JP5486524B2 - Skeletal angle control method, apparatus, and program - Google Patents

Description

  The present invention relates to control of the angle, that is, rotation (bending, twisting, etc.) of each bone in a skeleton model.

  When creating a pose or animation by moving a CG character or the like, a skeleton model (skeleton model) (FIG. 1A) in which a plurality of bones are connected by joints may be used. Such a skeleton model can be used not only in the virtual space but also in the case of controlling an object in a real space such as robot control.

  In the present specification, as shown in FIG. 1B, one of the bones on both sides of the joint is called a parent bone and the other is called a child bone. The state of rotation (bending, twisting) of the joint can be expressed by the orientation of the child bone relative 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. In the coordinate system of FIG. 1B, the x-axis rotation is twisted, and the y-axis rotation and the z-axis rotation are bending.

  In such a skeleton model used for controlling the posture (pose, movement, etc.) of a human body model, animal, structure, etc. in a three-dimensional (3D) virtual space or real space, it is shown in FIG. It is necessary to control the position, angle, rotation (bending, twisting), etc. of each bone and joint as intended. In the skeletal model shown in FIG. 1A, each bone is connected by a joint, and the movement of each bone affects the movement of other bones. Conventionally, various methods for modeling a skeleton and controlling the skeleton model have been proposed.

JP 2000-306116 A JP 2009-070340 A JP 2010-170279 A US Patent Publication No. 2010182329

  As an example, assume that the skeleton model of FIG. This skeleton model is a model of the human skeleton. Therefore, for example, the movable range of the arm beyond the elbow has a certain limitation. Making the arm bend at an angle beyond this limit (bending, twisting) is far from the model for real humans. In order to form a skeletal model that takes a natural pose as a human being, it is necessary to impose certain restrictions on the range of movement of the bone from the elbow joint, for example.

  Conventionally, as disclosed in Patent Document 1, for example, the orientation of a child bone relative to a parent bone is expressed by rotation angles around three axes (x, y, z) with respect to the parent bone coordinate system. is there.

  In Patent Document 2, a given point on a sphere centered on a joint is a focal point, a plane perpendicular to an axis connecting the central point corresponding to the joint and the focal point is a projection plane, and the movable range of the joint is A technique for setting on a projection surface is disclosed.

  In Patent Document 3 and Patent Document 4, when the orientation of the skeleton is not within the predetermined movable range, the skeleton with respect to the parent bone is satisfied so as to satisfy the condition according to the shape of the movable range. An invention for correcting the orientation of the above is disclosed.

  In the conventional skeleton control method as described above, a problem has been pointed out that the movable range is far from the intention of the designer. Further, as a technique for solving this problem to some extent, techniques disclosed in Patent Document 2, Patent Document 3 and Patent Document 4 have been proposed. However, although these techniques require a relatively small amount of calculation, when the articulated model is moved in real time, further reduction of the calculation amount is required.

  Therefore, it is desired to realize a technique that can flexibly cope with various skeleton models and controls the skeleton model with simpler processing. The object of the present invention is to solve the drawbacks of the prior art as described above. It should be noted that the problems taken up here and the object of the present invention are exemplary. Therefore, it should be noted that the object of the present invention is not limited to the above description.

  In this specification, for the sake of convenience, control of a skeleton model that can be formed in a virtual space will be described as an example. However, the application range of the present invention is not limited to the virtual space. For example, it is naturally intended to be used for controlling the position and angle (including bending, twisting, etc.) of a physical structure such as robot control. That is, it should be particularly noted that the present invention is intended for both virtual space and real space skeleton models. Further, the present invention is not limited to a specific skeleton model handled in this specification.

  The present invention is a method for controlling the operation of the skeleton in a skeletal model in which the skeleton is connected to the skeleton by limiting the angular range of the skeleton relative to the parent bone in the joint. , An angle range boundary circle setting step for setting an angle range boundary circle defining a boundary of the angle range of the skeleton on a spherical surface of a sphere centered on the joint, wherein the angle range boundary circle includes the joint Passing through the intersection point 1 between the spherical surface as a center and the extension line of the parent bone or the parent bone, and the position of the intersection point 2 of the spherical surface and the extension line of the child bone and the spherical surface is bisected by the angle range boundary circle The angle range of the skeleton is determined depending on which of the spheres should be present, the cone unit vector u formed by the center of the sphere and the angle range boundary circle and the half apex angle α of the cone, or Previous An angle range boundary circle setting step for setting the angle range boundary circle by storing a numerical value capable of deriving the axis unit vector u and the half apex angle α in a memory, and reading the numerical value stored in the memory To obtain the axis unit vector u and the half apex angle α, calculate the value of u · v-cos α as a test value using the unit vector v representing the orientation of the skeleton, and based on the test value And an angle range determination step of determining whether or not the angle of the skeleton is within the angle range. The calculation of the above equation (v · u) means the inner product of the vector v and the vector u.

The present invention further includes a three-dimensional orthogonal coordinate system (x, y, z) in which the joint is the origin, and the direction from the intersection 1 to the origin is the positive direction of the x axis.
μ = 2 tan ⁻¹ (−z / (x + 1))
ν = 2 tan ⁻¹ (y / (x + 1))
The angular range boundary circle satisfies at least one of μ = μ _e and ν = ν _e using the monopolar spherical coordinate system (μ, ν) having the following relationship (where μ _e and ν _e may be a constant).

In the present invention, the monopolar spherical coordinates of the intersection point 2 are (μ, ν), the lower and upper limits of μ are μ ₀ and μ _1, and the lower and upper limits of ν are ν ₀ and ν ₁ , respectively. In this case, the angle range boundary circle may be set so that the angle range satisfies μ ₀ <μ <μ ₁ and ν ₀ <ν <ν ₁ .

  The present invention may further use an auxiliary angle range boundary circle in which the angle range boundary circle does not pass through the intersection point 1. That is, this auxiliary angle range boundary circle may be used in the angle range determination step.

  In the present invention, in the angle range determination step, when it is determined that the angle of the skeleton is not within the angle range, the point from the intersection point 2 to a point on the spherical surface that satisfies the angle range. An angle correction step of correcting the angle of the child bone by moving the intersection point 2 to a point on the spherical surface where the distance measured along the spherical surface is the shortest may be further included.

  Further, the present invention may be realized as a program having instructions for executing the above method. Or this invention may be implement | achieved as an apparatus which has a means to implement | achieve the said method.

  According to the present invention, the angle of the skeleton can be controlled by a simple method. The effect of the present invention is not limited to this point.

