Abstract:
A system including a heavier-than-air vehicle (HTA), a lighter-than-air vehicle (LTA), and a tether is disclosed. The tether is coupled between the HTA and the LTA such that the LTA supports the HTA. The system may be further configured to suit the needs of the particular application. For example, the system may be configured to be controllable from a remote location or capable of autonomous operations. The system may also be configured such that the HTA comprises a mission payload and communications equipment. In an exemplary embodiment of the invention, the LTA is configured to provide lift for the system and the HTA is configured to provide station-keeping propulsion, the advantage being reduced fuel consumption and increased mission endurance.

Description:
TECHNICAL FIELD 
       [0001]    Embodiments of the present invention relate generally to composite air vehicles. More particularly, embodiments of the present invention relate to a composite air vehicle system comprising a heavier-than-air vehicle (HTA), a lighter-than-air vehicle (LTA), and a tether coupled between the HTA and the LTA such that the LTA supports the HTA. 
       BACKGROUND 
       [0002]    There is a current need for an airborne vehicle capable of performing surveillance, reconnaissance, communications, weapons delivery, or other missions, which can remain aloft for long periods of time. Because HTAs require propulsion to stay aloft, their endurance is limited by their fuel capacity and payload. The High Altitude Long Endurance (HALE) aircraft community has focused considerable effort over the past twenty years on designs to extend vehicle endurance up to one or two days, with future advanced concept designs targeting four to five days. However, airborne Intelligence, Surveillance, and Reconnaissance (ISR) mission requirements have lasted much longer than four to five days, as evidenced in recent theatres of conflict and other applications. Additionally, other missions such as communications relay, electronic warfare, and weapons delivery may also involve extended duration mission requirements. Furthermore, operational costs of carrying out these missions vary inversely as a function of aircraft endurance. 
         [0003]    On the other hand, LTAs often have adequate endurance and ample payload capacity but lack the propulsion and energy means required to keep a large LTA in one location, given the winds at high altitude. Additionally, LTAs are limited by materials technology. For example, ultraviolet radiation causes degradation at high altitude over periods of time. LTAs are further limited by operational constraints on takeoff and landing as well as survivability in military environments. 
       BRIEF SUMMARY 
       [0004]    An HTA and an LTA can be coupled together in order to utilize the positive characteristics of each while mitigating the negative factors. By coupling an HTA and an LTA using a tether such that the LTA supports the HTA, the system described herein can benefit from the lift provided by the LTA. At the same time, the system can exploit the station-keeping propulsion provided by the HTA. Such a system is desirable because the HTA can conserve fuel and remain aloft in one location for greater periods of time. The LTA can be simple, inexpensive, and expendable. Additionally, the system described herein can utilize currently available HTAs and LTAs with relatively minor design modifications. 
         [0005]    The above and other aspects of the invention may be carried out in one embodiment by a system comprising an HTA, an LTA, and a tether coupled between the HTA and the LTA such that the LTA can support the HTA. The HTA may include a mission payload. Also, the system may further comprise remote control or data collection subsystems for remote or autonomous control. 
         [0006]    Another embodiment is an LTA comprising a body having a lighter-than-air gas, an attachment mechanism, and a tether configured for supporting a heavier-than-air body and having an end configured for coupling to the attachment mechanism. 
         [0007]    Another embodiment is an HTA comprising an aircraft having a propulsion system, an attachment mechanism defining at least one attachment location on the aircraft, and a tether configured for supporting the aircraft and having an end configured for coupling to the attachment mechanism. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]    A more complete understanding of the present invention may be derived by referring to the detailed description and claims when considered in conjunction with the following figures, wherein like reference numbers refer to similar elements throughout the figures. 
           [0009]      FIG. 1  is a diagram of a system of one embodiment comprising an HTA, an LTA, and a tether coupled between the HTA and the LTA; 
           [0010]      FIG. 2  is a block diagram of a composite HTA/LTA system configured in accordance with an embodiment of the invention; 
           [0011]      FIG. 3  is a top view of an HTA of one embodiment showing possible tether attachment locations; 
           [0012]      FIG. 4  is a side view of a composite HTA/LTA system of one embodiment having a bladder for storing a consumable for the HTA; 
           [0013]      FIG. 5  is a top-view diagram showing thrust applied by an HTA to provide station-keeping against opposing forces; 
           [0014]      FIG. 6  is a side-view diagram showing thrust applied by an HTA to provide station-keeping against opposing forces, the HTA flying in a substantially straight path; and 
           [0015]      FIGS. 7-11  are diagrams related to a feasibility study for a composite HTA/LTA vehicle. 
       
