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QUIZ 3: Sampling, Motion Planning

Due date: 2026-04-02 23:59.
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Question 1

Suppose the configuration space is π’ž = [0, 1]ⁿ, and there are grid points at every (𝑖₁ Β· 0.1, 𝑖₂ Β· 0.1, …, 𝑖ₙ Β· 0.1) for 𝑖₁, 𝑖₂, …, 𝑖ₙ ∈ {0, 1, …, 10}.
Let 𝑁_π‘˜(π‘ž) for a grid point π‘ž be the set of grid points π‘žβ€² that are neighboring π‘ž in the following sense: at most π‘˜ coordinates in π‘ž and π‘žβ€² differ by 0.1. The rest must be identical. Also, π‘ž βˆ‰ 𝑁_π‘˜(π‘ž).
Which of the following are true?
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Question 2

Now let π’ž = (𝑆¹)ⁿ, in which each 𝑆¹ is represented as [0, 1] with 0 ∼ 1 (identification). The grid points and 𝑁_π‘˜ are defined as before, but differ by 0.1 uses the 𝑆¹ metric. For example, 0.9 and 0.0 differ by 0.1 on 𝑆¹.
Which of the following are true?
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Question 3

A unit quaternion π‘ž = (π‘Ž, 𝑏, 𝑐, 𝑑) with π‘ŽΒ² + 𝑏² + 𝑐² + 𝑑² = 1 represents a rotation of a rigid body by angle πœƒ about a unit axis 𝒗 = (𝑣₁, 𝑣₂, 𝑣₃) as:
q = (\cos(\theta/2), \sin(\theta/2) v_1, \sin(\theta/2) v_2, \sin(\theta/2) v_3). \\\
Which of the following are true?
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Question 4

The van der Corput sequence in base 𝑏 is constructed by writing the natural numbers 1, 2, 3, … in base 𝑏, then reflecting the digits about the decimal point.
For example, in base 2:
  • 1 β†’ 0.1β‚‚ = 1/2
  • 2 β†’ 0.01β‚‚ = 1/4
  • 3 β†’ 0.11β‚‚ = 3/4
  • ...
Which of the following are correct?
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Question 5

The dispersion of a finite point set 𝑃 in a metric space (𝑋, 𝜌) is defined as:
\delta(P) = \sup_{x \in X} \min_{p \in P} \rho(x, p) \\\
That is, the dispersion is the radius of the largest empty ball.
Consider the unit square 𝑋 = [0, 1]Β² and the following sequence of points: 𝑝₁ = (0, 0), 𝑝₂ = (1, 1), 𝑝₃ = (0, 1), 𝑝₄ = (1, 0), 𝑝₅ = (0.5, 0.5), 𝑝₆ = (0.25, 0.25), 𝑝₇ = (0.75, 0.75), π‘β‚ˆ = (0.25, 0.75), 𝑝₉ = (0.75, 0.25).
In the figure below you can visualise three sets formed by the above sequence:
𝑃₄ = {𝑝₁, 𝑝₂, 𝑝₃, 𝑝₄}, 𝑃₅ = {𝑝₁, …, 𝑝₅}, and 𝑃₉ = {𝑝₁, …, 𝑝₉}.
Which of the following are true?
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Question 6

Which of the following definitions and statements about motion planners are correct?
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Question 7

Consider a robot moving in π’ž = ℝ² with configuration π‘ž = (π‘₯, 𝑦). The robot's velocity (αΊ‹, ẏ) may be constrained as shown in the vizualizer below:
Which of the following are true?
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Question 8

The simple car model has configuration π‘ž = (π‘₯, 𝑦, πœƒ) and two inputs: 𝑒ₛ (speed) and π‘’πœ™ (steering angle). The equations of motion are:
\dot{x} = u_s \cos\theta, \quad \dot{y} = u_s \sin\theta, \quad \dot{\theta} = \frac{u_s}{L} \tan(u_\phi), \\\
in which 𝐿 is the distance between the front and rear axles.
You are welcome to use the tool below to visualize the car model equations of motion:
Which of the following are true?
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Authors

Steven M. LaValle, Anna LaValle.
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