## 1 Introduction  
Nature provides investigators with abundant insights and inspirations that aid in solving engineering problems and creating new inventions [1, 2]. With the advancement of science, engineering, and technology, a growing number of robots have been designed in different shapes and sizes to mimic biological systems [3, 4]. These biomimetic robots mimic animals, such as four-legged animals (quadrupeds) [5], monkeys [6], snakes [7], fish [8, 9], bats [10, 11], and various birds and insects [12, 13, 14]. These robots have diverse potential applications, ranging from simple toys to exploration robots in dangerous and hazardous environments. They have the potential to perform tasks that conventional robots cannot accomplish, such as climbing steep surfaces [15; 16], maneuvering through rubble [17], and walking on uneven terrains [5]. These unique capabilities that only animals can perform have motivated many researchers to develop biomimetic robots. The flapping-wing micro air vehicle (FWMAV) is one such type of robot. It has been popular for more than a decade [12; 18; 19; 20]. With birds and insects as the main source of inspiration for designing robots, FWMAVs are typically categorized into bird- or insect-inspired FWMAVs. Alternatively, they are differentiated into tailed or tailless FWMAVs based on the existence of the control surface on their tails, by which the differentiation agrees with the physical configurations of birds or insects, respectively [21; 22; 23; 24; 25]. In general, birds and insects have unique flight characteristics, which can be observed from the manner in which their wings are manipulated and modulated. Compared to birds, insects normally flap with higher flapping frequency and larger stroke amplitude to produce sufficient force to fly. The flapping stroke plane of insects is nearly horizontal, which causes similar lift force to be produced during a downstroke and an upstroke, whereas the flapping stroke plane of birds is practically vertical, which causes most of the lift force to be produced during downstroke [23; 24]. These flight characteristics are mimicked by FWMAVs using various approaches and engineering designs, especially for producing wing flapping motion and other kinematics to generate force and moment [19; 20; 26]. Unlike tailed FWMAVs, tailless FWMAVs can exhibit steady hover and high maneuverability during a flight [27], indicating their potential applications in restricted spaces. To date, several tailless FWMAV designs have successfully performed stable flight, including the Nano Hummingbird [27], RoboBee [28], Delfly Nimble [29], KUBeetle [30; 31; 32; 33; 34; 35], robotic hummingbird [36], Colibri [37], NUS-Robobird [38], and Purdue Hummingbird [39]. There are some similarities between each design. Nano Hummingbird, KUBeetle, robotic hummingbird, and Colibri use only one DC motor to generate thrust through one pair of wings and a few servos to manipulate the wing kinematics for control moment generation. They are mainly different in system integration and various transmission mechanisms, such as combination of string and crankshaft [27], combination of 4-bar linkage and pulley-string [30], and combination of slider-crank and 4-bar linkage [37]. In comparison, Delfly Nimble and NUS-Robobird use two DC motors to manipulate the flapping of two pairs of wings to generate thrust and roll moments. Servos are used to adjust the relative angle of two flapping mechanisms for pitch and yaw moment generation. A different control strategy was found in Purdue Hummingbird where each wing, one left and one right, is controlled by a DC motor to realize fully decoupled and independent wing kinematics manipulation. In this way, no servos are needed because control moments are generated by changing the wing flapping range and speed. While most tailless FWMAVs with successful flights use DC motors and servos, RoboBee uses piezoelectric actuators driven with high voltage.