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Senior Lecturer Dr. Eng. Sergiu-Dan Stan

Technical University of Cluj-Napoca Department of Mechanisms, Precision Mechanics and Mechatronics B-dul Muncii, Nr. 103-105, Corp C, Et. III, C304 400641 Cluj-Napoca, ROMANIA, EU Tel. +40-(0)264-401755 e-mail. [email protected] http://www.east.utcluj.ro/mec/mmfm/Colegi/stan.html http://www.sergiustan.ro/

What is a Parallel Manipulator?

A parallel manipulator consists of a moving platform and a fixed base, connected by several linkages (also called legs).

Figure 1. Classic Stewart platform (6 DOF) The Stewart Platform is characterized by high force-torque capacity, high structural rigidity, and low moving mass.

Figure 2. A typically PKM with 3 DOF designed for milling applications

As PKM are used for more difficult tasks, control requirements increase in complexity to meet these demands. The implementation for PKMs often differs from their serial counterparts, and the dual relationship between serial and parallel manipulators often means one technique which is simple to implement on serial manipulators is difficult for PKMs (and vice versa). Because parallel manipulators result in a loss of full constraint at singular configurations, any control applied to a parallel manipulator must avoid such configurations. The manipulator is usually limited to a subset of the usable workspace since the required actuator torques will approach infinity as the manipulator approaches a singular configuration. Thus, some method must be in place to ensure that the manipulators avoid those configurations. Parallel robots have many advantages comparing to the serial robots, such as high flexibility, high stiffness, and high accuracy. To achieve a higher accuracy the static and dynamic behavior must be better understood. The problems concerning kinematics and dynamics of parallel robots are as a rule more complicated than those of serial one.

Modeling and simulation of integrated environment for parallel robots

Modeling and simulation are important and essential stages in the engineering design and problem solving process because it allows to prevent the risks and to lower the costs that appear with the design, construction and testing stage of a new robot. In the design process of parallel robots different tools can be used to model and simulate a robot. There are however different modeling and simulation tools which can be used together with one another to help into the development of a model. These different tools can vary according to its functionality and according to its best use for Modeling. For example some tools are best suitable for first functionality model such as CAD software (SolidWorks). SolidWorks is mainly a CAD tool although one can also do some simulation in SolidWorks, it is not the best tool used for simulation. For that reason other simulation tools such as ADAMS can be used instead to perform the dynamical simulation of the robot. The same thing, ADAMS is again not the proper CAD tool so one cannot get detailed geometrical information about the parts and elements of the parallel Robots. ADAMS does not also perform the simulation of the functionality of the Robot, so it only shows how a robot would move in reality and one can get real physical information about the Robot and how it would perform dynamically in reality. For simulation and control of a parallel robot other software tool such as MATLAB can be used to simulate the parallel robot.

ROBOT MODEL

Figure 3. Integrated Environment for simulation and modeling of parallel robots

With the help of SolidWorks software we can make: 1) the CAD model of the parallel robot, 2) detailed modeling of the parallel robot components, 3) choose appropriate material for parts from the material library, 4) assembling parts and components, 5) get geometrical and physical information about the parts [2]. With ADAMS software we can determine: 1) the simulation of the dynamical movement of the parallel robot, 2) deal with joints and actuators, 3) simulate the real movement of the parallel robot. With MATLAB software we can elaborate and simulate the control of parallel robots. Unlike parallel robots, a serial robot is an open-chain structure consisting of several links connected in series. The advantages of parallel robots as compared to serial ones are: higher payload-to-weight ratio since the payload is carried by several links in parallel, higher accuracy due to non-cumulative joint error, higher structural rigidity, since the load is usually carried by several links in parallel and in some structures in compression-traction mode only, location of motors at or close to the base, simpler solution of the inverse kinematics equations. In the CAD model note that parallel robots usually are as standard design mechanism, in other words a design model for a parallel robot would only be the effort of designing one chain which is usually repeated symmetrically for the whole robot. Kinematical representation of parallel robots shows in a simple way the kinematical structure of a parallel robot. Mainly parallel robots can be represented with respect to the structure of their parallel mechanism. A parallel mechanics is a repetition of number of mechanical chains which are connecting the platform to the base (as previously defined). For example it was chosen the Hexaglide parallel robot with six degrees of freedom [3].

