Kinematic Transformation

Axes are connected via mechanical links providing movements of the ‘Tool Center Point’, TCP in space. TCP is a dis-tinguished point of the machine, sometimes also called ‘Point of Interest’, POI, or ‘effector’. The physical assembly of the axes and therefore the position of the TCP in MCS is described by a so called kinematic transformation. The kine-matic transformation connects ACS to MCS (forward conversion). By applying the kinematic transformation on a position related to ACS, this position can be transformed into a position in MCS. The other way round, applying the inverse kinematic transformation, a position related to MCS can be transformed into a position in ACS (backward conversion).

With simple cartesian machine constructions, in which axes are directly oriented in X-, Y-, and Z-directions of MCS, the kinematic transformation can easily be specified. One just has to define which axis is in the X-direction, which in Y, and which in the Z-direction. In the simplest case ACS is identically to MCS and one needn’t distinguish between both. But in praxis there are many non-cartesian structures, like SCARA robots or Tripods, where the kinematic transfor-mation is more complex.

Example for reaching the same position in space with a) a cartesian handling (2 linear axes) and b) a SCARA (2 rotary axes) with two possible configurations (elbow down and elbow up). (Note: the orientation is fixed in both examples)

Above example demonstrates how a position in space could be reached by a cartesian handling or a SCARA. Whereas the positions of the linear axes are more or less identical to the coordinates of the position in MCS, the positions of the axes of the SCARA are not that easy to calculate. Additionally there are two possible solutions of the backward kine-matic transformation, different configurations of the machine: elbow down and elbow up.