Abstract:
The present invention concerns a method for designing a deployment mechanism for a flight surface on an airborne body, including the steps of determining a stowed position of the flight surface, determining a deployed position of the flight surface, identifying a first rotation axis and respective rotation angle and a second rotation axis and respective rotation angle about and through which the flight surface is rotatable in sequence to move the flight surface from the stowed position to the deployed position, or vice versa, and using the identified first and second rotation axes and rotation angles to determine a single equivalent rotation axis and angle, about and through which the flight surface can be rotated from the stowed position to the deployed position, or vice versa.

Description:
FIELD OF THE INVENTION 
   The invention relates to a method for designing a deployment mechanism for a flight surface on a guided projectile, or a guided airborne body such as a bomb, dispenser, munition or missile. 
   BACKGROUND OF THE INVENTION 
   Guided airborne bodies such as munitions or missiles are routinely constructed with flight surface deployment systems to provide control and extended range capability. These systems include one or more control mechanisms for moving a foldable and/or retractable flight surface to move the flight surface from a stowed position to an active or deployed position. 
   Recently, because of volume constraints inside the airborne body, designs of airborne bodies have required the flight surface and/or control mechanism to be stored on, or partially on, the exterior of the airborne body. This has increased the complexity of deployment systems, in particular, those in which the flight surface is rotated, in sequence, about multiple axes to, for example, position the flight surface at a sweep angle or other obliquity relative to the airborne body axis. Such deployment systems include a complex arrangement of parts such as actuators and lock mechanisms to enable achievement of the desired active or deployed position of the flight surface. 
   To reduce the number of rotations and axes about which the surface is rotated and, consequently, the number of parts of the deployment system, attempts have been made to simulate the multiple axes rotations with a single equivalent axis and single rotation. Although satisfactory, such attempts have not been without difficulty. For example, according to at least one known method, numerous axes and angles are attempted on a trial and error basis until a suitable single axis and single rotation angle are obtained. Such a method is inefficient and tedious. 
   Thus, a need exists for an effective method for designing a deployment mechanism for a flight surface on an airborne body. Such a method preferably minimizes intrusive volume, the number of parts, and the complexity of the deployment mechanism. Moreover, such a method preferably simplifies the manner by which a single axis and single rotation angle may be obtained about and through which a flight surface is rotatable. 
   SUMMARY OF THE INVENTION 
   The present invention concerns a method for designing a deployment mechanism for a flight surface on an airborne body, including the steps of determining a stowed position of the flight surface, determining a deployed position of the flight surface, identifying a first rotation axis and respective rotation angle and a second rotation axis and respective rotation angle about and through which the flight surface is rotatable in sequence to move the flight surface from the stowed position to the deployed position, or vice versa, and using the identified first and second rotation axes and rotation angles to determine a single equivalent rotation axis and angle, about and through which the flight surface can be rotated from the stowed position to the deployed position, or vice versa. 
   In an embodiment, the using step determines the single equivalent rotation axis and angle in accordance with Euler&#39;s Rotation Theorem. 
   In an embodiment, the equivalent angle (ε r ) is obtained from the equation: 
         ɛ   r     =     2   ⁢       cos     -   1       ⁡     [       cos   ⁢     ɛ   2     ⁢   cos   ⁢       ɛ   ′     2       -     sin   ⁢     ɛ   2     ⁢   sin   ⁢       ɛ   ′     2     ⁢   cos   ⁢           ⁢   θ       ]             
 
and the equivalent rotation axis (k r ) is obtained from the equation: 
           k   r     =       k   ⁢       sin   ⁢     ɛ   2     ⁢   cos   ⁢       ɛ   ′     2         sin   ⁢       ɛ   r     2           +       k   ′     ⁢       sin   ⁢       ɛ   ′     2     ⁢   cos   ⁢     ɛ   2         sin   ⁢       ɛ   r     2           +       k   ′     ×   k   ⁢       sin   ⁢     ɛ   2     ⁢   sin   ⁢       ɛ   ′     2         sin   ⁢       ɛ   r     2               ,       
 
where ε is the first rotation angle, and ε′ is the second rotation angle, and θ is the angle between the rotation axes k and k′.
 
