Patent Publication Number: US-8528220-B2

Title: Six-direction indicator

Description:
TECHNICAL FIELD 
     The present invention relates to a six-direction indicator, and more specifically, to a six-direction indicator that can be used in a north finder that determines true north using a single-axis gyro. 
     BACKGROUND ART 
     Since magnetic north is slightly shifted from true north, true north cannot be measured using a magnetic compass. However, administrative maps are produced based on true north, and the Construction Standards Act is also based on true north. Therefore, in the field of civil engineering and construction, true north must be determined correctly. In underground tunnel construction, in particular, a magnetic compass does not function correctly because of the effect of a mineral vein or the like. 
     There is a known conventional gyro compass, as a true north detector that detects true north correctly, which detects the earth&#39;s rotational angular velocity to determine true north. As disclosed by JP 2002-296037 A, U.S. Pat. Nos. 6,594,911 and 6,918,186, gyro compasses are generally orthogonal three-axis type devices. These gyro compasses are large in size and costly to manufacture. 
     A number of one-axis or two-axis type devices have been suggested in order to reduce the size and cost. An azimuth indicator disclosed by JP 6-3149 A is a single-axis type device that has an angular velocity sensor (gyro sensor) rotated on a horizontal base at a fixed angular velocity. An azimuth indicator disclosed by JP 6-11350 A is a single-axis type device and has a gyro sensor rotated on a horizontal base to obtain provisional true north. True north is obtained by obtaining a bias based on a measurement value of an angular velocity in the provisional true north and separately input latitude. A true bearing measuring device disclosed by JP 11-160072 A is a two-axis type that does not need a horizontal plane and has a gyro sensor rotated for every 90° or 120° on two rotation bases to remove a bias. An azimuth indicator disclosed by JP 11-190633 A is a single-axis type device that does not need a horizontal plane and has a gyro sensor and an acceleration sensor indicate three directions on a rotating base. A latitude however must be input in this device. An azimuth detector disclosed by JP 2001-215121 A is a single-axis type device that has a gyro sensor rotated on a horizontal plane to obtain true north based on a sinusoidal gyro output. 
     Most of such gyro compasses have a gyro sensor and an acceleration sensor rotated on a rotation base, and a large space must be secured for its large rotation angle, which limits how compact the device can be. Most single-axis type gyro compasses need a horizontal plane, which make it cumbersome to handle them. Single-axis type gyro compasses that do not need a horizontal plane have been suggested, but their precision about azimuth finding is inferior to those of three-axis type compasses. 
     Therefore, the applicant has suggested a single-axis type azimuth measuring device that does not need a horizontal plane in the disclosure of JP 2008-215956 A. The azimuth measuring device disclosed by the document takes into account a UVW rectangular coordinate system in addition to an XYZ rectangular coordinate system. Six directions, +U, −U, +V, −V, +W, and −W are provided apart at intervals of 60° when they are projected orthogonally on an YZ plane. Elevation angles formed between the U, V, and W axes and the YZ plane are each 35.26°. The azimuth measuring device includes a rotational angular velocity sensor that detects rotational angular velocities ω U , ω V , and ω W  around the respective axes in the UVW rectangular coordinate system, a gravitational acceleration sensor that detects gravitational accelerations g U , g V , and g W  in the respective axial directions, a first stepping motor that rotates the rotational angular velocity sensor and the gravitational acceleration sensor 60×n° (n: a natural number) around the X-axis for positioning, and a second stepping motor that swings the rotational angular velocity sensor and the gravitational acceleration sensor ±35.26° around an axis orthogonal to the X-axis for positioning. The azimuth measuring device measures the rotational angular velocities ω U , ω V , and ω W  and the gravitational accelerations g U , g V  and g W  around the respective axes and coordinate-transforms the obtained rotational angular velocities ω U , ω V , and ω W  and the gravitational accelerations g U , g V , and g W  to produce ω X , ω Y , and ω Z  and the gravitational accelerations g X , g Y , and g Z  in the XYZ rectangular coordinate system. Then, an azimuth angle Ψ is calculated based on the obtained rotational angular velocities ω X , ω Y , and ω Z  and the gravitational accelerations g X , g Y , and g Z . Using the azimuth measuring device, the rotational angular velocity sensor and the gravitational acceleration sensor are swung only ±35.26°, so that the rotation angle is small and a necessary space is not larger than the space required by the conventional azimuth measuring device. Therefore, the size can be reduced as compared to the conventional single-axis azimuth measuring device. 
     DISCLOSURE OF THE INVENTION 
     However, the azimuth measuring device disclosed by JP 2008-215956 A includes two stepping motors as rotational driving sources. If the device can indicate the six directions only using a single rotational driving source, the device can be even more reduced in size. 
     It is an object of the present invention to provide a six-direction indicator that can indicate six directions using a single rotational driving source. 
     A six-direction indicator according to the present invention is provided in an XYZ rectangular coordinate system. The six-direction indicator includes an inclined crank mechanism, a driving source, and a guide member. The inclined crank mechanism includes a shaft that extends in an X-axis direction and a rotation member. The rotation member is rotatable around an inclined axis inclined with respect to said shaft and coupled to said shaft. The driving source rotates said shaft around the X-axis. The guide member has a zigzag circumferential edge provided around the X-axis of said inclined crank mechanism and bent alternately in a mountain-like shape and a valley-like shape at intervals of 60° around the X-axis. The rotation member includes a guide pin. The guide pin is provided upright around the inclined axis of said rotation member and in contact with said circumferential edge, moves around on said circumferential edge by rotation of said shaft and indicates one of positive and negative directions on U, V, and W axes crossing one another at intervals of 60° around the X axis. 
     In the six-direction indicator according to the present invention, the guide pin provided upright at the rotation member is in contact with the circumferential edges of the guide member. When the shaft is rotated, the rotation member rotates around the X axis and around the inclined axis as well. Therefore, the guide pin can move around the X-axis and indicate the ±directions on the U, V, and W directions (six directions in total) crossing one another while swinging in the X-direction depending on an angle formed between the X axis and the inclined axis. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a perspective view showing an outside structure of a six-direction indicator according to a first embodiment of the present invention. 
         FIG. 2  is a perspective view showing an outside structure of a crank mechanism shown in  FIG. 1 . 
         FIG. 3  is a perspective view showing an outside structure of a rotation member in  FIG. 2 . 
         FIG. 4A  shows a YZ plane showing a positional relation between the XYZ coordinate system in which the six-direction indicator shown in  FIG. 1  is provided and the UVW coordinate system used to measure a rotational angular velocity and a gravitational acceleration when viewed in the −X direction. 
         FIG. 4B  shows a ZX plane showing a positional relation between the XYZ coordinate system in which the six-direction indicator shown in  FIG. 1  is provided and the UVW coordinate system used to measure a rotational angular velocity and a gravitational acceleration when viewed in the +Y direction. 
         FIG. 5  is a side view of a guide in  FIG. 1 . 
         FIG. 6  is a side view of the guide on a side opposite to that shown in  FIG. 5 . 
         FIG. 7  is a perspective view of the guide and the rotation member in  FIG. 1 . 
         FIG. 8  is a view showing a relation between the rotation member and a slit shown in  FIG. 7 . 
         FIG. 9  is a view showing a relation between the rotation member and the slit when  FIG. 1  is viewed from the front of the device. 
         FIG. 10  is a side view of the rotation member when viewed from the obliquely upper left in  FIG. 9 . 
         FIG. 11  is a side view of the rotation member when viewed from the obliquely lower left in  FIG. 9 . 
         FIG. 12  is a side view of the rotation member when viewed from immediately below in  FIG. 9 . 
         FIG. 13  is a side view of the rotation member when viewed from the obliquely upper right in  FIG. 9 . 
         FIG. 14  is a side view of the rotation member when viewed from the obliquely upper right in  FIG. 9 . 
         FIG. 15  is a functional block diagram for illustrating a structure of the six-direction indicator shown in  FIG. 1 . 
         FIG. 16  shows a relation between a direction indicated by the six-direction indicator shown in  FIG. 1  and operation time. 
         FIG. 17  is a view showing the state of the rotation member in a first step in operation shown in  FIG. 16 . 
         FIG. 18  is a view showing the state of the rotation member in a second step in the operation shown in  FIG. 16 . 
         FIG. 19  is a view showing the state of the rotation member in a third step in the operation shown in  FIG. 16 . 
         FIG. 20  is a view showing the state of the rotation member in a fourth step in the operation shown in  FIG. 16 . 
         FIG. 21  is a view showing the state of the rotation member in a fifth step in the operation shown in  FIG. 16 . 
         FIG. 22  is a view showing the state of the rotation member in a sixth step in the operation shown in  FIG. 16 . 
         FIG. 23  is a perspective view showing an outside structure of another example of the six-direction indicator shown in  FIG. 1 . 
         FIG. 24  is a perspective view of an outside structure of another example of the six-direction indicator different from those in  FIGS. 1 and 23 . 
         FIG. 25  is a perspective view of an outside structure of another example of the six-direction indicator different from those in  FIGS. 1 ,  23 , and  24 . 
         FIG. 26  is a side view of the six-direction indicator shown in  FIG. 25 . 
         FIG. 27  is a sectional view of the rotation member in the vicinity of a guide pin in  FIG. 25 . 
         FIG. 28  is a graph showing a relation between an elevation angle α and a crossing angle β in XYZ and UVW coordinate systems shown in  FIGS. 4A and 4B . 
         FIG. 29  is a graph showing a relation between an error evaluation function f 1  in the X direction and an error evaluation function f 2  in the Z direction and the elevation angle α. 
         FIG. 30  shows a right triangle where α is 35.26°. 
         FIG. 31  is a graph showing a relation between f 2 /f 1 −1 and the elevation angle α. 
         FIG. 32A  is a view showing the azimuth angle Ψ of the six-direction indicator shown in  FIG. 1 . 
         FIG. 32B  is a view showing the pitch angle θ of the six-direction indicator shown in  FIG. 1 . 
         FIG. 32C  is a view showing the roll angle φ of the six-direction indicator shown in  FIG. 1 . 
         FIG. 33  is a graph showing a relation between an azimuth angle error evaluation function f and the elevation angle α when θ=0°. 
         FIG. 34  is a graph showing a relation between an azimuth angle error evaluation function f and the elevation angle α when 0 is 45°. 
         FIG. 35A  shows an envelope when a conventional azimuth indicator rotates a rotational angular velocity sensor and a gravitational acceleration sensor. 
         FIG. 35B  shows an envelope when the six-direction indicator shown in  FIG. 1  rotates the rotational angular velocity sensor and the gravitational acceleration sensor. 
         FIG. 36  is a graph showing a relation between L/R and the elevation angle α shown in  FIG. 35 . 
     