Figure 4. Hexaglide parallel robot (6 DOF)

3 DOF parallel robot

Figure 5. Kinematics scheme of the 3-RRR planar parallel robot To control the movement of the end-effector of the robot the program uses the inverse kinematics, the user will enter the position of the end-effector and the program will compute the angles of all three actuators. The computation of the angles is made in to steps, in the first step is determine the position of the points A, B, and C, all three points are situated in the corner of an equilateral triangle, the position of the mass center of the triangle is known Px and Py, also the angle q between the end-effector and the Ox axe is given, L3 represent the distance from P to A.

Ax Px L3 cos( )

Ay Py L3 sin( )

(1) (2)

Knowing the position of point A, B, C of the end-effector the problem is reduced to an inverse kinematics problem of a serial robot with 2 arms, fig. 3

Figure 6. Kinematics scheme of the 2-RR manipulator

K1 2( xB xO ) L1

(3)

K 2 2( yB yO ) L1

2 K3 L2 L1 ( xB xO ) 2 ( yB yO ) 2 2

(4) (5)

(6)

1 2[a tan 2( K 2 K12 K 22 K32 , K3 K1 )] 1 2[a tan 2( K 2 K12 K 22 K32 , K3 K1 )]

The angle have two solutions (Eq. 6), this is possible because the mechanism can have the point B in the same position in two situations when the elbow is up and when the elbow is down.

Figure 7. Planar parallel minirobot with 3 DOF The CAD model of the parallel minirobot with 3 dof is presented in figure 2.

Figure 8. CAD model of the planar parallel minirobot with 3 DOF

Figure 9. Configuration of the 3-RRR planar parallel minirobot for x=0.2, y=0.15, =45°

Figure 10. Planar 2 DOF mini parallel robot. The working envelope to machine size using variable length struts is dependent on the following factors: 1. The length of the extended and retracted actuator (Lmin, Lmax); 2. Limitations due to the joint angle range. Fig. 11 clearly illustrates the limiting effect of the joint limits.

Figure 11. Workspace of the micro parallel robot with variable length struts Kinematic analysis Robot kinematics deal with the study of the robot motion as constrained by the geometry of the links. Typically, the study of the robot kinematics is divided into two parts, inverse kinematics and forward (or direct) kinematics. The inverse kinematics problem involves a known pose (position and orientation) of the output platform of the robot to a set of input joint variables that will achieve that pose. The forward kinematics problem involves the mapping from a known set of input joint variables to a pose of the moving platform that results from those given inputs. However, the inverse and forward kinematics problems of our parallel robot can be described in closed form. The kinematics relation between x and q of this 2 DOF mini parallel robot can be expressed solving the: f(x, q)=0 (1)

Then the inverse kinematics problem of the parallel robot can be solved by writing the following equations:

2 2 q1 xP yP 2 q2 (b xP )2 yP

(2)

The TCP position can be calculated by using inverted transformation, from (2), thus the direct kinematics of the robot can be described as:

xP

2 q12 b 2 q2 2b

(3)

2 yP q12 xP

where the values of the xp, yP can be easily determined. The forward and the inverse kinematics problems were solved under the MATLAB environment and it contains a user friendly graphical interface. The user can visualize the different solutions and the different geometric parameters of the parallel robot can be modified to investigate their effect on the kinematics of the robot. This graphical user interface can be a valuable and effective tool for the workspace analysis and the kinematics of the parallel robots. The designer can enhance the performance of his design using the results given by the presented graphical user interface. The Matlab-based program is written to compute the forward and inverse kinematics of the parallel robot with 2 degrees of freedom. It consists of several MATLAB scripts and functions used for workspace analysis and kinematics of the parallel robot. A friendly user interface was developed using the MATLAB-GUI (graphical user interface). Several dialog boxes guide the user through the complete process.

Figure 12. Graphical User Interface (GUI) for solving inverse kinematics of the 2 DOF planar parallel robot in MATLAB environment. The user can modify the geometry of the 2 DOF parallel robot. The program visualizes the corresponding kinematics results with the new inputs.

