Autopilot design for small fixed wing aerial vehicles. Randy Beard Brigham Young University

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1 Autopilot design for small fixed wing aerial vehicles Randy Beard Brigham Young University

2 Outline Control architecture Low level autopilot loops Path following Dubins airplane paths and path management

3 Control Architecture destination, obstacles map Dubins path or waypoints path planner status path manager straight-line or spiral path tracking error airspeed, altitude, heading, commands path following autopilot position error servo commands wind unmanned aircraft state estimator on-board sensors ˆx(t)

4 Control Architecture (2) User input Dubins path or waypoints vision-based guidance status path manager straight-line or spiral path tracking error airspeed, altitude, heading, commands path following autopilot position error servo commands wind unmanned aircraft state estimator on-board sensors ˆx(t) camera

5 Outline Control architecture Low level autopilot loops Path following Dubins airplane paths and path management

6 Control Architecture destination, obstacles map Dubins path or waypoints path planner status path manager straight-line or spiral path tracking error airspeed, altitude, heading, commands path following autopilot position error servo commands wind unmanned aircraft state estimator on-board sensors ˆx(t)

7 Lateral Control loop PI control for outer loop PD control for inner loop Control course angle.

8 Sideslip Hold e If the airframe has a rudder, then the rudder is used to zero the side slip angle. Requires direct measurement or estimate of.

9 Lateral Autopilot In Flight Tuning If model is not known, and autopilot must be tuned in flight, then the following gains are tuned one at a time, in this specific order: Inner Loop (roll attitude hold) k d - Increase k d until onset of instability, and then back o by 20%. k p -Tunek p to get acceptable step response. Outer Loop (course hold) k p -Tunek p to get acceptable step response. k i -Tunek i to remove steady state error. Sideslip hold (if rudder is available) k p -Tunek p to get acceptable step response. k i -Tunek i to remove steady state error.

10 Longitudinal Autopilot Traditional Descend zone h c Altitude hold zone Climb zone Take-off zone c

11 Pitch Attitude Hold Pitch attitude hold used for most longitudinal modes. Note that DC gain is not unity. Adding an integrator on this inner loop is not a good strategy.

12 Longitudinal Autopilot Traditional Figure 6.20:

13 Longitudinal Autopilot In Flight Tuning If model is not known, and autopilot must be tuned in flight, then the following gains are tuned one at a time, in this specific order: Inner Loop (pitch attitude hold) k d - Increase k d until onset of instability, and then back o by 20%. k p -Tunek p to get acceptable step response. Altitude Hold Outer Loop k ph -Tunek ph to get acceptable step response. k ih -Tunek ih to remove steady state error. Airspeed Hold Outer Loop k pv2 -Tunek pv2 to get acceptable step response. k iv2 -Tunek iv2 to remove steady state error. Throttle hold (inner loop) k pv -Tunek pv to get acceptable step response. k iv -Tunek iv to remove steady state error.

14 Airspeed and Attitude Coupling Traditional approach assumes longitude and lateral dynamics are decoupled Altitude and airspeed does not influence course and position Valid assumption Airspeed and altitude are decoupled Invalid Assumption Altitude (m) 10 5 Airspeed (m/s) Time (s) 14

15 Airspeed and Attitude Coupling Assumed Response True Response Pitch Thrust Altitude Airspeed Altitude Airspeed * - * - * * - * - * * *

16 Total Energy Control System Developed in the 1980 s by Antonius Lambregts Based on energy manipulation techniques from the 19 s Control the energy of the system instead of the altitude and airspeed Thrust Total Energy Kinetic Energy Potential Energy Elevator Drag

17 Total Energy Control System Kinetic Energy: E K, 1 2 mv 2 a Potential Energy: Total Energy: E P, mgh E T, E P + E K Energy Di erence: E D, E P E K

18 Total Energy Control System Original TECS proposed by Lambregts is based on energy rates: T c = T D + k p,t Ė t + k i,t R t t 0 Ẽ t T D is thrust needed to counteract drag PI controller based on total energy rate c = k p, Ė d + k i, R t t 0 Ẽ d PI controller based on energy distribution rate Stability shown for linear systems We will show that the performance of this scheme is less than desirable.

19 Total Energy Control System Nonlinear re-derivation: Error Definitions Ẽ K = 1 2 m V d a 2 Va 2 Ẽ P = mg h d h Lyapunov Function V = 1 2Ẽ2 T + 1 2Ẽ2 D Controller T c = D + Ẽd T V a c =sin 1 + k T Ẽ T V a ḣ d V a + 1 2mgV a k 1 Ẽ K + k 2 Ẽ P!

20 Total Energy Control System Original: T c = D + k p,t Ė T mgv a + k i,t Ẽ T mgv a Nonlinear: T c = D + Ėd T V a + k T Ẽ T V a Similar if k p,t = mg and k i,t = mgk T. The nonlinear controller uses the desired energy rate.

21 Total Energy Control System Modified Original (Ardupilot): c = k p, V a mg k 2 [0, 2] (2 k) ĖP kėk + k i, V a mg ẼD Nonlinear: c =sin 1 ḣ d V a + 1 2mgV a k 1 Ẽ K + k 2 Ẽ P! k 1, k T k D k 2, k T + k D 0 <k T apple k D Lyapunov derivation suggests potential energy error should be weighted more than kinetic energy

22 Total Energy Control System If the drag is unknown, then we can add an adaptive estimate: T c = ˆD + c =sin 1 ˆ = T ẼT > ˆ + Ė d T V a ḣ d V a + + k T Ẽ T V a 1 2mgV a DẼD V a k 1 Ẽ K + k 2 Ẽ P!

