**3. Kinematic designs of robotic structures**

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 endeffector.

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, 2002).

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

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

Parallel, Serial and Hybrid Machine Tools and

*b*1

Robotics Structures: Comparative Study on Optimum Kinematic Designs 115

*b b*<sup>3</sup> <sup>2</sup>

*l*<sup>6</sup> *l*<sup>5</sup>

*l*<sup>2</sup> *l*<sup>3</sup>

*a*<sup>2</sup> *a*<sup>3</sup>

where **F** and **T** are respectively the resultants 3-D external force and torque systems applied to the movable platform. This result suggests that to support external force and torque along arbitrary directions, **J**1 and **J**2 must both have a rank three. Now to support these external force and torque resultants using bounded joint space forces, the condition numbers of **J**<sup>1</sup>

where *w* is a weighing factor in the range 0 1 which signify how much emphases is given to translational and rotational dexterities, and *PM*1 and *PM*2 are respectively performance measures for the translational motion and the rotational motion of the structure, and are

> 1 1 *V PM V dV*

> 2 2 *V PM V dV*

which is a subset of the total reachable space of the mechanism *V*. PM will then be in the

An overall local performance measure *PM* can be obtained from the following relation

*a*5 *a*6

*zC*

*a*1

*yM*

defined as (Harib and Sharif Ullah, 2008, Stoughton and Arai, 1993).

*zM*

Fig. 4. Typical Construction of a hexapodic machine tools.

and **J**2 must be both as close to unity as possible.

In the previous equations,

*xM*

*xC*

*l*1

*b*6

*b*5

*b*4

*a*4

*yC*

*l*4

1 2 *PM w PM w PM* (1 ) (8)

**<sup>J</sup>** (9)

**<sup>J</sup>** (10)

is the condition number function and *V* is the workspace

Fig. 3. A schematic of a 6-DOF Puma robot (serial kinematic structure)

#### **3.1 Parallel kinematic designs**

A main objective of the optimal design of parallel kinematic machines is to maintain consistent dexterity within the workable space of the machine. Dexterity of the mechanism is a measure of its ability to change its position and orientation arbitrarily, or to apply forces and torques in arbitrary directions. As such the Jacobian matrix of the mechanism is widely used in formulating the dexterity measure. For a six degrees-of-freedom hexapod mechanism (Harib and Srinivasan, 2003), shown in Figure 4, the Jacobian matrix J relates the translational and rotational velocity vectors of the moving platform to the extension rate of the legs as indicated below (Harib and Sharif Ullah, 2008).

$$\mathbf{J}\left[\dot{l}\_1\cdots\dot{l}\_6\right]^T = \mathbf{J}\_1\dot{\mathbf{c}} + \mathbf{J}\_2\mathbf{c}\mathbf{o} = \mathbf{J}\left[\dot{\mathbf{c}}^T \quad \mathbf{o}\mathbf{o}^T\right]^T\tag{4}$$

where **J** = [**J**<sup>1</sup> **J**2] is the Jacobian matrix of the hexapod, which consists of two 63 submatrices **J**1 and **J**2 that are given as

$$\mathbf{J}\_1 = \begin{bmatrix} \mathbf{u}\_1 \cdots \mathbf{u}\_6 \end{bmatrix}^T \tag{5}$$

$$\mathbf{J}\_2 = \left[ {}^M \mathbf{R}\_{\mathbb{C}} \, {}^{\mathbb{C}} \mathbf{a}\_1 \times \mathbf{u}\_1 \, \cdots \, {}^M \mathbf{R}\_{\mathbb{C}} \, {}^{\mathbb{C}} \mathbf{a}\_6 \times \mathbf{u}\_6 \right]^T \tag{6}$$

where **u** *i* and *C***a** *i* are respectively a unit vector along the *i*th leg and the position vector of its attachment point to the moving platform in the platform coordinate frame C, and M**R** *C* is the rotational matrix of the moving platform. The Jacobian matrix **J** relates also the external task space forces and torques and the joint space forces as indicated below.

$$\begin{bmatrix} \mathbf{F} & \mathbf{T} \end{bmatrix}^T = \mathbf{J}^T \begin{bmatrix} f\_1 \cdots f\_6 \end{bmatrix}^T \tag{7}$$

Fig. 3. A schematic of a 6-DOF Puma robot (serial kinematic structure)

16 1 2

space forces and torques and the joint space forces as indicated below.

the legs as indicated below (Harib and Sharif Ullah, 2008).

