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Neural Network Sliding-Mode-PID Controller Design for Electrically Driven Robot Manipulators

S. E. Shafiei1, M. R. Soltanpour2

1. Department of Electrical and Robotic Engineering, Shahrood University of Technology, Shahrood, Iran Tel:+989133187993, E-mail: sehshf@yahoo.com

2. Department of Electrical Engineering, Shahid Sattari University, Tehran, Iran Tel:+989124863071, E-mail: m_r_soltanpour@yahoo.com

Abstract: This paper addresses a neural-network-based chattering free sliding mode control (SMC) for robot manipulators including structured and unstructured uncertainties in both manipulator and actuator dynamics by incorporating a PID outer loop. The main idea is that the robustness property of SMC and good response characteristics of PID are combined to achieve more acceptable performance. Uncertainties in the robot dynamics and actuator model are compensated by a two-layer neural network. External disturbance and approximation error are counteracted by robust signal with adaptive gain. The stability of closed-loop system is guaranteed by developed control scheme. Finally, the proposed methodology is applied to a two-link elbow robot as a case of study. The simulation results show the effectiveness of the method and its robustness to uncertainties and disturbances.

Key Words: robot manipulators, sliding mode control, neural networks, PID control, uncertainties.

1. INTRODUCTION

Robot manipulators are well-known as nonlinear systems including high coupling between their dynamics. These characteristics, in company with structured uncertainties caused by model imprecision of link parameters, payload variation, etc., and unstructured uncertainties produced by un-modeled dynamics, such as nonlinear friction and external disturbances, make the motion control of rigid-link manipulators a complicated problem [1]. Additionally, one constraint in the robot controller designs is saturation nonlinearity of actuators which is less considered in control design of robot manipulators.

A well improved control method for coping with mentioned difficulties is sliding mode control which is capable in controlling wide range of different systems, such as nonlinear uncertain systems, MIMO systems and even, discrete time systems [2-3]. Moreover, it has a good deal of advantages such as insensitivity to parameter variations, disturbance rejection, and fast dynamic responses [4]. A good survey on this topic has been provided by Hung, et al [5]. Despite these merits, SMC suffers from some disadvantages. Actually, the sliding mode control law consists of two main parts [6]. The first part is the equivalent control law which involves inverse dynamics of model nonlinearities that demonstrates the dependency of SMC on the dynamical model of the plant. The second part is the robustifying term which has discontinuous nature and may employ unnecessary high control gain to overcome uncertainties and disturbances. However, this discontinuity may lead to chattering phenomenon that can excite un-modeled high-frequency plant dynamics and harm the overall control system. Also, using high control gain may cause saturating the actuators.

Since, calculating of what is known as the equivalent control law is difficult, SMC performance decreases in some applications where the mathematical modeling of the system is very hard (e.g. robotic

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systems), and where the system has a large range of parameter variations together with unexpected and sudden external disturbances. Accordingly, several methods have been developed for improving the SMC performance which the most significant of them is intelligent control approach [7] mainly includes fuzzy logic control [8-9] and neural network control [10-12].

Many neural network sliding mode control algorithms have been found for the robot control system without including the actuator dynamics [13-17], while actuator dynamics carry out a significant role in the complete robot dynamics and ignoring them may cause detrimental effects, especially in the case of high-velocity moment, highly varying loads, friction, and actuator saturations [18-21]. Since the electrical actuators are highly controllable in comparison with the other one, they are more convenient for driving manipulators. Also, in practical applications, the voltages or currents of the electrical actuators are accessible for applying control commands and consequently, torque-based control design confronts implementation problems when one intends to apply the torque control commands directly to actuators. Sliding-Mode-PID control for robot manipulator was explored by Shafiei and Ataei [22]. In their study, although, the uncertainties are considered but controller design is extremely model-dependent. Also, control command starts with high gain and actuator dynamics is neglected. Moreover, stability analysis is not investigated after incorporating fuzzy tuning system. A robust neural-fuzzy-network controller was designed in [20] for the position control of an n-link robot manipulator including actuator dynamics. Although, their control scheme does not require compensating auxiliary control design, but the employed network is more complicated and uses excess number of neurons. In addition, the second derivative of position angle is required as a part of controller inputs. Capisani et al., [23] presented an inverse dynamic-based second-order sliding mode controller to perform motion control of robot manipulators, but this method involves the higher order derivatives of the state variables.

