Position, Velocity and Acceleration Sensors — Robotics Tutorials

Rotary Encoders: The Foundation of Joint Sensing

Rotary encoders are the most common sensor for measuring joint angle in robot arms. They convert shaft rotation into electrical signals that a controller can count and interpret as angular position.

Incremental encoders produce a pulse train as the shaft rotates. A counter accumulates pulses from a known reference (home position). They are simple and inexpensive but lose position information on power loss — a homing routine is required at startup.

Absolute encoders output a unique digital code for every shaft position (Gray code or binary). Position is known immediately at power-on with no homing required. They are more expensive but essential for safety-critical applications.

Quadrature encoding uses two signals (A and B channels) offset by 90°. By monitoring which channel leads, the direction of rotation is determined. By counting all edges (four per cycle), the effective resolution is multiplied by four — this is called 4X decoding.

Resolvers are analog sensors using transformer-coupled windings. They output sine and cosine signals proportional to shaft angle: V_sin = V_ref × sin(θ), V_cos = V_ref × cos(θ). Resolvers are highly robust to vibration, temperature, and contamination — preferred in harsh industrial environments.

Hall effect sensors use the magnetic field of a permanent magnet attached to the shaft to produce a digital commutation signal. While lower resolution than encoders, they are compact and used in brushless DC motors for commutation and coarse position sensing.

Encoder Resolution and Velocity Calculation

For a quadrature encoder with N pulses per revolution (PPR), the angular resolution using 4X decoding is:

Resolution = 360° / (N × 4)

Angular velocity is computed by measuring the time between pulses or counting pulses over a fixed interval:

ω = Δθ / Δt = (pulses_counted × 360°) / (N × 4 × Δt)

Linear velocity at end-effector requires application of the robot Jacobian:

v = J(q) × q̇

Higher PPR improves resolution but increases counter frequency requirements. A 2500 PPR encoder with 4X decoding gives 10,000 counts/rev, or 0.036° resolution.

Comparison of Position Sensor Technologies

Type	Principle	Typical Resolution	Pros	Cons
Incremental Encoder	Optical grating or magnetic, pulse counting	100 – 100,000 PPR	Low cost, high speed, simple interface	Loses position on power loss, needs homing
Absolute Encoder	Optical Gray code or multi-turn magnetic	12 – 24 bits per revolution	Position known at power-on, no homing	Higher cost, more complex interface (SSI, BiSS)
Resolver	Transformer with sin/cos windings	12 – 16 bit (via R/D converter)	Highly robust, works at high temperature	Requires R/D converter, bulkier than encoder
Potentiometer	Resistive wiper on a track	10 – 12 bit (with ADC)	Very simple, inexpensive, absolute	Mechanical wear, limited life, lower accuracy
Hall Effect Sensor	Magnetic field switching	3 – 6 positions per revolution (commutation)	Compact, no mechanical contact, low cost	Very coarse resolution, mainly for commutation
Capacitive Encoder	Capacitive sensing of graduated scale	14 – 20 bit	Compact, immune to magnetic interference	Sensitive to contamination, higher cost

Inertial Measurement Units: Measuring Dynamics

An Inertial Measurement Unit (IMU) combines a gyroscope and an accelerometer in a single package, often alongside a magnetometer, to measure the dynamic motion state of a rigid body.

MEMS gyroscopes (Micro-Electro-Mechanical Systems) exploit the Coriolis effect: a vibrating proof mass experiences a force perpendicular to both its velocity and the rotation axis. Capacitive sensing detects the resulting deflection, yielding angular velocity (rad/s). MEMS gyroscopes are compact and inexpensive but exhibit bias drift — the zero-rate output slowly changes with temperature and time.

MEMS accelerometers measure the specific force (acceleration minus gravitational acceleration) on a proof mass suspended by spring-like structures. At rest, they measure gravity — from which tilt angle can be inferred. During motion, they measure the sum of kinematic and gravitational accelerations.

A 9-DoF IMU adds a magnetometer to provide a heading reference relative to Earth's magnetic north, enabling full 3D orientation (roll, pitch, yaw) estimation via sensor fusion algorithms such as the Madgwick or Mahony filter.