    
    
     DETAILED DESCRIPTION 
       [0016]    The following detailed description is merely illustrative in nature and is not intended to limit the embodiments of the invention or the application and uses of such embodiments. Furthermore, there is no intention to be bound by any expressed or implied theory presented in the preceding technical field, background, brief summary or the following detailed description. 
         [0017]      FIG. 1  shows a system of one embodiment  100  comprising a heavier-than-air vehicle (HTA)  102 , a lighter-than-air vehicle (LTA)  104 , and a tether  106  coupled between the HTA  102  and the LTA  104 . The tether  106  is configured such that the LTA  104  supports the HTA  102 . In a preferred embodiment, the system  100  is configured to be launched from a fixed or mobile surface launch facility and rise to an operational altitude appropriate for the mission and vehicle combination. 
         [0018]    The HTA  102  has a propulsion system which can be used to provide station-keeping for the system  100 , and which can be used for other maneuvering of the HTA and/or the LTA  104 . In a preferred embodiment, the HTA  102  is an unmanned aerial vehicle (UAV) of a fixed wing design. A preferred embodiment of the HTA  102  also has a payload sufficient to carry equipment necessary for Intelligence, Surveillance, and Reconnaissance (ISR) or other missions. One embodiment of the HTA  102  may be realized as an ALTAIR aircraft manufactured by GENERAL ATOMICS. The ALTAIR craft has adequate payload to carry the PREDATOR mission payload and the ALTAIR craft is powered by a 700 horsepower turboprop engine. In practice, the propulsion system of the HTA  102  may utilize other technologies such as, without limitation: jet engines; piston-powered propellers; hybrid gas/electric propulsion; or electric propulsion. Additionally, the ALTAIR craft is approximately 11 meters in length and has a mass of approximately 3200 kilograms. Another embodiment of the HTA  102  may be realized as an ALTUS II craft, also manufactured by GENERAL ATOMICS. The ALTUS II craft is approximately 7.3 meters in length and has a mass of approximately 725 kilograms. An embodiment of the HTA  102  may also include some modification of the ALTAIR or ALTUS II, such as increasing the propeller diameter or adding additional propeller blades. 
         [0019]    The LTA  104  may be a balloon, a bag, a blimp, an aerostat, a shell, or any suitable component having virtually any suitable shape that is filled with a lighter-than-air gas, such as helium or hydrogen. A preferred embodiment of the LTA  104  is realized as an aerostat having aerodynamic characteristics (shape, configuration, and/or other features that provide aerodynamic qualities). The LTA  104  relies on buoyancy for lift, not on dynamic lift. In a preferred embodiment, the LTA  104  is a low-cost, unpowered, unmanned, disposable component which provides lift for the system  100 . The LTA  104  may have a diameter (at its widest point) in the range of approximately 15 to 36 meters. The specific size, shape, and configuration of the LTA  104  will be dictated by the mass of the HTA, required operational altitude, cost constraints, weight restrictions, visibility considerations, stability considerations, and other practical conditions. The LTA  104  is configured for coupling to the tether  106  at attachment location  108 . Although a single attachment location  108  may be utilized as depicted in  FIG. 1 , an embodiment of the LTA  104  may include multiple attachment locations  108  for a tether  106  having a plurality of coupling features. 
         [0020]    A preferred embodiment of the tether  106  will have a first end configured for coupling to the attachment mechanism of the LTA  104  and a second end configured for coupling to the HTA  102 . In this example, the tether  106  is “dumb” because it does not carry power or communications. Rather, the tether  106  is primarily utilized as a load-bearing component; the tether  106  may be configured to support the selected HTA vehicle. Of course, the load rating of the tether  106  may be selected to accommodate the weight of the HTA  102  and to accommodate anticipated dynamic forces caused by environmental conditions, equipment located on the HTA  102 , and/or propulsion of the HTA  102 . In practice, the tether  106  can be inexpensive and lightweight compared to traditional ground-anchored tethers. The tether  106  may be formed from any appropriate material having the desired physical properties and load rating. For example, tether  106  may be formed from a flexible, high-strength, low-density material such as Zylon or Kevlar polymer products, or the like. In another embodiment, the tether  106  may be rigid or partially rigid. A length of the tether  106  may be within a wide range, depending on mission goals. A length of the tether between the HTA  102  and the LTA  104  may have a wide range, depending on mission goals. In one embodiment, the tether is less than 500 feet. However, embodiments with very short (e.g., a few feet) or very long (e.g., 20,000 feet) do not depart from the scope of this invention. Also, the length of the tether  106  may be changeable during system operation, such as with the use of a reel to bring the HTA and LTA closer together, farther apart, or into contact with each other. A thickness of the tether is selected to provide the desired load rating for supporting the HTA  102  and for towing the LTA  104  against wind forces. 
