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QUIZ 4: Sensor Fusion, Triangulation

Due date: 2026-02-07 23:59.
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Question 1

Suppose 𝑋 = ℝ³, corresponding to the position of a point, 𝑝, in a 3D environment. Consider a sensor that is a pinhole camera with known position and orientation, as defined in lecture 5.
Suppose the point 𝑝 is observed in the image (i.e., it lies in the field of view of the camera). Which statements are TRUE?
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Question 2

Using the same setup as in the previous question, suppose the point 𝑝 is not observed (i.e., it lies outside the field of view of the camera).
Which statements are TRUE?
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Question 3

Consider the state-time space 𝑍 defined as:
Z = X \times T, \qquad X = [-1,1], \qquad T = [0,10]. \\\
Define the sensor:
h(x,t)=t. \\\
Which statements are TRUE if the observation is ℎ(𝑧)=5?
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Question 4

Consider the preimages of a sensor ℎ : ℝ² → ℝ, defined as:
h(x)=\lVert x-(1,1)\rVert. \\\
Which statements are TRUE?
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Question 5

Consider a point robot with the state space 𝑋 = ℝ². Define a region 𝑉 ⊂ 𝑋 and a sensor:
h(x)= \begin{cases} 1, & x\in V\\ 0, & x\notin V. \end{cases} \\\
Which statements are TRUE?
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Question 6

Let 𝑋 = ℝ² be a state space. There are two sensors. One sensor, ℎ₁, measures the distance to a landmark located at coordinates (0, 0). The other sensor, ℎ₂, measures the distance to a landmark located at coordinates (0, 4). Consider a spatial triangulation approach that uses these sensors to solve the problem of estimating the state 𝑥 ∈ 𝑋.
Define the intersection of two preimages, 𝑦₁ and 𝑦₂, as Δ(𝑦₁, 𝑦₂) = ℎ₁⁻¹(𝑦₁) ∩ ℎ₂⁻¹(𝑦₂). Which statements are TRUE?
Note: Symbol ⟺ denotes "if and only if".
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Question 7

Consider the state–time space:
Z = X \times T, \quad X = \mathbb{R}^2, \quad T = [0,10]. \\\
Define three sensors on 𝑍:
  • Time sensor: ℎ₁(𝑥, 𝑡) = 𝑡.
  • Distance-to-landmark sensor: ℎ₂(𝑥, 𝑡) = ∥𝑥 − (0, 0)∥.
  • Detection sensor for a region 𝑉 ⊂ 𝑋:
h_3(x,t) = \begin{cases} 1, & x \in V, \\ 0, & x \notin V. \end{cases} \\\
For three observations, 𝑦₁, 𝑦₂, and 𝑦₃, define the 3-way triangulation as the intersection of three preimages:
\Delta(y_1, y_2, y_3) = h_1^{-1}(y_1) \cap h_2^{-1}(y_2) \cap h_3^{-1}(y_3). \\\
Which statements are TRUE?
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Question 8

Consider the state space 𝑋 = ℝ².
  • The distance-to-landmark at (1, 1) sensor is defined as:
ℎ_1(𝑥) = ∥𝑥 − (1, 1)∥. \\\
  • Now suppose the sensor is affected by a nondeterministic disturbance with known bound 𝜀 > 0, with the resulting nondeterministic sensor:
h_2(x,d)= ∥x − (1,1)∥ + d, \,\,\,\, d \in [-\varepsilon, \varepsilon]. \\\
Let 𝐻₂⁻¹ denote the nondeterministic preimage of ℎ₂(𝑥, 𝑑). Which statements are TRUE?
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Question 9

Consider the state-time space
Z = X \times T, \qquad X = \mathbb{R}^2, \qquad T = [0,10]. \\\
Now consider the time, distance, and detection sensors, as in Question 7, but assume that the time and distance sensors are affected by nondeterministic disturbances with known bounds. Let 𝜀₁ > 0 and 𝜀₂ > 0 denote the disturbance bounds for the time and distance sensors, respectively.
Define the following sensors on 𝑍:
  • Nondeterministic time sensor: ℎ₁(𝑥, 𝑡, 𝑑) = 𝑡 + 𝑑, 𝑑 ∈ [−𝜀₁, 𝜀₁].
  • Nondeterministic distance-to-landmark at (0,0) sensor: ℎ₂(𝑥, 𝑡, 𝑑) = ‖𝑥‖+ 𝑑, 𝑑 ∈ [−𝜀₂, 𝜀₂].
  • Detection sensor for a region 𝑉 ⊂ 𝑋 (no disturbance):
h_3(x,t) = \begin{cases} 1, & x \in V, \\ 0, & x \notin V. \end{cases} \\\
For observations 𝑦₁, 𝑦₂, 𝑦₃, define the 3-way triangulation as the intersection of the corresponding preimages:
\Delta(y_1, y_2, y_3) = H_{1}^{-1}(y_1) \cap H_{2}^{-1}(y_2) \cap h_3^{-1}(y_3). \\\
Which statements are TRUE?
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Question 10

Consider a point robot with the state space 𝑋 = ℝ². Let 𝑉 ⊂ 𝑋 be a detection region.
The robot is equipped with a probabilistic detection sensor that outputs 𝑦 = ℎ(𝑥) ∈ {0, 1}, where 𝑦 = 1 indicates 𝑥 ∈ 𝑉 and 𝑦 = 0 indicates 𝑥 ∉ 𝑉.
The sensor model is:
P(y=1 \mid x\in V)=\tfrac{9}{10}, \qquad P(y=0 \mid x\in V)=\tfrac{1}{10}, \\\ \\\ P(y=1 \mid x\notin V)=\tfrac{1}{10}, \qquad P(y=0 \mid x\notin V)=\tfrac{9}{10}. \\\
Assume the prior:
P(x\in V)=\tfrac{1}{2}. \\\
Suppose the observation is 𝑦 = 1. Which statements are TRUE?
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Authors

Anna LaValle, Steven M. LaValle.
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