# CS 452/652 Winter 2022 - Lecture 21

March 2, 2022 prev next

### Train Modelling - Acceleration

• 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
• 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

#### Measurement

• 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 = (v1 + v2) / 2

#### Processing

• 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 velocity 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
• t = t1 + t2
• va = d1 / t1
• v2 = d2 / t2
• can solve for d1, d2, t1, t2
• acceleration: (v2 - v1) / t1
• Scenario 2: acceleration not complete before 2nd sensor hit (t > d / va)
• actual average velocity during acceleration: vs = d / t
• velocity at 2nd sensor hit: vr = vs + (vs - v1)
• acceleration: (vr - v1) / t

#### Stop Distance

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

#### Stop Time

• 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

#### Start

• 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?

#### Short Moves

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

#### 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: