**2. Kinematic of parallel manipulator**

In this section, let us consider a 6-DOF parallel manipulator with actuators situated on the base. The mechanical architecture of the considered robot is illustrated in Fig. 1.

Singularity Analysis, Constraint Wrenches and Optimal Design of Parallel Manipulators 361

With this approach, the linear actuators can be firmly fixed on the base to reduce high acceleration movements because the power is not used to move heavy actuators but lightweight links. However, the obstacle is a smaller working space in comparison with a Stewart platform, due to the movement of the linear actuators. Moreover, forces acting on the actuator have a perpendicular component, whereas forces exerted upon Stewart

Let us consider an inverse kinematic problem of position of parallel manipulators, which has characteristic relation between the numbers of chains. The manipulator with six kinematic chains offers convenience in optimization of working space in terms of decreased

Likewise, the generalized coordinates of the *i-th* segment (the length) which are equal to the

when the length of the link *li* is known, the distance between the points *Ai* and *C'i* (Fig. 2, b):

*i Ai i*

It is the solution of the inverse kinematic problem. The inverse kinematics for parallel manipulator can be formulated to determine the required actuator heights for a given pose of the mobile platform with respect to the base. The pose consists of both position and orientation in the Cartesian system. Actuators are considered to act linearly in the vertical direction, parallel to the *z-*axis, in order to simplify the mathematics, although that needs not

Influence of singularities on parameters of the working space of the parallel manipulator is a significant factor worth investigating. In these singularity configurations, the system is out of control and that greatly affects its functionality. It is necessary to determine the extent of the lack of control to see how that affects the parameters of the working space. These

The constraint wrenches of zero pitch acting to the output link from the legs are located

*i=si*

is the Clifford factor,

*zi = sCxieCyi - sCyieCxi* and can be expressed by Plücker coordinates *Ei = (xi,* 

*i)*. These coordinates make form the 6 6 matrix *(E)*:

<sup>2</sup> <sup>2</sup>

By geometrical method, the distance between the points *Ci* and *C'i* (Fig. 2, b):

We could obtain the generalized coordinate *θi* (Fig. 2, c) as follows:

singularity configurations also affect the optimization results.

*eo*

2 2 <sup>2</sup> , 1,...,6 *i Ai Bi Ai Bi Ai Bi f xx yy zz i* (1)

2 2 ( ) ( ) , 1,...,6 *i i Ai g f zi* (2)

*ii i hl g* (3)

*z h* (4)

*<sup>i</sup>*, *(i=1,…,6)* where *ei* is the unit vector directed along the axis

*ei* corresponding with *eo*

*2=0* (for a vector, *ei eo*

*i=0*). *Ei*

*xi = sCyieCzi - sCzieCyi;*

actuators have a longitudinal component.

distance between the points *Ai* and *Bi*, can be expressed as:

rigidity and load-bearing capacity.

be the case.

*eo*

*yi, zi, xo*

**3. Multi-criteria optimization** 

along the unit screws: *Ei=ei+*

*yi = sCzieCxi - sCxieCzi; eo*

*i, yo i, zo*

of the line *CiAi* of the corresponding leg,

consists of the unit vector *ei* and its moment *eo*

Fig. 1. Parallel manipulator with linear actuators located on the base

The parallel manipulator as seen in Fig. 1 is composed of a mobile platform connected to a fixed base via six kinematic sub-chains (legs) comprising of one prismatic, one universal and one spherical pair (PUS pairs). Parameters of design of the platform and the base form an irregular hexagon positioned in the *(x-y)* plane. *Ai*, *Bi (i=1,…,6)* are coordinates of the points of the mobile platform (the output link) and of the base respectively. The points *A1A3A5* and *A2A4A6* make form equilateral triangles, the angle *ψp* determines their location and *Rp* is the radius of the circumscribed circle (Fig. 2, a). Similarly, the angle *ψb* and the radius *Rb* determines the location of the equilateral triangles *B1B3B5* and *B2B4B6* located on the base. Let the distance between the centers of the universal and spherical pairs *Ai* and *Ci* of the *i-th* leg be *li*. In addition, the generalized coordinates, which are equal to the distance between the points *Bi* and *Ci* are designated *θi*. The radius-vectors of the points *Ai* and *Ci* are *ri(xAi, yAi, zAi)* and *si(xCi, yCi, zCi)* respectively *(i=1,…,6)*. We could note that the coordinates of the points *Bi* and *Ci* are *xCi=xBi, yCi=yBi, zCi=θi.* 

Fig. 2. Parametrical and geometrical design of parallel manipulator

With this approach, the linear actuators can be firmly fixed on the base to reduce high acceleration movements because the power is not used to move heavy actuators but lightweight links. However, the obstacle is a smaller working space in comparison with a Stewart platform, due to the movement of the linear actuators. Moreover, forces acting on the actuator have a perpendicular component, whereas forces exerted upon Stewart actuators have a longitudinal component.

Let us consider an inverse kinematic problem of position of parallel manipulators, which has characteristic relation between the numbers of chains. The manipulator with six kinematic chains offers convenience in optimization of working space in terms of decreased rigidity and load-bearing capacity.

Likewise, the generalized coordinates of the *i-th* segment (the length) which are equal to the distance between the points *Ai* and *Bi*, can be expressed as:

$$f\_i = \sqrt{\left(\mathbf{x}\_{Ai} - \mathbf{x}\_{Bi}\right)^2 + \left(y\_{Ai} - y\_{Bi}\right)^2 + \left(z\_{Ai} - z\_{Bi}\right)^2} \text{ , i = 1,...,6\tag{1}$$

By geometrical method, the distance between the points *Ci* and *C'i* (Fig. 2, b):

$$\log\_i = \sqrt{{(f\_i)}^2 - {(z\_{Ai})}^2}, \text{ i } = 1, \dots, 6 \tag{2}$$

when the length of the link *li* is known, the distance between the points *Ai* and *C'i* (Fig. 2, b):

$$h\_i = \sqrt{\left(l\_i\right)^2 - \left(g\_i\right)^2} \tag{3}$$

We could obtain the generalized coordinate *θi* (Fig. 2, c) as follows:

$$
\theta\_i = z\_{Ai} - h\_i \tag{4}
$$

It is the solution of the inverse kinematic problem. The inverse kinematics for parallel manipulator can be formulated to determine the required actuator heights for a given pose of the mobile platform with respect to the base. The pose consists of both position and orientation in the Cartesian system. Actuators are considered to act linearly in the vertical direction, parallel to the *z-*axis, in order to simplify the mathematics, although that needs not be the case.

