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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Navigation function</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Navigation function</b> usually refers to a function of position, velocity, acceleration and time which is used to plan robot trajectories through the environment. Generally, the goal of a navigation function is to create feasible, safe paths that avoid obstacles while allowing a robot to move from its starting configuration to its goal configuration.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Potential_functions_as_navigation_functions">Potential functions as navigation functions</h2></div>

<p>Potential functions assume that the environment or work space is known. Obstacles are assigned a high potential value, and the goal position is assigned a low potential. To reach the goal position, a robot only needs to follow the negative <a href="Gradient" title="Gradient">gradient</a> of the surface.
</p><p>We can formalize this concept mathematically as following: Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> be the state space of all possible configurations of a robot. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{g}\subset X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X_{g}\subset X}</annotation>
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</math></span><img src="./a7e2296ce4a047b6f2d03dda30f35c04392a4da0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.024ex; height:2.843ex;" alt="{\displaystyle X_{g}\subset X}" loading="lazy"></span> denote the goal region of the state space.
</p><p>Then a potential function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation>
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</math></span><img src="./546b660b2f3cfb5f34be7b3ed8371d54f5c74227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}" loading="lazy"></span> is called a (feasible) navigation function if <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<ol><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)=0\ \forall x\in X_{g}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
<mtext>&nbsp;</mtext>
<mi mathvariant="normal">∀<!-- ∀ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \phi (x)=0\ \forall x\in X_{g}}</annotation>
</semantics>
</math></span><img src="./014ff68befdc9a979862ffc465634068275e3e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.774ex; height:3.009ex;" alt="{\displaystyle \phi (x)=0\ \forall x\in X_{g}}" loading="lazy"></span></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)=\infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \phi (x)=\infty }</annotation>
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</math></span><img src="./057423db75abd79696061a98d653e7a7eb2b6b85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.947ex; height:2.843ex;" alt="{\displaystyle \phi (x)=\infty }" loading="lazy"></span> if and only if no point in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {X_{g}}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle {X_{g}}}</annotation>
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</math></span><img src="./09fe1753d96f5486f38b96debbb54e440d676ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.946ex; height:2.843ex;" alt="{\displaystyle {X_{g}}}" loading="lazy"></span> is reachable from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>.</li>
<li>For every reachable state, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X\setminus {X_{g}}}">
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<annotation encoding="application/x-tex">{\displaystyle x\in X\setminus {X_{g}}}</annotation>
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</math></span><img src="./7fb2a1c8dcc02322e5e56914ef795d0d20056c5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.291ex; height:3.009ex;" alt="{\displaystyle x\in X\setminus {X_{g}}}" loading="lazy"></span>, the local operator produces a state <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mo>′</mo>
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<annotation encoding="application/x-tex">{\displaystyle x'}</annotation>
</semantics>
</math></span><img src="./0ac74959896052e160a5953102e4bc3850fe93b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}" loading="lazy"></span> for which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x')<\phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>&lt;</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (x')&lt;\phi (x)}</annotation>
</semantics>
</math></span><img src="./2549e3b68bd22a0734949218f80946fc89d444a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.832ex; height:3.009ex;" alt="{\displaystyle \phi (x')<\phi (x)}" loading="lazy"></span>.</li></ol>
<div class="mw-heading mw-heading2"><h2 id="Probabilistic_navigation_function">Probabilistic navigation function</h2></div>
<p>Probabilistic navigation function is an extension of the classical navigation function for static stochastic scenarios. The function is defined by permitted collision probability, which limits the risk during motion. The Minkowski sum used for in the classical definition is replaced with a convolution of the geometries and the Probability Density Functions of locations. Denoting the target position by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{d}}</annotation>
</semantics>
</math></span><img src="./c46388e6e01fcdbb8017eac0adc20dca1061af1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.422ex; height:2.009ex;" alt="{\displaystyle x_{d}}" loading="lazy"></span>, the Probabilistic navigation function is defined as:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\varphi }(x)={\frac {\gamma _{d}(x)}{{\left[{\gamma _{d}^{K}(x)+\beta \left(x\right)}\right]}^{\frac {1}{K}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi>φ<!-- φ --></mi>
