JPH10208072A - Skeleton model shape deforming method, picture composing device, and information storage medium - Google Patents

Abstract

PROBLEM TO BE SOLVED: To easily set a complicated restrictive condition by obtaining a solution of a fundamental variable which approximately meets a fundamental formula and makes the value of an evaluation formula minimum, maximum, or stationary. SOLUTION: Coordinates (x, y, z) of nodes 2 to 4 and components (u, v, w) of arc normal vectors 10 to 12 perpendicular to arcs 6 to 8 are used as fundamental variables representing a skeleton model 18. Formulas or the like which prescribe that lengths of arcs are given values are prepared as fundamental formulas having fundamental variables as unknown values as shown in a figure B. At this time, fundamental formulas in the figure B are nonlinear, and simultaneous equations to be solved are shown in a figure C. The evaluation formula to unequivocally specify the solution meeting the fundamental formulas is prepared, and a condition that the value of the evaluation formula is made approximately minimum, maximum, or stationary is added, and the Lagrange multiplier method is applied to obtain the solution of fundamental variables. Consequently, a formula to be solved is as shown in a figure D, and the solution can be unequivocally determined.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention relates to a method for deforming the shape of a skeleton model, an image synthesizing apparatus, and an information storage medium.

[0002]

2. Description of the Related Art A technique called inverse kinematics (inverse kinematics) has been known. In this technique, first, FIG.
A skeleton (skeleton) model 20 as shown in FIG. The user performs an operation of deforming the skeleton model 20 using an input device such as a mouse while looking at the shape of the skeleton model 20 displayed on a CRT or the like, and creating a motion or a posture intended by the user. In the following description, the skeleton model 20
Are referred to as nodes, and bones connecting the nodes are referred to as arcs.

[0003] As a conventional technique of such inverse kinematics, for example, "Human body shape modeling and animation" (Fukui Kazuo, Information Processing Society of Japan "person modeling and display technology" seminar material, September 1991) and the like. is there. In this prior art, as shown in FIG. 22B, angles θ ₁ , θ ₂ , and θ ₃ between arcs are used as basic variables representing the shape of a skeleton model. And the problem that conventional inverse kinematics should solve is the fixed node 40
When the position (x, y, z) of the node 42 at the other end of the linear skeleton having is specified, the angles between the arcs θ ₁ , θ ₂ , θ ₃ that realize this are obtained. This problem corresponds to _obtaining the inverse function f ^-1 (x, y, z) of the function f (θ _1x , θ _1y , θ _1z , θ _2x ...). However, since the function f is nonlinear, it is not easy to find the inverse function. Therefore, the small change amount ｘx, △ y, △ z of x, y, z is changed to the small change amount △ θ of the angle between arcs.
_1x , △ θ _1y , △ θ _1z , △ θ _2x ... Thus, the problem to be solved is reduced to solving the simultaneous equations shown in FIG. However, since the matrix M in FIG. 22C is not a square matrix and the number of variables is larger than the number of equations, there are countless solutions. Therefore, the simultaneous equation of FIG. 22C is solved by using a mathematical method of adding a condition that the sum of squares of the change amount between the arcs is minimized.

However, in the above-mentioned prior art, since the angle between arcs is used as a basic variable representing a shape for a linear skeleton fixed at one end, there are the following problems.

It is difficult to handle a multi-branch structure. In the prior art, in order to handle a skeleton model of a multi-branch structure (a structure in which three or more arcs are connected to one node), for example, FIG. As shown in FIG.
It is necessary to set a parent-child relationship as shown in FIG. For example, if one hand of the skeleton model 20 is to be moved, the node 30 is set to the root and the node 32 which is an effector is moved. If the other hand is to be moved, the node 30 is set to the root and the effector is set. Node 34 is moved. If you want to move one foot,
This time, the node 36 is set as the root, and the node 38 serving as an effector is moved. Therefore, it is necessary for the user to perform the operation while always minding which node is the root and which node is the effector, and the operation is complicated.

It is difficult to handle complicated constraints. For example, when constraints are set for a plurality of nodes, it is necessary to disassemble the skeleton into a linear structure in accordance with the constraints, and the processing becomes complicated. Become. In general, the constraint condition is expressed as an expression of node coordinates. For example, FIG.
In the case where the node 38 of FIG.
An expression that specifies that 8 is always located on the polygon will be used. However, in the prior art, since the basic variable is the angle between arcs, it is difficult to use the above equation, and it is difficult to handle the constraint condition.

It is difficult to move a plurality of nodes in different directions at the same time. That is, in the prior art, for example, when trying to move nodes 38 and 39 in different directions, the processing becomes extremely complicated.

The present invention has been made in view of the above technical problems, and an object of the present invention is to easily handle a skeleton model having a multi-branch structure and to easily set a complicated constraint condition. It is an object of the present invention to provide a skeleton model shape deformation method, an image synthesizing device, and an information storage medium that can be used.

[0009]

SUMMARY OF THE INVENTION In order to solve the above problems, the present invention relates to a method of deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including the coordinates of the nodes of the skeleton model. At least one of a basic expression including an expression that defines that the arc length is a given value and the basic variable as an unknown, and an evaluation expression for uniquely specifying a solution that satisfies the basic expression, A skeleton model is determined based on the determined solution by changing based on given information to obtain a solution of a basic variable that substantially satisfies the basic expression and makes the value of the evaluation expression substantially minimum, approximately maximum, or almost stationary. Is characterized by deforming the shape of.

According to the present invention, a skeleton model is represented by basic variables including node coordinates, and a solution that substantially satisfies the basic expression is uniquely identified by adding a condition for making the value of the evaluation expression substantially minimum. Then, the shape of the skeleton model is determined. Therefore, according to the present invention, it becomes possible to treat all nodes equally, and it becomes easy to handle a skeleton model having a multi-branch structure. Further, since the constraint condition is often defined based on the coordinates of the node, the present invention can easily cope with the setting of a complicated constraint condition.

Further, according to the present invention, the basic variable includes a component of an arc normal vector perpendicular to the arc, and the basic expression specifies that the length of the arc normal vector has a given value. An equation and an equation defining that the arc normal vector is perpendicular to the arc are included.

According to the present invention, since the arc normal vector is used as the basic variable, it is possible to treat all arcs equally, and it is easy to handle a skeleton model having a multi-branch structure.

Further, according to the present invention, the basic variable includes a component of an arc normal vector perpendicular to the arc, and the variation of the component of the arc normal vector is represented by a given variable to solve the basic variable. Is obtained.

According to the present invention, since the amount of change in the component of the arc normal vector is represented by one given variable, the number of unknowns to be solved can be reduced, and the processing can be speeded up. It is particularly desirable that the amount of change in the component of the arc normal vector is represented by a linear expression of one given variable.

Further, in the present invention, the basic variables include a component of an arc normal vector perpendicular to the arc, a change amount of an angle between the arcs, a distance between a node and a given point, and Basically, the value of the evaluation expression including at least one of the change amount of the direction of the arc having the fixed node, the change amount of the node coordinates, and the change amount of the speed of the node coordinates becomes one of substantially minimum, substantially maximum, and substantially stationary. It is characterized by finding the solution of a variable.

By including the variation of the angle between arcs in the evaluation formula, the skeleton model can be deformed in a well-balanced manner. By including the distance between a node and a given point, the rubber band Etc. can be set. Also, by including the change amount of the node coordinates and the speed change amount in the evaluation formula, it is possible to prevent the solution from being undefined and the skeleton model from being violent.

Further, the present invention provides a solution of the basic variables by iterative calculation, and synchronizes at least one of the basic formula and the evaluation formula with the end or start of an iterative loop and successively based on given information. It is characterized by changing.

When an iterative method such as the Newton's method is used, the known number and shape of the expression usually do not change until the calculation converges.
According to the present invention, before the calculation converges, for example, every one iteration loop or every two or three iteration loops, the basic formula and the evaluation formula change one after another. By doing so, for example, when given information is the coordinates of the cursor position of the mouse operated by the user, it is possible to deform the shape with sufficient followability to the user's input, Convenience can be improved. When the error becomes large, it is desirable to perform correction by scaling.

Further, the present invention is characterized in that the coordinates of the fixed nodes or the components of the fixed arc normal vector are known numbers.

In this way, the coordinates of the nodes and the direction of the arc can be easily fixed, and the convenience of the user can be improved. Also, the number of unknowns can be reduced, and the processing can be speeded up.

Further, according to the present invention, the evaluation formula may be a formula for constraining an angle formed between arcs to a given range, a formula for constraining a node to a given surface or a given line, and a skeleton model or an accompanying model. The method includes at least one of a skeleton model or a formula for constraining the first object to the second object when the first object moves and interferes with another second object.

As described above, according to the present invention, just by including the above-mentioned various expressions in the evaluation expression, the movable range of the angle between arcs can be limited, the node can be attached to a given surface or the like, and the skeleton model can be used. Constraints such as interference between the
It can be easily set.

The present invention also relates to a method for deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including coordinates of nodes of the skeleton model and components of an arc normal vector perpendicular to an arc of the skeleton model.
At least one of a change in angle between arcs, a distance between a node and a given point, a change in direction of an arc having a fixed node at an end, a change in node coordinates, and a change in speed of node coordinates. The shape of the skeleton model is modified so that the value of the expression including the two is substantially minimum, approximately maximum, or substantially stationary.

According to the present invention, since the components of the node coordinates and the arc normal vector are the basic variables, all nodes,
Arcs can be handled equally, making it easier to handle multi-branch structures.
In addition, by minimizing the expression including the amount of change in the angle between arcs, a well-balanced shape deformation can be achieved.

The present invention also relates to a method for deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including the coordinates of the nodes of the skeleton model, and the coordinates of a given point for controlling the shape deformation of the skeleton model are determined. In the case of a change, the shape of the skeleton model is modified so that the distance between the node and the given point is minimized.

According to the present invention, it is possible to set a so-called rubber band and the like, and the shape can be deformed with an appropriate shape as an initial state, so that a well-balanced shape deformation can be performed.

Further, the present invention is characterized in that the shape of the skeleton model is modified so that the distance between two or more nodes and two or more given points associated with each of the nodes is minimized. I do.

According to the present invention, it is possible to change the shape of the skeleton model using two or more given points as control points, thereby improving user convenience.

The present invention also relates to a method of deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including the coordinates of the nodes of the skeleton model, and the solution of the expression including the basic variables is obtained by iterative calculation. The formula is characterized in that it is changed one after another based on given information in synchronization with the end or start of the iterative loop.

According to the present invention, before the calculation converges, for example, the basic formula and the evaluation formula change one after another for each iteration loop or every two or three iteration loops, so that a quick shape deformation of the skeleton model is required. Even if it is done, it is possible to cope with this in a state where the appearance does not become so unnatural.

The present invention also relates to a method for deforming the shape of a skeleton model, wherein when a skeleton model or an object accompanying the skeleton model exceeds a given boundary, the skeleton model or the object is returned to the boundary side. One iterative loop of an iterative calculation for deforming the shape of the skeleton model under conditions and determining the shape of the skeleton model is performed with the constraints disabled and the skeleton model or object does not exceed the boundary. Performs the calculation in the next iteration loop with the constraints disabled, and when the first iteration loop is performed with the constraints disabled and the skeleton model or object exceeds the boundary In the next iterative loop, the above conditions are made valid and the calculation is performed.
If the first iteration loop is performed with the constraint conditions valid and the skeleton model or object is beyond the boundary, the calculation is performed with the constraint conditions also valid in the next iteration loop. If the one iteration loop is performed with the constraints enabled and the skeleton model or object does not cross the boundary, then in the next iteration loop the calculations are performed with the constraints disabled. It is characterized by performing.

