  Applying the nonrelativistic quantum chromodynamics factorization formalism
to the $\Upsilon(1S,2S,3S)$ hadroproduction, a complete analysis on the
polarization parameters $\lambda_{\theta}$, $\lambda_{\theta\phi}$,
$\lambda_{\phi}$ for the production are presented at QCD next-to-leading order.
With the long-distance matrix elements extracted from experimental data for the
production rate and polarization parameter $\lambda_{\theta}$ of $\Upsilon$
hadroproduction, our results provide a good description for the measured
parameters $\lambda_{\theta\phi}$ and $\lambda_{\phi}$ in both the helicity and
the Collins-Soper frames. In our calculations the frame invariant parameter
$\tilde{\lambda}$ is consistent in the two frames. Finally, it is pointed out
that there are discrepancies for $\tilde{\lambda}$ between available
experimental data and corresponding theoretical predictions.
