Robot Manipulators 2011 Part 7 pps

In this study, a fuzzy sliding mode controller based on RBFNN is proposed for robot manipulator.. Adaptive Neural Network Based Fuzzy Sliding Mode Control Sliding Mode Control If the d

In this study, a fuzzy sliding mode controller based on RBFNN is proposed for robot

manipulator Fuzzy logic is used to adjust the gain of the corrective control of the sliding

mode controller The weights of the RBFNN are adjusted according to some adaptive

algorithm for the purpose of controlling the system states to hit the sliding surface and then

slide along it

The paper is organized as follows: In section 2 model of robot manipulator is defined

Adaptive neural network based fuzzy sliding mode controller is presented in section 3

Robot parameters and simulation results obtained for the control of three link scara robot

are presented in section 4 Section 5 concludes the paper

2 Model of Robot Manipulator

The dynamic model of an n-link robot manipulator may be expressed in the following

Lagrange form:

)(),()(),()(q q C q q q G q F q q u t

M( )∈ denotes the inertia matrix; C( q,q)∈R nxn expresses the coriolis and

centrifugal torques, G(q)∈R n is the gravity vector; F( q,q)∈R nxn is the unstructured

uncertainties of the dynamics including friction and other disturbances; u(t)∈R nx1 is the

actuator torque vector acting on the joints

3 Adaptive Neural Network Based Fuzzy Sliding Mode Control

Sliding Mode Control

If the desired system states are available, the desired angle of the manipulator joint are

denoted by q d The control objective is to drive the joint position q to the desired position

d

q The tracking error equation can be written as follows:

d

q q

Define the sliding surface of the sliding mode control design should satisfy two

requirements, i.e., the closed-loop stability and performance specifications (Chen & Lin,

2002) A conventional sliding surface corresponding to the error state equation can be

represented as:

e e

where λ is a positive constant

In general, sliding mode control law can be represented as:

c

u

Adaptive Neural Network Based Fuzzy Sliding Mode Control of Robot Manipulator 203

where ueq is the equivalent control law for sliding phase motion and u c is the corrective

control for the reaching phase motion The control objective is to guarantee that the state

trajectory can converge to the sliding surface So, corrective control u c is chosen as follows:

00

01)(

s s

s s

Notice that (5) exhibits high frequency oscillations, which is defined as chattering

Chattering is undesired because it may excite the high frequency response of the system

Common methods to eliminate the chattering are usually adopting the following i) Using

the saturation function ii) Inserting a boundary layer (Tsai et al., 2004) In this paper, shifted

sigmoid function is used instead of sign function:

11

2)

e s

3.2 Fuzzy Sliding Mode Controller

Control gain K is fuzzified with the fuzzy system that shown in Fig 1 (Guo & Wuo, 2003)

Figure 1 Diagram for a fuzzy system

The rules in the rule base are in the following form:

IF s i is A i m, THEN K i is B i m

where A i m and B i m are fuzzy sets In this paper it is chosen that s i has membership

functions: NB, NM, NS, Z, PS, PM, PB and Ki has membership functions: Z, PS, PM, PB;

where N stands for negative, P positive, B big, M medium, S small, Z zero They are all

Gaussian membership functions defined as follows:

σ

α

i A

x

From the knowledge of the fuzzy systems, K i can be written as

)()

(

)(

1

1

i ki T ki M

m

i A

M m

i A m

s

s K

m

m i

ψ θ μ

m ki i ki i

ψ , and M is the amount of the rules

3.3 Radial Basis Function Network

A whole knowledge of the system dynamics and the system parameters is required to be

able to compute the equivalent control (Ertugrul & Kaynak, 1998) This is practically very

difficult To be able to solve this problem, neural network can used to compute the

equivalent control

A RBFNN is employed to model the relationship between the sliding surface variable, s, and

equivalent control, ueq In other words, sliding variable, s, will be used as the input signal

for establishing a RBFNN model to calculate the equivalent control

The Gaussian function is used as the activation function of each neuron in the hidden layer