    
    
     BEST MODE FOR CARRYING OUT THE INVENTION 
     Now, an embodiment of the present invention will be described in detail in conjunction with the accompanying drawings in which the same or corresponding portions are designated by the same reference characters and their description will not be repeated. 
     Structure of Six-Direction Indicator 
     Referring to  FIG. 1 , the six-direction indicator  10  includes a stepping motor  14  as a driving source, an inclined crank mechanism  22 , and a guide member  20 . The inclined crank mechanism  22  includes shafts  221  coupled to a motor shaft  16  of the stepping motor  14  and a rotation member  18  rotatable around an inclined axis  220  inclined with respect to the shaft  221 . The guide member  20  is a spherical outer shell provided circumferentially around the rotation member  18 . In the six-direction indicator  10 , the stepping motor  14  and the guide member  20  are fixed to a base  12 . In this example, the stepping motor is used as a driving source, while an ultrasonic motor may be used instead of the stepping motor and the kind of motor is not specifically limited. 
     The six-direction indicator  10  is provided in an XYZ rectangular coordinate system. The coordinate system is a rectangular coordinate system in which the X, Y, and Z axes are at right angles to one another. More specifically, the X-axis is allocated on the motor shaft  16  of the stepping motor  14  and an XY plane is allocated parallel to the surface of the base  12 . 
     Referring to  FIG. 2 , two shafts  221  of the inclined crank shaft  22  are both provided on the X-axis. The rotation member  18  is provided between the two shafts  221 . An arm member  224  is provided between each of the shafts  221  and the rotation member  18 . The rotation member  18  is coupled to each of the shafts  221  with a corresponding arm member  224 . As shown in  FIG. 1 , the two shafts  221  are supported rotatably at the guide member  20 . 
     Referring to  FIGS. 2 and 3 , the rotation member  18  includes a disk shaped substrate  184  and an inner shell  181 . The inner shell  181  is spherical and covers upper and lower surfaces of the substrate  184 . As shown in  FIG. 3 , two axial members  185  are provided on a side surface of the substrate  184 . The two axial members  185  are provided on the same straight line that passes through a center of the substrate  184 . 
     The arm member  224  includes a crank arm  222  and a boss  223 . The boss  223  is formed at an end of the crank arm  222 . The axial member  185  is fitted into the boss  223 . In this way, the substrate  184  is attached rotatably around the inclined axis  220  that is inclined α° with respect to the shaft  221 . 
     Two guide pins  24 B and  24 D are provided upright in positions on the side surface of the substrate  184  that are orthogonal to the axial members  185 . The guide pins  24 B and  24 D are rod-shaped members that extend in the lengthwise direction. The guide pins  24 B and  24 D are orthogonal to the inclined axis  220 . Two guide pins  24 A and  24 C are provided upright on the surface of the inner shell  181  on a normal to the substrate  184  that passes through the center of the substrate  184 . The guide pins  24 A and  24 C are rod-shaped members that extend in the lengthwise direction. 
     The guide pins  24 A to  24 D are each orthogonal to the inclined axis  220 . The guide pins  24 A and  24 C are provided on the same straight line and the guide pins  24 B and  24 D are provided on the same straight line. The guide pins  24 A to  24 D are all provided on the same plane. 
     A rotational angular velocity sensor  26  and a gravitational acceleration sensor  28  are attached on the substrate  184 . The indication directions of the sensors  26  and  28  match the normal direction to the substrate  184 . In other words, the directions match the axial direction of the guide pin  24 A. The rotational angular velocity sensor  26  detects a rotational angular velocity component around the indication direction (i.e., the axial direction of the guide pin  24 A). The gravitational acceleration sensor  28  detects a gravitational acceleration component in the indication direction. 
     As described above, the six-direction indicator  10  is provided in the XYZ rectangular coordinate system. According to the present embodiment, a UVW coordinate system is taken into account. The relation between the XYZ rectangular coordinate system and the UVW coordinate system is given in  FIGS. 4A and 4B .  FIG. 4A  is a view of a YZ plane as viewed in the −X direction and  FIG. 4B  is a view of a ZX plane as viewed in the +Y direction (a right side view of  FIG. 4A ). The elevation angle formed between the U, V, and W axes and the YZ plane is an angle α. More specifically, the angle α of the inclined axis  220  with respect to the shaft  221  matches the elevation angle. Furthermore, the crossing angle formed among the U, V, and W axes is β. 
     The six axes, in other words, the +U, −U, +V, −V, +W, and −W are always provided at intervals of 60° on the YZ plane. More specifically, the +U axis is inclined 60° with respect to the −Z axis when it is orthogonally projected on the YZ plane. The −U axis is inclined 60° with respect to the +Z axis when it is orthogonally projected on the YZ plane. The +V axis is inclined at 60° with respect to the −Z axis when it is orthogonally projected on the YZ plane. The −V axis is inclined at 60° with respect to the +Z axis when it is orthogonally projected on the YZ plane. The +W axis matches the +Z axis when it is orthogonally projected on the YZ plane. The −W axis matches the −Z axis when it is orthogonally projected on the YZ plane. In this case, the elevation angle α is preferably from 30° to 40°, most preferably 35.26°. The crossing angle β is most preferably 90°. The reason for the preference will be described and in the following description, α is 35.26° and β is 90°. 
       FIG. 5  is a side view of the guide member  20  and  FIG. 6  is a side view of the guide member on the opposite side to  FIG. 5 . A portion of the guide member  20  on the X-axis has holes  205  through which the two shafts  221  of the inclined crank mechanism  22  are passed. The two shafts  221  are each inserted in a corresponding one of the holes. Therefore, the rotation member  18  is supported by the guide member  20 . The holes  205  are provided with bearings that are not shown. 
     The guide member  20  further has a track slit  203  formed around the X axis along the entire circumference in the center. The track slit  203  is a zigzag slit bent alternately in mountain-like and valley-like shapes in the X direction at intervals of 60° around the X axis. 
     The guide member  20  has a first guide member  201  and a second guide member  202 . The first guide member  201  is provided on the side of the stepping motor  14  with respect to the track slit  203  as a boundary. The second guide member  202  is provided on the opposite side to the stepping motor  14  with respect to the track slit  203  as a boundary. The first guide member  201  includes a circumferential wall  2010  and a circumferential edge  41 . The circumferential wall  2010  is provided around the X-axis and has an opening in the X direction. The circumferential edge  41  is an end of the circumferential wall  2010  and has mountain-like portions MO 1  and valley-like portions VA 1  alternately formed at intervals of 60° around the X axis. 
     Similarly to the first guide member  201 , the second guide member  202  includes a circumferential wall  2020  and a circumferential edge  42 . The circumferential wall  2020  is provided around the X axis and has an opening in the X axis direction. The circumferential edge  42  is an end of the circumferential wall  2020  and has mountain-like portions MO 2  and valley-like portions VA 2  alternately formed at intervals of 60° around the X axis. 
     The track slit  203  is formed by the circumferential edge  41  of the first guide member and the circumferential edge  42  of the second guide portion  202 . The circumferential edges  41  and  42  are opposed to and apart from each other. The mountain-like portions MO 2  and the valley-like portions VA 2  of the circumferential edge  42  are provided around the X-axis as they are 60° shifted from the mountain-like portions MO 1  and the valley-like portions VA 1  of the circumferential edge  41 . Therefore, a zigzag gap is formed between the circumferential edges  41  and  42  to define the track slit  203 . 
     The tops of the mountain-like portions MO 1  and MO 2  and the bottoms of the valleys VA 1  and VA 2  correspond to the U, V, and W axes. The bottom VA 1   −V  and the top MO 1   +V  of the circumferential edge  41  and the top MO 2   −V  and the bottom VA 2   +V  of the circumferential edge  42  both correspond to the V axis. In other words, the V-axis passes a bent point formed by the bottom VA 1   −V  and the top MO 2   −V  and a bent point formed by the top MO 1   +V  and the bottom VA 2   +V  in the track slit  203 . Similarly, the top MO 1   +U , the bottom VA 2   +U , the top MO 2   −U  and the bottom VA 1   −U  correspond to the U axis. The top MO 1   +W , the bottom VA 2   +W , the top MO 2   −W , and the bottom VA 1   −W  correspond to the W axis. 
     Referring back to  FIG. 1 , the four guide pins  24 A to  24 D provided upright on the rotation member  18  are inserted to the track slit  203 . The guide pins  24 A to  24 D each contact the circumferential edges  41  and/or  42 . When the stepping motor  14  is driven and the shaft  221  starts to rotate, the rotation member  18  coupled to the shaft  221  also rotates. At the time, the guide pins  24 A to  24 D each move around on the track slit  203  while they are in contact with the circumferential edges  41  and  42 . Stated differently, the guide pins  24 A to  24 D move around on the circumferential edges  41  and/or  42 . 
     As the guide pins  24 A to  24 D move around on the track slit  203 , the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  provided at the rotation member  18  can indicate the six directions (±U, ±V, and ±W directions) on the U, V, and W axes. 
     The guide member  20  is fixed to the base  12  in such a position that the guide pins  24 A to  24 D do not contact the base  12  when the guide pins  24 A to  24 D move around on the track slit  203 . 
       FIG. 7  is a view (top view) of the inclined crank mechanism  22  and the guide member  20  of the six-direction indicator  10  shown in  FIG. 1  when viewed from immediately above (i.e., in the +Z direction) and  FIG. 8  is a top view showing only the rotation member  18  and the track slit  203  in  FIG. 7  for ease of illustration. In the following description, the bent points of the zigzag track slit  203  will be referred to as the tops MO 1   +W , MO 1   +V , and MO 1   +W , and the bottoms VA 1   −U , VA 1   −V , and VA 1   −W  of the circumferential edge  41  of the guide member  201 . In the drawings to be referred to in the following, the width of the track slit  203  is smaller than the guide pins  24 A to  24 D, but the width of the track slit  203  is actually greater than the guide pins  24 A to  24 D as described above. 
     Referring to  FIGS. 7 and 8 , the four guide pins  24 A to  24 D are all inserted to the track slit  203 . A tip end of the guide pin  24 A indicates the −W direction. More specifically, in the state shown in  FIGS. 7 and 8 , the rotational angular velocity sensor  26  and the gravitational velocity sensor  28  indicate the −W direction. 
       FIG. 9  is a view of the rotation member  18  and the track slit  203  when the six-direction indicator  10  is viewed from the front (i.e., in the −X direction).  FIGS. 10 to 14  are views of the rotation member  18  and the track slit  203  when viewed in a direction inclined at 60×n° (n: 1 to 6) around the X-axis as shown in  FIG. 9 . 
     Referring to  FIG. 15 , the six-direction indicator  10  further includes an information processor  30  including a computer and the like. The information processor  30  includes a motor controller  32 , a rotational angular velocity measuring unit  36 , a gravitational acceleration measuring unit  38 , a rotational angular velocity coordinate transformer  40 , a gravitational acceleration coordinate transformer  46 , and an azimuth angle calculator  44 . 
     The motor controller  32  controls the stepping motor  14 . The rotational angular velocity measuring unit  36  measures rotational angular velocity components ω U , ω V , and ω W  detected by the rotational angular velocity sensor  26 . The gravitational acceleration measuring unit  38  measures gravitational acceleration components g U , g V , and g W  detected by the gravitational acceleration sensor  28 . 
     The rotational angular velocity coordinate transformer  40  coordinate-transforms the angular velocity components ω U , ω V , and ω W  measured by the rotational angular velocity measuring unit  36  into rotational angular velocity components ω X , ω Y , and ω Z  by the following Expression (4): 
     
