CS 452/652 Winter 2022 - Lecture 21

Train Modelling - Acceleration

• velocity is the derivate of movement
• acceleration is the derivative of velocity
• deceleration is negative acceleration
• kinematic reality:
  • velocity is finite → movement must be continuous
    • alternative would be teleportation
  • acceleration is finite → velocity must be continous
    • alternative would be infinite forces tearing train apart
  • show with curves: movement, velocity, acceleration
• kinematic model: assume constant acceleration
  • approximate as average velocity during acceleration interval
  • or approximate as velocity step change in the middle of acceleration interval
  • both are ok, if low-quality location estimate is acceptable during acceleration

Measurement

• measure similar to velocity
• measure time between two sensors
• change speed level at first sensor
• compute acceleration based on known estimates for velocities
  • first detect whether train has reached target velocity at 2nd sensor?
  • constant acceleration → average velocity during acceleration = (v₁ + v₂) / 2

Processing

• assume acceleration from known velocity v₁ to v₂
  • experiment changes the speed at a sensor and measures the time to another sensor hit
  • measure times and estimate time averages first, as before!
  • t: average time; d: distance
  • average velocity during acceleration: v_a = (v₁ + v₂) / 2
• Scenario 1: acceleration complete before 2nd sensor hit (t < d / v_a)
  • split d into two segments:
  • d₁: acceleration, and d₂: stable velocity v₂
  • d = d₁ + d₂
  • t = t₁ + t₂
  • v_a = d₁ / t₁
  • v₂ = d₂ / t₂
  • can solve for d₁, d₂, t₁, t₂
  • acceleration: (v₂ - v₁) / t₁
• Scenario 2: acceleration not complete before 2nd sensor hit (t > d / v_a)
  • actual average velocity during acceleration: v_s = d / t
  • velocity at 2nd sensor hit: v_r = v_s + (v_s - v₁)
  • acceleration: (v_r - v₁) / t

Stop Distance

• special case of deceleration
• manual experiment
  • send stop command when sensor is triggered
  • manually measure stop distance

Stop Time

• compute using acceleration model
• experiment by trying to stop right after sensor
  • use search algorithm to find right time: could be automated
  • stop time + velocity → compute stop distance?
  • search algorithm (e.g., binary search) might be brittle and need many experiments

Start

• special case of acceleration
• measure based on known estimate for stop time/distance
• stop at known location, then start
• measure time to sensor
• can compute acceleration
  • similar caveat as acceleration measurement: did train reach target velocity?

Short Moves

• stop before reaching target velocity
• manual experiments with different (short) times between start and stop

Earlier Documents

Stopping (2016)
The Kinematics of Train Calibration (2017)
The Kinematics of Train Calibration (2015)
Reverse Engineering Acceleration/Deceleration(2011)