- 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

- 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

- 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

- velocity is finite → movement must be continuous
- 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

- assume acceleration from known velocity v
_{1}to v_{2}- 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}= (v_{1}+ v_{2}) / 2

- Scenario 1: acceleration complete before 2nd sensor hit (t < d / v
_{a})- split d into two segments
- d
_{1}: acceleration, and d_{2}: stable velocity v_{2} - d = d
_{1}+ d_{2} - t
_{1}= d_{1}/ v_{a} - t
_{2}= d_{2}/ v_{2} - t = t
_{1}+ t_{2} - can solve for d
_{1}, d_{2}, t_{1}, t_{2} - acceleration: (v
_{2}- v_{1}) / t_{1}

- 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}- v_{1}) - acceleration: (v
_{r}- v_{1}) / t

- average velocity during acceleration: v

Stopping (2016)

The Kinematics of Train Calibration (2017)

The Kinematics of Train Calibration (2015)

Reverse Engineering Acceleration/Deceleration(2011)