RollerCoaster Tycoon X Like the original, but safer Adriel Luo - PowerPoint PPT Presentation

Adriel Luo, Xue An Chuang RollerCoaster Tycoon X Like the original, but safer Adriel Luo Xue An Chuang 1

Adriel Luo, Xue An Chuang Who has taken a roller coaster? Who wants to be confident that they are safe? 2

Adriel Luo, Xue An Chuang The safety of coasters is verified by numerical simulation 3

Adriel Luo, Xue An Chuang But numerical simulation is subject to error { x’ = -y, y’ = x } { x’ = x } 4

Adriel Luo, Xue An Chuang Approach 5

Adriel Luo, Xue An Chuang We break a coaster down into track sections and prove safety individually ● Reduces complexity More generalizable ● ● Prove properties for individual sections ● Ending conditions for one section are the starting conditions for next ● Piece the different proofs back together to form a complete proof 6

Adriel Luo, Xue An Chuang Coasters can be modeled with straight lines and arcs 7

Adriel Luo, Xue An Chuang Coasters can be modeled with straight lines and arcs 8

Adriel Luo, Xue An Chuang Coasters can be modeled with straight lines and arcs 9

Adriel Luo, Xue An Chuang Coasters can be modeled with straight lines and arcs 10

Adriel Luo, Xue An Chuang Coasters can be modeled with straight lines and arcs 11

Adriel Luo, Xue An Chuang Coasters can be modeled with straight lines and arcs 12

Adriel Luo, Xue An Chuang Coasters can be modeled with straight lines and arcs 13

Adriel Luo, Xue An Chuang Coasters can be modeled with straight lines and arcs Straight lines: y = mx + c ● Arcs: (cx-x) 2 + (cy-y) 2 = r 2 ● 14

Adriel Luo, Xue An Chuang We want to prove several properties about roller coasters ● Train goes forward Energy is conserved ● ● Train stays on the track 15

Adriel Luo, Xue An Chuang Models 16

Adriel Luo, Xue An Chuang Modeling: Basic Assumptions ● Assumptions: zero friction, unit gravity, point mass, two dimensions 17

Adriel Luo, Xue An Chuang Modeling: Straight Line Dynamics ● For straight lines: x’ = v*dx ○ ○ y’ = v*dy v’ = g sin θ ○ 18

Adriel Luo, Xue An Chuang Modeling: Arc Dynamics ● For arcs (clockwise): x’ = v*(y-cy)/r ○ ○ y’ = -v*(x-cx)/r v’ = (x-cx)/r ○ 19

Adriel Luo, Xue An Chuang Proof 20

Adriel Luo, Xue An Chuang Proof approach ● If ODEs are solvable, solve! Identify which properties would be proved through similar means ● ● Proving properties that were essential in proving others ○ Positive velocity is a powerful property! Useful proof rules ● ○ Differential invariants ○ Differential ghosts 21

Adriel Luo, Xue An Chuang Proof Example: Arc motion 1. Stays on track and energy is conserved Proved with invariants ○ 2. Strictly positive velocity ○ Proved with ghosts as velocity is decreasing 3. Stays within the quadrant ○ Proved with invariants using information about velocity 22

Adriel Luo, Xue An Chuang RollerCoaster Tycoon X is a safe roller coaster design tool 23

Adriel Luo, Xue An Chuang We modeled the Top Thrill Dragster and proved that it is safe 24

Adriel Luo, Xue An Chuang 25