Image display apparatus, method and program based on rigid body dynamics

A computer readable medium storing a plurality of program code for controlling a data processor of a computer system, that when executed, cause the data processor to perform one or more tasks for simulating the motion of objects and displaying the results on a display screen, the computer system including a memory having memory locations for position parameters and for velocity parameters, the position and velocity parameters defining an initial state of a model system having a plurality of bodies, the plurality program code comprising:

• program code for storing, in the memory, constraint parameters defining at least one constraint function constraining the motion of the bodies in the model system, wherein the constraint parameters include at least one position constraint constraining the motion of the plurality of rigid bodies;
• program code for calculating the position and velocity parameters defining the state of the system after a predetermined time step based on rigid body dynamics, including:
• a) program code for performing a semi-implicit integration step subject to the constraints constraining the motion of the simulated objects to determine the velocity after the step;
• and
• b) program code for determining constraint forces that act to keep the system in compliance with the constraints by ensuring that the first derivative of the constraint function is zero;
• and
• program code for generating, at the end of the predetermined time step, a screen image of a series of screen images for display on the display screen, the screen image displaying a view of the simulated objects such that the viewer can perceive the motion of the simulated objects in compliance with the physics model, the screen image being based on the calculated position and velocity parameters defining the state of the simulated objects at the end of the predetermined time step;
• and
• program code for displaying the screen image on the display screen,
• wherein the model system includes a model of friction in which the frictional force between a pair of objects is independent of a normal force between the objects for both elastic and inelastic collisions.

Apparatus for simulating the motion of objects and displaying the results on a display screen comprising:

• a display screen,
• a memory,
• a processing unit,
• and a computer program stored in the memory for causing the apparatus to carry out the steps of:
• a) storing in the memory position and velocity parameters defining an initial state of a model system having a plurality of bodies,
• b) storing in the memory parameters defining at least one constraint function constraining the motion of the bodies in the model system, wherein the at least one constraint function defines at least one position constraint constraining the motion of the plurality of rigid bodies,
• c) storing in the memory parameters defining a bounded constraint force to simulate the effects of friction in which the constraint force acts in the plane of contact between a pair of objects to prevent sliding of one of the pair of objects over the other of the pair, wherein the constraint force is bounded to be not greater than a predetermined constant value to allow sliding of the objects over one another and thus include dynamic friction in the simulation, and wherein the bounded constraint force is independent of a normal force between the objects for both elastic and inelastic collisions;
• d) calculating in the processor the position and velocity parameters defining the state of the system after a predetermined time step based on rigid body dynamics, including
• i) carrying out a semi-implicit integration step subject to the constraints, to determine the velocity after the step;
• and
• ii) determining the constraint forces that act to keep the system in compliance with the constraints by ensuring that the first derivative of the constraint function is zero;
• and
• e) generating, at the end of the predetermined time step, a screen image of a series of screen images for display on the display screen, the screen image displaying a view of the simulated objects such that the viewer can perceive the motion of the simulated objects in compliance with the physics model, the screen image being based on the calculated position and velocity parameters defining the state of the simulated objects at the end of the predetermined time step, and
• f) displaying, on the display screen, the screen image.

A computer-implemented method, using a computer, for simulating movement of a plurality of rigid bodies according to a physics model by calculating motion of the plurality of the rigid bodies the method comprising:

• storing, into storage accessible to the computer, position parameters defining initial positions of a plurality of rigid bodies;
• defining at least one position constraint constraining the motion of the plurality of rigid bodies, the at least one position constraint defining constraints on positions of the plurality of rigid bodies represented by the position parameters read from the storage;
• providing the at least one position constraint to the computer;
• determining a bounded constraint force, using the computer, wherein the bounded constraint force represents a frictional force between a pair of objects of the plurality of rigid bodies, the frictional force acting in a plane of contact between the pair of objects and wherein at least one reaction force is bounded to be not greater than a predetermined value relative to the frictional force, and wherein the bounded constraint force is independent of a normal force between the objects for both elastic and inelastic collisions;
• calculating, using the computer, new position parameters for the plurality of rigid bodies using the computer, the new position parameters defining new positions of the plurality of rigid bodies after a predetermined time step, wherein the new position parameters depend on at least one reaction force that acts to keep the plurality of rigid bodies in compliance with the at least one position constraint, such that the new position parameters satisfy the at least one position constraint directly without an iterative error process;
• and
• generating, using the processor, at the end of the predetermined time step, a screen image of a series of screen images for display on a display device, the screen image displaying a view of the simulated objects such that the viewer can perceive the motion of the simulated objects in compliance with the physics model, the screen image being based on the calculated position and velocity parameters defining the state of the simulated objects at the end of the predetermined time step;
• and
• displaying the screen image on the display device.

