Abstract:
An apparatus for sensing rotation includes a plurality of motion sensors constructed in a substantially coplanar arrangement. The plurality of motion sensors is each configured to generate incremental movement data indicative of movement of the sensor in two dimensions. A rotation data generator generates rotation data based on the incremental movement data. The rotation data represents rotation of a first one of the motion sensors about a second one of the motion sensors.

Description:
REFERENCE TO RELATED PATENTS 
     This Application is related to the subject matter described in the following U.S. patents: U.S. Pat. No. 5,578,813, filed Mar. 2, 1995, issued Nov. 26, 1996, and entitled FREEHAND IMAGE SCANNING DEVICE WHICH COMPENSATES FOR NON-LINEAR MOVEMENT; U.S. Pat. No. 5,644,139, filed Aug. 14, 1996, issued Jul. 1, 1997, and entitled NAVIGATION TECHNIQUE FOR DETECTING MOVEMENT OF NAVIGATION SENSORS RELATIVE TO AN OBJECT; U.S. Pat. No. 5,786,804, filed Oct. 6, 1995, issued Jul. 28, 1998, and entitled METHOD AND SYSTEM FOR TRACKING ATTITUDE; U.S. Pat. No. 6,057,540, filed Apr. 30, 1998, issued May 2, 2000, and entitled MOUSELESS OPTICAL AND POSITION TRANSLATION TYPE SCREEN POINTER CONTROL FOR A COMPUTER SYSTEM; U.S. Pat. No. 6,151,015, filed Apr. 27, 1998, issued Nov. 21, 2000, and entitled PEN LIKE COMPUTER POINTING DEVICE; U.S. Pat. No. 6,281,882, filed Mar. 30, 1998, issued Aug. 28, 2001, and entitled PROXIMITY DETECTOR FOR A SEEING EYE MOUSE; and U.S. patent application Ser. No. 10/004,512, filed Oct. 26, 2001, and entitled APPARATUS AND METHOD FOR THREE-DIMENSIONAL RELATIVE MOVEMENT SENSING. 
    
