1. Multi-Disciplinary Challenges in Analysis and Design of Flapping-Wing MAVs
Vehicle Technology Directorate
Army Research Laboratory
Sep 10, 2015
Muhammad Hajj
Biomedical Engineering and Mechanics
Virginia Tech
Haithem Taha
Mechanical and Aerospace Engineering
University of California, Irvine

2. Hedrick & Daniel, J. Exp. Biol. 2006.
Hovering Insects
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Hedrick & Daniel, J. Exp. Biol

3. Taha et al., Nonlinear Dynam. 2012.
Challenges!
Taha et al., Nonlinear Dynam

4. Quasi-Steady Aerodynamics
1. Aerodynamic Modeling: Gaps in the Literature
Forward Flight
Hovering
𝑘 < 0.1
Quasi-Steady Aerodynamics
𝑘 > 0.1
𝛼<15−20°
2D:
* Theodorsen
* Wagner
* Schwarz and Sohngen
* Peters
3D:
* RT Jones
* Reissner
* ULLT
* UVLM
𝛼>25°
LEV Contribution
Unsteadiness
Proposed Model
Proposed
Model
There is a need to develop an aerodynamic model that captures the LEV along with unsteadiness with a feasible computational burden.

5. 1. Aerodynamic Modeling:
Extension of Duhamel Principle to Arbitrary Unconventional Lift Curves
Wagner Function (the Step Response):
Response to Arbitrary 𝛼 𝑡
Extended Duhamel’s Principle
Use of Finite State Aerodynamics
ℓ(𝑠) =ℓ 𝑠𝑡𝑎𝑡𝑖𝑐 𝑊(s)
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Taha et al., Aerosp. Sci Technol

6. 1. Aerodynamic Modeling:
Validation with Sun and Du DNS results
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Taha et al., Aerosp. Sci Technol

7. 2. Flight Dynamics
Longitudinal Flight Dynamics of Rigid Fixed-Wing Aircraft
Time-Invariant System
Autonomous System
Equilibrium/Balance:
Lift = Weight
Thrust = Drag
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Fixed Point
Stability Analysis: Relatively Easy
Inertial and
Gravitational Loads
Aerodynamic and
Propulsive Loads

8. 2. Flight Dynamics
Flight Dynamics of Hovering Insects/ Micro-Air-Vehicles
Neglect Wing flexibility.
Neglect Wing Inertia.
But !
Essentially Time-Periodic
Aerodynamic Loads
Time-Varying System
Non-autonomous System
Equilibrium/Balance: No Fixed Point can be achieved for all times.
Rather a periodic orbit.
Stability Analysis??
Remember that even for 𝒙 = 𝑨 𝑡 𝒙, eigenvalues of A cannot imply (un)stability.

9. 2. Flight Dynamics
Flight Dynamics of Hovering Insects/ Micro-Air-Vehicles
𝒈 𝑎 (𝒙,𝑡)= 𝒈 𝑓 𝜏 +[𝑮 𝜏 ]𝝌(t)
𝑢 (𝑡) 𝑤 (𝑡) 𝑞 (𝑡) 𝜃 (𝑡) = −𝑞(𝑡)𝑤(𝑡)−𝑔𝑠𝑖𝑛𝜃(𝑡) 𝑞(𝑡)𝑢(𝑡)+𝑔𝑐𝑜𝑠𝜃(𝑡) 0 𝑞(𝑡) 𝑋 0 (𝜏) 𝑍 0 (𝜏) 𝑀 0 (𝜏) 𝑋 𝑢 (𝜏) 𝑋 𝑤 (𝜏) 𝑋 𝑞 (𝜏) 0 𝑍 𝑢 (𝜏) 𝑍 𝑤 (𝜏) 𝑍 𝑞 (𝜏) 0 𝑀 𝑢 (𝜏) 0 𝑀 𝑤 (𝜏) 𝑀 𝑞 (𝜏) 𝑢(𝑡) 𝑤(𝑡) 𝑞(𝑡) 𝜃(𝑡)
𝜏: Fast time scale of flapping motion.
t: Slow time scale of body motion.
For the slowest flapping insect (Hawkmoth) 𝜔 𝐹 𝜔 𝑛 ≈30
 Averaging
Taha et al., JGCD 2013.

