A Robomech Class Parallel Manipulator With Three Degrees of Freedom

This paper presents the methods of structural-parametric synthesis and kinematic analysis of a parallel manipulator with three degrees of freedom working in a cylindrical coordinate system. This parallel manipulator belongs to a RoboMech class because it works under the set laws of motions of the end-effector and actuators, which simplifies the control system and improves its dynamics. Parallel manipulators of a RoboMech class work with certain structural schemes and geometrical parameters of their links. The considered parallel manipulator is formed by connecting the output point to a base using one passive and two active closing kinematic chains (CKC). Passive CKC have zero degree of freedom and it does not impose a geometrical constraint on the movement of the output point, so the geometrical parameters of the links of the passive CKC are freely varied. Active CKCs have active kinematic pairs and they impose geometrical constraints on the movement of the output point. The geometrical parameters of the links of the active CKCs are determined on the basis of the approximation problems of the Chebyshev and least-square approximations. For this, the equations of geometrical constraints are derived in the forms of functions of weighted differences, which are presented in the forms of generalized (Chebyshev) polynomials. This leads to linear iterative problems.<br><br>The direct and inverse problems of the kinematics of the investigated parallel manipulator are solved. In the direct kinematics problem, the coordinates of the output point are determined by the given position of the input links. In the inverse kinematics problem, the positions of the input links are determined by the coordinates of the output point. The direct and inverse problems of the kinematics of the investigated parallel manipulator are reduced to solving problems on the positions of Sylvester dyads. Numerical results of structural-parametric synthesis and kinematic analysis of the considered parallel manipulator are presented. The numerical results of the kinematic analysis show that the maximum deviation of the movement of the output point from the orthogonal trajectories is 1.65 %


Introduction
Existing methods of designing both serial and parallel manipulating robots are mainly reduced to solving the inverse kinematics problem, i. e. determining the laws of movements of manipulator actuators according to the specified laws of movements of end-effectors, followed by the development of control systems. At the same time, manipulator actuators usually operate in controlled modes of intensive accelerations and decelerations, which worsens their dynamics and reduces efficiency.
Parallel manipulators, i. e. manipulators with closed kinematic chains, which possess the property of manipulating robots as the manipulation of moving output objects in accordance with their laws of movements and possess the property of mechanisms as setting the laws of movements of actuators, are called parallel manipulators of a RoboMech class [1][2][3]. In simultaneously setting the laws of movement of end-effectors and actuators, parallel manipulators of a RoboMech class work under certain structural schemes and geometric parameters of the links.

