Composite Rotations

When a number of fundamental rotation matrix are multiplied together, the product matrix represents a sequence of rotations about the unit vectors. This multiple rotations is a form of composite rotations.

Step by step

Start by setting the cumulative rotation matrix $R$ to an identity matrix:

$$R=I$$

If the next rotation is by an angle $\theta$ about the $k$-th axis of the fixed frame $F$, update $R$ by pre-multiplying:

$$R\leftarrow R_k(\theta )\cdot R$$

If the next rotation is by an angle $\theta$ about the $k$-th axis of the mobile frame $M$, update $R$ by post-multiplying:

$$R\leftarrow R\cdot R_k(\theta )$$

• basically if the rotation is about the fixed frame, we pre-mulitply, which is adding the rotation in front of our initial identity $R$
• for rotation about its own mobile frame $M$, we post-multiply, which is adding the rotation behind the initial identity $R$

Worked Example

$$YPR(\theta )=R_3({\theta }_3)R_2({\theta }_2)R_1({\theta }_1)I$$ $$=R_3({\theta }_3)R_2({\theta }_2)\left[\begin{array}{ccc}1& 0& 0\\ 0& C_1& -S_1\\ 0& S_1& C_1\end{array}\right]$$ $$=R_3({\theta }_3)\left[\begin{array}{ccc}{C}_2& 0& S_2\\ 0& 1& 0\\ -S_2& 0& C_2\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& C_1& -S_1\\ 0& S_1& C_1\end{array}\right]$$ $$=R_3({\theta }_3)\left[\begin{array}{ccc}{C}_2& S_1{S}_2& C_1{S}_2\\ 0& C_1& -S_1\\ -S_2& C_2{S}_1& C_1{C}_2\end{array}\right]$$ $$=\left[\begin{array}{ccc}{C}_3& -S_3& 0\\ S_3& C_3& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}{C}_2& S_1{S}_2& C_1{S}_2\\ 0& C_1& -S_1\\ -S_2& C_2{S}_1& C_1{C}_2\end{array}\right]$$ $$=\left[\begin{array}{ccc}{C}_2{C}_3& C_3{S}_1{S}_2-S_3{C}_1& C_1{C}_3{S}_2+S_1{S}_3\\ C_2{S}_3& S_1{S}_2{S}_3+C_1{C}_3& C_1{C}_2{S}_3-S_1{C}_3\\ -S_2& C_2{S}_1& C_1{C}_2\end{array}\right]$$

Tip

Yaw is for Axis 1
Pitch is for Axis 2
Roll is for Axis 3

• Think of it as "You Practice Robotics!", where its 1, ,2, 3 respectively