Abstract:
An inclination calculation apparatus within a game machine computes an inclination of an input device having an acceleration sensor/detector capable of simultaneously detecting an acceleration in at least two axial directions. Programmed logic circuitry within the apparatus consecutively generates preliminary data which represents an inclination and which is uniquely determined from acceleration data consecutively output from the acceleration sensor/detector. Programmed logic circuitry within the apparatus also consecutively computes a new/updated inclination based on a previously computed inclination, an acceleration due to gravity and a current inclination represented by the preliminary data, wherein the new/updated inclination is computed to have a direction oriented between the previously computed inclination and a current inclination represented by the preliminary data by a degree to which the acceleration due to gravity contributes to the acceleration data.

Description:
CROSS REFERENCE OF RELATED APPLICATION 
   The disclosure of Japanese Patent Application No. 2006-79485 is incorporated herein by reference. 
   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to an inclination calculation apparatus and an inclination calculation program, and more specifically to an inclination calculation apparatus and an inclination calculation program for calculating an inclination of an input device using an output from an acceleration detection device included in the input device. 
   2. Description of the Background Art 
   Conventionally, technologies for calculating an inclination of a device including acceleration detection means have been proposed. For example, patent document 1 (Japanese Laid-Open Patent Publication No. 2001-159951) describes an information processing device including acceleration detection means. In this information processing device, an inclination of the device is calculated from an acceleration detected by the acceleration detection means (an angle of the device with respect to a direction of gravity), and an operation command is generated using the calculated inclination. 
   This information processing device has a built-in two-axial acceleration sensor. The inclination angle of the device is calculated from a component of the acceleration which is detected in two axial directions when the device is inclined. If the detected acceleration represents only an acceleration of gravity, the inclination of the device can be accurately calculated from the detected acceleration. However, the detected acceleration possibly includes an acceleration of a noise component provided by vibration of the hand of a user or the like in addition to the acceleration of gravity. Therefore, the device described in the above-mentioned publication executes processing of extracting a low-frequency component of the detected acceleration; for example, the device calculates a time average of the detected acceleration. In addition, this information processing device assumes that a change in the acceleration caused by a motion of a human being is smooth and slow. When the low-frequency component does not change during a predetermined pause period, the device determines that the user has performed some operation on the device and processes to change a value to be output as an inclination of the device (inclination operation information). By such processing, the acceleration of the noise component is eliminated from the acceleration detected by the acceleration detection means (acceleration sensor), and thus the inclination of the device can be calculated. 
   According to the technology described in patent document 1, the device executes the processing of calculating a time average and the processing of determining whether or not the detected acceleration has changed during a predetermined pause period, in order to calculate the inclination of the device. Such processing causes a delay in providing the inclination operation information. In other words, the above-described technology is not capable of calculating the inclination of the device in real time from the detected value of the acceleration. Also according to the technology described in patent document 1, characteristic information to be used for the above-mentioned processing for generating the inclination operation information needs to be prepared in accordance with the type of the operation. Therefore, the contents of the processing needs to be changed in accordance with the characteristic information. This restricts the range of applicable operations and complicates the processing. 
   SUMMARY OF THE INVENTION 
   Therefore, at least one aspect of the non-limiting illustrative exemplary implementation disclosed herein is to provide an inclination calculation apparatus and an inclination calculation program capable of computing an inclination of a device in real time using an acceleration detected an by acceleration detector sensor. 
   Reference numerals, additional explanations and the like shown within parentheses in this section of the specification indicate correspondence with elements within the Figures illustrating the non-limiting example implementations described herein. 
   In a first non-limiting aspect, applicant&#39;s illustrative example implementation disclosed herein is directed to an inclination calculation apparatus (game apparatus  3 ) for computing an inclination of an input device (controller  7 ) which includes an acceleration detector (acceleration sensor  37 ) that is capable of repeatedly detecting an acceleration in at least two axial directions. The inclination calculation apparatus comprises preliminary data generation means (CPU  10 , etc. for executing step S 13  or S 29 ; hereinafter, only the corresponding step numbers will be mentioned) and inclination calculation means (S 17  and S 18 , or S 33  and S 34 ). The preliminary data generation means sequentially generates preliminary data ( 531 ) which represents an inclination and is uniquely determined from acceleration data ( 521 ) output from the acceleration detection means. The inclination calculation means sequentially calculates a new inclination (i.e., repeatedly computes corresponding new/updated inclination values) by making an inclination previously calculated closer to an inclination represented by the preliminary data at a predetermined angle/degree (effectiveness k). (i.e., by using an inclination determined by using an “effectiveness” value “k” which represents a predetermined inclination vector angle/degree between a previously computed inclination vector and a current inclination represented by the preliminary data.) 
   In a second non-limiting aspect of applicant&#39;s illustrative example implementation disclosed herein, the preliminary data may represent an inclination of the input device under an assumption that the acceleration data represents only an acceleration of gravity. 
   In a third non-limiting aspect of applicant&#39;s illustrative example implementation disclosed herein, the acceleration detection means may be capable of detecting an acceleration in three axial directions (x-y-z directions). In this case, the inclination calculation means calculates an inclination in two axial directions (x-y directions) among the three axial directions. The preliminary data generation means generates preliminary data representing an inclination in two axial directions, corresponding to the inclination calculated by the inclination calculation means, from the acceleration data representing the acceleration in the three axial directions. 
   In a fourth non-limiting aspect of applicant&#39;s illustrative example implementation disclosed herein, the inclination calculation apparatus may further comprise magnitude calculation means (S 11  or S 21 ) for calculating a magnitude of the acceleration represented by the acceleration data. In this case, the inclination calculation means sequentially varies the degree in accordance with the magnitude of the acceleration, such that the degree is greater as the magnitude of the acceleration calculated by the magnitude calculation means is closer to a magnitude of the acceleration of gravity. 
   In a fifth non-limiting aspect of applicant&#39;s illustrative example implementation disclosed herein, the inclination calculation apparatus may further comprise magnitude calculation means (S 11  or S 21 ) for calculating a magnitude of the acceleration represented by the acceleration data. In this case, the inclination calculation means calculates an inclination only when a difference between the magnitude of the acceleration calculated by the magnitude calculation means and a magnitude of the acceleration of gravity is equal to or less than a predetermined threshold value. 
   In another non-limiting aspect, applicant&#39;s illustrative example implementation disclosed herein may be provided in the form of a computer-readable storage medium having stored thereon an inclination calculation program for causing a computer of an inclination calculation apparatus to execute the above-described invention. The present invention may be provided in the form of a game apparatus using an inclination calculated by the above invention as an operation input for a game, or in the form of a computer-readable storage medium having stored thereon a game program for causing a computer of the game apparatus to execute the above-described invention. 
   According to the first aspect, an inclination of the input device is calculated based on an inclination determined from the acceleration data and an inclination calculated previously, instead of regarding an inclination determined from the acceleration data as the inclination of the input device. Thus, even when the acceleration rapidly changes as a result of an influence of components other than the acceleration of gravity being added to the output from the acceleration detection means, the influence can be alleviated. Therefore, according to the first aspect, a reasonably accurate inclination can be obtained constantly. In addition, since the information used for calculating an inclination of the input device is an inclination determined from the acceleration data and an inclination calculated previously, the inclination of the input device can be calculated in real time. 
   According to the second aspect, a preliminary vector can be easily obtained from the output of the acceleration detection means. 
   According to the third aspect, an inclination in two axial directions from an acceleration in three axial directions detected by the acceleration detection means. 
   According to the fourth aspect, the inclination calculation means varies a predetermined degree in accordance with the closeness between the magnitude of the acceleration detected by the acceleration detection means and the magnitude of the acceleration of gravity. Thus, when the detected acceleration is significantly influenced by components other than the acceleration of gravity, i.e., when the inclination represented by the generated data is determined to be significantly offset from the accurate inclination, the inclination calculated by the inclination calculation means is close to the inclination calculated previously. By contrast, when the detected acceleration is not much influenced by components other than the acceleration of gravity, i.e., when the inclination represented by the generated data is determined to be close to the accurate inclination, the inclination calculated by the inclination calculation means is close to the inclination represented by the generated data. Thus, the degree at which the generated data influences the inclination calculated by the inclination calculation means can be varied in accordance with the determination result on whether or not the inclination represented by the generated data is close to the accurate inclination (inclination to be calculated). Therefore, the inclination of the input device can be more accurately calculated. 
   According to the fifth aspect, when the difference between the magnitude of the acceleration detected by the acceleration detection means and the magnitude of the acceleration of gravity is equal to or greater than a threshold value, the inclination is not calculated using the generated data and the inclination calculated previously. In the above case, the acceleration detected by the acceleration detection means includes many components other than the acceleration of gravity. Therefore, it is difficult to accurately calculate the inclination. According to the fifth aspect, the inclination is not calculated in such a case. Thus, an inaccurate inclination can be prevented from being calculated. 
   These and other objects, features, aspects and advantages of the present invention will become more apparent from the following detailed description of the present invention when taken in conjunction with the accompanying drawings. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is an external view of a game system including a game apparatus having an exemplary inclination calculation apparatus; 
       FIG. 2  is a functional block diagram of the game apparatus  3 ; 
       FIG. 3A  is an isometric view of a controller  7 ; 
       FIG. 3B  is another isometric view of the controller  7 ; 
       FIG. 4  is a front view of the controller  7 ; 
       FIG. 5A  is a view illustrating an internal structure of the controller  7 ; 
       FIG. 5B  is another view illustrating the internal structure of the controller  7 ; 
       FIG. 6  is a block diagram illustrating a structure of the controller  7 ; 
       FIG. 7  is a view illustrating the relationship between the inclination of the controller  7  and the output from an acceleration sensor; 
       FIG. 8  is another view illustrating the relationship between the inclination of the controller  7  and the output from the acceleration sensor; 
       FIG. 9  shows a general view of a game operation using the controller  7 ; 
       FIG. 10  illustrates an inclination calculation processing; 
       FIG. 11  shows the controller  7  in a still state; 
       FIG. 12  shows the controller  7  in the middle of being rotated counterclockwise from the state shown in  FIG. 11 ; 
       FIG. 13  shows main data stored on a main memory of the game apparatus in a first non-limiting example implementation; 
       FIG. 14  is a flowchart illustrating game processing executed by the game apparatus  3 ; 
       FIG. 15  is a flowchart illustrating a detailed flow of inclination computation processing in step S 3  shown in  FIG. 14  in the first example implementation; 
       FIG. 16  shows main data stored on the main memory  13  of the game apparatus  3  in a second non-limiting example implementation; 
       FIG. 17  is a flowchart illustrating a detailed flow of the inclination computation processing executed in the second non-limiting example implementation; and 
       FIG. 18  is a flowchart illustrating a detailed flow of the inclination computation processing executed in the second non-limiting example implementation. 
   

   DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   With reference to  FIG. 1 , a game system  1  including a game apparatus as an example of an inclination calculation apparatus according to a first non-limiting illustrative example implementation will be described.  FIG. 1  is an external view illustrating the game system  1 . In the following exemplary description, example the game apparatus described is of an installation type. 
   As shown in  FIG. 1 , the game system  1  includes an installation type game apparatus (hereinafter, referred to simply as a “game apparatus”)  3 , which is connected to a display (hereinafter, referred to as a “monitor”)  2  such as a home-use TV receiver including a speaker via a connection cord, and a controller  7  for giving operation data to the game apparatus  3 . Two markers  8   a  and  8   b  are provided in the vicinity of the monitor  2  (above the screen of the monitor  2  in  FIG. 1 ). The markers  8   a  and  8   b  are specifically infrared LEDs, and each outputs infrared light forward from the monitor  2 . The game apparatus  3  is connected to a receiving unit  6  via a connection terminal. The receiving unit  6  receives operation data which is wirelessly transmitted from the controller  7 . The controller  7  and the game apparatus  3  are connected to each other by wireless communication. On the game apparatus  3 , an optical disc  4  as an example of an exchangeable information storage medium is detachably mounted. The game apparatus  3  has, on a top main surface thereof, a power ON/OFF switch, a game processing reset switch, and an OPEN switch for opening a top lid of the game apparatus  3 . When a player presses the OPEN switch, the lid is opened, so that the optical disc  4  is mounted or dismounted. 
   On the game apparatus  3 , an external memory card  5  is detachably mounted when necessary. The external memory card  5  has a backup memory or the like mounted thereon for fixedly storing saved data or the like. The game apparatus  3  executes a game program or the like stored on the optical disc  4  and displays the result on the monitor  2  as a game image. The game apparatus  3  can also reproduce a state of a game played in the past using saved data stored on the memory card  5  and display the game image on the monitor  2 . The player playing with the game apparatus  3  can enjoy the game by operating the controller  7  while watching the game image displayed on the display screen of the monitor  2 . 
   The controller  7  wirelessly transmits operation data from a communication section  36  included therein (described later) to the game apparatus  3  connected to the receiving unit  6 , using the technology of, for example, Bluetooth®. The controller  7  is operation means for operating an operation target (an object displayed on the display screen of the monitor  2 ). The controller  7  includes an operation section having a plurality of operation buttons. As described later in detail, the controller  7  also includes an acceleration sensor  37  (described later) for detecting an acceleration in at least two axial directions perpendicular to each other. Data representing an acceleration detected by the acceleration sensor  37  is transmitted to the game apparatus  3  as a part of the operation data. The game apparatus  3  performs a predetermined calculation on data representing the acceleration to calculate an inclination of the controller  7  and executes processing in accordance with the inclination when necessary. The controller  7  further includes an imaging information calculation section  35  (described later) for taking an image seen from the controller  7 . The imaging information calculation section  35  takes an image of each of the markers  8   a  and  8   b  located in the vicinity of the monitor  2 . The game apparatus  3  executes processing in accordance with the position and the posture of the controller  7  by calculation processing based on the image. 
   With reference to  FIG. 2 , a structure of the game apparatus  3  will be described.  FIG. 2  is a functional block diagram of the game apparatus  3 . 
   As shown in  FIG. 2 , the game apparatus  3  includes, for example, a RISC CPU (central processing unit)  10  for executing various types of programs. The CPU  10  executes a start program stored in a boot ROM (not shown) to, for example, initialize memories including a main memory  13 , and then executes a game program stored on the optical disc  4  to perform game processing or the like in accordance with the game program. The CPU  10  is connected to a GPU (Graphics Processing Unit)  12 , the main memory  13 , a DSP (Digital Signal Processor)  14 , and an ARAM (Audio RAM)  15  via a memory controller  11 . The memory controller  11  is connected to a controller I/F (interface)  16 , a video I/F  17 , an external memory I/F  18 , an audio I/F  19 , and a disc I/F  21  via a predetermined bus. The controller I/F  16 , the video I/F  17 , the external memory I/F  18 , the audio I/F  19  and the disc I/F  21  are respectively connected to the receiving unit  6 , the monitor  2 , the external memory card  5 , a speaker  22  and a disc drive  20 . 
   The GPU  12  performs image processing based on an instruction from the CPU  10 . The GPU  12  includes, for example, a semiconductor chip for performing calculation processing necessary for displaying 3D graphics. The GPU  12  performs the image processing using a memory dedicated for image processing (not shown) and a part of the storage area of the main memory  13 . The GPU  12  generates game image data and a movie to be displayed on the display screen of the monitor  2  using such memories, and outputs the generated data or movie to the monitor  2  via the memory controller  11  and the video I/F  17  as necessary. 
   The main memory  13  is a storage area used by the CPU  10 , and stores a game program or the like necessary for processing performed by the CPU  10  as necessary. For example, the main memory  13  stores a game program read from the optical disc  4  by the CPU  10 , various types of data or the like. The game program, the various types of data or the like stored in the main memory  13  are executed by the CPU  10 . 
   The DSP  14  processes sound data or the like generated by the CPU  10  during the execution of the game program. The DSP  14  is connected to the ARAM  15  for storing the sound data or the like. The ARAM  15  is used when the DSP  14  performs predetermined processing (for example, storage of the game program or sound data already read). The DSP  14  reads the sound data stored in the ARAM  15  and outputs the sound data to the speaker  22  included in the monitor  2  via the memory controller  11  and the audio I/F  19 . 
   The memory controller  11  comprehensively controls data transfer, and is connected to the various I/Fs described above. The controller I/F  16  includes, for example, four controller I/Fs, and communicably connects the game apparatus  3  to an external device which is engageable via connectors of the controller I/Fs. For example, the receiving unit  6  is engaged with such a connector and is connected to the game apparatus  3  via the controller I/F  16 . As described above, the receiving unit  6  receives the operation data from the controller  7  and outputs the operation data to the CPU  10  via the controller I/F  16 . In other embodiments, the game apparatus  3  may include a receiving module for receiving the operation data transmitted from the controller  7 , instead of the receiving unit  6 . In this case, the operation data received by the receiving module is output to the CPU  10  via a predetermined bus. The video I/F  17  is connected to the monitor  2 . The external memory I/F  18  is connected to the external memory card  5  and is accessible to a backup memory or the like provided in the external card  5 . The audio I/F  19  is connected to the speaker  22  built in the monitor  2 , and is connected such that the sound data read by the DSP  14  from the ARAM  15  or sound data directly output from the disc drive  20  is output from the speaker  22 . The disc I/F  21  is connected to the disc drive  20 . The disc drive  20  reads data stored at a predetermined reading position of the optical disc  4  and outputs the data to a bus of the game apparatus  3  or the audio I/F  19 . 
   With reference to  FIG. 3A  through  FIG. 8 , the controller  7  will be described.  FIG. 3A  through  FIG. 5B  are external isometric views of the controller  7 .  FIG. 3A  is an isometric view of the controller  7  seen from the top rear side thereof.  FIG. 3B  is an isometric view of the controller  7  seen from the bottom rear side thereof.  FIG. 4  is a front view of the controller  7 . 
   As shown in  FIG. 3A ,  FIG. 3B  and  FIG. 4 , the controller  7  includes a housing  31  formed by plastic molding or the like. The housing  31  has a generally parallelepiped shape extending in a longitudinal or front-rear direction (the Z-axis direction shown in  FIG. 3A ). The overall size of the housing  31  is small enough to be held by one hand of an adult or even a child. The player can use the controller  7  to perform a game operation of pressing buttons provided thereon, a game operation of changing the inclination of the controller  7  itself (the angle of the controller  7  with respect to a direction of gravity), and a game operation of changing the position or direction of the controller  7  itself. For example, the player can change the inclination of the controller  7  to move an operation target (object) appearing in the game space. Also for example, the player can rotate the controller  7  with the longitudinal direction thereof as an axis to move the operation target through processing of the linear acceleration signals generated by the acceleration sensor  37 . The player can change the position indicated by the controller  7  on the display screen to move the object appearing in the game space. The “position indicated by the controller  7  on the display screen” is ideally a position at which a phantom straight line extending from a front end of the controller  7  in the longitudinal direction crosses the display screen of the monitor  2 . However, it is not necessary that the “position indicated by the controller  7  on the display screen” is strictly such a position. It is sufficient that the game apparatus  3  can calculate a position in the vicinity thereof. Hereinafter, a position indicated by the controller  7  on the display screen will be referred to as an “indicated position” or an “indicated position by the controller  7 ”. The longitudinal direction of the controller  7  (housing  31 ) will be sometimes referred to as an “indicated direction”. 
