Abstract: Applying an impulsive thrust during a close encounter with a celestial body can significantly improve the efficiency of the swing-by maneuver.In the existing literature, no analysis can be carried out when the impulse velocities are bigger than 1% of the orbital velocity of the spacecraft.To solve this problem, powered gravity assist is studied applying an arbitrary impulse with any magnitude and direction.The three-dimensional powered gravity assist maneuver based on the patched-conics approximation can be identified by eight independent parameters, in which five specify the three-dimensional gravity assist and the other three specify the magnitude and the direction of the impulse.Multiple reference frames are established to describe the trajectories before and after the impulse.Using the method of coordinate transformation and hyperbolic orbit dynamics, a set of new analytical equations are derived, including the variation in velocity, angular momentum, energy and inclination of the spacecraft due to the maneuver as a function of the eight parameters.These equations developed here are verified by numerical integrations, using the circular restricted threebody problem.Finally, the influences of the parameters on the orbit of spacecraft are discussed based on the above equations, and some conclusions about the optimal direction to apply the impulse are given.The results show that the optimal direction of the impulse is not parallel to the velocity of the spacecraft, and the orbital inclination is significantly influenced by the direction of the impulse.