Abstract:
A vehicle equipped with two or more propulsion units, each, for example, consisting of engine with propeller, with their thrust principally directed vertically along a z-axis, such vehicle characterized by that each propulsion unit can be controlled by rotation around two axes mainly perpendicular to the z-axis and that the propulsion units are positioned some distance apart in the z-direction enabling such control of the attitude of the propulsion units to obtain:
       a) lateral force without attendant moment around the center-of-gravity of the vehicle to control the lateral position of the vehicle, or   b) a moment around the center-of-gravity of the vehicle without attendant lateral force to control the lateral attitude of the vehicle, or   c) combination of (a) and (b)
 
thereby to simplify the implementation of a system to control the position and attitude of the vehicle whether such control system is manipulated manually, automatically or in a combination thereof.

Description:
BACKGROUND 
       [0001]    Control of a vehicle during vertical takeoff, flight and landing usually requires special means. This is because conventional aircraft control surfaces are rendered ineffective at low or zero flight speed. 
         [0002]    Several such special means already exist such as: 
         [0003]    (1) Cyclic variation of the angle-of attack for the rotor on helicopter and tilt-rotor aircraft (V-22 Osprey, Agusta Westland AW609, U.S. Pat. No. 5,381,985 Wechsler et al.). 
         [0004]    (2) Tail rotor on helicopter to control yaw 
         [0005]    (3) Aerodynamic control surfaces in the airflow from propeller, rotor or fan (U.S. Pat. No. 5,758,844 Cummings, U.S. Pat. No. 6,343,768 Muldoon, U.S. Pat. No. 3,049,320 Fletcher) 
         [0006]    (4) Small dedicated jets typically mounted at the front and rear of the fuselage and at the wingtips 
         [0007]    (5) Pivoting or rotable exhaust nozzle on combustion engines (AV8B, F-35B) 
         [0008]    (6) U.S. Pat. No. 8,256,704 to Lundgren, Vertical/Short Take-Off and Landing Aircraft discloses a thrust assembly comprising a pair of in-line counter-rotating fans and engines where the whole thrust assembly is rotatable about a lateral axis and a longitudinal axis to provide control of the location of the aircraft in the horizontal plane. Such rotation of the thrust assembly will create force components along the lateral axis and the longitudinal axis causing the aircraft to move in the fwd/aft and lateral direction. That concept requires that the Center of Gravity is located significantly below the axes of rotation of the thrust assembly.  FIG. 6  illustrates an application of the U.S. Pat. No. 8,256,704. 
         [0009]    The present invention is less dependant on the location of the Center of Gravity of the aircraft. While it can impose forces moving the aircraft in the horizontal plane, it can also control the attitude of the aircraft.  FIGS. 7 through 9  and  FIG. 14  shows both propulsion units aligned for steady state hover. 
         [0010]    This invention describes a new method to achieve control without the use of the special means mentioned above. 
         [0011]    Furthermore, this new method provides control of attitude and control of translation independent of each other for the pitch and roll axes. 
       DETAILED DESCRIPTION 
     List of Symbols 
       [0000]    
       
         Ftx x-axis component of the thrust force 
         Fty y-axis component of the thrust force 
         Ftz z-axis component of the thrust force 
         Ftxe xe-axis component of the thrust force 
         Ftye ye-axis component of the thrust force 
         Ftze ze-axis component of the thrust force 
         Lt x-axis component of the thrust moment around the center-of-gravity (rolling moment) 
         Mt y-axis component of the thrust moment around the center-of-gravity (pitching moment) 
         Nt z-axis component of the thrust moment around the center-of-gravity (yawing moment) 
         n subscript denoting propulsion unit number 
         T thrust force 
         W weight force 
         X,Y,Z coordinate system fixed to the vehicle with the x- and y-axes principally horizontal and with the z-axis positive downward 
         Xe,Ye,Ze coordinate system parallel to the earth 
         Xt,Yt,Zt distances of thrust force to the center-of-gravity 
         δ rotation angle of a propulsion unit around its pivot 
         θ the Euler angle between the x-axis and the horizontal plane (pitch angle) 
         φ the Euler angle between the y-axis and the horizontal plane (roll angle) 
         Ψ the Euler angle between the projection of the x-axis on the horizontal plane and a reference orientation for example North (heading angle) 
         p subscript denoting pitch 
         r subscript denoting roll 
           1 , 2 , 3  subscripts denoting propulsion unit number 
       