A main objective of the optimal design of parallel kinematic machines is to maintain consistent dexterity within the workable space of the machine. Dexterity of the mechanism is a measure of its ability to change its position and orientation arbitrarily, or to apply forces and torques in arbitrary directions. As such the Jacobian matrix of the mechanism is widely used in formulating the dexterity measure. For a six degrees-of-freedom hexapod mechanism (Harib and Srinivasan, 2003), shown in Figure 4, the Jacobian matrix J relates the translational and rotational velocity vectors of the moving platform to the extension rate of

where **J** = [**J**<sup>1</sup> **J**2] is the Jacobian matrix of the hexapod, which consists of two 63 sub-

where **u** *i* and *C***a** *i* are respectively a unit vector along the *i*th leg and the position vector of its attachment point to the moving platform in the platform coordinate frame C, and M**R** *C* is the rotational matrix of the moving platform. The Jacobian matrix **J** relates also the external task

1 6

*<sup>T</sup> <sup>T</sup> T T l l* **Jc J <sup>ω</sup> J c <sup>ω</sup>** (4)

2 11 66 [ ] *MC MC <sup>T</sup> C C* **J Rau Rau** (6)

<sup>116</sup> [ ]*<sup>T</sup>* **Juu** (5)

*T T <sup>T</sup>* **FT J** *f f* (7)

**3.1 Parallel kinematic designs** 

matrices **J**1 and **J**2 that are given as

where **F** and **T** are respectively the resultants 3-D external force and torque systems applied to the movable platform. This result suggests that to support external force and torque along arbitrary directions, **J**1 and **J**2 must both have a rank three. Now to support these external force and torque resultants using bounded joint space forces, the condition numbers of **J**<sup>1</sup> and **J**2 must be both as close to unity as possible.

An overall local performance measure *PM* can be obtained from the following relation

$$PM = \text{w } PM\_1 + (1 - w) \, PM\_2 \tag{8}$$

where *w* is a weighing factor in the range 0 1 which signify how much emphases is given to translational and rotational dexterities, and *PM*1 and *PM*2 are respectively performance measures for the translational motion and the rotational motion of the structure, and are defined as (Harib and Sharif Ullah, 2008, Stoughton and Arai, 1993).

$$PM\_1 = V \Big/ \int\_{V^\*} \kappa \left(\mathbf{J}\_1\right) dV \tag{9}$$

$$PM\_2 = V \left\langle \int\_{V^\*} \kappa \left( \mathbf{J}\_2 \right) dV \right. \tag{10}$$

In the previous equations, is the condition number function and *V* is the workspace which is a subset of the total reachable space of the mechanism *V*. PM will then be in the

Parallel, Serial and Hybrid Machine Tools and

Robotics Structures: Comparative Study on Optimum Kinematic Designs 117

Fig. 5. A 2-DOF Planner PKM System (Harib and Sharif Ullah, 2008)

Fig. 6. A 3-DOF Planner PKM System (Harib and Sharif Ullah, 2008)

range 0 1 , with 1.0 corresponding to the best possible performance, which in turn corresponds to a perfectly conditioned Jacobian matrix.

The workspace of PKMs is another design issue that needs careful attention due to the computational complexity involved. Algorithms proposed in the literature to determine the workspace of PKM structures use the geometric constraints of the structures, including maximum/minimum leg lengths, passive joint limits. The complexity of these computational methods varies depending on the constraints imposed. For example if the cross sectional variation hexapod legs is also considered as a factor to avoid leg collisions considerable computational requirement will be necessary (Conti et. al, 1998). If the considered design would ensure that the operation of the machine is far enough from possibility of leg collisions in the first place considerable design efforts could be saved.

Harib and Sharif Ullah (2008) used the axiomatic design methodology (Suh, 1990) to analyze the kinematic design of PKM structures. In terms of the kinematic functions of PKM structures and based on the aforementioned contemplation, the following basic Functional Requirements (FRs) were identified: (1) The mechanism should be able to support arbitrary 3-D system of forces i.e. PM1 should be as close to unity as possible. (2) The mechanism should be able to support arbitrary 3-D systems of torques i.e. PM2 should be as close to unity as possible. (3) The mechanism should be able to move the cutting tool through a desired workspace. (4) The mechanism should be able to orient the spindle at a desired range within the desired workspace. On the other hand, to achieve the FRs the following two Design Parameters (DPs) are often used: (1) Determine the lengths and strokes of the legs. (2) Determine the orientation of the legs relative to the fixed base and to the moving platform in the home position. From the perspective of AD this implies that the kinematic design of hexapodic machine tools is sort of coupled design. Therefore, gradual decomposition of FRs and DPs are needed to make the system consistent with the AD.

