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Convert dlux_global_planner documentation to use MathJax (#100)
* Convert to MathJax * Additional math
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| # Full Derivation | ||
| We start with our basic equation definitions. | ||
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| * $P(X) = \frac{-\beta + \sqrt{\beta^2 - 4 \gamma}}{2}$ | ||
| * $\beta = -\Big(P(C) + P(A)\Big)$ | ||
| * $\gamma = \frac{P(A)^2 + P(C)^2 - h^2}{2}$ | ||
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| ## Reformulate | ||
| First we want to reformulate in terms of our new  variable. | ||
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| First we want to reformulate in terms of our new $\delta$ variable. | ||
| * $\delta = \frac{P(C) - P(A)}{h}$ | ||
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| We can rewrite this so we have a new definition of  | ||
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| We can rewrite this so we have a new definition of $P(C)$ | ||
| * $P(C) = h\delta + P(A)$ | ||
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| This allows us to derive new values for the other equations. | ||
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| * $\beta = -\Big(P(C) + P(A)\Big)$ | ||
| * $\beta = -\Big(h\delta + P(A) + P(A)\Big)$ | ||
| * $\beta = -h\delta - 2 P(A)$ | ||
| * $\gamma = \frac{P(A)^2 + P(C)^2 - h^2}{2}$ | ||
| * $\gamma = \frac{P(A)^2 + \Big(P(A)+h\delta\Big)^2 - h^2}{2}$ | ||
| * $\gamma = P(A)^2 + h\delta P(A) + \frac{h^2\delta^2 - h^2}{2}$ | ||
| * $P(X) = \frac{-\beta + \sqrt{\beta^2 - 4 \gamma}}{2}$ | ||
| * $P(X) = \frac{-(-h\delta - 2 P(A)) + \sqrt{\Big(-h\delta - 2 P(A)\Big)^2 - 4 \Big(P(A)^2 + h\delta P(A) + \frac{h^2\delta^2 - h^2}{2}\Big)}}{2}$ | ||
| * $P(X) = \frac{h\delta + 2 P(A) + \sqrt{\Big(-h\delta - 2 P(A)\Big)^2 - 4 P(A)^2 -4 h\delta P(A) - 2h^2\delta^2 + 2h^2}}{2}$ | ||
| * $P(X) = \frac{h\delta + 2 P(A) + \sqrt{4P(A)^2 + 4h\delta P(A) + h^2\delta^2 - 4 P(A)^2 -4 h\delta P(A) - 2h^2\delta^2 + 2h^2}}{2}$ | ||
| * $P(X) = \frac{h\delta + 2 P(A) + \sqrt{-h^2\delta^2 + 2h^2}}{2}$ | ||
| * $P(X) = P(A) + \frac{h}{2} \Big(\delta + \sqrt{2-\delta^2}\Big)$ | ||
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| We're going to call this a new function , so now  is in terms of  | ||
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| We're going to call this a new function $f(\delta)$, so now $P(X)$ is in terms of $f(\delta)$ | ||
| * $f(\delta) = P(A) + \frac{h}{2} \Big(\delta + \sqrt{2-\delta^2}\Big)$ | ||
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| ## Taylor Series Expansion | ||
| For computational efficiency, we are going to compute the Taylor series expansion. So first, we'll need the derivatives of . | ||
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| For computational efficiency, we are going to compute the Taylor series expansion. So first, we'll need the derivatives of $f(\delta)$. | ||
| * $f'(\delta) = \frac{h}{2} (1 - \frac{\delta}{\sqrt{2-\delta^2}})$ | ||
| * $f''(\delta) = \frac{-h}{(2-\delta^2)^\frac{3}{2}}$ | ||
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| ### 0th Order Taylor | ||
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| * $f_0(\delta, a) = f(a)$ | ||
| * $f_0(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2}\Big)$ | ||
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| ### 1st Order Taylor | ||
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| * $f_1(\delta, a) = f(a) + f'(a) (\delta - a)$ | ||
| * $f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2}\Big) + \frac{h}{2} (1 - \frac{a}{\sqrt{2-a^2}}) (\delta - a)$ | ||
| * $f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2} + (1 - \frac{a}{\sqrt{2-a^2}}) (\delta - a)\Big)$ | ||
| * $f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2} + \delta - a - \frac{a\delta}{\sqrt{2-a^2}} + \frac{a^2}{\sqrt{2-a^2}}\Big)$ | ||
| * $f_1(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \sqrt{2-a^2} + \frac{a^2}{\sqrt{2-a^2}}\Big)$ | ||
| * $f_1(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}}\Big)$ | ||
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| ### 2nd Order Taylor | ||
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| * $f_2(\delta, a) = f(a) + f'(a) (\delta - a) + \frac{f''(a)}{2!}(\delta - a)^2$ | ||
| * $f_2(\delta, a) = f_1(\delta, a) + \frac{1}{2} \frac{-h}{(2-a^2)^\frac{3}{2}} (\delta - a)^2$ | ||
| * $f_2(\delta, a) = f_1(\delta, a) + \frac{h}{2} \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)$ | ||
| * $f_2(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}}\Big) + \frac{h}{2} \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)$ | ||
| * $f_2(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}} + \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)\Big)$ | ||
| * $f_2(\delta, a) = P(A) + h\Big(c_0(a) + c_1(a)\delta + c_2(a) \delta^2\Big)$ | ||
| * $c_0(a) = \frac{1}{2}\Big(\frac{2}{\sqrt{2-a^2}} - \frac{a^2}{(2-a^2)^\frac{3}{2}}\Big)$ | ||
| * $c_0(a) = \frac{4 - 3a^2}{2(2-a^2)^\frac{3}{2}}$ | ||
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| * $c_1(a) = \frac{1}{2}\Big(1 - \frac{a}{\sqrt{2-a^2}} + \frac{2a}{(2-a^2)^\frac{3}{2}} \Big)$ | ||
| * $c_2(a) = \frac{-1}{2(2-a^2)^\frac{3}{2}}$ | ||
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| ## Exact coefficients | ||
| Now that we have the general equations for the Taylor series, we can evaluate it at different values of a in the range `[0, 1]`. | ||
| Now that we have the general equations for the Taylor series, we can evaluate it at different values of a in the range $[0, 1]$. | ||
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| |  |  |  |  | | ||
| | ------ | ----------- | ----------- | ----------- | | ||
| | 0.0 | 0.7071 | 0.5000 | -0.1768 | | ||
| | 0.5 | 0.7019 | 0.5270 | -0.2160 | | ||
| | 1.0 | 0.5000 | 1.0000 | -0.5000 | | ||
| | $a$ | $c_0(a)$ | $c_1(a)$ | $c_2(a)$ | | ||
| | --- | -------- | -------- | -------- | | ||
| | 0.0 | 0.7071 | 0.5000 | -0.1768 | | ||
| | 0.5 | 0.7019 | 0.5270 | -0.2160 | | ||
| | 1.0 | 0.5000 | 1.0000 | -0.5000 | | ||
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| Historically, the values used by [`navfn`](https://github.com/ros-planning/navigation/blob/1f335323a605b49b4108a845c55a7c1ba93a6f2e/navfn/src/navfn.cpp#L509) are | ||
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| |  |  |  | | ||
| | ----------- | ----------- | ----------- | | ||
| | 0.7040 | 0.5307 | -0.2301 | | ||
| | $c_0$ | $c_1$ | $c_2$ | | ||
| | ------ | ------ | ------- | | ||
| | 0.7040 | 0.5307 | -0.2301 | | ||
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| You can see these values plotted [here](https://www.desmos.com/calculator/vbpkey1mt6). | ||
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| The historical values are pretty close to the values for , although the exact reason for the difference is unknown, but its close enough to not be overly concerning. | ||
| The historical values are pretty close to the values for $\delta=0.5$, although the exact reason for the difference is unknown, but its close enough to not be overly concerning. |
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