1. Objective: To develop a fully-autonomous control system for the Q-ball based on onboard IMU/Magnetometer/Ultrasound sensory information Summer Internship Research Project Dhruv Soni 3 rd year, BTech Electrical Engineering, IIT Bombay

2. Moving quad rotor helicopter in 3 different sinusoidal motions Experimental Results: Opti-track positional data of the x, y, z position of the helicopter x, y, z gyro data x, y, z accelerometer data which is very noisy due to propeller motion

3. X Control: Left-right periodic motion at a fixed height Y-axis Command Z-axis Command

4. This command signals are used to derive the Roll and Pitch Commands Roll and Pitch are back calculated using IMU Sensors and sent as a feedback signals X-Command

5. a. Roll signal Roll derived from command Signal Roll signal Calculated from Accelerometer output

6. Implementation of Complementary Filter available in QuarC for Roll Command Complementary Filter Output Roll signal Calculated from Accelerometer output

7. b. Pitch signal Pitch derived from command Signal Pitch signal Calculated from Accelerometer output

8. Implementation of Complementary Filter for Pitch Command Complementary Filter Output Pitch signal Calculated from Accelerometer output

9. Localization Position command LQRPID Q-ball. IMUFilter Pitch/Roll command Pitch/Roll observed Opti-track Position observed

10. Localization Position command LQRPID Q-ball. IMUFilter Pitch/Roll command Pitch/Roll observed Opti-track Position observed Replace this with IMU-in the loop control block

11. Derivation of the kinematic equations for calculating the position of the Q-ball Navigation with respect to the inertial frame:

12. The accelerometers usually provide a measurement of specific force in a body fixed axis set, denoted by. In order to navigate, it is necessary to resolve the components of the specific force in the chosen reference frame. In the event that the inertial frame is selected, this may be achieved by pre-multiplying the vector quantity by the direction cosine matrix using: Where represents the turn rate of the body with respect to the i- frame as measured by the Gyroscopes.

13. Algorithm for Localization

14. Calculation of Rotational Matrix C To solve : f (t)= C(t) * f b (t) Note: the solution will depend on initial attitude of the Q-ball On solving this differential equation we get Rotational matrix C (t) is calculated by solving this differential equation

15. Simplifying Rotational Matrix C On applying Taylor’s expansion : Here B = δ t * [ 0-w_zw_y w_z 0-w_x w_y w_x 0 ] And σ = | δ t * sqrt{(w_x)2 + (w_y)2 + (w_z)2}|;

16. Deriving Position of Q-ball f (t)= C(t) * f b (t) Note that here δ t is very small (order of m.sec) hence integration can be simplified to the following linear equations: V (t + δ t) = V(t) + δ t * {f(t + δ t) + g} here g= [0 0 9.8] S (t + δ t) = S(t) + δ t * V(t + δ t)

17. Experiment with Motors on (no propellers) (only noise signals) X optitrack X calculated

18. Experiment with Motors off (only noise signals) y optitrack y calculated

19. Experiment with Motors off (only noise signals) z optitrack z calculated

20. Sources of Error It was assumed that Initial attitude angles are zero i.e. C(0) = [ 1 0 0 0 1 0 0 0 1 ] which may not be the case because even a slight tilt in initial orientation of q-ball, gives components of gravitational force which gets integrated over time in the algorithm can give us the exponential growth (as observed)

21. Experiment with initial tilt Accelerometer reading : Accel_x = 8.9478 Accel_y = 0.4572 Accel_z = -3.7881 Gyroscope reading : Gyro_x = 0 Gyro_y = 2.1817e-004 Gyro_z = 0

22. Modification in algorithm 1.> Velocity reset : Velocity of Q-ball is reset to zero at each sampling time 2.> Initial orientation of Q-ball is calculated to derive exact elements of C(0) matrix 3.> Constant gravitational acceleration vector is back transformed to body frame and subtracted from the Accelerometer reading

23. Results after Modifications (Motors on propellers off) X (Modified Algo)X (Previous Algo)

24. Results after Modifications Y (Modified Algo)Y (Previous Algo)

25. Results after Modifications Z (Previous Algo) Z (Modified Algo)

26. Next Steps  Test the algorithm for linear motion of Q-ball (smooth noise free motion, motors off)  Implement an Attitude determination algorithm for exact calculation of Rotational matrix in the localization algorithm