JP2009070340A - Skeleton operation control system, program, and information storage medium - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a skeleton operation control system capable of securing a sufficient movable range of joints with a high degree of freedom to attain smooth operation expression of a character, and to provide a program and an information storage medium. <P>SOLUTION: The skeleton operation control system controls the operation of a skeleton model with a master bone and a slave bone connected by a joint. A movable range setting section 112 sets a given point on a spherical surface with the joint as a center point, as a focus, sets a plane orthogonal to an axis connecting the focus to the center point corresponding to the joint, as a projective plane, and sets the movable range of the joint on the projective plane. A coordinate conversion section 112B projects a point indicating the direction of the slave bone on the spherical surface, on the projective plane based on the focus. A skeleton operation computing section 112C computes the direction of the slave bone with respect to the master bone under the limitation of the movable range set by the movable range setting section 112A based on the position of the point projected on the projective plane. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a skeletal motion control system, a program, and an information storage medium.

  2. Description of the Related Art Conventionally, there has been known a method for calculating a motion of a character (for example, a person, an animal), a rope, a cloth, and the like in a virtual three-dimensional space (object space) using a skeleton model (skeleton model). The skeletal model is composed by connecting multiple bones with joints. Generally, a parent-child relationship is set between the bones, and the operation of determining the movement of the skeleton is performed so as to follow the movement of the parent bone. Yes.

  In such a skeleton model, the degree of freedom differs for each joint, and the movable range differs for each joint. For example, joints corresponding to elbows and knees in a human body model tend to have a low degree of freedom and the movable range tends to be narrow, and joints corresponding to shoulders and the like have a high degree of freedom and tend to be widened.

Further, when calculating the movement of the skeleton corresponding to the movement of the parent bone, a process for calculating rotation information such as the bending state of the skeleton relative to the parent bone and the twisting state of the skeleton relative to the parent bone at the joint is performed. In the technique, the orientation of the skeleton with respect to the parent bone is expressed by rotation angles around three axes (x, y, z) with respect to the parent bone coordinate system.
JP 2000-306116 A

  However, when the motion of the skeletal model is designed by simply applying the conventional method, if the motion range of the joint is set in a wide range, the motion range of the joint is far from the intention of the motion designer. However, there is a problem that it is difficult to express a smooth character motion.

  The present invention has been made in view of the above circumstances, and an object of the present invention is to provide a skeletal motion control system, a program, and an information storage medium that can ensure a smooth motion of a character by sufficiently securing a movable range of joints. There is to do.

  (1) The present invention is a program for controlling the operation of a skeletal model in which a parent bone and a child bone are connected by a joint, and focuses on a given point on a sphere centered on the joint, A plane perpendicular to the axis connecting the center point and the focal point is a projection plane, a movable range setting unit for setting the movable range of the joint on the projected plane, and a point representing the orientation of the skeleton on the spherical surface And a coordinate conversion unit that projects onto the projection surface based on the focal point, and a position of the point projected on the projection surface, and the orientation of the child bone relative to the parent bone is limited under the movable range. The present invention relates to a skeletal motion control system including a skeletal motion calculation unit that calculates. The present invention also relates to a program that causes a computer to function as each of the above-described units and a computer-readable information storage medium that stores such a program.

  The “point indicating the orientation of the skeleton on the spherical surface” in the present invention can be, for example, the intersection of the skeleton or the extension line of the skeleton and the spherical surface.

  According to the present invention, the range of motion of the joint can be reduced compared to the conventional method of setting the range of motion of the joint by converting the point on the spherical surface onto the projection surface using projection and setting the range of motion. It can be set in a wide range and orderly range, and an operation for smoothly operating the skeleton model can be realized. For example, for a joint with a narrow range of motion, the skeletal model can be operated with the same restrictions as a conventionally used method, and for a joint with a wide range of motion, it is more natural than the conventionally used method. It can express the behavior of a smooth skeleton model.

  (2) In the skeletal motion control system, the program, and the information storage medium of the present invention, the movable range setting unit may set the movable range so as to form a rectangle on the projection surface.

  In this way, the motion range of the joint can be easily grasped, and the motion design of the skeleton model can be designed with the same feeling as the conventionally used method, so the environment of the skeletal model motion design is improved. be able to.

  (3) In the skeletal motion control system, the program, and the information storage medium of the present invention, the movable range setting unit may set the movable range so as to form an ellipse on the projection plane.

  In this way, the motion range of the joint can be easily grasped, and the motion design of the skeleton model can be designed with the same feeling as the conventionally used method, so the environment of the skeletal model motion design is improved. be able to.

  (4) In the skeletal motion control system, the program, and the information storage medium of the present invention, the movable range setting unit may set the movable range so as to form an oval shape on the projection plane.

  In this way, the motion range of the joint can be easily grasped, and the motion design of the skeleton model can be designed with the same feeling as the conventionally used method, so the environment of the skeletal model motion design is improved. be able to.

(5) In the skeletal motion control system, the program, and the information storage medium of the present invention, the axis along the parent bone is set as the x axis, and the two axes orthogonal to the parent bone and orthogonal to each other are the y axis. The coordinate axes orthogonal to each other are set as the s-axis and the t-axis, respectively, on the projection plane, and the coordinate conversion unit s = −z / (X + 1) and t = y / (x + 1) {where x ² + y ² + z ² = 1} is used to project a point (x, y, z) on the spherical surface to a point on the projected surface You may make it convert into (s, t).

  In this way, since the transformation by projection is composed of only four arithmetic operations, the calculation load is small.

  (6) In the skeletal motion control system, the program, and the information storage medium of the present invention, the shape of the movable range formed on the projection plane for the joint of the skeleton model is preset as one of a rectangle, an ellipse, and an oval And an allowable range of the rotation angle of the skeleton around the y-axis and the rotation angle of the skeleton around the z-axis in the joint is set in advance, and the movable range setting unit is formed on the projection plane. The movable range of the joint is set on the projection plane based on the shape of the movable range and the allowable range of the rotation angle of the skeleton around the y axis and the rotation angle of the skeleton around the z axis in the joint. You may do it.

  In this way, it is possible to easily set the movable range of the joint by effectively utilizing the motion design assets of the skeleton model to which the conventional method is applied.

  (7) In the skeletal motion control system, the program, and the information storage medium of the present invention, the skeleton motion with respect to the master bone obtained by the skeleton motion calculation unit calculating the motion of the skeleton bone corresponding to the motion of the master bone. If the orientation of the child bone relative to the parent bone is not within the movable range, the orientation of the child bone is determined according to the shape of the movable range on the projection plane. The orientation of the child bone relative to the parent bone may be corrected so as to satisfy the following conditions.

  In this way, it is possible to obtain a bone orientation that can express a natural movement while suppressing a calculation load.

  Embodiments of the present invention will be described below. The embodiments described below do not unduly limit the contents of the present invention described in the claims. In addition, all the configurations described in the embodiments described below are not necessarily essential configuration requirements of the present invention.

1. Configuration FIG. 1 shows an example of a functional block diagram of a game system (an example of a skeleton motion control system) according to the present embodiment. Note that the game system of this embodiment may have a configuration in which some of the components (each unit) in FIG. 1 are omitted.

  The operation unit 160 is for a player to input operation data of a character (multi-joint model such as a person, animal, or robot), and functions as a lever, a button, a steering, a microphone, a touch panel display, or a housing. It can be realized by. The storage unit 170 serves as a work area for the processing unit 100, the communication unit 196, and the like, and its function can be realized by a RAM, a VRAM, or the like.

  The information storage medium 180 (computer-readable medium) stores programs, data, and the like, and functions as an optical disk (CD, DVD), magneto-optical disk (MO), magnetic disk, hard disk, and magnetic tape. Alternatively, it can be realized by a memory (ROM). The processing unit 100 performs various processes of the present embodiment based on a program (data) stored in the information storage medium 180. That is, the information storage medium 180 stores a program for causing a computer to function as each unit of the present embodiment (a program for causing a computer to execute processing of each unit).

  The display unit 190 outputs an image generated according to the present embodiment, and its function can be realized by a CRT, LCD, touch panel display, HMD (head mounted display), or the like. The sound output unit 192 outputs the sound generated by the present embodiment, and its function can be realized by a speaker, headphones, or the like.

  The portable information storage device 194 stores player personal data, game save data, and the like. Examples of the portable information storage device 194 include a memory card and a portable game device. The communication unit 196 performs various controls for communicating with the outside (for example, a host device or other image generation system), and functions thereof are hardware such as various processors or communication ASICs, programs, and the like. It can be realized by.