#### **3. Multi-criteria optimization**

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

*y' z'*

*O'*

*x'*

*dri Ai*

The parallel manipulator as seen in Fig. 1 is composed of a mobile platform connected to a fixed base via six kinematic sub-chains (legs) comprising of one prismatic, one universal and one spherical pair (PUS pairs). Parameters of design of the platform and the base form an irregular hexagon positioned in the *(x-y)* plane. *Ai*, *Bi (i=1,…,6)* are coordinates of the points of the mobile platform (the output link) and of the base respectively. The points *A1A3A5* and *A2A4A6* make form equilateral triangles, the angle *ψp* determines their location and *Rp* is the radius of the circumscribed circle (Fig. 2, a). Similarly, the angle *ψb* and the radius *Rb* determines the location of the equilateral triangles *B1B3B5* and *B2B4B6* located on the base. Let the distance between the centers of the universal and spherical pairs *Ai* and *Ci* of the *i-th* leg be *li*. In addition, the generalized coordinates, which are equal to the distance between the points *Bi* and *Ci* are designated *θi*. The radius-vectors of the points *Ai* and *Ci* are *ri(xAi, yAi, zAi)* and *si(xCi, yCi, zCi)* respectively *(i=1,…,6)*. We could note that the coordinates of the points *Bi*

(a) (b)

*B2*

*Bi*

*Ci*

*li*

*θ<sup>i</sup> Bi*

*Ai*

*fi*

*gi*

*hi*

*Ci '*

*'*

Fig. 2. Parametrical and geometrical design of parallel manipulator

*B3 B5*

*<sup>R</sup> <sup>ψ</sup><sup>b</sup> <sup>b</sup>*

*B1*

*B4*

Fig. 1. Parallel manipulator with linear actuators located on the base

*Bi*

*<sup>z</sup> <sup>y</sup>*

*O*

*x*

*si*

*ei*

*Ci*

*ri*

and *Ci* are *xCi=xBi, yCi=yBi, zCi=θi.* 

*B6*

Influence of singularities on parameters of the working space of the parallel manipulator is a significant factor worth investigating. In these singularity configurations, the system is out of control and that greatly affects its functionality. It is necessary to determine the extent of the lack of control to see how that affects the parameters of the working space. These singularity configurations also affect the optimization results.

The constraint wrenches of zero pitch acting to the output link from the legs are located along the unit screws: *Ei=ei+eo <sup>i</sup>*, *(i=1,…,6)* where *ei* is the unit vector directed along the axis of the line *CiAi* of the corresponding leg, is the Clifford factor, *2=0* (for a vector, *ei eo i=0*). *Ei* consists of the unit vector *ei* and its moment *eo i=siei* corresponding with *eo xi = sCyieCzi - sCzieCyi; eo yi = sCzieCxi - sCxieCzi; eo zi = sCxieCyi - sCyieCxi* and can be expressed by Plücker coordinates *Ei = (xi, yi, zi, xo i, yo i, zo i)*. These coordinates make form the 6 6 matrix *(E)*:

$$\mathbf{f}(E) = \begin{pmatrix} x\_1 & y\_1 & z\_1 & x\_1^o & y\_1^o & z\_1^o \\ x\_2 & y\_2 & z\_2 & x\_2^o & y\_2^o & z\_2^o \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ x\_6 & y\_6 & z\_6 & x\_6^o & y\_6^o & z\_6^o \end{pmatrix} \tag{5}$$

Singularity Analysis, Constraint Wrenches and Optimal Design of Parallel Manipulators 363

3.2. Determine *Npi, Dei, Nci*; assume *j= 1*, the criteria of optimal solution *K1=1, K2=0*. 3.3. Determine Npj, Dej, Ncj; if Npi > Npj or Dei> Dej or Nci> Ncj then K2=1; if Npi= Npj and

3.5. If *K1 = 1* then *k = k + 1*, *Par1k= H1i*, …, *Par7k= Nci*, *i=i+1*; if *i ≤ np* then go back to 3.2. Singularity of the manipulator is determined by closeness to zero of determinant of matrix

construction of the working space of the manipulator shows that the same results we can get without singularity constraint. Giving various values of the criterion of the singularity, we

Further, analysis influences of the criterion of singularity *ε*, │det*(E)* │≤*ε* on the value of the working volume. With the value of the criterion of singularity is equal to *ε=0.01* there exist *81* available solutions, but only *8* of them are Pareto-optimal. By the condition of the criterion of singularity is equal to *ε=0.01* and the condition *ε=0,* there Pareto-optimal set consists of *6* and *29* solutions correspondingly. Therefore, the value of *ε* influences on the

Value of the criterion that determines the proximity to singular configurations is equal to zero, we can assume that the constraints associated with the singularity in general, are not imposed in the analysis of each specific configuration. However, the criterion for determining the load capacity occurs. As a result, the number of Pareto-optimal variants

Limiting possible module of a determinant of matrix (*E*) to singularity configurations changes the Plücker coordinates of the wrenches transmitted on the output link. Methodology for analyzing the singularities on optimization appearing in the parallel manipulator and their impact in the working space is proposed. The practical significance from the fact is the results

The approach based on matrix *(E)* consisting of the Plücker coordinates of the constraint wrenches allows determining the twists of the platform inside singularity (Glazunov, 2006). Let us consider the increases of the Plücker coordinates of the unit screws *Ei* after an infinitesimal displacement *\$ = (dφ, dr) = (drx, drx, drx, dφx, dφy, dφz)T* of the platform corresponding to displacement *dri = (dxAi, dyAi, dzAi,)T* of the point *Ai* of the manipulator

> *Ai x y Ai z Ai Ai y z Ai x Ai Ai z x Ai y Ai*

*dx dr d z d y dy dr d x d z dz dr d y d x*

 

, ,

(6)

 

 

  ). If 

*>0* as a criterion of

*=0* then the

2.4. Determine the criteria Npi= Np , Dei= De / Nc, Nci= Nc / Np; assume Np = Nc = De= 0.

2.3. *j = j+1*, if *j ≤ nc* then go back to 2.2.

2.5. *i = i+1*, if *i ≤ np* then go back to 2.1.

Dei= Dej and Nci= Ncj then K2=1.

can get interval of the determinant of matrix (*E*).

results of optimization.

**4. Twist inside singularity** 

presented on the Fig. 1.

**Step 3.** Determine the Pareto-optimal solutions (matrix *(Par)*).

3.1. Assume i= 1, (by this Npi= Np1, Dei= De1, Nci= Nc1), k= 0.

3.4. If *K2≠1* then *K1=0*; *K2=0*; *j=j+1*; if *j ≤ np* then go back to 3.3.

*(E)* of Plücker coordinates of unit wrenches. Let us fix certain value

singularity (the manipulator is in singular position if │det(*E*)│≤

varies very much. Here, *29* variants satisfy the conditions of Pareto set.

obtained in this work increase the effectiveness of design automation.

