Unit 2: Force & Velocity Transducers

ECE246 — Sensors For Robotics 11 min read

I. Orientation

Force and velocity transducers convert mechanical variables into electrical signals suitable for measurement and robot control. Force is commonly inferred from deformation, while velocity is obtained from generated voltage, pulse frequency, or the time derivative of position. The measurement chain includes the sensing element, signal conditioning, analogue-to-digital conversion, and estimation.

Mechanical input: Force is measured in newtons (N), torque in newton-metres (N·m), displacement in metres (m), and velocity in metres per second (m/s) or revolutions per minute (rpm).
Transduction principle: A transducer changes one form of energy or physical quantity into another, usually an electrical voltage, current, resistance, frequency, or digital pulse train.
Static characteristics: Sensitivity, range, linearity, hysteresis, repeatability, resolution, and zero drift determine measurement quality.
Dynamic characteristics: Bandwidth, response time, damping, natural frequency, and phase delay determine whether rapid robot motion can be measured accurately.
Measurement convention: Sensor output should be calibrated against a known input, and its sign must be defined consistently with the robot coordinate system.
Practical assumption: The sensor should disturb the mechanical system as little as possible while providing an adequate signal-to-noise ratio.

II. Force Measurement Generalities — Mechanical input and sensor performance

Force measurement determines the magnitude and direction of an applied load from the mechanical effect produced in a sensing element.

A. Force measurement generalities

The purpose of force measurement generalities is to relate an applied force to a measurable physical change while controlling error sources.

Newton’s law: Force may be related to acceleration by
TEXT
```
  F = m a
```
where F is force in N, m is mass in kg, and a is acceleration in m/s².
Deformation method: An elastic member deflects under load; within its elastic range, deformation is approximately proportional to force:
TEXT
```
  F = k x
```
where k is stiffness in N/m and x is deflection in m.
Sensitivity: For an output voltage V_o, sensitivity is
TEXT
```
  S = ΔV_o / ΔF
```
expressed, for example, in mV/N.
Error sources: Offset, temperature variation, creep, hysteresis, overload, mounting stress, and electromagnetic interference can all alter the indicated force.
Robot application: Wrist force sensors commonly measure three forces (F_x, F_y, F_z) and three moments (M_x, M_y, M_z) for compliant gripping and collision detection.

III. Strain Gauges — Resistive sensing through elastic deformation

A strain gauge measures strain by changing its electrical resistance when bonded to a deforming surface.

A. Strain gauges

The operating principle is that mechanical strain changes the length and cross-sectional area of a conductor, producing a small resistance variation.

Strain definition: Normal strain is
TEXT
```
  ε = ΔL / L
```
where ε is dimensionless strain, ΔL is change in length, and L is original length.
Gauge factor: The resistance response is described by
TEXT
```
  GF = (ΔR / R) / ε
```
where R is nominal resistance and ΔR is resistance change. Metallic gauges typically have GF ≈ 2.
Stress relation: For an elastic material,
TEXT
```
  σ = E ε
```
where σ is stress in Pa and E is Young’s modulus in Pa.
Construction: A thin metal foil grid is bonded to the test member. The adhesive must transfer strain and remain stable over the operating temperature range.
Bridge arrangement: Four gauges in a Wheatstone bridge can increase sensitivity and compensate temperature. Tension and compression gauges are placed in adjacent or opposite arms according to the desired output polarity.
Limitation: Gauge outputs are small, often a few millivolts per volt of bridge excitation, so low-noise amplification and careful wiring are required.

IV. Non-Resistive Transducers — Alternative force-sensing principles

Non-resistive transducers produce force-related changes in charge, capacitance, inductance, or magnetic field rather than primarily changing resistance.

A. Non-resistive transducers

These devices are useful when high frequency response, non-contact operation, or very small displacement must be measured.

Piezoelectric sensor: A crystal or ceramic generates charge under force:
TEXT
```
  Q = d F
```
where Q is charge in C, d is the piezoelectric coefficient in C/N, and F is force in N.
- Strength: Excellent dynamic response and high stiffness.
- Limitation: Charge leakage prevents reliable static or very low-frequency force measurement.
Capacitive sensor: Force changes plate spacing or overlap, changing capacitance:
TEXT
```
  C = ε_r ε_0 A / d
```
where C is capacitance, ε_r is relative permittivity, ε_0 is vacuum permittivity, A is plate area, and d is separation.
Inductive sensor: Displacement caused by force changes inductance or mutual coupling. Linear variable differential transformers (LVDTs) provide robust displacement measurement with good repeatability.
Magnetic sensor: Hall-effect or magnetoresistive elements detect changes in magnetic field caused by motion or force.
Selection criterion: Piezoelectric sensors suit impacts, capacitive devices suit small static deflections, and inductive sensors suit industrial displacement measurement.