  A program (data) for causing a computer to function as each unit of the present embodiment is distributed from the information storage medium of the host device (server) to the information storage medium 180 or the storage unit 170 via the network and the communication unit 196. Also good. Use of the information storage medium of such a host device (server) can also be included in the scope of the present invention.

  The processing unit 100 performs processing such as game processing, image generation processing, or sound generation processing based on operation data and programs from the operation unit 160. Here, as the game process, a process for starting a game when a game start condition is satisfied, a process for advancing the game, a process for placing an object such as a character or a map, a process for displaying an object, and a game result are calculated. There is a process or a process of ending a game when a game end condition is satisfied. The processing unit 100 performs various processes using the storage unit 170 as a work area. The functions of the processing unit 100 can be realized by hardware such as various processors (CPU, GPU, DSP, etc.), ASIC (gate array, etc.), and programs.

  The processing unit 100 includes an object space setting unit 110, a movement / motion processing unit 112, a virtual camera control unit 114, a drawing unit 120, and a sound generation unit 130. Note that some of these may be omitted.

  The object space setting unit 110 is based on the object information stored in the object information storage unit 172A of the main memory 172, and includes a car, a character, a building, a stadium, a tree, a pillar, a wall, a course (road), and a map (terrain). Various kinds of objects (objects composed of primitives such as a polygon, a free-form surface, or a subdivision surface) representing a display object such as are arranged and set in the object space. Specifically, the position and rotation angle (synonymous with orientation and direction) of the object in the world coordinate system are determined based on the processing result of the movement / motion processing unit 112 described later, and the determined position (X, Y , Z), the objects are arranged at the rotation angles determined (the rotation angles around the X axis, the Y axis, and the Z axis).

  The movement / motion processing unit 112 performs movement / motion calculation (movement / motion simulation) of an object (such as a character). That is, based on operation data input by the player through the operation unit 160, a program (movement / motion algorithm), various data (motion data), or the like, the object is moved in the object space, or the object is moved (motion, animation). ) Process. Specifically, object movement information (position, rotation angle, speed, or acceleration) and motion information (position or rotation angle of each part that constitutes the object) are sequentially transmitted every frame (1/60 seconds). Perform the required simulation process. A frame is a unit of time for performing object movement / motion processing (simulation processing) and image generation processing.

  In particular, in the present embodiment, as shown in FIG. 2, a plurality of bones are connected by joints, one bone connected by the joints is a parent bone, and other bones are connected by the same joint as the parent bone. A process is performed in which the motion of the object is determined using a skeleton model in which the movement of each bone is determined so that the skeleton can follow the movement of the parent bone with the bone as the skeleton. In the example of the skeletal model shown in FIG. 2, in the joint A, the bone A corresponds to the parent bone, and the bone B corresponds to the child bone. The movement / motion processing unit 112 includes a movable range setting unit 112A, a coordinate conversion unit 112B, and a skeleton motion calculation unit 112C.

  The movable range setting unit 112A has a given point on a spherical surface with the joint as the central point as a focal point, and a plane perpendicular to an axis connecting the central point and the focal point as a projection plane, and the movable range of the joint is projected. Set on the surface. At this time, the movable range can be set so as to form a rectangle, an ellipse, or an oval on the projection plane.

  In the present embodiment, regarding the joint of the skeleton model, one of a rectangular shape, an elliptical shape, and an oval shape is preset as the shape of the movable range formed on the projection plane, and the skeleton of the skeleton around the y-axis in the joint is set. Allowable ranges for the rotation angle (y-axis rotation angle) and the rotation angle of the skeleton around the z-axis (z-axis rotation angle) are set in advance. Specifically, the shape control information indicating the shape of the movable range for each joint and the allowable range information indicating the allowable range of the y-axis rotation angle and the allowable range of the z-axis rotation angle for each joint are the motion control of the main memory 172. It is stored in the information storage unit 172B. Then, the movable range setting unit 112A determines the shape of the movable range of the calculation target joint formed on the projection plane obtained from the shape pattern information, and the y-axis rotation angle and the z-axis rotation angle of the calculation target joint obtained from the allowable range information. The movable range of the joint is set on the projection plane based on the allowable range.

The coordinate conversion unit 112B projects a point representing the orientation of the skeleton on the projection surface on the spherical surface with the joint as the center point based on the focal point. In the present embodiment, the axis along the parent bone is set as the x axis, and two axes orthogonal to the parent bone and orthogonal to each other are set as the y axis and the z axis, respectively, on the projection plane, The orthogonal coordinate axes are set as the s axis and the t axis, respectively, and the coordinate conversion unit 112B sets s = −z / (x + 1) and t = y / (x + 1) {however, x ² + y with reference to the focal point. By performing the projection by ² + z ² = 1}, the point (x, y, z) on the spherical surface is converted to the point (s, t) on the projected surface.

  Based on the position of the point projected on the projection plane, the skeletal motion calculation unit 112C calculates the orientation of the skeleton relative to the parent bone under the limitation of the movable range set by the movable range setting unit 112A. At this time, the skeleton motion calculation unit 112C determines the motion of the parent bone based on input information of the player, external force information due to collision (hit) between objects, other motion influence information (for example, gravity information, wind power information, and the like). It is determined whether or not the orientation of the skeleton relative to the parent bone obtained by calculating the operation of the corresponding skeleton is within the movable range. When the orientation of the skeleton is not within the movable range, the orientation of the skeleton relative to the parent bone is corrected so as to satisfy the condition according to the shape of the movable range on the projection plane, Decide the orientation of the skeleton so that it is within the range of motion of the joint. In the movement / motion processing unit 112, based on the calculation result of the skeleton motion calculation unit 112C, a calculation is performed to move the object by deforming a polygon or the like (surface shape) constituting the surface of the object.

  The virtual camera control unit 114 performs a virtual camera (viewpoint) control process for generating an image viewed from a given (arbitrary) viewpoint in the object space. Specifically, the process (viewpoint position, line-of-sight direction, or angle of view) for controlling the position (X, Y, Z) or rotation angle (rotation angles around the X axis, Y axis, and Z axis) of the virtual camera. Process to control).

  For example, when an object (for example, a car, a character, or a ball) is photographed from behind using a virtual camera, the position or rotation angle of the virtual camera (the direction of the virtual camera) ) To control. In this case, the virtual camera can be controlled based on information (an example of predetermined control information) such as the position, rotation angle, or speed of the object obtained by the movement / motion processing unit 112. Alternatively, control may be performed such that the virtual camera is rotated at a predetermined rotation angle or moved along a predetermined movement path. In this case, the virtual camera is controlled based on virtual camera data (an example of predetermined control information) for specifying the position (movement path) or rotation angle of the virtual camera. When there are a plurality of virtual cameras (viewpoints), the above control process is performed for each virtual camera.

  The drawing unit 120 performs drawing processing based on the results of various processing (game processing) performed by the processing unit 100, thereby generating an image and outputting the image to the display unit 190. When generating a so-called three-dimensional game image, first, object information (model information) including vertex data (vertex position coordinates, texture coordinates, color data, normal vector, α value, etc.) of each vertex of the object (model) ) Is input from the object information storage unit 172A of the main memory 172, and vertex processing (shading by a vertex shader) is performed based on the vertex information included in the input object information. When performing the vertex processing, vertex generation processing (tessellation, curved surface division, polygon division) for re-dividing the polygon may be performed as necessary. In the vertex processing, according to the vertex processing program (vertex shader program, first shader program), vertex movement processing, coordinate conversion (world coordinate conversion, camera coordinate conversion), clipping processing, perspective processing, and other geometric processing are performed. On the basis of the processing result, the vertex information given to the vertex group constituting the object is changed (updated or adjusted). Then, rasterization (scan conversion) is performed based on the vertex information after the vertex processing, and the surface of the polygon (primitive) is associated with the pixel. Subsequent to rasterization, pixel processing (shading or fragment processing by a pixel shader) for drawing pixels (fragments forming a display screen) constituting an image is performed. In pixel processing, according to the pixel processing program (pixel shader program, second shader program), various processes such as texture reading (texture mapping), setting / changing color information, translucent composition, anti-aliasing, etc. are performed. The final drawing color of the constituent pixels is determined, and the drawing color of the perspective-transformed object is output (drawn) to the frame buffer 174C (buffer capable of storing image information in units of pixels) of the video memory 174 as a rendering target. To do. That is, in pixel processing, per-pixel processing for setting or changing image information (color, normal, luminance, α value, etc.) in units of pixels is performed. Thereby, an image that can be seen from the virtual camera (given viewpoint) in the object space is generated. Note that when there are a plurality of virtual cameras (viewpoints), an image can be generated so that an image seen from each virtual camera can be displayed as a divided image on one screen.