         [0021]    In a preferred embodiment of the system  100 , the propulsion system on the HTA  102  can be throttled to produce a force in order to provide station-keeping for the system  100 . As used herein, the term “station-keeping” refers to maneuvers that maintain the system  100  within a specified area relative to a designated reference position. For example, the station-keeping area or distance for the system  100  may correspond to a range of up to 20,000 meters away from the reference location, depending on mission application. The actual station-keeping distance may vary from one deployment to another.  FIG. 1  shows the HTA  102  flying in a path  110  defined by the attachment location  108  and the tether  106 .  FIG. 1  depicts a substantially circular path  110 . However, the path  110  can be of any trajectory, size, or shape, including a substantially straight one. In practice, the path  110  may be dependent on opposing forces such as wind, as the HTA  102  provides station-keeping for the system  100 . Two possible flight paths  110  for the HTA  102  are illustrated in  FIGS. 5 and 6  and are described in more detail below. Alternatively, the HTA  102  may simply hang suspended at the end of the tether  106 . If the HTA  102  is equipped with a propulsion system having restart capability, the propulsion system can be shut down in order to conserve fuel (assuming the environmental conditions allow such operation). 
         [0022]      FIG. 2  shows a block diagram of a system  200 , similar to that described above, but further comprising various equipment and subsystems. In this regard, the system  200  generally includes an HTA  202 , an LTA  204 , and a tether  206 . An embodiment of the system  200  may have any combination of the equipment and subsystems shown in  FIG. 2 , including all or none of them, as well as the standard vehicle systems and subsystems. 
         [0023]    One embodiment of the system  200  may include an energy collection subsystem  208 . The energy collection subsystem  208  may be comprised of solar panels, low-cost solar cells, or any suitable energy collection device, mechanism, or apparatus, and may be located on the HTA  202  and/or the LTA  204 . For example, the energy collection subsystem  208  may utilize solar cells on the outer surface of the HTA  202  and/or the LTA  204 . A preferred embodiment of system  200  does not include an energy collection subsystem  208  at the LTA  204  because the LTA  204  is designed to be passive, inexpensive, and expendable. The system  200  might also have a suitably configured energy storage subsystem  210 , located on the HTA  202  and/or the LTA  204 ; the energy storage subsystem  210  may cooperate with the energy collection subsystem  208  to serve as a power source for the system  200 . A preferred embodiment of system  200  does not include an energy storage subsystem  210  at the LTA  204  because the LTA  204  is designed to be passive, inexpensive, and expendable. Accordingly, either an energy collection subsystem  208  or an energy storage subsystem  210  would be an enhancement and is not required for an HTA-LTA system, such as the system described above in conjunction with  FIG. 1 , to fulfill its mission. 
         [0024]    Also depicted in  FIG. 2  are mission payload  212  and communications equipment  214 . The mission payload  212  may be surveillance, reconnaissance, communications, weather sensors, electronic warfare, weapons, or other subsystem(s). In addition to the mission payload  212 , the HTA  202  may carry other sensors dedicated to supporting remote control and/or autonomous operations of the HTA  202 . Moreover, the HTA  202  may also comprise communications equipment  214  that is configured to establish data communication with one or more devices or subsystems external (or internal) to the HTA  202 . For example, the communications equipment  214  may be realized as a wireless data communication system that uses any suitable data transmission or protocol. In practice, the HTA  202  may also carry other vehicle subsystems as needed. 
         [0025]      FIG. 2  shows an attachment mechanism  216  on the LTA  204  for coupling the tether  206  to the LTA  204 . The attachment mechanism  216  for the LTA  204  may be a swiveling fixture on the bottom of the LTA  204 , such as a ball joint. Alternatively, the attachment mechanism may be a u-joint, gimbal, or other mechanism. Furthermore, an embodiment of the LTA  204  may utilize multiple attachment mechanisms  216  for a tether having a plurality of coupling features. Similarly, as shown in  FIG. 2 , the HTA  202  has an attachment mechanism  218 . In a preferred embodiment of the HTA  202 , the attachment mechanism  218  will couple the tether  206  to one or a plurality of attachment locations on the HTA  202 . 
         [0026]    One advantage of the composite HTA/LTA system is that at mission completion, when the LTA is no longer needed to provide or maintain lift for the system, the LTA can be jettisoned. Thus by carrying any costly equipment on the HTA and recovering the HTA at mission completion, the LTA can be made in the least costly manner possible. The HTA could be recovered by allowing it to return under its own power either remotely or autonomously controlled. If desired, the LTA could also be recovered, either apart from or with the HTA. In this regard,  FIG. 2  indicates a decoupling mechanism  220  on the HTA  202  for separating the HTA  202  and the LTA  204 . One possible embodiment of the decoupling mechanism  220  is a guillotine-type mechanism that severs the tether  206  as needed. Alternatively, the decoupling mechanism  220  may be realized as any of the following, without limitation: a pyrotechnic device; a solenoid-initiated quick release device; or a wide variety of other detachment mechanisms. 
