SYNTHESIS AND CONTROL 3-DOF PLANAR PARTLY DECOUPLED PARALLEL MANIPULATOR

This paper deals with a new planar parallel manipulator, kinetic energy of this mechanism, designing controller. The translation movements are decoupled from rotation. The control algorithm of parallel robot satisfying partial decoupling between translation and orientation degrees of freedom is synthesized. Simulation result of control algorithm in desired trajectory of end-effector of robot is shown. Here parallel manipulator control algorithm while intersecting singular zones is considered.


INTRODUCTION
The idea to design of a new partially decoupled parallel manipulator is following. We use well known planar five-bar mechanism and add the third kinematic chain containing two prismatic kinematic chains formed as planar parallelograms (Fig. 1, 2) [1]. Let us consider the degree of freedom of this planar parallel manipulator which is to be developed. Here we can see three kinematic chains [2]. Two of them are used for translation of the output link and one of them is used for rotation of this link [3].

Figure 1:
A new planar parallel mechanism [4] DOF of mechanism is determined by Chebyshev formula [6]: The main property corresponding to this structure is partial decoupling of translating and rotating degrees of freedom [7]. Please note that it is possible to develop this structure for another degrees of freedom and movements of the end-effector [8].
The most important task is that arising in modeling of the robot and control it, the problem about positions. We present the solution of a inverse problem. Coordinates of point A in (xy) are known A (XA, YA). It is necessary to find the generalized coordinates (1, 2). The initial equations are [9]: After transformations we obtain [10] For the second kinematic chain is similar relation.
Then kinematic modeling in term of speed of robot is considered. It is possible to express the solution of a problem on positions in the form of implicit functions. Then these functions are differentiated, we have[10]: On the left side we have the matrix A, on the right -a matrix B. These equations are used for the analysis singularities. The first type of singularity corresponds to the determinant of matrix B goes to zero. The second type of singularity corresponds to the degradation of matrix A [9].

Figure 4:
The type I singularity [5] The third type of singularity corresponds to the degradation of both matrices. In order to explain the third kinematic chain, matrices A and B must be converted [12]: The third chain provides rotation with a single gear ratio. For the analysis of previous unknown type of singularity of the fourth modification of the proposed matrix. Taking into account the possible relationship imposed by the third kinematic chain [13].
According to it the fourth type of singularity has been found.

KINETIC ENERGY OF 3-DOF PLANAR PARALLEL MANIPULATOR
Energy relations that takes place in the parallel robot structure are considered. This question is important for modeling dynamics of the robot and control.
The general energy is equal to the sum corresponding to a output link, input and intermediate links [14].

Figure 6:
The first kinematic chain [15] T = T1 + T2 + T3 To determine the energy we use schema (Fig. 6). In particular, for the intermediates links In these equations include inertial parameters of links and their speed. For studying dynamics of the robot it is necessary to have partial derivatives of these expressions on the generalized velocities and coordinates. These relations are quite complicated in nature. In particular, we find differentials from the following expressions [ Where the coefficients k1, …, k4 depend on manipulator parameters.

ANALYSIS MANIPULATOR DYNAMICS
This part deals with the dynamic analysis of the parallel robot structure. We use the type II Lagrange equations. First, we solve the inverse dynamic problem. The law of movement is known and it is necessary to find the control action in actuator. Corresponding algorithms and programs have been developed, the concrete example of the parallel robot (a Fig. 7) is considered. The simulation results are shown in Fig. 8, 9[17].  Then examined the direct dynamic problem -from the known laws of change of the generalized forces to find the corresponding motion of the robot. We designed corresponding algorithms and programs, the concrete example of a configuration of the parallel robot is considered.
The simulation results are shown in Fig. 10, 11.
Planning trajectory of the robot is important for control. In particular, we can choose from a variety of trapezoidal velocity curves (Fig. 12).

Figure 12: Trapezoidal velocity curves
Thus generalized coordinates should change as follows.

CONTROL ALGORITHMS
This section deals with control algorithm of parallel robot structure. The algorithm proposed by R.Paul and tested on robot with consistent structure. Idea behind the algorithm is that compensating mutual influence between degrees of freedom of the robot, and also introduce control algorithm of the factors that take into account weight led to drive mass and the moment of inertia of the manipulator.
It is necessary to notice that in paper the interferences caused by a configuration of the robot and the generalized speeds are compensated only. Compensation of the mutual influence caused by accelerations are not carried out because the accelerometers make big noise. In addition, influence of accelerations is supposed not so high because of small speeds. The structure of control system is shown in Fig. 13  We have parameters of the DC separate-excited Actuator as following:

PARALLEL MANIPULATOR CONTROL WHILE INTERSECTING SINGULARITY ZONES
It is known that there could be a parallel manipulators control loss in singular configuration that is why we suggest the use of additional actuator that is shown in Fig. 17. The point is that approaching singular configurations the system of equations becomes degenerate, and required generalized forces (in actuators) become too large. In this case an additional actuator should be used, which has to be taken into consideration in the control algorithm.
In parallel manipulator (Fig.2) the singular configuration is manifested by links AB and AC' forming one line. This singular zone is one-dimensional, as in case of constant relative position of the above-mentioned links the manipulator has only one degree of freedom.
We introduce the criterion of singular configurations: the overrun of the generalized moment's marginal tolerance value. It is necessary that the moment surpass the nominal value not more than two times. On reaching such configuration there should be a load transfer with taking extra actuator into account.
The singular configuration wouldn't be a singular one if the actuators are situated in B, C and H and B pairs (Fig.2). The algorithm could be realized this way: at initial stage the three main actuators B, C and H are in operation. When the moment of one of them reaches the surpass nominal value, the other extra actuator (Fig. 17) is put into operation.

CONCLUSION
A new planar parallel manipulator is considered. The translation movements are decoupled from rotation. Kinematic modeling of parallel robot is realized and the algorisms and criteria of its singularities are developed corresponding to loss of degree of freedom or loss of controllability, a new kind of singularity is found; Dynamic modeling of parallel robot is realized based on energy conditions, the direct and inverse dynamic problems of parallel robot.
The control algorithm of parallel robot satisfying partial decoupling between translation degrees of freedom is synthesized.
Suggestion solving for parallel manipulator control while intersecting singularity zones.