Unit 2: Force & Velocity Transducers - Subjective Questions

Force measurement in robotics is the process of detecting and quantifying an external or internal force acting on a robot. Important considerations include:\n\n- Range: The transducer must measure the expected minimum and maximum force without saturation.\n- Sensitivity: It should produce a measurable output for small changes in force.\n- Resolution: The smallest detectable force change must be suitable for the application.\n- Accuracy and linearity: The output should closely follow the actual force over the operating range.\n- Hysteresis: The difference between outputs for increasing and decreasing force should be small.\n- Repeatability: The same force should produce nearly the same output under identical conditions.\n- Response time: Fast robotic motion and contact control require a sufficiently high bandwidth.\n- Temperature stability: Variations in temperature should not significantly affect the measurement.\n- Mechanical compatibility: The sensor must withstand overloads, vibration, shock, and mounting stresses.\n\nForce sensors are used in gripping, assembly, collision detection, compliant motion, and force-controlled machining.

A bonded metallic strain gauge consists of a thin metallic wire or foil arranged in a grid and attached to the surface of an elastic member. When the member is deformed, the gauge deforms by the same amount. Its electrical resistance changes because its length, cross-sectional area, and resistivity change.\n\nFor a conductor,\n\n\n\nTaking small changes,\n\n\n\nThe longitudinal strain is . The gauge factor is defined as\n\n\n\nTherefore,\n\n\n\nFor metallic gauges, the gauge factor is typically close to 2. Strain gauges are commonly connected in a Wheatstone bridge to convert this small resistance variation into a measurable voltage.

A Wheatstone bridge converts small changes in strain-gauge resistance into a differential voltage. For four bridge resistances, the output voltage is\n\n\n\nWhen the bridge is balanced, . Applied force changes one or more resistances and produces a nonzero output.\n\n- Quarter bridge: Uses one active strain gauge and three fixed resistors. It is simple but has low sensitivity and is more affected by temperature.\n- Half bridge: Uses two active gauges, often arranged so that one experiences tension and the other compression. It provides higher sensitivity and partial temperature compensation.\n- Full bridge: Uses four active gauges. It gives the highest sensitivity, improved temperature compensation, and better rejection of common disturbances.\n\nThe bridge output is usually small, so it must be amplified before analog-to-digital conversion.

Non-resistive force transducers measure force through changes in capacitance, electric charge, inductance, or another physical quantity rather than through a change in resistance.\n\n- Capacitive transducer: Force causes the distance between capacitor plates or their overlapping area to change. Since\n \n a change in plate spacing produces a change in capacitance.\n- Piezoelectric transducer: A piezoelectric material generates charge when mechanically stressed. The charge is approximately proportional to applied force:\n \n where is the piezoelectric coefficient. These sensors are suitable for dynamic and impact forces but are poor for long-term static-force measurement because of charge leakage.\n- Inductive transducer: Force moves a magnetic core or changes the reluctance of a magnetic circuit. This changes the inductance or the differential output of an LVDT. Inductive sensors are rugged and suitable for displacement-based force measurement.\n\nEach type requires suitable excitation and signal-conditioning electronics.

A load cell is a force transducer designed to convert mechanical load into an electrical signal. A typical strain-gauge load cell contains:\n\n- An elastic metal element, such as a beam, column, or shear member.\n- Strain gauges bonded at locations where the desired strain is produced.\n- A Wheatstone bridge for electrical conversion.\n- Protective housing and mechanical interfaces for load application.\n\nWhen a load is applied, the elastic element deforms within its elastic limit. The bonded gauges experience tension or compression, changing their resistance. The bridge produces an output voltage approximately proportional to the applied load:\n\n\n\nwhere is the sensitivity and is the force. Load cells are used in robotic grippers, force-torque sensors, weighing systems, assembly robots, and robot wrist force control. Proper alignment is essential because off-axis loads can cause measurement errors.

Consider a quarter bridge in which one arm has an active resistance and the other three arms have resistance . The bridge output is\n\n\n\nFor small resistance changes, ,\n\n\n\nHence, the magnitude of the output is approximately\n\n\n\nUsing the strain-gauge relation\n\n\n\nwe obtain\n\n\n\nThe output is very small, which is why instrumentation amplifiers with high input impedance, low noise, and low offset are normally used.

Important error sources include:\n\n- Temperature effects: Temperature changes alter gauge resistance and the elastic element properties. Use dummy gauges, half or full bridges, temperature compensation, and temperature calibration.\n- Nonlinearity: The output may not be exactly proportional to force. It can be reduced by selecting a suitable elastic element and applying calibration correction.\n- Hysteresis and creep: The output may depend on loading history or change slowly under a constant load. Use stable materials and operate within the specified range.\n- Misalignment and side loads: Improper mounting introduces bending or transverse forces. Use accurate fixtures and mechanical alignment.\n- Electrical noise: Shielded cables, proper grounding, filtering, and differential measurement reduce interference.\n- Overload: Excessive force can permanently deform the sensing element. Mechanical overload stops and suitable safety margins are required.\n- Lead-wire resistance: Three-wire or four-wire connections and bridge compensation reduce the effect of cable resistance.\n\nRegular calibration with known loads is necessary to maintain accuracy.

