**1. Introduction**

24 Will-be-set-by-IN-TECH

108 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

[10] Deblaise, D., Hernot, X. & Maurine, P. (2006). A Systematic Analytical Method for PKM Stiffness Matrix Calculation, In: *Proceedings of the 2006 IEEE International Conference on*

[11] Gonçalves, R. S. & Carvalho, J. C. M. (2008). Stiffness analysis of parallel manipulator using matrix structural analysis, In: *EUCOMES 2008, 2-nd European Conference on*

[12] Wittbrodt, E., Adamiec-Wójcik, I. & Wojciech, S. (2006). *Dynamics of Flexible Multibody*

[13] Gosselin, C.M. (1990). Stiffness mapping for parallel manipulator, *IEEE Trans. on Robotics*

[14] Quennouelle, C. & Gosselin, C.M. (2008). Kinemato-Static Modelling of Compliant Parallel Mechanisms: Application to a 3-PRRR Mechanism, the Tripteron, In: *Proceedings of the Second International Workshop on Fundamental Issues and Future Research Directions*

[16] Yoon, W.K., Suehiro, T., Tsumaki, Y. & Uchiyama, M. (2004). Stiffness analysis and design of a compact modified Delta parallel mechanism, *Robotica*, 22(5), pp. 463-475. [17] Majou, F., Gosselin, C.M., Wenger, P. & Chablat, D. (2004). Parametric stiffness analysis of the orthoglide, In: *Proceedings of the 35th International Symposium on Robotics*, March,

[18] El-Khasawneh, B.S. & Ferreira, P.M. (1999). Computation of stiffness and stiffness bounds for parallel link manipulator, *Int. J. Machine Tools and manufacture*, vol. 39(2),

[19] Gosselin, C.M. & Zhang, D. (2002). Stiffness analysis of parallel mechanisms using a

[20] Pashkevich, A., Chablat, D. & Wenger, P. (2009). Stiffness analysis of overconstrained

[21] Shabana, A.A. (2008). *Computational continuum mechanics*, Cambridge University Press,

[22] Cammarata, A. & Angeles, J. (2011). The elastodynamics model of the McGill Schönflies Motion Generator, In: *Multibody Dynamics 2011*, 4-7 July, Bruxelles, Belgium. [23] Angeles, J. (2007). *Fundamentals of Robotic Mechanical Systems: Theory, Methods and*

[24] Salgado, O., Altuzarra, O., Petuya, V. & Hernández, A. (2008). Synthesis and design of a novel 3T1R fullyparallel manipulator, *ASME Journal of Mechanical Design*, Vol. 130, Issue

[25] Merlet, J.-P. (2006). *Parallel robots, Second Edition*, Kluwer Academic Publisher, ISBN-10

parallel manipulators, *Mechanism and Machine Theory*, Vol. 44, pp. 966-982.

lumped model, *Int. J. of Robotics and Automation*, vol. 17, pp 17-27.

*Algorithms, Third edition*, Springer, ISBN 0-387-29412-0, New York.

*for Parallel Mechanisms and Manipulators*, September 21-22, Montpellier, France. [15] Briot, S., Pashkevich, A. & Chablat, D. (2009). On the optimal design of parallel robots taking into account their deformations and natural frequencies, In: *Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE*, August 30 - September 2, San Diego,

*Robotics and Automation*, May, Orlando, Florida.

*Mechanism Science*, Cassino, Italy.

*and Automation*, vol. 6, pp 377-382.

*Systems*, Springer.

California, USA.

Paris, France.

pp 321-342.

4, pp. 1-8.

ISBN 978-0-521-88569-0, USA.

1-4020-4132-2(HB), the Netherlands.

After their inception in the past two decades as possible alternatives to conventional Computer Numerical Controlled (CNC) machine tools structures that dominantly adapt serial structures, Parallel Kinematic Machines (PKM) were anticipated to form a basis for a new generation of future machining centers. However this hope quickly faded out as most problems associated with this type of structures still persist and could not be completely solved satisfactorily. This especially becomes more apparent in machining applications where accuracy, rigidity, dexterity and large workspace are important requirements. Although the PKMs possess superior mechanical characteristics to serial structures, particularly in terms of high rigidity, accuracy and dynamic response, however the PKMs have their own drawbacks including singularity problems, inconsistent dexterity, irregular workspace, and limited range of motion, particularly rotational motion.

