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Compact objects inspiraling into supermassive black holes, known asextreme-mass-ratio inspirals, are an important source for future space-bornegravitational-wave detectors. When constructing waveform templates, usually theadiabatic approximation is employed to treat the compact object as a testparticle for a short duration, and the radiation reaction is reflected in thechanges of the constants of motion. However, the mass of the compact objectshould have contributions to the background. In the present paper, employingthe effective-one-body formalism, we analytically calculate the trajectories ofa compact object around a massive Kerr black hole with generallythree-dimensional orbits and express the fundamental orbital frequencies inexplicit forms. In addition, by constructing an approximate "constant" similarto the Carter constant, we transfer the dynamical quantities such as energy,angular momentum, and the "Carter constant" to the semilatus rectum,eccentricity, and orbital inclination with mass-ratio corrections. The linearmass-ratio terms in the formalism may not be sufficient for accurate waveforms,but our analytical method for solving the equations of motion could be usefulin various approaches to building waveform models.

Geometrized effective-one-body formalism for extreme-mass-ratio limits: Generic orbits