Gyroscope sensors

Figure 1: Single-axis MEMS yaw gyroscope.

  • A gyroscope measures angular velocity (the rate of change in orientation angle),not angular orientation itself.
  • Each gyroscope measures the rotation around one axis
  • Gxz – is the angle between the Rxz (projection of R on XZ plane) and Z axis
  • Gyz – is the angle between the Ryz (projection of R on YZ plane) and Z axis
  • Gyroscopes measure the rate of change of these angles.

Coriolis effect

  • Uses Coriolis effect to transform an angular velocity into a displacement
  • The Coriolis force acts perpendicular to the rotation axis and to the velocity of the body in the rotating frame

                                                     Fc= -2m Ω x v

  • The displacement induces a change in capacitance between the mass and the housing, thus transforming the angular rate input to the gyroscope into an electrical output
  • Must first initialize the sensor position with a known value (possibly from the gyroscope measures), then measure the angular velocity (ω) around the X, Y and Z axes at measured intervals (Δt)

                                          ω × Δt = change in angle 

  • The new orientation angle is the original angle plus this change
  • This is integrating – adding up many small computed intervals – to find orientation
  • Repeatedly adding up increments of  ω × Δt results in small systematic errors becoming magnified over time
  • Gyroscopic drift–over long timescales the gyroscope data will become increasingly inaccurate

A gyroscope sensor has the following basic specifications:

  • Measurement range
  • Number of sensing axes
  • Nonlinearity
  • Working temperature range
  • Shock survivability
  • Bandwidth
  • Angular Random Walk (ARW)
  • Bias
  • Bias Drift
  • Bias Instability

Measurement range This parameter specifies the maximum angular speed with which the sensor can measure, and is typically in degrees per second (˚/sec).Number of sensing axes Gyroscopes are available that measure angular rotation in one, two, or three axes. Multi-axis sensing gyros have multiple single-axis gyros oriented orthogonal to one another. Vibrating structure gyroscopes are usually single-axis (yaw) gyros or dual-axis gyros, and rotary and optical gyroscope systems typically measure rotation in three axes.

Nonlinearity – Gyroscopes output a voltage proportional to the sensed angular rate.

Nonlinearity is a measure of how close to linear the outputted voltage is proportional to the actual angular rate. Not considering the nonlinearity of a gyro can result in some error in measurement. Nonlinearity is measured as a percentage error from a linear fit over the full-scale range, or an error in parts per million (ppm).

Working temperature range – Most electronics only work in some range of temperatures. Operating temperatures for gyroscopes are quite large; their operating temperatures range from roughly -40˚C to anywhere between 70 and 200˚C and tend to be quite linear with temperature. Many gyroscopes are available with an onboard temperature sensor, so one does not need to worry about temperature related calibrations issues.

Shock Survivability – In systems where both linear acceleration and angular rotation rate are measured, it is important to know how much force the gyroscope can withstand before failing. Fortunately gyroscopes are very robust, and can withstand a very large shock (over a very short duration) without breaking. This is typically measured in g’s (1g = earth’s acceleration due to gravity), and occasionally the time with which the maximum g-force can be applied before the unit fails is also given.

Bandwidth The bandwidth of a gyroscope typically measures how many measurements can be made per second. Thus the gyroscope bandwidth is usually quoted in Hz.

Angular Random Walk (ARW) – This is a measure of gyro noise and has units of deg/hour1/2 or deg/sec1/2. It can be thought of as the variation (or standard deviation), due to noise, of the result of integrating the output of a stationary gyro over time. So, for example, consider a gyro with an ARW of 1°/sec1/2 being integrated many times to derive an angular position measurement: For a stationary gyro, the ideal result – and also the average result – will be zero. But the longer the integration time, the greater will be the spread of the results away from the ideal zero. Being proportional to the square root of the integration time, this spread would be 1° after 1 second and 10° after 100 seconds.

