Patent Publication Number: US-9428269-B1

Title: Methods and apparatus to achieve independent six degree control of flapping wing micro air vehicle

Description:
RIGHTS OF THE GOVERNMENT 
     The invention described herein may be manufactured and used by or for the Government of the United States for all governmental purposes without the payment of any royalty. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to aircraft and, more particularly, to biomimetic aircraft having insect-like features and maneuverability. 
     BACKGROUND OF THE INVENTION 
     Flapping wing micro air vehicles (FWMAV) have received a great deal of interest in the research community in the past decade. Researchers have developed and designed FWMAVs using numerous innovative materials and manufacturing techniques. 
     Control laws have been developed to govern the operation of flapping wing vehicles. Some designs use averaging theory to parameterize the varying wing kinematics on the vehicle forces and moments. This control strategy required at least four actuators. In those designs, two actuators were used to actively control the angle-of-attack, and two were required to drive each wing in the stroke plane. Other designs used a similar time-averaged approach to develop a longitudinal flight controller for a flapping-wing micro air vehicle. The control law was written in terms of the aerodynamic forces averaged over one wing-beat cycle. It was assumed that the mean forces could be controlled as desired; however, the relationships between the mean-aerodynamic forces and the wing kinematic parameters were not established. The Nano Hummingbird is an example of an operational flapping wing vehicle and control law that utilized three actuators. 
     A split-cycle constant period frequency modulation technique independently controls four degrees-of-freedom (DOF) with two physical actuators. By adding a bob-weight and third actuator, it is possible to independently control five DOF. The split-cycle technique may be modified to include a wing bias that provides independent five DOF control with only two physical actuators. It should be noted that the two actuator/five DOF technique is only applicable to configurations using actuators that are capable of being biased, such as piezoelectric devices, and is not applicable to direct current motor actuators. 
     Given that power to weight ratio is a critical element in FWMAV design, it is desirable to enable control over the greatest possible number of DOFs using the fewest actuators. One technique that has been proposed by researchers for controlling a FWMAV is split-cycle constant-period frequency modulation (SCCPFM). In this technique, each wing&#39;s upstroke and downstroke can have different velocities, resulting in the ability to generate non-zero cycle averaged drag forces. These drag forces provide the ability to control the fore/aft translation, roll, and pitch degrees-of-freedom of the vehicle. Unfortunately, by using this technique, it is not possible to generate fore/aft translation independent from pitch commands. Non-zero cycle averaged lift can also be produced by the wings, allowing control over the vertical and yaw degrees of freedom. Therefore, by utilizing SCCPFM, four independent degrees-of-freedom of the vehicle can be controlled. 
     As a result, there exists a need for an FWMAV activation and control method capable of yielding control over 6 independent DOFs. 
     SUMMARY OF THE INVENTION 
     The present invention overcomes the foregoing problems and other shortcomings, drawbacks, and challenges of requiring additional actuators to achieve all possible degrees of aircraft maneuverability. While the invention will be described in connection with certain embodiments, it will be understood that the invention is not limited to these embodiments. To the contrary, this invention includes all alternatives, modifications, and equivalents as may be included within the spirit and scope of the present invention. 
     According to one embodiment of the present invention in an aircraft including a first wing planform and a second wing planform driven by a corresponding first actuator and a corresponding second actuator, wherein each of the first wing planform and the second wing planform performs an upstroke and a downstroke over a complete wingbeat cycle, a method of configuring a wing velocity profile is provided. The method includes segmenting the complete wingbeat cycle into a set of quarter-strokes including a first quarter-stroke, a second quarter-stroke, a third quarter-stroke, and a fourth quarter-stroke. The method further includes generating time-varying quarter-stroke wing position commands for each of the first and second wing planforms to produce wing flapping trajectories that generate non-zero, cycle averaged wing lift and drag and alter a cycle average center of pressure of each of the first and the second wing planforms relative to the aircraft. The wing position commands include velocities and positions, with temporally symmetric and asymmetric quarter-stroke components, for the first wing planform and the second wing planform. The first wing planform and the second wing planform produce controlled six-degrees-of freedom vehicle motion without using actuators in excess of the first actuator and the second actuator. 
     Additional objects, advantages, and novel features of the invention will be set forth in part in the description which follows, and in part will become apparent to those skilled in the art upon examination of the following, or may be learned by practice of the invention. The objects and advantages of the invention may be realized and attained by means of the instrumentalities and combinations particularly pointed out in the appended claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the present invention and, together with a general description of the invention given above, and the detailed description of the embodiments given below, serve to explain the principles of the present invention. 
         FIG. 1  is an illustration depicting an apparatus suitable for being controlled according to embodiments of the disclosed invention. 
         FIG. 2  is an illustration depicting the apparatus of  FIG. 1  in several stopped motion flapping segments. 
         FIG. 3  is a graphical representation of a flapping motion having symmetric upstroke and downstroke elements. 
         FIG. 4  is graphical representation of a split-cycle flapping motion having asymmetric upstroke and downstroke elements. 
         FIG. 5  is a graphical representation of a quarter-cycle flapping motion having two independently adjustable upstroke elements and two independently adjustable downstroke elements. 
         FIG. 6  is a graphical depiction of wing velocity versus time of a quarter-cycle waveform in accordance with embodiments of the disclosed invention. 
         FIG. 7  is a block diagram illustrating components of a wing position waveform generator in accordance with embodiments of the disclosed invention. 
         FIG. 8  is a block diagram depicting a flight control system using quarter-cycle waveform modulation in accordance with embodiments of the disclosed invention. 
     
