Q2: Sensors, Sensor Mappings, State Spaces – Fundamentals of Sensing, Tracking, and Autonomy 1

Question 1¶

A linear sensor is calibrated using a linear model and least squares.

Which sensor property is directly improved by calibration?

Question 2¶

A linear sensor is modeled as:

r=as+b. \\\

What is the sensitivity of this sensor?

Question 3¶

TODO:

Add assumption that only humidity is the variable that affects the temperature readings, and it is an established fact.
Also add references to definitions of reliability, bias and ...
Add another question with ground truth for humidity, and calibration results.

A digital thermometer reports temperature in discrete steps of 0.1°C. Because of humidity variations, the thermometer’s readings vary from day to day. The true temperature and the reported value on each day are:

Day 1 (low humidity): true temperature 22.0°C, reported 21.9°C
Day 2 (medium humidity): true temperature 22.5°C, reported 22.1°C
Day 3 (high humidity): true temperature 22.6°C, reported 22.3°C
Day 4 (medium humidity): true temperature 22.0°C, reported 22.0°C
Day 5 (low humidity): true temperature 22.5°C, reported 21.8°C

Which of the following statements are DEFINITELY correct?

The sensor has low repeatability, because identical true temperatures produce different outputs.

The resolution of the sensor is 0.1°C.

The resolution of the sensor is 0.5°C.

The sensor exhibits a humidity-dependent bias.

Calibration alone is sufficient to fully correct the observed errors.

Measurement uncertainty is increased due to environmental effects.

The sensitivity of the sensor changes with humidity.

For the reported temperature of 22.0°C, the preimage has width at least 0.1°C.

The sensor is unreliable.

The observed variability is caused purely by random noise.

The sensor is repeatable but inaccurate.

A calibration model that includes humidity as an additional input could reduce the error.

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Question 4¶

A point-sized mobile robot moves in an environment:

E \subset \mathbb{R}^2. \\\

The robot position is given by coordinates (𝑥,𝑦) ∈ 𝐸 and it has no other degrees of freedom.

Let the state space be denoted by 𝑋. Which of the following statements correctly describe the state space 𝑋?

Question 5¶

A point-sized mobile robot moves in an environment:

E \subset \mathbb{R}^2. \\\

The robot position is (𝑥,𝑦) ∈ 𝐸 and the robot heading is 𝜃 ∈ 𝑆¹.

Let the state space be denoted by 𝑋. Which of the following statements correctly describe the state space 𝑋?

Question 6¶

Consider 10 point-sized mobile robots moving in a planar square environment

E = [-1,1]^2 \subset \mathbb{R}^2. \\\

Each robot has no heading and no additional degrees of freedom. Let the state space of the system be denoted by 𝑋, and the state of the robot 𝑖 is its position:

p_i = (x_i,y_i) \in E, \quad i = 1,\dots,10. \\\

Let the detection region be a fixed set 𝑉 ⊂ 𝐸.

Which of the following statements are TRUE?

Question 7¶

Consider a point-sized mobile robot with a heading that moves within a planar square environment:

E = [-1,1]^2 \subset \mathbb{R}^2. \\\

The robot state is (𝑥,𝑦,𝜃), in which (𝑥,𝑦) ∈ 𝐸 and 𝜃 ∈ 𝑆¹. Let the observation space be 𝑌 ⊂ ℝ.

A sensor is modeled as a virtual sensor mapping:

h : X \to Y, \\\

which maps each physical state to a sensor observation.

Which of the following are TRUE statements about a sensor mapping ℎ that measures the distance from the robot’s position to the closest wall?

ℎ(𝑥,𝑦,𝜃) = min{1 − |𝑥|, 1 − |𝑦|}.

ℎ(𝑥,𝑦,𝜃) = 𝑥² + 𝑦².

ℎ(𝑥,𝑦,𝜃) = |𝑥|cos𝜃 + |𝑦|sin𝜃.

The sensor reports the minimum Euclidean distance from the robot’s position to any wall of the square.

The sensor reports the distance to the wall in the direction the robot is facing.

The sensor reports the distance from the robot to the center of the environment.

The mapping ℎ depends only on the position (𝑥,𝑦), not on 𝜃.

The sensor output changes when the robot rotates in place.

The observation space 𝑌 = 𝐸.

The set of all possible observations is [0, 1].

The set of all possible observations is {0,1}.

The observation space 𝑌 ⊂ ℝ².

