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Stabilizing air dampers for hovering aerial robotics: design, insect-scale flight tests, and scaling

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Abstract

Most hovering aircraft such as helicopters and animal-inspired flapping-wing flyers are dynamically unstable in flight, quickly tumbling in the absence of feedback control. The addition of feedback loops can stabilize, but at the cost of additional sensing and actuation components. This can add expense, weight, and complexity. An alternative to feedback is the use of passive mechanisms such as aerodynamic drag to stabilize attitude. Previous work has suggested that small aircraft can be stabilized by adding air dampers above and below the center of mass. We present flight tests of an insect-scale robot operating under this principle. When controlled to a constant altitude, it remains stably upright while undergoing cyclic attitude oscillations. To characterize these oscillations, we present a nonlinear analytic model derived from first principles that reproduces the observed behavior. Using numerical simulation, we analyze how changing damper size, position, mass, and midpoint offset affect these oscillations, building on previous work that considered only a single configuration. Our results indicate that only by increasing damper size can lateral oscillation amplitude be significantly reduced, at the cost of increased damper mass. Additionally, we show that as scale diminishes, the damper size must get relatively larger. This suggests that smaller damper-equipped robots must operate in low-wind areas or in boundary-layer flow near surfaces.

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Notes

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    A detailed analysis suggests that if instead the damper is considered to be made of beams that must support the load of a impact landing, quadratic scaling is reasonable. We consider the damper support structure as a simply-supported beam. We neglect the mass of the polyester layer because it has similar density (\(\approx \)15  kg/m\(^{3}\)) to carbon fiber but a much lower thickness, at \(3~\upmu \) m compared to \(80~\upmu \) m for the carbon fiber composite. Assume the maximum force the beam is expected to support is \(F^{*}\), caused by, for example, crash landings. Then the greatest moment applied to the beam of length l occurs at its base, and is equal to \(M^{*}=F^{*}l\). Suppose the beam has a width w and thickness t (\(\approx \)80 \(\upmu \) m for our material). Tensile stress inside the beam at its base is \(\sigma =\frac{M^{*}y}{I}\), where \(I=\frac{wt^{3}}{12}\) is the moment of inertia of the rectangular beam and y is the distance from its centerline. Then maximum stress \(\sigma ^{*}\) in the material occurs at the top and bottom of the beam, and is equal to3 \(\sigma ^{*}=\frac{M^{*}t/2}{I}\).

    Our interest is in a scaling law that, for a constant load \(F^{*}\) and material strength \(\sigma ^{*}\), gives the mass of the damper. Substituting the above equations into each other, we find that, for constant t (that is, a fixed fabrication process), the width of the beam must be \(w=6\frac{F^{*}}{\sigma ^{*}}\frac{l}{t^{2}}\). The mass of a single beam is \(m=\rho lwt\), where \(\rho \) is the density of the material. Substituting, we find that the mass must be \(m=6\rho \frac{F^{*}}{\sigma ^{*}}\frac{l^{2}}{t}\), or, written more succinctly, \(m=k_{m}l^{2}\) for all other terms kept constant. To find the value of \(k_{m}\), rather than compute these terms, we simply calibrate it to a damper design that, after a few iterations, has been found to support the necessary loads. In this case, the damper consists of many separate beams, all of which are of the same thickness of carbon fiber and must support similar loads.

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Acknowledgements

This work was partially supported by the National Science Foundation (Award Nos. CCF-0926148 and CMMI-0746638) and the Wyss Institute for Biologically Inspired Engineering. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors wish to thank Longlong Chang for showing us that the original analysis was missing an inertial added mass term and Will Dickson for insightful discussions.

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Correspondence to Sawyer B. Fuller.

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Fuller, S.B., Teoh, Z.E., Chirarattananon, P. et al. Stabilizing air dampers for hovering aerial robotics: design, insect-scale flight tests, and scaling. Auton Robot 41, 1555–1573 (2017). https://doi.org/10.1007/s10514-017-9623-3

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Keywords

  • Micro aerial vehicle
  • Insect-scale vehicle
  • Hovering flight
  • Stability
  • Nonlinear dynamics
  • Limit cycle