Joining arcs of helices it is pretty easy to obtain closed curves with constant curvature and $C^2$ regularity (see http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0203.pdf). Joining arcs of Salkowski curves (constant curvature, non-constant torsion) this regularity can be increased to $C^3$ (see https://www.sciencedirect.com/science/article/pii/S0167839608000988).
Any nice (non-trivial) examples of explicitly-parametrized closed space curve with constant curvature and higher regularity?
Note: If a constant-curvature curve is arc-length parametrized, its tangent indicatrix is a constant-speed curve on the unit sphere. Up to integration, my question is equivalent to finding a regular constant-speed closed curve on the sphere whose barycenter coincides with the origin.