Abstract:
A control actuator system. The novel system includes a control surface mounted on a body and adapted to move in a first direction relative to the body, and a first mechanism for storing energy as the control surface moves in the first direction and releasing the stored energy to move the control surface in a second direction opposite the first direction. In an illustrative embodiment, the system is adapted to rotate an aerodynamic control surface of a rolling missile, and the first mechanism is a torsional spring arranged such that rotating the control surface in the first direction winds up the spring and releasing the spring causes the control surface to oscillate back and forth, alternating between the first and second directions. In a preferred embodiment, the spring has a spring constant such that the control surface oscillates at a natural frequency matching a roll rate of the missile.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to actuators. More specifically, the present invention relates to control actuator systems for rolling missiles. 
     2. Description of Related Art 
     Future concepts for highly maneuverable tactical missiles require high performance airframes controlled by very high performance control actuator systems (CAS). Missile maneuvering is traditionally controlled using a cruciform arrangement of four aerodynamic control surfaces (e.g., control fins) with four actuator motors and gear trains that independently control the aerodynamic control surfaces. Conventional missile control actuator systems, however, can have very high power requirements, especially for missiles with a rolling airframe. 
     Rolling airframe missiles are designed to roll or rotate about their longitudinal axes at a desired rate (typically about 5 to 15 revolutions per second), usually to gain various advantages in the design of the missile control system. Small, rolling airframes, however, exacerbate CAS power density requirements, as the control fins must be driven to large amplitudes at the roll frequency of the missile to produce large maneuvers. In contrast with standard non-rolling missiles, rolling airframe missiles require constant movement of the control fins, thus expending energy throughout the flight. The required power increases linearly with roll rate and deflection angle. In order to achieve the high maneuverability desired in new missile designs, conventional control actuator systems would require power densities that are beyond those fielded in current missile systems. 
     Most prior approaches for reducing the power requirements of a control actuator system in a rolling missile have centered around minimizing hinge moments (due to aerodynamic loads), minimizing inertias at the control surface, and optimizing CAS design parameters. High gear ratio designs require very high CAS motor accelerations and speeds, leading to high current, high voltage motor designs. As the gear ratios are reduced, CAS motor speeds are reduced but CAS torque requirements increase as the control surfaces have more influence (inertia and hinge moments) on the CAS motor. Attempts to minimize hinge moments through hinge line placement are not always realized as the control surface center of pressure moves around with mach number. The typical solution has been to design the CAS to meet the power (torque/speed) requirements, even if excessive, and size the flight battery/power supplies accordingly. 
     Hence, a need exists in the art for an improved control actuator system for rolling missiles that requires less power than prior approaches. 
     SUMMARY OF THE INVENTION 
     The need in the art is addressed by the control actuator system of the present invention. The novel system includes a control surface mounted on a body and adapted to move in a first direction relative to the body, and a first mechanism for storing energy as the control surface moves in the first direction and releasing the stored energy to move the control surface in a second direction opposite the first direction. In an illustrative embodiment, the system is adapted to rotate an aerodynamic control surface of a rolling missile, and the first mechanism is a torsional spring arranged such that rotating the control surface in the first direction winds up the spring and releasing the spring causes the control surface to oscillate back and forth, alternating between the first and second directions. In a preferred embodiment, the spring has a spring constant such that the control surface oscillates at a natural frequency matching a roll rate of the missile. The system may also include a servo motor for providing an initial torque to rotate the control surface in the first direction, and for periodically adding energy to the system such that the control surface continues oscillating to a desired angle and phase. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a three-dimensional view of a rolling airframe missile designed in accordance with an illustrative embodiment of the present teachings. 
         FIG. 2  is a simplified diagram of a control fin and control actuator system designed in accordance with an illustrative embodiment of the present teachings. 
         FIG. 3  is a three-dimensional view of a control actuator system designed in accordance with an illustrative embodiment of the present teachings. 
         FIG. 4  is a simplified block diagram representing a control actuator system designed in accordance with an illustrative embodiment of the present teachings. 
         FIG. 5  is a three-dimensional view of a control actuator system for four control fins designed in accordance with an illustrative embodiment of the present teachings. 
     
