Three Axis Inertial Measurement Unit — Robotics/Navigation Sensors

Three-axis Inertial Measurement Unit

Robotics / Navigation Sensors

A Three-axis Accelerometer and a Three-axis Gyroscope in one block, sharing one location, one update rate and one set of inputs. The accelerometer reads the specific force at the IMU location; the gyroscope reads the body rates, its g-sensitive bias driven by the body acceleration in g.

d = [1 −1 1] · (CG − location)
A = (A_b − g) + ω × (ω × d) + ω̇ × d
A_meas = sat_a( H_a( SFCC_a · A + bias_a ) )
ω_meas = sat_g( H_g( SFCC_g · ω + bias_g + g_s · A_b / g₀ ) ), with each H(s) = ω_n² / (s² + 2ζω_ns + ω_n²) on each axis and g₀ the standard gravity in the chosen units.

Ports

• Ab – A_b, the acceleration of the CG in body axes, a [3,1] column. It also drives the gyroscope's g-sensitive bias.
• w – ω, the body angular rates p, q, r in rad/s, a [3,1] column.
• dw/dt – ω̇, the body angular accelerations in rad/s², a [3,1] column.
• CG – the centre of gravity, a [3,1] column in the axes and units of the IMU Location: station (positive aft), buttline (positive right), waterline (positive up).
• g – the gravity vector in body axes, a [3,1] column, subtracted by the accelerometer.
• A_meas – the measured acceleration, always a [3,1] column.
• w_meas – the measured body rates, always a [3,1] column.

The port order is the Simulink block's own. Every input must be a [3,1] column.

Parameters

• Units – the unit system of the accelerations, and here it moves one number: the standard gravity g₀ the gyroscope's g-sensitive term divides by.
  • Metric (MKS) – m/s², g₀ = 9.80665 (default).
  • English – ft/s², g₀ = 9.80665 / 0.3048.
• IMU Location – where the IMU sits, three numbers in the station/buttline/waterline axes the CG port uses. Default [0 0 0].
• Update Rate (s) – how both sensors' dynamics are realized, and the Simulink block's own "Update rate":
  • 0 (default) – continuous. Under a continuous solver the responses are integrated as they stand; when the block is stepped – the discrete solver, and every code export – each is the exact zero-order-hold discretization at the block's period, which is what the continuous sensor computes from inputs that hold across each sample.
  • greater than 0 – discrete sensors updating every that many seconds, with the Tustin (bilinear, unwarped) discretization – exactly what the Simulink block builds for a positive rate. The block then runs at its own period, which must be this same value: set Sampling Time (s) to it, or leave that at zero with the model's sampling time equal to it. A mismatch is reported and stops the run.
• Accelerometer Second-order Dynamics – on (default) passes each accelerometer axis through H_a; off makes it instantaneous.
• Accelerometer Natural Frequency (rad/s) – ω_n of H_a, greater than zero. Default 190.
• Accelerometer Damping Ratio – ζ of H_a, zero or more. Default 0.707.
• Accelerometer Scale Factors and Cross-coupling – the [3,3] matrix SFCC_a; the identity (default) is a perfect sensor.
• Accelerometer Measurement Bias – three numbers added before the dynamics. Default [0 0 0].
• Accelerometer Lower and Upper Output Limits – six numbers, three lower then three upper, applied last; -inf/inf (default) mean no limit.
• Gyro Second-order Dynamics – on (default) passes each gyroscope axis through H_g; off makes it instantaneous.
• Gyro Natural Frequency (rad/s) – ω_n of H_g, greater than zero. Default 190.
• Gyro Damping Ratio – ζ of H_g, zero or more. Default 0.707.
• Gyro Scale Factors and Cross-coupling – the [3,3] matrix SFCC_g. Default the identity.
• Gyro Measurement Bias – three numbers added before the dynamics. Default [0 0 0].
• G-sensitive Bias – g_s, three numbers in rate per g, each multiplying its own axis of A_b / g₀. Default [0 0 0].
• Gyro Lower and Upper Output Limits – six numbers, three lower then three upper, applied last; -inf/inf (default) mean no limit.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Each lower limit must not exceed its upper one.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. Every parameter is baked in at export time, and so is the discretization: the section coefficients are computed for the block's period (or its update rate) and printed at full precision. All ten are rendered from the one description of the arithmetic the block's own simulation runs, so they perform the same operations in the same order.

The three HDL targets are simulation-only: they carry the whole model in real arithmetic and quantize only at the ports. At a period short against the sensors' response the sections' poles sit close to z = 1, where Q16.16 coefficients lose the filters' DC gain, and the default limits are infinite, which fixed point cannot hold. VHDL keeps the arithmetic in a function of its own, called once per value it returns.

Simulink bridge

Import and export, mapped to Aerospace Blockset's aerolibnav/Three-axis Inertial Measurement Unit (the library path carries a newline after "Inertial"). Units ↔ units (Metric (MKS)/English, 1:1), IMU Location ↔ imu, Update Rate (s) ↔ i_Ts, Accelerometer Second-order Dynamics ↔ dtype_a (on/off, 1:1), Accelerometer Natural Frequency (rad/s) ↔ w_a, Accelerometer Damping Ratio ↔ z_a, Accelerometer Scale Factors and Cross-coupling ↔ a_sf_cc, Accelerometer Measurement Bias ↔ a_bias, Accelerometer Lower and Upper Output Limits ↔ a_sat, Gyro Second-order Dynamics ↔ dtype_g (on/off, 1:1), Gyro Natural Frequency (rad/s) ↔ w_g, Gyro Damping Ratio ↔ z_g, Gyro Scale Factors and Cross-coupling ↔ g_sf_cc, Gyro Measurement Bias ↔ g_bias, G-sensitive Bias ↔ g_sens, Gyro Lower and Upper Output Limits ↔ g_sat, the numbers passing through unchanged.

i_rand ("Noise on") is always off (see Notes); a model importing it on is reported rather than silently changed. The block has no SampleTime, so Sampling Time (s) stays on the ICore side and the rate crosses as i_Ts instead – which is not the same thing: a rate of −1 in i_Ts makes both of the Simulink block's filters diverge.

Notes

• Stateful with either sensor's dynamics on: two states per axis per sensor, starting at zero. With Update Rate (s) at 0 a sensor with dynamics has no direct feedthrough; with a positive rate, or with its dynamics off, it does.
• The gyroscope's g-sensitive bias sees A_b, not the specific force: no gravity and no lever arm reach it, exactly as in the Simulink block.
• No sensor noise. The Simulink block can add white noise from its own random number generator; no generated code can reproduce that sequence, so no test could confirm a noisy export, and the option is not offered. To model noise, add a Band-Limited White Noise per axis after this block – it lands after the output limits rather than before them.
• Nonlinear – the lever-arm terms are products of the rates, and the limits clamp – so the block carries no state space and model reduction reports it as unmergeable.