     
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0034]      FIG. 1  shows the body x and z axes with pitch angle and two propulsion units 
           [0035]      FIG. 2  shows the body y and z axes with roll angle and two propulsion units 
           [0036]      FIG. 3  shows the body x and z axes with pitch angle and three propulsion units 
           [0037]      FIG. 4  shows the body y and z axes with roll angle and three propulsion units 
           [0038]      FIG. 5  shows the body x and y axes with yawing moment and three propulsion units 
           [0039]      FIG. 6  shows an example with two in-line propulsion units 
           [0040]      FIG. 7  shows the longitudinal and lateral axes on a vehicle with two in-line propulsion units 
           [0041]      FIG. 8  shows the lateral axis on a vehicle with two in-line propulsion units 
           [0042]      FIG. 9  shows the lateral axis in a front view on a vehicle with two in-line propulsion units 
           [0043]      FIG. 10  shows a negative lateral force component with zero rolling moment and two in-line propulsion units 
           [0044]      FIG. 11  shows a positive lateral force component with zero rolling moment and two in-line propulsion units 
           [0045]      FIG. 12  shows a negative rolling moment with zero lateral force component and two in-line propulsion units 
           [0046]      FIG. 13  shows a positive rolling moment with zero lateral force component and two in-line propulsion units 
           [0047]      FIG. 14  shows the longitudinal axis in a side view on a vehicle with two in-line propulsion units 
           [0048]      FIG. 15  shows a positive longitudinal force component with zero pitching moment and two in-line propulsion units 
           [0049]      FIG. 16  shows a negative longitudinal force component with zero pitching moment and two in-line propulsion units 
           [0050]      FIG. 17  shows a positive pitching moment with zero longitudinal force component and two in-line propulsion units 
           [0051]      FIG. 18  shows a negative pitching moment with zero longitudinal force component and two in-line propulsion units 
       
    
    
     EQUATIONS FOR THRUST, FORCES AND MOMENTS  
       [0052]    The components (F tx , F ty , F tz ) of the thrust vector for one propulsion unit in a coordinate system (x,y,z), with its origin at the vehicle center-of-gravity, fixed to the body of the vehicle, can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           Ftx 
                         
                       
                       
                         
                           Fty 
                         
                       
                       
                         
                           Ftz 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   p 
                                 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   p 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     δ 
                                      
                                     
                                         
                                     
                                      
                                     p 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   p 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   r 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     δ 
                                      
                                     
                                         
                                     
                                      
                                     r 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   r 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   r 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               T 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   1 
                   ] 
                 
               
             
           
         
       
     
         [0000]    where T is the thrust of the propulsion unit and δp and δr are the consecutive control rotations of the propulsion unit around its rotation axes in pitch and roll. Note that the above is only one example for the placement of the rotation axes. They need not be parallel to any of the body axes (x, y, z) or be perpendicular between themselves. In this example the following is obtained: 
         [0000]        Ftx/T =−sin(δ p )cos(δ r )   [2a]
 
         [0000]        Fty/T =sin(δ r )   [2b]
 
         [0000]        Ftz/T =−cos(δ p )cos(δ r )   [2c]
 
         [0053]    These components can be used to calculate the moments (Lt, Mt, Nt) of the thrust vector around the axes ( x, y, z) 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           Lt 
                         
                       
                       
                         
                           Mt 
                         
                       
                       
                         
                           Nt 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             0 
                           
                           
                             
                               - 
                               Zt 
                             
                           
                           