    
    
     DESCRIPTION OF THE INVENTION 
     Illustrative embodiments and exemplary applications will now be described with reference to the accompanying drawings to disclose the advantageous teachings of the present invention. 
     While the present invention is described herein with reference to illustrative embodiments for particular applications, it should be understood that the invention is not limited thereto. Those having ordinary skill in the art and access to the teachings provided herein will recognize additional modifications, applications, and embodiments within the scope thereof and additional fields in which the present invention would be of significant utility. 
       FIG. 1  is a three-dimensional view of a rolling airframe missile  10  designed in accordance with an illustrative embodiment of the present teachings. The missile  10  includes a missile body (or airframe)  12  and a plurality of control fins  14  for controlling the aerodynamic maneuvering of the missile  10  (four fins  14 A,  14 B,  14 C, and  14 D are shown in the illustrative embodiment of  FIG. 1 ). The missile is adapted to roll about its longitudinal axis at a predetermined rate. The missile roll rate may be controlled by the missile launcher and/or by the control fins  14  or by canted tail fins  21  (the illustrative embodiment of  FIG. 1  includes six tail fins  21 ). 
     The missile body  12  houses a seeker  16 , guidance system  18 , and a novel control actuator system  20 . The seeker  14  tracks a designated target and measures the direction to the target. The guidance system  16  uses the seeker measurements to guide the missile  10  toward the target, generating control signals that are used by the actuator system  20  to control the movement of the fins  14 . In the illustrative embodiment, the missile  10  includes four control fins  14  located in the middle of the missile  10 , spaced equally around the circumference of the missile  10  and arranged in a cross-like configuration. Each control fin  14  is controlled independently by a different actuator motor and gear train of the control actuator system  20 . 
     In a rolling missile, the control fins  14  are driven at the roll frequency of the missile  10  to produce a maneuver in a single plane. In a standard non-rolling missile, in order to move the missile in a particular direction, the control fins are held at a fixed deflection angle. For example, to move the missile left at an angle of 10°, the top and bottom fins  14 A and  14 C would be rotated to the left at an angle of 10° (i.e., fin  14 A rotated 10° counter-clockwise and fin  14 C rotated 10° clockwise). To perform the same maneuver in a rolling missile  10 , the control fins  12  are moved back and forth (between +10°and −10°) at the roll frequency of the missile  10 , so that when the missile  10  rolls upside-down the fins are pointed left (fin  14 A rotated 10° clockwise and fin  14 C rotated 10° counter-clockwise) and when the missile  10  rolls back to its original orientation (as depicted in  FIG. 1 ) the fins are again pointing left (fin  14 A rotated 10° counter-clockwise and fin  14 C rotated 10° clockwise). Thus, for a steady state maneuver, the control fins  14  are moved in a sinusoidal motion to produce the desired airframe motion. It is the acceleration term of this sinusoidal motion that drives the power requirements of a conventional rolling missile control actuator system. 
     The present invention employs the idea of a spring-mass system to store energy and restore the energy back into the system, greatly reducing the overall power requirements for the CAS and CAS battery in a rolling missile. The moments of inertia of the control fin, gears, and motor act as the “mass” of this system. In accordance with the teachings of the present invention, a torsional spring is added to provide a restoring torque such that the natural frequency of the spring-mass system matches the desired roll rate of the rolling missile. The torsional spring can be attached either to the output shaft (attached to the control surface) or to an adjunct gear. 
       FIG. 2  is a simplified diagram of a control fin  14  and associated control actuator system  20  designed in accordance with an illustrative embodiment of the present teachings.  FIG. 3  is a three-dimensional view of the actuator system  20  designed in accordance with an illustrative embodiment of the present teachings. For simplicity,  FIGS. 2 and 3  show an actuator system  20  for controlling only one fin  14 . The system  20  may also be adapted to control additional fins. 
