# Euler angle worksheet

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In this worksheet, we will use the python3 SymPy package to derive expressions for converting between euler angles and matrices.

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For this assignment, you do not need to run this notebook. It has been compiled for you and saved as a webpage. However, if you would like to play with it, start by running all the cells (from the menu: goto 'Cell' -> 'Run All'). 


```python
# python3 
from sympy import *
init_printing(use_latex='mathjax')
import math
```


```python
# Define symbols
cx,sx = symbols('cx sx')
cy,sy = symbols('cy sy')
cz,sz = symbols('cz sz')

Rx = Matrix([ 
        [1, 0, 0], 
        [0, cx,-sx], 
        [0, sx, cx]])

Ry = Matrix([
        [ cy, 0, sy], 
        [ 0,  1, 0], 
        [-sy, 0, cy]])

Rz = Matrix([ 
        [cz, -sz, 0], 
        [sz,  cz, 0], 
        [0,    0, 1]])
```

# Convert from ZYX euler angles to a matrix

We can compute the matrix Rzyx by multiplying matrixes corresponding to each consecutive rotation, e.g. 

$$
R_{zyx}(\theta_x, \theta_y, \theta_z) = R_z(\theta_z) * R_y(\theta_y) * R_x(\theta_x)
$$

In this file, we will use the [SymPy](https://www.sympy.org/en/index.html) to compute algebraic expressions for euler angle matrices. Using these expressions, we will be able to derive formulas for converting from matrices to euler angles.

In the following example, let

* cx = $cos(\theta_x)$
* sx = $sin(\theta_x)$
* cy = $cos(\theta_y)$
* sy = $sin(\theta_y)$
* cz = $cos(\theta_z)$
* sz = $sin(\theta_z)$



```python
Rzyx = Rz * Ry * Rx
pprint(Rzyx)
```

    ⎡cy⋅cz  -cx⋅sz + cz⋅sx⋅sy  cx⋅cz⋅sy + sx⋅sz⎤
    ⎢                                          ⎥
    ⎢cy⋅sz  cx⋅cz + sx⋅sy⋅sz   cx⋅sy⋅sz - cz⋅sx⎥
    ⎢                                          ⎥
    ⎣ -sy         cy⋅sx             cx⋅cy      ⎦


Now that we have a matrix expression for the ZYX euler angles, we have formulas which describe how matrices and euler angles relate to each. Specifically, suppose we have a 3x3 rotation matrix R with the following elements

$$
R = 
\begin{bmatrix}
r_{00} & r_{01} & r_{02} \\
r_{10} & r_{11} & r_{12} \\
r_{20} & r_{21} & r_{22} \\
\end{bmatrix}
$$

where each $r_{ij}$ represents a scalar value in $\mathbb{R}$. Usually math texts will use indexing at 1, but here let's use 0-based indexing so that it will be easier to use implement these formulas later.

Now suppose we wish to extract the euler angles from this matrix. We can get the Y rotation back from the term $r_{20}$.

$$
r_{20} = -\sin(\theta_y) \\
=> \theta_y = \sin(-r_{20})
$$

What about the rotations around X and Z? We can obtain these similarly using the terms from the first column and last row. A robust method involves using the fact that 

$$
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
$$

to form the following expression for obtaining $\theta_x$

$$
\frac{r_{21}}{r_{22}} = \frac{\sin(\theta_x)}{\cos(\theta_x)} = \tan(\theta_x) \\
=> \theta_x = \text{atan2}(r_{21}, r_{22})
$$

The expression for $\theta_z$ can be obtained similarly

$$
\frac{r_{10}}{r_{00}} = \frac{\sin(\theta_z)}{\cos(\theta_z)} = \tan(\theta_z) \\
=> \theta_z = \text{atan2}(r_{10}, r_{00})
$$

Using atan2 makes it easier to handle the cases when $\theta$ is near 0, 90, or 180 degrees, which makes sine and cosine close to zero and 1. Be careful when using acos and asin because values even *slightly* out of the range [-1,1] can lead to NaNs. The computer will not tolerate nansense!

