# Lecture 20 - Trains

## Pubilic Service Announcement

1. Exam: starts 09.00, Thursday, 11 August, 2011; ends 11.30, Friday, 12 August, 2011.
2. Stopping trains. There is a tricky method that stops a train instantly. You nust not use it. Why?
3. Doubling segments.

# Train Properties

A locomotive travels on the track at a given speed following the path created by directions of turn outs.

• As it travels it triggers sensors that give you feedback as to where it is.
• Actually, not quite where it is. There is a time lag.
• Train triggers sensor at t: x(t) = Sn + 0 cm
• Report of sensor is recorded (time-stamped) at t + dt. dt includes
• interval between time of triggering and next sensor query
• time for train controller to process query and return result
• time in your application between receiving bytes from train controller and packaging bytes into a time stamped event

You should be able to estimate each of these time intervals

• At t + dt: x(t + dt) = Sn + dx
• dx = \int_t^t+dt v(t') dt' ~= v(t) dt
• In the event time-stamped at t + dt the train appears to be at Sn, but it is actually at Sn + v(t) dt
• Does this matter?
• How fast do trains go? Estimate 20 cm/sec.
• If dt is 100 msec you are off by 2 cm.

How do you know where the locomotive is?

• intermittently, at a sensor
• between sensors, dead reckoning, which means you need to know the train's velocity

Velocity is controlled by changing the train's speed, BUT, the mapping between speed and velocity is complex.

• Velocity changes are not instantaneous.
• After the speed is changed the train speeds up and slows down gradually.
• Tricks' that make the train stop instantly are not acceptable because they wear out the trains.
• The velocity decreases when travelling over turn outs or around curves.
• The smaller the radius of curvature the slower the velocity.
• Different locomotives travel at different velocities when set to the same speed.
• Velocity of a given locomotive decreases over time
• As the track gets dirty.
• As the time since the locomotive's last lubrication increases

Important. Some of these effects matter; some don't. It's part of your task to find out which is which.

Furthermore, things can go wrong, such as

• A turn out switches while a locomotive is on top of it.
• You need to estimate where the train will be when the switch changes in order to know if it is safe to throw the switch
• Locomotives run off the ends of sidings.
• You need to know how far a train will travel between when you give the stop command and when the train stops.
• Locomotives stall because they pass over difficult parts of the track too slowly.
• Why? Friction increases when a train is on curved track.
• Sensors fail to trigger, or trigger in the absence of a locomotive
• You need to know when you expect the sensor to be triggered if you are to know that it has not been triggered.

Avoiding such failures, or responding sensibly to them, is possible only if you have a good enough' velocity calibration. (You get a perfect calibration only in the limit t->infinity. Failures like these also pollute your attempt to acquire reliable data for your calibration.

#### Factors that might effect a calibration.

In general the velocity of a locomotive may be a function of many variables

1. which locomotive it is
2. which speed is set
3. time since the last speed change
4. the velocity at which it was travelling before the last speed change
5. where it is on the track
• possibly on what type of track it is on
6. how long since the track was cleaned
7. how long since the locomotive was lubricated

Important. Some of these effects are matter; some don't. It's part of your task to find out which is which.

## How to Get Started

The simplest objective:

• know where the train stops when you give it a command to stop
• restrict the stop commands to just after the train passes a sensor
• only one train moving

Sequence of events

1. Train triggers sensor at t
• train at Sn + 0 cm
2. Application receives report at t + dt1
3. You give command at t + dt1 + dt2
4. Train receives and executes command at t + dt1 + dt2 + dt3
5. Train slows and stops at t + dt1 + dt2 + dt3 + dt4
• train at Sn + y cm
• (You measure y with a tape measure.)

• If you do this again, same sensor, same speed, will you get the same answer?
• If you do this again, different sensor, same speed, will you get the same answer?
• If you do this again, same sensor, different speed, will you get the same answer?
• If you do this again, different sensor, different speed, will you get the same answer?
• And all the other important ones in the list above.

This is very time-consuming!

• The only way to reduce the number of measurements is to eliminate factors that are unimportant.
• The only way to know that a factor is always unimportant is to measure it repeatedly. Developing the ability to estimate quickly, and to find the worst case quickly is the main way of being smart in tasks like this one.

Now make a table

 Sensor 1 Sensor 2 ... Speed 6 Speed 8 ...

There are enough measurements in each cell of the table that you can estimate the random error. (Check with other groups to make certain that your error is not too big.)

Based on calibrations I have seen in previous terms you will find substantial variation with speed setting and train, little variation with sensor.

Group across cells that have the same' value. Maybe all have the same value.

Hint. Interacting with other groups is useful to confirm that you are on track. Of course, simply using another group's calibration without saying so is academic dishonesty'.

## The Essence of Calibration

1. You measure the time interval between two adjacent sensor reports.
2. Knowing the distance between the sensors you calculate the velocity of the train
• velocity = distance / time interval
• measured in cm / sec.

Note that on average the lag mentioned above -- waiting for sensor read, time in train controller, time in your system before time stamp -- is unimportant.

• Sensor1 actually hit at t1.
• You record (S1, t1 + dt) as the first event.
• Sensor2 actually hit at t2
• You record (S2, t2 + dt) as the second event
• You compute the velocity as (S2 - S1) / (t2 + dt - (t1 + dt)) = (S2 - S1) / (t2 - t1), which is the correct answer.
• But the variation in dt from measurement to measurement adds noise to the measurement.
3. After many measurements you build a table
• Use the table to determine the current velocity
• Use the time since the last sensor report to calculate the distance beyond the sensor
• distance = velocity * time interval

## The Problems You Have to Solve

1. The table is too big.
• You need a ton of measurements
2. The values you measure vary randomly.
• You need to average and estimate error.
3. The values you measure vary systematically
• For example, each time you measure the velocity estimate is slower, presumably because the train is moving towards needing oiling.
• You need to make fewer measurements or use the measurement you make more effectively.