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J. Cent. South Univ. Technol. (2010) 17: 807815

DOI: 10.1007/s117710100560y

Dynamic surface control-backstepping based impedance control for

5-DOF flexible joint robots

XIONG Gen-liang()1, XIE Zong-wu()1, HUANG Jian-bin()1,

LIU Hong()1, 2, JIANG Zai-nan()1, SUN Kui()1

1. State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China;

2. Institute of Robotics and Mechatronics, German Aerospace Center (DLR), Wessling 82230, Germany

Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: A new impedance controller based on the dynamic surface control-backstepping technique to actualize the anticipant

dynamic relationship between the motion of end-effector and the external torques was presented. Comparing with the traditional

backstepping method that has explosion of terms problem, the new proposed control system is a combination of the dynamicsurface control technique and the backstepping. The dynamic surface control (DSC) technique can resolve the explosion of terms

problem that is caused by differential coefficient calculation in the model, and the problem can bring a complexity that will cause the

backstepping method hardly to be applied to the practical application, especially to the multi-joint robot. Finally, the validity of the

method was proved in the laboratory environment that was set up on the 5-DOF (degree of freedom) flexible joint robot. Tracking

errors of DSC-backstepping impedance control that were 2.0 and 1.5 mm are better than those of backstepping impedance control

which were 3.5 and 2.5 mm in directions X, Y in free space, respectively. And the anticipant Cartesian impedance behavior and

compliant behavior were achieved successfully as depicted theoretically.

Key words: Cartesian impedance control; dynamic surface control; backstepping; PPSeCo; flexible joint robots

1 Introduction

In contrast to traditional industrial robots, people in

robotics domain transferred their interest into the service

robot in the past several years, such as medical robots,

mobile robots and exploration robots. These robots will

work in the laboratory environments or practical

environments, even in the space. The compliant behavior

of the manipulator is predesigned whenever a robot is

supposed to perform some manipulation tasks such as

picking and placing operation in practice. In order to

achieve the compliant behavior by a control method, animpedance control method, a classical issue, which could

provide a unified framework for achieving compliant

behavior when robot contacted with an unknown

environment, was brought up.

Impedance control was theorized by HOGAN [1]

and experimentally applied by KAZEROONI et al [2].

Based on a singular perturbation approach, flexible joint

robot was controlled by impedance control [3], the

feedback of the joint torques was therein considered as

the control input of a fast inner control loop that received

its set point values from an outer impedance controller.

HUANG et al [4] also proposed an impedance controller

for the flexible joint robot based on the singular

perturbation theory by DSP (digital signal processing)/

FPGA (field programmable gate array) hardware

structure. However, the main drawback of the singular

perturbation method is lacking of theoretical justification

for proving the stability due to the limitation of

Tychonovs theorem [5]. Therefore, ALIN and GERD [6]

proposed Cartesian impedance control techniques for the

torque control of light-weight flexible joint robots, using

local stiffness control to enhance the impedance control.

CHRISTIAN et al [7] developed decoupling based

Cartesian impedance control of flexible joint robots anda formal analyzed the stability of the proposed controller.

CHRISTIAN et al [810] investigated the Cartesian

impedance control of the light-weight flexible joint

robots of DLR with torque feedback, gravity

compensation and complete static states feedback, and

proved asymptotical stability based on passivity theory.

OZAWA and KOBAYASHI [11] proposed a new

impedance control concept for elastic joint robots with

programmable passive impedance devices in the

transmission. The concept allows the user to use the

same index for free motion and contacting task, but it

Foundation item: Project(2006AA04Z228) supported by the National High-Tech Research and Development Program of China; Project(PCSIRT) supported

by Program for Changjiang Scholars and Innovative Research Team in University

Received date: 20091219; Accepted date: 20100401

Corresponding author: XIONG Gen-liang, PhD; Tel: +8645186412042; E-mail: [email protected]

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was applied to one-DOF elastic joint robots. GIANNI et

al [12] presented an impedance controller for elastic joint

industrial manipulators. Special attention was paid to all

aspects that qualify an industrial robot, including

decentralized proportional integral derivative positioncontrol, torsional flexibility, and friction at the joints, etc.

CHIEN and HUANG [13] proposed a regressor-free

adaptive impedance controller for an n-link flexible joint

robot, the function approximation technique (FAT) was

employed to transform the time-varying uncertainties

into finite combinations of orthogonal basis functions.