  Note that the vertex processing and pixel processing are realized by hardware that enables polygon (primitive) drawing processing to be programmed by a shader program written in a shading language, so-called programmable shaders (vertex shaders and pixel shaders). Programmable shaders can be programmed with vertex-level processing and pixel-level processing, so that the degree of freedom of drawing processing is high, and expressive power is greatly improved compared to conventional hardware-based fixed drawing processing. Can do.

  The drawing unit 120 performs geometry processing, texture mapping, hidden surface removal processing, α blending, and the like when drawing an object.

  In the geometry processing, processing such as coordinate conversion, clipping processing, perspective projection conversion, or light source calculation is performed on the object. Then, object information (positional coordinates of object vertices, texture coordinates, color data (luminance data), normal vector, α value, etc.) after geometry processing (after perspective projection conversion) is stored in the object information in the main memory 172. Stored in the part 172A.

  Texture mapping is a process for mapping a texture (texel value) stored in the texture storage unit 174A of the video memory 174 to an object. Specifically, the texture (surface properties such as color (RGB) and α value) is read from the texture storage unit 174A using the texture coordinates set (given) at the vertex of the object. Then, a texture that is a two-dimensional image is mapped to an object. In this case, processing for associating pixels with texels, bilinear interpolation, trilinear interpolation, etc. are performed as texel interpolation.

  As the hidden surface removal processing, hidden surface removal processing by a Z buffer method (depth comparison method, Z test) using a Z buffer 174B (depth buffer) in which a Z value (depth information) of a drawing pixel is stored may be performed. it can. That is, when drawing a pixel corresponding to the primitive of the object, the Z value stored in the Z buffer 174B is referred to. Then, the Z value of the referenced Z buffer is compared with the Z value at the drawing pixel of the primitive, and the Z value at the drawing pixel is a Z value (for example, a small Z value) on the near side when viewed from the virtual camera. If there is, the drawing process of the drawing pixel is performed and the Z value of the Z buffer 174B is updated to a new Z value.

As the α blending, a translucent synthesis process (usually α blending, addition α blending, subtraction α blending, or the like) based on the α value (A value) is performed. For example, in the case of normal α blending, the following processing (α1) to (α3) is performed.
R _Q = (1−α) × R ₁ + α × R ₂ (α1)
G _Q = (1−α) × G ₁ + α × G ₂ (α2)
B _Q = (1−α) × B ₁ + α × B ₂ (α3)

In addition, in the case of addition α blending, the following formulas (α4) to (α6) are performed. In the case of simple addition, α = 1 and the following formulas (α4) to (α6) are performed.
R _Q = R ₁ + α × R ₂ (α4)
G _Q = G ₁ + α × G ₂ (α5)
B _Q = B ₁ + α × B ₂ (α6)

In the case of subtractive α blending, the following formulas (α7) to (α9) are performed. In the case of simple subtraction, the following formulas (α7) to (α9) are performed with α = 1.
R _Q = R ₁ −α × R ₂ (α7)
G _Q = G ₁ −α × G ₂ (α8)
B _Q = B ₁ −α × B ₂ (α9)

Here, R ₁ , G ₁ and B ₁ are RGB components of an image (original image) already drawn in the frame buffer 174C of the video memory 174, and R ₂ , G ₂ and B ₂ are the frame buffer 174C. Are RGB components of the image to be drawn. R _Q , G _Q , and B _Q are RGB components of an image obtained by α blending. The α value is information that can be stored in association with each pixel (texel, dot), for example, plus alpha information other than color information. The α value can be used as mask information, translucency (equivalent to transparency and opacity), bump information, and the like.

  The sound generation unit 130 performs sound processing (sound processing) based on the results of various processes performed by the processing unit 100, generates game sounds such as BGM, sound effects, and sounds, and outputs a sound output unit 192 (speaker). ).

  Note that the image generation system of the present embodiment may be a system dedicated to the single player mode in which only one player can play, or may be a system having a multiplayer mode in which a plurality of players can play. Further, when a plurality of players play, game images and game sounds to be provided to the plurality of players may be generated using one terminal, or connected via a network (transmission line, communication line) or the like. Alternatively, it may be generated by distributed processing using a plurality of terminals (game machine, mobile phone).

2. In the following, the method of the present embodiment will be described in comparison with the conventional method.

2-1. Conventional method (3-axis rotation angle method)
2-1-1. Coordinate system One of the bones on either side of a given joint is called the parent bone and the other is the child bone. The state of rotation (bending, twisting) of the joint can be expressed by the orientation of the skeleton with respect to the parent bone. In order to express the direction of the bone, a coordinate system is defined for each bone as shown in FIG.

2-1-2. Expression by 3-axis rotation angle In the conventional method, the orientation of the child bone relative to the parent bone is expressed by the rotation angles of the 3 axes of the xyz coordinate system (respectively indicated as r _x , r _y , r _z ) The movable range is set according to the range that the rotation angle of the shaft can take. Various orders of rotating the skeleton are conceivable. In the following example, the order of the x-axis rotation, the z-axis rotation, and the y-axis rotation is set with reference to the coordinate system of the parent bone.

Here, if a spherical surface S having a radius 1 with the joint as a center point is created and the intersection point between the x-axis of the skeleton and the spherical surface S is P, the bending state of the joint can be expressed by the position of the point P. When the relationship between the rotation angle about the three axes and the position of the point P is illustrated, it becomes a form similar to the latitude and longitude of the globe as shown in FIG. In FIG. 4, the rotation angle is expressed in 10 ° intervals, r _y is longitude, r _z corresponds to the latitude, r _x will correspond to torsion at that location.

2-1-3. Setting the movable range In the conventional method, setting the movable range related to the bending angle of the joint is equivalent to setting an area corresponding to the movable range (hereinafter simply referred to as a movable area) on the spherical surface S. For example, when the y-axis rotation is −40 ° to 20 ° and the z-axis rotation is −10 ° to 30 °, the movable region set on the spherical surface S is as shown in FIG. When the center of the spherical surface S and the boundary of the movable area are connected and displayed like a movable area, it is as shown in FIG.

As in this example, there is no problem when the movable range related to the bending angle of the joint is relatively narrow. However, if the movable range of the z-axis rotation is set to exceed ± 90 °, a problem occurs. In general, the longitude value can be in the range of −180 ° to 180 °, while the latitude value cannot exceed the range of −90 ° to 90 °. A dare set itself movable area if a contrivance is possible, for example, _{r y} is -40 ° to 20 °, if _{r z} is -10 ° to 110 °, the movable area as shown in FIGS. 7 to 9 Is set. However, in each of the setting examples shown in FIGS. 7 to 9, it is difficult to adopt a problem in actual use in which the shape of the movable area becomes distorted or the setting work of the movable area becomes complicated.

2-2. Method of the present embodiment by the projection method 2-2-1. Projection from Spherical Surface S to Plane λ As shown in FIG. 10, in the method of the present embodiment, a point of coordinates (−1, ₀ , ₀ ) is a point P _0, and a plane of x = 0 is a plane λ (projection plane). ), An arbitrary point P on the spherical surface S is projected onto the plane λ using the point P ₀ as a focal point (light source) to obtain a point Q. In the present embodiment, the above-described projection is f, and the transformation from the point P on the spherical surface S to the point Q on the plane λ is represented as Q = f (P). In the present embodiment, the projection from the point Q on the plane λ to the point P on the spherical surface S is set to f ^−1, and the conversion from the point Q on the plane λ to the point P on the spherical surface S is P = f ^−1. (Q). In the present embodiment, the point P ₀ is referred to as the pole of the spherical surface S.

Under the above assumption, the projection f causes a point on the spherical surface S (except for the point P ₀ ) and a point on the plane λ to have a one-to-one correspondence (one-to-one mapping), and f (P ₀ ) Is a point at infinity on the plane λ. In the method of the present embodiment, by setting the movable region of the point Q on the plane λ, the position that the point P on the spherical surface S can take is limited, and the movable range that limits the bending angle of the joint is set. .