   The foregoing and other features of the invention are hereinafter more fully described and particularly pointed out in the claims, the following description and the annexed drawings setting forth in detail illustrative embodiments of the invention, such being indicative, however, of but a few of the various ways in which the principles of the invention may be employed. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a flow chart of a method for designing a deployment mechanism for a flight surface in accordance with the present invention. 
       FIGS. 2-7  schematically illustrate an embodiment of first and second rotation axes and respective rotation angles about and through which the flight surface is rotatable in sequence to move the flight surface from a stowed position to a deployed position. 
       FIGS. 8 and 9  schematically illustrate the coordinate systems of the first and second rotation axes shown in  FIGS. 2-7  transposed relative to one another. 
       FIGS. 10 and 11  schematically illustrate a single equivalent rotation axis and single equivalent rotation angle based on the first and second rotation axes and respective rotation angles of  FIGS. 2-7 , about and through which the flight surface may be rotated from the stowed position to the deployed position, or vice versa, in accordance with the invention. 
       FIGS. 12-15  schematically illustrate another embodiment of first and second rotation axes and respective rotation angles about and through which the flight surface is rotatable in sequence to move the flight surface from a stowed position to a deployed position. 
       FIG. 16  schematically illustrates the coordinate systems of the first and second rotation axes shown in  FIGS. 12-15  transposed relative to one another. 
       FIGS. 17 and 18  schematically illustrate a single equivalent rotation axis and single equivalent rotation angle based on the first and second rotation axes and respective rotation angles of  FIGS. 12-15 , about and through which the flight surface may be rotated from the stowed position to the deployed position, or vice versa, in accordance with the invention. 
   

