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 = ωr perp • 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 F t • F t = ma t = mr α • T = mr 2 α • Let I = mr 2 • 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 acceleration a mass m p = mv F = ma angular velocity ω angular acc. α moment of inertia I L = I ω T = I α

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

dxdydz -xy -xz x^2 + z^2 -yz -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 0 • I = 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 x(t) d/dt q(t) P(t) L(t)

=

v(t) ½ ωq(t) F(t) T(t)

P and L

• Note that – v = P/m (from P=mv) – ω = I -1 L (from L = I ω) • 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