    
    
     It should be understood that the appended drawings are not necessarily to scale, presenting a somewhat simplified representation of various features illustrative of the basic principles of the invention. The specific design features of the sequence of operations as disclosed herein, including, for example, specific dimensions, orientations, locations, and shapes of various illustrated components, will be determined in part by the particular intended application and use environment. Certain features of the illustrated embodiments have been enlarged or distorted relative to others to facilitate visualization and clear understanding. In particular, thin features may be thickened, for example, for clarity or illustration. 
     DETAILED DESCRIPTION OF THE INVENTION 
     The disclosed invention, includes a novel FWMAV and a method of control thereof, that enables independently controlled six degree-of-freedom flight of the fuselage of a flapping wing vehicle. The disclosed invention operates without the need for artificial motion constraints, and may do so with as few as two actuators. 
     Turning attention to  FIG. 1 , the aircraft  10  includes independently actuated wings  12   a  and  12   b  whose position and velocity profiles can be manipulated to generate controllable forces and moments that enable the aircraft to achieve insect-like maneuverability and hover capability. As discussed herein, components listed on the left side of  FIG. 1  shall be referred to as “first” while components on the right hand side shall be referred to as “second.” Further, components denoted “second” correspond to the left side of the aircraft  10 , and conversely with respect to “first.” A fuselage  14  includes a first and second brushless direct current (BLDC) motor,  16  and  18 , respectively, mated thereto. First BLDC motor  16  is the prime mover for first wing planform  20 , while second BLDC motor  18  is the prime mover for second wing planform  22 . Each of BLDC motor  16  and  18  include a corresponding first output shaft  24  and second output shaft  26 . First output shaft  24  is mated to first crank  28 , while the second output shaft  26  is attached to second crank  30 . Electrical potential is provided by the output of a control board  32  powered by a power supply  34 . In some embodiments, the power supply  34  may be onboard, while in other embodiments the power supply may be located remotely. The control board  32  receives input wingbeat position profile commands and provides a corresponding electrical commutation signal to each BLDC motor  16  and  18 . The rotary motion of BLDC motors  16  and  18  cause rotary motion of cranks  28  and  30 . In some embodiments gearing, pulleys, or the like may be used at various points throughout the mechanism to scale input and output angular velocities and torques. A first connecting rod  36  operably interconnects a first lobe  40  of the first crank  28  to a first rocker  44 . Likewise, a second connecting rod  38  operably interconnects a second lobe  42  of the second crank  30  to a second rocker  46 . Rotation of cranks  28  and  30  cause connecting rods  36  and  38  to move along a sector of a circle, which causes rockers  44  and  46  to oscillate. This provides the back and forth, or flapping motion, of wings  12   a  and  12   b . Rockers  44  and  46  may be cantilevered with respect to the fuselage  14  and provide attachment points for first wing spar  48  and second wing spar  50 . The first wing spar  48  is attached to the first rocker  44  such that the first wing planform  20  can passively rotate about the longitudinal axis of the first wing spar  48  to provide nonzero angle of attack with respect to the stroke plane of the first wing planform  20 . The stroke plane of the first wing spar  48  is in a plane that includes the first crank  28 , first connecting rod  36 , and first rocker  44 . Similarly, the stroke plane of the second wing spar  50  is in a plane that includes the second crank  30 , second connecting rod  38 , and second rocker  46 . 
     In furtherance of the passive rotation operation noted above, the second wing spar  50  is attached to the second rocker  46  by passing the circular cross-section of the second wing spar  50  through a second ball bearing  54   a  and  54   b . An interference joint is established such that, when wing  12  flapping occurs, the second wing planform  22  rotates and impacts the second wing fixed angle of attack stop  58   a , such that a prescribed wing angle of attack, a, is achieved. As shown, the wing is at 90° angle of attack. Likewise, when the second wing planform  22  is flapping in the opposite direction, the second wing planform  22  rotates and impacts the corresponding second fixed angle of attack stop  58   b , such that a prescribed wing angle of attack is achieved. Second fixed angle of attack stops  58   a  and  58   b  interfere with the second wing planform  22  at an angle α, selected by the designer, and prevents the second wing planform  22  from over rotating. Interference between the second wing planform  22  and the second wing fixed angle of attack stops  58   a  and  58   b  causes the wing  12   b  to hold a constant angle of attack relative to the stroke plane once a sufficient dynamic pressure is reached. The same operation holds true for the interaction between the first wing planform  20 , first ball bearings  52   a  and  52   b , first fixed angle of attack stops  56   a  and  56   b , but is reflected about the fuselage  14 . 
