EE3901/EE5901 Sensor Technologies

Week 2 Tutorial

Question 1

Let $x_{1}$ and $x_{2}$ be random variables that have variances $\sigma_{x_{1}}^{2}$ and $\sigma_{x_{2}}^{2}$ respectively. Let $y=f(x_{1},x_{2})$ be a scalar function. If $x_{1}$ and $x_{2}$ are uncorrelated then prove that the standard deviation of $y$ is

Hint: write expressions for the Jacobian and covariance matrix and then use $\sigma_{y}^{2}=J\Sigma J^{T}$.

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Question 2

You measure the current flowing through a circuit element and obtain $i=2.20\pm0.1$ mA, where the uncertainty is interpreted as the standard deviation of the measurement. Next, you measure the resistance of this element and obtain $R=500\pm20$ Ω, again where the uncertainty is interpreted as the standard deviation of the measurement. You believe that the current and resistance error are uncorrelated.

What is the voltage across the circuit element ($v=iR$) and its associated uncertainty?

Hint: use Eq. (1), above.

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Question 3

A thermistor (a temperature sensor) has a transfer function

where $R$ is the resistance, $T$ is the absolute temperature, $R_{0}$ is the resistance at a temperature $T_{0}$ and $k$ is a sensitivity constant.

(a) Suppose that there is a measurement uncertainty in $R$, modelled by a standard deviation $\sigma_{R}$. Assuming that all other parameters are exact, find an expression for the standard deviation in temperature.

(b) Now suppose that the uncertainty is instead found in the parameter $R_{0}$. All other parameters are exact but $R_{0}$ has a standard deviation $\sigma_{R_{0}}$. Find an expression for the standard deviation in temperature.

(c) Now suppose that both $R$ and $R_{0}$ are subject to uncertainty. You may assume that the two parameters are uncorrelated. Find an expression for standard deviation in temperature.

(d) If you change to a different thermistor with increased sensitivity (i.e. larger value of $k$), but keep everything else the same, would you expect the temperature precision to change?

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Question 4

A robot uses a wheel encoder to measure its velocity. There is software running on the robot that numerically differentiates velocity to obtain acceleration. The errors in velocity and acceleration are correlated because they derive from the same physical sensor.

The power consumption of the robot is estimated using the formula

where the first term represents drag and the second term represents the force required to change the acceleration.

When the robot is maintaining a constant velocity, the standard deviation of the velocity measurement is found to be $\sigma_{v}=0.1\ \text{m/s}$. Later, a constant acceleration is applied, and the standard deviation of the acceleration measurement is found to be $\sigma_{a}=0.2\ \text{m/s}^{2}$. The correlation in the errors is $\rho_{va}=0.3$.

At a given instant, the measurements are $v=1.7\ \text{m/s}$ and $a=0\ \text{m/s}^{2}$. What is the standard deviation in $P$?

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Question 5

A wearable inertial measurement unit is being used to monitor curvature of the spine during rowing. The sensor (Figure Q5) has a flat edge that is taped to the participant at specific vertebra of the spine. The parameter of interest is flexion and extension of the spine, meaning how far the person has bent forward or backward. This is characterised by the angle of the sensor with respect to the vertical, i.e. the angle between the sensor $y$ axis and the vertical direction in the lab’s reference frame.

The angle can be calculated using the formula

where $y$ and $z$ are the respective components of a unit vector that express the sensor’s Y axis in the reference frame of the laboratory.

Calculate the standard deviation in $\theta$ given the following parameters:

Hint: the derivative of arctan (when the angle is expressed in radians) is

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Question 6

A linear actuator drives a terminal device (e.g. gripper, hand, etc) of a robotic manipulator. The force exerted by the gripper is related to the displacement of the linear actuator by a function $f(x)$. For example, imagine a spring where the force depends upon how tightly it is compressed. However, the actual $f(x)$ may be different to that of a spring. The system is designed such that $x>0$ and $f(x)>0$ across the entire range of motion.

The terminal device is known to be perfectly accurate, but there is some error in the linear actuator. The error is expressed as a percentage by normalising to the standard deviation, e.g.

$$e_{x}=\frac{\sigma_{x}}{x}$$

and

$$e_{f}=\frac{\sigma_{f}}{f}.$$

(a) Show that the relative error in the force is given by:

$$e_{f}=\frac{x}{f}\frac{df}{dx}e_{x}.$$

(b) Find a suitable transfer function $f(x)$ such that the force error remains constant across the entire range of motion.

Hint: If the force error is constant then it must have no dependence upon f or x. Write an equation, separate the variables to obtain $dx$ on one side and $df$ on the other side, then integrate and solve for $f(x)$.

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Answer

(a) Outline of proof: use the result

$$\sigma_{f}^{2}=\left(\frac{\partial f}{\partial x}\right)^{2}\sigma_{x}^{2},$$

and use the definitions of $e_{f}$ and $e_{x}$ to eliminate $\sigma_{f}$ and $\sigma_{x}$ respectively. Also use the fact that standard deviations are always positive so you can take the positive square root only when simplifying the equation.

(b)

$$f(x)=ax^{e_{f}/e_{x}}$$

where a is an arbitrary constant.