Fuzzy MTEJ controller with integrator for control of underactuated manipulators

Robotics and Computer–Integrated Manufacturing 48 (2017) 93–101

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Robotics and Computer–Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim

Fuzzy MTEJ controller with integrator for control of underactuated manipulators

MARK

⁎

S. Reza Naghibia, , Ali A. Pirmohamadia, S. Ali A. Moosavianb a b

Department of Mechanical Engineering, University of Zanjan, Zanjan, Iran Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran

A R T I C L E I N F O

A BS T RAC T

Keywords: Underactuated manipulator MTEJ control Deadband in actuators Fuzzy control

This paper presents control of underactuated robot manipulators in task space utilizing novel algorithm called fuzzy modified transpose effective Jacobian (MTEJ) with integrator term which ensures tracking trajectory in task space with high-quality performance. Although non-model base MTEJ have been introduced before and evaluated for underactuated robots, this method has poor operation against non-linear factors in actuators and joints observed in practical tests like deadband, backlash or Coulomb damping. The contributions of this paper are in twofold. First, to introduce improved MTEJ algorithm with additional integrator term that is efficient to eliminate effects of mentioned factors or other source of steady state error (SSE). Second, using new fuzzy rules to manage its terms to stabilize system with better control properties, the controller is used to make the endeffector to both track a predefined trajectory or set on an exact point. Global stabilization of this control method is proved. Simulation and experimental results are offered which compared tracking performance of the improved fuzzy MTEJ with integrator to other methods for both tracking and point to point (P.T.P) control. Outcomes of these experiments reveal privileges of using fuzzy improved MTEJ in various areas like removing SSE with better control characteristics and low computational efforts.

1. Introduction In recent years, a significant research attempt has been made to the study of underactuated manipulators, i.e. articulated chains of bodies which are actuated with control inputs which are less than its dynamics equations. The source of underactuation in manipulators may be grippers if the number of actuators are reduced while keeping the hand capability to adapt its shape to the grasped object. Many techniques have been introduced to use this strategy for proper picking up and holding of objects of different shapes [1–3]. In space systems, it is extremely important to be able to tolerate element or subsystem failure, otherwise the total mission can be endangered, [4]. For instance, imagine the consequences of a failure in a joint of the space arm, therefore, it cannot be driven back into its stowed position. While these systems are often designed with surplus elements to give them higher reliability, it is also highly advantageous if they are able to operate after a failure, even with reduced capabilities. In the above example, the ability to utilize the working manipulator joints to control the system partially and stowing it is significant. Manipulators with passive joints are so multifaceted that they cannot be controlled by any regular control methods. Many approaches have been used for the

⁎

complex problem of controlling mechanical manipulators and robotic systems. Energy method in [5–8], nonlinear control method like sliding mode and back stepping in [9–11], dynamic feedback linearization in [12], and open loop methods in [13–15]. Recently, soft computing method has been employed to control underactuated systems. Some of these methods are fuzzy controls in [16–20], neuro-fuzzy and genetic fuzzy in [21]. Unknown nonlinear functions in dynamic model are the major motivation for using control law which can deal with uncertainties. Some free model controller can be achieved using a combination of soft computing and adaptive sliding mode control. Some of the most recent studies with a focus on underactuated systems are [22–24]. Nevertheless in practice, several problems may appear because ideal condition is not met. This has greater negative impact on the precise control of end effectors. Some of these unavoidable conditions are deadband in electrical actuators, Coulomb damping in joints and gravity forces. The authors encountered significant SSE in practical use of convenient MTEJ in an underactuated planar robot. The source of this SSE was deadband, undesirable torque in shaft due to misalignment, Coulomb damp in actuators and etc. Many approaches have been used to cope with these nonlinearities. Among them, learning methods especially neural networks (NN) require less

Corresponding author. E-mail address: [email protected] (S.R. Naghibi).

http://dx.doi.org/10.1016/j.rcim.2017.03.006 Received 20 March 2016; Received in revised form 20 March 2017; Accepted 21 March 2017 0736-5845/ © 2017 Elsevier Ltd. All rights reserved.