10. 2. Flight Dynamics Insects are unstable at hover. Direct Averaging
Time-Invariant System
Autonomous System
Equilibrium/Balance:
Average Lift = Weight
Fixed point at the origin: Hovering
Stability Analysis:
Taylor et al
Sun et al
Cheng and Deng 2011
Insects are unstable at hover.
Taha et al., JGCD 2013.

11. 2. Flight Dynamics But ! Averaging Theorem How small is small enough?!
Time-Periodic  Time-Invariant (Autonomous)
Periodic Orbit  Fixed Point
Thm: For small enough 𝝐, Stability of the fixed point Stability of the Periodic Orbit
𝜖≡ 1 𝜔 𝐹
i.e., for high enough flapping frequency 𝝎 𝑭 , direct averaging works.
But !
How small is small enough?!
Taha et al., JGCD 2013.

12. 2. Flight Dynamics Higher Order Averaging
First-order averaging (direct):
𝑋,𝑌 = 𝜕𝑌 𝜕𝑥 𝑋− 𝜕𝑋 𝜕𝑥 𝑌
Second-order averaging:
Thm: If the series 𝒊=𝟏 ∞ 𝛬 𝒊 converges to 𝛬, then
Stability of the fixed point Stability of the Periodic Orbit irrespective of 𝝐.
Agrachev & Gamkrelidze, Mathematics of the USSR 1979:
Exponential Representation of Flows and Chronological calculus
Sarychev, and Vela, 2003:
Complete Averaging

13. Perturbation Techniques (MMS) Higher-Order Averaging
2. Flight Dynamics
Necessity of Higher Order Techniques
Floquet
Perturbation Techniques (MMS)
Higher-Order Averaging
Numerical
Analytical
Vibrational Stabilization
Parametric
Excitation
- Taha et al., Nonlinear Dynamics 2014.
- Taha et al., Bioinspiration and Biomemtics 2015.

14. 2. Flight Dynamics 𝑞 = 𝜃 =−28.45𝜃 Spring Action
Vibrational Stabilization: Stabilizing Mechanism
First-order averaging (direct):
Second-order averaging:
𝑢 𝑤 𝑞 𝜃 = 𝐷 𝜖 𝜦 𝟏 + 𝜖 2 𝜦 2 (𝟎) 𝑢(𝑡) 𝑤(𝑡) 𝑞(𝑡) 𝜃(𝑡)
𝑞 = 𝜃 =−28.45𝜃
Spring Action
Taha et al., Bioinspiration and Biomemtics 2015.

15. 2. Flight Dynamics Vibrational Stabilization: Kapitza Pendulum
Similar Spring Action

16. Change in the stability characteristics
2. Flight Dynamics
Applicability of Direct Averaging
Insect
f (Hz)
𝝎 𝑭 𝝎 𝒏
Change in the stability characteristics
Hawkmoth
26.3
28.78 √
Cranefly
45.5
50.62
Bumblebee
155
144.46 x
Dragonfly
157
145.50
Hoverfly
160
113.98
𝝎 𝑭 𝝎 𝒏 <𝟓𝟎: Direct averaging is NOT applicable
𝝎 𝑭 𝝎 𝒏 >𝟏𝟎𝟎: Direct averaging is applicable
- Taha et al., Nonlinear Dynamics 2014.
- Taha et al., Bioinspiration and Biomemtics 2015.

