JPH07168952A - Method and device for simulating collision of object - Google Patents

Abstract

PURPOSE:To provide a system and a device for simulating the collision of an object in which the motion of objects at the time of the collision between the objects can be approximated with high precision. CONSTITUTION:A CPU 200 controls a whole data processor, and while reading out a simulation program stored beforehand in a memory 202, it interprets and executes it. In the memory 200, the simulation program of this embodiment and various necessary data are stored. A keyboard 203 and a pointing device 204 input the data related to the object to be simulated and an execution command, etc. A display monitor 201 displays the data of the result of the simulation and the motion image of the object and so on.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method and apparatus for simulating the collision of objects in computer graphics, particularly animation and virtual reality.

[0002]

2. Description of the Related Art When simulating a collision between virtual objects in a virtual three-dimensional space built inside a computer, the object shape is conventionally regarded as a mass point or a sphere, and the coefficient of restitution and friction coefficient between objects are taken into consideration. Instead, it was calculated as a rigid body without friction.

[0003]

The conventional method of performing collision simulation by regarding the shape of an object as a mass point or a sphere is easy to calculate, but cannot accurately calculate an object having a complicated shape. Is clear. Further, the method that does not consider the coefficient of restitution and the coefficient of friction between objects has a drawback that it cannot cope with various actual collision phenomena.

The present invention has been made in view of the above-mentioned conventional example, and an object thereof is to provide an object collision simulation method and apparatus capable of approximating the motion of the objects at the time of collision between the objects with high accuracy.

[0005]

In order to achieve the above object, an object collision simulation method and apparatus of the present invention have the following configurations. That is, based on the characteristic data of the virtual object and the state data of the virtual object at the time of collision, a calculation step of calculating the state data of the virtual object after the collision, based on the state data of the virtual object after the collision, A display step of displaying the state of the virtual object after the collision.

Further, another invention is to calculate the state data of the virtual object after the collision based on the characteristic data of the virtual object and the state data of the virtual object at the time of collision, and the virtual means after the collision. Display means for displaying the state of the virtual object after the collision based on the state data of the object.

[0007]

As described above, according to the present invention, the state data of the virtual object after the collision is calculated based on the characteristic data of the virtual object and the state data of the virtual object at the time of the collision, and the state data after the collision is calculated. The state of the virtual object after the collision is displayed based on the state data of the virtual object.

According to another aspect of the invention, the calculating means calculates the state data of the virtual object after the collision based on the characteristic data of the virtual object and the state data of the virtual object at the time of collision, and the state data after the collision is obtained. The display means displays the state of the virtual object after the collision based on the state data of the virtual object.

[0009]

[Embodiment] To automatically calculate the behavior of a virtual object group in a virtual three-dimensional space built in a computer,
This is a useful technique in animation and virtual reality. The present embodiment relates to a method of simulating the behavior when objects collide with each other, and shows a method of calculating the state of an object immediately after collision from the state of the object at the time of collision and the state of collision.

In this embodiment, when simulating a collision between virtual objects, the state of the virtual objects at the time of collision (position, inclination, velocity, angular velocity), collision position, collision direction (tangential plane direction of collision position) , Calculate the state of the virtual object after the collision. This method can perform calculations for arbitrary shapes if the mass and moment of inertia of the object are known. Further, the collision time, the collision position, and the collision direction are determined by the states and shapes of the two objects, and by using these, the same processing can be performed regardless of the object shapes.

The state of the virtual object after the collision is calculated by calculating the impulse generated by the collision.
By calculating in consideration of the coefficient of restitution and the coefficient of friction between two objects, it is possible to perform calculations corresponding to various collision phenomena. When the coefficient of restitution and the coefficient of friction between two objects are unknown, the coefficient of restitution (0 to 1) and coefficient of friction (0 to 1) unique to each object are defined and expressed by the product of the coefficients between the two objects. When the coefficient of restitution between two objects or the coefficient of friction is not 0 or 1, the impulses when the coefficients are 0 and 1 are calculated and interpolated.

When the friction coefficient between the objects is 0, slippage occurs at the collision position and the force acts in the collision direction. When the friction coefficient is 1, rolling occurs at the collision position, and a force acts so that the velocities in the collision tangent plane at the collision position become equal. When the coefficient of repulsion between objects is 0, no force acts in the collision direction, and the objects move at the same speed in the collision direction at the collision position. When the coefficient of restitution is 1, it is regarded as a collision between rigid bodies, and when the coefficient of friction is 0 or 1, the kinetic energy is saved.