Literature review and problem statement
The basis for the structural synthesis of planar mechanisms is proposed in Assur [4,5], according to which the mechanism is formed by connecting to the input link (actuator) and the base of structural groups with zero degree of freedom (DOF). These structural groups are then called Assur groups. The Assur groups or Assur kinematic chains based composition principle for planar mechanisms is extended for spatial mechanisms [6].
The structure of serial manipulators consists of a manipulator arm, which represents an open kinematic chain with three DOF, and a wrist mechanism with three DOF. The arm of the manipulator delivers the wrist mechanism to the specified position of space, and the wrist mechanism provides the orientation of the gripper.
In a parallel manipulator, the positioning and orientation of the end-effector are performed by a single mechanism, which is a complex closed kinematic chain with numerous types of kinematic pairs. The simplest and most common method of structural synthesis of parallel manipulators is to determine their structural schemes (architecture) by a given number of DOF, links, and kinematic pairs using the Grubler-Kutsbach formula or criterion [7]. This formula establishes a relationship between the number of DOF of the manipulator with the number of links, kinematic pairs, and their mobility, as well as the dimension of the space in which the manipulator operates. In general, the type synthesis of parallel manipulators requires a condition, including the moving characteristics and DOF of terminal components. Before that, it is essential to have a mathematical description of the motion of components. Only in this way, the DOF of the components (or the manipulator) can be acquired correctly. On this issue, several systematic mathematical approaches have been proposed for the type synthesis of parallel manipulators, such as the methods based on the screw theory, on the theory of differential geometry, on the theory of linear transformation, on the displacement group theory, on the single-open-chain units. A review of research on the synthesis of types of parallel robotic mechanisms was made in [8].
In kinematic synthesis (dimensional or parametric synthesis) of mechanisms, with their known structural schemes, the geometric parameters of the links are determined according to the given laws of motions (or discrete positions) of the input and output links. Depending on the type of movements of the output links, kinematic synthesis of mechanisms is divided into the kinematic synthesis of function-generating, path-generating and motion-generating mechanisms. The function-generating mechanism generates a required functional relationship between the displacements of its input and output links. The path-generating mechanism generates a given path of its output point on a floating link. The motion-generating (or rigid body guidance) mechanism generates the given motion of the output link.
Generation of the specified movements of output links can be performed exactly and approximately. Exact reproduction of the required movements of a rigid body by linkage mechanisms is possible with a limited number of positions, depending on the structural scheme of the mechanism-generator, while the possibility of their approximate reproduction is not limited to the number of specified positions.
Exact methods for synthesis of mechanisms or called geometric methods, for the synthesis of mechanisms are based on kinematic geometry. The fundamentals of kinematic geometry for finite positions of a rigid body in a plane motion were developed by Burmester and for finite positions of a rigid body in space were developed by Shoenflies. Burmester in [9] developed the theory of a moving plane having four and five positions on circles. Shoenflies in [10] formulated theorems on the geometrical places of points of a rigid body having seven positions on a circle and three positions on a line. The graphical methods of Burmester and Schoenflies theories received an analytical interpretation, which is summarized in the monograph [11].
Geometric methods of mechanism synthesis are clarity and simple. However, these methods are applicable only for a limited number of positions. Moreover, the algorithms for solving problems using these methods depend significantly on the number of specified positions, and their complexity increases with the number of positions. Approximation (algebraic) methods of mechanism synthesis are devoid of these disadvantages.
Problems of approximation synthesis of mechanisms were first formulated and solved in [12]. Least-square approximations are the most widely used in the approximation synthesis of mechanisms. For the development of this method, a new deviation function-a weighted difference with a parametric weight, proposed in [13], was important. In contrast to the actual deviation, the weighted difference can be reduced to linear forms (generalized polynomials). This makes it quite easy to apply linear approximation methods to the synthesis of mechanisms. This eliminates the limit on the maximum number of specified positions of the moving object.
Combining the main advantages of geometric and approximation methods, a new direction-approximation kinematic geometry of mechanism synthesis was formulated. It studies a special class of approximation problems related to the definition of points and lines of a rigid body describing the constraint of the synthesizing kinematic chains. In the works [14,15], the basics of approxima-tion kinematic geometry of the plane and spatial movements are presented, where circular square points [14] and points with approximately spherical and coplanar trajectories [15] are defined, which correspond to binary links of the type RR, SS, SP k . Further, in the works [16,17], the concept of discrete Chebyshev approximations was introduced for the kinematic synthesis of linkage mechanisms. Theorems characterizing the Chebyshev circle and straight line in plane motion [16] and the Chebyshev sphere and plane in spatial motion [17], as well as iterative algorithms for determining Chebyshev circular, spherical and other points based on minimizing the limit values of the weighted difference are formulated.
Similar studies on the kinematic geometry of the plane and spatial motions were given in [18]. In [19][20][21], six-bar linkages for function motion and path generation by means of polynomial homotopic continuation algorithms were synthesized.
The literature review shows that the structural and kinematic synthesis of the designed manipulator is carried out separately. At the same time, it is possible that the geometric parameters of the synthesizing parallel manipulator links of the considered structure may not provide the required laws of motions of the output object. Therefore, it is advisable to carry out kinematic synthesis in conjunction with the structural synthesis of the designed manipulator.

The aim and objectives of the study
The aim of the research is the structural-parametric synthesis of a RoboMech class parallel manipulator with three DOF, operating in a cylindrical coordinate system.
To achieve this aim, the following objectives are set: -to develop the method of structural synthesis; -to develop the method of parametric synthesis; -to solve the direct and inverse kinematics problems; -to carry out a numerical experiment to assess the reliability of the results.

Structural synthesis
Let be given N and M values of the coordinates X Pi (i= =1, 2,…, N), and Y Pj ( j=1, 2,…., М) of the point P along the axes X and Y, respectively, and K values of the angle θ k (k= =1, 2,…, K) around the axis Y in a cylindrical coordinate system (Fig. 1). It is necessary to determine the structural scheme and geometrical parameters of links of a RoboMech class parallel manipulator with three DOF, in which each actuator reproduces these three types of the end-effector movements. Fig. 2 shows the block structure of the formed parallel manipulator.
According to the developed principle of the formation of mechanisms and manipulators [1], this parallel manipulator is formed by connecting the output object (point P) to the base using three closing kinematic chains (CKC): one passive ABP and two active CDE and FGH in the following sequence: firstly, the point P is connected to the base using the passive СKC АВР, reaching all the specified positions of the point P along the OX and OY axes, then we connect the active CKC CDE whose active kinematic pair C reproduces the coordinates Y Pj along the vertical lines, lastly, we connect the next active CKC FGH, whose active kinematic pair F reproduces the coordinates X Pi along the horizontal lines. The rotation on the angle θ k is carried out by rotating the entire manipulator around the axis OY.

