Newton–Euler equations

The Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.^{[1][2] [3][4][5]} With respect to a coordinate frame whose origin coincides with the body's center of mass, they can be expressed in matrix form as:

$\left(\begin{matrix} {\bold f} \\ {\boldsymbol \tau} \end{matrix}\right) = \left(\begin{matrix} m {\bold I} & 0 \\ 0 & {\bold J}_c \end{matrix}\right) \left(\begin{matrix} \ddot {\bold q} \\ \dot {\boldsymbol \omega} \end{matrix}\right) + \left(\begin{matrix} 0 \\ {\boldsymbol \omega} \times {\bold J}_c \, {\boldsymbol \omega} \end{matrix}\right),$

where

$\mathbf{f}$ = total force acting on the center of mass

m = mass of the body

${\bold I}$ = the identity matrix

$\ddot{\bold q}$ = acceleration of the center of mass

$\boldsymbol \tau$ = total torque (or moment) acting about the center of mass

${\bold J}_c$ = moment of inertia about the center of mass

${\boldsymbol \omega}$ = angular velocity of the body

With respect to a coordinate frame that is not coincident with the center of mass, the equations assume the more complex form:

$\left(\begin{matrix} {\bold f} \\ {\boldsymbol \tau} \end{matrix}\right) = \left(\begin{matrix} m {\bold I} & - m [{\bold c}]\\ m [{\bold c}] & {\bold J}_c - m [{\bold c}][{\bold c}]\end{matrix}\right) \left(\begin{matrix} \ddot {\bold q} \\ \dot {\boldsymbol \omega} \end{matrix}\right) + \left(\begin{matrix} {m \boldsymbol \omega} \times \left({\boldsymbol \omega} \times {\bold c}\right) \\ {\boldsymbol \omega} \times ({\bold J}_c - m [{\bold c}][{\bold c}])\, {\boldsymbol \omega} \end{matrix}\right),$

where $\mathbf{c}$ is the location of the center of mass, and

$[\mathbf{c}] \equiv \left(\begin{matrix} 0 & -c_z & c_y \\ c_z & 0 & -c_x \\ -c_y & c_x & 0 \end{matrix}\right)$

denotes a skew-symmetric cross product matrix.

The inertial terms are contained in the spatial inertia matrix

$\left(\begin{matrix} m {\bold I} & - m [{\bold c}]\\ m [{\bold c}] & {\bold J}_c - m [{\bold c}][{\bold c}]\end{matrix}\right),$

while the fictitious forces are contained in the term

$\left(\begin{matrix} {m \boldsymbol \omega} \times \left({\boldsymbol \omega} \times {\bold c}\right) \\ {\boldsymbol \omega} \times ({\bold J}_c - m [{\bold c}][{\bold c}])\, {\boldsymbol \omega} \end{matrix}\right) .$^[6]

When the center of mass is not coincident with the coordinate frame (that is, when ${\bold c}$ is nonzero), the translational and angular accelerations ($\ddot {\bold q}$ and $\dot{\boldsymbol \omega}$) are coupled, so that each is associated with force and torque components.

The Newton–Euler equations are used as the basis for more complicated "multi-body" formulations 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.^[6][2][7]

References

1. ^ Hubert Hahn (2002). Rigid Body Dynamics of Mechanisms. Springer. p. 143. ISBN 3540423737. http://books.google.com/books?id=MqrN3KY7o6MC&pg=PA143&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U00jfE08smw1IqJt69QdcMSKvDIeA. 
2. ^ ^{a b} Ahmed A. Shabana (2001). Computational Dynamics. Wiley-Interscience. p. 379. ISBN 9780471371441. http://books.google.com/books?id=dGfcbOsm2PwC&pg=PA379&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U01BZBb84es37aiHVpdE33IdGze-A. 
3. ^ Haruhiko Asada, Jean-Jacques E. Slotine (1986). Robot Analysis and Control. Wiley/IEEE. pp. §5.1.1, p. 94. ISBN 0471830291. http://books.google.com/books?id=KUG1VGkL3loC&pg=PA94&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U3LiZyQRj0zYXQ8ON2zwuiiwQO7dA. 
4. ^ Robert H. Bishop (2007). Mechatronic Systems, Sensors, and Actuators: Fundamentals and Modeling. CRC Press. pp. §7.4.1, §7.4.2. ISBN 0849392586. http://books.google.com/books?id=3UGQsi6VamwC&pg=PT104&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U1DtQ2BGV_Q34yAj-WhnQ4tStxPCw#PPT104,M1. 
5. ^ Miguel A. Otaduy, Ming C. Lin (2006). High Fidelity Haptic Rendering. Morgan and Claypool Publishers. p. 24. ISBN 1598291149. http://books.google.com/books?id=lk0StvDRoEMC&pg=PA24&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U0iOPnq-nMrS34O40ZMt0EbJEqu6g#PPA24,M1. 
6. ^ ^{a b} Roy Featherstone (2008). Rigid Body Dynamics Algorithms. Springer. ISBN 978-0-387-74314-1. http://books.google.ca/books?id=UjWbvqWaf6gC&printsec=frontcover&dq=Rigid+Body+Dynamics+Algorithms. 
7. ^ Constantinos A. Balafoutis, Rajnikant V. Patel (1991). Dynamic Analysis of Robot Manipulators: A Cartesian Tensor Approach. Springer. Chapter 5. ISBN 0792391454. http://books.google.com/books?id=7BcpyUjmLpUC&pg=PT195&dq=%22Kane%27s+dynamical+equations%22&lr=&as_brr=0&sig=ACfU3U1m290WlCUy1101Oj9Z9w3j5a4Lww#PPT151,M1. 

See also

• Euler angles
• Inverse dynamics
• Centrifugal force
• Principal axes