Integration for physically based animation

CSE 3541Matt Boggus

What is motion?

Observe an object’s position over timeWe could say y = f(time)

Equations of motion – terms

• Position (x,y,z)– Point with respect to the origin

• Velocity (x,y,z)– Speed (vector magnitude)– Direction

• Acceleration (x,y,z)– Rate of change of velocity– Magnitude and direction

r

Equations of motion graph examples

• Initial conditions:– p = 0, v = 5

• If we have the function for acceleration, we can integrate it and use initial conditions to solve for the velocity and position functions

acceleration

velocity

position

Euler integrationFor arbitrary function f(t) with known derivative

)( itf)( itf

ttftftf iii )()()( 1

Step in the direction of the derivative

)( itf

Integration – derivative field

For arbitrary function, f(t)

The force acting on a point may vary in space

SamplingA fixed amount of time passes between frames.Approximate the continuous position curve with discrete samples.

Integration and step size

2.0x

5x

Inaccuracy and instability

Runge Kutta Integration: 2nd orderaka Midpoint Method

• Compute a “full” Euler step

• Evaluate f at midpoint

• Take a step from the original point using the midpoint f value

Runge Kutta Integration: 2nd orderaka Midpoint Method

For unknown function, f(t); known f ’(t)

)( itf

)( itf ttftftfi

ii

)()()(2

11

)(2

1

itf

ttftftf iii

)(2

1)()(

2

1

Step size 2t

Euler Integration

Midpoint Method

Integration comparison

Image from http://www.physics.drexel.edu/students/courses/Comp_Phys/Integrators/simple.html