608 lines
		
	
	
		
			19 KiB
		
	
	
	
		
			Mathematica
		
	
	
		
		
			
		
	
	
			608 lines
		
	
	
		
			19 KiB
		
	
	
	
		
			Mathematica
		
	
	
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								(*** Wolfram Notebook File ***)
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								(* http://www.wolfram.com/nb *)
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								Notebook[{
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								 "The \[OpenCurlyQuote]right trivialised\[CloseCurlyQuote] tangent of the \
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								exponential map, ",
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								  FormBox["dexpR", TraditionalForm]],
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								  FormatType->"TraditionalForm"],
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								 ", according to Iserles05an, formula 2.42, pg. 32 can be written as\n\t",
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								 Cell[BoxData[
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								         RowBox[{"-", "1"}]]}], "=", 
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								        SubscriptBox["dexpR", "\[Omega]"], "(", 
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								    }], TraditionalForm]],
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								  FormatType->"TraditionalForm"],
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								 "\nwhere ",
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								 Cell[BoxData[
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								  FormBox[
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								   RowBox[{"g", "=", 
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								    RowBox[{"exp", "(", "\[Omega]", ")"}]}], TraditionalForm]],
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								  FormatType->"TraditionalForm"],
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								 ", and ",
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								 Cell[BoxData[
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								  FormBox[
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								    RowBox[{"g", "'"}], "=", 
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								    RowBox[{
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								     RowBox[{"exp", "'"}], 
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								     RowBox[{"(", "\[Omega]", ")"}]}]}], TraditionalForm]],
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								  FormatType->"TraditionalForm"],
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								 ".\nCompare this to the left Jacobian matrix ",
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								  FormatType->"TraditionalForm"],
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								 " in Chirikjian11book2, pg. 26, we see that ",
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								 Cell[BoxData[
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								  FormBox["dexpR", TraditionalForm]],
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								  FormatType->"TraditionalForm"],
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								 " and ",
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								 Cell[BoxData[
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								  FormBox[
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								  FormatType->"TraditionalForm"],
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								 " are the same.\n\nHence, be careful, Iserles\[CloseCurlyQuote]s \
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								\[OpenCurlyQuote]",
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								 " trivialised\[CloseCurlyQuote] tangent of the exponential map ",
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								  FormatType->"TraditionalForm"],
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								 " is in fact Chirikjian\[CloseCurlyQuote]s ",
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								 StyleBox["left",
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								  FontWeight->"Bold"],
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								 " Jacobian matrix ",
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								 Cell[BoxData[
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								  FormatType->"TraditionalForm"],
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								 "!!!\n\nWe want to compute the s \[OpenCurlyQuote]",
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								 StyleBox["left",
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								  FontWeight->"Bold"],
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								 " trivialised\[CloseCurlyQuote] tangent of the exponential map, ",
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								 ", for SE2, hence, we need to use Chirikjian\[CloseCurlyQuote]s ",
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								 " Jacobian matrix ",
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								 " formula in Chirikjian11book2, pg. 36."
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