Fig 1.
Covert feathers deflect in response to vertical gusts.
Fig 2.
Electromechanical covert feather [22]. During gusts flap transfers motion to PZT through mechanical linkage and spring. PZT generates voltage and feds it to controller for further generation of control input which is fed through voice coil to flap which deflects to transpire gust.
Fig 3.
Bond graph model of a complete ornithopter.
Bond graph model of the proposed ornithopter is composed of bond graph models of subsystems namely the main body, motors, the flapping mechanism, rigid wings and GMS.
Table 1.
Parameters of the LTI Model.
Fig 4.
Open loop pole zero plot of the ornithopter shows that multiple poles are in right half plan and therefore depicts unstable internal dynamics of the ornithopter.
Fig 5.
Ornithopters’ open loop states response.
Open loop states response of the ornithopter is diverging and therefore depicts unstable internal dynamics of the ornithopter.
Fig 6.
Open loop step response of the multi input multi output (MIMO) system.
Open loop step response of the MIMO ornithopter system is diverging and therefore depicts unstable internal dynamics of the ornithopter.
Fig 7.
H₂ control block diagram showing ornithopter plant G regulated by controller K in feedback loop.
Table 2.
Closed-loop eigenvalues.
Fig 8.
All poles are present in left half plan and therefore depict that the H2 controller has successfully stabilized the unstable ornithopter.
Fig 9.
H2 Closed loop step response of the MIMO ornithopter system.
MIMO step responses under H₂ control show stable tracking of u, w, and θ with settling times <1.5 s and minimal cross-coupling.
Fig 10.
Closed loop state response of ornithopter at 20 m/s gust for reference tracking with r = [0.20 0 0]T.
Step response for forward velocity command (u-only) under H₂ control showing accurate tracking with settling times <1.3 s, and negligible impact on w, θ, and q.
Fig 11.
Closed loop state response of ornithopter at 20 m/s gust for reference tracking with r = [0 0.25 0]T.
Step response for vertical velocity command (w-only) under H₂ control showing smooth tracking with settling times <1.4 s, negligible cross-coupling and well-damped θ and q oscillations.
Fig 12.
Closed loop state response of ornithopter at 20 m/s gust for reference tracking with r = [0.20 0.25 0]T.
Simultaneous step responses for u and w under H₂ control showing accurate multi-input tracking with settling<1.4 s, negligible overshoot and well-damped θ and q dynamics.
Fig 13.
Control input response of ornithopter for reference tracking with r = [0.20 0 0]T.
Control inputs for u-only step showing dominant αₘ actuation (≈ 0.07 rad) with minimal φₒ and αₒ activity, confirming efficient control allocation.
Fig 14.
Control input response of ornithopter for reference tracking with r = [0 0.25 0]T. Control inputs for w-only step showing dominant αₒ (≈ −0.038 rad) with smaller φₒ contribution and negligible αₘ, indicating adaptive control allocation.
Fig 15.
Control input response of ornithopter for reference tracking with r = [0.20 0.25 0]T. Control inputs for simultaneous u–w step showing dominant αₘ (≈ 0.07 rad) with compensatory φₒ (≈ −0.01 rad) and αₒ (≈ −0.038 rad), confirming coordinated multi-axis actuation.
Fig 16.
Closed loop state response to sinusoidal gust for reference tracking with r = [0.20 0.25 0]T. Closed-loop state responses under sinusoidal gust (amplitude 10, frequency 1 Hz) with H₂ control, showing accurate tracking and well-damped oscillations.
Fig 17.
Control input response to sinusoidal gust for reference tracking with r = [0.20 0.25 0]T. Control inputs of ornithopter subjected to sinusoidal gust (amplitude 10, frequency 1 Hz) remain smooth and bounded, confirming robust actuation under continuous gust excitation.
Table 3.
Quantitative comparison of LQR vs H2.
Fig 18.
Closed loop states response of H2 controller vs LQR at 20 m/s step gust for reference tracking with r = [0.20 0.25 0]T.
Closed-loop responses under 20 m/s step gust show that H₂ control improves rise and settling times in u, w, and θ compared to LQR, with zero overshoot and stable convergence within ~1.1 s which is consistent with the experimental findings reported in [15].
Fig 19.
Tracking error of H2 controller vs LQR at 20 m/s step gust for reference tracking with r = [0.20 0.25 0]T.
Tracking error responses for u, w, θ, and q under LQR and H₂ control, showing faster error decay and improved robustness with H₂.
Fig 20.
Control effort of H2 controller vs LQR at 20 m/s step gust for reference tracking with r = [0.20 0.25 0]T.
Comparison of control input responses showing that the H₂ controller settles faster but demands higher initial control effort than the LQR controller.
Fig 21.
Gust mitigation system (GMS) installed ornithopter vertical displacement at 20 m/s gust.
Vertical displacement of ornithopter under 20 m/s step gust showing 32% reduction with GMS and H₂ control actuation, confirming effective gust alleviation by the proposed design.