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Nov 17, 2023 278 likes β€’456 views

A Framework for Multi-Vehicle Navigation using Feedback-Based Motion Primitives Marijan Vukosavljev, Zachary Kroeze, Mireille E. Broucke, and Angela P. Schoellig IROS, September 25, 2017 Motivation M. Vukosavljev, Z. Kroeze, M. E. Broucke, and

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SLIDE 1

A Framework for Multi-Vehicle Navigation using Feedback-Based Motion Primitives

Marijan Vukosavljev, Zachary Kroeze, Mireille E. Broucke, and Angela P. Schoellig IROS, September 25, 2017

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SLIDE 2

Motivation

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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SLIDE 3

Motivation

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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Path planning and control in known environments

goal

  • bstacle
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SLIDE 4

Problem Statement

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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  • Given:
  • dynamics 𝑦̇ = 𝑔 𝑦, 𝑣 , outputs 𝑧 = β„Ž 𝑦 ,

where 𝑦 ∈ ℝ,, 𝑧 ∈ ℝ-

  • goal and obstacle sets in output space
  • Find: feedback controller 𝑣 𝑦 and set of

initial conditions π‘Œ/ βŠ‚ ℝ, such that 𝑧(𝑒) eventually enters the goal set and always avoids the obstacle set

  • Can be posed as a reach-avoid problem for a

control system

y1 y2

  • bstacle

goal y(t)

y1 y2

  • bstacle

goal

  • Example: two double

integrators, n = 4, p = 2

  • 𝑦̇4 = 𝑦5

𝑦̇5 = 𝑣4 𝑦̇6 = 𝑦7 𝑦̇7 = 𝑣5 𝑦̇ = 𝑔 𝑦, 𝑣

  • 8𝑧4 = 𝑦4

𝑧5 = 𝑦6 𝑧 = β„Ž(𝑦)

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Framework Features

  • Feedback control
  • Wide range of initial conditions
  • Robust to disturbances
  • Requires no explicit path
  • Safety guarantees
  • Simultaneous motion
  • Computational efficiency
  • Symmetry
  • Lower dimensional spaces
  • Modularity
  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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y1 y2

  • bstacle

goal y(t)

Most related literature: Pappas Kumar Belta Frazzoli

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SLIDE 6

Proposed Framework

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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Given

Maneuver Automaton Problem

p outputs for

Output Transition System

Shortest path algorithm

Control Strategy

p-dim

feedback control, umi(x) discrete maneuver plan

Product

dynamics

Hybrid

low-level control Primitives, mi Motion Λ™ x = f(x, u)

  • bstacle and

goal regions

Automaton

motion capabilities in output space grid y = h(x)

Data

Gridding Box size

multi-robot system

  • f output

space

Partition of environment Path planning Control design Solution to problem Automated

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SLIDE 7

Maneuver Automaton

  • Formally a hybrid system
  • Hybrid state space: motion primitives and continuous state in ℝ,
  • Edges: concatenation constraints between motion primitives
  • Each motion primitives is implemented by a feedback controller over

a designated subset in ℝ,

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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H F B Οƒ0 Οƒ0, Οƒ+ Οƒ0, Οƒβˆ’ Οƒ0 Οƒ+ Οƒ0 Οƒβˆ’

v2 v3 v5 v4

Hold

v2 v3 v6 v5 v4

Forward

Οƒ+ v1 v2 v3 v5 v4

Backward

Οƒβˆ’

x1 x2

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Maneuver Automaton - Design

  • First focus on double integrator: 𝑦̇4 = 𝑦5, 𝑦̇5 = 𝑣, with 𝑧 = 𝑦4
  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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Hold Forward Backward

H F B Οƒ0 Οƒ0, Οƒ+ Οƒ0, Οƒβˆ’ Οƒ0 Οƒ+ Οƒ0 Οƒβˆ’

Hold Forward Backward

Output space behaviour 𝑧

v2 v3 v6 v5 v4

Forward

Οƒ+ v1 v2 v3 v5 v4

Backward

Οƒβˆ’ v2 v3 v5 v4

Hold

x1 x2

State space behaviour 𝑦5 𝑦4

Reach control

  • B. Roszak and M. E. Broucke, β€œNecessary and

sufficient conditons for reachability on a simplex,” 2006.

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SLIDE 9

Maneuver Automaton - Application

  • Quadrocopter model reduces to double integrator in each positional

direction

  • For the multi-quadrocopter model, stack all the double integrators
  • Choose Hold, Forward, and Backward in each output component
  • For example, one quadrocopter with planar motion:
  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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βœ“ F H β—† βœ“ F F β—† βœ“ B H β—† βœ“ H F β—† βœ“ H H β—† 𝑧4 𝑧5 𝑧4 𝑧5

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Control Policy on the Product Automaton

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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y1 y2

βœ“ F H β—† βœ“ H F β—† βœ“ F H β—† βœ“ F F β—† βœ“ F F β—† βœ“ F F β—† βœ“ F F β—† βœ“ F F β—† βœ“ F H β—† βœ“ F H β—† βœ“ F H β—† βœ“ H F β—† βœ“ H F β—† βœ“ H F β—† βœ“ H F β—† βœ“ H H β—†

y1 y2

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Experimental Results

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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Experimental Results - Nominal

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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Conclusion

  • Addressed a path planning and control

problem in known environments as a reach- avoid problem

  • Employed a modular framework consisting of

an output space partition, low-level feedback controllers, and a high-level feedback for selecting motion primitives

  • Highly robust control design that enables

simultaneous motion in a computationally feasible way

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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y1 y2

  • bstacle

goal y(t)

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SLIDE 15
  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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Comparison to Literature

Paper Feedback control Simultaneous motion Computational efficiency Frazzoli, Dahleh, and Feron; 2005 Kloetzer and Belta; 2008 Fainekos, Girard, Kress-Gazit, Pappas; 2009 Ayanian, Kumar; 2010 Raman, Kress-Gazit; 2014 Vukosavljev, Kroeze, Broucke, Schoellig, 2017

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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Stack multiple copies 𝑦̇4 = 𝑦5 𝑦̇5 = 𝑣4 𝑦̇6 = 𝑦7 𝑦̇7 = 𝑣5 𝑦̇ = 𝑔 𝑦, 𝑣 8𝑧4 = 𝑦4 𝑧5 = 𝑦6 𝑧 = β„Ž(𝑦)

  • M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig

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Symmetry Lower dimensions Modularity