# Lecture 19 - Acceleration/deceleration; Anthropomorphic Programming.

## Public Service Annoucements

1. On Friday the 17th we decided that the final exam will start at 12.30 on Thursday 6 April, 12.30 and end on Friday 7 April at 15.00.
2. You can download data from the terminal, such as the track graph, by putting a file onto the terminal program's output.
3. You can upload data to the terminal by sending its input to a file.
4. First Train Control Milestone: Tuesday 7 March.
5. Calibration as a process
• We make measurements and we check values we calculated in the past against those measurements.
• We revise as necessary. In some cases we revise our estimate of the train's location, in others our estimates of its operating parameters, in most cases both.
6. Here is a situation that reveals latent bugs in your interaction with the train controller: reversing the train.
1. You give a stop command and wait patiently until the train is completely stopped.
2. You give a reverse command.
3. You immediately give a speed command nd the train behaves as though it didn't receive the reverse command.
4. What happened?

# Calibration I

## 1. Calibrating Stopping Distance

It is important to know where the train is when it has stopped, which may not be the same as your estimate. Why? (Hint. The train will start again at the position where it stopped.) Therefore, keep reading sensors while the train is stopping.

• Once you have thought about acceleration and deceleration you will have estimates of when you expect to cross those sensors.
• Even without those estimates you will know if gross errors are occurring.

## 2. Calibrating Constant Velocity

To stop the train at any point on the track you must be able to give the stop command anywhere on the track. Knowing where you are when not at a sensor is possible only if you know your velocity.

### Calibrating Velocity

An implicit assumption you make 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.
Subtraction removes the systematic error. Calibration document describes in detail how to analyze random error, concluding that neglecting random error works just fine.
3. After many measurements you build a table. At its simplest this table has speed as its key and velocity as its value. A more comprehensive table would have a more complex key.
• Use the table to determine the current velocity.
• Use the time since the last sensor report to calculate the distance beyond the sensor
• location = location of previous sensor + velocity * time interval
• Correct your position every time you receive a sensor report.

### Using Resources Effectively

The scarce resources

1. Bandwidth to the train controller
2. Use of the train itself

How to make the most of them. We give you a requirement that you display on the terminal

• the time you expect to hit the next sensor,
• the time you actually hit the next sensor and the difference,
• the difference converted to distance.
We find these numbers very useful in knowing how well your train is following its calibration model. You should also use them in the same way.

Here is a concrete way you can use those numbers, which will make your demo improve as it runs, especially if your initial calibration is approximate, which it will be if your favourite train has gone to train heaven.

1. Squeeze every piece of information out of each measurement. For example
• You read the sensors to know where a train is by correcting your dead reckoning.
• Each time you detect a sensor you know the time spent travelling since the previous sensor hit.
• You can calculate the velocity and use the result to update the speed/velocity table.
• New velocity entry = a * new measurement + (1-a)*old velocity entry
• Allow the user (you) to adjust parameter values at the terminal. For example,
• You are likely to have some padding to make certain that you give switch commands early enough that the train will pass after switching is complete.
• When you are testing to see if you have enough padding and discover you have too little, you can try different values without having to go through setting up the train
This is a standard technique used, for example, when tuning games.
• ### Practical Problems You Have to Solve

1. The table is too big.
• You potentially need a ton of measurements
2. The values you measure vary randomly.
• Sometimes you need to average and estimate error.

### How Long does it Take to Stop?

You need to know how long it takes the train to stop. Why? (You usually assume that the train is at the stopping place before you give the next command.) 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 pick-up on the sensor
5. Calculate the time between when you gave the command and when the sensor triggered.
6. Look for regularities.

## 3. Calibrating Acceleration and Deceleration: short distances.

Trains often travel short distance, starting with the train stopped, and finishing with it stopped. When doing so the train spends its whole time either accelerating or decelerating. Your constant speed calibration is useless because the train doesn't travel at constant speed. Similarly your measured stopping distances are not useful.

Creating a perfect calibration of the train's position while it is accelerating is hard. But there is an easy and precise calibration that covers most of the moves the train makes where you need a good calibration. It's the subject of this section.