Optimization of parameters of the parallel manipulator with linear actuators located on the base is considered. Let us take in account three criteria: working volume, dexterity and stiffness of parallel manipulator. The first criterion *Np* is the quantity of the reachable points of the centre of the mobile platform. The second criterion *Nc* is the average quantity of orientations of the mobile platform in each reachable point. The third criterion *De* is the average module of the determinant │det(*E*)│ in each configuration. Determinant │det(*E*)│ constructed from coordinate axes of the drive kinematic couples is used as a third criterion of optimization. Since the value of this criterion is related to one of the important characteristics of the manipulator - its stiffness or load capacity. If determinant are more qualifiers, then the manipulator away from the singularity configuration and the stiffness of the above.

Let us consider optimization of the parameters of the manipulator for different values of the criterion of proximity to singular configurations, as well as the influence of this criterion in the optimization results. We set up four coefficients *H1, H2, H3* and *H4* expressed four parameters of optimization. The coefficient *H1* characterizes the length *l = li* of the links *AiCi (i=1,…,6)* (in Fig. 2, b). The coefficient *H2* characterizes the angle *ψp* (Fig. 2, a) determining the location of the triangles *A1A3A5* and *A2A4A6* of the mobile platform. The coefficient *H3* characterizes the angle *ψb* determining the location of the triangles *B1B3B5* and *B2B4B6* on the base. Moreover, the coefficient *H4* characterizes the relation between the radius *Rp* and the radius *Rb* of the circumscribe circles of the platform and of the base respectively.

The algorithm of determination of the Pareto-optimal solutions can be presented as follows:

**Step 1.** Establish the limits of the parameters of optimization.

*H1min ≤ H1 ≤ H1max, H2min ≤ H2 ≤ H2max, H3min ≤ H3 ≤ H3max, H4min ≤ H4 ≤ H4max*. The number of steps of scanning in the space of parameters is *np*. The limits of the scanned Cartesian coordinates of the centre of the moving platform and the limits of the scanned orientation angles of this platform in interval are *xmin ≤ x ≤ xmax, ymin ≤ y ≤ ymax, zmin ≤ z ≤ zmax, min ≤ ≤ max, min ≤ ≤ max, γ min ≤ γ ≤ γ max*. As well as the number *nc* of steps of scanning in the space of these coordinates and the limitation of changing of the generalized coordinates *θimin ≤ θ<sup>i</sup> ≤ θi max (i=1,…,6)*. The limit of the determinant │det(*E*)│ ≥ *ε*. At this step assume *i=0*, by this the parameters are *H10 = H1min, H20 = H2min, H30 = H3min, H40 = H4min , Np = Nc = De= 0*.

**Step 2.** Determine the values of the criteria for all the values of the parameters.

2.1. Determine the parameters H1i,…, H40, assume j=0, by this x0 = xmin, y0 = ymin, z0 = zmin, 0 = min, 0 = min, γ0 = γmin.

2.2. Determine θi (i=1,…,6) and │det(E)│; if all the θi min ≤ θ<sup>i</sup> ≤ θi max (i=1,…,6) and │det(E)│ ≥ ε then Nc = Nc+ 1, De = De+│det(E)│; if xj ≠ xj-1 or yj ≠ yj-1 or zj ≠ zj-1 then Np = Np+ 1.

2.3. *j = j+1*, if *j ≤ nc* then go back to 2.2.

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

...... ( ) ......

*E*

the above.

*zmax, min ≤ ≤ max, min ≤ ≤* 

Np+ 1.

1 111 11 2 222 22

*xyzxyz xyzxyz*

 

*o oo o oo*

*ooo*

(5)

6 666 66

*xyzxyz*

Optimization of parameters of the parallel manipulator with linear actuators located on the base is considered. Let us take in account three criteria: working volume, dexterity and stiffness of parallel manipulator. The first criterion *Np* is the quantity of the reachable points of the centre of the mobile platform. The second criterion *Nc* is the average quantity of orientations of the mobile platform in each reachable point. The third criterion *De* is the average module of the determinant │det(*E*)│ in each configuration. Determinant │det(*E*)│ constructed from coordinate axes of the drive kinematic couples is used as a third criterion of optimization. Since the value of this criterion is related to one of the important characteristics of the manipulator - its stiffness or load capacity. If determinant are more qualifiers, then the manipulator away from the singularity configuration and the stiffness of

Let us consider optimization of the parameters of the manipulator for different values of the criterion of proximity to singular configurations, as well as the influence of this criterion in the optimization results. We set up four coefficients *H1, H2, H3* and *H4* expressed four parameters of optimization. The coefficient *H1* characterizes the length *l = li* of the links *AiCi (i=1,…,6)* (in Fig. 2, b). The coefficient *H2* characterizes the angle *ψp* (Fig. 2, a) determining the location of the triangles *A1A3A5* and *A2A4A6* of the mobile platform. The coefficient *H3* characterizes the angle *ψb* determining the location of the triangles *B1B3B5* and *B2B4B6* on the base. Moreover, the coefficient *H4* characterizes the relation between the radius *Rp* and the

The algorithm of determination of the Pareto-optimal solutions can be presented as follows:

*H1min ≤ H1 ≤ H1max, H2min ≤ H2 ≤ H2max, H3min ≤ H3 ≤ H3max, H4min ≤ H4 ≤ H4max*. The number of steps of scanning in the space of parameters is *np*. The limits of the scanned Cartesian coordinates of the centre of the moving platform and the limits of the scanned orientation angles of this platform in interval are *xmin ≤ x ≤ xmax, ymin ≤ y ≤ ymax, zmin ≤ z ≤*

scanning in the space of these coordinates and the limitation of changing of the generalized coordinates *θimin ≤ θ<sup>i</sup> ≤ θi max (i=1,…,6)*. The limit of the determinant │det(*E*)│ ≥ *ε*. At this step assume *i=0*, by this the parameters are *H10 = H1min, H20 = H2min, H30 =* 

2.1. Determine the parameters H1i,…, H40, assume j=0, by this x0 = xmin, y0 = ymin, z0 =

2.2. Determine θi (i=1,…,6) and │det(E)│; if all the θi min ≤ θ<sup>i</sup> ≤ θi max (i=1,…,6) and │det(E)│ ≥ ε then Nc = Nc+ 1, De = De+│det(E)│; if xj ≠ xj-1 or yj ≠ yj-1 or zj ≠ zj-1 then Np =

*max, γ min ≤ γ ≤ γ max*. As well as the number *nc* of steps of

radius *Rb* of the circumscribe circles of the platform and of the base respectively.

