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Adriel Luo, Xue An Chuang
RollerCoaster Tycoon X
Like the original, but safer Adriel Luo Xue An Chuang
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Adriel Luo, Xue An Chuang
Who has taken a roller coaster? Who wants to be confident that they are safe?
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RollerCoaster Tycoon X Like the original, but safer Adriel Luo - - PowerPoint PPT Presentation
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
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Adriel Luo, Xue An Chuang
The safety of coasters is verified by numerical simulation
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Adriel Luo, Xue An Chuang
But numerical simulation is subject to error
{x’ = -y, y’ = x} {x’ = x}
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Adriel Luo, Xue An Chuang
Approach
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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
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Adriel Luo, Xue An Chuang
Coasters can be modeled with straight lines and arcs
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Adriel Luo, Xue An Chuang
Coasters can be modeled with straight lines and arcs
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Adriel Luo, Xue An Chuang
Coasters can be modeled with straight lines and arcs
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Adriel Luo, Xue An Chuang
Coasters can be modeled with straight lines and arcs
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Adriel Luo, Xue An Chuang
Coasters can be modeled with straight lines and arcs
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Adriel Luo, Xue An Chuang
Coasters can be modeled with straight lines and arcs
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Adriel Luo, Xue An Chuang
Coasters can be modeled with straight lines and arcs
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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 = r2
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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
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Adriel Luo, Xue An Chuang
Models
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Adriel Luo, Xue An Chuang
Modeling: Basic Assumptions
- Assumptions: zero friction, unit gravity, point mass, two dimensions
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Adriel Luo, Xue An Chuang
Modeling: Straight Line Dynamics
- For straight lines:
○ x’ = v*dx ○ y’ = v*dy ○ v’ = g sin θ
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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
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Adriel Luo, Xue An Chuang
Proof
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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
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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
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Adriel Luo, Xue An Chuang
RollerCoaster Tycoon X is a safe roller coaster design tool
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Adriel Luo, Xue An Chuang
We modeled the Top Thrill Dragster and proved that it is safe
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Adriel Luo, Xue An Chuang
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