Figure 13. The GUI for calculus of workspace for the planar 2 DOF mini parallel robot. Case I: Conditions: q1min q2 min b , q1 max b , q2 max b a) for y>0

Figure 14. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region. b) for y , there exist two regions of the workspace

Figure 15. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region. Case II: Conditions: q1min q2 min b , q1 max b , q2 max b a) for y>0

Figure 16. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region. b) for y , there exist two regions of the workspace

Figure 17. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region. Case III: Conditions: q1 min q2 min b , q1 max b , q2 max b

Figure 18. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region. Case IV: Conditions: q1min q2 min b , q1 max b , q2 max b

Figure 11. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region. Case V: Conditions: q1 min q2 min b , q1 max b q2 min , q2 max b q1 min

Figure 12. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region. Case VI:

q b q1min Conditions: q1 min q2 min b , q1 max b q2 min , 2 max

Figure 13. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region. Case VII: Conditions: q1 min b , q1 max b , q2 min b , q2 max b ,

q1 min q2 min b , q1 max q2 max b

Figure 14. The workspace of the planar 2 DOF mini parallel robot is shown as the shading region.

Description of the 2 DOF parallel structure

A planar parallel robot is formed when two or more planar kinematic chains act together on a common rigid platform. For simplicity, the origin of the fixed base frame {B} is located at base joint A with its x-axis towards base joint B, and the origin of the moving frame {M} is located in TCP, as shown in Fig. 15. The position of the moving frame {M} in the base frame {B} is x=(xP, yP)T and the actuated joint variables are represented by q=(q1, q2)T.

Figure 15. Kinematic scheme of the 2 DOF parallel robot The kinematics relation between x and q of this 2 DOF mini parallel robot can be expressed solving the: f(x, q)=0 (1)

Then the inverse kinematics problem of the parallel robot can be solved by writing the following equations:

2 2 2 q1 x p l12 y P ; q2 x p l2 yP

(2)

The designer can enhance the performance of his design using the results given by the presented graphical user interface. The Delphi-based program is written to compute the forward and inverse kinematics of the parallel robot with 2 degrees of freedom.

Figure 15. Assembly of the robot, CAD model of 2 DOF parallel mini robot A general planar micro parallel robot with two degrees of freedom activated by prismatic joints is shown in Fig. 1.

Fig. 16. Variable length struts micro parallel robot

Fig. 2. The general kinematic scheme of a 2 DOF parallel robot with translation actuators

Fig. 3. Assembly of the robot, CAD model of 2 DOF parallel mini robot

Fig. 5. Workspace limits of Biglide The mechanism PRRRP realizes a wide workspace as well as high-speed. Analysis, visualization of workspace is an important aspect of performance analysis. A numerical algorithm to generate reachable workspace of parallel manipulators is introduced.

This section presents the methodology to determine the workspace of the 2 DOF mini parallel robot. It consists of several MATLAB scripts and functions used for workspace analysis and kinematics of the parallel robot. A friendly user interface was developed using the MATLAB-GUI (graphical user interface). Several dialog boxes guide the user through the complete process.

Fig. 6. The GUI for calculus of workspace for the planar 2 DOF mini parallel robot. A fundamental characteristic that must be taken into account in the dimensional design of robot manipulators is the area of their workspace. It is crucial to calculate the workspace and its boundaries with perfect precision, because they influence the dimensional design, the manipulator's positioning in the work environment, and its dexterity to execute tasks.

a) Workspace for the planar 2 DOF mini parallel robot, case q1max q2 max 100 mm

b) Workspace for the planar 2 DOF mini parallel robot, case q1 max q2 max 200 mm

c) Workspace for the planar 2 DOF mini parallel robot, case q1max q2 max 400 mm Fig. 7. Different regions of workspace for different lengths of stroke of actuators a), b) and c) Factors that limit the workspace of a given parallel manipulator include actuator limits, leg interference, and singular configurations.

Fig. 8. Two different areas of workspace in two possible configurations C. Singularity analysis Because singularity leads to a loss of the controllability and degradation of the natural stiffness of manipulators, the analysis of parallel manipulators has drawn considerable attention. In some configurations, the robot cannot be fully controlled. Most parallel robots suffer from the presence of singular configurations in their workspace that limit the machine performances. In the case of parallel manipulators and closed-loop mechanisms, singularity analysis is much more difficult since such mechanisms contain unactuated joints and joints with more than one degree of freedom. In general, closed-form solutions for singular curves/surfaces for parallel manipulators of arbitrary architecture requires elimination of unwanted variables from several nonlinear transcendental equations, and this is quite difficult. Based on the forward and inverse Jacobian matrices, three kinds of singularities of parallel manipulators can be obtained [10]. Let q denote the actuated joint variables, and let x describe the location of the moving platform. The kinematic constraints imposed by the limbs are expressed as f(x,q) = 0. Differentiating with respect to time, a relation between the input joint rates and the end-effector output velocity is obtained as:

J x x Jqq

where J x

f f and J q . Hence, the overall jacobian matrix J, can be written as: dx dq

q Jx

(7)

(8)

where J J J .

