Fuzzy-Sliding Mode Control for Flexible-joint Manipulator based on Friction Compensation

Ni Fenglei and Liu Yechao Dang Jin State Key Laboratory of Robotics and system Guang Xi Yuchai Heavy Industry

Harbin Institute of Technology company Limited Harbin, 150001, China Guang Xi Province, China

Hit_lfn2002@yahoo.com.cn dangjinhit@yahoo.com.cn Abstract - For overcoming unknown external disturbance, the nonlinearity and uncertainty of the flexible joint manipulator, a novel robust controller is proposed based on fuzzy-sliding mode control and friction compensation. The fiction compensation based on model can effectively predict the friction amplitude and compensate the dynamic error. The external uncertain disturbance, nonlinear part of friction, parameters variation, modeling error and so on can be overcome by the fuzzy-sliding mode control. Experimental results on the platform of HIT four degrees of freedom flexible joint manipulator show that the proposed controller not only can realize good position tracking performance, independency to parameters fluctuation, but also have a strong anti-interference ability. Index Terms - Flexible joint; Friction compensation; Fuzzy-sliding mode control; Manipulator.

I. INTRODUCTION

In recent years, there has been growing interest in the emerging field of flexible joint manipulator due to its unique performance features such as compact size, low energy consumption, high load-to-weight ratio. Usually the main factors affecting the flexible joint manipulator dynamic accuracy and stability include the following aspects: flexibility issues, complicated nonlinear friction, variable system parameters, uncertainty disturbances and modeling errors. The requirements of accuracy and stability for the joint control cannot be realized by the traditional PID control rules. So much research on these questions have been done, and many effective methods have been proposed, such as adding sensors, variable structure control, singular perturbation method, feedback linearization, and adaptive control. Adding sensors is extracting more system information for control, this method is subject to the development of sensor technology [1]. Sliding mode control has recently attracted many researchers’ interest, but controlling a plant with sliding mode control method often requires high control gains, and easily results in a chattering phenomenon. The singular perturbation method possesses a fast joint torque control loop corresponding to the fast part of the manipulator dynamics, and a slower outer control loop corresponding to the rigid body dynamics of the robot. These control strategies use the assumption of a weak elasticity of the joints [2-3], but under conditions of considerable elasticity, noisy torque and torque derivative signals, the bandwidth of the resulting torque controller limits the overall bandwidth of the system. Feedback linearization method was proposed by Spong using a simplified robot model, and has been

implemented by De Luca [4]. However, the dynamics of the flexible joint robot is not feedback linearizable in practice. Adaptive control is a research focus in recent years [5], but it is difficult to suppress the uncertainty disturbance.

Because of these issues, the fuzzy sliding mode control controller is proposed by combining a friction compensator. The feasibility and effectiveness of the controller is experimentally demonstrated on a 4 degree-of-freedom (DOF) flexible joint robot. This paper is organized as follows: Section 2 outlines an overall description about the HIT 4-DOF flexible joint manipulator. Section 3 describes the dynamics of flexible joint robots and controller design. Section 4 gives experimental results. Finally, conclusions and future work are addressed in section 5.

II. HIT 4-DOF FLEXIBLE JOINT MANIPULATOR

To minimize the differences in joint structure, and enhance interoperability among joints, HIT 4-DOF manipulator joint adopted modular concept to design as illustrated in Fig. 1. Each joint consists of a highly integrated circuit board, a multi-sensory, a motor and a harmonic drive gear. In this section, mechanical design, sensory systems and hardware architecture of the joint are outlined.

End

effector

Joint 4

Joint 3

Joint 2

Joint 1

Fig. 1 HIT 4-DOF flexible joint robot and its coordination.

A. Mechanical design and sensory systems To reduce the weight of joint and increase the joint output torque, the structure of the motor adding harmonic drive gear

1868978-1-4673-1278-3/12/$31.00 ©2012 IEEE

Proceedings of 2012 IEEEInternational Conference on Mechatronics and Automation

August 5 - 8, Chengdu, China

is adopted to build the joint. The frameless motor and special short and light weight harmonic drive gear are used. Most mechanical parts are made of aluminum. The 3D model of one joint mechanical structure is shown in Fig.2. Each electronic module is of the same type and consists of a motor drive board, the FPGA control board, the power board and sensor conditioning board. All these circuit boards, motor and harmonic drive gear are placed inside the housing to save the space. All housings are made of aluminum and are designed to transfer thermal energy from motor and power electronics to themselves.

Fig. 2 One joint mechanical structure.

