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⎡ ⎛ 2 2⎞ ⎤ | |
⎢ 2⋅⎝gy + gz ⎠ 2⋅(-gw⋅gz + gx⋅gy) 2⋅(gw⋅gy + gx⋅gz) ⎥ | |
⎢- ───────────────────── + 1 ───────────────────── ───────────────────── ⎥ | |
⎢ 2 2 2 2 2 2 2 2 2 2 2 2 ⎥ | |
⎢ gw + gx + gy + gz gw + gx + gy + gz gw + gx + gy + gz ⎥ | |
⎢ ⎥ | |
⎡sin(pitch)⋅sin(roll)⋅sin(yaw) + cos(roll)⋅cos(yaw) sin(roll)⋅cos(pitch) sin(pitch)⋅sin(roll)⋅cos(yaw) - sin(yaw)⋅cos(roll)⎤ ⎢ ⎛ 2 2⎞ ⎥ | |
⎢ ⎥ ⎢ 2⋅(gw⋅gz + gx⋅gy) 2⋅⎝gx + gz ⎠ 2⋅(-gw⋅gx + gy⋅gz) ⎥ | |
⎢sin(pitch)⋅sin(yaw)⋅cos(roll) - sin(roll)⋅cos(yaw) cos(pitch)⋅cos(roll) sin(pitch)⋅cos(roll)⋅cos(yaw) + sin(roll)⋅sin(yaw)⎥ = ⎢ ───────────────────── - ───────────────────── + 1 ───────────────────── ⎥ | |
⎢ ⎥ ⎢ 2 2 2 2 2 2 2 2 2 2 2 2 ⎥ | |
⎣ sin(yaw)⋅cos(pitch) -sin(pitch) cos(pitch)⋅cos(yaw) ⎦ ⎢ gw + gx + gy + gz gw + gx + gy + gz gw + gx + gy + gz ⎥ | |
⎢ ⎥ | |
⎢ ⎛ 2 2⎞ ⎥ | |
⎢ 2⋅(-gw⋅gy + gx⋅gz) 2⋅(gw⋅gx + gy⋅gz) 2⋅⎝gx + gy ⎠ ⎥ | |
⎢ ───────────────────── ───────────────────── - ───────────────────── + 1⎥ | |
⎢ 2 2 2 2 2 2 2 2 2 2 2 2 ⎥ | |
⎣ gw + gx + gy + gz gw + gx + gy + gz gw + gx + gy + gz ⎦ |
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⎡⎛ ⎛ 2⋅(gw⋅gy - gx⋅gz) ⎞ ⎛ 2⋅(gw⋅gx + gy⋅gz) ⎞ ⎛ 2⋅(gw⋅gz - gx⋅gy) | |
⎢⎜π - asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟, asin⎜─────────────────────⎟ + π, π - asin⎜────────────────────────────────────────────────────────────────────────────────────────── | |
⎢⎜ ⎜ _______________________________________________________________________________________________ ⎟ ⎜ 2 2 2 2⎟ ⎜ ____________________________________________________________________________________ | |
⎢⎜ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟ ⎝gw + gx + gy + gz ⎠ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 | |
⎢⎜ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz | |
⎢⎜ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟ ⎜ ╱ ─────────────────────────────────────────────────────────────────────────────────── | |
⎢⎜ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 | |
⎣⎝ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy | |
⎞⎞ ⎛ ⎛ 2⋅(gw⋅gy - gx⋅gz) ⎞ ⎛ 2⋅(gw⋅gx + gy⋅gz) ⎞ ⎛ 2⋅ | |
───────────────────────────────────⎟⎟, ⎜π - asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟, asin⎜─────────────────────⎟ + π, asin⎜──────────────────────────────────────────────────────── | |
___________ ⎟⎟ ⎜ ⎜ _______________________________________________________________________________________________ ⎟ ⎜ 2 2 2 2⎟ ⎜ __________________________________________________ | |
2⎞ ⎟⎟ ⎜ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟ ⎝gw + gx + gy + gz ⎠ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ | |
+ gz ⎠ ⎛ 2 2 2 2⎞⎟⎟ ⎜ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝g | |
────────── ⋅⎝gw + gx + gy + gz ⎠⎟⎟ ⎜ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟ ⎜ ╱ ───────────────────────────────────────────────── | |
2 4 ⎟⎟ ⎜ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟ ⎜ ╱ 4 2 2 2 2 2 2 4 | |
⋅gz + gz ⎠⎠ ⎝ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2 | |
(gw⋅gz - gx⋅gy) ⎞⎞ ⎛ ⎛ 2⋅(gw⋅gy - gx⋅gz) ⎞ ⎛ 2⋅(gw⋅gx + gy⋅gz) ⎞ ⎛ | |
─────────────────────────────────────────────────────────────────────⎟⎟, ⎜asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟ + π, -asin⎜─────────────────────⎟, asin⎜───────────────────────── | |
_____________________________________________ ⎟⎟ ⎜ ⎜ _______________________________________________________________________________________________ ⎟ ⎜ 2 2 2 2⎟ ⎜ ___________________ | |
2 2 2 2⎞ ⎟⎟ ⎜ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟ ⎝gw + gx + gy + gz ⎠ ⎜ ╱ ⎛ 2 | |
w + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟⎟ ⎜ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟ ⎜ ╱ ⎝gw - 2⋅gw⋅gx | |
──────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟⎟ ⎜ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟ ⎜ ╱ ────────────────── | |
2 2 2 2 4 2 2 4 ⎟⎟ ⎜ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟ ⎜ ╱ 4 2 2 | |
⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠⎠ ⎝ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠ ⎝╲╱ gw + 2⋅gw ⋅gx + | |
2⋅(gw⋅gz - gx⋅gy) ⎞ ⎞ ⎛ ⎛ 2⋅(gw⋅gy - gx⋅gz) ⎞ ⎛ 2⋅(gw⋅gx + gy⋅gz) | |
────────────────────────────────────────────────────────────────────────────────────────────────────⎟ + π⎟, ⎜asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟ + π, -asin⎜─────────────────── | |
____________________________________________________________________________ ⎟ ⎟ ⎜ ⎜ _______________________________________________________________________________________________ ⎟ ⎜ 2 2 2 | |
2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟ ⎟ ⎜ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟ ⎝gw + gx + gy + g | |
+ gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟ ⎟ ⎜ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟ | |
─────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟ ⎟ ⎜ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟ | |
2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟ ⎟ ⎜ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟ | |
2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠ ⎠ ⎝ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠ | |
⎞ ⎛ 2⋅(gw⋅gz - gx⋅gy) ⎞⎞ ⎛ ⎛ 2⋅(gw⋅gy - gx⋅gz) | |
──⎟, -asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟⎟, ⎜-asin⎜──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── | |
2⎟ ⎜ _______________________________________________________________________________________________ ⎟⎟ ⎜ ⎜ _______________________________________________________________________________________________ | |
z ⎠ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟⎟ ⎜ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ | |
⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟⎟ ⎜ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2 | |
⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟⎟ ⎜ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz | |
⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟⎟ ⎜ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 | |
⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠⎠ ⎝ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz | |
⎞ ⎛ 2⋅(gw⋅gx + gy⋅gz) ⎞ ⎛ 2⋅(gw⋅gz - gx⋅gy) ⎞ ⎞ ⎛ ⎛ 2⋅(gw⋅gy - gx⋅gz) | |
─⎟, -asin⎜─────────────────────⎟, asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟ + π⎟, ⎜-asin⎜──────────────────────────────────────────────────────────────────────────────────────────── | |
⎟ ⎜ 2 2 2 2⎟ ⎜ _______________________________________________________________________________________________ ⎟ ⎟ ⎜ ⎜ ______________________________________________________________________________________ | |
⎟ ⎝gw + gx + gy + gz ⎠ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟ ⎟ ⎜ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 | |
⎞⎟ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟ ⎟ ⎜ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + | |
⎠⎟ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟ ⎟ ⎜ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────── | |
