Unit 1: Measurements in Robotics

ECE246 — Sensors For Robotics 11 min read

I. Measurement Foundations

Robotic measurement is the process of obtaining information about physical variables and converting it into a form that a controller can interpret. A measurement chain normally runs from a physical phenomenon through a sensing element, signal-processing circuitry, and software to a control decision.

Governing principle: A sensor produces an output related to an input measurand, such as position, temperature, force, pressure, or acceleration.
Measurement convention: The measured value is reported with a unit, range, resolution, and uncertainty; for example, 25.0 °C ± 0.2 °C.
Reference requirement: Every measurement is compared with a reference, such as a voltage reference, timing clock, calibration standard, or known geometric position.
Robotic requirement: Sensors must support perception, localization, motion control, safety, and interaction with the environment.
Measurement chain: The usual sequence is measurand → sensor → transducer → signal conditioning → ADC or communication interface → controller.

II. Definitions of Sensors and Transducers — Physical-to-Information Conversion

A sensor detects a physical quantity, while a transducer converts energy or information from one form into another. In robotics, the terms often overlap, but their functional emphasis differs.

A. Definitions of sensors and transducers

This subsection distinguishes detection from conversion and identifies their roles in a robotic measurement system.

Sensor: A device that responds to a physical measurand and provides a usable output. An incremental encoder senses shaft rotation and outputs electrical pulses.
Transducer: A device that converts one form of energy into another. A thermocouple converts a temperature difference into a small thermoelectric voltage.
Primary sensing element: The part directly affected by the measurand. In a strain-gauge load cell, the elastic beam deforms under force.
Secondary conversion: Additional circuitry may convert the primary response into voltage, current, frequency, or digital data.
Active transducer: Generates its own output energy, as a piezoelectric sensor produces charge when mechanically stressed.
Passive transducer: Requires external excitation, as an RTD changes resistance but needs a current or voltage source for measurement.
Input and output variables: A sensor model may be written as y = f(x), where x is the measurand and y is the sensor output. Ideally, f(x) = Sx, with sensitivity S.

III. Errors & Classification — Quality and Taxonomy of Measurements

Sensor error is the difference between the indicated value and the accepted true or reference value. Classification organizes sensors by operating principle, energy requirement, output form, and relationship to the measured variable.

A. Errors & Classification

This subsection explains how errors arise and how sensor categories help engineers select suitable devices.

Measurement error:
TEXT
```
  e = x_m - x_t
```
Here, e is error, x_m is the measured value, and x_t is the reference or true value.
Systematic error: A repeatable bias, such as an encoder mounting offset of +2°; calibration can often reduce it.
Random error: Unpredictable variation caused by electrical noise, vibration, or environmental fluctuations; repeated measurements can estimate its spread.
Gross error: A mistake caused by wiring, incorrect range selection, or software interpretation rather than sensor physics.
Accuracy and precision: Accuracy describes closeness to the reference; precision describes repeatability. A sensor can be precise but inaccurate if it has a constant offset.
Absolute and relative error: Relative error is |e|/|x_t|; a 0.1 N error is significant at 1 N but small at 100 N.
Classification by energy: Active sensors generate energy; passive sensors require excitation.
Classification by output: Analog sensors provide continuously varying voltage or current; digital sensors provide pulses, counts, or coded data.
Classification by contact: Contact sensors physically touch the object, while proximity, optical, and ultrasonic sensors operate without contact.
Classification by measurand: Internal sensors measure robot state, such as motor current or joint angle; external sensors measure the environment, such as camera images or range data.

IV. Other Generalities — Measurement-System Context

Sensor selection depends on the complete measurement task rather than on the sensing element alone. Mechanical installation, electronics, computation, and environment all influence the final result.

A. Other generalities

This subsection covers practical concepts that connect a sensor specification to actual robotic operation.