   DETAILED DESCRIPTION 
   Referring now in detail to the drawings, and initially to  FIG. 1 , there is shown a flow chart  100  of a method for designing a deployment mechanism for a flight surface in accordance with the present invention. Initially, at step  110 , a stowed position and a deployed position of the flight surface are identified.  FIGS. 2 and 12 , which are described in greater detail below, show exemplary stowed positions and  FIGS. 7 and 15 , which also are described in greater detail below, show exemplary deployed positions. 
   In step  120 , a first rotation axis and a respective rotation angle are identified about and through which the flight surface may be rotated from the stowed position towards the desired deployed position. In step  130 , a second, or successive, rotation axis and a respective rotation angle are identified about and through which the flight surface may be rotated to position the flight surface in the deployed position. In step  140 , a single equivalent rotation axis and angle are determined based on the first and second rotation axes and rotation angles, about and through which the flight surface may be rotated to achieve the deployed position. 
   Several advantages are realized by the method according to the present invention. Because two axes and successive rotations are replaced by a single equivalent axis and rotation, the amount of parts and, consequently, the volume occupied by the deployment mechanism, is minimized. Moreover, by minimizing the number of parts, the structure of the deployment mechanism may be simplified. Also, the method simplifies the manner by which a single axis and single rotation angle may be obtained about and through which a flight surface is rotatable. 
   The following examples demonstrate the method according to the present invention and the advantages associated therewith. 
   EXAMPLE 1 
     FIGS. 2-7  schematically illustrate an embodiment of a flight surface  204  rotated in sequence from a stowed position to a deployed position. To facilitate describing the invention, the flight surface  204  is shown transposed in a cube with three of its sides marked with, respectively, the letters A, B and C.  FIG. 2  shows the flight surface  204  in a stowed position. As is shown in  FIG. 2 , an orthogonal coordinate system (axes i, j and k) is assigned to the flight surface  204 , the coordinate system representing the flight surface  204  in its stowed position. When the flight surface  204  is in its stowed position, the flight surface  204  lies in a plane parallel to the i and j axes, and a proximal end  206  of the flight surface  204  is parallel to the j axis and perpendicular to the plane defined by the i and k axes. The coordinate system serves as a reference datum from which a first rotation of the flight surface  204  is measured. 
   Referring to  FIG. 3 , k and ε represent the first rotation axis and the respective first rotation angle about and through which the flight surface  204  is rotated from the stowed position to a position intermediate the stowed position and the desired deployed position. In the illustrated embodiment, the first rotation angle ε is measured counterclockwise about the k axis. Once rotated, an orthogonal coordinate system (axes i 1 , j 1  and k) is assigned to the flight surface  204 , wherein i 1  and j 1  represent unit vectors obtained by rotating i and j, respectively, through the angle ε. The i 1 , j 1  and k coordinate system represents the flight surface  204  in the first, or intermediate, rotated position. The first rotation is one of two successive rotations in which the flight surface  204  is moved from its stowed position ( FIG. 2 ) to its extended or deployed position (FIG.  7 ).  FIG. 4  shows the flight surface  204  rotated through an exemplary first rotation angle ε of 90 degrees counterclockwise about the k axis. 
   Referring now to  FIG. 5 , a second orthogonal coordinate system (axes i′, j′ and k′) is assigned to the flight surface  204 . The  FIG. 5  coordinate system also represents the flight surface  204  in its intermediate position but serves as a reference datum from which a second rotation of the flight surface  204  is measured. As is described in greater detail below, assigning the i′, j′ and k′ coordinate system to the intermediate position facilitates analyzing the relationship of the stowed and deployed positions relative to the intermediate position of the flight surface  204 . 
   In a manner similar to the identification of the first rotation axis k and first rotation angle ε, as shown in  FIG. 6 , a second rotation axis k′ and a respective second rotation angle ε′ about and through which the flight surface  204  is rotated from the intermediate position to the desired deployed position, are identified. In the illustrated embodiment, the second rotation angle ε′ is measured counterclockwise about the k′ axis. Once rotated, an orthogonal coordinate system (axes i′ 1 , j′ 1  and k′) ( FIGS. 6 and 7 ) is assigned to the flight surface  204 , wherein i′ 1  and j′ 1  represent unit vectors obtained by rotating i′ and j′, respectively, through the angle ε′. The i′ 1 , j′ 1  and k′ coordinate system represents the flight surface  204  in a second rotated position (i.e., the deployed position). The second rotation is the second of the two successive rotations in which the flight surface  204  is moved from its stowed position ( FIG. 2 ) to its extended or deployed position (FIG.  7 ). In  FIG. 7 , the flight surface  204  is shown rotated through an exemplary second rotation angle ε′ of 105 degrees counterclockwise about the k′ axis. 
   Having identified the first and second rotation axes k and k′ and respective rotation angles ε and ε′ about and through which the flight surface  204  is successively rotated, a single equivalent rotation axis and rotation angle may then determined. One way to determine the single axis and rotation angle is by application of Euler&#39;s Theorem, which provides that a single axis and rotation angle may be derived from two axes and respective successive rotations. To facilitate use of Euler&#39;s Theorem in the present embodiment, the aforementioned i 1 , j 1  and k intermediate coordinate system ( FIG. 4 ) and the i′, j′ and k′ intermediate coordinate system ( FIG. 5 ) are transposed relative to one another according to the manner shown in FIG.  8 . As is shown in  FIG. 8 , i 1  coincides with i′, and i 1  and i′ are perpendicular to k and k′. In this way, j 1  and j′ are in the plane of k and k′. The angle θ represents the angle between k and k′. 
   Referring to  FIG. 9 , assuming that the first and second rotations ε and ε′ are about mutually perpendicular axes, then the angle θ between k and k′ is 90 degrees (i.e., axes k and k′, are perpendicular). Applying Euler&#39;s Theorem, a single equivalent angle (ε r ) may be obtained from the equation: 
         ɛ   r     =     2   ⁢       cos     -   1       ⁡     [       cos   ⁢     ɛ   2     ⁢   cos   ⁢       ɛ   ′     2       -     sin   ⁢     ɛ   2     ⁢   sin   ⁢       ɛ   ′     2     ⁢   cos   ⁢           ⁢   θ       ]             
 
