8.3: Exercises

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Take-home lessons

Uncertainty can be expressed by means of a probability density function.
More often than not, the Gaussian distribution is chosen as it allows treating error with powerful analytical tools.
In order to calculate the uncertainty of a variable that is derived from a series of measurements, we need to calculate a weighted sum in which each measurement’s variance is weighted by its impact on the output variable. This impact is expressed by the partial derivative of the function relating input to output.

Exercises

1. Given two observations q̂₁ and q̂₂ with variances σ₁ and σ₂ of a normal distributed process with actual value q̂, an optimal estimate can be calculated by minimizing the expression

\[S=\frac{1}{\sigma _{1}^{2}}(q̂-q̂_{1})^{2}+\frac{1}{\sigma _{2}^{2}}(q̂-q̂_{2})^{2}\]

Calculate q̂ so that S is minimized.

2. An ultrasound sensor measures distance x = c∆t/2. Here, c is the speed of sound and ∆t is the difference in time between emitting and receiving a signal.

Let the variance of your time measurement ∆t be σ²_t . What can you say about the variance of x, when c is assumed to be constant? Hint: how does a change in ∆t affect x?
Now assume that c is changing depending on location, weather, etc. and can be estimated with variance σ²_c . What is the variance of x now?

3. Consider a unicycle that turns with angular velocity φ and has radius r. Its speed is thus a function of φ and r and is given by

\[v=f(\phi ,r)=r\phi \]

Assume that your measurement of φ is noisy and has a standard deviation σ_φ. Use the error propagation law to calculate the resulting variance of your speed estimate σ²_v.