[Uncertainty] Sensing

 

• The robot in the image is sending out a green dashed line toward a cylindrical object, representing a measurement (probably range + bearing).

 

• This is typical of range-bearing sensing, where you get both:

- Distance (range) to the object

- Direction (bearing) of the object

 

 

• However, uncertainty in these sensors makes localization of objects ambiguous.

 

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• Banana Shape Problem

 

 

- A common nonlinear uncertainty distribution in range-bearing sensing.

- When you only measure distance and angle to an object from one position, your uncertainty in object location is not symmetric — it forms a curved, banana-like shape in 2D.

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For example,

 

 

- An object is sensed 20 cm away at a 60° angle.

- But the sensors are imperfect:

Range error: ±1 cm

Bearing error: ±10°

- These errors create uncertainty in the actual object location.

- The triangle and green dashed line show the robot and the direction of measurement, while the shaded area suggests the range of possible actual positions.

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- To model uncertainty, generate 1,000 samples (particles) assuming:

Range r ∈ [19, 21] cm

Angle a ∈ [50°, 70°]

- Then transform to Cartesian coordinates: \( \begin{aligned} x &= r \cos(a) \\ y &= r \sin(a) \end{aligned} \)

 

- This produces a spread of points in 2D space that represent where the object might be.

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- The generated particles form a cloud in a banana-like shape.

- One of them is the actual object, but we don’t know which.

- This motivates using a particle filter — a probabilistic approach that updates the belief of where the object is based on measurements.

- The banana-shaped distribution is not symmetric.

- It is not Gaussian, but a particle cloud.

- A first-order approximation may be fitted to this, but it’s not perfect.

- The transformation from range and bearing to Cartesian is nonlinear.

- We must linearize to apply linear estimation methods.

 

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• Linearization of Nonlinear Equation

 

- Use first-order Taylor expansion around a point \( (r_0, a_0)\) \[ f(x, y) \approx f(r_0, a_0) + \begin{bmatrix} \frac{\partial f}{\partial r} & \frac{\partial f}{\partial a} \end{bmatrix} \begin{bmatrix} r - r_0 \\ a - a_0 \end{bmatrix} \]

- This lets us represent the uncertainty locally using a matrix A (Jacobian) even though the original transformation is nonlinear.

- Here, the linearization point is chosen: \( r_0 = 20\,\text{cm}, \quad a_0 = 60^\circ \)

 

 

- At this point, compute the Jacobian matrix A to approximate the local uncertainty.