The figure which shows a skeleton model, a master bone, a skeleton, and a coordinate system. The figure which shows a monopolar spherical-surface coordinate ((b) is the figure which looked at (a) from the other side). The figure which shows that the intersection of a plane group and a unit spherical surface becomes a vertical grid. The figure which shows that the intersection of a plane group and a unit spherical surface becomes a horizontal grid. The figure which shows the relationship between the direction of a plane, and (phi) and (theta). The figure which shows (phi) and (mu) of the point P on xz plane. The figure which shows (theta) and (nu) of the point P on xy plane. The figure which shows the angle restriction | limiting by a rectangular area. The figure which shows the angle restriction | limiting (horizontal shape) by an oval type | mold area | region. The figure which shows the angle restriction | limiting (vertically long shape) by an oval type | mold area | region. The figure which shows typically the inside / outside determination of a rectangular area. The figure which shows the auxiliary | assistant circle of an oval type | mold area | region (landscape). The figure which shows the auxiliary | assistant circle of an oval type | mold area | region (vertically long). The figure which shows the nearest point Pn from the point P with respect to the circular area | region C. FIG. The figure which shows four vertices of a rectangular area. Shows a 3 ¥ _{C 2, C 3, C t0} in contact with each other on the sphere. It shows an inscribed circle a spherical triangle _O 2 _O 3 _{O t0.} The figure which shows the procedure of the inside / outside determination of a rectangular area. The figure which shows the procedure of the inside / outside determination of a horizontally long oval type | mold area | region. The figure which shows the procedure of the inside / outside determination of a vertically long oval type | mold area | region. The figure which shows the procedure which calculates | requires the nearest point with respect to a rectangular area. The figure which shows the procedure which calculates | requires the nearest point with respect to a horizontally long oval type | mold area | region. The figure which shows the procedure which calculates | requires the nearest point with respect to a vertically long oval type | mold area | region. The figure which shows the procedure which determines whether an oval type | mold area shape is vertically long or horizontally long. The figure which shows the apparatus structure of the Example of this invention. The figure which shows the hardware constitutions of the Example of this invention. The figure which shows the procedure of the Example of this invention. The figure which shows the structure of the Example of this invention. The figure which shows the procedure of correction of bending and twist.

[1. Monopolar spherical coordinates]
As described above, in order to represent the bone direction of the skeleton model (FIG. 1 (a)), a coordinate system is defined for each bone as shown in FIG. 1 (b).

  If a unit spherical surface (spherical surface with a radius of 1) centering on the joint is made and the intersection point between the x-axis of the skeleton (x> 0 part) and the spherical surface is P, the bending state of the joint is the point P on the spherical surface. It can be expressed by position. One method of expressing joint rotation (bending, twisting) is to use a triaxial rotation angle (Euler angle). As an example, when the rotation is performed in the order of x-axis rotation, z-axis rotation, and y-axis rotation with reference to the coordinate system of the parent bone, paying attention to the relationship between the rotation angle of each axis and the position of the point P, the y-axis rotation angle Is equivalent to longitude, and the z-axis rotation angle is equivalent to latitude. Therefore, if a grid is formed on a spherical surface with the y-axis rotation angle and the z-axis rotation angle set at equal intervals, the shape is similar to the meridian / parallel of the globe. Since the two points corresponding to the North Pole and the South Pole are singular points in the angle expression, there is a problem that it is difficult to handle.

  On the other hand, as a technique for reducing the number of singular points to one, a method using a projection (Patent Document 2, Patent Document 3, and Patent Document 4) is known. The characteristics of this method are represented by a grid on a spherical surface as shown in FIG. This grid can be regarded as a coordinate system created on a spherical surface. Since there is one pole (singular point), in the present specification, this coordinate system is referred to as “monopolar spherical coordinate”.

The following items regarding the monopolar spherical coordinates are described below.
-Method of constructing a monopolar spherical coordinate system on a spherical surface without using projection-Method of limiting joint angles using monopolar spherical coordinates [2. Configuration of monopolar spherical coordinate system]
Generally, when a scale is added to a two-dimensional coordinate system, a grid is formed. In the following description, the term “grid” may be used for convenience, but the representation represents the meaning of a coordinate system, not a simple grid.

The point of coordinates (-1, 0, 0) is P _* and is called “pole”. Pole P _* a straight line parallel to the street y-axis and l _y, as shown in FIG. 3, to prepare a group of planes of equal angular interval containing a l _y. An intersection line between the plane group and the unit spherical surface is defined as a vertical line of the grid.

Similarly, let l _z be a straight line passing through the pole P _* and parallel to the z-axis, and a plane group having equiangular intervals including l _z is prepared as shown in FIG. The intersecting line between the plane group and the unit sphere is defined as a horizontal line of the grid.

As in Figure 5, represents the orientation of the plane containing the straight line l _y at an angle phi, it represents the orientation of the plane containing the straight line l _z at an angle theta. The signs of φ and θ are positive in the direction shown in the figure. The position of the point P on the spherical surface is determined as an intersection of the three surfaces of the plane determined by φ, the plane determined by θ, and the unit spherical surface. Let μ≡2φ and ν≡2θ, and use μ and ν to represent the position on the sphere.

  By using μ and ν without using φ and θ as they are, there are the following advantages. Consider the case where the point P is on the xz plane (ie, ν = 0) as shown in FIG. The orientation of the skeleton is OP. Let Q be the point at coordinates (1, 0, 0).

∠OP _* P = ∠OPP _* = φ
∠OP _* P + ∠OPP _* = ∠QOP
∴∠QOP = 2φ = μ
Therefore, when ν = 0, μ is the rotation angle around the y axis of the skeleton. Similarly, as shown in FIG. 7, when the point P is on the xy plane (that is, μ = 0), ν is the rotation angle around the z axis of the skeleton.

  In this way, if μ and ν are used, the state of rotation (the orientation of the skeleton) can be easily grasped sensuously. For example, FIG. 2 shows a grid formed with μ and ν values set at 10 ° intervals, but it can be seen that the grids are equally spaced by 10 ° on the circumference on the xy plane and the xz plane.

[3. Monopolar spherical coordinates and orientation of the skeleton]
A unit vector representing the orientation of the skeleton is represented by v≡ (x, y, z), and the unit vector represented by monopolar spherical coordinates is represented by (μ, ν). The relationship between the two is represented by the following equation (in the present specification, v represents a vector).

However, M, N, and T are as follows.

A method for deriving the expression will be described later (see [Expression Deriving Method A]).

  The specific angle (bending and twisting) control is described in detail below.

[4. Setting of movable area]
[4.1 Angle limitation by rectangular area]
By setting the upper and lower limits for the values of μ and ν, the movable region can be set as shown in FIG. Such a movable area restriction is called a rectangular restriction. Let μ ₁ and μ ₀ be the upper and lower limits of μ, and let ν ₁ and ν ₀ be the upper and lower limits of ν, respectively.

The expression μ = μ ₀ represents one circle on the unit sphere. This circle and _{C 0.} The surface of the sphere is divided into two regions by the circle C ₀ , and the movable region side (region where μ> = μ ₀ ) is called the “inside” of the circle C ₀ . When determining inside / outside of a circle on a spherical surface, the determination is performed using a cone as described later. Therefore, the cone parameter is obtained as follows.