First, an st coordinate system composed of an s-axis and a t-axis orthogonal to each other on the plane λ is taken, and the position of the point Q is represented by the st coordinate system. At this time, if f (P ₁ ) obtained by converting the point P ₁ that is the coordinates (1, 0, 0) on the spherical surface S is the origin of the st coordinate system, f (P ₁ ) is an xyz coordinate system that defines the spherical surface S. Matches the origin of.

  In the present embodiment, the positive direction (+ s) of the s axis is the negative direction (−z) of the z axis, and the positive direction (+ t) of the t axis is the positive direction (+ y) of the y axis. With such a setting, the movement direction of the point Q when the y-axis rotation is applied to the point P in the zx plane is the s direction, and the point Q when the z-axis rotation is applied to the point P in the xy plane. The moving direction is the t direction.

Then, the transformation from the point P (x, y, z) to the point Q (s, t) by the projection f is as shown in Expression (1). However, x ² + y ² + z ² = 1.

Further, the conversion from the point Q to the point P by the projection f ⁻¹ is as shown in Expression (2). At this time, since the result obtained by the conversion by the projection f ⁻¹ satisfies x ² + y ² + z ² = 1, the process for normalizing the coordinate values is not necessary.

2-2-2. Correspondence with the bending angle of the joint First, when the angle φ and the angle θ are taken as shown in FIG. 10, the correspondence between the coordinates (s, t) of the point Q and the angle φ and the angle θ 3). The ranges of the angle φ and the angle θ are −90 ° <φ <90 ° and −90 ° <θ <90 °, respectively.

Here, when the angle φ and the angle θ are changed from −85 ° to 85 ° at intervals of 5 ° to form a grid (lattice line) on the plane λ, and this is projected onto the spherical surface S with f ⁻¹ , FIG. 11 shows. Such an orthogonal grid is obtained.

Next, the correspondence between the conventional three-axis rotation angle method and the projection method of this embodiment will be described. The correspondence relationship between the y-axis rotation angle r _y and the angle φ in the case of the _z -axis rotation angle r _z = 0 is as shown in FIG. 12, and can be expressed by Expression (4).

Similarly, the correspondence relationship between the z-axis rotation angle r _z and the angle θ when the _y -axis rotation angle r _y = 0 is θ = r _z / 2. Therefore, when setting the movable range, if s, t and r _y , r _z are made to correspond as shown in the equation (5), the y-axis rotation angles r _y and z as shown in FIG. 4 and FIG. In a region where the absolute value of the axial rotation angle r _z is small, the movable region by the conventional triaxial rotational angle method and the movable region by the projection method of the present embodiment are substantially the same shape.

Next, consider a case where the position of the point P on the spherical surface S is represented by an angular coordinate α as shown in FIG. When the angle coordinate α corresponds to the latitude on the spherical surface S, that is, when the angle coordinate α corresponds to the z-axis rotation angle r _z in the conventional three-axis rotation angle method, the coordinate of the point P is α ₁ . Also when the angle coordinates alpha corresponds to the longitude of the spherical surface S, that is, if the angle coordinates alpha corresponds to the y-axis rotation angle r _y in the conventional triaxial rotation angle method, the coordinates of the point P is alpha ₃ . In contrast, in the projection method of the present embodiment, the coordinates of the point P is alpha _2. In this way, the joint angle expression by the projection method takes an intermediate value between the latitude expression and the longitude expression by the conventional three-axis rotation angle method. For this reason, the projection method of this embodiment has an advantage that the method can be smoothly changed without greatly deviating from the conventional joint angle expression by the three-axis rotation angle method.

2-2-3. Comparison between the conventional triaxial rotation angle method and the projection method of the present embodiment In the projection method of the present embodiment, the range of s is tan-20 ° to tan10 °, the range of t is tan-5 ° to tan15 °, FIG. 14 shows a case where the movable region is restricted to a rectangle (compare FIG. 6). Details of the rectangle restriction will be described later in “2-3-1. Region setting method by rectangle restriction”.

Next, an example corresponding to a case where the z-axis rotation angle r _z exceeds ± 90 ° will be described. FIG. 15 shows a case where the range of s is set to tan-20 ° to tan10 °, the range of t is set to tan-5 ° to tan55 °, and the movable region is restricted to a rectangle by the projection method of the present embodiment (FIGS. 7 and FIG. Compare with 8).

  Further, when the movable area is limited to an ellipse within the same range as described above, the result is as shown in FIG. 16, and when the movable area is limited to an oval shape, the result is as shown in FIG. 17 (compare with FIG. 9). Details of the ellipse restriction and the oval restriction will be described later in “2-3-3. Area setting method by oval restriction” and “2-3-3. Area setting technique by oval restriction”, respectively.

As described above, if the projection method of this embodiment is used, as shown in FIGS. 15 to 17, even if a wide movable region exceeding ± 90 ° that cannot be handled by the conventional three-axis rotation angle method is set, It is possible to express the joints smoothly. Further, as shown in FIGS. 6 and 14, when a relatively narrow movable region in which the y-axis rotation angle r _y and the z-axis rotation angle r _z are about 0 ° to ± 50 ° is set, the conventional 3 Since a movable region having substantially the same shape as that of the shaft rotation angle method is obtained, it is easy to shift from the conventional method, and the correspondence between both can be easily set using Equation (5).

2-3. Setting method according to the shape of the movable region The movable region setting method according to the projection method of this embodiment sets the movable region related to the bending angle of the joint by limiting the range of s and t on the plane λ. Hereinafter, unless otherwise specified, the possible range of s and t is assumed to be set as in Expression (6).

2-3-1. Region Setting Method by Rectangle Restriction Rectangle restriction is a method of setting a rectangular movable region on the plane λ as shown in FIG. That is, a region where both s and t satisfy Expression (6) is a movable region. The actual movable region of the joint is set as shown in FIGS. 14 and 15.

2-3-2. Region Setting Method by Ellipse Restriction Ellipse restriction is a method of setting an ellipse movable region on the plane λ as shown in FIG. That is, the inside of the ellipse (the axis of the ellipse is parallel to the s-axis and the t-axis) that is inscribed in the s and t regions set by Expression (6) is the movable region. The actual movable region of the joint is as shown in FIG. 16 or FIG.

Here, the elliptical restriction is expressed by an equation (7).
However, s _a , s _b , t _a , and t _b are as in Expression (8).

  The ellipse restriction makes it easy to determine the inside / outside of the area, but requires a device different from the case where the rectangular restriction is adopted for correction when the area exceeds the movable area, and will be described in detail later.

2-3-3. Region Setting Method by Oval Restriction Oval shape limitation is a method for setting an oval movable region on a plane λ as shown in FIG. The boundary of the movable region is composed of a semicircle and a straight line. Figure 21 is a case of _{s a> t a, t =} t 0, t = linear portion of the _{t 1} has a laterally long shape with a _{(s a, t} a is as illustrated in formula (8)). For s _{a <t} a, becomes a vertically long shape that the movable area is rotated 90 ° as shown in FIG. 21, in the case of s _{a =} t a, the length of the straight section is zero, the movable area The shape is a circle.

  The actual joint movable region has a shape as shown in FIGS. 17 and 22, and its boundary is composed of four small circles (for details, see “2-4-2. Characteristics of Projection f” described later). "reference). The small circle is a name that classifies the circles on the sphere, and is a circle formed by the intersection of the sphere and a plane that does not pass through the center of the sphere. For example, parallel lines other than the equator are small circles. It corresponds to.

2-3-4. Region setting method based on super-elliptical restriction According to the method of the present embodiment, the above-mentioned elliptic restriction can be generalized to be super-elliptical restriction. Whereas the ellipse limit is expressed as a sum of squares, the super ellipse limit is expressed as a sum of m-th power and n-th power as shown in Equation (9).
In Expression (9), for example, when m = n = 4 (sum of the fourth power), the movable region is as shown in FIG. 23 (compare with FIG. 20 and FIG. 22). The above-mentioned elliptical restriction corresponds to the case where m = n = 2 in Expression (9), and the above-described rectangular restriction corresponds to the case where m = n = ∞ in Expression (9).

2-3-5. Region setting method by combination of arc and straight line Instead of the super ellipse restriction and the more complicated shape described above, it is also possible to set a movable region on the plane λ by combining an arc and a straight line. When correcting a value when a value outside the movable range is obtained, the optimal solution cannot be obtained algebraically with the super-elliptical restriction, but if the region shape is a combination of an arc and a straight line, the optimal solution is algebraic. It is possible to ask for.

