Abstract:
The aim of the invention is to compensate for the position-dependent length changes caused by the effect of weight in a variety of closed kinematic chains (K 1  . . . Kn), for connecting a stationary first element (E 1 ) to a movable second element (E 2 ). Said aim is achieved, by using a back transformation (Λ −1 ), which determines a compensation value for each length change (dq 1 , dq 2  . . . dqn), resulting from the application of the weight (Fg) impinging on the movable element (E 2 ) in each kinematic chain.

Description:
BACKGROUND OF THE INVENTION 
   The present invention relates to a method and a device for compensating for position-dependent length changes caused by the effect of weight in a plurality of closed kinematic chains for connecting a stationary first element to a movable second element, on which a weight acts, respective actuators being assigned to the kinematic chains, via which a relative movement between the first and the second element is predefined in a freely definable sequence of control instructions, by the respective positions of the respective actuators required for desired positions of the movable element being determined via a back transformation Λ −1 , in particular for parallel kinematic systems for use in numerically controlled machine tools and robots. 
   Such aforementioned parallel kinematic systems are being employed to an increasing extent in machine tools and industrial robots. In this case, a parallel kinematic system constitutes a three-dimensional coupling mechanism which comprises at least one closed kinematic chain. The kinematic chain, for its part, is used to connect a fixed frame platform to a movable working platform. Kinematic chains in this case comprise either links and connecting elements of constant or variable length. In the illustrations according to  FIGS. 2 and 3 , two different embodiments of parallel kinematic systems according to the prior art are shown, which are used in machine tools and robots. Here,  FIG. 2  shows a parallel kinematic system with invariable-length connecting elements, while  FIG. 3  shows one such with variable-length connecting elements, which are in each case driven by associated actuators. Shown in each case is a frame platform P 1 , which is connected to a movable working platform P 2  via links G 1  . . . Gn and connecting elements V 1  . . . Vn. In each case a weight Fg acts on the working platform P 2 , which represents an inertial mass. 
   As compared with conventional machine tools and serial robots, parallel kinematics are distinguished by their high dynamics and stiffness. The advantage of parallel kinematic systems consists inter alia in the fact that forces act predominantly axially on the movable connecting elements. Nevertheless, even in parallel kinematic systems, the problem arises that the connecting elements and links change their length as a result of the effect of weight or processing forces, which leads to positioning errors of the working point (working platform P 2 ). 
   These length changes of the kinematic chains lead to position errors of the working point and impair the production quality. In the case of conventional machine tools, this leads to a positioning accuracy of about 10 μm. The aforementioned length changes are of approximately the same order of magnitude. 
   In the case of conventional machines with mutually perpendicular actuators, linear axes as they are known, the length changes caused by a constant weight are taken into account axis by axis in the form of a measured correction table and are compensated for in this way. Hitherto, however, there has been no remedy for the problem outlined in parallel kinematic systems and in general in closed kinematic chains for connecting a stationary first element to a movable second element, on which a weight acts. 
   It is therefore an object of the present invention to permit compensation of the position-dependent length change caused by the effect of weight in kinematic chains. 
   SUMMARY OF THE INVENTION 
   According to the present invention, this object is achieved in that the known method cited at the beginning is developed by the following method steps according to the invention, in that
         by using the transformation rules, the weight acting on the movable element is distributed to the respective kinematic chains and in this way the forces caused therein are determined,   in accordance with the respective compliance of the kinematic chains, the resultant respective length changes are determined and   the respective actuators are acted on with a respective compensation value which compensates for the respective length change determined.       

   According to a first advantageous refinement of the method of compensating for the position-dependent length changes caused by the effect of weight in a plurality of closed kinematic chains, according to the present invention the respective compensation values determined are incorporated into the sequence of control instructions for the relative movement between the first and the second element. 
   A further advantageous configuration of the method of compensating for the position-dependent length changes caused by the effect of weight in a plurality of closed kinematic chains according to the present invention is distinguished by the fact that each kinematic chain comprises links and connecting elements of constant length. 
   An alternative advantageous configuration of the method of compensating for the position-dependent length changes caused by the effect of weight in a plurality of closed kinematic chains according to the present invention is distinguished by the fact that each kinematic chain comprises links and connecting elements of variable length. 
   A further advantageous configuration of the method of compensating for the position-dependent length changes caused by the effect of weight in a plurality of closed kinematic chains according to the present invention is distinguished by the fact that this is used in a three-dimensional coupling mechanism, in particular in a parallel kinematic system. 
   Furthermore, according to the invention, the object of the invention is achieved in that a known three-dimensional coupling mechanism having a plurality of closed kinematic chains for connecting a stationary frame platform to a movable working platform, each kinematic chain being associated with at least one driven axis, for carrying out a relative movement between the stationary frame platform and the movable working platform, having a control unit which has computing means for the back transformation Λ −1  of the respective positions of the respective axes required for desired positions of the movable working platform, is developed by the following device features according to the invention: 
   the computing means are configured in such a way that 
   
       
       
         
           by using these transformation rules, the weight acting on the moved element can be distributed to the respective kinematic chains and in this way the forces caused therein can be determined, 
           in accordance with a respective compliance of the kinematic chains, the resultant respective length changes can be determined and 
           the respective axes can be acted on with a respective compensation value which compensates for the respective length change determined. 
         
       
     
  
   A three-dimensional coupling mechanism according to the present invention can be particularly advantageously employed in a parallel kinematic system. 
   A parallel kinematic system of this type according to the present invention can in particular be a kinematic chain which comprises links and connecting elements of constant length. 
   Another advantageous parallel kinematic system according to the present invention can in particular be a kinematic chain which comprises links and connecting elements of variable length. 
   Such a parallel kinematic system according to the present invention can particularly advantageously be used in a numerically controlled machine tool or a numerically controlled robot. 
   By means of the invention, a non-negligible influence on the positioning accuracy of parallel kinematic systems can be compensated for, for example, by the force Fg acting on the working platform as a result of the inherent weight being distributed to the kinematic chains by an algorithm sketched in more detail at a later point. Given known compliance of the kinematic chains, these length changes can be calculated and taken into account axis by axis as a compensation value, in a manner analogous to temperature compensation. 
   In this case, the result, as advantages of the invention, is primarily an increase in the static positioning accuracy, the fact that no additional measuring devices are needed, the fact that the method can be applied to any desired parallel kinematic system, and the fact that transformations already implemented can be used. 
   Further advantages and details of the present invention are given by using the presentation which now follows of preferred embodiments and in conjunction with the figures. In this case, elements with a constant functionality are identified by the same designations. 

   
     BRIEF DESCRIPTION OF THE DRAWING 
     FIG.  1 : shows weight and force distribution to two connecting elements at different working points, 
     FIG.  2 : shows a parallel kinematic system with invariable-length rods and 
     FIG.  3 : shows a parallel kinematic system with variable-length connecting elements. 
   

   DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
   The illustrations according to  FIGS. 2 and 3  have already been explained at the beginning, in order to present the general construction of a parallel kinematic system and the different possible constructional configurations. 
   For the purpose of illustration,  FIG. 1  shows a two-axis, flat parallel kinematic system having a stationary first element E 1  and a movable second element E 2  in two positions E 2   A  (left-hand figure) and E 2   B  (right-hand figure). Here, the stationary element E 1  can for example represent the frame platform P 1 , and the movable second element E 2  can represent the working platform P 2 . The movable second element E 2  is connected to the stationary first element E 1  via two connecting elements V 1 , V 2  and two links G 1 , G 2  (via a kinematic chain K 1  and K 2  in each case). The weight Fg acts in the stationary second element E 2  and is divided up into partial forces F 1  and F 2  onto the respective connecting elements V 1  and V 2 . This is illustrated in the left-hand figure for a first position and in the right-hand figure for a second position. 
   At an obtuse angle α A  (left-hand figure), the forces acting on the connecting elements are low and, accordingly, result in only low loading of the connecting elements V 1  and V 2 . At an acute angle α B  (right-hand figure), by contrast, very high forces act on the connecting elements V 1  and V 2 , and can exceed the weight Fg acting at the working point by many times. 
   In parallel kinematic systems, the connecting elements V 1 , V 2  are always loaded in the axial direction. In this direction, these connecting elements generally exhibit desired high stiffness caused by the construction. The resultant stiffness of the parallel kinematic systems, however, can be below this, depending on the position (E 2   A  or E 2   B ). As a result of the respective partial forces F 1 , F 2  acting on each connecting element V 1 , V 2 , even with a particularly high stiffness in the axial direction, there are respective length changes dq 1  and dq 2 , which are all the greater the higher the partial forces F 1  and F 2  become, which is the case with an increasingly more acute angle (compare the left-hand and right-hand figures). However, these undesired length deviations dq 1  and dq 2  in turn result in position deviations of the element E 2  from the desired ideal working point. 
   In order to carry out, for example, a processing task on a machine tool or an industrial robot, the relative movement of the two platforms P 1  or E 1  and P 2  or E 2  is predefined in a freely definable sequence of instructions. With the aid of a mathematical algorithm implemented in the control unit (e.g. CNC=computerized numeric control), the back transformation Λ −1 , as it is known, the control unit calculates the positions q of the n connecting elements V 1  . . . Vn (for example driven axes A 1  . . . An) needed for the desired positions x of the working platform P 2  or E 2  in the case of n kinematic chains K 1  . . . Kn (in the exemplary embodiment, n=2). In additional mathematical algorithms implemented in the control unit, these transformation formulas Λ, Λ −1  are used to distribute the weight Fg acting on the moved platform P 2  or E 2  to the kinematic chains K 1  . . . Kn. This procedure is independent of the respective kinematic system and can therefore be used as a universal module for all parallel kinematic systems. 
   According to the invention, additional mathematical algorithms (which are stored for example in the software of the control unit) are now implemented in the control unit, for example a suitably programmed microcomputer. These algorithms will be presented below. 
   Via a forward transformation Λ corresponding to the back transformation Λ −1 , the given positions q of the driven axes A 1  . . . An are used to calculate the Cartesian position and possibly orientation x of the working platform P 2  or E 2 . The back transformation Λ −1  calculates the associated positions q of the driven axes A 1  . . . An from x. 
   This results in the following basic calculation relationship:
 
Λ: D⊂IR   M   →IR   R ,Λ( q )= x  and Λ −1   :E⊂IR   R   →IR   M ,Λ −1 ( x )= q   (1)
 
   The first mathematical derivative of Λ is the function known to those skilled in the art as the “Jacobian matrix”). Given simple transformation relationships, it can be specified in the form of a closed formula or approximated numerically in the general case, which makes the implementation easier in suitable computing means R of a control unit for an appropriately compensated three-dimensional coupling mechanism. 
   The following further calculation rule applies to carrying out the transformation for all the kinematic chains K 1  . . . Kn: 
             J   =         ⅆ   Λ       ⅆ   q       =     (             ⅆ     Λ   1         ⅆ     q   1             ⋯           ⅆ     Λ   1         ⅆ     q   n                 ⋮       ⋰       ⋮               ⅆ     Λ   n         ⅆ     q   1             ⋯           ⅆ     Λ   n         ⅆ     q   n               )               (   2   )             
 
   The weight Fg=m*g acts on the inertial mass m, which represents the working platform P 2  or E 2 , g representing the acceleration due to gravity of 9.81. By utilizing the transformation formulas of the forward transformation Λ shown, the respective partial forces F 1  . . . Fn or τ1 . . . τn caused in the actuators A 1  . . . An by the weight Fg are determined. 
   Here, it is true that τ=J T *Fg, J T  being the transposed matrix of J. The latter relationship is known to those skilled in the art and, inter alia, can be gathered from the following reference: “Roboter mit Tastsinn” [Robots with a sense of feel], Matthias Müller, Vieweg-Verlag, Braunschweig, 1993. The corresponding relevant text is to be deemed to be incorporated by reference. 
   Given a known compliance G of the actuators A 1  . . . An, the respective length changes dq 1  . . . dqn can be determined in accordance with the following calculation rule
 
 dq=G*q*τ   (3)
 
and, as compensation values, can be compensated for in a manner analogous to a temperature-induced length change, for example by the respective compensation values determined being taken into account in a sequence of control instructions.