     The novel control actuator system  20  includes an output fin shaft  22 , servo motor  24 , gear train  26 , and spring  28 . The control fin  14  is attached to the fin shaft  22  such that when the shaft  22  rotates (about the longitudinal axis of the shaft  22 ), the fin  14  also rotates. The shaft  22  is normal to the longitudinal axis of the missile. A servo motor  24  provides a torque to rotate the shaft  22  in response to control signals from the guidance system. The gear train  26  couples the motor to the fin shaft  22 . 
     In accordance with the present teachings, the control actuator system  20  also includes a torsional spring  28 . One end  30  of the spring  28  is attached to the missile body  12 , or some other component of the missile  12  such that the spring end  30  is fixed and does not rotate with the shaft  22 . The other end  32  of the spring  28  is attached to the fin shaft  22  such that rotating the shaft  22  winds or unwinds the spring  28 . In the illustrative embodiment, the spring  28  is in a neutral position (no tension) when the fin  14  is in line with the missile body  12 . Rotating the fin  14  in a first direction winds the spring  28 , and rotating the fin  14  in the opposite direction unwinds the spring  28 . 
     The present invention takes advantage of the fact that in a rolling missile  10 , the control fins  14  move in a cyclical fashion, moving back and forth at the roll frequency of the missile  10 . In a conventional actuator system, the servo motor requires a large amount of power to constantly rotate the fins  14  back and forth in this manner. In accordance with the teachings of the present invention, a spring  28  is added to the actuator system  20  to store some of the energy used to rotate the fin  14  in the first direction. The stored energy is then released to rotate the fin  14  back in the opposite direction, causing the fin  14  to oscillate back and forth at the natural frequency of the system. By choosing a spring  28  with an appropriate spring constant, the natural frequency of the system can be made to match the roll frequency of the missile  10 . 
     An actuator system  20  designed in accordance with the present teachings can therefore control the fins  14  of a rolling missile  10  with reduced power requirements than prior approaches. With this actuator system  20 , it may take a little more energy from the motor  24  to rotate the fin  14  (and wind up the spring  28 ) the first time, but the fin  14  will then continue to oscillate with very little additional energy from the motor  24  (a little energy may need to be added periodically to compensate for friction). The servo motor  24  may include a feedback system adapted to measure the output angle of the fin  14  and add additional torque as needed to keep the fin  14  oscillating to the desired deflection angles. 
       FIG. 4  is a simplified block diagram representing a control actuator system  20  designed in accordance with an illustrative embodiment of the present teachings. The block diagram shown is a mathematical model of the system  20 , showing the signal flow from an input current I m  applied to the servo motor  24  to the resultant rotational angle θ of the fin  14  (where the angle θ is measured with respect to the centerline of the missile  10 ). 
     In the mathematical model of  FIG. 4 , a current I m  is input to the motor  24 , which is represented by its motor constant K T , resulting in the motor  24  generating a torque T A . Additional torque contributions due to friction  48  (represented by the friction constant K f ) and the torsional spring  28  (represented by the spring constant K s ) are subtracted from the applied torque T A  at a summing node  40  to form the total torque T m  in the system. The total torque T m  is applied to the overall moment of inertia J m  of the system, represented by block  42 , resulting in the angular acceleration {umlaut over (θ)} of the fin  14 . The overall moment of inertia J m  includes the moments of inertia of the control fin  14 , shaft  22 , gear train  26 , and motor  24 . Integration of the angular acceleration {umlaut over (θ)} at block  44  results in the rotational rate {dot over (θ)} of the fin  14 . The torque contribution due to friction  48  is a function of the rotational rate {dot over (θ)}. Integration of the rotational rate {dot over (θ)} at block  46  results in the output angle θ of the fin  14 . The torque contribution due to the spring  28  is a function of the angle θ. 
     The dotted line in  FIG. 4  represents the addition of the torsional spring  28  in accordance with the present teachings. The system without the block  28  representing the torsional spring will be referred to as the “baseline design”. The transfer function of the system of the baseline design can be written as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         θ 
                         
