Newton–Euler equations
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In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.[1][2] [3][4][5]
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:
where
- F = total force acting on the center of mass
- m = mass of the body
- I3 = the 3×3 identity matrix
- acm = acceleration of the center of mass
- vcm = velocity of the center of mass
- τ = total torque acting about the center of mass
- Icm = 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:
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/567cf49319502eaae7ae434408d3674c3bffbdd6)
where c is the vector from P to the center of mass of the body expressed in the body-fixed frame, and
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982b7fcda4970bd5ad646d59044c3cb2c6ba3038)
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
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f92d8453c00659bb35e9c07511eb9282de33f01d)
while the fictitious forces are contained in the term:[6]
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c64b63994d85159297cd5093df4a39ad5c43abbf)
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]- Euler's laws of motion for a rigid body.
- Euler angles
- Inverse dynamics
- Centrifugal force
- Principal axes
- Spatial acceleration
- Screw theory of rigid body motion.
References
[edit]- ^ Hubert Hahn (2002). Rigid Body Dynamics of Mechanisms. Springer. p. 143. ISBN 3-540-42373-7.
- ^ 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.