**Step 2.** Determine the values of the criteria for all the values of the parameters.

**Step 1.** Establish the limits of the parameters of optimization.

*H3min, H40 = H4min , Np = Nc = De= 0*.

zmin, 0 = min, 0 = min, γ0 = γmin.

......

2.4. Determine the criteria Npi= Np , Dei= De / Nc, Nci= Nc / Np; assume Np = Nc = De= 0.

2.5. *i = i+1*, if *i ≤ np* then go back to 2.1.

**Step 3.** Determine the Pareto-optimal solutions (matrix *(Par)*).

3.1. Assume i= 1, (by this Npi= Np1, Dei= De1, Nci= Nc1), k= 0.

3.2. Determine *Npi, Dei, Nci*; assume *j= 1*, the criteria of optimal solution *K1=1, K2=0*.

3.3. Determine Npj, Dej, Ncj; if Npi > Npj or Dei> Dej or Nci> Ncj then K2=1; if Npi= Npj and Dei= Dej and Nci= Ncj then K2=1.

3.4. If *K2≠1* then *K1=0*; *K2=0*; *j=j+1*; if *j ≤ np* then go back to 3.3.

3.5. If *K1 = 1* then *k = k + 1*, *Par1k= H1i*, …, *Par7k= Nci*, *i=i+1*; if *i ≤ np* then go back to 3.2.

Singularity of the manipulator is determined by closeness to zero of determinant of matrix *(E)* of Plücker coordinates of unit wrenches. Let us fix certain value *>0* as a criterion of singularity (the manipulator is in singular position if │det(*E*)│≤). If *=0* then the construction of the working space of the manipulator shows that the same results we can get without singularity constraint. Giving various values of the criterion of the singularity, we can get interval of the determinant of matrix (*E*).

Further, analysis influences of the criterion of singularity *ε*, │det*(E)* │≤*ε* on the value of the working volume. With the value of the criterion of singularity is equal to *ε=0.01* there exist *81* available solutions, but only *8* of them are Pareto-optimal. By the condition of the criterion of singularity is equal to *ε=0.01* and the condition *ε=0,* there Pareto-optimal set consists of *6* and *29* solutions correspondingly. Therefore, the value of *ε* influences on the results of optimization.

Value of the criterion that determines the proximity to singular configurations is equal to zero, we can assume that the constraints associated with the singularity in general, are not imposed in the analysis of each specific configuration. However, the criterion for determining the load capacity occurs. As a result, the number of Pareto-optimal variants varies very much. Here, *29* variants satisfy the conditions of Pareto set.

Limiting possible module of a determinant of matrix (*E*) to singularity configurations changes the Plücker coordinates of the wrenches transmitted on the output link. Methodology for analyzing the singularities on optimization appearing in the parallel manipulator and their impact in the working space is proposed. The practical significance from the fact is the results obtained in this work increase the effectiveness of design automation.

## **4. Twist inside singularity**

The approach based on matrix *(E)* consisting of the Plücker coordinates of the constraint wrenches allows determining the twists of the platform inside singularity (Glazunov, 2006). Let us consider the increases of the Plücker coordinates of the unit screws *Ei* after an infinitesimal displacement *\$ = (dφ, dr) = (drx, drx, drx, dφx, dφy, dφz)T* of the platform corresponding to displacement *dri = (dxAi, dyAi, dzAi,)T* of the point *Ai* of the manipulator presented on the Fig. 1.

$$\begin{aligned} dx\_{Ai} &= dr\_{\chi} + d\rho\_y z\_{Ai} - d\rho\_z y\_{Ai} \\ dy\_{Ai} &= dr\_y + d\rho\_x x\_{Ai} - d\rho\_x z\_{Ai} \\ dz\_{Ai} &= dr\_z + d\rho\_x y\_{Ai} - d\rho\_y x\_{Ai} \end{aligned} \tag{6}$$

Singularity Analysis, Constraint Wrenches and Optimal Design of Parallel Manipulators 365

*d E*

0.158 0.148 0.148 0.158 0.148 0.148

 

0 0.148 0.148 0 0.148 0.148

0.988 0.978 0.978 0.988 0.978 0.978

0 1.572 1.572 0 1.572 1.572

The determinant consisting of the Plücker coordinates of the unit screws is det(*E*) = *0*. Their

det( ) det( ) det( ) 0.02, 0, 0, *x yz E EE r rr*

det( ) det( ) det( ) 0, 0.002, <sup>0</sup> *xy z EE E*

Using the approach presented above, we find five independent twists inside singularity: *\$1(1, 0, 0, 0, 0, 0), \$2(0, 0, 1, 0, 0, 0), \$3(0, 0, 0, 0, 1, 0), \$4(0, 0, 0, 0, 0, 1), \$5(0, 4.433, 0, 1, 0, 0)*. The twist-gradient is calculated to be *\$\*(-0.02, 0, 0, 0, 0.002, 0)*. This twist-gradient is practically

In this section, let we consider the reduction of the dynamical coupling of the motors of the parallel manipulator with linear actuators located on the base. The basic idea is to represent the kinetic energy as the quadratic polynomial including only the squares of the generalized velocities (Glazunov, & Kraynev, 2006). The kinetic energy can be expressed by means of the

Let *m* be the mass and *Jx, Jy, Jz* be the inertia moments of the platform. Assuming that the mass of the platform is much more than the masses of the legs and using the Eqs. (6)-(8), the kinetic energy *T* can be expressed as follows (Dimentberg, 1965; Kraynev, & Glazunov,

1.618 1.083 1.083 1.618 1.083 1.083

0 0.074 0.074 0 0.074 0.074

> 

 

platform inside singularity.

partial derivatives are:

0.305, 0.305). Matrix (E) is determined as:

important as it offers the highest speed.

**5. Dynamical decoupling** 

matrix *(E)*.