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<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
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<mi>γ<!-- γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
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<mi>γ<!-- γ --></mi>
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<mi>K</mi>
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<mo stretchy="false">(</mo>
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<mo>+</mo>
<mi>β<!-- β --></mi>
<mrow>
<mo>(</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>K</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\varphi }(x)={\frac {\gamma _{d}(x)}{{\left[{\gamma _{d}^{K}(x)+\beta \left(x\right)}\right]}^{\frac {1}{K}}}}}</annotation>
</semantics>
</math></span><img src="./b537c8d25afdebd4378d740ff6b7c0106e4b8fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.598ex; height:8.009ex;" alt="{\displaystyle {\varphi }(x)={\frac {\gamma _{d}(x)}{{\left[{\gamma _{d}^{K}(x)+\beta \left(x\right)}\right]}^{\frac {1}{K}}}}}" loading="lazy"></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
</semantics>
</math></span><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> is a predefined constant like in the classical navigation function, which ensures the Morse nature of the function. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{d}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>γ<!-- γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
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</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma _{d}(x)}</annotation>
</semantics>
</math></span><img src="./9182e8272e807fc61fd2e645d84f30d971be9ac1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.435ex; height:2.843ex;" alt="{\displaystyle \gamma _{d}(x)}" loading="lazy"></span> is the distance to the target position <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {||x-{x_{d}}|{|^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi>x</mi>
<mo>−<!-- − --></mo>
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<annotation encoding="application/x-tex">{\displaystyle {||x-{x_{d}}|{|^{2}}}}</annotation>
</semantics>
</math></span><img src="./2e5f77b25f4b0b5b2556c4d940d4cb3ec7177800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.233ex; height:3.343ex;" alt="{\displaystyle {||x-{x_{d}}|{|^{2}}}}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \left(x\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
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<mo>(</mo>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta \left(x\right)}</annotation>
</semantics>
</math></span><img src="./83da728a5c8fe4cf38d9d51da86b0ee85bbd62b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.858ex; height:2.843ex;" alt="{\displaystyle \beta \left(x\right)}" loading="lazy"></span> takes into account all obstacles, defined as
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \left(x\right)=\prod \limits _{i=0}^{N_{o}}{{\beta _{i}}\left(x\right)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
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<mo>=</mo>
<munderover>
<mo movablelimits="false">∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>o</mi>
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</msub>
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</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta \left(x\right)=\prod \limits _{i=0}^{N_{o}}{{\beta _{i}}\left(x\right)}}</annotation>
</semantics>
</math></span><img src="./dd97523ff4329d6b15b7155fdc0e4aa6bc8eef39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.955ex; height:7.343ex;" alt="{\displaystyle \beta \left(x\right)=\prod \limits _{i=0}^{N_{o}}{{\beta _{i}}\left(x\right)}}" loading="lazy"></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{i}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{i}(x)}</annotation>
</semantics>
</math></span><img src="./d4d4690628d6e7da071edabccdf66d74a503d01b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.254ex; height:2.843ex;" alt="{\displaystyle \beta _{i}(x)}" loading="lazy"></span> is based on the probability for a collision at location <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>. The probability for a collision is limited by a predetermined value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>
</semantics>
</math></span><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span>, meaning:
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{i}(x)=\Delta -p^{i}\left(x\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo>−<!-- − --></mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{i}(x)=\Delta -p^{i}\left(x\right)}</annotation>
</semantics>
</math></span><img src="./905eec72a8ac5b73f3d54c3f26a80be8cbde090b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.624ex; height:3.176ex;" alt="{\displaystyle \beta _{i}(x)=\Delta -p^{i}\left(x\right)}" loading="lazy"></span>
and,
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{0}(x)=-\Delta +p^{0}\left(x\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo>+</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{0}(x)=-\Delta +p^{0}\left(x\right)}</annotation>
</semantics>
</math></span><img src="./7825680fc4590ce01ef752b0d5ec175145de0caa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.942ex; height:3.176ex;" alt="{\displaystyle \beta _{0}(x)=-\Delta +p^{0}\left(x\right)}" loading="lazy"></span>
</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{i}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p^{i}(x)}</annotation>
</semantics>
</math></span><img src="./eb0d07b66d731940aaaf6cb003785c25a497b82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.198ex; height:3.176ex;" alt="{\displaystyle p^{i}(x)}" loading="lazy"></span> is the probability to collide with the i-th obstacle.