    
     THE FIELD OF THE INVENTION 
     This invention relates generally to motion sensor devices. This invention relates more particularly to a motion sensor device for sensing rotation. 
     BACKGROUND OF THE INVENTION 
     The use of a hand operated pointing device for use with a computer and its display has become almost universal. By far the most popular of the various devices is the conventional (mechanical) mouse, used in conjunction with a cooperating mouse pad. Centrally located within the bottom surface of the mouse is a hole through which a portion of the underside of a rubber-surfaced steel ball extends. Interior to the mouse are rollers, or wheels, that contact the ball at its equator and convert its rotation into electrical signals representing orthogonal components of mouse motion. These electrical signals are coupled to a computer, where software responds to the signals to change by a ΔX and a ΔY the displayed position of a pointer (cursor) in accordance with movement of the mouse. 
     In addition to mechanical types of pointing devices, such as a conventional mouse, optical pointing devices have also been developed. In one form of an optical pointing device, rather than using a moving mechanical element like a ball in a conventional mouse, movement between an imaging surface, such as a finger or a desktop, and photo detectors within the optical pointing device, is optically sensed and converted into movement information. 
     The photo detectors in optical pointing devices are typically implemented in a flat, two-dimensional array. The array of photo detectors is capable of measuring absolute two-dimensional movement. As the array moves across an image, or the image moves across a stationary array, motion can be detected by comparing successive images. The sensed motion is in terms of the number of pixels that the image on the pixel array has moved. The array is typically at a fixed distance and a fixed angle from the surface being imaged, so the motion that is sensed is absolute (within the error tolerance of the system). 
     Existing optical sensors, such as those used in optical pointing devices, sense movement in an X and Y direction, but do not sense rotation. It would be desirable to provide a sensing apparatus using multiple two-dimensional photo detector arrays for sensing rotation of the apparatus. 
     SUMMARY OF THE INVENTION 
     One form of the present invention provides an apparatus for sensing rotation. The apparatus includes a plurality of motion sensors constructed in a substantially coplanar arrangement. The plurality of motion sensors is each configured to generate incremental movement data indicative of movement of the sensor in two dimensions. A rotation data generator generates rotation data based on the incremental movement data. The rotation data represents rotation of a first one of the motion sensors about a second one of the motion sensors. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a top view of an optical mouse, which is suitable for incorporating one embodiment of the present invention. 
         FIG. 2  is an electrical block diagram illustrating major components of the optical mouse shown in FIG.  1 . 
         FIG. 3  is a diagram of a general rotation sensor orientation, three special rotation sensor orientations, and mirror rotation sensor orientations. 
         FIG. 4A  is a diagram illustrating translation of a rotation sensor in the X direction. 
         FIG. 4B  is a diagram illustrating translation of a rotation sensor in the X and Y directions. 
         FIG. 4C  is a diagram illustrating translation of a rotation sensor in the Y direction. 
         FIG. 4D  is a diagram illustrating rotation of a rotation sensor. 
         FIG. 4E  is a diagram illustrating rotation and translation of a rotation sensor. 
         FIG. 5  is a diagram illustrating a positive angle of rotation on an X-Y axis. 
         FIG. 6A  is a diagram illustrating various translations and positive rotations of a rotation sensor in the general orientation. 
         FIG. 6B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 6A  have been eliminated. 
         FIG. 7A  is a diagram illustrating various translations and negative rotations of a rotation sensor in the general orientation. 
         FIG. 7B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 7A  have been eliminated. 
         FIG. 8A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a horizontal mirror of the general orientation. 
         FIG. 8B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 8A  have been eliminated. 
         FIG. 9A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a horizontal mirror of the general orientation. 
         FIG. 9B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 9A  have been eliminated. 
         FIG. 10A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a vertical mirror of the general orientation. 
         FIG. 10B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 10A  have been eliminated. 
         FIG. 11A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a vertical mirror of the general orientation. 
         FIG. 11B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 11A  have been eliminated. 
         FIG. 12A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a horizontal and vertical mirror of the general orientation. 
         FIG. 12B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 12A  have been eliminated. 
         FIG. 13A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a horizontal and vertical mirror of the general orientation. 
         FIG. 13B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 13A  have been eliminated. 
         FIG. 14  is a diagram of an isosceles triangle representing rotation of a rotation sensor in the general orientation after translation has been eliminated. 
         FIG. 15A  is a diagram illustrating rotation of a rotation sensor in the general orientation prior to a coordinate transformation. 
         FIG. 15B  is a diagram illustrating rotation of a rotation sensor in the general orientation after a coordinate transformation. 
         FIG. 16A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a first special orientation. 
         FIG. 16B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 16A  have been eliminated. 
         FIG. 17A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a first special orientation. 
         FIG. 17B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 17A  have been eliminated. 
         FIG. 18A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a vertical mirror of the first special orientation. 
         FIG. 18B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 18A  have been eliminated. 
         FIG. 19A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a vertical mirror of the first special orientation. 
         FIG. 19B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 19A  have been eliminated. 
         FIG. 20A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a second special orientation. 
         FIG. 20B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 20A  have been eliminated. 
         FIG. 21A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a second special orientation. 
         FIG. 21B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 21A  have been eliminated. 
         FIG. 22A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a horizontal mirror of the second special orientation. 
         FIG. 22B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 22A  have been eliminated. 
         FIG. 23A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a horizontal mirror of the second special orientation. 
         FIG. 23B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 23A  have been eliminated. 
         FIG. 24A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a third special orientation. 
         FIG. 24B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 24A  have been eliminated. 
         FIG. 25A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a third special orientation. 
         FIG. 25B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 25A  have been eliminated. 
         FIG. 26A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a horizontal mirror of the third special orientation. 
         FIG. 26B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 26A  have been eliminated. 
         FIG. 27A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a horizontal mirror of the third special orientation. 
         FIG. 27B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 27A  have been eliminated. 
         FIG. 28A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a vertical mirror of the third special orientation. 
         FIG. 28B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 28A  have been eliminated. 
         FIG. 29A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a vertical mirror of the third special orientation. 
         FIG. 29B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 29A  have been eliminated. 
         FIG. 30A  is a diagram illustrating various translations and positive rotations of a rotation sensor in a horizontal and vertical mirror of the third special orientation. 