(10) The excitation values of these Gaussian functions are distances between the input

values of the sliding surface value, s, and the central positions of the Gaussian functions

j j

c s x

g

where j is the j neuron of the hidden layer, c j is the central position of neuron j σj is the

spread factor of the Gaussian function

Weightings between input and hidden layer neurons are specified as constant 1 Weightings

between hidden and output layer neurons (w j) are adjusted based on adaptive rule

The output of the network is:

∑

=

j j j

Based on the Lyapunov theorem, the sliding surface reaching condition is s< 0 If control

input chooses to satisfy this reaching condition, the control system will converge to the

Adaptive Neural Network Based Fuzzy Sliding Mode Control of Robot Manipulator 205

origin of the phase plane Since a RBFNN is used to approximate the non-linear mapping

between the sliding variable and the control law, the weightings of the RBFNN should be

regulated based on the reaching condition, s< 0 An adaptive rule is used to adjust the

weightings for searching the optimal weighting values and obtaining the stable converge

property The adaptive rule is derived from the steep descent rule to minimize the value of

swith respect to wj Then the updated equation of the weighting parameters is (Huang et

al., 2001):

)(

)()(

t w

t s t s w

where η is the learning rate parameter The structure of RBFNN is shown in Fig 2

Figure 2 Radial Basis Function Neural Network (RBFNN)

3.3 Proposed Controller

The configuration of proposed controller is shown in Fig 3 The control law for proposed

controller is as (4) form K gain of the corrective control u c is adjusted with fuzzy logic and

is the equivalent control ueq is computed by RBFNN The structure of RBFNN using this

study has three inputs and three outputs The number of hidden nodes is equal to 5 The

weights of the RBFNN are changed with adaptive algorithm in adaptive law block Outputs

of the corrective control and RBFNN are summed and then applied to robotic manipulator

Figure 3 Block diagram of the proposed control system

4 Application

4.1 Robot Parameters

A three link scara robot parameters are utilized in this study to verify the effectiveness of the proposed control scheme The dynamic equations which derived via the Euler-Lagrange method are presented as follows (Wai & Hsich, 2002):

23 22 21

13 12 11

) sin(

q q q C C C C C C C C C q l q q q M M M

M M M

M M M

1 3

t u t t q f g m

=++

11

3)cos(

2

m l q m m l m m m l M

0

32 31 23

13=M =M =M =

M

21 3 2 2 2 2 3 2 2 1

m l q m m l

22

m l M

3

33 m

M =

) 2

Adaptive Neural Network Based Fuzzy Sliding Mode Control of Robot Manipulator 207

⎟⎟

⎞

⎜⎜

⎛+

−

= 2 2 3

m q

0

33 32 31 23 22

C

in which q1,q2,q3, are the angle of joints 1,2 and 3; m1,m2,m3 are the mass of the links 1,2

and 3; l1,l2,l3 are the length of links 1,2 and 3; g is the gravity acceleration Important

parameters that affect the control performance of the robotic system are the external

disturbance t1(t), and friction term f  (q)

The system parameters of the scara robot are selected as following:

5

)2sin(

5

)2sin(

5)1

t t

t t

++

++

=

)3sin(

3)(2.012

)3sin(

3)(2.012

)3sin(

3)(2.012)(

3 3

2 2

1 1

t q

sign q

t q

sign q

t q

sign q

q f

Central positions of the Gaussian function c j are selected from -2 to 2 Spread factors σj

are specified from 0.1 to 0.7 Initial weights of network are adjusted to zero

The proposed fuzzy SMC based on RBFNN in Fig 3 is applied to control the Scara robot

manipulator

The desired trajectories for three joint to be tracked are,

)

* sin(

1 0 1 )

Tracking position, tracking error, control torque and sliding surface of joint 1 is shown in

Fig 4 a, b, c and d respectively Tracking position, tracking error, control torque and sliding

surface of joint 2 is shown in Fig 5 a, b, c and d respectively Fig 6 a, b, c and d shows the

tracking position, tracking error control torque and sliding surface of joint 3 Position of joint

1, 2 and 3 is reached the desired position at 2s, 1.5s and 0.5s, respectively

In Fig 4c, 5c and 6c it can be seen that there is no chattering in the control torques of joints