A cone formed by connecting the circle C ₀ and the center O of the sphere is OC _0, and an axial unit vector of the cone OC ₀ is u ₀ ≡ (a ₀ , b ₀ , c ₀ ) (in this specification) u represents a vector). The direction of u ₀ is the movable region side. Further, the half apex angle (angle formed between the axis u ₀ and the conical surface) of the cone OC ₀ is α ₀ . These values are obtained by the following equation.

Similarly, the circles represented by the expressions μ = μ ₁ , ν = ν ₀ , and ν = ν ₁ are C ₁ , C ₂ , and C ₃ , respectively. The central axes of the cones OC ₁ , OC ₂ , and OC ₃ are denoted by u ₁ ≡ (a ₁ , b ₁ , c ₁ ), u ₂ ≡ (a ₂ , b ₂ , c ₂ ), u ₃ ≡ (a ₃ , b ₃ , c ₃ ), and the half apex angles are α ₁ , α ₂ , and α ₃ , respectively. These values are obtained by the following equation.

Since the range of values is −π <μ ₀ <μ ₁ <π and −π <ν ₀ <ν ₁ <π, the following equation holds.

As for other movable areas to be handled in the future, the circle on the spherical surface has the movable area side “inside” and the direction of the cone axis vector is the movable area side.

[4.2 Angle limitation by oval area]
Moving the skeletal model using the rectangle restriction may get caught in the corner of the movable area, resulting in non-smooth movement. In order to improve this behavior, an oval movable area is set as follows.

[4.2.1 Creation of corner circles]
As shown in FIGS. 9 and 10, corner circles in contact with the three sides of the rectangular area are created, and the corners are removed. Depending on the shape of the original rectangular area, a part of C ₂ and C ₃ remains as shown in FIG. 9 and a part of C ₀ and C ₁ remains as shown in FIG. The former shape is called “horizontally long” and the latter shape is called “vertically long”.

In the case of the horizontally long shape, as shown in FIG. 9, the corner circle on the μ ₀ side (circle contacting C ₀ , C ₂ , C ₃ ) is C _t0, and the corner circle on the μ ₁ side (C ₁ , C ₂ , C _3). C _t1 is a circle in contact with.

In the case of the vertically long shape, as shown in FIG. 10, the corner circle on the ν ₀ side (circle contacting the C ₀ , C ₁ , C ₂ ) is C _t2, and the corner circle on the ν ₁ side (C ₀ , C ₁ , C _3). C _t3 is a circle in contact with

[4.2.2 Conic parameters of corner circles]
The cone parameter of the corner circle is obtained by the following calculation. First, k ₀ , k ₁ , k ₂ , and k ₃ are defined as follows.

Since −π <μ ₀ <μ ₁ <π and −π <ν ₀ <ν ₁ <π, k ₀ <k ₁ , k ₂ > k ₃ . The vertical and horizontal length of the area shape is determined by the following formula.

If D _a <0, it is horizontally long, and if D _a > 0, it is vertically long. When D _a = 0, the two corner circles become one continuous circle. The cone parameter of the corner circle is obtained by the following procedure.

[When D _a ≦ = 0 (Landscape or Circular)]
The unit vector in the axial direction of the cone OC _t0 is set to u _t0 ≡ (a _t0 , b _t0 , c _t0 ), and the half apex angle is set to α _t0 . These values are obtained by the following calculation.

The axial unit vector u _t1 ≡ (a _t1 , b _t1 , c _t1 ) and the half apex angle α _t1 of the cone OC _t1 are obtained by the following calculation.

[When D _a > 0 (vertically long shape)]
The axial unit vector u _t2 ≡ (a _t2 , b _t2 , c _t2 ) and the half apex angle α _t2 of the cone OC _t2 are obtained by the following calculation.

The axial unit vector u _t3 ≡ (a _t3 , b _t3 , c _t3 ) and the half apex angle α _t3 of the cone OC _t3 are obtained by the following calculation.

The method for deriving the above cone parameter calculation formula will be described later (see “Formula derivation method B”). Further, a method of deriving a determination formula D _a will be described later (see the derivation method D of Expression).

[5. Inside / outside judgment for movable area]
"5.1 Internal / external judgment using a cone"
The shape of the movable area is a combination of multiple circles for both rectangular and oval restrictions. Therefore, the inside / outside determination for the movable region is a combination of the inside / outside determination for the circular region. The inside / outside judgment for a circular region can be realized using a cone as follows.

  A unit vector representing the orientation of the skeleton (direction of the x axis) is represented by v≡ (x, y, z). It is assumed that a circle C on the spherical surface is given, the axial unit vector of the cone OC is u≡ (a, b, c), and the half apex angle is α. The inside / outside determination with respect to the cone OC of the vector v can be realized as follows.

  It is a necessary and sufficient condition for v to be inside the cone OC that the angle formed by v and u is less than α. This condition is expressed as follows.

Since cos monotonously decreases in the angle range [0, π], it can be transformed as follows.

When the determination formula D is defined as follows, the inside / outside of the cone OC can be determined by the sign of D.

If D> = 0, it is inside the cone, otherwise it is outside the cone.

  This judgment formula can also be viewed as judgment using a plane. A plane whose normal vector is (a, b, c) and whose distance from the origin is cos α is defined as a plane κ. The value of the determination formula D means a signed distance (the u direction is positive) from the plane κ of the point (x, y, z). Therefore, the determination based on whether D is positive or negative means whether the point (x, y, z) is on the front or back side of the plane κ.

  The interpretation of the judgment formula D as a plane formula can be understood as follows. In general, a circle on a sphere is regarded as a line of intersection between a sphere and a “plane containing the circle”. Therefore, the inside / outside judgment for the circular area on the spherical surface can be realized by using the judgment by plane. The plane represented by the formula D = 0 is a plane including the circle C.

[5.2 Determination of inside / outside of rectangular area]
As shown in FIG. 11, the skeleton is within the rectangular region only when the orientation v is inside the circles C ₀ , C ₁ , C ₂ , and C ₃ . As shown in FIG. 18, the inside / outside determination for the region is performed according to the following procedure.

In S1801, and out the determination by the circle _{C 0,} it performs one of determination whether the inside. If this determination is “No”, it is determined that the region is out (S1811). If this determination is “Yes”, the process proceeds to S103.

In S1803, and out the determination by the circle _{C 1,} it performs one of determination whether the inside. If this determination is “No”, it is determined that the region is out (S1811). If this determination is “Yes”, the process proceeds to S105.

In S1805, and out the determination by the circle _{C 2,} it performs one of determination whether the inside. If this determination is “No”, it is determined that the region is out (S1811). If this determination is “Yes”, the process proceeds to S107.

In S1807, and out the determination by the circle _{C 3,} it performs one of determination whether the inside. If this determination is “No”, it is determined that the region is out (S1811). If this determination is “Yes”, it is determined to be within the area (S1809).

[5.3 Determination of inside / outside of oval area]
[5.3.1 Creating auxiliary circles]
For the inside / outside determination of the oval area, not only the circles (C _{0 to} C ₃ , C _t0 , to C _t3 ) constituting the area shape, but also the following auxiliary circles are used.