         FIG. 30B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 30A  have been eliminated. 
         FIG. 31A  is a diagram illustrating various translations and negative rotations of a rotation sensor in a horizontal and vertical mirror of the third special orientation. 
         FIG. 31B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 31A  have been eliminated. 
         FIG. 32  is a diagram of a sensor array divided into four sub arrays, which are suitable for implementing various embodiments of the present invention. 
         FIG. 33  is a block diagram of a rotation sensor according to one embodiment of the present invention. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In the following detailed description of the preferred embodiments, reference is made to the accompanying drawings, which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following detailed description, therefore, is not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims. 
     I. Motion Sensing With a Single Optical Motion Sensor 
       FIG. 1  is a top view of an optical mouse  10 , which is suitable for incorporating one embodiment of the present invention. Mouse  10  includes plastic case  12 , left mouse button (LB)  14 A, right mouse button (RB)  14 B, and optical motion sensor chip  16 . Sensor chip  16  is covered by plastic case  12 , and is therefore shown with dashed lines in FIG.  1 . 
       FIG. 2  is an electrical block diagram illustrating major components of optical mouse  10 . Optical mouse  10  includes light source  2 , lenses  4  and  8 , and optical motion sensor  16 . Optical motion sensor  16  includes photo detector array  148 , electronic shutter  150 , a plurality of sense capacitors  154 A- 154 C (collectively referred to as sense capacitors  154 ), multiplexer  156 , amplifier  157 , analog to digital (A/D) converter  158 , correlator  160 , system controller  162 , shutter controller  164 , and light controller  166 . 
     The operation of optical motion sensor  16  is primarily controlled by system controller  162 , which is coupled to multiplexer  156 , A/D converter  158 , correlator  160 , shutter controller  164 , and light controller  166 . In operation, according to one embodiment, light source  2  emits light that is projected by lens  4  onto surface  6 , which is a desktop or other suitable imaging surface. Light source  2  is controlled by signals from light controller  166 . Reflected light from surface  6  is directed by lens  8  onto photo detector array  148 . Each photo detector in photo detector array  148  provides a current that varies in magnitude based upon the intensity of light incident on the photo detector. 
     Electronic shutter  150  is controlled by a shutter signal from shutter controller  164 . When electronic shutter  150  is “open,” charge accumulates on sense capacitors  154 , creating a voltage that is related to the intensity of light incident on the photo detectors in array  148 . When electronic shutter  150  is “closed,” no further charge accumulates or is lost from sense capacitors  154 . Multiplexer  156  connects each sense capacitor  154  in turn to amplifier  157  and A/D converter  158 , to amplify and convert the voltage from each sense capacitor  154  to a digital value. Sense capacitors  154  are then discharged through electronic shutter  150  so that the charging process can be repeated. 
     Based on the level of voltage from sense capacitors  154 , A/D converter  158  generates a digital value of a suitable resolution (e.g., one to eight bits) indicative of the level of voltage. The digital values for the photo detector array  148  represent a digital image or digital representation of the portion of the desktop or other imaging surface under optical mouse  10 . The digital values are stored as a frame into corresponding locations within an array of memory within correlator  160 . 
     The overall size of photo detector array  148  is preferably large enough to receive an image having several features. Images of such spatial features produce translated patterns of pixel information as optical mouse  10  moves over a surface. The number of photo detectors in array  148  and the frame rate at which their contents are captured and digitized cooperate to influence how fast optical mouse  10  can be moved across a surface and still be tracked. Tracking is accomplished by correlator  160  by comparing a newly captured sample frame with a previously captured reference frame to ascertain the direction and amount of movement. In one form of the invention, motion tracking is accomplished using techniques disclosed in the related patents identified above in the Reference to Related Patents section. 
     In one embodiment, the entire content of one of the frames is shifted by correlator  160  by a distance of one pixel successively in each of the eight-directions allowed by a one pixel offset trial shift (one over, one over and one down, one down, one up, one up and one over, one over in the other direction, etc.). That adds up to eight trials. Also, since there might not have been any motion, a ninth trial “null shift” is also used. After each trial shift, those portions of the frames that overlap each other are subtracted by correlator  160  on a pixel by pixel basis, and the resulting differences are preferably squared and then summed to form a measure of similarity (correlation) within that region of overlap. Larger trial shifts are possible, of course (e.g., two over and one down), but at some point the attendant complexity ruins the advantage, and it is preferable to simply have a sufficiently high frame rate with small trial shifts. The trial shift with the least difference (greatest correlation) can be taken as an indication of the motion between the two frames. That is, it provides raw movement information that may be scaled and or accumulated to provide movement information (ΔX and ΔY) of a convenient granularity and at a suitable rate of information exchange. 
     In addition to providing digital images to correlator  160 , A/D converter  158  also outputs digital image data to shutter controller  164 . Shutter controller  164  helps to ensure that successive images have a similar exposure, and helps to prevent the digital values from becoming saturated to one value. Controller  164  checks the values of digital image data and determines whether there are too many minimum values or too many maximum values. If there are too many minimum values, controller  164  increases the charge accumulation time of electronic shutter  150 . If there are too many maximum values, controller  164  decreases the charge accumulation time of electronic shutter  150 . 
     II. Rotation Sensor Overview 
     As described above, optical mouse  10  uses a single optical motion sensor  16  for generating ΔX and ΔY movement data. One embodiment of the present invention generates rotation data based on ΔX and ΔY data generated by two optical motion sensors  16  (also referred to as optical motion sensors A and B, which are shown in FIG.  3 ). The two sensors A and B are collectively referred to as a rotation sensor. 
     The two sensors A and B are positioned at a known distance apart, and in a known orientation. There are a variety of possible orientations of the two sensors, as described below with reference to FIG.  3 . The output from each sensor A and B is a Δx and a Δy count since the last position report. In one embodiment, sensors A and B measure position once a frame, which can occur at any defined interval. For current navigation sensors, the frame rate is typically 1500 to 2000 frames per second, either with the sensor reporting position once each frame, or via an I/O port, 100 to 200 times per second. In one embodiment, sensors A and B are operated at the same frame rate, and “re-reference” at the same time. The term “re-reference” refers to the storing of a new reference frame, which can occur when the sensor has moved from the original reference frame and the overlap between the current frame and the reference frame is decreasing. Re-referencing can also occur after times of no movement, so that the current reference frame is recent. 
     III. Rotation Sensor Orientation and Alignment 
       FIG. 3  is a diagram of a general rotation sensor orientation, three special rotation sensor orientations, and mirror rotation sensor orientations. The orientations shown in  FIG. 3  are divided into four columns and four rows. The first row shows a general orientation (first column), a first special orientation (second column), a second special orientation (third column), and a third special orientation (fourth column). The second, third, and fourth rows show a horizontal mirror, a vertical mirror, and a combined horizontal and vertical mirror orientation, respectively, for each of the four orientations shown in the first row. 
     In each orientation, the two sensors (A and B) are placed a known distance, d, apart. In one embodiment, sensor A is at the origin coordinates (0,0), and sensor B is at different coordinates, which depend on the orientation and alignment of, and distance between, the two sensors. The following Table I provides coordinates for sensor B for the various orientations illustrated in  FIG. 3 , assuming a distance of “d” between sensors A and B. 
     
