Draft:Holonomic (robotics)

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{{Short description|Concept in optimal control and differential topology}}

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In control theory (which formalizes robotics) and differential topology, a path in the tangent bundle of the manifold of states is holonomic if the tangent components correspond to the derivative of the projection of the path to the manifold. More generally, the term "holonomic" can be used to describe a section of the bundle of jets of order $k>0$ that corresponds to the derivatives of order $k$ of some $k$-times differentiable map, as opposed to a section that assigns some jets that may not correspond to the derivatives of any map, which would be called "non-holonomic".{{cite book |last1=Eliashberg |first1=Y. |last2=Mishachev |first2=Nikolai M. |title=Introduction to the h-principle |date=2002 |publisher=American Mathematical Society |location=Providence, RI |isbn=0821832271 |page=11}}

For example, if the path $c\backslash colon[a,b]\backslash to\; \backslash mathbb\; R^n\backslash times\; \backslash mathbb\; R^n$ has the two components $c(t)=(x(t),v(t))$ where $x(t)$ is the state and $v(t)$ is tangent to the manifold of states (in this case, $\backslash mathbb\; R^n$), then $c$ is holonomic if $x\text{'}(t)=v(t)$ for each $t\backslash in[a,b]$. In the same vein, an example of a path that is not holonomic is, with $n=2$, $c(t)=((t,0),(0,1))$, since $(t,0)\text{'}=(1,0)\backslash neq(0,1)$.