[Landscape]
When the area shape is horizontally long, an auxiliary circle is created as shown in FIG. The contact of the circle _{C t0} and _{C 2} and _{P 02,} the contact point of the circle _{C t0} and _{C 3} and _{P 03.} A circle passing through the three points P _* , P ₀₂ and P ₀₃ is _defined as an auxiliary circle C _s0 . The “inside” of the circle C _s0 is the center side of the oval area.

The axial unit vector of the cone OC _s0 is (a _s0 , b _s0 , c _s0 ), and the half apex angle is α _s0 . These values are obtained by the following calculation.

Similarly, the contacts of the circle _{C t1} and _{C 2} and _{P 12,} the contact point of the circle _{C t1} and _{C 3} to _{P 13.} An auxiliary circle passing through the three points P _* , P ₁₂ and P ₁₃ is _defined as C _s1 . The axis unit vector (a _s1 , b _s1 , c _s1 ) and the half apex angle α _s1 of the cone OC _s1 are obtained by the following calculation.

[For vertically long shape]
If the area shape is vertically long, an auxiliary circle is created as shown in FIG. The contact of the circle _{C t2} and _{C 0} and _{P 20,} the contact point of the circle _{C t2} and _{C 1} to _{P 21.} An auxiliary circle passing through the three points P _* , P ₂₀ and P ₂₁ is _defined as C _s2 . The axis unit vector (a _s2 , b _s2 , c _s2 ) and the half apex angle α _s2 of the cone OC _s2 are obtained by the following calculation.

The contact of the circle _{C t3} and _{C 0} and _{P 30,} the contact point of the circle _{C t3} and _{C 1} to _{P 31.} An auxiliary circle passing through the three points P _* , P ₃₀ and P ₃₁ is _defined as C _s3 . The axis unit vector (a _s3 , b _s3 , c _s3 ) and the half apex angle α _s3 of the cone OC _s3 are obtained by the following calculation.

A method for deriving the above cone parameter calculation formula will be described later (see [Formula derivation method C]).

[5.3.2 Internal / external determination using an auxiliary circle]
[Landscape]
Six circles of C _t0 , C _t1 , C ₂ , C ₃ , C _s0 , and C _s1 are used for the inside / outside determination of the horizontally long oval area. As shown in FIG. 19, the inside / outside determination for a region is performed according to the following procedure.

In S1901, and out the determination by the circle _{C 2,} it performs one of determination whether the inside. If this determination is “No”, it is determined that the region is out (S1913). If this determination is “Yes”, the process proceeds to S1903.

In S1903, and out the determination by the circle _{C 3,} it performs one of determination whether the inside. If this determination is “No”, it is determined that the region is out (S1913). If this determination is “Yes”, the process proceeds to S1905.

In S1905, the inside / outside determination using the circles _Cs0 and _Cs1 determines whether both are inside. If this determination is “Yes”, it is determined that the area is present (S1911). If this determination is "No", the process proceeds to S1907.

In S1907, and out the determination by the circle _{C t0,} it performs one of determination whether the inside. If this determination is “Yes”, it is determined that the area is present (S1911). If this determination is "No", the process proceeds to S1909.

In S1909, and out the determination by the circle _{C t1,} it performs one of determination whether the inside. If this determination is “Yes”, it is determined that the area is present (S1911). If this determination is “No”, it is determined that the area is out (S1913).
[For vertically long shape]
Six circles of C _t2 , C _t3 , C ₀ , C ₁ , C _s2 , and C _s3 are used for the inside / outside determination of the vertically long oval area. As shown in FIG. 20, the inside / outside determination for the region is performed according to the following procedure.

In S2001, and out the determination by the circle _{C 0,} it performs one of determination whether the inside. If this determination is “No”, it is determined that the region is out of the area (S2013). If this determination is “Yes”, the process proceeds to S2003.

In S2003, and out the determination by the circle _{C 1,} it performs one of determination whether the inside. If this determination is “No”, it is determined that the region is out of the area (S2013). If this determination is “Yes”, the process proceeds to S2005.

In S2005, the inside / outside determination by the circles C _s2 and C _s3 determines whether both are inside. If this determination is “Yes”, it is determined that the area is inside (S2011). If this determination is “No”, the process proceeds to S2007.

In S2007, and out the determination by the circle _{C t2,} it performs one of determination whether the inside. If this determination is “Yes”, it is determined that the area is inside (S2011). If this determination is “No”, the process proceeds to S2009.

In S2009, and out the determination by the circle _{C t3,} it performs one of determination whether the inside. If this determination is “Yes”, it is determined that the area is inside (S2011). If this determination is “No”, it is determined that the area is out (S2013).

[6. Twist angle]
[6.1 Order of bending and twisting]
The torsion angle of the skeleton is λ, and the monopolar spherical coordinates including the torsion are expressed by λ, μ, and ν. The order of bending and twisting can be considered as follows.
・ After bending, twist around the x-axis of the skeleton.
-Twist around the x axis and then bend.
The former is called “original bending” and the latter is called “original twist”.

[6.2 Definition of torsion angle]
In order to handle the twist angle, it is necessary to define a state where the twist angle λ = 0. First, the original bending method will be described. The bending state of the joint can be represented by the position of a point P on the spherical surface (intersection of the x-axis of the skeleton and the unit spherical surface). On the other hand, twist is expressed as rotation around the axis OP. A state in which the y-axis and z-axis of the skeleton are aligned with the grid directions (+ ν direction and −μ direction, respectively) is defined as a twist angle of zero.

  In the case of the original twist method, the grid itself is rotated by λ around the x axis (the x axis of the parent bone) by twisting. Even if bending is applied after twisting, the y-axis and z-axis of the skeleton will always coincide with the orientation of the rotated grid (+ ν direction and −μ direction, respectively).

[Relationship between 6.3 matrix and monopolar spherical coordinates]
Let M be a matrix that represents the orientation of the skeleton. The matrix elements are arranged in such a way that the matrix is multiplied from the left to the column vector (M represents a matrix in this specification).

The matrices M _b and M _t are defined as follows. However, M, N, and T are as shown in equations (6) and (7).

M _b is a bend, and M _t is a matrix representing torsion. Let v _b ≡ (x _b , y _b , z _b ) be a unit vector that represents the orientation of the skeleton when only bending is applied. Since v _b is the first column (the leftmost column) of M _b , the following equation is obtained.

v _b determined T, M, and N from the equation becomes:.

The reason why another formula is used in the case of x≈−1 is to avoid a decrease in accuracy due to a digit loss.

[6.3.1 Original bending method]
The procedure for obtaining λ, μ, ν from M is as follows. The reason why another formula is used in the case of m ₀₀ ≈−1 is to avoid a decrease in accuracy due to a digit loss.

Conversely, to obtain M from λ, μ, and ν, M _b and M _t are obtained using equations (64) and (65), and these are substituted into the following equations.

In order to obtain λ and v _b from M, Equation (70) and the following equation are used.

Conversely, to obtain M from λ and v _b , T, M, and N are obtained using equations (67), (68), and (69), and these are substituted into equation (64) to obtain equation (65 ), (74).

