Kinematics of Exechon PKM

Updated: 2008/9/30


      The kinematics transformation "Tripod with hybrid kinematics" is a 5-axis transformation with parallel-serial kinematics, see figures below.

      It allows a rotationally symmetric tool (milling cutter, laser beam) to be orientated in relation to the workpiece in the machining space. The path and path velocity are programmed analogously to the 3-axis transformation. The tool alignment is programmed as an additional parameter in motion blocks. The real-time transformation calculates the resultant movement of all 5 axes. As a consequence, machining programs are not machine-specific. There is no need for kinematics-specific postprocessors for 5-axis machining operations. Tool length compensation is included in the machining calculations.

      By virtue of the explicit, programmed storage (TOFRAME) of the tool-to-workpiece-surface orientation in a separate FRAME, it is possible to retract the tool vertically from the workpiece.

Figure1 Equivalent kinematics diagram of the kinematics module in XZ plane


Figure 2: Equivalent kinematics diagram of the kinematics module in YZ plane



      Each corner of a triangular base platform is connected to a linear axis by a universal joint. These linear axes determine the position of a triangular, mobile platform relative to the base platform. Each linear axis is connected to the moving platform by a swiveling axis. The second linear axis has an additional degree of freedom, i.e. rotation around the direction of the linear axis.

      To the moving platform a rotating and a swiveling joint is linked. Both are used to orientate the spindle in space.

      The kinematical linkage is parallel regarding the first three axes, to which two rotary axes are linked serially. Thus the machine possesses three spatial and two orientational degrees of freedom.


      We use small letters for one dimensional quantities, e.g. lengths, capitals for spatial points and bold letters for direction vectors. In the following the index i runs from 1 to 3.

BCS     basic coordinate system

ICS       internal coordinate system

MPS     moving platform coordinate system

OICS    origin of the base platform

OMPS  origin of the moving platform, i.e. intersection of axis 4 with the perpendicular of axis 5 to axis 4

ui          rotary axis of the universal joint fixed at the base platform ("outer joint")

vi          rotary axis of the universal joint fixed at the linear axis ("inner joint")

di          distance between both rotaries of a universal joint

e          offset perpendicular to axis 2 and v2

f           offset parallel to v2

Ai         intersection of the perpendicular between the axes of a universal joint at the outer joint

λ          distance between A1 and A3 along the X direction of the ICS

φ          Angle between the second linear axis about its longitudinal direction, i.e. the angle between v2 and w2

qi          length of the perpendicular between vi and wi. This matches the value of the corresponding linear axis.

q4         Angle of the fourth rotary axis

q5         Angle of the fifth rotary axis

wi         rotary axis at the moving platform

Bi         intersection of the perpendicular between vi and wi at the rotary near the moving platform. This point is unique by the requirement that the perpendicular point from the outer to the inner joint of the universal joint is on that perpendicular.

r           direction vector of the fourth axis with respect to the moving platform

H          hand point, i.e. intersection between axis five and the perpendicular from OMPS

c          distance between OMPS and H

tool       tool direction

t1         tool dimension along axis 5

t2         tool dimension perpendicular to the tool direction and axis 5

t3         tool dimension along the tool direction

      Without loss of generality all direction vectors are supposed to be of unit length.


      The OEM transformation "Tripod with hybrid kinematics" has three spatial and two orientation degrees of freedom, allowing a rotationally symmetric tool (milling cutter, laser beam) to be oriented freely in relation to the workpiece at any point within the machining space.

      The workpiece is always programmed in the rectangular coordinate system (WCS), while any programmed or set frames rotate and shift the latter in relation to the basic system (BCS), also rectangular. The kinematic transformation then converts this information into motion commands to the real machine axes.