A computer system comprising a processor and instruction memory, the instruction memory comprising instructions that, when executed, calculate simulated movement of a plurality of rigid bodies according to a physics model from stored representations of the plurality of rigid bodies, the computer system comprising:

• a data memory configured to store position parameters defining initial positions of the plurality of rigid bodies;
• a constraint definer that, when executed, defines at least one position constraint constraining motion of the plurality of rigid bodies, the at least one position constraint defining constraints on positions of the plurality of rigid bodies represented by the stored position parameters read from the storage;
• a friction modeler operable to determine a bounded constraint force, wherein the bounded constraint force represents a frictional force between a pair of objects of the plurality of rigid bodies, the frictional force acting in a plane of contact between the pair of objects, wherein at least one reaction force is bounded to be not greater than a predetermined value relative to the frictional force, and wherein the bounded constraint force is independent of a normal force between the objects for both elastic and inelastic collisions;
• and
• a position parameter calculator that, when executed, calculates new position parameters for the plurality of rigid bodies defining new positions of the plurality of rigid bodies after a predetermined time step, wherein the new position parameters depend on at least one reaction force that acts to keep the plurality of rigid bodies in compliance with the at least one position constraint, such that the new position parameters satisfy the at least one position constraint directly without an iterative error process;
• a image generator that when executed at the end of the predetermined time step, generates a screen image of a series of screen images for display on a display device, the screen image displaying a view of the simulated objects such that the viewer can perceive the motion of the simulated objects in compliance with the physics model, the screen image being based at least in part on the position parameters for the plurality of rigid bodies at the end of the predetermined time step;
• wherein the processor is configured, when executed, to perform processing operations that implement the constraint definer and the position parameter calculator in accordance with computer executable instructions stored in the instruction memory.

Processing apparatus for calculating simulated movement of a plurality of rigid bodies according to a physics model, the processing apparatus comprising:

• means for storing position parameters defining initial positions of a plurality of rigid bodies;
• means for defining at least one position constraint constraining the motion of the plurality of rigid bodies, the at least one position constraint defining constraints on positions of the plurality of rigid bodies represented by the position parameters read from the means for storing;
• means for determining a bounded constraint force, wherein the bounded constraint force represents a frictional force between a pair of objects of the plurality of rigid bodies, the frictional force acting in a plane of contact between the pair of objects, wherein at least one reaction force is bounded to be not greater than a predetermined value relative to the frictional force, and wherein the bounded constraint force is independent of a normal force between the objects for both elastic and inelastic collisions;
• means for calculating new position parameters for the plurality of rigid bodies defining new positions of the plurality of rigid bodies after a predetermined time step, wherein the new position parameters depend on at least one reaction force that acts to keep the plurality of rigid bodies in compliance with the at least one position constraint, such that the new position parameters satisfy the at least one position constraint directly without an iterative error process;
• and
• means for generating, at the end of the predetermined time step, a screen image of a series of screen images for display on a display device, the screen image displaying a view of the simulated objects such that the viewer can perceive the motion of the simulated objects in compliance with the physics model, the screen image being based on the calculated position and velocity parameters defining the state of the simulated objects at the end of the predetermined time step;
• and
• means for displaying the screen image on the display device.

A data carrier tangibly embodying a program of machine readable instructions executable by a digital processing apparatus having stored thereon instructions that, when executed, calculate simulated movement of a plurality of rigid bodies according to a physics model, the program comprising:

• program code for storing, into storage accessible to the computer, position parameters defining initial positions of a plurality of rigid bodies;
• program code for defining at least one position constraint constraining the motion of the plurality of rigid bodies, the at least one position constraint defining constraints on positions of the plurality of rigid bodies represented by the position parameters read from the storage;
• program code for determining a bounded constraint force, wherein the bounded constraint force represents a frictional force between a pair of objects of the plurality of rigid bodies, the frictional force acting in a plane of contact between the pair of objects, wherein at least one reaction force is bounded to be not greater than a predetermined value relative to the frictional force, and wherein the bounded constraint force between the pair of objects is independent of a normal force between the objects for both elastic and inelastic collisions;
• program code for calculating new position parameters for the plurality of rigid bodies defining new positions of the plurality of rigid bodies after a predetermined time step, wherein the new position parameters depend on at least one reaction force that acts to keep the plurality of rigid bodies in compliance with the at least one position constraint, such that the new position parameters satisfy the at least one position constraint directly without an iterative error process;
• and
• program code for generating, at the end of the predetermined time step, a screen image of a series of screen images for display on a display device, the screen image displaying a view of the simulated objects such that the viewer can perceive the motion of the simulated objects in compliance with the physics model, the screen image being based on the calculated position and velocity parameters defining the state of the simulated objects at the end of the predetermined time step;
• and
• program code for displaying the screen image on the display device.