1 q x

Singular configurations should be avoided. The singular configurations (also called singularities) of a parallel robot may appear inside the workspace or at its boundaries. The first kind of singularity The first kind of singularity occurs when the following condition is satisfied: det(Jx ) 0, det(Jq ) 0 This kind of singularity corresponds to the limit of the workspace. The second kind of singularity The second kind of singularity occurs when we have following: det(Jx ) 0, det(Jq ) 0

(9)

(10)

The physical interpretation of this kind of singularity is that even if all of the input velocities are zero, there are still be instantaneous motion of the end-effector. In this configuration, the manipulator loses stiffness and becomes uncontrollable. This kind of singularity is located inside the workspace of the manipulator. Such a singularity is very difficult to locate only by analyzing and expanding the equation det(Jq ) 0 . A numerical method is thus a good selection for solving this problem. The third kind of singularity The third kind of singularity occurs when both:

det(Jq ) det(Jx ) 0

(11)

This kind of singularity corresponds to the first and second type of singularity occurring simultaneously. This singularity is both configuration and architecture dependent. Parallel singularities are particularly undesirable because they cause the following problems: a high increase of forces in joints and links, that may damage the structure, a decrease of the mechanism stiffness that can lead to uncontrolled motions of the tool though actuated joints are locked.

Fig. 9. Singular configuration for the planar 2 DOF mini parallel robot

Fig. 10. Singular configuration for the planar 2 DOF mini parallel robot

Fig. 11. Singular configuration for the planar 2 DOF mini parallel robot

Fig. 13 Transmission quality index for PRRRP mini parallel robot

Fig. 3 CAD model of the parallel robot The forward and the inverse kinematics problems were solved under the MATLAB environment and it contains a user friendly graphical interface.

Fig. 4. Graphical User Interface for solving IKP

Fig. 5. Robot configuration for XP=25 mm YP=60 mm

Fig. 6. Robot configuration for XP=35 mm YP=60 mm

Fig. 2. Six degrees-of-freedom micro parallel robot

Fig. 3. Workspace views of the HEXAPOD micro parallel robot with six degrees-of-freedom

Fig. 4. Workspace views of the HEXAPOD micro parallel robot with six degrees-of-freedom The possible workspace of the robot is of a great importance for optimization of the parallel robots. Without the ability to solve the workspace is impossible to state that the robot can fulfill any work task. The general analysis of the workspace consists in workspace determination using the described discretization method.

Fig. 5. Graphical User Interface for determining the shape of the HEXAPOD micro parallel robot with six degrees-of-freedom The workspace is the volume in the space case where the tool centre point (TCP) can be controlled and moved continuously and unobstructed. The workspace is limited by singular configurations. At singularity poses it is not possible to establish definite relations between input and output coordinates. Such poses must be avoided by the control.

Fig. 6. Top view of the workspace of the HEXAPOD micro parallel robot with six degrees-offreedom The robotics literature contains various indices of performance [19], [20], such as the workspace index W.

Fig. 7. Lateral view of the workspace of the HEXAPOD micro parallel robot with six degreesof-freedom

C. Performance evaluation Beside workspace which is an important design criterion, transmission quality index is another important criterion. The transmission quality index couples velocity and force transmission properties of a parallel robot, i.e. power features [21]. Its definition runs:

T

E

2

J J 1

(3)

where E is the unity matrix. T is between 0<T<1; T=0 characterizes a singular pose, the optimal value is T=1 which at the same time stands for isotropy [22].

Fig. 8. Transmission quality index for HEXAPOD micro parallel robot with six degrees-offreedom (3D view)

Fig. 9. Transmission quality index for HEXAPOD micro parallel robot with six degrees-offreedom (top view)

As it can be seen, the micro parallel robot presents better performances in the middle of its workspace, as presented in Fig. 6-7. 2. 3-RRR PARALLEL ROBOT AND WORKSPACE ANALYSIS The planar micro 3 DOF parallel robot is shown in Fig.1. This structure is also known as 3RRR robot. Since mobility of this micro parallel robot is three, three actuators are required to control this robot.