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method is presented and its global stability is studied. Finally, experimental and simulation results are provided for the new method as well as regular one for both tracking and P.T.P problems. Results reveal the advantages of using fuzzy MTEJ with integrator method in controlling of underactuated robot with better control properties.

information of the system dynamics. In [25] adaptive control base on NN (ANN) has been studied for a system with unknown deadband. This method is further extended in [26] to control a flexible manipulator with input deadband. The approach eliminates the evaluation of inverse dynamic model as well as the time-consuming training process. Besides that, the controller is robust and easy for real-time application. The main method is used to control other mechanical systems, [27– 29]. However since selection of higher number of nodes of NN is necessary to have satisfactory control performance, the method is more time consuming. In [30], ANN is used to control a robot manipulator in task space. Since in underactuated manipulators, the Jacobean which relates task and joint spaces is dynamic based, its estimation should be considered in learning process which makes the control response unsatisfactory. Transposed Jacobian (TJ) control is one of the simplest algorithms employed to control motion of robotic manipulators. According to [31], the TJ algorithm has been arrived at intuitively, and is similar to classic PD-action controllers. For underactuated manipulators, this algorithm turns to TEJ where effective Jacobian is used instead of Jacobian matrix. Both experimental and simulation results demonstrate the behavior of the TJ algorithm in controlling of highly non-linear and complex systems with multiple degrees of freedom (DOF), thereby encouraging additional work on this algorithm. Nevertheless, since the TJ is not dynamics-based, poor performance may appear in fast trajectory tracking. Use of high gains can make the performance critically worse in the existence of feedback measurement noise. Another disadvantage is that there is no prescribed method of selecting its control gains, and also a heuristic selection of gains makes it difficult to apply. In [32], Modified Transpose Jacobian (MTJ) algorithm is developed which uses stored data of the control command in the prior time step, as a learning tool to yield a better performance. The gains of this new algorithm can be selected more methodically, and do not need to be large; hence the noise rejection characteristics of the algorithm are enhanced. The main method is extended to control other mechanical systems, [33,34]. In [35], this idea is extended to the problematical control of underactuated robots in Cartesian space based on the notion of Transpose Effective Jacobian as Modified Transpose Effective Jacobian (MTEJ) algorithm. In [36] a tuned MTJ control algorithm is presented that improves and facilitates the implementation of the MTJ algorithm by proper tuning of the switching gain matrix. This improves the performance in the presence of significant disturbances and noises. However, it is inefficient against unknown deadzone. To conquer the mentioned drawbacks, in this paper, an additional integrator term is employed to eliminate SSE by changing MTEJ gains in proper way using a new fuzzy method so, control properties like settling time and overshoot are enhanced. The main contributions of this method with respect to existing works, [13–36], are emphasized below:

2. Dynamic modeling in workspace In this section, a brief review of the dynamics modeling of underactuated manipulator is presented. Considering a n-DOF robot manipulator in a m-degree workspace. The dynamics of this rigid underactuated robot arm with ‘a′ active and ‘p′ passive joint can adequately be described using the Euler–Lagrange equations of motion. Resulting in: (1)

H (q ) q + ̈ C (q , q )̇ = τ = Bτa T [qaT qpT ] ∈R n

is the vector of generalized joint coordinates, where q = qa∈R a , vector of active joints, qp∈Rp , vector of passive joints, H ∈R n × n is inertia matrix, C ∈R n is the vector of Carioles, centrifugal and gravity torques, B = [Ia × a oaT× p]∈R n × a is the matrix of inputs, τa∈R a is the vector of torques acting at the joints. Since the manipulator has ‘a′ active and ‘p′ passive joints, the above equation can be divided by actuated and underactuated parts as follows:

⎡ Haa Hap ⎤ ⎡ qä ⎤ ⎡Ca ⎤ ⎡ τa ⎤ ⎢ ⎥ ⎢ ⎥ +⎢ ⎥ = Bτa = ⎢ 0 ⎥ ⎣ p×l ⎦ ⎣ Hpa Hpp ⎦ ⎣ qp̈ ⎦ ⎣Cp ⎦ Haa∈R a × a ,Hap∈R a × p

،Hpa∈Rp × a ،Hpp∈Rp × p ,

(2)

Ca∈R a ,

where Jacobian matrix, task velocities can be written as:

qˆ ̇ = J (q ) q ̇ ,

J ∈R m × n

Cp∈Rp ,

using (3)

The Jacobian matrix can be divided into actuate and underactuated parts like dynamic model:

