Decision and Control, Florence, Italy, December, 2013.

[9] M. Deghat et. al., “Multi-target localization and circumnavigation by a single agent usingbearing measurements”, Int. J. Robust Nonlinear Control , 2014.

[10] H. Khalil, “Nonlinear systems”, 3nd ed., Prentice-Hall, 2002.
http://find/

Cisda2014 Minh

Documents

Transcript of Cisda2014 Minh