     The aircraft  10  is designed to be controlled and maneuvered in the vicinity of hover and, for the purposes of analysis, the local velocity of the wind relative to the wing planform  20  or  22  is taken to be V=rip, where r is the distance from the rocker pivot points  60  or  62  of the wing spar  48  or  50  to an arbitrary chord line on the wing planform  20  or  22 , φ is the angular position of the wing planform  20  or  22 , and is the time derivative of the wing planform  20  or  22  angular position. 
       FIG. 2  shows a top view of the aircraft  10 , with only part of the mechanism shown (notably, the second connecting rod  38  and second rocker  46 ) along with the second wing spar  50 . This figure is to be read in a counterclockwise fashion, starting from the upper left corner (which represents the beginning of the upstroke). Second wing spar  50  starts at an angular position of 1 rad, measured from the horizontal, and moves, over a time period of k 1 Δt, to the midpoint angular position of 0 rad, where Δt is the fundamental period of one complete wingbeat cycle. Following this, second wing spar  50  moves over a time period of Δt (k 2 −k 1 ) to an angular position of −1 rad. At this point, the upstroke ends and the downstroke begins. During the downstroke, second wing spar  50  moves from an angular position of −1 rad to 0 rad over a time period of Δt (k 3 −k 2 ), then moves from 0 rad angular position to 1 rad angular position over a time period of Δt (1−k 3 ). This completes the downstroke and another upstroke begins. The motion shown in  FIG. 2  comprised one complete wingbeat cycle. 
     In the discussion that follows, it should be noted that variations may be made to the mechanism of the aircraft  10 , to the prime movers (i.e., BLDC motors  16  and  18 ), or fixed angle of attack stops  56   a ,  56   b ,  58   a , or  58   b . The physical aircraft  10  apparatus shown herein is intended to provide a frame-of-reference to describe various disclosed methods for controlling the wings  12   a  and  12   b  so as achieve six DOF control of the aircraft. It should be noted that some embodiments of the disclosed invention may employ reconfigurable versions of the fixed angle of attack stops  56   a ,  56   b ,  58   a , or  58   b . In such reconfigurable embodiments, the angle stops may be relocated in one of a plurality of alternate locations by placing pins or threaded members into bores. Likewise, the angle stops may slide within an arcuate track, and thereafter be clamped into a desired location. Further, automated means may be employed to change the position of the angle of attack stops  56   a ,  56   b ,  58   a ,  58   b  so as to provide dynamic adjustments of the angle of attack stop  56   a ,  56   b ,  58   a ,  58   b  positions while in active flight, or to accommodate different flight configurations. 
     Control of the aircraft  10  is achieved by independently driving actuators (BLDC motors  16  and  18  or other prime mover) so that specifically designed angular displacements of rockers  44  and  46  in the stroke plane occur. In order for each wing  12  to produce independently controlled cycle-averaged body forces in both the vertical and horizontal directions, wherein the angle of attack of each wing  12  is passively regulated by the interference joints and fixed angle of attack stops  56   a ,  56   b ,  58   a , and  58   b  previously described, the wing  12  velocities must be temporally asymmetric over the wingbeat cycle. In the disclosed invention, this is accomplished by parameterizing the waveforms that define the motion of each wing  12  in the stroke-plane in terms of four components. The method may be referred to as Quarter-Cycle Constant-Period Frequency Modulation. It is assumed that the position of the wing can be described by a triangular waveform. 
     The quarter-cycle constant period frequency modulation method includes piecing together triangular waves of different frequencies over one complete wing beat cycle such that the resulting waveform is continuous in position at all transition points. Each wingbeat cycle consists of four strokes, namely, first, second, third, and fourth quarter-strokes. The period of the cycle is fixed by the symmetric or fundamental frequency, while the speed of each stroke within the cycle is governed by three additional quarter-cycle parameters. The nominal waveform is symmetric such that each quarter-stroke operates over identically ¼ of the fundamental period. With quarter-cycle constant period frequency modulation, each quarter-stroke can occur over a longer or shorter period than nominal, provided that the overall cycle period equals the given fundamental period. Quarter-cycle modulation allows one to change the duration of each quarter-stroke to a value different than ¼ of the fundamental period, which provides the ability to independently control all six rigid body degrees-of-freedom. 
     In order to describe the quarter-cycle frequency modulation technique, a symmetric waveform is first described, followed by an example of a split-cycle waveform, and lastly a quarter-cycle waveform. In the waveform examples, it is assumed that the period of a full wingstroke cycle is 1 sec, the amplitude of wing motion is 1 rad, and the wingstroke starts at 0 sec. Additionally, it is assumed that the wing position waveforms are triangular waves, but could be replaced by a different waveform if desired. Two independently generated waveforms define the motion of the first wing planform  20  and second wing planform  22 . 