1991):

det( ) det( ) det( ) det( )

Using (10) the criterion of the singularity locus can be presented as d[det(*E*)]=*0*. This condition imposes only one constraint. Therefore, there exist five twists of motions of the

For example, let us obtain five inside singularity of manipulator. Set up the coordinates of the vectors be ri are r1(-1, 0, 4), r2(-0.5, 1, 4), r3(0.5, 1, 4), r4(1, 0, 4), r5(0.5, -1, 4), r6(-0.5, -1, 4); si are s1(-1.5, 0, 0.866), s2(-1, 1.5, 0.707), s3(1, 1.5, 0.707), s4(1.5, 0, 0.866), s5(1, -1.5, 0.707), s6(-1, - 1.5, 0.707). From here, we can see the generalized coordinates as (0.136, 0.305, 0.305, 0.136,

*ddd EEE r dr r dr r dr*

 

 

det( ) det( ) det( ) *xx yy zz*

*EEE*

 

(10)

*xx yy zz*

The generalized coordinate after mentioned infinitesimal displacement *(dxi, dyi, dzi)* is:

$$d\theta\_i + d\theta\_i = z\_{Ai} + dz\_{Ai} - \sqrt{l\_i^2 - \left(x\_{Ai} + dx\_{Ai} - x\_{Ci}\right)^2 - \left(y\_{Ai} + dy\_{Ai} - y\_{Ci}\right)^2} \tag{7}$$

After transformations the increase of the generalized coordinate is:

$$d\theta\_i = \frac{\left[ (\mathbf{x}\_{Ai} - \mathbf{x}\_{Ci})d\mathbf{x}\_{Ai} + (y\_{Ai} - y\_{Ci})dy\_{Ai} + (z\_{Ai} - z\_{Ci})dz\_{Ai} \right]}{\left(z\_{Ai} - z\_{Ci}\right)} \tag{8}$$

The unit screw *Ei* can be rewritten as *Ei+dEi* or as *ei+dei* and *eo i + deo i* . Using (6), (7), and (8) the coordinates of the *dei* and *deo <sup>i</sup>* can be expressed as:

$$dx\_i = \frac{\partial \mathbf{x}\_i}{\partial \boldsymbol{\rho}\_x} d\boldsymbol{\rho}\_x + \frac{\partial \mathbf{x}\_i}{\partial \boldsymbol{\rho}\_y} d\boldsymbol{\rho}\_y + \frac{\partial \mathbf{x}\_i}{\partial \boldsymbol{\rho}\_z} d\boldsymbol{\rho}\_z + \frac{\partial \mathbf{x}\_i}{\partial \boldsymbol{r}\_x} d\mathbf{r}\_x + \frac{\partial \mathbf{x}\_i}{\partial \boldsymbol{r}\_y} d\mathbf{r}\_y + \frac{\partial \mathbf{x}\_i}{\partial \boldsymbol{r}\_z} d\mathbf{r}\_z$$
 
$$\text{where}$$
 
$$dz\_i^o = \frac{\partial \mathbf{z}\_i^o}{\partial \boldsymbol{\rho}\_x} d\boldsymbol{\rho}\_x + \frac{\partial \mathbf{z}\_i^o}{\partial \boldsymbol{\rho}\_y} d\boldsymbol{\rho}\_y + \frac{\partial \mathbf{z}\_i^o}{\partial \boldsymbol{\rho}\_z} d\boldsymbol{\rho}\_z + \frac{\partial \mathbf{z}\_i^o}{\partial \boldsymbol{r}\_x} d\mathbf{r}\_x + \frac{\partial \mathbf{z}\_i^o}{\partial \boldsymbol{r}\_y} d\mathbf{r}\_y + \frac{\partial \mathbf{z}\_i^o}{\partial \boldsymbol{r}\_z} d\mathbf{r}\_z$$

where

$$\frac{\partial \mathbf{\hat{x}}\_i}{\partial \boldsymbol{\phi}\_x} = \mathbf{0}, \frac{\partial \mathbf{x}\_i}{\partial \boldsymbol{\phi}\_y} = \frac{\mathbf{z}\_{Ai}}{l\_i}, \frac{\partial \mathbf{x}\_i}{\partial \boldsymbol{\phi}\_z} = -\frac{y\_{Ai}}{l\_i}, \frac{\partial \mathbf{x}\_i}{\partial r\_x} = \frac{1}{l\_i}, \frac{\partial \mathbf{x}\_i}{\partial r\_y} = \mathbf{0}, \frac{\partial \mathbf{x}\_i}{\partial r\_z} = \mathbf{0},$$

$$\frac{\partial \mathbf{x}\_i^o}{\partial \boldsymbol{\phi}\_x} = \frac{\left\{ \left( y\_{Ai} - y\_{Ci} \right) \left[ \frac{\left( y\_{Ai} - 2y\_{Ci} \right) z\_{Ai}}{\left( z\_{Ai} - z\_{Ci} \right)} - y\_{Ai} \right] + z\_{Ci} z\_{Ai} \right\}}{l\_i},$$

$$\begin{split} \frac{\partial \boldsymbol{\alpha}\_{i}^{o}}{\partial \rho\_{y}} &= \frac{\left[ \frac{\left( \mathbf{x}\_{Ai} - \mathbf{x}\_{Ci} \right) \left( 2y\_{Ci} - y\_{Ai} \right) z\_{Ai}}{\left( z\_{Ai} - z\_{Ci} \right)} - \mathbf{x}\_{Ai} \left( y\_{Ai} - y\_{Ci} \right) \right]}{l\_{i}}, \\ \frac{\partial \boldsymbol{\alpha}\_{i}^{o}}{\partial \rho\_{z}} &= \frac{\left[ \frac{\left[ \left( y\_{Ai} - y\_{Ci} \right) \mathbf{x}\_{Ai} - \left( x\_{Ai} - \mathbf{x}\_{Ci} \right) y\_{Ai} \right] \left( 2y\_{Ci} - y\_{Ai} \right)}{\left( z\_{Ai} - z\_{Ci} \right)} - \mathbf{x}\_{Ai} z\_{Ci} \right]}{l\_{i}}, \\ \frac{\partial \boldsymbol{\alpha}\_{i}^{o}}{\partial \mathbf{x}\_{i}} &= \frac{\left( \mathbf{x}\_{Ai} - \mathbf{x}\_{Ci} \right) \left( 2y\_{Ci} - y\_{Ai} \right)}{\left( z\_{Ai} - z\_{Ci} \right) l\_{i}}, \frac{\partial \boldsymbol{\alpha}\_{i}^{o}}{\partial \mathbf{r}\_{y}} = \frac{\left( y\_{Ai} - y\_{Ci} \right) \left( 2y\_{Ci} - y\_{Ai} \right)}{\left( z\_{Ai} - z\_{Ci} \right)} - z\_{Ci} \\ \frac{\partial \boldsymbol{\alpha}\_{i}^{o}}{\partial \mathbf{r}\_{z}} &= \frac{\left( y\_{Ci} - y\_{Ai} \right)}{l\_{i}}, \; i = 1, \ldots, 6 \end{split}$$

Other partial derivatives also can be obtained from Eqs. (6), (7), and (8). By means of the properties of linear decomposition of determinants d[det*(E)*] can be obtained as the sum of *36* determinants (Glazunov, 2006). From this, d[det*(E)*] can be presented as:

$$\begin{split} d\left[ \det(E) \right] &= \frac{\hat{\sigma} \left[ \det(E) \right]}{\hat{\circ} \varphi\_x d\varphi\_x} + \frac{\hat{\sigma} \left[ \det(E) \right]}{\hat{\circ} \varphi\_y d\varphi\_y} + \frac{\hat{\sigma} \left[ \det(E) \right]}{\hat{\circ} \varphi\_z d\varphi\_z} + \\ &+ \frac{\hat{\sigma} \left[ \det(E) \right]}{\hat{\circ} r\_x dr\_x} + \frac{\hat{\sigma} \left[ \det(E) \right]}{\hat{\circ} r\_y dr\_y} + \frac{\hat{\sigma} \left[ \det(E) \right]}{\hat{\circ} r\_z dr\_z} \end{split} \tag{10}$$

Using (10) the criterion of the singularity locus can be presented as d[det(*E*)]=*0*. This condition imposes only one constraint. Therefore, there exist five twists of motions of the platform inside singularity.

