From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> VI. SCALING LAW, RESULTS AND DISCUSSION <s> Flight is one of the most demanding adaptations found in nature because of the physical problems in remaining airborne. Therefore, fliers in nature have been subjected to strong selection for optimal morphology. Different flight modes are associated with distinct wing design, dictated by the animal’s niche and habitat selection. This paper focuses on vertebrate fliers. A main problem for flying animals and aircraft is to obtain enough lift in slow flight. Flapping flight always includes phases with unsteady flow, and in this situation the flexible animal wing has superior performance than a fixed aeroplane wing. Animals can manoeuvre better and change wing form to meet different flight conditions and requirements and they can also make compensating wing movements to avoid stall. Flying vertebrates have evolved several sophisticated lift-enhancing wing adaptations similar to those developed for flying machines, such as slots, slats, flaps and turbulence generators of the boundary layer. Dimensional analysis is a valuable tool in studies of the biomechanics and energetics of animal flight. Empirical allometric relationships allow exploration of how different mechanical, physiological and ecological constraints change in importance with size and wing form. Flight parameters, such as wing loading (weight/wing area) and aspect ratio (wingspan 2 /wing area), are widely used both in aircraft engineering and in studies of animal flight. But the shape of the tail is also important; the tails of birds vary more than their wings. Regression of wing morphology and aerodynamic characteristics are also given for various vertebrate fliers, with a few comparisons with aircraft. <s> BIB001 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> VI. SCALING LAW, RESULTS AND DISCUSSION <s> Flapping wing micro air vehicles (MAVs) take inspiration from natural fliers, such as insects and hummingbirds. Existing designs manage to mimic the wing motion of natural fliers to a certain extent; nevertheless, differences will always exist due to completely different building blocks of biological and man-made systems. The same holds true for the design of the wings themselves, as biological and engineering materials differ significantly. This paper presents results of experimental optimization of wing shape of a flexible wing for a hummingbird-sized flapping wing MAV. During the experiments we varied the wing 'slackness' (defined by a camber angle), the wing shape (determined by the aspect and taper ratios) and the surface area. Apart from the generated lift, we also evaluated the overall power efficiency of the flapping wing MAV achieved with the various wing design. The results indicate that especially the camber angle and aspect ratio have a critical impact on the force production and efficiency. The best performance was obtained with a wing of trapezoidal shape with a straight leading edge and an aspect ratio of 9.3, both parameters being very similar to a typical hummingbird wing. Finally, the wing performance was demonstrated by a lift-off of a 17.2 g flapping wing robot. <s> BIB002 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> VI. SCALING LAW, RESULTS AND DISCUSSION <s> Flying animals with flapping wings may best exemplify the astonishing ability of natural selection on design optimization by excelling both stability and maneuverability at insect/hummingbird scale. Flapping Wing Micro Air Vehicle (FWMAV) holds great promise in bridging the performance gap between engineering system and their natural counterparts. Designing and constructing such a system is a challenging problem under stringent size, weight and power (SWaP) constraints. In this work, we presented a systematic approach for design optimization and integration for a hummingbird inspired FWMAV. Our formulation covers aspects of actuation, dynamics, flight stability and control, which was validated by experimental data for both rigid and flexible wings, ranging from low to high wing loading. The optimization yields prototypes with onboard sensors, electronics, and computation units. The prototype flaps at 30Hz to 40Hz, with 7.5 to 12 grams of system weight and 12 to 20 grams of maximum lift. Liftoff was demonstrated with added payloads. Flapping wing platforms with different requirements and scales can now be designed and optimized with minor modifications of proposed formulation. <s> BIB003
In this subsection, the scaling laws induced by regression analysis of species of hummingbirds are established based on the concept of geometric similarity. The relation between weights and parameters of wing performance can be predicted and determined before being applied in designing the hummingbird-size FWMAD. In other words, this concept provides rules to design and compare flying objects within different sizes and weights through several orders of magnitude. Summarized plots of the relationship among wing surface, flapping frequency and wing length against body mass are firstly presented. The fitting curve is provided in blue color. Fig. 9 exhibits that the figures are intensive with an excellent general tendency. Wing surface and wing length are proportional to body mass in a log-log domain, whereas the flapping frequency is inversely proportional to the body mass, shown in Fig. 9 (right) . The fitted functions of wing surface and mass, wing length and mass, and flapping frequency and mass are given in the Fig. 9 . The relation of wing loading, lumped parameter and aspect ratio with body mass are shown in Fig. 10 . From this figure, we observe that the three relations do not follow the rules induced by geometrical similarity. The wing loading is inversely proportional to weight. The similar results are achieved in BIB001 . Actually, the wing loading generally rises with the size increase for natural flyers excluding hummingbirds. Regarding k and AR, both slightly vary with weight. Therefore, it is roughly considered that AR is independent of size, a similar result also stated in BIB001 . The fitted functions of wing loading and mass, lumped parameter and mass, and aspect ratio and mass are present in the Fig. 10 . In this study, the statistical mean value of AR is presented, which is expressed as a mean value with standard deviation, AR=7.86±0.56 and the statistical mean value of the lumped parameter is about 0.31±0.04. At present, certain FWMADs inspired by hummingbird are developed, which is summarized in Table 2 . Since the wing loading and aspect ratio are widely applied to quantify the size and shape in aircraft engineering and in studies of animal flight, the plot of aspect ratio and lumped parameters against wing loading are given for comparing the wing performance of natural hummingbirds and hummingbird-like MADs. And wing length and surface as basic and useful parameters are also utilized for comparison. The compared figures are shown in Fig. 11 . This figure displays that the data are intensive and existing FWMADs (except Maryland-hummingbird) are nearly in the range of species and are following these scale rules. The size of Maryland-hummingbird is actually in the scope of bats. Meanwhile, the KUBeetle inspired by beetles, its size is closed to the hummingbird, so it is added for comparison. In addition, the wings of Colibri (hummingbird-like MAV) are designed based on the above-studied scale laws, and the efficiency of wings is verified by flight experiment, shown in Fig. 12 . In this figure, the flapping mechanism is shown in Fig. 12 a) , which is composed of two stages 1) slider crank with a rocker 2) four-bar mechanism. The wings construction is given in Fig. 12 b) (Details of wing manufacture can be seen the study of Nan et al. BIB002 ), the certain selected wings are presented in Fig. 12 c) , and the integrated MAV with sail is given in in Fig. 12 e) , and flying demonstration is shown in in Fig. 12 d) . Additional details of wing designs based on different materials and methods are summarized in Appendix Table 6 . Besides, Zhang did the similar jobs on wing design, optimization and system integration of robotic hummingbird BIB003 .