     In the following figures, only a single waveform is shown, but it should be understood that this waveform could apply to either the first wing planform  20  or second wing planform  22 . A symmetric waveform has an upstroke and downstroke that each comprise exactly ½ of a wingstroke period.  FIG. 3  shows a symmetric waveform. Here, the velocities of the wing during the upstroke and downstroke are identical in magnitude and opposite in sign. 
     A split-cycle waveform allows modulation of the portions of the wingstroke period that are occupied by the upstroke and downstroke. An example split-cycle waveform is shown in  FIG. 4 . In this example, the upstroke is retarded while the downstroke is advanced. Therefore, the magnitude of the wing  12  velocity during the upstroke will be less than the magnitude of the wing  12  velocity during the downstroke. The split-cycle parameter defines when the upstroke ends and the downstroke begins, while the frequency determines the period of the wingstroke (set to 1 sec in this example). Hence, there are two split-cycle control parameters that control the durations of the upstroke and downstroke within a period (one for each of the first and second wings) and two symmetric frequencies that set the period of the wingstroke (one for each of the first wing planform  20  and second wing planform  22 ). Thus, split-cycle constant period frequency modulation uses four control parameters and provides independent control of four degrees-of-freedom of the aircraft  10 . 
     A quarter-cycle waveform breaks the positions of wings  12   a  and  12   b  into four distinct parts for each wing  12   a  and  12   b .  FIG. 5  shows a quarter-cycle wing position waveform. In this case, the complete wing position for each wing is described by four distinct parts, namely, the first quarter, second quarter, third quarter, and fourth quarter-strokes. By doing this, the wing velocities during each of the four portions can be varied, as shown in  FIG. 6 . For a quarter-cycle waveform, there are eight control parameters: two symmetric frequencies that set the period of the wingstroke (one for each of the first wing planform  20  and second wing planform  22 ), two k 1  parameters that define the relative time at which the first quarter-stroke ends (one for each of the first wing planform  20  and second wing planform  22 ), two k 2  parameters that define the relative time at which the second quarter-stroke ends (one for each of the first wing planform  20  and second wing planform  22 ), and two k 3  parameters that define the relative time at which the third quarter-stroke ends (one for each of the first wing planform  20  and second wing planform  22 ). With eight control parameters, six independent DOF can be controlled with only two physical actuators. It should be noted that the quarter-cycle parameters have theoretical limits of 0&lt;k 1 &lt;k 2 &lt;k 3 ′&lt;1. Additionally, there are other, more constrained limits imposed on these variables by the physical limitations of the prime movers and flapping mechanism. 
     Each waveform in the quarter-cycle constant period frequency scheme has the following forms: 
                                               ϕ   1     ⁡     (   t   )       =     (         -   t         k   1     ⁢   Δ   ⁢           ⁢   t       +   1     )             0   ≤   t   &lt;       k   1     ⁢   Δ   ⁢           ⁢   t                                 ϕ   2     ⁡     (   t   )       =         -   1       Δ   ⁢           ⁢     t   ⁡     (       k   2     -     k   1       )           ⁢     (     t   -       k   1     ⁢   Δ   ⁢           ⁢   t       )                   k   1     ⁢   Δ   ⁢           ⁢   t     ≤   t   &lt;       k   2     ⁢   Δ   ⁢           ⁢   t                                       ϕ   3     ⁡     (   t   )       =         1     Δ   ⁢           ⁢     t   ⁡     (       k   3     -     k   2       )           ⁢     (     t   -       k   2     ⁢   Δ   ⁢           ⁢   t       )       -   1                 k   2     ⁢   Δ   ⁢           ⁢   t     ≤   t   &lt;       k   3     ⁢   Δ   ⁢           ⁢   t                                 ϕ   4     ⁡     (   t   )       =       1     Δ   ⁢           ⁢     t   ⁡     (     1   -     k   3       )           ⁢     (     t   -       k   3     ⁢   Δ   ⁢           ⁢   t       )                   k   3     ⁢   Δ   ⁢           ⁢   t     ≤   t   &lt;     Δ   ⁢           ⁢   t                         Equation   ⁢           ⁢   1                 
where it is assumed that the amplitude of wing  12  motion is 1 rad, Δt is the period of the full cycle wingbeat, and k 1 , k 2 , and k 3  define the end of each quarter-stroke. Previously, it was assumed that the full cycle period was 1 sec, whereas the full period is now defined by Δt, which can be value other than 1 sec. Additionally, the amplitude of wing  12  motion is set to 1 rad in this example, but the technique is applicable to any flapping motion amplitude. The time variable, t, in Equation 1 is reset to zero at the beginning of each stroke and has a duration of Δt sec. Nominally, for a symmetric waveform, k 1 =¼, k 2 =½, and k 3 =¾. Equation 1 is a general expression which is not specific to either the first or second wing. In order to make Equation 1 apply to the right wing, simply replace Δt, k 1 , k 2 , k 3  by Δt RW , k RW     1   , k RW     2   , k RW     3    and for the left wing replace Δt, k 1 , k 2 , k 3  by Δt RW , k LW     1   , k LW     2   , k LW     3   . The parameters that will be used to control the wingbeat positions are the wingstroke periods, Δt RW , Δt LW , and the quarter-cycle points, k RW     1   , k RW     2   , k RW     3   , k LW     1   , k LW     2   , k LW     3   .
 