Figure 5 shows a 2-DOF planar parallel kinematics structure. The structure includes two extendable legs with controllable leg lengths *l*1 and *l*2 and three revolute joints *a*1, *a*2, and *c*. The controlled extension of the two legs places the end-effectors point c at an arbitrary position (x, y) in the x-y plane.

The way the function requirements are fulfilled is this design is by assembling the mechanism such that the two legs are orthogonal to each other at the central position of the workspace as shown in Figure 5. This result is coherent with the isotropic configuration that could be obtained for this mechanism (Huang et al., 2004). Away from that position the mechanism is not expected to deviate much from this condition for practical configuration if the limits of the leg lengths are appropriately selected. It is clear that arbitrary strokes and average lengths of the two legs can be selected while maintaining leg orthogonally condition by adjusting the position of *b*1 and *b*2.

The reachable space of the 2-DOF PKM of Figure 5 is bounded by four circular arc segments with radii *l1-max*, *l1-min*, *l2-max* and *l2-min* and centers *b*1 and *b*2. With the two legs normal to each other the workspace can be modified along any of the two orthogonal directions independent from the other.

An extension of the previous design method to three DOF planner PKM structures is shown in Figure 6. Selecting the reference point of the mechanism to be the concurrent attachment point of the two orthogonal legs serves the purpose of showing the validity of the previously established result of uncoupled design in terms of the previously defined FRs and DPs. As indicated on Figure 6, with this choice of reference point, the same workspace of the 2-DOF structure is obtained.

range 0 1 , with 1.0 corresponding to the best possible performance, which in turn

The workspace of PKMs is another design issue that needs careful attention due to the computational complexity involved. Algorithms proposed in the literature to determine the workspace of PKM structures use the geometric constraints of the structures, including maximum/minimum leg lengths, passive joint limits. The complexity of these computational methods varies depending on the constraints imposed. For example if the cross sectional variation hexapod legs is also considered as a factor to avoid leg collisions considerable computational requirement will be necessary (Conti et. al, 1998). If the considered design would ensure that the operation of the machine is far enough from possibility of leg collisions in the first place considerable design efforts could be saved. Harib and Sharif Ullah (2008) used the axiomatic design methodology (Suh, 1990) to analyze the kinematic design of PKM structures. In terms of the kinematic functions of PKM structures and based on the aforementioned contemplation, the following basic Functional Requirements (FRs) were identified: (1) The mechanism should be able to support arbitrary 3-D system of forces i.e. PM1 should be as close to unity as possible. (2) The mechanism should be able to support arbitrary 3-D systems of torques i.e. PM2 should be as close to unity as possible. (3) The mechanism should be able to move the cutting tool through a desired workspace. (4) The mechanism should be able to orient the spindle at a desired range within the desired workspace. On the other hand, to achieve the FRs the following two Design Parameters (DPs) are often used: (1) Determine the lengths and strokes of the legs. (2) Determine the orientation of the legs relative to the fixed base and to the moving platform in the home position. From the perspective of AD this implies that the kinematic design of hexapodic machine tools is sort of coupled design. Therefore, gradual decomposition of FRs and DPs are needed to make the system consistent with the AD. Figure 5 shows a 2-DOF planar parallel kinematics structure. The structure includes two extendable legs with controllable leg lengths *l*1 and *l*2 and three revolute joints *a*1, *a*2, and *c*. The controlled extension of the two legs places the end-effectors point c at an arbitrary

The way the function requirements are fulfilled is this design is by assembling the mechanism such that the two legs are orthogonal to each other at the central position of the workspace as shown in Figure 5. This result is coherent with the isotropic configuration that could be obtained for this mechanism (Huang et al., 2004). Away from that position the mechanism is not expected to deviate much from this condition for practical configuration if the limits of the leg lengths are appropriately selected. It is clear that arbitrary strokes and average lengths of the two legs can be selected while maintaining leg orthogonally condition