       
         
           
             
               
                 
                   
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     The gravitational acceleration coordinate transformer  46  coordinate-transforms the gravitational acceleration components g U , g V , and g W  measured by the gravitational acceleration measuring unit  38  into gravitational acceleration components g X , g Y , and g Z  by the following Expression (5): 
     
       
         
           
             
               
                 
                   
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     The azimuth angle calculator  44  calculates an azimuth angle Ψ according to the following expression (6) based on the rotational angular velocity components ω X , ω Y , and ω Z  obtained by the rotational angular velocity coordinate transformer  40  and the gravitational accelerator components g X , g Y , and g Z  obtained by the gravitational acceleration coordinate transformer  46 . 
     
       
         
           
             
               
                 
                   
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     Operation of Six-Direction Indicator 
     The operation of the six-direction indicator  10  having the above-described structure will be described. 
       FIG. 16  shows an example of a sequence of the operation of the six-direction indicator  10 . Referring to  FIG. 16 , the motor controller  32  controls the stepping motor  14  so that the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  are sequentially directed to the −W, +V, −U, +W, −V, and +U in this order (S 1  to S 6 ). The motor controller  32  rotates the shaft  221  240° anti-clockwise when viewed from the front of the device (i.e., in −X direction), the shaft  221 , the rotation member  18 , and the guide member  20  interact with one another, so that each of the sensors  26  and  28  can be directed to a corresponding direction in each of the steps. More specifically, the sensors  26  and  28  can be made to indicate the six directions (±U, ±V, and ±W directions) simply by rotating the one shaft using one driving source. Now, referring to  FIGS. 16 to 22 , the operation of the six-direction indicator  10  will be described in detail. In  FIGS. 17 to 22 , a tip end of the guide pin  24 A that matches the direction indicated by each of the sensors  26  and  28  is circled for better understanding of the operation of the six-direction indicator  10 . 
     Referring to  FIG. 17 , in step S 1 , the guide pin  24 A is positioned at the bottom VA 1   −W  of the guide member  201  and its tip end indicates the −W direction. As described above, since the guide pin  24 A matches the direction indicated by the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28 , the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  indicate the −W direction in  FIG. 17 . In this state, the rotational angular velocity measuring unit  36  samples a rotational angular velocity component ω W  detected by the rotational angular velocity sensor  26  in a prescribed sampling cycle (for example from 1 msec to 10 msec) and stores the values sequentially in a memory that is not shown. The gravitational acceleration measuring device  38  similarly samples a gravitational acceleration component g W  detected by the gravitational acceleration sensor  28  in a prescribed sampling cycle (for example 1 msec to 10 msec) and sequentially stores the values in the memory. 
     Then, in step S 2 , the guide pins  24 A to  24 D are moved around on the track slit  203  in order to sample a rotational angular velocity component ω V  and a gravitational acceleration component g V  in the +V direction, and the guide pin  24 A is moved from the bottom VA 1   −W  to MO 1   +V . At the time, the motor controller  32  controls the stepping motor  14  to rotate the shaft  221  240° anti-clockwise when viewed from the front of the six-direction indicator  10 . Then, as shown in  FIG. 18 , the guide pin  24 A moves on the track slit  203  to reach the top MO 1   +V  and the sensors  26  and  28  each indicate the +V direction. Now, the operation will be described in detail. 
     In the state shown in  FIG. 17 , the inclined axis  220  is orthogonal to the W axis. When the shaft  221  rotates 240° anti-clockwise from the state when viewed from the front of the device, the inclined axis  220  stops in a state where it is orthogonal to the V axis (see  FIG. 18 ). 
     Here, if the guide member  20  does not exist and the rotation member  18  does not rotate around the inclined axis  220  and is fixed to the shaft  221 , the rotation member  18  rotates 240° anti-clockwise around the X axis together with the shaft  221 . In this case, the tip end of the guide pin  24 A is directed to the −V direction. 
     However, the rotation member  18  can rotate around the inclined axis  220 . Therefore, the guide pin  24 A that is orthogonal to the inclined axis  220  rotates around the inclined axis  220 . The inclined axis  220  is inclined at α° (35.26° in this example) with respect to the X axis (shaft  221 ), and therefore as the shaft  221  rotates, the guide pin  24 A can swing in the range of ±α° in the X direction. 
     In short, the guide pin  24 A can rotate freely around the X axis by the rotation of the shaft  221  and can swing in the range of ±α° in the X direction by rotating around the inclined axis  220 . Therefore, the guide pin  24 A can be made to indicate the + and − directions of the U, V, and W axes that have a relation with the XYZ rectangular coordinate system as shown in  FIGS. 4A and 4B . 
     The track slit  203  made of the circumferential edges  41  and  42  of the guide member  20  serves to have the guide pin  24 A move on the U, V, and W axes described above and to position the tip end of the pin toward the six directions (±U, ±V, and ±W directions). If for example the track slit were in a straight line shape parallel to the X axis instead of the zigzag shape as in the present embodiment, the guide pin  24 A would merely swing along the X axis by the rotation of the shaft  221 . 
     In contrast, the track slit  203  according to the present embodiment is in a zigzag arrangement that bends in mountain-like shapes (tops MO 1   +U , MO 1   +V , and MO 1   +W ) and valley-like shapes (bottoms VA 1   −U , VA 1   −V , and VA 1   −W ) alternately at intervals of 60° around the X axis. More specifically, the track is formed obliquely with respect to the X axis. Therefore, the guide pin  24 A can not only swing in the X axis direction but also go around the X-axis. The top MO 1   +U  and the bottom VA −U  correspond to the U axis, the top MO 1   +V  and the bottom VA −V  correspond to the V axis, and the top MO 1   +W  and the bottom VA −W  correspond to the W axis. Therefore, as the shaft  221  rotates, the guide pin  24 A swings in the range of ±α° in the X direction while moving around the X axis and is provided on the U, V, and W axes when it reaches the tops and bottoms. 
     Referring to  FIGS. 17 and 18 , in step S 2 , when the shaft  221  rotates, the guide pin  24 A contacts the circumferential edge  41  while moving on the track slit  203  from the bottom VA 1   −W  toward the top MO 1   +V . At the time, the guide pin  24 A rotates anti-clockwise around the X axis when viewed from the front of the device (in the −X direction) as the shaft  221  rotates. The guide pin  24 A contacts the circumferential edge  41  that is part of the track slit  203  and therefore receives force from the circumferential edge  41 . Therefore, the rotating member  18  rotates around the inclined axis  220  clockwise when viewed from the front of the device, so that the tip end of the guide pin  24 A swings from the −X direction to the +X direction. More specifically, the guide pin  24 A swings in the X direction while rotating around the X axis by the slit  203 . 
     When the shaft  221  rotates 240° anti-clockwise, the guide pin  24 A rotates 60° anti-clockwise around the X axis by the track slit  203 . 
     Furthermore, as the guide pin  24 A contacts the circumferential edge  41 , the rotation member  18  is provided with force to rotate clockwise around the inclined axis  220 , so that the rotation member rotates 180° clockwise around the inclined axis  220 . As a result, the guide pin  24 A reaches the top MO 1   +V  as shown in  FIG. 18 . At the time, the guide pin  24 A is provided on the V axis and its tip end indicates the +V direction. More specifically, the sensors  26  and  28  indicate the +V direction. 
     When the guide pin  24 A reaches the top MO 1   +V , the rotational angular velocity measuring unit  36  samples a rotational angular velocity component ω V  and stores the data in the memory similarly to step S 1 . The gravitational acceleration measuring unit  38  also samples a gravitational acceleration component g V  and stores the sampled data in the memory. 
     Similarly to the above described step S 2 , the shaft  221  rotates 240° anti-clockwise each time in steps S 3  to S 6 , so that the tip end of the guide pin  24 A indicates each direction (±U, ±V, and ±W directions) on the UVW coordinates. In step S 3 , the shaft  221  rotates 240° anti-clockwise from the state shown in  FIG. 18  when viewed from the front of the device. Then, the guide pin  24 A swings 2α(=70.52)° in the −X direction and rotates 60° anti-clockwise around the X axis when viewed from the front of the device. As a result, as shown in  FIG. 19 , the guide pin  24 A reaches the bottom VA 1   −U  of the track slit  203 . At the time, the guide pin  24 A is provided on the U axis and its tip end is directed to the −U direction. In this state, the rotational angular velocity measuring unit  36  samples a rotational angular velocity component ω U  and the gravitational acceleration measuring unit  38  samples a gravitational acceleration component g U . 
