A Novel Pole-Placement Controller Design

for a Robot

 

I.Hassanzadeh and S.Mobayen

 

Control Engineering Department, Faculty of Electrical & Computer Engineering,

University of Tabriz, Tabriz, Iran

 

Izadeh@tabrizu.ac.ir , Smobayen@gmail.com

 

 

 

 

Abstract- This research presents a pole-placement method for designing of a feedback controller for a robot. The idea of proposed method is derived from the fact that the internal signals of a feedback control system can be eliminated by executing row operations. The design method can be implemented easily and insightfully on Diophantine equations after choosing proper closed-loop poles. In this paper, a 5-bar-linkage manipulator is considered as a case study and the controller has been designed based on the governing Euler-Lagrange equations on the robot. The superior features of this method are its simplicity and flexibility to apply on various control systems. 

 

Keywords- Pole-placement design, Diophantine equation, 5-bar-linkage manipulator.

 

I.     INTRODUCTION

He feedback design method is very prevalent rather than other control methods in nature and technology. Feedback modifies the dynamics of the system and decreases the system sensitivity to signal and model uncertainty in control processes [1-2].

Many precursors have accomplished significant efforts to develop design approaches for feedback control systems. The linear quadratic Gaussian technique (LQG) has been established in the 1960's [1-3] and some modifications of the LQG are shown in the recent researches [4]. In 1970, Rosenbrock proposed the Nyquist-Array method in which the compensator can be obtained as a series connection of three types of matrices: permutation matrices, product of elementary matrices, and diagonal matrices [5]. MacFarlane and Kouvaritakis innovated the characteristic-locus method in 1977. The main step of the proposed method consists of designing an estimated commutative compensator. The information of the method is demonstrated in [6]. Based on Edmund's algorithm that was suggested in 1979 [7], the optimization design of controller parameters is executed in feedback control systems. An advanced way of using this algorithm was developed in 1988 [8]. Generally speaking, Edmunds's algorithm has allowed the designer to get a simpler and more accurate controller than the approaches resulting from other design methods such as LQG. An influential paper was presented in 1981 which through the use of singular values, it showed how the classical loop-shaping feedback design ideas could be generalized to multivariable systems. Because the closed-loop stability requirements cannot be satisfied simultaneously, feedback design is therefore a trade-off over the frequency of conflicting objectives. This is the basic motivation for exchange in the MIMO feedback design. Numerous application and improvement works can be seen in [7-10].

With simpler ideas and less theoretical mathematics and graphics than the methods mentioned above, the pole-placement method is a remarkable and flexible control design method for industrial control processes and it is now a well-established technique for controller design. The proposed pole-placement design approach is the same as solving a Diophantine equation. Solving of a Diophantine equation can be converted into solving a linear algebraic equation [2, 11]. If n is the degree of the plant, degree of controller m must be n-1 or higher. If the condition m=n-1 is satisfied, the Diophantine equation has one unique solution. If m<n-1, some set of poles and not all poles of the controller are assigned [10-12]. This paper is focused on pole-placement technique for a 5-bar-linkage manipulator where using simplifications on dynamic equations, it is a SISO system. Although, based on the analytical and operator-based design procedure of this method, it is possible to extend this approach to the control of MIMO systems.

The paper is organized as follows: section 2 contains a description about the system and presents dynamic equations of the robot. Section 3 presents an overview of Diophantine equation solution. In section 4, controller design procedure for the robot is proposed. Section 5 implements the proposed method on the plant followed by conclusion in section 6.

 

II.      Five-Bar-Linkage Manipulator Robot

 

Fig.1 shows the 5-bar-linkage manipulator built in robotics research lab in our department. Also, fig.2 indicates the robot’s schematic view which the links form a parallelogram [13]. It is clear from this figure that even though there are four links being moved, there are in fact only two degrees of freedom, identified as q₁ and q₂. In this section, the dynamic equations of a 5-bar-linkage manipulator are presented.  Let q_i, T_i and I_hi be the joint variable, torque and hub inertia of the i^th motor. Also, let I_i, l_i, l_Ci and m_i be the inertia matrix, length, distance to the center of gravity and mass of the i^th link, respectively. The manipulator specification consisting of mass, length and center of gravity of links are represented in Table I.

Fig. 1 Planar presentation of robot

 

Fig. 2   Planar presentation of robot

 

 

The coordinates of the centers of mass of the four links are calculated as a function of the generalized coordinates [14-17]. This gives

 

                                                                 (1)

                                                                 (2)

                                                 (3)

                                                     (4)

 

By differentiating the above equations, the velocities of the various centers of mass are assigned as a function of and . The result is

 

                                                           (5)

                                                           (6)

                                          (7)

                                          (8)

 

The dynamic equations of this manipulator are [4]:

 

                                     (9)

                                       (10)

 

where g is the gravitational constant. The inertia matrix is given by

 

                                   (11)

where

 

                             (12)

                            (13)

                     (14)

 

 we note from the above equations that if the equation

 

                                                                 (15)

 

is satisfied, then M₁₂ and M₂₁ are zero, i.e., the inertia matrix is diagonal and constant. Therefore the dynamic equations of this manipulator will be

 

                    (16)

                      (17)

 

Notice that T₁ depends only on q₁ but not on q₂ and similarly T₂ depends only on q₂ but not on q₁. This subject helps to explain the popularity of the parallelogram configuration in industrial robots. If the condition (15) is satisfied, then we can adjust two angles independently, without any interaction between them.

