Dynamics and control analysis of a single flexible link robot with translational joints

Use your smartphone to scan this QR code and download this article ABSTRACT Modern design always aims at reducing mass, simplifying the structure, and reducing the energy consumption of the system especially in robotics. These targets could lead to lowing cost of the material and increasing the operating capacity. The priority direction in robot design is optimal structures with longer lengths of the links, smaller and thinner links, more economical still warranting ability to work. However, all of these structures such as flexible robots are reducing rigidity and motion accuracy because of the effect of elastic deformations. Therefore, taking the effects of elastic factor into consideration is absolutely necessary for kinematic, dynamic modeling, analyzing, and controlling flexible robots. Because of the complexity of modeling and controlling flexible robots, the single-link and two-link flexible robots with only rotational joints are mainly mentioned and studied by most researchers. It is easy to realize that combining the different types of joints of flexible robots can extend their applications, flexibility, and types of structure. However, the models consisting of rotational and translational joints will make the kinematic, dynamic modeling, and control becomes more complex than models that have only rotational joints. This study focuses on the dynamics model and optimal controller based on genetic algorithms (GA) for a single flexible link robot (FLR) with a rigid translational joint. The motion equations of the FLR are built based on the Finite Element Method (FEM) and Lagrange Equations (LE). The difference between flexible manipulators that have only rotational joints and others with the translational joint is presented through boundary conditions. A PID controller is designed with parameters that are optimized by the GA algorithm. The cost function is established based on errors signal of translational joint, elastic displacements of the End-Point (EP) of the FLR. Simulation results show that the errors of the joint variable, the elastic displacements (ED) are destructed in a short time when the system is controlled following the reference point. The results of this study can be basic to research other flexible robots with more joint or combine joint styles.


INTRODUCTION
Dynamics and control are fundamental problems in the robotics field. The modeling and building of the motion equations problems meet challenges with the robot which has many degrees of freedom. Especially the robots take into account the ED factor of links. There are many studies on the FLR. However, the robot model mentioned in such works has only rotational joints (R joint). The translational joint (T joint) has almost not been considered yet. The inclusion of the T joint into the FLR increases flexibility, further enriching the application of this robot type. Of course, the modeling complexity also increases. Few authors have studied the FLR with only T joint. A single FLR with T joint is presented in   1 . The Galerkin method is used to model the robot. The authors also proposed a feedback control law in   2 . Kwon and Book 3 present a single link robot which is described and modeled by using assumed modes method (AMM). Stable inversion method is studied for the same robot configuration but the nonlinear effect is taken into account 4 . A new method to solve the inverse dynamics of a FLR is described in 5 . Some FLR with a T joint combining a R joint are considered in Pan et al. (1990) 6 , Yuh and Young (1994) 7 , Bedoor and Khulief (1997) 8 . The main approach was based on the Assumed mode method (AMM) and a few works were based on FEM to model the system. A FLR with R and T joints was presented in Pan et al. (1990) 6 based the FEM approach. The Newmark method was used to solve the motion equations of the robot. The Partial Differential Equations was established in Yuh and Young (1994) 7 for a FLR with R-T joint by using AMM. A general dynamic model for R-T robot was introduced in Bedoor and Khulief (1997) [8] based on FEM and LE approach. A Fuzzy Logic controller was designed in Farid and Salimi (2001) 9 to track the motion of a FLR with R- T joint. An adaptive control of a single FLR with T joint with the ball-screw mechanism was studied in Qiu (2012) 10 . This controller is also mentioned in Zhao and Hu (2015) 11 with AMM and LE approach. In this study, the dynamics model and optimal controller of a single FLR with a T joint are considered based on the FEM and LE approach. The T joint variable is defined which is distance from element at the origin coordinate system to the EP of link. The value of depends on time. The position of other elements is changed too. Number of elements can be optionally selected and enough huge. A PID controller is designed with parameters which are optimized by using GA algorithm. Cost function is built based on errors signal of T joint variable, ED of the EP of the FLR.

