NASCAR Refueling Challenges: Connie Wang Outline Background Model - - PowerPoint PPT Presentation

NASCAR Refueling Challenges:

Connie Wang

Outline

• Background
• Model Highlights
• Safety and Efficiency
• Proof Highlights
• Conclusion

The Strategy Behind a Pit Stop by Connie Wang

Background

• NASCAR races
• 36 total races
• 34 oval tracks
• .526 – 2.66 miles long
• 188 – 500 laps
• Refueling rules
• No sensors to monitor exact gas level
• 24 gallons per pit stop

The Strategy Behind a Pit Stop by Connie Wang

Model Highlights

• Controls
• 𝑗𝑔 𝑔𝑣𝑓𝑚 > 𝑔𝑑 ∗ 𝑤 ∗ 𝑈; 𝑑𝑝𝑜𝑢𝑗𝑜𝑣𝑓;
• 𝑗𝑔 𝑔𝑣𝑓𝑚 ≤ 𝑔𝑑 ∗ 𝑤 ∗ 𝑈; 𝑔𝑣𝑓𝑚 = 𝑛𝑏𝑦;
• ODEs
• 𝑦5 = 𝑤 ∗ 𝑒𝑦
• 𝑧5 = 𝑤 ∗ 𝑒𝑧
• 𝑒𝑦5 = −𝑒𝑧
• 𝑒𝑧5 = 𝑒𝑦
• 𝑔𝑣𝑓𝑚5 = −𝑔𝑑 ∗ 𝑤 (linear)
• 𝑔𝑣𝑓𝑚5 = −(𝑔𝑑 ∗ 𝑤 ∗ 𝑢 + 𝑑) (quadratic)

The Strategy Behind a Pit Stop by Connie Wang

Safety and Efficiency

• Stay on track
• 𝑦< + 𝑧< = 𝑠𝑏𝑒<
• Sufficient fuel
• 𝑔𝑣𝑓𝑚 ≥ 0
• Do not stop unnecessarily
• 𝑗𝑔 𝑔𝑣𝑓𝑚 > 𝑔𝑑 ∗ 𝑤 ∗ 𝑈; 𝑑𝑝𝑜𝑢𝑗𝑜𝑣𝑓;

The Strategy Behind a Pit Stop by Connie Wang

Proof Highlights (on track)

• Loop invariants
• 𝑦< + 𝑧< = 𝑠𝑏𝑒<
• 𝑒𝑦< + 𝑒𝑧< = 1
• 𝑒𝑦 ∗ 𝑤 = −𝑧
• 𝑒𝑧 ∗ 𝑤 = 𝑦
• 𝑠𝑏𝑒 ≥ 0
• Differential Cuts
• 𝑒𝑦 ∗ 𝑤 = −𝑧
• 𝑒𝑧 ∗ 𝑤 = 𝑦

The Strategy Behind a Pit Stop by Connie Wang

Proof Highlights (sufficient fuel)

• Loop Invariants
• 𝑔𝑑 > 0
• 𝑈 > 0
• 𝑔𝑣𝑓𝑚𝑗𝑜𝑗𝑢 > 𝑔𝑑 ∗ 𝑤 ∗ 𝑈 (linear)
• 𝑔𝑣𝑓𝑚𝑗𝑜𝑗𝑢 > 𝑔𝑑 ∗ 𝑤 ∗ 𝑈< + 𝑑 ∗ 𝑈 (quadratic)
• 𝑛𝑏𝑦 > 𝑤𝑑 ∗ 𝑤 ∗ 𝑈
• Differential Cuts
• 𝑔𝑣𝑓𝑚 = 𝑔𝑣𝑓𝑚𝑗𝑜𝑗𝑢 − 𝑔𝑑 ∗ 𝑤 ∗ 𝑈 (linear)
• 𝑔𝑣𝑓𝑚 = 𝑔𝑣𝑓𝑚𝑗𝑜𝑗𝑢 − (𝑔𝑑 ∗ 𝑤 ∗ 𝑈< + 𝑑 ∗ 𝑈) (quadratic)

The Strategy Behind a Pit Stop by Connie Wang

Conclusion

• Can CPS models help NASCAR teams?
• Proof helps devise strategies
• Use of algorithmic CPS controllers
• Future work
• Acceleration/deceleration
• Time constraints
• Multiple cars
• Tire degradation

The Strategy Behind a Pit Stop by Connie Wang

Thanks!

The Strategy Behind a Pit Stop by Connie Wang