EE3901/EE5901 Sensor Technologies
Week 2 Tutorial

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College of Science and Engineering, James Cook University
Last updated: 14 January 2022

Question 1

Let x1x_{1} and x2x_{2} be random variables that have variances σx12\sigma_{x_{1}}^{2} and σx22\sigma_{x_{2}}^{2} respectively. Let y=f(x1,x2)y=f(x_{1},x_{2}) be a scalar function. If x1x_{1} and x2x_{2} are uncorrelated then prove that the standard deviation of yy is

σy=(fx1)2σx12+(fx2)2σx22.\begin{equation} \sigma_{y}=\sqrt{\left(\frac{\partial f}{\partial x_{1}}\right)^{2}\sigma_{x_{1}}^{2}+\left(\frac{\partial f}{\partial x_{2}}\right)^{2}\sigma_{x_{2}}^{2}}. \end{equation}

Hint: write expressions for the Jacobian and covariance matrix and then use σy2=JΣJT\sigma_{y}^{2}=J\Sigma J^{T}.

Question 2

You measure the current flowing through a circuit element and obtain i=2.20±0.1i=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±20R=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=iRv=iR) and its associated uncertainty?

Hint: use Eq. (1), above.

Question 3

A thermistor (a temperature sensor) has a transfer function


where RR is the resistance, TT is the absolute temperature, R0R_{0} is the resistance at a temperature T0T_{0} and kk is a sensitivity constant.

(a) Suppose that there is a measurement uncertainty in RR, modelled by a standard deviation σR\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 R0R_{0}. All other parameters are exact but R0R_{0} has a standard deviation σR0\sigma_{R_{0}}. Find an expression for the standard deviation in temperature.

(c) Now suppose that both RR and R0R_{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 kk), but keep everything else the same, would you expect the temperature precision to change?

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 σv=0.1 m/s\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 σa=0.2 m/s2\sigma_{a}=0.2\ \text{m/s}^{2}. The correlation in the errors is ρva=0.3\rho_{va}=0.3.

At a given instant, the measurements are v=1.7 m/sv=1.7\ \text{m/s} and a=0 m/s2a=0\ \text{m/s}^{2}. What is the standard deviation in PP?

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 yy axis and the vertical direction in the lab’s reference frame.

Figure Q5: The sensor being considered in this question. Image from Vicon IMeasureU BlueThunder data sheet. Zoom:

The angle can be calculated using the formula


where yy and zz 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 θ\theta given the following parameters:

y=0.3z=0.4σy=0.05σz=0.05ρyz=0.2.\begin{align*} y & =0.3\\ z & =0.4\\ \sigma_{y} & =0.05\\ \sigma_{z} & =0.05\\ \rho_{yz} & =0.2. \end{align*}

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

ddxarctanx=1x2+1.\frac{d}{dx}\arctan x=\frac{1}{x^{2}+1}.

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)f(x). For example, imagine a spring where the force depends upon how tightly it is compressed. However, the actual f(x)f(x) may be different to that of a spring. The system is designed such that x>0x>0 and f(x)>0f(x)>0 across the entire range of motion.

Figure Q6: The relationship between the linear actuator and terminal device. Zoom:

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.




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


(b) Find a suitable transfer function f(x)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 dxdx on one side and dfdf on the other side, then integrate and solve for f(x)f(x).

Question 7

You are calibrating a low-cost temperature sensor using the one point calibration method. With the sensor on the laboratory bench, you measure the temperature five times and obtain the following results: 26.86 °C, 26.91 °C, 28.04 °C, 27.99 °C, 27.99 °C.

You also have a reference thermometer, which you believe is substantially more accurate than the cheap sensor you are calibrating. It gives you the following measurements: 24.91 °C, 25.21 °C, 25.05 °C, 24.99 °C, 25.05 °C.

(a) Calculate the expected value of the measurements according to the sensor and the reference thermometer.

(b) Calculate the standard deviation of the measurements according to the sensor and the reference thermometer.

(c) Calculate the bias in the cheap sensor, assuming that the better thermometer is a reliable reference.

(d) Calibrate the cheap sensor, i.e. obtain an expression that can be used to correct for the bias.

(e) Suppose that you decide to combine information from both sensors by averaging their measurements. In other words, you will take both sensors into an environment to measure, you will obtain two measurements, and then average the result. What is the standard deviation of the average?

(f) You notice from part (e) that the standard deviation of the average is worse than the standard deviation of just using the better sensor, i.e. there was no benefit obtained from the cheap sensor at all. Intuitively, there must be one way to extract information even from low quality sensors, so you are motivated to try a weighted average method of the form


where 0w10 \leq w \leq 1, y1y_1 is the measurement according to the cheap sensor, and y2y_2 is the measurement from the better sensor. Find a value for ww that minimises σf2\sigma_{f}^{2}, and the value of σf\sigma_{f} at this minimum value.

Note: this principle (of giving more weight to those measurements which have lower variance) is a key concept behind the Kalman filter, which you will study in Weeks 3 - 4.