Abstract:
A servo track has identical servo pattern frames including two pairs of parallel magnetic transitions, the transitions of each pair spaced apart an equal distance, the transitions of the first pair at an azimuth angle to the longitudinal axis of the tape, the transitions of the second pair at an opposite azimuth angle. A servo channel receives signals at first times corresponding to when a servo read head detects the first and second transitions of the first pair of parallel transitions of a servo pattern frame, and second times corresponding to when the servo read head detects the first and second transitions of the second pair of parallel transitions. The servo channel determines a relative lateral movement and velocity between the tape and the tape head based on the azimuth angle and the ratio of the difference of the first and the difference of the second times.

Description:
FIELD OF THE INVENTION 
     The present invention relates generally to tape drive systems having a timing-based servo for positioning a head, and more particularly to a system for determining lateral head movement and velocity based on servo track timing measurements. 
     BACKGROUND 
     Densities for linear tape storage systems are at a point where precision lateral positioning of the tape heads perpendicular to the longitudinal direction of motion of the tape is a requirement. Timing-based servo (TBS) is a technology developed in the mid-1990s for linear tape drives to specifically address this issue. In TBS systems, recorded servo patterns consist of transitions with two different azimuthal slopes, and head lateral-position relative to the servo track is derived from the relative timing of pulses generated by a narrow servo tracking head reading the pattern. 
     A popular tape drive technology that has adopted the TBS standard is Linear Tape Open (LTO). Linear Tape Open and LTO are registered trademarks of Hewlett-Packard Company, International Business Machines Corporation, and Quantum Corporation. In LTO, the tape width is divided into four data bands sandwiched between five narrow servo bands or tracks. Each servo band has a TBS pattern that is written to the servo band during the tape manufacturing process. The tape head assembly straddles two adjacent servo bands, with two or more servo read heads and 8 or 16 data read/write heads. Each data head moves up and down within its own data sub-band the same width as the servo band. 
     As the servo track deviates from the ideal centerline positioning relative to the servo tracking head, the servo control will activate and move the servo tracking head to follow the servo track. The actuator that enables precise positioning of the read head, utilizing the servo system, can involve an arrangement in which the head actuator assembly is suspended using a spring system that possesses mass and stiffness. Such an actuator suspension and servo system has resonant frequencies with the first natural resonance mode typically having a frequency below the closed loop bandwidth. In other arrangements, resonance modes may occur in various shafts, cantilevered arms, and other moving and fixed parts of the actuator assembly. Thus, another issue involving tape heads is effective damping of the tape head actuator. A factor in determining such effective damping is the velocity of the tape head actuator in the lateral direction. 
     SUMMARY 
     Embodiments of the present invention provide a servo system to determine a relative lateral movement and velocity between a tape and a tape head, the tape having at least one longitudinal defined servo track, the servo track including a longitudinal series of identical servo pattern frames, each servo pattern frame including two pairs of non-overlapping parallel magnetic transitions, the transitions of each pair being spaced apart an equal distance, the transitions of the first pair forming an azimuth angle η to the longitudinal axis of the tape, the transitions of the second pair forming the azimuth angle η to the longitudinal axis of the tape but at an opposite slope about the lateral axis of the tape. The servo system includes a tape head including a servo read head configured to read the servo pattern frames in the at least one servo track and produce servo signals at times T A  and T B , corresponding to the times that the servo read head detects the first and second transitions of the first pair of parallel transitions of a servo pattern frame, and times T C  and T D , corresponding to the times that the servo read head detects the first and second transitions of the second pair of parallel transitions of the servo pattern frame. The servo system also includes a servo channel configured to receive the servo signals and determine a relative lateral movement LM AB  or LM CD  between the tape and the tape head between times T A  and T B , or times T C  and T D , respectively, according to the relationships 
               α   =     Arctan   ⁡     (       [       (       T   B     -     T   A       )     -     (       T   D     -     T   C       )       ]       [       (       T   B     -     T   A       )     +     (       T   D     -     T   C       )       ]       )         ,         
and
 
               LM   AB     =     α   *     d       sin   ⁡     (   η   )       -     (     α   *     cos   ⁡     (   η   )         )                 
or
 
                 LM   CD     =     α   *     d       sin   ⁡     (   η   )       +     (     α   *     cos   ⁡     (   η   )         )             ,         
respectively.
 
