# Newton–Euler equations

Classical mechanics |
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Core topics |

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

With respect to a coordinate frame whose origin coincides with the body's center of mass, they can be expressed in matrix form as:

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

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:

where **c** is the location of the center of mass expressed in the body-fixed frame,
and

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

while the fictitious forces are contained in the term:^{[6]}

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

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

- 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

- ↑ Hubert Hahn (2002).
*Rigid Body Dynamics of Mechanisms*. Springer. p. 143. ISBN 3-540-42373-7. - 1 2 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 0-8493-9258-6. - ↑ Miguel A. Otaduy, Ming C. Lin (2006).
*High Fidelity Haptic Rendering*. Morgan and Claypool Publishers. p. 24. ISBN 1-59829-114-9. - 1 2 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.