J(q) = [Ja (q) Jp (q) ] , Ja∈Rm×a, Jp∈Rm×p

(4) ∼ H͠ aa∈R a × a and Ja∈R m × a are called inertia matrix and effective Jacobian matrix, respectively and are achieved by the following relations:

∼ −1 Ja = Ja−Jp H pp Hpa −1 H͠ aa = Haa−Hap H pp Hpa

(5)

It was shown in [35] that in the case where m=a, the relationship between actuated torques (τa ) and the torque in workspace (τˆ) is:

∼T τa = J a τˆ

(6)

Using the following notations, the model of robot in workspace can be written as:

̈ Cˆ = τˆ ˆ ˆ+ Hq 1. A novel Fuzzy MTEJ controller with an integrator is introduced to control the underactuated manipulators in the task space in the presence of both uncertain dynamics and external disturbances. The new algorithm can compensate practical nonlinearity such as friction effects, which leads to accurate tracking. 2. The controller does not require dynamics modeling, and is not model-based. 3. Comparing to the sliding mode methods, switching is avoided so that lower energy consumption while an accurate tracking is achieved. Besides, it doesn’t require boundary of uncertainties. 4. Considering the obtained results, various control characteristics such as SSE and settling time are improved. 5. This new controller requires low computation efforts, which is an important issue for real-time implementations.

(7)

where

∼−T ∼−1 Hˆ = J a H͠ aa J a ∼−T ∼−1 Cˆ = J a H͠ aa J a (JH−1C −Jq̇ )̇ ∼−T τˆ = J a τa

(8)

3. Control of the robot 3.1. Regular MTEJ controllers In order to put our contribution in perspective, we will briefly review convenient MTEJ control for system (1). The task is to control the output variables qˆ and their time rates to track the time varying ̇ ]. It has been assumed that the dynamic characteristic vector [ qˆdes qˆdes of manipulator is unknown. For the case that the last link is passive, the effective Jacobian matrix became Non-Model-Based if the last link be

The rest of the paper is organized as follows: first, the procedure of finding the relationship between task space and control variables along with regular MTEJ control method are reviewed, next, the new control 94

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3.3. Tuning MTEJ gains using fuzzy

uniform. Therefore using effective Jacobian from (5) and MTJ method which has been introduced in [32], MTEJ method was introduced in [35]:

∼T τa = J a (kD e+ ̇ kP e+kt τˆ t −∆t )

This section describes a new method for tuning of controller parameters in improved MTEJ controllers. As earlier mentioned, large value of kI can make system instable especially when initial error is high. A solution to this problem is restricting integrator to act only when transient errors are removed. In fact, it should integrate only permanent errors due to factors earlier mentioned and be passive about transient errors. The key to distinguish these errors is the fact that in tracking control, when transient errors are eliminated, first and second derivatives of errors are close to zero simultaneously, fuzzy block interprets the error in steady state, and then it can make the controller to start integrating the errors. Moreover, if it is restricted to integrate just permanent errors, the integrator gains can be set to be larger; and this can eliminate them faster. Also, derivative term can be useful for reducing overshoot but can make response slow. To fix this problem, when error and its derivatives have different marks, the fuzzy controller make the kD lower than when they are the same. So the fuzzy has three inputs, task error, its rate and acceleration. Consequently, the controller (11) will be modified as:

(9)

Where e is tracking error defined as e =qˆdes−qˆ. τˆ t − ∆t is workspace torque at a previous small time step. kt is a switching coefficient to prevent outsized torques when e and ė are high. To achieve softer switching, kt is obtained by the following equation:

⎛ ⎛ e ė ⎞⎞ kii = exp ⎜⎜ −⎜ i + i ⎟ ⎟⎟ ̇ i ⎠⎠ ⎝ ⎝ emaxi emax

(10)

3.2. MTEJ algorithm with integrator TEJ/MTEJ algorithms have several benefits in comparison with Model-Based methods (MBA) in that they do not require the inverse of the Jacobian matrix and its time derivative, which are necessary for implementing the MBA. Therefore, despite the MBA controllers, it is evident that the TEJ and MTEJ controllers do not get stuck at singular points. Nevertheless, it was observed in practices, that the accuracy of these controllers is strongly affected because of practical issues like deadband in electrical actuators and friction in joints especially Coulomb damping. Mentioned issues are inevitable in manipulators and are usually unknown; without taking them into consideration; it will not be possible to achieve accurate control in task space and eliminating SSE. To resolve this issue, using an additive integrator term is offered in this work. Therefore, we introduce the following law instead of (9):