The reachable space of the 2-DOF PKM of Figure 5 is bounded by four circular arc segments with radii *l1-max*, *l1-min*, *l2-max* and *l2-min* and centers *b*1 and *b*2. With the two legs normal to each other the workspace can be modified along any of the two orthogonal directions

An extension of the previous design method to three DOF planner PKM structures is shown in Figure 6. Selecting the reference point of the mechanism to be the concurrent attachment point of the two orthogonal legs serves the purpose of showing the validity of the previously established result of uncoupled design in terms of the previously defined FRs and DPs. As indicated on Figure 6, with this choice of reference point, the same workspace

corresponds to a perfectly conditioned Jacobian matrix.

position (x, y) in the x-y plane.

by adjusting the position of *b*1 and *b*2.

independent from the other.

of the 2-DOF structure is obtained.

Fig. 5. A 2-DOF Planner PKM System (Harib and Sharif Ullah, 2008)

Fig. 6. A 3-DOF Planner PKM System (Harib and Sharif Ullah, 2008)

Parallel, Serial and Hybrid Machine Tools and

along these directions.

schematically in Figure 8.

freedom).

**3.2.2 Alternative hybrid kinematic mechanism** 

**3.2.1 The Exechon mechanism** 

Robotics Structures: Comparative Study on Optimum Kinematic Designs 119

to provide arbitrary motion along three directions and to support associated force system

The Exechon machining center is based on a hybrid five degrees-of-freedom mechanism that consists of parallel and serial kinematic linkages (Zoppi et al., 2010). The parallel kinematic structure of the Exechon is an overconstrained mechanism with eight links and a total of nine joints; three prismatic joints with connectivity one, three revolute joints with connectivity one, and three universal joints with connectivity two. This mechanism is shown

The number of overconstraint (geometrical) conditions *s* is 3. These conditions require that the two prismatic joints *l*1 and *l*2 form a plane, and that the two axes of the joints *a*1 and *a*2 to be perpendicular to this plane, and the axis of joint *a*3 be perpendicular to the axes of joints *a*1 and *a*2. The parameters of the underlying mechanism can be identified as: *L* = 8, *j* = 9, *fi* =12 for all the nine revolute, prismatic and universal joints. The mobility of this mechanism is erroneously calculated by Equation 1 as *M* = 0, which indicates that the mechanism is a structure. Nevertheless, if the geometrical constraints involved in this mechanism are considered and Equation 3 is applied, the mobility is correctly calculated as *M* = 3. These three degrees of freedom obviously correspond to the three actuating linear motors. The overconstraints in this mechanism considerably reduce the required joints, which obviously improves the rigidity of the mechanism. However, the geometric constraints that result in reducing the mobility to three require structural design for the joints to bear the transmitted bending moments and torque components. This requirement is more stringent in the case of the prismatic joints of the three legs. These legs will not be two-force members as in the six

DOF hexapodic mechanism and have to be designed to hold bending moments.

The parallel kinematic part can be viewed as a 2-DOF planner mechanism formed by the two struts *l*1 and *l*2 and the platform, which could be revolved about an axis (the axes of the base joints *b*1 and *b*2, shown as dashed line in Figure 8) via the actuation of the third strut *l*3. To achieve 2-DOF in the planner mechanism, three overconstraints are required. As indicated before these overconstraints come as requirements on the axes of the revolute joints *a*1 and *a*2 to be normal to the plane formed by *l*1 and *l*2, and on the third revolute joint *a*3 to be normal to the other two joints. Thus the projection of this strut onto the plane is constraining the rotational degree-of-freedom of the moving platform in the plane. This situation resembles the 2-DOF planner mechanism of Figure 2. When this projection onto the plane vanishes (i.e. when the angle between the third strut and the plane made by other two struts is 90 degree), the mechanism becomes singular (attains additional degree-of-

In this section we demonstrate employoing the Axiomatic Design to evaluate a potential design of a 5-axes alternative hybrid kinematic machine tools mechanism consisting of a 3- DOF parallel kinematic structure and a 2-DOF wrest joint. Axiomatic design is a structured design methodology which is developed to improve design activities by establishing criteria on which potential designs may be evaluated and enhanced (Suh, 1990). The general function requirements (FRs) for the proposed hybrid mechanism can be listed as follows. The mechanism should 1) provide required positioning and orientation capabilities, 2) have

The previous 3-DOF PKM design of Figure 6 suggests extending the idea to a 6-DOF structure, as shown in Figure 7. The six legs of the suggested structure are arranged such that the idea remains the same (two parallel legs connected by a link and one orthogonal leg) in each of three mutually orthogonal planes. The purpose of the design is to support an arbitrary 6-DOF force and torque system.