Multiple distinct states in 𝑋 can produce the same sensor observation.

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Question 8¶

Consider a point-sized mobile robot with a heading that moves within a planar disc environment:

E = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \le 1\} \subset \mathbb{R}^2. \\\

The robot state is (𝑥,𝑦,𝜃), where (𝑥,𝑦) ∈ 𝐸 and 𝜃 ∈ 𝑆¹. Let the observation space be 𝑌 ⊂ ℝ.

A sensor is modeled as a virtual sensor mapping:

h : X \to Y. \\\

Which of the following are TRUE statements about a sensor mapping ℎ that measures the distance from the robot’s position to the closest wall of the environment?

ℎ(𝑥,𝑦,𝜃) = 𝑥² + 𝑦².

ℎ(𝑥,𝑦,𝜃) = |𝑥|cos𝜃 + |𝑦|sin𝜃.

The sensor reports the minimum Euclidean distance from the robot’s position to the circular boundary.

The sensor reports the distance to the boundary in the direction the robot is facing.

The sensor reports the distance from the robot to the center of the environment.

The mapping ℎ depends only on the position (𝑥,𝑦), not on 𝜃.

The sensor output changes when the robot rotates in place.

The set of all possible observations is [0, 1].

The set of all possible observations is {0, 1}.

The set of all possible observations is [0, √2].

Multiple distinct states in 𝑋 can produce the same sensor observation.

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Question 9¶

Consider a point-sized mobile robot with a heading that moves within a planar square environment:

E = [-1,1]^2 \subset \mathbb{R}^2. \\\

The robot state is (𝑥,𝑦,𝜃), where (𝑥,𝑦) ∈ 𝐸 and 𝜃 ∈ 𝑆¹. Let the observation space be 𝑌 ⊂ ℝ.

A sensor is modeled as a virtual sensor mapping:

h : X \to Y. \\\

Which of the following are TRUE statements about a sensor mapping ℎ that measures the distance to the wall along the robot’s heading?

ℎ(𝑥,𝑦,𝜃) = min{𝑡 ≥ 0 ∣ (𝑥,𝑦) + 𝑡(cos𝜃, sin𝜃) ∈ ∂𝐸}.

ℎ(𝑥,𝑦,𝜃) = |𝑥| + |𝑦|.

The sensor reports the distance from the robot to the first wall it would hit if it moved straight forward.

The sensor reports the minimum distance to any wall.

The sensor reports the distance to the center of the environment.

The mapping ℎ depends on both (𝑥,𝑦) and 𝜃.

The sensor output is unchanged when the robot rotates in place.

The set of all possible observations is {0, 1}.

The set of all possible observations is [0, 1].

The set of all possible observations is [0, √2].

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Question 10¶

Consider a point-sized mobile robot with a heading that moves within a planar circular environment:

E = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \le 1\}. \\\

The robot state is (𝑥,𝑦,𝜃), where (𝑥,𝑦) ∈ 𝐸 and 𝜃 ∈ 𝑆¹. Let the observation space be 𝑌 ⊂ ℝ.

A sensor is modeled as a virtual sensor mapping:

h : X \to Y. \\\

Which of the following are TRUE statements about a sensor mapping ℎ that measures the distance to the wall along the robot’s heading?

ℎ(𝑥,𝑦,𝜃) = min{𝑡 ≥ 0 ∣ (𝑥,𝑦) + 𝑡(cos𝜃, sin𝜃) ∈ ∂𝐸}.

ℎ(𝑥,𝑦,𝜃) = 𝑥² + 𝑦².

ℎ(𝑥,𝑦,𝜃) = 1 − (𝑥² + 𝑦²).

The sensor reports the distance from the robot to the point where a ray in the heading direction intersects the boundary circle.

The sensor reports the minimum distance to the boundary.

The sensor reports the distance to the center of the circle.

The mapping ℎ depends on both (𝑥,𝑦) and 𝜃.

The sensor output is unchanged when the robot rotates in place.

The set of all possible observations is {0, 1}.

The set of all possible observations is [0, 1].

The set of all possible observations is [0, 2].

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Lovelace

QUIZ 2: Sensors, Sensor Mappings, State Spaces¶

Sensor Characteristics¶

Question 1¶

Question 2¶

Question 3¶

State Spaces¶

Question 4¶

Question 5¶

Question 6¶

Sensor Mappings¶

Question 7¶

Question 8¶

Question 9¶

Question 10¶

Authors¶

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