2-4. Correction Method When Exceeding Movable Region When the bending state of the joint exceeds the set movable region, correction is required to return it to the movable region, and the correction method will be described.

2-4-1. Spherical perpendicular and optimal solution When a movable area A is set on the spherical surface S and a point P outside the area A is given, a point obtained by correcting the position of the point P within the area A (including the boundary) is defined as a point P _m To do. Among the points in the area A, is considered to be the best to the point where the distance from the point P (a distance on the sphere S) is minimized and the point P _m, the optimum satisfy this condition point P _m I will call it the solution.

  Here, for convenience of explanation, the term “spherical perpendicular” is defined as follows. When a point P and a curve L are given on the sphere S, a great circle passing through the point P and perpendicular to the curve L is called a spherical perpendicular drawn from the point P to the curve L, and the intersection of the curve L and the spherical perpendicular is the intersection It is a leg of a spherical perpendicular. The great circle is a name that classifies the circles on the sphere, and refers to a circle formed by the intersection of the sphere and the plane that passes through the center of the sphere. In the case of the Earth, the equator and meridians are equivalent to the great circle. To do.

As shown in FIG. 24, the optimal solution of the point P _m is a foot sphere perpendicular dropped to the boundary of the movable area from the point P. However, when the area shape as rectangular limit has a vertex, sometimes this vertex is optimal solution of the point P _m.

2-4-2. Features of Projection f The projection f from the spherical surface S to the plane λ has the following features (A1) and (A2).

  (A1) The angle at which the two lines intersect does not change with the projection f (f is an equiangular mapping).

(A2) The circle on the spherical surface S is projected onto a circle or a straight line on the plane λ, and vice versa. The detailed correspondence between the circle on the spherical surface S and the circle or straight line of the plane λ, which are in a projecting relationship with each other, is as shown in the table below. The condition C1 in Table 1 is that the equation (10) is satisfied when the radius of the circle is r and the distance from the origin (0, 0) to the center of the circle is d as shown in FIG. .

Therefore, the spherical perpendicular drawn from the point P to the boundary of the movable region on the spherical surface S is either of the following (B1) and (B2) on the plane λ.
(B1) A circle perpendicular to the movable region boundary (condition C1 is satisfied)
(B2) A straight line orthogonal to the boundary of the movable region (passes the origin (0, 0))
By utilizing this property, depending on the shape of the movable area, it is possible to find an optimal solution of P _m.

2-4-3. Optimal solution for rectangular restriction In the rectangular restriction, a rectangular movable region as shown in FIG. 18 is set on the plane λ. First, consider that the spherical perpendicular PP _f is lowered from the point P to one side of the movable region (for example, the side corresponding to s = s ₀ ) on the spherical surface S. The coordinates of f (P) are set to (s _p , t _p ) (f (P) is a point obtained by projecting the point P onto the plane λ).

Here, when t _p ≠ 0, obtaining the spherical perpendicular PP _f is equivalent to obtaining a circle that satisfies the following three conditions (D1) to (D3) on the plane λ.
(D1) It passes through the point f (P).
(D2) perpendicular to the straight line s = _{s 0.}
(D3) Expression (10) is satisfied.

Then, as shown in FIG. 26, the center coordinates of the circle to determine _{(s c, t} c), when the radius is _{r c,} which are obtained by Equation (11).

In FIG. 26, a point f (P _f ) is an intersection of the circle obtained by the equation (11) and the straight line s = s _0, and the coordinates of the point f (P _f ) are (s _f , t _f ). (Necessarily s _f = s ₀ ). At this time, if t ₀ ≦ t _f ≦ t ₁ , it means that the spherical perpendicular is lowered with respect to the side that becomes the boundary of the movable region set to the rectangle. However, as shown in FIG. 27, the spherical perpendicular is not necessarily the shortest distance to the side that becomes the boundary of the movable region. In the example shown in FIG. 27, the side _B 1 _{B 2} to the spherical perpendicular _P a _{P f} drawn from the point _{P a,} which is the shortest distance from the point _{P a} to the side _B 1 _{B 2.} However, the shortest distance from the point _Pb to the side B ₁ B ₂ is not the spherical perpendicular P _b P _f but the great circle P _b B ₂ .

Here, a center on the spherical surface of the side B ₁ B ₂ (small circle B ₁ B ₂ ) is set as a point P _z . The spherical perpendicular drawn from the point P on the sphere in the side B ₁ B ₂ is the shortest distance to the side B ₁ B ₂ is only the case if the point P is located within the area P _z B ₁ B ₂ . Furthermore, point P if the interior of the region _P z _B 1 _{B 2,} it is possible to lower the spherical perpendicular always from the point P to the side _B 1 _{B 2.} In the present embodiment, the region P _z B ₁ B ₂ is referred to as the closest point region of the side B ₁ B ₂ . FIG. 28 shows an example of the nearest point area with respect to the four sides of the movable area on the plane λ.

In Figure 26, where the t coordinate of the point f _{(P f)} becomes a _t f = _{t c} ± _{r c,} become effective within the decoding ± is only the same as the sign of _{t p.} Otherwise, f (P) is outside the nearest point area. If this effective t _f is t ₀ ≦ t _f ≦ t ₁ , P _f may be an optimal solution.

When t _p = 0, the projection (extension) of the spherical perpendicular PP _f onto the plane λ is a straight line connecting f (P) and the origin (0, 0). Therefore, if t ₀ ≦ 0 ≦ t ₁ , the spherical perpendicular is lowered. Furthermore, only when s ₀ − (s ₀ ² +1) ^1/2 ≦ s _p ≦ s ₀ , the point P is included in the nearest point region, and the foot of this spherical perpendicular (s _f = s ₀ , t _f = 0) ) Is the optimal solution. This judgment formula can be rewritten as shown in formula (12).

Further, when the side of s = s ₁ is targeted, Expression (13) is used instead of the above-described determination expression (Expression (12)).

If the point f (P) is not included in any of the nearest point areas corresponding to each of the four sides of the movable area set to a rectangle, the spherical perpendicular cannot be lowered from the point P to the boundary of the movable area. Means. In this case, since the optimum solution P _m is one of the four vertices of the movable region, the distance between the point P and the four vertices of the movable region on the spherical surface S is obtained, and the vertex having the smallest obtained distance is the optimum solution P. _m .

2-4-4. Simplified solution for rectangle limitation The following simple solution can be considered as a correction method for rectangle limitation. This is a method in which a point in the movable region that minimizes the distance from the point f (P) on the plane λ is used as a solution. The s coordinates of the point f (P) and _{s p,} the s coordinates after the correction when the _{s m,} so that the equation (14). The same applies to the t coordinate.

However, this simple solution can be used as an approximate solution for the optimal solution only when the movable region is narrow (the absolute values of s ₀ , s ₁ , t ₀ , and t ₁ are small) or when the correction amount is small. If this condition is not met, a problem occurs as in the example shown in FIG. In the example shown in FIG. 29, while the optimal solution for the point _{P a} is a point _{P ma,} simple solutions a point _{P ma} '. Also, for optimal solution to the point P _b that is P _mb, simple solutions becomes a point P _{mb 'around} from the opposite side of the sphere S.

2-4-5. Elliptic restriction In the case of elliptical restriction, finding the optimal solution corresponds to finding a circle orthogonal to the boundary of the elliptical movable region on the plane λ. However, the equation for finding the solution is a high-order equation and cannot be solved algebraically.

  As an approximate solution of the optimal solution, there can be considered a method in which the solution is a perpendicular foot drawn on the boundary of the elliptical movable region on the plane λ. In this case, the equation to be solved is a quartic equation and a solution can be obtained algebraically. If mathematical expression processing software or the like is used, an equation for the solution can be actually obtained, but it becomes a very complicated equation and is not practical. As a similar problem, a method for obtaining a perpendicular to an ellipsoid is described in “John C. Hart, Distance to an Ellipsoid. Graphics Gems IV, pp113-119, Academic Press Inc., 1994”.

Therefore, in this embodiment, although the solution is completely inaccurate, as shown in FIG. 30, a method is adopted in which f (P) is corrected toward the center of the ellipse to obtain a simple solution f (P _m ). Yes.

The coordinates (s _m , t _m ) of the point f (P _m ) can be easily obtained as in the following equation (15). However, s _a , s _b , t _a , and t _b are as in Expression (8).

In general, the simple solution in the elliptical restriction of the present embodiment obtained by the equation (15) is completely inaccurate. However, when s ₀ = −s ₁ = t ₀ = −t ₁ , the movable region is Since it is a circle centered on the x-axis, it matches the optimal solution. Therefore, if the movable region meets this condition or is close to this condition, there is no problem even if the simple solution is used.