A strain-gauge bridge normally produces a low-level differential voltage, so the interface includes several stages:\n\n1. Bridge excitation: A stable voltage or current source supplies the bridge. Ratiometric excitation can reduce errors caused by supply variation.\n2. Instrumentation amplifier: A high-input-impedance differential amplifier provides accurate gain while rejecting common-mode voltage.\n3. Filtering: A low-pass filter removes high-frequency noise and prevents unwanted signals from entering the controller.\n4. Offset and temperature compensation: Bridge imbalance and thermal drift can be corrected using calibration or compensation circuits.\n5. Isolation and protection: Input protection, shielding, and electrical isolation may be required in industrial environments.\n6. Analog-to-digital conversion: The ADC must have sufficient resolution, sampling rate, and input range.\n7. Digital calibration: The measured code is converted to force using an offset and scale factor, often obtained from calibration data.\n\nThe interface bandwidth should be high enough for the intended force-control loop but limited enough to avoid excessive noise.

Velocity measurement determines the rate and direction of change of a robot joint or end-effector position. For linear motion,\n\n\n\nand for rotational motion,\n\n\n\nVelocity feedback is important because it:\n\n- Improves damping and reduces oscillation in position-control systems.\n- Enables accurate trajectory tracking.\n- Supports torque, force, and impedance control.\n- Allows detection of overspeed and abnormal motion.\n- Helps coordinate multiple robot joints.\n- Provides information for estimating acceleration and dynamic forces.\n\nVelocity may be measured directly using a tachogenerator or encoder, or estimated by differentiating position measurements. Direct measurement generally avoids some numerical differentiation problems, while estimation can reduce hardware cost.

A DC tachogenerator is a small generator mechanically coupled to a rotating shaft. Its main parts are a permanent-magnet or separately excited stator, a rotating armature, a commutator, and brushes. When the shaft rotates, electromagnetic induction generates an output voltage.\n\nIdeally, the generated voltage is proportional to angular velocity:\n\n\n\nwhere is the tachogenerator voltage, is the voltage constant, and is angular velocity. Reversing the direction of rotation reverses the polarity of the output.\n\nIn practice, errors arise from brush voltage drop, commutator ripple, magnetic nonlinearity, temperature variation, loading by the interface circuit, and friction. A high-input-impedance amplifier is therefore used. Tachogenerators provide continuous velocity information and are useful in analog servo systems, although they require mechanical coupling and regular maintenance of brushes.

Both devices measure rotational motion, but they differ in construction and output:\n\n- Operating principle: A tachogenerator produces an analog voltage proportional to speed, whereas an incremental encoder produces digital pulses as a coded disk rotates.\n- Direction: A tachogenerator indicates direction through voltage polarity. An encoder uses phase-shifted channels, commonly A and B, to determine direction.\n- Accuracy: Encoders generally provide better resolution and digital repeatability.\n- Low-speed performance: A tachogenerator can provide continuous output, while an encoder may require pulse counting over a time interval or reciprocal-period measurement at low speed.\n- Maintenance: Tachogenerators contain brushes and commutators; optical encoders have fewer wearing electrical parts but require clean optical paths.\n- Interface: Tachogenerators need analog amplification and filtering. Encoders require digital counters, decoding logic, and possibly signal-level conversion.\n\nEncoders are common in modern robots because they can provide both position and velocity information with high resolution.

An optical incremental encoder contains a light source, a slotted or patterned rotating disk attached to the shaft, photodetectors, and signal-conditioning electronics. As the disk rotates, the slots alternately transmit and block light, producing a pulse train.\n\n- Channel A: Produces periodic pulses corresponding to shaft motion.\n- Channel B: Produces a similar pulse train shifted by approximately electrical from channel A. The phase relationship identifies the direction of rotation.\n- Channel Z or index: Produces one pulse per revolution and establishes a reference position.\n\nIf the encoder has pulses per revolution, the angular displacement associated with one pulse is approximately\n\n\n\nQuadrature decoding can count rising and falling edges to obtain one, two, or four counts per pulse cycle. Velocity can be obtained by counting pulses over a fixed time or measuring the time between successive pulses.

Suppose an incremental encoder produces pulses per revolution and the controller counts pulses during a sampling interval . The angular displacement is\n\n\n\nTherefore, the average angular velocity is\n\n\n\nThis frequency or pulse-counting method works well at high speed, but at low speed few pulses occur in one sample, causing quantization and delay.\n\nAlternatively, measure the period between successive pulses. Then\n\n\n\nThis period-measurement method gives better low-speed resolution, but it becomes sensitive to timing jitter and may be computationally demanding at high speed.\n\nPractical systems switch or blend both methods, apply filtering, and use quadrature decoding to preserve direction and improve resolution.