     Other embodiments of the present invention provide a method to determine a relative lateral movement and velocity between the tape and the tape head in a servo system for positioning a tape head laterally to follow lateral motion of a longitudinal tape moving in a substantially longitudinal direction with respect to the tape head, the tape having at least one longitudinal defined servo track, the servo track including a longitudinal series of identical servo pattern frames, each servo pattern frame including two pairs of non-overlapping parallel magnetic transitions, the transitions of each pair being spaced apart an equal distance, the transitions of the first pair forming an azimuth angle η to the longitudinal axis of the tape, the transitions of the second pair forming the azimuth angle η to the longitudinal axis of the tape but at an opposite slope about the lateral axis of the tape, said servo system including an actuator configured to move the tape head laterally with respect to the longitudinal tape, the tape head including a servo read head configured to read the servo pattern frames in the servo track and produce servo signals, a servo channel configured to receive and process the servo signals, a position error signal loop configured to sense the servo signals, to determine position error between the servo read head and a desired centerline position of the at least one defined servo track based on the servo signals, and to operate the actuator to move the tape head laterally to reduce the determined position error. The servo read head reads a servo pattern frame in the servo track, the relative movement of the servo head with respect to the tape forming a trajectory angle α with respect to the center-line of the at least one defined servo track, the trajectory of the servo read head intersecting the first and second transitions of the first pair of parallel transitions of the servo pattern frame at times T A  and T B , respectively, and intersecting the first and second transitions of the second pair of parallel transitions of the servo pattern frame at times T C  and T D , respectively, whereby the servo read head produces servo signals at times T A , T B , T C , and T D . The servo channel determines a relative lateral movement LM AB  or LM CD  between the tape and the actuator between times T A  and T B , or times T C  and T D , respectively, according to the relationships 
               α   =     Arctan   ⁡     (       [       (       T   B     -     T   A       )     -     (       T   D     -     T   C       )       ]       [       (       T   B     -     T   A       )     +     (       T   D     -     T   C       )       ]       )         ,         
and
 
               LM   AB     =     α   *     d       sin   ⁡     (   η   )       -     (     α   *     cos   ⁡     (   η   )         )                 
or
 