   The housing  31  has a plurality of operation buttons. Provided on a top surface of the housing  31  are a cross key  32   a , an X button  32   b , a Y button  32   c , a B button  32   d , a select switch  32   e , a menu switch  32   f , and a start switch  32   g . On a bottom surface of the housing  31 , a recessed portion is formed. On a rear slope surface of the recessed portion, an A button  32   i  is provided. These buttons and switches are assigned various functions in accordance with the game program executed by the game apparatus  3 , but this will not be described in detail because the functions are not directly relevant to the present invention. On the top surface of the housing  31 , a power switch  32   h  is provided for remotely turning on or off the game apparatus  3 . 
   The controller  7  has the imaging information calculation section  35  ( FIG. 5B ). As shown in  FIG. 4 , a light incident opening  35   a  of the imaging information calculation section  35  is provided on a front surface of the housing  31 . On a rear surface of the housing  31 , a connector  33  is provided. The connector  33  is, for example, a 32-pin edge connector, and is used for connecting the controller  7  to another device. In a rear part of the top surface of the housing  31 , a plurality of LEDs  34  are provided. The controller  7  is assigned a controller type (number) so as to be distinguishable from the other controllers  7 . The LEDs  34  are used for informing the player of the controller type which is currently set to controller  7  that he/she is using. Specifically, when the controller  7  transmits the operation data to the game apparatus  3 , one of the plurality of LEDs  34  corresponding to the controller type is lit up. 
   With reference to  FIG. 5A ,  FIG. 5B  and  FIG. 6 , an internal structure of the controller  7  will be described.  FIG. 5A  and  FIG. 5B  illustrate an internal structure of the controller  7 .  FIG. 5A  is an isometric view illustrating a state where an upper casing (a part of the housing  31 ) of the controller  7  is removed.  FIG. 5B  is an isometric view illustrating a state where a lower casing (a part of the housing  31 ) of the controller  7  is removed.  FIG. 5B  shows a reverse side of a substrate  300  shown in  FIG. 5A . 
   As shown in  FIG. 5A , the substrate  300  is fixed inside the housing  31 . On a top main surface of the substrate  300 , the operation buttons  32   a  through  32   h , the acceleration sensor  37 , the LEDs  34 , a quartz oscillator  46 , a wireless module  44 , an antenna  45  and the like are provided. These elements are connected to a microcomputer  42  (see  FIG. 6 ) via lines (not shown) formed on the substrate  300  and the like. The wireless module  44  and the antenna  45  allow the controller  7  to act as a wireless controller. The quartz oscillator  46  generates a reference clock of the microcomputer  42  described later. 
   As shown in  FIG. 5B , at a front edge of a bottom main surface of the substrate  300 , the imaging information calculation section  35  is provided. The imaging information calculation section  35  includes an infrared filter  38 , a lens  39 , an imaging element  40  and an image processing circuit  41  located in this order from the front surface of the controller  7 . These elements are attached to the bottom main surface of the substrate  300 . At a rear edge of the bottom main surface of the substrate  300 , the connector  33  is attached. The operation button  32   i  is attached on the bottom main surface of the substrate  300  rearward to the imaging information calculation section  35 , and cells  47  are accommodated rearward to the operation button  32   i . On the bottom main surface of the substrate  300  between the cells  47  and the connector  33 , a vibrator  48  is attached. The vibrator  48  may be, for example, a vibration motor or a solenoid. The controller  7  is vibrated by an actuation of the vibrator  48 , and the vibration is conveyed to the player holding the controller  7 . Thus, a so-called vibration-responsive game is realized. 
     FIG. 6  is a block diagram showing the structure of the controller  7 . The controller  7  includes the acceleration sensor  37  mentioned above. The acceleration sensor  37  detects an acceleration of the controller  7  (including an acceleration of gravity). Namely, the acceleration sensor  37  detects a force applied to the controller  7  (including gravity) and outputs the detected force as an acceleration.  FIG. 7  and  FIG. 8  show the relationship between the inclination of the controller  7  and the output of the acceleration sensor  37 . As shown in  FIG. 7  and  FIG. 8 , the acceleration sensor  37  detects an acceleration in each of three axial directions regarding the controller  7 , i.e., the up-down direction (y-axis direction in  FIG. 7 ), the left-right direction (x-axis direction in  FIG. 7 ), and the front-rear direction (the z-axis direction in  FIG. 7 ). Namely, the acceleration sensor  37  detects an acceleration in a linear direction along an axis, and therefore an output from the acceleration  37  represents an a value of an acceleration in each axis. Therefore, the detected acceleration is represented as a three-dimensional vector in an x-y-z coordinate system (see  FIG. 7  and  FIG. 8 ) which is set based on the controller  7 . Herein, the upward direction regarding the controller  7  is set as a positive y-axis direction, the frontward direction regarding the controller  7  is set as a positive z-axis direction, and the leftward direction regarding the controller  7  in the case where the controller  7  is viewed from the rear end thereof toward the front end thereof is set as a positive x-axis direction. 
   As explained above, the controller  7  preferably includes a three-axis, linear acceleration sensor  37  that detects linear acceleration in each of the three axial directions described above. Alternatively, a two axis linear accelerometer that only detects linear acceleration along each of the X-axis and Y-axis (or other pair of axes) may be used in another embodiment depending on the type of control signals desired. As a non-limiting example, the three-axis or two-axis linear accelerometer  37  may be of the type available from Analog Devices, Inc. or STMicroelectronics N.V. Preferably, the acceleration sensor  37  is an electrostatic capacitance or capacitance-coupling type that is based on silicon micro-machined MEMS (microelectromechanical systems) technology. However, any other suitable accelerometer technology (e.g., piezoelectric type or piezoresistance type) now existing or later developed may be used to provide the three-axis or two-axis acceleration sensor  37 . 
   As one skilled in the art understands, linear accelerometers, as used in acceleration sensor  37 , are only capable of detecting acceleration along a straight line corresponding to each axis of the acceleration sensor. In other words, the direct output of the acceleration sensor  37  is limited to signals indicative of linear acceleration (static or dynamic) along each of the two or three axes thereof. As a result, the acceleration sensor  37  cannot directly detect movement along a non-linear (e.g. arcuate) path, rotation, rotational movement, angular displacement, tilt, position, attitude or any other physical characteristic. 
   However, through additional processing of the linear acceleration signals output from the acceleration sensor  37 , additional information relating to the controller  7  can be inferred or calculated, as one skilled in the art will readily understand from the description herein. For example, by detecting static, linear acceleration (i.e., gravity), the linear acceleration output of the acceleration sensor  37  can be used to infer or calculate tilt or inclination of the object relative to the gravity vector by correlating tilt angles with detected linear acceleration. In this way, the acceleration sensor  37  can be used in combination with the micro-computer  42  (or another processor) to determine tilt, attitude or position of the controller  7 . Similarly, various movements and/or positions of the controller  7  can be calculated or inferred through processing of the linear acceleration signals generated by the acceleration sensor  37  when the controller  7  containing the acceleration sensor  37  is subjected to dynamic accelerations by, for example, the hand of a user. In another embodiment, the acceleration sensor  37  may include an embedded signal processor or other type of dedicated processor for performing any desired processing of the acceleration signals output from the accelerometers therein prior to outputting signals to micro-computer  42 . For example, the embedded or dedicated processor could convert the detected acceleration signal to a corresponding tilt angle when the acceleration sensor is intended to detect static acceleration (i.e., gravity). 