                             Yt 
                           
                         
                         
                           
                             Zt 
                           
                           
                             0 
                           
                           
                             
                               - 
                               Xt 
                             
                           
                         
                         
                           
                             
                               - 
                               Yt 
                             
                           
                           
                             Xt 
                           
                           
                             0 
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             Ftx 
                           
                         
                         
                           
                             Fty 
                           
                         
                         
                           
                             Ftz 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   3 
                   ] 
                 
               
             
           
         
       
     
         [0054]    In the above example the following is obtained: 
         [0000]        Lt =−( Zt )( Fty )+( Yt )( Ftz )
 
         [0000]        Mt =( Zt )( Ftx )−( Xt )( Ftz )
 
         [0000]        Nt =−( Yt )( Ftx +( Xt )( Fty )
 
         [0000]        Lt/T =(− Zt )(sin(δ r )−( Yt )(cos(δ p )(cos(δ r ))   [4a]
 
         [0000]        Mt/T =(−( Zt )(sin(δ p )+( Xt )(cos(δ p ))(cos(δ r )   [4b]
 
         [0000]        Nt/T =( Yt )(sin(δ p ))(cos(δ r ))+( Xt )(sin(δ r ))   [4c]
 
         [0055]    The transformation of the (Ftx, Fty, Ftz) in the body fixed coordinate system (x, y, z) to an earth fixed coordinate (x e , y e , z e ) through the Euler angles ψ, θ, and φ obtained through consecutive rotation is: 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           Ftxe 
                         
                       
                       
                         
                           Ftye 
                         
                       
                       
                         
                           Ftze 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           
                             
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   ψ 
                                   ) 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                           
                           
                             
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             Ftx 
                           
                         
                         
                           
                             Fty 
                           
                         
                         
                           
                             Ftz 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   6 
                   ] 
                 
               
             
           
         
       
     
         [0056]    The choice of heading angle is arbitrary and we choose ψ=0 for continued analysis. In the above example the following is obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         ( 
                         
                           
                             
                               Ftxe 
                             
                           
                           
                             
                               Ftye 
                             
                           
                           
                             
                               Ftze 
                             
                           
                         
                         ) 
                       
                       = 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     sin 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     φ 
                                     ) 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       φ 
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     φ 
                                     ) 
                                   
                                 
                               
                               
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     φ 
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               
                                 Ftx 
                               
                             
                             
                               
                                 Fty 
                               
                             
                             
                               
                                 Ftz 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         ( 
                         
                           
                             
                               Ftxe 
                             
                           
                           
                             
                               Ftye 
                             
                           
                           
                             
                               Ftze 
                             
                           
                         
                         ) 
                       
                       = 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     sin 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               
                                 Ftx 
                               
                             
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       φ 
                                       ) 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       Fty 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       φ 
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       Ftz 
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       φ 
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       Fty 
                                       ) 
                                     
                                   
                                   + 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       φ 
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       Ftz 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         ( 
                         
                           
                             
                               Ftxe 
                             
                           
                           
                             
                               Ftye 
                             
                           
                           
                             
                               Ftze 
                             
                           
                         
                         ) 
                       
                       = 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     Ftx 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                          
                                         Fty 
                                       
                                       + 
                                       
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             φ 
                                             ) 
                                           
                                         
                                          
                                         
                                           ( 
                                           Ftz 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     φ 
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     Fty 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     φ 
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     Ftz 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     ( 
                                     Ftx 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             φ 
                                             ) 
                                           
                                         
                                          
                                         
                                           ( 
                                           Fty 
                                           ) 
                                         
                                       
                                       + 
                                       
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             φ 
                                             ) 
                                           
                                         
                                          
                                         
                                           ( 
                                           Ftz 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   7 
                   ] 
                 
               
             
             
               
                 
                   
                     Ftxe 
                      
                     
                       / 
                     
                      
                     T 
                   
                   = 
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         ( 
                         
                           δ 
                            
                           
                               
