Mood Brightener: ...more from Stay Homas. (Confination IV)

 

Questions/Answers from Reflection on Previous Days:

Perhaps, upon reflection, you have some more questions as a result of previous days discussions?

[Placeholder for Q/A]

Translating Movement into Position (Location)

The math we will be using whenever we are relating how fast we are going to how far we have moved is one that you already know and use in your daily life even if you don't think of it as math. If you have 70 km to travel and the speed limit on the road is 35 km/h how long do you expect it to take you to reach your destination? The math you just did in your head is based on the following mathematical statement:

Distance (traveled) = rate (speed) * time

From a physics and science perspective the point I want to make here is that each of the quantities in this expression have specific units that give them meaning. Distance is measured in meters, or cm, or km. Time can be measures in seconds, minutes, or hours. Finally the rate at which the object is traveling (usually called the speed) has units of distance divided by time: km/h, m/s, or perhaps cm/min. What is particularly important is that in any calculation we do using this expression we need to make sure the units make sense BEFORE we worry about the numbers.

distance in km = rate in km/h * time in h or km = (km/h)*h

You'll notice that the last two terms are km/h and h which have hours upstairs and downstairs in the fractions (h = h/1 as fraction). That means the h's (hours) cancel and we are left with km on both sides of the equation. This is as it should be since the equal sign says that the stuff on the left is the same as the stuff on the right which means they must ultimately have the same units. This is an important way that science checks what is happening by being sure that the units on the left of the = sign are the same as the units on the right of the = sign.

distance in m = rate in m/s * time in h or m = (m/s)*h

This expression doesn't meet our test since the units on the left are m and the units on the right are (m*h/s) which are not the same. If I convert my time into s then things will work out but until I do that the calculation is not really valid.

New Symbols:

In physics and science we tend to use symbols as a shorthand for various quantities to be clearer and consistent. Distance is usually represented as x (or sometimes y depending on direction). Time is indicated with t and rate (speed) is represented by v. V is used because velocity and speed are very closely related. Rewriting this expression as we would in physics or engineering leads to...

x = v*t

which a physicist would read as position equals speed times time. There are some subtleties that I am glossing over here but that will do for now. From some previous algebra course you know that I can rearrange this expression to solve for any of the three quantities (assuming the units are correct)

x = v*t; v = x/t; and t = x/v

Thunder and Lightning:

So....take a moment and think about what you were taught about thunder and lightning and how to tell how close the lightning is. Nearly everybody has been taught something in their family. If we have anyone in the class from other parts of the world we may hear that they were taught a different rule. All I want you to do is remember what you were taught and write it down. I will ask you all to share when we have class.

I'm not going to 'tell you the answer' here because that's no fun. Let's approach this by asking how the whole thunder and lightning process works. Most people would say that they were told to notice when the lightning flashes and then count seconds until the thunder arrives. So how do we actually calculate this? I apologize for Darth Vader-esque voice in this video.

 

At some point you will be asked to figure out the thunder and lightning rule that your friends in Italy should teach their kids when lightning is 1 km away.

GPS:

GPS (Global Positioning System) works very much the same way you know how far away the lightning is (if you do your math right). There is no compelling reason for me to explain this when there are web resources that do it better than I can. Go read this description (all 5 pages for sure) to get a sense for how this works. When you are done reading you should be able to explain to me (in class) why it takes 4 satellites to figure this out. This is not a simple math problem but conceptually it should feel comfortable for you.

What about the precision of GPS? Knowing where you are with GPS depends on knowing how long (time of flight) it took the signal to reach you. If we know that time of flight to 1 ns (what is a ns? -- a nanosec) then that sets limits on how precisely we can know where we are. Light (radio signals are a form of light) travels at 3^.10⁸ m/s so how far does it go in 1 ns? Try it yourself first and check out the video if you need support.

 

 

The result is 30 cm or .3 m which is about a foot. If we can measure the GPS times from the satellites with an accuracy of 1 ns then we can determine where we are within about 30 cm (1 ft). From a historical perspective you should be aware of Admiral G Hopper who famously handed out 'nanoseconds' to trainees in her classes.

 

 

GPS is a reasonable system for keeping track of where a car is or even a person outside. Why is it less useful for figuring out where a robot exploring a hallway is or where the tip of a robotic welder is?

Ultrasonic Ranger:

Because light moves so fast it's hard to use it as a distance measurement for short distances. It can be done as you may have seen on the DARPA Grand Challenge (the rotating LIDAR on top of one of the cars) but it is a challenging technical problem. For our robots we will use sound waves to measure distances much like bats do with their echo locations. Imagine you are some distance from a wall and you send a pulse of sound towards the wall. If it returns to you in 420 μs (remember those prefixes?) how far away is the wall?

 

Dead Reckoning:

So far we have talked about how to describe where you are (the previous Breadcrumb) and various ways of measuring and calculating where you are. GPS is a wonderful modern tool that allows us to simply ask where we are and get an answer. As you are well aware this only works if you can receive signals from the GPS satellites. What did people do before we had GPS? If we have maps we and know how to read them we can look at the map and determine where we are relative to some feature in the landscape much like the Ultrasonic Ranger can determine how far away from the wall it is. But what if you don't have known reference points that you can 'see'? This problem is relevant to mariners at sea (where the ocean looks pretty much the same in all directions) or if you're a pilot who has no instruments for whatever reason. Think of this from the perspective of Amelia Earhart who had to determine where she was relative to a featureless sky and ocean knowing only where she started and where she needed to get to. In the context of autonomous vehicles having a secondary way of tracking where the vehicle is seems thoughtful in the event the GPS system is unavailable. Here's an interesting discussion of the strengths and weaknesses of the various global positioning systems.

Dead Reckoning is the process of determining how far you've gone from your starting point (using the x=v*t math from earlier) and tracking it using some location system like an x-y plot. If someone says to you walk forward 10 steps, turn left and walk 5 steps, turn left and walk 3 steps, turn left and walk 5 steps many of you can visualize where you are now relative to where you started. If not trying drawing it on a piece of paper. For a variety of reasons autonomous vehicles are using a form of dead reckoning to keep track of where they are relative to the last GPS measurement they were able to make.

Assignment Breadcrumb Reading: Bb Assignment

Thunder and Lightning!

Knowing that sound travels at 343 m/s on earth and light travels at 3^.10⁸ m/s calculate the delay time between the arrival of the light from the lightning and the arrival of the sound when the lightning is 1600 m (1 mile) away. Show each calculation and articulate a simplified rule to teach your kids and your family members to determine how far away lighting is.

Before Next Class:

Assignment HW: Bb Quiz

Dead Reckoning

Here is your dead reckoning problem. It may remind you of your list of waypoints from a previous HW problem.

i) Go forward for 5 s at 20 cm/s, turn R (all turns will be 90 degrees).

ii) Go forward (new direction) for twice the previous time at 1/4 the speed, Turn R.

iii) Go forward 120 cm in 12 s, Turn R.

iv) Go forward for 9 s at your most recent speed.

v) Back up 40 cm at half your previous speed.

You will be asked how far you are from where you started and how long the entire trip took (ignoring the time to change direction)

Assignment HW: Bb Quiz

Ultrasound Distance

In this quiz you will be given a time of flight for a sound pulse bouncing off the wall and you will be asked to calculate how far away the wall is. The time of flight will be randomly generated so do it until you get it correct 2x in a row.

Looking Ahead:

Look ahead to the next Breadcrumb: Moving?

Assignment: