ISSUES.md

Issue Tracker

World's Okayest Group

Table of Contents

• Prelude
• Jerk
• Precision
• NOTE

Prelude

This is the issue tracker for the project. Issues can be theorized or actual, and solutions should be proposed as they come along.

Jerk

As a result of the equations used to map velocity over time, we have found that it's expected to have a harsh jump that levels off as the vehicle approaches the set speed. For acceleration, we are utilizing the solution to a seperable ODE of the form:

$V_{1}(t) = \frac{V_{s}}{1+\frac{V_{s}-V_0}{V_0} *e^{-t\alpha_{1}}}$

With $\alpha_{1}$ defined as:

$\alpha_{1} = -\frac{1}{t}ln((\frac{V_{s}}{V_{s}-\varepsilon}-1)(\frac{V_{0}}{V_{s}-V_{0}}))$

For deceleration, we are using:

$V_{2}(t) = \frac{V_{s}}{1+\frac{V_{s}-V_{0}}{V_0} *e^{-t\alpha_{2}}}$

With $\alpha_{2}$ defined as:

$\alpha_{2} = -\frac{1}{t}ln((\frac{V_{s}}{V_{s}+\varepsilon}-1)(\frac{V_{0}}{V_{s}-V_{0}}))$

Where $t$ is time in seconds, $V_{s}$ is the set velocity, $V_{0}$ is the initial velocity, and $\varepsilon$ is the single precision machine epsilon.

These equations, when plugged into graphing calculators, yield:

With the green function representing deceleration, and the black function representing acceleration over a $t$ of 2 seconds.

As a result, we expect extreme jerk to occur. We believe an effective solution will be to quadratically increase time allotments based on magnitude of change between set and inital velocity, with a point set at the initial velocity and the graph interpolated with that point at at a discretization rate of 0.1.

Precision and Singularity

We are currently encountering an issue with the ego vehicle having a singularity at $t$ = 0.25, where the derivative becomes non-finite. We believe this to be because it is being fed an input of -0.5, which will in turn cause a asymptote at the point. We may also be encountering subtractive cancellation in our precision, which could lead to a massive amount of error. We are pursuing further advice regarding a solution to this, as we are unsure of the problem and unable to effectively diagnose it ourselves.

Note

Algorithmically populate the mesh in the form of a logistic function over TIME Interpolate for the desired interval, then take the derivative of the resulting function for acceleration over time. Integrate subroutine SAFE such that:

SAFE polls TIME-GAP TAU

IF TAU < 0
	RETURN TRUE
ELSE
	RETURN FALSE

Where the returned value either allows for continued acceleration (0) or breaks acceleration (1), calling for deceleration from current point. This nets - A smooth polynomial such that V(t) exists and is valid over time 0-n (thus avoiding Runge phenomenon complications) in the desired form, where V'(t) shows the acceleration in m/s. A feedback loop contained in the subroutine to ensure safe acceleration.

Issues - If time-gap is very small, acceleration still permitted. Hard code in a minimum bound.

Graphically, we are establishing a dynamic linear upper bound on velocity with the subroutine such that if our velocity will cause a crash (time-gap becomes negative), it quits and decelerates back to a safe speed such that time-gap is approximately 0.

This is a new approach to this problem, rather than using the analytically solved equation or the temporary piecewise solution.

MATLAB has an integrated function interpn, with arg pchip that yields an interpolating polynomial.