     In step S 4 , the shaft  221  rotates 240° anti-clockwise further from the state in  FIG. 19 . Then, the guide pin  24 A swings 2α(=70.52)° in the +X direction and rotates 60° anti-clockwise around the X axis when viewed from the front of the device. Therefore, as shown in  FIG. 20 , the guide pin  24 A reaches the top MO 1   +W  and is provided on the W axis, so that its tip end indicates the −W direction. In this state, the rotational angular velocity measuring unit  36  again samples a rotational angular velocity component ω W  and the gravitational acceleration measuring unit  38  also samples a gravitational acceleration component g W . More specifically, the six-direction indicator  10  measures the rotational angular velocity component ω W  and the gravitational acceleration component g W  both in the +W and −W directions on the W axis. 
     Similarly, in step S 5 , the shaft  221  rotates 240° anti-clockwise when viewed from the front of the device from the state in  FIG. 20 . At the time, the guide pin  24 A swings 2α° in the −X direction and rotates 60° anti-clockwise around the X-axis when viewed from the front of the device. Therefore, as shown in  FIG. 21 , the guide pin  24 A moves from the top MO 1   +W  to the bottom VA 1   −V  along the track slit  203 . Upon reaching the bottom VA 1   −V , the guide pin  24 A is provided on the V axis and its tip end is directed to the −V direction. In this state, the rotational angular velocity measuring unit  36  again samples a rotational angular velocity component ω V  and the gravitational acceleration measuring unit  38  samples a gravitational acceleration component g V . 
     In step S 6 , the shaft  221  rotates 240° anti-clockwise from the state shown in  FIG. 21 . At the time, the guide pin  24 A rotates 60° anti-clockwise around the X axis and swings 2α° in the +X direction. As a result, as shown in  FIG. 22 , the guide pin  24 A reaches the top MO 1   +U  from the bottom VA 1   −V . At the time, the guide pin  24 A is provided on the U axis and its tip end indicates the +U direction. In this state, the rotational angular velocity measuring unit  36  samples a rotational angular velocity component ω U  and the gravitational acceleration measuring unit  38  samples a gravitational acceleration component g U . 
     Using the sampled data in steps S 1  to S 6 , the rotational angular velocity measuring unit  36  calculates rotational angular velocities ω U , ω V , and ω W  in the UVW rectangular coordinate system by a method such as the method of least squares or arithmetic mean. The gravitational acceleration measuring unit  38  calculates g U , g V , and g W  in the UVW rectangular coordinate system by a method such as the method of least squares or arithmetic mean. As described above, in any of the U, V, and W axes, the rotational angular velocities (ω U , ω V , and ω W ) and the gravitational accelerations (g U , g V , and g W ) measured both in the + and − directions are averaged. Therefore, biases unique to the six direction indicator  10  are cancelled, so that measuring errors can be reduced. 
     The rotational angular velocity coordinate transformer  40  coordinate-transforms the rotational angular velocities (ω U , ω V , and ω W ) calculated by the rotational angular velocity measuring unit  36  into rotational angular velocities (ω X , ω V , and ω Z ) in the XYZ rectangular coordinate system. The gravitational acceleration coordinate transformer  46  coordinate-transforms the gravitational accelerations (g U , g V , and g W ) calculated by the gravitational acceleration measuring unit  38  into gravitational accelerations (g X , g Y , and g Z ) in the XYZ rectangular coordinate system. 
     Finally, the azimuth angle calculator  44  calculates an azimuth angle Ψ based on the rotational angular velocities (ω X , ω Y , and ω Z ) and gravitational accelerations (g X , g Y , and g Z ) obtained by the rotational angular velocity coordinate transformer  40  and the gravitational acceleration coordinate transformer  46 . 
     As in the foregoing, by the function of the rotation member  18 , the guide pin  24 A, and the circumferential edges  41  and  42  that forms the zigzag track slit  203 , the six direction indicator  10  allows the guide pin  24 A to indicate the six directions (±U, ±V, and ±W directions) only by one rotation driving source, so that the sensors  26  and  28  can indicate the six directions. 
     The six direction indicator  10  uses the U, V and W axes, and therefore the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  need only swing ±α° (70.25°) in order to indicate the six directions, so that the rotation angle is small and a large space is not necessary to allow the rotation. Furthermore, the device can operate using only one rotation driving source as described above and therefore the device can be even more reduced in size. 
     According to the above-described embodiment, the elevation angle α is 35.26° but the elevation angle is not limited to the angle and needs only be in the range from 30° to 40°. The angle β needs only be in the range from 80° to 100° 
     The six-direction indicator  10  does not have to include both the first and second guide members  201  and  202 . For example, as shown in  FIG. 23 , the six-direction indicator  10  may include only the first guide member  201  without having the second guide member  202  or the other way around. In this case, the circumferential edge  41  of the first guide member  201  forms a zigzag track. More specifically, the guide pins  24 A to  24 D are in contact with the circumferential edge  41  and rotate around the X axis. 
     In the arrangement shown in  FIG. 23 , as far as the rotation member  18  includes the guide pins  24 A and  24 C both provided on the same straight line, the guide pin  24 A and the sensors  26  and  28  can indicate the six directions (±U, ±V, and ±W directions) only by the rotation of the shaft  221 . Note that as shown in  FIG. 1 , when the guide member  20  includes the first and second guide members  201  and  202 , at least one guide pin is necessary. In other words, even when only the guide pin  24 A is provided upright at the rotation member  18  while the guide pins  24 B to  24 D are not provided, the guide pin  24 A can move on the track slit  203  as it is in contact with the circumferential edges  41  and  42 . 
     When the rotation member  18  has four guide pins (guide pins  24 A to  24 D), the tops of the guide members  201  and  202  do not have to be provided on the U, V, and W axes. For example, when the six direction indicator  10  includes only the first guide member  201  as the guide member  20  but not the second guide member  202 , and the rotation member  18  includes the four guide pins  24 A to  24 D, the tops MO 1   +U , MO 1   +V , and MO 1   +W  of the mountain-like portions MO 1  at the circumferential edge  41  of the first guide member  201  may be lower than those in  FIG. 1  and do not have to correspond to the U, V, and W axes. In  FIG. 24 , the mountain-like portions MO 1  each have a trapezoidal shape having its width gradually reduced from the valley-like portion VA 1  toward the top. Note however that the bottoms VA 1   −U , VA 1   −V , and VA 1   −W  correspond to the U, V, and W axes. Even in this arrangement, the guide pin  24 A can indicate the six directions (±U, ±V, and ±W directions). 
     In  FIG. 24 , the circumferential edge  41  of the first guide member  201  has three mountain-like portions MO 1  and three valley-like portions VA 1  but there may be two mountain-like portions and two valley-like portions. The two mountain-like portions need only be arranged 60° apart from each other and the two valley-like portions also need only be arranged 60° apart from each other. 
     In  FIG. 2 , the shaft  221  is coupled to the rotation member  18  with the crank arm  222  to form the inclined crank mechanism  22  but the crank mechanism  22  may be formed by a crankshaft including the shaft  221 , the crank arm  222  and a crank pin provided on the inclined axis  220  and the rotation member  18  having the crank pin of the crankshaft inserted therein and rotatable around the crank pin (i.e., the inclined axis  220 ). 
     In the above described embodiment, the inclined crank mechanism  22  includes the two shafts  221  but only one of the shafts  221  may be provided. 
     Referring to  FIGS. 25 and 26 , a six-direction indicator  10 A includes a stepping motor  14 , an inclined crank mechanism  22 A, and a guide member  20 A. 
     The stepping motor  14  is stored in a motor case  500  and fixed to an inner surface of the motor case  500 . The motor case  500  is fixed to the base  12 . The shaft  221  is attached at an end of the stepping motor  14 . 
     A guide member  20 A is provided on an end side opposite to the stepping motor  14  with the inclined crank mechanism  22  therebetween. The guide member  20 A is cylindrical and has openings at both ends. The guide member  20 A is fixed to the base  12  similarly to the guide member  20 . The guide member  20 A has a cylindrical circumferential wall  2020 A and a circumferential edge  42 A as an end of the circumferential wall  2020 A on the side of the inclined crank mechanism  22 . The circumferential edge  42 A has mountain-like portions MO 2  and valley-like portions VA 2  formed alternately at intervals of 60° around the X axis. 
     An opening  2030  is formed at an end of the circumferential wall  2020 A on the opposite side to the circumferential edge  42 A. A plurality of wirings connected to the inclined crank mechanism  22 A are passed through the opening  2030 . 
     The guide member  20 A corresponds to the second guide member  202  described above. The tops of the three mountain-like portions MO 2  of the circumferential edge  42 A do not correspond to the U, V, and W axes and have trapezoidal shapes. On the other hand, the three valley-like portions VA 2  of the circumferential edge  42 A correspond to the U, W, and V axes. 
     The inclined crank mechanism  22 A includes the shaft  221 , an arm member  224 , and a rotation member  18 A. 