FIG. 1 shows a state at the moment when two objects, that is, a rectangular parallelepiped object 1 (1) and a spherical object 2 (2) collide with each other. Here, each mass is M
1, M2 and inertia tensors are [I1] and [I2]. The state of each object at the time of collision is expressed by the following parameters. That is, the barycentric coordinates of the object A: O1, the barycentric coordinates of the object B: O2, the speed of the object A: V1, the speed of the object B: V2, the tilt of the object A: [S1] the tilt of the object B: [S2
], Angular velocity of object A: W1 Angular velocity of object B: W2, where O1, O2, V1 and V2 are vector quantities in the inertial system, and [S1] and [S2] are converted from the object coordinate system to the inertial system 3 A rotation matrix of × 3, W1 and W2 are vector quantities in the object coordinate system, and the inertial system and the object coordinate system are [S
1] and [S2] are linked and can be converted mutually. Further, if the collision position P and the collision direction N are given to the inertial system, the collision positions R1 and R2 with respect to the center of gravity in each object coordinate system can be immediately obtained by using [S]. That is,

[Equation 1] (Equation 1)

[Equation 2] (Equation 2) Here, [S] -1 is an inverse matrix of [S]. In this embodiment, the object coordinate system is selected so as to match the direction of the principal axis, and the inertia tensor [I] is as follows. Is diagonalized in the object coordinate system.

[0014]

[Equation 3] FIG. 2 shows the state of the object after the collision. The impact states F and −F act on the two objects by the collision, and the motion state changes. The collision force is based on the law of action and reaction,
The sizes are equal and the directions are opposite. Immediately after the collision, the barycentric coordinates and inclination of each object do not change, and as an ideal object,
The velocities V'1 and V'2 and the angular velocities W'1 and W'2 immediately after the collision are obtained. From the changes in momentum and angular momentum before and after the collision, the following equation is obtained.

[0015]

[Equation 4] (Formula 3)

[Equation 5] (Equation 4)

[Equation 6] (Equation 5)

[Equation 7] (Equation 6) Equations (3) and (4) show changes in momentum of each object with respect to the inertial system, and Equations (5) and (6) show changes in angular momentum of each object with respect to the object coordinate system. Shows. Ft is the impulse generated at the collision position at the time of collision, and F1, F
2 is a transformation of Ft into the object coordinate system. The impulses are connected by the following relations, and the above equations satisfy the momentum conservation law and the angular momentum conservation law.

[0016]

[Equation 8] (Equation 7) From Equations (3) to (7), V′1, V′2, W′1 and W′2 can be expressed as follows using the impulse Ft.

[0017]

[Equation 9] (Equation 8)

[Equation 10] (Equation 9)

[Equation 11] (Equation 10)

[Equation 12] (Equation 11) Here, [C1] and [C2] are inertia tensors [I1],
[I2], rotation matrices [S1], [S2], and a 3 × 3 angular velocity conversion matrix determined by the collision position R with respect to the center of gravity,

[Equation 13]

[Equation 14]

[Equation 15] When,

[Equation 16] (Equation 12) is calculated.

As described above, if the impulse Ft is determined, the velocities V'1 and V'2 and the angular velocities W'1 and W'2 immediately after the collision are determined.
Can be asked. The method for obtaining the impulse Ft will be described below.

The kinetic energies E and E'before and after the collision are

[Equation 17] (Equation 13)

[Equation 18] (Equation 14) is obtained. The relative velocity dV at the contact point immediately after the collision is calculated by taking the moving velocity and the rotation velocity into consideration.

[Formula 19] (Equation 15) In order to express all the speeds in the inertial system, the rotation speed is converted from the object coordinate system to the inertial system using [S]. Expressions (14) and (15) are expressions (8) to (1).
Using 1), it can be expressed using the impulse Ft as follows.

[0020]

[Equation 20] (Equation 16)

[Equation 21] (Equation 17) Here, E0 is a constant, Ef (Ft) is a quadratic expression regarding Ft, V0 is a vector quantity, and [Vf] is a 3 × 3 relative velocity conversion matrix. Ft is determined using both of these equations.

In the determination of Ft, the repulsive force and the frictional force between the objects are also taken into consideration in the calculation. The frictional force gives a friction coefficient μ (0 to 1) to each object, and the frictional force between the objects is expressed by the product (0 to 1) of the frictional coefficient between the objects. When the coefficient of friction between the objects is 0, slippage occurs at the collision position and the force acts in the collision direction. When the coefficient of friction is 1, rolling occurs at the collision position, and a force acts so that the velocities in the collision tangent plane at the collision position become equal.

Similar to the frictional force, the repulsive force is given a repulsion coefficient ε (0 to 1) to each object, and the repulsive force between the objects is expressed by the product (0 to 1) of the repulsive coefficients between the objects. When the restitution coefficient between objects is 0, no force acts in the collision direction, and after the collision, the objects move at the same speed in the collision direction. When the coefficient of restitution is 1, it is regarded as a collision between rigid bodies, and when the coefficient of friction is 0 or 1, kinetic energy is stored.

