# Newton- Euler Rigid Body

Hello i am busy using Newton-Euler equations to get the dynamic of a quad drone. However, as I am looking online at papers I see mostly 2 different kinds of equations.

$$\begin{bmatrix} mI & 0 \\ 0 & J \end{bmatrix} \begin{bmatrix} \dot{v}_G \\ \dot{\omega}_G \end{bmatrix} + \begin{bmatrix} m\omega_G\times v_G \\ \omega_G\times J\omega_G \end{bmatrix} = \begin{bmatrix} f \\ \tau_G \end{bmatrix} \tag1$$

$$\begin{bmatrix} mI & 0 \\ 0 & J \end{bmatrix} \begin{bmatrix} \dot{v}_G \\ \dot{\omega}_G \end{bmatrix} + \begin{bmatrix} 0 \\ \omega_G\times J\omega_G \end{bmatrix} = \begin{bmatrix} f \\ \tau_G \end{bmatrix} \tag2$$

Even equation 2 is found on wikipedia but most papers use equation 1 and my question is, what is the difference and when are both of them used??

• Don't the online papers/wikipedia tell you how to use these equations? – sammy gerbil May 1 '18 at 23:32
• It would be helpful if you could link to some of the papers that you have been reading. – rob May 1 '18 at 23:51
• It depends if $\dot{v}_G$ is material or spatial acceleration. – ja72 May 2 '18 at 1:00
• Related question – ja72 May 7 '18 at 0:00

The proper equation at the center of mass is

\begin{aligned} \hat{f}_G &= \hat{\rm J}_G\, \dot{\hat{v}_G} + \hat{v}_G \times \hat{\rm J}_G\,\hat{v}_G \\ \begin{bmatrix}f\\ \tau_{G} \end{bmatrix} & =\begin{bmatrix}m\\ & J \end{bmatrix}\begin{bmatrix}\dot{v}_{G}\\ \dot{\omega} \end{bmatrix}+\begin{bmatrix}\omega\times & 0\\ v_{G}\times & \omega\times \end{bmatrix}\begin{bmatrix}m\\ & J \end{bmatrix}\begin{bmatrix}v_{G}\\ \omega \end{bmatrix} \\ & = \begin{bmatrix}m\\ & J \end{bmatrix}\begin{bmatrix}\dot{v}_{G}\\ \dot{\omega} \end{bmatrix} + \begin{bmatrix}\omega\times m\,v_{G}\\ \omega\times J\omega \end{bmatrix} \end{aligned}

where $v_G$ is the velocity at the center of mass, and $\dot{v}_G$ is the spatial acceleration at the center of mass.

The material acceleration of the center of mass is

$$a_G = \dot{v}_G + \omega \times v_G$$ as well as the identity $$\alpha = \dot{\omega}$$

Proof

The standard form of the equations is

\begin{aligned} f & = m a_G \\ \tau_G & = J \alpha + \omega \times J \omega \end{aligned}

and $$f = m (\dot{v}_G + \omega \times v_G) = m \dot{v}_G + \omega \times (m v_G)$$

At some other location A, other than the center of mass, where $c$ is vector from that location to the CM the NE equations of motion are

\begin{aligned} \hat{f}_A &= \hat{\rm J}_A\, \dot{\hat{v}_A} + \hat{v}_A \times \hat{\rm J}_A\,\hat{v}_A \\ \begin{bmatrix}f\\ \tau_{A} \end{bmatrix} & =\begin{bmatrix}m & -m [c \times] \\ m [c \times] & J-m [c\times][c\times] \end{bmatrix}\begin{bmatrix}\dot{v}_{A}\\ \dot{\omega} \end{bmatrix}+\begin{bmatrix}\omega\times & 0\\ v_{A}\times & \omega\times \end{bmatrix}\begin{bmatrix}m & -m [c \times] \\ m [c \times] & J-m [c\times][c\times] \end{bmatrix}\begin{bmatrix}v_{A}\\ \omega \end{bmatrix} \end{aligned}

I leave it up to the reader the prove this, based on the standard transformation equations for torque, velocity and spatial acceleration.