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Nonlinear Control of UAVs Using

Dynamic Inversion

Alejandro Osorio

Department of Aerospace Engineering

Cal Poly Pomona

AIAA Aerospace Systems and Technology (ASAT) Conference

May 3, 2014

Overview

• Unmanned Aerial Vehicles

• Motivations

• Research Objectives

• Twin-Engine Airplane

• Nonlinear Flight Dynamics Model

• Flight Test for Data Acquisition

• Nonlinear Dynamic Inversion

• Future Work

2

Advantages of Unmanned Aerial Vehicles

• Do not contain or need a qualified pilot on board

• Can enter environments that are dangerous to human life

• UAVs are indispensable for military and civilian

applications

• Military

• Reconnaissance, battlefield damage assessment, strike

capabilities, etc.

• Civilian

• Infrastructure maintenance, agriculture management, disaster

relief, etc.

• Significantly lower operating costs

3

Motivations

• Existing UAVs have a high acquisition cost and are

limited to restricted airspace

• Cost effective operations require

• Increased autonomy, reliability, and availability

• Most existing autopilots are designed using linearized

flight dynamics model and lack robustness

• Nonlinear controllers can work for entire flight envelope,

thereby helping increase UAV autonomy

4

Research Objectives

• Develop and validate nonlinear flight dynamics models for

Cal Poly Pomona UAVs

• Use Dynamic Inversion Technique for the design of

nonlinear controllers for the UAVs

• Verify the controllers in software and hardware-in-the-loop

simulations

• Validate the design in flight tests

• Use H∞ (H-Infinity) control system design technique along

for the design and implementation of robust nonlinear

controllers

5

Cal Poly Pomona UAV Lab

• Dedicated to research on advanced topics in

flight dynamics and control

• The lab consists of airplane and helicopter UAVs

of various sizes and payload capacity

• Sensors and associate equipment

• Internal measurement units

• Differential GPS

• Air data probes

• Commercial-off-the-shelf autopilots

• Laser altimeter

6

CPP Research UAVs

Sig Kadet Airplane

SR-100 Helicopter

12’ Telemaster Airplane

Raptor-90 Helicopter

Twin-Engine Airplane

7

Twin-Engine Airplane

• DA 50 Gasoline engine powered

• Length- 95 inches, wing span- 134 inches

• Empty weight- 42 lbs, payload- up to 25 lbs

• Equipped with Piccolo II autopilot for autonomous flight

and data acquisition

8

Nonlinear Flight Dynamics Model

9

Force Equations:

𝑈 = 𝑅𝑉 +𝑊𝑄 − 𝑔 sin 𝜃 +𝐹𝑋

𝑚

𝑉 = −𝑈𝑅 +𝑊𝑃 + 𝑔 sin ϕ cos 𝜃 +𝐹𝑌

𝑚

𝑊 = 𝑈𝑄 − 𝑉𝑃 + 𝑔 cos ϕ cos 𝜃 +𝐹𝑧

𝑚

Kinematic Equations:

ϕ = 𝑃 + 𝑡𝑎𝑛𝜃(𝑄𝑠𝑖𝑛ϕ + Rcosϕ)

𝜃 = 𝑄𝑐𝑜𝑠ϕ + Rsinϕ

𝛹 =𝑄𝑠𝑖𝑛ϕ + 𝑅𝑐𝑜𝑠ϕ

𝑐𝑜𝑠𝜃

Moment Equations:

P = 𝑐1𝑅 + 𝑐2𝑃 𝑄 + 𝑐3𝐿 + 𝑐4𝑁

𝑄 = 𝑐5𝑃𝑅 + 𝑐6 𝑃2 − 𝑅2 + 𝑐7𝑀 𝑅 = 𝑐8𝑃 − 𝑐2𝑅 𝑄 + 𝑐4𝐿 + 𝑐9𝑁

Navigation Equations:

𝑥 = 𝑈𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛹 + 𝑉 −𝑐𝑜𝑠ϕsin𝛹 + 𝑠𝑖𝑛ϕsinθ𝑠𝑖𝑛𝛹+𝑊 sinϕsin𝛹 + cosϕsinθ𝑐𝑜𝑠𝛹