According to the present invention, since the skeleton model or the like is constrained by the boundary as a spring, for example, whether the user is moving the skeleton model toward the boundary or away from the boundary Can be easily determined. Since the repetitive calculation for determining the constraint condition is not required, the processing speed can be remarkably improved.

Further, according to the present invention, the value of the expression including the distance between the skeleton model or the object and the boundary is substantially minimized,
It is characterized in that the shape of the skeleton model is deformed under a constraint condition of being substantially maximal or substantially stationary.

By doing so, it becomes easy to realize the function of interfering with the skeleton model and the object.

Further, the present invention provides a virtual boundary at a position separated by a predetermined distance from the boundary, and when the skeleton model or the object exceeds the boundary, returns the skeleton model or the object to the virtual boundary. It is characterized in that the validity / invalidity of the constraint condition is determined using the virtual boundary instead of the boundary.

By doing so, for example, the problem at the time of multiple collision between a node or the like and an object can be easily solved.

The present invention also relates to a method for deforming the shape of a skeleton model, wherein the domain of at least one first variable defining the shape of the skeleton model is represented by an inequality including the first variable. , An equation in which all the expressions and numbers of the inequality are transposed to one of the left side and the right side, and the other side is replaced by a function of the second variable that is constantly positive or negative or non-negative or non-positive. And defining the domain of the first variable by the equation in place of the inequality, thereby deforming the shape of the skeleton model under discontinuous constraints.

For example, the setting of the domain of the first variable such as the angle between arcs, the node coordinates, the component of the arc normal vector, etc.
Usually this can be done using inequalities. According to the present invention, this inequality is replaced by an equation having the same effect as the inequality in setting the domain. Then, the domain of the first variable is defined based on this equation, and a discontinuous constraint condition is given to the shape deformation of the skeleton model. By replacing the inequalities with the equations in this way, it is possible to save the time and labor required when using the inequalities and to speed up the processing. If the equation is an equation, for example, it can be easily incorporated into an equation for imposing a constraint condition on the skeleton model, and the processing can be simplified.

Further, the present invention is characterized in that the shape of the skeleton model is deformed such that the value of the expression including the variation of the second variable is one of substantially minimum, substantially maximum, and substantially stationary. .

According to the present invention, it is possible to prevent the value of the second variable from greatly changing. Thus, it is possible to effectively prevent a situation in which the second variable is rampant and the first variable is not restricted within the domain.

Further, the present invention is characterized in that when the second variable has a value near a given convergence value, a given value is added to the second variable.

If the second variable converges to a given convergence value and takes much time to get out of the convergence state,
For example, the response speed of the skeleton model to a user input is reduced. According to the present invention, a given minute value is added to the second variable before the second variable converges to a convergence value. Therefore, it is possible to prevent the second variable from converging to the convergence value, and it is possible to solve the above-described problem of a decrease in response speed. It is desirable to use a random number as the given value to be added to the second variable.

The present invention also relates to a method for deforming the shape of a skeleton model, wherein in one iterative loop of an iterative calculation for obtaining the shape of the skeleton model, the skeleton model or an object moving therewith is constrained to a given boundary. Performing a first calculation in a state in which the constraint is invalid, and in the first iteration loop, after the first calculation, when the skeleton model or an object and the boundary have a given positional relationship, the skeleton model or A second calculation for drawing an object to the boundary is performed.

According to the present invention, first, the first calculation is performed in a state where the constraint condition is invalid, and then, the second calculation for drawing the skeleton model or the like to the boundary is performed. Therefore,
The skeleton model or the like can be constrained to the boundary by performing only two calculations in one iteration loop. Therefore, the processing speed at the time of iterative calculation can be remarkably improved as compared with the conventional method in which one iterative loop includes another iterative loop. The second calculation for drawing the skeleton model or the like to the boundary may be performed as necessary, using the positional relationship between the skeleton model or the like and the boundary as a judgment material.

Further, according to the present invention, in the first calculation,
Calculation is performed such that the value of the expression including the amount of movement of the skeleton model or object in the boundary direction is substantially minimum, approximately maximum, or substantially stationary, and in the second calculation, the skeleton model or object and the The calculation is performed so that the value of the expression including the distance to the boundary is substantially minimum, approximately maximum, or substantially stationary.

According to the present invention, it is possible to prevent a skeleton model or the like to be constrained by a boundary from being too far away from the boundary to be unable to be pulled back by making the expression including the amount of movement in the boundary direction substantially minimal. it can. Further, by making the expression including the distance to the boundary substantially equal to the minimum, it is possible to easily return the skeleton model or the like to the boundary.

Further, the present invention is characterized in that the boundary is a finite surface or a finite line.

As described above, since the skeleton model can be constrained to a finite surface or a finite line (for example, a polygon having a finite size, a polygon edge having a finite length, etc.), user convenience can be improved. In the case of constraining to an infinite surface or an infinite line, for example, the size and the length of the edge of the polygon can be sufficiently increased.

The present invention also relates to a method of deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including the coordinates of the nodes of the skeleton model, and a first simultaneous equation in which the amount of change of the basic variables is unknown. Is determined to be in a state of degeneracy or close to it, and if it is determined that the first simultaneous equation is in a state of degenerate or close to it, the first simultaneous equation is converted to an independent It is characterized in that the shape of the skeleton model is deformed by reconstructing the second simultaneous equation and obtaining a solution of the reconstructed second simultaneous equation.

For example, when the first system of equations is in a degenerate state including dependent equations, a situation arises in which the solution of the first system of equations becomes indefinite. In such a case, according to the present invention, an independent second simultaneous equation with a reduced number of equations is reconstructed. Therefore, even if the above-mentioned uncertain solution situation occurs, the solution can be uniquely determined, and the skeleton model including the node coordinates as a basic variable can be appropriately deformed.

Further, according to the present invention, the first simultaneous equation is:
It is an equation for finding a solution of an equation that defines that the arc length of the skeleton model has a given value.

For example, a first terminal having one end connected to a fixed node
And a second equation defining the length of a second arc having one end connected to the fixed node and the other end connected to the other end of the first arc. Consider a case. In this case, both the first and second equations define the coordinates of the first node at the connection point of the first and second arcs. However, the definition by the first expression and the definition by the second expression may have the same meaning, and in this case, the first expression and the second expression are dependent on each other.
When the first and second equations have a dependent relationship, these first,
The first simultaneous equation for finding the solution of the second equation is in a degenerated state. According to the present invention, even in such a case, by reconstructing an independent second simultaneous equation,
An appropriate solution can be obtained.

The image synthesizing apparatus according to the present invention further comprises means for outputting the shape of the skeleton model deformed by any one of the above-described shape deformation methods, and operation means for allowing the operator to instruct the skeleton model to be deformed. It is characterized by including.

This makes it possible to realize a shape data creation tool and the like using the shape deformation method of the present invention.

Further, the image synthesizing apparatus according to the present invention is characterized by including means for outputting an image of a moving object accompanying a skeleton model deformed by any one of the above-mentioned shape deforming methods.

Thus, it is possible to realize a game device or the like using the shape data obtained by the shape deformation method of the present invention.

[0057]

BEST MODE FOR CARRYING OUT THE INVENTION Hereinafter, embodiments of the present invention will be described.
This will be described in detail with reference to the drawings.

1. Overview of Embodiment Basic Principle In the present embodiment, as shown in FIG. 1A, coordinates (x, y, z) of nodes 2, 3, 4 and arcs 6, 7 are used as basic variables representing a skeleton model 18. , 8, the components (u, v, w) of the arc normal vectors 10, 11, 12 are used. For example, FIG. 22 using the angle between arcs as a basic variable
In the conventional examples (A), (B), and (C), it is necessary to set a parent-child relationship, and nodes and arcs cannot be treated equally.
In the present embodiment using node coordinates and arc normal vectors as basic variables, all nodes and arcs can be treated as equal. For example, it is also possible to substitute the arc normal vector with another variable.

Next, as shown in FIG. 1 (B), as a basic expression in which the above-mentioned basic variables are unknowns, an expression for defining that the arc length has a given value, the length of the arc normal vector Is provided, and an equation is provided that specifies that the arc normal vector is perpendicular to the arc.
For example, the basic expressions g = 0 and h = 0 can be replaced with other expressions.

Here, the basic equation shown in FIG. 1B is nonlinear. Therefore, in this embodiment, a solution is obtained by an iterative calculation method such as the Newton method. By using the Newton method or the like in this manner, it is possible to express the basic expression by a linear expression of the amount of change of the basic variable. The simultaneous equations to be solved by this are
(C).

However, in the equation of FIG. 1C, the number of variables is larger than the number of equations. Therefore, in this embodiment, an evaluation formula for uniquely specifying a solution satisfying the basic formula is prepared.
Then, a condition that the value of the evaluation expression is substantially minimum, approximately maximum, or substantially stationary is added, and the solution of the basic variable is obtained by applying the Lagrange multiplier method. By using the Lagrange multiplier method, the equation to be solved is as shown in FIG. 1D, and the number of variables matches the number of equations, so that the solution can be uniquely determined.

In the present embodiment, the skeleton model is deformed based on the coordinates of the cursor position of the mouse moved by the user. At this time, the basic formula and the evaluation formula change based on the given information such as the coordinates of the cursor position. Specifically, based on the coordinates of the cursor position and the like, the known number included in the basic expression and the evaluation expression, and the shape of the expression itself change, thereby changing the shape of the skeleton model. As the given information, not only the coordinates of the cursor position but also various kinds of data such as data prepared in advance can be used.

First-Order Approximation of Variation of Arc Normal Vector In this embodiment, as shown in FIG. 2A, the variation of the arc normal vector n (u, v, w) is given by 1 The solution of the basic variable is obtained by expressing the two variables as ρ (see Algorithm 2 described later). More specifically,
The variation of the arc normal vector n is approximated by a linear expression of a variable ρ. Although the arc normal vector n is represented by three variables u, v, and w, since n is perpendicular to the arc 50 and has a constant size as shown in FIG. The degree is one. Therefore, the amount of change in n can be represented by one variable ρ, whereby the number of basic variables to be solved can be reduced, and the processing can be speeded up.

Evaluation Formula In this embodiment, in order to obtain a preferable deformation of the skeleton model, the amount of change in the angle between the arcs, the distance between the node and a given point, and the direction of the arc having a fixed node at the end are obtained. The change amount, the change amount of the node coordinates, the speed change amount of the node coordinates, and the like are included in the evaluation formula. More specifically, FIG.
Various conditions are added to the evaluation formula as shown in FIG.
For example, by including the change amount of the angle between the arcs in the evaluation formula and minimizing the sum of squares of the change amount of the angle between the arcs, it is possible to prevent a situation such that only a certain set of arcs protrudes and bends. The bending between the arcs can be made generally well balanced. In addition, by including the distance between the node and a given point in the evaluation expression and setting a rubber band or the like described later, it is possible to start deformation with an appropriate shape in an initial state, and to achieve a good balance. The shape deformation of the skeleton model can be realized. Also, by including the change amount of the direction of the arc having the fixed node at the end and the change amount of the node coordinates in the evaluation formula, and setting the arc rotation damper around the fixed node, the node damper, etc., the solution becomes indefinite. Can be prevented. In addition, by including the change amount of the node coordinates and the speed change amount of the node coordinates in the evaluation formula and setting the node damper and the node inertia, it is possible to prevent a ramp of an arc due to unstable calculation convergence.

Updating of Basic Expression and Evaluation Expression for Each Iterative Loop Both real-time and non-real-time can be considered as the usage of this embodiment. In the case of non-real-time use, the shape of the skeleton model may be updated after the calculation is sufficiently converged. On the other hand, in the case of real-time use, sufficient followability to user input is required.

Therefore, in this embodiment, the solution of the basic variable is obtained by iterative calculation such as the Newton method shown in FIG. 3A, and at least one of the basic expression and the evaluation expression is synchronized with the end or start of the iterative loop. Are sequentially changed based on given information.