         [0027]      FIG. 2  also shows two subsystems which are remote from the system  200 . These two subsystems may be ground-based, aircraft-based, space-based, or otherwise, and in a preferred embodiment, will communicate with the HTA  202 . One of the subsystems is a remote control subsystem  222 . The remote control subsystem  222  may comprise components and logic for controlling adjustments in position, altitude, and attitude of the system  200 . In particular, the remote control subsystem  222  may be configured to remotely control the operation of the HTA  202 , including propulsion maneuvers, station-keeping maneuvers, and/or landing maneuvers. However, the system  200  may be configured to control itself autonomously and may not require a remote control subsystem  222 . The other remote subsystem in this example is a data collection subsystem  224 . The data collection subsystem  224  may include components and logic for sensing position, altitude, and attitude of the system  100 . In practice, the data collection subsystem  224  and/or the remote control subsystem  222  may cooperate with the communications equipment  214  onboard the HTA  102  to support data transfer to the system  200 . 
         [0028]      FIG. 2  further shows a consumable storage subsystem  226  on the HTA  202  and a consumable bladder or other storage mechanism  228  on the tether  206 . The consumable handled by these components may be, for example, fuel, coolant, lubricant, or hydraulic fluid for the HTA  202 . Each of the consumable storage subsystems ( 226  and  228 ) are possible enhancements and are not required for an HTA-LTA system, such as the system described above in conjunction with  FIG. 1 , to fulfill its mission. The consumable storage subsystem  228  is described further below in conjunction with  FIG. 4 . 
         [0029]    It should be appreciated that  FIG. 2  depicts an embodiment of the system  200  that includes several optional features. In practice, the system  200  need not (and preferably will not) be deployed with all of features and components shown in  FIG. 2 . In this regard, a preferred embodiment of system  200  may utilize a fully passive, low-cost, and disposable LTA  204  that carries no energy collection subsystem, no energy storage subsystem, no communication equipment, etc. Of course, the preferred embodiment of LTA  204  will utilize some type of attachment mechanism  216  for the tether  206 . Moreover, a preferred embodiment of the HTA  202  need not include the energy collection subsystem  208  or the energy storage subsystem  210 . In certain deployments, the HTA  202  may only require the attachment mechanism  218  and the mission payload  212 . 
         [0030]      FIG. 3  depicts a top view of an HTA  300 , which may be configured as described above.  FIG. 3  illustrates possible attachment locations  302  on the HTA  300  for an attachment mechanism coupled to a tether. The attachment mechanism may define one attachment location  302  on the HTA  300  or a plurality of attachment locations  302 , as shown in  FIG. 3 . A compatible tether may include a plurality of coupling elements corresponding to these attachment mechanisms. In this example, the attachment locations  302  are arranged to provide stability for the HTA  300  during station-keeping maneuvers. In particular, the attachment locations  302  correspond to a fore location, and aft location, a left wing location, and a right wing location. The specific attachment locations may vary from that depicted in  FIG. 3 , depending upon the particular system deployment, the configuration of the HTA  300 , and other practical considerations. 
         [0031]      FIG. 4  shows a system  400  comprising an HTA  402 , an LTA  404 , a tether  406 , and consumable storage  408  coupled to the tether  406 . The consumable storage  408  coupled to the tether  406  may be an enhancement to an HTA-LTA system and is not required for an HTA-LTA system, such as the system described in conjunction with  FIG. 1 , to fulfill its mission. The consumable storage  408  may have an outlet  410  configured to deliver a consumable (or any substance) to the HTA  402 . The outlet  410  can be designed to mate with a receptacle of the HTA  402 . The consumable may, for example, be fuel, coolant, lubricant, or hydraulic fluid for the HTA  402 . In one embodiment, the consumable is gravity-fed from a bladder  408  to the HTA  402 . In such an embodiment, the bladder  408  is suitably configured to gravitationally deliver the consumable to the HTA  402 . Also, the consumable storage  408  can be formed from a flexible material and can pressure feed the consumable through the outlet  410  to the HTA  402 . A preferred embodiment of the system  400  would comprise a long, cylindrical bladder  408  attached to the tether  406  at a plurality of locations near the end of the tether  406  that is closest to the HTA  402 . One embodiment of the outlet  410  may be a flexible fuel feed line from the bladder  408  to the HTA  402 . As described above, the HTA  402  may include a consumable storage subsystem that receives the consumable delivered by the consumable storage  408  (see  FIG. 2 ). 
         [0032]    An HTA in a composite aircraft system as described herein may utilize its propulsion system to maintain the system within a desired station-keeping area. The methodology of how this is done depends on the type of station-keeping required and the forces such as winds causing the system to displace from its desired location. Since winds are perceived as the most disruptive force, the following analysis focuses on that factor, but other forces may be present. Two disparate mission requirements are possible. In one, the system is to maintain a placement in the sky to within a few hundred meters. In the other, the system could move around in a certain area (e.g., a “box” that is ten kilometers on a side). The station-keeping requirements will be different for these scenarios. In addition, there may be three types of disruptive winds: no wind, light wind, or high wind. Table 1 describes possible station-keeping regimes against different wind and location tolerances. 
         [0000]    
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Station-Keeping Operations Versus Mission Requirements and Winds 
               