𝑦 = 𝑈𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝛹 + 𝑉 𝑐𝑜𝑠ϕcos𝛹 + 𝑠𝑖𝑛ϕsinθ𝑠𝑖𝑛𝛹+𝑊 −sinϕcos𝛹 + cosϕsinθ𝑠𝑖𝑛𝛹

ℎ = 𝑈𝑠𝑖𝑛𝜃 − 𝑉𝑠𝑖𝑛ϕcosθ − 𝑐𝑜𝑠ϕcosθ

Aerodynamic Model

10

Aerodynamic Forces and Moments

𝐿𝐴 = 𝑞 𝑆𝐶𝑙𝑏

𝑀 = 𝑞 𝑆𝐶𝑚𝐶 𝑁 = 𝑞 𝑆𝐶𝑛𝑏

Aerodynamic Coefficients

𝐶𝐷 = 𝐶𝐷𝑜 + 𝐶𝐷𝛼𝛼 + 𝐶𝐷𝑞𝑄𝐶

2𝑉𝑜+ 𝐶𝐷𝛼 𝛼

𝐶

2𝑉𝑜+ 𝐶𝐷𝑢

𝑢

𝑉𝑜+ 𝐶𝐷𝛿𝑒𝛿𝑒

𝐶𝑌 = 𝐶𝑌𝛽𝛽 + 𝐶𝑌𝑝𝑃𝑏

2𝑉𝑜+ 𝐶𝑌𝑟𝑅

𝑏

2𝑉𝑜+ 𝐶𝑌𝛿𝑎𝛿𝑎 + 𝐶𝑌𝛿𝑟𝛿𝑟

𝐶𝐿 = 𝐶𝐿𝑜 + 𝐶𝐿𝛼𝛼 + 𝐶𝐿𝑞𝑄𝐶

2𝑉𝑜+ 𝐶𝐿𝛼 𝛼

𝐶

2𝑉𝑜+ 𝐶𝐿𝑢

𝑢

𝑉𝑜+ 𝐶𝐿𝛿𝑒𝛿𝑒

𝐶𝑙 = 𝐶𝑙𝛽𝛽 + 𝐶𝑙𝑝𝑃𝑏

2𝑉𝑜+ 𝐶𝑙𝑟𝑅

𝑏

2𝑉𝑜+ 𝐶𝑙𝛿𝑎𝛿𝑎 + 𝐶𝑙𝛿𝑟𝛿𝑟

𝐶𝑚 = 𝐶𝑚𝑜+ 𝐶𝑚𝛼

𝛼 + 𝐶𝑚𝑞𝑄

𝐶

2𝑉𝑜+ 𝐶𝑚𝛼

𝛼 𝐶

2𝑉𝑜+ 𝐶𝑚𝑢

𝑢

𝑉𝑜+ 𝐶𝑙𝛿𝑒𝛿𝑒

𝐶𝑛 = 𝐶𝑛𝛽𝛽 + 𝐶𝑛𝑝𝑃𝑏

2𝑉𝑜+ 𝐶𝑛𝑟𝑅

𝑏

2𝑉𝑜+ 𝐶𝑛𝛿𝑎𝛿𝑎 + 𝐶𝑛𝛿𝑟𝛿𝑟

𝐷 = 𝑞 𝑆𝐶𝐷

𝐿 = 𝑞 𝑆𝐶𝐿

Y = 𝑞 𝑆𝐶𝑌

Flight Test

• The airplane flown for doublet inputs in aileron, rudder,

and elevator

• The data is used for the model validation

• Validated model is then used for control system design

11

662 664 666 668 670 672 674-10

-5

0

5

Aile

ron

(d

eg

)

Roll Doublet

662 664 666 668 670 672 674-50

0

50

100

Ro

ll A

ng

le (

de

g)

Time (sec)

Model Validation

12

14 16 18 20 22 24-10

0

10

A (

de

g)

Airplane Lateral-Directional Response

14 16 18 20 22 24-100

0

100

p (

de

g/s

ec)

Flight Data

Simulation

14 16 18 20 22 24-50

0

50

r (

de

g/s

ec)

Time (sec)