For example, in the calculation process, the process from obtaining the coordinate value of the mouse cursor, deforming the skeleton model based on the coordinate value, and displaying the result is referred to as one event cycle. Since the user moves the position of the picked node by dragging the mouse from time to time, it is necessary to reduce the amount of calculation in one event cycle as much as possible to deform the skeleton model following this . Therefore, in this embodiment,
During mouse dragging, a method of performing an iterative calculation such as the Newton method during one event cycle, for example, only one iterative loop is used. Specifically, as shown in FIG. 3B, when the position coordinates of the mouse cursor are obtained, the known number of the basic expression and the evaluation expression and the shape of the expression itself are changed based on the position coordinates.
For example, an iterative calculation is performed for one iterative loop.

When this method is adopted, the calculation does not converge during mouse dragging, and the basic formula is not satisfied. For example, there is a possibility that the arc length is longer than the original length.
However, the elongation is at a level where there is no practical problem. In addition, when the shape of the skeleton model is finely adjusted, the movement of the mouse is slowed, so that the skeleton model deforms in a state close to the convergent shape. When the movement of the mouse stops, the calculation converges and an accurate skeleton model shape can be obtained.

In this embodiment, when the amount of change of the variable in one iteration loop of the iterative calculation is large, the scaling correction is performed. This can prevent the skeleton model from having an unnatural shape even when the above-described method of changing the basic expression and the evaluation expression for each iteration loop is used. In the above, 1
Although the basic expression and the like are changed for each iteration loop, the present embodiment is not limited to this. For example, the basic expression and the like may be changed for every two iteration loops and every three iteration loops. That is, it is sufficient that the basic formulas and the like are changed one after another in synchronization with at least the end or start of the repetitive loop.

Fixed Node, Fixed Arc Normal Vector In this embodiment, the coordinates of the fixed node or the components of the fixed arc normal vector are treated as known numbers. For example, FIG. 4 shows a skeleton model 52 according to this embodiment.
2 shows an example of the shape deformation. Here, the nodes 54 and 56 are fixed nodes. On the other hand, the node 58 is a node picked by the user, and the shape of the skeleton model is deformed when the user moves the node 58 by dragging the mouse. The coordinates of the fixed nodes 54 and 56 are known numbers such as basic expressions, and the coordinates of the other nodes and components of the arc normal vector are unknown numbers to be obtained. Therefore, when the node 58 is moved, not only the portion of the skeleton model 52 at the node 58 but also the entire skeleton model 52 is smoothly deformed.

In FIG. 4, the nodes are fixed.
It is also possible to fix the direction of the arc, that is, fix the direction of the arc normal vector. In this case, the component of the fixed arc normal vector becomes a known number.

In the conventional example shown in FIGS. 22A, 22B, and 22C, the fixed node is limited to the root and the movable node is limited to the effector. A desired node or arc can be freely fixed or moved without concern for a relationship or the like. This is because in the present embodiment, node coordinates and arc normal vectors are adopted as basic variables, and all nodes and arcs can be treated equally.

Rubber Band In this embodiment, the shape of the skeleton model is deformed by introducing the concept of a rubber band. That is, when the coordinates of a given point that controls the shape deformation of the skeleton model are changed, the shape of the skeleton model is deformed so that the distance between the node and the given point is minimized. For example, in FIG.
Consider the case where 0 is moved. The rubber band has a root and a tip, and the root moves according to the movement of the mouse, and the tip is fixed to the picked node. If the user clicks the mouse and the mouse cursor has not yet moved from point 62, the root and tip of the rubber band are at node 60.
In the position. As shown in FIG. 5B, when the user drags the mouse and moves the mouse cursor from point 62 to point 64, the root of rubber band 66 moves from the position of node 60 to point 68. At this time, the tip of the rubber band 66 remains fixed at the position of the node 60. The rubber band 66 has a property of shrinking, that is, a property of trying to minimize the distance between the node 60 and a given point 68. As a result, as shown in FIG.
0 moves to the point 68, and the shape of the skeleton model is deformed.

When the movement of the node 58 in FIG. 4 is performed using a rubber band, the coordinates of the node 58 are unknown. On the other hand, when the user moves the node 58 directly without going through the rubber band, or when the coordinates of the node 58 are changed based on data prepared in advance, the coordinates of the node 58 become a known number.

In this embodiment, as shown in FIG.
More than one node, eg, nodes 58, 59, can be moved simultaneously. In this case, FIG.
As shown in (E), two rubber bands 66, 67 are generated, and the rubber bands 66, 67 are deformed so that the distance between the nodes 60, 61 and the given points 68, 69 is minimized.

FIG. 7 shows an example of the shape deformation of the skeleton model 70 having a more complicated multi-branch structure. Here, the nodes 72 and 74 and the arc 73 are fixed, and the node 76 is moved via a rubber band. As described above, according to the present embodiment, unlike the conventional example, the skeleton model can be smoothly deformed without depending on the complexity of the multi-branch structure.

Constraint Conditions -1 Kinds of Constraint Conditions In this embodiment, various constraint conditions can be set for the shape deformation of the skeleton model as described below.

First, in the present embodiment, it is possible to set a constraint condition for fixing the position of the node and the direction of the arc, and as described above, the coordinates of the fixed node and the arc normal vector of the fixed arc can be set. It can be realized by treating it as a known number.

Second, in the present embodiment, it is possible to set a constraint condition for providing a movable range in the angle between the arcs of the skeleton model. For example, in FIGS. 8A and 8B,
A discontinuous constraint condition is set such that the angle θ between the arcs 78 and 80 of the skeleton model 77 is 180 degrees or less. In other words, it is unnatural if the elbow joint is bent at the time of shape deformation of the skeleton model, and the user's convenience can be improved by setting such a constraint condition. Note that this constraint condition can be realized, for example, by including in the evaluation expression an expression that restricts the angle between the arcs within a given range.

Third, in this embodiment, it is possible to set conditions for restricting the nodes of the skeleton model to a given surface or line, for example, a polygon or its edge. Hereinafter, an operation of restraining a specific node on a specified polygon or edge is referred to as “attach”. For example, FIG.
In (A), a node 82 of a skeleton model 81 is attached to a polygon 84 by a user's specification.
In FIG. 9B, the node 82 is attached to the edge 86. Here, a feature of this embodiment is that the node 82 can be discontinuously bound to the polygon 84 or the edge 86. That is, in FIG. 9A, the node 82 is the polygon 8
4 can be moved only within the plane of the polygon 84.
Can not move beyond the edge of. FIG.
In (B), the node 82 can move only within the line of the edge 86, and cannot move beyond the vertex. Since the nodes can be discontinuously bound to polygons and edges in this way, user convenience can be improved.

FIG. 10 shows an example in which the skeleton model climbs a wall using the above-mentioned attach function. Nodes 90, 91, 92, 93 (corresponding to the tips of the limbs) of the skeleton model 88 are attached to the first wall object 96. On the other hand, the node 94 of the skeleton model 88,
95 (corresponding to the back) is attached to a second wall object 98 that is placed at a given distance from the first wall object 96. By providing the second wall object 98 in this manner, the skeleton model 88 can be deformed and moved while keeping the body at a moderate distance from the first wall object 96. FIG. 11 shows a diagram for clarifying the positional relationship between the skeleton model 88 and the first and second wall objects 96 and 98. In FIG.
The node 91 is attached to the first wall object 96 and its coordinates are fixed. The coordinates of the node 89 are also fixed. Then, when the user picks and moves the nodes 87 and 90, the skeleton model 88 climbs the wall while deforming the shape in accordance with the movement. At this time,
Since the nodes 92, 93, 94, and 95 are attached to the first and second wall objects 96 and 98, they move while being constrained within the plane of these wall objects.

Fourth, in this embodiment, it is possible to set a constraint condition for preventing the skeleton model from invading or passing through the object when the skeleton model and the object interfere with each other. For example, as shown in FIG. 12, when a skeleton model 100 representing a whip and a cylindrical object 102 interfere with each other, the skeleton model 10
0 is deformed so as to wind around the cylindrical object 102 without entering the cylindrical object 102. Such a constraint condition can be realized by including, in the evaluation expression, an expression for restricting the skeleton model to the object when the skeleton model and the object interfere with each other.

-2 Improvement of Discontinuous Constraint Conditions When handling such discontinuous constraint conditions, there are the following problems. That is, in the general method so far,
As shown in FIG. 13A, first, the calculation is performed in a state in which the constraint condition is invalid (free). For example, when it is determined that the node of the skeleton model and the object have collided, the condition in which the constraint condition is valid (the node is And completely repeat the calculation. As a result of the calculation, if another node that hits the object is detected, the calculation is performed again. In this method, the node is completely constrained to the object when the constraining condition is valid. Therefore, in order to determine whether the user has moved the skeleton model in the direction of the object or in the direction away from the object, it is necessary to first perform the calculation with the constraint conditions invalid. Conversely, if the calculation is first performed in a state where the constraint condition is valid, the node is completely constrained by the object, and it is not possible to determine in which direction the user is moving the object.

As described above, according to the above-described method, calculation in a state where the constraint condition is invalid and iterative calculation for determining the constraint condition are required in one iteration loop. That is, 1
One iterative loop contains another iterative loop,
Since the iteration loop is duplicated, the processing speed is greatly reduced.

Therefore, in this embodiment, discontinuous constraint conditions are handled by a method as shown in FIG. First, when the skeleton model (or a moving object accompanying the skeleton model) crosses a given boundary, the skeleton model is deformed under the constraint of returning the skeleton model to the boundary side. That is, in a state where the constraint condition is valid, the nodes and the like are not completely restricted to the boundary but are so-called spring (spring) restricted. By restraining to the boundary by spring restraint,
It is possible to determine whether the user is moving the skeleton model in the direction of the boundary of the object or the like or in the direction away from the boundary.

When the iterative loop for obtaining the shape of the skeleton model is performed in a state where the constraints are invalid and the nodes 106 and the arcs of the skeleton model do not exceed the boundary 104, the constraints are invalidated in the next iterative loop as well. The calculation is performed with the state as follows.

On the other hand, when the iterative loop is performed in a state where the constraint condition is invalid and the node 106 or the like is beyond the boundary 104, it can be determined that the user is moving the skeleton model in the direction of the boundary 104. In the next iteration loop, the calculation is performed with the constraint conditions being valid.
In this case, scale the calculation as needed,
A correction for returning the node 106 to the position 107 is performed.

When the iterative loop is performed in a state where the constraint condition is valid and the node 106 or the like exceeds the boundary 104, the calculation is performed in the next iterative loop with the constraint condition being valid. In a state where the constraint condition is valid, a strong spring constraint is added, so that the penetration of the node 104 is so small that the user does not notice.

On the other hand, when the iterative loop is performed in a state where the constraint condition is valid and the node 106 or the like does not exceed the boundary 104, it can be determined that the user moves the skeleton model in a direction away from the boundary 104. Therefore, in the next iterative loop, the calculation is performed with the constraint condition being invalid. In this case, re-calculation is performed with the constraint condition invalid. However, when the skeleton model is moving in a direction away from the boundary 104, an appropriate calculation result can be obtained in the next iteration loop, so that this recalculation can be omitted.

By treating discontinuous constraints in the manner described above, the amount of calculation in the iterative loop can be reduced as shown in FIG.
Compared with (A), the number can be significantly reduced, and the processing speed can be increased.