     The rotation member  18 A rotates around an inclined axis that is α° inclined with respect to the shaft  221  similarly to the rotation member  18 . The rotation member  18 A is coupled to the shaft  221  with the arm  224 . 
     The rotation member  18 A includes a disk-shaped substrate  184  similarly to the rotation member  18 . The rotation member  18 A includes an annular member  181 A in place of the inner shell  181 . The annular member  181 A is formed on the substrate  184  and orthogonal to the substrate  184 . 
     As shown in  FIG. 26 , one axial member  185  is formed at a side surface of the substrate  184 , and the axial member  185  is inserted into a hole formed at a tip end of the arm member  224 . In this way, the rotation member  18 A is supported rotatably around the inclined axis. 
     Two guide pins  24 B and  24 D are provided upright at a side surface of the substrate  184 . The guide pins  24 B and  24 D are provided on a straight line passing the center of the substrate  184 . The guide pins  24 B and  24 D are arranged orthogonally to the inclined axis  220 . At the side surface of the annular member  181 A, two guide pins  24 A and  24 D are provided upright on a normal to the substrate  184  that passes through the center of the substrate  184 . 
     Similarly to  FIG. 2 , the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  are attached on the substrate  184 . The indication directions of these sensors  26  and  28  match the axial direction of the guide pin  24 A. More specifically, the guide pin  24 A serves as an indication pin that indicates the indication directions of the sensors  26  and  28 . 
     The inclined crank mechanism  22 A includes only a single shaft  221 . More specifically, the inclined crank mechanism  22 A has the shaft  221  only on the side of the stepping motor  14  and does not have the shaft  221  on the side of the guide member  20 A. 
     The six-direction indicator  10 A further includes a receiving base  510  and an elastic member  520 . The receiving base  510  is provided at an end face of the motor case  500 . The receiving base  510  is a cylinder into which the shaft  221  is inserted. The receiving base  510  is fixed to the motor case  500 . Therefore, when the shaft  221  is rotated by the stepping motor  14 , the receiving base  510  does not rotate. 
     The elastic member  520  is a spiral spring provided between the receiving base  510  and the arm member  224 . The shaft  221  is inserted in the elastic member  520 . The elastic member  520  provides the rotation member  18 A with force toward the guide member  20 A through the arm member  224 . The elastic member  520  pushes the rotation member  18 A toward the guide member  20 A, so that the rotation member  18 A is supported stably using the single shaft  221  without providing another shaft  221  on the side of the guide member  20 A. 
     The rotation member  18 A rotates as it is pressed against the guide member  20 A by the elastic member  520 . Therefore, the rotation member  18 A is supported stably by the single shaft  221 . Since only one shaft  221  and only one guide member are necessary, the six-direction indicator  10 A can be made compact as compared to the six-direction indicator  10 . Furthermore, a space for providing wirings to be connected to the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  can be secured easily. 
     As shown in  FIGS. 25 and 26 , the circumferential edge  42 A has trapezoidal mountain-like portions MO and valley-like portions VA provided alternately at intervals of 60°. The circumferential edge  42 A further has depressions  410  at ends of the upper side portions (i.e., the tops) of the mountain-like portions MO. The guide members  24 B and  24 D are fitted into the depressions  410 . 
     The guide pin  24 D and the guide pin  24 B (which is not shown in FIGS.  25  and  26 ) coaxially provided with the guide pin  24 D serve as auxiliary pins to allow the guide pin  24 A to be provided correctly on the U, V, and W axes. Hereinafter, the guide pins  24 B and  24 D will be referred to as “auxiliary guide pins  24 B and  24 D” and the guide pin  24 A will be referred to as “indication pin  24 A.” 
     The depression  410  is formed in a position where any of the auxiliary pins  24 B and  24 D contacts the circumferential edge  42 A when the indication pin  24 A is provided on any of the U, V, and W axes. More specifically, the depression  410  is formed in a position where any of the auxiliary pins  24 B and  24 D contacts the circumferential edge  42 A when the indication pin  24 A indicates any of the +U, −U, +V, −V, +W, and −W axes. 
     When the six-direction indicator  10 A starts to operate, the rotation member  18 A rotates around the X axis and the inclined axis  220 . When for example a tip end of the indication pin  24 A indicates the +U direction, one of the auxiliary pins  24 B and  24 D is fitted in the depression  410 . Therefore, the indication pin  24 A can indicate the +U axis correctly. Similarly, when the indication pin  24 A is provided on any of the U, V, and W axes, one of the auxiliary pins  24 B and  24 D is fitted into the depression  410  by force applied by the elastic member  520 . Therefore, the indication precision of the indication pin  24 A improves. 
     Since the depression  410  is shallow, the auxiliary pins  24 B and  24 D move easily out of the depression  410  as the stepping motor  14  rotates the shaft  221 . The auxiliary pins  24 B and  24 D then move around on the circumferential edge  42 A again. 
     As shown in  FIG. 27 , a root part of the auxiliary pin  24 D is inserted in a bearing  530 . The bearing  530  is fixed at an inner portion of a side surface of the substrate  184 . In short, the auxiliary pin  24 D is attached to the rotation member  18 A in a rotatable manner around the axis of the auxiliary pin  24 D. While the six-direction indicator  10 A operates, the auxiliary pin  24 D moves on the circumferential edge  42 A as it is in contact with the circumferential edge  42 A. Since the auxiliary pin  24 D rotates around the axis of the auxiliary pin  24 D, friction between the auxiliary pin  24 D and the circumferential edge  42 A can be reduced. Similarly to the auxiliary pin  24 D, the auxiliary pin  24 B has its root portion inserted in a bearing and is attached at the rotation member  18 A in a rotatable manner around the axis of the auxiliary pin  24 B. 
     The outside structure and operation of the six-direction indicator are as described above. In the embodiment described above, the circumferential edges  41  and  42  of the guide member  20  are slit type tracks but the track may have any shape other than the slit. For example, in place of the guide member  20  having the track slit  203 , a guide without a slit may be provided with a zigzag groove similar to the track slit  203  around the X axis at the inner surface of the guide and the groove may be used as the track. Even in this case, when the tip end of the guide pin  24 A contacts the circumferential edge of the groove (track), the above-described operation can be achieved. A rail may be used in place of the track slit  203  and a circumferential edge formed at the top of the rail may be used as a track. More specifically, the guide member  20  is not specifically limited as long as the guide member has a circumferential edge contacted by the guide pin  24  while the pin goes around the X axis. 
     According to the embodiment described above, the shape of the track slit  203  when viewed from the front of the device (i.e., the shape of circumferential edge  41  of the guide member  201 ) is circular, but the shape of the track slit  203  may have any other shapes. For example, the shape of the track slit  203  when viewed from the front of the device may be a hexagonal shape or any other shape as far as a bent track having an alternate arrangement of mountain like shapes and valley like shapes at intervals of 60° around the X axis. 
     In the six-direction indicator  10 A shown in  FIGS. 25 and 26 , the elastic member may be a plate spring or rubber in place of the spiral spring. The elastic member may be provided on the side of the guide member  20 A and the rotation member  18 A may be pulled by the elastic member to be pressed against the circumferential edge  42 A. In short, the elastic member is not particularly specified as far as it provides the rotation member with force in the direction of said guide member. 
     In the six direction indicator  10 A, the depression  410  is formed where one of the auxiliary pins  24 B and  24 D contacts the circumferential edge  42 A when the indication pin  24 A is provided on any of the U, V, and W axes. However, the position of the depression  410  is not limited to the position in this example. For example, when the circumferential edge  42 A of the six-direction indicator  10 A has mountain-like portions MO 2  having tops corresponding to the U, V, and W axes and the guide pin  24 A is provided on one of the U, V, and W axes, a depression may be formed in a position where the guide pin  24 C provided 180° apart from the guide pin  24 A around the inclined axis  220  contacts the circumferential edge  42 A. When the guide pin  24 A is provided on any of the U, V, and W axes, a depression may be formed in a position where the guide pin  24 A and the circumferential edge  42 A contact with each other. In short, the above described effects are provided when the rotation member  18  includes a plurality of guide pins  24 A to  24 D provided 90° or 180° apart from one another around the inclined axis  220 , the guide pin  24 A indicates any of the U, V, and W axes, and a depression is formed in a position where any of the guide pins  24 A to  24 D contacts the circumferential edge  42 A. 
     Angles α and β 
     Now, the reason why the elevation angle α is preferably from 30° to 40°, most preferably 35.26° and the crossing angle β is most preferably 90° will be described. 
     The tip end positions i U , i V , and i W  of the U, V, and W axes are represented by the following Expression (7) in the XYZ rectangular coordinate system. 
     