V. Load Cells — Calibrated force-measuring assemblies

A load cell is a mechanically packaged force transducer designed to produce a calibrated electrical output proportional to load.

A. Load cells

Load cells combine an elastic structure with one or more sensing elements so that an applied force produces a controlled strain field.

Strain-gauge load cell: A metal element such as a beam, column, or shear web carries the load; gauges bonded at high-strain regions measure its deformation.
Common forms: Single-point, bending-beam, shear-beam, compression, tension, and S-type load cells are selected according to mounting and force direction.
Rated capacity: A load cell might be specified as 0–1 kN; applying force beyond its safe overload rating can cause permanent deformation.
Bridge output: A typical specification is 2 mV/V. With 10 V excitation at full load:
TEXT
```
  V_full-scale = 2 mV/V × 10 V = 20 mV
```
Calibration: Known reference loads establish offset and scale. A linear calibration model is
TEXT
```
  F = (V_o - V_zero) / S
```
where F is force, V_zero is unloaded output, and S is volts per newton.
Installation: Load must follow the intended axis; side loads, uneven mounting, cable forces, and structural friction create measurement errors.

VI. Interfacing of Force Transducers — Conditioning small analogue signals

Interfacing of force transducers converts a weak sensor output into a stable, digitized signal for a controller.

A. Interfacing of force transducers

The interface must provide excitation, differential amplification, filtering, protection, and conversion without adding significant offset or noise.

Bridge excitation: A regulated voltage supplies the Wheatstone bridge. Ratiometric ADC measurement reduces error because sensor output and ADC reference vary together.
Instrumentation amplifier: A high common-mode rejection ratio (CMRR) amplifier measures the bridge difference while rejecting shared interference:
TEXT
```
  V_o = G (V_+ - V_-)
```
where G is gain and V_+, V_- are differential input voltages.
Gain selection: If a bridge produces 20 mV full scale and the ADC accepts 0–5 V, the ideal gain is approximately 250, subject to headroom and offset requirements.
Filtering: An analogue low-pass filter suppresses vibration and aliasing; the cutoff frequency must be below half the sampling frequency:
TEXT
```
  f_c < f_s / 2
```
where f_c is filter cutoff and f_s is sampling frequency.
ADC requirements: Resolution, reference stability, sampling rate, and input range determine the smallest detectable force.
Protection and grounding: Shielded twisted-pair wiring, star grounding, input protection, and isolation may be needed in motor-driven robots.

VII. Velocity Measurements — Translational and rotational motion

Velocity measurements determine how quickly position changes, either directly through a velocity sensor or indirectly from position observations.

A. Velocity measurements

Velocity is the time derivative of position and may be measured as a continuous analogue quantity or estimated from digital counts.

Definition:
TEXT
```
  v(t) = dx(t) / dt
```
where v is translational velocity in m/s and x is position in m.
Rotational form:
TEXT
```
  ω(t) = dθ(t) / dt
```
where ω is angular velocity in rad/s and θ is angular position in rad.
Direct measurement: A tachogenerator produces voltage proportional to angular speed.
Incremental measurement: An encoder produces pulses; pulse frequency or counted edges gives speed.
Numerical estimation: For sampled positions,
TEXT
```
  v[k] ≈ (x[k] - x[k-1]) / T_s
```
where T_s is sample period. Differentiation amplifies high-frequency noise.
Design concern: Sampling rate must capture the highest expected speed and preserve sufficient resolution at low speed.

VIII. Tacho-Generator — Analogue rotational velocity sensing

A tacho-generator is a small electric generator whose output voltage is approximately proportional to shaft angular velocity.

A. Tacho-generator

The device uses electromagnetic induction: shaft rotation produces an output voltage whose magnitude indicates speed and whose polarity indicates direction.

