A stabilization device for stabilizing an attachment component relative to movements of a basic component which occur outside a permitted plane of movement. The attachment component and the basic component are connected via a stabilization arrangement that includes: a first compensation arrangement for compensating rotational movements of the basic component with respect to an intermediate component, about rotational axes lying in the plane of movement, including a first compensation device which can be actuated actively, a second compensation arrangement for compensating residual linear movements of the intermediate component in a compensation direction, perpendicular to the plane of movement, in relation to the attachment component, having a second compensation device which can be actuated actively, a plurality of inertial sensors assigned to the first and/or second compensation arrangement, and a control device for actuating the compensation devices for movement compensation as a function of sensor data of the inertial sensors.