       
         
           
             
               
                 
                   
                     
                       i 
                       U 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   
                                     3 
                                   
                                   ⁢ 
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                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       i 
                       V 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   3 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               2 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       i 
                       W 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The crossing angle β on the U, V, and W axes, which is derived from the inner product of the above described column vectors, is represented by the following Expression (8): 
     
       
         
           
             
               
                 
                   
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     β 
                   
                   = 
                   
                     
                       
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           α 
                         
                         - 
                         
                           
                             
                               cos 
                               2 
                             
                             ⁢ 
                             α 
                           
                           2 
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ∴ 
                       β 
                     
                     = 
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               sin 
                               2 
                             
                             ⁢ 
                             α 
                           
                           - 
                           
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               α 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
       FIG. 28  is a plot in which the abscissa represents the elevation angle α and the ordinate represents the crossing angle β. As can be clearly understood from  FIG. 28 , when the elevation angle α is 0°, the crossing angle β is 120°, and when the elevation angle α is 90°, the crossing angle β is 0°. 
     Here, from Expression (7), rotational angular velocity components ω U , ω V , and ω W  around the U, V, and W axes are represented by the following Expression (9) using rotational angular velocity components ω X , ω Y , and ω Z  around the X, Y, and Z axes. 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 ω 
                                 U 
                               
                             
                           
                           
                             
                               
                                 ω 
                                 V 
                               
                             
                           
                           
                             
                               
                                 ω 
                                 W 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         A 
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   ω 
                                   X 
                                 
                               
                             
                             
                               
                                 
                                   ω 
                                   Y 
                                 
                               
                             
                             
                               
                                 
                                   ω 
                                   Z 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       A 
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     
                                       3 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                             
                             
                               
                                 
                                   
                                     3 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 2 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                             
                             
                               0 
                             
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                          
                         A 
                          
                       
                       = 
                       
                         
                           
                             
                               
                                 3 
                               
                               2 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                           + 
                           
                             
                               
                                 3 
                               
                               4 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             αsin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                           + 
                           0 
                           + 
                           
                             
                               
                                 3 
                               
                               4 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             αsin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                           + 
                           
                             
                               
                                 3 
                               
                               4 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             αsin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                           - 
                           0 
                         
                         = 
                         
                           
                             
                               3 
                               ⁢ 
                               
                                 3 
                               
                             
                             2 
                           
                           ⁢ 
                           
                             cos 
                             2 
                           
                           ⁢ 
                           αsin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     In Expression (9), A is a coordinate transformation matrix from the XYZ rectangular coordinate system to the UVW coordinate system and |A| is the determinant of the coordinate transformation matrix A. 
     Now, a rotational angular velocity ω will be described by way of illustration while the same applies to a gravitational acceleration g. More specifically, if gravitational acceleration components in the X, Y, and Z directions are g X , g Y , and g Z , gravitational acceleration components g U , g V , and g W  in the U, V, and W directions are represented similarly by Expression (9). 
     When α=0 or 90, |A|=0 results, and therefore there is no inverse matrix for the matrix A. Otherwise, the inverse matrix C of the matrix A is represented by the following Expression (10): 
     
       
         
           
             
               
                 
                   
                     
                       
                         C 
                         ≡ 
                           
                         ⁢ 
                         
                           A 
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           2 
                           
                             3 
                             ⁢ 
                             
                               3 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             αsin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     
                                       3 
                                     
                                     ⁢ 
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     α 
                                   
                                   2 
                                 
                               
                               
                                 
                                   
                                     
                                       3 
                                     
                                     ⁢ 
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     α 
                                   
                                   2 
                                 
                               
                               
                                 
                                   
                                     
                                       3 
                                     
                                     ⁢ 
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     α 
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     
                                       3 
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       αsin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     3 
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   2 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     
                                       
                                         3 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       αsinα 
                                     
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     
                                       
                                         3 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       αsin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     3 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   αsin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   1 
                                   
                                     3 
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                               
                                 
                                   1 
                                   
                                     3 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                               
                                 
                                   1 
                                   
                                     3 
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     1 
                                     
                                       
                                         3 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                               
                               
                                 
                                   1 
                                   
                                     
                                       3 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     1 
                                     
                                       3 
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     1 
                                     
                                       3 
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                               
                               
                                 
                                   2 
                                   
                                     3 
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     As for a normal coordinate transformation matrix between rectangular coordinate systems, an inverse matrix is the transposed matrix of the original matrix, but in this case, unlike a normal coordinate transformation matrix, the inverse matrix C is not the transposed matrix of the matrix A. 
     From Expressions (9) and (10), the rotational angular velocity components ω X , ω Y , and ω Z  around the X, Y, Z, axes are represented by the following Expression (11): 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               ω 
                               x 
                             
                           
                         
                         
                           
                             
                               ω 
                               Y 
                             
                           
                         
                         
                           
                             
                               ω 
                               Z 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     1 
                                     
                                       3 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                                 
                                   
                                     1 
                                     
                                       3 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                                 
                                   
                                     1 
                                     
                                       3 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     1 
                                     
                                       
                                         3 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                                 
                                   
                                     1 
                                     
                                       
                                         3 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       
                                         3 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         α 
                                       
                                     
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       1 
                                       
                                         3 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         α 
                                       
                                     
                                   
                                 
                                 
                                   
                                     2 
                                     
                                       3 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       α 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     ω 
                                     U 
                                   
                                 
                               
                               
                                 
                                   
                                     ω 
                                     v 
                                   
                                 
                               
                               
                                 
                                   
                                     ω 
                                     W 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ∴ 
                         
                           ω 
                           X 
                         
                       
                       = 
                       
                         
                           
                             ω 
                             U 
                           
                           + 
                           
                             ω 
                             V 
                           
                           + 
                           
                             ω 
                             W 
                           
                         
                         
                           3 
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       ω 
                       Y 
                     
                     = 
                     
                       
                         
                           - 
                           
                             ω 
                             U 
                           
                         
                         + 
                         
                           ω 
                           V 
                         
                       
                       
                         
                           3 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       ω 
                       Z 
                     
                     = 
                     
                       
                         
                           - 
                           
                             ω 
                             U 
                           
                         
                         - 
                         
                           ω 
                           V 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             ω 
                             W 
                           
                         
                       
                       
                         3 
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Now, measurement errors by the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  will be described. 
     A differentiation of Expression (11) results in the following Expression (12). 
     
       
         
           
             
               
                 
                   
                     
                       δω 
                       X 
                     
                     = 
                     
                       
                         
                           δω 
                           U 
                         
                         + 
                         
                           δω 
                           V 
                         
                         + 
                         
                           δω 
                           W 
                         
                       
                       
                         3 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       δω 
                       Y 
                     
                     = 
                     
                       
                         
                           - 
                           
                             δω 
                             U 
                           
                         
                         - 
                         
                           δω 
                           V 
                         
                       
                       
                         
                           3 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       δω 
                       Z 
                     
                     = 
                     
                       
                         
                           - 
                           
                             δω 
                             U 
                           
                         
                         - 
                         
                           δω 
                           V 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             δω 
                             W 
                           
                         
                       
                       
                         3 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Since δω X , δω Y , and δω Z  are independent from one another, the expectations of square mean of the left sides of Expression (12) &lt;δω X   2 &gt;, &lt;δω Y   2 &gt;, and &lt;δω Z   2 &gt; are represented by the following Expression (13): 
     
       
         
           
             
               
                 
                   
                     
                       〈 
                       
                         δω 
                         X 
                         2 
                       
                       〉 
                     
                     = 
                     
                       
                         
                           〈 
                           
                             δω 
                             U 
                             2 
                           
                           〉 
                         
                         + 
                         
                           〈 
                           
                             δω 
                             V 
                             2 
                           
                           〉 
                         
                         + 
                         
                           〈 
                           
                             δω 
                             W 
                             2 
                           
                           〉 
                         
                       
                       
                         9 
                         ⁢ 
                         
                           sin 
                           2 
                         
                         ⁢ 
                         α 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       〈 
                       
                         δω 
                         Y 
                         2 
                       
                       〉 
                     
                     = 
                     
                       
                         
                           〈 
                           
                             δω 
                             U 
                             2 
                           
                           〉 
                         
                         + 
                         
                           〈 
                           
                             δω 
                             V 
                             2 
                           
                           〉 
                         
                       
                       
                         3 
                         ⁢ 
                         
                           cos 
                           2 
                         
                         ⁢ 
                         α 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       〈 
                       
                         δω 
                         Z 
                         2 
                       
                       〉 
                     
                     = 
                     
                       
                         
                           〈 
                           
                             δω 
                             U 
                             2 
                           
                           〉 
                         
                         + 
                         
                           〈 
                           
                             δω 
                             V 
                             2 
                           
                           〉 
                         
                         + 
                         
                           〈 
                           
                             δω 
                             W 
                             2 
                           
                           〉 
                         
                       
                       
                         9 
                         ⁢ 
                         
                           cos 
                           2 
                         
                         ⁢ 
                         α 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Provided that errors are the same among the U, V, and W axes, if &lt;δω U   2 &gt;, &lt;δω V   2 &gt;, &lt;δω W   2 &gt;≡&lt;δω 2 &gt; in Expression (13), the following Expression (14) is obtained. 
     