In this note, the motion tracking control of multiple-link robot manipulators actuated by permanent magnet DC motors is addressed. Sliding-mode-PID tracking controller is designed such that all the states and signals of the closed loop system remain bounded in the presence of unknown parameters and uncertainties. Also, neural network universal approximation property is employed for compensating uncertainties. Furthermore, the proposed controller contains an outer PID-loop that enhances the approximation performance during the initial period of weight adaptations, and provides designing a simple NN with lower amount of layers and neurons. Adaptation laws are applied to adjust the NN weights on-line. In order to avoid high gain control, the gain factor of robustifying term is designed adaptively.

The reminder of this paper is organized as follows. In section 2, the mathematical models of both electrically and mechanically parts of an n-link electrically driven robot manipulator are given and the tracking problem is considered. Section 3 presents the design of the SMC including PID loop to which is denoted as SMC-PID and explains the NN structure. The design of NNSMC-PID is developed in section 4. In section 5, the simulation results carried out on a two-link elbow robot is provided and finally, some conclusions are drawn in section 6.

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2. PLANT DYNAMICS AND PROBLEM FORMULATION

The mathematical equations describing electrical and mechanical dynamics of a permanent magnet DC motor are as follows [1]:

(1)

τ (2)

τ K i (3)

where is the armature voltage of the motor, and are armature equivalent resistance and inductance, respectively, is the back electromotive force constant, is the armature current and denotes the rotor position, is the total moment of inertia, is the damping coefficient, and τ represent the generated motor torque and the load torque, respectively, and is the diagonal matrix of motor torque constant.

The dynamical equation of an n-link robot manipulator in the standard form is describing by [1]:

, (4)

where ∈ is the completed inertia matrix, the vectors , , ∈ are the position, velocity and angular acceleration of the robot joints, respectively. Moreover, the matrix , ∈ is the matrix of Coriolis and centrifugal forces and ∈ is the gravity vector. Also, ∈ stands for the dynamic friction vector, ∈ denotes the vector of disturbance and un-modeled dynamics, and finally, is the torque vector.

With the purpose of increasing the motion speed of the manipulators, motors are equipped by the high reduction gears as follows:

(5)

and

(6)

where is the diagonal matrix of reduction ratio. In the following, some conventional properties of the robot manipulators and a practical constraint are considered.

Property 1– The inertia matrix is symmetric and positive definite, .

Property 2– The matrix of 2 is skew-symmetric, i.e. for any vector of , we have

2 0.

Constraint 1– The maximum voltage that joint actuator can supply is . So, we have: | | , 1,⋯ ,

It should be noted that, the applicable control input for driving robot arm is the armature voltage of the motors, here. So, by using equations (1)-(6) and neglecting inductance , because of its tiny amount, the following equation is achieved.

(7)

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The previous equation can be expressed in a compact form as:

(8)

with is the control command and the other parameters are

(9)

(10)

(11)

(12)

Remark 1– By noting that the parameters, , , and are positive definite diagonal matrices, the matrix is symmetric and positive definite.

Remark 2– From relations (9) and (11), and property 2, the matrix 2 is skew-symmetric too.

If one considers the desired trajectory which joint position must be held on it as , then the tracking error could be defined as:

(13)

Here, the tracking problem refers to define a control law such that the error would be driven toward the inside of an arbitrary small region around zero, with maintaining the armature voltage within the constraint 1. This aim will be attained in the next section.

3. SMC-PID DESIGN AND NN DESCRIPTION

A key step in designing sliding mode controller is to introduce a proper sliding surface so that tracking errors and output deviations can be reduced to a satisfactory level [24]. Accordingly, the sliding surface is considered as (14), containing the integral part in addition to the derivative term.

(14)

where is diagonal positive definite matrix. Hence, 0 is a stable sliding surface and → 0 as → ∞. Only defining the sliding surface as (14) is not adequate to claim that SMC-PID is designed, but

the control effort must contain the independent PID part. For this purpose, the robot dynamic equations can be rewritten based on the sliding surface (in term of filtered error) as follows:

(15)

where

(16)

where , and are given by (9), (11) and (12) respectively, and

(17)

Note that the input vector of includes linear combination of and , (i.e. ) which they comprise , and , , too, respectively. The input dimension of the two-layer NN designed here is

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less than that of given by [13], and thus the proposed method is more desirable from an implementation point of view. Sliding mode control strategy consists of designing a two-part controller.