⎟ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟ ⎟ ⎜ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 | |
⎠ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠ ⎠ ⎝ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅g | |
⎞ ⎛ 2⋅(gw⋅gx + gy⋅gz) ⎞ ⎛ 2⋅(gw⋅gz - gx⋅gy) ⎞⎞ ⎛ ⎛ 2⋅(gw⋅gy - | |
─────────────────────────────────⎟, -asin⎜─────────────────────⎟, -asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟⎟, ⎜asin⎜──────────────────────────────────────────────────────────────── | |
_________ ⎟ ⎜ 2 2 2 2⎟ ⎜ _______________________________________________________________________________________________ ⎟⎟ ⎜ ⎜ __________________________________________________________ | |
2⎞ ⎟ ⎝gw + gx + gy + gz ⎠ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟⎟ ⎜ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 | |
gz ⎠ ⎛ 2 2 2 2⎞⎟ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟⎟ ⎜ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅g | |
──────── ⋅⎝gw + gx + gy + gz ⎠⎟ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟⎟ ⎜ ⎜ ╱ ───────────────────────────────────────────────────────── | |
2 4 ⎟ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟⎟ ⎜ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 | |
z + gz ⎠ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠⎠ ⎝ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy | |
gx⋅gz) ⎞ ⎛ 2⋅(gw⋅gx + gy⋅gz) ⎞ ⎛ 2⋅(gw⋅gz - gx⋅gy) ⎞⎞ ⎛ ⎛ | |
─────────────────────────────────────────────────────────────⎟, asin⎜─────────────────────⎟ + π, π - asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟⎟, ⎜asin⎜────────────────────────────── | |
_____________________________________ ⎟ ⎜ 2 2 2 2⎟ ⎜ _______________________________________________________________________________________________ ⎟⎟ ⎜ ⎜ ________________________ | |
2 2 2⎞ ⎟ ⎝gw + gx + gy + gz ⎠ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟⎟ ⎜ ⎜ ╱ ⎛ 2 2 | |
w⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟⎟ ⎜ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx | |
──────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟⎟ ⎜ ⎜ ╱ ─────────────────────── | |
2 2 4 2 2 4 ⎟ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟⎟ ⎜ ⎜ ╱ 4 2 2 2 | |
+ 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠⎠ ⎝ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw | |
2⋅(gw⋅gy - gx⋅gz) ⎞ ⎛ 2⋅(gw⋅gx + gy⋅gz) ⎞ ⎛ 2⋅(gw⋅gz - gx⋅gy) ⎞⎞⎤ | |
───────────────────────────────────────────────────────────────────────────────────────────────⎟, asin⎜─────────────────────⎟ + π, asin⎜─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────⎟⎟⎥ | |
_______________________________________________________________________ ⎟ ⎜ 2 2 2 2⎟ ⎜ _______________________________________________________________________________________________ ⎟⎟⎥ | |
2 2⎞ ⎛ 2 2 2 2⎞ ⎟ ⎝gw + gx + gy + gz ⎠ ⎜ ╱ ⎛ 2 2 2 2⎞ ⎛ 2 2 2 2⎞ ⎟⎟⎥ | |
+ gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟ ⎜ ╱ ⎝gw - 2⋅gw⋅gx + gx + gy - 2⋅gy⋅gz + gz ⎠⋅⎝gw + 2⋅gw⋅gx + gx + gy + 2⋅gy⋅gz + gz ⎠ ⎛ 2 2 2 2⎞⎟⎟⎥ | |
────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟ ⎜ ╱ ───────────────────────────────────────────────────────────────────────────────────────────── ⋅⎝gw + gx + gy + gz ⎠⎟⎟⎥ | |
2 2 2 4 2 2 2 2 4 2 2 4 ⎟ ⎜ ╱ 4 2 2 2 2 2 2 4 2 2 2 2 4 2 2 4 ⎟⎟⎥ | |
⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠ ⎝╲╱ gw + 2⋅gw ⋅gx + 2⋅gw ⋅gy + 2⋅gw ⋅gz + gx + 2⋅gx ⋅gy + 2⋅gx ⋅gz + gy + 2⋅gy ⋅gz + gz ⎠⎠⎦ |
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