∼T ⎛ τc = τa+τs sign (qṪ ) = J a ⎜kD e+ ̇ kP e+kI ⎝

∫0

t

⎞ edt +kt τˆ t −∆t ⎟ ⎠

t ∼ ⎞ ∼ ∼T ⎛ τc = J a ⎜Kp e+kD e+ kI edt +kτˆ t −∆t ⎟ ̇ ⎝ ⎠ 0 (12) ∼ ∼ In (12), kD and kI are non-fixed derivative and integrator gains and tuned by fuzzy rules: ∼ kI = (0. 5 + ∆1) kI (13)

∫

∼ kD = (1 + ∆2 ) kD Where −0.5 ≤ ∆1≤0.5 and −1<∆2 < 1, are outputs of fuzzy block. The fuzzy sets task error has two members; EN (error-negative) and EP (error-positive); the fuzzy set ∆1 has two members; ON1 (output1negative) and OP1 (output1-positive); the fuzzy set ∆2 has two members; ON2 (output2-negative) and OP2 (output2 positive) as shown in Fig. 1. The fuzzy set rate of error has four members; RBN (rate-big-negative), RN (rate-negative), RP (rate-positive) and RBP (rate-big- positive); the fuzzy set acceleration of error has four members; ABN (acceleration-big-negative), AN (acceleration-negative), AP (acceleration-positive) and ABP (acceleration-big-positive) as shown in Fig. 2. Proposed fuzzy rule sets are presented in Tables 2, 3. The control rule set comprises 24 fuzzy control rules.

(11)

where kI is the additive integration gain. Although using integrator is a method of dealing with SSE in robot control, in practical use of (11), many difficulties emerged. It seemed small integrator gain did not have enough ability to encounter deadband and other nonlinearity sources and tracking performance was weak. On the other hand, increasing integration gain can cause other problems and make control characters away from favorable target. Given that improved MTEJ parameters perform the same task of PID parameters, it can inspire one to perform similar tuning works for improved MTEJ method. Some well methods like the ‘Ziegler-Nichols method’ are employed to find suitable values for kP, kI , kD in PID controllers. The aim of the controller tuning is to obtain both fast responses, and good stability. The effects of increasing each of the controller parameters are summarize in Table 1: Since in underactuated manipulators, the possibility of singularity is more, it is necessary to decrease the task-space error as soon as possible to avoid singularity. This requires reducing kD and increasing kI as a solution to resolving deadband and friction effects. Nevertheless, it causes increasing overshoot and settling time. In experiments, it was observed that changing these two parameters as earlier mentioned cause instability. The reason will be determined during the study of improved MTEJ stability in the next session. To access practical control, finding a new tuning based on state parameters is necessary.

Theorem.. With invertible effective Jacobian and a=m in (4), if the control law described by (12) is applied to the underactuated manipulator system (2), then the tracking errors are globally asymptotically stable. Proof.. The steady state value of the integral in the PID control can be described as:

Is =

∫0

ts

∼ kI (qˆdes−qˆ) dτ

where ts implies the time in steady state. Iˆ is defined as an error of the integral to the steady state value of the integral

Table 1 The effects of increasing each of the improved MTEJ controller parameters. Parameter

Rise time

Overshoot

Settling time

SSE

kP kI kD

Decrease Decrease NT

Increase Increase Decrease

NT Increase Decrease

Decrease Eliminate NT

(14)

Fig. 1. membership functions plot for ‘e’ and incremental outputs of fuzzy.

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while the second term denotes an artificial potential energy. It is easy to show that steady state value of integral is equal to equivalent friction torque which contains deadband effects and friction torques in active joints (Is = −τˆs sign (qˆ)). Differentiating with respect to time and reminding that the rate of the kinetic energy in a mechanical system is equal to the power provided by the external forces leads to:

∼ T T V ̇ = (qˆ, qˆ )̇ = qˆ ̇ τˆ+qˆ ̇ kP qˆ + qˆT kI Iˆ

(17)

where T

τˆ = τˆc−τˆs sign (qˆ ̇ )

is motor output torques minus effective equivalent friction torques in active joints. Substituting (12) into (8) and the result into (17) for ̇ ≡ 0 yields: qˆdes, qˆdes

Fig. 2. membership functions plot for e,̇ e ̈. Table 2 Fuzzy rules for fixing ∆1 . Rate or task error

