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Tait-Bryan angles
 
 
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Yaw, pitch, and roll, also known as Tait–Bryan rotations, named after Peter Guthrie Tait
Peter Guthrie Tait

Peter Guthrie Tait was a Scotland Mathematical physics, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with William Thomson, 1st Baron Kelvin....
 and George Bryan, are a specific sequence of Euler angles
Euler angles

The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in dimension Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles....
 very often used in aerospace applications to define the relative orientation of a vehicle. The three angles specified in this formulation are defined as the roll angle, pitch angle, and yaw angle.

Tait-Bryan rotations are used in aerospace to define a rotation between a reference axis system and a vehicle-fixed axis system.






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Yaw, pitch, and roll, also known as Tait–Bryan rotations, named after Peter Guthrie Tait
Peter Guthrie Tait

Peter Guthrie Tait was a Scotland Mathematical physics, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with William Thomson, 1st Baron Kelvin....
 and George Bryan, are a specific sequence of Euler angles
Euler angles

The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in dimension Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles....
 very often used in aerospace applications to define the relative orientation of a vehicle. The three angles specified in this formulation are defined as the roll angle, pitch angle, and yaw angle.

Tait-Bryan rotations are used in aerospace to define a rotation between a reference axis system and a vehicle-fixed axis system. Consider an aircraft-body coordinate (ABC) system which is fixed to the vehicle (rotates and translates with the vehicle). The origin of the ABC system is located at the vehicle's center of gravity, the x-axis points forward along some convenient reference line along the body, the y-axis points to the right of the vehicle along the wing, and the z-axis points downward to form an orthogonal right-handed system. Consider a local horizontal and local vertical reference frame (LHLV) that shares the same origin as the ABC system but is always aligned with x pointing in the direction of true north, y-axis pointing to true east, and the z-axis pointing down towards the center of gravity of the earth.

Given this definition, the rotation sequence from LHLV to ABC is defined as follows: (Stevens, 26)

1. Right-handed rotation about the z-axis by the yaw angle.
2. Right-handed rotation about the new (once-rotated) y-axis by the pitch angle.
3. Right-handed rotation about the new (twice-rotated) x-axis by the roll angle.

This rotation sequence can be represented mathematically by the following equation: (Stevens, 26)

Rlhlv to abc = Rx(roll) × Ry(pitch) × Rz(yaw)


and

Uabc = Rlhlv to abc × ULHLV


where Rx(a), Ry(a), and Rz(a), is shorthand notation for the planar rotation matrices of a positive rotation by angle a about x-, y-, and z-axes, respectively, and U is a column vector of Cartesian coordinates. The inverse rotation, from ABC to LHLV, is represented by the transpose of this matrix.

The composite rotation matrix, from the LHLV system to the ABC system, is defined as follows (Stevens, 26):



It is important to notice that sometimes, informally, these names are used not to refer to the three angles, but to the rotations associated to each one of them.

See also


  • Yaw angle
    Yaw angle

    The yaw angle is the angle between a vehicle's heading and a reference heading . One of the Tait-Bryan angles. In aeronautics, robotics and marine control, it is typically assigned the shorthand notation ....
  • Euler angles
    Euler angles

    The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in dimension Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles....
  • flight dynamics
    Flight dynamics

    Flight dynamics is the science of aircraft and spacecraft vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of mass, known as pitch, roll and yaw ....
  • attitude control
  • Moving frame
    Moving frame

    In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the differential geometry of smooth manifolds embedded in a homogeneous space....