Device for assisting the manipulation of an instrument, comprising: a hinged arm intended to be fixed to a frame and manipulable by an operator, to which an instrument can be attached at an attachment point of said hinged arm forming a passive ball joint between the hinged arm and the instrument, the passive ball joint transmitting no moment to the attachment point of the instrument and allowing free handling of the instrument by the operator, the hinged arm comprising motors (M1, M2,....) for displacing the attachment point in a reference frame bound to the frame, the instrument being manipulable around a Fulcrum having a known and fixed position in the reference frame, a processing unit comprising a processor configured to control the motors (M1, M2,...); the device being configured to implement a preliminary step of determining a position of the fulcrum in the reference frame consisting in the steps of: - Obtaining, in the reference frame, a plurality of straight lines (Δ) defined by the instrument axis (X-X'), the straight lines corresponding to a plurality of positions of the said instrument about the Fulcrum and passing through the attachment point, each straight line being obtained after a given displacement of the attachment point; - Estimating of an intersection zone (V ol) of said plurality of straight lines (Δ), - Obtaining the position of the center of said zone (V ol), which corresponds to the fulcrum, the device being configured to determine data relating to a position and/or velocity of the attachment point in the reference frame; the processing unit being configured for: determining (E2, E3, E4) a force ( F P) to be applied to the attachment point so as to apply a given impedance at a point located on the instrument axis (X-X') connecting the attachment point to the Fulcrum, the point being located at a known distance from the attachment point, the force to be applied being a function of said data relating to the attachment point, the position of the Fulcrum, the known distance from the attachment point to the point, and of the given impedance to be applied to the point; the device also being configured to control motors (M1, M2,...) to transmit the force ( F P) determined at the instrument at the attachment point, so as to produce the impedance at the point. - Obtaining, in the reference frame, a plurality of straight lines (Δ) defined by the instrument axis (X-X'), the straight lines corresponding to a plurality of positions of the said instrument about the Fulcrum and passing through the attachment point, each straight line being obtained after a given displacement of the attachment point; - Estimating of an intersection zone (V ol) of said plurality of straight lines (Δ), - Obtaining the position of the center of said zone (V ol), which corresponds to the fulcrum, the device being configured to determine data relating to a position and/or velocity of the attachment point in the reference frame; the processing unit being configured for: determining (E2, E3, E4) a force ( F P) to be applied to the attachment point so as to apply a given impedance at a point located on the instrument axis (X-X') connecting the attachment point to the Fulcrum, the point being located at a known distance from the attachment point, the force to be applied being a function of said data relating to the attachment point, the position of the Fulcrum, the known distance from the attachment point to the point, and of the given impedance to be applied to the point; the device also being configured to control motors (M1, M2,...) to transmit the force ( F P) determined at the instrument at the attachment point, so as to produce the impedance at the point. the device being configured to determine data relating to a position and/or velocity of the attachment point in the reference frame; the processing unit being configured for: determining (E2, E3, E4) a force ( F P) to be applied to the attachment point so as to apply a given impedance at a point located on the instrument axis (X-X') connecting the attachment point to the Fulcrum, the point being located at a known distance from the attachment point, the force to be applied being a function of said data relating to the attachment point, the position of the Fulcrum, the known distance from the attachment point to the point, and of the given impedance to be applied to the point; the device also being configured to control motors (M1, M2,...) to transmit the force ( F P) determined at the instrument at the attachment point, so as to produce the impedance at the point. determining (E2, E3, E4) a force ( F P) to be applied to the attachment point so as to apply a given impedance at a point located on the instrument axis (X-X') connecting the attachment point to the Fulcrum, the point being located at a known distance from the attachment point, the force to be applied being a function of said data relating to the attachment point, the position of the Fulcrum, the known distance from the attachment point to the point, and of the given impedance to be applied to the point; the device also being configured to control motors (M1, M2,...) to transmit the force ( F P) determined at the instrument at the attachment point, so as to produce the impedance at the point.