Fig. 1. 3-DOF micro parallel robot of type 3-RRR The workspace of a robot is defined as the set of all endeffector configurations which can be reached by some choice of joint coordinates. As the reachable locations of an endeffector are dependent on its orientation, a complete representation of the workspace should be embedded in a 6-dimensional workspace for which there is no possible graphical illustration; only subsets of the workspace may therefore be represented. The knowledge of the workspace of a 3 DOF micro parallel robot is very important in planning a dexterous manipulation task. The workspace is one of the most important kinematic properties of robots, even by practical viewpoint because of its impact on robot design. In this section, the workspace of the proposed robot will be discussed systematically. Here, we propose an approach to compute and visualize the workspace of a 3 DOF micro parallel robot. Micro parallel robots are good candidates for microminiaturization into a microdevice. Case I. Workspace can be determined by using discretization method. There were identified several cases of workspace.

(1)

The main disadvantage of parallel robots is their small workspace in comparison to serial arms of similar size. Despite the advantages of parallel manipulators there are certain disadvantages to be encountered such as complicated kinematics and dynamics, many singular configurations, and poor workspace availability. It is very important to analyze the area and the shape of workspace for parameters given robot in the context of industrial application.

Fig. 2. Workspace of the 3-RRR parallel micro robot

Fig. 3. Workspace of the 3-RRR parallel micro robot

Case II. , , (2)

Singular configurations were identified as it is presented in the following figures.

Fig. 4. Singular configuration, =0°

Fig. 5. Singular configuration,

Fig. 6. Singular configurations,

Fig. 2. CAD model of the 3-DOF UPU micro parallel robot

Fig. 3. The GUI for calculus of workspace for the 3 DOF UPU micro parallel robot. Resolution of the latter is very important for any manipulator design. If we restrict ourselves to a 3-DOF parallel robot (a UPU parallel robot for example) we will find that the link lengths limit the workspace. On the other hand, there will necessarily be a design limitation of some sort on the link lengths. Also one must have a compact design which is capable of full manoeuvring. The workspace is primarily limited by the boundary of solvability of inverse kinematics. Then the workspace is limited by the reachable extent of drives and joints, occurrence of singularities and by the link and platform collisions.Analysis, visualization of workspace is an important aspect of performance analysis. A numerical algorithm to generate reachable workspace of parallel manipulators is introduced. This section presents the methodology to determine the workspace of the 3 DOF micro parallel robot. It consists of several MATLAB scripts and functions used for workspace analysis and kinematics of the parallel robot. A friendly user interface was developed using the MATLABGUI (graphical user interface). Several dialog boxes guide the user through the complete process. In the followings are presented the workspace of the UPU parallel robot for different values of diameter of the circle for the lower platform, d.

Fig. 4. Workspace of the 3-DOF UPU parallel robot, d=100mm

Fig. 5. Workspace of the 3-DOF UPU micro parallel robot, d=110mm

Fig. 6. Workspace of the 3-DOF UPU micro parallel robot, d=120mm

Fig. 7. Workspace of the 3-DOF UPU micro parallel robot, d=130mm This graphical user interface (Fig. 2) can be a valuable and effective tool for the workspace analysis and the kinematics of the parallel robots. The designer can enhance the performance of his design using the results given by the presented graphical user interface. The 2 degree of freedom Parallel Kinematic Machine The Parallel Kinematic Machine considered in this paper is the 2-dof planar parallel mechanism shown in Figure 1. The PKM consists of a five-bar mechanism connected to a base by two rotary actuators, which control the two output degrees of freedom of the endeffector. The actuators are joined to the base and platform by means of revolute joints identified by the letters A­D. It will be assumed that AO=OC=AC/2. The coordinates of point P, the end-effector point, are (xP, yP). In more general terms, the actuator joint angles are the input variables, i.e. v=[q1, q2]T 2. The global coordinates of the working point P form the output coordinates, i.e. u=[xP, yP]T 2. The 2-dof PKM may be used only for positioning P in

the x-y plane. It is evident that this manipulator thus has 2-dof. Thus, the generalized coordinates for this kind of PKM are therefore given by: q=[uT, vT]T=[xP, yP, q1, q2]T4 (1)

