# Robotics Kinematics and Dynamics/Serial Manipulator Dynamics

## Acceleration of a Rigid Body[edit]

The linear and angular accelerations are the time derivatives of the linear and angular velocity vectors at any instant:

- ,

and:

The linear velocity, as seen from a reference frame , of a vector , relative to frame of which the origin coincides with , is given by:

Differentiating the above expression gives the acceleration of the vector :

The equation for the linear velocity may also be written as:

Applying this result to the acceleration leads to:

In the case the origins of and do not coincide, a term for the linear acceleration of , with respect to , is added:

For rotational joints, is constant, and the above expression simplifies to:

The angular velocity of a frame , rotating relative to a frame , which in itself is rotating relative to the reference frame , with respect to , is given by:

Differentiating leads to:

Replacing the last term with one of the expressions derived earlier:

## Inertia Tensor[edit]

The inertia tensor can be thought of as a generalization of the scalar moment of inertia:

## Newton's and Euler's equation[edit]

The force , acting at the center of mass of a rigid body with total mass, causing an acceleration , equals:

In a similar way, the moment , causing an angular acceleration , is given by:

- ,

where is the inertia tensor, expressed in a frame of which the origin coincides with the center of mass of the rigid body.