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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
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Outline
1 Introduction
2 Preliminaries and problem formulation
3 The proposed control law and stability analysis
4 Simulation and hardware experiment
5 Conclusion
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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
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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.
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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
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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.
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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 experiment
SimulationHardware experiment with quadrotors
5 ConclusionReferences
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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
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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.
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The subtended bearing
(a) Case 1: α3 = ϑ3 (b) Case 2: α3 = 2π − ϑ3
Let ϑk = |β k −1 − β k +1|, 0 ≤ ϑk
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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.
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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.
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Outline
1 IntroductionMotivationLiterature review
2 Preliminaries and problem formulationPreliminariesProblem formulation
3 The proposed control law and stability analysisProposed control lawStability Analysis
4 Simulation and hardware experiment
SimulationHardware experiment with quadrotors
5 ConclusionReferences
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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.
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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.
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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.
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Th b d d b i ’ d i
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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,
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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.
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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
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Outline
1 IntroductionMotivationLiterature review
2 Preliminaries and problem formulationPreliminariesProblem formulation
3 The proposed control law and stability analysisProposed control lawStability Analysis
4 Simulation and hardware experiment
SimulationHardware experiment with quadrotors
5 ConclusionReferences
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Si l ti
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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
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Quadrotor platform
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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
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Hardware experiment
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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
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Outline
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Outline
1 IntroductionMotivationLiterature review
2 Preliminaries and problem formulationPreliminariesProblem formulation
3 The proposed control law and stability analysisProposed control lawStability Analysis
4 Simulation and hardware experiment
SimulationHardware experiment with quadrotors
5 ConclusionReferences
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Conclusion
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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.
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Q & A
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Q. & A.
Thank you!
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References
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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.
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