# CS 452/652 Winter 2020 - Lecture 20

February 26, 2020
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### 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

#### Goals / Objectives

- location tracking
- collision avoidance
- accurate stopping

- train efficiency: complete trips as fast as possible

#### Velocity

#### 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 (?)

#### Uncertainty

- timestamp uniformly distributed across duration of polling loop (~70ms)
- sample mean/variance are good estimates of real mean/variance
- sum of uniform distributions: triangle distribution
*but*: measured data typically forms bimodal distribution

- 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

#### 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

#### 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)