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Multiple AMV Trajectory Planning using the PRONTO Toolkit
- 1. Multiple AMV Trajectory Planning Using the PRONTO TK Andreas J. Häusler Instituto Superior Técnico Institute for Systems and Robotics Dyanamical Systems and Ocean Robotics Laboratory Lisbon, Portugal
- 2. About me From Bavaria, but living in Portugal for almost 5 years PhD Student in Marine Robotics at Instituto Superior Técnico (Lisbon, Portugal) Currently working on Trajectory Planning and Generation for Multiple Marine Vehicles CU Boulder, May 4, 2012 2A. Häusler
- 3. Some Not-So-Important Trivia Arrived at Boulder in January with wife and daughter will return to Lisbon in roughly a week had a great time at CU and in Colorado CU Boulder, May 4, 2012 A. Häusler 3 The Häusler’s coat of arms (1632)
- 4. Introduction Efficient algorithms for multiple vehicle trajectory planning are crucial for cooperative control systems Need to take into account vehicle dynamics, mission parameters and external influences Aim for minimum energy usage CU Boulder, May 4, 2012 4A. Häusler
- 5. Path Planning System MULTIPLE VEHICLE PATH PLANNING SYSTEM Initial Positions Initial Velocities Final Positions Final Velocities Cost Criterion (e.g. weighted sum of energies, maneuvering time) Vehicle Model & Hydrodynamics Collision Avoidance Constraints Mission Parameters (e.g. obstacles, bathymetric map) Feasible Trajectories CU Boulder, May 4, 2012 5A. Häusler
- 6. What it Really Looks Like CU Boulder, May 4, 2012 A. Häusler 6 Initial State & Input Final State & Input Initial Guess Dynamics Trajectory Revised Guess Expected Energy Trajectories Safety Distance Comm. Constraint Cost Criterion Current Obstacles Desired Trajectory BatymetricData AMV Data Projection Operator “Magic” Newton Method Mission Specifications CoordinatedTrajectory TrackingController Pre- Planner Environmental Constraints SupervisingMission Operator
- 7. The MEDUSA ASV Semi-Submersible Two Thrusters (Seabotix HPDC 1507) 1035mm × 300mm × 840mm total Max. Speed 1.5 m/s Originally designed for diver assistance/harbor patrolling scenarios CU Boulder, May 4, 2012 A. Häusler 7
- 8. Dynamic Model Assumption: since the MEDUSA is a surface craft, we can safely consider only 2D dynamics (i.e. ignore roll and pitch dynamics) The vessel is controlled with common and differential thrust (two thrusters, no rudders) This gives us 3 kinematic states + 3 dynamic states + 2 inputs CU Boulder, May 4, 2012 A. Häusler 8
- 9. Dynamic Model Kinematics 𝑥 𝑦 𝜓 = cos 𝜓 − sin 𝜓 0 sin 𝜓 cos 𝜓 0 0 0 1 𝑥 𝑦 𝜓 Dynamics 𝑀𝑟𝑏 + 𝑀 𝑎 𝜈 + 𝐶𝑟𝑏 𝜈 − 𝐶 𝑎 𝜈 𝜈 + 𝐷 + 𝐷 𝑛 𝜈 𝜈 = 𝜏 where 𝜈 = 𝑢 𝑣 𝑟 and 𝜏 = 𝑇𝑝𝑠 + 𝑇𝑠𝑏 0 𝑙 𝑇𝑝𝑠 − 𝑇𝑠𝑏 Inputs 𝑛 𝑝𝑠 and 𝑛 𝑠𝑏 CU Boulder, May 4, 2012 A. Häusler 9
- 10. Thruster Model Due to design: negligible propeller-hull interaction Four-quadrant propeller model 𝑇 = 1 2 𝜌𝑐 𝑇 𝛽 𝑉𝑎 2 + 𝑉𝑝 2 𝜋𝑅2 𝑄 = 1 2 𝜌𝑐 𝑄 𝛽 𝑉𝑎 2 + 𝑉𝑝 2 𝜋𝑅2 𝑑 where 𝑉𝑎 𝑝𝑠 = sin atan2 −𝑝 𝑦, 𝑝 𝑥 𝑙𝑟 + 𝑢 = −𝑝 𝑦 𝑟 + 𝑢 𝑉𝑝 𝑝𝑠 = 0.7𝑅𝜔 𝑝𝑠 = 0.7𝑅2𝜋𝑛 𝑝𝑠 CU Boulder, May 4, 2012 A. Häusler 10
- 11. DC Motor Model Standard DC motor equations 𝐿 𝑎 𝑑𝐼 𝑑𝑡 + 𝑅 𝑎 𝐼 = 𝑉 − 𝐾𝑒 𝜔 𝐽 𝑚 𝜔 + 𝑏𝜔 = 𝐾𝑡 𝐼 𝑎 − 𝑄 For now, assume sufficiently fast dynamics → steady state equation for voltage 𝑉 = 𝑅 𝑎 𝐼 + 𝐾𝑒 𝜔 = 𝑅 𝑎 𝐼 + 𝐾𝑒2𝜋𝑛 Current map obtained from measurements 𝐼 𝑝𝑠 𝑛 𝑝𝑠 = 𝑎𝑛 𝑝𝑠 3 + 𝑏𝑛 𝑝𝑠 2 + 𝑐𝑛 𝑝𝑠 + 𝑑 CU Boulder, May 4, 2012 A. Häusler 11
- 12. Cost and Constraints Using the motor model, the instantaneous power requirement is 𝑃𝑡𝑜𝑡 = 𝑉𝑝 𝑠 𝐼 𝑝𝑠 + 𝑉𝑠𝑏 𝐼𝑠𝑏 + 𝑃𝑝 Additionally: desired traj. (using 𝐿2 minimization) Additionally: inter-vehicle collision and obstacle avoidance constraints (using log barrier function) 𝑥 𝑖 𝑡 − 𝑥 𝑗 𝑡 2 2𝑟𝑐 2 + 𝑦 𝑖 𝑡 − 𝑦 𝑗 𝑡 2 2𝑟𝑐 2 − 1 ≥ 0 CU Boulder, May 4, 2012 A. Häusler 12
- 13. Results Scenario: 3 Medusas and 2 obstacles Desired trajectories passing through obstacles & causing collisions Result: collision-free trajectories through narrow passage CU Boulder, May 4, 2012 A. Häusler 13
- 14. Illustration: Armijo “Exploration” CU Boulder, May 4, 2012 A. Häusler 14
- 15. Illustration: “Morphing” Trajectories CU Boulder, May 4, 2012 A. Häusler 15
- 16. Illustration: Trajectory “Tracking” CU Boulder, May 4, 2012 A. Häusler 16
- 17. Open Issues & Outlook Issues with the four-quadrant model still prevent 2nd order descent Gains need to be tweaked Incorporate mission performance metrics à la Naomi Leonard Use a (propably topology-based) pre-planning method for global optimal (in some discrete sense) initial guess CU Boulder, May 4, 2012 A. Häusler 17
- 18. Questions? Comments? Suggestions? CU Boulder, May 4, 2012 A. Häusler 18