game physics – part i dan fleck coming up: rigid body dynamics
TRANSCRIPT
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
€
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?
€
acm =F T
M
€
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
€
a =−v
m
€
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.
€
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)
€
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
€
€
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