The coefficient of restitution or the coefficient of friction between two objects is 0.
If not, the impulses when the respective coefficients are 0 and 1 are obtained and interpolated. Hereinafter, how to obtain Ft will be described for each combination of the values of the coefficient of restitution and the coefficient of friction. [Case 1] When Friction Coefficient μ = 0 and Repulsion Coefficient ε = 1 From the friction coefficient μ = 0, the direction of the impulse Ft coincides with the collision direction N because the contact point is smooth and there is no friction. Also, since there is no friction and the coefficient of restitution ε = 1, kinetic energy is preserved before and after the collision. Ft = kN (Equation 18) E ′ = E (Equation 19) Here, by substituting the equation (18) into the energy equation of the equation (16), E ′ is represented by k, and from the equation (19), Find k. Ft at this time is set to F01. [Case 2] When the coefficient of friction μ = 1 and the coefficient of restitution ε = 1 From the coefficient of friction μ = 1, rolling occurs without slipping at the contact point,
The velocities of the two objects after the collision on the collision tangent plane become equal.
That is, the direction of the relative speed of the two objects matches the collision direction.
In addition, since the rolling repulsion coefficient ε = 1 without slipping, kinetic energy is preserved before and after the collision. dV = kN (Equation 20) E '= E (Equation 21) Here, Ft is represented by k by substituting the equation (20) into the equation of the relative speed of the equation (17).

[0024]

[Equation 22] (Equation 22) Substituting this into the energy equation of Equation (16), E ′
Is expressed using k, and k is calculated from the equation (21). Ft at this time is set as F11. [Case 3] 3 When coefficient of friction μ = 0 and coefficient of restitution ε = 0 From the coefficient of friction μ = 0, since the contact point is smooth and there is no friction, the direction of the impulse Ft coincides with the collision direction N. Also, from the coefficient of restitution ε = 0, the velocity in the collision direction after collision will match,
The direction of relative velocity is perpendicular to the collision direction. Ft = kN (Equation 23) dV · N = 0 (Equation 24) Here, by substituting the relative speeds of the equations (23) and (24) into the equation, k is obtained. k = -V0.N / ([Vf] N) .N (Equation 25) Let Ft be Ft at this time. [Case 4] When friction coefficient μ = 1 and repulsion coefficient ε = 0 From the friction coefficient μ = 1, rolling occurs without slipping at the contact point,
The velocities of the two objects after the collision on the collision tangent plane become equal.
Further, from the coefficient of restitution ε = 0, the velocities in the collision direction match after the collision. That is, the velocities of the two objects after the collision match at the collision position. dV = 0 (Equation 26) Here, Ft is obtained by substituting the relative speed of the equation (17) into the equation (26).

[0025]

[Equation 23] (Formula 27) Ft at this time is set to F10.

[Case 5] When the coefficient of friction μ (0 to 1) and the coefficient of restitution ε (0-1) When the coefficient of friction μ and the coefficient of restitution ε are in the range of 0 to 1,
The impulse Ft is the impulses F00, F01, F10, F obtained so far.
Calculation is performed by using 11 as follows. Ft = F00 + (F10−F00) μ + (F01−F00) ε + (F11−F01−F10 + F00) εμ (Equation 28) Thus, the impulse Ft considering the friction coefficient and the restitution coefficient.
Therefore, the velocities V′1 and V′2 and the angular velocities W′1 and W′2 immediately after the collision can be calculated from the equations (8) to (11).

Next, a concrete example of collision of various objects will be described.

Specific Example 1 Collision between Spheres FIG. 3 shows a state in which spheres traveling on the same straight line while rotating in the same direction and at the moment when sphere 3 and sphere 4 collide. Here, 10 indicates a three-dimensional basic Cartesian coordinate axis that defines the coordinates of these spheres, and is composed of an x axis 5, ay axis 7, and az axis 6. The traveling directions of the sphere 1 (3) and the sphere 2 (4) are parallel to the x-axis 5, and their rotation axes are parallel to the z-axis 6. The two objects have a radius L = 1, a mass M = 1, and a rotation speed W = 1.
Then, the sphere 1 (3) has a velocity V1 = 1 in the positive direction on the x-axis,
Sphere 2 (4) is moving in the negative direction on the x-axis at a speed V2 = -1. At this time, the collision position R seen from the center of the sphere 1 (3)
1 is (1, 0, 0), the collision position R2 seen from the center of the sphere 2 (4) is (-1, 0, 0), and the collision direction is the x-axis 5 direction.