HUANG et al [14] proposed an adaptive impedance

controller for flexible joint robot based on the friction

model. The friction model includes viscous friction,

payload and motor position based friction. The closed

loop stability was investigated. LIU et al [15]investigated the Cartesian impedance control and

nonlinear compensation for a harmonic drive robot based

on joint torque sensors, and the imperfect Cubic model

for harmonic drive friction was detected according to

friction identification experiments. A Cartesian

impedance control law was introduced by virtual

decomposition [16] to realize the compliance control

which incorporated with three means to make the

flexible manipulator come into compliant contact with

the objects.

Compared with the above singular perturbations,passivity theories and decoupling methods, the

backstepping technique was barely applied to the flexible

joint robot control except the bcakstepping method on

Cartesian impedance control of flexible joint

manipulators [17] even though it represented a complete

solution of impedance control problem for flexible joint

robot model including the tracking case and inertia

shaping. On the contrary, the backstepping technique was

widely used to design output or state feedback controller

for flexible joint robots in the tracking case [1826].

Because of these issues, a new impedance controllerbased on dynamic surface control-backstepping to

actualize the anticipated dynamical relationship between

the motion of end-effector and the external torques was

presented. The dynamic surface control technique was

able to resolve the explosion of terms problem in

backstepping controller. The Lyapunov function

guaranteed global stability of the proposed controller.

Moreover, a DSP/FPGA-FPGA hardware structure was

established to support the proposed impedance controller.

2 Dynamics of flexible joint robots

Generally, the reduced dynamic model of an n-link

flexible joint robot refers to robot dynamics and actuator

dynamics, which could be described as the following

form proposed by SPONG [27]:

ext( ) ( , ) ( )+ + = +M q q C q q q g q&& & & (1)m

+ =B

&& (2)( )= K q (3)

where nq R and n R denote the link and motor

side positions, respectively; ( ) ,n n

M q R ( , ) C q q&

Rn and ( ) ng q R denote the inertia matrix, centripetal

and Coriolis vector, and gravity vector, respectively;n

R denotes the joint torque; extn

R denotes the

external torque that shown by the manipulators

environment; mn

R denotes the motor torque served

as the control input; the constant positive definite

diagonal matrices n nK R and n nB R represent

the joint stiffness and the actuator inertia, respectively.

Moreover, two well-known properties of the robot model

utilized in the following sections are described as.

Property 1: Link inertia matrix M(q) is symmetric

and positive definite:

T T( ) ( ) , ( )=M q M q y M q y0, , 0 n q y R (4)

Property 2: Matrix ( ) 2 ( , )M q C q q& & is skew

symmetric, if ( , )C q q& is chosen suitably using the

Christoffel symbol, then:

T ( ( ) 2 ( , )) 0, , , n = y M q C q q y y q q& & & R (5)

These properties were proved by SCIAVICCO and

SICILIAND [28].

3 Backstepping based Cartesian impedancecontrol

The most important contribution of this work was to

develop a design of impedance controller. Similar to the

state transformation, Eqs.(1)(3) could be rewritten in

the Cartesian coordinates x=f(q) with the Jacobian

J(q)=f(q)/q, Cartesian velocities ( ) ,=x J q q& &

accelerations ( ) ( )= +x J q q J q q&&& && & and torque variables

and & as:

T Text( ) ( , ) ( ) ( ) ( ) ( )

+ + = +x x x x x J q g q J q && & & (6)

1m

+ = BK Bq&&&& (7)

where matrices ( )x and ( , ) x x& were given in

Ref.[17].

Considering that only the non-redundant and

non-singular case was treated in this work, thus it isassumed that the manipulators Jacobian J(q) has full

row rank in the considered region of the workspace.

External torques ext should be related to the vector

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1 2

T T1 2 1 1 2 3 ext

3 4

14 m 3

( ) ( , ) ( ) ( ) ( ) ( )

( )

=

= + +

=

=

x x

x x x x x J q g q J q x

x x

x KB Bq x

&

& &

&

& &&

(15)

First, define the multiple dynamic surfaces S1, S2, S3

and S4 as follows:

=

=

=

=

4d44

3d33

2d22

d111

xxS

xxS

xxS

xxS

(16)

where x1d denotes the desired trajectory; and x2d, x3d and

x4d denote the stabilizing functions for the subsystem

consisting of dynamic surfaces S2, S3 and S4, respectively.

According to Eq.(15), the derivatives of Eq.(16) could be

expressed as follows:

1 1 1d 2 2d

1 T2 2 2d 1 1 1 2

T3 ext 2d

3 3 3d 4 3d

14 4 4d m 3 4d

= ( )[ ( , ) ( ) ( )

( ) ( )]

= ( )

= =

= +

+

= =

=

S x x x x

S x x x x x x J q g q

J q x x

S x x x x

S x x KB Bq x x

& & & &

& & & &

&

& & & &

& & & && &

(17)

To stabilize dynamic system (17), i.e. S1, S2, S3 and

S4 0, the following control algorithms could be

obtained:

2 1 1 1d

T3 1 1 2 ext

1 2 2 2d

4 3 3 3d

1m 3 4 4 4d

= ( ){ ( , ) ( ) [ ( ) ]

( )( )}

( )

q

= +

+ +

+

= + = + + +

x S x

x J q x x x J q g

x S x

x S x

Bq x BK S x

&

&

&

&

&& &

(18)

where 1, 2, 3 and 4 are positive design parameters.

However, distinct from the approach of backstepping

that was described in the previous section, x2d was

obtained by 2x through a first-order filter with time

constant t2,

2 2d 2d 2

2d 2(0) (0)

t + =

=

x x x

x x

&

(19)

Similar to x2d, x3d and x4d could be obtained. So,

actual control vector m could be described as

1 4 4dm 3 4 4

4t = + + +

x xBq x BK S&& (20)

Substituting expression x3= into Eq.(20) leads to

1 4 4dm 4 4

4t

= + + +

x xBq BK S&& (21)

Comparing controller (14) with controller (21), the

proposed impedance controller (21) for flexible joint

robots could resolve the explosion of terms problem ofbackstepping impedance controller (14) by evaluating

the repeated derivatives of the virtual controllers. That is,

the proposed DSC-backstepping control system for

flexible joint robots used the outputs of first-order filters

with 32 , xx and 4x as the inputs, which were

required in place of 2 3,x x& & and 4x

& in the

backstepping design procedure. Thus, the proposed

controller design procedure based on DSC-backstepping

technique could be very simple.

4.2 Stability analysis

In this section, the stability analysis of the proposed

control approach on the 5-DOF flexible joint robots was

given. First, choose the Lyapunov function as follows:4 3

T T

1 1

, , , ) ( )1

(2

i i i i i i ii i

V= =

+= + e x t S S e eS %

T T T T1 1 2 1 2 3 3 4 4[ ( ) ]

1

2+ + +x x x x x x x x x% % % % % % % % (22)

where Si denotes the surface error; ei denotes the

boundary layer error; ( )i i i i= x x x x% % denotes the

observation error; and ix denotes the the estimation

value ofxi.

Lyapunov function (22) was differentiated with

respect to time, and Si, ei, ,~

ix triangle inequality and

property P2 were utilized. The derivative could be

obtained as follows:

T1 2 1 1 1 2

T 1 T2 1 3 2 2 2 1 2

T T3 4 3 3 3 4 4 4 4 2 4

3T T T T

1 1 2 1 2 3 311

, , , ) ( )

[ ( ) ( ) ( ) ]

( ) ( )

( ) ( )

(

i i i

ii i

ii

k

k

V

+=

+ + +

+ + +

+ + + + +

+ + + + +

=

e x t S e S x

x J q S e S x

S e S x S x

ee x x x x x x x

t

S S

S

S S

% %

%

% %

& & &% % % % % %

&

T T4 4 2 1 2

1 ( )2

+x x x x x& &% % % % 4 32 2

1 1

i i i i

i i

= =

S e

322 2 2 2

1 1 2 2 3 3 4 41

ii

l l l l c=

+ =x x x x% % % % V + (23)

where l1, l2, l3, l4, k1 and k2 represent design parameters;

, i and ci represent positive constants; and , V

represent design function.

The derivative implied that , , , )( i i iV e x tS %& 0

when V=p and /p. Therefore, Vp was an invariant

value, in other words, ifV(0)p, then V(t)p fort0.And the surface errorSi could be arbitrarily small to

ensure stability of the controller.

The proposed DSC-backstepping Cartesian

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J. Cent. South Univ. Technol. (2010) 17: 807815 811

impedance control scheme for flexible joint manipulators

could be summarized by the DSC system. Accordingly,

our control system could be designed more easily and

simply than the Cartesian impedance control system via

the backstepping technique.