To overcome joint flexibility, increase joint sensory

capability and reliability, earch joint is equipped with multi-sensor as shown in Table I. To protect the mechanical structure of robot arm and measure the actually exerted torque to each joint, the torque sensor is used and placed between the output of harmonic drive gear and the link. The deformation of radial beams is measured by strain gauges. By using eight strain gauges and temperature sensor, transverse forces and temperature effects can be compensated. The structure of torque sensor and I-DEAS analysis is shown in Fig 3.a. In the presence of the elasticity and hysteresis of the transmission system, absolute joint angle sensor is needed. To reduce the joint weight and increase sensor’s system integration, conventional contact potentiometer is used as absolute joint angle, which is shown in Fig.3.b. Torque sensor and potentiometer are integrated together, which is shown in Fig.3.c. Motor position information is fed by digital Hall sensor and magnetic encoder. Moreover, limit position protection can be realized by only one digital Hall sensor assembled in the potentiometer. At the same time, to avoid the system overheating, 1-wire digital thermometer is assembled in every board.

a. Torque sensor and I-DEAS analysis

b. Contact potentiometer c. Torque sensor and potentiometer

Fig. 3 Main sensor in the joint

TABLE I SENSOR EQUIPMENT OF ONE JOINT

Sensory type Quantity PrincipleTorque Sensor 1 Strain gauge Joint Angle Sensor 1 Potentiometer Limit position switch 1 Hall Effect Motor Angle Sensor 1 Magnetic Encoder Digital Hall Sensor 3 Hall Effect Temperature Sensor 4 Digital Thermometer

B. Joint control system To realize the proposed controller, the FPGA is used as joint controller, and the DSP chip is used as the top controller. The structure of the hardware control system is shown in Fig. 4. In order to minimize cabling and weight of the 4-DOF flexible joint manipulator, a fully mechatronic design methodology was introduced into developing the hardware system. All the analog signals were converted into proper digital signals and serially transmitted into joint FPGA board and further to PCI-based central processor. The hardware system consisted of PCI-based DSP/FPGA board configured as a higher level, joint FPGA board for four joints control configured as a joint level. Joint’s FPGA board (Slave) took charge of the joint level controller, and a PCI-based DSP/FPGA board (Master) executed as the higher level. In the joint control level, the FPGA technology was chosen to achieve a more flexible implementation of the joint controller with a high control rate and a small sized joint electronics.

To implement real time control of the robot, the higher level needed the feedback information of positions, velocities and other information of the joints and calculated immediately. At the same time, the joint level should update the input data in time especially for the transient state. Therefore, a high speed data bus of point-to-point serial communication (PPSeCo) was designed for this requirement, the cycle time less than 200us and communication rate up to 25Mbps. The communication and other control programs for FPGA were written in VHDL and run in FPGA.

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{{

{

Fig. 4 Hardware system based on DSP/FPGA-FPGA

III. DYNAMICS AND CONTROLLER DESIGN

A. Dynamics of flexible joint robots The typical flexible joint is powered by electro-mechanical drives, which consists of permanent magnet synchronous motor and harmonic drive gear. The main sources of flexibility concentrate in the joint, link can be treated as a rigid component. Therefore, by the assumption of Spong [6], the dynamic equation of flexible joint robot model can be considered as follows:

���

���

�

−=

++=

+=++

)(

)(),()(

qK

B

qgqqqCqqM

j

Fjm

extj

θτττθτ

ττ��

����

(1)

In the equations, )(qM represents the inertia matrix, ),( qqC � is the centrifugal/Coriolis term, and )(qg is the gravity

term. The vector of the joint torques is given by )( qKj −= θτ ,

whereθ , q indicate the vector of the motor angle divided by the gear ratio and the link side joint angle respectively, and K and B are diagonal matrices which contain the joint stiffness and the motor inertias multiplied by the gear ratio squared. In addition, extτ is the external torques and mτ is the generalized

motor torques vector which is regarded as input variables. B. Friction compensation There are many approaches about friction identification in harmonic drive. To describe the relation between the friction and the joint velocity, high-gain closed-loop experiment have been carried out where a PD controller is used for joint position tracking to achieve good velocity-tracking. The friction model of velocity-dependent and position dependent based on PWM are described in detail in the following sections. B.1 Velocity-Dependent Friction

To identify the friction-velocity relationship, each joint was commanded to move at a constant velocity and the mean torque was taken to be the friction for this velocity. In order to investigate the friction model accurately, data for joint velocities between 0.5 and 30deg/s were collected, which were recorded in different intervals with 1deg/s increment between 1 and 6deg/s, 2deg/s increments between 6 and 20deg/s, 5deg/s increments between 20 and 30deg/s respectively. Three

trials were performed for each velocity value in both the positive and negative directions, for a total of 96 measurements per joint. After collecting data for all four joints, we fit three different friction models using the MATLAB toolbox, that is, 1) the kinetic plus viscous friction model, 2) the cubic polynomial model, and 3)the Stribeck curve model were used to model friction. The results of this analysis are shown in Table 2 and comparisons were made for different models.

