22-Agric-A5 Principles of Instrumentation · Undated paper
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 04-Agric-A5 Principles of Instrumentation, National Exam (printed exam date May 2019) — a three-hour open-book exam; any non-communicating calculator is permitted. Questions 1 and 2 are compulsory (20 marks each); candidates then choose any three (3) of Questions 3-7 (20 marks each) for a 100-mark paper. All seven questions are worked here.
Reference texts. E.O. Doebelin, Measurement Systems: Application and Design, 5th ed. (calibration, noise, dynamic sensor response, sampling and ADCs); J.P. Bentley, Principles of Measurement Systems, 4th ed. (error propagation, signal conditioning, bridge circuits); P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (op-amp circuits, precision rectifiers, instrumentation amplifiers, shot/Johnson noise); J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 5th ed. (photodetectors, gas sensors, Hall-effect and thermal sensors); F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer (forced-convection/King's-Law correlations for the hot-wire bridge).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
a) Given. An accelerometer outputs the instantaneous linear acceleration $a(t)$ along its sensitive axis. Find. How velocity and position are recovered. Velocity is obtained by integrating the sensed acceleration once with respect to time, and position by integrating a second time, each starting from a known initial condition: $$v(t)=v_0+\int_0^t a(\tau)\,d\tau,\qquad x(t)=x_0+\int_0^t v(\tau)\,d\tau.$$ For example, a constant sensed acceleration $a_0=2\text{ m/s}^2$ held for $t=4\text{ s}$ from rest ($v_0=x_0=0$) integrates in closed form to $v=a_0t=\boxed{8\text{ m/s}}$ and $x=\tfrac12a_0t^2=\boxed{16\text{ m}}$; a step-by-step numerical (trapezoidal-rule) integration of the same acceleration record reproduces these same values, which is exactly what an IMU's onboard processor does with sampled accelerometer data in practice.
b) Because position comes from a double integration of the raw signal, errors have no restoring term and accumulate (drift) rather than average out over time. A constant bias/offset in the accelerometer reading integrates into a linearly-growing velocity error and a quadratically-growing (accelerating) position error — even a very small, constant bias eventually dominates. Other contributors are scale-factor error, axis misalignment/cross-axis coupling, random noise (which integrates into a random-walk velocity error), and quantization/sampling error; all of them compound with time because nothing in a pure dead-reckoning integration corrects them.
c) Orientation in three-dimensional space has three independent rotational degrees of freedom (rotation about each of three mutually orthogonal axes, e.g. roll, pitch and yaw), so three single-axis rate gyroscopes — one per axis — are required to fully determine attitude (or a single tri-axial gyroscope package providing the equivalent three independent rate outputs). A one- or two-axis gyroscope set can only resolve rotation about that axis or that plane and leaves at least one rotational degree of freedom completely unobserved.
d) Both accelerometers and gyroscopes measure rates or derivatives (linear acceleration, angular rate) rather than absolute position or orientation. Integrating a rate signal only ever yields a change relative to whatever the system started at — the constant of integration is mathematically undetermined unless it is supplied externally. An initial reference (a known starting position/velocity for the accelerometer chain, a known starting orientation for the gyroscope chain) supplies that missing constant of integration; without it, the IMU's output is only ever a correct measure of the object's change in state, never its absolute state.
