Robot-assisted rehabilitation for three-degree-of-freedom joints, such as hip and ankle, is significant for patients with motor function injuries. The control of such robots involves attitude control. To adapt to different disease stages, multimode hybrid control is considered to be one of the best choices. Passive mode is based on trajectory tracking control, whereas active mode is based on field-based assist-as-needed (AAN) control. The key to AAN control is the solution of the closest attitude point. However, the attitude point belongs to a special orthogonal group SO(3), and its topology is completely different from Euclidean space, which causes difficulties in the solution. Both passive and active control methods are affected by the inaccuracy of model parameters and external disturbances. Therefore, this article proposes a multimode hybrid control method on SO(3). First, the expressions of trajectory tracking and contour tracking errors are proposed. To solve the contour tracking error, a feedback linearization algorithm based on a sliding surface is used. A radial basis function neural network is used for adaptive compensation. Subsequently, a controller for different modes is designed, and its stability is analyzed. Experiments are conducted using a hip exoskeleton, and the results verify the effectiveness of the proposed control method.