In this work, we propose a free-breathing magnetic resonance fingerprinting
method that can be used to obtain $B_1^+$-robust quantitative maps of the
abdomen in a clinically acceptable time. A three-dimensional MR fingerprinting
sequence with a radial stack-of-stars trajectory was implemented for
quantitative abdominal imaging. The k-space acquisition ordering was adjusted
to improve motion-robustness. The flip angle pattern was optimized using the
Cramér-Rao Lower Bound, and the encoding efficiency of sequences with 300,
600, 900, and 1800 flip angles was evaluated. To validate the sequence, a
movable multicompartment phantom was developed. Reference multiparametric maps
were acquired under stationary conditions using a previously validated MRF
method. Periodic motion of the phantom was used to investigate the
motion-robustness of the proposed sequence. The best performing sequence length
(600 flip angles) was used to image the abdomen during a free-breathing
volunteer scan. When using a series of 600 or more flip angles, the estimated
$T_1$ values in the stationary phantom showed good agreement with the reference
scan. Phantom experiments revealed that motion-related artefacts can appear in
the quantitative maps, and confirmed that a motion-robust k-space ordering is
essential in preventing these artefacts. The in vivo scan demonstrated that the
proposed sequence can produce clean parameter maps while the subject breathes
freely. Using this sequence, it is possible to generate $B_1^+$-robust
quantitative maps of proton density, $T_1$, and $B_1^+$ under free-breathing
conditions at a clinically usable resolution within 5 minutes.