1. University of California, Irvine Craig A Woolsey and Muhammad R Hajj
 A Geometric Control Approach for the Longitudinal Flight Stability of Hovering Insects/FWMAVs
Haithem E Taha
University of California, Irvine
Craig A Woolsey and Muhammad R Hajj
Virginia Tech
AIAA Science & Technology Forum & Exposition, 5-9 Jan 2015, Kissimmee, Florida
Haithem Taha

2. Flight Dynamics of Hovering Insects: Two Major Assumptions
Neglecting the Wing Inertial Effects
Averaging the Dynamics over the Cycle
𝑚 w 𝑚 v ~5%
𝝌 (𝑡)=𝒇 𝝌 𝑡 + 𝒈 𝑎 (𝝌(𝑡),τ)
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
Essentially Time-Periodic
Aerodynamic Loads
Taylor et al
Sun et al
Cheng and Deng 2011
Taha et al., JGCD 2013.
Remember that even for 𝒙 = 𝑨 𝑡 𝒙, eigenvalues of A cannot imply (in)stability.
Insects are unstable at hover.
Haithem Taha

3. Objective
Rigorously assess the flight stability of hovering insects/FWMAVs while including the wing inertial effects and relaxing the direct averaging assumption, using an analytical approach.
𝝌= 𝑢 𝑤 𝑞 𝜃 𝜑 𝜑
Haithem Taha

4. Geometric Control Theory
Equations of Motion
Multi-body Problem
Taha et al., AIAA 2013.
𝜏 𝜑 =𝑈cos𝜔𝑡
Nonlinear, Time-Periodic, Mechanical System (NLTP)
Geometric Control Theory
Haithem Taha

5. Geometric Control Remedies for Direct Averaging Variation of Constants Formula: high-frequency, high-amplitude, periodic forcing
ad 𝒈 𝒇= 𝒈,𝒇 = 𝜕𝒇 𝜕𝒙 𝒈− 𝜕𝒈 𝜕𝒙 𝒇
Agrachev & Gamkrelidze, Mathematics of the USSR 1979:
Exponential Representation of Flows and Chronological calculus

6. Smooth Approximation of the Hovering Dynamics
Dependence on | 𝜑 | and sign( 𝜑 )
𝜑 ≅ 𝜑 2 𝜀 and sign( 𝜑 )≅ 2 𝜑 𝜀 near the origin 𝜑 ∈[−𝜀,𝜀]

7. Trim/Balance Trim/Balance 𝑔+ 𝑈 2 4 𝜔 2 𝑔 𝑐 (4) =0 𝝌 =𝒇(𝝌)+g(𝝌)Ʋ(t)
Ʋ(t)=𝑈cos𝜔𝑡
𝝌 =𝑭( 𝝌 )=𝒇( 𝝌 )+ 𝑈 2 4 𝜔 2 [g,[g,f]] 𝒈 𝑐 ( 𝝌 )
Trim/Balance
Hovering ( 𝑢 =0, 𝑤 =0, 𝑞 =0, 𝜑 =0), with Symmetric Flapping ( 𝜑 =0)
𝑔+ 𝑈 2 4 𝜔 2 𝑔 𝑐 (4) =0

8. Trim/Balance
𝑔+ 𝑈 2 trim 4 𝜔 2 𝑔 𝑐 (4) =0
Versus

9. Original and Averaged Dynamics
𝝌= 𝜃 𝜑 𝑢 𝑤 𝑞 𝜑
𝝌 =𝒇(𝝌)+g(𝝌)Ʋ(t)
𝝌 =𝒇( 𝝌 )+ 𝑈 2 4 𝜔 2 [g,[g,f]] 𝒈 𝑐 ( 𝝌 )
𝒈(𝟎)= 1 ∆ − 𝑚 w 𝑟 𝑐𝑔 𝐼 𝑦,tot 0 𝑚 w 𝑟 𝑐𝑔 𝑑 sin 𝛼 𝑚 𝑚 v 𝐼 𝑦,tot − 𝑚 w 2 𝑑 2 sin 2 𝛼 𝑚 sin 2 𝛼 𝑚
𝒈 𝑐 (𝟎)= 𝑔 𝑐 (4) 0 0

10. Stability Analysis NLTP 𝝌 =𝒇(𝝌)+g(𝝌)Ʋ(t) Averaging Theory
𝝌 =𝒇( 𝝌 )+ 𝑈 2 4 𝜔 2 [g,[g,f]] 𝒈 𝑐 ( 𝝌 )
NLTI
Lyapunov Indirect Method
𝝌 =[ 𝑨 𝑑 + 𝑈 2 4 𝜔 2 𝑨 𝑐 ] 𝝌
LTI

11. Stability Analysis Inertial Aerodynamic Total Pitch Stiffness -ve Zero
Pitch Damping
+ve
u-Damping
w-Damping
Su and Cesnik, AIAA 2009.
Taylor et al., AIAA 2006.

12. Stabilization Using Simple Pitch Feedback
𝐾 𝜃 does not affect pitch stiffness because
of lack of pitch control authority at hover
when using symmetric Flapping.
𝐾 𝜃 >0.0186
𝐾 𝜃 <0.0065
Positive pitch stiffness: 𝐾 𝜃 >0.0023
Doman et al., JGCD 2010.

13. Thank You! Haithem E Taha
 A Geometric Control Approach for the Longitudinal Flight Stability of Hovering Insects/FWMAVs
Thank You!
Haithem E Taha
Mechanical and Aerospace Engineering
University of California, Irvine