23 Total Energy Control System Step in Altitude, Constant Airspeed Airspeed (m/s) Altitude (m) Time (s) Original TECS Nonlinear TECS Adaptive TECS SLC TECS PID ArduPilot PID

24 Total Energy Control System Step in Airspeed, Constant Altitude Altitude (m) Airspeed (m/s) Time (s) Original TECS Nonlinear TECS Adaptive TECS SLC TECS PID ArduPilot PID

25 Total Energy Control System Step in Altitude and Airspeed Altitude (m) Airspeed (m/s) Time (s) Original TECS Nonlinear TECS Adaptive TECS SLC TECS PID ArduPilot PID

26 Total Energy Control System Observations TECS seems to work better than successive loop closure. Still need separate mode for take-off Nonlinear TECS seems to work better, but the Ardupilot controller works very well.

27 Outline Control architecture Low level autopilot loops Path following Dubins airplane paths and path management

28 Control Architecture destination, obstacles map Dubins path or waypoints path planner status path manager straight-line or spiral path tracking error airspeed, altitude, heading, commands path following autopilot position error servo commands wind unmanned aircraft state estimator on-board sensors ˆx(t) Focus on following straight lines and helices

29 Dubins Airplane Model i i (north) j i (east) k i (down) ṙ n = V cos cos v horizontal'component' of'airspeed'vector' i i ṙ d = V sin ṙ e = V sin cos Dubins airplane model: ṙ n = V cos cos c ṙ e = V sin cos c ṙ d = V sin c g = V tan c. with the constraint that V cos Flight'path'projected'onto'ground' c apple c apple.

30 3D Vector Field Path Following Path defined as intersection of two 2D manifolds in R 3 : 1 (r) =0 2 (r) =0 The path given by the intersection is connected and one-dimensional. Define the function V (r) = (r) (r). Consider the velocity command u 0 = where K 1 > 0 and K 2 1 K 1 + K 2 } 2, } Forces cross track convergence Forces along track convergence Method based on: Goncalves, et. al., Vector Fields for Robot Navigation Along Time-Varying Curves in n-dimensions, IEEE Transactions on Robotics, vol. 26, pp , 2010.

31 3D Vector Field Path Following To ensure that the magnitude of the velocity vector is V, normalize u 0 as u = V u0 ku 0 k. The commanded flight-path angle c, and the desired heading angle d are therefore given by h c = sat sin 1 u i 3 V d = atan2(u 2,u 1 ). Method based on: Goncalves, et. al., Vector Fields for Robot Navigation Along Time-Varying Curves in n-dimensions, IEEE Transactions on Robotics, vol. 26, pp , 2010.

32 3D Vector Field Path Following Straight-line Paths lon (r) 4 = n > lon(r c`) =0 lat (r) 4 = n > lat(r c`) =0. lon (r) =0 n lon n lat c` q` Parameterized by lat (r) =0 Start Position c` P line (r, q`) Method based on: Goncalves, et. al., Vector Fields for Robot Navigation Along Time-Varying Curves in n-dimensions, IEEE Transactions on Robotics, vol. 26, pp , 2010.

33 3D Vector Field Path Following Helical Paths cyl (r) = pl (r) = rn c n R h rd c d R h 2 + re + tan h h R h 2 c e 1 tan 1 re r n c e c n h. Parameterized by Spiral Center (c n,c e,c d ) Radius R h Direction h Initial heading h Method based on: Goncalves, et. al., Vector Fields for Robot Navigation Along Time-Varying Curves in n-dimensions, IEEE Transactions on Robotics, vol. 26, pp , 2010.

34 Outline Control architecture Low level autopilot loops Path following Dubins airplane paths and path management

35 Control Architecture destination, obstacles map Dubins path or waypoints path planner status path manager straight-line or spiral path tracking error airspeed, altitude, heading, commands path following autopilot position error servo commands wind unmanned aircraft state estimator on-board sensors ˆx(t)

36 Options for Dubins car Dubins Airplane c ls z s s c ls s z s c rs RSR z e c re c rs e e c le c c le re z e RSL c ls z s s c ls z s s c rs LSR z e c re e c le c rs LSL cre z e e c le For Dubins airplane, also need to address altitude

37 Dubins Airplane Low altitude case: z de z ds apple L car (R min ) tan, Can achieve altitude gain without modifying Dubins car path RSR z e z s RSL z e z s 1 0 North East North East 200 North z e LSR LSL z e z s 0 z s East 1 0 North East

38 Dubins Airplane High altitude case: z de z ds > [L car (R min )+2 R min ] tan. Can achieve altitude gain by spiraling at beginning or end of Dubins car path RSR z e RSL z e z s 0 0 North North East z s zs 0 1 East z s East z e 1 LSR 0 North 1 North 0 z e 1 LSL 0 East

39 Dubins Airplane Medium altitude case: L car (R min ) tan < z de To achieve altitude gain, must add path deviation. z ds apple [L car (R min )+2 R min ] tan RLSR z e RLSL ze 1 1 z s 0 North 1 0 East 1 1 North 2 0 z s 0 East 1 z s z s z e LSLR North East North 1 0 z e LSRL 1 0 East

40 Path Management H` q` c e e e =1 z e = w e Follow start helix Cross H s w` q e H e Intermediate arc at beginning no yes Follow intermediate arc Cross H i Follow straight line Cross H` i =1 w s c i q i i H i w i Intermediate arc at end no Follow end helix yes Follow intermediate arc Cross H i s = 1 c s s q s H s Cross H e END z s

41 Summary Autopilot Successive Loop Closure for Lateral Control Total Energy Control System for Longitudinal Control Path Following 3D Vector field method Path Management Dubins airplane paths Methods are computationally simple, and easily fit on embedded processors. Methods have flown extensively on small fixed wing UAS.

42 Questions?

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