Measurand: The quantity intended to be measured, such as wheel speed in rad/s or end-effector force in N.
Reference frame: A measurement must be associated with a coordinate frame. A camera detects an object in camera coordinates, which may need transformation into the robot base frame.
Static versus dynamic measurement: Static measurement concerns slowly changing inputs; dynamic measurement concerns time-varying inputs such as vibration or acceleration.
Loading effect: A sensor can alter the system being measured. A force probe with excessive stiffness may change contact behavior.
Environmental influence: Temperature, humidity, dust, electromagnetic interference, and mechanical shock can change sensor output.
Interface compatibility: Voltage levels, current limits, sampling rate, communication protocol, and connector standards must match the controller.
Redundancy: Two independent sensors measuring joint position can improve fault detection, although they increase cost and integration effort.
Observability: A robot state is observable when available sensor measurements contain enough information to estimate it, such as estimating velocity from position samples.

V. Definition and Architecture of Smart Sensors — Embedded Measurement

A smart sensor combines a sensing element with local electronics and computation. It can compensate, diagnose, process, and communicate measurements instead of sending only a raw analog signal.

A. Definition and architecture of smart sensors

This subsection describes the functional blocks that transform a conventional sensor into an intelligent measurement node.

Definition: A smart sensor is a sensor system containing sensing, signal conditioning, processing, and communication functions, often with memory and self-diagnostic capability.
Sensing element: Detects the physical quantity, such as a MEMS accelerometer responding to inertial force.
Excitation and conditioning: Provides bridge excitation, amplification, filtering, linearization, or isolation before conversion.
Data conversion: An ADC converts an analog signal into a code. For an ideal n-bit ADC with reference V_ref, the approximate voltage resolution is:
TEXT
```
  q = V_ref / 2^n
```
q is the least significant bit size.
Processor or microcontroller: Executes calibration compensation, unit conversion, limit checking, and diagnostic routines.
Memory: Stores calibration coefficients, serial number, configuration, and sometimes a sensor identification file.
Communication interface: CAN, I²C, SPI, UART, Ethernet, or wireless links transfer processed measurements to the robot controller.
Self-diagnostics: The sensor can detect open circuits, saturation, implausible values, or communication failure and report a fault status.

VI. Calibration Techniques — Relating Output to a Reference

Calibration establishes the relationship between sensor output and a known reference input. It is performed before deployment and may be repeated when accuracy requirements or environmental conditions demand it.

A. Calibration techniques

This subsection compares common calibration procedures and shows how calibration data is applied.

Static calibration: Known constant inputs are applied at several points. A pressure sensor may be tested at 0, 50, and 100 kPa.
Two-point calibration: Offset and gain are estimated from two references. For output y and input x, a linear model is:
TEXT
```
  y = ax + b
```
a is sensitivity or gain, and b is offset.
Multipoint calibration: Three or more reference points reveal nonlinearity and support a polynomial or lookup-table correction.
Comparison calibration: The test sensor is compared with a traceable reference instrument under the same conditions, such as comparing two thermometers in a controlled bath.
Zero or tare calibration: With the measurand set to zero, the measured offset is stored and subtracted during operation. A load cell may be tared before lifting a payload.
Dynamic calibration: The sensor is exposed to time-varying inputs to determine bandwidth, phase delay, and transient behavior.
Temperature compensation: Calibration coefficients may depend on temperature. A smart sensor can measure its own temperature and apply a correction table.
In-situ calibration: Calibration occurs while the sensor remains installed, useful for robot joint encoders or force sensors whose mounting affects the zero point.
Traceability: Calibration is meaningful when reference values are linked through an unbroken chain to recognized standards.

VII. Sensor Characteristics — Interpreting Specifications

Sensor characteristics describe how accurately, quickly, and reliably a sensor responds to its input. They are commonly divided into static characteristics for steady inputs and dynamic characteristics for changing inputs.

A. Sensor characteristics

This subsection defines the principal specifications used to compare robotic sensors.

