Gyroscope

A gyroscope is a sensor that measures the rotation speed of an object, or angular velocity, in radians per second. In a smartphone, this MEMS sensor provides the angular velocity around the three axes.

Discover FizziQ

How to measure it in class

FizziQ displays the angular velocity of the smartphone around its three axes, in radians per second. It really is a rotation speed that is measured, not an angle.

Steps:

• Place the smartphone motionless on the table and start the gyroscope acquisition: the three angular velocities should stay close to zero, within a few thousandths of a rad/s. This residual offset is the sensor’s bias.
• Rotate the smartphone a quarter turn around a single axis, then bring it back to rest: observe that the curve forms a peak during the rotation and returns to zero as soon as it stops, even though the orientation has changed.
• Place the smartphone on a swivel stool or a turntable platter and rotate it steadily: record the constant value of the angular velocity.
• Time one complete turn and check that ω = 2π/T matches the value displayed by the sensor.
• Calculate the area under the angular velocity curve during a quarter-turn rotation and check that you recover π/2 rad: it is the integration that gives the angle.
• Repeat this integration over one minute of successive manipulations and observe that the calculated angle progressively drifts away from the actual angle: this is the gyroscope’s drift.

Scientific activities on this topic

By studying the motion of the Perseverance rover on Mars, you will cover linear motion and see how Perseverance can use its gyroscope to hold its course (see the activity)

By measuring the rotation speed of a smartphone placed in a salad spinner, you can verify the relation between centripetal acceleration and rotation speed. Would an astronaut placed in the salad spinner survive the force exerted on them? (see the activity)

By repeating the same rotation and comparing the angle obtained by integrating the gyroscope with the actual angle, you address the notions of measurement uncertainty and systematic error (see the activity)

Learn more

What a smartphone gyroscope really measures

This is the most frequent confusion: the gyroscope does not give an orientation, it gives an ANGULAR VELOCITY, in radians per second. With the smartphone motionless, it reads zero, whatever position it was left in. To obtain an angle, you must integrate the angular velocity over time. But every measurement carries a small bias, and the integration accumulates this bias: the calculated angle gradually drifts away from the actual angle, this is the drift. A bias of 0.01 rad/s already produces a deviation of 34° after one minute. This is why phones never rely on the gyroscope alone: they fuse its data with that of the accelerometer, which locates the vertical through the direction of g, and of the magnetometer, which locates magnetic north. These two sensors do not drift but are noisy; the gyroscope is accurate in the short term but drifts. Fusing the three yields an orientation that is both stable and responsive.

Foucault’s gyroscope was based on the principle of conservation of angular momentum, which means that any rotating object tends to keep its direction of rotation unchanged, unless an external force acts on it. He designed a spinning rotor mounted on a frame, and by spinning the rotor he was able to observe that the gyroscope maintained its initial direction of rotation, even when the frame was moved.

There are several types of gyroscopes, each with its own design approach. The main types of gyroscopes include:

• Mechanical gyroscopes: These gyroscopes use a spinning rotor to maintain stability. They are made of mechanical parts, such as a disk or a spinning top, rotating at high speed.
• Optical gyroscopes (Sagnac effect): Two light beams travel in opposite directions around the same loop, either a coil of optical fiber or a laser cavity. If the device rotates, the two paths no longer take the same time and a phase shift appears, proportional to the angular velocity. These are the most accurate gyroscopes, used in aeronautical and space inertial navigation.
• MEMS gyroscopes: MEMS (Micro-Electro-Mechanical Systems) are small miniaturized mechanical and electronic devices manufactured using microfabrication techniques. They contain no rotor: a micro-mass is set vibrating, and when the sensor rotates, the Coriolis force deflects this vibration perpendicularly. The deflection, measured through a change in electrical capacitance, is proportional to the angular velocity. These MEMS gyroscopes have become commonplace in smartphones because of their small size and low power consumption.

Gyroscopes have many applications in various fields, including:

• Navigation: Gyroscopes are used in inertial navigation to determine the orientation and motion of objects such as aircraft, ships and missiles. They help maintain stability and provide accurate navigation data.
• Consumer electronics: Gyroscopes are commonly built into smartphones and tablets to detect fast rotational movements, essential for gaming, augmented reality and video stabilization. Automatic screen rotation, on the other hand, relies on the accelerometer, which detects the direction of the vertical.
• Camera and drone stabilization: Gyroscopes are used to stabilize video cameras, still cameras, drones and binoculars in order to minimize shakes and jolts when shooting.
• Aerospace: Gyroscopes are used in satellites, space telescopes and scientific instruments to maintain their orientation relative to the stars.
• Military equipment: Gyroscopes are found in missile guidance systems, night vision systems and other military equipment.