On the other hand, it is in the following cases that the simple solution causes a problem clearly. The movable region of the joint in the three-dimensional space is schematically represented as shown in FIG. 31, and the joint is assumed to receive an external force (such as gravity) in the downward direction of the figure. In this case the joint should settle at the position of A ₀ (the lowest position), the position of A _n When correction is performed using a simple solution. For example, once Even the position of A _0, A ₀ by an external force _'movement, the correction simply solution A _1, A ₁ by an external _force' movement, the correction in a simple solution A _2, through the stages of the final settle down to the position of _{a n} in.

2-4-6. Oval restriction In oval restriction, an oval movable region as shown in FIG. 21 is set on the plane λ. The straight line portion of the movable region boundary is called “side”, and the semicircular portion is called “arc”. Further, the projection of these onto the spherical surface S will be called in the same way. The center coordinates of the arc _{(s e, t e),} the radius is _{r e.} At this time, when s _a ≦ t _a (the shape of the movable region is vertically long on the plane λ), Expression (16) is obtained. In the equation (16), the two values for t _e is sequentially directed against the lower side of the arc and an upper arc.

Next, when s _a ≧ t _a (the shape of the movable region is horizontally long on the plane λ), the equation (17) is obtained. In equation (17), the two values relating to s _e are for the left arc and the right arc in order.

The following describes a correction method in the case of _{s a} ≦ _{t a.} First, on the spherical surface S, _let us consider dropping the spherical perpendicular PP _f from the point P to the side of the boundary of the movable region. This is exactly the same as the case where a spherical perpendicular is drawn on one side of the boundary of the movable region in the rectangular restriction. Therefore, whether or not the spherical perpendicular can be lowered and whether or not the lowered spherical perpendicular is the shortest distance to the movable region can be determined by the same method as in the case of the rectangular restriction. If optimal solutions on the two sides of the movable area is not present, the foot P _f of the spherical perpendicular drawn to the arc of the movable area is optimal solution. The point _Pf is obtained for two arcs, and the one having the smallest distance from the point P on the spherical surface S is determined as the optimum solution.

In the following, an example the lower arc of the two arcs of the movable region of the oval limits (case of s _a ≦ t _a), describes a method for determining the foot spherical perpendicular. As described above, the great circle on the spherical surface S becomes a circle or a straight line on the plane λ.

First, consider a case where the spherical perpendicular PP _f is a circle on the plane λ. Obtaining this spherical perpendicular corresponds to obtaining a circle that satisfies the following three conditions (E1) to (E3) on the plane λ.
(E1) Pass through the point f (P).
(E2) It is orthogonal to the arc of the movable region.
(E3) Expression (10) is satisfied.

Here, if the center coordinates of the circle to be obtained are (s _c , t _c ) and the radius is r _c , the coordinates (s _f , t _f ) of the point f (P _f ) can be obtained by the following procedure.

First, s _c and t _c are obtained from equation (18). In Equation (18), k ₂ = 0 when the point f (P), the arc center point (s _e , t _e ), and the origin (0, 0) exist on a straight line. This is the case where the spherical perpendicular PP _f becomes a straight line on the plane λ. Therefore, here k ₂ ≠ 0.

Then, determine the _{r c} by the equation (19).

Then, determine the _{k 4} by the equation (20).
Here, s _f and t _f obtained by the equation (22) are valid only when k ₄ obtained by the equation (20) satisfies the equation (21). This is because when k ₄ > 0, it means that the spherical normal is lowered not in the arc but in its extension (inside the movable region), so that it is invalid as a solution.

However, Formula (20) and Formula (22) are in the same order of compound numbers. In addition, the condition k ₄ ≦ 0 must be noted because it may hold simultaneously for both of the complex signs in equation (20). This means that two spherical perpendiculars can be drawn from the point P with respect to one arc of the movable region. This is because the arc of the movable region is a semicircle (center angle 180 °) on the plane λ, but the center angle may exceed 180 ° on the spherical surface S, so two spherical perpendiculars may exist. .

The above description is a case where a spherical normal is drawn on the lower arc of the movable region. In the case of the upper arc, equation (21) becomes k ₄ ≧ 0.

Next, consider a case where the spherical perpendicular PP _f is a straight line on the plane λ. This is a case where k ₂ = 0 in Equation (18). Obtaining the spherical perpendicular corresponds to obtaining a straight line that satisfies the following three conditions (F1) to (F3) on the plane λ.
(F1) Passes through the point f (P).
(F2) orthogonal to the arc of the movable region (this is equivalent to passing through the point (s _e , t _e )).
(F3) Pass through the origin (0, 0).

At this time, the coordinates (s _f , t _f ) of the optimum solution P _f are obtained by the equation (23). However, to become effective within the decoding ±, if the upper arc the same as the sign of the t _h, in the case of lower arc are only those of opposite sign of t _h. Otherwise, it means that a spherical normal is drawn not on the arc but on its extension (inside the movable area), so it is invalid as a solution.

2-4-7. Super-elliptic restriction In the case of super-elliptical restriction, as in the case of elliptic restriction, it is necessary to solve a higher-order equation to obtain an optimal solution, which is not practical. Similar to the simple solution in the ellipse restriction, a simple solution can be obtained if correction is made toward the center of the super-elliptical movable region on the plane λ. However, different from the case of elliptic restriction, it should be noted that the simple solution does not match the optimal solution even in the case of s ₀ = −s ₁ = t ₀ = −t ₁ .

2-4-8. Limitation by a combination of arc and straight line Instead of the super elliptical limit, if a movable area is set by a combination of an arc and a straight line as shown in Fig. 32, an optimum solution can be obtained by the correction method almost the same as the case of oval limitation. Can do.

2-5. Handling of torsion angle 2-5-1. Definition of Twist Angle As already described, the bending state of the joint can be expressed by the position of the point P on the spherical surface S. On the other hand, twist is expressed as rotation around the axis OP (O is the center of the spherical surface S). In order to handle the twist angle, when a bent state is given, it is necessary to predefine the direction of the twist angle of 0 ° in the bent state.

First, a state in which the directions of the coordinate axes of the parent bone and the child bone are coincident (all coincident with the three axes) is defined as a reference state, and the twist angle in this state is defined as 0 °. This is a state where the twist angle is 0 ° when a point indicating a bending state exists on the spherical surface S shown in FIG. 10 at a point P ₁ (point of coordinates (1, 0, 0)).

  In this embodiment, “simple rotation” is defined as follows.

Specifically, a rotation that satisfies the following conditions (G1) and (G2) with respect to arbitrary unit vectors v ₁ and v ₂ sharing the start point is defined as a simple rotation from v ₁ to v ₂ . In addition, when v ₁ = v ₂ , no rotation is performed (unit matrix when expressed as a rotation matrix), and when v ₁ = −v ₂ , the definition is impossible.
(G1) Rotation around an axis perpendicular to the plane formed by v ₁ and v ₂ .
(G2) v ₁ matches v ₂ by rotation.

A state in which the simple rotation from OP ₁ to OP with respect to the reference state is defined as a state where the twist angle in the bending state P is 0 °. The orientation of the y-axis and the z-axis of the skeleton in the state where the twist angle is 0 ° is the same as the grid in FIG. When P = P ₀ , the direction with a twist angle of 0 ° cannot be defined.

When defining the twist angle of the joint as described above, the rotation matrix M of the joint shown in equation (24), equation (25) bent as shown in to Formula (27) decomposed into a matrix M _b and torsion matrix M _t it can. However, the matrix type (the order of multiplication, the arrangement of elements, etc.) is the type of multiplication by column vectors.

However, d ₀ in formula (26) and formula (27) is as shown in formula (28). Note that equation (28) is equivalent even when d ₀ = 1 / (m ₀₀ +1), but in order to avoid a loss of digits when m ₀₀ ≈−1 (corresponding to P≈P ₀ ), equation (28) ) Is better.

2-5-2. A description will be given of a case where the twist angle of the joint is always 0 ° while correcting the bending of the joint within the movable range. The matrix representing rotation states of a given joint is M of the formula (24), the matrix obtained by correcting it to the movable range and M _1.

At this time, assuming that a point on the spherical surface S representing the given bending state is a point P, the coordinates (x, y, z) of the point P are obtained by Expression (29).

This point P is corrected within the movable region by the method described above. If a point on the spherical surface S corresponding to the correction result is a point P _m and its coordinates are (x _m , y _m , z _m ), M ₁ is obtained by the equations (30) and (31).

Further, when M ₁ is expressed using the st coordinate (s _m , t _m ) of the point f (P _m ), the equations (32) and (33) are obtained.

By using the equations (32) and (33), after correcting the bending state on the plane λ, it is possible to directly perform M without going through the calculation for obtaining the coordinates (x _m , y _m , z _m ) corresponding to the point P _{m. Since 1} is obtained, the amount of calculation can be reduced.

2-5-3. Setting method of movable range of torsion angle A case where both bending and twisting of a joint are corrected within the set movable range will be described. The matrix that represents a given rotational state is M of the formula (24), the matrix obtained by correcting it to the movable range and M _2.

  Here, M can be decomposed into bending and twisting as shown in Expression (25), Expression (26), Expression (27), and Expression (28). About bending, it correct | amends using Formula (29), Formula (30), and Formula (31).

Assuming that the twist angle before correction is ψ, ψ can be obtained from equation (34).
If this twist angle ψ is outside the movable range, this is corrected to ψ _m . The twist matrix M _t2 by ψ _m is as shown in Equation (35), and M ₂ is obtained by Equation (36).

2-6. Representation by quaternion 2-6-1. Bending and twisting decomposition Consider the case where quaternions are used to represent joint rotation. As with the previous matrix, quaternions can be broken down into bends and twists. Quaternion representing a rotation of the joint q is decomposed into a quaternion bent as follows q _b and twist quaternion q _t.

If the elements of q are (q _x , q _y , q _z , q _w ), the elements (q _bx , q _by , q _bz , q _bw ) of the bending quaternion q _b are as follows.

The elements of the twisted quaternion q _t (q _tx , q _ty , q _tz , q _tw ) are as follows.

2-6-2. Bending and twisting correction method When a point on the spherical surface S representing a given bending state is a point P, the coordinates (x, y, z) of the point P are obtained by the equation (40).
This point P is corrected within the movable region by the method described above. If the point on the spherical surface S corresponding to the correction result is a point P _m and the coordinates are (x _m , y _m , z _m ), the corrected bending quaternion q ₁ = (q _1x , q _1y , q _1z , q _1w ) is obtained by the equation (41).

If the coordinates (s, t) of the point f (P) are used without using the coordinates (x, y, z) of the point P, the expressions (42) and (43) are obtained.

If the twist angle before correction is ψ, ψ can be obtained from the equation (44).
If this ψ is outside the movable range, this is corrected to be ψ _m . The twisted quaternion q ₂ = (q _2x , q _2y , q _2z , q _2w ) due to ψ _m is as shown in Expression (45).

Thus, the quaternion q ₃ corrected for both bending and twisting is obtained by the equation (46).

2-6-3. Bending angle limitation using quaternion Since the quaternion in the above equation (38) represents bending of a joint, a method for limiting the bending angle by setting a movable range with respect to its elements q _by and q _bz is described. Conceivable. However, as described below, this method is inferior to the method using projection.

q _by = sin (η / 2), q _bz = sin (ζ / 2), and values of η and ζ are changed from −180 ° to 180 ° at intervals of 10 °, and a grid (lattice line) is formed on the spherical surface S. When made, it becomes as shown in FIG. However, the range of q _by and q _bz is q _by ² + q _bz ² ≦ 1.

Here, when the case where the quaternion shown in FIG. 33 is adopted and the case where the projection method shown in FIG. 11 is adopted are compared, in a region where the bending angle is small, both have similar shapes. However, in the projection method, the grids are orthogonal, whereas the grids are not orthogonal when the angle is limited by q _by and q _bz . In particular, in the polar hemisphere, as shown in FIG. 34, when the angle is limited _by the quaternion (q _by , q _bz ), the _crossing angle of the grid is greatly away from the right angle, which is not suitable for setting the movable region. However, when setting the axisymmetric circular movable area with respect to the x-axis, q _By, With the q _bz 2 sum limit, it is possible to set the movable area accurately.

2-7. Effects of the Method of the Present Embodiment According to the projection method of the present embodiment, the transformation by projection and its inverse transformation are configured only by four arithmetic operations, so the calculation load is small.

  Further, according to the projection type correction processing (rectangular limitation, oval limitation) of the present embodiment, an optimal solution can be obtained with a practical calculation amount.

  In addition, the correction processing (rectangular limitation, oval limitation) of the projection system of this embodiment is composed of only four arithmetic operations except for a small part of the square root calculation, so the calculation load is small and the vector calculator (or SIMD) ) Can be accelerated.

3. Processing of this embodiment Next, a processing example for realizing the method of this embodiment will be described with reference to the flowchart of FIG.

  First, as an initial setting, the movable range of each joint constituting the skeleton model is set on the plane λ based on the shape of the movable range given in advance for each joint and the allowable range of the bending angle (step S10). .

  When the frame update timing (for example, when 1/60 second has elapsed from the previous frame) has arrived (Y in step S11), player input information is detected (step S12).

  Next, a joint to be calculated is selected from a plurality of joints included in the skeleton model related to the character to be operated by the player (step S13), input information of the player, external force information by hit between objects, and other Based on the motion influence information, the bending / twisting of the joint to be calculated is calculated (step S14).

  Next, with respect to the parent bone and the phalanx connected at the joint to be calculated, a point P on the spherical surface S centering on the joint and indicating the orientation of the skeleton relative to the parent bone is projected onto the plane λ by the projection f. The point f (P) is obtained (step S15), and it is determined whether the point f (P) on the plane λ is within the movable range (step S16).

If the point f (P) is outside the movable range (N in step S16), the optimum solution f (P _m ) is obtained by a correction equation corresponding to the shape of the movable range of the joint to be calculated (step S16). S17), and calculates the orientation of the child bone after correction by inversely converted on a spherical surface S the point f _{(P m)} by the projection ^{f -1} finding the point _{P m} on the sphere S (step S18).

  When the calculation for all the joints constituting the skeleton model is completed (Y in step S19), the player's operation target character is operated according to the movement (change in orientation) of each bone of the skeleton model (step S20). .

4). Hardware Configuration Example An example of a hardware configuration for realizing the game system of the present embodiment will be described with reference to FIG. Note that FIG. 36 shows only the main configuration, and hardware (such as a memory controller and a speaker) not shown in FIG. 36 can be provided as necessary.

  The main processor 10 operates based on a program stored on the optical disc 72 (information storage medium such as CD, DVD, Blu-ray disc), a program transferred via the communication interface 80, a program stored on the hard disk 60, or the like. The main memory 40 accessible via the internal bus b4 is used as a work area (work area) to execute various processes such as game processing, image processing, and sound processing.

  The main processor 10 includes a single processor 12 and a plurality of vector processors 14. The processor 12 executes various processes such as OS execution, hardware resource management, game processing, and operation management of the vector processor 14. The vector processor 14 is a processor specialized for vector operations, and mainly executes processing such as geometry processing and codec processing of image data and sound data.

  The drawing processor 20 is configured to be accessible to the video memory 30 via the internal bus b1. The drawing processor 20 is connected to the main memory 40 via an internal bus b2 connecting the drawing processor 20 and the main processor 10, a bus b3 inside the main processor 10, and an internal bus b4 connecting the main processor 10 and the main memory 40. Is also made accessible. That is, in the game system of the present embodiment, the drawing processor 20 can use the video memory 30 as a basic rendering target, and can use the main memory 40 as an exceptional rendering target as needed. ing.

  The drawing processor 20 performs geometric processing such as coordinate transformation, perspective transformation, light source calculation, and curved surface generation, has a product-sum calculator and a divider capable of high-speed parallel computation, and performs matrix computation (vector computation). Run fast. For example, when processing such as coordinate transformation, perspective transformation, and light source calculation is performed, a program operating on the main processor 10 instructs the drawing processor 20 to perform the processing.

  The drawing processor 20 executes a rendering process of an image of an object after geometry processing (an object composed of a primitive surface such as a polygon or a curved surface) at high speed. During the multi-pass rendering process, the drawing processor 20 performs hidden surface removal using the Z buffer 34 and texture mapping using the texture storage unit 36 based on drawing information (vertex information and other parameters). The image is rendered in the frame buffer 32 of the video memory 30. At this time, the rendering processor 20 can perform filter processing such as glare filter processing, motion blur processing, depth-of-field processing, and fog processing as multi-pass rendering processing. When the image is rendered in the frame buffer 32 provided in the video memory 30 in the final rendering pass, the image is output to the display 50.

