Robotics Kinematics and Dynamics/Serial Manipulator Dynamics

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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:

_a \dot{v} = \dfrac{d \, _a v}{dt} = \lim_{\Delta t \rightarrow 0} \dfrac{_a v(t + \Delta t) - \,_a v(t)}{\Delta t}


_a \dot{\omega} = \dfrac{d \, _a \omega}{dt} = \lim_{\Delta t \rightarrow 0} \dfrac{_a \omega(t + \Delta t) - \,_a \omega(t)}{\Delta t}

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

_a v_q = \, ^b _a R \, _b v_q + \, _a \omega_b \times \, ^b _a R \, _b q

Differentiating the above expression gives the acceleration of the vector q:

_a \dot{v}_q = \dfrac{d}{dt} \, ^b _a R \, _b v_q + \, _a \dot{\omega}_b \times \, _a ^b R \, _b q + \, _a \omega_b \times \dfrac{d}{dt} \, ^b _a R \, _b q

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

_a v_q = \dfrac{d}{dt} \, ^b _a R \, _b q = \, ^b _a R \, _b v_q + \, _a \omega_b \times \, ^b _a R \, _b q

Applying this result to the acceleration leads to:

_a \dot{v}_q = ^b _a R \, _b \dot{v}_q + \, _a \omega_b \times \, ^b _a R \, _b v_q + \, _a \dot{\omega}_b \times \, _a ^b R \, _b q + \, _a \omega_b \times \left( ^b _a R \, _b v_q + \, _a \omega_b \times \, ^b _a R \, _b q \right)

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

_a \dot{v}_q = \, _a \dot{v}_{b,org} + \, ^b _a R \, _b \dot{v}_q + \, _a \omega_b \times \, ^b _a R \, _b v_q + \, _a \dot{\omega}_b \times \, _a ^b R \, _b q + \, _a \omega_b \times \left( ^b _a R \, _b v_q + \, _a \omega_b \times \, ^b _a R \, _b q \right)

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

_a \dot{v}_q = \, _a \dot{v}_{b,org} + \, _a \dot{\omega}_b \times \, _a ^b R \, _b q + \, _a \omega_b \times \left( _a \omega_b \times \, ^b _a R \, _b q \right)

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

_a \omega_c = \, _a \omega_b + \, _a ^b R \, _b \omega_c

Differentiating leads to:

_a \dot{\omega}_c = \, _a \dot{\omega}_b + \dfrac{d}{dt} \, _a ^b R \, _b \omega_c

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

_a \dot{\omega}_c = \, _a \dot{\omega}_b + \, _a \omega_b \times \, _a ^b R \, _b \omega_c

Inertia Tensor[edit]

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

_a I = \begin{pmatrix}
I_{xx} & -I_{xy} & -I_{xz} \\
I_{xy} & I_{zz} & -I_{yz} \\
I_{xz} & -I_{yz} & I_{zz} \\

Newton's and Euler's equation[edit]

The force F, acting at the center of mass of a rigid body with total massm, causing an acceleration \dot{v}_{com}, equals:

F = m \dot{v}_{com}

In a similar way, the moment N, causing an angular acceleration \dot{\omega}, is given by:

N = \, _c I \dot{\omega} + \omega \, _c I \omega

where _c I is the inertia tensor, expressed in a frame \{c\} of which the origin coincides with the center of mass of the rigid body.