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Computer Graphics and Imaging UC Berkeley CS184 Summer 2020
Simulation and Kinematics
Agenda
• Simulation
- Particle Systems
- Mass-spring system
- Euler’s Method
- Verlet Integration
- Forward and Inverse kinematics
• Animation
Particle SystemMass-spring
Euler’s method
Verlet Integration
Particle System — ODE
• Many dynamic systems can be described via an ordinary differential equation (ODE)
- “ordinary” means it involves derivatives in time but not in space
• Goal: Find a function based on information about or relationships between its derivatives
F = ma·x = v·v = − kx
Mass-spring System
Mass-spring System
E =12
k(L − L0)2
• Simple and common particle connection
• Hooke’s Law
• Newton’s laws of motion
Rest lengthCurrent length
Stiffness (Spring constant)
Potential Energy
F = k(L − L0) = maForce
Mass-spring SystemF = k(L − L0) = ma
Mass-spring SystemF = k(L − L0) = ma
In a cloth simulation (spring connecting particles):
• What happens if I make the spring more ‘stiff’
• What happens if I have a lighter cloth
Fabric under wind
Numerical Integration
• Key idea: replace derivatives with differences
- ODE only considers derivative in time
ddt
q(t) = f(q(t))
qk+1 − qk
τ= f(q)
Current, known
Unit timestamp, (update frequency)
Future, unknown
Euler’s Method — Newton’s laws
• Approximate the velocity as constant between timeframes
- Acceleration from the net force:
- Velocity from the acceleration:
- Position from the velocity:
a =Fm
v(t + Δt) = v(t) + a(t)Δt
x(t + Δt) = x(t) + v(t)Δt
Above is an example for forward Euler, different methods use different time stamp for evaluation. But the principle of
acceleration —> velocity —> position follows
Euler’s MethodForward Euler
xt+Δt = xt + Δt ·x(t)
• Simple but unstable
- Accumulating errors
- May diverge
Euler’s MethodBackward Euler == Implicit Euler
• Implicit — based on information from the future!
• Hard to solve but stable
- Use given future force to solve for
- What if we can’t anticipate the forces?
·x(t + Δt)
xt+Δt = xt + Δt ·x(t + Δt)
Verlet Integration
• Derived from Taylor expansion
• Velocity-less representation
• Instead store current and previous positions
Position-based Integration
x(t + Δt) = x(t) + ·x(t)Δt +12
··x(t)Δt2
v(t)
=
a(t)
=
x(t − Δt) = x(t)− ·x(t)Δt +12
··x(t)Δt2
How we cancel out velocity
Verlet IntegrationPosition-based Integration
x(t + Δt) = x(t) + ·x(t)Δt +12
··x(t)Δt2
x(t − Δt) = x(t)− ·x(t)Δt +12
··x(t)Δt2
x(t + Δt) = 2x(t) − x(t − Δt) + ··x(t)Δt2
+
=
Known, Previous position
Unknown, future position
Known, current position
Stability• Damping
-
• Modified Euler
- Average velocities
• Adaptive step size
- Recursively adjusting step size, avoid diverging
• Implicit Euler, Position-based Verlet Integration, …
x(t + Δt) = 2x(t)−kx(t − Δt) + ··x(t)Δt2
Project 4 (The last one!)
• Verlet integration to update positions of each particle
Animation
Kinematics• Motion of points, bodies, and systems of bodies
without considering the forces that cause them to move
- Referred to as the "geometry of motion"
Forward Kinematics
• Forward and backward
- Recall forward and backward Euler
Unknown
Known
Known
Backward Kinematics
Known
Unknown
Unknown
• Optimization problem (no constraints!)
- May have 0, 1 or many solutions
Motion appears mechanical
Motion Capture
• Physically-based and data-driven animation
Motion Capture — Triangulation
3D Point
dr
dl
Baseline
Graphics in Films
Pirates of the CaribbeanDawn of the Planet of the Apes
Alita Battle AngelAvatar
Physical Simulation and Animation
• Physical simulation is realistic because the motion follows the laws of physics, but may appear mechanical and unnatural.
• Animation from motion capture preserves realistic human motion but makes it hard to do on-line interaction.
• Researchers combine physical modeling (simulation following biomechanics studies and robot control theory) and motion capture data (enrich the styles of motion).
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