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 = ω x r

• 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 = 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

Intuition for Torque

• Larger the larger from the centre

• Lever action: small force yields equivalent torque far from fulcrum

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

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 = r x p

• 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 overbut 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) dxdydz

y^2 + z^2 -xy -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:

• I =

• where, e.g., Ixx = m(y*y + z*z)• Off-diagonal entries called "products of

inertia"

Ixx 0 0

0 Iyy 0

0 0 Izz

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

• 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)

Rotation Differentiation

Using quaternions gives

Rigid Body Equations of Motion

x(t)

q(t)

P(t)

L(t)

v(t)

½ ωq(t)

F(t)

T(t)

d/dt =

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