(18)

with is equivalent control part which is applied to cancel the uncertain nonlinear function , and

specifies robust control term. Considering unknown parameter, uncertainties and disturbances indicates that the function is not accessible. Briefly speaking, neural networks incorporate to reconstruct the

part by approximating the function , here. According to universal approximation property of neural networks [11], there is a two-layer NN with sufficient number of neurons, and sigmoid or RBF activation function for hidden layer and linear activation function for output layer (see Fig.1) such that:

(19)

where ∈ is the input vector computed by (17), ∈ and ∈ represents the NN weights for hidden and output layers, respectively, ∙ denotes activation function of the hidden layer and is NN approximation error. Choosing activation function is arbitrary provided that the function satisfies an approximation property and it and its derivative are bounded [11], consequently the sigmoid activation function is considered, here.

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(20)

Succeeding section explains complete controller design and investigates stability content.

Fig. 1. Two-layer NN structure

4. NNSMC-PID CONTROLLER DESIGN

Note that the utilized weights in (19) are optimum and is approximated ideally, over there.

Estimation of is accomplished by the estimated weights and , respectively. So, the NN controller is designed as:

(21)

where is estimation of and and are updated adaptively. The estimation errors are defined as follows:

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, W W W (22)

also, the hidden layer output error for a given input is

(23)

Consider the as its Taylor series expansion as

σ′ (24)

where ∙ denotes higher order terms in Taylor series and

′ (25)

From (23) and (24), we have:

(26)

Now, one can obtain overall error between optimum function and its estimation as:

(27)

where

(28)

is the uncertain term and is supposed to be bounded by as demonstrated in (29).

‖ ‖ ‖ ‖ ‖ ‖ (29)

Design of the control system is provided in the following theorem and is illustrated in Fig.2 schematically.

Theorem 1– Robot manipulator including actuator dynamics represented by equation (8) is considered, and the sliding surface is defined by (14). If the control input is designed as (30) together with adaptation laws of NN controller as (31)-(33), then the asymptotic stability of the dynamical system is guaranteed.

sgn (30)

′ (31)

′ (32)

sgn (33)

where is a positive definite diagonal matrix, is the estimated value of . Also, , and are positive constants and sgn ∙ denotes sign function.

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Proof: consider the following Lyapunov function candidate

12

12

12

12

(34)

where ∙ denotes the trace operator and . Differentiating of the relation (34) gives

12

1 1 1 (35)

By substituting (30) in to the first part of (35) and by using (27) one can obtain

sgn s

sgn s (36)

Some useful relations for manipulating last tow equations are provided in the following.

Replacing (36) in (35) and using above relations, produce

12

21

1sgn s

1

(37)

Note that , , , and remark 2 yields 2 0. Also, if adaptive

laws (31) and (32) are taken in to account, then we have

sgn s1

sgn s

(38)

substituting (33) in (38) and adopting (29), yields

‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ⋯ ‖ ‖ ‖ ‖ ‖ ‖ 0 (39)

where is minimum singular value of . Since 0, the stability in the sense of Lyapunov is

guaranteed which implies that the parameters , , and (and consequently , , ) are bounded. In addition

lim→

∞

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whereas, is bounded, hence Barbalat’s Lemma [3] indicates that lim → 0. Note that ‖ ‖ 0, as a result → 0 as → ∞. Therefore, the proposed control system is

asymptotically stable.

Fig. 2. Block diagram of the control system structure

Remark 3– The PID term in the above control effort, makes Lyapunov derivative more negative, so it makes the transient response faster and also ensures the performance efficiency during the initial period of weights adaptations.

Remark 4– In practical systems, however, it is impossible to achieve infinitely fast switching control, because of finite time delays for the control computation and limitation of physical actuators. For that reason, the sign function is replaced by saturation function here, and the stability matter is investigated analytically.