Fig. 7. A schematic of a 6-DOF spatial PKM (Harib and Sharif Ullah, 2008)

While the FRs' and DPs' of the axiomatic design methods are difficult to be decoupled here, this design of the 6 DOF mechanism is shown to be a logical extension from planner mechanisms designed with such design methodology.

#### **3.2 Hybrid kinematic designs**

Similar to the serial kinematic robotic design strategy, hybrid kinematic structures could be designed such the first three links and joints, forming the parallel structure, handle the gross positioning of the end-effector. The rest of joints and links could be made to form a concurrent serial kinematic structure that is responsible for orienting the end-effector. Thus this strategy decoupled two main functional requirements (FRs) of the mechanism and their design parameters (DPs). Now, while the serial kinematic part, which is responsible for the orientation of the end-effector, could be a standard wrest joint consisting concurrent revolute joints, the focus could bow be directed on the design of the parallel kinematic part which still requires considerable design attention. The decoupling of the design requirements reduces the design problem to a design of a three-degrees-of-freedom parallel kinematic spatial structure that position the concurrent wrest joint along the x, y and z axes. Although the design requirements on the orientation are not part of the design requirements of the parallel part of the mechanism, ability to support a system of transmitted torque is still part of the design requirements. This is in addition to the requirements of having ability to provide arbitrary motion along three directions and to support associated force system along these directions.

### **3.2.1 The Exechon mechanism**

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

The previous 3-DOF PKM design of Figure 6 suggests extending the idea to a 6-DOF structure, as shown in Figure 7. The six legs of the suggested structure are arranged such that the idea remains the same (two parallel legs connected by a link and one orthogonal leg) in each of three mutually orthogonal planes. The purpose of the design is to support an

*b*<sup>6</sup>

*l*<sup>6</sup>

*a*<sup>6</sup>

*a*<sup>3</sup>

*a*<sup>4</sup>

*b*<sup>5</sup>

*x* 

*a*<sup>5</sup>

*b*3

*b*<sup>4</sup>

*l*<sup>3</sup>

*l*<sup>4</sup>

*z* 

While the FRs' and DPs' of the axiomatic design methods are difficult to be decoupled here, this design of the 6 DOF mechanism is shown to be a logical extension from planner

Similar to the serial kinematic robotic design strategy, hybrid kinematic structures could be designed such the first three links and joints, forming the parallel structure, handle the gross positioning of the end-effector. The rest of joints and links could be made to form a concurrent serial kinematic structure that is responsible for orienting the end-effector. Thus this strategy decoupled two main functional requirements (FRs) of the mechanism and their design parameters (DPs). Now, while the serial kinematic part, which is responsible for the orientation of the end-effector, could be a standard wrest joint consisting concurrent revolute joints, the focus could bow be directed on the design of the parallel kinematic part which still requires considerable design attention. The decoupling of the design requirements reduces the design problem to a design of a three-degrees-of-freedom parallel kinematic spatial structure that position the concurrent wrest joint along the x, y and z axes. Although the design requirements on the orientation are not part of the design requirements of the parallel part of the mechanism, ability to support a system of transmitted torque is still part of the design requirements. This is in addition to the requirements of having ability

*a*<sup>2</sup>

*l*<sup>5</sup>

*a*<sup>1</sup>

*y* 

Fig. 7. A schematic of a 6-DOF spatial PKM (Harib and Sharif Ullah, 2008)

*l*<sup>2</sup>

*b*<sup>2</sup>

mechanisms designed with such design methodology.

**3.2 Hybrid kinematic designs** 

*l*<sup>1</sup>

arbitrary 6-DOF force and torque system.

*b*<sup>1</sup>

The Exechon machining center is based on a hybrid five degrees-of-freedom mechanism that consists of parallel and serial kinematic linkages (Zoppi et al., 2010). The parallel kinematic structure of the Exechon is an overconstrained mechanism with eight links and a total of nine joints; three prismatic joints with connectivity one, three revolute joints with connectivity one, and three universal joints with connectivity two. This mechanism is shown schematically in Figure 8.