  The hard disk 60 stores system programs, save data, personal data, and the like.

  The optical drive 70 drives an optical disc 72 (information storage medium) in which programs, image data, sound data, and the like are stored, and enables access to these programs and data.

  The communication interface 80 is an interface for transferring data to and from the outside via a network. In this case, as a network connected to the communication interface 80, a communication line (analog telephone line, ISDN), a high-speed serial bus, etc. can be considered. By using a communication line, data transfer via the Internet becomes possible. Further, by using the high-speed serial bus, data transfer with other image generation systems becomes possible.

  As described above, in the game system (computer) of this embodiment, the program for causing the main processor 10 to perform coordinate conversion, movable range setting, and skeletal motion calculation is the optical disc 72 (an example of an information storage medium) or the hard disk 60 (information storage medium). In the example).

  Note that all the units (units) of the present embodiment may be realized only by hardware, or only by a program stored in an information storage medium or a program distributed via a communication interface. May be. Alternatively, it may be realized by both hardware and a program.

  And when each part of this embodiment is implement | achieved by both hardware and a program, the program for functioning hardware (computer) as each part of this embodiment is stored in an information storage medium. Become. More specifically, the program instructs each processor 10, 20, etc., which is hardware, and passes data if necessary. Each of the processors 10 and 20 implements each unit of the present invention based on the instruction and the passed data.

  The present invention is not limited to that described in the above embodiment, and various modifications can be made. For example, terms cited as broad or synonymous terms in the description in the specification or drawings can be replaced with broad or synonymous terms in other descriptions in the specification or drawings. Further, the method for operating the skeleton model is not limited to that described in the present embodiment, and methods equivalent to these are also included in the scope of the present invention.

  The present invention can also be applied to various games (such as fighting games, shooting games, robot fighting games, sports games, competitive games, role playing games, music playing games, dance games, etc.). Further, the present invention can be applied to various game systems such as a business game system, a home game system, a large attraction system in which a large number of players participate, a simulator, a multimedia terminal, a system board for generating game images, and a mobile phone. .

The functional block diagram of the game system of this embodiment. The figure which shows the structure of a skeleton model typically. The figure which shows the coordinate system set to a master bone and a child bone. R _y by triaxial rotation angle _method, shows the relationship between the position on the r _z and spherical S. The figure which shows the movable area | region of the joint set by the 3 axis | shaft rotation angle system. The figure which shows the movable area | region of the joint set by the 3 axis | shaft rotation angle system. The figure which shows the movable area | region of the joint set by the 3 axis | shaft rotation angle system. The figure which shows the movable area | region of the joint set by the 3 axis | shaft rotation angle system. The figure which shows the movable area | region of the joint set by the 3 axis | shaft rotation angle system. The figure for demonstrating the projection from the point P on the spherical surface S to the point Q on the plane λ. The figure for demonstrating the result of having projected the grid on plane (lambda) on the spherical surface. diagram showing the relationship between the y-axis rotation angle r _y and the angle phi. Comparison diagram of projection method and latitude / longitude expression. The figure which shows the movable area | region of the joint set by the projection system. The figure which shows the movable area | region of the joint set by the projection system. The figure which shows the movable area | region of the joint set by the projection system. The figure which shows the movable area | region of the joint set by the projection system. The figure which shows the movable area | region where the rectangle restrictions were carried out on plane lambda. The figure which shows the movable area | region by which the elliptical shape was restricted on plane (lambda). The figure which shows the movable area | region of the joint set by the projection system. The figure which shows the movable area | region by which the oval shape restriction | limiting was carried out on the plane (lambda). The figure which shows the movable area | region of the joint set by the projection system. The figure which shows the movable area | region of the joint set by the projection system. It shows the foot P _m spherical perpendicular drawn to the boundary of the movable area from the point P. The figure which shows the conditions from which the circle | round | yen on plane (lambda) turns into a great circle on the spherical surface S. FIG. The figure which shows the circle | round | yen obtained by projecting a spherical perpendicular to the plane (lambda). The figure which shows the spherical perpendicular drawn down to the edge of a rectangular area. The figure which shows the nearest point area | region with respect to each edge | side of a rectangular area. The figure which shows the malfunction which generate | occur | produces with the correction method by the simple solution in a rectangle restriction | limiting. The figure which shows the simple solution with respect to elliptical restrictions. The figure which shows the malfunction which generate | occur | produces with the correction method by the simple solution in elliptical restrictions. The figure which shows the example of a setting of the movable area by the combination of a circular arc and a straight line. q _By, shows a grid made on a sphere S with different values of _{q bz.} The figure which compares the state of the pole side of a projection system (left) and a quaternion system (right). The flowchart which shows the specific process example of this embodiment. Hardware configuration example.

Explanation of symbols

100 processing unit,
110 Object space setting unit, 112 Movement / motion processing unit,
112A movable range setting unit, 112B coordinate conversion unit, 112C skeleton movement calculation unit,
114 virtual camera control unit,
120 drawing units, 130 sound generation units,
160 operation unit, 170 storage unit,
172 main memory,
172A Object information storage unit, 172B Motion control information storage unit,
174 video memory,
174A texture storage unit, 174B Z buffer, 174C frame buffer,
180 information storage medium, 190 display unit, 192 sound output unit,
194 Portable information storage device, 196 communication unit

Claims (9)

A program for controlling the movement of a skeletal model in which a parent bone and a skeleton are connected by a joint,
A given point on a sphere centered on a joint is a focal point, a plane orthogonal to the axis connecting the central point and the focal point is a projection plane, and the movable range of the joint is set on the projection plane A movable range setting section;
A coordinate conversion unit that projects a point representing the orientation of the skeleton on the spherical surface onto the projection plane based on the focal point;
As a skeletal motion calculation unit that calculates the orientation of the child bone relative to the parent bone under the limitation of the movable range, based on the position of the point projected on the projection plane,
A program characterized by causing a computer to function. In claim 1,
The movable range setting unit is
A program characterized in that the movable range is set so as to form a rectangle on the projection plane. In claim 1,
The movable range setting unit is
A program characterized in that the movable range is set so as to form an ellipse on the projection plane. In claim 1,
The movable range setting unit is
A program characterized in that the movable range is set so as to form an oval on the projection surface. In any one of Claims 1-4,
An axis along the parent bone is set as an x axis, and two axes orthogonal to the parent bone and orthogonal to each other are set as a y axis and a z axis, respectively.
On the projection plane, coordinate axes orthogonal to each other are set as an s axis and a t axis, respectively.
The coordinate conversion unit is
Based on the focus,
s = −z / (x + 1) and t = y / (x + 1) {where x ² + y ² + z ² = 1}
A program characterized by converting a point (x, y, z) on the spherical surface into a point (s, t) on the projected surface by performing a projection according to. In claim 5,
Regarding the joint of the skeletal model, either a rectangle, an ellipse, or an oval is preset as the shape of the movable range formed on the projection plane, and the rotation angle of the skeleton around the y axis at the joint and The allowable range of the rotation angle of the skeleton around the z axis is preset,
The movable range setting unit is
Based on the shape of the movable range formed on the projection plane and the allowable range of the rotation angle of the skeleton around the y-axis and the rotation angle of the skeleton around the z-axis in the joint, the movable range of the joint is A program characterized in that it is set on the projection surface. In any one of Claims 1-6,
The skeletal motion calculation unit is
It is determined whether or not the orientation of the bone relative to the parent bone obtained by calculating the motion of the skeleton corresponding to the motion of the parent bone is within the movable range, and the orientation of the skeleton relative to the parent bone Is not within the movable range, the program corrects the orientation of the child bone relative to the parent bone so as to satisfy a condition according to the shape of the movable range on the projection plane.   An information storage medium readable by a computer, wherein the program according to any one of claims 1 to 7 is stored. A skeletal motion control system for controlling the motion of a skeletal model in which a parent bone and a skeleton are connected by a joint,
A given point on a sphere centered on a joint is a focal point, a plane orthogonal to the axis connecting the central point and the focal point is a projection plane, and the movable range of the joint is set on the projection plane A movable range setting section;
A coordinate conversion unit that projects a point representing the orientation of the skeleton on the spherical surface onto the projection plane based on the focal point;
Based on the position of the point projected on the projection plane, under the limitation of the movable range, a skeletal motion calculation unit that calculates the orientation of the child bone relative to the parent bone;
A skeletal motion control system comprising: