CS 452/652 Winter 2023 - Lecture 15

Train Modelling - Velocity

• Kinematics: area of Mechanics (and thus Physics)
• studies "motion of objects without reference to the forces which cause the motion."
• model trains emulate real trains
• we model the model trains
• velocity is inherently an average: distance / time

Goals / Objectives

• location tracking
  • collision avoidance
  • accurate stopping
• train efficiency: complete trips as fast as possible

Experiments and Data

• how many speed levels to investigate?
• determine minimum speed (per segment?) to avoid getting stuck
• stop from lower speed is more accurate
• how much data can you feasibly collect?
• document and justify decisions!

Measurement

• recommendation: use tight polling loop (no kernel)
• sensor → timestamp
• measurement errors
  • signal delivery, Märklin processing: constant (?)
    • constant errors cancel out when subtracting timestamps
  • delivery to software: variable (polling loop)
    • statistical modelling (see below)
  • processing in software: small (?)
    • ignore small errors

Uncertainty

• assume actual sensor times uniformly distributed across duration of polling loop (~70ms)
• time interval between two sensors is sum of uniform distributions
  • general case: Irwin-Hall distribution
  • special case (N=2): triangular distribution
  • sample mean/variance are good estimates of real mean/variance
  • could also use min/max to estimated midpoint?
• but velocity (distance/time) is non-linear in time!
• average of time is straightforwarded; average of velocity not so much
  • consider simple example: 100m distance, 2 time samples 10s, 20s
  • speed: 10m/s, 5m/s → average would be 7.5m/s
  • compute average time first, then speed: 100m/15s = 6.66m/s
• observation: measured data typically forms bimodal distribution
  • why? is that a problem?
  • low-frequency sampling (70ms): sample distribution bimodal (corner cases trimodal)
• general recommendation: keep experiment raw data as much as possible
  • bug in processing → repeat only processing
  • new approach to data processing → possible with raw data

Dynamic/Continuous Calibration

• verify validity of and/or update current estimates
• long-term variability: track degradation? (or improvement)
• real-world variations (wear and tear): difficult to model
• need window of recent measurements
• basic technique: Exponentially Weighted Moving Average (EWMA)
  • c: current estimate; d: next data sample, a: weighting factor
  • c := c * (1 - a) + d * a
  • no need to store array of samples
  • with appropriate choice of a, can use bitshift instead of division
  • can use similar approximation for standard deviation

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 graphs: movement, velocity, acceleration
• simplified 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 somewhat lower-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 (d/t > v_a)
  • split d into two segments (d₁ acceleration, d₂ stable velocity v₂): d = d₁ + d₂
  • split t into corresponding segments: 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 (d/t < 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