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Here the total length $L\\geq 0$ is free, $s$ denotes the arclength parameter, $\\kappa$ denotes the absolute curvature of $\\mathbf{x}$, and $\\xi>0$ is constant. We lift problem $\\mathbf{P_{curve}}$ on $\\mathbb R^3$ to a sub-Riemannian problem $\\mathbf{P_{mec}}$ on $\\operatorname{SE(3)}\\nolimits/(\\{\\mathbf{0}\\}\\times \\operatorname{SO(2)}\\nolimits)$. 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