                           
                            
                           p 
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             δ 
                              
                             
                                 
                             
                              
                             r 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               ( 
                               φ 
                               ) 
                             
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
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                               ) 
                             
                           
                         
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                   [ 
                   
                     8 
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                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Ftye 
                        
                       
                         / 
                       
                        
                       T 
                     
                     = 
                     
                       
                         cos 
                          
                         
                             
                         
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                           ) 
                         
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                           ( 
                           δr 
                           ) 
                         
                       
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                         sin 
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                   [ 
                   
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                     b 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     Ftze 
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       Control of Vehicle Attitude with Two Propulsion Units 
       [0057]    Reference:  FIGS. 1 and 2 . 
         [0058]    Both the lower and upper propulsion units are deflected together such as to give zero net force in the x and y directions of the body fixed coordinate system. Equations (2) then give 
         [0000]        Ftx =−( T 1)(sin(δ p 1)cos(δ r 1))−( T 2)(sin(δ p 2)cos(δ r 2))=0
 
         [0000]        Fty =( T 1)(sin(δ r 1))+( T 2)(sin(δ r 2))=0
 
         [0000]    or 
         [0000]      sin(δ p 1)cos(δ r 1)=−( T 2 /T 1)(sin(δ p 2)cos(δ r 2))   [9a]
 
         [0000]      sin(δ r 1)=−( T 2/ T 2)(sin(δ r 2))   [9b]
 
         [0059]    It is assumed that the thrust values T1 and T2 can be calculated, for example, from parameters such as throttle position, RPM and so on. For particular control inputs δp2, δr2 to the upper propulsion unit then the control defletions δp1, δr1 for the lower unit can be calculated from equations (9) above to yield moments around the center-of-gravity of the vehicle without attendant force in the xy-plane of the body fixed coordinate system. 
         [0060]    The method of control described above lends itself to control the vehicle attitude to given or desired values of pitch angle θ and roll angle φ through the use of an automatic feedback control system. Such a system can be expected to hold the attitude of the vehicle close to desired values. If these values are chosen to be θ=φ=0 then the resulting forces in horizontal plane per equations (8) will be small. 
         [0061]    It is assumed that the two propulsion units are counter rotating to cancel out the propulsion torque. It is then possible to control the heading, in the horizontal plane, by varying the trust or rpm for the two units to give a differential torque. The incurred difference in thrust between the two units is accounted for in equations (9) above. Control of attitude around ALL three axes is thus attained. 
         [0062]      FIGS. 12 and 13  exemplifies the attitude control around the longitudinal or roll axis, while  FIGS. 17 and 18  illustrates attitude control around the lateral or pitch axis. 
       Control of Vehicle Position with Two Propulsion Units 
       [0063]    Reference:  FIGS. 1 and 2 . 
         [0064]    A deflection of the lower propulsion unit will create a lateral force but may also cause an undesirable moment around the center-of-gravity of the vehicle, see  FIG. 1 . The upper propulsion unit is therefore deflected to counter that moment to give a total net moment equal to zero. 
         [0065]    Equations (4) give: 
         [0000]        Lt =−( T 1)(( Y 1)(cos(δ p 1)cos(δ r 1))+( Z 1)(sin(δ r 1)))−( T 2)(( Y 2)(cos(δ p 2)cos(δ r 2))+( Z 2)(sin(δ r 2)))
 
         [0000]        Mt =( T 1)(( X 1)(cos(δ p 1))−( Z 1)(sin(δ p 1)cos(δ r 1)))+( T 2)(( X 2)(cos(δ p 2))−( Z 2)(sin(δ p 2)cos(δ r 2)))
 
         [0000]    or 
         [0000]      ( Y 2)(cos(δ p 2)cos(δ r 2))+( Z 2)(sin(δ r 2))=−( T 1 /T 2)(( Y 1)(cos(δ p 1)cos(δ r 1))+( Z 1)(sin(δ r 1)))   [10a]
 