     FIG. 7  shows a state where an acceleration of gravity (vector Va in  FIG. 7 ) is directed downward regarding the controller  7 . In this state, the value Va of an acceleration detected by the acceleration sensor  37  (hereinafter, referred to as an “acceleration vector”) is in a negative y-axis direction. In  FIG. 7  and  FIG. 8 , it is assumed that the controller  7  is in a still state. In the state shown in  FIG. 7 , only the y coordinate value of the acceleration vector Va is not zero, and both the x coordinate value and the z coordinate value of the acceleration vector Va are zero.  FIG. 8  shows a state in which the controller  7  is inclined as a result of being rotated from the state shown in  FIG. 7  around the z axis. In the state shown in  FIG. 8 , the direction of the acceleration vector Va is changed from the state in  FIG. 7 . The x coordinate value and the y coordinate value of the acceleration vector Va are not zero, and the z coordinate value of the acceleration vector Va is zero because the controller  7  has been rotated around the z axis. As shown in  FIG. 7  and  FIG. 8 , the acceleration sensor  37  can detect a value of an acceleration having three axial directions regarding the controller  7  as components. Thus, a calculation handling the value of the acceleration as an acceleration vector having the three axial components is performed by software processing using a computer such as the microcomputer  42  or the CPU  10 , and thus an inclination of the controller  7  can be calculated. Data representing the acceleration detected by the acceleration sensor  37  (acceleration data) is output to the communication section  36 . In the first embodiment, the acceleration sensor  37  outputs a value in accordance with the acceleration sequentially (specifically, frame by frame). The game apparatus  3  performs a predetermined calculation handling the value as an acceleration vector to calculate the inclination (posture) of the controller  7 , and executes game processing in accordance with the inclination. 
   In this embodiment, the magnitude of an acceleration which is detected when the controller  7  is in a still state, i.e., the magnitude of an acceleration which represents only an acceleration of gravity, is set as 1. For example, the values of the components of the acceleration vector Va detected in the state shown in  FIG. 7  are (0, 1, 0). 
   In the first embodiment, it is intended to calculate an inclination of the controller  7  in the x-y directions. Therefore, in the first embodiment, an acceleration sensor for detecting an acceleration in only two axial directions (x-y directions) may be used instead of the acceleration sensor  37  for detecting an acceleration in three axial directions. The acceleration sensor  37  is typically of a static capacitance type, but may be of any other system. 
   The controller  7  includes the operation section  32  (operation buttons), the imaging information calculation section  35 , and the communication section  36  in addition to the acceleration sensor  37 . In this embodiment, the controller  7  only needs to include acceleration detection means (the acceleration sensor  37 ) and may not absolutely need to include the operation section  32  or the imaging information calculation section  35 . 
   Returning to  FIG. 6 , the imaging information calculation section  35  is a system for analyzing image data taken by imaging means and detecting the position of the center of gravity, the size and the like of an area having a high brightness in the image data. The imaging information calculation section  35  has, for example, a maximum sampling period of about 200 frames/sec., and therefore can trace and analyze even a relatively fast motion of the controller  7 . 
   Specifically, the imaging information calculation section  35  includes the infrared filter  38 , the lens  39 , the imaging element  40  and the image processing circuit  41 . The infrared filter  38  allows only infrared light to pass therethrough, among light incident on the front surface of the controller  7 . The markers  8   a  and  8   b  located in the vicinity of the display screen of the monitor  2  are infrared LEDs for outputting infrared light forward from the monitor  2 . Therefore, the provision of the infrared filter  38  allows the image of each of the markers  8   a  and  8   b  to be taken more accurately. The lens  39  collects the infrared light which has passed through the infrared filter  38  and outputs the infrared light to the imaging element  40 . The imaging element  40  is a solid-state imaging device such as, for example, a CMOS sensor or a CCD. The imaging element  40  takes an image of the infrared light collected by the lens  39 . Accordingly, the imaging element  40  takes an image of only the infrared light which has passed through the infrared filter  38  and generates image data. Hereinafter, an image taken by the imaging element  40  will be referred to as a “taken image”. The image data generated by the imaging element  40  is processed by the image processing circuit  41 . The image processing circuit  41  calculates the positions of the imaging targets (the markers  8   a  and  8   b ) in the taken image. The positions are represented in a coordinate system (x′-y′ coordinate system) in which the downward direction of the taken image is a positive y′-axis direction and the rightward direction of the taken image is a positive x′-axis direction. The image processing circuit  41  outputs coordinate values indicating the respective positions of the markers  8   a  and  8   b  in the taken image to the communication section  36  as imaging data. Since these coordinate values vary in accordance with the direction or position of the controller  7  itself, the game apparatus  3  can calculate the direction and position of the controller  7  using these coordinate values. 
   The communication section  36  includes the microcomputer  42 , a memory  43 , the wireless module  44  and the antenna  45 . The microcomputer  42  controls the wireless module  44  for wirelessly transmitting the data obtained by the microcomputer  42  while using the memory  43  as a storage area during processing. 
   Data which is output from the operation section  32 , the acceleration sensor  37  and the imaging information calculation section  35  to the microcomputer  42  is temporarily stored in the memory  43 . The wireless transmission from the communication section  36  to the receiving unit  6  is performed at a predetermined time interval. Since game processing is generally performed at a cycle of 1/60 sec., the wireless transmission needs to be performed at a cycle of a shorter time period. At the transmission timing to the receiving unit  6 , the microcomputer  42  outputs the data stored in the memory  43  to the wireless module  44  as operation data. The wireless module  44  uses, for example, the Bluetooth (registered trademark) technology to modulate a carrier wave of a predetermined frequency with the operation data and radiate the resultant very weak electric signal from the antenna  45 . Namely, the operation data is modulated into a very weak electric signal by the wireless module  44  and transmitted from the controller  7 . The very weak electric signal is received by the receiving unit  6  on the side of the game apparatus  3 . The received very weak electric signal is demodulated or decoded, so that the game apparatus  3  can obtain the operation data. The CPU  10  of the game apparatus  3  executes the game processing based on the obtained operation data and the game program. 
   The shape of the controller  7 , and the shape, number, position or the like of the operation buttons and switches shown in  FIG. 3A  through  FIG. 5B  are merely exemplary, and may be altered without departing from the scope of the present invention. The position of the imaging information calculation section  35  in the controller  7  (the light incident opening  35   a  of the imaging information calculation section  35 ) does not need to be on the front surface of the housing  31 , and may be on another surface as long as light can enter from the outside of the housing  31 . In this case, the “indicated direction” is a direction vertical to the light incident opening, i.e., the direction in which the imaging element  40  takes images of the imaging targets. 
   By using the controller  7 , the player can perform a game operation of changing the inclination of the controller  7 , of changing the position of the controller  7  itself, or of rotating the controller  7 , in addition to the conventional game operation of pressing the operation buttons or switches. Hereinafter, the game operations using the controller  7  will be described. 
     FIG. 9  is a general view of a game operation using the controller  7 . As shown in  FIG. 9 , when playing the game using the controller  7  with the game system  1 , the player holds the controller  7  with one hand. The markers  8   a  and  8   b  are located parallel to the transverse or width direction of the monitor  2 . The player holds the controller  7  such that the front surface of the controller  7  (having the light incident opening  35   a  by which the imaging information calculation section  35  takes the image of each of the markers  8   a  and  8   b ) faces the markers  8   a  and  8   b . In this state, the player performs a game operation of changing the inclination of the controller  7 , of changing the position indicated by the controller  7  on the display screen (indicated position), or of changing the distance between the controller  7  and the markers  8   a  and  8   b.    
