Rigid Body Motion
Game Physics
• “Linear physics”– physics of points
– particle systems, ballistic motion…
– key simplification: no orientation
• “Rotational physics”
– orientation can change
Rigid Bodies
No longer points: distribution of mass
instead.
Rigid bodies: distances between mass
elements never change.
Orientation of body can change over time.
Rigid Body Translation
• Can treat translational motion of rigid
bodies exactly the same as points
• Single position (position of center of mass)
• F=ma (external forces)
• v = ∫a dt
• x = ∫v dt
• momentum conservation
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?
Angular velocity
• Infinitesimally small rotations do commute
• Suppose we have a rigid body rotating
about an axis
• Can use a notion of angular velocity:
• ω = dθ/dt
Angular velocity
• Connection between linear and angular
velocity
• Magnitudes: v = ωrperp
• 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 = ω x r
Angular velocity
• v=ωxr
• Or, ω = r x v / |r|2
• Note: ω, r, v vectors
• Angular velocity defined this way so that
constant angular velocity behaves sensibly
– spinning top has constant ω
Applying force
• What happens when you push on a
spinning object? (exert force)
• F=ma, so we know the movement of the
centre of mass
• How does the force affect orientation?
Torque
• T=rxF
• 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
Intuition for Torque
• Larger the larger from the centre
• Lever action: small force yields equivalent
torque far from fulcrum
Direction of Torque
• T=rxF
• 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
single particle
•
•
•
•
•
•
T = r F sinθ
T = r Ft
Ft = mat = mrα
T = mr2α
Let I = mr2
T = Iα
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)
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
Angular momentum
• Define angular momentum similarly to
torque:
• L=rxp
• Note that with this definition, T = dL/dt, just
as F = dp/dt
Force and Torque
• Note: a force is a force and a torque
• Moves body linearly: F=ma, changes
linear momentum
• Rotates body: produces torque, changes
angular momentum
Linear vs. Angular
linear quantity
angular quantity
velocity v
angular velocity ω
acceleration a
angular acc. α
mass m
moment of inertia I
p = mv
L = Iω
F = ma
T = Iα
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 doesn’t fall
Moment of Inertia
• Said that moment of inertia of a point
particle is mr^2
• In the general case, I = ∫ ρ r^2 dV where r
is the distance perpendicular to the axis of
rotation
• Don't know the axis of rotation beforehand
Moment of Inertia
• I = ∫ρ(x,y,z) y^2 + z^2 -xy
dxdydz
-xz
-xy
x^2 + z^2 -yz
-xz
-yz
x^2+y^2
Diagonalized Moment of Inertia
• Luckily, we can choose axes (principal
axes of the body) so that the matrix
simplifies:
Ixx 0
• I=
0
0
Iyy 0
0
0
Izz
• where, e.g., Ixx = m(y*y + z*z)
• Off-diagonal entries called "products of
inertia"
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
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)
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
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
Case in 2D
• In 2D, the vectors T, ω, α become scalars
(their direction is known – only magnitude
is needed)
• Moment of inertia becomes a scalar too:
• I = ∫prdA
Single planar rigid body
• state contains x, y, θ, vx, vy, ω
• Have
– F = ma (2 equations)
– T = Iω
– x = ∫vx dt
– y = ∫vy dt
– θ = ∫ω dt
• Integrate to obtain new state, and proceed
Rigid body in 3D
• Need some way to represent general
orientation
• Need to be able to compose changes in
orientation efficiently
Quaternions
• Quaternion: structure for representing
rotation
– unit vector (axis of rotation)
– scalar (amount of rotation)
– recall, store (cos(θ/2), v sin(θ/2) )
• Can represent orientation as quaternion,
by interpreting as rotation from canonical
position
Quaternions
• Rotation of θ about axis v:
– q = (cos(θ/2), v sin(θ/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)
Quaternion Rotation
• To rotate a vector r by θ about axis v:
– take q = (cos(θ/2), v sin(θ/2)
– Let p = (0,r)
– obtain p' from the quaternion resulting from
qpq-1
– p' = (0, r')
– r' is the rotated vector r
Rotation Differentiation
• Note:
– q(t) = (s(t), v(t))
– q(t) = [ cos(θ(t)/2), u sin(θ(t)/2) ]
– For a body rotating with constant angular
velocity ω, it can be shown
• q’(t) = [0, ½ ω] q(t)
• Summarize this ½ ω q(t)
Rigid Body Equations of Motion
Using quaternions gives
d/dt
x(t)
v(t)
q(t)
½ ωq(t)
P(t)
L(t)
=
F(t)
T(t)
P and L
• Note that
– v = P/m (from P=mv)
– ω = I-1L (from L = Iω)
• Often useful to use momentum variables
as main variables, and only compute v and
ω (auxiliary variables) as needed for the
integration
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