The saturation function is selected as

sats

φ

s

φ‖s‖ φ

sgns

φ‖s‖ φ

(40)

where is a thin boundary layer such that 0 1. The adaptive law (33) must be replaced by

; So, the equation (38) is changed to

sat (41)

Now, there are two situations; a) if ‖ ‖ , then

‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ⋯ ‖ ‖ ‖ ‖ 0 (42)

b) if ‖ ‖ , then

GAIN ADAPTATION

WEIGHT ADAPTATIONS

NN

PID

ROBUST COMPENSATO

R

ELECTRICALLY DRIVEN ROBOT MANIPULATOR

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‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ⋯ ‖ ‖ ‖ ‖ 0 (43)

Note that, since 0 1, therefore ‖ ‖. Both situations imply that 0, and

consequently, the control system remains stable after replacing saturation function. It is worth mentioning that,

Remark 5– The sliding gain is chosen dynamically and its dynamic depends on sliding surface. When

the states go far from the sliding manifold, the absolute value of increases to force them back to

sliding manifold, and when the states are close to the sliding manifold, the absolute value of decreases accordingly. This feature beside the replacing saturation function, act as what is heuristically designed by fuzzy system in [22]. Furthermore, the system stability is addressed here.

5. THE CASE STUDY AND SIMULATION RESULTS

In order to show the effectiveness of the proposed control method, it is applied to a two-link elbow robot (see Fig.3) driven by permanent magnet DC motors with the following parameters:

2 cos coscos

,sin sinsin 0

cos cos coscos

(44)

where is the angle of joint , is the mass of link , is the total length of link , is center-of-gravity length of link , and 9.8 m/s2 is gravity acceleration.

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Fig. 3. Two-Link Elbow Robot Manipulator

The detailed parameters of this robot manipulator and permanent magnet DC motor actuators are provided in table 1 [20]. According to the actuator manufacturer, the DC motors are able to accept input voltages within the following bounds:

| | 12 [V]

| | 12 [V] (45)

For example, one can use 12V DC servo motors for actuating joints. In practice, also, a servo control card is required which should include multichannels of digital/analog (D/A) and encoder interface circuits.

TABLE 1 PARAMETERS OF TWO-LINK ELBOW ROBOT AND ACTUATORS

Two-Link Elbow Robot Permanent-Magnet DC Motors

3.55 kg 0.75 kg 3.7 10 kg.m2 1.47 10 kg.m2

205 mm 210 mm 1.3 10 N.m/s 2 10 N.m/s

154.8 mm 105 mm 2.8 Ω 4.8 Ω

0.21 N.m/A 0.23 N.m/A 3 mH 2.4 mH

1/60 1/30 2.42 10 s/rad.V 2.18 10 s/rad.V

The external disturbances can be considered as external forces injected into the robotic system, and are supposed to have following expression.

sin 4 sin 4 (46)

Also, the friction term is considered here as [20]:

20 0.8sgn 4 0.16sgn (47)

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In order to show the effectiveness of proposed controller in tracking of desired trajectory, it is assumed to have the sinusoidal shape in this simulation.

sin sin (48)

The design parameters are given in table 2. The gain matrices and are selected such that the roots of the characteristic polynomial 0 lie strictly in the open left half of the complex plane when the system is in sliding mode ( 0). The neural network designed here has four neurons as hidden layer and two neurons as output layer, and its weights are totally initialized at zero.

Remark 6– For a two-layer NN designed here with the input vector given by (17), we have 6, 4 and 2, for a two-link manipulator. Accordingly, the numbers of adaptive weights are 24

and 8 for input-to-hidden layer weights and output layer weights, respectively. So, only 32 weight parameters must be adaptively updated here while using the NN given in [13], with 10, 10 and 2, this number increases to 120. If the network size is chosen to large, the improvement of control performance is limited and the computation burden for the CPU is significantly increased.

The gain matrix which acts as the gain of the PID term is determined large enough to improve transient response in the initial period of weight adaptations. On the other hand, choosing to a large extent increases the overall controller gain and may exceed the permissible voltages of the actuators that are regarded in constraint 1. So, there is a tradeoff between fast response and practical limitations.