4 to 8 show the impulse F generated when the balls collide with each other under various conditions of the coefficient of friction and the coefficient of restitution.
1 and F2, velocities V'1 and V'2 after the collision, angular velocities W'1 and W'2 after the collision, and the states of two objects when the time Δt = 1 has elapsed. For impulses, velocities, and angular velocities, the direction is indicated by an arrow, the magnitude is indicated by the length of the arrow and a numerical value, and the numerical value is added with the sign when the values in the x, y, and z directions are added. There is.

In the following description, the value of the z-axis 6 component is 0.
In the case of, only the (x, y) component is shown. Furthermore, when the value of the y-axis 7 component is also 0, only the x component is shown.

FIG. 4 shows a friction coefficient μ = 0 and a restitution coefficient ε = 1.
The state after the collision at the time of is shown. In this state, since there is no friction, the impulse is directed in the collision direction (the direction of the x-axis 5) and the ball 1
The impulses of (3) and sphere 2 (4) in the object coordinate system are
F1 = -2 and F2 = 2. Since the impulse is toward the center of the sphere, the angular velocity does not change before and after the collision, and W'1 = 1, W'2
= 1. Further, the vehicle travels on the x-axis 5 at the speeds of V'1 = -1, V'2 = 1. That is, the angular velocities of the two objects do not change, and the two objects travel at the same magnitude as before the collision and in the opposite direction.

FIG. 5 shows the state after the collision when the friction coefficient μ = 1 and the restitution coefficient ε = 1. In this state,
Because of friction, the impulse also includes the y-axis 7-direction component, and the sphere 1
The impulses of (3) and sphere 2 (4) in the object coordinate system are
F1 = (-2.13, -0.29), F2 = (2.13,
0.29). Since the impulse is not directed to the center of the sphere, the magnitude of the angular velocity changes before and after the collision, and the angular velocities of sphere 3 and sphere 4 are W'1 = 0.29 and W'2 = 0.29, respectively. Including the y-direction component, V'1 = (-1.13, -0.2)
9), V'2 = (1.13, 0.29). That is, the directions of the angular velocities of both objects do not change, but the size decreases by the same amount, and the two objects move at the same speed and in opposite directions.

FIG. 6 shows a friction coefficient μ = 0 and a restitution coefficient ε = 0.
It shows the state after the collision at the time. In this state, because there is no friction, the impulse is in the collision direction (x-axis 7 direction).
And the impulses of the sphere 1 (3) and the sphere 2 (4) in the object coordinate system are F1 = -1 and F2 = 1 respectively. Since the impulse is directed toward the center of the sphere, the angular velocities do not change before and after the collision, and W ′ 1 = 1 and W ′ 2 = 1 respectively. Further, the velocity does not repel in the collision direction, so both objects become 0. That is, after the collision, the two objects only rotate at the collision position and the positions do not change.

FIG. 7 shows that the coefficient of friction μ = 1 and the coefficient of restitution ε = 0.
It shows the state after the collision at the time. In this state, because of friction, the impulse also includes the y-axis 7 component, and the sphere 1
The impulses of (3) and sphere 2 (4) in the object coordinate system are
F1 = (-1, -0.29) and F2 = (1, 0.29). Since the impulse is not directed toward the center of the sphere, the magnitude of the angular velocity changes before and after the collision, and W'1 = 0.29,
W'2 = 0.29, the velocity does not repel in the collision direction, so there are only 7 components on the y-axis, and V'1 = -0.29, V'2 =
It becomes 0.29. That is, the two objects moving on the x-axis 5 decrease in rotational speed by the same amount and move in parallel to the y-axis 7 and in opposite directions.

FIG. 8 shows the coefficient of friction μ = 0.5 and the coefficient of restitution ε.
It shows the state after the collision when = 0.5. In this state, the impulse is F1 = (-1.53, -0.1
4), F2 = (1.53, 0.14). The magnitudes of the angular velocities change, W'1 = 0.64 and W'2 =, respectively.
0.64, the velocity is V ′ 1 = (-0.53,-
0.14) and V'2 = (0.53, 0.14). That is, the two objects decrease in angular velocity by the same amount, and the velocity is equal in magnitude and travel in opposite directions. [Specific Example 2] Collision of a cuboid and a sphere FIG. 9 shows a sphere 9 traveling while rotating on a stationary cuboid 8.
Shows the state at the moment of collision. Here, the rectangular parallelepiped 8 is arranged so that the sides of the rectangular parallelepiped 8 corresponding to the x-axis 5, the y-axis 7, the z-axis 6 and the respective axes are parallel to each other. The direction of the rotation axis of the sphere 9 is the z-axis 6 direction, and the traveling direction is parallel to the x-axis 5. Box 8
The length of each side is Lx = 2, Ly = 3, Lz = 1 in each of the x-, y-, and z-axis directions, and the mass is M1 = 1.
Is. On the other hand, the sphere 9 has a radius L2 = 1, a mass M2 = 1, an angular velocity W2 = 1, and a velocity V2 = -1. At this time, the collision position R1 seen from the center of the rectangular parallelepiped is (1, 1, 0),
The collision position R2 viewed from the center of the sphere is (-1, 0, 0), and the collision direction is the x-axis direction.

