Lecture

Navigation Equations: Part 1

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In course

ENV-548: Sensor orientation

Determination of spatial orientation (i.e. position, velocity, attitude) via integration of inertial sensors with satellite positioning. Prerequisite for applications related to remote sensing, environmental monitoring, mobile mapping, robotics, space expl ...

Description

This lecture covers the navigation equations, including body frame orientation, symmetric and orthonormal matrices, skew symmetric matrix, and attitude determination. The instructor explains the calculations step by step.

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Instructor

Jan Skaloud

Expertise Satellite positioning, inertial and integrated navigation Sensor orientation and calibration, attitude determination Mobile mapping, airborne laser scanning Adjustment methods, Kalman Filtering Mission Teaching in 3 EPFL sections and 2 faculties, director of ESO lab. Books / Monographs J. Skaloud and M. Cramer, «Data Capture,» in GIS Handbook, W. Kresse and D. Danko, Eds. Springer, 2011. D. Lichti and J. Skaloud, «Registration and Calibration,» in Airborne and terrestrial laser scanning, H.-G. Ma

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if denotes the entry in the th row and th column then for all indices and Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.

Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: or in matrix form: Hermitian matrices can be understood as the complex extension of real symmetric matrices.

Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to The matrix is skew-symmetric because Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2.

Orthogonal matrix

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers.

Normal matrix

In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A^: The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.

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