Game Physics – Part I

Dan Fleck

Coming up: Rigid Body Dynamics

Rigid Body Dynamics Kinematics is the study of movement over time Dynamics is the study of force and masses

interacting to cause movement over time (aka kinematic changes).

Example: How far a ball travels in 10 seconds at 50mph is

kinematics How far the same ball travels when hit by a bat and under

the force of gravity is dynamics

Additionally for simplification we’re going to model rigid bodies – ones that do not deform (not squishy)

We can model articulated rigid bodies – multiple limbs connected with a joint

Coming up: Bring on calculus

Bring on calculus Calculus was invented by Newton (and

Leibniz) to handle these problems

Newton’s Laws 1. An object at rest stays at rest and an object in

uniform motion stays in the same motions unless acted upon by outside forces (conservation of inertia)

2. Force = Mass * Acceleration 3. For every action there is an equal and opposite

reaction

Coming up: F=ma

F=ma r=Position, v=Velocity, a=acceleration

Velocity is equal to the change in position over time.

Acceleration is equal to the change in velocity over time.

Coming up: Intuitive Understanding

Intuitive Understanding If every second my position changes by 5m,

what is my velocity?

Acceleration is the change in velocity over time. If I am traveling at 5m/s at time t=1, and 6m/s at t=2, my acceleration is 1m/s^s

Coming up: Integration

Integration Integration takes you backwards

Integrating acceleration over time gives you velocity

Intuition: If you are acceleration at 5m/s^s, then every second you

increase velocity by 5. Integrating ‘sums’ up these changes, so your velocity is

What is “C”? At time t=0, what is velocity? C… so C is initial velocity

So, if you are accelerating at 5m/s^s, starting at 7m/s what is your velocity at time t=3 seconds?

Coming up: Integration

Integration Similarly, integrating velocity over time gives you

position Example: If you’re accelerating at a constant 5m/s^s, then:

So, given you have traveled for 5 seconds starting from point 0, where are you?

Plug in the values:

So, given initial position, initial velocity, and acceleration you can find the new position, velocity.

We will do this every frame, using values from the previous frames. €

r(t) =5

252 + 0t + 0 = 62.5

Coming up: Forces

Forces But wait… how do we find the acceleration to begin

with?

Linear momentum is denoted as p which is:

To change momentum, we need a force.Newton says:

So, given a force on a point mass, we can find the acceleration and then we can find position, velocity… whew, we’re done… but…..

€

p = mv

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p = mv

Coming up: Finding Momentum

Finding Momentum On a rigid body, we have mass spread over an

area We compute momentum by treating each

point on the object discretely and summing them up:

Lets try to simplify this by introducing the center of mass (CM). Define CM as (where M is the total mass of the body):

Coming up: Center of Mass

Center of Mass Using this equation, multiply both

sides by M and take the derivative

Aha.. .now we have total momentum on the right, but what is on the left?

Because M is a constant it comes out of the derivative and then we have change in position over time of the center of mass… or velocity of CM!

Coming up: Acceleration of CM

Acceleration of CM

Total linear momentum can be found just using the velocity of the CM (no summation needed!)

So, finally the acceleration of the entire body can be calculated by assuming the forces are all acting on the CM and computing the acceleration of CM

Coming up: Partial Summary

Partial Summary We now know, that given an object’s acceleration we

can compute it’s velocity and position by integrating:

And to determine acceleration, we can sum forces acting on the center of mass (CM) and divide by total mass

Current challenge: Integrating symbolically the find v(t) and t(t) is very hard! Remember differential equations?

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acm =F T

M

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acm =F T

M

Coming up: Differential Equations

Differential Equations These equations occur when the dependent variable

and it’s derivative both appear in the equation. Intuitively this occurs frequently because it means the rate of change of a value depends on the value.

Example: air friction.. the faster you are going, the more force it applies to slow you down:

f = -v = ma (solve for a) but a is the derivative of v, so

Solving this analytically is best left to you and your differential equations professor

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a =−v

m

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dv

dt=

−v

m

Coming up: Numerical Integration of Ordinary Differential Equations

(ODEs)

Numerical Integration of Ordinary Differential Equations (ODEs) Analytically solving these is hard, but solving

them numerically is much simpler. Many methods exist, but we’ll use Euler’s method.

Integration is simply summing the area under the curve, and the derivative is the slope of the curve at any point. Euler says:

t=3 t=5

Integrating from t=3 to

5 is summing the y values

for that section.

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yn +1 = yn + hdyn

dt

Coming up: Euler’s Approximation

Euler’s Approximation

Euler numerical integration is an

approximation (src: Wikipedia)

Numerically integrating velocity and position we

get these equations:

Coming up: Final Summary of Equations

Final Summary of Equations Sum up the forces acting on the body at the

center of mass to get current acceleration

To get new velocity and position, use your current acceleration, velocity, position and numerical integration over some small time step (h)

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acm =F T

M

Coming up: Now we can code!

Now we can code!ForceRegistry: stores which forces are being applied to

which objectsForceGenerator: virtual (abstract) class that all Forces

implement

Mainloopfor each entry in Registry

add force to accumulator in object for each object compute acceleration using resulting total force

compute new velocity using accelerationcompute new position using velocity

reset force accumulator to zero

Coming up: ForceRegistry

ForceRegistry√

Coming up: ForceGenerator

ForceGenerator

Coming up: ImpulseForceGenerator

ImpulseForceGenerator

Coming up: DragForce generator

Warning: This code is actually changing the

acceleration, it should just update the forces and the

acceleration should be computed at the end of all

forces

DragForce generator In order to slow an object down, a drag force can

be applied that works in the opposite direction of velocity.

typically a simplified drag equation used in games is:

k1 and k2 are constants specifying the drag force, and the direction is in the opposite direction of velocity.

Coming up: DragForce Generator

€

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fdrag = −ˆ v (k1 v + k2 v2)

DragForce Generator

Coming up: Mainloop – Updating Physics Quantities

Add force to current forces upon the player

Mainloop – Updating Physics Quantities

Coming up: What’s next?

After the forces have been updated, you must then apply

the forces to create acceleration and update velocity and

position.

Inside mainloop

What’s next? Other forces

Spring forces – push and pull Bungee forces – pull only Anchored springs/bungees

Rotational forces forces instead of moving the force can also induce

rotations on the object

Collisions Conversion from 2D to 3D

Coming up: References

References These slides are mainly based on Chris

Hecker’s articles in Game Developer’s Magazine (1997). The specific PDFs (part 1-4) are available at:

http://chrishecker.com/Rigid_Body_Dynamics

Additional references from: http://en.wikipedia.org/wiki/Euler_method Graham Morgan’s slides (unpublished)

End of presentation