         [0000]      ( X 2(cos(δ p 2))− Z 2(sin(δ p 2)))(cos(δ r 2))=−( T 1 /T 2)( X 1(cos(δ p 1))− Z 1(sin(δ p 1)))(cos(δ r 1))   [10b]
 
         [0066]    The geometry values (X1, Y1, Z1) and (X2, Y2, Z2) are assumed known. 
         [0067]    It is also assumed that the thrust values T1 and T2 can be calculated, for example, from parameters such as trottle position, RPM and so on. For particular control inputs δp1, δr1 to the lower propulsion unit then the control deflections δp2, δr2 for the upper unit can be calculated from equations (10a), (10b) above to yield lateral forces without attendant moment around the center-of-gravity of the vehicle. Control of the vertical position of the vehicle is obtained through changing the thrust of both propulsion units. Control of vehicle position along ALL three axes is thus attained. 
         [0068]      FIGS. 10 and 11  exemplifies the position control in the lateral direction, while  FIGS. 15 and 16  illustrates position control in the longitudinal direction. 
       Control with Three Propulsion Units 
       [0069]    Reference:  FIGS. 3 to 6   
         [0070]    An arrangement with propulsion units has the following advantages over two units placed in tandem depending on control authority and design:
       a) Control of heading through varying the direction of thrust instead of using differential torque   b) Capability to control a malfunctioning propulsion unit       
 
         [0073]    All three propulsion units are deflected together such as to give zero net force along the x and y directions of the body fixed coordinate system to control vehicle attitude in pith and roll. Equations (2) then give: 
         [0000]        Ftx=−Σ   n=1   3 ( Tn )(sin(δ pn ))(cos(δ rn ))=0   [11]
 
         [0000]        Fty=Σ   n=1   3 ( Tn )(sin(δ rn ))=0   [12]
 
         [0074]    Control of the vehicle attitude around the x-axis (the roll axis) is illustrated in  FIG. 4 . The deflection of propulsion unit 2 is considered a control input, δr2 
         [0075]    Equation (12) then gives: 
         [0000]      ( T 1)(sin(δ r 1))+( T 3)(sin(δ r 3))=−( T 2)(sin(δ r 2))   [13]
 
         [0076]    If, for example, δr3 is set equal to δr1, then equation (13) gives: 
         [0000]      sin(δ r 1)=sin(δ r 3)=−( T 2/( T 1 +T 3)) (sin(δ r 2))   [14]
 
         [0077]    It is assumed that the thrust values T1, T2 and T3 can be determined Equation (14) then gives the values δr1 and δr3 which propulation units 1 and 3 need to be deflected given the control input δr2 and the thrust values to yield a rolling moment around the x-axis without attendent force Fty. 
         [0078]    Control of the vehicle attitude around the y-axis (the pitch axis) is shown in  FIG. 3 . The deflection of propulsion unit 2 is considered a control input, δp2. The corresponding deflection δpp1 and εpp3 of propulsion units 1 and 3 are to be determined. In addition, propulsion units 1 and 3 are deflected in opposite directions, δpy1 and δpy3, for control of the vehicle attitude around the z-axis (the yaw axis), see  FIG. 5 . The deflections δpy1 and δpy3 are considered known control inputs. 
         [0079]    If, for example, δpp3 is set equal to δpp1 and δpy3 is set equal to −δpy1, then equation (11) gives: 
         [0000]      ( T 1)(sin(δ pp 1 +δpy 1))(cos(δ r 1))+( T 3)(sin(δ pp 1 −δpy 1))(cos(δ r 3))=−( T 2)(sin(δ p 2))(cos(δ r 2))   [15]
 
         [0080]    With thrust values T1, T2 and T3 known, then δpp1=δpp3 can be determined from equation (15). 
         [0081]    To illustrate, as one example, assume that T1=T2=T3, 
         [0000]    and that δr3 is set equal to δr1, as in the example above, then equation (15)becomes: 
         [0000]      (sin(δ pp 1))(cos(δ py 1))+(cos(δ pp 1))(sin(δ py 1))+(sin(δ pp 1))(cosδ py 1))−(cos(δ pp 1))(sin(δ py 1))=−(sin(δ p 2))(cos(δ r 2))/(cos(δ r 1))
 