   Next, processing of calculating an inclination of the controller  7  using an output from the acceleration sensor  37  will be described. In a still state of controller  7 , the acceleration vector which is output from the acceleration sensor  37  is directed in the direction of the acceleration of gravity (see  FIG. 7  and  FIG. 8 ). Therefore, the angle of the controller  7  with respect to the direction of the acceleration of gravity, i.e., the inclination of the controller  7 , can be represented as a difference between the direction of the acceleration vector and the negative y-axis direction. Namely, in a still state of the controller  7 , the value of the acceleration vector itself can be used as the inclination of the controller  7 . In a state where the controller  7  is moving or rotating, the acceleration vector does not match the direction of the acceleration of gravity. Therefore, the value of the acceleration vector itself cannot be used as the inclination of the controller  7 . In this embodiment, the acceleration is sequentially detected by the acceleration sensor  37 , and also the inclination of the controller  7  is sequentially calculated. The inclination of the controller  7  at the current time is calculated using the output from the acceleration sensor  37  detected at the current time and the inclination of the controller  7  calculated at the previous time. 
   First with reference to  FIG. 10 , an overview of inclination calculation processing will be described. In the first embodiment, an inclination of the controller  7  regarding the rotation around the z axis will be calculated.  FIG. 10  illustrates the inclination calculation processing. The inclination calculation processing is executed as follows. The game apparatus  3  first calculates a preliminary vector vh from the acceleration vector Va detected by the acceleration sensor  37 . The preliminary vector vh indicates an inclination of the controller  7  represented by the acceleration vector itself. Specifically, the preliminary vector vh is obtained by extracting an x-axis component and a y-axis component of the acceleration vector Va and performing predetermined coordinate set conversion on the extracted two-dimensional vector. The preliminary vector vh is represented in the x′-y′ coordinate system shown in  FIG. 10 , and has the origin of the x′-y′ coordinate system as a start point. The preliminary vector vh is a unit vector having a length of 1. The preliminary vector vh is uniquely determined from the acceleration vector. The preliminary vector vh represents an inclination of the controller  7  under an assumption that the acceleration vector represents the acceleration of gravity (an assumption that the acceleration vector is directed in the direction of the acceleration of gravity). 
   The reason why only the x-axis component and the y-axis component of the acceleration vector Va are extracted is as follows. In the first embodiment, it is intended to calculate an inclination of the controller  7  regarding the rotation around the z axis (in the x-y directions), and therefore a z-axis component is not necessary. The reason why predetermined coordinate set conversion is performed on the extracted two-dimensional vector is that in the first embodiment, an inclination of the controller  7  is represented in a coordinate system different from the x-y-z coordinate system (represented in the x′-y′ coordinate system shown in  FIG. 10 ). In the x′-y″ coordinate system, the positive x′-axis direction corresponds to the negative y-axis direction of the x-y-z coordinate system, and the positive y-axis direction corresponds to the negative x-axis direction of the x-y-z coordinate system. The x′-y′ coordinate system is for representing positions of the imaging targets (the markers  8   a  and  8   b ) in a taken image which are calculated by the imaging information calculation section  35 . By using the same coordinate system to process the inclination of the controller  7  calculated from the output of the acceleration sensor  37  and the positions of the imaging targets obtained from the imaging information calculation section  35 , processing of reflecting both the inclination and the positions on the game operation is facilitated. In other embodiments, it is not absolutely necessary to use the images of the markers  8   a  and  8   b  for the game operation, in which case the above-mentioned coordinate set conversion is not necessary. 
   After calculating the preliminary vector vh, the game apparatus  3  calculates an inclination vector vah (see  FIG. 10 ) based on the preliminary vector vh and a previous vector vbh. The inclination vector represents an inclination of the controller  7  to be calculated and is used for the game operation. The previous vector vbh is an inclination vector calculated previously. In this specification, the term “previous” means “immediately previous”. In the case where, for example, the game apparatus  3  calculates an inclination vector frame by frame, the previous vector vbh is an inclination vector calculated one frame before. Like the preliminary vector vh, the inclination vector vah and the previous vector vbh are both a unit vector having a length of 1, and have the origin of the x′-y′ coordinate system as a start point. 
   As shown in  FIG. 10 , the inclination vector vah is obtained by making the direction of the previous vector vbh closer to the direction of the preliminary vector vh at a predetermined degree. In the following description, the predetermined degree will be represented as an effectiveness k (0≦k≦1). Specifically, the inclination vector vah is directed from the origin toward point P and has a length of 1. Point P divides a line segment connecting the end point of the previous vector vbh and the end point of the preliminary vector vh at a ratio of k:(1−k). In the first embodiment, the effectiveness k is calculated based on the length of the preliminary vector vh. A method for calculating the effectiveness k will be described in detail later. 
   In the state where the controller  7  is being moved or rotated by the operation of the player, acceleration components other than the acceleration of gravity are detected. Therefore, the acceleration vector itself cannot be regarded as the inclination of the controller  7 . An inclination of the controller  7  calculated only based on the acceleration vector is not accurate. With reference to  FIG. 11  and  FIG. 12 , this will be described. 
     FIG. 11  shows the controller  7  in a still state.  FIG. 11  and  FIG. 12  shows the controller  1  seen from the rear end thereof. In the state shown in  FIG. 11 , the direction of an acceleration vector Va 1  detected by the acceleration sensor  37  matches a direction G of an acceleration of gravity. Therefore, the acceleration vector Va 1  itself can be regarded as the inclination of the controller  7 . The inclination thus calculated is accurate.  FIG. 12  shows the controller  7  in the middle of being rotated counterclockwise from the state in  FIG. 11 . In the state shown in  FIG. 12 , an acceleration vector Va 2  which is offset clockwise (in the opposite direction to the moving direction of the controller  7 ) from the direction G of the actual acceleration of gravity is detected. As shown in  FIG. 12 , the direction of the acceleration vector changes more drastically than the inclination of the controller  7 . The length of the acceleration vector Va 2  is different from the length of the acceleration vector Va 1  which matches the acceleration of gravity (in the example of  FIG. 12 , the acceleration vector Va 2  is longer than the acceleration vector Va 1 ). (It is considered that such changes are caused by, for example, a centrifugal force or an influence of an acceleration by the destabilization of the hand of the like). An inclination of the controller  7  calculated only based on the acceleration vector Va 2  is not accurate. As described above, in the state of the controller  7  in the middle of being moved or rotated by the player, the inclination of the controller  7  cannot be accurately calculated because the acceleration detected by the acceleration sensor  37  often includes components other than the acceleration of gravity. It may be possible to wait until the controller  7  gets still and calculate the inclination thereof using an acceleration vector obtained after the controller  7  gets still. This method, however, cannot calculate an inclination of the controller  7  in real time and thus cannot be easily applied to a game apparatus or the like which requires a quick responsiveness for operations thereof. 
   According to the first embodiment, the current inclination of the controller  7  is calculated using the direction of the current acceleration vector (the direction of the preliminary vector) and also the direction of the previous inclination vector (see  FIG. 10 ), instead of in the direction of the current acceleration vector as the inclination of the controller  7 . With this method, even when the direction of the acceleration vector rapidly changes from the direction of the previous acceleration vector, the inclination vector is calculated such that the direction thereof is changed more slowly than the direction of the acceleration vector. In the example of  FIG. 12 , the direction of the inclination vector is between the direction of the previous acceleration vector Va 1  and the preliminary vector Va 2  (for example, the inclination vector is a vector Va 3 ). With this method, an inclination vector representing an inclination closer to an accurate inclination (i.e., closer to the actual instantaneous inclination) of the controller  7  can be obtained as compared to the case where the direction of the acceleration vector itself (i.e., as currently detected) is used as the inclination of the controller  7 . According to the first embodiment, the inclination of the controller  7  can be calculated more accurately. 