and a single equivalent rotation axis (k r ) may be obtained from the equation: 
           k   r     =       k   ⁢       sin   ⁢     ɛ   2     ⁢   cos   ⁢       ɛ   ′     2         sin   ⁢       ɛ   r     2           +       k   ′     ⁢       sin   ⁢       ɛ   ′     2     ⁢   cos   ⁢     ɛ   2         sin   ⁢       ɛ   r     2           +       k   ′     ×   k   ⁢       sin   ⁢     ɛ   2     ⁢   sin   ⁢       ɛ   ′     2         sin   ⁢       ɛ   r     2               ,       
 
where ε is the first rotation angle, ε′ is the second rotation angle, and θ is the angle between the first and second rotation axes k and k′.
 
   Based on the foregoing, for a first rotation angle ε of 90 degrees, a second rotation angle ε′ of 105 degrees, and an angle θ of 90 degrees between the first and second rotation axes k and k′, the single equivalent rotation is:
 
ε r =129 degrees
 
and the single equivalent rotation axis is:
 
 k   r =0.477 k +0.622 k ′+0.622 k′×k 
 
   However, from the relationships of the above coordinate systems, the following equations are applicable:
 
 j   1  sin θ= k  cos θ− k′, 
 
  j ′ sin θ= k−k ′ cos θ, and
 
 i  sin θ= k×k′ 
 
   Thus, transposing the equivalent axis to the coordinate system i, j and k of the stowed position results in the following single equivalent axis:
 