Parametric synthesis
According to the block structure of the formed parallel manipulator, its parametric synthesis is carried out on the basis of the parametric synthesis of the CKC in the following sequence: firstly, the passive CKC ABP, secondly, the active CKC CDE, and thirdly, the active CKC FGH. Since the passive CKC ABP does not impose geometrical constraints where ρ is the variable distance between the points A and P, l AB and l BP are the lengths of the links AB and BP.
For the parametric synthesis of the active CKC CDE, the coordinate system 3 3 Сх у is fixed to the active joint C, whose 3 Сх axis direction shows the angle 3 j ϕ of rotation of the input link 3. The coordinate system 2 2 Bx y is fixed to the point B of the passive CKC ABP, whose Вх of the coordinate system 2 2 Вх у are determined by the equations The signs "+" and "-" in the equation (4) are chosen depending on the assembly of the CKC ABP. Then the synthesis parameters of the active CKC CDE are , , E x (2) y E are the coordinates of the joints C, D, E in the absolute coordinate system AXY and in the local coordinate systems 3 3 Сх у and 2 2 , Вх у respectively, DE l is the length of the link DE.
Let denote these synthesis parameters by the vector and write a CDEBC vector loop-closure equation , , , , Eliminating the unknown angle ( ) ij DE ϕ from the equation (5) we obtain (6) Equation (6) is the equation of the geometrical constraint imposed by the active CKC CDE on the motion of the output point P. The problem of determining the geometrical parameters of the links at which such geometrical constraint is approximately realized is the problem of parametric synthesis of the active CKC CDE.
The left side of the equation (6) is denoted by 1 , ij q ∆ which is a weighted difference function (7) The geometrical interpretation of function (7) is the deviation of the trajectories of the joints D and E from circles with centers in the joints D and E and the radius , DE l and the minimization of this function is the connection of the planes 3 3 Cx y and 2 2 Bx y by the binary link DE of type RR, where R is a revolute joint. After transformation the equation (7) and the next change of variables the function 1ij q ∆ is represented as the linear forms by groups of synthesis parameters ( ) The linear representability of function (9) allows to formulate and solve the Chebyshev and least-square approx-( ) ( ) imations for parametric synthesis [11]. In the Chebyshev approximation problem, the vectors of synthesis parameters are determined from the minimum of the functional In the least-square approximation problem, the synthesis parameter vectors are determined from the minimum of the functional The linear representability of the equations (7) in the form (9) allows to use the kinematic inversion method, which is an iterative process, at each step of which one group of synthesis parameters ( ) 2 k p is determined to solve the Chebyshev approximation problem (17). In this case, the linear programming problem is solved. To do this, we introduce a new variable , p = ε ′ where ε is the required approximation accuracy. Then the minimax problem (17) is reduced to the following linear programming problem: determine the minimum of the sum with the following constraints where [ ] The sequence of the obtained values of the function (k) S will decrease and have a limit as a sequence bounded below, because Solving these systems of linear equations for each group of synthesis parameters for given values of the remaining groups of synthesis parameters, we determine their values then the revolute kinematic pair is replaced by a prismatic kinematic pair.
It is easy to show that the Hessian of the matrix ( ) k H is positively defined together with the main minors. Then the solutions of the systems of linear equations (25) correspond to the minimums of the functions S (k) . Thus, the least-square approximation problem (18) can be solved by the linear iteration method, at each step of which one group of synthesis parameters ( ) 2 k p is determined. The sequence of values of the functions S (k) will be decreasing and have a limit as a sequence bounded.
Let consider the parametric synthesis of the next active CKC FGH. To do this, the coordinate system 5 5 Fx y is fixed to the active joint F whose axis 5 Fx shows the angle 5i ϕ of rotation of the input link 5, and the coordinate system 4 4 Dx y is fixed to the point D of the first active CKC CDE, whose axis 4 Dx is directed along the link DE. Then the synthesis parameters of the active CKC FGH are , y are the coordinates of the joints F,G,H in the absolute coordinate system AXY, in the local coordinate systems 5 5 , Fx y 4 4 , Dх у respectively, GH l is the length of the link GH. We denote these synthesis parameters by the vector , , Equation (29) is the equation of the geometrical constraint imposed by the active CKC FGH on the motion of the output point P. The problem of determining the geometrical parameters of the links at which such geometrical constraint is approximately realized is the problem of the parametric synthesis of the active CKC FGH.
The left side of the equation (29)  .
The geometrical interpretation of function (30) is the deviation of the trajectories of the joints G and H from circles with centers in the joints G and H and the radius , GH l and the minimization of this function is the connection of the planes 5 5 Fx y and 4 4 Dx y by the binary link GH of type RR.
After transformation the equation (30) and the next change of variables Further, the parametric synthesis of the active CKC FGH is carried out similarly to the parametric synthesis of the considered active CKC CDE by the Chebyshev and leastsquare approximations.