                           I 
                           m 
                         
                       
                        
                     
                     Baseline 
                   
                   = 
                   
                     
                       
                         K 
                         T 
                       
                       
                         J 
                         m 
                       
                     
                     
                       s 
                       · 
                       
                         ( 
                         
                           s 
                           + 
                           
                             
                               K 
                               f 
                             
                             
                               J 
                               m 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   1 
                   ] 
                 
               
             
           
         
       
     
     The transfer function of the system  20  with the added torsional spring  28  can be written as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         θ 
                         
                           I 
                           m 
                         
                       
                        
                     
                     Spring 
                   
                   = 
                   
                     
                       
                         K 
                         T 
                       
                       
                         J 
                         m 
                       
                     
                     
                       
                         s 
                         2 
                       
                       + 
                       
                         
                           
                             K 
                             f 
                           
                           
                             J 
                             m 
                           
                         
                         ⁢ 
                         s 
                       
                       + 
                       
                         
                           K 
                           S 
                         
                         
                           J 
                           m 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   2 
                   ] 
                 
               
             
           
         
       
     
     The ratio of the motor currents in the system  20  of the present invention (with the torsional spring  28 ) relative to the baseline design can therefore be found by dividing Eqn. 2 into Eqn. 1: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             θ 
                             
                               I 
                               m 
                             
                           
                            
                         
                         Baseline 
                       
                       
                         
                           
                             θ 
                             
                               I 
                               m 
                             
                           
                            
                         
                         Spring 
                       
                     
                     = 
                     
                       
                         
                           
                             K 
                             T 
                           
                           
                             J 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                             
                           
                         
                         
                           s 
                           · 
                           
                             ( 
                             
                               s 
                               + 
                               
                                 
                                   K 
                                   f 
                                 
                                 
                                   J 
                                   m 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             K 
                             T 
                           
                           
                             J 
                             m 
                           
                         
                         
                           
                             s 
                             2 
                           
                           + 
                           
                             
                               
                                 K 
                                 f 
                               
                               
                                 J 
                                 m 
                               
                             
                             ⁢ 
                             s 
                           
                           + 
                           
                             
                               K 
                               S 
                             
                             
                               J 
                               m 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         I 
                         
                           m_Sprin 
                           ⁢ 
                           g 
                         
                       
                       
                         I 
                         m_Baseline 
                       
                     
                     = 
                     
                       
                         
                           s 
                           2 
                         
                         + 
                         
                           
                             
                               K 
                               f 
                             
                             
                               J 
                               m 
                             
                           
                           ⁢ 
                           s 
                         
                         + 
                         
                           
                             K 
                             S 
                           
                           
                             J 
                             m 
                           
                         
                       
                       
                         s 
                         · 
                         
                           ( 
                           
                             s 
                             + 
                             
                               
                                 K 
                                 f 
                               
                               
                                 J 
                                 m 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   3 
                   ] 
                 
               
             
           
         
       
     
     In accordance with the present teachings, the spring constant, K S , is chosen to set the natural frequency of the system  20  to the desired operating frequency of the system  20 . In the case of a rolling airframe missile  10 , the operating frequency is the roll frequency of the airframe, denoted ω roll . The natural frequency of the torsional-spring-mass system is given by: 
     
       
         
           
             
               
                 
                   
                     ω 
                     natural 
                   
                   = 
                   
                     
                       
                         
                           K 
                           S 
                         
                         
                           J 
                           m 
                         
                       
                     
                     = 
                     
                       ω 
                       roll 
                     
                   
                 
               
               
                 
                   [ 
                   4 
                   ] 
                 
               
             
           
         
       
     
     With this condition set, the transfer function in Eqn. 3 can be evaluated at the operating frequency, s=jω roll , resulting in: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           I 
                                           m_Spring 
                                         
                                         
                                           I 
                                           m_Baseline 
                                         
                                       
                                        
                                     
                                     
                                       s 
                                       = 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ω 
                                           roll 
                                         
                                       
                                     
                                   
                                   = 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             K 
                                             S 
                                           
                                         
                                         
                                           J 
                                           m 
                                         
                                       
                                       + 
                                       
                                         
                                           
                                             K 
                                             f 
                                           
                                           
                                             J 
                                             m 
                                           
                                         
                                         ⁢ 
                                         s 
                                       
                                       + 
                                       
                                         
                                           K 
                                           S 
                                         
                                         
                                           
                                             J 
                                             m 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                         
                                       
                                     
                                     
