I wondered if a nice spherical coordinate parametrization was known for something a bit like children expanding balls known as Hoberman spheres.
a bit like means a map $$ (t,\theta,\phi)->f(t,\theta,\phi) $$ with $f$ continuous on $(0,1]\times[0,2\pi)\times[0,\pi]$ and $f(1,\theta,\phi)=1$ so that the domain $$ \{(x,y,z)=(r\cos(\theta)\sin(\phi),r\sin(\theta)\sin(\phi),r\cos(\phi))~:~ 0\leq r\leq f(0+,\theta,\phi)\} $$ (where the notation "$0+$" means a positive arbitrarily small small value for t), looks like a spiky ball (more like a lot of spikes than a ball really).
nice means something explicit enough that one can do computations with it : in fact, if it was a coordinate change, it would be fab'.
Unless I am missing something, you can get such a parametrization on $t\in[0,1]$ as follows. Choose your favorite spherical parametrization $g(\theta,\phi)$ of your favorite "spiky ball" contained in the unit ball. For example, using a regular octahedron as a crude spiky ball, you could choose the standard spherical parametrization thereof $$c(\theta,\phi)=\frac{1/2}{|\cos\theta\sin\phi| + |\sin\theta\sin\phi| + |\cos\phi|}$$ (where the 1/2 is chosen to make the longest diagonals of the octahedron have length 1 and fit the entire octahedron inside the unit ball). Then simply take the parametrization $f$ to be $$f(t,\theta,\phi) = t + (1-t)g(\theta,\phi).$$ I think that satisfies the desired constraints.
Now of course you may find a regular octahedron to be an insufficiently spiky ball. You might prefer something more like the rhombic hexecontahedron, or something even spikier. But the same process should provide what you want: just start with a polar parametrization of the desired "spiky ball" and linearly interpolate from there to the final unit ball.