-3 Replacement of Inequalities that Define Discontinuity Constraints with Equations In this embodiment, details of algorithm 1 (setting of angle range between arcs) and FIGS. 18A to 18C will be described in detail. Thus, the inequality (50) defining the discontinuous constraint condition is replaced by the equation (48) having the same effect. For example, the domain of the first variable such as the angle between arcs can be defined by the inequality (50). However, if the inequality is used as a basic expression or an evaluation expression because it needs to be classified, the processing becomes complicated. Therefore, in the present embodiment, the equation and the number of the inequality (50) are shifted to the left side (or the right side), and a new second variable (t _X or the like) is added.
An equation that replaces the right-hand side (or the left-hand side) with a function (such as t _X ² ) that is always positive or negative or non-negative or non-positive.
(48) is introduced. Then, the domain of the first variable is defined by the equation (48), and the shape of the skeleton model is deformed. Since the equations are easier to incorporate into the basic equations and the evaluation equations than the inequalities, according to the present embodiment, the processing can be simplified.

In order to prevent the second variable from greatly fluctuating and the constraint condition from functioning, the algorithm 1
It is desirable to introduce an equation including the amount of change of the second variable as shown in equation (58) of -2. In order to prevent the second variable from converging to a given convergence value and reducing the response speed of the deformation of the skeleton model, it is desirable to add the given value to the second variable before convergence. Furthermore, this method of replacing inequalities with equations is not only a discontinuous constraint condition that sets the movable range of the angle between arcs, but also other techniques such as attaching nodes to polygons and edges, and interference between skeleton models and objects. It can also be applied to discontinuous constraints.

-4 Drag back Method In this embodiment, the discontinuous constraint condition is handled by employing a method called the Drag back method which will be described in detail in Algorithm 1 described later. In this method, in one iterative loop, the first calculation is performed in a state in which a constraint condition for restricting a node, an arc, or the like of a skeleton model to a boundary such as a polygon or an edge is invalid. Next, in the same one iterative loop, after the above-described first calculation, a second calculation for drawing nodes, arcs, and the like of the skeleton model to the boundary is performed. By doing so, it is possible to perform the calculation required for one iteration loop twice, and the conventional method in which one iteration loop includes another iteration loop (FIG. 13 (A) The processing can be remarkably speeded up as compared with)).

In the above first calculation, it is desirable to use a surface-perpendicular damper or a line-perpendicular damper as shown in equations (62) and (65) of -1 and -2 of Algorithm 1. That is, the expression including the amount of movement of the nodes and arcs of the skeleton model in the boundary direction is minimized. In this way, it is possible to prevent a situation in which nodes, arcs, and the like are too far from the boundary and cannot be pulled back. In the second calculation, it is desirable to use a face magnet, a line magnet, and a point magnet as shown in -3, -4, and -5 of Algorithm 1. That is, the expression including the distance between the node, arc, etc. and the boundary is minimized. By doing so, it is possible to easily return the nodes, arcs, and the like to the boundaries. Furthermore, the Drag back method is applicable not only to discontinuous constraint conditions when attaching nodes to polygons and edges, but also to other discontinuous constraint conditions such as setting the movable range of the angle between arcs and interference between the skeleton model and the object. Applicable.

Singular Value Decomposition (Reconstruction of Simultaneous Equations) In this embodiment, as will be described in detail later with reference to Algorithm 1 (singular states and corresponding methods) and FIGS. 17B and 17C, simultaneous equations Is a degenerate state or a state close to the degenerate state, a method of reconstructing an independent new simultaneous equation to obtain a solution is adopted. If the simultaneous equations are in a degenerate state including dependent equations, the solution becomes indefinite, and proper shape deformation cannot be performed. In particular, there is a possibility that such a solution indefinite may occur in an equation for finding a solution of an equation that defines the arc length, such as F _ij in Equation (4) of Algorithm 1, but according to the present embodiment, Such an indeterminate situation can be effectively prevented.

Next, two examples of the detailed algorithm of this embodiment will be described. The meaning of the sum code used in the following description is as follows. In the following, the angle between arcs is called an angle, and the two arcs that make up the angle and the three nodes that are the endpoints are called angle components.

(Equation 1) The relationship between the subscript order and the sign is as follows.

(Equation 2) 2. In algorithm 1 basic diagram 14, the coordinates of the node _{N i (x i, y i} , z i)
And The arc is provided with an arc normal vector to represent rotation in a three-dimensional space. Arc A _ij normal vectors (the node N _i, arc connecting the node N _j) n
_ij (u _ij , v _ij , w _ij ). In the present embodiment, the shape of the skeleton model is represented using these basic variables. The following basic equation holds for each arc.

(Equation 3) The first expression means that the length of the arc A _ij is L _ij . The second expression means that the magnitude of the normal vector is 1. The third equation means that the arc normal vector and the arc are perpendicular. Further, a scale factor λ is used so that the equation is invariant with respect to the similar deformation of the skeleton model. That is, when the shape of the first skeleton model is deformed, and when the shape of the second skeleton model having twice the size of the first skeleton model is deformed by the above-described twice, the obtained results are similar. I do. As the value of λ, an average value of the arc length of each skeleton model is used.

Solving the Basic Equations In order to obtain the skeleton movement by inverse kinematics, when the user picks and moves a node, the coordinates of the other nodes and the value of the normal vector of the arc change with the picking and moving of the node. Must be asked. These values are obtained as a solution of equation (1). However, equation (1) is a non-linear equation, and the number of equations is less than the number of variables. Therefore, the Newton method and the Lagrange multiplier method are combined to solve.

When it is not necessary to indicate otherwise, the variable is omitted from the suffix.

-1 Application of Newton's Method The amount of change of each variable for each loop is defined as follows.

(Equation 4) By applying the Newton method to the above basic equation (1), the following equation is obtained.

(Equation 5) This can be rearranged into the following equation.

(Equation 6) In equation (4), the number of equations is smaller than the number of variables, so that this alone cannot uniquely determine a solution. Therefore, as described below, a solution is obtained by adding a minimization condition and applying the Lagrange multiplier method.

In this embodiment, as already described with reference to FIG. 3B, the basic expression and the like are changed for each repetition loop based on the mouse cursor position and the like.

-2 Application of Lagrange's Multiplier Method Evaluation expressions U ₁ , U ₂ , U ₃ ,... Are prepared, and a condition for minimizing the sum is added to expression (4). Ξ, η, ζ, p, q, and r are obtained by using the method. Each U _k is defined later. The undetermined multipliers α _ij , β _ij , and γ _ij are introduced, and U is defined as in the following equation.

(Equation 7) Equation (4) and the following equation (6) are combined to obtain ξ, η, ζ, p, q, r
To get the solution.

(Equation 8) Based on the obtained solution, new x, y,
z, u, v, w are obtained, and the calculation for one iteration loop of the Newton method is completed.

(Equation 9) Creating an evaluation formula As mentioned earlier, to apply Lagrange's multiplier method,
It is necessary to prepare an evaluation formula to be minimized. Here, an evaluation formula is created such that the skeleton model is preferably deformed.

-1 Sum of squares of change in angle between arcs Let U ₁ be the sum of squares of change in angle between arcs (joint angle). By minimizing U _1, so that the joints bend good balance. Also, by assigning different weights to each component of the angle between arcs, it is possible to make a difference in the ease of bending of the joint.

-1-1 Inter-Angle Angle and Rotation Matrix The rotation matrix M _ij from the world coordinate system to the arc A _ij coordinate system is represented by the following equation. However, as shown in FIG. 15, the arc A _ij coordinate system is a right-handed system in which the direction of the arc normal vector is the x axis, the direction from node _Ni to node N _j is the y axis. The arrangement of the matrix elements follows a method of multiplying a column vector by a matrix from the left.

(Equation 10) A rotation matrix from the arc A _ij coordinate system to the arc A _jk coordinate system is _{defined as Mijk} .

[Equation 11] Note that the subscript ijk is omitted when it is not necessary to specify otherwise.

M _ijk is expressed by the following equation.

(Equation 12) On the other hand, if the rotation angles from the arc A _ij coordinate system to the arc A _jk coordinate system are θ _x , θ _y , and θ _z , and the rotation order is the order of the y-axis, x-axis, and z-axis, the matrix _Mijk becomes Can be expressed as

(Equation 13) Therefore, the following equation holds from equations (11), (12), and (14).

[Equation 14] -1-2 The linear equations θ _x , θ _y , and θ _z relating to the angle change are expressed as φ _x , φ _y ,
φ _z is defined.

(Equation 15) Here, φ _x , φ _y , φ _z and ξ _i , η _i , ζ _i , p _ij , q _ij ,
An attempt is made to represent the relationship with r _ij by a first-order approximation. However, as described below, to define the amount of change in a _mn and b _mn.

(Equation 16) Obtain phi _x by modifying the equation (15) differentiation.

[Equation 17] Similarly, φ _y and φ _z are obtained as follows.

(Equation 18) b ₀₂ , b ₁₀ , b ₁₁ , b ₁₂ , and b ₂₂ are obtained as follows by differentiating Expression (15).

[Equation 19]

(Equation 20)

(Equation 21)

(Equation 22)

(Equation 23) As described above, the angle change amounts φ _x , φ _y , φ _z are represented by ξ,
It could be expressed by a linear expression of η, 表 す, p, q, and r.

-1-3 Partial Differentiation of Sum of Squares of Angle Change Amount U ₁ is expressed by the following equation as the sum of squares of angle change between arcs. Here, W _x , W _y , and W _z are weighting factors for adjusting the ease of bending of the joint, and are x-axis, y-axis,
Corresponds to z-axis rotation.

(Equation 24) Since it is necessary when calculating equation (6), φ _x ² and φ _y
^2. Find the partial derivative of φ _z ² . These are ξ, η, ζ,
It becomes a linear expression of p, q, and r. Where ω is the variables ξ, η,
ζ, p, q, r.

(Equation 25) Moving -2 directly the need picked node N _i rubberband settings -2-1 rubber band, because the skeleton model to deform the shape shown in FIG. 16 (A) as an initial state, FIG. 16 (B) The result is an undesirable shape as shown. Therefore, the nodes N _i, to be moved through the rubber band explained in FIG. 5 (A) ~ (E) . The rubber band is a virtual arc having a property of contracting, one end of which is connected to a node N _i (x _i , y _i , z _i ) and the other end (x _ci , y _ci , z _ci). ) Is moved according to the movement of the mouse cursor. Thus, the skeleton model is deformed with the shape of FIG. 16C as an initial state, so that a preferable shape can be obtained as shown in FIG. 16D. In FIGS. 16A and 16B, iterative calculation is started assuming that the picked node is at the point 110 in the initial state, whereas in FIGS. 16C and 16D, the point 112 is in the initial state. , And the iterative calculation is started.

-2-2 Setting of Rubber Band By adding the following U ₂ to the evaluation formula U, the contractility of the rubber band can be expressed. W _r is a weighting factor for representing the strength of contractility, and N _pick is the number of picked nodes.

(Equation 26) By adding U ₂ to the evaluation formula, the picked node and
2 of the distance from the other end of the rubber band ( _xci , _yci , _zci )
The skeleton model can be transformed so that the sum of squares is minimized. Further, as already described with reference to FIGS. 5 (D), (E) and FIG. 6, or as is clear from the equation (28), in this embodiment, a plurality of nodes can be moved via a rubber band.

The partial differentiation of U _2i is as follows.

[Equation 27] -2-3 Setting of rubber band limit If the mouse is dragged greatly and the rubber band is pulled too much, the calculation of Newton's method does not converge and the skeleton model may run wild. Therefore, if the length l _rub of the rubber band has exceeded a given value l ^* _rub, in W _r, l ^*
_{Apply rub} / l _rub to correct the rubber band tension. In addition, when a plurality of nodes are picked, the tension of the rubber band as a whole increases. Therefore, in order to correct this, Expression (28) is divided by N _pick .

