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Optimal Path Planning on Matrix Lie Groups

Optimal Path Planning on Matrix Lie Groups. Mechanical Engineering and Applied Mechanics Sung K. Koh. U n i v e r s i t y o f P e n n s y l v a n i a. Motivation. Providing trajectories for airplanes near airports.

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Optimal Path Planning on Matrix Lie Groups

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  1. Optimal Path Planning on Matrix Lie Groups Mechanical Engineering and Applied Mechanics Sung K. Koh U n i v e r s i t y o f P e n n s y l v a n i a

  2. Motivation • Providing trajectories for airplanes near airports. • Optimal path between given initial position, orientation, and final position, orientation to be made with a final time T. • Optimal path is obtained by minimizing the cost which is the sum square of the inputs. • Control tower problem : • The airplane from some initial position and orientation is assigned a final position and orientation plus a final time. U n i v e r s i t y o f P e n n s y l v a n i a

  3. Landing Tower Problem U n i v e r s i t y o f P e n n s y l v a n i a

  4. Maximization of Hamiltonian U n i v e r s i t y o f P e n n s y l v a n i a

  5. Co-state and Invariant • Evolution of costate p • Invariant U n i v e r s i t y o f P e n n s y l v a n i a

  6. Example 1 : Path Planning on SE(2) • Given that the car always dives forward at a fixed velocity, finding the steering • controls so that the robot, starting from an initial position and orientation, arrives • at some final goal position and orientation at a fixed time. • Dynamics U n i v e r s i t y o f P e n n s y l v a n i a

  7. Example 1 : Path Planning on SE(2) U n i v e r s i t y o f P e n n s y l v a n i a

  8. Input dynamics U n i v e r s i t y o f P e n n s y l v a n i a

  9. Example 2 : Path Planning on SO(3) U n i v e r s i t y o f P e n n s y l v a n i a

  10. Example 2 : Path Planning on SO(3) U n i v e r s i t y o f P e n n s y l v a n i a

  11. Example 2 : Path Planning on SO(3) U n i v e r s i t y o f P e n n s y l v a n i a

  12. Example 2 : Path Planning on SO(3) U n i v e r s i t y o f P e n n s y l v a n i a

  13. Conclusions • The problems formulated as an optimal control problem of a left invariant control system on the Lie group are considered. • Through the use of Pontryagin’s maximum principle and the techniques of numerical optimization, the solutions of problems on SE(2) and on SO(3) are presented. U n i v e r s i t y o f P e n n s y l v a n i a

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