In general, factors imposed by the physical construction of the planar parallel manipulator, which limit the workspace, may be related to the input variables or a combination of input, output and intermediate variables. An example of former type for the planar parallel manipulator are joint angles limits, and of the latter, limits on the angular displacement of the revolute joints connecting the legs to the ground and to the platform. These limiting factors are described by means of inequality constraints and may, respectively, take the general forms: vmin v vmax g

min

(2)

max

g(u, v) g

(3)

The above general definitions are necessary in order to facilitate the mathematical description of kinematics and workspaces of the 2-dof planar PKM. In this study mixed constraints, represented by (3), are not taken into consideration.

Figure 1. Parallel Kinematic Machine with 2 degrees of freedom

3. The kinematics and condition number of the manipulator Kinematic analysis of five-bar mechanism is needed before carrying out derivations for the mathematical model. It is considered the five-bar mechanism with revolute joints as in Figure 1. It is known the length of the links as well as the fixed joints coordinates. The fivebar mechanism is symmetric toward Oy-axis, thus la = ld = l respectively lb = lc = L. Actuators are placed in A and C. Attaching to each link a vector, on the OABPO respectively OCDPO, we can write successively the relations:

OP OA AB BP; OP OC CD DP. Based on the above relations, the coordinates of the point P have the following forms: d d xP l cos q1 L cos q3 l cos q2 L cos q4 ; 2 2 yP l sin q1 L sin q3 l sin q2 L sin q4 .

(4)

(5)

3.1 Direct Kinemtics Problem Analysis In this part, kinematics of a planar POM articulated with revolute type joints has been formulated to solve direct kinematics problem, where the position, velocity and acceleration of the PKM end effector are required for a given set of joint position, velocity and acceleration. The Direct Kinematic Problem (DKP) of PKM is an important research direction of mechanics, which is also the most basic task of mechanic movement analysis and the base such as mechanism velocity, mechanism acceleration, force analysis, error analysis, workspace analysis, dynamical analysis and mechanical integration. For this kind of PKM solving DKP is easy. Coordinates of point P in the case when values of joint angles are known q1 and q2 are obtained from relations:

xP

D D 2 4 BC , 2C

yP

A x P ( xB x D ) . yB yD

(6)

where: 1 A ( x 2 y 2 x 2 y 2 L2DP L2BP ); B ( y B y D ) 2 ( x 2 y 2 L2DP ) A 2 2 y D ( y B y D )A D D B B D D 2 (7) C ( y B y D ) 2 ( x B x D ) 2 ; D 2 y D ( y B y D )( x B x D ) 2x D ( y B y D ) 2 2A( x B x D ) d d x D l cos q 2 ; y D l sin q 2 ; x B l cos q1 ; y B l sin q1 2 2 The speed of the point P is obtained differentiating the relations (1). Thus results:

Vx JA JB 1 3 Vy

where J A

L cos q3 L cos q4

L sin q3 0 l L sin(q1 q3 ) . ; J B L sin q4 0 l L sin(q2 q4 ) (8)

or where

J J B J A1

Vx 1 V J . 3 y sin q4 sin(q1 q3 ) sin q3 sin(q2 q4 ) L . sin(q4 q3 ) cos 4 sin(q1 q3 ) cos q3 sin(q2 q4 )

(9)

(10)

And J represents the Jacobian matrix. Acceleration of the point P are obtained by differentiating of relation (9), as it yealds:

Ax A J y

15 y 10

d 1 J 1. 3 dt 3

15 y 10

(11)

5

5

0 O x

0 O x

-5 -8

-6

-4

-2

0

2

4

6

8

-5 -8

-6

-4

-2

0

2

4

6

8

Figure 2. The two forward kinematic models: (a) the up-configuration and (b) the downconfiguration 3.2 Inverse Kinematics Problem Analysis Based on the inverse kinematics analysis are determined the motion lows of the actuator links function of the kinematics parameters of point P. The values of joint angles qi , (i = 1...4 ) knowing the coordinates xP, yP of point P, may be computed with the following relations: 2 2 2 2 B B 2 (C 2 A 2 ) i ; q arctg M N P ) arctg N ; 2 q1 2arctg M 3 CA P 2 2 2 B B 2 (f 2 e 2 ) i ; q 2arctg b b (F E ) , = 1 or - 1 (12) q 2 2arctg 4 i f e FE

