Dimensional Synthesis and Analysis of the 2-UPS-PU Parallel Manipulator

The purpose of this is to encourage and inspire the continual invention of robot manipulators for science and the good of humanity. The concepts of artificial intelligence combined with the engineering and technology of feedback control, have great potential for new, useful and exciting machines. The concept of eclecticism for the design, development, simulation and implementation of a real time controller for an intelligent, vision guided robots is now being explored. The dream of an eclectic perceptual, creative controller that can select its own tasks and perform autonomous operations with reliability and dependability is starting to evolve. We have not yet reached this stage but a careful study of the contents will start one on the exciting journey that could lead to many inventions and successful solutions.


Introduction
Compare with conventional serial robots, parallel manipulators has the advantages of higher accuracy, higher stiffness, and higher ratio of force-to-weight, so it has been intensively researched and evaluated by industry and intuitions over the last two decades [1]. It is well known that a main drawback of parallel manipulator is their reduced workspace. Furthermore computing this workspace is not an easy task as, at the opposite of classical serial robot, the translational and orientation workspace are coupled [2]. A number of authors have described the workspace of a parallel mechanism by discretizing the Cartesian workspace [3]. In the case of three degree of freedom (3-DOF) manipulators, the workspace is limited to a region of the three-dimensional Cartesian space. A more challenging problem is designing a parallel manipulator for a given workspace. Merlet [4] propounded an algorithm to determine all the possible geometries of Gough-type 6-DOF parallel manipulators whose workspace must include a desired one. Boudreau and Gosselin [5] proposed an algorithm that allows for the determination of some parameters of the parallel manipulators using a genetic algorithm method in order to obtain a workspace as close as possible to a prescribed one. Kosinska et al. [6] presented a method for the determination of the parameters of a Delta-4 manipulator, where the prescribed workspace has been given in the form of a set of points. Snyman et al. [7] proposed an algorithm for designing the planar 3-RPR manipulator parameters, for a prescribed two-dimensional physically reachable output workspace. Laribi et al. [8] presented an optimal dimensional synthesis method of the DELTA parallel manipulator for a prescribed workspace. This problem was generally solved numerically, and none of the authors mentioned above took driving force into account. In this paper, the 2-UPS-PU Parallel manipulator is designed to have a specified workspace. The algorithm is proposed to solve the optimization problem which not only takes into account the leg-length limits, the mechanical limits on the passive joints, and interference between links, but also the driving forces of the three legs.

Displacement analysis
The 2-UPS-PU parallel manipulator is shown in Fig. 1. This manipulator consists of three kinematic chains, including two UPS legs with identical topology and one PU leg, connecting the fixed base to a moving platform. In this parallel manipulator, the UPS legs, from base to platform, consist of a fixed Universal joint, an actuated prismatic joint and a spherical joint attached to the platform. The PU leg connecting the base center to the platform consists of a prismatic joint attached to the base, a universal joint attached to the platform. This branch is used to constrain the motion of the platform to the three degrees of freedom. The reference frame O-X b Y b Z b is fixed on the base and mobile frame U 0 -X p Y p Z p is fixed on the moving platform (see Fig. 1). At the initial position, the X p -axis and Y p -axis of mobile frame are coincidence with the axes of the Universal joints U 0 respectively. The orientation of the first axis of U 0 is fixed. The orientation of the mobile frame can be represented by θ 1 and θ 2 shown in Fig. 2, which are two Euler angles about two axes of U 0 , respectively. Such Euler angles are defined by first rotating the mobile frame about the base x p -axis (first axis of U 0 ) by an angle 1  , then about the mobile y p -axis by an angle. For this choice of Euler angles, the rotation matrix is defined as ¶(9pt) are the basic rotation matrices.
The position of the mobile frame can be represented by l 0 , which is the distance between the Universal joint U 0 and reference point O. The homogeneous transform matrix T, which represents the orientation and position of the mobile frame, is ¶(9pt)     Initial orientation

Fig. 2. The 2-UPS-PU parallel manipulator
The spherical joints (S 1 and S 2 ) of the UPS legs are arranged on the moving platform and their distances to the Universal joint U 0 on the moving platform is r. The Universal joints (U 1 and U 2 ) are fixed on the base platform and the distances to the reference point O on the base is R. The coordinate of U 0 , S 1 and S 2 in mobile frame and the coordinate of U 1 and U 2 in reference frame are expressed as:

Inverse kinematics
For a given position and orientation of the mobile platform, we can compute the related link lengths, denoted by l i , using the following relation:  (4) is the solution of the so-called inverse kinematics problem.

Direct kinematics
If set the second axis of Universal joint U 0 pass through either S 1 or S 2 ( 0 Then, we can calculate θ 1 and θ 2 by ¶(9pt)     (6pt) The workloads can be simplified as a wrench F w applied onto moving platform at U 0 .