 

 

 

TABLE I. The 5-bar-linkage manipulator data

 

 

III.      Diophantine Equation Description

 

Pole-placement is an important controller design method for linear time-invariant control systems. This method is established on the fact that several performance requirements can be met by using output feedback control to sufficiently put closed-loop poles in the complex plane [18-19]. The solution of the classical pole-placement problem for systems represented by proper transfer functions can be reduced, under appropriate conditions, to the solution of an algebraic equation known as Diophantine equation.

Consider the plant given by g(s)=b(s)a^-1(s) of typical control loop in fig.3 where a(s) and b(s) are polynomials in the laplace variable s. The signals r(s), e(s), w(s), u(s) and y(s) are reference signal, error signal, measurement signal, input to the plant, and output signal, respectively. The polynomials d(s), Δ(s), and c(s) are controller components and the purpose here is to design a controller such that the closed-loop poles are specified.

 

Fig. 3 Control loop representation

 

 

The equations of the control loop in fig.3 are

 

                           (18)

 

where the expression λ(s)=a^-1(s)u(s) is the “partial state” of the plant [18]. The purpose is to perform elementary row operations on equation (18) and arrive to the following simplified equations:

 

 (d(s)a(s)+c(s)b(s)).λ(s)=Δ(s).r(s)                                         (19)

  y(s)=b(s)λ(s)                                                                         (20)

 

where f(s)=d(s)a(s)+c(s)b(s) and output of the closed-loop system is expressed by

 

y(s)=b(s)f^-1(s)Δ(s)r(s).                                                            (21)

 

 

IV.     Design Procedure of a Controller for the Robot

 

Now, consider the dynamic equations of the first motor of robot as follows:

 

                                        (22)

 

The laplace transform of the above equation can be derived as follows:

 

                                                            (23)

 

namely, a(s)=s⁴+s²+38.387s, b(s)=18.868(s²+1) and the degree of a(s) is r=4. Design procedure of the controller can be represented in there steps as follows: 

 

Step1- Diophantine Equation Assignment

 

Diophantine equation is represented as Follows

 

                                                         (24)

 

 The unknown coefficients c’s and d’s can be solved from the following equation if the square matrix is nonsingular.

 

                (25)

 

Step2- Choice of f(s)

 

If the choice of f(s)=Δ(s)p(s) is made, then the closed-loop equation will be

 

y(s)=b(s)p^-1(s)r(s)                                                                   (26)

 

Controller components d^-1(s)Δ(s) and Δ^-1(s)c(s) must be proper. Since d(s) and c(s) are of degrees 3, Δ(s) can be selected based on the given desired poles.

 

Δ(s)=(s+2)[(s+1)²+1]                                                             (27)

 

the degree of f(s) must be sum of the degrees of a(s) and d(s). Then, we can place four other poles in f(s), say,

 

f(s)=(s+2)[(s+1)²+1][(s+7.348)²+133.1][(s+0.0022)²+1]         (28)

 

 

Step3- Diophantine Equation Solution

 

Using attained polynomial for f(s) in the previous stage and inverting the Diophantine equation, controller components c(s) and d(s) are achieved,

 

c(s)=13.297s³+41.465s²+24.402s+39.7                                 (29)

d(s)=s³+18.691s²+0.895s+18.719                                          (30)

 

and the closed-loop equation is

 

                               (31)

 

Note that to comply with the steady-state value equal to 1, a gain is multiplied to the main system, i.e., 18.868. Fig.4 shows the obtained control loop of this design procedure for two motors of the robot.

 

Fig. 4  Block diagram of two motors with pole-placement controllers.

 

V.      Simulation Results

 

 Fig.5 illustrates the step response without controllers for two motors. Fig.6 shows that the proposed controller stabilizes the original robot system and the input signals u(t) are shown in fig.7. A swing in direction of the input signals perceived in fig.7, and that’s why the robot system is non-minimum-phase. 

 

Fig. 5  Step response of the motors without any controllers

 

 

Fig. 6  The square wave response of the closed-loop control system using pole-placement controllers (for both of the motors).

Fig. 7  The input signal u(t) of the closed-loop control system for both of the motors.

 

The block diagram of the system with load disturbance and measurement noise is shown in Fig. 8. Here the load disturbance and measurement noise are added using Pulse generator and Band-limited white noise blocks. Disturbances may arise from external sources or internal load variations must be rejected. Therefore, a good control system should be able to track reference input and to eliminate the effects of disturbances and noises. Responses to a square wave in the input, band-limited white noise (noise power=0.00001, sample time=0.001) and step disturbance with a magnitude of 1 at time t=60s are shown in Figs. 9 and 10. It is seen from the results that the system is fairly robust under both the load disturbance and measurement noise. Besides, it is seen that the settling time is almost unaffected. Figs. 11 and 12 illustrate the input signals of the system where the measurement noise and load disturbance are applied to the system.