In this article we show that the vector-Apodizing Phase Plate (vAPP) coronagraph can be designed such that the coronagraphic point spread functions (PSFs) can act as a wavefront sensor to measure and correct the (quasi-)static aberrations, without dedicated wavefront sensing holograms nor modulation by the deformable mirror. The absolute wavefront retrieval is performed with a non-linear algorithm. The focal-plane wavefront sensing (FPWFS) performance of the vAPP and the algorithm are evaluated with numerical simulations, to test various photon and read noise levels, the sensitivity to the 100 lowest Zernike modes and the maximum wavefront error (WFE) that can be accurately estimated in one iteration. We apply these methods to the vAPP within SCExAO, first with the internal source and subsequently on-sky. In idealised simulations we show that for $10^7$ photons the root-mean-square (RMS) WFE can be reduced to $simlambda/1000$, which is 1 nm RMS in the context of the SCExAO system. We find that the maximum WFE that can be corrected in one iteration is $simlambda/8$ RMS or $sim$200 nm RMS (SCExAO). Furthermore, we demonstrate the SCExAO vAPP capabilities by measuring and controlling the lowest 30 Zernike modes with the internal source and on-sky. On-sky, we report a raw contrast improvement of a factor $sim$2 between 2 and 4 $lambda/D$ after 5 iterations of closed-loop correction. When artificially introducing 150 nm RMS WFE, the algorithm corrects it within 5 iterations of closed-loop operation. FPWFS with the vAPPs coronagraphic PSFs is a powerful technique since it integrates coronagraphy and wavefront sensing, eliminating the need for additional probes and thus resulting in a $100%$ science duty cycle and maximum throughput for the target.
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