                 LM   CD     =     α   *     d       sin   ⁡     (   η   )       +     (     α   *     cos   ⁡     (   η   )         )             ,         
respectively.
 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         FIG. 1  is a simplified component view of a time-based servo system, in accordance with embodiments of the present invention. 
         FIG. 2  is a simplified block diagram of a typical time-based servo control system. 
         FIG. 3  illustrates a frame of a TBS servo pattern in accordance with embodiments of the invention. 
         FIG. 4  illustrates a frame of a second TBS servo pattern in accordance with embodiments of the invention. 
         FIG. 5  illustrates the servo pattern frame of  FIG. 3 , including a servo head trajectory. 
         FIG. 6  illustrates a detail of the servo pattern frame of  FIG. 3 , including a servo head trajectory. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the present invention will now be described in detail with reference to the accompanying drawings. 
       FIG. 1  is a simplified component view of a time-based servo system  100 , in accordance with embodiments of the present invention. Tape head actuator  102  includes narrow servo read heads  104  and data read/write head  106 . Base plate  110  supports actuator shaft  108 . Tape head actuator  102  moves along actuator shaft  108  in the lateral Y direction via a servo motor or electromagnet (not shown). Typically, tape head actuator  102  includes or is connected to a stepper motor arrangement for gross movements, and a voice coil arrangement for fine movements. For simplicity, these details are not shown. Tape  112  represents a portion of a linear tape medium that is ideally moving in the longitudinal X direction. Tape  112  includes a data track  114 , shown with eight sub-tracks, sandwiched between two servo tracks  116 A and  116 B that have been imprinted during the tape manufacturing process with a magnetic servo pattern  118  that consists of transitions with two different azimuthal angles, which will be described in greater detail below. Although only a single data track  114  is shown, a tape  112  typically has several data tracks separated by servo tracks. In addition, each data track typically includes several sub-tracks, and data read/write head  106  will include several read/write heads. 
     In operation, tape  112  moves in the X direction past tape head actuator  102 . Servo read heads  104 , which are small in the lateral dimension in comparison to servo tracks  116 , detect servo patterns  118  in servo tracks  116 A and  116 B. Based on the timing of pulses generated by servo read heads  104  reading servo patterns  118 , the position in the lateral Y direction of servo read heads  104  relative to the position of the servo tracks in the lateral Z direction can be determined. Typically, there is some movement of tape  112  in the lateral Z direction relative to the ideal longitudinal X direction of travel, as indicated in  FIG. 1  by the slight “wave” shape of tape  112 . To keep data read/write head  106  in good alignment with data track  114 , a state variable feedback system controls the servo that moves tape head actuator  102  along actuator shaft  108  in the Y direction based on the relative position of servo read heads  104  and the ideal position relative to servo tracks  116 A and  116 B, which may be the centerline of servo tracks  116 A and  116 B or may be a lateral offset to that centerline. 
       FIG. 2  is a simplified block diagram of a typical time-based servo control system  200 . The servo control system  200  is based on a position error signal loop utilizing a proportional-integral-derivative (PID) controller  202 . The servo control system  200  includes PID controller  202 , actuator  204 , a head module  206 , at least one servo read head  208  located in or on the head module  206 , a servo channel  210 , and a subtractor  212 .  FIG. 2  also shows various disturbances that are often present in tape drive systems (e.g., shocks, vibrations, stack shifts, and narrowband disturbances).  FIG. 2  further shows a reference signal r(t), which is the reference signal associated with, for example, the centerline of servo tracks  116  to which servo read head  208  should be tracking, a position error signal (PES) e(t), and a control signal u control , a signal s(t) provided by servo read head  208  to servo channel  210 , a tape velocity estimate signal v(t), and a lateral position estimate signal y(t). PES e(t) corresponds to the difference between reference signal r(t) and lateral position estimate signal y(t). With regard to  FIG. 1 , actuator  204  and head module  206  correspond generally to tape head actuator  102 , and servo read head  208  corresponds to servo read heads  104 . Servo channel  210  may be implemented, for example, as a microprocessor with microcode instructions stored either inside servo channel  210  or in a separate EPROM (not shown), or as a field-programmable gate array (FPGA), or as an application-specific integrated circuit (ASIC), or as a combination of the foregoing, or any other computing device capable of performing the functionality required in embodiments of the invention. 
     In operation, servo control system  200  uses the PES e(t) as an input to PID controller  202 . PID controller  202  outputs control signal u control  to actuator  204 . Based on the control signal u control , the actuator  204  adjusts the position of the head module  206 , which in turn determines the position of servo read head  208  and corresponding read/write heads (not shown). The read/write heads are maintained at a desired “on track” position via motion of the actuator and also via feedback provided by the servo read head  208 . Specifically, servo read head  208  provides a signal s(t) to the servo channel  210 . The servo channel  210  processes the signal s(t) to generate a lateral position estimate signal y(t) and a tape velocity estimate signal v(t), which indicates an estimate of the longitudinal velocity of the tape being read/written. Lateral position estimate signal y(t) along with reference signal r(t) is input to subtractor  212 , which outputs the PES difference signal e(t). 
     In the embodiments shown in  FIGS. 1 and 2 , actuator  204  typically experiences vibrational resonances that must be controlled. The mechanical behavior of actuator  204  may be approximated by a simple spring-damper-mass model. As is known in the art, a state-space form of the differential equations representing a spring-damper-mass model is as follows: 
                     [             ⅆ   y       ⅆ   t                     ⅆ   2     ⁢   y       ⅆ     t   2               ]     =         [         0       1               -   k     m             -   c     m           ]     ⁡     [         y               ⅆ   y       ⅆ   t             ]       +         [         0       0           Kf       Cf         ]     ⁡     [           z   -   y                   ⅆ   z       ⅆ   t       -       ⅆ   y       ⅆ   t               ]       .               (   1   )               
In equation (1), all elements are known, except for z-y, and
 
                 ⅆ   z       ⅆ   t       -         ⅆ   y       ⅆ   t       .           
In equation (1), m is the mass of tape head actuator  102  in kilograms, including any additional mass attributed to, for example, head cables and servo motors to be overcome when accelerating tape head actuator  102  in the Y direction; k is the mechanical spring rate of tape head actuator  102  in the Y direction, in Newtons per meter; and c is the mechanical damping experienced by tape head actuator  102  in the Y direction, in Newton-seconds per meter. Additionally, Kf is the feedback coefficient with units of seconds −2  and Cf is the feedback coefficient with units of second −1 .
 