   According to the first embodiment, information used for calculating the inclination vector is the current acceleration vector and the previous inclination vector (i.e., the previous vector). Thus, according to the first embodiment, the inclination of the controller  7  can be calculated using only the information owned by the game apparatus  3  at the time when the acceleration sensor  37  obtains the acceleration vector. Therefore, the inclination of the controller  7  can be calculated in real time. 
   Next, game processing executed by the game apparatus  3  will be described in detail. First, main data used for the game processing will be described with reference to  FIG. 13 .  FIG. 13  shows main data stored on the main memory  13  of the game apparatus  3 . As shown in  FIG. 13 , the main memory  13  has stored thereon a game program  51 , operation data  52 , calculation processing data  53  and the like. In addition to the above-mentioned data, the main memory  13  has stored thereon image data of characters appearing in the game, data representing various parameters of the characters, and other data necessary for the game processing. 
   The game program  51  is partially or entirely read from the optical disc  4  at an appropriate time after the game apparatus  3  is powered on and stored on the main memory  13 . The game program  51  includes an inclination calculation program  511 . The inclination calculation program  511  is a program for executing the processing of calculating an inclination of the controller  7  (the inclination calculation processing) using an output from the acceleration sensor  37 . The game program  51  includes programs necessary for the game processing in addition to the inclination calculation program  511 . 
   The operation data  52  is transmitted from the controller  7  to the game apparatus  3  and stored on the main memory  13 . The operation data  52  includes acceleration data  521 . The acceleration data  521  represents an acceleration vector Va detected by the acceleration sensor  37 . Herein, the acceleration data  521  represents an acceleration in each of three axial directions (x-y-z directions) shown in  FIG. 7 . In addition to the acceleration data  521 , the operation data  52  includes data representing positions of the imaging targets (the markers  8   a  and  8   b ) in a taken image and data representing operations performed on the buttons and switches of the operation section  32 . 
   The calculation processing data  53  is used for the inclination calculation processing. The calculation processing data  53  includes preliminary data  531 , inclination data  532 , previous data  533 , two-axial length data  534 , effectiveness data  535 , and first variable data  536 . 
   The preliminary data  531  represents a preliminary vector vh mentioned above. More specifically, the preliminary data  531  represents an inclination uniquely determined from the acceleration data  521 , i.e., an inclination of the controller  7  calculated under an assumption that the acceleration data  521  represents an acceleration of gravity. The inclination data  532  represents an inclination vector vah. The inclination data  532  is basically calculated frame by frame, but may not be calculated when the acceleration vector Va represented by the acceleration data  521  fulfills a condition described later (step S 12 ). The previous data  533  represents a previous vector vbh. When the inclination data  532  is newly calculated (i.e., updated) and thus the main memory  13  is updated, the newly calculated inclination data is also stored on the main memory  13  as updated “previous data”  533  to be used in the next frame for calculating an inclination vector. 
   The two-axial length data  534  represents a length L 1  of the acceleration vector Va represented by the acceleration data  521  in two axial directions (x-y directions) (i.e., the length of a vector having an x component and a y component of the acceleration vector Va). The length L 1  is used for, for example, calculating an effectiveness k. The effectiveness data  535  represents an effectiveness k, which indicates a degree at which the previous vector vbh is made closer to the preliminary vector vh. As described later in more detail, the effectiveness k is a variable representing a degree at which the acceleration of gravity contributes to the data of the acceleration vector (a degree at which the acceleration vector is directed in the direction of the acceleration of gravity). 
   The first variable data  536  represents a first variable d 1  which indicates, in the case where the magnitude of the acceleration of gravity detected by the acceleration sensor  37  is 1, how close to 1 the length L 1  of the acceleration vector is. The first variable d 1  is used for calculating the effectiveness k. 
   Next, the game processing executed by the game apparatus  3  will be described in detail with reference to  FIG. 14  and  FIG. 15 .  FIG. 14  is a flowchart illustrating a flow of the game processing executed by the game apparatus  3 . When the game apparatus  3  is turned on, the CPU  10  of the game apparatus  3  executes a start program stored on the boot ROM (not shown) to initialize each unit such as the main memory  13 . The game program stored on the optical disc  4  is read into the main memory  13 , and the CPU  10  starts the execution of the game program. The flowchart shown in  FIG. 14  illustrates the game processing after the above-described processing is completed. With reference to  FIG. 14 , the game processing for calculating an inclination of the controller  7  from an acceleration detected by the acceleration sensor  37  will be explained in detail, and other game processing not directly relevant to the present invention will be omitted. 
   First in step S 1 , a game space is constructed and displayed on the monitor  2 . The CPU  10  constructs, for example, a three-dimensional game space (or a two-dimensional game space) and locates objects appearing in the game space at predetermined initial positions. A game image representing the game space thus constructed is generated and displayed on the monitor  2 . After this, the processing loop of steps S 2  through S 6  is repeated frame by frame, and thus the game proceeds. 
   In step S 2 , the CPU  10  obtains operation data from the controller  7 . More specifically, the controller  7  transmits the operation data to the game apparatus  3  at a predetermined time interval (for example, frame by frame), and the CPU  10  stores the transmitted operation data on the main memory  13 . The operation data includes the acceleration data. 
   Next in step S 3 , the CPU  10  executes the inclination calculation program  511  to execute the inclination calculation processing. In the inclination calculation processing, an inclination of the controller  7  is calculated based on the acceleration data  521  included in the operation data  52  stored on the main memory  13  in step S 2 . Hereinafter, with reference to  FIG. 15 , the inclination calculation processing will be described in detail. 
     FIG. 15  is a flowchart illustrating a detailed flow of the inclination calculation processing in step S 3  shown in  FIG. 14 . The inclination calculation processing is executed as follows. First in step S 11 , a length L 1  regarding x and y components of the acceleration data Va is calculated. When the acceleration vector Va is (ax, ay, az), the length L 1  is calculated in accordance with the following expression.
   L 1=( ax   2   +ay   2 ) 1/2   
   The CPU  10  stores data representing the calculated length L 1  on the main memory  13  as the two-axial length data  534 . 
   Next in step S 12 , it is determined whether or not the length L 1  calculated in step S 11  is within a predetermined range. The predetermined range is determined in advance, and is 0&lt;L 1 ≦2 in this non-limiting example implementation. When it is determined in step S 12  that the length L 1  is within the predetermined range, processing in step S 13  is executed. When it is determined in step S 12  that the length L 1  is not within the predetermined range, processing in steps S 13  through S 18  is skipped and the CPU  10  terminates the inclination calculation processing. Namely, when the result of the determination in step S 12  is negative, the inclination of the controller  7  is not calculated. 
   The processing in step S 12  is executed in order to prevent an inclination of the controller  7  from being calculated when the length L 1  calculated in step S 1  is not encompassed in a predetermined range (in this embodiment, 0&lt;L 1 ≦2) around the magnitude of the acceleration of gravity. It is considered that the length L 1  is not encompassed in the predetermined range when the acceleration detected by the acceleration sensor  37  includes many component other than the acceleration of gravity as a result of the controller  7  being violently moved by the player or the like. In such a case, it is difficult to obtain an accurate inclination of the controller  7  from the output of the acceleration sensor  37 . Therefore, in this embodiment, an inaccurate inclination is prevented from being obtained in such a case by executing the determination processing in step S 12 . 
   In step S 13 , a preliminary vector vh is calculated. The preliminary vector vh can be calculated from the acceleration vector Va. The CPU  10  refers to the acceleration data  521  and the two-axial length data  534  stored on the main memory  13  to calculate components (hx, hy) of the preliminary vector vh in accordance with the following expressions.
 