[6.3.2 Original twist method]
The procedure for obtaining λ, μ, ν from M is as follows.

Conversely, to obtain M from λ, μ, and ν, M _b and M _t are obtained using equations (64) and (65), and these are substituted into the following equations.

In order to obtain λ and v _b from M, Equation (76) and the following equation are used.

Conversely, to obtain M from λ and v _b , T, M, and N are obtained using equations (67), (68), and (69), and these are substituted into equation (64) to obtain equation (65 ), (80).

  The method for deriving the calculation formula will be described later (see [Formula derivation method E]).

[6.4 Relationship between quaternion and monopolar spherical coordinates]
A quaternion representing the orientation of the skeleton is assumed to be q as shown in the following equation.

The quaternions q _b and q _t are defined as follows.

q _b is a quaternion representing bending and q _t is twisting.

[6.4.1 Original bending method]
The equation for obtaining λ, μ, ν from q is as follows.

Conversely, to obtain q from λ, μ, and ν, q _b and q _t are obtained using equations (83) and (84), and these are substituted into the following equations.

In order to obtain λ and v _b from q, equation (85) and the following equation are used.

Conversely, in order to obtain q from λ and v _b , T, M, and N are obtained using equations (67), (68), and (69), and these are substituted into equation (83). ), (88).

[6.4.2 Original twist method]
The equation for obtaining λ, μ, ν from q is as follows.

Conversely, to obtain q from λ, μ, and ν, q _b and q _t are obtained using equations (83) and (84), and these are substituted into the following equations.

In order to obtain λ and v _b from q, equation (85) and the following equation are used.

Conversely, in order to obtain q from λ and v _b , T, M, and N are obtained using equations (67), (68), and (69), and these are substituted into equation (83). ), (93).

[7. Correction method outside the movable area]
A description will be given of a method for correcting the bending of the joint outside the movable region into the movable region. The bending state of the joint is represented by the position of the point P on the unit sphere. Let P _m be the point where the position of P is corrected within the movable region (including the boundary).

Given a region A on the sphere and a point P on the sphere outside the region A, the point on the region boundary that has the smallest distance from P (the distance on the sphere) is It will be referred to as “P's closest point for region A”. If only a movable region and a point P are given, it is considered best to set P _m as the closest point of P with respect to the movable region. Then, the correction method using the latest point is described.

[7.1 Nearest point to circle area]
The shape of the movable area is a combination of multiple circles for both rectangular and oval restrictions. Therefore, in order to obtain the nearest point for the movable region, it is necessary to obtain the nearest point for the circular region.

As shown in FIG. 14, the unit vector representing the orientation of the skeleton (direction of the x axis) is v≡ (x, y, z), and the coordinates of the point P are (x, y, z). It is assumed that a circle C on the spherical surface is given, the axial unit vector of the cone OC is u≡ (a, b, c), and the half apex angle is α. _Let P _n be the closest point of P with respect to the circle region C, and let w be the vector OP _n . w is obtained by the following equation.

However, β is an angle formed by v and u, and 0 <β <= π. Since sin β> = 0, sin β and tan β are obtained by the following equations.

sin β = 0 when v = −u, in which case all the points on the circle C are the closest points.

[Distance to 7.2 circle area]
It is assumed that the point P is outside the circular area C on the unit spherical surface. Let γ be the distance from P to circle C (distance on the unit sphere). Using the symbols described above (Section 7.1), γ = ∠POP _n = β−α, so the following equation is obtained.

Since 0 <γ <π, cos γ is monotonically decreasing with respect to γ. Therefore, cos γ can be used in place of γ for determining the distance to the circular region (however, the magnitude relationship is reversed).

[7.3 Nearest point for rectangular area]
Nearest point P _m of the point P with respect to the rectangular region, as shown in FIG. 21, is obtained by the following procedure. Let P be outside the rectangular area. Further, as shown in FIG. 15, the four vertices of the corners of the rectangular area are denoted as P _a , P _b , P _c , and P _d , respectively.

In S2101, in and out determination of P with a circle _{C 0,} it performs one of determination whether the inside. If the determination is “Yes”, the process proceeds to S2107. If the determination is “No”, the process proceeds to S2103.

In S2103, obtains the nearest point P with respect to the circle _{C 0,} the process proceeds to S2105 and _{P n.}

In S2105, whether or not both are inside is determined by the inside / outside determination of _{Pn using} the circles C ₂ and C ₃ . If the determination is “yes”, P _{n is set} to P _m (S2127), and the process ends. If the determination is “No”, the process proceeds to S2113.

In S2107, in and out determination of P with a circle _{C 1,} it performs one of determination whether the inside. If the determination is “Yes”, the process proceeds to S2113. If the determination is “No”, the process proceeds to S2109.

In S2109, obtains the nearest point P with respect to the circle _{C 1,} the process proceeds to S2111 and _{P n.}

In S2111, it is determined whether or not both are inside by the inside / outside determination of _Pn by the circles C ₂ and C ₃ . If the determination is “Yes”, P _{n is set} to P _m (S2127), and the process ends. If the determination is “No”, the process proceeds to S3113.

In S2113, in and out determination of P with a circle _{C 2,} it performs one of determination whether the inside. If the determination is “Yes”, the process proceeds to S2119. If the determination is “No”, the process proceeds to S2115.

In S2115, obtains the nearest point P with respect to the circle _{C 2,} the flow proceeds to S2117 and _{P n.}

In S2117, it is determined whether or not both are inside by the inside / outside determination of P _n by the circles C ₀ and C ₁ . If the determination is “Yes”, P _{n is set} to P _m (S2127), and the process ends. If the determination is “No”, the process proceeds to S2125.

In S2119, it is determined P is outside the circle _{C 3.} The process proceeds to S2121.

In S2121, obtains the nearest point P with respect to the circle _{C 3,} the process proceeds to S2123 and _{P n.}

In S2123, it is determined whether or not both are inside by the inside / outside determination of P _n by the circles C ₀ and C ₁ . If the determination is “Yes”, P _{n is set} to P _m (S2127), and the process ends. If the determination is “No”, the process proceeds to S2125.

In S2125, among the four points P _a , P _b , P _c , and P _d , the one closest to P (the minimum distance on the spherical surface) is set as P _m , and the process ends.

[7.4 Recent points for the oval area]
[Landscape]
Nearest point P _m of the point P with respect to a horizontally long oval region, as shown in FIG. 22, is obtained by the following procedure. Let P be outside the oval area.

In S2201, in and out determination of P with a circle _{C 2,} it performs one of determination whether the inside. If the determination is “Yes”, the process proceeds to S2207. If the determination is “No”, the process proceeds to S2203.

In S2203, the process proceeds to S2205 and _{P n} seek the nearest point P with respect to the circle _{C 2.}

In S2205, in and out determination of _{P n} with a circle _{C s0, C s1,} it performs both of determining whether it is inside. If the determination is “Yes”, P _n is set as P _m (S2215), and the process ends. If the determination is “No”, the process proceeds to S2213.