Furthermore, sliding surfaces in Fig 4d, 5d, 6d converge to zero It is obvious that the

chattering in the sliding surfaces is eliminated

5 Conclusion

The dynamic characteristics of a robot manipulator are highly nonlinear and coupling, it is difficult to obtain the complete model precisely A novel fuzzy sliding mode controller based on RBFNN is proposed in this study To verify the effectiveness of the proposed control method, it is simulate on three link scara robot manipulator

RBFNN is used to compute the equivalent control An adaptive rule is employed to on-line adjust the weights of RBFNN On-line weighting adjustment reduces data base requirement Adaptive training algorithm were derived in the sense of Lyapunov stability analysis, so that system-tracking stability of the closed-loop system can be guaranteed whether the uncertainties or not Using the RBFNN instead of multilayer feed forward network trained with back propagation provides shorter reaching time

In the classical SMC, the corrective control gain may choose larger number, which causes the chattering on the sliding surface Or, corrective control gain may choose smaller number, which cause increasing of reaching time and tracking error Using fuzzy controller to adjust the corrective control gain in sliding mode control, system performance is improved Chattering problem in the classical SMC is minimized

It can be seen from the simulation results, the joint position tracking response is closely follow the desired trajectory occurrence of disturbances and the friction forces

Simulation results demonstrate that the adaptive neural network based fuzzy sliding mode controller proposed in this paper is a stable control scheme for robotic manipulators

(b) 0 1 2 3 4 5 6 7 8 9 10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Adaptive Neural Network Based Fuzzy Sliding Mode Control of Robot Manipulator 209

(b) 0 1 2 3 4 5 6 7 8 9 10-1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

(b) 0 1 2 3 4 5 6 7 8 9 10-1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

t(s)Figure 6 Simulation results for joint 3: (a) tracking response; (b) tracking error; (c) control torque; (d) sliding surface

6 References

Guo, Y., Woo, P (2003) An Adaptive Fuzzy Sliding Mode Controller for Robotic

Manipulator IEEE Trans on System, Man and Cybernetics-Part A: Systems and Humans, Vol.33, NO.2

Utkin, V.I., (1997) Variable Structure Systems with Sliding Modes IEEE Trans on Automatic

Control AC-22 212-222

Edwards, C and Spurgeon, K (1998) Sliding Mode Control Taylor&Fransis Ltd

Hussain, M.A., Ho, P.Y (2004) Adaptive Sliding Mode Control with Neural Network Based

Hybrid Models Journal of Process Control 14 157-176

Lin, S and Chen, Y (1994) RBF-Network-Based Sliding Mode Control System, Man and

Cybernetics, ‘Humans, Information and Technology’ IEEE Conf Vol.2

Ertugrul, M., Kaynak, O (1998) Neural Computation of the Equivalent Control in Sliding

Mode for Robot Trajectory Control Applications IEEE International Conference on Robotics&Automation

Tsai, C., Chung, H.and Yu, F (2004) Neuro-Sliding Mode Control with Its Applications to

Seesaw Systems IEEE Trans on Neural Networks, Vol.15, NO1

Morioka, H., Wada, K., Sabanociv, A., Jezernik, K., (1995) Neural Network Based Chattering

Free Sliding Mode Control SICE’95 Proceeding of the 34th Annual Conf

Huang, S., Huang, K., Chiou, K (2001) Development and Application of a Novel Radial

Basis Function Sliding Mode Controller Mechatronics pp 313- 329

Lin, C., Mon, Y (2003) Hybrid adaptive fuzzy controllers with application to robotic

systems Fuzzy Set and Systems 139 pp 151-165

Chen, C and Lin, C (2002) A Sliding Mode Approach to Robotic Tracking Problem IFAC

Wai, R.J and Hsich, K.Y (2002) Tracking Control Design for Robot Manipulator via Fuzzy