5 Experiment analysis

5.1 Structure and physical parameters of 5-DOF

flexible joint robots

In order to demonstrate the effectiveness of the

proposed DSC-backstepping Cartesian impedance

controller, firstly, the structure of 5-DOF flexible joint

manipulators is shown in Fig.1 and the kinematic

parameters and frame assignments are shown in Fig.2.

The flexible joints of the robot are identical in the macro

structure, which are driven by brushless DC andharmonic drive gear combined planet gear (gear ratio

1:12 000). A potentiometer and three hall sensors were

equipped to measure the absolute angular position of the

joint and the relative angular position of the motor. Eight

strain gauges were fixed crossly to the output shaft of the

harmonic drive gear to construct two full-bridges that

measured the joint torques. The measurements of the

angular position and the torque were fed to the Cartesian

impedance controller. Moreover, the five joints were

connected in series from the base toward the tip, joint i,

connected link i with link i1. Five frames, {Li}, i=1,

2, , 5, were defined in a way that frame {Li} was fixed

to linki with itsZ-axis coincident with the ith joint axis,

then the rotations by X-, Y-axes were constrained to the

coordinate origin.

The robot parameters of kinematics and dynamics

Fig.1 Five-DOF flexible joint robot

Fig.2 Kinematic parameters and frame assignments for 5-DOF

flexible joints of robot

were very precisely computed using 3D mechanical

CAD programs. Finally, K could be calculated by Eq.(3)

in the joint impedance control when each joint contacted

a rigid environment in which q was constant. The

manipulator parameters are listed in Table 1. Where ai, i,

di and i represent Denavit-Hartenberg (D-H) parameters;

mi represents the joint quality; Bi represents the joint

damp; andKi represents the joint stiffness.

5.2 DSP/FPGA-FPGA based hardware system

The hardware system based on DSP/FPGA-FPGA

as shown in Fig.3 was given to realize the proposed

controller. In order to minimize cabling and weight of the

5-DOF flexible joint manipulator, a fully mechatronic

design methodology was introduced to develop the

hardware system. All the analog signals were converted

into proper digital signals and serially transmitted into

joint FPGA board and further to PCI (peripheral

component interconnect)-based central processor. The

hardware system consisted of PCI-based DSP/FPGA

board configured as a Cartesian level and joint FPGA

board for five-joint control configured as a joint level.

The control algorithm is illustrated in Fig.4. Joints

FPGA board (Slave) took charge of the joint level

controller, and a PCI-based DSP/FPGA board (Master)executed as Cartesian 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

Table 1 Manipulator parameters

Frame ai/mm i/() di/mm i/() mi/kg Bi/(kgm2) Ki/(Nmrad

1)

{L1} 0 90 110.76 0 0.800 0.85 82.175

{L2} 530 0 0 0 1.090 0.85 68.236

{L3} 470 0 0 0 1.069 0.85 68.236

{L4} 0 90 135.66 0 0.700 0.85 68.236

{L5} 0 90 75.26 0 0.700 0.85 68.236

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Fig.3 Hardware system based on DSP/FPGA-FPGA

Fig.4 Block diagram of controller

Cartesian level needed the feedback information of

positions, velocities and torques of the joints and

calculated the required torques swiftly. 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, in which the cycle

time is less than 200 s and communication rate is up to

25 Mb/s. The communication and other control programs

for FPGA were written in VHDL and run in FPGA.

5.3 Experiments

To illustrate the validity of the proposed methods,

the following three experiments on manipulator were

carried out.

The first experiment was to use the 5-DOF flexible

joint robot to track sine curve inX- and Y-directions, and

remain static in Z-direction for the case of free motion

(Fext=0) by backstepping impedance control and

DSC-backstepping impedance control in Cartesian space.

Fig.5 shows Cartesian coordinate positions and tracking

errors in different directions under the methods of

backstepping impedance control (BIC) and DSC-

backstepping impedance control (DSC-BIC). From

Figs.5(b) and (d), it can be seen that the tracking errorswere 3.5 and 2.5 inX- and Y-directions by backstepping

impedance control, while the tracking errors were 2.0

and 1.5 mm inX- and Y-directions by DSC-backstepping

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Fig.5 Cartesian coordinate positions and tracking errors in different directions: (a) Positions in X-direction; (b) Errors inX-direction;

(c) Positions in Y-direction; (d) Errors in Y-direction

impedance control, respectively. The control parameters

of two controllers are listed in Tables 2 and 3. According

to Fig.5, the accuracy of the tracking errors was

improved evidently by using DSC-backstepping

impedance control.