The number of overconstraint (geometrical) conditions *s* is 3. These conditions require that the two prismatic joints *l*1 and *l*2 form a plane, and that the two axes of the joints *a*1 and *a*2 to be perpendicular to this plane, and the axis of joint *a*3 be perpendicular to the axes of joints *a*1 and *a*2. The parameters of the underlying mechanism can be identified as: *L* = 8, *j* = 9, *fi* =12 for all the nine revolute, prismatic and universal joints. The mobility of this mechanism is erroneously calculated by Equation 1 as *M* = 0, which indicates that the mechanism is a structure. Nevertheless, if the geometrical constraints involved in this mechanism are considered and Equation 3 is applied, the mobility is correctly calculated as *M* = 3. These three degrees of freedom obviously correspond to the three actuating linear motors. The overconstraints in this mechanism considerably reduce the required joints, which obviously improves the rigidity of the mechanism. However, the geometric constraints that result in reducing the mobility to three require structural design for the joints to bear the transmitted bending moments and torque components. This requirement is more stringent in the case of the prismatic joints of the three legs. These legs will not be two-force members as in the six DOF hexapodic mechanism and have to be designed to hold bending moments.

The parallel kinematic part can be viewed as a 2-DOF planner mechanism formed by the two struts *l*1 and *l*2 and the platform, which could be revolved about an axis (the axes of the base joints *b*1 and *b*2, shown as dashed line in Figure 8) via the actuation of the third strut *l*3. To achieve 2-DOF in the planner mechanism, three overconstraints are required. As indicated before these overconstraints come as requirements on the axes of the revolute joints *a*1 and *a*2 to be normal to the plane formed by *l*1 and *l*2, and on the third revolute joint *a*3 to be normal to the other two joints. Thus the projection of this strut onto the plane is constraining the rotational degree-of-freedom of the moving platform in the plane. This situation resembles the 2-DOF planner mechanism of Figure 2. When this projection onto the plane vanishes (i.e. when the angle between the third strut and the plane made by other two struts is 90 degree), the mechanism becomes singular (attains additional degree-offreedom).

#### **3.2.2 Alternative hybrid kinematic mechanism**

In this section we demonstrate employoing the Axiomatic Design to evaluate a potential design of a 5-axes alternative hybrid kinematic machine tools mechanism consisting of a 3- DOF parallel kinematic structure and a 2-DOF wrest joint. Axiomatic design is a structured design methodology which is developed to improve design activities by establishing criteria on which potential designs may be evaluated and enhanced (Suh, 1990). The general function requirements (FRs) for the proposed hybrid mechanism can be listed as follows. The mechanism should 1) provide required positioning and orientation capabilities, 2) have

Parallel, Serial and Hybrid Machine Tools and

*M* = 3.

Robotics Structures: Comparative Study on Optimum Kinematic Designs 121

two prismatic joints *l*1 and *l*2 form a plane and that the axis of joint *a*2 to be perpendicular to this plane. Calculating the mobility using Equation 1 yields *M* = 1. However considering the overconstraints (*s* = 2), the mobility of the mechanism, as calculated by Equation 3, will be

Fig. 9. A schematic of a proposed hybrid machine tools mechanism

addressed using the following design parameters: DP121*i*: the type of the *i*th platform end joint *ai* DP122*i*: the type of the *i*th base end joint *bi* DP123*i*: the stroke of *i*th leg (*li-max – li-min*)

DP124*i*: average lengths of the *i*th leg (*li-max* + *li-min*)/2

In order to reach an optimum design, the Axiomatic Design FRs and DPs are grouped hierarchically. The design problem is also formulated such that the FRs are independent from each other (to fulfill the Independence Axiom), and the DPs are uncoupled at least partially (to fulfill the Information Axiom). Thus, the design strategy is directed to fulfill the FRs using uncoupled DPs first. Figure 10 shows main FRs for a hybrid kinematic mechanism design arranged hierarchically. The fundamental function requirement (FR1 = positioning and orientation capabilities) is split into two independent function requirements (FR11 and FR12) which can be addressed using independent design parameters. FR12 is split into three function requirements (FR121, FR122, FR123). For a given configuration of the parallel kinematic mechanism, the function requirements (FR121, FR122, FR123) can be

It is worth mentioning here that the joint axes resemble the five axes of the machine tools at the center of the workspace, and could be maintained to be close to this situation by proper

adequate and consistent dexterity throughout the workspace, 3) have good structural rigidity, and 4) have a large and well shaped workspace. The design parameters (DPs) that could be used to achieve the function requirements concerning the parallel kinematic part of the mechanism include 1) the configuration of the wrest joint, 2) the configuration of the parallel kinematic mechanism, 3) the types of the end joints, and 4) the strokes and average lengths of the legs.