For example, let us obtain five inside singularity of manipulator. Set up the coordinates of the vectors be ri are r1(-1, 0, 4), r2(-0.5, 1, 4), r3(0.5, 1, 4), r4(1, 0, 4), r5(0.5, -1, 4), r6(-0.5, -1, 4); si are s1(-1.5, 0, 0.866), s2(-1, 1.5, 0.707), s3(1, 1.5, 0.707), s4(1.5, 0, 0.866), s5(1, -1.5, 0.707), s6(-1, - 1.5, 0.707). From here, we can see the generalized coordinates as (0.136, 0.305, 0.305, 0.136, 0.305, 0.305). Matrix (E) is determined as:

$$
\begin{pmatrix}
0.158 & 0 & 0.988 & 0 & 1.618 & 0 \\
0.148 & -0.148 & 0.978 & 1.572 & 1.083 & -0.074 \\
0.148 & 0.148 & 0.978 & -1.572 & 1.083 & 0.074
\end{pmatrix}
$$

The determinant consisting of the Plücker coordinates of the unit screws is det(*E*) = *0*. Their partial derivatives are:

$$\begin{aligned} \frac{\partial \left[ \det(E) \right]}{\partial r\_x} &= -0.02, \; \frac{\partial \left[ \det(E) \right]}{\partial r\_y} = 0, \; \frac{\partial \left[ \det(E) \right]}{\partial r\_z} = 0, \\\\ \frac{\partial \left[ \det(E) \right]}{\partial \varphi\_x} &= 0, \; \frac{\partial \left[ \det(E) \right]}{\partial \varphi\_y} = 0.002, \; \frac{\partial \left[ \det(E) \right]}{\partial \varphi\_z} = 0 \end{aligned}$$

Using the approach presented above, we find five independent twists inside singularity: *\$1(1, 0, 0, 0, 0, 0), \$2(0, 0, 1, 0, 0, 0), \$3(0, 0, 0, 0, 1, 0), \$4(0, 0, 0, 0, 0, 1), \$5(0, 4.433, 0, 1, 0, 0)*. The twist-gradient is calculated to be *\$\*(-0.02, 0, 0, 0, 0.002, 0)*. This twist-gradient is practically important as it offers the highest speed.

### **5. Dynamical decoupling**

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

*i i Ai Ai i Ai Ai Ci Ai Ai Ci*

 *Ai Ci Ai Ai Ci Ai Ai Ci Ai*

*x x dx y y dy z z dz <sup>d</sup> z z*

................................................................................

*x x x xxx dx d d d dr dr dr rrr*

 

*o o o oo i i i ii x y zx x y z xy*

2

*i i i i ii x y z x yz*

*l ll*

*y yz y y y zz z z l*

 

*x x x x xx*

 

*i ii Ai Ci Ai Ai Ci Ai Ci Ai <sup>o</sup> Ai Ci <sup>i</sup>*

*i*

 

*Ai Ci Ai Ai Ci Ai Ci Ai Ai Ci <sup>o</sup> Ai Ci <sup>i</sup>*

*i*

*l*

*o o Ci*

*Ai Ci i i*

By means of the properties of linear decomposition of determinants d[det*(E)*] can be obtained as the sum of *36* determinants (Glazunov, 2006). From this, d[det*(E)*] can be

*i*

*z z*

*l y yx x xy y y*

*x x y yz xy y*

*i i Ai Ci Ci Ai Ai Ci*

*xx yy z z z zl l*

*z z z zz dz d d d dr d r r*

*Ai Ai*

*z y*

 

*Ai Ai Ci <sup>o</sup> Ai Ci <sup>i</sup>*

 

2

*z z*

*Ai Ci Ci Ai Ai*

 

*x x r r*

 

*x y*

Other partial derivatives also can be obtained from Eqs. (6), (7), and (8).

, 1,...,6 *Ci Ai*

*i y y <sup>i</sup> l*  *Ai Ci*

*i i i iii i x y zxyz x y z xyz*

<sup>1</sup> 0, , , , 0, 0,

<sup>2</sup> <sup>2</sup> <sup>2</sup>

(8)

*i + deo i* . 

*<sup>i</sup>* can be expressed as:

*o i y z z <sup>z</sup> r dr <sup>r</sup>* 

,

*x z*

,

,

*r rr*

 

2

*yy yy <sup>z</sup>*

*Ai Ci Ci Ai*

2

<sup>2</sup> , ,

(9)

*d z dz l x dx x y dy y* (7)

The generalized coordinate after mentioned infinitesimal displacement *(dxi, dyi, dzi)* is:

After transformations the increase of the generalized coordinate is:

The unit screw *Ei* can be rewritten as *Ei+dEi* or as *ei+dei* and *eo*

Using (6), (7), and (8) the coordinates of the *dei* and *deo*

 

*i*

*o i*

*x*

*x*

 

*y*

*x*

 

*z*

*x*

 

*z*

*o i*

*x r*

 

presented as:

where

In this section, let we consider the reduction of the dynamical coupling of the motors of the parallel manipulator with linear actuators located on the base. The basic idea is to represent the kinetic energy as the quadratic polynomial including only the squares of the generalized velocities (Glazunov, & Kraynev, 2006). The kinetic energy can be expressed by means of the matrix *(E)*.