Direct kinematics
In direct kinematics, values of the angles φ 3j and φ 5i of the manipulator input links 3 and 5 are given, it is necessary to determine the coordinates To determine the unknown direction of the vector Let consider the position analysis of the group II (1,2). To do this, we derive an ABE vector loop-closure equation To determine the unknown direction of the vector ( ) , ij AB we transfer the vector BE ij l e to the right side of the equation (48) and square its both sides ( ) and obtain Direction of the vector ( ) ij BE is determined by the Then the coordinates of the output point P are determined by the equation where (2) ( ) (2) tg .
ij ij In the equation (58) Therefore, in the inverse kinematics, the position analysis of the group II (1,2), II (3,4), and II (5,6) are successively solved. To solve the position analysis of the group II (1,2), we derive an ABP vector loop-closure equation ij To solve the position analysis of the group II (3,4), we derive a CDE vector loop-closure equation 3 4 where the module and direction of the vector ( ) ij CE are determined by the expressions ( ) ( ) Coordinates of the joint E in the absolute coordinate system AXY in the equations (68) and (69) are determined by the expressions To determine the unknown direction of the vector ( ) i CD in the equation (67)

Direction of the vector ( ) ij DE
is determined by the equation 4 tg , To solve the position analysis of the group II (5,6), we derive an FGH vector loop-closure equation where the module and direction of the vector ( ) ij FH are determined by the expressions Coordinates of the joint H in the absolute coordinate system AXY in the equations (77) To determine the unknown direction of the vector ( ) i FG in the equation (76) (5) G y are the coordinates of the joint G in the local coordinate systems 5 5 . Fx y

Discussion of the research results
The fulfilled studies show that three types of the output point's movements of the synthesized parallel manipulator in a cylindrical coordinate system, i. e. movements in two orthogonal directions and rotations around a vertical axis are carried out by separate drives. Therefore, in this case, there is a functionally independent operation of the drives, which leads to their kinematic and dynamic independents. In the existing methods for designing manipulating robots, control of the laws of motions of the drives is determined by solving the inverse kinematics problem with the simultaneous operation of all three drives. Therefore, an approach for the design of parallel manipulators proposed in this paper simplifies the control system and improves dynamic characteristics. However, at the same time, manipulators work with certain structural schemes and geometrical parameters of links, and this requires special methods of structural and parametric synthesis. The developed methods of structural-parametric synthesis allow dividing the problem of synthesis of parallel manipulators of complex structure into subproblems for the synthesis of their separate structural modules. The de-velopment of these methods in relation to spatial platform parallel manipulators is proposed.

Conclusions
1. Method of structural synthesis of a RoboMech class parallel manipulator with three DOF operating in a cylindrical coordinate system has been developed. This manipulator is formed by connecting the output point to the base using one passive and two active CKC.
2. Active CKC impose geometrical constraints on the movements of the output point, therefore they work at certain values of the geometrical parameters of their links. CKC synthesis parameters are determined on the base of Chebyshev and least-square approximations.
3. The direct and inverse kinematics problems of the synthesized manipulator are solved. Numerical results showed that the maximum deviation of the output point movement from the orthogonal trajectories is 1.65 %.
4. On the base of the numerical results analysis of the direct and inverse kinematics problems, it is found that there are functionally independent drives, i. e. orthogonal trajectories of the output point are reproduced by separate drives.