Tactical-grade and navigation-grade IMUs (ring-laser gyroscopes, fiber-optic gyroscopes) offer orders-of-magnitude lower drift and are used in aerospace and autonomous vehicles, but cost thousands to tens of thousands of dollars.

IMU Integration and Drift

Velocity is obtained by integrating accelerometer output; position by double integration:

v(t) = v₀ + ∫a(τ)dτ

x(t) = x₀ + ∫v(τ)dτ

Orientation is obtained by integrating gyroscope output:

θ(t) = θ₀ + ∫ω(τ)dτ

Angular acceleration (for torque estimation) can be obtained by differentiating encoder velocity:

α = Δω / Δt

Integration amplifies noise. A constant bias b in the accelerometer reading produces position error growing as ½bt². After 60 seconds, a 1 mg (0.0098 m/s²) bias produces ~17.6 m of position error — illustrating why IMU-only navigation is unsuitable for long durations.

IMU Drift and the Case for Sensor Fusion

Dead reckoning using an IMU alone is fundamentally limited by drift. The gyroscope bias causes orientation error to grow linearly with time; the accelerometer bias causes position error to grow quadratically.

This is why modern robots always fuse IMU data with an absolute reference:

Visual-inertial odometry (VIO): IMU + camera features provide drift-corrected pose at high rate
LiDAR-inertial odometry: IMU + LIDAR scan matching (e.g., LIO-SAM, FAST-LIO)
GNSS/INS fusion: IMU + GPS corrects drift in outdoor navigation

The IMU provides high-rate (200–1000 Hz), low-latency motion estimates between slower sensor updates, while the external reference corrects accumulated drift. This complementary relationship is the foundation of robust robot state estimation.

Key Principles of Motion Sensing

01
Incremental encoders require homing at startup; absolute encoders provide position immediately — the choice depends on application safety requirements.
02
Quadrature decoding (4X) multiplies effective encoder resolution by four by detecting all rising and falling edges of both A and B channels.
03
Resolvers are preferred in harsh environments (high temperature, vibration, contamination) despite requiring a Resolver-to-Digital (R/D) converter.
04
MEMS IMUs are ubiquitous in robotics due to size and cost, but exhibit bias drift that must be compensated through calibration and sensor fusion.
05
Double integration of accelerometer data to obtain position amplifies noise quadratically — IMU-only positioning drifts rapidly and must be corrected by external references.
06
Sensor fusion of encoders and IMU provides robust velocity and acceleration estimates that outperform either sensor alone.
07
For collaborative robots, joint torque sensors (measuring motor current or using strain gauges) enable safe human-robot interaction by detecting unexpected contact forces.

Key terms

Rotary encoder: A sensor that converts shaft rotation to electrical pulse signals, used to measure joint angle and angular velocity in robot joints.
Incremental encoder: An encoder that generates pulses relative to a reference position; position is known by counting pulses from a known home position.
Absolute encoder: An encoder that outputs a unique digital code for every shaft angle, providing position information immediately at power-on without homing.
Quadrature encoding: A technique using two phase-shifted signals (A and B channels) to determine both position and direction of rotation, multiplying resolution by 4 via edge detection.
Resolver: An analog rotary position sensor using transformer-coupled sine and cosine windings, valued for robustness in harsh industrial environments.
MEMS: Micro-Electro-Mechanical Systems — microscale mechanical structures integrated with electronics on silicon chips, used in modern gyroscopes and accelerometers.
Gyroscope: A sensor that measures angular velocity (rate of rotation) about one or more axes; MEMS gyroscopes use the Coriolis effect on a vibrating proof mass.
Accelerometer: A sensor that measures the specific force (kinematic acceleration minus gravitational acceleration) acting on a proof mass, used to detect linear motion and tilt.
IMU (Inertial Measurement Unit): A sensor package combining gyroscope and accelerometer (and optionally magnetometer) to measure 3D orientation, angular velocity, and linear acceleration.
Dead reckoning: Estimating position by integrating velocity or acceleration measurements from a known starting point, without external position fixes; errors accumulate over time.
Tachometer: A device that measures the rotational speed (angular velocity) of a shaft, often via back-EMF measurement or pulse counting from an encoder.
Hall effect sensor: A sensor that detects magnetic field strength using the Hall effect, used in brushless DC motors for commutation and coarse position sensing.

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