     Hence, there are a total of 8 control parameters. The derivatives of the wing position functions, with respect to time, are: 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         
                           
                             
                               ϕ 
                               · 
                             
                             1 
                           
                           = 
                           
                             
                               - 
                               1 
                             
                             
                               
                                 k 
                                 1 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               t 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ϕ 
                               · 
                             
                             2 
                           
                           = 
                           
                             
                               - 
                               1 
                             
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 t 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       k 
                                       2 
                                     
                                     - 
                                     
                                       k 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ϕ 
                               · 
                             
                             3 
                           
                           = 
                           
                             1 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 t 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       k 
                                       3 
                                     
                                     - 
                                     
                                       k 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ϕ 
                               · 
                             
                             4 
                           
                           = 
                           
                             1 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 t 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       k 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
               
             
           
         
       
     
     The general block diagram that describes the generation of the wingbeat command profile is shown in  FIG. 7 . Note that at the beginning of each upstroke, Δt, k 1 , k 2 , k 3  are computed and used to generate the wingbeat commands for each wing. Block  301  provides as input the wingbeat parameters Δt, k 1 , k 2 , k 3 . A resettable timer  311  provides a time, t, that is used to generate the time varying wing position commands. The time, t, is subtracted from the wingbeat period, Δt, by summing block  315 . Comparator block  313  takes as input Δt−t and checks if this value is greater than zero. Comparator block  313  produces a binary output of 1 if Δt−t&gt;0 and an output of 0 if Δt−t≦0. When the output of comparator block  313  transitions from a 1 to a 0, timer block  311  is reset and switch  303  is closed, thus producing another sample of input parameters Δt, k 1 , k 2 , k 3 . Hold block  305  takes as input the input parameters and holds these values constant until a new vector of input parameters becomes available. Separator block  307  takes as input a vector which contains Δt, k 1 , k 2 , k 3  and provides two outputs, a vector with k 1 , k 2 , k 3  and a scalar signal Δt. The separation is used to obtain Δt so that it can be compared with the current time, t, so that the timer can be reset. Block  309  simply combines its input signals into a single vector. Combine block  317  takes as input a vector signal Δt, k 1 , k 2 , k 3  and a scalar time, t, and provides a single vector signal output of Δt, k 1 , k 2 , k 3 , t. Comparator block  319  outputs a binary value of 1 if k 1 Δt−t&gt;0 and a value of 0 if k 1 Δt−t≦0. This determines whether or not the current time is contained in the first quarter-stroke. Comparator block  321  outputs a binary value of 1 if k 2 Δt−t&gt;0 and a value of 0 if k 2 Δt−t≦0. This determines whether or not the current time is contained in the first or second quarter-strokes. Comparator block  323  outputs a binary value of 1 if k 3 Δt−t&gt;0 and a value of 0 if k 3 Δt−t≦0. This determines whether or not the current time is contained in the first, second, or third quarter-strokes. When the output of comparator block  319  is high, switch  343  is closed. Block  325  generates the wingbeat command profile for the first quarter-stroke and is an input to summing block  351 . When switch  343  is