             
          
           
               
                   
                 Station-Keeping Requirements 
               
               
                   
                   
               
             
          
           
               
                 Winds 
                 Maintain close requirements 
                 Allowable movement within a large box 
               
               
                   
                 within a one hundred meter box. 
                 of approximately 10,000 meters on a 
               
               
                   
                   
                 side. 
               
               
                 High Wind 
                 The HTA points into the wind 
                 The HTA points into the wind and holds 
               
               
                   
                 and holds the LTA against this 
                 the LTA against this force. There is 
               
               
                   
                 force. There is enough airflow 
                 enough airflow over the control surfaces 
               
               
                   
                 over the control surfaces to 
                 to maintain control authority. 
               
               
                   
                 maintain control authority. 
               
               
                 Low Wind 
                 The HTA flies in a circular 
                 The HTA hangs limply under the LTA 
               
               
                   
                 pattern as prescribed by the 
                 and drifts back against the wind until it 
               
               
                   
                 tether. There are times when the 
                 reaches its station-keeping boundary. 
               
               
                   
                 LTA is blown back and other 
                 Then, the engine throttles up, and the 
               
               
                   
                 times when the LTA is pulled 
                 HTA pulls the LTA back against the 
               
               
                   
                 against the wind. The vehicle 
                 wind to the other side of the box. The 
               
               
                   
                 stays within its station-keeping 
                 combination of wind and movement 
               
               
                   
                 box. The circular pattern keeps 
                 provides enough control flow over the 
               
               
                   
                 flow over the HTA&#39;s surfaces and 
                 wings. There it will throttle down and 
               
               
                   
                 maintains control authority. 
                 be slowly blown back by the wind to 
               
               
                   
                   
                 repeat the process. 
               
               
                 No Wind 
                 The HTA powers down and 
                 The HTA powers down and hangs 
               
               
                   
                 hangs beneath the LTA. 
                 beneath the LTA. 
               
               
                   
               
             
          
         
       
     