To alleviate the PKMs' limitations, considerable research efforts were directed to solve these problems. Optimum design methods are among the various methods that are attempted to improve the dexterity as well as to maximize the workspace (Stoughton and Arai, 1993, Huang et al., 2000). Various methods to evaluate the workspace were suggested (Gosselin, 1990; Luh et al., 1996; Conti et al., 1998; Tsai et al., 2006). Workspace optimization is also addressed (Wang and Hsieh, 1998). A new shift in tackling the aforementioned problems came when researchers start to look at hybrid structures, consisting of parallel and serial linkages as a compromise to exploit the advantageous characteristics of the serial and parallel structures. This shift creates new research and development needs and founded new ideas.

Among the early hybrid kinematic designs, the Tricept was considered as the first commercially successful hybrid machine tools. This hybrid machine which was developed by Neos Robotics, has a three-degrees-of-freedom parallel kinematic structure and a

Parallel, Serial and Hybrid Machine Tools and

structures with open kinematics chains.

**2.1 Mobility of planner mechanisms** 

equation (Gogu , 2005, Norton, 2004).

Robotics Structures: Comparative Study on Optimum Kinematic Designs 111

where *fi* is the DOF associated with joint *i*. Equation 1 is used to calculate the mobility of spatial robotic mechanisms as most industrial robots and machine tools structures are serial

To gain an insight into the effect of mobility on the kinematic analysis and design of serial, parallel and hybrid kinematic structures, we will also look at the mobility of planner mechanisms, which can be obtained from the following planner Kutzbach-Gruebler's

where *M*, *L*, *j* and *fi* are as defined before in Equation 1. As shown in Figure 1, the robotic structures are arranged in serial, parallel and hybrid kinematic chains, and thus have different number of links and joints. Using Equation 2, all the three structures in Figure 1 have three degrees of freedom, or mobility three. This gives the end-effector two translational degrees-of-freedom to position it arbitrarily in the x-y plane, and one rotational degree-of-freedom to orient it about the z-axis. In the serial kinematic structure all three joints are actuated, whereas for the parallel and hybrid structures only the three prismatic joints are actuated whereas the revolute joints are passive. The parallel kinematic part of the hybrid structure in Figure 1.c, has two degrees of freedom, which is achieved by reducing

Figure 2 shows an alternative way to reduce the degrees of freedom of the parallel kinematic mechanism, and hence to reduce the number of actuated prismatic joints. In this example this is done by eliminating one of the revolute joints which connect the legs to the platform. The corresponding leg has a passive prismatic joint to constraint one of the degrees-offreedom. By removing the revolute joint though the leg becomes a three-force member and hence it will be carrying bending moments necessitating considerable design attention to maintain desired stiffness levels and accuracy. This concept of reducing the degrees of freedom is adopted in the spatial Tricept mechanism to reduce the degrees-of-freedom from six to three. Compared to the mechanism in 1.c, this mechanism has more joints and links,

Fig. 1. Schematics of planner 3 degrees-of-freedom robotic structures with a) fully serial (*L*=4, *j*=3, *fi* =3), b) fully parallel (*L*=8, *j*=9, *fi* =9), and c) hybrid topologies (*L*=6, *j*=6, *fi* =6)

(a) (b) (c)

*M Lj f*

3 1

the number of legs to two and eliminating one of the passive revolute joints.

which is not desirable from the design point of view.

1

(2)

*j i i*

standard two-degrees-of-freedom wrest end-effector holding joint. The constraining passive leg of the machine has to bear the transmitted torque and moment between the moving platform and the base (Zhang and Gosselin, 2002). Recently the Exechon machine is introduced as an improvement over the Tricept design. The Exechon adopts a unique overconstrained structure, and it has been improved based on the success of the Tricept (Zoppi, et al., 2010, Bi and Jin, 2011). Nonetheless, regardless of the seemingly promising prospect of the hybrid kinematic structures, comprehensive study and understanding of the involved kinematics, dynamics and design of these structures are lacking. This paper is attempting to provide a comparative study and a formulation for the kinematic design of hybrid kinematic machines. The remainder of this paper is as follows: Section 2 provides a discussion on the mobility of serial, parallel and hybrid kinematic structures and the involved effects of overconstrain on the mobility of the mechanism. Section 3 provides a discussion on kinematic design for hybrid machines and the implication of the presented method. Concluding remarks are presented in Section 4.