During the comparisons from different models, we have found that the Stribeck model provided a reasonable and intrinsic description for harmonic driving friction. It is worth to notice that the residual variance for the cubic model presented by Tuttle [7] is similar than the Stribeck model for the robot’s joints. But we have found that the differences are remarkable between models fit by whole data and by partial data when using the cubic model. For example, we have acquired 16 experimental data to describe the relationship between friction torque and joint velocity, and find that the cubic model parameters fit by the first 14 data are obviously distinct from those fit by the whole 16 data. The cubic model is invalid beyond the velocity zone where the identification experiments are executed. It seems that this model just makes the fit curve “looks” good, but not exposes the intrinsic rule of the harmonic friction. This inconsistent phenomena is not shown so remarkable in Stribeck model.

Fig. 5 Curves for different friction models

As is shown in Fig.5, curve 1 is the cubic curve approximation fit by the first 14 data and curve 2 is that fit by the whole 16 data; curve 3 is the Stribeck curve approximation fit by the first 14 data and curve 4 is that fit by the whole 16 data. It is very clear that the Stribeck model is better than the cubic model in model the harmonic friction. So, in our experiments we adopt the Stribeck model as friction model of harmonic drive. For convenience, the expression for this model is given by:

dvcvbavTvf +⋅+⋅−⋅= )exp()( (2) where a, b, c, d are parameters to be identified for the model and the best-fit approximation of experimental data by a Stribeck curve for joint2 is shown in Fig.6.

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Fig. 6 Velocity-dependent friction for joint 2

B.2 Position-Dependent Friction

As is referred in [7,8], friction in the HDT can be separated into two parts: velocity-dependent friction and position-dependent friction. Position-dependent friction is caused because of kinematic error in the transmission. The error is caused by a number of factors such as tooth-placement errors on both the circular spline and flexspline, out-of-roundness in the three transmission components, and misalignment during assembly. According to [9], we know that compensating for Coulomb friction can eliminate almost all the effects of kinematic error at low velocity. During experiments, we have found that the joint torque value measured by the joint torque sensor will vary periodically and only depend on the joint position even no gravity. This value was recorded and compensated by lookup table. So it is enough for us to develop impedance control by neglecting the position-dependent friction.

C. Fuzzy sliding mode controller

The sliding mode control has recently attracted the interest of many researchers. This control method can be applied in the presence of model uncertainties, variations and external disturbances ensuring the robustness and stability of the system [10–13]. However, high control gains and chattering is the main problem of sliding mode control. In order to utilize the robustness and stability of the sliding mode control, and to suppress the chattering[14-16], this paper proposed a fuzzy sliding mode controller scheme to realize the flexible-joint control based on friction compensation. C.1 Sliding surface

Because measuring all state variables of the system during practical implementation is difficult, one strategy is to select two main state variables, namely, position tracking error and its differentiation, for defining a sliding surface on the phase plane

0,)( >+= λλeetS � (3) The sliding surface variable )(tS and its differentiation

)(tS� would be employed as the input variables for constructing a fuzzy logic control system to approximate the specified equivalent control law. Following the equivalent control law, the closed-loop control system has a dynamic behavior of asymptotical stability

0)( =+ StS λ� (4) Since λ is a strictly positive value, the sliding surface

variable )(tS and its differentiation )(tS� will gradually converge to zero. According to the defined sliding surface variable )(tS and its differentiation )(tS� in Eqs.(3) and (4), the system’s error states e and e� will also asymptotically converge to zero. C.2 Fuzzy logic control

In this paper define S , S� as fuzzy input variable, and FSU (output PWM for motor control) as the fuzzy output variable. Each fuzzy input variable used seven equal-span triangular membership functions,

},,,,,,{ PBPMPSZONSNMNBS =

},,,,,,{ PBPMPSZONSNMNBS =� we used seven equal-span triangular membership functions for the fuzzy output variable, },,,,,,{ PBPMPSZONSNMNBU FS = , corresponding to 49 fuzzy rules, listed in Table II.

TABLE II FUZZY CONTROL RULES

UFS

S NB NM NS ZO PS PM PB

S�

NB PB PB PB PM PM PM PSNM PB PM PM PM PS PS PSNS PM PM PS PS ZO NS NSZO PM PS PS ZO NS NS NMPS PS PS ZO NS NS NM NMPM NS NS NS NM NM NM NBPB NS NM NM NM NB NB NB

Finally, we employed a center of area method to

defuzzify the output variables in order to gain a control law FSU for this system. The center of area method can be

described as follows:

�� ⋅

= mi

mii

FSw

cwU

1

1 (5)

Where m is the number of rules and ][ 21 mcccC �= is the fuzzy consequent parameter vector; ic is a fuzzy consequent parameter; iw is the weight of the corresponding fuzzy rule that has been activated.