-3 Rotation of Arc around Fixed Node In the case of FIG. 17A, there are countless solutions around the axis of the arc. Accordingly, the arc A _ij for a fixed node N _i and the end point, adds the rotation resistance. The following U ₃ is evaluated by U
, This resistance can be expressed. Swing rotation (swi
The weight coefficient W _s for ng), the weighting factor for twisting rotation (twist) and W _t.

[Equation 28] By adding U ₃ to the evaluation formula, the skeleton model can be deformed so that the sum of squares of the change in the direction of the arc having the fixed node at the end is minimized.

The partial differentiation of U _3ij is as follows.

(Equation 29) -4 Setting node inertia Rampage may occur depending on the shape of the skeleton model. Rampage can be reduced by adding inertia (inertia) and a damper described later to the node. U below
_By adding ₄ to the evaluation expression U, the inertia of the node can be expressed.

[Equation 30] U _4i corresponds to the inertia of the node N _i. (Ξ ",
eta ", zeta") is a movement vector of the node N _i in the previous event cycle. W _I is a weighting coefficient representing the strength of inertia. By adding U ₄ to the evaluation formula can be modified to skeleton model as the sum of squares of speed variation of the node coordinates are minimized.

The partial differentiation of U _4i is as follows.

(Equation 31) By adding configure U ₅ below -5 node damper evaluation formula U, it can be expressed nodes damper.

(Equation 32) U _5i corresponds to the damper of the node N _i. W _d is a weight coefficient representing the strength of the damper. By adding U ₅ in the evaluation formula can be modified to skeleton model as the square sum of the node coordinates of the variation is minimized.

The partial differentiation of U _5i is as follows.

[Equation 33] Singular state and its correspondence method -1 Occurrence of singular state Depending on the shape of the skeleton model, subordinate formula may be included in formula (4). This is shown in FIG.
This is the case as shown in (C), where equation F _ij is dependent.
The expressions G _ij and H _ij do not depend.

For example, in the case of FIG. 17B, Expression (4) includes Expressions F _ij (hereinafter, referred to as Expressions F ₁ and F ₂ , respectively) regarding the arcs 120 and 122. Here, the equation F ₁ is the arc 12
The first-order approximation specifies that the length of 0 is a constant value, and specifies that the node 124 is in a plane 126 perpendicular to the arc. On the other hand, Formula F ₂ is for the length of the arc 122 defines a first approximation to the constant value, in the same manner as F _1, node 124 provides that in the plane 126. That is, F ₁ and F ₂ have the same meaning,
F ₁ and F ₂ are mutually dependent equations. If Equation (4) includes a dependent equation, the solution cannot be uniquely determined even if the number of equations is equal to the number of variables using the Lagrange multiplier method. Similarly, in the case of FIG. 17C, the position of the node 136 is uniquely determined by the expression F _ij relating to the arcs 128, 130, and 132, for example.
The expression F _ij regarding 4 becomes a dependent expression.

As described above, when the equation dependent on the equation (4) is included, the solution is not uniquely determined even if the Lagrange multiplier method is applied as it is. Therefore, the simultaneous equations F _ij are reconstructed into independent simultaneous equations by using a singular value decomposition technique described below.

Singular Value Decomposition Singular value decomposition is a method of decomposing an m × n matrix A as shown in the following equation.

(Equation 34) Where U is an m × n matrix, V is an n × n matrix, and W is an n × n matrix.
n is a diagonal matrix, and U and V satisfy the following equation.

(Equation 35) By using singular value decomposition, as described below, dependent simultaneous equations can be reconstructed into independent simultaneous equations with a reduced number of equations. For example, assume that the original equation was represented by the following equation.

[Equation 36] When m ≧ n, the following equation can be obtained by decomposing and transforming the matrix A into the form of equation (41).

(37) On the other hand, in the case of m <n, singular value decomposition as following equation matrix A ^t (45). Here, U is an n × m matrix, V is an m × m matrix, W is an m × m diagonal matrix, and U and V satisfy Expression (42).

(38) As a result, equation (43) is transformed into the following equation.

[Equation 39] In Equations (44) and (46), by removing rows where the diagonal element of W is 0, independent simultaneous equations can be obtained.

Whether the simultaneous equations include subordinate equations,
That is, whether or not the simultaneous equations are in a degenerate state or a state close to it can be determined based on whether or not the diagonal components of the diagonal matrix W include 0 or a value close to 0.

Setting of movable range of arc angle -1 addition of basic formula As described with reference to FIGS. 8A and 8B, the movable range of the angle between arcs is limited. Arc angles θ _x , θ _y , θ
Set the lower and upper limits of _z as follows.

(Equation 40) New variable t _x, t _y, introducing t _z, equation e _x below,
By adding e _y and _ez to the basic equation (1), θ _x , θ _y , θ _z
Can be limited.

[Equation 41] Here, each coefficient of the equation (48) is as follows.

(Equation 42) The meaning of equation (48) is as follows. FIG.
As shown in FIG. 8A, the following expression (50) limits the range of existence of cos θ _x and sin θ _x to the shaded area. Therefore, θ _x is
It is limited to the range of θ _x0 ≦ θ _x ≦ θ _x1 .

[Equation 43] Since t _x ² ≧ 0, the first expression of Expression (48) has the same effect as Expression (50). Similarly, the second and third expressions of Expression (48) are
This has the effect of limiting the range of θ _y and θ _z .

[0117] variable t _x, t _y, the variation of t _z is defined as follows.

[Equation 44] When the Newton method is applied to the basic equation, the following equation is added to the equation (4).

[Equation 45] The following equation is added to the portion corresponding to the basic equation in equation (5).
Ψ _x , Ψ _y , Ψ _z are newly introduced undetermined multipliers.

[Equation 46] Partial differentiation of U _AL is as follows. Here, ω _i represents variables ξ _i , η _i , ζ _i , Ω _i represents variables x _i , y _i , z _i , and ω _i
≡ △ Ω _i . Ω _ij is a variable p _ij , q _ij , r
_ij , Ω _ij represents variables u _ij , v _ij , w _ij , and ω _ij ≡ △ Ω
_ij .

[Equation 47] e _x, e _y, partial differential of e _z is expressed by the following equation. Where Ω
Represents the variable x, y, z, u, v, w, t x, t y, t z.

[Equation 48] -2 Divergence of variable values With only the above settings, the movable range limitation of the angle between arcs does not function at all. This is because the values of the variables t _x , t _y , and t _z vary greatly and do not converge. _{Therefore, t x, t y, t} z variables tau _x is the amount of change, tau _y, to converge the calculated by the action of damper tau _z. That is, the damper is activated by adding the following U ₆ to the evaluation formula U.

[Equation 49] The partial derivative of U ₆ is as follows:

[Equation 50] -3 Sticking to the upper and lower limits With the above settings, if the movable range of the angle between arcs is activated, the following sticking phenomenon occurs. If the skeleton model is deformed in the direction in which the angle between arcs of a certain joint increases, and is kept pressed against the upper limit of the movable range for a while, and then moved in the direction in which the angle between arcs decreases, the angle between arcs changes immediately. Instead, it behaves like it sticks to the upper limit angle for a while. The sticking time is almost proportional to the time you were pressing on the upper limit. The same applies to the lower limit of the angle limit. This is for the following reason. Variable t _x, t _y, t _z is the convergence becomes slow as the value approaches zero. Therefore, once it converges to a value sufficiently close to 0, many calculation loops are required to return to the original value again.

Therefore, in order to avoid this problem, if the variables t _x , t _y , t _z become smaller than a certain value,
t _x, t _y, the small amount of modification of the value of t _z is added, so as not to converge to 0. Specifically, random numbers are generated and t _x , t _y ,
Add to t _z.

-4 Setting of Multiple Conditions When the movable range (upper limit value-lower limit value) of the arc angle is smaller or larger than a certain value, the calculation does not converge and the skeleton model becomes unsteady. The range in which the calculation converges stably is limited to a given range. Therefore, when the movable range is narrower or wider than the given range, the following method is used.

-4-1 Correspondence when the movable range is narrow As shown in FIG. 18B, two movable range limits A and B
Use at the same time. The movable range of each of A and B is a ₁ degree.

-4-2 Correspondence when the movable range is wide As shown in FIG. 18C, two movable range limits A and B are prepared, and A, B and Free (no angle limit) depending on the state.
Switch to use. A or B when the arc-to-arc angle is within a ₂ degrees from the upper limit and the lower limit, and Free otherwise. A, B respectively of the movable range is a ₃ degrees. Method of Attaching Node to Polygon or Edge As shown in FIGS. 9A to 9D, FIGS. 10 and 11, a specific node is attached on a specified polygon or edge. By adding the following equation to the basic formula (1), can attach the node N _i to the plane ax + by + cz + d = 0.

(Equation 51) In addition, since a straight line can be expressed as an intersection of two planes, nodes can be attached to a straight line by preparing and combining two plane expressions.

With the above method, a node can be attached to an arbitrary infinite plane or an infinite straight line, but the movement range of the node cannot be limited to a polygon having a finite size or an edge having a finite length. When the number of basic equations increases, the size of the simultaneous equations increases when the Lagrange multiplier method is applied, which is not preferable in terms of calculation speed.
Therefore, instead of the method described here, the drag back method described below is used as a method for realizing the attachment.

Drag back method To attach a node on a polygon or an edge,
The drag back method described here is used. In this method, 1
During the event cycle, the original node movement operation (drag
forward) and an operation of dragging back the attached node (drag back).

[0124] Calculation in the case of attaching the node N _i to the polygon F is as follows. First, the characteristics of the node damper node N _i, set so that only the resistance to the movement in the direction perpendicular to the polygon F increases. This damper is called a face-perpendicular damper. The reason why such a damper works is that if the node is too far away from the polygon, the subsequent behavior will be poor due to the influence of the error. With this setting, calculation of normal node movement, that is, calculation in which the constraint condition is invalidated, is performed, and the distance L _dist between the node _Ni and the polygon F is obtained. If L _dist is equal to or less than a given value L ^* , the node coordinates and the arc normal vector are changed as calculated. When L _dist > L ^* , a value obtained by multiplying the change amount of all node coordinates and the arc normal vector by the scale L ^* / L _dist is used.

Next, the constraint condition is validated, and the node N _i
(Drag back) is returned to the polygon F.
At this time, three methods are properly used depending on the positional relationship between the node and the polygon F.

[0126] and P the point closest to the located nodes N _i on the polygon F. Depending on the positional relationship between the node _Ni and the polygon F, the point P is one of the following.

[0127] (A) one of the vertices of the node N _i perpendicular foot drawn down to the polygon F from (B) the node N _i from the polygon F edge of one the down was perpendicular foot (C) polygon F (C ), Pull node N _i back toward its vertex. This retraction force is called a point magnet.

[0128] In the case of (B) is pulled back in the direction toward the node N _i to the edge. This pullback force is line magnet
Call.

[0129] In the case of (A) is pulled back in the direction toward the node N _i to the polygon plane. Face magne
Call it t.

For example, in the state of FIG.
Works, and in the state of FIG. 9B, the linemagnet works. Point magnet if the point closest to node 82 is the vertex
Works. By doing so, the movement range of the node can be limited to a polygon having a finite area, and user convenience can be improved. The attachment of the node to the infinite plane can be easily dealt with by making the polygon sufficiently large.

As shown in FIG. 9B, when attaching the node Ni to the edge of the polygon, a line vertical damper is used instead of the surface vertical damper. This is a damper that works only for movement in the direction perpendicular to the edge. At the time of dragback, either point magnet or line magnet is used depending on the positional relationship between the node and the edge. Thereby, the movement range of the node can be limited to an edge having a finite length.

-1 Face Vertical Damper It is assumed that a polygon plane is represented by the following equation.