Airplane Longitudinal Response

13

16 18 20 22 24 26 28 30-10

0

10

Time (sec)

E (

de

g)

Airplane Longitudinal Response

16 18 20 22 24 26 28 30-100

-50

0

50

q (

de

g/s

ec)

Time (sec)

Flight Data

Simulation

FlightGear Model

14

Nonlinear Dynamic Inversion

• The nonlinear dynamic system can be represented as the

first order model

• 𝑥 = 𝑓 𝑥 + 𝑔 𝑥 𝑢

• Both functions f(x) and g(x) are nonlinear in x

• If the system is affine in the controls, then solving

explicitly for the control vector yields

• 𝑢 = 𝑔−1 𝑥 𝑥 − 𝑓 𝑥

• Replacement of the inherent dynamics with the desired

dynamics results in the control that will produce the

desired dynamics

• 𝑢 = 𝑔−1 𝑥 𝑥 𝑑𝑒𝑠 − 𝑓 𝑥

15

Time-Scale Separation

• Standard nonlinear equations of motion cannot be directly

used because the A matrix (system matrix) is not square

• The original dynamic model is formulated as two lower-

order systems

• Translational mechanics

• Rotational dynamics

• Four control inputs = four variables in each time-scale

• Dynamics are separated into slow and fast dynamics

• Slow controlled states are the angle of attack, climb angle, bank

angle and sideslip angle (α, γ, φ, β)

• The fast controlled states are the three angular rates plus the

forward speed (V, P, Q, R).

16

𝛼 = 𝑄 − tan 𝛽 𝑃𝑐𝑜𝑠𝛼 + 𝑅𝑠𝑖𝑛𝛼 +1

𝑚𝑉𝑐𝑜𝑠𝛽−𝐿 +𝑚𝑔𝑐𝑜𝑠𝛾𝑐𝑜𝑠𝜙 − 𝑇𝑠𝑖𝑛𝛼

𝛾 =1

𝑚𝑉𝐿𝑐𝑜𝑠𝜙 − 𝑚𝑔𝑐𝑜𝑠𝛾 − 𝑌𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝛽 +

𝑇

𝑚𝑉, 𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝛽𝑐𝑜𝑠𝛼 + 𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝛼

Φ = 𝑃 + tan 𝜃(𝑄 sinΦ + 𝑅 cosΦ)

𝛽 = 𝑃 sin 𝛼 − 𝑅 cos 𝛼 +1

𝑚𝑉cos 𝛾 sinΦ + 𝑌 cos 𝛽 − 𝑇 sin 𝛽 cos𝛼

Nonlinear Coupled Differential Equations of Motion

17

Slow Dynamics

Fast Dynamics

𝑃 = 𝑐1𝑅 + 𝑐2𝑃 𝑄 + 𝑐3𝐿 + 𝑐4𝑁 𝑄 = 𝑐5𝑃𝑅 − 𝑐6 𝑃2 − 𝑅2 + 𝑐7𝑀

𝑅 = 𝑐8𝑃 − 𝑐2𝑅 𝑄 + 𝑐4𝐿 + 𝑐9𝑁

𝑉 =1

𝑚[−𝐷 + 𝑌𝑠𝑖𝑛𝛽 −𝑚𝑔𝑠𝑖𝑛𝛾 + 𝑇𝑐𝑜𝑠𝛽𝑐𝑜𝑠𝛼]

Time-Scale Separation Cont.

• The outer-loop involves the translational dynamics

• In response to position and velocity commands, it

produces the δ command for the inner-loop to track

• The inner-loop involves the rotational dynamics

• Tracks the attitude reference by determining the δT, δE,

δA, and δR commands

18

Nonlinear Dynamic Inversion Model

19

Future Work

• Further refine the flight dynamics model

• Use flight data for the development of flight dynamics

models using Parameter Identification techniques

• Design nonlinear controllers using dynamic inversion

techniques for complete autonomous missions

• Use H technique to design robust controllers

• Take into account modeling uncertainties

20

Acknowledgements

• NSF Award No. 1102382

• Hovig Yaralian

• Matthew Rose

• Nigam Patel

• Luis Andrade

• Dr. Subodh Bhandari, Mentor

21

Questions?

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