Euler, Gimbal lock

• Gimbal lock

 

 

The loss of one degree of freedom in a multi-dimensional mechanism at certain alignments of the axes. In a three-dimensional three-gimbal mechanism, gimbal lock occurs when the axes of two of the gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space.

When the red and blue gimbals become aligned.

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The Gimbal Lock occurs in various forms depending on how it rotates 90 degrees on each axis.

 

https://en.wikipedia.org/wiki/Gimbal_lock

https://www.youtube.com/watch?v=vHr77Dre25Q

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• Euler angle

 

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The middle(green) can transform the innermost(blue),
the outermost(magenta) can transform the two inner(blue, green) rotation axes.

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• Euler formula

 

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$$\begin{array}{}& e^{i\mathrm{\theta <!--\; \theta \; -->}}=\cos <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})+i\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})\end{array}$$ $e$ : The base of the natural logarithm
$i$ : The imaginary unit

θ : A real number and is measured in radians.

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• The right side of an equation

 

$$\begin{array}{}& \cos <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})+i\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})\end{array}$$ $\begin{array}{}& =\end{array}$

$\begin{array}{}& x+iy\end{array}$

∴ $e^{i\mathrm{\theta <!--\; \theta \; -->}}$ = Complex Number

means the bar which has length 1 is rotated by θ radian from the x-axis.

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◦ $e$ : The base of the natural logarithm $$\begin{array}{}& e=\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{(1+\frac{1}{n})}^n\end{array}$$

- If The base of the natural logarithm e has an exponent x-squares : $$\begin{array}{}& e^x=\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{(1+\frac{x}{n})}^n={\left[\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{(1+\frac{x}{n})}^{n/x}\right]}^x\end{array}$$

- x to i : $$\begin{array}{}& e^i=\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{(1+\frac{i}{n})}^n={\left[\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{(1+\frac{i}{n})}^{n/i}\right]}^i\end{array}$$

- if n = 1 $$\begin{array}{}& \text{n=1 }\Rightarrow <!--\; \Rightarrow \; -->{(1+\frac{i}{1})}^1\end{array}$$

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- if n=2

$$\begin{array}{}& \text{n=2 }\Rightarrow <!--\; \Rightarrow \; -->{(1+\frac{i}{2})}^2=(1+\frac{i}{2})(1+\frac{i}{2})=0.75+i\end{array}$$

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- if n=3

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- if n=10

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- As n increases, the length of the hypotenuse of a right-angled triangle approaches 1.

 

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Therefore

$e^{i\mathrm{\theta <!--\; \theta \; -->}}$ means a point on an arc with a radius of 1 and an angle of radian(θ).

 

 

https://gaussian37.github.io/math-calculus-euler_formula/

https://angeloyeo.github.io/2020/07/07/Euler_Formula.html