# Lecture 22 - Trains

## Public Service Announcements

1. Open house, November 9, 2010

# 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
• At t + \delta: x(t + dt) = Sn + dx
• dx = \int_t^t+dt v(t') dt' ~= v(t) dt
• How fast do trains go? Estimate 20 cm/sec
• If dt is 100 msec you are off by 2 cm.
• What determines the size of dt? Work it out.
1. Poll correct sensor.

How do you know where the locomotive is?

• sensor

Things can go wrong, such as

• A turn out switches while a locomotive is on top of it.
• Locomotives run off the ends of sidings.
• Locomotives stall because they pass over difficult parts of the track too slowly.
• Sensors fail to trigger, or trigger in the absence of a locomotive

The mapping between speed controls and train 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 are matter; some don't. It's part of your task to find out which is which.

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

1. which locomotive you have
2. which speed you set
3. time since the last speed change
4. the speed it was travelling at 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?

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.)

Group across cells that have the same' value.

## The Essence of Calibration

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

The values you measure vary.

You need to average and estimate error.