History

The word “gyroscope” comes from the Greek words gyros (“circle”) and scope (“to observe”). The invention of the first three-axis gyroscope is attributed to the German astronomer Bohnenberger in 1810. However, it was not until 1852 that the French physicist and astronomer Léon Foucault improved the device and gave it its current name. Foucault, known for his famous pendulum, discovered that the spinning disk inside the gyroscope keeps its initial direction in space, even though the Earth rotates on its axis. He also found that the gyroscope can indicate geographic north and the latitude of a place if its rotation axis is parallel to the Earth’s.

At the end of the 19th century, gyroscopes were motorized, which allowed the appearance of the gyrocompass, capable of pointing to geographic north rather than magnetic north. Gyroscopic compasses replaced the metal compasses on ships. In the early 20th century, gyroscopes became a key element of the military industry and are present in many vehicles and weapons.

Orders of magnitude

Measurement range of a smartphone gyroscope: about ±35 rad/s, that is ±2,000 °/s. Noise at rest: a few thousandths of a rad/s. Typical drift: several degrees per minute. Earth’s rotation: 7.3 × 10⁻⁵ rad/s, a hundred times too weak to be detected by a smartphone, but perfectly measurable by an optical gyroscope. A bicycle wheel at 20 km/h: about 16 rad/s. A 33 rpm vinyl record: 3.5 rad/s. A salad spinner: 30 to 60 rad/s.

Formula

The angular velocity relates the angle swept to the duration of the rotation:

ω = Δθ / Δt

where:

• ω: angular velocity (rad/s)
• Δθ: angle swept (rad)
• Δt: duration (s)

For a uniform rotation of period T:

ω = 2π / T = 2πf

where:

• T: rotation period (s)
• f: rotation frequency (Hz or rev/s)

The orientation can only be obtained by integrating the angular velocity measured by the gyroscope:

θ(t) = θ₀ + ∫ ω(t) dt

where:

• θ₀: initial orientation, which the gyroscope alone does not know (rad)

A constant bias ε on the measurement of ω produces an orientation error that grows with time:

error = ε × t

For a point located at distance r from the axis, the speed and the centripetal acceleration are:

v = ω × r and a = ω² × r

Application examples

• The image stabilization of a camera or a drone: the gyroscope detects vibrations within a few milliseconds and the optics or the motors compensate for them in real time.

• Game controllers and virtual reality headsets track head movements thanks to the gyroscope, the only sensor fast enough to avoid a feeling of latency.

• An airliner’s inertial navigation unit combines three optical gyroscopes and three accelerometers to know its position without GPS, with a drift on the order of one kilometer per hour of flight.

• A Mars rover like Perseverance holds its course with its gyroscope, because it has neither GPS nor a global magnetic field to orient itself.

• A smartphone in a salad spinner rotating at 40 rad/s at 10 cm from the axis experiences a centripetal acceleration of 0.10 × 40² = 160 m/s², about 16 g.

• A ship’s gyrocompass indicates geographic north, not magnetic north: it is therefore not disturbed by the steel hull or by magnetic declination.

FAQ

Q: Does the gyroscope give me the orientation of my phone? A: No. It measures an angular velocity, in rad/s. If the phone is motionless, it reads zero, whether it is lying flat or standing against a wall. The orientation can only be obtained by integrating this velocity over time, and provided the starting orientation is known.

Q: What is gyroscope drift and why is it unavoidable? A: The sensor always measures ω with a small offset. When you integrate to obtain the angle, this offset adds up at every instant instead of canceling out, and the error grows proportionally to time. A bias of 0.01 rad/s gives 0.6 rad, that is 34°, after one minute.

Q: Why does my phone use three sensors to know its orientation? A: Because none is sufficient on its own. The gyroscope is fast and accurate in the short term but drifts. The accelerometer gives the vertical without drifting, but it is disturbed as soon as you move. The magnetometer gives north without drifting, but it is sensitive to metal masses. A fusion algorithm combines the three and keeps the best of each.

Q: Does a MEMS gyroscope contain a spinning top? A: No, it has no rotating part. A micro-mass vibrates, and when the sensor rotates, the Coriolis force deflects this vibration sideways. It is this deflection, on the order of a nanometer, that is measured electrically.

Q: Can the Earth’s rotation be detected with a smartphone gyroscope? A: No. The Earth rotates at 7.3 × 10⁻⁵ rad/s, a value well below the noise of a MEMS sensor, which is a few thousandths of a rad/s. An optical laboratory gyroscope is needed to achieve this, as Foucault did with his mechanical gyroscope in 1852.

Related concepts

Accelerometer - Magnetometer - Orientation - Coriolis Force - Uniform Circular Motion