TABALE 2 DESIGN PARAMETERS

10 00 10

24 00 24

1/60 00 1/30

5 00 5

5 5

2 0.05

The mass variation of second link, the external disturbance and the friction are the major factors that affect the control performance of the robotic system. In the reminder of this section, two simulation cases are carried out to show the improvement due to the NNSM_PID control method proposed in this paper. In both cases, the simulation results of applying presented method are compared with the related results of the fuzzy sliding mode_PID (FSM_PID) control method proposed in [22]. In the first case, the disturbance (46) and mass variation are injected and in the second case, the friction term is exerted too. The mass variation condition is that 1 kg weight is added to the mass of 2nd link (i.e. 1.75 kg). For the FSM_PID case, the control law is as following [22]:

sgn (49)

(50)

12

(51)

where, is the control input, is of fuzzy system output and and are the scaling gain of

the fuzzy system output. Here, it is assumed that only manipulator parameters could be estimated and

actuator parameters are still unknown. So, is chosen as [22]:

(52)

where , and are achieved from nominal value of manipulator parameters. However, all of the manipulator parameters are considered with 10% uncertainty. The design parameters of the FSM_PID controller are

3.2 00 3.5

, 0.8 00 0.7

(53)

Simulation 1 In this case, the friction term is neglected, mass variation occurs at 3 sec and external disturbance is injected at 6 sec. The desired trajectory is depicted in Fig.4. The vectors of tracking errors of FSM_PID and NNSM_PID are shown in Fig.5 (a) and (b), respectively. Both diagrams of Fig.5 are plotted in the same scaled axes to achieve fairly comparison. The FSM_PID controller does not meet the tracking purpose in the unknown actuator parameters and mass variation conditions. On the contrary, the method proposed in this paper provides swift and precise tracking responses. Fig.6 displays the control efforts (i.e. input armature voltages of motors). The FSM_PID associated control commands are jagged to some extent, while, the NNSM_PID case produces smooth control commands with slowly variation and lower voltage amplitude. Lower voltage commands are more protected toward actuator saturations. The NN outputs are shown in Fig.7 and it indicates that the designed neural network can approximate nonlinear terms with unknown parameters, smoothly and boundedly.

Simulation 2 With the purpose of showing robustness of our designed controller against uncertainties and un-modeled dynamics, the friction term (47) is added here. The vectors of tracking errors of FSM_PID and NNSM_PID are shown in Fig.8 (a) and (b), respectively. However, the response of the FSM_PID case is further undesirable in this condition, on the other hand, the NNSM_PID control remains robust and its response is satisfactory, as well as previous simulation case. Control efforts of this case are demonstrated in Fig.9. Because of exerting friction term, the input voltage commands are higher than previous case but the NNSM_PID control commands are still smooth and vary slowly. The NN output is shown in Fig.8. Finally, as can be seen from Fig.9, matrix norm of the adaptive weights,

and , have bounded value, less than 3, that it verifies what was claimed in the Theorem 1 about

boundedness of these signals.

6. CONCLUSION

Whenever, fast and high-precision position control is required for a system which has high nonlinearity and unknown parameters, and also, suffers from uncertainties and disturbances, such as robot manipulators, in that case, necessity of designing a developed controller that is robust and has self-learning ability is appeared. For this purpose, an efficient combination of sliding mode control, PID

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control and neural network control for position tracking of robot manipulators driven by permanent magnet DC motors was addressed in this note. SMC is robust against uncertainties, but it is extremely dependent on model and uses unnecessary high control gain; So, NN control approach is employed to approximate major part of the model. A PID part was added to make the response faster, and to assure the reaching of sliding surface during initial period of weight adaptations. Moreover, four practical aspects of robot manipulator control such as actuator dynamics, restriction on input armature voltage of actuators due to saturation of them, friction and uncertainties were considered. In spite of these features, the controller was designed based on Lyapunov stability theory and it could carry out the position control with fast transient and high-precision response, successfully. Finally, three-step simulation results of applying the proposed methodology to an electrically driven two-link elbow robot were provided, and they confirmed the success of presented approach. However, the presented design was performed in the joint space of robot manipulator and kinematic uncertainty was not considered. In the future work, we will expand this method to work space design with uncertain kinematics.

Fig. 4. Desired input trajectory

(a) (b)

Fig. 5. (sim1) Tracking error of joints, (a) FSM_PID (b) NNSM_PID

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(a) (b)

Fig. 6. (sim1) Control commands (a) FSM_PID (b) NNSM_PID

Fig. 7. (sim1) NN control effort

(a) (b)

Fig. 8. (sim2) Tracking error of joints (a) FSM_PID (b) NNSM_PID

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(a) (b)

Fig. 9. (sim2) Control commands (a) FSM_PID (b) NNSM_PID

Fig. 10. (sim2) NN control effort

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Fig. 11. (sim2) Matrix norm of adaptive weights ‖ ‖ and ‖ ‖ REFERENCES

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