Ingeometry, theorientation,attitude,bearing orangular position of an object – such as aline,plane orrigid body – is part of the description of how it is placed in thespace it occupies.[1]More specifically, it refers to the imaginaryrotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement, in which case it may be necessary to add an imaginarytranslation to change the object'sposition (or linear position). The position and orientation together fully describe how the object is placed in space. The above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its position does not change when it rotates.
Euler's rotation theorem shows that in three dimensions any orientation can be reached with a singlerotation around a fixed axis. This gives one common way of representing the orientation using anaxis–angle representation. Other widely used methods includerotation quaternions,rotors,Euler angles, orrotation matrices. More specialist uses includeMiller indices in crystallography,strike and dip in geology andgrade on maps and signs.Aunit vector may also be used to represent an object'snormal vectordirection or therelative direction between two points.
Typically, the orientation is given relative to aframe of reference, usually specified by aCartesian coordinate system.
In general the position and orientation in space of arigid body are defined as the position and orientation, relative to the main reference frame, of another reference frame, which is fixed relative to the body, and hence translates and rotates with it (the body'slocal reference frame, orlocal coordinate system). At least three independent values are needed to describe the orientation of this local frame. Three other values describe the position of a point on the object.All the points of the body change their position during a rotation except for those lying on the rotation axis. If the rigid body hasrotational symmetry not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. For example, the orientation in space of aline,line segment, orvector can be specified with only two values, for example twodirection cosines. Another example is the position of a point on the Earth, often described using the orientation of a line joining it with the Earth's center, measured using the two angles oflongitude and latitude. Likewise, the orientation of aplane can be described with two values as well, for instance by specifying the orientation of a linenormal to that plane, or by using the strike and dip angles.
Further details about the mathematical methods to represent the orientation of rigid bodies and planes in three dimensions are given in the following sections.
Intwo dimensions the orientation of any object (line, vector, orplane figure) is given by a single value: the angle through which it has rotated. There is only one degree of freedom and only one fixed point about which the rotation takes place.
When there ared dimensions, specification of an orientation of an object that does not have any rotational symmetry requiresd(d − 1) / 2 independent values.
Several methods to describe orientations of a rigid body in three dimensions have been developed. They are summarized in the following sections.
The first attempt to represent an orientation is attributed toLeonhard Euler. He imagined three reference frames that could rotate one around the other, and realized that by starting with a fixed reference frame and performing three rotations, he could get any other reference frame in the space (using two rotations to fix the vertical axis and another to fix the other two axes). The values of these three rotations are calledEuler angles.
These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles. Mathematically they constitute a set of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane. In aerospace engineering they are usually referred to as Euler angles.
Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis (Euler's rotation theorem). Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed.
Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore, any orientation can be represented by a rotation vector (also called Euler vector) that leads to it from the reference frame. When used to represent an orientation, the rotation vector is commonly called orientation vector, or attitude vector.
A similar method, calledaxis–angle representation, describes a rotation or orientation using aunit vector aligned with the rotation axis, and a separate value to indicate the angle (see figure).
With the introduction of matrices, the Euler theorems were rewritten. The rotations were described byorthogonal matrices referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is commonly called orientation matrix, or attitude matrix.
The above-mentioned Euler vector is theeigenvector of a rotation matrix (a rotation matrix has a unique realeigenvalue).The product of two rotation matrices is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe.
Theconfiguration space of a non-symmetrical object inn-dimensional space isSO(n)×Rn. Orientation may be visualized by attaching a basis oftangent vectors to an object. The direction in which each vector points determines its orientation.
Another way to describe rotations is usingrotation quaternions, also called versors. They are equivalent to rotation matrices and rotation vectors. With respect to rotation vectors, they can be more easily converted to and from matrices. When used to represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions.
The attitude of a rigid body is its orientation as described, for example, by the orientation of a frame fixed in the body relative to a fixed reference frame. The attitude is described byattitude coordinates, and consists of at least three coordinates.[2] One scheme for orienting a rigid body is based upon body-axes rotation; successive rotations three times about the axes of the body's fixed reference frame, thereby establishing the body'sEuler angles.[3][4] Another is based uponroll, pitch and yaw,[5] although these terms also refer toincremental deviations from the nominal attitude
...the attitude of a plane or a line — that is, its orientation in space — is fundamental to the description of structures.
Euler angle rigid body attitude.
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