A map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> is said to be a probabilistic navigation function if it satisfies the following conditions:
</p>
<ol><li>It is a navigation function.</li>
<li>The probability for a collision is bounded by a predefined probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>
</semantics>
</math></span><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span>.</li></ol>
<div class="mw-heading mw-heading2"><h2 id="Navigation_Function_in_Optimal_Control">Navigation Function in Optimal Control</h2></div>
<p>While for certain applications, it suffices to have a feasible navigation function, in many cases it is desirable to have an optimal navigation function with respect to a given <a href="Cost_functional" class="mw-redirect" title="Cost functional">cost functional</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J}</annotation>
</semantics>
</math></span><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span>. Formalized as an <a href="Optimal_control" title="Optimal control">optimal control</a> problem, we can write
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{minimize }}J(x_{1:T},u_{1:T})=\int \limits _{T}L(x_{t},u_{t},t)dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>minimize&nbsp;</mtext>
</mrow>
<mi>J</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>:</mo>
<mi>T</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>:</mo>
<mi>T</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</munder>
<mi>L</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{minimize }}J(x_{1:T},u_{1:T})=\int \limits _{T}L(x_{t},u_{t},t)dt}</annotation>
</semantics>
</math></span><img src="./c7e39b61d1b5762361efefc8e15e8b04bd3604f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:40.407ex; height:7.343ex;" alt="{\displaystyle {\text{minimize }}J(x_{1:T},u_{1:T})=\int \limits _{T}L(x_{t},u_{t},t)dt}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{subject to }}{\dot {x_{t}}}=f(x_{t},u_{t})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>subject to&nbsp;</mtext>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{subject to }}{\dot {x_{t}}}=f(x_{t},u_{t})}</annotation>
</semantics>
</math></span><img src="./ca00d15323fdd9f8d7aba30b73f799542ee6519b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.098ex; height:2.843ex;" alt="{\displaystyle {\text{subject to }}{\dot {x_{t}}}=f(x_{t},u_{t})}" loading="lazy"></span></dd></dl>
<p>whereby <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> is the state, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> is the control to apply, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> is a cost at a certain state <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> if we apply a control <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> models the transition dynamics of the system.
</p><p>Applying <a href="Bellman_equation" title="Bellman equation">Bellman's principle of optimality</a> the optimal cost-to-go function is defined as
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \phi (x_{t})=\min _{u_{t}\in U(x_{t})}{\Big \{}L(x_{t},u_{t})+\phi (f(x_{t},u_{t})){\Big \}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">min</mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>U</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.623em" minsize="1.623em">{</mo>
</mrow>
</mrow>
<mi>L</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.623em" minsize="1.623em">}</mo>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \displaystyle \phi (x_{t})=\min _{u_{t}\in U(x_{t})}{\Big \{}L(x_{t},u_{t})+\phi (f(x_{t},u_{t})){\Big \}}}</annotation>
</semantics>
</math></span><img src="./bfbe5970da47b0bca961fad744697f035050ca07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.951ex; height:5.509ex;" alt="{\displaystyle \displaystyle \phi (x_{t})=\min _{u_{t}\in U(x_{t})}{\Big \{}L(x_{t},u_{t})+\phi (f(x_{t},u_{t})){\Big \}}}" loading="lazy"></span>
</p><p>Together with the above defined axioms we can define the optimal navigation function as
</p>
<ol><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)=0\ \forall x\in X_{g}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)=\infty }">
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<annotation encoding="application/x-tex">{\displaystyle \phi (x)=\infty }</annotation>
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</math></span><img src="./057423db75abd79696061a98d653e7a7eb2b6b85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.947ex; height:2.843ex;" alt="{\displaystyle \phi (x)=\infty }" loading="lazy"></span> if and only if no point in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {X_{G}}}">
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<annotation encoding="application/x-tex">{\displaystyle {X_{G}}}</annotation>
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</math></span><img src="./43161d80973637fb46b55af2cc08e6851dfdd2cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.448ex; height:2.509ex;" alt="{\displaystyle {X_{G}}}" loading="lazy"></span> is reachable from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
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<li>For every reachable state, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X\setminus {X_{G}}}">
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<annotation encoding="application/x-tex">{\displaystyle x\in X\setminus {X_{G}}}</annotation>
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</math></span><img src="./a5052be720680179fd5c12fef3e21ed53af2f700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.793ex; height:2.843ex;" alt="{\displaystyle x\in X\setminus {X_{G}}}" loading="lazy"></span>, the local operator produces a state <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}">
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</math></span><img src="./0ac74959896052e160a5953102e4bc3850fe93b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}" loading="lazy"></span> for which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x')<\phi (x)}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \phi (x')&lt;\phi (x)}</annotation>