 hx=−ay/L 1
 
 hy=−ax/L 1
 
   In the above expressions, ax is a value of the x component of the acceleration vector Va, and ay is a value of the y component of the acceleration vector Va. The reason why −ay is used for calculating hx and −ax is used for calculating hy in the above expressions is that coordinate set conversion from the coordinate system of the acceleration vector (x-y-z coordinate system) into the coordinate system of the preliminary vector (x′-y′ coordinate system) is to be performed. The reason why −ay and −ax are each divided by the length L 1  is that the length of the preliminary length is to be 1. The CPU  10  stores data representing the calculated preliminary vector vh (=(hx, hy)) on the main memory  13  as the preliminary data  531 . 
   By a series of processing in steps S 14  through S 16 , a first variable d 1  is calculated based on the length L 1 . In the series of processing, the first variable d 1  is calculated such that the value of the first variable d 1  is greater within the range of 0≦d 1 ≦1 as the length L 1  is closer to 1. First in step S 14 , it is determined whether or not the length L 1  is less than 1. The CPU  10  can find the value of the length L 1  by referring to the two-axial length data  534  stored on the main memory  13 . When it is determined in step S 14  that the length L 1  is less than 1, processing in step S 15  is executed. When it is determined in step S 14  that the length L 1  is equal to or greater than 1, processing in step S 16  is executed. 
   In step S 15 , the value of the length L 1  is set as the value of the first variable d 1 . In step S 16 , the first variable d 1  is calculated in accordance with the following expression.
 
 d 1=2 −L 1
 
   Namely, the first variable d 1  is represented as a closeness of the length L 1  to 1. Data representing the first variable d 1  obtained in step S 15  or S 16  is stored on the main memory  13  as the first variable data  536 . After step S 15  or S 16 , processing in step S 17  is executed. 
   In step S 17 , an effectiveness k is calculated based on the first variable d 1 . As described above, the effectiveness k is a variable representing a degree at which the direction of the previous vector vbh is made closer to the direction of the preliminary vector vh for calculating an inclination vector vah. Specifically, the CPU  10  calculates the effectiveness k in accordance with the following expression.
 
 k=d 1 2   ×A 
 
   In the above expression, A (&gt;0) is a constant predetermined in the inclination calculation program  511 . Data representing constant A is stored on the main memory  13  in advance. The CPU  10  stores data representing the calculated effectiveness k on the main memory  13  as the effectiveness data  535 . As can be appreciated from the above expression, the effectiveness k is greater in the range of 0≦k≦1 as the value of the first variable d 1  is greater. 
   Next in step S 18 , an inclination vector vah is calculated. In this embodiment, the inclination vector vah is calculated using the preliminary vector vh, the previous vector vbh, and the effectiveness k. Specifically, the CPU  10  first calculates a vector (ahx′, ahy′) in accordance with the following expressions.
 
 ahx ′=( hx−bhx )× k+bhx 
 
 ahy ′=( hy−bhy )× k+bhy 
 
   In the above expressions, the preliminary vector vh is (hx, hy) and the previous vector vbh is (bhx, bhy). The vector (ahx′, ahy′) calculated by the above expressions is directed in the same direction as the inclination vector vah. Next, the CPU  10  corrects the above-calculated vector into a unit vector in accordance with the following expressions, thus to calculate the inclination vector vah (=(ahx, ahy)).
 
 ahx=ahx ′/(( ahx′   2   +ahy′   2 ) 1/2 )
 
 ahy=ahy ′/(( ahx′   2   +ahy′   2 ) 1/2 )
 
   The inclination vector vah is calculated by the above expressions. The CPU  10  stores data representing the calculated inclination vector on the main memory  13  as the updated inclination data  532 . The CPU  10  also stores the post-update inclination data  532  on the main memory  13  as the previous data  533  to be used in the next frame for calculating an inclination vector. After step S 18 , the CPU  10  terminates the inclination calculation processing. 
   As the controller  7  is moved more violently, the value of the length L 1  is farther from the magnitude of the acceleration of gravity (1 in this embodiment). Based on the length L 1 , the moving state of the controller  7 , i.e., whether the controller  7  is being moved violently, being moved slowly, or being still, can be found. As the value of the length L 1  is closer to 1, the effectiveness k is greater. Therefore, in this embodiment, the value of the effectiveness k is smaller as the controller k is moved more violently, whereas the value of the effectiveness k is greater as the controller k is closer to being still. When the controller  7  is being moved violently, it can be determined that the acceleration sensor  37  cannot accurately detect the acceleration of gravity. Therefore, the effectiveness k is smaller, and the inclination vector is closer to the previous vector. By contrast, when the controller  7  is closer to being still, it can be determined that the acceleration sensor  37  can accurately detect only the acceleration of gravity. Therefore, the effectiveness k is greater, and the inclination vector is closer to the preliminary vector. The inclination vector can be accurately calculated in step S 18  by varying the ratio between the preliminary vector and the previous vector (the effectiveness k) in accordance with how reliable the output of the acceleration sensor  37  is (how accurately the output of the acceleration sensor  37  represents the acceleration of gravity). 
   Returning to  FIG. 14 , in step S 4  after step S 3 , the game processing based on the inclination of the controller  7  calculated in step S 3  is executed. Specifically, the inclination data  532  stored on the main memory  13  is transferred (output) to the program for executing the game processing, and the game processing is executed in accordance with the program. The game processing is, for example, processing of moving a player character appearing in the game space in accordance with the inclination. When the inclination is not calculated in the inclination calculation processing (i.e., when the determination result in step S 12  shown in  FIG. 15  is negative), the CPU  10  may execute the game processing based on an inclination represented by the previous inclination vector (previous vector) or execute the game processing assuming that the player did not provide any an input. 
   Next in step S 5 , a game image reflecting the result of the game processing executed in step S 4  is generated and displayed on the monitor  2 . Next in step S 6 , the CPU  10  determines whether or not to terminate the game. The determination in step S 6  is made in accordance with, for example, whether or not the player has cleared the game, or when a time limit is provided for the game, whether or not the time has passed. When the result of determination in step S 6  is negative, the processing returns to step S 2  and the processing loop in steps S 2  through S 6  is repeated until it is determined that the game is to be terminated. When the result of determination in step S 6  is positive, the CPU  10  terminates the game processing shown in  FIG. 14 . So far, the game processing has been described. 
   As described above, according to this embodiment, an inclination vector is calculated using a preliminary vector and a previous vector. Thus, the inclination of the controller  7  can be calculated in real time. By varying the degree at which the previous vector is made closer to the preliminary vector (the effectiveness k) in accordance with the magnitude of the detected acceleration, the inclination can be more accurately calculated. 
   In the first embodiment, an inclination of the controller  7  in the x-y directions is calculated as an example. The present invention is applicable to calculating an inclination of the controller  7  in the x-y-z directions. In this case, the length of the acceleration vector is used instead of the length L 1 , and the preliminary vector and the inclination vectors are calculated as three-dimensional vectors. 
   Second Embodiment 
   Next, a game system including a game apparatus as an example of an inclination calculation apparatus according to a second embodiment of the present invention will be described. The hardware structure of the game system according to the second embodiment is similar to the game system  1  according to the first embodiment. In the second embodiment, the contents of the inclination calculation processing are different from those of the first embodiment. Hereinafter, the second embodiment will be described mainly regarding the differences thereof from the first embodiment. 
     FIG. 16  shows main data stored on the main memory  13  of the game apparatus  3  in the second embodiment. In  FIG. 16 , data identical as that in  FIG. 13  each bear an identical reference numeral and detailed descriptions thereof will be omitted. 
   In the second embodiment, the calculation processing data  53  includes second acceleration data  61 , three-axial length data  62 , and second variable data  63  in addition to the data shown in  FIG. 13 . In the second embodiment, the acceleration data  521  will be referred as a “first acceleration data  521 ” to be distinguished from the second acceleration data  62 . The first acceleration data  521  represents the same content as that of the acceleration data  521 . 
   The second acceleration data  61  represents an acceleration detected by the acceleration sensor  37  in the x-y directions. Namely, the second acceleration data  61  is represented by a two-dimensional vector. In the second embodiment, a three-dimensional vector detected by the acceleration sensor  37  will be referred to as a “first acceleration vector Va” (=(ax, ay, az)), and the acceleration vector representing the acceleration in the x-y directions will be referred to as a “second acceleration vector Vc” (=(cx, cy)). 
   The three-axial length data  62  represents a length L 2  of the first acceleration vector Va (a three-dimensional vector having an x component, a y component and a z component of the first acceleration vector Va). 
   The second variable data  63  represents a second variable d 2 , which indicates, in the case where the magnitude of the acceleration of gravity detected by the acceleration sensor  37  is 1, how close to 1 the length L 2  of the first acceleration vector Va is. The second variable d 2  is used for calculating the effectiveness k together with the first variable d 1 . 
     FIG. 17  and  FIG. 18  are flowcharts illustrating a detailed flow of the inclination calculation processing executed in the second embodiment. Except for the inclination calculation processing shown in  FIG. 17  and  FIG. 18 , the game processing in the second embodiment is substantially the same as that of the first embodiment. 
   In the second embodiment, the inclination calculation processing is executed as follows. First in step S 21 , a length L 2  in the three axial directions of the first acceleration vector Va is calculated. When the first acceleration vector Va is (ax, ay, az), the length L 2  is calculated in accordance with the following expression.
 
