Nonlinear Control of UAVs Using Dynamic Inversion Conference... · Nonlinear Control of UAVs Using Dynamic Inversion Alejandro Osorio Department of Aerospace Engineering Cal Poly

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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