       
         
           
             
               
                 
                   
                     
                       〈 
                       
                         δω 
                         X 
                         2 
                       
                       〉 
                     
                     = 
                     
                       
                         〈 
                         
                           δω 
                           2 
                         
                         〉 
                       
                       
                         3 
                         ⁢ 
                         
                           sin 
                           2 
                         
                         ⁢ 
                         α 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       〈 
                       
                         δω 
                         Y 
                         2 
                       
                       〉 
                     
                     = 
                     
                       
                         2 
                         ⁢ 
                         
                           〈 
                           
                             δω 
                             2 
                           
                           〉 
                         
                       
                       
                         3 
                         ⁢ 
                         
                           cos 
                           2 
                         
                         ⁢ 
                         α 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       〈 
                       
                         δω 
                         Z 
                         2 
                       
                       〉 
                     
                     = 
                     
                       
                         2 
                         ⁢ 
                         
                           〈 
                           
                             δω 
                             2 
                           
                           〉 
                         
                       
                       
                         3 
                         ⁢ 
                         
                           cos 
                           2 
                         
                         ⁢ 
                         α 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     By making Expression (14) dimensionless, error evaluation functions f 1  and f 2  in the X, Y, and Z directions are represented by the following Expression (15): 
     
       
         
           
             
               
                 
                   
                     
                       
                         f 
                         1 
                       
                       ≡ 
                       
                         
                           
                             〈 
                             
                               δω 
                               X 
                               2 
                             
                             〉 
                           
                           
                             〈 
                             
                               δω 
                               2 
                             
                             〉 
                           
                         
                       
                     
                     = 
                     
                       1 
                       
                         
                           3 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         f 
                         2 
                       
                       ≡ 
                       
                         
                           
                             〈 
                             
                               δω 
                               Y 
                               2 
                             
                             〉 
                           
                           
                             〈 
                             
                               δω 
                               2 
                             
                             〉 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             〈 
                             
                               δω 
                               Z 
                               2 
                             
                             〉 
                           
                           
                             〈 
                             
                               δω 
                               2 
                             
                             〉 
                           
                         
                       
                       = 
                       
                         
                           2 
                         
                         
                           
                             3 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     From Expression (15), the error in the Y direction and the error in the Z direction are equal. This can be expected from the symmetry between the Y axis and the Z axis shown in  FIGS. 4A and 4B . 
       FIG. 29  is a plot in which the abscissa represents the elevation angle α and the ordinate represents the error evaluation functions f 1  and f 2 . As can be clearly understood from  FIG. 29 , if α=0, f 1 =∞ results. This is because measurement in the Y and Z directions is disabled in this case. 
     From Expression (15), the elevation angle α when f 1 =f 2  is represented by the following Expression (16): 
     
       
         
           
             
               
                 
                   
                     1 
                     
                       
                         3 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                     
                   
                   = 
                   
                     
                       
                         
                           2 
                         
                         
                           
                             3 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ∴ 
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                       
                     
                     = 
                     
                       
                         tan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       = 
                       
                         
                           
                             1 
                             
                               2 
                             
                           
                           ⁢ 
                           
                             
 
                           
                           ∴ 
                           α 
                         
                         = 
                         
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 
                                   2 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             35.26 
                             ⁢ 
                             
                               ( 
                               deg 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     The relation of Expression (16) is shown in  FIG. 30 . The following Expression (17) is derived from  FIG. 30 . 
     
       
         
           
             
               
                 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                     
                     = 
                     
                       
                         2 
                       
                       
                         3 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                     
                     = 
                     
                       1 
                       
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the coordinate transformation matrix C in this case is represented by the following Expression (18): 
     
       
         
           
             
               
                 
                   C 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             1 
                             
                               3 
                             
                           
                         
                         
                           
                             1 
                             
                               3 
                             
                           
                         
                         
                           
                             1 
                             
                               3 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               1 
                               
                                 2 
                               
                             
                           
                         
                         
                           
                             1 
                             
                               2 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             
                               1 
                               
                                 6 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               1 
                               
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               2 
                               3 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     From Expression (8), the crossing angle β in this case is represented by the following Expression (19): 
     
       
         
           
             
               
                 
                   β 
                   = 
                   
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               sin 
                               2 
                             
                             ⁢ 
                             α 
                           
                           - 
                           
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               α 
                             
                             2 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           cos 
                           
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               1 
                               3 
                             
                             - 
                             
                               
                                 1 
                                 2 
                               
                               × 
                               
                                 2 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               cos 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             0 
                           
                           ⁢ 
                           
                             
 
                           
                           ∴ 
                           β 
                         
                         = 
                         
                           90 
                           ⁢ 
                           
                             ( 
                             deg 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Since the crossing angle β is 90°, the UVW coordinate system in this case is a rectangular coordinate system. 
     When the elevation angle α is shifted from 35.26°, how much the error evaluation functions f 1  and f 2  of Expression (15) are shifted is represented by the following Expression (20): 
     
       
         
           
             
               
                 
                   
                     
                       
                         f 
                         2 
                       
                       
                         f 
                         1 
                       
                     
                     - 
                     1 
                   
                   = 
                   
                     
                       
                         
                           
                             2 
                           
                           
                             
                               3 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           1 
                           
                             
                               3 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                       
                       - 
                       1 
                     
                     = 
                     
                       
                         
                           2 
                         
                         ⁢ 
                         tan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       - 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
       FIG. 31  is a plot where the abscissa represents the elevation angle α and the ordinate represents the value of Expression (20). As can be clearly understood from  FIG. 31 , when the elevation angle α is from 30° to 40°, the errors among the U, V, and W axes are not more than about 20%. 
     Now, effects on azimuth angle errors will be described. 
     The rotational angular velocity Ω and the gravitational acceleration G of the earth are represented by the following Expression (21):
 
Ω=√{square root over (ω X   2 +ω Y   2 +ω Z   2 )}
 
 G =√{square root over ( g   X   2   +g   Y   2   +g   Z   2 )}
 
     The attitude of the six direction indicator  10  is represented by an Eulerian angle (an azimuth angle Ψ, a pitch angle θ and a roll angle φ) shown in  FIG. 32 . As shown in  FIG. 32A , if rotation is positive in the clockwise direction toward the vertical direction Down (or the +Z direction), the azimuth angle Ψ is formed by the north direction North and the +X direction (or the east direction East and the +Y direction). As shown in  FIG. 32B , if rotation is positive in the clockwise direction toward the east direction East (or the +Y direction), the pitch angle θ is formed by the north direction North and the +X direction (or the vertical direction Down and the +Z direction). As shown in  FIG. 32C , if rotation is positive in the clockwise direction toward the north direction North (or the +X direction), the roll angle φ is formed by the east direction East and the +Y direction (or the vertical direction Down and the +Z direction). 
     The roll angle φ, the pitch angle θ, and the azimuth angle Ψ are represented by the following Expression (22): 
     
       
         
           
             
               
                 
                   
                     ϕ 
                     = 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             g 
                             y 
                           
                           
                             g 
                             Z 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     θ 
                     = 
                     
                       - 
                       
                         
                           sin 
                           
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               g 
                               X 
                             
                             G 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     ψ 
                     = 
                     
                       - 
                       
                         
                           tan 
                           
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   - 
                                   
                                     ω 
                                     Y 
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ϕ 
                               
                               + 
                               
                                 
                                   ω 
                                   Z 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ϕ 
                               
                             
                             
                               
                                 
                                   ω 
                                   X 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               - 
                               
                                 
                                   ω 
                                   Y 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ϕsin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               + 
                               
                                 
                                   ω 
                                   Z 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θsin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     The latitude λ of the location where the six direction indicator  10  is provided is represented by the following Expression (23): 
     
       
         
           
             
               
                 
                   λ 
                   = 
                   
                     - 
                     
                       
                         sin 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 ω 
                                 X 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             - 
                             
                               
                                 ω 
                                 Y 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕcos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             - 
                             
                               
                                 ω 
                                 Z 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕcos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                           Ω 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the information processor  30  shown in  FIG. 15  may further include a roll angle calculator, a pitch angle calculator, and a latitude calculator. 
     From Expressions (21) to (23), the relation between an error of the azimuth angle Ψ and a measurement error of the rotational angular velocity sensor  26  is represented by the following Expression (24): 
     
       
         
           
             
               
                 
                   
                     〈 
                     
                       δψ 
                       2 
                     
                     〉 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θsin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ψ 
                                   
                                   ) 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 〈 
                                 
                                   δω 
                                   X 
                                   2 
                                 
                                 〉 
                               
                             
                             + 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ϕcos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ψ 
                                     
                                     + 
                                     
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ϕsin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       θsin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ψ 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 〈 
                                 
                                   δω 
                                   Y 
                                   2 
                                 
                                 〉 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ϕcos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ψ 
                                   
                                   - 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ϕsin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θsin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ψ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             ⁢ 
                             
                               〈 
                               
                                 δω 
                                 Z 
                                 2 
                               
                               〉 
                             
                           
                         
                       
                     
                     
                       
                         ( 
                         
                           Ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           c 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           λ 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Substituting Expression (14) in Expression (24) yields the following Expression (25): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             〈 
                             
                               δψ 
                               2 
                             
                             〉 
                           
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 〈 
                                 
                                   δω 
                                   2 
                                 
                                 〉 
                               
                               
                                 
                                   ( 
                                   
                                     Ωcos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     λ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             [ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       θsin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ψ 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   3 
                                   ⁢ 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   α 
                                 
                               
                               + 
                             
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ϕcos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ψ 
                                       
                                       + 
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ϕsin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θsin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ψ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               
                                 3 
                                 ⁢ 
                                 
                                   cos 
                                   2 
                                 
                                 ⁢ 
                                 α 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ϕcos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ψ 
                                     
                                     - 
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ϕsin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       θsin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ψ 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             
                               3 
                               ⁢ 
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               α 
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 〈 
                                 
                                   δω 
                                   2 
                                 
                                 〉 
                               
                               
                                 
                                   ( 
                                   
                                     Ωcos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     λ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     
                                       θsin 
                                       2 
                                     
                                     ⁢ 
                                     ψ 
                                   
                                   
                                     3 
                                     ⁢ 
                                     
                                       sin 
                                       2 
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                                 + 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             cos 
                                             2 
                                           
                                           ⁢ 
                                           ψ 
                                         
                                         + 
                                         
                                           
                                             sin 
                                             2 
                                           
                                           ⁢ 
                                           
                                             θsin 
                                             2 
                                           
                                           ⁢ 
                                           ψ 
                                         
                                       
                                     
                                   
                                   
                                     3 
                                     ⁢ 
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           ∴ 
                           
                             
                               f 
                               ⁢ 
                               
                                 ( 
                                 
                                   α 
                                   , 
                                   θ 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                             ≡ 
                               
                             ⁢ 
                             
                               
                                 
                                   〈 
                                   
                                     δψ 
                                     2 
                                   
                                   〉 
                                 
                                 
                                   
