## Interpolation: Yes but Beware!

In the movie version of the book “Charlotte’s Web” there is a song “Zuckermans Famous Pig” which features the lyrics

Fine swine wish he was mine

What if he’s not so big

Seeing this cartoon while in graduate school(1) and while taking a numerical methods course led to the parody song

Fine Spline, coefficients are prime
What if it grows too big?

## Interpolation: When to use it

There are three general categories when interpolation is used

• Sampled real-world data does not cover the full range (regression case)
If the interpolation covers points inside of the data set this is generally a “safe” scenario e.g., the interpolated data will be smooth within the range. If the interpolation goes outside of the data set then the values predicted by the interpolation should be checked against real-world expectations. In general, 10~15% beyond the sampled range (for smooth data) is reasonable to interpolate.
• To reduce calculation cost (speed or memory)
Some calculations are memory or FLOPs intensive; interpolations (especially polynomial interpolation) can show a significant reduction in the total number of operations.(2)
• Handle discontinuities (piecewise interpolation)
For equations with discontinuities, an interpolation can be used to provide a non-infinite transition between the operating realms.

## Interpolation: When not to use it

The “when not to use” is the mirror image of the “when to use.”

• Mister toad’s wild ride:(3) (Sampled and Discontinuities):
In some instances, the curvature of the equations is so severe that interpolations cannot accurately capture the data
• Flip(4)side: The real thing is cheaper:
Depending on the equation, and your target processor, the real calculation may be less intensive. In general when I hit a polynomial of order 6 or greater I start to question the value; (Taylor series after 3 terms).
• Integer data: Gear 1.3
The class interpolation failure is when integer data is interpolated to floating-point values. My first exposure to this was when I interpolated a non-CVT vehicle into the 1.3rd gear.(5)

Follow these tips and you will know if you can “Pig out or Pig In” with your interpolation.

## Footnotes

1. Back at this time, campus television had a limited number of channels. I would estimate that about 50% of my classmates, like me, had it on in the background the day before.
2. When performing polynomial interpolation save your powers, e.g.
x2 = x*x;
x3 = x2 * x;
x4 = x2 * x2;
3. Continuing with the children’s story theme
4. I hope these puns don’t get you off on the wrong foot with my Flip-FLOPS.
5. Interestingly enough it was seeing that (a good decade before CVT’s were common) that I understood the impact that a CVT could have on fuel economy. If you are interested in fuel economy take a look at this series I’m writing.

## Everyday engineering: MBD & MPG (Part-3 F=MA)

Sometimes simple is good enough.  Let’s start by reviewing what goes into fuel consumption for a car; e.g., what are the forces acting on it?

• Acceleration: getting the car from 0 to 55 (and above) requires force
• Aerodynamic drag: the faster you go the higher it is
• Gravity: uphill or down, it has a way of changing your speed

Using these three forces acting on the vehicle, we can (when we add in losses) calculate the energy needed to get from point A to point B. (And if you are curious about how the data for A to B is collected check out this previous post)

Our simple model uses the Lat / Lon / Elevation and Speed data we downloaded as part of the last blog post for our points A to B.

In a frictionless, lossless world, my car with regenerative braking(1) could reach 100% efficiency.  However, your car-not (2) able to do this in the real world. Our first pass of “driving the route” will make the following assumptions:

• We hit every stoplight(3)
• We drive at the speed limit(4)
• There is no traffic(5)
• Standard profile acceleration and deceleration between speed zones(6)
• 20% energy recapture on braking.

## The first route: To the Tech Center!

My first working commute was from Farmington Hills Michigan to GM’s Warren(7) Tech Center.  If we break it down by distance and stops we get the following table.(8)

Between each point, there is a deceleration to stop(9) followed by an acceleration to the target speed.  If we put this information into the Simulink model we get the following energy usage profile.  There is an interesting modeling point between points 1, 2 and 3; it is a short stretch of the road section of road where the car does not have time to get up to speed before you have to slow down.  I’ve included the Stateflow chart that I created to solve this look ahead in the footnotes.(10)

Because this trip was mainly highway there was very little chance for regenerative breaking;(11) in contrast, my ADI (Applied Dynamics International) commute had many more start-stop moments with more regenerative braking events.

Reviewing the data from these two routes reinforces some basic knowledge:

• Total distance is only one factor in energy usage
• Energy usage goes up with rate of travel
• Start-stop events (with acceleration) have a large impact on energy use (e.g. steady speed is better)
• Yellow is an odd default choice for plotting color.

In next week’s post we will add in a “human” driver model to improve the accel and decel behavior of these models.