       
         
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE I 
               
             
             
               
                   
                   
               
               
                   
                 Orientation 
               
             
          
           
               
                 Alignment 
                 General 
                 #1 
                 #2 
                 #3 
               
               
                   
               
               
                 Normal 
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               - 
                               d 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               N 
                             
                           
                           , 
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               N 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 (0, d) 
                 (d, 0) 
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 - 
                                 d 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               45 
                             
                             , 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               45 
                             
                           
                           ) 
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 - 
                                 d 
                               
                               
                                 2 
                               
                             
                             , 
                             
                               d 
                               
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 Horizontal Mirror 
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               - 
                               d 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               H 
                             
                           
                           , 
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               H 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                   
                 (0, d) 
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 - 
                                 d 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               315 
                             
                             , 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               315 
                             
                           
                           ) 
                         
                         = 
                         
                           ( 
                           
                             
                               d 
                               
                                 2 
                               
                             
                             , 
                             
                               d 
                               
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 Vertical Mirror 
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               - 
                               d 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               V 
                             
                           
                           , 
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               V 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 (0, −d) 
                   
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 - 
                                 d 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               135 
                             
                             , 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               135 
                             
                           
                           ) 
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 - 
                                 d 
                               
                               
                                 2 
                               
                             
                             , 
                             
                               
                                 - 
                                 d 
                               
                               
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 Horizontal and Vertical Mirror 
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               - 
                               d 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               HV 
                             
                           
                           , 
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               HV 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                   
                   
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 - 
                                 d 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               225 
                             
                             , 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               225 
                             
                           
                           ) 
                         
                         = 
                         
                           ( 
                           
                             
                               d 
                               
                                 2 
                               
                             
                             , 
                             
                               d 
                               
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
     As shown in  FIG. 3 , the alignment angle for the general orientation in the normal alignment is θ N . For the general orientation,  FIG. 3  also shows the horizontal mirror alignment angle (θ H ), vertical mirror alignment angle (θ V ), and horizontal and vertical mirror alignment angle (θ HV ). These alignment angles are related to the normal alignment angle (θ N ) by the relationships shown in the following Table II: 
                                     TABLE II                       Alignment   Angle   Equation                           Horizontal Mirror   θ H     θ H  = 360 − θ N             Vertical Mirror   θ V     θ V  = 180 − θ N             Horizontal and Vertical   θ HV     θ HV  = 180 + θ N              Mirror                        
IV. Motion of Rotation Sensor
 
     As shown in  FIGS. 4A-4E , motion of sensors A and B can include translation, rotation, or a combination of the two. It is assumed in the following description that rotation is around sensor A. Rotation about sensor B can be achieved via mirroring. The initial position of the two sensors is designated by the letters A and B, and the position of the two sensors after being moved is designated by A′ and B′. 
       FIG. 4A  is a diagram illustrating translation of sensors A and B in the positive X direction.  FIG. 4B  is a diagram illustrating translation of sensors A and B in the positive X and negative Y directions.  FIG. 4C  is a diagram illustrating translation of sensors A and B in the negative Y direction.  FIG. 4D  is a diagram illustrating rotation of sensors A and B about the center of sensor A.  FIG. 4E  is a diagram illustrating three different combinations of rotation and translation of sensors A and B. 
       FIG. 5  is a diagram illustrating a positive angle of rotation on an X-Y axis. As shown in  FIG. 5 , the angle of rotation (α) is defined to be positive for counterclockwise rotations. The angle of rotation for clockwise rotations is negative. In an alternative embodiment, the angle of rotation could be defined as negative for counterclockwise rotations, and positive for clockwise rotations. 
       FIGS. 6A-13A  and  16 A- 31 A are diagrams illustrating various translations and rotations of sensors A and B in the various orientations and alignments shown in FIG.  3 . Each of these Figures includes three columns and three rows of motion diagrams. The first column illustrates movements that include translations in the negative X direction. The second column illustrates movements with no translation in the X direction. The third column illustrates movements that include translations in the positive X direction. The first row illustrates movements that include translations in the positive Y direction. The second row illustrates movements with no translation in the Y direction. The third row illustrates movements that include translations in the negative Y direction. Each of these Figures is described in further detail below. 
     A. General Orientation 
       FIG. 6A  is a diagram illustrating various translations and positive rotations of sensors A and B in the general orientation. In the general orientation, sensors A and B are orientated so that the “x” and “y” motion reports from both sensors are in the same direction. Sensor B is located at a random angle from sensor A, but is at a known distance, d, from sensor A. When the normal alignment is mirrored, the orientations of sensor A and B are changed so that they are in the same direction as the normal alignment. In one embodiment, the X and Y motion reports are always reported in the same direction. 
       FIG. 6B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 6A  have been eliminated. The translations shown in  FIG. 6A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 6B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 7A  is a diagram illustrating various translations and negative rotations of sensors A and B in the general orientation.  FIG. 7B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 7A  have been eliminated. The translations shown in  FIG. 7A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 7B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 8A  is a diagram illustrating various translations and positive rotations of sensors A and B in a horizontal mirror of the general orientation.  FIG. 8B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 8A  have been eliminated. The translations shown in  FIG. 8A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 8B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 9A  is a diagram illustrating various translations and negative rotations of sensors A and B in a horizontal mirror of the general orientation.  FIG. 9B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 9A  have been eliminated. The translations shown in  FIG. 9A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 9B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 10A  is a diagram illustrating various translations and positive rotations of sensors A and B in a vertical mirror of the general orientation.  FIG. 10B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 10A  have been eliminated. The translations shown in  FIG. 10A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 10B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 11A  is a diagram illustrating various translations and negative rotations of sensors A and B in a vertical mirror of the general orientation.  FIG. 11B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 11A  have been eliminated. The translations shown in  FIG. 11A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 11B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 12A  is a diagram illustrating various translations and positive rotations of sensors A and B in a horizontal and vertical mirror of the general orientation.  FIG. 12B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 12A  have been eliminated. The translations shown in  FIG. 12A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 12B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 13A  is a diagram illustrating various translations and positive rotations of sensors A and B in a horizontal and vertical mirror of the general orientation.  FIG. 13B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 13A  have been eliminated. The translations shown in  FIG. 13A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 13B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
     1. Determining the Angle of Rotation 
     For the general orientation, after the translation of sensor A is subtracted out, the resulting figure is an isosceles triangle.  FIG. 14  is a diagram of an isosceles triangle representing rotation of sensors A and B in the general orientation after translation has been eliminated. As shown in  FIG. 14 , two of the sides are length “d”, the third side has a length of [(BΔx−AΔx) 2 +(BΔy−AΔy) 2 ] (1/2) , and the angle of rotation is represented by α. 
     A formula for a general triangle that relates the sides to the angle α is provided in the following Equation I: 
               cos   ⁢           ⁢   α     =       (       b   2     +     c   2     -     a   2       )       2   ⁢   b   ⁢           ⁢   c               Equation  I             
 
