# Lecture 19 - Calibration II

## Public Service Annoucements

1. Final exam: 16.00, 14 April, 2015.
2. First Train Control Demo, Wednesday, 11 March, 2015.
3. Don't throw away the implementation you did for kernel 4. Use parts of it to get turn-outs into alignment and get the train going when doing the first train control demo.
4. Keep reading the sensors as the train slows to a stop, and use to readings to update your model. You need this
• because we will ask you where the train thinks it is when stopping is complete, and
• because the train must know where it is when you give it the next command to drive.
5. We expect the train to find itself for milestone 2. But if it can find itself correctly the tracking software should be able to follow it as you drive it manually.
6. When you have your tracking working it should include a prediction of the time each train expects to trigger the next sensor, and how big the error is, measured in distance.

# Calibration

## 1. Calibrating Stopping Distance

• 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 ...

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.

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'.

In previous terms some groups have given not a `tr 0`, but instead a `tr 2` command. Then,

• as the train slows they read the times at which they cross sensors, <\li>
• use the times to update the model, and
• crawl the last few cm.

### Measuring the time to stop

In addition to the stopping distance you will want to know the time it takes to stop. Why might this be useful?

• When you give the `tr 0` your application's model of the state of the track know where the train will stop, but does not know how long it will take the train to stop. The next command should be given only after the train has stopped.
An obvious and 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.
At worst this will give you a worst case upper limit for very conservative train driving.

A better method is available if you find that the stopping distance is the same everywhere on the track. If so,

• Measure a point one stopping distance before a sensor.
• Give the `tr 0` command and measure the time when the train passes that point.
• When the sensor triggers, measure the time. The time difference is the stopping time.

With a velocity calibration you can do the method above with better precision.

## 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.

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 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)
• 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 most plentiful resource

• CPU

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

### 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.
• 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.

### How much time 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.

## 3. Calibrating Acceleration and Deceleration

### How Long does it Take to Start and then Stop a Train? How Far does it go?

Most of the your train project can get away with ignoring acceleration and decelleration. 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 for that series of commands.

#### Procedure to calibrate short moves.

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.
6. Repeat for a range of times.
7. Build a table that tells you how far the train will travel for a given t.
8. Build an inverse table given the time needed for a move of a known distance.

Now that you know how far the train moves for a sequence of commands you can start the train that distance short of a sensor and measure the time between the first command and the triggering of the sensor.

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 mant 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.
• 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?

• When the velocity change begins? The model gets ahead of the train.
• When the velocity change ends? The model falls behind the train.
• Somewhere in between? There is a time when the train gets ahead of the model for a while, then falls behind exactly the same amount so that at the end of the speed change the model exactly matches the train position.

You can improve this by constructing a linear velocity change model, a bilinear velocity change model, a quadratic velocity change model, or whatever. Oral comments in class give possibly helpful, possibly extraneous suggestions.

When you drive a train there are four things you care about, all of which are functions of time. Particularly you care about the effects of discontinuities in them.

1. Location on the track, specified by x(t), usually anchored at the previous sensor.
• A discontinuity is teleportation, which requires infinite velocity.
• Trains do not provide infinite velocity (teleportation)
2. Velocity along the track, specified by v(t) = x'(t).
• A discontinuity is an instantaneous change in velocity, which requires an infinite force.
• Trains do not provide infinite force
3. Acceleration, specified as a(t) = v'(t) = x''(t).
• A discontinuity in acceleration, which feels like a jerk. Think of what it feels like when you are tied to a rope that is being pulled. First the rope exerts no force on you, then suddenly there is a jerk and you feel a force. Not nice.
• Do trains provide infinite jerk?
4. Jerk, specified as j(t) = a'(t) = v''(t) = x'''(t)
• We can avoid discontinuities in acceleration by keeping jerk constant.
• Thus, j'(t) = x''''(t) = 0.