Why torus-unstable solar filaments experience failed eruption?

To investigate the factors that control the success and/or failure of solar eruptions, we study the magnetic field and 3-Dimensional (3D) configuration of 16 filament eruptions during 2010 July - 2013 February. All these events, i.e., erupted but failed to be ejected to become a coronal mass ejection (CME), are failed eruptions with the filament maximum height exceeding $100 Mm$. The magnetic field of filament source regions is approximated by a potential field extrapolation method. The filament 3D configuration is reconstructed from three vantage points by the observations of STEREO Ahead/Behind and SDO spacecraft. We calculate the decay index at the apex of these failed filaments and find that in 7 cases, their apex decay indexes exceed the theoretical threshold ($n_{crit} = 1.5$) of the torus instability. We further determine the orientation change or rotation angle of each filament top during the eruption. Finally, the distribution of these events in the parameter space of rotation angle versus decay index is established. Four distinct regimes in the parameter space are empirically identified. We find that, all the torus-unstable cases (decay index $n > 1.5$), have a large rotation angles ranging from $50^circ - 130^circ$. The possible mechanisms leading to the rotation and failed eruption are discussed. These results imply that, besides the torus instability, the rotation motion during the eruption may also play a significant role in solar eruptions.