Let *m* be the mass and *Jx, Jy, Jz* be the inertia moments of the platform. Assuming that the mass of the platform is much more than the masses of the legs and using the Eqs. (6)-(8), the kinetic energy *T* can be expressed as follows (Dimentberg, 1965; Kraynev, & Glazunov, 1991):

$$\begin{aligned} T &= \frac{m}{2} \left[ \left( \sum\_{i=1}^{6} p\_i^o \dot{\theta}\_i \mathbf{G}\_{ii} \right)^2 + \left( \sum\_{i=1}^{6} q\_i^o \dot{\theta}\_i \mathbf{G}\_{ii} \right)^2 + \left( \sum\_{i=1}^{6} r\_i^o \dot{\theta}\_i \mathbf{G}\_{ii} \right)^2 \right] + \\\\ &+ \frac{I\_x}{2} \left( \sum\_{i=1}^{6} p\_i \dot{\theta}\_i \mathbf{G}\_{ii} \right)^2 + \frac{I\_y}{2} \left( \sum\_{i=1}^{6} q\_i \dot{\theta}\_i \mathbf{G}\_{ii} \right)^2 + \frac{I\_z}{2} \left( \sum\_{i=1}^{6} r\_i \dot{\theta}\_i \mathbf{G}\_{ii} \right)^2 \end{aligned} \tag{11}$$

where *i* are the generalized velocities, *pi, qi, ri, po i, qo <sup>i</sup>*, *ro <sup>i</sup>* are the components of the matrix *(E)-1*, *Gii* are the components of the diagonal matrix *(G) (i=1,…,6)*.

The Lagrange equation of motion for a parallel manipulator can be written as:

$$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{\theta}\_i}\right) - \frac{\partial T}{\partial \theta\_i} = Q\_i \tag{12}$$

Singularity Analysis, Constraint Wrenches and Optimal Design of Parallel Manipulators 367

000 0 0

*x*

*y*

*y y*

*J m J m*

*J*

0 0000 00 0 0 0 ( ) ( ) 000 0 0

*m*

matrix *(E)* is proposed:

*m*

0000 0

Let the axes of the links *AiCi* be parallel to the axes of the links *BiCi* and the matrix *(G)* is unit matrix. Then in order to satisfy the condition of orthogonal columns of the matrix *(D)* the rows of the inverse matrix *(D)-1 = (E)(M)-1* are to be orthogonal. From this, the following

> 001 0 0 00 1 0 0

010 0 0 0 10 0 0

*J m J m*

The matrix (15) corresponds to the Fig. 3. Here the center of the mass of the platform coincides with the center of the coordinate system xyz and the axes of the links of the legs

x y

z *J <sup>y</sup> m*

*x x*

*J m <sup>E</sup>*

 

0000 0

100 0 0 ( )

The determinant of the matrix *(E)* (15) can be written as:

Fig. 3. Parallel manipulator with dynamical decoupling

10 0 0 0

*m D E G J*

<sup>1</sup>

<sup>1</sup> () () *D ME G* (14)

*z z*

<sup>3</sup> det( ) 8 *E JJJ m xyz* (16)

*J m*

(15)

*z*

*J*

where *Qi* are the generalized forces *(i=1,…,6)*.

The dynamical coupling can be determined using the Eq. (11). The expression of each generalized force comprises all other generalized velocities and accelerations. In order to reduce the dynamical coupling we represent the kinetic energy as follows:

$$\begin{aligned} T &= \frac{\begin{bmatrix} \sum\_{i=1}^{6} \left( p\_{i}^{o} \dot{\theta}\_{i} \mathbf{G}\_{ii} \right)^{2} + 2 \sum\_{i=1, j=1, i \neq j}^{6} p\_{i}^{o} p\_{j}^{o} \dot{\theta}\_{i} \mathbf{G}\_{ii} \dot{\theta}\_{j} \mathbf{G}\_{jj} + \\\\ + \dots + \sum\_{i=1}^{6} \left( r\_{i}^{o} \dot{\theta}\_{i} \mathbf{G}\_{ii} \right)^{2} + 2 \sum\_{i=1, j=1, i \neq j}^{6} r\_{i}^{o} r\_{j}^{o} \dot{\theta}\_{i} \mathbf{G}\_{ii} \dot{\theta}\_{j} \mathbf{G}\_{jj} \end{bmatrix} + \\\\ &+ \frac{I\_{\times}}{2} \left[ \sum\_{i=1}^{6} \left( p\_{i} \dot{\theta}\_{i} \mathbf{G}\_{ii} \right)^{2} + 2 \sum\_{i=1, j=1, i \neq j}^{6} p\_{i} p\_{j} \dot{\theta}\_{i} \mathbf{G}\_{ii} \dot{\theta}\_{j} \mathbf{G}\_{jj} \right] + \\\\ &+ \dots + \frac{I\_{\times}}{2} \left[ \sum\_{i=1}^{6} \left( r\_{i} \dot{\theta}\_{i} \mathbf{G}\_{ii} \right)^{2} + 2 \sum\_{i=1, j=1, i \neq j}^{6} r\_{i} r\_{j} \dot{\theta}\_{i} \mathbf{G}\_{ii} \dot{\theta}\_{j} \mathbf{G}\_{jj} \right] \end{aligned} \tag{13}$$

According to the Eq. (13) dynamical decoupling can be satisfied if the columns of the following matrix *(D)* are orthogonal:

$$(D) = \begin{pmatrix} p\_1^\diamond G\_{11}\sqrt{m} & \cdot & \cdot & \cdot & p\_6^\diamond G\_{66}\sqrt{m} \\ q\_1^\diamond G\_{11}\sqrt{m} & \cdot & \cdot & \cdot & q\_6^\diamond G\_{66}\sqrt{m} \\ r\_1^\diamond G\_{11}\sqrt{m} & \cdot & \cdot & \cdot & r\_6^\diamond G\_{66}\sqrt{m} \\ p\_1 G\_{11}\sqrt{I\_x} & \cdot & \cdot & \cdot & p\_6 G\_{66}\sqrt{I\_x} \\ q\_1 G\_{11}\sqrt{I\_y} & \cdot & \cdot & \cdot & q\_6 G\_{66}\sqrt{I\_y} \\ r\_1 G\_{11}\sqrt{I\_z} & \cdot & \cdot & \cdot & r\_6 G\_{66}\sqrt{I\_z} \end{pmatrix}$$

*ooo i i ii i i ii i i ii*

 

111

*iii*

*<sup>m</sup> T pG qG rG*

2

where *Qi* are the generalized forces *(i=1,…,6)*.

2

2

2

*D*

*<sup>m</sup> <sup>T</sup>*

following matrix *(D)* are orthogonal:

where *i* *(E)-1*, *Gii* are the components of the diagonal matrix *(G) (i=1,…,6)*.

are the generalized velocities, *pi, qi, ri, po*

1 11

*i i dT T <sup>Q</sup> dt* 

The dynamical coupling can be determined using the Eq. (11). The expression of each generalized force comprises all other generalized velocities and accelerations. In order to

 

*o o <sup>o</sup> i i ii i j i ii j jj*

*p G pp G G*

2

*<sup>x</sup> i i ii i j i ii j jj*

 

*<sup>J</sup> p G pp G G*

*<sup>z</sup> i i ii i j i ii j jj*

*<sup>J</sup> r G rr G G*

*o o <sup>o</sup> i i ii i j i ii j jj*

   

 

> 

*r G rr G G*

*<sup>y</sup> x z i i ii i i ii i i ii i ii*

 

*<sup>J</sup> J J pG qG rG*

222

The Lagrange equation of motion for a parallel manipulator can be written as:

reduce the dynamical coupling we represent the kinetic energy as follows:

6 6 2

... 2

.... ( ) ....

