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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
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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
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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
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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
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CPP Research UAVs
Sig Kadet Airplane
SR-100 Helicopter
12’ Telemaster Airplane
Raptor-90 Helicopter
Twin-Engine Airplane
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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
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Nonlinear Flight Dynamics Model
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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
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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
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-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
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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
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0
50
r (
de
g/s
ec)
Time (sec)
Airplane Longitudinal Response
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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
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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
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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).
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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
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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
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Nonlinear Dynamic Inversion Model
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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
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Acknowledgements
• NSF Award No. 1102382
• Hovig Yaralian
• Matthew Rose
• Nigam Patel
• Luis Andrade
• Dr. Subodh Bhandari, Mentor
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Questions?
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