     In this case:
         b=d;   c=d; and   a=√{square root over ((BΔx−AΔx) 2 +(BΔy−AΔy) 2 )}{square root over ((BΔx−AΔx) 2 +(BΔy−AΔy) 2 )}       

     Inserting these values for a, b, and c into Equation I, and solving for the angle of rotation, α, results in the following Equation II: 
             α   =       cos     -   1       ⁢           ⁢         2   ⁢     d   2       -       (       B   ⁢           ⁢   Δ   ⁢           ⁢   x     -     A   ⁢           ⁢   Δ   ⁢           ⁢   x       )     2     -       (       B   ⁢           ⁢   Δ   ⁢           ⁢   y     -     A   ⁢           ⁢   Δ   ⁢           ⁢   y       )     2         2   ⁢     d   2                   Equation  II             
 
     Equation II always results in a positive value for the angle of rotation. To identify the direction of rotation, the sign of α is determined as described below. 
     2. Determining the Sign of α 
     To determine the sign of α, the general orientation undergoes a coordinate transformation to rotate the initial position back to the Y-axis, which allows an easy determination of the sign of the rotation.  FIG. 15A  is a diagram illustrating rotation of sensors A and B in the general orientation prior to a coordinate transformation. The initial position of sensor B is (−d sin θ, d cos θ), and the position of sensor B after sensor rotation is (−d sin θ+Δx, d cos θ+Δy), where Δx is (BΔx−AΔx) and Δy is (BΔy−AΔy). 
       FIG. 15B  is a diagram illustrating rotation of sensors A and B in the general orientation after a −θ coordinate transformation. The formula for the coordinate transformation is given in the following Equation III: 
               (       x   ′     ,     y   ′       )     =       [           cos   ⁢           ⁢   α             -   sin     ⁢           ⁢   α               sin   ⁢           ⁢   α           cos   ⁢           ⁢   α           ]     ⁢     (     x   ,   y     )               Equation  III                       where:
           (x, y) is the position of sensor B after rotation and before the coordinate transformation; and   (x′, y′) is the position of sensor B after rotation and after the coordinate transformation.   
               
     After performing the matrix multiplication in Equation III, the following Equations IV and V are obtained:
 
 x′=x  cos α− y  sin α  Equation IV
 
 y′=x  sin α+ y  cos α  Equation V
 
     Substituting α=−θ into Equations IV and V results in the following Equations VI and VII:
 
 x′=x  cos(−θ)− y  sin(−θ)  Equation VI
 
 y′=x  sin(−θ)+ y  cos(−θ)  Equation VII
 
     Substituting cos(−θ)=cos θ and sin(−θ)=−sin θ into Equations VI and VII results in the following Equations VIII and IX:
 
 x′=x  cos θ+ y  sin θ  Equation VIII
 
 y′=−x  sin θ+ y  cos θ  Equation IX
 
     Substituting x=(−d sin θ+Δx) and y=(d cos θ+Δy) into Equations VIII and IX results in the following Equations X and XI:
 
 x ′=(− d  sin θ+Δ x ) cos θ+( d  cos θ+Δ y )sin θ  Equation X
 
 y ′=−(− d  sin θ+Δ x )sin θ+( d  cos θ+Δ y )cos θ  Equation XI
 
     Rearranging terms in Equations X and XI results in the following Equations XII and XIII:
 
 x′=Δx  cos θ− d  sin θ cos θ+ d  sin θ cos θ+Δ y  sin θ  Equation XII
 
 y′=d  sin 2    θ−Δx  sin θ+ d  cos 2    θ+Δy  cos θ  Equation XIII
 
     Combining terms in Equations XII and XIII results in the following Equations XIV and XV:
 
 x′=Δx  cos θ+Δ y  sin θ  Equation XIV
 
 y′=d (sin 2  θ+cos 2  θ)−Δ x  sin θ+Δ y  cos θ  Equation XV
 
     Applying the Pythagorean identity, sin 2  θ+cos 2  θ=1, to Equation XV, results in the following Equation XVI:
 
 y′=d−Δx  sin θ+Δ y  cos θ  Equation XVI
 
     After solving Equation XIV for x′ using the appropriate θ from Table II above, the sign of α is determined, which is the inverse of the sign of x′. In an alternative embodiment, y′ from Equation XVI could be used to determine the sign of α. 
     B. First Special Orientation 
       FIG. 16A  is a diagram illustrating various translations and positive rotations of sensors A and B in the first special orientation. In the first special orientation, sensors A and B are orientated so that the “x” and “y” motion reports from both sensors are in the same direction. Sensor B is located above sensor A at a Y distance of “d”. Since horizontal mirroring has no effect on the first special orientation, only the vertical mirror will be discussed. The first special orientation in the normal alignment is the same as the general orientation in the normal alignment with an alignment angle, θ N , equal to zero degrees. 
       FIG. 16B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 16A  have been eliminated. The translations shown in  FIG. 16A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 16B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 17A  is a diagram illustrating various translations and negative rotations of sensors A and B in the first special orientation.  FIG. 17B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 17A  have been eliminated. The translations shown in  FIG. 17A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 17B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 18A  is a diagram illustrating various translations and positive rotations of sensors A and B in a vertical mirror of the first special orientation. The first special orientation in the vertical mirror alignment is the same as the general orientation in the normal alignment with an alignment angle, θ N , equal to 180 degrees.  FIG. 18B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 18A  have been eliminated. The translations shown in  FIG. 18A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 18B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 19A  is a diagram illustrating various translations and negative rotations of sensors A and B in a vertical mirror of the first special orientation.  FIG. 19B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 19A  have been eliminated. The translations shown in  FIG. 19A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 19B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
     1. Determining the Angle of Rotation 
     Since the starting axis between sensor A and sensor B is vertical in the first special orientation, the general formula given in Equation II for the angle of rotation, α, can be simplified. The distance (BΔx−AΔx) is perpendicular to the axis between sensors A and B. Since the distance between A and B is known to be d, the angle α is determined from the following Equation XVII: 
             α   =       sin     -   1       ⁡     (         B   ⁢           ⁢   Δ   ⁢           ⁢   x     -     A   ⁢           ⁢   Δ   ⁢           ⁢   x       d     )               Equation  XVII             
 