MATERIALS AND METHODS
The structure of a single FLR with T joint is considered in Figure 1. The XOY is the fixed frame. A flexible link is divided into n elements. Each element has 2 nodes. Each node j has the flexural (u 2 j−1 , u 2 j+1 ) and the slope displacements (u 2 j , u 2 j+2 ). The length of an element is l e = L n with is the total length of flexible link. The T joint variable is d (t) which is changed by driven force F T (t). The F y is external force at the EP of the link. Element k is inside the hub of the T joint It is noteworthy to mention that value of k is the number of position element. So, it must be taken the raw value of k in computing process. Total elas- Coordinate r j of element j is given by Elastic kinetic energy (KE) of element j is determined as Where m is mass per meter. Mass matrix M j of element j is calculated as s, e = 1, 2..., 5 Where Q js and Q je are the s th , e th element of Q j vector. It can be shown that M j is of the form The potential energy (PE) of element j includes two components: elastic P j and gravity G j potential energy.
Where, E and I are Young's modulus and inertial moment of link. The stiffness matrix K j is shown as PE due to gravity can be given as The total elastic KE and the PE of link are described as The LE with Lagrange function L = T − P is shown as d dt When KE and PE are known, the Eq. (10) can be rewritten as

M(Q)
.. Vector The matrices M and K are established from matrices M j and K j . Vector G * = ∂ G ∂ Q(t) can be determined by partial derivative G(t) = ∑ n j=1 G j .Structural damping matrix D = αM + β K is calculated as in Ge et al. (1997) 12 . Symbols α and β are the damping ratios of the system which are determined by experience. The size of M,K and D matrices is (2n × 3) × (2n + 3).

BOUNDARY CONDITIONS
The displacements of element k are zero because assumed that the translational joint hub is treated as rigid. The rows and columns (2k − 1) th ; (2k) th of matrices M, K, D, G * and F(t), Q (t) vectors are eliminated following FEM theory and values of these are continually changed depending on time because of changing of element k position. It is noteworthy to mention that value of k depends on time. This boundary condition is clearly different point between FLR with only R joints and FLR with combine T joint and R joint. So now, size of matrices M, K, D, G * is (2n + 1) × (2n + 1) and size of F (t) and Q (t) is (2n + 1) × 1. The k variable is continuously updated for each time step in solving process. Vector generalized coordinate Q (t) is rebuilt after each loop too because of changing value k variable ( Figure 2).

SYSTEM CONTROL
In this paper, a PID controller is constructed to control the position of an FLR. The position error of EP of the elastic stitch is continuously reflected and minimized. GA is applied to find the optimal parameters of the PID control system. The error values are used to assess the fitness of each chromosome in the GA. There are some main steps in this algorithm as selection, mutation, crossover, and reproduction. The reproduction step is stopped when an optimum solution is found 13 . The sequences of operations involved in GA are described in Figure 3.
Structural control of dynamic system is designed as Cost function J is the linear quadratic regulator (LQR) 14 . The optimum target is finding the minimum cost of J function with values of respective parameters which are changed from lower bound to upper bound values. It means to reduce minimum driving energy of joint, error of T joint variable and elastic displacements at the end-effector point. The optimum parameters K P , K I , K D are given when minimum value of J is found.

SIMULATION RESULT AND DISCUSSIONS
Parameters of the FLR are shown in Table 1.
It is noteworthy to mention that system has an elastic displacement value at initial time as static state. This initial displacement value is w (t=0) (0) = 3.E.I . Parameters are used in GA following Table 2. The reference point and optimum parameters are shown in Table 3.
The cost values of J function are described in Figure 5. The minimum cost value is 0.0594. This value shows  that the control quality is effective. The generation can be reduced.
The simulation results of T joint, error of joint is shown in Figure 6 and   The minimum cost value of J function 0.0594

CONCLUSION
A dynamic model of a single FLR with T joint is considered based on FEM and LE approach. Considering joint variable which is distance from element k at the origin coordinate system to the EP of link is effective. The difference between flexible manipulators which have only R joint and others with T joint is presented through boundary conditions. Control system is pro-   ACKNOWLEDGMENT I am extremely grateful to anonymous reviewers for valuable comments that helped to improve this article.

CONFLICT OF INTEREST
There are no potential conflicts of interest with respect to the research, authorship, and publication of this article.

AUTHOR CONTRIBUTION
The author is the only person who made the presented study, conducted the numerical simulation, wrote the manuscript, as well as formulated the statement to the problem.