(18)

T∼ ̇ qˆṪ V ̇ = −qˆ ̇ kD qˆ −

Acceleration of task error

t

∫

∼ ˆ − qˆṪ τs+qˆṪ kt τˆc kI qdτ

RBN RN RP RBP

AN

AP

ABP

N N N N

N P P N

N P P N

N N N N

Or at t=tn , (n ≥ 2), T∼ T Vṅ = − qˆṅ k Dn qˆṅ − qˆṅ

EN EP

P N

∼

n −1 ⎛ ⎞ ∼ ∼ T τs+ qˆṅ k n τˆcn −1+ qˆnT k I n qˆn ⎜⎜Is+ ∑ kI i qˆi ∆ti⎟⎟ ⎝ ⎠ i =1

(20) To be assured of Lyapunov stability, it must be shown that V ̇ is negative semidefinite function.To achieve this, a ‘mathematical induction’ approach similar to [35] is followed. First, steady state error of j th coordinate of task space due to τˆs is defined as:

Rate of task error RBN

n −1

∑ kI i qˆi ∆ti− qˆnṪ i =1

Table 3 Fuzzy rules for fixing ∆2 . Task error

∼ˆ Tk II (19)

0

ABN

t −∆ t + qˆ

RN

RP

P N

N P

Eˆ sj =

RBP N P

τˆsj k Pj

,( j = 1. .m ) (21)

j th

diagonal element of kP . Therefore, the proof includes Where k Pj is two parts: qˆ > Eˆs and qˆ ≤Eˆs . Part (a). For qˆ > Eˆs . In this case, endeffector is not in stable equilibrium and proportional part of controller can produce adequate force to reduce error, therefore fuzzy commentator make integral part of controller inactive and (20) becomes: T T T∼ Vṅ = −qˆṅ k Dn qˆṅ − qˆṅ τs+qˆṅ k n τˆcn−1

(22)

For n =1, k is equal to zero so: T∼ T V1̇ = −qˆ1̇ k D1 qˆ1̇ − qˆ1̇ τs

Therefore, V1̇ is negative semi-definite. In [35], it has been shown for convenient MTEJ, if Vṅ is negative semi-definite function, with the right choice of kD and kP that is lower gain kP compared to kD , and small ̇ will also be semi-definite. enough time steps, Vn+1 Part (b). For qˆ < Eˆs . Suppose task error is gradually reduced until n=r, transient error for some task coordinates will be removed. For these coordinates at this moment, (qˆr̈ , qˆṙ )≈0 and Eq. (20) at t= tn+1 becomes:

Fig. 3. Underactuated manipulator.

Iˆ = Is−

∫0

t

∼ kI (qˆdes−qˆ) dτ

(15)

To investigate the stability of improved MTEJ algorithm at the ̇ ≡ 0 , the following positive definite function is introorigin, qˆdes, qˆdes duced as a Lyapunov function candidate:

V (qˆ, qˆ )̇ =

1 ̇ˆ ̇ T T ˆ ˆ +qˆ kp qˆ+Iˆ Iˆ ) (qHq 2

(23)

n

∼ ∼ T T T T ∼ Vṅ +1 = −qˆṅ +1 k Dn +1 qˆṅ +1− qˆṅ +1 τs−qˆṅ +1 ∑ kI i qˆi ∆ti+qˆṅ +1 k n +1 τˆcn+qˆnT+1 kI n +1 Iˆn +1 i=r

(16)

(24)

where the first term represents the kinetic energy of the manipulator

From (15), it is evident that:

Fig. 4. The structure of Fuzzy tuning MTEJ with integrator.

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Fig. 5. Detecting each joint using vision.

fuzzy controller, nevertheless, since integrator just performs when τs j qˆj ≤ k ( j =1. .m ), for each task coordinate, the amount of summations is

Table 4 Controller gains.

pj

Control type

kP

kI

kD

Fuzzy MTEJ with integrator MTEJ with big integrator gain MTEJ with small integrator MTEJ