     The general formula given in Equation II can also be used to determine the angle of rotation for the first special orientation. 
     2. Determining the Sign of α 
     The appropriate sign of a can be determined for the first special orientation in either the normal or vertical mirror alignment from the following Table III: 
     
       
         
               
               
               
             
           
               
                   
                 TABLE III 
               
               
                   
                   
               
               
                   
                 Alignment 
                 Sign of α 
               
               
                   
                   
               
             
             
               
                   
                 Normal 
                 inverse sign of (BΔx − AΔx) 
               
               
                   
                 Vertical Mirror 
                 sign of (BΔx − AΔx) 
               
               
                   
                   
               
             
          
         
       
     
     C. Second Special Orientation 
       FIG. 20A  is a diagram illustrating various translations and positive rotations of sensors A and B in the second special orientation. In the second special orientation, sensors A and B are orientated so that the “x” and “y” motion reports from both sensors are in the same direction. Sensor B is located to the right of sensor A at an X distance of “d”. Since vertical mirroring has no effect on the second special orientation, only the horizontal mirror will be discussed. The second special orientation in the normal alignment is the same as the general orientation in the normal alignment with an alignment angle, θ N , equal to 270 degrees. 
       FIG. 20B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 20A  have been eliminated. The translations shown in  FIG. 20A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 20B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 21A  is a diagram illustrating various translations and negative rotations of sensors A and B in the second special orientation.  FIG. 21B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 21A  have been eliminated. The translations shown in  FIG. 21A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 21B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 22A  is a diagram illustrating various translations and positive rotations of sensors A and B in a horizontal mirror of the second special orientation. The second special orientation in the horizontal mirror alignment is the same as the general orientation in the normal alignment with an alignment angle, θ N , equal to 90 degrees.  FIG. 22B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 22A  have been eliminated. The translations shown in  FIG. 22A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 22B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 23A  is a diagram illustrating various translations and negative rotations of sensors A and B in a horizontal mirror of the second special orientation.  FIG. 23B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 23A  have been eliminated. The translations shown in  FIG. 23A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 23B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
     1. Determining the Angle of Rotation 
     Since the starting axis between sensor A and sensor B is horizontal in the second special orientation, the general formula given in Equation II for the angle of rotation can be simplified. The distance (BΔy−AΔy) is perpendicular to the axis between sensors A and B. Since the distance between A and B is known to be d, the angle α can be determined from the following Equation XVIII: 
             α   =       sin     -   1       ⁡     (         B   ⁢           ⁢   Δ   ⁢           ⁢   y     -     A   ⁢           ⁢   Δ   ⁢           ⁢   y       d     )               Equation  XVIII             
 
     The general formula given in Equation II can also be used to determine the angle of rotation for the second special orientation. 
     2. Determining the Sign of α 
     The appropriate sign of α can be determined for the second special orientation in either the normal or horizontal mirror alignment from the following Table IV: 
     
       
         
               
               
               
             
           
               
                   
                 TABLE IV 
               
               
                   
                   
               
               
                   
                 Alignment 
                 Sign of α 
               
               
                   
                   
               
             
             
               
                   
                 Normal 
                 sign of (BΔy − AΔy) 
               
               
                   
                 Horizontal Mirror 
                 inverse sign of (BΔy − AΔy) 
               
               
                   
                   
               
             
          
         
       
