gamma[t_] := {Sin[t] - t Cos[t], Cos[t] + t Sin[t], t^2}
unitTangent[t_] := (t/(Abs[t] Sqrt[5])) {Sin[t], Cos[t], 2}

unitNormal[t_] := (t/Abs[t]){Cos[t], -Sin[t], 0}
biNormal[t_] := Cross[unitTangent[t], unitNormal[t]]

movingFrame[s_] := ParametricPlot3D[{gamma[s] + t unitTangent[s], gamma[s]  + t unitNormal[s], gamma[s] + t biNormal[s]},{t,0,3}, PlotRange->{{-10,10},{-10,10},{0,50}},PlotStyle->{Thickness[0.02],Thickness[0.02],Thickness[0.02]}]

curvePlot := ParametricPlot3D[gamma[s], {s,-10Pi, 10 Pi}, PlotRange->{{-10,10},{-10,10},{0,50}}]

Manipulate[Show[{movingFrame[t],curvePlot}],{t,-6.5,6.5},ControlPlacement->Bottom]