[<< wikibooks] 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

v
˙

=

d

a

v

d
t

=

lim

Δ
t
→
0

a

v
(
t
+
Δ
t
)
−

a

v
(
t
)

Δ
t

{\displaystyle _{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

ω
˙

=

d

a

ω

d
t

=

lim

Δ
t
→
0

a

ω
(
t
+
Δ
t
)
−

a

ω
(
t
)

Δ
t

{\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}}}
The linear velocity, as seen from a reference frame

{
a
}

{\displaystyle \{a\}}
, of a vector

q

{\displaystyle q}
, relative to frame

{
b
}

{\displaystyle \{b\}}
of which the origin coincides with

{
a
}

{\displaystyle \{a\}}
, is given by:

a

v

q

=

a

b

R

b

v

q

+

a

ω

b

×

a

b

R

b

q

{\displaystyle _{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

{\displaystyle q}
:

a

v
˙

q

=

d

d
t

a

b

R

b

v

q

+

a

ω
˙

b

×

a

b

R

b

q
+

a

ω

b

×

d

d
t

a

b

R

b

q

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

a

v

q

=

d

d
t

a

b

R

b

q
=

a

b

R

b

v

q

+

a

ω

b

×

a

b

R

b

q

{\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}
Applying this result to the acceleration leads to:

a

v
˙

q

=

a

b

R

b

v
˙

q

+

a

ω

b

×

a

b

R

b

v

q

+

a

ω
˙

b

×

a

b

R

b

q
+

a

ω

b

×

(

a

b

R

b

v

q

+

a

ω

b

×

a

b

R

b

q

)

{\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)}
In the case the origins of

{
a
}

{\displaystyle \{a\}}
and

{
b
}

{\displaystyle \{b\}}
do not coincide, a term for the linear acceleration of

{
b
}

{\displaystyle \{b\}}
, with respect to

{
a
}

{\displaystyle \{a\}}

a

v
˙

q

=

a

v
˙

b
,
o
r
g

+

a

b

R

b

v
˙

q

+

a

ω

b

×

a

b

R

b

v

q

+

a

ω
˙

b

×

a

b

R

b

q
+

a

ω

b

×

(

a

b

R

b

v

q

+

a

ω

b

×

a

b

R

b

q

)

{\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)}
For rotational joints,

b

q

{\displaystyle _{b}q}
is constant, and the above expression simplifies to:

a

v
˙

q

=

a

v
˙

b
,
o
r
g

+

a

ω
˙

b

×

a

b

R

b

q
+

a

ω

b

×

(

a

ω

b

×

a

b

R

b

q

)

{\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)}
The angular velocity of a frame

{
c
}

{\displaystyle \{c\}}
, rotating relative to a frame

{
b
}

{\displaystyle \{b\}}
, which in itself is rotating relative to the reference frame

{
a
}

{\displaystyle \{a\}}
, with respect to

{
a
}

{\displaystyle \{a\}}
, is given by:

a

ω

c

=

a

ω

b

+

a

b

R

b

ω

c

{\displaystyle _{a}\omega _{c}=\,_{a}\omega _{b}+\,_{a}^{b}R\,_{b}\omega _{c}}

a

ω
˙

c

=

a

ω
˙

b

+

d

d
t

a

b

R

b

ω

c

{\displaystyle _{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

ω
˙

c

=

a

ω
˙

b

+

a

ω

b

×

a

b

R

b

ω

c

{\displaystyle _{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
=

(

I

x
x

−

I

x
y

−

I

x
z

I

x
y

I

y
y

−

I

y
z

I

x
z

−

I

y
z

I

z
z

)

{\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}}}

== Newton's and Euler's equation ==
The force

F

{\displaystyle F}
, acting at the center of mass of a rigid body with total mass

m

{\displaystyle m}
, causing an acceleration

v
˙

c
o
m

{\displaystyle {\dot {v}}_{com}}
, equals:

F
=
m

v
˙

c
o
m

{\displaystyle F=m{\dot {v}}_{com}}
In a similar way, the moment

N

{\displaystyle N}
, causing an angular acceleration

ω
˙

{\displaystyle {\dot {\omega }}}
, is given by:

N
=

c

I

ω
˙

+
ω
×

c

I
ω

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

c

I

{\displaystyle _{c}I}
is the inertia tensor, expressed in a frame

{
c
}

{\displaystyle \{c\}}
of which the origin coincides with the center of mass of the rigid body.