         [0033]    Propulsion may be applied in a controlled manner to compensate for wind, turbulence, thermal currents, and other environmental conditions. In this regard,  FIG. 5  is a top-view diagram that illustrates thrust applied by an HTA  502  to provide station-keeping against opposing forces  504  (such as wind).  FIG. 5  depicts the HTA  502  flying in a path  506  that is substantially circular (or other closed figure). A substantially circular flight path  506  by the HTA  502  may be desirable in order to conserve fuel. A substantially elliptical or other flight path  110  by the HTA  102  may be similarly advantageous. In such a situation, thrust may be provided by the HTA  502  only during half of each rotation about the attachment mechanism on the LTA  508 . The arrow  510  in  FIG. 5  represents the flight path. Such a sequence of applying thrust can produce a net force  512  to provide station-keeping by countering opposing forces  504 . Opposing forces  504  may be caused by wind or other factors. 
         [0034]      FIG. 6  is a side-view diagram showing thrust  602  applied by an HTA  604  to provide station-keeping against opposing forces  606 . This formation is used against the high wind scenario and in the power forward/drift back formation. 
         [0035]    Feasibility Study 
         [0036]    The low wind/tight station-keeping scenario presents the requirement of maintaining control within a constrained area. In that case, the HTA is required to fly in tight circles, pulling the LTA against the wind force and maintaining enough airflow over the control surfaces. The forces involved can be complicated, but they all must balance to ensure that the HTA/LTA system remains within its required airspace. 
         [0037]    A composite HTA/LTA vehicle can be deployed as a practical working embodiment, as demonstrated in this section, which refers to  FIGS. 7-11 . Referring to  FIG. 7 , assume that the system includes an aerostat with an aircraft hanging underneath it. The airplane is under power and flying in a tight circle. The airplane is maintaining a constant velocity of Vp (m/s). The length of the tether is defined by the symbol P (meters). The plane has a weight of W (Newtons) and a lift of L (Newtons). Since the plane is traveling around in a circle it has a centripetal motion of C (Newtons). The aircraft is also supported in part by the tether which has a tension T (Newtons). 
         [0038]    Referring to  FIG. 8 , the following relationships apply: 
         [0000]    
       
         
           
             
               φ 
               2 
             
             = 
             
               
                 
                   φ 
                   3 
                 
                 + 
                 
                   φ 
                   1 
                 
               
               2 
             
           
         
       
     
         [0039]    R=P sin(φ 1 −φ 2 ); R is the circular radius corresponding to the tethered flight path of the HTA. 
         [0040]    The angle φ 2  is defined as the angle from the vertical that defines the center of the circle that the aircraft is moving around. β is the angle that defines the aircraft&#39;s path within the circle, and γ is the angle that defines the angle away from the centerline. The following relationships apply: 
         [0000]      Speed=Vp 
         [0000]      Circle=2π R= 2π P  sin(φ 1 −φ 2 ) 
         [0000]    
       
         
           
             
               Time 
               = 
               
                 t 
                 = 
                 
                   
                     Circle 
                     / 
                     Speed 
                   
                   = 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       P 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               φ 
                               1 
                             
                             - 
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     Vp 
                   
                 
               
             
             ; 
           
         
       
     
         [0000]    this is the time required for the HTA to complete one rotation. 
         [0000]    
       
         
           
             
               β 
               = 
               
                 
                   ω 
                    
                   
                       
                   
                    
                   t 
                 
                 = 
                 
                   
                     Vp 
                     
                       P 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               φ 
                               1 
                             
                             - 
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   t 
                 
               
             
             ; 
           
         
       
     
         [0041]    β is the radial angle away from the LTA/HTA centerline. 
         [0000]    
       
         
           
             0 
             ≤ 
             t 
             ≤ 
             
               
                 2 
                  
                 
                     
                 
                  
                 π 
                  
                 
                     
                 
                  
                 P 
                  
                 
                     
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     
                       
                         φ 
                         1 
                       
                       - 
                       
                         φ 
                         2 
                       
                     
                     ) 
                   
                 
               
               Vp 
             
           
         
       
       
         
           
             
               
                 
                   φ 
                   = 
                   
                     
                       φ 
                       2 
                     
                     + 
                     
                       arc 
                        
                       
                           
                       
                        
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               R 
                                
                               
                                   
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               β 
                             
                             
                               P 
                                
                               
                                   
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       φ 
                                       1 
                                     
                                     - 
                                     
                                       φ 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       φ 
                       2 
                     
                     + 
                     
                       arc 
                        
                       
                           
                       
                        
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               P 
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       φ 
                                       1 
                                     
                                     - 
                                     
                                       φ 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   β 
                                 
                                 ) 
                               
                             
                             
                               P 
                                
                               
                                   
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       φ 
                                       1 
                                     
                                     - 
                                     
                                       φ 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             φ 
             = 
             
               
                 φ 
                 2 
               
               + 
               
                 arc 
                  
                 
                     
                 
                  
                 
                   tan 
                    
                   
                     ( 
                     
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               φ 
                               1 
                             
                             - 
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           cos 
                            
                           
                               
                           
                            
                           β 
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             φ 
             = 
             
