Wingbeat Shape Modulation for Flapping-Wing Micro-Air-Vehicle Control During Hover

Abstract

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The need for higher-order averaging in the stability analysis of hovering, flapping-wing flight

“…However, many simulations such as the one shown in figure 4 indicate that the system (8) is stable. Figure 4 shows that direct application of the averaging theorem is not sufficient to analyze the stability of the system (8). It should be noted that the timeinvariant systems obtained through direct averaging of (8) and the original hovering flight dynamics (4) are the same.…”

“…In this subsection, we show that second-order averaging is able to more accurately capture the stability characteristics of the example considered in the last section (hovering dynamics with fixed point), which is presented in equation (8).…”

“…Doing so yields a system with a fixed point, equation (8), that has been shown to be stabilized for the hawkmoth case due to the high-amplitude, periodic terms. The full dynamics (4) can be regarded as the system (8) subjected to a bounded, zero-mean, periodic forcing g t ( ) 1 .…”

“…It should be noted that such comparison will not be appropriate if it is performed using the full flight dynamics (with periodic orbit), because second-order averaging requires different flapping parameters to achieve balance than those required by direct averaging, as shown in section 4.3. Thus, to perform the comparison having the same equilibrium (using the same set of trim flapping parameters), the flight dynamics with fixed point is considered; i.e., equation (8). Table 1 shows the ratios of flapping frequency to the body natural frequency for the five insects, and the eigenvalues revealing the stability of the system (8) using first-order and second-order averaging.…”

“…Even simple models of flapping flight dynamics involve multi-body, nonlinear, non-autonomous (time-varying) dynamics. One common assumption used in flapping flight dynamic models is that the effect of the wing inertial forces on the body dynamics may be neglected [2][3][4][5][6][7][8][9][10]. The validity of this assumption is a subject of continuing debate; see the recent review of Taha et al [11] or Sun [12].…”