(Equation 52) Plane perpendicular damper for node N _i can be realized by adding the following equation U _fd evaluation formula U. U _fd is proportional to the square of polygon planes vertical movement of the node N _i. Therefore, by the evaluation formula U Add U _fd, the mobile node N _i in the direction perpendicular to the polygon plane will experience resistance. On the other hand, the movement of the node N _i to a direction parallel to the polygon plane is not subject to any resistance. W _fd is a weight coefficient representing the strength of the damper.

(Equation 53) The partial derivative of U _fd is as follows.

(Equation 54) -2 line vertical damper (line damper) It is assumed that an edge straight line passes through a point (x _p , y _p , z _p ) and its direction is represented by a vector (a, b, c). However,

[Equation 55] And Line vertical damper for node N _i can be realized by adding the following equation U _ld evaluation formula U. W _ld is a weight coefficient representing the strength of the damper.

[Equation 56] D of the above equation _i0, d _i1 respectively, before the movement, means the distance between the node N _i and the edge straight after the movement. Therefore, the evaluation expression U
A By adding U _ld, so that the movement of the node N _i in the direction perpendicular to the edge straight line resisted. On the other hand, the movement of the node N _i to a direction parallel to the edge line is not subject to any resistance. The partial differential of U _ld is as follows.

[Equation 57] -3 Face magnet Suppose the polygon plane is expressed by the following equation.

[Equation 58] The face magnet for the node N _i is calculated by the following expression U _fm
Can be realized by adding W _fm is a weight coefficient representing the strength of the magnet.

[Equation 59] D _{i in the} above equation means the distance between the moved node _Ni and the polygon plane. Therefore, by adding U _fm to the evaluation expression U,
The square of the distance between the node N _i and the polygon plane after the movement are minimized, the node N _i after movement are attracted to the polygon plane. The partial derivative of U _fm is as follows.

[Equation 60] -4 Line magnet Suppose that an edge straight line passes through a point (x _p , y _p , z _p ) and its direction is represented by a vector (a, b, c). However,

[Equation 61] And The line magnet is expressed by adding the following expression U _lm to the evaluation expression U. W _lm is a weight coefficient representing the strength of the magnet.

(Equation 62) D _{i in the} above equation means the distance between the moved node N _i and the edge straight line. Therefore, by the evaluation formula U Add U _lm, the distance between the node N _i and the edge straight after the movement are minimized, the node N _i after movement are attracted to the edge line. U _lm
Is as follows.

[Equation 63] -5 Point magnet Point toward the point (x _p , y _p , z _p ) for the node N _i
Make magnet work. A point magnet is similar to a rubber band, but does not extend as long as a rubber band, so there is no limit.

[0133] point magnet can be realized by adding the following equation U _pm on the evaluation formula U. W _pm is a weight coefficient representing the strength of the magnet.

[Equation 64] The partial derivative of U _pm is as follows.

[Equation 65] 3. Algorithm 2 Unlike algorithm 1 described above, algorithm 2 uses a single variable ρ to first-order approximate the amount of change in variables u, v, and w, as shown in FIG. The discontinuous constraint condition is
This is handled by the method described with reference to FIG.

Basic Formula The basic formula is the same as in Algorithm 1, and is as follows.

[Equation 66] Solution of basic equation -1 Application of Newton's method By applying Newton's method to basic equation (75), the following equation is obtained.

[Equation 67] This can be rearranged into the following equation.

[Equation 68] For equations (76) and (77), the following first-order approximation using one variable ρ is used without using the Newton method in order to reduce the unknowns of the equations to be solved.

-2 First-Order Approximation of Variation of Arc Normal Vector Since the arc normal vector ni _j is perpendicular to the arc A _ij and has a magnitude of 1, the components u _ij , v _ij and w _ij are represented by three variables. However, the degree of freedom of the change is 1 as shown in FIG. So, to reduce the unknowns of the equations to be solved
Performs first-order approximation using one variable.

As shown in FIG. 14, the unit vector in the axial direction of the arc A _ij is a _ij, and a vector s _ij ≡n _ij × a _{ij is} created.

[Equation 69] The following equation is obtained by differentiating equations (76) and (77).

[0137]

[Equation 70] Δu _ij , Δv _ij , and Δw _ij are restricted by the equations (82) and (83). Considering a three-dimensional space in which _{Δu ij} , △ v _ij and △ w _ij are orthogonal coordinate axes, △ u _ij , △ v _ij and △ w _ij are represented by a plane expressed by the equation (82) and an equation (83) ) Can be taken only on the intersection line L of the plane represented by Therefore, △ u _ij , △
v _ij and △ w _ij can be represented by the following equations.

[Equation 71] (K ₁ , k ₂ , k ₃ ) is a vector representing the direction of the intersection line L, and is a normal vector (u _ij , v
_ij , w _ij ) and perpendicular to the normal vector (x _j −x _i , y _j −y _i , z _j −z _i ) of the plane represented by equation (83).
That is, the direction of the vector (k ₁ , k ₂ , k ₃ ) matches the direction of the vector s _{ij in} equation (81). Thus, k ₁ in equation (84),
By solving k ₂ , k ₃ , l ₁ , l ₂ , and l ₃ geometrically, the following equation (85) is obtained. That is, n _ij (u _ij , v _ij , w _ij )
_{変 化 ij} (△ u _ij , △ v _ij , △ w _ij ) can be approximated as a linear expression of a newly introduced unknown variable ρ _ij .

[0138]

[Equation 72] n _ij is, the value to take in the next event cycle with n _'ij. Although n ′ _ij should originally have a size of 1 perpendicular to the arc A _ij , an error occurs in the order of the square of the change amount. Therefore, first, the direction is corrected by the first expression of the following expression (86),
The value obtained by correcting the size to 1 in the second equation is defined as n ′ _ij . Here, a ′ _ij is a value that a _ij takes in the next event cycle.

[Equation 73] -3 Application of Lagrange's multiplier method In equation (79), since the number of equations is smaller than the number of variables and does not include the variable ρ, the deformation shape of the skeleton model cannot be uniquely determined by this alone. Therefore, a condition of minimizing the evaluation expression including ρ is added, and a solution is obtained by applying the Lagrange multiplier method. U ₁ , U ₂ ,
U ₃ ,... Are prepared, and the sum of these is minimized. An undetermined multiplier α _ij is introduced, and U is defined as in the following equation.

[Equation 74] Equation (79) and the following equation (88) are made simultaneous to obtain a solution ξ, η, ζ, ρ.

[Equation 75] Based on the obtained solution, new values x ′, y ′, z ′ of x, y, z are obtained by the following equation (89), and n is obtained by the above equation (86).
By calculating a new value n ′ (u ′, v ′, w ′) of (u, v, w), the calculation for one event cycle is completed.

[Equation 76] -4 Setting of limit for arc deflection angle When the user moves the mouse fast, the shape deformation of the skeleton model in one event cycle increases.
Accordingly, the calculation error increases. In this embodiment, FIG.
This is because, as shown in (B), a method of updating the basic expression and the evaluation expression for each iteration loop is employed. When the arc is translated by the deformation, there is no error in extending the length of the arc. An error occurs when the direction of the arc changes. That is, the magnitude of the error depends on the deflection angle of the arc, and appears as an error in the arc length. Therefore, an allowable limit value is set for the arc deflection angle, and scaling is performed on the calculation result.

The swing angle Φ _ij of the arc A _ij is represented by the following equation.

[Equation 77] The maximum value of Φ _ij max (Φ _ij) is, if it exceeds the allowable limit value [Phi ^* determines a scaling factor mu ₁ by the following equation,
All values of μ, η, ζ, ρ are multiplied by μ ₁ and used.

[Equation 78] Thereby, the maximum deflection angle of the arc can be suppressed to Φ ^* or less. In practice, equation (90) can be substituted with equation (93). The limit setting for the arc deflection angle is
Algorithm 1 is also used.

[Expression 79] Similar to the sum of squares algorithm first evaluation formula created -1 arc between angle variation, when the square sum of the arc between the angular variation to U _1, good balance each joint by minimizing U ₁ Can be bent.

-1-1 Angle between arcs and rotation matrix From equations (8) to (14) shown in Algorithm 1, the following equation (94) holds.

[Equation 80] -1-2 Linear Expression Regarding the Angle Change Amount The relationship of Expressions (18), (19), and (20) holds for the angle change amounts φ _x , φ _y , and φ _z as in Algorithm 1. The following equations (95) to (99) are obtained by differentiating equation (94).

[0141]

[Equation 81]

(Equation 82)

[Equation 83]

[Equation 84]

[Equation 85] As is clear from the above, the angle change amounts φ _x , φ _y , φ
_{Let z} be the eleven variables ξ _i , _j , _k , η _i , η _j , η _k ,
It can be represented by a linear combination of ζ _i , ζ _j , ζ _k , ρ _ij , and ρ _jk .

-1-3 Partial Differentiation of Sum of Squares of Angle Change A partial sum of squares of angle change U ₁ and partial derivatives of φ _x ² , φ _y ² , φ _z ² can be obtained by Equation (26) of Algorithm 1, Same as (27). However, in this case, ω in equation (27) represents ρ in addition to ξ, η, and ζ.

-2 Setting of rubber band Setting of the rubber band is the same as that of the algorithm 1, and therefore the description is omitted.

-3 Rotation of arc around fixed node In order to add resistance to the rotation of the arc around the fixed node,
Add U ₃ below the formulas.

[Equation 86] Unlike Equation (32) of Algorithm 1, in Equation (101), the portion of twist rotation (twist) is represented by a variable ρ _ij . Also the formula
Based on the same idea as approximating (90) to equation (93), run-out rotation (s
wing). The partial derivative of U _3ij is as follows.

[Equation 87] -4 Setting of node inertia and node damper The setting of the node inertia and node damper is the same as that of the algorithm 1, and the description is omitted.

Singular States and Their Corresponding Methods The singular states and their corresponding methods are the same as in Algorithm 1 and will not be described. Setting the movable range of the angle between arcs -1 Limiting the movable range by the angle spring Set the movable range of the angle between the arcs, and move only within this range. Generally, when solving a deformation problem under such a discontinuous constraint condition, iterative calculation is required to determine the constraint condition as shown in FIG. The following method is used to minimize the amount of calculation. In the following description, only the upper limit of the movable range will be described, but the same applies to the lower limit.

When the value θ of the angle between arcs exceeds the upper limit of the movable range θ ^* , a strong spring is applied so as to make θ close to θ ^* . Since it is not a complete constraint but a spring constraint, θ = θ ^* is not strictly set.However, setting the spring strength to a sufficiently large value has the same effect as fixing practically θ to θ ^*. Can be

According to the value of θ, the condition is set such that the spring is valid when θ ≧ θ ^* and the spring is invalid when θ <θ ^* , and the calculation is performed. The combination of the state of the spring and the calculation result is 4
FIG. 13B in each case.
The processing is performed as follows. (A) Calculate with the spring disabled and the result is within the movable range: Use the result as is. (B) Calculate with the spring disabled and the result is outside the movable range: Scale and use the result. (C) Calculate with the spring enabled and the result Within the movable range: re-calculate with the spring disabled. (D) Calculate with the spring valid and the result is outside the movable range: use the result as it is. In case (B) above, scaling is performed by the following method. An overshoot amount よ う な as shown in FIG. 19 and the following equation is defined for the change amount φ of θ obtained as a result of the calculation.

[Equation 88] The allowable limit value of the overshoot [psi ^*, may be set in advance as shown in FIG. 19, it obtains a scaling factor mu ₂ by the following equation. By allowing the overshoot to some extent in this way, it is possible to prevent a reduction in the deformation speed of the skeleton model due to excessive scaling.

[Equation 89] All ξ, η, ζ, use over the mu ₂ to the value of [rho. As a result, the amount of overshoot can be suppressed to Ψ ^* or less. When the overshoot occurs at a plurality of positions of the skeleton model is used as the smallest mu ₂ values ones of the mu ₂ values of each location. Also, to suppress the skeleton model from rampaging after overshoot occurs,
It is effective to operate the angle damper.

