Force Analysis of Lower-mobility Parallel Mechanisms with Over-constrained Couples

Lower-mobility parallel mechanisms often have over-constraints,and then statically indeterminate problem arises in force analysis.When the statically indeterminate mechanism is analyzed,not only force and moment equilibrium equations of all the bodies,but also corresponding complementary equations should be built to solve all the unknowns.4-UPU parallel mechanism which has over-constrained couples and whose statically indeterminate number is two is taken as an example.Its degree of freedom and inverse displacement analysis are obtained.When 4-UPU mechanism is in a certain pose,its constraint reactions of kinematic pairs and active forces,which are equilibrated with the existent external forces acting on the mechanism and the inertia force of every component,are solved by building force and moment equilibrium equations of the bodies and the complementary equations.A set of numerical results including active forces and constraint reactions of main pairs is given.Let the kinematic pairs connecting the moving platform and the limbs as main kinematic pairs.Firstly,active forces and constraint reactions of main pairs are determined by some force and moment equilibrium equations and complementary equations.Then,the equilibrium of each body is considered from up to down one by one,and constraint reactions of other pairs are solved by the residual force and moment equilibrium equations.That is this method can realize partial force decoupling in force analysis.Over-constraints can be divided into over-constrained forces and over-constrained couples.This principle is generalized and can be applied to other statically indeterminate parallel mechanisms including those with over-constrained forces.