CS 452/652 Winter 2022 - Lecture 20

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 sensor timestamps 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