By the above method, in cases other than the above (C), recalculation becomes unnecessary. Also in the case of the above (C), good behavior may be obtained even if the result is used as it is without recalculation.

[0149] By adding the following equation U ₆ to evaluate expression evaluation formula U -2 angle limiting, the angle limiting is enabled. Where θ ^* _x , θ ^* _y , θ ^* _z are θ _x , θ _y ,
Indicates the upper limit or lower limit of θ _z .

[Equation 90] W _as and W _ad are weight coefficients representing the strength of the spring and the damper, respectively. The partial derivatives of U _ax , U _ay , and U _az are as follows. Here, ω represents variables ξ, η, ζ, and ρ.

[Equation 91] Attaching Nodes to Polygons and Edges As shown in FIGS. 9A and 9B to 11, in order to attach a node to a polygon or an edge, an evaluation formula for drawing the node to an attachment target is prepared. Add this to U. Although the node is not strictly restricted on the object to be attached, a practically sufficient attachment effect can be obtained by setting the attraction strength to a sufficiently large value.

When a node is attached to a polygon, the following three methods are selectively used depending on the positional relationship between the two.
That is, the point magnet, line mag described in Algorithm 1
Use net and face magnet properly.

When the attracting condition changes with the movement of the node, an overshoot may occur as in the case where the movable range of the angle between arcs is limited. In such a case,
The calculation result is scaled so that the node _Ni is not too far from the polygon.

Note that point magnet, line magnet, face
Since the magnet has already been described in detail in the algorithm 1, the description is omitted here.

Interference Function with Polygon Object-1 Interference Function As shown in FIG. 12, when the skeleton model collides with the polygon object, the skeleton model does not enter or pass through the polygon object, and behaves naturally. Add such functions. Such a function,
Let's call it interference function.

The following three types of interference functions can be considered.

(A) Interference between nodes and polygon objects.

(B) Interference between an arc and a polygon object (c) Interference between a polygon object moving accompanying a skeleton model and another polygon object.

Here, the node-to-polygon object interference function (a) will be described.

-2 Collision Detection In order to realize the interference function, it is necessary to first detect a collision between the skeleton model and the polygon object. The following two methods can be considered for collision detection.

(A-1) Detection by Node Position If the node is inside the polygon object, it is determined that the collision occurs.

(A-2) Detection based on the locus of the node. If the locus of the node intersects the polygon object, it is determined that the collision occurs.

When the node 142 moves with respect to the polygon object 140 as shown in FIG. 20A, a collision cannot be detected in the above (a-1). Then, the collision is detected by determining the intersection of the trajectory 144 of the node 142 and the polygon object 140 by employing the method (a-2). In addition, according to this method, an object that is not closed can be handled, and collision can be detected regardless of the front and back of the polygon.

-3 Setting of Restriction Conditions In order to operate the interference function, it is necessary to restrict the node on the polygon plane when the node is pressed against the polygon, and to release the restriction when trying to move away from the polygon. . For the constraint on the polygon plane, the face magnet already described in Algorithm 1 is used. Further, the switching of the constraint condition uses the same method as that for limiting the movable range of the arc-to-arc angle, that is, the method described with reference to FIG. In other words, facemagnet is switched on / off as follows. (A) Calculate with face magnet OFF and result is no collision: Use result as it is (B) Calculate with face magnet OFF and result collision occurs: Scale and use result (C) Calculate with face magnet ON and result is non- Collision state: face ma
Turn off gnet and recalculate (D) face magnet ON and the result is a collision state: Use the result as it is The surface on the side where the node was located immediately before the collision out of both sides of the polygon is the collision surface, and the node is the collision surface Is located on the opposite side of the collision surface of the polygon as a collision state.

-4 Handling of Multiple Collision As described above, since the face magnet is used as the constraint condition, the node penetrates through the polygon, albeit slightly. By seeing this penetration, ON / OFF of the face magnet
Switching OFF. However, as shown in FIG. 20B, when the node 150 passes through the polygon 152, a problem occurs when the node 150 moves in the direction of the arrow 156. That is, node 15 should be
Although 0 should collide with the two polygons 152 and 154, a collision with the polygon 154 cannot be detected.

Therefore, as shown in FIG. 20C, a virtual surface (virtual boundary) 158 is created by offsetting the polygon (boundary) 152 by a minute amount d toward the collision surface. If the node 150 is in a collision state, the node 150 is perpendicularly projected onto the virtual plane 158, and is set as a new node 160 position. The virtual surface 158 is used for the face magnet and the collision detection instead of the surface of the polygon 152. This method can cope with multiple collisions of nodes.

[0165] 4. Image Synthesizing Apparatus and Information Storage Medium FIGS. 21A, 21B and 21C show various embodiments of an image synthesizing apparatus and an information storage medium using the skeleton model shape deformation method of this embodiment.

FIG. 21A shows an example in which this embodiment is applied to a shape data creation tool which is one of image synthesizing apparatuses. The input of the initial shape data of the skeleton model, the designation of a fixed node or a fixed arc, the designation of the movable range of the angle between arcs, the designation of the attachment of a node to a polygon or an edge, the instruction of picking and moving a node, etc. , Mouse 2 as a pointing device
02. The processing unit 210 includes a shape data generation unit 212 that deforms the shape of the skeleton model by the shape deformation method of the present embodiment to generate shape data, and a skeleton model based on the shape data generated by the shape data generation unit 212. Image combining unit 214 for combining images
And The processing unit 210 has a hardware
It is composed of a U, a DSP, a memory, and if necessary, an image synthesis IC. An information storage medium 216 including an FD, a CDROM, a DVD, an IC card, a memory, and the like includes:
A variety of information such as a shape data creation program for realizing the shape deformation method of the present embodiment and data necessary for executing the program are stored. The processing unit 210 performs processing based on input information from the keyboard 200 and the mouse 202 and information stored in the information storage medium 216, and thereby displays a skeleton model image 219 on the display unit 218.
Are displayed. According to this shape data creation tool,
The shape data can be created while confirming the shape of the displayed skeleton model image 219, and the design work can be made more efficient.

FIG. 21B shows an example in which this embodiment is applied to a game device which is one of the image synthesizing devices. The game controllers 220 and 222 are for the player to input operation information. The processing unit 230 includes a game calculation unit 232 that performs calculation for synthesizing a game image based on operation information from the player,
And an image synthesizing unit 234 for synthesizing an image based on the calculation result from the CPU 32.
P, a memory, and if necessary, an IC dedicated to image synthesis. The information storage medium 236 stores shape data created by the shape deformation method according to the present embodiment, various information generated based on the shape data, a game program, and the like. On the display unit 238, images 239 and 240 of the game character operating based on the shape data and the like are displayed. If the game device is for business use, the shape data and the game program are stored in ROM
If the game device is for home use, the shape data and the game program are stored in C
It is stored in an information storage medium such as a D-ROM and a game cassette. In a game device of a type in which a plurality of terminals are connected by a communication line via a host device and a game program or the like is distributed, the shape data and the game program are stored in an information storage medium of the host device, for example, a magnetic disk device, a memory, or the like. Is stored.

On the other hand, FIG.
2 is an example of a game device including a shape data generation unit 243. In this case, instead of the shape data, the information storage medium 246 stores a shape data generation program for realizing the shape deformation method of the present embodiment, and the shape data is generated by the shape data generation unit 243 incorporated in the game device. Do in real time. For example, the game controller 220,
The player causes the skeleton model to perform a desired operation by 222 or the like. As a result, the images 239 and 240 of the moving object accompanying the skeleton model are displayed on the display unit 23.
8 will be displayed.

The present invention is not limited to the embodiments described above, and various modifications can be made.

For example, the invention in which a skeleton model is represented by basic variables including components of node coordinates and arc normal vectors, and an expression including a change amount of an angle between arcs is minimized includes the Newton method, the Lagrange multiplier, and the like. In addition to the method, for example, it can be realized using various mathematical methods such as a method obtained by improving the Newton method or the Lagrange multiplier method, and various modifications can be made. In particular, when the equation to be solved is represented by a linear equation of an unknown variable as in the present embodiment,
The solution can also be obtained by the method of linear algebra.

Similarly, it is possible to obtain a solution of an expression including basic variables such as the coordinates of nodes by iterative calculation, and to change the expression one after another based on given information in synchronization with the end or start of the iterative loop. , The skeleton model is spring-bound to the boundary, and the constraint conditions for the next iteration loop are determined based on whether the constraint conditions were enabled or disabled in one iteration loop and whether the skeleton model, etc., exceeded the boundary. The invention that determines validity / invalidity can be realized using various mathematical methods other than the combination of the Newton method and the Lagrange multiplier method, and various modifications can be made.

[0172]

[Brief description of the drawings]

FIGS. 1A to 1D are diagrams for explaining the principle of the present embodiment.

FIG. 2A is a diagram for explaining a method of expressing a change amount of an arc normal vector by one variable, and FIG. 2B is a diagram showing various examples of an evaluation expression; is there.

FIG. 3A is a diagram for explaining the Newton method, and FIG. 3B is a diagram for explaining a method of changing a basic expression or the like for each iteration loop.

FIG. 4 is a diagram illustrating an example in which a skeleton model is deformed by picking a node;

FIGS. 5A to 5E are views for explaining a rubber band; FIG.

FIG. 6 is a diagram illustrating an example in which two nodes are picked and the skeleton model is deformed in shape.

FIG. 7 is a diagram illustrating an example in which a skeleton model having a complicated multi-branch structure is deformed in shape.

FIGS. 8A and 8B are diagrams for explaining setting of a movable range of an angle between arcs.

FIGS. 9A and 9B are diagrams for explaining a function of attaching nodes to polygons and edges.

FIG. 10 is a diagram for describing an example in which a skeleton model climbs a wall using an attach function.

FIG. 11 is a diagram for explaining an example in which a skeleton model climbs a wall using an attach function.

FIG. 12 is a diagram for explaining an interference function between a skeleton model and an object.

FIGS. 13A and 13B are diagrams for explaining a method of handling a constraint condition.

FIG. 14 is a diagram for explaining the relationship among arcs, nodes, arc normal vectors, and the like.

FIG. 15 is a diagram for describing an arc coordinate system.

FIGS. 16A to 16D are diagrams for explaining the effectiveness of a rubber band.

17A is a diagram for explaining an arc rotation around a fixed node, and FIGS. 17B and 17C are diagrams for explaining occurrence of a singular state.

FIGS. 18A, 18B, and 18C are diagrams for explaining setting of a movable range of an angle between arcs.

FIG. 19 is a diagram for describing overshooting of an angle between arcs and setting of an allowable limit value thereof.

FIGS. 20A, 20B, and 20C are diagrams for explaining an interference function between a skeleton model and an object;

FIGS. 21A, 21B, and 21C are diagrams showing various embodiments of an image synthesizing apparatus and an information storage medium.

FIGS. 22A, 22B, and 22C are diagrams for explaining a conventional skeleton model shape deformation method;

FIGS. 23A and 23B are diagrams for describing setting of a parent-child relationship.

