The linear and angular accelerations are the time derivatives of the linear and angular velocity vectors at any instant:
,
and:
![{\displaystyle _{a}{\dot {\omega }}={\dfrac {d\,_{a}\omega }{dt}}=\lim _{\Delta t\rightarrow 0}{\dfrac {_{a}\omega (t+\Delta t)-\,_{a}\omega (t)}{\Delta t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/537511aab9216930e0d8c16df083f75c9c87b9c6)
The linear velocity, as seen from a reference frame
, of a vector
, relative to frame
of which the origin coincides with
, is given by:
![{\displaystyle _{a}v_{q}=\,_{a}^{b}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfcd637b1f88778173f2d3b497fda03e20fb7346)
Differentiating the above expression gives the acceleration of the vector
:
![{\displaystyle _{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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b33a7eac5fde27941459b7746bf39524921b143a)
The equation for the linear velocity may also be written as:
![{\displaystyle _{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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fed7c583ddbf3bf2b1a30572f47b8fc3a00a504)
Applying this result to the acceleration leads to:
![{\displaystyle _{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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e3c2d2bbaeb39be1061dc9d8548ef577270287)
In the case the origins of
and
do not coincide, a term for the linear acceleration of
, with respect to
, is added:
![{\displaystyle _{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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5a0ed323c37c3dfdb44798e28f185ef4fff831)
For rotational joints,
is constant, and the above expression simplifies to:
![{\displaystyle _{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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8514d1238b5d78f62409c79433f7d304c1a46ac)
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:
![{\displaystyle _{a}\omega _{c}=\,_{a}\omega _{b}+\,_{a}^{b}R\,_{b}\omega _{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc9691f9a61b4aafb3afa1af4be08cde902d178)
Differentiating leads to:
![{\displaystyle _{a}{\dot {\omega }}_{c}=\,_{a}{\dot {\omega }}_{b}+{\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}\omega _{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ae4f64b32d0d9e142d45e57b9120bd9a5a68fc)
Replacing the last term with one of the expressions derived earlier:
![{\displaystyle _{a}{\dot {\omega }}_{c}=\,_{a}{\dot {\omega }}_{b}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}\omega _{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/595d57389cba09b8b239f6af88eccb89c26f0db8)
The inertia tensor can be thought of as a generalization of the scalar moment of inertia:
![{\displaystyle _{a}I={\begin{pmatrix}I_{xx}&-I_{xy}&-I_{xz}\\I_{xy}&I_{yy}&-I_{yz}\\I_{xz}&-I_{yz}&I_{zz}\\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c01698f3714c22a294d6108092380f62eceea82e)
The force
, acting at the center of mass of a rigid body with total mass
, causing an acceleration
, equals:
![{\displaystyle F=m{\dot {v}}_{com}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81a86898f54a66812b57b0ca9e4f808e378d299a)
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.