In addition, the trajectory tracking accuracy of

impedance control was lower than that of position

control because the impedance control sacrificed some

Table 2 Parameters of backstepping impedance control

Joint No. KP,i KD,i

1 33.955 3.480

2 27.354 2.804

3 27.354 2.257

4 27.354 2.325

5 27.354 2.325

Table 3 Parameters of DSC-backstepping impedance control

Filter No. Positive design parameter (i) ti/s

1 14.0

2 0.8 0.01

3 1.5 0.01

4 16.0 0.01

tracking accuracy. Moreover, the more flexible the joint,

the lower the accuracy of track, even not meeting the

requirements. The tracking accuracy would be improved

when the joint with little flexibility was regarded as a

rigid joint, but the impedance performance would be

down. Therefore, the above experiment compromised the

impedance performance and the tracking accuracy. And

that the phenomenon of the error was asymmetric about

X-axis in Fig.5 was caused by the imperfect gravity and

friction compensation.

The second experiment was Cartesian impedance

experiment made on a 5-DOF flexible joint manipulator.

Firstly, the robot was placed on a virtual equilibrium

position CD=[0, 0, 0]. The anticipant stiffness (Ki) and

damping (Di) in Table 4 were used. Then, the robot was

pulled in different directions shown in Fig.6. Finally, the

robot overcame the gravity and returned to the CD as

soon as the force was released. Fig.6 shows the

corresponding Cartesian forces along with the Cartesian

position. It could be concluded that the theoretic

Cartesian impedance behavior was achievedsuccessfully.

In Cartesian impedance control, the desired

Cartesian impedance stiffness could be set a small value

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Table 4 Impedance parameters in workspace

Coordinate Ki/(Nm1) Di/(Nsm

1)

X-axis 2 000 482.6

Y-axis 2 000 482.6

Z-axis 2 000 78.3

Fig.6 Variation of Cartesian coordinates in DSC-backstepping

Cartesian impedance control: (a) Position; (b) Force

(smaller than 20 N/m). Under this condition, the robot

could be pushed freely using a small force, and when the

force was removed, the robot could stay at the last

position stably, which was called zero-force control.

The third experiment, which was based on DSC-

backstepping Cartesian impedance controller, was to test

whether the end-effector could process a compliant

behavior under a constrained space. The end-effector of

the manipulator anticipant position was to make a circle

motion with a cylindrical obstacle during the trajectory

in Fig.7. During the motion, robot maintained a larger

contact force in the tangent direction of the obstacle,

while a smaller retention in the normal direction. Then,

by the geometric calculation, the forces of the tangent

direction and normal direction were converted to the

contact force of directions X and Y on X-Y plane.According to Fig.7, it could be concluded that the

DSC-backstepping based impedance controller realized

the compliant behavior of flexible joint robot when it

contacted with stiffness environment. The control

parameters are listed in Table 5, whereFi represents the

external force.

Fig.7 Circle tracking in X-Y plane constrained space by

DSC-backstepping based impedance controller

Table 5 Impedance parameters in constrained space

Coordinate Ki/(Nm1) Di/(Nsm

1) Fi/N

X-axis 1 500 20 5

Y-axis 1 500 20 5

Z-axis 1 500 20 5

Therefore, if expectation torques of the tangent

direction and normal direction of end-effector were kept

within an acceptable range, not only the robot itself and

the object could be protected, but also the smooth surface

obstacles could be avoided effectively.

6 Conclusions

(1) The DSP/FPGA-FPGA based special hardware

system with PPSeCo is established. The hardware

structure is designed not only to achieve thecommunication between joint level and Cartesian level,

but also to calculate acceleration q&& and the jerk q&&&

which cannot be measured directly, and actualize the

control algorithm.

(2) Two Cartesian impedance controllers based on

backstepping and DSC-backstepping technique are

proposed, the design of the latter is easier and simpler

than that of the former because of the introduction of

dynamic surface control technique which resolves the

explosion of terms problem of backstepping-based

impedance controller. And aformal stability of the DSC-backstepping impedance controller is proved based on

Lyapunov function.

(3) Experimental results justify that the proposed

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DSC-backstepping impedance controller gives

satisfactory tracking performance. And the impedance

performance in free space and the compliant behavior in

constrained space are achieved on the 5-DOF flexible

joint robot.

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