Ideal relationship:
TEXT
```
  V_t = K_t ω
```
where V_t is tachogenerator voltage in V, K_t is tachometer constant in V·s/rad, and ω is angular velocity in rad/s.
Direction: Reversing rotation reverses the polarity of V_t, enabling signed velocity feedback.
Example: If K_t = 0.05 V·s/rad and ω = 100 rad/s, then V_t = 5 V.
Dynamic effects: Armature inductance, friction, brush noise, ripple, and loading cause deviation from the ideal proportional relation.
Advantages: Continuous output, simple feedback, and high update rate make tachogenerators useful in motor speed control.
Limitations: They require mechanical coupling, add inertia, wear through brushes, and may have poor accuracy near zero speed.

IX. Optical Incremental Encoder Systems — Pulse-based position and speed sensing

An optical incremental encoder system converts shaft rotation into digital pulses using a patterned disc, light source, and photodetectors.

A. Optical incremental encoder systems

The encoder measures relative motion through transitions rather than directly reporting absolute shaft position.

Construction: An indexed disc interrupts an LED beam; photodiodes or phototransistors detect transparent and opaque regions.
Quadrature signals: Channels A and B are phase-shifted by 90°. Their lead-lag relationship determines direction.
Index channel: A once-per-revolution Z pulse provides a reference position for homing or count correction.
Resolution: If the encoder has N pulses per revolution and quadrature decoding counts four edges per pulse:
TEXT
```
  counts/revolution = 4N
```
Position calculation:
TEXT
```
  θ = 2π C / C_rev
```
where C is accumulated count and C_rev is counts per revolution.
Speed calculation: Counting ΔC counts during interval Δt gives
TEXT
```
  ω = 2π ΔC / (C_rev Δt)
```
Limitations: Missed pulses, optical contamination, electrical noise, and quantization errors affect accuracy, especially at very low speed.

X. Interfacing of Velocity Transducers — From analogue voltage or pulses to control data

Interfacing of velocity transducers adapts tachogenerator outputs and encoder signals to a robot controller.

A. Interfacing of velocity transducers

The interface must preserve direction, timing, and signal amplitude while converting sensor information into a reliable velocity estimate.

Tachogenerator input: A differential amplifier scales V_t, while a low-pass filter removes commutator ripple before ADC sampling.
Encoder input: Schmitt-trigger receivers convert noisy transitions into logic-level edges. Pull-ups, line drivers, or differential RS-422 receivers may be used for long cables.
Timer/counter method: Hardware counters accumulate encoder edges; a timer reads the count at fixed intervals, avoiding software timing uncertainty.
Period method: At low speed, measuring time T_p between pulses can provide better resolution:
TEXT
```
  f = 1 / T_p
  ω = 2π f / N
```
where f is pulse frequency and N is pulses per revolution.
Aliasing and latency: Sampling and filtering must balance noise reduction against control delay; excessive delay can destabilize a feedback loop.
Fault detection: Invalid quadrature transitions, missing index pulses, overspeed, and sensor disconnection should generate diagnostic flags.

XI. Kalman Filter — Model-based estimation under noise

A Kalman filter estimates hidden states such as position and velocity by combining a motion model with noisy sensor measurements.

A. Kalman filter

The discrete linear Kalman filter is optimal for minimum-variance estimation when the system is linear and process and measurement noises are zero-mean Gaussian with known covariances.

State model:
TEXT
```
  x_k = A x_{k-1} + B u_k + w_k
  z_k = H x_k + v_k
```
where x_k is the state, u_k is control input, z_k is measurement, A and B are system matrices, H maps state to measurement, and w_k, v_k are process and measurement noise.
Prediction step:
TEXT
```
  x̂_k^- = A x̂_{k-1} + B u_k
  P_k^- = A P_{k-1} Aᵀ + Q
```
Here x̂^- is predicted state, P^- is predicted covariance, and Q is process-noise covariance.

Update step:

TEXT

  K_k = P_k^- Hᵀ (H P_k^- Hᵀ + R)^-1
  x̂_k = x̂_k^- + K_k(z_k - H x̂_k^-)
  P_k = (I - K_k H)P_k^-

K_k is Kalman gain, R is measurement-noise covariance, and I is the identity matrix.

Robot example: A state [position, velocity]ᵀ can combine encoder position with tachogenerator velocity, producing a smoother velocity estimate than direct differentiation.
Tuning effect: Larger Q makes the filter trust new measurements and model changes more; larger R makes it trust measurements less and rely more on prediction.
Limitations: Incorrect models, nonlinear motion, poor covariance choices, and unmodelled bias can produce lag or misleading estimates.