FIGS. 10 to 14 show the respective impulses F1 and F2 generated when the rectangular parallelepiped 8 and the sphere 9 collide and the respective velocities V'1 after the collision under various conditions of the friction coefficient and the restitution coefficient. , V'2, the respective angular velocities W'1, W'2 after the collision,
And the states of the two objects when the time Δt = 1 has elapsed. For impulse, velocity, and angular velocity, the direction is indicated by an arrow, and the size is indicated by the length and numerical value of the arrow. The numerical values are the values in the x-axis 5, y-axis 7, and z-axis 6 directions. Is added.

In the following description, when the value of the z-axis component is 0, only the (x, y) component is shown. Furthermore, when the value of the y-axis component is 0, only the x component is shown.

FIG. 10 shows that the friction coefficient μ = 0 and the coefficient of restitution ε =
It shows the state after the collision at the time of 1. In this state, since there is no friction, the impulse is in the collision direction (x-axis 5 direction).
And F1 = -0.68 and F2 = 0.68. Since the impulse is directed toward the center with respect to the sphere 9, the angular velocity does not change before and after the collision, and W′2 = 1, but a rotational force is applied to the rectangular parallelepiped 8 and the angular velocity of the z-axis 6 W'1 = 0.6 around
It becomes 3. The velocities of both objects proceed in the negative direction in parallel with the x-axis, and V'1 = -0.68 and V'2 = -0.32, respectively.

FIG. 11 shows a friction coefficient μ = 1 and a restitution coefficient ε = 1.
It shows the state after the collision at the time. In this state, because of friction, the impulse also includes the y-axis 7-direction component, and F
1 = (-0.92, -0.34), F2 = (0.92,
0,34). Since the direction of the impulse is off the center of both objects, the angular velocity changes before and after the collision, and W'1 =
0.53, W'2 = 0.15, the speed is V'1 = (-
0.92, -0.34), V'2 = (-0.08, 0.3
4).

FIG. 12 shows a friction coefficient μ = 0 and a restitution coefficient ε = 0.
It shows the state after the collision at the time. In this state, since there is no friction, the impulse is in the collision direction (x-axis 5 direction).
And F1 = -0.34, F2 = 0.34. Since the impulse is directed toward the center with respect to the sphere 9, the angular velocity does not change before and after the collision and W′2 = 1, but a rotational force is applied to the rectangular parallelepiped 8 and the angular velocity is around the z-axis. W'1 = 0.32
Becomes The velocities of both objects proceed in the negative direction in parallel with the x-axis 5, and V'1 = -0.34 and V'2 = -0.66. Here, both objects of the sphere 9 and the rectangular parallelepiped 8 are displayed in an overlapping manner, but this indicates that the sphere 9 continues to push the rectangular parallelepiped 8 for a certain period of time because it does not repel in the collision direction. In order to calculate accurately, it is necessary to obtain the collision state of the rectangular parallelepiped 8 and the sphere 9 after a short time, and to repeat the collision calculation. For example, when the minute time is set to 1 in this embodiment, the collision direction is N = (0.95,0.31), the collision position is R1 = (0.73,0,69) when viewed from the center of the rectangular parallelepiped, It may be calculated by setting R2 = (-0.95, -0.32) as viewed from the center of the sphere.

In FIG. 13, the coefficient of friction μ = 1 and the coefficient of restitution ε =
It shows the state after the collision at 0. In this state, because of friction, the impulse also includes the y-axis 7 component, and F1 =
(-0.42, -0.26), F2 = (0.42, 0,
26). Since the direction of the impulse is off the center of both objects, the magnitude of the angular velocity changes before and after the collision, and W'1 = 0.
15, W'2 = 0.36, and the speed is V'1 = (-0.4
2, -0.26) and V'2 = (-0.58, 0.26).

FIG. 14 shows the state after the collision when the friction coefficient μ = 0.5 and the restitution coefficient ε = 0.5. In this state, the impulse is F1 = (-0.59, -0.15),
F2 = (0.59, 0.15). The magnitudes of the angular velocities change, W′1 = 0.41, W′2 = 0.63, and the velocities V′1 = (− 0.59, −0.15), V′2 = (− 0. Four
1, 0.15).

Finally, the coefficient of friction μ and the coefficient of restitution ε are 0 to 1
In order to check the validity of the interpolation performed when the value is between, the change in kinetic energy before and after the collision is examined. The kinetic energies E and E ′ before and after the collision are given by equation (13),
Equation (14) can be used to obtain the mass of two objects,
It is determined by the moment of inertia, velocity, and angular velocity.

