A question is whether it is possible to integrate them to get an equivalent accelerometer without using angular movement information.

After consideration, I think the answer for that question is 'YES'.

If there are

*n*accelerometers with accelerations,

$$ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \ldots, \mathbf{a}_n, \,$$

at locations,

$$ \mathbf{P}_1, \mathbf{P}_2, \mathbf{P}_3, \ldots, \mathbf{P}_n, \,$$

respectively, the value of the equivalent integrated accelerometer, \( \mathbf{a}_{eq} \), is

$$ \mathbf{a}_{eq}= \frac{1}{n} \sum_{i=1}^{n} \mathbf{a}_{i} \,$$

and its location, \( \mathbf{P}_{eq} \), is at

$$ \mathbf{P}_{eq}= \frac{1}{n} \sum_{i=1}^{n} \mathbf{P}_{i}. \,$$

To check the equations, I have made a simulation in Octave. The source code for the simulation is available at https://github.com/yan9a/Accelerometer_Integration.

The name of the main file to run is

*acc_i.m*. It uses

*GenerateSin.m*function to generate sinusoidal waveform for given peak and frequency. The function

*CalculateAcc2.m*calculates the acceleration at location {j} using an acceleration at location {i} and angular motion of the rigid body.

*AddWhiteNoise.m*function adds normally distributed noise with a given standard deviation to an input signal. Finally, it compares the result of the proposed equations with the simulated values and plots in 3 dimensions.

The following excerpts from an article by Latt

*et al.*[1] explains the equation in

*CalculateAcc2.m*. The total acceleration at location {i}, \(A_i\), of a rigid body can be represented by inertial acceleration of the body, \(A_{IN}\), the Gravity, G, and the rotation induced accelerations. The rotation induced accelerations are the centripetal accelerations, \(A_C\), and the tangential accelerations, \(A_T\). $$ A_i = A_{IN} + G + A_C + A_T $$

$$ A_i = A_{IN} + G + \Omega \times (\Omega \times R) + \alpha \times R $$

where all the variables are relative to the body frame {B}, \( \Omega = \begin{bmatrix} \omega_x & \omega_y & \omega_z \end{bmatrix} \) is the angular velocity vector, R is the vector from the unknown instantaneous center of rotation to the point of sensing, and \( \alpha = \begin{bmatrix} \alpha_x & \alpha_y & \alpha_z \end{bmatrix} \) is the angular acceleration vector. The symbol \( \times \) represents the cross product operation.

Finding the acceleration at location {j} using similar method, and taking the difference gives the following equation. $$ A_j = A_{j} + \Omega \times (\Omega \times R_{ij}) + \alpha \times R_{ij} $$ where \( R_{ij} = \begin{bmatrix} R_x & R_y & R_z \end{bmatrix} \) is the vector from {i} to {j}.

Reference:

[1] Win Tun Latt, U-Xuan Tan, Cameron N. Riviere, Wei Tech Ang, Placement of accelerometers for high sensing resolution in micromanipulation, Sensors and Actuators A: Physical, Volume 167, Issue 2, June 2011, Pages 304-316, ISSN 0924-4247, http://dx.doi.org/10.1016/j.sna.2011.03.001. (http://www.sciencedirect.com/science/article/pii/S0924424711001282)

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