Zahariev and Cuadrado [8] presented an approach to solve both kinematics and dynamics of over-constrained mecha- nisms。 Their main idea is to replace the geometrical constraint

by the force constraint。 As such, the over-constraint (which is indeed in the sense of geometrical constraint) disappears,   and

is handled at the force and acceleration level。 Their   approach 3

falls into the category of the first situation as classified in the above with consideration of link flexibility。 One shortcoming with this approach is the need for a transformation process from the geometrical constraint to the force constraint, and when the number of links of mechanism is large, say over 10,

the transformation process can be lengthy and prone to   error。 A

Besides, this approach seems to be difficult to make    system- F

atic。

Zhao and Huang [9] proposed an approach to solving the dynamic force analysis problem (the first situation with consid- eration of the link flexibility) with the idea to provide a sup- plemental equation or equations (depending on the degree of over-constraints in the mechanisms)。 However, their approach is preliminary with an ad-hoc process, and thus they restricted their approach to the mechanism which has one degree of free- dom only。 This also implies that the approach cannot be made systematic to over-constrained mechanisms with more than one degree of freedom。 Most recently, Jiang [10] extended the approach of Zhao and Huang to a more systematic one in the area of how to generate the supplemental equation。 With this extension, the dynamic force analysis of over-constrained mechanisms can be made more systematic, and for this reason they demonstrated the effectiveness of their approach to an over-constrained parallel robotic press machine。 The impor- tant shortcoming of both the studies of Zhao and Huang and Jiang is that they only consider the axial deformation, while the link may be subjected to transversal load and thus there is a bending deformation in the link。

We present a new approach to include the bending deforma- tion。 Based on the deformation compatibility analysis of a typical over-constrained mechanism, a general procedure to generate the supplemental equation is developed for the prob- lem of the dynamic force analysis of planar over-constrained mechanisms or robots。 Combined with  the  equations gained by the D’Alembert principle, a dynamic model of an over- constrained mechanism is established。 A more complex press machine is used as a case for illustrating our approach and testing the accuracy of the calculation of the joint force and moment。 The choice of this relatively complex  example is with the intention to enhance the validation of the proposed approach。 The experiment shows that the approach is valid and the accuracy is acceptable。 Several conclusions on the design of this press machine are also drawn to help future improvement of the design of the machine。

2。 The approach of dynamic modeling

Let us consider a typical over-constrained mechanism in Fig。 1。 This mechanism has over-constraint introduced by the link  2 at the point A。 The link 4 (slider) receives an input force (F), and the  link  3 (slider)  is the  end-effector,  where  there  is  a

Fig。 1。 Typical example of the over-constrained mechanism。

Fig。 2。 Deformation compatibility for one member。

condition lAB = lBC = lBD。 Kinematic analysis can be done by removing the link 2, though for the dynamic force analysis, one cannot ignore the link 2。 In the following, we assume that the motions of all the links in the mechanism are known。

For the dynamic force analysis of the mechanism, the num- ber of unknown force and moments is twelve (two forces in each pivot joint and one moment and one force in the each sliding joint) plus one driving force (F)。 The number of equa- tions is twelve (each link has three equations for force and moment equilibrium)。 In the following, the notion of the sup- plemental equation based on the deformation compatibility analysis is introduced to make the number of equations equal the number of unknown variables。