17. Current and Future Interests
1- Unsteady Nonlinear Aerodynamics
Theodorsen 1938
Zhimiao, Taha and Hajj
Aerosp. Sci Technol. 2014

18. Current and Future Interests
1- Unsteady Nonlinear Aerodynamics
Zhimiao, Taha and Hajj Aerosp. Sci Technol. 2014

19. Current and Future Interests
1- Unsteady Nonlinear Aerodynamics
Zakaria, Taha and Hajj Under Review

20. Geometric Control = Differential Geometry + Control Theory
Current and Future Interests
2- Geometric Control Theory
Geometric Control = Differential Geometry + Control Theory
Differential Geometry: Mathematical tool to perform calculus on curvy spaces.
“Mechanics is the paradise of mathematical sciences, because by means of it one comes to the fruits of mathematics.”
Leonardo da Vinci

21. Current and Future Interests
2- Geometric Control Theory
Controllability: The ability to steer(drive) a system from point A to point B.
Controllability of Linear Systems
𝒙 =𝑨𝒙+𝑩𝒖
The system is controllable if and only if the controllability matrix
𝑩 𝑨𝑩 𝑨 2 𝑩 … 𝑨 𝑛−1 𝑩
has rank n.
Controllability of Nonlinear Systems: Linearize!
If the linearized system is controllable, then the nonlinear system is controllable.
If the linearized system is NOT controllable, the nonlinear system may still be controllable.
But !

22. Current and Future Interests
2- Geometric Control Theory
Kinematic Car Example
Linearization:
𝑥 =𝑉 cos 𝜃
𝑦 =𝑉 sin 𝜃
𝜃 =𝜔 y
𝑉= 𝑢 1
𝜔= 𝑢 2 𝜃 x
𝑥 𝑦 𝜃 = cos 𝜃 sin 𝜃 𝒈 1 𝑢 𝒈 2 𝑢 2
𝑥 𝑦 𝜃 = cos 𝜃 sin 𝜃 0 𝑢 𝑢 2
𝑥 𝑦 𝜃 = cos 𝜃 0 sin 𝜃 rank 2 𝑢 1 𝑢 2
𝑥 𝑦 𝜃 =0+ cos 𝜃 0 sin 𝜃 𝑢 1 𝑢 2
Linearly Uncontrollable!

23. Motion Generation in Unactuated Directions!
Current and Future Interests
2- Geometric Control Theory
Kinematic Car Example Lie Bracket
𝑥 𝑦 𝜃 = cos 𝜃 sin 𝜃 𝒈 1 𝑢 𝒈 2 𝑢 2 y
𝑉= 𝑢 1
𝜔= 𝑢 2 𝜃 x
𝒈 1 , 𝒈 2 = 𝜕 𝒈 2 𝜕𝒙 𝒈 1 − 𝜕 𝒈 1 𝜕𝒙 𝒈 2 = −sin 𝜃 cos 𝜃 0
Nonlinearly Controllable!
Motion Generation in Unactuated Directions!

24. Spacecraft Attitude Control:
Current and Future Interests
2- Geometric Control Theory
Spacecraft Attitude Control
Three pairs of gas jets
Two pairs of gas jets
One pair of gas jets
Geometric Nonlinear Analysis:
Crouch, IEEE Transactions on Automatic Control, 1984.
The system is still controllable with only one pair of gas jets.
 Linearly Controllable.
 Linearly Uncontrollable.
 Linearly Uncontrollable.
Unconventional Force Generation and Stabilization Mechanisms in Bio-Locomotion Systems

25. Acknowledgment UCI Virginia Tech: Students: Prof. Kenneth Mease
Prof. Ali Nayfeh
Prof. Craig Woolsey
Prof. John Burns
Prof. Robert Canfield
Prof. Mayuresh Patil
Students:
Sevak Tahmasian
Zhimia Yan
Mohammed Zakaria
Ahmed Roman
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26. Multi-Disciplinary Challenges in Analysis and Design of Flapping-Wing MAVs
Thank You!
Muhammad Hajj
Biomedical Engineering and Mechanics
Virginia Tech
Haithem Taha
Mechanical and Aerospace Engineering
University of California, Irvine