Chapter 9. Nonlinear Design Models. Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 1

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1 Chapter 9 Nonlinear Design Models Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 1

2 Architecture Destination, obstacles Waypoints Path Definition Airspeed, Altitude, Heading, commands Servo commands Wind Path planner Path manager Path following Autopilot Unmanned Vehicle Map Status Tracking error Position error State estimator On-board sensors Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 2

3 Simplifying Dynamic Models When designing higher level autopilot functions, we need models that are easier to analyze and simulate Models must capture the essential behavior of system We will derive reduced-order, reduced-complexity models suitable for design of higher-level guidance strategies Two types of guidance models: Kinematic utilize kinematic relationships, do not consider aerodynamics, forces directly Dynamic apply force balance relations to pointmass models Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 3

4 Autopilot Models (transfer function) Airspeed hold and roll hold loops can be modeled as: (s) = b s + b Va V c a (s) (s) = b s + b c (s) Altitude and course hold loops can be modeled as: h(s) = b ḣ s + b h s 2 + bḣs + b h h c (s) (s) = b s + b s 2 + b s + b c (s) Alternatively, the heading hold loop can be modeled as: (s) = b s + b s 2 + b s + b c (s) Flight-path angle and load factor loops can be modeled as: (s) = b s + b c (s) n lf (s) = b n s + b n n c lf (s) Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 4

5 Autopilot Models Airspeed hold and roll hold loops can be modeled as: = b Va (Va c ) = b ( c ) Altitude and course hold loops can be modeled as: ḧ = bḣ(ḣc ḣ)+b h (h c h) = b ( c )+b ( c ) Alternatively, the heading hold loop can be modeled as: = b ( c )+b ( c ) Flight-path angle and load factor loops can be modeled as: = b ( c ) ṅ lf = b n (n c lf n lf ) Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 5

6 Kinematic Model of Controlled Flight The velocity vector can be written as 0 cos cos 1 Vg i = V sin cos sin A which gives: ṗ ṗ e A = V cos cos sin cos ḣ sin 1 A horizontal component of groundspeed vector Flight path projected onto ground Alternatively, using the Wind Triangle: ṗ ṗ e A cos cos 1 0 a sin cos a A w 1 n w e A ḣ sin a w d Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 6

7 Kinematic Model of Controlled Flight Beginning with ṗ n cos cos ṗ e A sin cos a A + ḣ sin a w n w e w d 1 A and assuming level flight, i.e., a = 0, and no down component of wind, i.e., w d = 0: ṗ n ṗ e ḣ 1 A = cos sin 0 1 A + w 1 n w e A 0 This equation is typically called the Dubin s car model. horizontal component of groundspeed vector Flight path projected onto ground Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 7

8 From Chapter 2 we have Coordinated Turn = g V g tan cos( ) i b V w For constant-altitude flight with no down component of wind: a V g V g = cos( ) + V g tan( ) = Va tan( )+ V g cos( ) Differentiate both sides of vector wind-triangle equation (2.9). Solve resulting messy matrix equation. If airspeed is constant, then we get the standard coordinated turn condition = g tan which is true even if when w n,w e 6= 0. Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 8

9 Accelerating Climb Summing forces gives F lift cos = mv g + mg cos Solving for : = g V g Flift mg cos cos The load factor (often expressed in g s) is defined as n lf 4 = Flift /mg. Note that for wings level, horizontal flight n lf = 1. Therefore = g V g (n lf cos cos ) In a constant climb where = 0, we have n lf = cos cos. Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 9

10 Kinematic Guidance Models Several guidance models can be derived, with varying levels of fidelity Choice of model depends on application Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 10

11 Kinematic Guidance Model - #1a The simplest model (and the one used in Chapters 10, 11, 12) is given by ṗ n = cos + w n ṗ e = sin + w e = b ( c )+b ( c ) ḧ = bḣ(ḣc ḣ)+b h (h c h) = b Va (V c a ) where is given by (equation (2.12)) = sin 1 1 wn w e > sin cos! Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 11

12 Kinematic Guidance Model - #1b If the autopilot is designed to command heading instead of course, then the model becomes ṗ n = cos + w n ṗ e = sin + w e = b ( c )+b ( c ) ḧ = bḣ(ḣc ḣ)+b h (h c h) = b Va (V c a ) Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 12

13 Kinematic Guidance Models - #2a A more accurate model is to command the roll angle and use the coordinated turn model for : ṗ n = cos + w n ṗ e = sin + w e = g V g tan cos( ) ḧ = bḣ(ḣc ḣ)+b h (h c h) V a = b Va (Va c ) = b ( c ) where and V g are given by (equations (2.10) and (2.12)) = sin 1 1 wn w e > sin cos Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 13! V g = w n cos + w e sin + p (w n cos + w e sin ) 2 + V 2 a w 2 n w 2 e

14 Kinematic Guidance Models - #2b Or, in terms of heading we have ṗ n = cos + w n ṗ e = sin + w e = g tan ḧ = bḣ(ḣc ḣ)+b h (h c h) V a = b Va (Va c ) = b ( c ) where and V g are computed from the wind triangle if needed by the autopilot Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 14

15 Kinematic Guidance Models - #3 More accurate still is to command the flight path angle ṗ n = cos cos a + w n ṗ e = sin cos a + w e ḣ = sin a g = tan = b ( c ) w d V a = b Va (Va c ) = b ( c ) where a is given by equation (2.11) a =sin 1 Vg sin + w d and and V g are computed from the wind triangle if needed by the autopilot Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 15

16 Kinematic Guidance Models - #4 More accurate still is to command the load factor ṗ n = cos cos a + w n ṗ e = sin cos a + w e ḣ = sin a g = tan w d = g V g (n lf cos cos ) V a = b Va (Va c ) = b ( c ) ṅ lf = b n (n c lf n lf ) where, V g, and a are computed from the wind triangle Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 16

17 Dynamic Guidance Models The ground speed and course dynamics can be expressed as V g = F thrust m F drag m g sin and = g V g tan cos( ) = g sin cos( ) n V g cos = F lift sin cos( ) mv g cos Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 17

18 Dynamic Guidance Models The dynamic guidance model is therefore ṗ n = V g cos cos F lift = SC L ṗ e = V g sin cos F drag = SC D 2 ḣ = V g sin C D = C D0 + KC L V g = F thrust F drag 4 Flift g sin E max = m m F drag = F lift sin cos( ) mv g cos = F lift mv g cos g V g cos K = Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide E 2 maxc D0 where and are computed from the wind triangle as v 0 1> 0 1 u cos cos w n = tvg 2 2V sin cos w e A + Vw 2 sin w d >! = sin 1 1 wn sin. cos a w e cos max

19 Dynamic Guidance Models In the no wind case we have ṗ n = cos cos F lift = 1 2 V 2 a SC L ṗ e = sin cos F drag = 1 2 V 2 a SC D 2 ḣ = sin C D = C D0 + KC L V a = F thrust F drag 4 Flift g sin E max = m m F drag F lift sin 1 = K = m cos = F lift m cos g cos 4E 2 maxc D0 max Inputs are thrust, lift coe cient, and bank angle. Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 19