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:

$_{a}{\dot {v}}={\dfrac {d\,_{a}v}{dt}}=\lim _{\Delta t\rightarrow 0}{\dfrac {_{a}v(t+\Delta t)-\,_{a}v(t)}{\Delta t}}$,

and:

$_{a}{\dot {\omega }}={\dfrac {d\,_{a}\omega }{dt}}=\lim _{\Delta t\rightarrow 0}{\dfrac {_{a}\omega (t+\Delta t)-\,_{a}\omega (t)}{\Delta t}}$

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:

$_{a}v_{q}=\,_{a}^{b}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q$

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

$_{a}{\dot {v}}_{q}={\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times {\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}q$

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

$_{a}v_{q}={\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}q=\,_{a}^{b}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q$

Applying this result to the acceleration leads to:

$_{a}{\dot {v}}_{q}=_{a}^{b}R\,_{b}{\dot {v}}_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(^{b}_{a}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)$

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:

$_{a}{\dot {v}}_{q}=\,_{a}{\dot {v}}_{b,org}+\,_{a}^{b}R\,_{b}{\dot {v}}_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(^{b}_{a}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)$

For rotational joints, $_{b}q$ is constant, and the above expression simplifies to:

$_{a}{\dot {v}}_{q}=\,_{a}{\dot {v}}_{b,org}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)$

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

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

$_{a}I={\begin{pmatrix}I_{xx}&-I_{xy}&-I_{xz}\\I_{xy}&I_{yy}&-I_{yz}\\I_{xz}&-I_{yz}&I_{zz}\\\end{pmatrix}}$

Newton's and Euler's equation

The force $F$, acting at the center of mass of a rigid body with total mass$m$, 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:

$N=\,_{c}I{\dot {\omega }}+\omega \times \,_{c}I\omega$,

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.