## Footnote

1. For a short introduction to the efficiency of regenerative braking I recommend this link.  In short, there are two limitations to capturing energy from regenerative braking. First, a portion of the brake force is applied through conventional brake pads. Second, the torque/speed of the wheels at braking cannot be tuned for optimal energy capture.  As a first pass approximation, we will assume that 20% of the energy is re-captured during brake implementation.
2. To instructors of Thermodynamics courses, please feel free to use this joke under a GPL Open Source License.
3. It only ever seems this way when you are driving.
4. Generally this is true with the exception of 25 MPH zones which always seem way slower than anyone drives.
5. Ok, so that never happens, but one could dream.
6. The first pass approximation of this is 10 mph/sec on acceleration, and 20 mph/sec on deceleration.
7. GM Technical Center is a large complex with tunnels connecting all the buildings; I often thought that I was working in a “rabbit’s warren.”
8. For the first pass, we are assuming a “flat” drive.  In southeast Michigan, this is generally true.
9. The stops require a “look ahead” model, e.g., we have to know when to start stopping.
10. I implemented this as a Stateflow chart with the intention that additional logic will be added to account for the driver behavior model in subsequent updates.  For now, it is a simple accel / deccel / hold calculation.
1. And because this was in 1995 there was zero chance for regenerative braking.  GM had the EV1 then but I did not own one.

## Everyday engineering: MBD & MPG (Part 2: Routes)

Getting from point A to point B sounds simple enough. Pull out of your driveway, turn left, turn right, turn left…(1) Yet as many a traveling salesperson has taught us, finding the best route is not a simple task. When you add in multiple competing factors, evaluating the cost of the route and determining the “best” path is not an easy task. Today we are going to take the first step in getting those routes by extracting map data.

## Map data: What do we care about?

For any given route(2) what data do we care about and how do we characterize it? The primary categories are physical data (distance, elevation) and road characteristics (speed limits, stop signs, turns). Additional data such as traffic patterns will be modeled in a future human factors post.

## Example routes

To start this off I use my first professional commute from Farmington Hills, MI to Warren, MI at the General Motors Technical Center.(3) This is primarily a highway drive and because of that, easiest to get data on. So how are we going to get this data?

## MAP APIs

• USGS: A United States Government service that contains a wealth of information
• Open Streets: An open-source map source
• Google Maps: Comes with a MATLAB friendly API(5)

Getting the data from the Google Maps API requires three queries:

• Distances: Translating Lat and Lon into meters or miles
• Elevations: Those ups and downs
• Roads: Used to get both the speed limits and the signage

## From tables to algorithms

For my starting point I used a static query, e.g. I downloaded the full route in one batch. The result is a 10,000+ entry point table set at 2-meter resolution.(6)

For my eventual model I added in a category of “ahead.” Ahead is calculated based on the position of a stop sign or turn, the posted speed limit, and a human factor of “driver aggression.” For safe drivers the “Ahead” will toggle early, leading to a reasonable deceleration.

## More routes different data

Several years later, my wife and I moved to Ann Arbor, MI, a small college town (home of the University of Michigan). During our Ann Arbor days, I had 2 commutes; first, an “in-town” route to ADI (Applied Dynamics International), the other a mix of highway and surface streets to Ford Research. The in-town transit will be of interest for the high number of “starts and stops” while the highway commute to Ford Research, where I worked on full vehicle simulations to predict fuel economy, will act as a validation point of the earlier GM commute.

## Next post

In the next post I will build up the physics model and then run it in a traffic free version of these maps.

## Footnotes

1. When my wife and I lived in the Boston area we joked that directions between any two points could be given just by referencing Dunkin Donuts. E.g. turn right at the first DD, turn left at the 3rd DD on your left; this was not far from the truth.
2. We will cover multiple routes in a future post.
3. I worked on my first full vehicle simulation H.I.L. system for GM on a project known as SimuCar. This is also the time period when I met my wife, Deborah, a great start to a wonderful life.
4. This post is centered around the US; there are other download sites for the rest of the world.
5. I selected Google Maps due to familiarity with the API; the others listed could be used with equal success. For other examples, these demos will give you a good start.
6. I selected 2 meters resolution for processing speed, based on a 1/2 car length estimate. In later posts I will examine how the map resolution effects the MPG estimates.

## Novel input and Safety-Critical failures in ML / AI / DL

Machine Learning (ML), Artificial Intelligence (AI), and Deep Learning (DL) have moved from the research end of embedded systems development into real-world applications. These methodologies are in use to support the solution to problems that traditional control algorithms either cannot solve or solve at a high computational cost. Adopting a new technology means understanding the advantages and disadvantages of the technology. Today I want to talk about the novel input problem and how it can result in critical system faults.

## Name that tune

ML/AI/DL systems have made huge strides in their ability to classify inputs into reasoned guesses. Recently Google announced a “name that tune” feature that allows users to provide a snippet and google will try to name that tune. In my 5 test cases it got 4 out of 5 right. The one it got wrong was John Cage’s 4’33”. It got it wrong because it is a song with no sound, something that falls outside all of the classification categories.

## Conceptually understanding a black box

ML/AI/DL systems are notoriously black box systems, e.g. what goes on inside is not known or in a functional sense, knowable. But, there is a way of thinking of them that can help with the design of the complete controls system. ML/AI/DL systems can be thought of as “self taught” expert systems.

They “ask” themselves a set of questions, and for each question they guide down to a set of probabilities to the final “answer.”