�=m

im wwwwW121 /][ � is the weight of each singleton

fuzzy subset for establishing a control law FSU . Block diagram of the controller is depicted in Fig. 7.

dq�

dq

e�

asbs

+

q

q�

λe eλS

SΔ FSu

fcu

Se

�e=

+ �

Fig. 7 Block diagram of the controller

IV. EXPERIMENTS

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To illustrate the validity of the proposed controller, the following three experiments on manipulator were carried out, and conducted on the 2nd joint which is affected seriously by the uncertain factors. The first experiment is used to verify the position tracking performance of the proposed controller. In the experiment proceeding, joint tracked the desired trajectory with a speed of 1deg/s. The desired trajectory is shown as Fig. 8(a). As the Fig. 8(b) showed, with the traditional PD control to track the desired trajectory, the tracking error is ±0.055°, while the tracking error of the controller combined friction compensation and fuzzy sliding controller can achieve ± 0.014°. From Fig. 8(c), the velocity curve of the new controller was submerged in the velocity curve of PD controller, namely, with the proposed controller joint can track the desired trajectory more smoothly. Analyzing these results, the non-linear part of friction, speed fluctuations have been effectively suppressed by fuzzy-sliding mode controller, and make the joint track the desired trajectory more accurately and smoothly.

0 1 2 3 4 5 6 7

x 104

-30

-20

-10

0

10

20

30

time(ms)

traj

ecto

ry(d

eg)

desired trajectory

FC&FS

PD

(a)trajectory curve

0 1 2 3 4 5 6 7

x 104

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

time(ms)

trac

king

err

or(d

eg)

FC&FS

PD

(b) position error curve

0 1 2 3 4 5 6 7

x 104

-1.00

-0.75

-0.50

-0.25

0

0.25

0.50

0.75

1.00

time(ms)

velo

city

(deg

/s)

FC&FS

PD

(c) velocity curve

Fig. 8 velocity and position error curves tracking the desired trajectory The following experiment is used to validate the

controller suppression performance and robustness to uncertainty disturbance. In the process, joint moved along a

desired smooth trajectory, and imposed two random disturbances during its moving. The first random disturbance torque is 3Nm, the duration is 240ms; the second random disturbance torque is 3Nm, the duration is 40ms, as shown in Fig. 9(a).

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

x 104

0

0.5

1

1.5

2

2.5

3

3.5

time(ms)

dist

urba

nce(

Nm

) 3Nm, 240ms

3Nm, 40ms

(a) random disturbances

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

-3000

-2000

-1000

0

1000

2000

3000

time(ms)

join

t an

gle(

0.01

deg)

9500 9600 9700 9800

-1720

-1700

-1680

-1660

-1640

2.64 2.65 2.66 2.67

x 104

1680

1700

1720

1740

PD

FC&FSdesired trajectory

(b) tracking curve

0.5 1 1.5 2 2.5 3 3.5

x 104

-0.5

0

0.5

1

1.5

2

2.5

3

time(ms)

velo

city

(deg

/s)

2.625 2.65 2.675 2.7

x 104

1.4

1.6

1.8

2

2.2

2.4

9005 9405 9805 10205

1.5

2

2.5

FC&FS

PD

(c) velocity curve Fig. 9 position and velocity curves when subjecting to random disturbances

Fig. 9(b) and Fig. 9(c) demonstrates that joint position and velocity can track the desired trajectory with high precision if there is no disturbance. When a random uncertainty disturbance exits, the external disturbance can reflect to the fuzzy sliding mode control section of the proposed controller, fuzzy sliding mode control can control motor speed more effectively, thus the proposed controller has a faster response time to track the desired trajectory and the vibration amplitude of the joint velocity can be reduced significantly. The second and third experimental results has embodied the characteristics of the fuzzy sliding mode control, namely, the sliding motion is independent with system parameters fluctuations, external disturbances, etc. The controller combined friction compensation and the fuzzy sliding mode inherits the robustness of fuzzy sliding mode control.

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V. CONCLUSIONS

This paper analyzed the main issues to be considered in the design of flexible joint robot controller. For the exiting of nonlinear friction and uncertainty disturbance in the control system, this paper proposed a novel robust controller which combines friction compensation and fuzzy-sliding mode controller. The design idea of the controller is clear, avoids the complex calculation of nonlinear compensation, and has independency and robustness for the system parameters variation, external random disturbances, modeling error, etc. In the relevant experiments on the platform of HIT 4-DOF flexible joint manipulator has been carried out. Experimental results show that the controller has good position tracking performance, is independency to parameters fluctuation and disturbance, and have a strong anti-interference ability.

ACKNOWLEDGMENT

This work is supported by the National High Technology Research and Development Program of China.

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