FIG. 15 shows the kinetic energy after the collision when the friction coefficient μ and the restitution coefficient ε are variously changed when the balls shown in FIG. 3 collide with each other. Further, FIG. 16 shows kinetic energy after the collision when the cuboid and the sphere shown in FIG. 9 collide.

Referring to FIG. 15, the kinetic energy before the balls collide with each other is 2.8, and the kinetic energy is stored when the coefficient of restitution ε is 1 and the coefficient of friction μ is 0 or 1,
It can be seen that the kinetic energy is most lost when ε = 0 and μ = 1. Even if the coefficient of restitution ε is 1, the coefficient of friction μ is 0.
For values between 1 and 1, the kinetic energy decreases before and after the collision. This is in good agreement with a natural phenomenon because it does not store kinetic energy when the sphere actually rolls while sliding.

Referring to FIG. 16, the kinetic energy before the cuboid and the sphere collide is 1.4, and the kinetic energy is saved when the coefficient of restitution ε is 1 and the coefficient of friction μ is 0 or 1. It can be seen that the kinetic energy is most lost when ε = 0 and μ = 1. Further, even if the coefficient of restitution ε is 1, and the coefficient of friction μ is a value between 0 and 1, the kinetic energy decreases before and after the collision. This is in good agreement with a natural phenomenon because it does not store kinetic energy when the sphere actually rolls while sliding.

Next, an outline of the data processing device for performing the simulation of this embodiment will be described below with reference to FIG.

The CPU 200 controls the entire data processing device, interprets and executes the simulation program stored in the memory 202 while reading it out. The memory 200 stores the simulation program of this embodiment and various required data in advance. The keyboard 203 and the pointing device 204 input data regarding an object to be simulated, execution command commands, and the like. The display monitor 201 displays simulation result data, a motion image of an object, and the like.

Next, the procedure for calculating the motion state immediately after the collision of two objects will be described with reference to the flowcharts of FIGS.

In step S100, the keyboard 203 receives various data representing the two objects themselves and their motion states.
Or pointing device 204 or the like.
As data of the body, masses M1 and M2, inertia tensor [I
1], [I2], the barycentric coordinate of the object 1 is O1, the barycentric coordinate of the object 2 is O2, the speed of the object 1 is V1, the speed of the object 2 is V2,
The inclination of the object 1 is [S1], the inclination of the object 2 is [S2], the angular velocity of the object 1 is W1, the angular velocity of the object B is W2, and the data of the friction coefficient μ and the coefficient of restitution ε are input.

In step S101, the collision positions R1 and R2 with respect to the center of gravity of two objects are calculated based on the equations (1) and (2).

A step S102 checks the data values of the friction coefficient μ and the restitution coefficient ε. And the friction coefficient μ =
If 0 and the coefficient of restitution ε = 1, the process proceeds to step 103. If the friction coefficient μ = 1 and the restitution coefficient ε = 1, step 106
Go to. If the friction coefficient μ = 0 and the restitution coefficient ε = 0, the process proceeds to step 109. If the friction coefficient μ = 1 and the restitution coefficient ε = 0, the process proceeds to step 112. If the coefficient of friction μ is a value between 0 and 1 and the coefficient of restitution ε is a value between 0 and 1, the routine proceeds to step 115.

In step S103, since the friction coefficient μ = 0, the contact point is smooth and there is no friction, so the direction of the impulse Ft coincides with the collision direction N. There is also no friction and the coefficient of restitution ε = 1
From the condition that the kinetic energy is preserved before and after the collision, the formula (18) is substituted into the energy formula of the formula (16), k is calculated from the formula (19), and the impulse at this time is calculated. Calculate F01.

In step S104, the impulse F01 is calculated by the equation 8
By substituting this into equation 9, the velocities V′1 and V′2 immediately after the collision are calculated.

In step S105, the impulse F01 is calculated by the equation 1
By substituting 0 into Equation 11, the velocities W'1 and W'2 immediately after the collision are calculated.

In step S106, from the friction coefficient μ = 1, rolling occurs at the contact point without slipping, the velocities of the two objects on the collision tangential plane after collision are equal, and the direction of the relative speed of the two objects is the collision direction. And the coefficient of restitution ε = 1,
From the condition that the kinetic energy is preserved before and after the collision, Equation 22 is substituted into the energy equation of Equation 16 to obtain Equation 21
Then k is calculated, and the impulse F11 at this time is calculated.

In step S107, the impulse F11 is calculated by the equation 8
By substituting this into equation 9, the velocities V′1 and V′2 immediately after the collision are calculated.