In S2207, in and out determination of P with a circle _{C 3,} it performs one of determination whether the inside. If the determination is “Yes”, the process proceeds to S2213. If the determination is “No”, the process proceeds to S2209.

In S2209, the process proceeds to S2211 and _{P n} seek the nearest point P with respect to the circle _{C 3.}

In S2211, it is determined whether or not both are inside by the inside / outside determination of P _n by the circles C _s0 and C _s1 . If the determination is “Yes”, P _{n is set} to P _m (S2215) and the process ends. If the determination is “No”, the process proceeds to S2213.

In S2213, the closest point of P with respect to the one of the circles C _t0 and C _t1 having the smaller distance from P is set as P _m , and the process ends.

[For vertically long shape]
As shown in FIG. 23, the closest point of the point P with respect to the vertically long oval area is obtained by the following procedure. Let P be outside the oval area.

In S2301, in and out determination of P with a circle _{C 0,} it performs one of determination whether the inside. If the determination is “Yes”, the process proceeds to S2307. If the determination is “No”, the process proceeds to S2303.

In S2303, the _{P n} seek the nearest point P with respect to the circle _{C 0,} the process proceeds to S2205.

In S2305, it is determined whether or not both are inside by the inside / outside determination of _Pn by the circles _Cs2 and _Cs3 . If the determination is “Yes”, P _{n is set} to P _m (S2315), and the process ends. If the determination is “No”, the process proceeds to S2313.

In S2307, in and out determination of P with a circle _{C 1,} it performs one of determination whether the inside. If the determination is “Yes”, the process proceeds to S2313. If the determination is “No”, the process proceeds to S2309.

In S2309, the _{P n} seek the nearest point P with respect to the circle _{C 1.}

In S2311, in and out determination of _{P n} with a circle _{C s2, C s3,} it performs both of determining whether it is inside. If the determination is “Yes”, P _{n is set} to P _m (S2315) and the process ends. If the determination is “No”, the process proceeds to S2313.

In S2313, among the circle _{C t2, C t3,} the nearest point P and _{P m} for better distance from P is small, it ends.

[7.5 How to correct bending and twisting]
The state of joint rotation (bending and twisting) is given by a matrix (or quaternion), and the procedure for correcting this within the angle limit range is as follows, as shown in FIG.

In S2901, λ and v _b are obtained from the matrix (or quaternion) (see sections 6.3 and 6.4).

In S2903, _{v b} investigate whether the movable region (Section 5.2, see section 5.3). If outside the movable region to correct _{v b} (Section 7.3, see Section 7.4) (S2905).

  In S2907, if an upper and lower limit is set for λ, it is checked whether the value is within this range. If it is out of this range, λ is corrected so that λ is within the upper and lower limits (S2909).

In S2911, a matrix (or quaternion) is obtained from λ and v _b (see Sections 6.3 and 6.4).

[Expression Derivation Method A] Derivation of Relational Expression between Direction Vector and Monopolar Spherical Coordinates The unit vector representing the orientation of the skeleton is (x, y, z), and this is expressed in monopolar spherical coordinates (μ, v). According to the construction method of the monopolar spherical coordinate system, the point of coordinates (x, y, z) is on the following two planes.

Equation (100) is a plan where the xy plane is rotated in a straight line _{l y} around an angle mu / 2, equation (101) is a plane xz-plane is rotated in a straight line _{l z} around an angle [nu / 2 . The range of values that μ and ν can take are −π <μ <π and −π <ν <π. Since (x, y, z) is a unit vector,

It is. Solve the above three equations simultaneously.

  Since cos (ν / 2) ≠ 0, the following equation is obtained from equation (101). However, N≡tan (ν / 2).

an (μ / 2).

Substituting equations (103) and (104) into equation (102) yields a quadratic equation for x. Solving this, and selecting the solution that is not x = −1, x is obtained as in the following equation. However, T≡2 / (M ² + N ² +1).

Substituting this into equations (103) and (104) yields y and z as in the following equations.

[Formula derivation method B] Derivation of conical parameters of the oval-shaped area corner circle A circle C _t0 in contact with the circles C ₀ , C ₂ , C ₃ is obtained. An axial unit vector of the cone OC _t0 is an unknown vector w≡ (x, y, z), and a half apex angle is an unknown τ (in this specification, w represents a vector).

The axis unit vector of the cone OC ₀ is u ₀ ≡ (a ₀ , b ₀ , c ₀ ), and the half apex angle is α ₀ . Since the circle C _t0 is in contact with the inner side (movable region side) of the circle C _0, the angle formed by u ₀ and w is α ₀ −τ. Therefore,

It becomes. Substituting cosα ₀ = −a ₀ , sinα ₀ = −c ₀ , b ₀ = 0

If k ₀ ≡a ₀ / c ₀ , the following equation is obtained.

Similarly, if cos α ₂ = −a ₂ , sin α ₂ = b ₂ , and c ₂ = 0 are used for the cones OC ₂ and OC _t0 , and k ₂ ≡a ₂ / b ₂ , the following equation is obtained. .

Similarly, using cos α ₃ = −a ₃ , sin α ₃ = −b ₃ , c ₃ = 0 for the cones OC ₃ and OC _t0 , and setting k ₃ ≡a ₃ / b ₃ , the following equation is obtained. It is done.

From the equations (109) and (110), the following equation is obtained.

From the equations (108) and (112), the following equation is obtained.

From formulas (111), (113), and (112)

If x ² + y ² + z ² = 1 is used to organize the equations and the right side is set to k ₄ , the following equation is obtained.

From equations (112) and (114)

It becomes. Since sin τ> 0 and k ₂ > k ₃ , x + cos τ> 0 from the equation (112). Therefore, the following equation is obtained from the above equation (115).

From the equations (114) and (116), the following equation is obtained.

However, l ₄ is as follows.

Using the obtained cos τ, x is obtained from the equation (114) as in the following equation.

Using the obtained x and cos τ, y and z can be obtained from the equations (111) and (113) as follows.

Thus, the corner cone OC _t0 is obtained.

For the remaining three corner circles C _t1 , C _t2 , C _t3 , the cone parameter can be obtained by the same method.

[Expression Derivation Method C] Derivation of Conical Parameters of Auxiliary Circle As shown in FIG. 16, the three circles C ₂ , C ₃ , C _t0 on the spherical surface are in contact with each other, and the contact points of the circles C ₂ , C ₃ are The contact point between the pole P _* and the circles C ₂ and C _t0 is P ₀₂ , and the contact point between the circles C ₃ and C _t0 is P ₀₃ . A circle passing through the three points P _* , P ₀₂ and P ₀₃ is _defined as C _s0 . An axial unit vector of the cone OC _s0 is an unknown vector w≡ (x, y, z), and a half apex angle is an unknown τ.

