# 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: va = (v1 + v2) / 2
• Scenario 1: acceleration complete before 2nd sensor hit (t < d / va)
• split d into two segments
• d1: acceleration, and d2: stable velocity v2
• d = d1 + d2
• t1 = d1 / va
• 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 / va)
• average velocity during acceleration: vs = d / t
• velocity at 2nd sensor hit: vr = vs + (vs - v1)
• acceleration: (vr - 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: