TNB=(({{0, -1, 0}, {(1/(Sqrt[2])), 0, -(1/(Sqrt[2]))}, {(1/(Sqrt[2])), 0, (1/(Sqrt[2]))}}));
α[s_]={{Cos[(s/(Sqrt[2]))]-1},{Sin[(s/(Sqrt[2]))]},{(s/(Sqrt[2]))}};

TNB

A=Inverse[TNB]

β[s_]:=A.α[s]//Expand

β[s]

Series[β[s],{s,0,3}]

ParametricPlot3D[{s-((s^3)/24),((s^2)/4),((s^3)/24)},{s,-2.5,2.5}]

a0=ParametricPlot3D[{s-((s^3)/24),((s^2)/4),((s^3)/24)},{s,-2.5,2.5}];
a1=ParametricPlot3D[{s-((s^3)/24),((s^2)/4),0},{s,-2.5,2.5}];
a2=ParametricPlot3D[{s-((s^3)/24),0,((s^3)/24)},{s,-2.5,2.5}];
a3=ParametricPlot3D[{0,((s^2)/4),((s^3)/24)},{s,-2.5,2.5}];
Show[{a0,a1,a2,a3},ViewPoint->{100,100,100}]