244
4 Navigations from Ground to Space
system to the Earth. Usually, using the attitude sensors, such as the Sun sensors, Earth
sensors, star sensors and inertial gyroscopes, carried with the spacecrafts, the attitude
parameters can be measured and estimated autonomously. The attitude estimation
accuracy is closely related to the performances of the sensors and the methods of data
processing. In general, there are three ways for expressing the spacecraft’s attitudes:
attitude matrix, Euler angles and quaternion.
When the spatial reference coordinate system (r-system) and the spacecraft’s body
coordinate system (b-system) are all right-handed orthogonal coordinate systems,
there are nine direction cosines between the two sets of coordinate axes. Using the
nine direction cosines, a direction cosine matrix can be formed to determine the
geometric direction of the b-system relative to the r-system. The direction cosine
matrix represents the coordinate rotation transformation relationship from the rsystem to the b-system, called attitude matrix. Through the attitude matrix, the same
direction vector can be represented, respectively, in the spacecraft body coordinate
system and the reference coordinate system, connected with each other, and thus
the basic orientation relationship of the b-system relative to the r-system can be
determined completely. The attitude matrix is an orthogonal matrix, that is, there are
six constraints among the elements of the attitude matrix. In other words, only three
of the nine direction cosine elements in the attitude matrix are independent.
It is well known that the orientation of the b-system relative to the r-system has
nothing to do with the relative displacements of their coordinate origins. Thereby,
supposed that the origins of two coordinate systems coincide, the motion of the bsystem relative to the r-system can be regarded as the fixed-point rotation of a rigid
body. In the terms of Euler’s rotation theorem, the general displacement of a rigid
body (or a coordinate frame) with a fixed point is equivalent to a single rotation
about a certain axis that passes through the fixed point, where this axis is called
Euler axis. It also means that the composition of two rotations is also one rotation,
and any rotation may be described using three angles, which are called Euler angles.
Therefore, as long as the r-system has continuously made three rotations about its
coordinate axes, it can be transformed into the b-system, and the angle of rotation
each time is an Euler angle. Furthermore, the attitude matrix determined by the Euler
angles is the product of the matrixes from three rotations. Apparently, the attitude
matrix is related to the order of three rotations in the coordinate system. The rotation
sequence of the coordinate system can usually be divided into two categories: one is
the first and third rotations are about the same coordinate axis, the second rotation
is about one of the other two coordinate axes, and the other is that each rotation is
about a different coordinate axis. In this way, there are 12 kinds of combinations
of the Euler angle rotation sequence for the two categories of rotations. The most
commonly used Euler angles are obtained by the two kinds of rotation sequences,
Z-X-Z and Z-X-Y, and correspondingly two sets of the Euler angles are labeled as ψ,
θ and ϕ, as well as ψ, ϕ and θ, respectively.
In Fig. 4.7, both the r-system O-X r Y r Z r and the b-system O-X b Y b Z b are established with the same fixed point O as the coordinate origins, and their Euler angles
are represented in accordance with the rotation sequence of Z-X-Z. When t = 0,
the two coordinate systems are completely coincident. According to the right-hand
4 Navigations from Ground to Space
system to the Earth. Usually, using the attitude sensors, such as the Sun sensors, Earth
sensors, star sensors and inertial gyroscopes, carried with the spacecrafts, the attitude
parameters can be measured and estimated autonomously. The attitude estimation
accuracy is closely related to the performances of the sensors and the methods of data
processing. In general, there are three ways for expressing the spacecraft’s attitudes:
attitude matrix, Euler angles and quaternion.
When the spatial reference coordinate system (r-system) and the spacecraft’s body
coordinate system (b-system) are all right-handed orthogonal coordinate systems,
there are nine direction cosines between the two sets of coordinate axes. Using the
nine direction cosines, a direction cosine matrix can be formed to determine the
geometric direction of the b-system relative to the r-system. The direction cosine
matrix represents the coordinate rotation transformation relationship from the rsystem to the b-system, called attitude matrix. Through the attitude matrix, the same
direction vector can be represented, respectively, in the spacecraft body coordinate
system and the reference coordinate system, connected with each other, and thus
the basic orientation relationship of the b-system relative to the r-system can be
determined completely. The attitude matrix is an orthogonal matrix, that is, there are
six constraints among the elements of the attitude matrix. In other words, only three
of the nine direction cosine elements in the attitude matrix are independent.
It is well known that the orientation of the b-system relative to the r-system has
nothing to do with the relative displacements of their coordinate origins. Thereby,
supposed that the origins of two coordinate systems coincide, the motion of the bsystem relative to the r-system can be regarded as the fixed-point rotation of a rigid
body. In the terms of Euler’s rotation theorem, the general displacement of a rigid
body (or a coordinate frame) with a fixed point is equivalent to a single rotation
about a certain axis that passes through the fixed point, where this axis is called
Euler axis. It also means that the composition of two rotations is also one rotation,
and any rotation may be described using three angles, which are called Euler angles.
Therefore, as long as the r-system has continuously made three rotations about its
coordinate axes, it can be transformed into the b-system, and the angle of rotation
each time is an Euler angle. Furthermore, the attitude matrix determined by the Euler
angles is the product of the matrixes from three rotations. Apparently, the attitude
matrix is related to the order of three rotations in the coordinate system. The rotation
sequence of the coordinate system can usually be divided into two categories: one is
the first and third rotations are about the same coordinate axis, the second rotation
is about one of the other two coordinate axes, and the other is that each rotation is
about a different coordinate axis. In this way, there are 12 kinds of combinations
of the Euler angle rotation sequence for the two categories of rotations. The most
commonly used Euler angles are obtained by the two kinds of rotation sequences,
Z-X-Z and Z-X-Y, and correspondingly two sets of the Euler angles are labeled as ψ,
θ and ϕ, as well as ψ, ϕ and θ, respectively.
In Fig. 4.7, both the r-system O-X r Y r Z r and the b-system O-X b Y b Z b are established with the same fixed point O as the coordinate origins, and their Euler angles
are represented in accordance with the rotation sequence of Z-X-Z. When t = 0,
the two coordinate systems are completely coincident. According to the right-hand