The ideal system works like this:

1. I know what this is: It is a hot dog!
2. I know the type of thing it is: It is a type of sausage
3. I know the general class of things: it is food
4. I don’t know what it is: but I think it is safe
5. I don’t know what it is: it may not be safe

Depending on the type of system you are designing you may need to reject the results for anything that comes back 3 or lower in the list.

## I don’t know: it may not be safe

Crying wolf, that is to say alerting when there is not a good reason to alert, is the curse of all ML/AI/DL systems. At the same time, as ML/AI/DL systems are often used in safety critical systems they need to have a better safe then sorry approach to input response. The first answer? A graduated response.

In most cases the warning can be responded to in stages. First “slow down” the rate of the event happening; use that time to re-evaluate the alert. If the conditions continue to merit, then “stop.”

The second approach? Stop. If you have a safety critical system, “fail safe” should always be prioritized over continued operation. Ideally the data gathered can be used as part of a DevOps workflow so that in the future the “correct” approach can be followed.

## I can use a slide rule…

I learned to use a slide rule in 11th grade AP physics. To be clear, at the time there was no need to learn how to use a slide rule as digital calculators were common enough and not too expensive. So why did I learn how to use one?

## Clarity of ideas

Having learned how to do multiplication and division with a slide rule I never forgot the fundamental properties of logarithms. The tool encoded an idea. This is often how we incorporate fundamental concepts, through using them. This is the argument for working with “primitive” tools; they can ground us in topics so we can use them going forward.

## You are grounded!

But there is a “tipping point.” Somewhere along the line, you will hit a point where further use of the “basic tools” has a diminishing return on investment. I never learned how to solve roots with a slide rule and I don’t think I would have benefited from it. How do you determine when you have “tipped”?

## How to tip

Unlike a restaurant, there is no set “tip” (15% for basic service in the US). Moreover you can easily miss it if you are not going into the exercise with the correct attitude. To identify the tipping point you need to:

1. Actively think about “what am I learning (Y) as I do X?”
2. Think about how what you learn could be applied to other situations.
3. Recognize when you are are just “getting better at X and not learning about Y.”

Follow those guidelines and you will slide into new areas with ease; after you slide for a while you will get the feel for when you are ready to dive in.

## Everyday engineering: MBD and MPG: Part 1

Every engineering project has a starting point where you map out what you want to realize and how you will enact your vision. For complicated projects there are iterations at each stage as the initial design process is honed in on and leads toward the final design. With this series of blog posts I am going to walk you through my design process for optimizing a set of my “historical daily commutes”(1) for fuel-efficiency.

## The cost function

Optimization requires a function (or set of functions) to optimize against. Our first task then is to outline what will go into our “cost function.”(2)

• Vehicle physics
• Vehicle efficiency: engine & driveline
• Vehicle aerodynamics: the effect of speed on MPG
• Environmental factors
• Route topology: the ups and downs that make your MPG go up or down
• Weather: The rain, the sleet, and the snow that changes traction and heating you know…
• Human factors(3)
• Driver behavior: lead-footed devil or soft-coasting angel?
• Traffic conditions: what are the other drivers like?

## Types of modeling

As I’ve written about before, one of the keys to creating a system is selecting the correct level of fidelity for your models. To that end I will consider:

• Vehicle physics: This can be done with basic “table lookup” models. Past experience shows that there are dimension returns on higher fidelity models/
• Environmental factors: The environment directly impacts the vehicle physics; modeling this can be done using real-world data (maps) and historical data (weather). These models will give us the chance to explore data-based modeling.
• Human factors: I will be drawing on network theory models for traffic flow and human/traffic interactions.

## Optimizing the cost function

This is where things become interesting; this cost function is highly nonlinear. How we navigate (optimize) the cost function is an open question. I will try and consider which of the following is the best approach:

• Segmentation: decompose the optimization into sub-optimization problems and run integration analysis?
• Neural network: global optimization of the system?
• Model linearization: create a linearized version of the models enabling linear optimizations?

## Footnote

1. I will be selecting commuting routes from my early years of work when I was part of the auto industry. Travel to the GM Tech Center in Warren, Ford’s Scientific Research Labs, and the Milford Proving grounds. Each route will be analyzed individually and then the goodness-of-fit of the algorithm will be compared between the results.
2. The objective with a cost function is to minimize the cost of the tasks. I’ve also seen this formulated as a “satisfaction quotient” where the objective is to maximize satisfaction. While I like the latter concept more, the minimization algorithms are simpler to implement so we will be using those.
3. When driving in rush hour it can seem like you are the only human on the road until someone is nice and lets you merge over to that lane you’ve needed to for the past 10 minutes.

## Everyday Model-Based Design

Over the next few months, I will be running a series of blog posts that I am calling “Every Day Model-Based Design.” In them, I will be creating physical and controls models to understand and explore everyday events. The first will be…

• Driving style MPG Optimization: Taking a look at my daily commutes throughout my career to determine an optimal driving style. Featuring:
• Big data analysis (e.g. traffic analysis)
• Physical modeling (e.g. road and vehicle behavior)
• Human modeling (e.g. how do I drive)
• TBD: What the world shows me…

I plan on having fun with these topics, going wide and deep as I explore how mathematics and systems engineering can describe the world around us.