Robotics Kinematics and Dynamics/Serial Manipulator Dynamics

< Robotics Kinematics and Dynamics

Acceleration of a Rigid BodyEdit

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 TensorEdit

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

 

Newton's and Euler's equationEdit

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.