       FIGS. 3 and 4  each illustrate a frame of a TBS servo pattern in accordance with embodiments of the invention by which the terms z-y, and 
                 ⅆ   z       ⅆ   t       -       ⅆ   y       ⅆ   t             
can be derived from the relative timing of pulses generated by a servo read head  104  reading the servo pattern, such as servo pattern  118  in  FIG. 1 . Servo pattern frames  300  and  400  each comprise two sets of parallel transitions, each set having equal azimuth angles to the centerline of the TBS servo track but opposite to the other set, and which no transitions cross each other. Although for ease of explanation the azimuth angles are stated with respect to the servo track centerline, any parallel to the centerline can be used. In alternative embodiments, servo tracks  116 A and  116 B include either a longitudinal series of servo pattern frames  300  or servo pattern frames  400 . In  FIG. 3 , servo pattern frame  300  comprises parallel transitions  302  and  306 , having an azimuth angle η  310  with respect to the servo track centerline X, and parallel transitions  304  and  308 , having an equal azimuth angle η 312 , but in the opposite direction as azimuth angle  310 . Parallel transitions  302  and  306  are separated by a distance  314  of length d, and parallel transitions  304  and  308  are separated by an equal distance d  316 . In the arrangement shown in  FIG. 3 , parallel transition pair  302  and  306  is interleaved with parallel transition pair  304  and  308 , forming a double chevron, or “M” shape.  FIG. 4  shows an alternative arrangement with a servo pattern frame  400  in which parallel transition pair  302  and  306  are not interleaved with parallel transition pair  304  and  308 . For purposes of the invention, embodiments can use either arrangement. For purposes of explanation, the interleaved pattern frame  300  of  FIG. 3  will be used.
 
       FIG. 5  illustrates the servo pattern frame  300  of  FIG. 3 , including a servo head trajectory  518 . Servo head trajectory  518  represents, for example, the path over servo pattern frame  300  that a servo head  104  would follow when tape  112  is experiencing movement in the negative lateral Y direction as it moves in the longitudinal X direction. Servo head trajectory  518  forms a positive angle α 520  with the X direction. As illustrated, servo head trajectory  518  also forms an angle (η−α)  522  with parallel transition pair  302  and  306 , and an angle (η+α)  524  with parallel transition pair  304  and  308 . Servo head trajectory  518  crosses parallel transition pair  302  and  306  at points A and B, respectively, and crosses parallel transition pair  304  and  308  at points C and D, respectively. In practice, as adjustments are made by servo control system  200  to keep the read/write heads of tape head actuator  102  on track, trajectory angle α 520  will change. However, within a servo pattern frame  300 , servo head trajectory  518  can be considered to be linear, and trajectory angle α 520  as constant. 
     In equation (1), the term z-y represents a relative movement of tape  112  in the lateral Z direction with respect to a movement of tape head actuator  102  in the lateral Y direction (see  FIG. 1 ). This may be most easily understood as a movement of tape  112  from an observational frame of reference tied to tape head actuator  102 .  FIG. 6  shows a detail of  FIG. 5  relating to parallel transition pair  302  and  306 . With reference to  FIG. 6 , (z-y)  602  represents the lateral movement of a servo read head  104  in the Y direction as a servo read head  104  traverses a path over servo pattern frame  300  between points A and B along servo head trajectory  518 . As can be seen from  FIG. 6 , 
                       sin   ⁡     (   α   )       =         (     z   -   y     )     AB     AB       ,           (   2   )               
where AB is the length of the segment between points A and B that a servo read head  104  traverses along servo head trajectory  518 , and (z-y) AB  is the lateral movement in the Y direction of tape head actuator  102  as it traverses segment AB. The length of segment AB can be expressed in terms of tape velocity by the equation:
 
 AB=V   TapeAB *( T   B   −T   A ),  (3)
 
where V TapeAB  is the velocity of the tape as detected by a servo read head  104  along segment AB, and (T B −T A ) is the time it takes a servo read head  104  to traverse segment AB. Expressing equation (2) in terms of (z-y) and using the identity of equation (3), gives:
 
( z−y ) AB =sin(α)* V   TapeAB *( T   B   −T   A ).  (4)
 
As can be seen from  FIG. 6 ,
 
                     V   TapeAB     =       d     sin   ⁡     (     η   -   α     )         *       1     (       T   B     -     T   A       )       .               (   5   )               
Substituting the identity of equation (5) into equation (4) gives:
 
