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Force required to lift an object Description Find the smallest amount of force required to lift an object, grasping it at a set of possible grasping points. Further explanation of this example:
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grasp.mos (!********************************************************************* Mosel NL examples ================= file grasp.mos `````````````` Find the smallest amount of normal force required to "grasp" an object given a set of possible grasping points. SOCP formulation. Based on grasp.mod, gasp_exp.mod, grasp_nonconvex.mod Source: http://www.orfe.princeton.edu/~rvdb/ampl/nlmodels/grasp/ Reference: "Applications of Second-Order Cone Programming", M.S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, 1998 (c) 2013 Fair Issac Corporation author: S. Heipcke, Nov. 2005, rev. Sep. 2013 *********************************************************************!) model "grasping (NL)" uses "mmxnlp" parameters N = 6 ! Number of lifting points MU = 0.3 ! Friction coefficient end-parameters declarations RN = 1..N ! Set of lifting points DIM = 1..3 ! Set of dimensions P: array(RN,DIM) of real ! Contact point GRAD_NORM: array(RN) of real ! Auxiliary term V: array(RN,DIM) of real ! Unit normal vector at contact point f_ext: array(DIM) of real ! Externally applied force torq_ext: array(DIM) of real ! Externally applied torque force: array(RN,DIM) of mpvar ! Contact force at point nforce: array(RN) of mpvar ! Normal force at point munforce: array(RN) of mpvar ! Aux. var for SOCP reformulation tforce: array(RN,DIM) of mpvar ! Tangential force at point torq: array(RN,DIM) of mpvar ! Torque at point pressure: mpvar ! Objective variable, maximum of nforce Friction: array(RN) of nlctr ! Friction relation end-declarations ! Defining bounds and start values pressure is_free forall (d in DIM, i in RN) do force(i,d) <= 10 tforce(i,d) <= 2 torq(i,d) <= 10 force(i,d) >= -10 tforce(i,d) >= -10 torq(i,d) >= -10 setinitval(force(i,d), 1.0) end-do forall (i in RN) nforce(i) >= 0 f_ext :: [0.0, 0.0, -1.0] forall(d in DIM) torq_ext(d) := 0.0 ! Calculate parameters forall(i in RN) do ! P(i) is a contact point on a parabolic "nose cone" to be lifted P(i,1) := 0.3 + cos(2*M_PI*i/N) P(i,2) := sin(2*M_PI*i/N) P(i,3) := P(i,1)^2 + P(i,2)^2 GRAD_NORM(i) := sqrt( (2*P(i,1))^2 + (2*P(i,2))^2 + 1 ) ! V(i) is the unit normal vector at contact point P(i) V(i,1) := -2*P(i,1)/GRAD_NORM(i) V(i,2) := -2*P(i,2)/GRAD_NORM(i) V(i,3) := 1/GRAD_NORM(i) end-do ! Constraints: forall(i in RN) do ! Normal force at point P(i) nfDef(i):= nforce(i) = sum(d in DIM) V(i,d)*force(i,d) ! Tangential force at point P(i) forall(d in DIM) tfDef(i,d):= tforce(i,d) = force(i,d) - V(i,d)*nforce(i) ! Torq about (0,0,0) at point P(i) torq1Def(i):= torq(i,1) = P(i,2)*force(i,3) - force(i,2)*P(i,3) torq2Def(i):= torq(i,2) = P(i,3)*force(i,1) - force(i,3)*P(i,1) torq3Def(i):= torq(i,3) = P(i,1)*force(i,2) - force(i,1)*P(i,2) ! Objective function definition t_bnds(i) := nforce(i) <= pressure ! Definition of friction munforce(i)=MU*nforce(i) Friction(i):= sum(d in DIM) tforce(i,d)^2 <= munforce(i)^2 end-do ! Force balances forall(d in DIM) f_Balance(d) := sum(i in RN) force(i,d) = -f_ext(d) forall(d in DIM) t_Balance(d) := sum(i in RN) torq(i,d) = -torq_ext(d) ! Solving the problem setparam("xnlp_verbose", true) minimize(pressure) ! Solution display setparam("REALFMT", "%7.4f") forall(i in RN) do write("force(",i,") = ") forall(d in DIM) write(getsol(force(i,d)), " ") write("\ntorque(",i,") = ") forall(d in DIM) write(getsol(torq(i,d)), " ") writeln("\nnormal force(",i,") = ", getsol(nforce(i))) write("tangential force(",i,")= ") forall(d in DIM) write(getsol(tforce(i,d)), " ") writeln end-do writeln("\n Pressure = ", getsol(pressure)); end-model | |||||||||||||
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