[Explanation of symbols]

 2, 3, 4 nodes 6, 7, 8 arc 10, 11, 12 arc normal vector 18 skeleton model

Claims (33)

[Claims] 1. A method for deforming a shape of a skeleton model, wherein the skeleton model is represented by basic variables including coordinates of nodes of the skeleton model, and defines that the arc length of the skeleton model has a given value. At least one of a basic expression including an expression and the basic variable as an unknown and an evaluation expression for uniquely identifying a solution satisfying the basic expression is changed based on given information, and the basic expression is almost satisfied. A shape deformation of the skeleton model characterized by finding a solution of a basic variable that makes the value of the evaluation formula almost minimum, almost maximum, or almost stationary, and deforming the shape of the skeleton model based on the obtained solution. Method. 2. The method according to claim 1, wherein the basic variable includes a component of an arc normal vector perpendicular to the arc, and the basic expression indicates that the length of the arc normal vector is a given value. A method for deforming the shape of a skeleton model, comprising: a formula that defines a formula; and a formula that defines that the arc normal vector is perpendicular to an arc. 3. The method according to claim 1, wherein the basic variable includes a component of an arc normal vector perpendicular to an arc, and a variation of the component of the arc normal vector is represented by a given one variable. A method for deforming the shape of a skeleton model, wherein a solution of a basic variable is obtained. 4. The method according to claim 1, wherein the basic variables include a component of an arc normal vector perpendicular to the arc, a change amount of an angle between the arcs, and a difference between a node and a given point. The values of the evaluation formulas including at least one of the distance between them, the amount of change in the direction of the arc having a fixed node at the end, the amount of change in the node coordinates, and the amount of change in the speed of the node coordinates are substantially minimum, substantially maximum, and substantially stationary. A method for deforming the shape of a skeleton model, wherein a solution of a basic variable is obtained so as to be any one of them. 5. The method according to claim 1, wherein a solution of the basic variable is obtained by iterative calculation, and at least one of the basic expression and the evaluation expression is given in synchronization with the end or start of an iterative loop. A method for deforming the shape of a skeleton model, wherein the shape is changed one after another based on the information of 6. The method for deforming a shape of a skeleton model according to claim 1, wherein the coordinates of the fixed nodes or the components of the fixed arc normal vector are known numbers. 7. The expression according to claim 1, wherein the evaluation expression is an expression that constrains an angle between arcs to a given range, and an expression that constrains a node to a given plane or a given line. And at least one of a formula for constraining the skeleton model or the first object to the second object when the skeleton model or the first object moving therewith interferes with another second object. Shape deformation method of the skeleton model. 8. A method for deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including a coordinate of a node of the skeleton model and a component of an arc normal vector perpendicular to an arc of the skeleton model. An expression including at least one of an angle change, a distance between a node and a given point, a change in the direction of an arc having a fixed node at an end, a change in node coordinates, and a change in speed of node coordinates. Wherein the shape of the skeleton model is deformed so that the value is substantially minimum, approximately maximum, or substantially stationary. 9. A method for deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including the coordinates of nodes of the skeleton model, and the coordinates of a given point controlling the shape deformation of the skeleton model are changed. A method of deforming the shape of the skeleton model, wherein the shape of the skeleton model is deformed so that the distance between the node and the given point is minimized. 10. The method of claim 9, wherein deforming the shape of the skeleton model such that the distance between the two or more nodes and the two or more given points associated with each of them is minimized. Shape deformation method of the skeleton model that is the feature. 11. A method for deforming a shape of a skeleton model, wherein the skeleton model is represented by basic variables including coordinates of nodes of the skeleton model, and a solution of an expression including the basic variables is obtained by iterative calculation, and A method for deforming the shape of a skeleton model, characterized in that it is changed one after another based on given information in synchronization with the end or start of an iterative loop. 12. A method for deforming the shape of a skeleton model, comprising: when a skeleton model or a moving object accompanying the skeleton model exceeds a given boundary, the skeleton model or the object is returned to the boundary side under a constraint condition. When the shape of the skeleton model is deformed by, and one iterative loop of the iterative calculation for obtaining the shape of the skeleton model is performed in a state where the constraint conditions are invalid and the skeleton model or the object does not exceed the boundary, the following is performed. Also in the iterative loop, the calculation is performed with the constraints in an invalid state.If the one iterative loop is performed in a state in which the constraints are invalid and the skeleton model or object is beyond the boundary, In the next iteration loop, the calculation is performed with the constraint conditions being valid, and the first iteration loop is When the bundle condition is performed in a valid state and the skeleton model or the object exceeds the boundary, the calculation is performed in the next iterative loop with the constraint condition being valid, and the one iterative loop is executed. When the constraint is performed in a valid state and the skeleton model or the object does not exceed the boundary, the calculation is performed with the constraint in an invalid state in a next iteration loop. How to deform the shape of the model. 13. The skeleton model according to claim 12, wherein the shape of the skeleton model under a constraint condition that a value of an expression including a distance between the skeleton model or an object and the boundary is substantially minimum, approximately maximum, or substantially stationary. A shape deformation method of a skeleton model characterized by being deformed. 14. The virtual boundary according to claim 12, wherein a virtual boundary is provided at a predetermined distance from the boundary, and when the skeleton model or the object exceeds the boundary, the virtual skeleton is moved to the virtual boundary. And determining whether the constraint condition is valid or invalid by using the virtual boundary instead of the boundary. 15. A method for deforming a shape of a skeleton model, wherein the domain of at least one first variable that defines the shape of the skeleton model is represented by an inequality including the first variable. Introduce an equation in which all the expressions and numbers in are transposed to either one of the left and right sides, and the other side is replaced by a function of the second variable that is constantly positive or negative or non-negative or non-positive. A method for deforming the shape of the skeleton model under discontinuous constraint conditions by defining the domain of the first variable by the equation instead of the inequality. 16. The skeleton model according to claim 15, wherein the shape of the skeleton model is modified such that the value of the expression including the amount of change of the second variable is one of substantially minimum, substantially maximum, and substantially stationary. Shape deformation method of the skeleton model. 17. The method according to claim 15, wherein a given value is added to the second variable when the second variable becomes a value near a given convergence value. Shape deformation method of skeleton model. 18. A method for deforming the shape of a skeleton model, wherein in one iteration loop of an iterative calculation for obtaining the shape of the skeleton model, a constraint condition for constraining the skeleton model or an accompanying moving object to a given boundary. Perform a first calculation in a disabled state, and in the one iteration loop, after the first calculation,
When the boundary between the skeleton model or the object and the boundary has a given positional relationship, a second calculation for attracting the skeleton model or the object to the boundary is performed. 19. The method according to claim 18, wherein in the first calculation, a value of an expression including an amount of movement of the skeleton model or the object in the boundary direction is one of substantially minimum, substantially maximum, and substantially stationary. Performing a calculation, wherein in the second calculation, the calculation is performed such that a value of an expression including a distance between the skeleton model or an object and the boundary is one of substantially minimum, substantially maximum, and substantially stationary. Shape deformation method of skeleton model. 20. The method according to claim 18, wherein the boundary is a finite surface or a finite line. 21. A method for deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including the coordinates of nodes of the skeleton model, and a first simultaneous equation in which the amount of change of the basic variables is unknown is degenerated or It is determined whether or not it is in a state close to it, and if it is determined that the first simultaneous equation is in a degenerate state or a state close to it, the first simultaneous equation is converted into an independent second A shape transformation method of the skeleton model by reconstructing the simultaneous equations and solving the second simultaneous equation. 22. The method according to claim 21, wherein the first simultaneous equation is an equation for finding a solution of an equation that specifies that the arc length of the skeleton model has a given value. Skeleton model shape deformation method. 23. Means for outputting a shape of a skeleton model deformed by the shape deforming method according to any one of claims 1 to 22, and operation means for an operator to instruct deformation of the skeleton model. Characteristic image synthesis device. 24. An image synthesizing apparatus, comprising: means for outputting an image of a moving object accompanying a skeleton model deformed by the shape deforming method according to any one of claims 1 to 22. 25. An information storage medium for storing shape data of a skeleton model deformed by the shape deforming method according to claim 1. Description: 26. An information storage medium for deforming the shape of a skeleton model, wherein information for representing the skeleton model by basic variables including the coordinates of the nodes of the skeleton model and the length of the arc of the skeleton model are provided. Based on given information, at least one of a basic expression including an expression defining a given value and the basic variable as an unknown, and an evaluation expression for uniquely specifying a solution satisfying the basic expression, based on given information Information to obtain a solution of a basic variable that substantially satisfies the basic expression and that makes the value of the evaluation expression substantially minimum, approximately maximum, or almost stationary. An information storage medium including information for deforming a shape of a skeleton model. 27. An information storage medium for deforming the shape of a skeleton model, wherein the skeleton model is represented by basic variables including a coordinate of a node of the skeleton model and a component of an arc normal vector perpendicular to an arc of the skeleton model. Information, the amount of change in the angle between the arcs, the distance between the node and a given point, the amount of change in the direction of the arc with a fixed node at the end, the amount of change in the node coordinates, and the speed of the node coordinates An information storage medium comprising information for deforming the shape of a skeleton model such that a value of an expression including at least one of the amounts of change is substantially minimum, approximately maximum, or substantially stationary. 28. An information storage medium for deforming the shape of a skeleton model, comprising: information for representing the skeleton model by basic variables including coordinates of nodes of the skeleton model; and a place for controlling the shape deformation of the skeleton model. An information storage medium including information for deforming the shape of the skeleton model so that the distance between the node and the given point is minimized when the coordinates of the given point change . 29. An information storage medium for deforming the shape of a skeleton model, comprising: information for representing the skeleton model by basic variables including coordinates of nodes of the skeleton model; and a solution of an expression including the basic variables. An information storage medium which is obtained by iterative calculation and includes information for changing said expression one after another based on given information in synchronization with the end or start of an iterative loop. 30. An information storage medium for deforming the shape of a skeleton model, wherein when a skeleton model or an object moving therewith crosses a given boundary, the skeleton model or the object is moved to the boundary side. Information for deforming the shape of the skeleton model under the constraint to be returned, and one iterative loop of an iterative calculation for determining the shape of the skeleton model is performed in a state where the constraint is invalid and the skeleton model or the object If not, the information for performing the calculation with the constraint condition being invalid even in the next iteration loop, and the one iteration loop is performed with the constraint condition invalid and the skeleton model Or, if the object is beyond the boundary, in the next iteration loop, the constraint will be enabled. And information for performing the calculation, if the one iterative loop is performed in a state where the constraint is valid and the skeleton model or the object is beyond the boundary, the constraint is also set in the next iterative loop. Information for performing the calculation with the condition being valid; and if the one iteration loop is performed with the constraint being valid and the skeleton model or object does not exceed the boundary, the next iteration An information storage medium including information for performing a calculation with the constraint condition being invalid in a loop. 31. An information storage medium for deforming a shape of a skeleton model, wherein a domain of at least one first variable defining a shape of the skeleton model is represented by an inequality including the first variable. In the case, all the expressions and numbers of the inequality were transferred to one of the left side and the right side, and the other side was replaced with a function of the second variable, which is constantly positive or negative or non-negative or non-positive. Introducing an equation and defining the domain of the first variable by the equation instead of the inequality includes information for performing shape deformation of the skeleton model under discontinuous constraints. Characteristic information storage medium. 32. An information storage medium for deforming the shape of a skeleton model, wherein in one iteration loop of an iterative calculation for determining the shape of the skeleton model, a skeleton model or an object moving therewith is provided with a given boundary. And information for performing the first calculation in a state where the constraint condition for restraining is invalid. In the first iteration loop, after the first calculation,
An information storage medium comprising: a skeleton model or an object and information for performing a second calculation for drawing the skeleton model or the object to the boundary when the boundary has a given positional relationship. 33. An information storage medium for deforming the shape of a skeleton model, comprising: information for representing the skeleton model by basic variables including coordinates of nodes of the skeleton model; Information for determining whether the first simultaneous equation is in a degenerate state or a state close to the first simultaneous equation, and when it is determined that the first simultaneous equation is in a degenerate state or a state close to the first simultaneous equation, Reconstructing the simultaneous equations into independent second simultaneous equations with a reduced number of equations, and including information for performing a shape deformation of the skeleton model by obtaining a solution of the reconstructed second simultaneous equations. An information storage medium characterized by the following.