Body Dynamics

myPhysicsLab

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

  • $\dot v=\frac{1}{m}f_{cm}$
  • $\dot \omega=I^{-1}(\tau_{cm}-\omega\times I\cdot\omega)$

• Inverse Dynamics: Computing forces and torques required to generate desired motion

Physical Simulation

Particles Systems

Mass, Momentum, and Force

• Mass: $m$

• Momentum: $\vec p=m\vec v$

• Force: $\vec f=\frac{d\vec p}{dt}=m\frac{d\vec v}{dt}+\frac{dm}{dt}\vec v=m\vec a+\dot m \vec v$

  • If $m$ is constant
    • $\vec f=m\vec a$

• Acceleration: $\vec a$

• Given constant acceleration, $\vec a_0$ over time $\Delta t$

  • Velocity: $\vec v(\Delta t)=\int_0^{\Delta t}\vec adt=\vec v_0+\vec a_0\Delta t$
  • Position: $\vec x(\Delta t)=\int_0^{\Delta t}\vec vdt=\vec x_0+\vec v_0\Delta t+\frac{1}{2}a_0(\Delta t)^2$

$\ddot x=\frac{1}{m}\sum \vec f_i$

Newton's Laws

• 1st Law：$x=x_0+v_0t$
• 2nd Law：$f=ma$
• 3rd Law：$f_{AB}=-f_{BA}$

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

$f_{total}=\sum f_i$

Typical Particle Forces

Gravity

$f=mg$

$f=G\frac{m_1m_2}{r^2}$

$f_{gravity}=G\frac{m_1m_2}{\|d^2\|}\hat d$

$\hat d=\frac{d}{\|d\|}=\frac{x_1-x_2}{\|x_1-x_2\|}$

Springs

A simple spring force can be

$\vec f_{spring}=-k\vec x$

where $k$ is a spring constant

$x=\|x_1-x_2\|-l$

Dampers

$\vec f_{damp}=-k_d\vec v$

Opposite to velocity

Spring Dampers

$\vec f=-k_s\vec x-k_d\vec v$

Friction

$\vec f=\mu$

Static friction: $f\ge N\mu_s$

Dynamic friction: $f=N\mu_k$

$f=$force tangent to surface

$\mu_k=$coefficient of static friction

$\mu_k=$coefficient of dynamic (kinetic) friction

Aerodynamic Forces

Hydrodynamic

$f_{drag}=\frac{1}{2}\rho\|v\|^2c_dA\hat v$

where $\hat v=\frac{v}{\|v\|}$

$f_{lift}=\frac{1}{2}\rho\|v\|^2c_LS(\alpha-\alpha_0)$

Force Fields

$f_{field}=f(x)$

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)

$\ddot x=\frac{1}{m}f(x,\dot x)$

$\dot v=\frac{1}{m}f(x,v)$

Phase Space (State Space)

State Position (6 * 1)

Solver Interface

Diffeq Solver

Derivative

Rotational Dynamics of Particles

Angular Momentum

$L=r\times p$

Moment of Force (Torque)

扭矩

Angular Momentum

$L=r\times p=r\times mv$

Rate of change of Angular Momentum

$\vec\tau=\frac{dL}{dt}=\vec v\times m\vec v+\vec r\times m\vec a=r\times f$

Rotational Inertia

转动惯量

$L=r\times p=r\times(mv)=mr\times v=mr\times(\omega\times r)$

$L=I\cdot\omega$

Derivative of rotating vector

centripetal acceleration

• $\vec v=\dot r=\omega\times \vec r$

• $a=\dot \omega\times r+\omega\times(\omega\times r)$

$f_{centrifugal}=-m\omega\times(\omega\times r)$

derivative of rotation matrix

$R=[\begin{array}{c:c} u_x&u_y&u_z \end{array}]$

Rotational Inertia

$L=I\cdot\omega$

$I=-m\Tau\cdot\Tau$

$$\Tau=\begin{bmatrix} 0&-r_z&r_y\\ r_z&0&-r_x\\ -r_y&r_x&0 \end{bmatrix}$$

$\vec L=$

Rigid Bodies

Rigid Body Mass

Rigid Body Simulation Variables

Equation of Motion

$I$

body axis

rigid body object

Mass: $m=\sum m_i$

Center of mass: $m=\int \rho d\Omega$

force

Newton-Euler Equs

Translation: $m\vec a^0=m\dot {\vec v^0}=m\ddot{\vec x}=\sum f_i^0= f_{total}^0$

Rotation: $I\dot{\vec \omega}+\vec\omega\times I\vec \omega^0=\sum\tau_i=\sum r\times f$

I=moment of xxx (3*3 matrix)

translation: $\dot{\vec v}=\frac{1}{m}\sum f_i^0$

rotation: $\dot\omega=I^{-1}(\sum\tau-\omega\times I\omega)$

Euler Integration