A point with coordinates −u ₂ ≡ (−a ₂ , −b ₂ , −c ₂ ) is defined as O ₂ . Similarly, the points at coordinates −u ₃ and u _t0 are O ₃ and O _t0 , respectively. O ₂ , O ₃ , and O _t0 are the centers on the spherical surfaces of the circles C ₂ , C ₃ , and C _t0 , respectively.

As shown in FIG. 17, a spherical triangle O ₂ O ₃ O _t0 is created. The points P _* , P ₀₂ and P ₀₃ are on the side O ₂ O ₃ , the side O ₂ O _t0 and the side O ₃ O _t0 , respectively. Coordinates -w≡ (-x, -y, -z) points to _{O s0.} O _s0 is the center of a circle C _s0 on the spherical surface as shown in the figure.

The circle C _s0 is a circle that passes through the three points P _* , P ₀₂ , and P ₀₃ , and at the same time is an inscribed circle of the spherical triangle O ₂ O ₃ O _t0 .

As described above, the inscribed circle is apparent from the following points.
· The three sides phases _{like, △ O s0 P * O 3} ≡ △ O s0 P 03 O 3, thus a _{∠O s0 P * O 3 = ∠O} s0 P 03 O 3. _{Similarly, ∠O s0 P 03 O t0 =} ∠O s0 P 02 O t0, the _{∠O s0 P 02 O 2 = ∠O} s0 P * O 2.
· In _{addition, ∠O s0 P 03 O 3 =} π-∠O s0 P 03 O t0, ∠O s0 P 02 O t0 = π-∠O s0 P 02 O 2, ∠O s0 P * O 2 = π-∠ O is a _s0 P _* O _3.
When ∠O _s0 P ₀₃ O ₃ = ρ and applying this to the above relationship, ρ = π−ρ is obtained, and ρ = π / 2 is obtained. _Therefore, it is _{O s0 P * ⊥O 2 O 3} , O s0 P 02 ⊥O 2 O t0, O s0 P 03 ⊥O 3 O t0.
Therefore, the circle C _s0 is in contact with the spherical triangle O ₂ O ₃ O _t0 .

Therefore, the conical _{OC s0} is in contact with the plane _OO 2 _{O 3} and the plane _OO 2 _{O t0.} Therefore, the angle formed by the vector −w and the plane OO ₂ O ₃ is equal to the angle formed by the vector −w and the plane OO ₂ O _t0 (both are π−τ).

The plane OO ₂ O ₃ is an xy plane. Therefore, the angle formed by the vector −w and the z-axis is equal to the angle formed by the normal line of the vector −w and the plane OO ₂ O _t0 (both are τ−π / 2). This is expressed as follows. However, e _z ≡ (0, 0, 1) (in the present specification, e represents a vector).

The plane OO _s0 P _* passes through the point P _* ≡ (−1, 0, 0) and is perpendicular to the plane OO ₂ O ₃ . Therefore, the vector w≡ (x, y, z) is in the xz plane.

And w = (x, 0, z). Since u ₂ = (a ₂ , b ₂ , 0) and u _t0 = (a _t0 , b _t0 , c _t0 ), Expression (122) is as follows.

On the other hand, | u ₂ × u _t0 | is expressed as follows.

Furthermore, since k ₂ ≡a ₂ / b ₂ , the equation (124) is as follows.

Since sin α _t0 > 0 and k ₂ > k ₃ , the following equation is obtained from equation (21).

Substituting equations (21) and (126) into equation (125) and rearranging results in the following equation. Let this equation be l ₀ .

| W | = 1 to x ² + z ² = 1. Since z <0, x and z are as follows.

Since the conical surface _{C s0} passes through _{P *, cosτ} is obtained as follows. However, e _x ≡ (1, 0, 0).

For the remaining three auxiliary circles C _s1 , C _s2 , and C _s3 , the cone parameter can be obtained by the same method.

[Formula Derivation Method D] Derivation of Oval Shape Vertical / Horizontal Judgment Formula Consider the moment when the shape of the oval shape is changed to switch from landscape to portrait. In this state, since the circles C _t0 and C _t1 coincide, u _t0 = u _t1 and α _t0 = α _t1 .
If u _t0 = u _t1 , then b _t0 = b _t1 , l ₄ = l ₅ from equations (19) and (25), and k ₄ = k ₅ from equations (17) and (23). .
On the contrary, if k ₄ = k ₅ , l ₄ = l ₅ from equations (17) and (23), u _t0 = u _t1 and α _t0 = α _t1 .

Therefore, the necessary and sufficient condition for the circles C _t0 and C _t1 to coincide is k ₄ = k ₅ . Substituting the formulas (16) and (22) into this, the following formula is obtained. This equation is a determination formula D _a.

From equation (13), k ₀ and k ₁ monotonically increase with respect to μ ₀ and μ ₁ , respectively. Therefore, when extending the movable range in the horizontal (mu ₀ reduction, or mu ₁ increases), the value of D _a is reduced.

Similarly, k ₂ and k ₃ are monotonically decreasing with respect to ν ₀ and ν ₁ from the equation (14). Therefore, when extending the movable range in the vertical ([nu ₀ reduction, or [nu ₁ increases), the value of D _a is increased.

Thus, using the value of D _a, as shown in FIG. 24, can determine portrait-landscape oval area shape as follows.

If D _a <0 holds in S2401, it is determined to be horizontally long (S2405).

If D _a > 0 holds in S2403, it is determined to be vertically long (S2407).

  If “NO” in S2403, it is determined to be one continuous circle (S2409).

[Expression Derivation Method E] Derivation of the relational expression between the matrix and the monopolar spherical coordinate The unit vectors in the y-axis and z-axis directions of the skeleton in the state where the torsion is 0 are respectively v _y and v _z . v _y and v _z are the + ν direction and −μ direction of the monopolar grid, respectively.
A unit vector representing the orientation of the skeleton is assumed to be v≡ (x, y, z). _Let v _μ and v _ν be partial derivatives of v as follows.

v _{y, v z} are each v _[nu, since the -v _mu things normalized, is expressed by the following equation.

In obtaining v _y and v _z , first, partial differentials of M and N are calculated as follows.

Using these, the partial differentiation of T≡2 / (M ² + N ² +1) is obtained as follows.

Using these, partial differentials of x = T-1, y = TN, and z = -TM are obtained as follows.

When calculating using T (M ² + N ² +1) = 2, the following equation is obtained.

Similarly, the following equation is obtained.

Therefore, v _y and v _z are as follows.

Let M be the matrix that represents the orientation of the skeleton.

Matrixes representing bending and twisting are denoted as M _b and M _t , respectively. The matrix elements are arranged by multiplying the column vector from the left by the matrix.

Elements of the bending matrix _{M b} has the formula (3), (4), (5), (150), as follows from (151).

The elements of the twist matrix M _t are as follows: However, S _λ ≡sin _λ and C _λ ≡cos _λ .

[For the original bending method]
Since a twist is applied after bending, M is as follows. If this equation is used, M can be obtained from λ, μ, and ν.

Conversely, the procedure for obtaining λ, μ, ν from M is as follows. First, T is calculated.

