Lecture 03 - Kinematics 1

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Reference Frame Rotation Transformation Principal Rotation Matrix

Mobile Crane Example

Goal: Express the position $P$ at the tip of the crane relative to the point 0 (Origo in the 0-Frame).

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This vector can be written as

$${\vec{r}}_{P/0}={\vec{r}}_{1/0}+{\vec{r}}_{2/1}+{\vec{r}}_{P/2}$$

Where ${\vec{r}}_{1/0}$ is based on the 0-frame (${\mathbf{r}}_{1/0}^0$), etc…

We must now find $r_{2/1}^0$ represented in the 0-frame ($r_{2/1}^0$) instead of the 1-frame ($r_{2/1}^1$), same for the third vector.

Remember Principal Rotation Matrix

$$\begin{array}{rl}{r}_{2/1}^0& =R_z(q_1)r_{2/1}^1\\ r_{2/1}^0& =R_1^0(q_1)r_{2/1}^1\end{array}$$

And

$$\begin{array}{rl}{r}_{P/2}^0& =R_1^0(q_1)r_{P/2}^1\\ r_{P/2}^0& =R_1^0(q_1)R_y(q_2)r_{P/2}^2\\ r_{P/2}^0& =R_1^0(q_1)R_2^1(q_2)r_{P/2}^2\end{array}$$