 k   r =0.622 i −0.622 j +0.477 k 
 
     FIG. 10  shows the single equivalent axis k r  transposed in the i, j and k coordinate system with the flight surface  204  in a stowed position.  FIG. 11  shows the flight surface  204  rotated through the single equivalent rotation angle ε r  of 129 degrees counterclockwise about the single equivalent axis k r . As can be seen by comparing  FIGS. 7 and 11 , the same result obtains if the flight surface  204  is successively rotated about and through first and second rotation axes and respective rotation angles (FIGS.  2 - 7 ), than if the flight surface  204  is rotated about and through a single equivalent rotation axis and rotation angle (FIGS.  10  and  11 ). 
   EXAMPLE 2 
     FIGS. 12-15  schematically illustrate another embodiment of a flight surface  304  rotated in sequence from a stowed position to a deployed position. To facilitate describing the invention, the flight surface  304  is shown transposed in a cube with three of its sides marked with, respectively, the letters A, B and C.  FIG. 12  shows the flight surface  304  in a stowed position. As is shown in  FIG. 12 , an orthogonal coordinate system (axes i, j and k) is assigned to the flight surface  304 , the coordinate system representing the flight surface  304  in its stowed position. When the flight surface  304  is in its stowed position, the flight surface  304  lies in a plane parallel to the i and j axes, and a proximal end  306  of the flight surface  304  is parallel to the j axis and perpendicular to the plane defined by the i and k axes. The coordinate system serves as a reference datum from which a first rotation of the flight surface  304  is measured. 
   Referring to  FIG. 13 , k and ε represent the first rotation axis and the respective first rotation angle about and through which the flight surface  304  is rotated from the stowed position to a position intermediate the stowed position and the desired deployed position. Here, the flight surface  304  is rotated through an exemplary first rotation angle ε of 75 degrees counterclockwise about the k axis. An orthogonal coordinate system (axes i 1 , j 1  and k) is assigned to the flight surface  304 , and represents the flight surface  304  in the first, or intermediate, rotated position. 
   As shown in  FIG. 14 , a second orthogonal coordinate system (axes i′, j′ and k′) is assigned to the flight surface  304  which represents the flight surface  304  in its intermediate position but serves as a reference datum from which a second rotation of the flight surface  304  is measured. Referring to  FIG. 15 , a second rotation axis k′ and a respective second rotation angle ε′ about and through which the flight surface  304  is rotated from the intermediate position to the desired deployed position, are identified. Here, the flight surface  304  is shown rotated through an exemplary second rotation angle ε′ of 50 degrees counterclockwise about the k′ axis. Once rotated, an orthogonal coordinate system (axes i′ 1 , j′ 1  and k′) ( FIG. 15 ) is assigned to the flight surface  304 , wherein i′ 1  and j′ 1  represent unit vectors obtained by rotating i′ and j′, respectively, through the angle ε′. The i′ 1 , j′ 1  and k′ coordinate system represents the flight surface  304  in a second rotated position (i.e., the deployed position). 
   Having identified the first and second rotation axes k and k′ and respective rotation angles ε and ε′ about and through which the flight surface  304  is successively rotated, a single equivalent rotation axis and rotation angle may then determined. To facilitate use of Euler&#39;s Theorem, the i 1 , j 1  and k coordinate system ( FIG. 13 ) and the i′, j′ and k′ coordinate system ( FIG. 14 ) are transposed relative to one another according to the manner shown in FIG.  16 . As is shown in  FIG. 16 , i 1  coincides with i′, and i 1  and i′ are perpendicular to k and k′. In this way, j 1  and j′ are in the plane of k and k′. The angle θ represents the angle between k and k′. 
   Referring to  FIG. 16 , assuming that the first and second rotations ε and ε′ are about mutually perpendicular axes, then the angle θ between k and k′ is 90 degrees (i.e., axes k and k′, are perpendicular). Applying Euler&#39;s Theorem, for a first rotation angle ε of 75 degrees, a second rotation angle ε′ of 50 degrees, and an angle θ of 90 degrees between the first and second rotation axes k and k′, the single equivalent rotation is:
 
ε r =88 degrees
 
and the single equivalent rotation axis is:
 
 k   r =0.794 k +0.482 k ′+0.370 k′×k 
 
   From the relationships of the above coordinate systems, transposing the equivalent axis to the coordinate system i, j and k of the stowed position results in the following single equivalent axis:
 
 k   r =0.482 i −0.370 j +0.794 k 
 
     FIG. 17  shows the single equivalent axis k r  transposed in the i, j and k coordinate system with the flight surface  304  in a stowed position.  FIG. 18  shows the flight surface  304  rotated through the single equivalent rotation angle ε r  of 88 degrees counterclockwise about the single equivalent axis k r . As can be seen by comparing  FIGS. 15 and 18 , the same result obtains if the flight surface  304  is successively rotated about and through first and second rotation axes and respective rotation angles (FIGS.  12 - 15 ), than if the flight surface  304  is rotated about and through a single equivalent rotation axis and rotation angle (FIGS.  17  and  18 ). 
   Although the invention has been shown and described with respect to certain preferred embodiments, equivalent alterations and modifications will occur to others skilled in the art upon reading and understanding this specification and the annexed drawings. In particular regard to the various functions performed by the above described integers (components, assemblies, devices, compositions, etc.), the terms (including a reference to a “means”) used to describe such integers are intended to correspond, unless otherwise indicated, to any integer which performs the specified function of the described integer (i.e., that is functionally equivalent), even though not structurally equivalent to the disclosed structure which performs the function in the herein illustrated exemplary embodiment or embodiments of the invention. In addition, while a particular feature of the invention may have been described above with respect to only one of several illustrated embodiments, such feature may be combined with one or more other features of the other embodiments, as may be desired and advantageous for any given or particular application.