Neural Network Proceedings of the IEEE International Conference on Fuzzy Systems

Volume 2, pp 1422–1427

Regarding force control of the flexible robot, a main body of the research has concentrated

on the position/force hybrid control A force control law was presented based on a linearized motion equation[l] A quasi-static force/position control method was proposed for a two-link planar flexible robot [2] For serial connected flexible-macro and rigid-micro robot arm system, a position/force hybrid control method was proposed [3] An adaptive hybrid force/position controller was resented for automated deburring[4] Two-time scale position/force controllers were proposed using perturbation techniques[5],[6] Furthermore,

a hybrid position-force control method for two cooperated flexible robots was also presented [7] The issue on impedance control of the flexible robot attracts many attentions

as well But very few research results have been reported An impedance control scheme was presented for micro-macro flexible robot manipulators [8] In this method, the controllers of the micro arm (rigid arm) and macro arm (flexible arm) are designed separately, and the micro arm is controlled such that the desired local impedance characteristics of end-effector are achieved An adaptive impedance control method was proposed for n-link flexible robot manipulators in the presence of parametric uncertainties

in the dynamics, and the effectiveness was confirmed by simulation results using a 2-link flexible robot [9]

In this chapter, we deal with the issue of impedance control of flexible robot manipulators For a rigid robot manipulator, the desired impedance characteristics can be achieved by changing the dynamics of the manipulator to the desired impedance dynamics through nonlinear feedback control[13] This method, however, cannot be used to the flexible robot because the number of its control inputs is less than that of the dynamic equations so that the whole dynamics including the flexible links cannot be changed simultaneously as desired by using any possible control methods To overcome this problem, we establish an impedance control system using a trajectory tracking method rather than the abovementioned direct impedance control method We aimed at desired behavior of the end-effector of the flexible robot If the end-effector motion precisely tracks the impedance trajectory generated by the impedance dynamics under an external force, we can say that the desired impedance property of the robot system is achieved Prom this point of view, the control design process can be divided into two steps First, the impedance control objective

is converted into the tracking of the impedance trajectory in the presence of the external force acting at the end-effector Then, impedance control system is designed based on any possible end-effector trajectory control methods to guarantee that end-effector motion converses and remains on the trajectory In this chapter, detail design of impedance control

is discussed Stability of the system is verified using Lyapunov stability theory As an application example, impedance control simulations of a two-link flexible robot are carried out to demonstrate the usage of the control method

2 Kinematics and dynamics

We consider a flexible robot manipulator consisted of n flexible links driven by n rigid joints Let be the joint variables, be the vector of the link flexible

displacements We assume that there is no redundant degree of freedom in the joints, and define vector to describe the end-effector position and orientation in the n- dimension workspace The kinematical relationship among , and is non-linear, and

can be given as

(1)

The relationship among the velocities , , and is a linear function given as

(2)where and are the Jacobian matrices of f with respect to and They are defined as , and By taking derivation of (2) with respect to time, we have

(3)The dynamics of the flexible robot can be derived using Lagrange approach and FEM(Finite Element Method) In the case that there is an external force acting at the end-effector, the approximated model with FEM can be given as

(4)

Impedance Control of Flexible Robot Manipulators 213

(5)where, (4) and (5) are the motion equations with respect to the joints and links of the robot

Coriolis, centrifugal, forces; and denote elastic forces and gravity; presents the input torque produced by the joint actuators

System representation (4) and (5) can be expressed in a compact form as follows:

(6)

, and (i,j = 1,2) respectively is defined as

matrix, and is skew symmetric These properties will be used in stability analysis

of the control system in Section 4

3 Objective of impedance control and problem formulation

3.1 Desirable impedance dynamics

When an external force acts on the end-effector of the flexible robot, the objective of impedance control is to cause the end-effector to respond to the force according to some desirable impedance dynamics The desirable impedance dynamics can be designed as following second-order equation

(7) where, are the commanded end-effector position, velocity and acceleration in

the workspace; are the responses to the external force and commanded position, velocity and acceleration; and are moment of inertia, damping, and stiffness matrices defined by user so that (7) possesses the desired dynamic characteristics These matrices are usually defined as constant and diagonal ones

3.2 The Main idea of impedance control using trajectory tracking approach

For a rigid robot manipulator, the desired impedance described above can be achieved by using impedance control to change dynamics of the robot manipulator [13] This method, however, is difficult to be used to a flexible robot because one cannot change the whole dynamics of the robot including the flexible links at same time using any possible control method To overcome this problem, the impedance control system is implemented using a trajectory tracking method rather than using the direct method-the abovementioned impedance control method Let's consider behavior of the end-effector of the flexible robot

