【正文】 a6 are related to each other by 2222 ayx BB ?? (5) ? ? 24221 ayax DD ??? (6) By substituting (1)－ (4) into (5)－ (6) the response equations of the support are obtained as ? ? ? ? 0s i nc o s 222525 ????? aayax ?? (7) ? ?? ? ? ?? ? 0s i nc os 2426216 ???????? aayaax ???? (8) This equation represents the base of the mathematical model for calculating the optimal values of parameters a1， a2， a4. Mathematical model The mathematical model of the system will be formulated in the form proposed by Haug and Arora (1979) : min f(u， v)， (9) subject to constraints ? ? 0?vugi ， i=1， 2，?， l (10) and response equations ? ? 0?vuhj ， j=l， 2，?， m (11) The vector u=[u1? un]T is called the vector of design variables， v=[v1? vm]T is the vector of response variables and f in (9) is the objective function. To perform the optimal design of the leading fourbar mechanism AEDB， the vector of design variables is defined as u=[ a1 a2 a4]T， (12) and the vector of response variables as v=[x y]T． (13) The dimensions a3， a5， a6 of the corresponding links are kept fixed. The objective function is defined as some “measure of difference” between the trajectory L and the desired trajectory K as f(u， v) =max[g0(y) － f0(y)]2 (14) where x= g0(y) is the equation of the curve K and x= f0(y) is the equation of the curve L. Suitable limitations for our system will be chosen. The system must satisfy the wellknown Grasshoff conditions ? ? ? ? 02143 ???? aaaa (15) ? ? ? ? 04132 ???? aaaa (16) Inequalities (15) and (16) express the property of a fourbar mechanism， where the links a2， a4 may only oscillate. The condition uuu ?? (17) prescribes the lower and upper bounds of the design variables. The problems (9)(11) is not be directly solvable with the usual gradientbased optimizations methods. This could be circumvented by introducing an artificial design variable un+1 as proposed by Hsieh and Arora (1984). The new formulation exhibiting a more convenient form may be written as min un+1 (18) subject to ? ? 0v ?，ugi ， i=1， 2?， 1 (19) ? ? 0v 1 ?? ?nuuf ， ， (20) and response equations ? ? 0v ?，uhj j=l， 2，? m (21) where u=[u1… un un+1]T and v=[v1… vm]T . A nonlinear programming problem of the leading fourbar mechanism AEDB can therefore be defined as min 7a ， (22) subject to constraints ? ? ? ? 02143 ???? aaaa (23) ? ? ? ? 04132 ???? aaaa (24) 111 aaa ??，222 aaa ??，444 aaa ?? (25) ? ? ? ?? ? ? ?yy07200 ， yayfyg ???? (26) and response equations ? ? ? ? 0s i nc o s 222525 ????? aayax ?? (27) ? ?? ? ? ?? ? 0s i nc os 2426216 ???????? aayaax ???? (28) The formulation enables the minimization of the different between the transversal displacement of the point C and the prescribed trajectory K. The result is the optimal values of the parameters a1， a2， a4. stochastic model of the hydraulic support The mathematical model (22) (28) may be used to calculate such values of the parameters a1， a2， a4， that the “difference between trajectories L and K” is minimal. However， the real trajectory L of the point C could deviate from the calculated values because of different influences. The suitable mathematical model deviation could be treated dependently on tolerances of parameters a1， a2， a4. The response equations (27) – (28) allow us to calculate the vector of response variables v in dependence on the vector of design variables u. This implies ? ?uhv ~? . The function h is the base of the mathematical model (22) – (28)， because it represents the relationship between the vector of design variables u and response v of our mechanical system. The same function h can be used to calculate the maximal allowed values of the tolerances 321 aaa ??? ， of parameters a1， a2， a4. In the stochastic model the vector u=[u1… un]T of design variables is treated as a random vector U=[U1… Un]T， meaning that the vector v=[v1… vm]T of response variables is also a random vector V=[V1? Vm]T， ? ?uhv ~? (29) It is supposed that the design variables U1? Un are independent from probabil