         [0000]    or 
         [0000]      sin(δ pp 1)=−((½)(sin(δ p 2))(cos(δ r 2))/(cos(δ r 1)))/(cos(δ py 1))   [16]
 
         [0082]    The deflections δp2 and δr2 are given control inputs. The deflections δr1=δr3 are given by equation (14). The deflections δpy1=−δpy3 are also given control inputs. The desired deflections δpp1=δpp3 can then be determined from equation (16) under the above assumptions. This yields moments around all three body axes to control the vehicle attitude without attendent forces Ftx and Fty. 
         [0083]    All three propulsion units are deflected together in such a manner as to give zero moment around all three axes in the body fixed coordinate system, in order to control the vehicle sideways position. 
         [0084]    Equations (4) then give: 
         [0000]        Xtn=−Σ   n=1   3 ( Tn )(( Zn )(sin(δ rn ))+( Yn )(cos(δ pn (cos(δ rn )))=0   [17]
 
         [0000]        Ytn=Σ   n=1   3 ( Tn )(−( Zn )(sin(δ pn ))+( Xn )(cos(δ pn )cos(δ rn )))=0   [18]
 
         [0000]        Ztn=Σ   n=1   3 ( Tn )(( Yn )(sin(δ pn ))(cos(δ rn ))+( Xn )(sin(δ rn )))=0   [19]
 
         [0085]    As an example, assume that the vehicle attitude angles θ and φ are controlled to values close to zero as described above. The forces Ftx and Fty can then be used to control the vehicle lateral position. It is desirable to keep the moments close to zero per equations (17) through (19) during such position control. 
         [0086]    To illustrate, as an example, consider the lateral force Fty with the requirement that Lt=0 per equation (17). Also assume that T1=T2=T3 and that the deflections of propulsion units 1 and 3, denoted δr1 and δr3 are known control inputs and that they are made equal, i.e. δr1=δr3. For clarity, assume symmetry such that Y1=−Y3 and Y2=0 in  FIG. 4 . Equation (17) then gives: 
         [0000]      ( Z 1)(sin(δ r 1))+( y 1)(cos(δ p 1))(cos(δ r 1))+( Z 3)(sin(δ r 1))−( Y 1)(cos(δ p 3))(cos(δ r 1))+( Z 2)(sin(δ r 2)=0
 
         [0000]    or 
         [0000]    
       
         
           
             
               
                 
                   
                     sin 
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         δ 
                          
                         
                             
                         
                          
                         r 
                          
                         
                             
                         
                          
                         2 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           ( 
                           
                             
                               
                                 Z 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               + 
                               
                                 Z 
                                  
                                 
                                     
                                 
                                  
                                 3 
                               
                             
                             
                               Z 
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                                
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         
                           sin 
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                            
                           
                             ( 
                             
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                         ) 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             Y 
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                             1 
                           
                           
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                             2 
                           
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               δ 
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                               1 
                             
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                         ) 
                       
                        
                       
                         ( 
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               ) 
                             
                           
                           - 
                           
                             cos 
                              
                             
                               ( 
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
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                                  
                                 
                                     
                                 
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                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   20 
                   ] 
                 
               
             
           
         
       
     
         [0000]    δp1 and δp3 are here also assumed known control inputs for the force Ftx. The geometry is known. The desired deflection angle for propulsion unit 2 is thus given by equation (20). 
         [0087]    The treatment of the longitudinal force Ftx with Mt=0 and Nt=0 is similar. 
         [0088]    Altitude is controlled with thrust on the three propulsion units such as to give zero moment. 
         [0089]    Control of the vehicle position is thus achieved without incurring attendant moments around the vehicle center-of-gravity.