EE3901/EE5901 Sensor Technologies
Week 2 Tutorial

College of Science and Engineering, James Cook University

Last updated 14 January 2022

Question 1

Let and be random variables that have variances and respectively. Let be a scalar function. If and are uncorrelated then prove that the standard deviation of is

Hint: write expressions for the Jacobian and covariance matrix and then use .

Answer

Since and are uncorrelated, the covariance matrix has zeros in the off-diagonal positions. Hence

Therefore

Question 2

You measure the current flowing through a circuit element and obtain mA, where the uncertainty is interpreted as the standard deviation of the measurement. Next, you measure the resistance of this element and obtain 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 () and its associated uncertainty?

Hint: use Eq. (1), above.

Answer

Question 3

A thermistor (a temperature sensor) has a transfer function

where is the resistance, is the absolute temperature, is the resistance at a temperature and is a sensitivity constant.

(a) Suppose that there is a measurement uncertainty in , modelled by a standard deviation . 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 . All other parameters are exact but has a standard deviation . Find an expression for the standard deviation in temperature.

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

Answer

(a) .

(b) .

(d) Yes, a more sensitive thermistor will result in less variance in temperature. Intuitively, the more the resistance changes, the easier it is to characterise the underlying temperature because the spread of resistances corresponds to a smaller spread of temperatures. (If you cannot see this, draw the transfer function with two different slopes and imagine the impact of a given amount of resistance error.)

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 . Later, a constant acceleration is applied, and the standard deviation of the acceleration measurement is found to be . The correlation in the errors is .

At a given instant, the measurements are and . What is the standard deviation in ?

Answer

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

The angle can be calculated using the formula

where and 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 given the following parameters:

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

Answer

Given we have

Hence the Jacobian is

The covariance matrix is

Hence the standard deviation of is

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 . For example, imagine a spring where the force depends upon how tightly it is compressed. However, the actual may be different to that of a spring. The system is designed such that and 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.

and

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

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

Answer

(a) Outline of proof: use the result

and use the definitions of and to eliminate and 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)

where a is an arbitrary constant.

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.

(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. Repeating 5 times as in the example here, you obtain a total of 10 measurements. What is the standard deviation obtained using this method?

(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 , is the measurement according to the cheap sensor, and is the measurement from the better sensor. Find a value for that minimises , and the value of 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.

Answer

(a) °C; °C.

(b) °C; °C.

Note that you must use the sample estimator, i.e.

with the on the denominator because this is just a sample, not the entire population.

(d) °C.

(e) Assuming uncorrelated errors, °C.

(f)

The standard deviation of the combined measurement is

i.e. smaller than the result in part (b). This shows that there was a benefit in combining the information from both sensors.