Modeling Constrained Vector Path

00:00 10 marks

Consider a scenario where an autonomous drone must navigate a planar environment while avoiding a designated obstacle region. The drone’s path is modeled as a parametric vector function. Answer the following parts regarding the path’s parameterization and the condition required for obstacle avoidance.

Part a

7 minutes 5 marks

Question

Derive a parametric representation for the drone’s path if it starts at the origin and moves with constant speed \(v\) in a direction given by an angle \(\theta\) relative to the positive x-axis. Express the vector function \(\vec{r}(t)\) in terms of \(v\), \(t\), and \(\theta\).

Hint Hint

Hint

Remember that constant speed along a straight line can be modeled using \(\cos(\theta)\) and \(\sin(\theta)\) for the x and y components respectively.

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Complete Answer

\(\vec{r}(t) = \langle v\,t\cos(\theta),\; v\,t\sin(\theta)\rangle\)

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Part b

8 minutes 5 marks

Question

Given an obstacle represented by the region between the parallel lines \(y = \tan(\alpha)x + d\) and \(y = \tan(\alpha)x + d + L\) (with \(L > 0\)), determine the condition on \(\theta\) such that the drone’s path does not intersect the obstacle. Assume the drone starts from a position below the obstacle.

Hint Hint

Hint

Compare the slope of the drone’s path, \(\tan(\theta)\), with the slope of the obstacle lines, \(\tan(\alpha)\). Consider the relative positions based on the y-intercepts if slopes are equal.

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Complete Answer

\(\tan(\theta) < \tan(\alpha)\)

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