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</math></span><img src="./2549e3b68bd22a0734949218f80946fc89d444a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.832ex; height:3.009ex;" alt="{\displaystyle \phi (x')<\phi (x)}" loading="lazy"></span>.</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \phi (x_{t})=\min _{u_{t}\in U(x_{t})}{\Big \{}L(x_{t},u_{t})+\phi (f(x_{t},u_{t})){\Big \}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \phi (x_{t})=\min _{u_{t}\in U(x_{t})}{\Big \{}L(x_{t},u_{t})+\phi (f(x_{t},u_{t})){\Big \}}}</annotation>
</semantics>
</math></span><img src="./bfbe5970da47b0bca961fad744697f035050ca07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.951ex; height:5.509ex;" alt="{\displaystyle \displaystyle \phi (x_{t})=\min _{u_{t}\in U(x_{t})}{\Big \{}L(x_{t},u_{t})+\phi (f(x_{t},u_{t})){\Big \}}}" loading="lazy"></span></li></ol>
<p>Even if a navigation function is an example for reactive control, it can be utilized for optimal control problems as well which includes planning capabilities.<sup id="cite_ref-SavkinMatveev2015_3-0" class="reference"><a href="#cite_note-SavkinMatveev2015-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Stochastic_Navigation_Function">Stochastic Navigation Function</h2></div>
<p>If we assume the transition dynamics of the system or the cost function as subjected to noise, we obtain a <a href="Optimal_control" title="Optimal control">stochastic optimal control</a> problem with a cost <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x_{t},u_{t})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
<mo stretchy="false">(</mo>
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<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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<mi>u</mi>
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<mi>t</mi>
</mrow>
</msub>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J(x_{t},u_{t})}</annotation>
</semantics>
</math></span><img src="./92edb73e1ba82cdf665db4d140ddea022b8c7cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.626ex; height:2.843ex;" alt="{\displaystyle J(x_{t},u_{t})}" loading="lazy"></span> and dynamics <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>. In the field of <a href="Reinforcement_learning" title="Reinforcement learning">reinforcement learning</a> the cost is replaced by a reward function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x_{t},u_{t})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
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<annotation encoding="application/x-tex">{\displaystyle R(x_{t},u_{t})}</annotation>
</semantics>
</math></span><img src="./b4da3cf5c5a732e29bf3443e987a62c877195725.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.919ex; height:2.843ex;" alt="{\displaystyle R(x_{t},u_{t})}" loading="lazy"></span> and the dynamics by the transition probabilities <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{t+1}|x_{t},u_{t})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
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<mi>x</mi>
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<mo stretchy="false">|</mo>
</mrow>
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<mi>x</mi>
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<mi>t</mi>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(x_{t+1}|x_{t},u_{t})}</annotation>
</semantics>
</math></span><img src="./cad45fc37807cfd0f59350211a09bdd40963fdb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.803ex; height:2.843ex;" alt="{\displaystyle P(x_{t+1}|x_{t},u_{t})}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Control_Theory" class="mw-redirect" title="Control Theory">Control Theory</a></li>
<li><a href="Optimal_Control" class="mw-redirect" title="Optimal Control">Optimal Control</a></li>
<li><a href="Robot_control" title="Robot control">Robot Control</a></li>
<li><a href="Motion_planning" title="Motion planning">Motion Planning</a></li>
<li><a href="Reinforcement_Learning" class="mw-redirect" title="Reinforcement Learning">Reinforcement Learning</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Lavalle, Steven, <a rel="nofollow" class="external text" href="http://planning.cs.uiuc.edu/node369.html">Planning Algorithms Chapter 8</a></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFHacohenShovalShvalb2019" class="citation journal cs1">Hacohen, Shlomi; Shoval, Shraga; Shvalb, Nir (2019). "Probability Navigation Function for Stochastic Static Environments". <i>International Journal of Control, Automation and Systems</i>. <b>17</b> (8): 2097–2113(2019). <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs12555-018-0563-2">10.1007/s12555-018-0563-2</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&nbsp;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:164509949">164509949</a>.</cite></span>
</li>
<li id="cite_note-SavkinMatveev2015-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-SavkinMatveev2015_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFAndrey_V._SavkinAlexey_S._MatveevMichael_Hoy2015" class="citation book cs1">Andrey V. Savkin; Alexey S. Matveev; Michael Hoy (25 September 2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yny8BwAAQBAJ&amp;pg=PA47"><i>Safe Robot Navigation Among Moving and Steady Obstacles</i></a>. Elsevier Science. pp.&nbsp;47–. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-12-803757-7</bdi>.</cite></span>
</li>
</ol></div></div>
<dl><dt>Sources</dt></dl>
<ul><li><cite id="CITEREFLaValle2006" class="citation cs2">LaValle, Steven M. (2006), <a rel="nofollow" class="external text" href="http://planning.cs.uiuc.edu/"><i>Planning Algorithms</i></a> (First&nbsp;ed.), Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-521-86205-9</bdi></cite></li>
<li><cite id="CITEREFLaumond1998" class="citation cs2">Laumond, Jean-Paul (1998), <a rel="nofollow" class="external text" href="http://homepages.laas.fr/jpl/book-toc.html"><i>Robot Motion Planning and Control</i></a> (First&nbsp;ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>3-540-76219-1</bdi></cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://github.com/johnyf/nfsim">NFsim</a>: MATLAB Toolbox for motion planning using Navigation Functions.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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