Most of the your train project can get away with ignoring acceleration and deceleration. The one place you can't is when you are doing a short move, giving a speed command followed by a stop command before it gets up to speed. How far will the train go? How long will it be before the train is fully stopped?

Short moves are common when the train is changing direction, which you need to increase the number of possible paths from one point to another.

The general idea is to give the train a carefully timed series of commands knowing how far and for how long the train moves during the series of commands.

#### A procedure to calibrate short moves.

Write a small application that performs the following sequence of actions.

1. Place the train on the track in the sort of location where you expect to make short moves.
2. Give the train a `speed n` command, where n is big enough to get the train moving reliably.
3. Wait `t` seconds.
4. Give the train a `speed 0` command.
5. Measure how far the train travelled from its initial location to where it stops.
6. You how far the train will travel for the chosen values of `n` and `t`.
Experiment with different values of `t` and `n` until you have a reasonable set of distances you can travel.

You now know how far the train moves for a given sequence of commands.

1. Position the train that distance ahead of a sensor.
2. Read the time and give a `speed n` command.
3. After `t` seconds give a `speed 0` command.
4. When the train triggers the sensor read the time again.
The distance between the two readings is the time it takes to make that short move.

Together with knowing when and where the train will stop if given the speed 0 command when running at a constant velocity, this will provide most projects with all the calibration they need. But you can do better.

## 4. Calibrating Acceleration and Deceleration: Doing Better

At this point you can do most of the things you will want to do for your project. But some things can only be done from a standing stop. It's more elegant to keep the train moving, speeding up and slowing down as required. To do so it's necessary fully to calibrate velocity during the act of accelerating and decelerating. Keeping a train at a pre-determined velocity, for example, requires changing from one speed to another frequently.

To explain velocity changes we must introduce models. On the track the train has a real location, so many cm past sensor S. In your program the train has a position, so many cm past sensor S'. The model is linked to the real train by the calibration. Neither the number of cm nor even the sensor is necessarily the same in the model and in reality because no calibration is perfect. The performance of a project, such as whether trains collide or not, depends on the difference between the model and reality. The remainder of this section is based on minimizing different measures of discrepancies beteen a model and reality.

Back to real trains. When you give the train a command to change speed, we know roughly how the velocity changes.

1. slowly at first
2. increasing
3. reaching a maximum, possibly for a non-zero time
4. decreasing
5. more and more slowly as the new velocity is approached

How should we model the process of speed changes?

• One might carefully measure the function that gives velocity as a function of time or distance after the command is given. How could one measure the instantaneous velocity?
• Video the train driving beside a tape measure, calculating mm/frame by differencing the positions in successive frames.
• Measure the time it takes for the train to travel at constant speed from one sensor to another. Then give the change speed command at varying distances before the sensor and measure the differences in time to the sensor trigger. Then construct a linear approximation.
This gives the experimental data that we need to approimate a model.
• Start with the crudest possible approximation, and improve it as you require more precision. An example is below.

The simplest possible model is a step change from the initial velocity to the final velocity. When should the change occur?

• If the train is changing to a higher speed, then the train in the model travels more slowly than the accelerating real train. The real train gets ahead of the train in the model.
• When the step change occurs in the model, the train in the model is travelling faster than the real train. The model starts to catch up to reality.
• The step change in the model is placed so that the train in the model is at the same position as the real train when the velocity change is complete.
• Is the amount that the train in the model falls behind acceptable? If it is not you need a more complex model.
How much does the train in the model fall behind the real train? It depends on what the real train is doing.
• Suppose the real train undergoes constant acceleration.
• The velocity of the real train increases linearly; the step in the model occurs at exactly the mid-point of the change.
• Before the step, the real train travels faster than the model train by
(t - t0) * (v1 - v0) / (t1 - t0)
• The distance the real train gets ahead is
(t - t0)^2 * (v1 - v0) / (2 * (t1 - t0))
• The step occurs at t = t0 + (t1 - t0) / 2 where the real train is ahead by
(t1 - t0) * (v1 - v0) / 8
• Whether this is good enough for your project depends on what your project is.