Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 1

Chapter 9

Nonlinear Design Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 2

Architecture

Path planner

Path manager

Path following

Autopilot

Unmanned Vehicle

Waypoints

On-board sensors

Position error

Tracking error

Status

Destination, obstacles

Servo commands State estimator

Wind

Path Definition

Airspeed, Altitude, Heading,

commands

Map

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 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 point-mass models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 4

Autopilot Models (transfer function) Airspeed hold and roll hold loops can be modeled as:

Va(s) =bVa

s+ bVa

V ca (s) �(s) =

b�s+ b�

�c(s)

Altitude and course hold loops can be modeled as:

h(s) =bhs+ bh

s2 + bhs+ bhhc(s) �(s) =

b�s+ b�s2 + b�s+ b�

�c(s)

Alternatively, the heading hold loop can be modeled as:

(s) =b s+ b

s2 + b s+ b c

(s)

Flight-path angle and load factor loops can be modeled as:

�(s) =b�

s+ b��c(s) nlf (s) =

bns+ bn

nclf (s)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 5

Autopilot Models Airspeed hold and roll hold loops can be modeled as:

˙Va = bVa(Vca � Va)

˙� = b�(�c � �)

Altitude and course hold loops can be modeled as:

¨h = bh(˙hc � ˙h) + bh(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 � �)

nlf = bn(nclf � nlf )

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 6

Kinematic Model of Controlled Flight

Flight  path  projected  onto  ground  

horizontal  component  of  groundspeed  vector  

The velocity vector can be written as

Vig = Vg

0

@cos� cos �sin� cos �� sin �

1

A

which gives:

0

@pnpe˙h

1

A= Vg

0

@cos� cos �sin� cos �

sin �

1

A

Alternatively, using the Wind Triangle:

0

@pnpe˙h

1

A= Va

0

@cos cos �asin cos �a

sin �a

1

A+

0

@wn

we

�wd

1

A

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 7

Kinematic Model of Controlled Flight

Flight  path  projected  onto  ground  

horizontal  component  of  groundspeed  vector  

Beginning with

0

@pnpe˙h

1

A= Va

0

@cos cos �asin cos �a

sin �a

1

A+

0

@wn

we

�wd

1

A

and assuming level flight, i.e., �a = 0,

and no down component of wind,

i.e., wd = 0:

0

@pnpe˙h

1

A= Va

0

@cos sin 0

1

A+

0

@wn

we

0

1

A

This equation is typically called the Du-

bin’s car model.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 8

Coordinated Turn

Vg

VwVa

�a

�

↵✓

ib

Differentiate both sides of vector wind-triangle equation (2.9). Solve resulting messy matrix equation.

From Chapter 2 we have

� =

g

Vgtan� cos(�� )

For constant-altitude flight with no down

component of wind:

˙Vg =

˙Va

cos(�� )+ Vg� tan(�� )

˙ =

˙Va

Vatan(�� ) +

Vg�

Va cos(�� )

If airspeed is constant, then we get the

standard coordinated turn condition

˙ =

g

Vatan�

which is true even if when wn, we 6= 0.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 9

Accelerating Climb Summing forces gives

Flift

cos� = mVg� +mg cos �

Solving for �:

� =

g

Vg

✓Flift

mgcos�� cos �

◆

The load factor (often expressed in g’s) is definedas

nlf4= F

lift

/mg.

Note that for wings level, horizontal flight nlf = 1.

Therefore

� =

g

Vg(nlf cos�� cos �)

In a constant climb where � = 0, we have nlf =

cos �cos� .

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 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 11

Kinematic Guidance Model - #1a

The simplest model (and the one used in Chapters 10, 11, 12) is given by

pn = Va cos + wn

pe = Va sin + we

� = b�(�c � �) + b�(�

c � �)

¨h = bh(˙hc � ˙h) + bh(h

c � h)

˙Va = bVa(Vca � Va)

where is given by (equation (2.12))

= �� sin

�1

1

Va

✓wn

we

◆>✓� sin�cos�

◆!

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 12

Kinematic Guidance Model - #1b

If the autopilot is designed to command heading instead of course, then the

model becomes

pn = Va cos + wn

pe = Va sin + we

¨ = b (˙ c � ˙ ) + b (

c � )

¨h = bh(˙hc � ˙h) + bh(h

c � h)

˙Va = bVa(Vca � Va)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 13

Kinematic Guidance Models - #2a A more accurate model is to command the roll angle and use the coordinated

turn model for �:

pn = Va cos + wn

pe = Va sin + we

� =

g

Vgtan� cos(�� )

¨h = bh(˙hc � ˙h) + bh(h

c � h)

˙Va = bVa(Vca � Va)

˙� = b�(�c � �)

where and Vg are given by (equations (2.10) and (2.12))

= �� sin

�1

1

Va

✓wn

we

◆>✓� sin�cos�

◆!

Vg = wn cos�+ we sin�+

p(wn cos�+ we sin�)2 + V 2

a � w2n � w2

e

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 14

Kinematic Guidance Models - #2b Or, in terms of heading we have

pn = Va cos + wn

pe = Va sin + we

=g

Vatan�

h = bh(hc � h) + bh(h

c � h)

Va = bVa(Vca � Va)

� = b�(�c � �)

where � and Vg are computed from the wind triangle if needed by the autopilot

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 15

Kinematic Guidance Models - #3 More accurate still is to command the flight path angle

pn = Va cos cos �a + wn

pe = Va sin cos �a + we

˙h = Va sin �a � wd

˙ =

g

Vatan�

� = b�(�c � �)

˙Va = bVa(Vca � Va)

˙� = b�(�c � �)

where �a is given by equation (2.11)

�a = sin

�1

✓Vg sin � + wd

Va

◆

and � and Vg are computed from the wind triangle if needed by the autopilot

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 16

Kinematic Guidance Models - #4 More accurate still is to command the load factor

pn = Va cos cos �a + wn

pe = Va sin cos �a + we

˙h = Va sin �a � wd

˙ =

g

Vatan�

� =

g

Vg(nlf cos�� cos �)

˙Va = bVa(Vca � Va)

˙� = b�(�c � �)

nlf = bn(nclf � nlf )

where �, Vg, and �a are computed from the wind triangle

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 17

Dynamic Guidance Models The ground speed and course dynamics can be

expressed as

˙Vg =

Fthrust

m� Fdrag

m� g sin �

and

� =

g

Vgtan� cos(�� )

=

g

Vg

sin� cos(�� )

cos �n

=

Flift

mVg

sin� cos(�� )

cos �

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 18

Dynamic Guidance Models The dynamic guidance model is therefore

pn = Vg cos� cos � Flift

=

1

2

⇢Va2SCL

pe = Vg sin� cos � Fdrag

=

1

2

⇢Va2SCD

˙h = Vg sin � CD = CD0 +KCL2

˙Vg =

Fthrust

m� F

drag

m� g sin � E

max

4=

✓Flift

Fdrag

◆

max

� =

Flift

mVg

sin� cos(�� )

cos �K =

1

4E2

max

CD0

� =

Flift

mVgcos�� g

Vgcos �

where Va and are computed from the wind triangle as

Va =

vuuutV 2

g � 2Vg

0

@cos� cos �sin� cos �� sin �

1

A>0

@wn

we

wd

1

A+ V 2

w

= ��sin

�1

1

Va cos �a

✓wn

we

◆>✓�sin�cos�

◆!.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 19

Dynamic Guidance Models

In the no wind case we have

pn = Va cos cos � Flift

=

1

2

⇢V 2

a SCL

pe = Va sin cos � Fdrag

=

1

2

⇢V 2

a SCD

˙h = Va sin � CD = CD0 +KCL2

˙Va =

Fthrust

m� F

drag

m� g sin � E

max

4=

✓Flift

Fdrag

◆

max

˙ =

Flift

mVa

sin�

cos �K =

1

4E2

max

CD0

� =

Flift

mVacos�� g

Vacos �

Inputs are thrust, lift coe�cient, and bank angle.