Fig. 8. A schematic of the Exechon hybrid kinematic machine tools mechanism

Based on the discussion in the previous sections and the axiomatic design formulation previously used for planner parallel kinematic structures (Harib and Sharif Ullah, 2008) a kinematic design of an alternative design for a hybrid kinematic machine tools mechanism is proposed. A schematic of the proposed mechanism is depicted in Figure 9 below. The parallel kinematic part has three perpendicular struts when the mechanism is at the center of the workspace, and consists of movable platform and three extendable struts. As shown in Figure 1, the first strut is rigidly connected to the platform, which in turn is connected to other two struts via revolute and universal joints. The struts are connected respectively to the machine frame via universal joints and a spherical joint with connectivity three. The number of overconstraint (geometrical) conditions s is 2. These conditions require that the

adequate and consistent dexterity throughout the workspace, 3) have good structural rigidity, and 4) have a large and well shaped workspace. The design parameters (DPs) that could be used to achieve the function requirements concerning the parallel kinematic part of the mechanism include 1) the configuration of the wrest joint, 2) the configuration of the parallel kinematic mechanism, 3) the types of the end joints, and 4) the strokes and average

Fig. 8. A schematic of the Exechon hybrid kinematic machine tools mechanism

Based on the discussion in the previous sections and the axiomatic design formulation previously used for planner parallel kinematic structures (Harib and Sharif Ullah, 2008) a kinematic design of an alternative design for a hybrid kinematic machine tools mechanism is proposed. A schematic of the proposed mechanism is depicted in Figure 9 below. The parallel kinematic part has three perpendicular struts when the mechanism is at the center of the workspace, and consists of movable platform and three extendable struts. As shown in Figure 1, the first strut is rigidly connected to the platform, which in turn is connected to other two struts via revolute and universal joints. The struts are connected respectively to the machine frame via universal joints and a spherical joint with connectivity three. The number of overconstraint (geometrical) conditions s is 2. These conditions require that the

lengths of the legs.

two prismatic joints *l*1 and *l*2 form a plane and that the axis of joint *a*2 to be perpendicular to this plane. Calculating the mobility using Equation 1 yields *M* = 1. However considering the overconstraints (*s* = 2), the mobility of the mechanism, as calculated by Equation 3, will be *M* = 3.

Fig. 9. A schematic of a proposed hybrid machine tools mechanism

In order to reach an optimum design, the Axiomatic Design FRs and DPs are grouped hierarchically. The design problem is also formulated such that the FRs are independent from each other (to fulfill the Independence Axiom), and the DPs are uncoupled at least partially (to fulfill the Information Axiom). Thus, the design strategy is directed to fulfill the FRs using uncoupled DPs first. Figure 10 shows main FRs for a hybrid kinematic mechanism design arranged hierarchically. The fundamental function requirement (FR1 = positioning and orientation capabilities) is split into two independent function requirements (FR11 and FR12) which can be addressed using independent design parameters. FR12 is split into three function requirements (FR121, FR122, FR123). For a given configuration of the parallel kinematic mechanism, the function requirements (FR121, FR122, FR123) can be addressed using the following design parameters:

DP121*i*: the type of the *i*th platform end joint *ai*

DP122*i*: the type of the *i*th base end joint *bi*

DP123*i*: the stroke of *i*th leg (*li-max – li-min*)

DP124*i*: average lengths of the *i*th leg (*li-max* + *li-min*)/2

It is worth mentioning here that the joint axes resemble the five axes of the machine tools at the center of the workspace, and could be maintained to be close to this situation by proper

Parallel, Serial and Hybrid Machine Tools and

**5. Acknowledgment** 

2011, 186-193

2170: 331-337.

20(3), 538-543

Grant No. 2008/052.

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