The objective of the inverse kinematic calculation is to compute the six displacements Di of the actuators required in order to achieve a desired motion of the mobile part: Di = f (x y z Rx Ry Rz) [1] Let us define the desired displacement of the mobile part u =(x y z Rx Ry Rz) with respect to a zero position in terms of the homogenous transformation matrix R (4x4) relative to the axes of the M2 (origin on vertex). The homogenous transformation method and the computation of the matrix are described in several books on robotics and serial links in particular. If the command is given with respect to an offset system of axes (i.e. the telescope axes), the matrix RT will be multiplied by an offset matrix R0: R = RT x R0 Let us set the matrices PF (6x3) and PM (6x3) with the coordinates (x,y,z) of the virtual hinge points of the leg flexures in the given zero position, respectively attached to the actuators (PF) and to the mobile part (PM). PM and PF must be referenced to an axes origin given by the mirror vertex. Consider the leg i and define the vector PMi :
The conditions of the mobile part hinge after the displacement are computed as: PMinew = R x Pmi, where the first 3 terms of PMinew are the x,y,z coordinates of the moved hinge point. Recalling that with the given geometry the base points PF can only move in axial direction (z), the displacement ZI is computed from the condition that the leg length L between the hinges remains constant: ZI = PMinew(3) - PFi,3 + D, where
By setting very small displacements, one degree of freedom at the time, we obtain a approximation linear transformation matrix T with which actuator displacements can be estimated as: DI = u * T The matrix T can be inverted and we have: u= DI * T-1 [2]
The forward (or direct) kinematics calculation is used to evaluate the position of the mobile part from the knowledge of the six actuator displacements: (x y z Rx Ry Rz) = f (Di ) This is done by solving the implicit equation [1] by an iterative method: Di - f (x y z Rx Ry Rz) = 0 Where the initial approximation is taken by equation [2].