## What happens to Rzyx when y is +/- 90 degrees?

When the middle euler angle is 90 degrees, we need to look to the non-zero terms to values for the first and last angles. For example, for ZYX euler angles, we need to handle the case when Y is either positive or negative 90 degrees. 


```python
# Compute Rzyx when y is +90
Ry90 = Matrix([
    [ 0, 0, 1], 
    [ 0, 1, 0], 
    [-1, 0, 0]])

Rzyx = Rz * Ry90 * Rx
pprint(Rzyx)
```

    ⎡0   -cx⋅sz + cz⋅sx  cx⋅cz + sx⋅sz⎤
    ⎢                                 ⎥
    ⎢0   cx⋅cz + sx⋅sz   cx⋅sz - cz⋅sx⎥
    ⎢                                 ⎥
    ⎣-1        0               0      ⎦


So now we have the above expression. We know that the Y rotation is 90 but what about the X and Z rotations? We need to look at the upper part of the matrix to figure these out.

Let's apply the sine and cosine [addition rules](https://en.wikipedia.org/wiki/List_of_trigonometric_identities)

$$
\sin(z + x) = \sin(z) \cos(x) + \cos(z) \sin(x) \\
\sin(z - x) = \sin(z) \cos(x) - \cos(z) \sin(x) \\
\cos(z + x) = \cos(z) \cos(x) - \sin(z) \sin(x) \\
\cos(z - x) = \cos(z) \cos(x) + \sin(z) \sin(x) \\
$$

Another useful property of sine and cosine is the following

$$
\sin(-\theta) = -\sin(\theta) \\
\cos(-\theta) = \cos(\theta)
$$

Let's try to simplify the above matrix using these rules. For example, the term in position $r_{12}$ has two terms containing both sine and cosine, so it corresponds to one of the sine rules. It also has a negative, so its the difference between two angles X and Z.

$$
\begin{bmatrix}
0 & s(x-z) & c(x-z) \\
0 & c(x-z) & s(z-x) \\
-1 & 0 & 0
\end{bmatrix}
$$
which can be rewritten so every term has angle $x-z$
$$
\begin{bmatrix}
0 & s(x-z) & c(x-z) \\
0 & c(x-z) & -s(x-z) \\
-1 & 0 & 0
\end{bmatrix}
$$

Therefore, we can use atan2($r_{12}$, $r_{13}$) to get the $\theta$ angle corresponding to the difference between $x-z$. Many values for X and Z could combine to be $\theta$. Let's choose one of X or Z to be zero and then the other can be $\theta$.  


```python
# Compute Rzyx when y is -90
Ry90_Minus = Ry90.T

Rzyx = Rz * Ry90_Minus * Rx
pprint(Rzyx)
```

    ⎡0  -cx⋅sz - cz⋅sx  -cx⋅cz + sx⋅sz⎤
    ⎢                                 ⎥
    ⎢0  cx⋅cz - sx⋅sz   -cx⋅sz - cz⋅sx⎥
    ⎢                                 ⎥
    ⎣1        0               0       ⎦


$$
\begin{bmatrix}
0 & -s(x+z) & c(x+z) \\
0 &  c(z+x) & -s(x+z) \\
1 & 0 & 0
\end{bmatrix}
$$


# Convert from all euler angles to a matrix

The other five euler angle combinations can be derived similarly. 

# XYZ


```python
print("Rxyz")
pprint(Rx * Ry * Rz)
print()
print()

print("Y = 90")
pprint(Rx * Ry90 * Rz)
print()
print()

print("Y = -90")
pprint(Rx * Ry90_Minus * Rz)
print()
print()

```

    Rxyz
    ⎡      cy⋅cz             -cy⋅sz         sy  ⎤
    ⎢                                           ⎥
    ⎢cx⋅sz + cz⋅sx⋅sy   cx⋅cz - sx⋅sy⋅sz  -cy⋅sx⎥
    ⎢                                           ⎥
    ⎣-cx⋅cz⋅sy + sx⋅sz  cx⋅sy⋅sz + cz⋅sx  cx⋅cy ⎦
    
    
    Y = 90
    ⎡      0               0        1⎤
    ⎢                                ⎥
    ⎢cx⋅sz + cz⋅sx   cx⋅cz - sx⋅sz  0⎥
    ⎢                                ⎥
    ⎣-cx⋅cz + sx⋅sz  cx⋅sz + cz⋅sx  0⎦
    
    
    Y = -90
    ⎡      0              0         -1⎤
    ⎢                                 ⎥
    ⎢cx⋅sz - cz⋅sx  cx⋅cz + sx⋅sz   0 ⎥
    ⎢                                 ⎥
    ⎣cx⋅cz + sx⋅sz  -cx⋅sz + cz⋅sx  0 ⎦
    