If the end-effector motion correctly tracks the desired impedance trajectories, we can say that the desired end-effector impedance is achieved Prom this point of view, impedance control can be achieved by tracking the desired impedance trajectory generated by the

designed impedance dynamics (7) in the presence of the external force acting at the effector Therefore, the impedance control design can be changed to the trajectory tracking control design Accordingly, impedance control is implemented using end-effector trajectory control algorithms Based on this methodology, the design of the control system is divided into three stages In the first stage, impedance control of the robot is converted into tracking

end-an on-line impedend-ance trajectory generated by the impedend-ance dynamics designed to possess with terms of the desired moment of inertia, damping, and stiffness as (7) On this stage, on-line impedance trajectory generating algorithms is established, and are calculated numerically at every sampling time in response to the external force which may vary The second stage is concerned on the designing of a manifold to prescribe the desirable performance of impedance trajectory tracking and link vibration damping The third stage is to derive a control scheme such that the end-effector motion of the robot precisely tracks the impedance trajectory Figure 1 illustrates the concept of impedance control via trajectory tracking

Figure 1 Impedance trajectory generated by the impedance model and tracked by the flexible robot

3.3 The manifold design

In order to achieve good performance in tracking the desired impedance trajectory while keeping the whole system stable, the control system has to include two basic functions One

is link vibration suppressing function and the other one is end-effector trajectory tracking function We design an ideal manifold

to formulate the desired performance of the system with respect to these two functions In

so doing, let II be the ideal manifold, and divide it into two sub-manifolds HI and n2

according to the functions abovementioned, and design each of them separately Now, we define error vectors as follows:

(8) (9)

We design the first submanifold to prescribe the desirable performance of end-effector trajectory tracking as follows

Impedance Control of Flexible Robot Manipulators 215

(10)where is a constant positive definite matrix

To describe a desired link flexural behavior, the second submanifold is specified as

(11) where is a constant positive definite matrix

By combining and together, we have the complete manifold as follows

(12)

In order to formulate in a compact form, we define a new vector as

(13)Taking (2), (8), and (9) into consideration, we have

(14)Obviously, the ideal manifold is equivalent to

(15)

We define a transformation matrix as

(16)and rewrite (14) as

(17) where

(18)Taking time differentiation of , we have

(19) where

(20)

Remark 2: It should be pointed out that the condition for existence of submanifold (10) is the

Condition for Compensability of Flexible Robots

The concept and theory of compensability of flexible robots are introduced and systematically analyzed by [10], [11], and [12] The sufficient and necessary condition of compensability of a flexible robot is given as

condition of existence of (24) is the condition of existence of such From [10], [11], and

[12] we know the condition of existence of is the compensability condition given by (22)

4 Impedance Control Scheme

In the following, we discuss the derivation of impedance control scheme and the verification

of stability of the system From (17) and (19), we have

(25) (26) Using the transformation defined above, the system representation (6) can be rewritten as

(27) where

(28) and

(29)

Impedance Control of Flexible Robot Manipulators 217

In (27), contains Coriolis, centrifugal and stiffness forces and the nonlinear forces resulting from the coordinate transformation from to w needed to be compensated in the end-effector trajectory tracking control However, some elements of with respect to the dynamics of the flexible links cannot be compensated by the control torque directly

To deal with them, we design an indirect compensating scheme such that stability of the system would be maintained To do so, we partition into the directly compensable term and not directly compensable term as follows

(30)

Note that in (27) the ideal manifold is expressed as the origin of the system Eventually, the purpose of the control design is to derive such a control scheme to insure the motion of the system converging to the origin We design the control scheme with two controllers

as

(31) where is designed for tracking the desired end-effector impedance trajectories, and is for link vibration suppressing In detail, and are given as follows

(32) (33)where and are the feedback gain matrices to be chosen as constant positive definite ones; is an arbitrary non-zero vector, , and

are the elements of two new vectors defined as bellow

(34)

(35)

From the control design expressed above, the following result can be obtained

Theorem For the flexible robot manipulator that the dynamics is described by (6), the

motion of the robot uniformly and asymptotically converges to the ideal manifold (15) under control schemes given by (31) ~ (33)

is also a positive definite matrix Thus, we choose the following positive scalar function as a Lyapunov function candidate

(36)Taking differentiation of with respect to time yields