The interface depends on whether the velocity transducer has an analog or digital output.\n\nFor a tachogenerator:\n\n- Use a high-input-impedance differential amplifier.\n- Apply gain scaling so the voltage matches the ADC range.\n- Use a low-pass filter to reduce commutator ripple and electrical noise.\n- Add polarity handling or sign detection for direction.\n- Calibrate the relation .\n\nFor an incremental encoder:\n\n- Connect channels A and B to quadrature decoder inputs.\n- Use channel Z for once-per-revolution referencing.\n- Apply level shifting, line receivers, or differential inputs when required.\n- Use hardware counters or interrupt-capable inputs to count edges.\n- Convert counts or pulse periods into velocity.\n- Include debouncing, noise rejection, and loss-of-signal detection.\n\nThe sampling rate, interface bandwidth, and latency must be compatible with the robot servo loop.

Direct measurement uses a sensor whose output is related directly to velocity, such as a tachogenerator or a velocity-sensitive sensor. Estimation obtains velocity from position samples, commonly by numerical differentiation or a state estimator.\n\n- Direct measurement:\n - Provides immediate velocity information.\n - Avoids amplification of position quantization during differentiation.\n - Requires an additional sensor and mechanical or electrical interface.\n - May contain sensor-specific noise, drift, or hardware wear.\n- Position-based estimation:\n - Can use an existing encoder and avoid a separate tachogenerator.\n - Numerical differentiation amplifies high-frequency noise.\n - Requires filtering, which introduces phase delay.\n - Can estimate unmeasured states such as acceleration when a model is available.\n\nA Kalman filter improves position-based velocity estimation by combining noisy measurements with a motion model.

A Kalman filter is a recursive state-estimation algorithm that combines a mathematical model of a system with noisy sensor measurements. For velocity estimation, a simple state vector may be\n\n\n\nwhere is position and is velocity at sample . The filter has two stages:\n\n- Prediction: The previous state is propagated using a motion model.\n- Correction: The prediction is adjusted using the new sensor measurement, weighted according to model uncertainty and measurement noise.\n\nFor a linear system, the state model is\n\n\n\nand the measurement model is\n\n\n\nwhere and represent process and measurement noise. The filter produces a smoothed velocity estimate, reduces noise, and can provide useful estimates even when measurements are incomplete or irregular.

Consider the discrete state model\n\n\n\n\n\nwhere has covariance and has covariance . The Kalman filter operates as follows.\n\nPrediction step:\n\n\n\n\n\nHere, is the predicted state and is its error covariance.\n\nCorrection step:\n\nThe innovation is\n\n\n\nThe innovation covariance is\n\n\n\nThe Kalman gain is\n\n\n\nThe corrected state and covariance are\n\n\n\n\n\nThe gain gives greater weight to the sensor when measurement noise is low and greater weight to the model when sensor noise is high.

Let the state be\n\n\n\nAssume constant velocity over a sampling period . The continuous relationships are approximated by\n\n\n\n\n\nThus, the state-transition matrix is\n\n\n\nIf acceleration is used as an input,\n\n\n\nIf the encoder measures position only, the measurement equation is\n\n\n\nThe filter predicts both position and velocity, then corrects them using the measured position. The process covariance represents uncertainty in the constant-velocity assumption, while represents encoder measurement noise.

A complete force-transducer interface performs the following functions:\n\n- Excitation: Supplies a stable voltage or current to a bridge or passive transducer. Excitation stability directly affects measurement stability.\n- Amplification: Increases the small sensor output to a useful level. Instrumentation amplifiers are preferred because they offer high input impedance and high common-mode rejection.\n- Filtering: Removes noise, electromagnetic interference, and unwanted vibration components. The cutoff frequency should preserve the useful force dynamics.\n- Calibration: Establishes the relationship between sensor output and force. A linear calibration may use\n \n while nonlinear systems may use a polynomial or lookup table.\n- Analog-to-digital conversion: Samples the conditioned signal for processing by the robot controller. ADC resolution and reference stability determine quantization performance.\n\nThe interface must also provide overload protection, grounding, shielding, and suitable sampling synchronization with the control loop.

The two categories differ in their sensing mechanisms and interface requirements.\n\n- Resistive transducers: Strain gauges and piezoresistive elements change resistance under force. They are relatively simple, compact, and suitable for static measurements. Their output is often small, and they require bridge excitation and amplification. Temperature and lead resistance can affect accuracy.\n- Capacitive transducers: Provide high sensitivity and low power consumption, but parasitic capacitance, humidity, and electromagnetic interference can affect them.\n- Piezoelectric transducers: Have high stiffness and wide bandwidth, making them excellent for impacts and vibration. They are less suitable for static loads because the generated charge decays.\n- Inductive transducers: Are rugged and reliable, but may be larger and require AC excitation and demodulation.\n\nResistive load cells are widely used for weighing and static force control. Piezoelectric devices are preferred for dynamic force measurement, while capacitive and inductive sensors are selected when their particular resolution, robustness, or displacement characteristics are advantageous.