                                       s 
                                       · 
                                       
                                         ( 
                                         
                                           s 
                                           + 
                                           
                                             
                                               K 
                                               f 
                                             
                                             
                                               J 
                                               m 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
 
                                 
                                 ⁢ 
                                 
                                   
                                     I 
                                     m_Spring 
                                   
                                   
                                     I 
                                     m_Baseline 
                                   
                                 
                               
                                
                             
                             
                               s 
                               = 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   roll 
                                 
                               
                             
                           
                           = 
                           
                             
                               
                                 
                                   K 
                                   f 
                                 
                                 
                                   J 
                                   m 
                                 
                               
                               ⁢ 
                               s 
                             
                             
                               s 
                               · 
                               
                                 ( 
                                 
                                   s 
                                   + 
                                   
                                     
                                       K 
                                       f 
                                     
                                     
                                       J 
                                       m 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                           
                             I 
                             m_Spring 
                           
                           
                             I 
                             m_Baseline 
                           
                         
                       
                        
                     
                     
                       s 
                       = 
                       
                         jω 
                         roll 
                       
                     
                   
                   = 
                   
                     
                       
                         K 
                         f 
                       
                       
                         J 
                         m 
                       
                     
                     
                       
                         j 
                         ⁢ 
                         
                           
                             
                               K 
                               S 
                             
                             
                               J 
                               m 
                             
                           
                         
                       
                       + 
                       
                         
                           K 
                           f 
                         
                         
                           J 
                           m 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   5 
                   ] 
                 
               
             
           
         
       
     
     The magnitude of the function can be taken as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                                
                               
                                 
                                   I 
                                   m_Spring 
                                 
                                 
                                   I 
                                   m_Baseline 
                                 
                               
                                
                             
                             
                               s 
                               = 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   roll 
                                 
                               
                             
                           
                            
                         
                         = 
                           
                         ⁢ 
                         
                            
                           
                             
                               
                                 K 
                                 f 
                               
                               
                                 J 
                                 m 
                               
                             
                             
                               
                                 j 
                                 ⁢ 
                                 
                                   
                                     
                                       K 
                                       S 
                                     
                                     
                                       J 
                                       m 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   K 
                                   f 
                                 
                                 
                                   J 
                                   m 
                                 
                               
                             
                           
                            
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               K 
                               f 
                             
                             
                               J 
                               m 
                             
                           
                           
                             
                               
                                 
                                   K 
                                   S 
                                 
                                 
                                   J 
                                   m 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       K 
                                       f 
                                     
                                     
                                       J 
                                       m 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   6 
                   ] 
                 
               
             
           
         
       
     
     The power dissipated in the servo motor  24  is proportional to the square of the motor current I m . Therefore, the ratio of power dissipated in the torsional-spring-mass design of the present invention versus the baseline design can be expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         Power 
                         Spring 
                       
                       
                         Power 
                         Baseline 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               K 
                               f 
                             
                             
                               J 
                               m 
                             
                           
                           
                             
                               
                                 
                                   K 
                                   S 
                                 
                                 
                                   J 
                                   m 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       K 
                                       f 
                                     
                                     
                                       J 
                                       m 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Power 
                         Spring 
                       
                       
                         Power 
                         Baseline 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               K 
                               f 
                             
                             
                               J 
                               m 
                             
                           
                           ) 
                         
                         2 
                       
                       
                         
                           
                             K 
                             S 
                           
                           
                             J 
                             m 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 K 
                                 f 
                               
                               
                                 J 
                                 m 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Power 
                         Spring 
                       
                       
                         Power 
                         Baseline 
                       
                     
                     = 
                     
                       1 
                       
                         
                           
                             
                               K 
                               S 
                             
                             ⁢ 
                             
                               J 
                               m 
                             
                           
                           
                             K 
                             f 
                             2 
                           
                         
                         + 
                         1 
                       
                     
                   
                 
               
               
                 
                   [ 
                   7 
                   ] 
                 
               
             
           
         
       
     
     The term K S J m /K f   2  is typically greater than one. Therefore, a torsional-spring-mass system designed in accordance with the present teachings should consume less power than the baseline system. 
     As a numerical example, consider a system with the following parameters:
 
