1. Nicholas Lawrance | ICRA 20111 1 Minimum Snap Trajectory Generation for Control of Quadrotors (Best Paper ICRA 2011) Daniel Mellinger and Vijay Kumar GRASP Lab, UPenn

2. Nicholas Lawrance | ICRA 20112 2 (Very) Brief Outline Goal is to develop planning and control techniques for control of an autonomous quadrotor Experimental setup –Ascending Technologies Hummingbird 200g payload 20min –Vicon Motion Capture system IR and visual cameras to capture position and orientation from markers on the target 200Hz

3. System Model Control of total body force and moments Nicholas Lawrance | ICRA 20113 L

4. Differential Flatness A complicated thing I don’t really understand 1 - Select a set of ‘flat’ outputs 2 - The inputs must be able to be written as a function of the flat outputs and a (limited number of) their derivatives 3 - ? 4 – Planning and control (profit) Nicholas Lawrance | ICRA 20114

5. For the quadrotor, the full state is: (position, velocity, orientation, angular velocity) Flat output selection Then allowable trajectories are smooth functions of position and yaw angle Then, need to find equations expressing the inputs (engine speeds) as a function of the flat outputs Nicholas Lawrance | ICRA 20115

6. Orientation as a function of flat outputs Body z-axis unit vector is the acceleration direction Further rotation by the yaw angle gives the x unit vector (in the intermediate yaw frame) Nicholas Lawrance | ICRA 20116

7. Angular velocity as a function of flat outputs Body acceleration Differentiate (u 1 is the total force from the motors) Isolate angular rate components Nicholas Lawrance | ICRA 20117

8. Angular accelerations as a function of flat outputs Angular acceleration x and y components (differantiate and dot-product with the x and y body vectors) z-component Nicholas Lawrance | ICRA 20118

9. Step 3 - ? Then, force is a function of the flat outputs Angular velocity and acceleration are functions of the flat outputs Then, u = f(x, y, z, ψ) Nicholas Lawrance | ICRA 20119

10. Control From a defined trajectory Control law to find desired force to travel to target trajectory Now need desired rotation matrix Nicholas Lawrance | ICRA 201110

11. As before, get the intermediate yaw frame, then the body axis unit vectors Orientation error Angular velocity error Control variables Nicholas Lawrance | ICRA 201111

12. Trajectories Keyframes (defined trajectory points) with ‘safety corridors’ between each keyframe Link keyframes with polynomial paths in the flat output space Minimise curvature for smooth achievable paths Nicholas Lawrance | ICRA 201112

13. i.e. “The cost function [...] is similar to that used by Flash and Hogan who showed human reaching trajectories appear to minimize the integral of the square of the norm of the jerk (the derivative of acceleration, k r = 3). In our system, since the inputs u 2 and u 3 appear as functions of the fourth derivatives of the positions, we generate trajectories that minimize the integral of the square of the norm of the snap (the second derivative of acceleration, k r = 4). Nicholas Lawrance | ICRA 201113

14. Components are decoupled which means they can be solved separately Re-consider the general problem for w Can also consider time scaling to determine the temporal length of the path segments Nicholas Lawrance | ICRA 201114

15. Nicholas Lawrance | ICRA 201115

16. Nicholas Lawrance | ICRA 201116

17. Nicholas Lawrance | ICRA 201117

18. Notes on the paper At the page limit –efficient use of space –reference previous work and theories Nice graphical representations of video data Nicholas Lawrance | ICRA 201118