               
                 φ 
                 2 
               
               + 
               
                 arc 
                  
                 
                     
                 
                  
                 
                   tan 
                    
                   
                     ( 
                     
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               φ 
                               1 
                             
                             - 
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             Vp 
                             
                               P 
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       φ 
                                       1 
                                     
                                     - 
                                     
                                       φ 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                            
                           t 
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             0 
             ≤ 
             t 
             ≤ 
             
               
                 2 
                  
                 
                     
                 
                  
                 π 
                  
                 
                     
                 
                  
                 P 
                  
                 
                     
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     
                       
                         φ 
                         1 
                       
                       - 
                       
                         φ 
                         2 
                       
                     
                     ) 
                   
                 
               
               Vp 
             
           
         
       
       
         
           
             γ 
             = 
             
               
                 arc 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                   R 
                   P 
                 
               
               = 
               
                 
                   arc 
                    
                   
                       
                   
                    
                   
                     sin 
                      
                     
                       ( 
                       
                         
                           P 
                            
                           
                               
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   φ 
                                   1 
                                 
                                 - 
                                 
                                   φ 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         P 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   ( 
                   
                     
                       φ 
                       1 
                     
                     - 
                     
                       φ 
                       2 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0042]    With reference also to  FIG. 9 , the following relationships apply: 
         [0000]    
       
         
           
             
               
                 
                   α 
                   = 
                   
                     arc 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             R 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             β 
                           
                           P 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     arc 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             P 
                              
                             
                                 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     φ 
                                     1 
                                   
                                   - 
                                   
                                     φ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             β 
                           
                           P 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     arc 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   φ 
                                   1 
                                 
                                 - 
                                 
                                   φ 
                                   2 
                                 
                               
                               ) 
                             
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           β 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             β 
             = 
             
               
                 ω 
                  
                 
                     
                 
                  
                 t 
               
               = 
               
                 
                   Vp 
                   
                     P 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             φ 
                             1 
                           
                           - 
                           
                             φ 
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
                  
                 t 
               
             
           
         
       
       
         
           
             α 
             = 
             
               arc 
                
               
                   
               
                
               
                 sin 
                  
                 
                   ( 
                   
                     sin 
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         
                           φ 
                           1 
                         
                         - 
                           
                          
                         
                           φ 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             Vp 
                             
                               P 
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       φ 
                                       1 
                                     
                                     - 
                                     
                                       φ 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                            
                           t 
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             0 
             ≤ 
             t 
             ≤ 
             
               
                 2 
                  
                 
                     
                 
                  
                 π 
                  
                 
                     
                 
                  
                 P 
                  
                 
                     
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     
                       
                         φ 
                         1 
                       
                       - 
                       
                         φ 
                         2 
                       
                     
                     ) 
                   
                 
               
               Vp 
             
           
         
       
     
         [0043]    Here, α defines the angle away from the x-y plane that the aircraft is making as it spins around the circle. 
         [0000]    
       
         
           
             
               ∑ 
               
                 F 
                 x 
               
             
             = 
             0 
           
         
       
       
         
           
             
               
                 
                   M 
                   p 
                 
                  
                 
                   
                     
                        
                       2 
                     
                      
                     x 
                   
                   
                      
                     
                       t 
                       2 
                     
                   
                 
               
               + 
               
                 C 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 φ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 β 
               
               - 
               
                 L 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 θ 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 φ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 α 
               
               - 
               
                 T 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 φ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 α 
               
             
             = 
             0 
           
         
       
       
         
           
             
               ∑ 
               
                 F 
                 y 
               
             
             = 
             0 
           
         
       
       
         
           
             
               
                 
                   M 
                   p 
                 
                  
                 
                   
                     
                        
                       2 
                     
                      
                     y 
                   
                   
                      
                     
                       t 
                       2 
                     
                   
                 
               
               - 
               
                 
                   M 
                   p 
                 
                  
                 G 
               
               + 
               
                 C 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 φ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 β 
               
               + 
               
                 L 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 θ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 φ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 α 
               
               - 
               
                 T 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 φ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 α 
               
             
             = 
             0 
           
         
       
       
         
           
             
               ∑ 
               
                 F 
                 z 
               
             
             = 
             0 
           
         
       
       
         
           
             
               
                 
                   M 
                   p 
                 
                  
                 
                   
                     
                        
                       2 
                     
                      
                     z 
                   
                   
                      
                     
                       t 
                       2 
                     
                   
                 
               
               - 
               
                 C 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 φsin 
                  
                 
                     