     
     D. Third Special Orientation 
       FIG. 24A  is a diagram illustrating various translations and positive rotations of sensors A and B in the third special orientation. In the third special orientation, sensors A and B are orientated so that the “x” and “y” motion reports from both sensors are in the same direction. Sensor B is located to the left of sensor A at an X distance of −d/(2) (1/2) , and up from sensor A at a Y distance of d/(2) (1/2) . The third special orientation in the normal alignment is the same as the general orientation in the normal alignment with an alignment angle, θ N , equal to 45 degrees. 
       FIG. 24B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 24A  have been eliminated. The translations shown in  FIG. 24A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 24B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 25A  is a diagram illustrating various translations and negative rotations of sensors A and B in the third special orientation.  FIG. 25B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 25A  have been eliminated. The translations shown in  FIG. 25A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 25B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 26A  is a diagram illustrating various translations and positive rotations of sensors A and B in a horizontal mirror of the third special orientation. The third special orientation in the horizontal mirror alignment is the same as the general orientation in the normal alignment with an alignment angle, θ N , equal to 315 degrees. 
       FIG. 26B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 26A  have been eliminated. The translations shown in  FIG. 26A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 26B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 27A  is a diagram illustrating various translations and negative rotations of sensors A and B in a horizontal mirror of the third special orientation.  FIG. 27B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 27A  have been eliminated. The translations shown in  FIG. 27A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 27B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 28A  is a diagram illustrating various translations and positive rotations of sensors A and B in a vertical mirror of the third special orientation. The third special orientation in the vertical mirror alignment is the same as the general orientation in the normal alignment with an alignment angle, θ N , equal to 135 degrees. 
       FIG. 28B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 28A  have been eliminated. The translations shown in  FIG. 28A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 28B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 29A  is a diagram illustrating various translations and negative rotations of sensors A and B in a vertical mirror of the third special orientation.  FIG. 29B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 29A  have been eliminated. The translations shown in  FIG. 29A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 29B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 30A  is a diagram illustrating various translations and positive rotations of sensors A and B in a horizontal and vertical mirror of the third special orientation. The third special orientation in the horizontal and vertical mirror alignment is the same as the general orientation in the normal alignment with an alignment angle, θ N , equal to 225 degrees. 
       FIG. 30B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 30A  have been eliminated. The translations shown in  FIG. 30A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 30B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
       FIG. 31A  is a diagram illustrating various translations and negative rotations of sensors A and B in a horizontal and vertical mirror of the third special orientation.  FIG. 31B  is a diagram illustrating rotation of the rotation sensor after the translations shown in  FIG. 31A  have been eliminated. The translations shown in  FIG. 31A  can be eliminated by subtracting the delta motion of sensor A (i.e., AΔx, AΔy) from the delta motion of sensor B (i.e., BΔx, BΔy).  FIG. 31B  also shows the distance, d, between sensors A and B, along with the angle of rotation, α. 
     1. Determining the Angle of Rotation 
     The general formula given in Equation II can be used to determine the angle of rotation for the third special orientation. 
     2. Determining the Sign of α 
     The appropriate sign of α, can be determined for the third special orientation in the normal, horizontal mirror, vertical mirror, and horizontal and vertical mirror alignments from the following Table V: 
                                 TABLE V                       Alignment   Sign of α                           Normal   inverse sign of (BΔx − AΔx)           Horizontal Mirror   inverse sign of (BΔx − AΔx)           Vertical Mirror   sign of (BΔx − AΔx)           Horizontal and Vertical   sign of (BΔx − AΔx)           Mirror                        
V. Center of Rotation
 
     The center of rotation of sensors A and B can be determined using the angle of rotation, and the beginning and final positions of sensors A and B. For a beginning position, (x, y), that is rotated by an angle a to a new position (x′, y′) around an arbitrary center of rotation point (x 0 , y 0 ), the following Equation XIX provides a relationship for the rotation and translation of a set of Cartesian coordinates: 
               (       x   ′     ,     y   ′       )     =       (       x   0     ,     y   0       )     +       [           cos   ⁢           ⁢   α             -   sin     ⁢           ⁢   α               sin   ⁢           ⁢   α           cos   ⁢           ⁢   α           ]     ⁢     (       x   -     x   0       ,     y   -     y   0         )                 Equation  XIX             
 
     Expanding the terms in Equation X results in the following Equations XX and XXI for x′ and y′, respectively, which is the final position:
 
 x′=x   0   +x  cos α− x   0  cos α− y  sin α+ y   0  sin α  Equation XX
 
 y′=y   0   +x  sin α− x   0  sin α+ y  cos α− y   0  cos α  Equation XXI
 
     Since the beginning point,(x, y), the final point, (x′, y′), and the angle α are known, the rotation point (x 0 , y 0 ) can be determined by rearranging Equations XX and XXI to arrive at the following Equations XXII and XXIII: 
               x   0     =       y   -     y   ′     +       x   ′     ⁢   sin   ⁢           ⁢   α     +     x   ⁢           ⁢   sin   ⁢           ⁢   α     +     y   ⁢           ⁢   cos   ⁢           ⁢   α     -       y   ′     ⁢   cos   ⁢           ⁢   α         2   ⁢           ⁢   sin   ⁢           ⁢   α               Equation  XXII             
               y   0     =         x   ′     -   x   -     x   ⁢           ⁢   cos   ⁢           ⁢   α     +       x   ′     ⁢           ⁢   cos   ⁢           ⁢   α     +     y   ⁢           ⁢   sin   ⁢           ⁢   α     +       y   ′     ⁢   sin   ⁢           ⁢   α         2   ⁢           ⁢   sin   ⁢           ⁢   α               Equation  XXIII             
 
     Since the original position, (x, y), of sensor A is the origin, (0,0), Equations XXII and XXIII can be simplified to the following Equations XXIV and XXV, respectively: 
               x   0     =         -     y   ′       +       x   ′     ⁢   sin   ⁢           ⁢   α     +       -     y   ′       ⁢   cos   ⁢           ⁢   α         2   ⁢           ⁢   sin   ⁢           ⁢   α               Equation  XXIV                 y   0     =         x   ′     +       x   ′     ⁢   cos   ⁢           ⁢   α     +       y   ′     ⁢   sin   ⁢           ⁢   α         2   ⁢           ⁢   sin   ⁢           ⁢   α               Equation  XXV             
 
VI. Rotation Sensor Implementations
 
     The two navigation sensors A and B can be implemented as two separate sensors, oriented in the same direction, separated by a distance that is equal to or greater than the sensor package size. Increasing the distance between sensors A and B will result in a larger system sensor, but greater sensitivity to slower rotations. The first and second special orientations are the easiest to implement. 
     The two sensors A and B can also be integrated into one sensor die, with sensor A and sensor B being subsets of the entire sensor. For example,  FIG. 32  is a diagram of a sensor array  200  divided into four sub arrays, which are suitable for implementing various embodiments of the present invention. As shown in  FIG. 32 , sensor array  200  has a length and width of magnitude “m,” and is divided into four sub arrays, numbered 1, 2, 3 and 4. The distance, d, between the centers of the sub arrays is m/2 for the 1-2, 1-3, 2-4 and 3-4 sub array combinations. The distance, d, is 1.414*(m/2) for the 1-4 and 3-2 combinations. Thus, the third special orientation, which would use either sub arrays 1 and 4, or sub arrays 2 and 3, provides the greatest separation between sensors in this embodiment. 
     Table VI below shows all of the possible two sub array combinations of sensor array  200 , along with the corresponding special orientations and alignments of sensors A and B. 
     
