CS 452/652 Fall 2019 - Lecture 21

October 28, 2019 prev next

Train Modelling

velocity: distance over time
average of time is straightforwarded
- both start and stop time are uniformly distributed
- use sample average as estimated mean
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

Dynamic Calibration

verify validity of current estimates
long-term variability: track degradation? (or improvement)
- minimum measured time (or maximum) might not change!
- need window of recent measurements
Exponentially Weighted Moving Average (EWMA)
- x: estimated average; m: next sample, a: weighting factor
- x * (1 - a) + m * 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

Acceleration

velocity is the derivate of movement
acceleration is the derivative of velocity
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 velocity step change in the middle of acceleration interval
- or approximate as average velocity during acceleration interval
- both are ok, if low-quality location estimate is acceptable during acceleration

Acceleration Measurements

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 speed 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
- t₁ = d₁ / v_a
- t₂ = d₂ / v₂
- t = t₁ + 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)
- 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

Earlier Documents

The material covered in class should be sufficient for train modelling. However, during the past years, various documents have described train modelling and calibration in the context of CS 452/652. The various versions are made available below with the caveat that they might or might not be helpful:

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