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

- measure similar to velocity
- measure time between two sensors
- change speed level at first sensor
- compute acceleration based on known estimates for velocities
- first detect whether train has reached target velocity at 2nd sensor?
- constant acceleration → average velocity during acceleration =
(v
_{1}+ v_{2}) / 2

- 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 velocity 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 = t
_{1}+ t_{2} - v
_{a}= d_{1}/ t_{1} - v
_{2}= d_{2}/ 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})- actual 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

- actual average velocity during acceleration: v

- special case of deceleration
- manual experiment
- send stop command when sensor is triggered
- manually measure stop distance

- compute using acceleration model
- experiment by trying to stop right after sensor
- use search algorithm to find right time: could be automated
- stop time + velocity → compute stop distance?
- search algorithm (e.g., binary search) might be brittle and need many experiments

- special case of acceleration
- measure based on known estimate for stop time/distance
- stop at known location, then start
- measure time to sensor
- can compute acceleration
- similar caveat as acceleration measurement: did train reach target velocity?

- stop before reaching target velocity
- manual experiments with different (short) times between start and stop

Stopping (2016)

The Kinematics of Train Calibration (2017)

The Kinematics of Train Calibration (2015)

Reverse Engineering Acceleration/Deceleration(2011)