 K   T =0.028Nm/A
 
 J   m =284 e   −6 Nm-s 2  
 
K f =0.0089Nm-s
 
ω roll =2π10rad/s
 
     To satisfy the condition that the natural frequency of the system is equal to the roll frequency of the airframe, the spring constant K S  is chosen to be: 
     
       
         
           
             
               
                 
                   
                     
                       
                         K 
                         S 
                       
                       
                         J 
                         m 
                       
                     
                   
                   = 
                     
                   ⁢ 
                   
                     ω 
                     roll 
                   
                 
               
             
             
               
                 
                   
                     K 
                     S 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       J 
                       m 
                     
                     · 
                     
                       ω 
                       roll 
                       2 
                     
                   
                 
               
             
             
               
                 
                   
                     K 
                     S 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             284 
                             ⁢ 
                             ⅇ 
                           
                           - 
                           6 
                         
                         ) 
                       
                       · 
                       
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               π 
                               · 
                               10 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Nm 
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     rad 
                   
                 
               
             
             
               
                 
                   
                     K 
                     S 
                   
                   = 
                     
                   ⁢ 
                   
                     1.12 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Nm 
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     rad 
                   
                 
               
             
           
         
       
     
     Plugging these values into Eqn. 7 gives the result that the power dissipation in the actuator system  20  with the addition of the torsional spring  28  relative to the baseline design is: 
     
       
         
           
             
               
                 Power 
                 Spring 
               
               
                 Power 
                 Baseline 
               
             
             = 
             
               1 
               
                 
                   
                     
                       K 
                       S 
                     
                     ⁢ 
                     
                       J 
                       m 
                     
                   
                   
                     K 
                     f 
                     2 
                   
                 
                 + 
                 1 
               
             
           
         
       
       
         
           
             
               
                 Power 
                 Spring 
               
               
                 Power 
                 Baseline 
               
             
             = 
             
               1 
               
                 
                   
                     
                       ( 
                       1.12 
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           284 
                           ⁢ 
                           ⅇ 
                         
                         - 
                         6 
                       
                       ) 
                     
                   
                   
                     0.0089 
                     2 
                   
                 
                 + 
                 1 
               
             
           
         
       
       
         
           
             
               
                 Power 
                 Spring 
               
               
                 Power 
                 Baseline 
               
             
             = 
             0.2 
           
         
       
     
     Thus, in the numerical example, the addition of a torsional spring  28  (with an appropriate spring constant K S ) to the control actuator system  20  should reduce the power dissipation by 80%. 
       FIGS. 2-4  showed an actuator system  20  for controlling only one fin  14 . In the illustrative embodiment of  FIG. 1 , the missile  10  includes four fins  14 A- 14 D.  FIG. 5  is a three-dimensional view of a control actuator system  20  for four control fins designed in accordance with an illustrative embodiment of the present teachings. In this embodiment, each fin  14 A- 14 D is controlled independently by a separate actuator  20 A- 20 D, respectively. Each individual actuator  20 A- 20 D includes a servo motor  24 , gear train  26 , fin shaft  22 , and torsional spring  28 , as shown in  FIGS. 2 and 3 . The actuator system  20  may also include electronics  50  for providing the drive currents I m  for the servo motors  24 . 
     Alternatively, a single actuator (as shown in  FIG. 3 ) may be used to control multiple fins simultaneously. For example, a missile having only two control fins may include two separate actuators for independently controlling the two fins, or it may include only one actuator for rotating one fin shaft that is coupled to both fins (in this embodiment, the two fins would move together in unison). Other implementations may also be used without departing from the scope of the present teachings. 
     Thus, the present invention has been described herein with reference to a particular embodiment for a particular application. Those having ordinary skill in the art and access to the present teachings will recognize additional modifications, applications and embodiments within the scope thereof. For example, while the invention has been described with reference to a rolling missile, the present teachings may also be applied to other applications such as a rocket or other air or space vehicle or projectile, a torpedo or other watercraft, or a high speed ground vehicle. 
     It is therefore intended by the appended claims to cover any and all such applications, modifications and embodiments within the scope of the present invention.