2.5 The TNB Frame

The TNB Frame

Introduction

Direction of travel

The Direction of Turning

The Binormal Vector

As CEO and Head of Engineering at Steer Clear, you decide that you are almost ready to start testing your self-driving car out on the road. Your navigation and telemetry software will use the location tracking system (LTS) to determine all of the important information about how the car is moving and how adjustments need to be made. Before you start programming your software, you decide to drive on a quiet, country road and collect data from the LTS to use as test data for your programming. In other words, you will drive on a section of road you already have mapped out in order to check that your software is calculating the correct information. The map in below shows the path you plan to take on the country road (with the direction given by the arrows on the plot). Notice that at points $B$ and $F$, the road crosses itself to go in a different direction.

At each of the labeled points on the curve, draw a vector in the direction of travel. Write a sentence about how you are determining this vector at the various points.

Work through this exercise and explain your reasoning step by step.

Find $\vv$, speed, and $\vT$ as functions of $t$ for the line parameterized as $\vr(t)=\langle 3t-1,2-2t,5+t\rangle$. Write a few sentences about why your results make sense and why in for this particular curve $\vT$ does not vary based on the parameter value.

Find $\vv$, speed, and $\vT$ for the curve with parameterization $\vr(t)=\langle t,t^2,t^3\rangle$.

Consider a curve for which we do not know the parameterization. However, we do know that at a point $P$, the tangent line to the curve through $P$ can be parameterized as $\langle 2-7t, 3t+1,-4t-1\rangle$. What are you able to say about $\vT$ for this curve at the point $P$?

Draw an example of a curve such that $\vT$ does not exist at a point on your curve. Identify the point where $\vT$ fails to exist. Explain both why the direction of travel does not exist at that point based on your plot and why the direction of turning will not exist at the same point.

Draw an example of a curve such that $\vT\, '=\vec{0}$ at a point on your curve. Explain why $\vT\, '=\vec{0}$ based on your plot and explain why $\vN$ does not exist at the same point.

Work through this exercise and explain your reasoning step by step.

Calculate velocity $\vv(t)$ and $\text{speed}(t)$ for the parameterization $\vr(t) =\langle t,t^2,t^3\rangle$.

Now calculate $\vT(t)$ for the path given by $\vr(t) =\langle t,t^2,t^3\rangle$.

Calculate the first component of $\vT\, '=\frac{d\vT}{dt}$.

Calculate the second component of $\vT\, '$.

Calculate the third component of $\vT\, '$.

Put together your work for the three components of $\vT\, '$.

In the previous part, you likely decided that trying to simplify your formulas for $\frac{d\vT}{dt}$ was more complicated than you had space for on your paper, and note that we still would need to calculate $\vecmag{\frac{d\vT}{dt}}$. Instead of doing more intricate algebraic computations, describe how you would calculate $\vN$ if you had nice formulas for $\frac{d\vT}{dt}$ and $\vecmag{\frac{d\vT}{dt}}$.

Work through this exercise and explain your reasoning step by step.