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system-design/problem.md: bug fixes and gimbal locks
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‎system-design/problem.md‎

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@@ -102,7 +102,7 @@ $$
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$$
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* The rotated vector around the X-axis with Z-convention expressed in direction cosines; could be further rewritten using matrix algebra into a **matrix product of a rotation matrix and the vector column matrix**; where $R_x$ is the rotation matrix around X-Axis with Z-convetion:
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The rotated vector around the X-axis with Z-convention expressed in direction cosines; could be further rewritten using matrix algebra into a **matrix product of a rotation matrix and the vector column matrix**; where $R_x$ is the rotation matrix around X-Axis with Z-convetion:
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$$
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\vec{V}^{'} = R_x(\pm \hat{\Phi_R}) \times \vec{V} \\
@@ -113,42 +113,23 @@ $$
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}\|\cos(\phi_{(\vec{V}, \vec{V}_x)}) \\
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\|\vec{V}_{yz}\|\sin(\phi_{(\vec{V}_{yz}, \vec{V}_z)}) \\
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\|\vec{V}_{yz}\|\cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\|\vec{V}\|.\cos(\phi_{(\vec{V}, \vec{V}_x)}) \\
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\|\vec{V}_{yz}\|.\sin(\phi_{(\vec{V}_{yz}, \vec{V}_z)}) \\
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\|\vec{V}_{yz}\|.\cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\end{pmatrix}
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$$
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$$
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= \begin{pmatrix}
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1 & 0 & 0 \\
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0 & \cos(\phi_{R}) & \pm \sin(\phi_{R}) \\
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0 & \mp \sin(\Phi_R) & \cos(\phi_{R})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\cos(\phi_{(\vec{V}, \vec{V}_x)}) \\
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\sin(\phi_{(\vec{V}_{yz}, \vec{V}_z)}) \\
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\cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}\| \\
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\|\vec{V}_{yz}\| \\
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\|\vec{V}_{yz}\|
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\end{pmatrix}^\top
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$$
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Therefore, the rest of rotation operations could be expressed using rotation matrices as follows:
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* Therefore, the rest of rotation operations could be expressed using rotation matrices as follows:
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* Rotation around Y-axis (**X-convention**):
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* Rotation around Y-axis (**X-convention**):
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$$
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\vec{V}^{'} = R_y(\pm \hat{\Phi_R}) \times \vec{V} \\
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= \begin{pmatrix}
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\|\vec{V}_{xz}\|(\cos(\phi_{R})\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) \mp
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\|\vec{V}_{xz}\|.(\cos(\phi_{R})\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) \mp
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\sin(\phi_{R})\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)})) \\
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\|\vec{V}\|\cos(\phi_{(\vec{V}, \vec{V}_y)}) \\
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\|\vec{V}_{xz}\|(\pm \sin(\phi_{R})\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) + \cos(\phi_{R})\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)}))
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\|\vec{V}\|.\cos(\phi_{(\vec{V}, \vec{V}_y)}) \\
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\|\vec{V}_{xz}\|.(\pm \sin(\phi_{R})\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) + \cos(\phi_{R})\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)}))
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\end{pmatrix}
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$$
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@@ -160,27 +141,21 @@ $$
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) \\
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\cos(\phi_{(\vec{V}, \vec{V}_y)}) \\
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\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)})
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\|\vec{V}_{xz}\|.\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) \\
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\|\vec{V}\|.\cos(\phi_{(\vec{V}, \vec{V}_y)}) \\
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\|\vec{V}_{xz}\|.\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}_{xz}\| \\
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\|\vec{V}\| \\
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\|\vec{V}_{xz}\|
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\end{pmatrix}^\top
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$$
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* Rotation around Z-axis (**X-Convention**) using direction cosines:
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* Rotation around Z-axis (**X-Convention**) using direction cosines:
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$$
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\vec{V}^{'} = R_z(\pm \hat{\Phi_R}) \times \vec{V} \\
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= \begin{pmatrix}
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\|\vec{V}_{xy}\|(\cos(\phi_{R})\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \mp
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\|\vec{V}_{xy}\|.(\cos(\phi_{R})\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \mp
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\sin(\phi_{R})\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)})) \\
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\|\vec{V}_{xy}\|(\pm \sin(\phi_{R})\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) + \cos(\phi_{R})\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)})) \\
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\|\vec{V}\|\cos(\phi_{(\vec{V}, \vec{V}_z)})
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\|\vec{V}_{xy}\|.(\pm \sin(\phi_{R})\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) + \cos(\phi_{R})\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)})) \\
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\|\vec{V}\|.\cos(\phi_{(\vec{V}, \vec{V}_z)})
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\end{pmatrix}
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$$
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \\
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\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \\
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\cos(\phi_{(\vec{V}, \vec{V}_z)}) \\
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\|\vec{V}_{xy}\|.\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \\
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\|\vec{V}_{xy}\|.\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \\
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\|\vec{V}\|.\cos(\phi_{(\vec{V}, \vec{V}_z)}) \\
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}_{xy}\| \\
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\|\vec{V}_{xy}\| \\
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\|\vec{V}\|
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\end{pmatrix}^\top
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$$
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* A set of rotations could be achieved by multiplying the rotation matrices to get the final rotation matrix, and then multiply it with the vector coordinates.
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* Mathematical Modelling: Errors of Rotation and Error Handling Techniques.
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**Gimbal Locks**: is a state of orientation in $R^3$ spaces created by rotating around an arbitrary by angle of $\frac{\pi}{2}$ (e.g., $R_x(\frac{\pi}{2})$ ) which results in a loss of a degree of freedom, in which two rotation matrices before and after performing this erroneous operation align in that coordinate system (e.g., $R_y(\hat{\beta})$ and $R_z(\hat{\gamma})$ ); under the conditions that one of them precedes the erroneous rotation, and the other proceeds them, and both the axes of the rotation matrices are orthogonal (i.e., $V_x = V_y \times V_z$ ).
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**Unlocking Gimbals**: there are various unlocking algorithm; one common algorithm is to **dynamically remap** rotation matrices of the two orthogonal axes, the other algorithm entails using **clamp functions** to prevent the literal value of $\frac{\pi}{2}$; the clamp functions increment or decrement the value of the angle by a very small amount approaching Zero; so it entails finding the angle in terms of the $\lim_{x=0}(\theta + x)$.
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> [!IMPORTANT]
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> Error Handling techniques (Anti-failure Measures) for Gimbal Locks:
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> (1) **Dynamic Remapping of rotation matrices** for the orthogonal axes that their cross product is the vector for which the erroneous rotation was performed about.
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> (2) **Clamping of the angle value using limits of the angle**.
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* Mathematical-Physical Model of angular (or rotational) motion in R(2) and R(3) vectorspaces:
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(WIP)

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