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Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices . These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments ) acting on the rigid body.
Center of mass frame [ edit ]
With respect to a coordinate frame whose origin coincides with the body's center of mass for τ (torque ) and an inertial frame of reference for F (force ), they can be expressed in matrix form as:
(
F
τ
)
=
(
m
I
3
0
0
I
c
m
)
(
a
c
m
α
)
+
(
0
ω
×
I
c
m
ω
)
,
{\displaystyle \left({\begin{matrix}{\mathbf {F} }\\{\boldsymbol {\tau }}\end{matrix}}\right)=\left({\begin{matrix}m{\mathbf {I} _{3}}&0\\0&{\mathbf {I} }_{\rm {cm}}\end{matrix}}\right)\left({\begin{matrix}\mathbf {a} _{\rm {cm}}\\{\boldsymbol {\alpha }}\end{matrix}}\right)+\left({\begin{matrix}0\\{\boldsymbol {\omega }}\times {\mathbf {I} }_{\rm {cm}}\,{\boldsymbol {\omega }}\end{matrix}}\right),}
where
F = total force acting on the center of mass
m = mass of the body
I 3 = the 3×3 identity matrix
a cm = acceleration of the center of mass
v cm = velocity of the center of mass
τ = total torque acting about the center of mass
I cm = moment of inertia about the center of mass
ω = angular velocity of the body
α = angular acceleration of the body
Any reference frame [ edit ]
With respect to a coordinate frame located at point P that is fixed in the body and not coincident with the center of mass, the equations assume the more complex form:
(
F
τ
p
)
=
(
m
I
3
−
m
[
c
]
×
m
[
c
]
×
I
c
m
−
m
[
c
]
×
[
c
]
×
)
(
a
p
α
)
+
(
m
[
ω
]
×
[
ω
]
×
c
[
ω
]
×
(
I
c
m
−
m
[
c
]
×
[
c
]
×
)
ω
)
,
{\displaystyle \left({\begin{matrix}{\mathbf {F} }\\{\boldsymbol {\tau }}_{\rm {p}}\end{matrix}}\right)=\left({\begin{matrix}m{\mathbf {I} _{3}}&-m[{\mathbf {c} }]^{\times }\\m[{\mathbf {c} }]^{\times }&{\mathbf {I} }_{\rm {cm}}-m[{\mathbf {c} }]^{\times }[{\mathbf {c} }]^{\times }\end{matrix}}\right)\left({\begin{matrix}\mathbf {a} _{\rm {p}}\\{\boldsymbol {\alpha }}\end{matrix}}\right)+\left({\begin{matrix}m[{\boldsymbol {\omega }}]^{\times }[{\boldsymbol {\omega }}]^{\times }{\mathbf {c} }\\{[{\boldsymbol {\omega }}]}^{\times }({\mathbf {I} }_{\rm {cm}}-m[{\mathbf {c} }]^{\times }[{\mathbf {c} }]^{\times })\,{\boldsymbol {\omega }}\end{matrix}}\right),}
where c is the vector from P to the center of mass of the body expressed in the body-fixed frame ,
and
[
c
]
×
≡
(
0
−
c
z
c
y
c
z
0
−
c
x
−
c
y
c
x
0
)
[
ω
]
×
≡
(
0
−
ω
z
ω
y
ω
z
0
−
ω
x
−
ω
y
ω
x
0
)
{\displaystyle [\mathbf {c} ]^{\times }\equiv \left({\begin{matrix}0&-c_{z}&c_{y}\\c_{z}&0&-c_{x}\\-c_{y}&c_{x}&0\end{matrix}}\right)\qquad \qquad [\mathbf {\boldsymbol {\omega }} ]^{\times }\equiv \left({\begin{matrix}0&-\omega _{z}&\omega _{y}\\\omega _{z}&0&-\omega _{x}\\-\omega _{y}&\omega _{x}&0\end{matrix}}\right)}
denote skew-symmetric cross product matrices .
The left hand side of the equation—which includes the sum of external forces, and the sum of external moments about P —describes a spatial wrench , see screw theory .
The inertial terms are contained in the spatial inertia matrix
(
m
I
3
−
m
[
c
]
×
m
[
c
]
×
I
c
m
−
m
[
c
]
×
[
c
]
×
)
,
{\displaystyle \left({\begin{matrix}m{\mathbf {I} _{3}}&-m[{\mathbf {c} }]^{\times }\\m[{\mathbf {c} }]^{\times }&{\mathbf {I} }_{\rm {cm}}-m[{\mathbf {c} }]^{\times }[{\mathbf {c} }]^{\times }\end{matrix}}\right),}
while the fictitious forces are contained in the term:[6]
(
m
[
ω
]
×
[
ω
]
×
c
[
ω
]
×
(
I
c
m
−
m
[
c
]
×
[
c
]
×
)
ω
)
.
{\displaystyle \left({\begin{matrix}m{[{\boldsymbol {\omega }}]}^{\times }{[{\boldsymbol {\omega }}]}^{\times }{\mathbf {c} }\\{[{\boldsymbol {\omega }}]}^{\times }({\mathbf {I} }_{\rm {cm}}-m[{\mathbf {c} }]^{\times }[{\mathbf {c} }]^{\times })\,{\boldsymbol {\omega }}\end{matrix}}\right).}
When the center of mass is not coincident with the coordinate frame (that is, when c is nonzero), the translational and angular accelerations (a and α ) are coupled, so that each is associated with force and torque components.
Applications [ edit ]
The Newton–Euler equations are used as the basis for more complicated "multi-body" formulations (screw theory ) that describe the dynamics of systems of rigid bodies connected by joints and other constraints. Multi-body problems can be
solved by a variety of numerical algorithms.[2] [6] [7]
See also [ edit ]
References [ edit ]
^ a b
Ahmed A. Shabana (2001). Computational Dynamics . Wiley-Interscience. p. 379. ISBN 978-0-471-37144-1 .
^
Haruhiko Asada, Jean-Jacques E. Slotine (1986). Robot Analysis and Control . Wiley/IEEE. pp. §5.1.1, p. 94. ISBN 0-471-83029-1 .
^
Robert H. Bishop (2007). Mechatronic Systems, Sensors, and Actuators: Fundamentals and Modeling . CRC Press. pp. §7.4.1, §7.4.2. ISBN 978-0-8493-9258-0 .
^
Miguel A. Otaduy, Ming C. Lin (2006). High Fidelity Haptic Rendering . Morgan and Claypool Publishers. p. 24. ISBN 1-59829-114-9 .
^ a b
Roy Featherstone (2008). Rigid Body Dynamics Algorithms . Springer. ISBN 978-0-387-74314-1 .
^
Constantinos A. Balafoutis, Rajnikant V. Patel (1991). Dynamic Analysis of Robot Manipulators: A Cartesian Tensor Approach . Springer. Chapter 5. ISBN 0-7923-9145-4 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Newton–Euler_equations&oldid=1227529522 "
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