    


Y = 90
$$
\begin{bmatrix}
0 & 0 & 1 \\
s(x+z) & c(x+z) & 0\\
-c(x+z) & s(x+z) & 0 \\
\end{bmatrix}
$$

Y = -90
$$
\begin{bmatrix}
0 & 0 & -1 \\
s(z-x) & c(z-x) & 0 \\
c(z-x) & -s(z-x) & 0 \\
\end{bmatrix}
$$


# YXZ


```python
print("Ryxz")
pprint(Ry * Rx * Rz)
print()
print()

Rx90 = Matrix([ 
        [1, 0, 0], 
        [0, 0,-1], 
        [0, 1, 0]])
print("+90")
pprint(Ry * Rx90 * Rz)
print()
print()

Rx90_Minus = Rx90.T
print("-90")
pprint(Ry * Rx90.T * Rz)
print()
print()

```

    Ryxz
    ⎡cy⋅cz + sx⋅sy⋅sz  -cy⋅sz + cz⋅sx⋅sy  cx⋅sy⎤
    ⎢                                          ⎥
    ⎢     cx⋅sz              cx⋅cz         -sx ⎥
    ⎢                                          ⎥
    ⎣cy⋅sx⋅sz - cz⋅sy  cy⋅cz⋅sx + sy⋅sz   cx⋅cy⎦
    
    
    +90
    ⎡cy⋅cz + sy⋅sz  -cy⋅sz + cz⋅sy  0 ⎤
    ⎢                                 ⎥
    ⎢      0              0         -1⎥
    ⎢                                 ⎥
    ⎣cy⋅sz - cz⋅sy  cy⋅cz + sy⋅sz   0 ⎦
    
    
    -90
    ⎡cy⋅cz - sy⋅sz   -cy⋅sz - cz⋅sy  0⎤
    ⎢                                 ⎥
    ⎢      0               0         1⎥
    ⎢                                 ⎥
    ⎣-cy⋅sz - cz⋅sy  -cy⋅cz + sy⋅sz  0⎦
    
    


X = 90
$$
\begin{bmatrix}
c(y-z) & s(y-z) & 0\\
0 & 0 & -1 \\
-s(y-z) & c(y-z) & 0 \\
\end{bmatrix}
$$

X = -90
$$
\begin{bmatrix}
c(y+z) & -s(y+z) & 0 \\
0 & 0 & 1 \\
-s(y+z) & -c(y+z) & 0 \\
\end{bmatrix}
$$


# ZXY


```python
print("Rzxy")
pprint(Rz * Rx * Ry)
print()
print()

print("+90")
pprint(Rz * Rx90 * Ry)
print()
print()

print("-90")
pprint(Rz * Rx90.T * Ry)
print()
print()
```

    Rzxy
    ⎡cy⋅cz - sx⋅sy⋅sz  -cx⋅sz  cy⋅sx⋅sz + cz⋅sy ⎤
    ⎢                                           ⎥
    ⎢cy⋅sz + cz⋅sx⋅sy  cx⋅cz   -cy⋅cz⋅sx + sy⋅sz⎥
    ⎢                                           ⎥
    ⎣     -cx⋅sy         sx          cx⋅cy      ⎦
    
    
    +90
    ⎡cy⋅cz - sy⋅sz  0  cy⋅sz + cz⋅sy ⎤
    ⎢                                ⎥
    ⎢cy⋅sz + cz⋅sy  0  -cy⋅cz + sy⋅sz⎥
    ⎢                                ⎥
    ⎣      0        1        0       ⎦
    
    
    -90
    ⎡cy⋅cz + sy⋅sz  0   -cy⋅sz + cz⋅sy⎤
    ⎢                                 ⎥
    ⎢cy⋅sz - cz⋅sy  0   cy⋅cz + sy⋅sz ⎥
    ⎢                                 ⎥
    ⎣      0        -1        0       ⎦
    
    


X = 90
$$
\begin{bmatrix}
c(y+z) & 0 & s(y+z) \\
s(y+z) & 0 & -c(y+z) \\
0 & 1 & 0 \\
\end{bmatrix}
$$

X = -90
$$
\begin{bmatrix}
c(y-z) & 0 & s(y-z) \\
-s(y-z) & 0 & c(y-z) \\
0 & -1 & 0 \\
\end{bmatrix}
$$


# XZY


```python
print("Rxzy")
pprint(Rx * Rz * Ry)
print()
print()

Rz90 = Matrix([ 
        [0, -1, 0], 
        [1,  0, 0], 
        [0,  0, 1]])

print("+90")
pprint(Rx * Rz90 * Ry)
print()
print()

print("-90")
pprint(Rx * Rz90.T * Ry)
print()
print()


```

    Rxzy
    ⎡      cy⋅cz         -sz        cz⋅sy      ⎤
    ⎢                                          ⎥
    ⎢cx⋅cy⋅sz + sx⋅sy   cx⋅cz  cx⋅sy⋅sz - cy⋅sx⎥
    ⎢                                          ⎥
    ⎣-cx⋅sy + cy⋅sx⋅sz  cz⋅sx  cx⋅cy + sx⋅sy⋅sz⎦
    
    
    +90
    ⎡      0         -1        0      ⎤
    ⎢                                 ⎥
    ⎢cx⋅cy + sx⋅sy   0   cx⋅sy - cy⋅sx⎥
    ⎢                                 ⎥
    ⎣-cx⋅sy + cy⋅sx  0   cx⋅cy + sx⋅sy⎦
    
    
    -90
    ⎡      0         1        0       ⎤
    ⎢                                 ⎥
    ⎢-cx⋅cy + sx⋅sy  0  -cx⋅sy - cy⋅sx⎥
    ⎢                                 ⎥
    ⎣-cx⋅sy - cy⋅sx  0  cx⋅cy - sx⋅sy ⎦
    
    


Z = 90
$$
\begin{bmatrix}
0 & -1 & 0 \\
c(x-y) & 0 & -s(x-y) \\
s(x-y) & 0 & c(x-y) \\
\end{bmatrix}
$$

Z = -90
$$
\begin{bmatrix}
0 & 1 & 0 \\
-c(x+y) & 0 & -s(x+y) \\
-s(x+y) & 0 & c(x+y) \\
\end{bmatrix}
$$

# YZX


```python
print("Ryzx")
pprint(Ry * Rz * Rx)
print()
print()

print("+90")
pprint(Ry * Rz90 * Rx)
print()
print()

print("-90")
pprint(Ry * Rz90.T * Rx)
print()
print()

```

    Ryzx
    ⎡cy⋅cz   -cx⋅cy⋅sz + sx⋅sy  cx⋅sy + cy⋅sx⋅sz⎤
    ⎢                                           ⎥
    ⎢  sz          cx⋅cz             -cz⋅sx     ⎥
    ⎢                                           ⎥
    ⎣-cz⋅sy  cx⋅sy⋅sz + cy⋅sx   cx⋅cy - sx⋅sy⋅sz⎦
    
    
    +90
    ⎡0  -cx⋅cy + sx⋅sy  cx⋅sy + cy⋅sx⎤
    ⎢                                ⎥
    ⎢1        0               0      ⎥
    ⎢                                ⎥
    ⎣0  cx⋅sy + cy⋅sx   cx⋅cy - sx⋅sy⎦
    
    
    -90
    ⎡0   cx⋅cy + sx⋅sy   cx⋅sy - cy⋅sx⎤
    ⎢                                 ⎥
    ⎢-1        0               0      ⎥
    ⎢                                 ⎥
    ⎣0   -cx⋅sy + cy⋅sx  cx⋅cy + sx⋅sy⎦
    
    


Z = 90
$$
\begin{bmatrix}
0 & c(x-y) & -s(x-y) \\
1 & 0 & 0 \\
0 & s(x-y) & c(x-y) \\
\end{bmatrix}
$$

Z = -90
$$
\begin{bmatrix}
0 & c(y-x) & s(y-x) \\
-1 & 0 & 0 \\
0 & -s(y-x) & c(y-x) \\
\end{bmatrix}
$$

e6851d2a32d1cee8e3520944e094d933b8bfb427
ipynb
Jupyter Notebook
Labs/EulerAngles.ipynb
isaacwasserman/website
c052e1e8b28b9a600623589768691585eeda774d
MIT
Labs/EulerAngles.ipynb
isaacwasserman/website
c052e1e8b28b9a600623589768691585eeda774d
MIT
Labs/EulerAngles.ipynb
isaacwasserman/website
c052e1e8b28b9a600623589768691585eeda774d
MIT
2021-09-28T20:41:54.000Z
2021-09-28T20:41:54.000Z
Qwen/Qwen-72B
1. YES
2. YES