However, the accuracy of this equation deteriorates due to a digit loss when m ₀₀ ≈-1. In such cases,

Using

T is obtained by μ and ν are obtained by the following equations.

Next, the following equation is used to obtain λ.

Then

The following equation is obtained.

Therefore, λ is obtained as in the following equation.

[In the case of the original twist method]
Since bending is applied after twisting, M is as follows. If this equation is used, M can be obtained from λ, μ, and ν.

Expressions for obtaining λ, μ, and ν from M are obtained by the same method as in the original bending method.

[Processing and Configuration Example of the Embodiment of the Present Invention]
FIG. 25 is a diagram showing a device configuration of the embodiment of the present invention. To the computer 2500, a keyboard 2505, a mouse 2506, a tablet 2507 including a pen 2508, and a display 2501 are connected. A canvas 2502 is displayed on the display 2501, and a figure is drawn on the canvas 2502 (2503).

  FIG. 26 is a diagram showing a hardware configuration of the embodiment of the present invention. A bus 2607 is connected to the CPU 2601. A memory 2602, various input devices 2603, a display device 2604 such as a display, and a printer 2605 are connected to the bus 2607.

  FIG. 27 shows an embodiment of the present invention. Referring to FIG. 27, first, a boundary circle constituting the angle range of the skeleton is set by the monopolar spherical coordinate system (S2701). Specifically, for example, upper and lower limits of μ, ν, shape (rectangular or oval type), etc. are designated. Then, it is determined whether or not the angle of the skeleton is within the angle range (that is, inside) (S2703). When it is determined that the angle of the skeleton is out of the angle range, the angle of the skeleton is corrected using the above-described method for obtaining the nearest point (S2705). Then, the skeleton model is displayed on the screen. Note that when controlling an object in real space such as a robot, control is performed to direct the skeleton to a desired angle (S2707).

  FIG. 28 shows an embodiment of a control device for controlling the skeleton. First, a boundary circle constituting the angle range of the skeleton is set by the monopolar spherical coordinate system (2801). Then, it is determined whether or not the angle of the skeleton is within the angle range (that is, inside) (2803). If it is determined that the angle of the skeleton is outside the angle range, the angle of the skeleton is corrected using the above-described method for obtaining the nearest point (2805). Then, the skeleton model is displayed on the screen (2807). Note that when controlling an object in real space such as a robot, control is performed to direct the skeleton to a desired angle (2808). Further, input means (2806) for setting various numerical values may be provided.

  In the specification of the present application, the control of the angle of the phalanx relative to the parent bone has been described as an example. Needless to say, the present invention can be applied to a skeleton model in which three or more bones are connected by joints.

  Although the present invention has been described above, the scope of the present invention is not limited to the above-described embodiments in the present specification. Those skilled in the art can understand variations of the present invention from the description of the present specification. Needless to say, these modifications and equivalents also belong to the technical scope according to the claims.

  Further, the steps defined in the claims do not have to be executed in the order described, and the steps defined in the claims can be executed in a different order as long as no contradiction arises. Needless to say, changes in the order of the steps of the claims are also included in the technical scope of the claims.

P _* Unit vector P _m nearest point of pole v of monopolar spherical coordinate system

Claims (7)

In a skeletal model in which a parent bone and a skeleton are connected by a joint, a method for controlling the movement of the skeleton by limiting an angle range in the joint of the skeleton relative to the parent bone,
An angular range boundary circle setting step for setting an angular range boundary circle defining a boundary of the angular range of the skeleton on a spherical surface of a sphere centered on the joint, the angular range boundary circle being centered on the joint; The spherical surface, which passes through the intersection point 1 of the spherical surface and the parent bone or the extension line of the parent bone, and the position of the intersection point 2 of the spherical surface and the extension line of the child bone and the spherical surface is bisected by the angular range boundary circle The angle range of the skeleton is determined depending on which of the cones, and the cone unit vector u formed by the center of the sphere and the angle range boundary circle and the half apex angle α of the cone, or the An angular range boundary circle setting step for setting the angular range boundary circle by storing in a memory numerical values from which an axis unit vector u and the half apex angle α can be derived;
The axis unit vector u and the half apex angle α are obtained by reading the numerical values stored in the memory, and the value of u · v-cos α is inspected using the unit vector v representing the orientation of the skeleton. Calculating an angle range, and determining whether or not the angle of the skeleton is within the angle range based on the inspection value; and
Having a method. From the three-dimensional orthogonal coordinate system (x, y, z), where the joint is the origin, and the direction from the intersection 1 to the origin is the positive direction of the x axis,
μ = 2 tan ⁻¹ (−z / (x + 1))
ν = 2 tan ⁻¹ (y / (x + 1))
Using a monopolar spherical coordinate system (μ, ν) having the relationship
The angular range boundary circle satisfies at least one of μ = μ _e and ν = ν _e ,
(Where μ _e and ν _e are constants),
The method of claim 1. When the monopolar spherical coordinates of the intersection 2 are (μ, ν), the lower and upper limits of μ are μ ₀ and μ _1, and the lower and upper limits of ν are ν ₀ and ν ₁ , respectively, the angular range But,
μ ₀ <μ <μ ₁ and ν ₀ <ν <ν ₁
Set the angle range boundary circle to satisfy
The method of claim 2. The angle range boundary circle further uses an auxiliary angle range boundary circle that does not pass through the intersection point 1;
4. A method according to any one of claims 1 to 3. In the angle range determination step,
If it is determined that the angle of the bone is not within the angular range,
By moving the intersection point 2 from the intersection point 2 to a point on the sphere that satisfies the angle range, the point measured on the sphere has the shortest distance on the sphere. An angle correction step to correct the angle of the bone,
The method according to any one of claims 1 to 4, further comprising:   A program comprising instructions for causing a computer to execute the method according to any one of claims 1 to 5. In a skeletal model in which a master bone and a skeleton are connected by a joint, an apparatus for controlling the operation of the skeleton by limiting an angle range of the joint to the parent bone in the joint,
Angle range boundary circle setting means for setting an angle range boundary circle defining a boundary of the angle range of the skeleton on a spherical surface of a sphere centered on the joint, wherein the angle range boundary circle is centered on the joint The spherical surface, which passes through the intersection point 1 of the spherical surface and the parent bone or the extension line of the parent bone, and the position of the intersection point 2 of the spherical surface and the extension line of the child bone and the spherical surface is bisected by the angular range boundary circle The angle range of the skeleton is determined depending on which of the cones, and the cone unit vector u formed by the center of the sphere and the angle range boundary circle and the half apex angle α of the cone, or the An angle range boundary circle setting means for setting the angle range boundary circle by storing in a memory a numerical value capable of deriving the axis unit vector u and the half apex angle α;
The axis unit vector u and the half apex angle α are obtained by reading the numerical values stored in the memory, and the value of u · v-cos α is inspected using the unit vector v representing the orientation of the skeleton. An angle range determination means for calculating as a value and determining whether the angle of the skeleton is within the angle range based on the inspection value;
Having a device.