1. Motion Primitives for an Autorotating Helicopter Sanjiban Choudhury

2. Problem Growing a state tree towards a goal involves solving a Boundary Value Problem Solve: actuator rate limits, saturation limits safety operation limits Solution: Frame an Optimal Control Problem Problem: Solving Optimal Control for real time systems is non- trivial Computationally expensive Numerical Stability Cannot be computed on the fly

3. Maneuver Automaton Reduction in control complexity : Finite number of state and control trajectories Motion primitives are a class of trajectories 2 special class of primitives: Trim: Velocity and control constant Maneuver: Joins one trim to another Finite State + Finite Trajectories = Discrete reachable set? e.g Hover, Steady left turn, Steady climb Library of trim and maneuver is Maneuver Automaton 1 Motion planning can be done on this Hybrid System 2 2. E. Frazzoli, “Robust hybrid control for autonomous vehicle motion planning,” Ph.D. dissertation, Massachusetts Institute of Technology,May 2001 1. Frazzoli, E., Dahleh, M. A., and Feron, E. (2005). Maneuver-based motion planning for nonlinear systems with symmetries. IEEE Transactions on Robotics, 21(6):1077–1091 State graph must be strongly connected

4. Dynamics of Autorotation u w Equations Coefficients S.Tierney and J. W. Langelaan. “Autorotation Path Planning Using Backwards Reachable Set and Optimal. Control”. Aerospace Engineering, Penn State

5. Designing the MA Trim State: Yaw rate: {0.0, 0.04, 0.08, 0.12, 0.16, 0.2, -0.04, -0.08, -0.12, -0.16, - 0.2} Each trim repeated for min descent and max glide slope(11x2 = 22) Maneuvers: OR s.t and Left, D Right, GR Left, GR Right, GR Straight, D Straight, GR 2x(4x10+2x5+4x5+1)=142 s.t and Solved using: Differential Dynamic Programming General Pseudospectral Optimal Control Software Rao, A. V., Benson, D. A., Darby, C. L., Patterson, M. A., Francolin, C., and Huntington, G. T., “Algorithm 902: GPOPS, A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using The Gauss Pseudospectral Method,” ACM Transactions on Mathematical Software, Vol. 37, No. 2Algorithm 902: GPOPS, A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using The Gauss Pseudospectral Method

6. Results 0.20.160.120.080.0400.20.160.120.080.040 0.20 0.44 976 0.30 746 0.33 188 0.20 353 0.12 909 16.5 27 00000 0.16 0.89 139 0 0.18 973 0.33 8 0.76 031 0.53 423 0 15.21 1 0000 0.12 0.43 324 0.21 373 0 0.43 35 0.34 523 2.41 9 00 14.6 75 000 0.08 5.69 94 4.84 2 3.67 8 0 1.10 29 1.24 59 000 12.0 3 00 0.04 5.81 48 3.86 35 2.09 02 1.32 7 000000 12.7 6 0 0 4.65 15 2.77 52 1.87 25 0.98 182 0000000 18.8 18 0.2 16.8 95 000000 0.44 714 0.27 095 0.42 98 0.28 005 0.19 352 0.160 15.8 46 0000 0.38 708 0 0.26 625 0.21 363 0.13 288 3.56 82 0.1200 14.4 8 000 0.94 967 0.43 697 0 0.29 2 0.25 9143 2.97 29 0.08000 13.3 87 00 0.46 26 0.17 97 3.51 53 0 0.22 057 1.98 12 0.04000011.050 0.24 801 4.09 34 2.70 85 2.00 82 0 0.34 064 000000 16.0 66 0.15 049 4.49 17 2.39 93 1.48 79 0.66 384 0 Trim trajectoriesManeuver time for left turns

7. Conclusion No need to solve Optimal Control on fly Easy to use library for sampling-based motion planners May lead to sub-optimal plans / Harder to prove resolution completeness Pragmatic approach with good reachability.