In step S108, the impulse F11 is calculated by the equation 1
By substituting 0 into Equation 11, the velocities W'1 and W'2 immediately after the collision are calculated.

In step S109, since the contact point is smooth and there is no friction from the friction coefficient μ = 0, the direction of the impulse Ft coincides with the collision direction N, and from the restitution coefficient ε = 0, the collision after collision occurs. Under the condition that the directional velocities match and the relative velocity direction becomes perpendicular to the collision direction, k is calculated from Equation 25, and the impulse F00 at this time is calculated.

In step S110, the impulse F00 is calculated by the equation 8
By substituting this into equation 9, the velocities V′1 and V′2 immediately after the collision are calculated.

In step S111, the impulse F00 is calculated by the equation 1
By substituting 0 into Equation 11, the velocities W'1 and W'2 immediately after the collision are calculated.

In step S112, from the friction coefficient μ = 1, rolling occurs without slipping at the contact point, the velocities of the two objects on the collision tangent plane after the collision become equal, and the restitution coefficient ε =
From 0, the impulse F10 is calculated by substituting the relative velocity of the equation (17) into the equation on the condition that the velocities in the collision direction after the collision match.

In step S113, the impulse F10 is calculated by the equation 8
By substituting this into equation 9, the velocities V′1 and V′2 immediately after the collision are calculated.

In step S114, the impulse F10 is calculated by the equation 1
By substituting 0 into Equation 11, the velocities W'1 and W'2 immediately after the collision are calculated.

In step S115, the impulse F00,
Find F01, F10, and F11. The method to obtain is
Step S109, Step S104, Step S11
2. The method is the same as the method for obtaining each impulse in step S106. Then, the obtained impulses F00, F01, F10, F11 are calculated by the equation 2
The impulse Ft is calculated by substituting it into the interpolation formula of FIG.

In step S116, the impulse Ft is calculated by the equation 8
By substituting this into equation 9, the velocities V′1 and V′2 immediately after the collision are calculated.

In step S114, the impulse Ft is calculated by the equation 1
By substituting 0 into Equation 11, the velocities W'1 and W'2 immediately after the collision are calculated.

When the CPU 200 executes the procedure described above, the motion state of the two objects after the collision can be easily obtained. Then, the calculated motion state of the two objects after the collision is displayed on the display monitor.

The present invention may be applied to a system composed of a plurality of devices or an apparatus composed of a single device. Further, it goes without saying that the present invention can be applied to the case where it is achieved by supplying a program to a system or an apparatus.

As described above, when simulating a collision between virtual objects in a virtual three-dimensional space built inside a computer, the state of the virtual objects at the time of collision, the collision position, the collision direction, By calculating the state of the virtual object, the calculation can be performed for any object shape. Also, by calculating the state of a virtual object after a collision by taking into account the coefficient of restitution between two objects and the coefficient of friction, and calculating the impulse generated by the collision, calculations corresponding to various collision phenomena can be performed. It can be carried out. As a result, when the coefficient of restitution is 1 but the coefficient of friction is between 0 and 1 (when rolling while sliding), actual natural phenomena such as kinetic energy not being stored are well represented.

[0071]

As described above, according to the present invention, it is possible to highly accurately approximate the motion of objects at the time of collision between objects.

[0072]

[Brief description of drawings]

FIG. 1 is a diagram illustrating a state at a moment when two objects collide with each other.

FIG. 2 is a diagram for explaining the movement of two objects after a collision.

FIG. 3 is a diagram showing a state at the moment when balls collide with each other.

FIG. 4 is a diagram illustrating a state after the balls have collided with each other under the conditions of a coefficient of friction = 0 and a coefficient of restitution = 1.

FIG. 5 is a diagram illustrating a state after the balls have collided with each other under the condition of the friction coefficient = 1 and the restitution coefficient = 1.

FIG. 6 is a diagram illustrating a state after the balls have collided with each other under the conditions of a coefficient of friction = 0 and a coefficient of restitution = 0.

FIG. 7 is a diagram illustrating a state after the balls have collided with each other under the condition of a friction coefficient = 1 and a restitution coefficient = 0.

FIG. 8 is a diagram illustrating a state after the balls have collided with each other under the conditions of a coefficient of friction = 0.5 and a coefficient of restitution = 0.5.

FIG. 9 is a diagram showing a state at a moment when a rectangular parallelepiped and a sphere collide with each other.

FIG. 10 is a diagram illustrating a state after a rectangular parallelepiped and a sphere collide with each other under the conditions of a coefficient of friction = 0 and a coefficient of restitution = 1.

FIG. 11 is a diagram illustrating a state after a rectangular parallelepiped and a sphere collide with each other under the condition of friction coefficient = 1 and repulsion coefficient = 1.

