CS452 - Real-Time Programming - Winter 2018

Lecture 16 - Trains

Public Service Annoucements

  1. Kernel 4 due in class on Friday, 16 February.
  2. First train control demo is in the trains lab on Tuesday, 8 March.
  3. March break open house: March 10.
  4. Exam scheduled for Monday, April 23.
  5. PDF documents containing mathematics.

The Train Project


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

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

You can probably think of lots more things that would make the velocity increase. A lot of what's hard about dealing with trains, as about anything in the real world, is figuring out which set of many possibilities matters in practice.

Note on precision

We are going to be doing arithmetic. Should we do it in fixed point, which requires thought, software floating point or hardware floating point, which has the inconvenience of increased state to save, not to mention compiler incompatibilities? Answer. Check the amount of time your idle task runs.

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 (3 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

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".
For you project you choose landmarks 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:

Sequence of events

  1. Train triggers sensor n at t.
  2. Somewhat later a task sends 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.)
Compared to what you probably thought at first, the train was not at the sensor when you sent the command, but at Sn + v * (t1 + t2 + t3). Nor was the train at the sensor when it received the command to stop. Think. What do you want to call the "stopping distance"? This question amounts to "Why are you stopping?"

Presumably you are stopping the train because you want it to be in a particular location: before the end of a siding, before you run into another train, etc. You control only the time at which you send the stop command, which you will do after receiving some signal from the train. You are always t1 + t2 + t3 behind sensor reports; you must be t4 + t5 ahead of when you want the train to start decelerating.

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.

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.

Questions: some of which need answering.

Which ones? How do you get to an answer? You build up some intuitions about what belongs in a model of the train.


What you are doing is building a model of the train. There are a few different ways you can reason in doing so.

Building a model is as much an art as a science. The following sentence should be read carefully. The intuition you need, in order to know that you don't need to measure without measuring, can only be built my measuring!

The sequence of events described above has a whole lot of small delays that get added together

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.

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 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.
  3. After many measurements you build a table

Return to: