User:Admazzeo/sandbox

Denavit and Hartenberg Matrices Applied to a Single Pendulum
For a single pendulum, we can write out the D-H transformation matrix is as follows:$$^1_0 T = M_{0,1} \left[ \begin{array}{ccc|c} \cos\theta_1 & -\sin\theta_1 & 0 & a_1 \cos\theta_1 \\ \sin\theta_1 & \cos\theta_1 & 0 & a_1 \sin\theta_1 \\ 0 & 0 & 1 & 0 \\   \hline 0 & 0 & 0 & 1 \end{array} \right] $$

where "body" 0 is the ground reference frame and body 1 is link 1.

The velocity matrix of body 1 with respect to the ground reference frame is:

$$W_{0,1(0)} = \left[ \begin{array}{ccc|c} 0 & -\dot\theta_1 & 0 & 0 \\ \dot\theta_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\   \hline 0 & 0 & 0 & 0 \end{array} \right] $$

The acceleration matrix of body 1 with respect to the ground reference frame becomes:

$$H_{0,1(0)} =\dot W_{0,1(0)}+ W^2_{0,1(0)} = \left[ \begin{array}{ccc|c} 0 & -\ddot\theta_1 & 0 & 0 \\ \ddot\theta_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\   \hline 0 & 0 & 0 & 0 \end{array} \right] + \left[ \begin{array}{ccc|c} -\dot\theta_1 ^2 & 0 & 0 & 0 \\ 0 & -\dot\theta_1 ^2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\   \hline 0 & 0 & 0 & 0 \end{array} \right] $$

We can describe the inertia $$J_1 $$ with the following matrix:

$$ J_{1(1)}=\left[ \begin{array}{ccc|c} 0 & 0  & 0  & -\frac{1}{2}mL\ \\ 0  & 0  & 0 & 0 \\ 0  & 0  & \frac{1}{12}mL^2  & 0 \\ \hline -\frac{1}{2}mL\ & 0 & 0 & m \end{array}\right] $$

We can express an action matrix $$ \Phi_1 $$:

$$ \Phi_{1(1)}=\left[ \begin{array}{ccc|c} 0 & 0  & 0  & -F_{base} \ \\ 0  & 0  & 0 & -mg\sin\theta_1 \\ 0  & 0  & 0  & 0 \\ \hline F_{base} & mg\sin\theta_1 & 0 & 0 \end{array}\right] $$

where $$ F_{base}=\frac{1}{2}mL\dot\theta_1^2-mg\cos\theta_1 $$ to account for both the centripetal acceleration of the rod and the weight of the bar.

Using Newton's Law in the form $$ \Phi_{k(0)} = H_{0,k} J_{k(0)} - J_{k(0)} H_{0,k}^T \, $$, we should arrive at the following:

Testing 1 2 3
How is this working?

$$ 1 +1 =2 $$

{Type the question here... + The correct answer. - Wrong or misleading answer. - Wrong or misleading answer. - Wrong or misleading answer.
 * type=""}

{Type the question here... + The correct answer. - Wrong or misleading answer. - Wrong or misleading answer. - Wrong or misleading answer.
 * type=""}

Testing sgature Admazzeo (discuss • contribs) 19:55, 21 January 2020 (UTC)

Another signature Admazzeo (discuss • contribs) 19:57, 21 January 2020 (UTC)