FIG. 12 is a diagram illustrating a state after a rectangular parallelepiped and a ball collide with each other under the conditions of a coefficient of friction = 0 and a coefficient of restitution = 0.

FIG. 13 is a diagram illustrating a state after a cuboid and a sphere collide with each other under the condition of friction coefficient = 1 and repulsion coefficient = 0.

FIG. 14 is a diagram illustrating a state after a rectangular parallelepiped and a ball collide with each other under the conditions of a coefficient of friction = 0.5 and a coefficient of restitution = 0.5.

FIG. 15 is a diagram showing kinetic energy after balls collide with each other.

FIG. 16 is a diagram showing kinetic energy after a rectangular parallelepiped and a sphere collide.

FIG. 17 is a diagram showing a configuration of a data processing device of this embodiment.

FIG. 18 is a diagram illustrating a simulation processing flow of this embodiment.

FIG. 19 is a diagram illustrating a simulation processing flow of the present embodiment.

FIG. 20 is a diagram illustrating a simulation processing flow of the present embodiment.

FIG. 21 is a diagram illustrating a simulation processing flow of this embodiment.

FIG. 22 is a diagram illustrating a simulation processing flow according to the present embodiment.

FIG. 23 is a diagram illustrating a simulation processing flow according to the present embodiment.

[Explanation of symbols]

1 object 1 2 object 2 3 sphere 1 4 sphere 2 5 x-axis 6 y-axis 7 z-axis 8 rectangular parallelepiped 9 sphere 10 basic Cartesian coordinate axis P collision position N collision direction F center of gravity O1 and O2 center of gravity of each object R1, R2 Collision position with respect to the center of gravity of each object M1, M2 Mass of each object [I1], [I2] Matrix representing the moment of inertia of each object [S1], [S2] Matrix representing the inclination of each object L1, L2 Radius of sphere, Lx, Ly, Lz Length of each side of rectangular parallelepiped, V1, V2 Velocity of each object before collision, W1, W2 Angular velocity of each object before collision F1, F2 Impulsive force for each object generated during collision V'1, V'2 Velocity of each object after collision W'1, W'2 Angular velocity of each object after collision μ Friction coefficient ε Repulsion coefficient.

Claims (12)

[Claims] 1. A method for simulating a collision between virtual objects in a virtual three-dimensional space, wherein the state data of the virtual object after the collision is based on characteristic data of the virtual object and state data of the virtual object at the time of the collision. And a display step of displaying the state of the virtual object after the collision based on the state data of the virtual object after the collision, the collision simulation of the virtual three-dimensional object. Method. 2. The state data of the virtual object at the time of collision is
The collision simulation method for a virtual three-dimensional object according to claim 1, further comprising a motion position, an inclination, a motion velocity, a rotational motion angular velocity, or a collision direction. 3. The virtual three-dimensional object according to claim 1, wherein the calculating step further calculates a state of the virtual object after the collision based on a predetermined coefficient of restitution between the virtual objects. Collision simulation method. 4. The state of the virtual object after the collision is calculated by calculating the impulse generated by the collision based on the coefficient of restitution, in the calculating step. The virtual three-dimensional object collision simulation method described. 5. The virtual three-dimensional object according to claim 1, wherein the calculation step further calculates a state of the virtual object after collision based on a predetermined coefficient of friction between the virtual objects. Collision simulation method. 6. The method according to claim 5, wherein the calculating step further calculates a state of the virtual object after the collision by calculating an impulse generated by the collision based on the friction coefficient. The virtual three-dimensional object collision simulation method described. 7. A device for simulating a collision between virtual objects in a virtual three-dimensional space, wherein the state data of the virtual object after the collision is based on characteristic data of the virtual object and state data of the virtual object at the time of the collision. And a display unit for displaying the state of the virtual object after the collision based on the state data of the virtual object after the collision. apparatus. 8. The state data of the virtual object at the time of collision is
The collision simulation apparatus for a virtual three-dimensional object according to claim 7, further comprising a movement position, an inclination, a movement speed, a rotational movement angular velocity, or a collision direction. 9. The virtual three-dimensional object according to claim 7, wherein the calculation means further calculates a state of the virtual object after collision based on a predetermined coefficient of restitution between the virtual objects. Collision simulator. 10. The calculation means further calculates a state of the virtual object after the collision by calculating an impulse generated by the collision based on the restitution coefficient. The virtual three-dimensional object collision simulation device described. 11. The virtual three-dimensional object according to claim 7, wherein the calculation means further calculates a state of the virtual object after collision based on a predetermined friction coefficient between the virtual objects. Collision simulator. 12. The calculation means further calculates the state of the virtual object after the collision by calculating the impulse generated by the collision based on the friction coefficient. The virtual three-dimensional object collision simulation device described.