Bias The bias, or bias error, of a rate gyro is the signal output from the gyro when it is NOT experiencing any rotation. Even the most perfect gyros in the world have error sources and bias is one of these errors. Bias can be expressed as a voltage or a percentage of full scale output, but essentially it represents a rotational velocity (in degrees per second). Again, in a perfect world, one could make allowance for a fixed bias error. Unfortunately bias error tends to vary, both with temperature and over time. The bias error of a gyro is due to a number of components:

  • calibration errors
  • switch-on to switch-on
  • bias drift
  • effects of shock (g level)

Individual measurements of bias are also affected by noise, which is why a meaningful bias measurement is always an averaged series of measurements.

Bias Drift – This refers specifically to the variation of the bias over time, assuming all other factors remain constant. Basically this is a warm-up effect, caused by the self heating of the gyro and its associated mechanical and electrical components. This effect would be expected to be more prevalent over the first few seconds after switch-on and to be almost non-existent after (say) five minutes.

Bias Instability – Bias Instability is a fundamental measure of the ‘goodness’ of a gyro. It is defined as the minimum point on the Allan Variance curve, usually measured in °/hr. It represents the best bias stability that could be achieved for a given gyro, assuming that bias averaging takes place at the interval defined at the Allan Variance minimum 9).


Loading effects.  Sensors (and hence instruments) work by removing energy from the system they are connected to.  As a result of this they change the state of the system.  This is known as a loading effect.  In other words, the parameter that is being measured is changed in some way by the method of measurement.  Flow measurement is often implemented by introducing a restriction in a pipe which changes the flowrate.  Another example is the electrical loading effect where an electrical meter which has an internal resistance affects the voltage it is measuring.

 Offset, Drift : a measurement must be made with respect to a known datum or base line.  It is very common and convenient to adjust the output of the instrument to zero at the datum.  For example, a thermometer is set up to display zero at the freezing point of water.  A pressure gauge is adjusted to read zero when open to atmosphere.

However, the output signal may be offset from zero by some amount.  It is often possible to adjust the instrument to remove this offset.  For example, a bathroom weighing scales can be adjusted to display zero when no one is standing on it.

A common problem with instruments is that the output at the datum drifts and introduces an error to the measurement.  All sensors are affected by drift to some extent whether it be short term or long term.  Short term drift is usually associated with changes in temperature or electronics stablising.  Long term drift is usually associated with aging of the transducer.

 Precision: the reproducibility with which repeated measurements of the same variable can be made under identical conditions.  An instrument can be precise but inaccurate and, likewise, it is possible to have an accurate but imprecise instrument.  See below:


Which is preferable – precise but inaccurate or accurate but not precise?

Range: The total range of values an instrument is capable of measuring.  For a standard thermometer this is 0 to 100°C.  This is the same as the full scale.

Repeatable: An instrument is repeatable if it produces a particular output when a certain input is applied in spite of the passage of time.

Resolution : smallest change in input signal needed to produce a change in the output signal

Sensitivity is defined as the output over the input (this is also called the gain of the instrument).  Static sensitivity is the sensitivity exhibited during steady state conditions.  Sensitivity has units which depend on the measuring system.  For example, a temperature measuring system that uses a platinum resistance temperature device (RTD) produces a change in resistance as the temperature changes.  The input is temperature and the output is resistance.  The output over the input is therefore,


Sensitivity = DR/DT    Units = W/°C

Overall sensitivity – a measurement system consists of a number of devices.  If the sensitivities of these devices are K1, K2, K3, etc., then the overall sensitivity is the product of the individual sensitivities, K1xK2xK3xetc.

Span : range of input signals corresponding to range of output signal (i.e. largest input signal which can be displayed/indicated – smallest input signal which can be displayed/indicated).  For a standard thermometer this is 100 – 0 = 100°C.  If the thermometers range is -30 to 220°C, then the span is equal to 250°C.

Stability: Output is constant when the input is constant.

Tolerance is the maximum error.

Repeatable: An instrument is repeatable if it produces a particular output when a certain input is applied in spite of the passage of time.

Resolution : smallest change in input signal needed to produce a change in the output signal

Sensitivity is defined as the output over the input (this is also called the gain of the instrument).


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