d d d 2 a 2 L xP ; b 2 y P L; c xP y P l 2 L2 ; e 2l xP ; 2 2 2 d d 2 f xP y P l 2 L2 ; A 2l xP ; B 2 y P l ; 2 2 d d d 2 2 C xP y P l 2 L2 ; E 2 L xP ; F xP y P l 2 L2 ; 2 2 2 2 2 M 2 L ; N 2 L ; P ; l cos q1 l cos q2 d ; l sin q1 l sin q2 . From Eq. (15), one can see that there are four solutions for the inverse kinematics problem of the 2-dof PKM. These four inverse kinematics models correspond to four types of working modes (see Fig. 3).

2 2 2 2

15

15

15

y 10

y 10

y 10

5

5

5

0 O x

0 O x

0 O x

-5 -8

-6

-4

-2

0

2

4

6

8

-5 -8

-6

-4

-2

0

2

4

6

8

-5 -8

-6

-4

-2

0

2

4

6

8

15

y 10

5

0 O x

-5 -8

-6

-4

-2

0

2

4

6

8

a) b) c) d) Figure 3. The four inverse kinematics models: (a)"+-" model; (b)" -+" model; (b)" --" model; (d)"++" model. 4. Singularities analysis of the planar 2-dof PKM In the followings, vector v is used to denote the actuated joint coordinates of the manipulator, representing the vector of kinematic input. Moreover, vector u denotes the Cartesian coordinates of the manipulator gripper, representing the kinematic output. The velocity equations of the PKM can be rewritten as:

Au Bv 0

(13)

T T Where v q1 , q2 , u xP , yP , and where A and B are square matrices of dimension 2, called Jacobian matrices, with 2 the number of degrees of freedom of the PKM. Referring to Eq. (13), (Gosselin and Angeles, 1990), have defined three types of singularities which occur in parallel kinematics machines.

(I) The first type of singularity occurs when det(B)=0. These configurations correspond to a set of points defining the outer and internal boundaries of the workspace of the PKM. (II) The second type of singularity occurs when det(A)=0. This kind of singularity corresponds to a set of points within the workspace of the PKM. (III) The third kind of singularity when the positioning equations degenerate. This kind of singularity is also referred to as an architecture singularity (Stan, 2003). This occurs when the five points ABCDP are collinear.

15

15

y 10

10

y

5

5

0 O x

0 O x

-5 -5

0

5

10

15

-5 -5

0

5

10

15

a)

b)

15

15

y 10

y 10

5

5

0 O x

0 O x

-5 -5

0

5

10

15

-5 -5

0

5

10

15

c)

d)

Figure 4. Some configurations of singularities: (a) the configuration when lb and lc are completely extended (b) both legs are completely extended; (c) the second leg is completely extended and (d) the first leg is completely extended. In this paper, it will be used to analyze the second type of singularity of the 2-dof PKM introduced above in order to find the singular configuration with this type of PKM. For the first type of singularity, the singular configurations can be obtained by computing the boundary of the workspace of the PKM. Furthermore, it is assumed that the third type of singularity is avoided by a proper choice of the kinematic parameters. For this PKM, we can use the angular velocities of links lc and lb as the output vector. Matrix A is then written as:

l cos(q4 ) lb cos(q3 ) Ac lc s in (q4 ) lb sin(q3 )

From Eq. (14), one then obtains:

(14)

det( A) lc lb sin(q4 q3 ).

(15)

From Eq. (15), it is clear that when q4 q3 n , n 0, 1, 2, ...., then det( A) 0. In other words if the two links lc and lb are along the same line, the PKM is in a configuration which corresponds to be second type of singularity. 5. Optimal design of the planar 2-dof PKM The performance index chosen corresponds to the workspace of the PKM. Workspace is defined as the region that the output point P can reach if q1 and q2 changes from 2 without the consideration of interference between links and the singularities. There were identified five types of workspace shapes for the 2-dof PKM. Each workspace is symmetric about the x and y axes. Workspace was determined using a program made in MATLABTM.

