Rigid Body Motion

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Rigid Body Motion
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Game Physics

• Linear physics physics of points
• particle systems, ballistic motion
• key simplification no orientation
• Rotational physics
• orientation can change

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Rigid Bodies

• No longer points distribution of mass instead.
• Rigid bodies distances between mass elements
  never change.
• Orientation of body can change over time.

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Rigid Body Translation

• Can treat translational motion of rigid bodies
  exactly the same as points
• Single position (position of center of mass)
• Fma (external forces)
• v ?a dt
• x ?v dt
• momentum conservation

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Rotation

• Rigid bodies also have orientation
• Treating rotation properly is complicated
• Rotation is not a vector (rotations do not
  commute, i.e., order of rotations matters)
• No analog to x, v, a in rotations?

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Angular velocity

• Infinitesimally small rotations do commute
• Suppose we have a rigid body rotating about an
  axis
• Can use a notion of angular velocity
• ? d?/dt

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Angular velocity

• Connection between linear and angular velocity
• Magnitudes v ?rperp
• Want vector relation
• Nice to have angular velocity about axis of
  rotation (so it doesn't have to change all the
  time for an object spinning in place)
• Let v ? x r

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Angular velocity

• v ? x r
• Or, ? r x v / r2
• Note ?, r, v vectors
• Angular velocity defined this way so that
  constant angular velocity behaves sensibly
• spinning top has constant ?

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Applying force

• What happens when you push on a spinning object?
  (exert force)
• Fma, so we know the movement of the centre of
  mass
• How does the force affect orientation?

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Torque

• T r x F
• r is vector from origin to location where force
  applied
• for convenience, often take origin to be center
  of mass of object
• F is force
• Magnitude proportional to force, proportional to
  distance from origin

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Intuition for Torque

• Larger the larger from the centre
• Lever action small force yields equivalent
  torque far from fulcrum

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Direction of Torque

• T r x F
• Perpendicular to both location and force vectors
• Direction is along axis about which rotation is
  induced
• Right hand rule thumb along axis, fingers curl
  in direction of rotation

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single particle

• T r F sin?
• T r Ft
• Ft mat mra
• T mr2a
• Let I mr2
• T Ia

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Many particles

• Real objects are (pretty much) continuous
• Game objects distribution of point masses
• not always, but common
• Can get reasonable behaviour with (e.g.) four
  point masses per rigid body
• Single orientation for body
• Single centre of mass (of course)

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Changing Coordinate Systems

• We dealt with changing coordinate systems all the
  time before
• Rigid bodies are much simpler if we treat them in
  a natural coordinate system
• origin at the centre of mass of the body
• or, some other sensible origin hinge of door
• Need to transform forces into body coordinate
  system to calculate torque
• Transform motion back to world space

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Angular momentum

• Define angular momentum similarly to torque
• L r x p
• Note that with this definition, T dL/dt, just
  as F dp/dt

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Force and Torque

• Note a force is a force and a torque
• Moves body linearly Fma, changes linear
  momentum
• Rotates body produces torque, changes angular
  momentum

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Linear vs. Angular
linear quantity angular quantity
velocity v angular velocity ?
acceleration a angular acc. a
mass m moment of inertia I
p mv L I?
F ma T Ia
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Conservation of Angular Momentum

• Consequence of T dL/dt
• If net torque is zero, angular momentum is
  unchanged
• Responsible for gyroscopes' unintuitive behaviour

The gyroscope is tipped over but it doesnt fall
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Moment of Inertia

• Said that moment of inertia of a point particle
  is mr2
• In the general case, I ? ? r2 dV where r is
  the distance perpendicular to the axis of
  rotation
• Don't know the axis of rotation beforehand

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Moment of Inertia

• I ??(x,y,z)
  dxdydz

y2 z2 -xy -xz
-xy x2 z2 -yz
-xz -yz x2y2
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Diagonalized Moment of Inertia

• Luckily, we can choose axes (principal axes of
  the body) so that the matrix simplifies
• I
• where, e.g., Ixx m(yy zz)
• Off-diagonal entries called "products of inertia"

Ixx 0 0
0 Iyy 0
0 0 Izz
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Avoiding products of inertia

• Do calculations in inertial reference frame whose
  axes line up with the principal axes of your
  object
• Transform the results into worldspace
• Moment of inertia of a body fixed, so can be
  precomputed and used at run-time

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Moment of Inertia

• In general, the more compact a body is, the
  smaller the moments of inertia, and the faster it
  will spin (for the same torque)

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Fake I

• Not doing engineering simulation (prediction of
  how real objects will behave)
• Can invent I rather than integrating
• Large values hard to rotate about this axis
• Avoid off-diagonal elements

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Fake constants

• For that matter, can fake lots of stuff
• Different gravity for different objects
• e.g., slow bullets in FPS
• e.g., fast falling in platformer
• fake forces, approximate bounding geometry

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Case in 2D

• In 2D, the vectors T, ?, a become scalars (their
  direction is known only magnitude is needed)
• Moment of inertia becomes a scalar too
• I ?prdA

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Single planar rigid body

• state contains x, y, ?, vx, vy, ?
• Have
• F ma (2 equations)
• T I?
• x ?vx dt
• y ?vy dt
• ? ?? dt
• Integrate to obtain new state, and proceed

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Rigid body in 3D

• Need some way to represent general orientation
• Need to be able to compose changes in orientation
  efficiently

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Quaternions

• Quaternion structure for representing rotation
• unit vector (axis of rotation)
• scalar (amount of rotation)
• recall, store (cos(?/2), v sin(?/2) )
• Can represent orientation as quaternion, by
  interpreting as rotation from canonical position

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Quaternions

• Rotation of ? about axis v
• q (cos(?/2), v sin(?/2))
• "Unit quaternion" q.q 1 (if v is a unit
  vector)
• Maintain unit quaternion by normalizing v
• Arbitrary vector r can be written in quaternion
  form as (0, r)

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Quaternion Rotation

• To rotate a vector r by ? about axis v
• take q (cos(?/2), v sin(?/2)
• Let p (0,r)
• obtain p' from the quaternion resulting from
  qpq-1
• p' (0, r')
• r' is the rotated vector r

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Rotation Differentiation

• Note
• q(t) (s(t), v(t))
• q(t) cos(?(t)/2), u sin(?(t)/2)
• For a body rotating with constant angular
  velocity ?, it can be shown
• q(t) 0, ½ ? q(t)
• Summarize this ½ ? q(t)

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Rigid Body Equations of Motion
Using quaternions gives
x(t)
q(t)
P(t)
L(t)
v(t)
½ ?q(t)
F(t)
T(t)

d/dt
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P and L

• Note that
• v P/m (from Pmv)
• ? I-1L (from L I?)
• Often useful to use momentum variables as main
  variables, and only compute v and ? (auxiliary
  variables) as needed for the integration

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Impulse

• Sudden change in momentum
• also, angular momentum (impulsive torque)
• Collision resolution using impulse
• new angular momentum according to conditions of
  collision
• algorithmic means available for resolving