gyro-calibration.cpp

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#include <old_calibrationn.h>
#include <Eigen/Cholesky>
#include <Eigen/SVD>
#include <Eigen/QR>

/*
 * Compute basic calibration parameters for a three axis gyroscope.
 * The measurement equation is
 * gyro_k = accelSensitivity * \tilde{accel}_k + bias + omega
 *
 *      where, omega is the true angular rate (assumed to be zero)
 *              bias is the sensor bias
 *              accelSensitivity is the 3x3 matrix of sensitivity scale factors
 *              \tilde{accel}_k is the calibrated measurement of the accelerometer
 *                      at time k
 *
 * After calibration, the optimized gyro measurement is given by
 * \tilde{gyro}_k = gyro_k - bias - accelSensitivity * \tilde{accel}_k
 */
void gyroscope_calibration(Vector3f &bias, Matrix3f &accelSensitivity, Vector3f gyroSamples[],
                           Vector3f accelSamples[], size_t n_samples)
{
    // Assume the following measurement model:
    // y = H*x
    // where x is the vector of unknowns, and y is the measurement vector.
    // the vector of unknowns is
    // [a_x a_y a_z b_x]
    // The measurement vector is
    // [gyro_x]
    // and the measurement matrix H is
    // [accelSample_x accelSample_y accelSample_z 1]

    // Note that the individual solutions for x are given by
    // (H^T \times H)^-1 \times H^T y
    // Everything is identical except for y and x.  So, transform it
    // into block form X = (H^T \times H)^-1 \times H^T Y
    // where each column of X contains the partial solution for each
    // column of y.

    // Construct the matrix of accelerometer samples.  Intermediate results
    // are computed in "twice the precision that the source provides and the
    // result deserves" by Kahan's thumbrule to prevent numerical problems.
    Matrix<double, Dynamic, 4> H(n_samples, 4);
    // And the matrix of gyro samples.
    Matrix<double, Dynamic, 3> Y(n_samples, 3);
    for (unsigned i = 0; i < n_samples; ++i) {
        H.row(i) << accelSamples[i].transpose().cast<double>(), 1.0;
        Y.row(i) << gyroSamples[i].transpose().cast<double>();
    }

#if 1
    Matrix<double, 4, 3> result;
    // Use the cholesky-based Penrose pseudoinverse method.
    (H.transpose() * H).ldlt().solve(H.transpose() * Y, &result);

    // Transpose the result and return it to the caller.  Cast back to float
    // since there really isn't that much accuracy in the result.
    bias = result.row(3).transpose().cast<float>();
    accelSensitivity = result.block<3, 3>(0, 0).transpose().cast<float>();
#else
    // TODO: Compare this result with a total-least-squares model.
    size_t n = 4;
    Matrix<double, Dynamic, 7> C;
    C << H, Y;
    SVD<Matrix<double, Dynamic, 7>> usv(C);
    Matrix<double, 4, 3> V_ab = usv.matrixV().block<4, 3>(0, n);
    Matrix<double, Dynamic, 3> V_bb = usv.matrixV().corner(BottomRight, n_samples - 4, 3);
    // X = -V_ab/V_bb;
    // X^T = (A * B^-1)^T
    // X^T = (B^-1^T * A^T)
    // X^T = (B^T^-1 * A^T)
    // V_bb is orthgonal but not orthonormal.  QR decomposition
    // should be very fast in this case.
    Matrix<double, 3, 4> result;
    V_bb.transpose().qr().solve(-V_ab.transpose(), &result);

    // Results only deserve single precision.
    bias = result.col(3).cast<float>();
    accelSensitivity = result.block<3, 3>(0, 0).cast<float>();

#endif
}