open, a first quarter-stroke value of zero is input to summing block  351 . NOT gate  333  takes as input the output of comparator  319  and produces a binary signal of 1 if k 1 Δt−t≦0 and a value of 0 if k 1 Δt−t&gt;0. AND gate  335  takes as input the outputs of NOT gate  333  and comparator block  321 . The output of AND gate  335  is a binary signal of 1 when the current time is contained in the second quarter-stroke and a value of 0 otherwise. When the output of AND gate  335  is high, switch  345  is closed. Block  327  generates the wingbeat command profile for the second quarter-stroke and is an input to summing block  351 . When switch  345  is open, a second quarter-stroke value of zero is input to summing block  351 . NOT gate  337  takes as input the output of comparator  321  and produces a binary signal of 1 if k 2 Δt−t≦0 and a value of 0 if k 2 Δt−t&gt;0. AND gate  339  takes as input the outputs of NOT gate  337  and comparator block  323 . The output of AND gate  339  is a binary signal of 1 when the current time is contained in the third quarter-stroke and a value of 0 otherwise. When the output of AND gate  339  is high, switch  347  is closed. Block  329  generates the wingbeat command profile for the third quarter-stroke and is an input to summing block  351 . When switch  347  is open, a third quarter-stroke value of zero is input to summing block  351 . NOT gate  341  takes as input the output of comparator  323  and produces a binary signal of 1 if k 3 Δt−t≦0 and a value of 0 if k 3 Δt−t&gt;0. When the output of NOT gate  341  is high, the current time is contained in the fourth quarter-stroke and switch  349  is closed. Block  331  generates the wingbeat command profile for the fourth quarter-stroke and is an input to summing block  351 . When switch  349  is open, a fourth quarter-stroke value of zero is input to summing block  351 . Summing block  351  takes as inputs the first, second, third, and fourth quarter-stroke wingbeat commands and adds them together to produce the desired wingbeat command, φ(t), in block  357 . Differentiator block  353  differentiates the wingbeat command signal with respect to time, producing {dot over (φ)}(t) in block  355 . Two instances of waveform generator  399  are contained in control board  32  (one for the first wing planform  20  and one for the second wing planform  22 ). The wings&#39; position commands are converted to BLDC motor commutation sequences by control board block  32  (see  FIG. 1 ) and are applied to BLDC motor  16  if generating a position command for the first wing planform  20  or are applied to BLDC motor  18  if generating a position command for the second wing planform  22 . The elements contained in the waveform generator  399  ensure continuity of the wingbeat position commands at all times. The sample, hold, and logic elements prevent the waveform generator from responding to high frequency variations in the control system commands that can distort the waveforms in inter-cycle. 
     The general method used to compute the cycle-averaged aerodynamic forces and their sensitivities to variations in the control input parameters is presented below. Define G(t) to be a generalized force aligned with a principal body-axis direction that may represent either a force or a moment. In order to compute the cycle-averaged generalized force associated with each wing, an integral of the following form is evaluated: 
                     G   _     =       1     Δ   ⁢           ⁢   t       ⁢       ∫   0     Δ   ⁢           ⁢   t       ⁢       G   ⁡     (   t   )       ⁢     ⅆ   t                   Equation   ⁢           ⁢   3               
where the overline notation, i.e.  G , denotes a cycle averaged quantity. Since the introduction of the quarter cycle parameters change the triangular waveform at the points k 1 Δt, k 2 Δt, k 3 Δt, it is convenient to split the integral as follows:
 
                     G   _     =         1     Δ   ⁢           ⁢   t       ⁢       ∫   0     Δ   ⁢           ⁢   t       ⁢       G   ⁡     (   t   )       ⁢     ⅆ   t           =       1     Δ   ⁢           ⁢   t       ⁢       (         ∫   0       k   1     ⁢   Δ   ⁢           ⁢   t       ⁢         G   1     ⁡     (   t   )       ⁢     ⅆ   t         +       ∫       k   1     ⁢   Δ   ⁢           ⁢   t         k   2     ⁢   Δ   ⁢           ⁢   t       ⁢         G   2     ⁡     (   t   )       ⁢     ⅆ   t         +       ∫       k   2     ⁢   Δ   ⁢           ⁢   t         k   3     ⁢   Δ   ⁢           ⁢   t       ⁢         G   3     ⁡     (   t   )       ⁢     ⅆ   t         +       ∫       k   3     ⁢   Δ   ⁢           ⁢   t       Δ   ⁢           ⁢   t       ⁢         G   4     ⁡     (   t   )       ⁢     ⅆ   t           )     .                 Equation   ⁢           ⁢   4               
where G 1 (t), G 2 (t), G 3 (t), G 4 (t) denote generalized forces for the first, second, third, and fourth quarter-strokes.
 
