Newton-Euler equations

In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.

Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments) acting on the rigid body.

With respect to a coordinate frame whose origin coincides with the body's center of mass, they can be expressed in matrix form as:

where

With respect to a coordinate frame located at point P that is fixed in the body and not coincident with the center of mass, the equations assume the more complex form:

where c is the location of the center of mass expressed in the body-fixed frame, and

denote skew-symmetric cross product matrices.

The left hand side of the equation—which includes the sum of external forces, and the sum of external moments about P—describes a spatial wrench, see screw theory.

The inertial terms are contained in the spatial inertia matrix

while the fictitious forces are contained in the term:

When the center of mass is not coincident with the coordinate frame (that is, when c is nonzero), the translational and angular accelerations (a and α) are coupled, so that each is associated with force and torque components.

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