**2.2 Over-constrained mechanisms**

Formula for a quick calculation of mobility is an explicit relationship between structural parameters of the mechanism: the number of links and joints, the motion/constraint parameters of the joints and of the mechanism. Usually, these structural parameters are easily determined by inspection without a need to develop a set of kinematic constraint equations. However, not all known formulas for a quick calculation of mobility fit for many classical mechanisms and in particular parallel robotic manipulators (Ionescu, 2003). Special geometric conditions play a significant role in the determination of mobility of such mechanisms, which are called paradoxical mechanisms, or overconstrained, yet mobile linkage (Waldron and Kinzel, 1999). However, as mentioned above, there are overconstrained mechanisms that have full range mobility and therefore they are mechanisms even though they should be considered as rigid structures according to the mobility criterion (i.e. the mobility *M* < 1 as calculated from Equation 1. The mobility of such mechanisms is due to the existence of a particular set of geometric conditions between the mechanism joint axes that are called overconstraint conditions.

Overconstrained mechanisms have many appealing characteristics. Most of them are spatial mechanisms whose spatial kinematic characteristics make them good candidates in modern linkage designs where spatial motion is needed. Another advantage of overconstrained mechanisms is that they are mobile using fewer links and joints than it is expected.

In fact, the planner mechanisms in Figures 1 and 2 can also be viewed as overconstrained spatial mechanisms, and thus the spatial version of Kutzbach-Gruebler's equation (Equation 1), does not work for some of these planner mechanisms. In particular, for the parallel and hybrid kinematic planner mechanisms, Equation 1 will result in negative mobility values suggesting that these mechanisms are rigid structures, although they are not. Since this is

Fig. 2. Schematics of a 2-degrees-of-freedom planner parallel robotic structure, *L*=7, *j*=8, *fi*

It should be noted here that the planner mechanisms are realized by necessitating that the involved revolute joints to be perpendicular to the plane and the prismatic joints to be confined to stay in the plane. As such these mechanisms can also be viewed as special cases of spatial mechanisms that are confined to work in a plane through overconstrains and thus Equation 1, with proper modification, rather than Equation 2 could be use, as discussed in

Formula for a quick calculation of mobility is an explicit relationship between structural parameters of the mechanism: the number of links and joints, the motion/constraint parameters of the joints and of the mechanism. Usually, these structural parameters are easily determined by inspection without a need to develop a set of kinematic constraint equations. However, not all known formulas for a quick calculation of mobility fit for many classical mechanisms and in particular parallel robotic manipulators (Ionescu, 2003). Special geometric conditions play a significant role in the determination of mobility of such mechanisms, which are called paradoxical mechanisms, or overconstrained, yet mobile linkage (Waldron and Kinzel, 1999). However, as mentioned above, there are overconstrained mechanisms that have full range mobility and therefore they are mechanisms even though they should be considered as rigid structures according to the mobility criterion (i.e. the mobility *M* < 1 as calculated from Equation 1. The mobility of such mechanisms is due to the existence of a particular set of geometric conditions between the

Overconstrained mechanisms have many appealing characteristics. Most of them are spatial mechanisms whose spatial kinematic characteristics make them good candidates in modern linkage designs where spatial motion is needed. Another advantage of overconstrained mechanisms is that they are mobile using fewer links and joints than it is

In fact, the planner mechanisms in Figures 1 and 2 can also be viewed as overconstrained spatial mechanisms, and thus the spatial version of Kutzbach-Gruebler's equation (Equation 1), does not work for some of these planner mechanisms. In particular, for the parallel and hybrid kinematic planner mechanisms, Equation 1 will result in negative mobility values suggesting that these mechanisms are rigid structures, although they are not. Since this is

=8. The prismatic joint of the middle leg is passive (unactuated)

mechanism joint axes that are called overconstraint conditions.

the next section,

expected.

**2.2 Over-constrained mechanisms** 

not true, it should be concluded that Equation 1 cannot be used for these over-constraint mechanisms (Mavroidis and Roth, 1995). The overconstraint in planner parallel and hybrid kinematic mechanisms is due to the geometrical requirement on the involved joint-axes in relation to each other. To solve the problem when using the spatial version of the Kutzbach-Gruebler's equation for planner mechanisms or for over-constraint mechanisms in general, Equation 1 has to be modified by adding a parameter reflecting the number of overconstaints existing in the mechanism (Cretu, 2007). The resulting equation is called the universal Somo-Malushev's mobility equation. For the case of mechanisms that do not involve any passive degrees of freedom it is written as

$$M = 6\left(L - 1 - j\right) + \sum\_{i=1}^{j} f\_i + s \tag{3}$$

where *s* is the number of overconstraint (geometrical) conditions. For example, the parallel kinematic mechanism in Figure 1.b has *L* = 8, *j*= 9, and *fi* = 9. Using these parameters in Equation 1 gives *M* = -3. However using Equation (3) and observing that there are 6 overconstraints in this mechanism, the mobility will amount to *M* = 3. The overconstraints in this mechanism are due to the necessity for confining the axes of the three prismatic joints to form a plane or parallel planes (two overcosntraints), and for the axes of the three revolute joints of the moving platform (two overconstraints), and the three revolute joints of the base to be perpendicular to the plane formed by the prismatic joints.
