Body Dynamics
Body Dynamics
Kinematics: The study of motion without consideration of underlying forces
- Forward Kinematics: Computing body motion as a function of joint angles
- Inverse Kinematics: Computing joint angles as a function of body motion
Dynamics: Study of physical motion due to the application of forces and torques
Forward Dynamics: Computing motion resulting from applied forces and torques
Inverse Dynamics: Computing forces and torques required to generate desired motion
Physical Simulation
Particles Systems
Mass, Momentum, and Force
Mass:
Momentum:
Force:
- If is constant
- If is constant
Acceleration:
Given constant acceleration, over time
- Velocity:
- Position:
Newton's Laws
- 1st LawοΌ
- 2nd LawοΌ
- 3rd LawοΌ
Conservation of Momentum
Total momentum in a closed system will remain constant
When particles interact, any gain of momentum by one particle must be met by an equal and opposite loss of momentum by another particle.
Force a particle
Typical Particle Forces
Gravity
Springs
A simple spring force can be
where is a spring constant
Dampers
Opposite to velocity
Spring Dampers
Friction
Static friction:
Dynamic friction:
force tangent to surface
coefficient of static friction
coefficient of dynamic (kinetic) friction
Aerodynamic Forces
Hydrodynamic
where
Force Fields
acceleration
velocity
0
Particle System
Simulation
Particles can be rendered using various techniques
- Points
- Lines (form last position to current position)
- Sprites (textured quad's facing the camera)
- Geometry (small objects...)
- Or other approaches...
Dynamics
State Space
2nd order Ordinary Differential Equation (ODE)
Phase Space (State Space)
State Position (6 * 1)
Solver Interface
Diffeq Solver
Derivative
Rotational Dynamics of Particles
Angular Momentum
Moment of Force (Torque)
ζη©
Angular Momentum
Rate of change of Angular Momentum
Rotational Inertia
转ε¨ζ―ι
Derivative of rotating vector
centripetal acceleration
derivative of rotation matrix
Rotational Inertia
Rigid Bodies
Rigid Body Mass
Rigid Body Simulation Variables
Equation of Motion
body axis
rigid body object
Mass:
Center of mass:
force
Newton-Euler Equs
Translation:
Rotation:
I=moment of xxx (3*3 matrix)
translation:
rotation:
Euler Integration