A Framework for Multi-Vehicle Navigation using Feedback-Based Motion - PowerPoint PPT Presentation

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 A. P. Schoellig 2

Motivation Path planning and control in known environments obstacle goal M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 3

Problem Statement Given : • • Example: two double dynamics 𝑦̇ = 𝑔 𝑦, 𝑣 , outputs 𝑧 = ℎ 𝑦 , • integrators, n = 4, p = 2 where 𝑦 ∈ ℝ , , 𝑧 ∈ ℝ - 𝑦̇ 4 = 𝑦 5 goal and obstacle sets in output space • 𝑦̇ 5 = 𝑣 4 y ( t ) goal goal 𝑦̇ = 𝑔 𝑦, 𝑣 • Find : feedback controller 𝑣 𝑦 and set of • 𝑦̇ 6 = 𝑦 7 initial conditions 𝑌 / ⊂ ℝ , such that 𝑧(𝑢) 𝑦̇ 7 = 𝑣 5 eventually enters the goal set and always obstacle obstacle • 8𝑧 4 = 𝑦 4 y 2 𝑧 = ℎ(𝑦) avoids the obstacle set y 2 𝑧 5 = 𝑦 6 Can be posed as a reach-avoid problem for a • control system y 1 y 1 M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 4

Framework Features • Feedback control Most related • Wide range of initial conditions y ( t ) goal literature : • Robust to disturbances • Requires no explicit path Pappas • Safety guarantees Kumar • Simultaneous motion obstacle y 2 Belta • Computational efficiency Frazzoli • Symmetry • Lower dimensional spaces y 1 • Modularity M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 5

Proposed Framework Partition of environment Path planning Output Product Given Automaton Automated Transition System Gridding p -dim of output motion capabilities Problem space in output space grid Data p outputs for multi-robot system Shortest path Maneuver Automaton y = h ( x ) algorithm Motion Primitives, m i obstacle and Hybrid goal regions Box size Control Strategy dynamics discrete maneuver plan x = f ( x, u ) ˙ low-level control feedback control, u m i ( x ) Solution to problem Control design M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 6

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 , Hold Forward Backward v 3 v 3 v 6 v 3 σ + σ 0 , σ − σ 0 , σ + σ 0 x 2 v 2 v 5 v 2 v 5 v 2 v 5 x 1 σ + σ 0 σ − B H F σ − σ 0 v 4 v 4 v 1 v 4 M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 7

Maneuver Automaton - Design • First focus on double integrator: 𝑦̇ 4 = 𝑦 5 , 𝑦̇ 5 = 𝑣 , with 𝑧 = 𝑦 4 σ 0 , σ − σ 0 , σ + σ 0 Output space Hold Forward Backward behaviour σ + σ 0 𝑧 B H F Hold Forward Backward σ − σ 0 v 3 v 3 v 6 v 3 Backward Hold Forward State space σ + behaviour x 2 Reach control v 2 v 5 v 2 v 5 v 2 v 5 𝑦 5 x 1 B. Roszak and M. E. Broucke, “Necessary and sufficient conditons for reachability on a σ − simplex,” 2006. 𝑦 4 v 4 v 4 v 1 v 4 M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 8

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: 𝑧 5 𝑧 4 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ H F H F B 𝑧 5 H H F F H 𝑧 4 M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 9

Control Policy on the Product Automaton ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ F F F H H H H H ✓ ◆ ✓ ◆ ✓ ◆ F H H F F F ✓ ◆ H ✓ ◆ ✓ ◆ ✓ ◆ F F H F ✓ ◆ F F H F H y 2 y 2 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ F F F H F F F F y 1 y 1 M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 10

Experimental Results M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 11

Experimental Results - Nominal M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 12

M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 13

Conclusion • Addressed a path planning and control problem in known environments as a reach- avoid problem y ( t ) goal • Employed a modular framework consisting of an output space partition, low-level feedback controllers, and a high-level feedback for obstacle y 2 selecting motion primitives • Highly robust control design that enables y 1 simultaneous motion in a computationally feasible way M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 14

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

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 16

Stack multiple copies 𝑦̇ 4 = 𝑦 5 𝑦̇ 5 = 𝑣 4 𝑦̇ = 𝑔 𝑦, 𝑣 𝑦̇ 6 = 𝑦 7 𝑦̇ 7 = 𝑣 5 8𝑧 4 = 𝑦 4 𝑧 = ℎ(𝑦) 𝑧 5 = 𝑦 6 Lower dimensions Symmetry Modularity M. Vukosavljev, Z. Kroeze, M. E. Broucke, and A. P. Schoellig 17