       
         
               
               
               
               
             
           
               
                 TABLE VI 
               
               
                   
               
               
                 Sensor “A” 
                 Sensor “B” 
                 Orientation 
                 Alignment 
               
               
                   
               
             
             
               
                 1 
                 2 
                 #2 
                 Normal 
               
               
                 2 
                 1 
                 #2 
                 Horizontal mirror 
               
               
                 1 
                 3 
                 #1 
                 Vertical mirror 
               
               
                 3 
                 1 
                 #1 
                 Normal 
               
               
                 1 
                 4 
                 #3 
                 Horizontal and Vertical mirror 
               
               
                 4 
                 1 
                 #3 
                 Normal 
               
               
                 2 
                 3 
                 #3 
                 Vertical mirror 
               
               
                 3 
                 2 
                 #3 
                 Horizontal mirror 
               
               
                 2 
                 4 
                 #1 
                 Vertical mirror 
               
               
                 4 
                 2 
                 #1 
                 Normal 
               
               
                 3 
                 4 
                 #2 
                 Normal 
               
               
                 4 
                 3 
                 #2 
                 Horizontal mirror 
               
               
                   
               
             
          
         
       
     
       FIG. 33  is a block diagram of a rotation sensor  300  according to one embodiment of the present invention. Rotation sensor  300  includes two optical motion sensors  16  (described above), and a rotation data generator  302 . The two optical motion sensors  16  correspond to sensors A and B, and may be positioned in any of the various orientations and alignments shown in FIG.  3 . 
     The two optical motion sensors  16  each output Δx and Δy data to rotation data generator  302 . Based on the Δx and Δy data received from the two sensors  16 , and on stored information regarding the particular orientation, alignment and separation of the two sensors  16 , rotation data generator  302  calculates rotation data and center of rotation data as described above, and outputs this data. In one embodiment, the rotation data represents rotation about the sensor  16  corresponding to sensor A (with positive rotation being defined as counterclockwise), and the center of rotation data represents the (x, y) coordinates of the center of rotation of the two sensors  16  relative to the origin, wherein the origin is the original location of the sensor  16  corresponding to sensor A. 
     In one form of the invention, the frame of reference should not rotate more than about ten degrees between frames for good correlation to determine Δx and Δy movement. In one embodiment, the equations shown above are valid for a range of ±90 degrees or 0 to 180 degrees, but the maximum rotation between frames should be less than 10 degrees. 
     Current optical navigation sensors  16  typically operate in the range of 1500 to 2000 frames per second. The actual report rate to a computer or other device is typically between 100 to 200 reports per second. Due to the difference between the measurement rate and the report rate, the Δx and Δy information is accumulated and then output. 
     The maximum rotation speed of rotation sensor  300  is determined by the angle that the sensor  300  can be rotated before the correlation between frames degrades to the point where good (X, Y) navigation begins to fail. Assuming that a rotation of 10 degrees occurs between frames, with a frame rate of 1500 frames per second, the maximum rotation is 41.6 revolutions per second, or 2500 rpm. 
     The minimum rotation speed is the speed that results in a minimum Δx and Δy count between the sensors  16 . This is dependent upon the frame rate and the distance between sensors  16 . If the distance between sensors  16  is increased, the minimum rotation speed that can be seen is lowered. Since the navigation sensors  16  typically only report data between 100 to 200 times per second, Δx and Δy counts should be accumulated over a number of frames. 
     Due to the fact that, in one embodiment, the rotation data is determined from the Δx and the Δy values, and the fact that the Δx and Δy values typically have noise in them, the rotation data is filtered (i.e., averaged, smoothed, weighed filters) in one form of the invention to dampen the changes in the rotation data and provide improved noise performance. 
     It will be understood by a person of ordinary skill in the art that functions performed by rotation sensor  300  may be implemented in hardware, software, firmware, or any combination thereof. The implementation may be via a microprocessor, programmable logic device, or state machine. Components of the present invention may reside in software on one or more computer-readable mediums. The term computer-readable medium as used herein is defined to include any kind of memory, volatile or non-volatile, such as floppy disks, hard disks, CD-ROMs, flash memory, read-only memory (ROM), and random access memory. 
     The two sensors  16  and rotation data generator  302  can be implemented as a single integrated circuit package or as separate packages. In alternative embodiments, rotation data generator  302  may be incorporated in an external device, such as a computer or other electronic device. 
     Although an optical motion sensor  16  has been discussed above in the context of an optical mouse, it will be understood that embodiments of the present invention are not limited to an optical mouse, and that the techniques described herein are also applicable to other devices where rotation sensing is desired, such as in game controllers, gestural controllers, personal digital assistant (PDA) devices, cellular telephones, or other devices. 
     Although specific embodiments have been illustrated and described herein for purposes of description of the preferred embodiment, it will be appreciated by those of ordinary skill in the art that a wide variety of alternate and/or equivalent implementations may be substituted for the specific embodiments shown and described without departing from the scope of the present invention. Those with skill in the chemical, mechanical, electro-mechanical, electrical, and computer arts will readily appreciate that the present invention may be implemented in a very wide variety of embodiments. This application is intended to cover any adaptations or variations of the preferred embodiments discussed herein. Therefore, it is manifestly intended that this invention be limited only by the claims and the equivalents thereof.