## **2. Mobility of robotic structures**

Mobility is a significant structural attribute of mechanisms assembled from a number of links and joints. It is also one of the most fundamental concepts in the kinematic and the dynamic modeling of mechanisms and robotic manipulators. IFToMM defines the mobility or the degree of freedom as the number of independent co-ordinates needed to define the configuration of a kinematic chain or mechanism (Gogu, 2005, Ionescu, 2003). Mobility, M, is used to verify the existence of a mechanism (M > 0), to indicate the number of independent parameters in the kinematic and the dynamic models and to determine the number of inputs needed to drive the mechanism.

The various methods proposed in the literature for mobility calculation of the closed loop mechanisms can be grouped in two categories (Ionescu, 2003): (a) approaches for mobility calculation based on setting up the kinematic constraint equations and their rank calculation for a given position of the mechanism with specific joint location, and (b) formulas for a quick calculation of mobility without need to develop the set of constraint equations. The approaches for mobility calculation based on setting up the kinematic constraint equations and their rank calculation are valid without exception. The major drawback of these approaches is that the mobility cannot be determined quickly without setting up the kinematic model of the mechanism. Usually this model is expressed by the closure equations that must be analyzed for dependency. There is no way to derive information about mechanism mobility without performing kinematic analysis by using analytical tools. For this reason, the real and practical value of these approaches is very limited in spite of their valuable theoretical foundations.

Many formulas based on approach (b) above have been proposed in the literature for the calculation of mechanisms' mobility. Many of these methods are reducible to the Cebychev-Grubler-Kutzbach's mobility formula given by Equation 1 below (Gogu, 2005). Using this formula, the mobility M of a linkage composed of L links connected with j joints can be determined from the following equation.

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

where *fi* is the DOF associated with joint *i*. Equation 1 is used to calculate the mobility of spatial robotic mechanisms as most industrial robots and machine tools structures are serial structures with open kinematics chains.

#### **2.1 Mobility of planner mechanisms**

110 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

standard two-degrees-of-freedom wrest end-effector holding joint. The constraining passive leg of the machine has to bear the transmitted torque and moment between the moving platform and the base (Zhang and Gosselin, 2002). Recently the Exechon machine is introduced as an improvement over the Tricept design. The Exechon adopts a unique overconstrained structure, and it has been improved based on the success of the Tricept (Zoppi, et al., 2010, Bi and Jin, 2011). Nonetheless, regardless of the seemingly promising prospect of the hybrid kinematic structures, comprehensive study and understanding of the involved kinematics, dynamics and design of these structures are lacking. This paper is attempting to provide a comparative study and a formulation for the kinematic design of hybrid kinematic machines. The remainder of this paper is as follows: Section 2 provides a discussion on the mobility of serial, parallel and hybrid kinematic structures and the involved effects of overconstrain on the mobility of the mechanism. Section 3 provides a discussion on kinematic design for hybrid machines and the implication of the presented

Mobility is a significant structural attribute of mechanisms assembled from a number of links and joints. It is also one of the most fundamental concepts in the kinematic and the dynamic modeling of mechanisms and robotic manipulators. IFToMM defines the mobility or the degree of freedom as the number of independent co-ordinates needed to define the configuration of a kinematic chain or mechanism (Gogu, 2005, Ionescu, 2003). Mobility, M, is used to verify the existence of a mechanism (M > 0), to indicate the number of independent parameters in the kinematic and the dynamic models and to determine the number of

The various methods proposed in the literature for mobility calculation of the closed loop mechanisms can be grouped in two categories (Ionescu, 2003): (a) approaches for mobility calculation based on setting up the kinematic constraint equations and their rank calculation for a given position of the mechanism with specific joint location, and (b) formulas for a quick calculation of mobility without need to develop the set of constraint equations. The approaches for mobility calculation based on setting up the kinematic constraint equations and their rank calculation are valid without exception. The major drawback of these approaches is that the mobility cannot be determined quickly without setting up the kinematic model of the mechanism. Usually this model is expressed by the closure equations that must be analyzed for dependency. There is no way to derive information about mechanism mobility without performing kinematic analysis by using analytical tools. For this reason, the real and practical value of these approaches is very limited in spite of

Many formulas based on approach (b) above have been proposed in the literature for the calculation of mechanisms' mobility. Many of these methods are reducible to the Cebychev-Grubler-Kutzbach's mobility formula given by Equation 1 below (Gogu, 2005). Using this formula, the mobility M of a linkage composed of L links connected with j joints can be

*M Lj f*

6 1

1

(1)

*j i i*

method. Concluding remarks are presented in Section 4.