                 
                  
                 β 
               
               + 
               
                 L 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 θ 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 α 
               
               + 
               
                 T 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 φsin 
                  
                 
                     
                 
                  
                 α 
               
             
             = 
             0 
           
         
       
     
         [0044]    With reference also to  FIG. 10 , the following relationships apply: 
         [0000]    
       
         
           
             C 
             = 
             
               
                 
                   
                     V 
                     p 
                     2 
                   
                    
                   W 
                 
                 RG 
               
               = 
               
                 
                   
                     V 
                     p 
                     2 
                   
                    
                   W 
                 
                 
                   P 
                    
                   
                       
                   
                    
                   
                     sin 
                      
                     
                       ( 
                       
                         
                           φ 
                           1 
                         
                         - 
                         
                           φ 
                           2 
                         
                       
                       ) 
                     
                   
                    
                   G 
                 
               
             
           
         
       
     
         [0000]    
       
         
           
             
               L 
               = 
               
                 
                   
                     V 
                     p 
                     2 
                   
                    
                   S 
                    
                   
                       
                   
                    
                   ρ 
                    
                   
                       
                   
                    
                   
                     C 
                     L 
                   
                 
                 2 
               
             
             ; 
           
         
       
     
         [0000]    the coefficient of lift C L  is a function of the bank angle that the aircraft makes as it spins around the circle. 
         [0045]    With reference to  FIG. 11 , at the aerostat, which jerks around as the plane circles, the following relationships apply: 
         [0000]    
       
         
           
             
               ∑ 
               
                 F 
                 x 
               
             
             = 
             0 
           
         
       
       
         
           
             
               
                 
                   M 
                   b 
                 
                  
                 
                   
                     
                        
                       2 
                     
                      
                     x 
                   
                   
                      
                     
                       t 
                       2 
                     
                   
                 
               
               + 
               
                 T 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 φ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 α 
               
               - 
               Drag 
             
             = 
             0 
           
         
       
       
         
           
             
               
                 
                   M 
                   b 
                 
                  
                 
                   
                     
                        
                       2 
                     
                      
                     x 
                   
                   
                      
                     
                       t 
                       2 
                     
                   
                 
               
               + 
               
                 T 
                  
                 
                     
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 φ 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 α 
               
               - 
               
                 
                   1 
                   2 
                 
                  
                 
                   
                     C 
                     D 
                   
                    
                   
                     ( 
                     balloon 
                     ) 
                   
                 
                  
                 
                   A 
                    
                   
                     ( 
                     balloon 
                     ) 
                   
                 
                  
                 ρ 
                  
                 
                     
                 
                  
                 
                   V 
                   w 
                   2 
                 
               
             
             = 
             0 
           
         
       
       
         
           
             
               ∑ 
               
                 F 
                 y 
               
             
             = 
             
               
                 
                   0 
                    
                   
                     
 
                   
                    
                   
                     M 
                     p 
                   
                    
                   
                     
                       
                          
                         2 
                       
                        
                       y 
                     
                     
                        
                       
                         t 
                         2 
                       
                     
                   
                 
                 + 
                 B 
                 - 
                 
                   T 
                    
                   
                       
                   
                    
                   cos 
                    
                   
                       
                   
                    
                   φ 
                    
                   
                       
                   
                    
                   cos 
                    
                   
                       
                   
                    
                   α 
                 
               
               = 
               
                 
                   0 
                    
                   
                     
 
                   
                    
                   
                     ∑ 
                     
                       F 
                       z 
                     
                   
                 
                 = 
                 
                   
                     
                       0 
                        
                       
                         
 
                       
                        
                       
                         M 
                         p 
                       
                        
                       
                         
                           
                              
                             2 
                           
                            
                           z 
                         
                         
                            
                           
                             t 
                             2 
                           
                         
                       
                     
                     + 
                     
                       T 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       φsin 
                        
                       
                           
                       
                        
                       α 
                     
                   
                   = 
                   0 
                 
               
             
           
         
       
     
         [0046]    While at least one example embodiment has been presented in the foregoing detailed description, it should be appreciated that a vast number of variations exist. It should also be appreciated that the example embodiment or embodiments described herein are not intended to limit the scope, applicability, or configuration of the invention in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing the described embodiment or embodiments. It should be understood that various changes can be made in the function and arrangement of elements without departing from the scope of the invention, where the scope of the invention is defined by the claims, which includes known equivalents and foreseeable equivalents at the time of filing this patent application.