Statics analysis
When ignoring the friction in all the joints and the mass of all the parts, the wrench F w is balanced by three active forces f i (i = 1,2, 3), two constrained forces f x and f y , and a constrained torque m z . Each of f i is applied on and along the axes of three legs; f x , f y and m z are applied on the moving platform about X p , Y p , Z p , respectively. the formulae for solving active forces and constrained forces are derived as ¶(9pt) where www.intechopen.com

Mechanical constraints
There are four main mechanical constraints that limit the workspace of a parallel manipulator The theoretical workspace of the manipulator is surrounded by singular surface. In the theoretical workspace, the determinant of jacobian matrix (|q|) should be always greater or less than zero. The relationship between the determinant of jacobian matrix and parameters of θ 1 θ 2 , for r =150, R =200, α=π/2, β=π/6 and l 0 =300, is shown in Fig. 3. If we set θ 1 =θ 2 =0 as the initial oriention, the determinant of jacobian matrix should be always greater than zero. ¶(9pt) | | 0 q  (9) ¶(9pt)

Actuators' stroke
The limited stroke of actuator i imposes a length constraint on link i, such that ¶(9pt) max min min 0.8 0,1, 2 where l imin and l imax are, respectively, the minimum and maximum lengths of leg i. ¶(9pt)

Range of the passive joints
Each passive joint has a limited range of motion. Let the maximum misalignment angle of the Universal joint U i be θ iumax , Then, the limits on Universal joint U i impose a constraint, such that ¶(9pt) max iU iU    (11) Similarly, Let the maximum misalignment angle of the spherical joint S i be θ ismax , Then, the limits on spherical joint S i impose a constraint, such that ¶(9pt)

Link interference
Let us assume that the links can be approximated by cylinders of diameter D. This imposes a constraint on the relative position of all pairs of links, such that or the minimum distance between every two line segments corresponding to the links of the parallel manipulator should be greater than or equal to D. The minimum distance between two line segments is not given by a simple formula but can be obtained through the application of a multi-step algorithm. Due to space limitations, we will not present that algorithm here but refer the reader to the well-detailed one given in [9].

Fig. 4. Cylindrical coordinate system
According to the characteristics of the 2-UPS-PU parallel manipulator, we present a cylindrical coordinate system (see Fig. 4), where φ and γ are exactly the polar coordinates representing the orientation of the moving platform and l 0 is the z-coordinate. The unit vector of Z p -axis in reference frame is expressed as: www.intechopen.com

Algorithm for the workspace ¶(6pt)
Step 1: Initialize double arrays A, B and Z, with dimensions (n+2)×m, where n is the number of equally spaced planes l 0 between l 0min and l 0max at which the workspace will be computed, and m is the number of points to be computed at each plane l 0 =const. These arrays will store, respectively, the values of ,   and l 0 for the points defining the workspace boundary.
Step 5: For the current l 0 , construct a polar coordinate system at   Step 7: Repeat steps 5-6 until l 0 becomes greater than l 0max .
Step 8: Transfer A and B into X and Y, so that

Optimal design
The aim of this section is to develop and to solve the multidimensional optimization problem of selecting the geometric design variables for the 2-UPS-PU parallel manipulator having a prescribed workspace with better driving capability. The prescribed workspace of the parallel manipulator is defined as a cylinder with radius γ and height h (see Fig.7) and the actual workspace is convex (see Fig.5). The prescribed workspace is inside the workspace of the parallel manipulator if all the points in the boundary curves are inside the actual workspace. ¶(9pt) Step 12: Repeat steps 4-11 until the cycle time is n. The proposed algorithm was again implemented in MATLAB. The optimal solution obtained for this example is presented in Table 1 for 60 u    and 100 h  . Fig. 8 shows the workspace of the 2-UPS-PU parallel manipulator. Fig. 9 shows the relationship between the minimum of γ and l 0 . One can notice the actual workspace include the prescribed workspace. ¶(9pt) β R r l 1,2min l 1,2max l 0min l 0max 27° 186 124 396 712 450 650 Table 1. The optimal dimensions of 2-UPS-PU parallel manipulator ¶(9pt)

Conclusion
This paper proposed a novel 2-UPS-PU parallel manipulator with simple structure, high rotational capability, high load carrying capacity. Based on the cylindrical coordinate system, an algorithm for computing three-dimensional workspace of the manipulator has been proposed in this paper. The boundary of the workspace on the specific plan is found out quickly by step-searching along the selected ray line. An optimal dimensional synthesis method was presented to optimize this manipulator for a prescribed workspace. The driving force parameters were introduced into the object function. All the dimensional parameters, including the length of the legs, were included into the optimizing algorithm. The methods proposed in this paper can be adopted universally. The results of this paper provide solid theoretical basis for further theoretical studies and practical application of this manipulator. ¶(14pt)