40 40 40 40

350 250 100 –

70 70 70 70

less. Also, since amount of k n+1 is less than one, in |RHS|, amount of fifth term is approximately equal or less than the second; the same argument is true for amount of fourth term in |RHS| and second term in |LHS|. Choosing small values for ∆tn results in smaller values for the |RHS|. Therefore, by selecting small time steps and lower gain kP, in ∼ comparison to gains kD , the |LHS| can be made smaller than the |RHS| in a defined region, resulting in satisfaction of the condition. Thus, (28) introduces a criterion for choosing gains and time steps. Since V (qˆ, qˆ )̇ ̇ vanishes only at the origin; in addition V (qˆ, qˆ )→∞ as qˆ, qˆ ̇ →∞, thus V (qˆ, qˆ )̇ is a negative semi-definite function, according to Lyapunov's theorem for global stability, the algorithm is globally asymptotically stable. The proposed controller's scheme is depicted in Fig. 4.

n

∼ Iˆn +1 = −τs sign (qˆ)+ ∑ kI i qˆi ∆ti

(25)

i=r

It can be concluded that ( Iˆj ≤τs j,j = 1. .m) . (Is = −τs sign (qˆn)), it can be shown that:

Due to the fact that

n

∼ T T T ∼ Vṅ +1 = −qˆṅ +1 k Dn +1 qˆṅ +1− qˆṅ +1 τs−qˆṅ +1 ∑ kI i qˆi ∆ti

4. Experimental results

i=r n −1 ⎛ ⎞ ∼ ∼ T +qˆṅ +1 k n +1 ⎜⎜ −k pqˆn −k Dn qˆṅ − ∑ kI i qˆi ∆ti+k n τˆcn −1⎟⎟ ⎝ ⎠ i=r T ∼ +qˆn +1 kI n +1 Iˆn +1

The validity of the theoretically proposed control method was experimentally investigated by employing the experimental equipment demonstrated in Fig. 3.The first link is actuated by a DC motor-gearbox with a rotary encoder [(Belt faulhaber-2342L012)] while the second one is passive. Position of two side of each link is determined by vision as demonstrated in Fig. 5. The lengths of the system are set to be equal to 0.4 m for all links. The friction coefficient of joints and deadband effect of the actuator are unknown. From Section 2, the goal is to control y-direction of endeffector in task space. Both tracking and P.T.P control are verified for this apparatus. So the following studies are planned:

(26)

where T T qˆṅ +1 = qˆṅ +∆tn +1 qˆn̈ +O (∆tn2+1)

(27)

Considering the fact that Vṅ is negative semi definite, neglecting higher-order terms and substituting (27) into (26) and taking (20) into consideration, using the same procedure as in [35] for convenient MTEJ, gives:

Case 1:. The task is forcing y-direction of endeffector tracking a desired trajectory defined by:

T ∼ T − qˆṅ +1 k Dn +1 qˆṅ +1− qˆṅ +1 τs n

∼ ∼ T T T ≤qˆṅ +1 ∑ k Ii q ∆ti− qˆnT+1 k In +1 Iˆn +1++ k n +1 qˆṅ k p qˆn − k n +1 qˆṅ τs i=r

∼ −1 + k n +1 qˆnT k In Iˆn+∆tn +1 k n +1 (kP + kD + kI + Cˆ n + sign (qˆT ) τˆs )T Hˆ n (kP+ kD+ kI )

Case 2:. To compare the ability of methods for P.T.P control of endeffector, an exponential trajectory is defined:

(28)

y = 0. 2(1−e−10t )

where

kP = kp (qˆn +k n qˆ n −1+…+k n…k2 qˆ1) ∼ ∼ ∼ kD = k Dn qˆṅ +k n k Dn −1 qˆ ṅ −1+…+k n…k2 k D1 qˆ1̇ kI =

(30)

y = 0. 1sin (t )+0. 016sin (0. 2t )

i

n −1

n −2

i=r

i=r

∑ k∼Ii qˆi ∆ti+kn ∑ k∼Ii qˆi ∆ti+…+kn…kr+2 k∼Ir qˆr ∆tr

(31)

The initializations for joint angles and measured joint speeds are:

(q1 (0), q2 (0), q1̇ (0),

q2̇ (0)) = (−0. 68,

1. 57,

0,

0)