                             (     z   -   y     )     AB     =       ⁢       sin   ⁡     (   α   )       *     [       d     sin   ⁡     (     η   -   α     )         *     1     (       T   B     -     T   A       )         ]     *     (       T   B     -     T   A       )                   =       ⁢       sin   ⁡     (   α   )       *     d     sin   ⁡     (     η   -   α     )                       =       ⁢       sin   ⁡     (   α   )       *       d         sin   ⁡     (   η   )       ⁢     cos   ⁡     (   α   )         -       cos   ⁡     (   η   )       ⁢     sin   ⁡     (   α   )             .                     (   6   )               
Assuming trajectory angle α 520  to be a small angle, sin(α) can be approximated as α, and cos(α) can be approximated as 1. Thus, equation (6) can be expressed as the following, which defines lateral movement LM AB :
 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         z 
                         - 
                         y 
                       
                       ) 
                     
                     AB 
                   
                   = 
                   
                     
                       LM 
                       AB 
                     
                     = 
                     
                       α 
                       * 
                       
                         
                           d 
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 η 
                                 ) 
                               
                             
                             - 
                             
                               ( 
                               
                                 α 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     η 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Similarly, with reference to  FIG. 5 , lateral movement LM CD  is defined as: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         z 
                         - 
                         y 
                       
                       ) 
                     
                     CD 
                   
                   = 
                   
                     
                       LM 
                       CD 
                     
                     = 
                     
                       α 
                       * 
                       
                         
                           d 
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 η 
                                 ) 
                               
                             
                             + 
                             
                               ( 
                               
                                 α 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     η 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     In equation (1), 
                 ⅆ   z       ⅆ   t       -       ⅆ   y       ⅆ   t             
represents the relative velocity of tape  112  in the lateral Z direction with respect to tape head actuator  102  in the lateral Y direction. This, too, may be most easily understood as the movement of tape  112  from an observational frame of reference tied to tape head actuator  102 . The term
 
                 ⅆ   z       ⅆ   t       -       ⅆ   y       ⅆ   t             
can be derived from equation (7) or equation (8) by dividing both sides of these equations by the time it takes a servo read head  104  to traverse segment AB or CD, respectively. Thus, lateral velocities LV AB  and LV CD  are defined as follows:
 
                         (         ⅆ   z       ⅆ   t       -       ⅆ   y       ⅆ   t         )     AB     =       LV   AB     =     α   *     d       sin   ⁡     (   η   )       -     (     α   *     cos   ⁡     (   η   )         )         *     1     (       T   B     -     T   A       )             ,           (   9   )               
and
 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             ⅆ 
                             z 
                           
                           
                             ⅆ 
                             t 
                           
                         
                         - 
                         
                           
                             ⅆ 
                             y 
                           
                           
                             ⅆ 
                             t 
                           
                         
                       
                       ) 
                     
                     CD 
                   
                   = 
                   
                     
                       LV 
                       CD 
                     
                     = 
                     
                       α 
                       * 
                       
                         d 
                         
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               η 
                               ) 
                             
                           
                           + 
                           
                             ( 
                             
                               α 
                               * 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   η 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       * 
                       
                         
                           1 
                           
                             ( 
                             
                               
                                 T 
                                 D 
                               
                               - 
                               
                                 T 
                                 C 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Equations (7) and (8), and (9) and (10) express the terms z-y, and 
                   ⅆ   z       ⅆ   t       -       ⅆ   y       ⅆ   t         ,         
respectively, from equation (1) in terms of trajectory angle α 520 , the angle between servo head trajectory  518  and direction X of a servo pattern frame  300  or  400 . All other terms of these equations are known or can be empirically measured during operation of time based servo system  100 .
 
     With reference to  FIG. 5 , trajectory angle α 520  can be expressed as a function of the ratio of the difference in servo head transit times over segments AB and CD. With reference to  FIG. 5 , and as stated above, 
                     V   TapeAB     =       d     sin   ⁡     (     η   -   α     )         *       1     (       T   B     -     T   A       )       .               (   5   )               
Similarly,
 
     
       
         
           
             
               
                 
                   
                     V 
                     TapeCD 
                   
                   = 
                   
                     
                       d 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             η 
                             + 
                             α 
                           
                           ) 
                         
                       
                     
