Mark Nelson mjas@itu.dk

Movement and physics

Fall 2013 www.itu.dk

Movement

You will probably want some things moving

Lots of ways to do that

General tradeoff between physics and animation And hybrid approaches

Basic movement model

Recall Pong Velocity per object Update position per frame

Control by changing an object’s velocity: Rotate Speed up / slow down

Smoothed movement

Instantaneous velocity changes aren’t always great

”Natural” motion is often non-linear Mario moving platforms speed up and slow down at path ends Even blinking lights aren’t always linear

MacBook sleep indicator LED curve

Simple smoothing

Send velocity changes via an API

Now, each game-loop iteration: Update position based on velocity Update velocity based on curve

Can be pre-programmed animation (platform movement) Or, reactive (e.g. with some lag)

Physics-based smoothing

The reactive version is basically velocity+acceleration: Objects have velocity Positions are updated based on velocity (1st derivative of position) Velocities are updated based on acceleration (2nd derivative of position)

Acceleration can be the basic unit, or computed from force F = ma, so equivalent for a constant-mass object

Physics versus animation

Why not always physics?

Inverse kinematics

Kinematics: How objects move in reaction to forces

Inverse kinematics: How to apply forces to get objects to move in specific ways

Hard in general

Curve-following

Say we want to follow a specific curved path

Robotics-style reactive controller: If we’re off the path, apply corrective velocity Produces smooth, physically realistic, but ”laggy” path following

Can follow more closely: Predictive controller, e.g. start turning a bit before a turn Analytical solution, like calculating a satellite’s thrust schedule

Some more pros/cons

When not physics: Weird emergent effects to debug Want effects you don’t want to fully simulate, e.g. tractor-beam motion

When physics: Want unscripted interaction between animations and other objects E.g. Platform on a pre-animated path, but can be knocked off-course by player

colliding with it

Some animation approaches

How do we follow a path?

Waypointing Animator gives specific points on a path Animation system moves objects between them Simplest is just drawing straight lines, but often want smoothing

Parametric curves

General representation of paths

Function X(t) gives object position from t_start to t_end

Bezier curves

Probably the most common parameterized curve

Quadratic Bezier: X(t) = (1-t)2P0 + 2(1-t)P1+t2P2

Some facts about parametric curves

speed(t) = |X’(t)|

distance(t) = integrate speed(t) from t_start to t

Plus, lots more useful relationships, e.g. curvature

Constant-speed motion

Often want to move at constant speed along a curve Or at least, want to control our speed separately, by varying t Not always easy to define a curve so that it comes with constant speed(t)

Solve by noticing: Constant speed means constant distance traveled

Constant-speed motion

Reparameterize by arc length

Choose desired arc position p we want to be at Solve distance(t) = p Done by finding root of distance(t) – p = 0

(Requires numeric integration, usually Newton’s method)

Bezier paths

Bezier curves

Notice that almost all Bezier curves don’t start out as constant-speed curves!

Also, can sometimes have weird artifacts Loops, backwards travel More common in higher-degree curves

May be inappropriate, or need more constraints, if you have obstacles

Physics engine

[see other slides]

2d platformer physics

Similar general principles But some things can be simplified

Grid-based methods Pixel-based methods

Collision response Bouncing off, stopping Special-case being ”on” a surface

2d platformer physics