... 2

6 6 2

6 6 2 1 1, 1,

2

1 1, 1,

*i i j i j*

6 6 2 1 1, 1,

*i i j ij*

According to the Eq. (13) dynamical decoupling can be satisfied if the columns of the

1 11 6 66 1 11 6 66 1 11 6 66 1 11 6 66 1 11 6 66 1 11 6 66

*o o o o o o*

.... ....

*p G m pG m qG m qG m rG m rG m*

 

> .... ....

*p G J pG J qG J qG J rG J rG J*

*x x y y z z*

*i i j ij*

1 1, 1,

*i i j i j*

<sup>222</sup> <sup>666</sup>

2 22 6 66

*i, qo <sup>i</sup>*, *ro*

*i*

(12)

(11)

(13)

*<sup>i</sup>* are the components of the matrix

 <sup>1</sup> 000 0 0 0 0000 00 0 0 0 ( ) ( ) 000 0 0 0000 0 0000 0 *x y z m m m D E G J J J* <sup>1</sup> () () *D ME G* (14)

Let the axes of the links *AiCi* be parallel to the axes of the links *BiCi* and the matrix *(G)* is unit matrix. Then in order to satisfy the condition of orthogonal columns of the matrix *(D)* the rows of the inverse matrix *(D)-1 = (E)(M)-1* are to be orthogonal. From this, the following matrix *(E)* is proposed:

$$
\begin{pmatrix}
\begin{matrix}
0 & 0 & 1 & 0 \\
0 & 0 & -1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 1 & 0 & -\sqrt{I\_x/m} \\
0 & 1 & 0 & -\sqrt{I\_x/m} \\
0 & -1 & 0 & -\sqrt{I\_x/m} \\
\end{matrix}
\end{pmatrix}
\tag{15}
$$

The determinant of the matrix *(E)* (15) can be written as:

$$\det(E) = 8 \sqrt{J\_{\chi} J\_{\chi} J\_{z} / m^3} \tag{16}$$

The matrix (15) corresponds to the Fig. 3. Here the center of the mass of the platform coincides with the center of the coordinate system xyz and the axes of the links of the legs

Fig. 3. Parallel manipulator with dynamical decoupling

are parallel to the main central inertia axes of the platform. The proposed approach can be applicable for manipulators characterized by small displacements and high speeds. Moreover, this architecture causes partial kinematic decoupling because if the generalized coordinates corresponding to the opposite legs are equivalent then the moving platform keeps constant orientation.

#### **6. Pressure angles**

The parallel manipulators have singularity configurations in which there is an uncontrolled mobility because some of the wrenches acting on the output link are linearly dependent. The local criterion of singular configurations is the singular matrix of the screw coordinates of the wrenches, such as:

$$\det(E) = z^\* \tag{17}$$

Singularity Analysis, Constraint Wrenches and Optimal Design of Parallel Manipulators 369

In Fig. 4, (a, b) the six-DOFs parallel mechanisms and their sub-chain has a parallel connection of links and actuators are shown in (Glazunov, et al. 1999) which were invented by (Kraynev, & Glazunov, 1991). Such mechanisms may be utilized to manipulate the corrosive medium at all actuators that are located out of the working space. Existence of several sub-chains and many closed loops determine the essential complication of the mathematical description of these mechanisms. Screw calculus using screw groups is universal and effective for parallel mechanisms analysis. Here, *1* describes the fixed base, *2* describes the output link and *3* describes the actuators. Addition, *Ai* expresses the spherical joint center situated on the fixed base; *Bj* expresses the center of the spherical joints combined with translational; *Cj* expresses the output link spherical joint centers; *li, dj* expresses the generalized coordinates and *sj, fj* expresses the link lengths *(i=1,…,6; j=1,…, 3)*.

*<sup>3</sup> <sup>3</sup> <sup>B</sup> <sup>3</sup> <sup>B</sup>*

*A*

*B*

(a)

*<sup>A</sup> <sup>A</sup>*

*D*

*<sup>B</sup> <sup>B</sup>*

*A*

*3*

*A*

*D*

*2*

*D*

*2*

*1*

*1*

*A*

*C C*

*C C*

(b)

*D*

In general, the wrench axis corresponding to *i-th* stalled actuator is located in the plane (*АiВjСj*), passes through center joint *Cj* and is directed perpendicular to its possible

Fig. 4. The six-DOFs parallel mechanisms

*3*

*A*

*A*

*D*

*C*

*A*

*D*

*C*

**7. Manipulator for external conditions** 

where *\** is the preassigned minimal determinant value. The pressure angle of the linear dependent sub-chain is equal to */2*, as a reciprocal twist to five-member group of screws has a perpendicular moment at about any points of the axis. All stalled actuators but one the manipulator has *DOF=1* and its output link can move along some twist = + 0 (*2 = 0*) reciprocal to five-member group of the wrenches corresponding to stalled actuators. We can find this twist from the reciprocity condition:

$$
gamma(\Omega, R\_i) = 0, \ i = 1, \ldots, 5 \tag{18}$$

In general the six-member group of the unit wrenches of zero parameter *Ri(ri, ri o) (i=1,…,6)* is acting on the output link of the such manipulators, determinant composed of the screw coordinates of these wrenches as given in (5).

The velocity of any point *Ai (i=1,…,6)* of the mobile platform can be found as a twist moment relative to this point:

$$V\_{A\_i} = \alpha^0 + \alpha \times r\_{A\_i}, \text{ i } = 1, \dots, 6 \tag{19}$$

where *rAi* is radius-vectors of the points *Ai.*

The pressure angle *<sup>i</sup>* for the stalled actuator *i-th* of the parallel manipulators (Fig.1) can be determined as:

$$\alpha\_{i} = \arccos\left(\frac{V\_{A\_{i}} F\_{i}}{\left|V\_{A\_{i}}\right|.\left|F\_{i}\right|}\right)' \text{ } i = 1, \ldots, 6 \tag{20}$$

where *Fi* is the force vector on the actuator axis. For normal functions of the manipulator it is necessary that working space be limited by positions:

$$a\_i \le a\_{KP'} \text{ i = 1,...,6} \tag{21}$$

where *КР* is maximum pressure angle is defined by friction coefficient.

The manipulator control system must be provided by algorithm testing the nearness to singular configurations based on the analysis of singular matrix (5) or on the pressure angle determination.
