A Practical Approach to Robotic Swarms

1
A Practical Approach to Robotic Swarms

• IASTED Conference on Control and Applications
• May 2008
• Howard M. Schwartz and Sidney N. Givigi Jr.

2
Objectives

• Develop a practical approach to robotic swarms.
• Must be easy to implement and tractable.
• Must appeal to the control engineers sense of
  performance.

3
Literature Review

• Olfati-Saber, R., Flocking for Multi-Agent
  Dynamic Systems Algorithms and Theory, IEEE
  Trans. Auto. Contr. 2006.
• Tanner, H.G., Jadbabaie, A., and Pappas G.J.,
  Stable Flocking of Mobile Agents, Part I Fixed
  Topology, Proc. of CDC, 2003.
• These methods require one design an attraction
  and repulsive function. Designing this function
  is not clear. Loss of control engineers
  intuition. Is the system working correctly?

4
Our Method

• We use an inertial model

• Define Connected and Unconnected Sets

Connected
Unconnected
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The Forces on the Robots

• The force on unconnected robots is a type of
  gravity force.

Where, is the unit vector from i to
j And rij is the distance from i to j

• The force on the connected robots is a type of
  spring damper force

• The total force on a given robot is

6
Simulation Results

• 20 Robots, 100x100 grid, kp4, kv4, d010,
  kg100, and r12.

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Swarming with obstacle avoidance

• Define a potential field.

• Forces act along negative gradient of field

• Then the complete force acting on each robot is

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Simulation of robots swarming with obstacle
avoidance

• kf 200 all other terms are the same as before.

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Swarm robots with constant motion and obstacle
avoidance.

• Define specified velocity vxd 1.0, then the
  force becomes,

10
Stability Analysis

• Why does this work.

Substituting for kv 4 and kp 4, we get the
eigenvalues, ?1 -1.17, ?2 -6.82, ?3 0.
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Stability of 3 Connected Robots

• Linearize for small motions about the equilibrium
  point.

The force on robot 1 due to robot 3 due to small
motions is,
The force in the x direction then becomes,
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Stability of 3 robots

• The acceleration of robot i in the x direction
  is,

In the case of 3 connected robots we have 12
states and we can write the linearized equations
in the form
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Stability of 20 Robots

• Using a computer to evaluate the configuration
  and recognizing only 3 distinct relationships
  between robots, we get the following maximum and
  minimum eigenvalues for the linearized system,
• ?max -19.86, ?min -0.120.47j
• Therefore the origin is asymptotically stable.

14
Experimental Results

• The robots are given positions over bluetooth
  link.
• The robots are controlled by a HC11 Handyboard.
• Web cameras installed in the ceiling track the
  robots.

15
Robots Following each other and doing obstacle
avoidance
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Conclusion

• Practical approach to swarm robots
• Connected and unconnected sets, gravity and
  spring/damper forces

• Potential fields define obstacles
• The swarm is locally stable
• Experimental results validate the method.