                     * 
                     
                       
                         1 
                         
                           ( 
                           
                             
                               T 
                               D 
                             
                             - 
                             
                               T 
                               C 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Within the same servo pattern frame  300  or  400 , V TapeAB  and V TapeCD  can be approximated as being equal, especially for the overlapping “M” configuration shown in  FIG. 3 . Thus, 
                   d     sin   ⁡     (     η   -   α     )         *     1     (       T   B     -     T   A       )         =       d     sin   ⁡     (     η   +   α     )         *     1     (       T   D     -     T   C       )           ,         
or
 
                   (       T   D     -     T   C       )       (       T   B     -     T   A       )       =       sin   ⁡     (     η   -   α     )         sin   ⁡     (     η   +   α     )           ,         
and
 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           T 
                           D 
                         
                         - 
                         
                           T 
                           C 
                         
                       
                       ) 
                     
                     
                       ( 
                       
                         
                           T 
                           B 
                         
                         - 
                         
                           T 
                           A 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               η 
                               ) 
                             
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                         - 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               η 
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               η 
                               ) 
                             
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                         + 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               η 
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     For the special case where azimuth angle η 310 / 312  is π/4 radians, or 45 degrees, cos(η)=sin(η), and equation (12) can be expressed as: 
                       (       T   D     -     T   C       )       (       T   B     -     T   A       )       =           cos   ⁡     (   α   )       -     sin   ⁡     (   α   )             cos   ⁡     (   α   )       +     sin   ⁡     (   α   )           .             (   13   )               
Multiplying the right-hand side of equation (13) by [1/cos(α))/(1/cos(α))] gives:
 
                       (       T   D     -     T   C       )       (       T   B     -     T   A       )       =         1   -     tan   ⁡     (   α   )           1   +     tan   ⁡     (   α   )           .             (   14   )               
Expressing equation (14) in terms of tan(α) gives:
 
                 tan   ⁡     (   α   )       =       [     1   -       (       T   D     -     T   C       )       (       T   B     -     T   A       )         ]     /     [     1   +       (       T   D     -     T   C       )       (       T   B     -     T   A       )         ]         ,         
or
 
                     tan   ⁡     (   α   )       =         [       (       T   B     -     T   A       )     -     (       T   D     -     T   C       )       ]       [       (       T   B     -     T   A       )     +     (       T   D     -     T   C       )       ]       .             (   15   )               
Thus,
 
     
       
         
           
             
               
                 
                   α 
                   = 
                   
                     
                       Arctan 
                       ⁡ 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     B 
                                   
                                   - 
                                   
                                     T 
                                     A 
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 ( 
                                 
                                   
                                     T 
                                     D 
                                   
                                   - 
                                   
                                     T 
                                     C 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     B 
                                   
                                   - 
                                   
                                     T 
                                     A 
                                   
                                 
                                 ) 
                               
                               + 
                               
                                 ( 
                                 
                                   
                                     T 
                                     D 
                                   
                                   - 
                                   
                                     T 
                                     C 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     For general azimuth angles η 310 / 312 , a table look-up based on equation (12) can be implemented within or called by, for example, servo channel  210  to determine trajectory angle α 520  based on a calculated ratio of the times (T D −T C )/(T B −T A ). In such a scheme, azimuth angle η 310 / 312  is known. For the special case where azimuth angle η 310 / 312  is π/4 radians, or 45 degrees, the table can be based on equation (16). After trajectory angle α 520  has been determined, values for z-y, and 
                 ⅆ   z       ⅆ   t       -       ⅆ   y       ⅆ   t             
can be determined with a second table look-up in a table based on equations (7) or (8), and (9) or (10). Alternatively, a single table encompassing equation (7) or (8), (9) or (10), and (15) or (16), can be used in a table look-up. For example, a table can be populated with entries that span possible values of the ratio of the times (T D −T C /(T B −T A ), and an interpolation routine can determine appropriate values LM AB , LM CD , LV AB , and/or LV CD .
 
     As will be appreciated by one skilled in the art, aspects of the present invention may be embodied as a system or method. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit,” “module,” or “system.” 
     Any flowcharts and block diagrams in the Figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present invention. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions. 
     The foregoing description of various embodiments of the present invention has been presented for purposes of illustration and description. It is not intended to be exhaustive nor to limit the invention to the precise form disclosed. Many modifications and variations are possible. Such modifications and variations that may be apparent to a person skilled in the art of the invention are intended to be included within the scope of the invention as defined by the accompanying claims.