Which make some initial position and velocity errors. Next, the performance of the proposed fuzzy MTEJ with integrator control law is investigated and compared to that of regular MTEJ control law and the MTEJ with integrator without fuzzy tuning. The selected gains for the each algorithm are shown in Table 4. The sensitivity thresholds for the MTEJ algorithm are taken to be equal to 3 m and 30 m/s, respectively. Membership function for outputs of fuzzy are selected so that coefficient −0.5 < ∆1 <0.5 and −0.6 < ∆2 <0.6 . By experiment, amount of L in Fig. 1 is selected as 0.001. Also, amounts of L1 and L 2 in Fig. 2 for rate and acceleration of error are (0.01, 0.005) and (0.1,0.2), respectively. The Mamdani method is used as fuzzy interface system. Control signals are passed via a low-pass filter to eliminate short term fluctuations. Results for endeffector errors and estimated motor torques in task space are shown in Fig. 6 for Case 1 and Fig. 7 for

(29)

3.3.1. Discussion T In (28), if qˆṅ +1≠0,

the left-hand side (LHS) is always negative and ∼ its amount can be increased by selecting large damping gain, kD . For kj =0, which describes the standard TEJ algorithm, the right-hand side (RHS) consists of just the first and second terms were the second is positive definite; in this situation, the condition is satisfied by choosing ∼ ∼ large damping gains kD with respect to integration term kI . For kj ≠ 0, if (RHS)≥0 then fulfillment of the above condition is assured. To satisfy this condition, when RHS < 0, the |RHS| has to be smaller than the | LHS|. Although integrator gains are selected to be larger than none 97

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Fig. 6. Tracking error in task space (right) and control torque (left) for Case 1. a) MTEJ, b) MTEJ with additive low integrator gain, c) MTEJ with additive high integrator gain, d) fuzzy tuning MTEJ with additive integrator term.

error are close to zero and are placed in the range where fuzzy interprets that there is just errors due to Coulomb effects and orders the start of integrating task error.

Case 2 for each controller. As shown, using convenient MTEJ, there is always a steady state error in task space, adding integrator without tuning regardless of the gain magnitude, caused overshoot. Nevertheless, the nonlinear fuzzy MTEJ with integrator control system shows a good tracking and regulating control with minimum overshoot and less settling time and the steady state error is eliminated. Figs. 8 and 9 show rate and acceleration of task error and integrator command for both experimental cases. As can be seen in the beginning of motion when there is transient error, the integral command is approximately inactive. When transient error is eliminated, rate and acceleration of

5. Comparison with learning controllers For further investigation of the performance and effectiveness of the proposed method in compare with other advanced methods based on soft computing, Adaptive Neural Network Control Method based on [30] has been developed for task space control of underactuated 98

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Fig. 7. Tracking error in task space (right) and control torque (left) for Case 2. a) MTEJ, b) MTEJ with additive low integrator gain, c) MTEJ with additive high integrator gain, d) fuzzy tuning MTEJ with additive integrator term.

Fig. 8. (a) Rate and acceleration of tracking error, (b) ∫ ki (0. 5 + ∆1) edt for Case 1.

99

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Fig. 9. (a) Rate and acceleration of task error, (b) ∫ ki (0.5 + ∆1) edt for Case 2. T T T τˆc = Wˆ H ξ (qd̈ +Λe)̇ +Wˆ C η+Wˆ τ ζ+χr

Table 5 Parameters of simulated underactuated manipulator.

(37)

Parameter, Description

Value (unit)

where χ is a symmetric positive-definite matrix, and the parameters are updated by

L1,2,links length I1,2,links moment of inertia

0.5,0.5 (m)

WˆḢ = Γξ (q ) xr̈ rWˆĊ =Qη (q , q )̇ rWˆτ̇ =Nζ (τ , qa, qȧ ) r

d1,2,center of gravity m1,2,mass of links

0.01,0.01,(kg.m2 ) 0.25,0.25 (m) 0.5 (kg)

(38)

where Γ,Q, N are dimensional compatible symmetric positive-definite matrices, which yields

V ̇ = −r T χr

(39)

∼T If it is assumed J a is known, using (6), the joint control torque vector will be obtained. For illustrative purposes, a simulation has been carried out using a planar two-link underactuated manipulator, where its properties are illustrated in Table 5. The desired trajectory in task space is chosen as yd = 0. 2

sin (0. 5t )

(40)

