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Euler angles Euler angles, one of the possible ways to describe an orientation. The first attempt to represent an orientation is attributed to Leonhard 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 other to fix the other two axes.

The values of these three rotations are called Euler 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. A rotation represented by an Euler axis and angle. Axis-angle representation 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, called axis-angle representationdescribes a rotation or orientation using a unit vector aligned with the rotation axis, and a separate value to indicate the angle see figure.

Rotation matrix With the introduction of matrices the Euler theorems were rewritten. The rotations were described by orthogonal 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 the eigenvector of a rotation matrix a rotation matrix has a unique real eigenvalue. 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.

Orientation may be visualized by attaching a basis of tangent vectors to an object. The direction in which each vector points determines its orientation.

Quaternions and spatial rotation Another way to describe rotations is using rotation quaternionsalso 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.

Newton's second law in three dimensions[ edit ] To consider rigid body dynamics in three-dimensional space, Newton's second law must be extended to define the relationship between the movement of a rigid body and the system of forces and torques that act on it.

Newton formulated his second law for a particle as, "The change of motion of an object is proportional to the force impressed and is made in the direction of the straight line in which the force is impressed.5 Dynamics of Rigid Bodies A rigid body is an idealization of a body that does not deform or change shape.

Formally it is defined as a collection of particles with the property that the distance between particles remains unchanged during the course of motions of the body.

A rigid body is an object with a mass that holds a rigid shape, such as a phonograph turntable, in contrast to the sun, which is a ball of gas.

Many of the equations for the mechanics of rotating objects are similar to the motion equations for linear motion. and is measured in radians/second/second or rad/s 2.

The kinematics equations for. Mechanics of Rigid Body 1.B Kinetics, Dynamics Introduction Kinematics.

Types of Rigid Body Motion: Translation, Select one adequate reference system- Kinematics relationships- Calculating torques (or Moment of forces about a point) Computing cross-product- Calculating Moment of Inertia.

The parallel-axis theorem. In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces exerted on it. A rigid body is usually considered as a continuous distribution of mass.

Kinematics Page 2 of 4 Fixed Axis Rotation The description is simple: the rigid body has a hinge, joint, or pivot which is connected to a non-moving foundation.

The rigid body rotates about a stationary axis passing through this fixed point. There is one point on the rigid body that has zero velocity, and it is of course this fixed point. In mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained (or desired) motion that is the mathematical model for a mechanical system.

As in the familiar use of the word chain, the rigid bodies, or links .

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EN4 Notes: Kinematics of Rigid Bodies