     The control derivatives associated with a generalized cycle-averaged force are calculated by computing the partial derivative of G with respect to each control input variable about the hover condition. At hover, the average lift over a complete wingbeat cycle is equal to the vehicle&#39;s weight, while the average sideforce, drag, rolling moment, pitching moment, and yawing moment over a complete wingbeat cycle arezero. At the hover condition, k 1 =¼, k 2 =½, k 3 =¾, and the wingbeat period is a nominal value, Δt 0 , such that the cycle averaged lift equals the vehicle&#39;s weight. The control derivatives are: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               ∂ 
                               
                                 G 
                                 _ 
                               
                             
                             
                               ∂ 
                               
                                 k 
                                 
                                   RW 
                                   1 
                                 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             hover 
                           
                         
                         , 
                         
                           
                             
                               ∂ 
                               
                                 G 
                                 _ 
                               
                             
                             
                               ∂ 
                               
                                 k 
                                 
                                   RW 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             hover 
                           
                         
                         , 
                         
                           
                             
                               ∂ 
                               
                                 G 
                                 _ 
                               
                             
                             
                               ∂ 
                               
                                 k 
                                 
                                   RW 
                                   3 
                                 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             hover 
                           
                         
                         , 
                         
                           
                             
                               ∂ 
                               
                                 G 
                                 _ 
                               
                             
                             
                               
                                 ∂ 
                                 Δ 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 t 
                                 RW 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             hover 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               ∂ 
                               
                                 G 
                                 _ 
                               
                             
                             
                               ∂ 
                               
                                 k 
                                 
                                   LW 
                                   1 
                                 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             hover 
                           
                         
                         , 
                         
                           
                             
                               ∂ 
                               
                                 G 
                                 _ 
                               
                             
                             
                               ∂ 
                               
                                 k 
                                 
                                   LW 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             hover 
                           
                         
                         , 
                         
                           
                             
                               ∂ 
                               
                                 G 
                                 _ 
                               
                             
                             
                               ∂ 
                               
                                 k 
                                 
                                   LW 
                                   3 
                                 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             hover 
                           
                         
                         , 
                         
                           
                             
                               ∂ 
                               
                                 G 
                                 _ 
                               
                             
                             
                               
                                 ∂ 
                                 Δ 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 t 
                                 LW 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             hover 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     These derivatives or sensitivities are used as design parameters for the control system for the aircraft. 
     The control derivatives are utilized in a control effectiveness matrix that provides information on how changes in the control variables affect the cycle averaged forces and moments. This matrix formulation is as follows: 
                     [             X   _     des                 Y   _     des                 Z   _     des                 RM   _     des                 PM   _     des                 YM   _     des           ]     ⁢           =       B   ⁡     [           k     RW   1                 k     RW   2                 k     RW   3                 Δ   ⁢           ⁢     t   RW                 k     LW   1                 k     LW   2                 k     LW   3                 Δ   ⁢           ⁢     t   LW             ]       =     B   ⁢           ⁢   δ               Equation   ⁢           ⁢   6               
where  X   des  is the desired cycle-averaged vertical force that acts along the x-body axis of the fuselage,  Y   des  is the desired cycle-averaged side force acting along the y-body axis of the fuselage,  Z   des  is the desired cycle-averaged horizontal force acting along the z-body axis of the fuselage,  RM   des  is the desired cycle-averaged rolling moment about the x-body axis of the vehicle,  PM   des  is the desired cycle-averaged pitching moment about the y-body axis of the vehicle, and  YM   des  is the desired cycle-averaged yawing moment about the z-body axis of the vehicle. These desired body-axes forces and moments are produced by an outer loop control law. In Equation 6, δ is the control vector given by:
 
                   δ   =       [                 k     RW   1             k     RW   2             k     RW   3             Δ   ⁢           ⁢     t   RW             k     LW   1             k     LW   2             k     LW   3                   Δ   ⁢           ⁢     t   LW             ]     T             Equation   ⁢           ⁢   7               
and B is the control effectiveness matrix. More specifically, the B matrix is:
 
     
       
         
           
             
               
                 
                   B 
                   = 
                   
                       
                     
                       
                         [ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   ∂ 
                                   
                                     X 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     X 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     X 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     X 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     RW 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     X 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     X 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     X 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     X 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     LW 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∂ 
                                   
                                     Y 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Y 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Y 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Y 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     RW 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Y 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Y 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Y 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Y 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     LW 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∂ 
                                   
                                     Z 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Z 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Z 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Z 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     RW 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Z 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Z 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Z 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     Z 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     LW 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∂ 
                                   