                                     〈 
                                     
                                       δω 
                                       2 
                                     
                                     〉 
                                   
                                   / 
                                   
                                     
                                       ( 
                                       
                                         Ωcos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         λ 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     
                                       θsin 
                                       2 
                                     
                                     ⁢ 
                                     ψ 
                                   
                                   
                                     3 
                                     ⁢ 
                                     
                                       sin 
                                       2 
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                                 + 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             cos 
                                             2 
                                           
                                           ⁢ 
                                           ψ 
                                         
                                         + 
                                         
                                           
                                             sin 
                                             2 
                                           
                                           ⁢ 
                                           
                                             θsin 
                                             2 
                                           
                                           ⁢ 
                                           ψ 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     3 
                                     ⁢ 
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     As can be clearly understood from the azimuth angle error evaluation function f in Expression (25), the azimuth angle error δΨ changes depending on the attitude (the azimuth angle Ψ and the pitch angle θ) of the six direction indicator  10 .  FIGS. 33 and 34  show a plot where the abscissa represents the elevation angle α and the ordinate represents the azimuth angle error evaluation function f.  FIG. 33  shows a case in which θ=0 and  FIG. 34  shows a case in which θ=45. As can be clearly understood from  FIGS. 33 and 34 , when the elevation angle α is within the range from 30° to 40°, the error is relatively small. When α is 35.26, in particular, the following Expression (26) is obtained from Expression (25). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             〈 
                             
                               δψ 
                               2 
                             
                             〉 
                           
                         
                         -&gt; 
                           
                         ⁢ 
                         
                           
                             
                               
                                 〈 
                                 
                                   δω 
                                   2 
                                 
                                 〉 
                               
                               
                                 
                                   ( 
                                   
                                     Ωcos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     λ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     
                                       θsin 
                                       2 
                                     
                                     ⁢ 
                                     ψ 
                                   
                                   
                                     3 
                                     × 
                                     
                                       1 
                                       3 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             cos 
                                             2 
                                           
                                           ⁢ 
                                           ψ 
                                         
                                         + 
                                         
                                           
                                             sin 
                                             2 
                                           
                                           ⁢ 
                                           
                                             θsin 
                                             2 
                                           
                                           ⁢ 
                                           ψ 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     3 
                                     × 
                                     
                                       2 
                                       3 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 〈 
                                 
                                   δω 
                                   2 
                                 
                                 〉 
                               
                               
                                 
                                   ( 
                                   
                                     Ωcos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     λ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     cos 
                                     2 
                                   
                                   ⁢ 
                                   
                                     θsin 
                                     2 
                                   
                                   ⁢ 
                                   ψ 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       
                                         cos 
                                         2 
                                       
                                       ⁢ 
                                       ψ 
                                     
                                     + 
                                     
                                       
                                         sin 
                                         2 
                                       
                                       ⁢ 
                                       
                                         θsin 
                                         2 
                                       
                                       ⁢ 
                                       ψ 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 〈 
                                 
                                   δω 
                                   2 
                                 
                                 〉 
                               
                               
                                 
                                   ( 
                                   
                                     Ωcos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     λ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           cos 
                                           2 
                                         
                                         ⁢ 
                                         θ 
                                       
                                       + 
                                       
                                         
                                           sin 
                                           2 
                                         
                                         ⁢ 
                                         θ 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   ψ 
                                 
                                 + 
                                 
                                   
                                     cos 
                                     2 
                                   
                                   ⁢ 
                                   ψ 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 〈 
                                 
                                   δω 
                                   2 
                                 
                                 〉 
                               
                               
                                 
                                   ( 
                                   
                                     Ωcos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     λ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   ψ 
                                 
                                 + 
                                 
                                   
                                     cos 
                                     2 
                                   
                                   ⁢ 
                                   ψ 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               〈 
                               
                                 δω 
                                 2 
                               
                               〉 
                             
                           
                           
                             ( 
                             
                               Ωcos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               λ 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     As can be clearly understood from Expression (26), the azimuth angle error δΨ in this case is constant without being affected by the attitude of the six-direction indicator  10 . 
     As in the foregoing, the elevation angle α in the UVW coordinate system is preferably from 30° to 40°. The crossing angle β in this case is from about 100° to 80° from  FIG. 28 . The elevation angle α is most preferably 35.26°. At the time, the crossing angle β is 90° and the UVW coordinate system is a rectangular coordinate system. 
     When the UVW coordinate system is a rectangular coordinate system, measurement errors are the same regardless of its positional relation with the XYZ coordinate system. Now, an envelope in the lengthwise direction (X direction) in this example will be described. 
     When the elevation angle is α, the tip end positions i U , i V , and i W  of the unit vectors of the U, V, and W axes are represented by the following Expression (27) in the XYZ coordinate system. 
     
       
         
           
             
               
                 
                   
                     
                       i 
                       U 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       i 
                       V 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       i 
                       W 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     As shown in  FIGS. 35A and 35B , when an envelope in the X direction is L and the radius of gyration of a vector is R, the ratio L/R is represented by the following Expression (28) in consideration of six directions including the positive and negative directions on the U, V, and W axes. 
     
       
         
           
             
               
                 
                   
                     L 
                     R 
                   
                   = 
                   
                     2 
                     × 
                     
                       max 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                             
                               2 
                             
                           
                           , 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
       FIG. 36  is a plot where the abscissa represents the elevation angle α and the ordinate represents L/R in Expression (28). As can be clearly understood from  FIG. 36 , when α=0, the W axis is in a YZ plane and matches the Z axis. When α=90, the W axis is orthogonal to the YZ plane and matches the X axis, while the U and W axes are within the YZ plane. 
     From Expression (28) and  FIG. 36 , L/R is minimized when α=35.26 as in the following Expression (29): 
     
       
         
           
             
               
                 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                     
                     
                       2 
                     
                   
                   = 
                   
                     
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       ⁢ 
                       
                         
 
                       
                       ∴ 
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                       
                     
                     = 
                     
                       
                         tan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       = 
                       
                         
                           
                             1 
                             
                               2 
                             
                           
                           ⁢ 
                           
                             
 
                           
                           ∴ 
                           α 
                         
                         = 
                         
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 
                                   2 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             35.26 
                             ⁢ 
                             
                               ( 
                               deg 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     From Expression (27), the tip end positions i U , i V , and i W  of the unit vectors on the U, V, and W axes are represented by the following Expression (30). They correspond to the first, second, and third columns of Expression (18). 
     
       
         
           
             
               
                 
                   
                     
                       i 
                       U 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               1 
                               
                                 3 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       i 
                       V 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               1 
                               
                                 3 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       i 
                       W 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               1 
                               
                                 3 
                               
                             
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 2 
                                 3 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     As in the foregoing, according to the present embodiment, the shaft  221 , the rotation member  18 , and the guide member  20  interact with one another, so that the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  can be made to indicate all the axial directions of the UVW coordinate system (±U, ±V, and ±W directions: six directions in total) by rotating the shaft  221  using a single driving source (the stepping motor  14  in this embodiment). Furthermore, since the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  swing only ±35.26°, the rotation angle is small and a large space is not necessary for rotation. This allows the device to be compact. 
     When the UVW coordinate system is a rectangular coordinate system and the elevation angle α is 35.26°, measurement errors for rotational angular velocity components ω X , φ Y , and ω Z  in the XYZ rectangular coordinate system are equal among one another, and measurement errors for gravitational velocity components g X , g Y , and g Z  in the XYZ rectangular coordinate system are equal among one another. An measurement error for the azimuth angle Ψ becomes smaller, independent of the attitude of the six-direction indicator  10 . Since the six-direction indicator  10  can be set in an arbitrary attitude, a reference such as a horizontal plane is not necessary and an azimuth can be measured easily. 
     Not only the azimuth angle Ψ but also the latitude of a place where the six-direction indicator  10  is provided can be measured correctly. In addition, not only the azimuth angle but also a roll angle φ and a pitch angle θ are calculated, so that the attitude of the provided six-direction indicator  10  can be measured. 
     In any of the U, V, and W axes, the rotational angular velocities (ω U , ω V , and ω W ) and the gravitational accelerations (g U , g V , and g W ) measured in the + and − directions are averaged, so that a bias unique to the six-direction indicator  10  is cancelled and measurement errors can be reduced. 
     According to the above-described embodiment, the six-direction indicator samples a rotational angular velocity component and a gravitational acceleration component in each of the U, V, and W axes. However, the six-direction indicator may sample these components only in two or one of the U, V, and W axes. 
     In  FIG. 16 , in each of steps S 1  to S 6 , the six-direction indicator  10  has the shaft  221  rotate anti-clockwise when viewed from the front of the device (i.e., in the −X direction) but the shaft may be rotated clockwise. 
     According to the above-described embodiment, the six-direction indicator includes the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28  and serves as an azimuth measuring device. However, the six-direction indicator can be used for purposes other than the azimuth measuring device. 
     For example, if a CCD camera arranged to shoot in the direction of the guide pin  24 A is provided on the rotation member  18  in place of the rotational angular velocity sensor  26  and the gravitational acceleration sensor  28 , the six-direction indicator can be used as a surveillance camera capable of covering the six directions. The six-direction indicator can be used as a receiving antenna. In short, the six-direction indicator according to the present invention can be used for all the purposes that necessitate a device provided on a rotation member such as a sensor, a camera and an antenna to indicate the six directions. 
     Furthermore, as a device that generates a pseudo gravity free state on the earth, there has been a known two-axis gimbal mechanism that creates a pseudo gravity-free state by constantly changing the gravity direction by rotation around two axes. The six-direction indicator according to the present invention may be used as a pseudo gravity-free state generator in place of the two-axis gimbal mechanism. 
     Although the present invention has been described and illustrated, it is clearly understood that the same is by way of illustration and example only. The invention is not limited to the above-described embodiments and can be subject to variations and modifications without departing the scope and spirit of the present invention.