The longitudinal steady-state control for going from hovering to small speed flight of a model insect is studied, using the method of computational fluid dynamics to compute the aerodynamic derivatives and the techniques based on the linear theories of stability and control for determining the non-zero equilibrium points. Morphological and certain kinematical data of droneflies are used for the model insect. A change in the mean stroke angle （δФ） results in a horizontal forward or backward flight; a change in the stroke amplitude （δФ） or a equal change in the down- and upstroke angles of attack （δα1） results in a vertical climb or decent; a proper combination of δФ and δФ controls （or δФ and δα1 controls） can give a flight of any （small） speed in any desired direction.
In the present paper, the lateral dynamic flight stability properties of two hovering model insects are predicted by an approximate theory based on the averaged model, and computed by numerical simulation that solves the complete equations of motion coupled with the Naviertokes equations. Comparison between the theoretical and simulational results provides a test to the validity of the assumptions made in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency （164Hz） and the other is a model hawkmoth which has relatively low wingbeat frequency （26 Hz）. The following conclusion has been drawn. The theory based on the averaged model works well for the lateral motion of the dronefly. For the hawkmoth, relatively large quantitative differences exist between theory and simulation. This is because the lateral non-dimensional eigenvalues of the hawkmoth are not very small compared with the non-dimensional flapping frequency （the largest lateral non-dimensional eigenvalue is only about 10% smaller than the non-dimensional flapping frequency）. Nevertheless, the theory can still correctly predict variational trends of the dynamic properties of the hawkmoth＇s lateral motion.
Our previous study shows that the lateral disturbance motion of a model drone fly does not have inherent stability （passive stability）,because of the existence of an unstable divergence mode.But drone flies are observed to fly stably.Constantly active control must be applied to stabilize the flight.In this study,we investigate the lateral stabilization control of the model drone fly.The method of computational fluid dynamics is used to compute the lateral control derivatives and the techniques of eigenvalue and eigenvector analysis and modal decomposition are used for solving the equations of motion.Controllability analysis shows that although inherently unstable,the lateral disturbance motion is controllable.By feeding back the state variables （i.e.lateral translation velocity,yaw rate,roll rate and roll angle,which can be measured by the sensory system of the insect） to produce anti-symmetrical changes in stroke amplitude and/or in angle of attack between the left and right wings,the motion can be stabilized,explaining why the drone flies can fly stably even if the flight is passively unstable.
In the present paper, the longitudinal dynamic flight stability properties of two model insects are predicted by an approximate theory and computed by numerical sim- ulation. The theory is based on the averaged model （which assumes that the frequency of wingbeat is sufficiently higher than that of the body motion, so that the flapping wings＇ degrees of freedom relative to the body can be dropped and the wings can be replaced by wingbeat-cycle-average forces and moments）; the simulation solves the complete equations of motion coupled with the Navier-Stokes equations. Comparison between the theory and the simulation provides a test to the validity of the assumptions in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency （164 Hz） and the other is a model hawkmoth which has relatively low wingbeat frequency （26 Hz）. The results show that the averaged model is valid for the hawkmoth as well as for the dronefly. Since the wingbeat frequency of the hawkmoth is relatively low （the characteristic times of the natural modes of motion of the body divided by wingbeat period are relatively large） compared with many other insects, that the theory based on the averaged model is valid for the hawkmoth means that it could be valid for many insects.
The aerodynamic interaction between the contralateral wings and between the body and wings of a model insect are studied, by using the method of numerically solving the Navier-Stokes equations over moving overset grids, under typical hovering and forward flight conditions. Both the interaction between the contralateral wings and the interaction between the body and wings are very weak, e.g. at hovering, changes in aerodynamic forces of a wing due to the present of the other wing are less than 3% and changes in aerodynamic forces of the wings due to presence of the body are less than 2%. The reason for this is as following. During each down- or up-stroke, a wing produces a vortex ring, which induces a relatively large jet-like flow inside the ring but very small flow outside the ring. The vortex rings of the left and right wings are on the two sides of the body. Thus one wing is outside vortex ring of the other wing and the body is outside the vortex rings of the left and right wings, resulting in the weak interactions.
We have examined the aerodynamic effects of corrugation in model wings that closely mimic the wing movements of a forward flight bumblebee using the method of computational fluid dynamics. Various corrugated wing models were tested （care was taken to ensure that the corrugation introduced zero camber）. Advance ratio ranging from 0 to 0.57 was considered. The results shown that at all flight speeds considered, the time courses of aerodynamic force of the corrugated wing are very close to those of the flat-plate wing. The cornlgation decreases aerodynamic force slightly. The changes in the mean location of center of pressure in the spanwise and chordwise directions resulting from the corrugation are no more than 3% of the wing chord length. The possible reason for the small aerodynamic effects of wing corrugation is that the wing operates at a large angle of attack and the flow is separated： the large angle of incidence dominates the corrugation in determining the flow around the wing, and for separated flow, the flow is much less sensitive to wing shape variation.
The longitudinal dynamic flight stability of a bumblebee in forward flight is studied. The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are employed for solving the equations of motion. The primary findings are as the following. The forward flight of the bumblebee is not dynamically stable due to the existence of one （or two） unstable or approximately neutrally stable natural modes of motion. At hovering to medium flight speed [flight speed Ue = （0-3.5）m s^-1; advance ratio J = 0-0.44], the flight is weakly unstable or approximately neutrally stable; at high speed （Ue = 4.5 m s^-1; J = 0.57）, the flight becomes strongly unstable （initial disturbance double its value in only 3.5 wingbeats）.
We describe a 2D fringe projection method that involves projecting two groups of sharp comb fringes onto a free-flying hawk-moth from different directions and recording the images of distorted fringes by two high speed cameras from two orthogonal views. By calculating the 3D coordinates of the points on the hawk-moth and three-dimensional reconstruction of the wing, the flight trajectory, body attitude and wing kinematics including flapping angle, elevation angle, torsion angle, and camber deformation are obtained.
Our previous study shows that the hovering and forward flight of a bumblebee do not have inherent stability （passive stability）. But the bumblebees are observed to fly stably. Stabilization control must have been applied. In this study, we investigate the longitudinal stabilization control of the bumblebee. The method of computational fluid dynamics is used to compute the control derivatives and the techniques of eigenvalue and eigenvector analysis and modal decomposition are used for solving the equations of motion. Controllability analysis shows that at all flight speeds considered, although inherently unstable, the flight is controllable. By feedbacking the state variables, i.e. vertical and horizontal velocities, pitching rate and pitch angle （which can be measured by the sensory system of the insect）, to produce changes in stroke angle and angle of attack of the wings, the flight can be stabilized, explaining why the bumblebees can fly stably even if they are passively unstable.
Yan Xiong Mao Sun Institute of Fluid Mechanics, Beihang University,Beijing 100083, China
