# Lecture 16 - Trains

## Public Service Annoucements

1. Kernel 4 due in class on Friday, 17 February.
2. First train control demo is in the trains lab on Tuesday, 7 March.
3. PDF documents containing mathematics.

## The Train Project

### Terminology

Note. I try to be consistent in distinguishing between two closely related concepts: speed and velocity.

• A train's speed is the value you send to the train controller, an integer between 0 and 14.
• A train's velocity is the rate at which it moves along the physical tracks, a real number measured in centimetres per second.
• Some trains actually have more than fourteen velocities: they go at different velocities when you decelerate to a speed than when you accelerate to it. (Not many of these trains remain.)

Velocity is controlled by changing the train's speed, BUT, the mapping between speed and velocity is not simple.

• Speed changes are effectovely instantaneous, but velocity changes are not.
• After the speed is changed the train's velocity changes gradually: whether increasing or decreasing.
• `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
• As the locomotive gradually wears out

Important . Some of these effects matter; some don't. It's part of your task to find out which effects matter and which don't. (If you don't figure out which is which you will spend an unlimited amount of time.)

#### Note on precision

We are going to be doing arithmetic. Should we do it in fixed point, which requires thought, or floating point, which has the inconvenience of increased state to save, not to mention compiler incompatibilities?

The biggest fixed point number is 2^31. How big is this? 2^10 = 10^3 = 1000, 2^30 = 10^9 = 1 000 000 000, 2^31 = 2 000 000 000.

Suppose that the smallest distance you care about is 0.1 mm. 2*10^9 of them is 200 000 metres or 200 Km, twice the distance to Toronto. At 50 cm/sec, about as fast as a train can go, a train travels about 1 m per minute, 60 m per hour, 1.5 km per day. It will take about 100 days, 30 months to travel 200 Km.

A very successful final demo runs for 15 min, 0.25 of an hour, 0.01 of a day. You have a factor of 10 000 before you will start to see round-off error.

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 turn-out switches in order to know if it is safe to execute a switch command.
• 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.
• The pickup lifts as the train travels over a sensor.
You need to know how fast trains must move to avoid stalling.
• 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, and the train you are calibrating falls over dead long before that.)

Such failures like these also pollute your attempt to acquire reliable data for your calibration.

# Train Properties

## Where is a train?

There are two methods of knowing where you are:

1. Being at a landmark: "I am at the big tree." I know that's true because I can see the big tree right beside me.
2. Knowing where you started and how far, in what direction you have travelled: "I am three blocks north of the big tree." Calculating how far you have travelled is usually done by "dead reckoning".
• Count the blocks as you walk.
• The odometre on the car integrates the velocity by counting revolutions of the wheels.
• Sailing ships threw a log over the side and counted knots.
For you project you choose landmarks
• sensors, turn-outs, etc.
• Remember the importance of fiducial marks: on the track, on the train.
You then know when the train is at a given landmark, and find a way -- most likely by integrating velocity -- to know how far it is past the landmark at any given time. If so, you need to know each train's velocity.

## 1. Calibrating Stopping Distance

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 n at t.
• The train is at Sn + 0 cm.
2. Somewhat later you ( = one of your tasks) send a command to the train controller asking it to poll the sensors. The time is t + t1. (t1 could be negative, but not too negative. How negative could it be.)
3. You receive the reply from the train controller: sensor n has been triggered. The time is t + t1 + t2.
4. You send the speed zero command. The time is t + t1 + t2 + t3.
5. The train controller receives the command and forwards it to the train. The time is t + t1 + t2 + t3 + t4.
6. Train receives the command and starts slowing down. The time is t + t1 + t2 + t3 + t4 + t5.
7. The train stops at t + t1 + t2 + t3 + t4 + t5 + t6.
• The train is at Sn + y cm.
• (You measure y with a tape measure.)

#### Fiducial Marks

When you pull out the tape measure to measure y you need to know the two locations between which you are to measure.

• It's pretty obvious where the sensor is,
• but where is the train?
• the pickup? front or back?
• the front bumper?
• the back bumper?
It doesn't matter which place you choose, but it matters a lot that you always use the same place.

You have just chosen a "fiducial mark", a location you can recognize so that you can make a comparable measurement the next time. Some landmarks, switches, for example, also require you to choose fiducial points carefully.

• 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?
• Or a different train, or different track condition, or ...

1. The sequence of events above has a whole lot of small delays that get added together
• Each one has a constant part and a random part. Try to use values that are differences of measurements to eliminate the constant parts.
• Some delays can be eliminated a priori because they are extremely small compared to other delays. The more you figure this out in advance the less measurement you have to do.
2. Knowing where you stop is very important when running the train on routes that require reversing
• Why are reversing routes important?
3. Clearly, knowing when you stop is equally important.

This is very time-consuming!

• The simplest 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. 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, with or without saying so, is `academic dishonesty'. Like most academic dishonesty it simply doesn't work anyway.

### Measuring the time to stop

In addition to the stopping distance you will want to know the time it takes to stop. A simple way to do so is

1. Start a stopwatch when you give the stop command.
2. Stop the stopwatch when you see that the train is stopped.

This might not be accurate enough for you. When you have calibrated the velocity and can stop anywhere on the track there's a better way.

1. Give the stop command so that the train will stop with its pickup on a sensor, recording the time you when you give the command.
2. When the sensor triggers, check the time.

## 2. Calibrating Constant Velocity

At this point there are a few places on the track where you can stop with a precision of a train length or better. However, suppose you want to stop not sitting on a switch.

• You want to be close to the switch, clear of the switch, and on the right side of the switch when you stop.
• You want to know when the train has stopped because until then you cannot give the command to throw the switch.
• You want to know when the switch-throwing is complete because until then you cannot start the train running in reverse.
To do this successfully you have to be able to give the stop command anywhere on the track and know how long it will take the train to stop.

### Knowing the Current Velocity

An implicit assumption you are making is that the future will closely resemble the past.

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 ${t}_{1}$.
• 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)
• 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

### Using Resources Effectively

The most scarce resources

• Bandwidth to the train controller
• Use of the train itself.

The two most plentiful resources

• CPU

Any time you can use a plentiful resource to eliminate use of a scarce one you have a win. Two examples

1. Squeeze all the functionality you can from every measurement. Every time you pass a sensor you can use the measurement to improve your velocity calibration.
2. Use your time with the trains efficiently. There's a lot of setup time each time you start using the train. make as many measruements as you can to economize on set-up time effectively. make it possible to change program parameters from the terminal.

### Practical Problems You Have to Solve

1. The table is too big. (Actually both tables!)
• You potentially need a ton of measurements
2. The values you measure vary randomly.
• You need to average and estimate error.

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.

## 3. Calibrating Acceleration and Deceleration

### How Long does it Take to Stop?

Try the following exercise.

1. Choose a sensor.
2. Put the train on a course that will cross the sensor.
3. Run the train up to a constant speed.
4. Give the speed zero command at a location that stops the train with its contact on the sensor
5. Calculate the time between when you gave the command and when the sensor triggered.
6. Look for regularities.