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317:. 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 called
110:(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.
467:, 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.
373:
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
203:). 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 has
343:
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
498:, and consists of at least three coordinates. 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's
414:). 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.
101:
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 imaginary
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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's
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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 by
403:
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.
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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 a
223:. 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 of
254:) 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.
370:). 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.
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Further details about the mathematical methods to represent the orientation of rigid bodies and planes in three dimensions are given in the following sections.
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Several methods to describe orientations of a rigid body in three dimensions have been developed. They are summarized in the following sections.
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603:...the attitude of a plane or a line â that is, its orientation in space â is fundamental to the description of structures.
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Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis (
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This article is about the orientation or attitude of an object or a shape in a space. For the orientation of a space, see
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dimensions, specification of an orientation of an object that does not have any rotational symmetry requires
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to represent an orientation, the rotation vector is commonly called orientation vector, or attitude vector.
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Quaternions and
Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality
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With the introduction of matrices, the Euler theorems were rewritten. The rotations were described by
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can be described with two values as well, for instance by specifying the orientation of a line
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aligned with the rotation axis, and a separate value to indicate the angle (see figure).
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to an object. The direction in which each vector points determines its orientation.
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airplane. In aerospace engineering they are usually referred to as Euler angles.
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shows that in three dimensions any orientation can be reached with a single
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630:. American Institute of Aeronautics and Astronautics. p. 71.
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Euler angles, one of the possible ways to describe an orientation
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The first attempt to represent an orientation is attributed to
688:. American Institute of Aeronautics and Astronautics. p.
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The orientation of a rigid body is determined by three angles
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it occupies. More specifically, it refers to the imaginary
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of a rotation matrix (a rotation matrix has a unique real
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445:. Orientation may be visualized by attaching a basis of
93:â is part of the description of how it is placed in the
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TaitâBryan angles. Other way for describing orientation
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can be specified with only two values, for example two
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to that plane, or by using the strike and dip angles.
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In general the position and orientation in space of a
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351:A rotation represented by an Euler axis and angle.
170:Typically, the orientation is given relative to a
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508:incremental deviations from the nominal attitude
250:the orientation of any object (line, vector, or
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381:, describes a rotation or orientation using a
583:Robert J. Twiss; Eldridge M. Moores (1992).
722:Modelling and Control of Robot Manipulators
716:Lorenzo Sciavicco; Bruno Siciliano (2000).
653:"Figure 4.7: Aircraft Euler angle sequence"
463:Another way to describe rotations is using
659:. Princeton University Press. p. 85.
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406:The above-mentioned Euler vector is the
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591:(2nd ed.). Macmillan. p. 11.
549:Rotation formalisms in three dimensions
287:Rotation formalisms in three dimensions
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724:(2nd ed.). Springer. p. 32.
680:Bong Wie (1998). "§5.2 Euler angles".
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331:Euler angles § TaitâBryan angles
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124:. Other widely used methods include
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227:. Likewise, the orientation of a
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519:Angular displacement
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465:rotation quaternions
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476:Rigid body
425:object in
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