Non-Hermitian systems can exhibit unconventional spectral singularities called exceptional points (EPs).Various EP sensors have been fabricated in recent years,showing strong spectral responses to extemal signals.Here we propose how to achieve a nonlinear anti-parity-time (APT) gyroscope by spinning an optical resonator.We show that,in the absence of any nonlinearity,the sensitivity or optical mode splitting of the linear device can be magnified up to 3 orders compared to that of the conventional device without EPs.Remarkably,the APT symmetry can be broken when including the Kerr nonlinearity of the materials and,as a result,the detection threshold can be significantly lowered,i.e.,much weaker rotations which are well beyond the ability of a linear gyroscope can now be detected with the nonlinear device.Our work shows the powerful ability of APT gyroscopes in practice to achieve ultrasensitive rotation measurement.