The robot is initially rested with its end-effector positioned at the (x (0),y (0)) = (0.55,0.1) and (x ̇ (0), y ̇ (0)) = (0,0). A coulomb friction is considered for active joint as Fc=−0.1sign (q1̇ ) . Control torque is limited at ± 0.5N . m. For each HˆN , CˆN , Δτˆ N , a 256-node neural network is used. The centers of radial based functions are in the area of [−1,1] and their radiuses are fixed at 2. The gains of controller are Λ=3,χ =20 . Adaptive law parameters are selected Γ = 0. 0005 and Q = N =0.001. FMTEJ parameters are similar to Table 4. As shown in Fig. 10, for both methods, the tracking error converges to small neighborhood around zero, however FMTEJ has excellent performance against friction effect and results in less settling time. Since time is needed for adaptive neural network for learning process. On the other hand, the simulation ∼T time for FMTEJ is less than ANN. It should be mentioned if J a is considered unknown, ANN performance will be unsatisfactory and the controller may fail to track desired trajectory. Accordingly, the FMTEJ results in better response in less time.

Fig. 10. Comparison of ANN and FMTEJ controller in simulation.

manipulators. First consider task space dynamic of underactuated ∼−T manipulator which is described by (7), (8). In this case, Hˆ and J a ˆ are functions of q while C is a function of q ,q .̇ In general, for deadzone nonlinearity, let τˆc = τˆ a+∆τˆ , ∆τ is error and it is a function of τc, qa, qȧ . Assume HˆN , CˆN , τˆN can be modeled as T T HˆN = Wˆ H ξ (q ) = Hˆ +EH CˆN = Wˆ C η (q , q )̇ = Cˆ +EC Δτˆ N T = Wˆ τ ζ(τ , qa, qȧ )=Δτˆ+Eτ

(32)

where WˆH , WˆC , Wˆτ are the weights of the neural networks, ξ , η , ζ are corresponding Gaussian basis functions with input vectors consisting of q ,q ,̇ τ . EH , EC , EC are the modeling errors.

6. Conclusions While the MTEJ method is a non-model-based controller to control underactuated manipulators, in practice, it was observed that its performance is not satisfactory in eliminating steady state errors due to unknown nonlinear forces like frictions for controlling underactuated manipulator in task space. In this paper, a fuzzy tuned MTEJ algorithm with additional integrator was introduced to remove steady state errors in underactuated manipulators. Fuzzy tuning of controller coefficients enhances control characteristics like steady state errors and settling time. The fuzzy set has three inputs; task error, its rate and acceleration and two outputs which are used for tuning damping and integrating gains. The global stability of controller is investigated using a Lyapunov function. The experimental results reveal that the fuzzy MTEJ controller has satisfactory performance in eliminating the effect of unknown perturbation and nonlinear forces with lower computation efforts.

5.1. Neural network Controller design Defining

e (t ) = xd (t )−x (t )

(33)

xṙ (t ) = xḋ (t )+Λe (t )

(34)

r (t ) = xṙ (t )−x ̇ (t ) = e ̇ (t )+Λe (t )

(35)

where xd (t ) is the desired trajectory in task space and Λ is a symmetric ∼T positive definite matrix. As mentioned before, J a in (6) is a function of manipulators dynamic. Defining

V=

1 Tˆ r Hr 2

(36)

similar to process in [30], using 100

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S. Reza Naghibi received his BS in mechanical Engineering from Iran University of Science and Technology (TehranIran), in 2008, and MS in Mechanical Engineering from Ferdowsi university in 2011. He is currently working towards his PhD in Mechanical Engineering at University of Zanjan. His research interest is in the areas of control, robotics and automation.

Ali. A.Pirmohamadi received his BS in mechanical Engineering from Bu- Ali Sina University, in 1993, and his MS and PhD both from Tarbiat Modaress University in 1996 and 2002 respectively. He is professor with mechanical engineering department at Zanjan university. He teaches courses in areas of Dynamics, Vibration, Control and FEM. His research interests are in the areas of Robotic, Vibration and Control.

S. Ali A. Moosavian received his BS, in 1986, from Sharif University of Tech-nology and MS, in 1990, from Tarbiat Modaress University (both in Tehran) and his PhD, in 1996, from McGill University (Montreal, Canada), all in Mechanical Engineering. He is Professor with the Mechanical Engineering Departmentat K. N. Toosi University of Technology. He teaches courses in the areas of robotics, dynamics, automatic control, analysis and synthesis of mechanisms. His research interests are in the areas of dynamics modeling and motion/impedance control of terrestrial and space robotic systems. He is one of the three founders of the ARAS Research Center for Design, Manufacturing and Control of Robotic Systems, and Automatic.

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