                                     RM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     RM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     RM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     RM 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     RW 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     RM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     RM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     RM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     RM 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     LW 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∂ 
                                   
                                     PM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     PM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     PM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     PM 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     RW 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     PM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     PM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     PM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     PM 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     LW 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∂ 
                                   
                                     YM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     YM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     YM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       RW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     YM 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     RW 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     YM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     YM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     YM 
                                     _ 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     k 
                                     
                                       LW 
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∂ 
                                   
                                     YM 
                                     _ 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     LW 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         | 
                         hover 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     Analysis has shown that the quarter-cycle waveforms as described in this disclosure are effective at generating controllable values of the desired cycle averaged forces and moments, namely,  X   des ,  Y   des ,  Z   des ,  RM   des ,  PM   des , and  YM   des . With appropriate choices of vehicle parameters, the rank of the B matrix is six, thus indicating the ability to independently control six rigid body degrees of freedom using quarter-cycle constant period requency modulation. 
       FIG. 8  shows a block diagram of the flight control system for the aircraft. An operator command processor  209  provides instantaneous values of the desired aircraft state to a flight control computer  205  in the form of inertial positions x c , y c , and z c  and vehicle attitude, which can be expressed in Euler angles, φ c , θ c , and ψ c . A sensor package  207  consisting of an inertial measurement unit, GPS receiver, and air data system, provides the actual position and orientation of the fuselage to said flight control computer. Said flight control computer compares the operator commands to the actual state of the aircraft and generates a vector of desired cycle-averaged force and moment commands  X   des ,  Y   des ,  Z   des ,  RM   des ,  PM   des , and  YM   des . The flight control laws in said flight control computer can be designed using any convenient method. The desired cycle averaged forces and moments are transformed into left and right wing waveform parameter commands by a control allocator  203 . Said control allocator can be implemented using any number of techniques developed for control allocation. One method is computing the pseudo-inverse, B # =(B T B) −1 B T  of the control effectiveness matrix B in order to compute: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             k 
                             
                               RW 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             k 
                             
                               RW 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             k 
                             
                               RW 
                               3 
                             
                           
                         
                       
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               RW 
                             
                           
                         
                       
                       
                         
                           
                             k 
                             
                               LW 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             k 
                             
                               LW 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             k 
                             
                               LW 
                               3 
                             
                           
                         
                       
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               LW 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       B 
                       # 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 X 
                                 _ 
                               
                               des 
                             
                           
                         
                         
                           
                             
                               
                                 Y 
                                 _ 
                               
                               des 
                             
                           
                         
                         
                           
                             
                               
                                 Z 
                                 _ 
                               
                               des 
                             
                           
                         
                         
                           
                             
                               
                                 RM 
                                 _ 
                               
                               des 
                             
                           
                         
                         
                           
                             
                               
                                 PM 
                                 _ 
                               
                               des 
                             
                           
                         
                         
                           
                             
                               
                                 YM 
                                 _ 
                               
                               des 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   9 
                 
               
             
           
         
       
     
     The waveform parameters for the first wing planform  20  and second wing planform  22  that are generated by said control allocator form the input to control board  32 . Inside control board  32  are two instantiations of waveform generator  399 . The left wing waveform generator, contained within control board  32 , supplies a time varying commutation signal to second BLDC motor  18  that ultimately creates the required flapping motion of second wing planform  22 . The right wing waveform generator, contained within control board  32 , supplies a time varying commutation signal to first BLDC motor  16  that ultimately creates the required flapping motion of first wing planform  20 . Flapping planforms  20  and  22  generate aerodynamic forces and moments that cause the fuselage to be attracted to a desired position and attitude set point specified by said operator command processor. 
     Although the disclosed invention makes use of BLDC motors to impart motion to the flapping wings of the aircraft, other prime movers may be utilized. Linear electric motors or piezoelectric actuators coupled with gearing and linkages could be used to impart quarter-cycle motion to the wings of the aircraft. Furthermore, wingbeat amplitude modulation could be exchanged for frequency modulation. The control effectiveness matrix could be computed based on a constant wingbeat frequency that was tuned for a particular amplitude associated with quasi-hover. The wingbeat amplitude would then appear as a control input in the control effectiveness matrix and the control law would otherwise be implemented as described herein. 
     While the present invention has been illustrated by a description of one or more embodiments thereof and while these embodiments have been described in considerable detail, they are not intended to restrict or in any way limit the scope of the appended claims to such detail. Additional advantages and modifications will readily appear to those skilled in the art. The invention in its broader aspects is therefore not limited to the specific details, representative apparatus and method, or illustrative examples shown and described. Accordingly, departures may be made from such details without departing from the scope of the general inventive concept.