Lecture V: The game-engine loop & Time Integration
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1 Lecture V: The game-engine loop & Time Integration
2 The Basic Game-Engine Loop Previous state: " #, %(#) ( #, )(#) Forces -(#) Integrate velocities and positions Resolve Interpenetrations Per-body change " # + #, %(# + #) ( # + #,,(# + #) Position correction Resolve Collisions Velocity correction
3 Kinematics: continuous motion in continuous time. Events are local and always valid. Computer simulation: Challenges Discrete time steps ". Discrete Space (mesh, particles, grids)
4 Updating Position Force induces acceleration. When mass is constant:! ", $ = & ' ((", $) Derivatives: +, $ = (($) and ", $ = +($) Thus:! ", $ = & ' ",, ($). A differential equation. Often impossible to solve analytically. More often, ( = ( +, $ (velocity dependence). Discretization: stability and convergence issues. 4
5 Taylor Approximation A function at! +! can be approximated by a polynomial centered at! with arbitrary precision: $! +! $! + t ' $ (! +!) 2 $((! + +!, -! $,! We do not usually use (or have) more than 2 nd derivatives. 5
6 First-Order Approximation If " is small enough, we approximate linearly: # " + " # " + t ' # ( " Euler s Method: approximating forward both velocity and position within the same step: ) " + " = ) " + + " " = ) " +,(") / " #(" + ") = #(") + )(") " Unknown for next time step Assumed known for this time step 6
7 Euler s Method Note: we approximate the velocity as constant between frames. We compute the acceleration of the object from the net force applied on it:! " = $ " % We compute the velocity from the acceleration: & " + " = & " +! " " We compute the position from the velocity: ) * " + " = ) * " + &(") " 8
8 Issues with Linear Dynamics A mere sequence of instants. Without the precise instant of bouncing. Trajectories are piecewiselinear. Constant velocity and acceleration in-between frames. 9
9 Time Step When " 0, we converge to % " = ( ) * +,+ (Im)possible solution: reducing " to convergence levels? First-order method, piecewise-constant velocity: not very stable. Our objective: make the most with every " you get. 10
10 Time Step First-order assumption: the slope at! as a good estimate for the slope over the entire interval!. The approximation can drift off the function. Farther drifting ó tangent approximation worse. # & # % errors # $ 11
11 Midpoint Method Estimating tangent in Half step:! " + " " =! " + ' ",! 2 2 Full step:! " + " =! " + ' " + ),! " + ) * * " 2 nd -order approximation. Compute position similarly with!. 12
12 Midpoint Method Approximating the tangent in mid-interval. Applying it to initial point across the entire interval. Error order: the square of the time step! # $. Better than Euler s method (!( #)) when # < 1. Approximating with a quadratic curve instead of a line. ) + ) +/$ can still drift off the function. ) + ) * 13
13 Improved Euler s Method Considers the tangent lines to the solution curve at both ends of the interval. Velocity to the first point (Euler s prediction):! " =! $ + $ ' (($,!) Velocity to the second point (correction point):!, =! $ + $ ' ( $ + $,! " Improved Euler s velocity! $ + $ =! " +!, 2 Compute position similarly with! ",!, instead of (. 14
14 Improved Euler s Method The order of the error is +( # & ). The final derivative is still inaccurate.! &! # + # =! % +! & 2!(#)! % # 15
15 Runge-Kutta Method There are methods that provide better than quadratic error. The Runge-Kutta order-four method (RK4) is!( $ % ). A combination of the midpoint and modified Euler s methods, with higher weights to the midpoint tangents than to the endpoints tangents. 16
16 RK4 Computing the four following tangents (note dependence of acceleration on velocity):! " = % & '(%,!(%))! + = % & ' % + % 2,! % + 1 2! "! / = % & ' % + % 2,! % + 1 2! +! 0 = % & ' % + %,! % +! / Blend as follows:! % + % =! % +! " + 2! + + 2! / +! 0 6 Compute position similarly with! values. 17
17 RK4! % = #/ 0(#,!(#))! * = #/ 0 # + # 2,! # + 1 2! %! + = #/ 0 # + # 2,! # + 1 2! *!, = #/ 0 # + #,! # +! +! *! # + #! +! % + 2! * + 2! + +!, 6! %!,!(#) #/2 #/2 Exercise! 18
18 Leapfrog Integration ( a cavallina ) vel pos
19 Leapfrog Integration t (in dt) Pos Vel Pos Vel Pos Vel
20 Leapfrog Integration first step t (in dt) Pos 0 Vel Vel 0
21 Leapfrog Integration t (in ") Pos Vel Pos Vel Pos Vel Pos $ 1 = $ 0 + )(0.5) " $ 2 = $ 1 + )(1.5) " $ 3 = $ 2 + )(2.5) " ) 1.5 = ) " ) 2.5 = ) "
22 More accurate than Euler-based methods Residue of!( $ % ) Leapfrog Method But at the same cost as Euler s method! Major advantage: fully reversible!
