Robotics Kinematics and Dynamics/Serial Manipulator Dynamics

Acceleration of a Rigid Body 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 The inertia tensor can be thought of as a generalization of the scalar moment of inertia: Newton's and Euler's equation 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.