 L 2=( ax   2   +ay   2   +az   2 ) 1/2 
 
   The CPU  10  stores data representing the calculated length L 2  on the main memory  13  as the three-axial length data  62 . 
   Next in step S 22 , it is determined whether or not the length L 2  calculated in step S 21  is within a predetermined range. The predetermined range is determined in advance, and is 0&lt;L 2 ≦2 in this embodiment. When it is determined in step S 22  that the length L 2  is within the predetermined range, processing in step S 23  is executed. When it is determined in step S 22  that the length L 2  is not within the predetermined range, processing in steps S 23  through S 36  is skipped and the CPU  10  terminates the inclination calculation processing. Namely, when the result of the determination in step S 22  is negative, the inclination of the controller  7  is not calculated. The processing in step S 22 , like the processing in step S 12  in the first embodiment, is executed in order to prevent an inaccurate inclination of the controller  7  from being calculated when it is difficult to obtain an accurate inclination from the output of the acceleration sensor  37 , i.e., when the acceleration detected by the acceleration sensor  37  includes many components other than the acceleration of gravity. 
   By a series of processing in steps S 23  through S 25 , a second variable d 2  is calculated based on the length L 2 . In the series of processing, the second variable d 2  is calculated such that the value of the second variable d 2  is greater within the range of 0≦d 2 ≦1 as the length L 2  is closer to 1. First in step S 23 , it is determined whether or not the length L 2  is less than 1. The CPU  10  can find the value of the length L 2  by referring to the three-axial length data  62  stored on the main memory  13 . When it is determined in step S 23  that the length L 2  is less than 1, processing in step S 24  is executed. When it is determined in step S 23  that the length L 2  is equal to or greater than 1, processing in step S 25  is executed. 
   In step S 24 , the value of the length L 2  is set as the value of the second variable d 2 . In step S 25 , the second variable d 2  is calculated in accordance with the following expression.
 
 d 2=2 −L 2
 
   Data representing the second variable d 2  obtained in step S 24  or S 25  is stored on the main memory  13  as the second variable data  63 . After step S 24  or S 25 , processing in step S 26  is executed. 
   In step S 26 , a second acceleration vector Vc is calculated based on the first acceleration vector Va. When the first acceleration vector Va is (ax, ay, az), the second vector Vc=(cx, cy) can be calculated in accordance with the following expressions.
 
cx=ax
 
cy=ay
 
   The CPU  10  stores data representing the calculated second acceleration vector Vc on the main memory  13  as the second acceleration data  61 . 
   Next in step S 27 , a length L 1  of the second acceleration vector Vc in the two axial directions is calculated. The length L 1  can be calculated in accordance with the following expression.
 
 L 1=( cx   2   +cy   2 ) 1/2 
 
   The CPU  10  stores data representing the calculated length L 1  on the main memory  13  as the two-axial length data  534 . 
   Next in step S 28 , it is determined whether or not the length L 1  calculated in step S 27  is 0. When it is determined in step S 28  that the length L 1  is not 0, processing in step S 29  is executed. When it is determined in step S 28  that the length L 1  is 0, processing in step S 29  through S 34  is skipped and the CPU  10  terminates the inclination calculation processing. Namely, when the result of the determination in step S 28  is positive, the inclination of the controller  7  is not calculated. 
   It is considered that the length L 1  is 0 in step S 28  when, for example, the controller  7  is in the up-down direction. In this case, an inclination to be calculated, i.e., an inclination in the x-y directions cannot be calculated. Therefore, in the second embodiment, the processing after step S 28  is skipped. 
   In step S 29 , a preliminary vector vh is calculated. In step S 29 , the preliminary vector vh can be calculated in accordance with the following expressions.
 
 hx=−cy/L 1
 
 hy=−cx/L 1
 
   The CPU stores data representing the calculated preliminary vector vh (=(hx, hy)) on the main memory  13  as the preliminary data  531 . 
   In the first and second embodiments, it is intended to calculate an inclination of the controller  7  in the x-y directions in the x-y-z coordinate system. This is why the preliminary vector vh is calculated as in steps S 13  and S 29 . In the case where it is intended to calculate an inclination in a coordinate system based on a space in which the controller  7  exists, the second acceleration vector Vc (=(cx, cy)) may be calculated as follows. For example, in step S 26 , the second acceleration vector may be calculated in accordance with the following expressions.
 
cx=ax
 
 cy =( ay   2   +az   2 ) 1/2 ( ay&lt; 0),  cy =−( ay   2   +az   2 ) 1/2 ( ay≧ 0)
 
   By calculating the second acceleration vector Vc in this manner, an absolute inclination in the space can be calculated. 
   Next, by a series of processing in steps S 30  through S 32 , a first variable d 1  is calculated based on the length L 1 . In the series of processing, the first variable d 1  is calculated such that the value of the first variable d 1  is greater within the range of 0≦d 1 ≦1 as the value of the length L 1  is closer to 1. The processing in steps S 30  through S 32  is the same as the processing in steps S 14  through S 16 . Data representing the first variable d 1  obtained in steps S 31  or S 32  is stored on the main memory  13  as the first variable data  53 . After step S 31  or S 32 , processing in step S 33  is executed. 
   In step S 33 , an effectiveness k is calculated based on the first variable d 1  and the second variable d 2 . Specifically, the CPU  10  calculates the effectiveness k in accordance with the following expression.
 
 k=d 1 2   ×d 2 2   ×A 
 
   In the above expression, A (&gt;0) is a constant predetermined in the inclination calculation program  511 . Data representing constant A is stored on the main memory  13  in advance. The CPU  10  stores data representing the calculated effectiveness k on the main memory  13  as the effectiveness data  535 . As can be appreciated from the above expression, the effectiveness k is greater in the range of 0≦k≦1 as the value of the first variable d 1  is greater and as the value of the second variable d 2  is greater. 
   Next in step S 34 , an inclination vector vah is calculated. In the second embodiment, the inclination vector vah is calculated in the same manner as in the first embodiment. Namely, the processing in step S 34  is the same as the processing in step S 18 . After step S 34 , the CPU  10  terminates the inclination calculation processing. 
   As described above, according to the second embodiment, an acceleration in the x-y-z directions is used in order to calculate an inclination of the controller  7  in the x-y directions. Specifically, in the second embodiment, the game apparatus  3  varies the effectiveness k based on the magnitude of the acceleration in the x-y-z directions (length L 2 ). This magnitude represents the acceleration of the controller  7  more accurately than the length L 1 . Therefore, the moving state of the controller  7  can be more accurately determined using this magnitude. By varying the effectiveness kin accordance with this magnitude, the inclination of the controller  7  can be calculated more accurately. In the second embodiment, the effectiveness k is calculated using both the length L 1  and length L 2 . In other embodiments, the effectiveness k may be calculated using either the length L 1  or length L 2 . 
   In the first and second embodiments, the inclination of the controller  7  is represented as a vector. Alternatively, the inclination of the controller  7  may be represented by another element, for example, an angle with respect to a certain direction. 
   As described above, the present invention is usable for, for example, a game apparatus or a game program as described above for calculating an inclination of a device in real time using an acceleration detected by acceleration detection means. 
   While the invention has been described in detail, the foregoing description is in all aspects illustrative and not restrictive. It is understood that numerous other modifications and variations can be devised without departing from the scope of the invention.