Figure 5. Workspace of the 2-dof Parallel Kinematic Machine Three DOF parallel robots The parallel robots are with 3 degrees-of-freedom parallel manipulator comprising a fixed base platform and a payload platform, linked together by three independent, identical, open kinematic chains (Fig. 1-2). The TRIGLIDE parallel robot consists of a spatial parallel structure with three translational degrees of freedom and is driven by three linear actuators. The platform is connected with each drive by two links forming a parallelogram and allowing only translational movements of the platform and keeping the platform parallel to the base plane. An additional rotational axis can be mounted on the working platform to adjust the orientation of the end-effector. The three drives of the structure are arranged in the base plane at intervals of 120 degrees star-shaped. Thus the structure has a workspace which is nearly round or triangle-shaped.

Fig. 1. TRIGLIDE parallel robot with 3 DOF. The DELTA linear parallel robot with 3 DOF is shown in Fig. 2 and the geometric parameters are presented in Fig. 3, where the moving platform is connected to the base platform by three identical serial chains. Each of the three chains contains one spatial parallelogram. The parallelogram is actually the vertices of which are four spherical joints

Fig. 2. DELTA parallel robot with linear actuators. Mathematical model To analyze the kinematics model of the parallel robots, two relative coordinate frames are assigned, as shown in Fig. 3. A static Cartesian coordinate frame XYZ is fixed at the center of the base while a mobile Cartesian coordinate frame XPYPZP is assigned to the center of the mobile platform. Ai, (i = 1, 2, 3) and Bi, (i = 1, 2, 3) are the joints that are located at the center of the base and the platform passive joints respectively. A middle link L 2 is installed between the mobile and fixed platform.

Fig. 3. Schematic diagram of TRIGLIDE parallel robot. Let L1, L2, L3 be the link's lengths as expressed in Eq. (1).

(1)

Fig. 4. Schematic diagram of mobile and fixed platform for TRIGLIDE parallel robot. It is known (i=1, 2, 3), afterwards is computed rAi Bi rBi rAi and then:

rBi rP rPBi x P L3 cos i y P L3 sin i zP

(2)

where i is computed as i (i 1) 120. It's obtained also rAi from Eq. (3):

qi cos i rAi qi sin i 0

(3)

where i is computes as i (i 1) 120. From Eq. (3) yields fi:

(

)

|

| (4)

(5) From Eq. (4) we obtain Eq. (5) and by reformulating Eq. (5) is obtained:

(6) and (7) by substitution from Eq. (7) we obtain the inverse kinematics problem of the TRIGLIDE parallel robot from Fig. 1 by solving the following equations: (8) The forward and the inverse kinematics problems were solved under the MATLAB environment and it contains a user friendly graphical interface. For DELTA linear the closed-form solutions for both the inverse and forward kinematics have been developed in [14]. Here, for convenience, we recall the inverse kinematics briefly.

a) 3 DOF DELTA linear parallel robot

b) Fixed platform

c) Mobile platform Fig. 5. Schematic diagram of mobile and fixed platform for DELTA linear parallel robot. Let R and r be the radii of the base and the platform passing though joints Pi and Bi (i = 1, 2, 3), respectively.

P1

P2

R P3 0 0

R 2 3 R 2 0

R 2 3 R 2 0

(9)

B1

B2

r r r 2 2 0 r 3 r 3 B3 2 2 0 0 0 0 0 0 u1 u 2 u3 0 0 0 1 1 1

(10)

(11)

After computing the position of joints Pi and Bi the inverse kinematics of the DELTA parallel robot with linear actuators can be solved by writing following equations: ( (( ) ) ) (( )

) (12)

((

)

)

((

)

)

Eq. (12) represents the inverse kinematics problem of the DELTA linear parallel robot. Workspace evaluation In this section, the workspace of the proposed robots will be discussed systematically. It is very important to analyze the area and the shape of workspace for parameters given robot in the context of industrial application. The workspace is primarily limited by the boundary of solvability of inverse kinematics. Then the workspace is limited by the reachable extent of drives and joints, occurrence of singularities and by the link and platform collisions. The parallel robots TRIGLIDE and DELTA linear realize a wide workspace. Analysis, visualization of workspace is an important aspect of performance analysis. A numerical algorithm to generate reachable workspace of parallel manipulators is introduced. Other design specific factors such as the end-effector size, drive volumes have been neglected for simplification.

Fig. 6. The GUI for calculus of workspace for the TRIGLIDE 3 DOF parallel robot.

Fig. 7. The GUI for calculus of workspace for the DELTA linear 3 DOF parallel robot

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