# Lecture 23 - Calibration I

## Public Service Announcements

1. Open house, November 9, 2010
2. Velocity/speed

# Train Properties

## The Velocity of a Train

In general the velocity 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.

## 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

## Terminology

Independent Variable. A quantity the value of which is determined solely by the application.

Dependent Variable. A quantity the value of which is determined directly by the values of the independent variables.

Calibration Table. A set of rows, one for each combination of values of all relevant independent variable, together with an estimate of all relevant dependent variables for the combination of values. You find out the current value of an independent variable by looking it up in the table.

Data Point. Measured value(s) of dependent variable(s) for a combination of independent variables.

Cell. A bin in which data points for the same set of dependent variables were used. Estimated values of dependent variables are usually the result of averaging the data points in a cell.

Collapsing a Table. Merging cells across the different values of a dependent variable. This is normally done after you have determined that there is no variation of dependent variable value that you care about when an independent variable is changed.

## The Problems You Have to Solve

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

# Building a Calibration Table

#### Remember the goal

• We measure to build a calibration.
• The calibration is used to predict the future.
• The calibration needs only to be as precise as our need for future knowledge.
• Building a calibration table that is not precise enough is an error.
• Building a calibration table that is overly precise is also an error.

## Measurement

Each measurement produces a data record

 Train Speed Section Previous Speed Time since speed change (seconds) Time since locomotive maintenance (hours) Time since track cleaning (hours) Section type Velocity (cm/sec) 25 8 31 10 23 76 36 curved 8.9

#### Note

1. Some data is nominal:
• train number is just the name of the train;
• section number is just the name of an edge in your graph;
• section type is just the name of a type of section.
2. Some data is ordinal:
• speed is ordered.
• You use the ordering of speed when you determine if the previous speed was higher or lower than the current one
3. Some data is numerical:
• all times and the velocity.
• You may add and subtract these values.
• You may interpolate between these values.
4. Data can be coded differently to move it from one category to another
• For example, time since locomotive maintenance (or track cleaning) could be coded as
• just done
• recent
• eons ago

which is ordinal, not numerical.

## Dealing with Data

We call the givens, the

• the givens, train, speed, ..., section type, the independent variables
• what we measure, velocity, the dependent variable.

The idea is that we expect what we measure, velocity, to be different when the dependent variables are different.

#### Remember the goal

We measure to build a calibration. The calibration is used to predict the future.

The basic method of data handling is simple.

1. Gather together all measurements that have the same dependent variables.
2. Calculate the mean and standard deviation of the measurements.
3. Use the mean when estimating how far the train travels in a given time.
4. Use the standard deviation to calculate how far away from your estimate the train might be.
5. The more measurements in your collection the smaller the standard deviation.

The basic method in its full implementation is not feasible!

• Why?

To make it more reasonable we sort the dependent variables into two categories:

1. the ones that matter, and
2. the ones that don't matter.

We omit the variables that don't matter. There are two ways of `not mattering'.

1. Error. Some independent variables have effects that are smaller than the standard deviation measured above.
2. Effect on your project. Sometimes the effect greated by an independent variable is too small to change the results of your project.

In either case the independent variable is omitted.

#### In a perfect world, which has infinite computing in zero time, and infinite availability of the tracks,

We do the following off-line.

1. We measure using as many different values of each independent variable as possible.
2. On the measurements we do an analysis of variance (ANOVA), including interactions.
3. We collapse the analysis across the variables that have no statistically significant effect
4. We then get the estimated values for each cell, and look at differences across different values of a variable
5. The differences map into differences of estimated position.
6. Differences in estimated position that are smaller than what you need for your project are used as a criterion for dropping more independent variables.

This provides a reasonable initial calibration table.

At the beginning of each run of the project, we run the locomotives we will be using all over the track for a while

• we drive the locomotives all over the track for a while to update the static calibration.

To update

1. In each cell of the table have
• the average velocity
• the variance of the average velocity
2. After each measurement calculate
• new average velocity = \alpha * old average velocity + (1 - \alpha) * measurement
• new variance = \alpha * old variance + (1 - \alpha) * measurement * measurement
• 0.0 < \alpha < 1.0
• \alpha near 0.0 ignores old measurements
• \alpha near 1.0 makes old measurements dominant

Alternatively, you could average in a circular buffer.

While the project continues to run,

• use every measurement you make to update the table.

### Practical Issues

You might want to consider

1. Using floating point for calculation. The easiest way to do this is to have a single calibration task that
• receives measurements in fixed point,
• calculates internally in floating point, and
• provides current calibration parameters in fixed point.

If more than one task uses fixed point you must change your context switch if any access to the floating point processor is non-atomic.

2. Turning on optimization, but be careful .
• There are places where you have done register allocation by hand.
• Previously undiscovered critical races could appear, and even critical races associated with bus clocks.
3. Size & align calibration tables by size & alignment of cache lines