Range: The interval over which specified performance is maintained, such as -10 to +10 g for an accelerometer.
Sensitivity: Output change per input change:
TEXT
```
  S = Δy / Δx
```
For a potentiometer producing 0–5 V over 0–180°, nominal sensitivity is 27.8 mV/°.
Resolution: Smallest detectable input change. A 12-bit ADC over 0–5 V has an ideal step of approximately 1.22 mV.
Linearity: Closeness of the calibration curve to a straight line; it is often specified as a percentage of full scale.
Hysteresis: Different outputs for the same input depending on whether the input is increasing or decreasing, common in magnetic and mechanical sensors.
Repeatability: Agreement among repeated measurements under unchanged conditions.
Response time: Time required to reach a specified fraction of the final output after an input step, often 63.2% for a first-order time constant.
Bandwidth: Frequency range over which amplitude and phase remain within specified limits. A force-control loop needs bandwidth high enough to capture contact changes.
Drift: Slow output change with time or temperature even when the measurand is constant.
Dead zone and threshold: Input interval producing no output change, or minimum input required to produce a detectable response.
Reliability and robustness: Ability to maintain operation under vibration, shock, contamination, and repeated duty cycles.

VIII. Signal Conditioning Basics — Preparing Sensor Signals

Signal conditioning modifies a raw sensor output so that it can be sampled, transmitted, and interpreted reliably. It may occur in analog electronics, digitally in a processor, or in both domains.

A. Signal conditioning basics

This subsection identifies the main operations between the sensing element and the controller.

Amplification: An instrumentation amplifier increases a small differential signal, such as the millivolt output of a strain-gauge bridge.
Attenuation: A voltage divider reduces a signal that exceeds ADC input limits; resistor tolerances affect its accuracy.
Level shifting: An offset circuit maps bipolar signals such as -2.5 to +2.5 V into a unipolar ADC range of 0–5 V.
Bridge completion: Wheatstone-bridge circuitry converts small resistance changes into a differential voltage and can provide temperature compensation.
Isolation: Optical, magnetic, or differential isolation reduces ground-loop and high-voltage risks.
Linearization: Software or analog circuitry corrects nonlinear response using a polynomial or lookup table.
Sampling and quantization: An ADC samples at frequency f_s and maps amplitude to discrete codes. The sampling rate must satisfy the signal-bandwidth requirement.
Anti-alias filtering: A low-pass filter before the ADC attenuates frequencies above the usable band so they do not appear as false low-frequency signals.
Protection: Clamps, fuses, series resistors, and transient suppressors protect inputs from electrostatic discharge and wiring faults.
Worked example: A 0–10 V sensor connected to a 0–5 V ADC requires a divider ratio of 1/2; the conditioned output is V_ADC = V_sensor/2.

IX. Noise and Filtering — Improving Measurement Quality

Noise is unwanted variation superimposed on the useful signal. Filtering reduces noise by exploiting differences between signal and noise in frequency, time, or statistical behavior, but filtering always introduces trade-offs.

A. Noise and filtering

This subsection explains common noise sources and the principal filtering methods used in robotic systems.

Thermal noise: Random voltage generated by resistive components; its power increases with resistance, temperature, and bandwidth.
Shot noise: Current fluctuation associated with discrete charge flow in semiconductor devices and photodetectors.
Electromagnetic interference: Motor switching, PWM drives, and long cables can couple unwanted signals into sensor wiring.
Quantization noise: ADC rounding produces an error generally bounded by approximately ±q/2, where q is the ADC step size.
Mechanical noise: Vibration and structural resonance can appear as acceleration or force signals, especially near the robot’s natural frequencies.
Low-pass filtering: Passes slow changes and attenuates rapid noise. A first-order filter has:
TEXT
```
  H(s) = 1 / (1 + sτ)
```
H(s) is transfer function, s is complex frequency, and τ is the time constant.
Moving-average filter: Averages the latest N samples:
TEXT
```
  y[k] = (1/N) Σ x[k-i],  i = 0 to N-1
```
x is the input sample and y is the filtered output; larger N gives more smoothing but more delay.
Median filter: Replaces a sample with the median of a neighborhood and is effective against isolated spikes in range measurements.
High-pass and band-pass filtering: High-pass filters remove slow drift; band-pass filters retain a chosen frequency interval, useful for vibration monitoring.
Trade-off: Greater attenuation usually means greater phase delay or slower response. Excessive filtering can destabilize a fast position or force-control loop.
Practical mitigation: Use twisted or shielded cables, differential inputs, proper grounding, separated power wiring, mechanical isolation, suitable sampling rates, and a filter cutoff above the control bandwidth but below dominant noise frequencies.