23 Verlet integration Based on the Taylor expansion series of the previous time step and the next one:! " # + # +! " # #! " # + # (! " ) # + #* 2 (! " )) # + +! " # # (! " ) # + #* 2! " )) # Cancels out! 24
24 Verlet integration Approximating without velocity:! " # + # = 2! " #! " # # + # )! " ** # ++( # - )! " (# + #)! " (# #)! " (#) # ) 0 # 2! " (#)! " (# #) # # 25
25 Verlet integration An!( $ % ) order of error. Very stable and fast without the need to estimate velocities. We need an estimation of the first ' ( ($ $) Usually obtained from one step of Euler s or RK4 method. Difficult to manage velocity related forces such as drag or collision. Introduction to position-based dynamics. 26
26 So far: computing current position!(#) and velocity %(#) for the next position (forward). Those are denoted as explicit methods. In implicit methods, we make use of the quantities from the next time step!! # =! # + # # * %(# + #) This particular one: backward Euler. Computing in inverse: Implicit methods Finding position! # + # which produces! # if simulation is run backwards. 27
27 Implicit methods Not more accurate than explicit methods, but more stable.! $! # Especially for a damping of the position (e.g. drag force or kinetic friction).! " 28
28 Backward Euler How to compute the velocity from the future? Given the forces applied, extracting from the formula: Example: a drag force! " = % & ' is applied: ' ( + ( ' ( = % & '(( + () ( And therefore ' ( + ( = '(() 1 + ( & % 29
29 Backward Euler Often not knowing the forces in advance (likely case in a game). Or that the backward equation is not easy to solve. We use a predictor-corrector method: one step of explicit Euler s method use the predicted position to calculate!(# + #) More accurate than explicit method but twice the amount of computation. 30
30 Semi-Implicit Method Combines the simplicity of explicit Euler and stability of implicit Euler. Runs an explicit Euler step for velocity and then an implicit Euler step for position:! " + " =! " + " ' " =! " + " ((")/, - " + " = -(") + "! " = -(") + "!(" + ") "!(") '(")!(" + ") " + " 31
31 Semi-Implicit Method The position update in the second step uses the next velocity in the implicit scheme. Good for position-dependent forces. Conserves energy over time, and thus stable. Not as accurate as RK4 (order of error is still!( $)), but cheaper and yet stable. Very popular choice for game physics engine. 32
32 Angular Integration Examples: Forward Euler!"!# = 1 2 0, ) " becomes: " # + # " # = 1 # 2 0, ) # " # " # + # = " # + 1 # 0, )(#) "(#) 2 The rest of the formulations follow suit for angular displacement 0, velocity ) and acceleration 1. 33
33 Summary First-order methods Implicit and Explicit Euler method, Semi-implicit Euler, Exponential Euler Second-order methods Verlet integration, Velocity Verlet, Trapezoidal rule, Beeman s algorithm, Midpoint method, Improved Euler s method, Heun s method, Newmark-beta method, Leapfrog integration. Higher-order methods Runge-Kutta family methods, Linear multistep method. Position-based methods Leapfrog, Verlet. 34
34 Dimension Concluding remarks Integration methods shown for 1D variables. Generalization using vector-based structures (e.g., Jacobians). Evaluation of all dimensions and variables should be done for the same simulation time ". 35
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