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Advanced Computer GraphicsRigid Body Simulation
Spring 2002
Professor Brogan
Upcoming Assignments
• Who wants a midterm instead of an assignment?
• Final will be take home
• Cloth/water/parallel particle sim presentations– Volunteers?– Papers selected by Thursday
Physical Simulation
• References– Text book (4.3 and Appendix B)– Physics for Game Developers (Bourg)– Chris Hecker Game Developer articles
• http://www.d6.com/users/checker/dynamics.htm
Equations of Motion
• The physics-based equations that define how objects move– Gravity– Turbulence– Contact forces with objects– Friction– Joint constraints
Equations of Motion
Current State
Position and velocity
Accelerations
Forces
Velocities
Equations ofMotion
Integrate
Integrate
Integrate
Equations of Motion
• Linear motions:
• Example:– Constant
acceleration of 5 m/s2
dttatv
dttvtr
)()(
)()(
002
0
0
0
2
55)()(
5)(
)0(5)0(
55)(
rtvtdtvtdttvtr
vttv
Cv
Cv
Ctdtdtatv
Linear Momentum
• Mass times velocity = linear momentum, p
• Newton’s Second Law
amvmdt
vmd
dt
pdpF
)(
vmp
Ceasing to identify vectors…
Rigid Bodies
• Imagine a rigid body as a set of point masses• Total momentum, pT, is sum of momentums of
all points:
• Center of Mass (CM) is asingle point. Vector to CMis linear combination ofvectors to all points in rigidbody weighted by their masses,divided by total mass of body
i
iiT vmp
M
rmr i
ii
CM
Total Momentum
• Rewrite total momentum in terms of CM
• Total linear momentum equals total mass times the velocity of the center of mass
(For continuous rigid bodies, all summations turn into integrals over the body, but CM still exists)
• We can treat all bodies as single point mass and velocity
dt
Mrd
dt
rmdvmp
CM
i
ii
i
iiT )()(
Total Force
• Total force is derivative of the total momemtum– Again, CM simplifies total force equation of a rigid
body
– We can represent all forces acting on a body as if their vector sum were acting on a point at the center of mass with the mass of the entire body
CMCMTT MavMpF
Intermediate Results
• Divide a force by M to find acceleration of the center of mass
• Integrate acceleration over time to get the velocity and position of body
• Note we’ve ignored where the forces are applied to the body
• In linear momentum, we don’t keep track of the angular terms and all forces are applied to the CM.
Ordinary Differential Equations
• A DifEq is an equation with– Derivatives of the dependent variable– Dependent variable– Independent variable
• Ex:
• v’s derivative is a function of its current value• Ordinary refers to ordinary derivatives
– As opposed to partial derivatives
m
v
dt
dva
Integrating ODEs
• Analytically solving ODEs is complicated
• Numerically integrating ODEs is much easier (in general)– Euler’s Method– Runge-Kutta
Euler’s Method
• Based on calculus definition of first derivative = slope
Euler’s Method
• Use derivative at time n to integrate h units forward
dx
dyhyy n
nn 1
Euler Errors
• Depending on ‘time step’ h, errors will accumulate
SIGGRAPH Course Notes
Accumulating Errors
SIGGRAPH Course Notes
Recap
Angular Effects
• Let’s remain in 2-D plane for now• In addition to kinematic variables
– x, y positions
• Add another kinematic variable – angle– CCW rotation of object
axes relative to world axes
Angular Velocity
• is the angular velocity
• is the angular acceleration
dt
d
dt
d2
2
Computing Velocities
• How do we combine linear and angular quantities?
• Consider velocity of a point, B, of a rigid body rotating about its CM
• r_perp is perpendicular to r vector from O to B• Velocity is -scaled perpendicular vector from
origin to point on body
OBB rv
More Details
• Point B travels radians• Point B travels C units• Radius of circle is r• C= r
– By definition of radians,where circumference = 2r
• B’s speed (magnitude of velocity vector)– Differentiate C= r w.r.t. time
rrdt
d
dt
rd
)(
More Details
• The direction of velocity is tangent to circle == perpendicular to radius
• Therefore, linearvelocity is angularvelocity multiplied by tangent vector
Chasles’ Theorem
• Any movement is decomposed into:– Movement of a single point on body– Rotation of body about that point
• Linear and Angular Components
OBOB rvv
Angular Momentum
• The angular momentum of point B about point A (we always measure angular terms about some point)
• It’s a measure of how much of point B’s linear momentum is rotating around A
BABAB prL
Angular Momentum
• Check: If linear momentum, pB, is perpendicular to r_perp, then dot product of two will cause angular momentum will be zero
Torque
• Derivative of Angular Momentum– Remember force was derivative of linear
momentum
– This measures how much of a force applied at point B is used to rotate about point A, the torque
BBBAB
BABAB
Frmar
dt
prd
dt
dL
A
)(
Total Angular Momentum
• Total angular momentum about point A is denoted LAT
• But computation can be expensive to sample all points
• Sampling of a surface would require surface integration
i
iiAi
i
iAiAT
vmr
prL
Moment of Inertia
• Remember– An alternate way of representing the
velocity of a point in terms of angular velocity
• If A is like the origini is like B, then substitute
OBB rv
i
iiAi
i
iAiAT
vmr
prL
A
i
Aii
i
AiAii
i
AiiAiAT
I
rm
rrm
rmrL
2)(
Moment of Inertia, IA
• The sum of squared distances from point A to each other point in the body, and each squared distance is scaled by the mass of each point
• This term characterizes how hard it is to rotate something– Ipencil_center will be much less than Ipencil_tip
A
i
Aii Irm 2)(
Total Torque
• Differentiate totalangular momentumto get total torque
• This relates total torque and the body’s angular acceleration through the scalar moment of inertia
AA
AAT
AT
II
Iddt
dL
)(
Planar Dynamics
• Calculate COM and MOI of rigid body• Set initial position and linear/angular velocities• Figure out all forces and their points of application• Sum all forces and divide by mass to find COM’s linear
acceleration• For each force, compute perp-dot-product from COM to point of
force application and add value into total torque of COM• Divide total torque by the MOI at the COM to find angular
acceleration• Numerically integrate linear/angular accelerations to update the
position/orientation and linear/angular velocities• Draw body in new position and repeat
Upcoming Topics
• Collisions
• 3-dimensional rigid bodies (inertia tensors)
• Forces (centripetal, centrifugal, viscosity, friction, contact)
• Constrained dynamics (linked bodies)