**2. Mobility of robotic structures** 

inputs needed to drive the mechanism.

their valuable theoretical foundations.

determined from the following equation.

To gain an insight into the effect of mobility on the kinematic analysis and design of serial, parallel and hybrid kinematic structures, we will also look at the mobility of planner mechanisms, which can be obtained from the following planner Kutzbach-Gruebler's equation (Gogu , 2005, Norton, 2004).

$$M = \Re(L - 1 - j) + \sum\_{i=1}^{j} f\_i \tag{2}$$

where *M*, *L*, *j* and *fi* are as defined before in Equation 1. As shown in Figure 1, the robotic structures are arranged in serial, parallel and hybrid kinematic chains, and thus have different number of links and joints. Using Equation 2, all the three structures in Figure 1 have three degrees of freedom, or mobility three. This gives the end-effector two translational degrees-of-freedom to position it arbitrarily in the x-y plane, and one rotational degree-of-freedom to orient it about the z-axis. In the serial kinematic structure all three joints are actuated, whereas for the parallel and hybrid structures only the three prismatic joints are actuated whereas the revolute joints are passive. The parallel kinematic part of the hybrid structure in Figure 1.c, has two degrees of freedom, which is achieved by reducing the number of legs to two and eliminating one of the passive revolute joints.

Figure 2 shows an alternative way to reduce the degrees of freedom of the parallel kinematic mechanism, and hence to reduce the number of actuated prismatic joints. In this example this is done by eliminating one of the revolute joints which connect the legs to the platform. The corresponding leg has a passive prismatic joint to constraint one of the degrees-offreedom. By removing the revolute joint though the leg becomes a three-force member and hence it will be carrying bending moments necessitating considerable design attention to maintain desired stiffness levels and accuracy. This concept of reducing the degrees of freedom is adopted in the spatial Tricept mechanism to reduce the degrees-of-freedom from six to three. Compared to the mechanism in 1.c, this mechanism has more joints and links, which is not desirable from the design point of view.

Fig. 1. Schematics of planner 3 degrees-of-freedom robotic structures with a) fully serial (*L*=4, *j*=3, *fi* =3), b) fully parallel (*L*=8, *j*=9, *fi* =9), and c) hybrid topologies (*L*=6, *j*=6, *fi* =6)

Parallel, Serial and Hybrid Machine Tools and

joints.

effector.

2002).

involve any passive degrees of freedom it is written as

**3. Kinematic designs of robotic structures** 

Robotics Structures: Comparative Study on Optimum Kinematic Designs 113

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

*M L j fs*

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

A widely used kinematic design strategy for serial kinematic robotic structures to optimize the workspace is to use the first group of links and joints to position the end-effector and the remaining links and joints to orient the end-effector, and thus breaking the design problem into two main tasks. For the 6-DOF Puma robot schematic shown in Figure 3, the first three links and joint are responsible for positioning the end-effector at the desired position, while the last three joints and links form a 3-DOF concurrent wrest joint that orient the end-

Conventional five axis machining centers achieve similar decoupling by splitting the five axes (three translational axes and two rotational axes) into two groups of axes. One group of serially connected axes is responsible for positioning/orienting the worktable which is holding the workpiece, while the other group of axes moves/orients the spindle (Bohez,

Unfortunately this strategy cannot be adopted for parallel kinematic structures due to the similarity of the legs and their way of working in parallel. As such decoupling of the two functions (positioning and orienting the end-effector) is not straightforward to do for parallel kinematic structures if not impossible. Partial decoupling has been attempted by

On the other hand, it should be noted here that parallel structures, and to some extent hybrid structures, can be built from identical parts and modules, and thus lend themselves well to adaptation as reconfigurable machines (Zhang, 2006). This attribute is not strongly relevant to serial structures which consist of axes that are stacked on each other making the

Harib and Sharif Ullah (2008) using the axiomatic design approach.

links and joints differ considerably in terms of size and shape.

6 1

1

(3)

*j i i*

Fig. 2. Schematics of a 2-degrees-of-freedom planner parallel robotic structure, *L*=7, *j*=8, *fi* =8. The prismatic joint of the middle leg is passive (unactuated)

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 the next section,
