3。3 Deformation compatibility equation

Based on the material mechanics, the deformation of com- ponent can be expressed with constraint force, material prop- erties and structural attributes。 So the relation between con- straint force of components and position error of end-effector (main slider 8) can be established。 Because the number of position errors of end-effector is independent of whether the over-constrained is introduced in mechanism, and the number of dynamic equations obtained is equal with that of position errors of the end-effector。

During the run time of the mechanism, the position of pivot points O51, O61, O52 and O62 of statically indeterminate sub- mechanism is determined by kinematic constraint of the origi- nal mechanism。 Therefore, the constraint forces F51, F52, F53 and F54 are considered as the input driver of the system。 The link produces micro-deformation, in terms of the assumption in Sec。 3。1 (the main slider 8 is assumed as a rigid body), the slider produces only displacement and deflection angle with- out deformation under the state of the workload。

The base coordinate system O-xy which is attached to the frame is established; the x axis is horizontal, the y axis is verti- cal, and they are all through the pivot joint O0 on the basis of the characteristics of the mechanism。 A moving coordinate system T xyis fixed on the center of mass of the main slider 8, which is parallel to the coordinate system O-xy。 The posi- tion vector diagram of a link is shown in Fig。 6。

The rotation matrix of coordinate system T xywith re- spect to the base coordinate system O-xy can be expressed as

RO  I    。 ` (12)

In terms of Sec。 2 and Fig。 6, the closed-loop constraint equation associated with the ith kinematic chain (connectivity path) can be written as

r RT    Ai     aie1   bie2   Liui , (13)

where r is the position vector of the origin T with respect to

the coordinate system O-xy, which is defined as  r (x, y, 0)T  ,

Ai  is the position vector of the pivot joint Ai with respect to the

coordinate system T xywhich is defined as   A (x , y  , 0)T  ,

ai is the x coordinate of Bj with respect to the coordinate sys- tem O-xy, bi is the y coordinate of Bj with respect to the coor- dinate system O-xy, L is the length of link AB , and e = (1,0)T, e  = (0,1)T。

Taking the derivative of Eq。 (13) with respect to time, the equations can be written as

Fig。 7。 Force analysis of under link。

According to the material mechanics, the  axial deforma- tion li of link li can be express as

where Fi is axial force acting on link, li is the effective length of link, Ei is elastic modulus, and Si is section area of link。

In terms of Eq。 (19), the axial deformation li of the link can

The link produces elastic deformation at runtime under the action of the workload, which will make the  end-effector (main  slider  8) produce  the  position  errorsx andy in the

be derived by unknown force Fi。 Then, the relation between position error of main slider 8 and force Fi can be established。

Thus, the axial deformation li of the kinematic  chain can be expressed as

direction of the x axis and the y axis and the deflection angle

error, which are vertical to the moving plane。 Then the Eq。 (14) can be changed towherer  (x,y)T  。

Taking the dot product with ui on both sides of Eq。 (15) leads to