Advances in Robot Manipulators Part 6 ppt

However, the following simulations are going to show that even in the presence of tracking errors, the contouring performance will not be degraded by using the proposed contouring contro

Therefore, the modified system dynamics in VCS becomes

t t t n

  2 Λ  (16.a)

n n 3 2 1

    Λ  (16.b) For (16), contouring controller design will be addressed in the next section

4 Contouring Controller Design

In this section, design of contouring controller is considered separately for tangential and

modified normal dynamics For demonstration purpose, a general proportional-derivative

(PD) controller is applied in tangential error equation It is well known that the PD

controller is capable of achieving stabilization and improving transient response, but is not

adequate for error elimination Consequently, tangential tracking errors exist unavoidably

Under this circumstance, we are going to show that precise contouring performance can still

be achieved by applying the CI approach Building on the developed contouring control

framework, the tangential and normal control objects can be respectively interpreted as

stabilization and regulation problems

4.1 Design of tangential control effort

Considering the tangential dynamics of (16.a), a PD controller with the form

t n t Pt t Vt

      2 (17)

is applied Substituting (17) into (16.a) results in

t t Pt t Vt

tK  K  Λ

  (18) where KVt and KPt are positive real The selection of control gains should guarantee the

criterion t R Eq (18) indicates that the tangential tracking error cannot be eliminated

very well due to the existence of Λ However, it will be shown that the existence of t t

causes no harm to contouring performance with the aid of CI

4.2 Design of normal control effort

In the following, an integral type sliding controller for the modified normal dynamics is

developed by using backstepping approach Firstly, let IndInd 1 and define an internal

state w Then the system (16.b) can be represented as

1 Ind

w  (19.a)

2 Ind 1 Ind 

  (19.b)

n n 3 2 1 2

 (20) where k10 Then, consider as a Lyapunov function

2/w

1 (21) The derivative of (21) is

0wkw

1 1

z

w   (23.a) and

1 2 Ind 1

z   (23.b) Second, in a similar manner, consider Ind2 as a virtual control input and let

1 1 2 2

2

      (24) Selecting as a Lyapunov candidate

2/z2/w

2  (25) and taking its time derivative gives

0zkwk

zz

w

zz

wV

2 1 2 2 1

1 2 1 1 1

1 2 Ind 1 1 1 2

1 1

z

w   (27.a)

Coordinate Transformation Based Contour Following Control for Robotic Systems 193

Therefore, the modified system dynamics in VCS becomes

t t

t n

   2 Λ  (16.a)

n n

3 2

1

    Λ  (16.b) For (16), contouring controller design will be addressed in the next section

4 Contouring Controller Design

In this section, design of contouring controller is considered separately for tangential and

modified normal dynamics For demonstration purpose, a general proportional-derivative

(PD) controller is applied in tangential error equation It is well known that the PD

controller is capable of achieving stabilization and improving transient response, but is not

adequate for error elimination Consequently, tangential tracking errors exist unavoidably

Under this circumstance, we are going to show that precise contouring performance can still

be achieved by applying the CI approach Building on the developed contouring control

framework, the tangential and normal control objects can be respectively interpreted as

stabilization and regulation problems

4.1 Design of tangential control effort

Considering the tangential dynamics of (16.a), a PD controller with the form

t n

t Pt

t Vt

      2 (17)

is applied Substituting (17) into (16.a) results in

t t

Pt t

Vt

tK  K  Λ

  (18) where KVt and KPt are positive real The selection of control gains should guarantee the

criterion t R Eq (18) indicates that the tangential tracking error cannot be eliminated

very well due to the existence of Λ However, it will be shown that the existence of t t

causes no harm to contouring performance with the aid of CI

4.2 Design of normal control effort

In the following, an integral type sliding controller for the modified normal dynamics is

developed by using backstepping approach Firstly, let IndInd 1 and define an internal

state w Then the system (16.b) can be represented as

1 Ind

w  (19.a)

2 Ind

1 Ind 

  (19.b)

n n

3 2

1 2

 (20) where k10 Then, consider as a Lyapunov function

2/w

1 (21) The derivative of (21) is

0wkw

1 1

z

w   (23.a) and

1 2 Ind 1

z   (23.b) Second, in a similar manner, consider Ind2 as a virtual control input and let

1 1 2 2

2

     (24) Selecting as a Lyapunov candidate

2/z2/w

2  (25) and taking its time derivative gives

0zkwk

zz

w

zz

wV

2 1 2 2 1

1 2 1 1 1

1 2 Ind 1 1 1 2

1 1

z

w   (27.a)

1 2 2

1 z

z    (27.b)

2 n n 3 2 1 2

z    Λ   (27.c) Suppose that the parameter uncertainties and external disturbances satisfy the inequality

 3 n

max Λ , where  is an unknown positive constant, then the final control object is

to develop a controller that provides system robustness against 

Design a control law in the following form

n 12z1k3z22sgn(z2) (28) where  denotes the switching gain Select a Lyapunov candidate as

2/z2/z2/w

2 2 1 2

3   (29) From (28) and (29), one can obtain

 

2 3

2 2 2 2 3 2 2 2 1

2 n 2 2 1 2 1 2 2 1 1 1 3

kV2z

kV2

zzsgnzzkzkwk

zz

zzz

wV

where k1k2k3k is applied Therefore, system (27) is exponentially stable by using the

control law (28) when the selected  satisfies  Since the upper bound of  may not

be efficiently determined, the following well known adaptation law (Yoo & Chung, 1992),

which dedicates to estimate an adequate constant value a, is applied

),z(satzˆ

2 2

  (31) where ˆa is denoted as an estimated switching gain and

0

, (32)

stands for an adaptation gain, where the use of dead-zone is needed due to the face that the

ideal sliding does not occur in practical applications For chattering avoidance, the

discontinuous controller (28) is replaced by

n12z1k3z22ˆasat(z2,) (33) The saturation function is described as follows

2

z/zzsgn:zsat , , (34)

where  is relative to the thickness of the boundary layer

Let the estimative error be ~a, i.e., ~aaˆa and then select a Lyapunov function as

/zzkV2

z,zsatzkV2

,zsatz

~z,zsatˆzzkzkwk

/

~

~zzzzwwV

4

2 a 2 4

2 2

a 2 4

2 2 a 2 2

a 2 2 3 2 2 2 1

a a 2 2 1 1 4

where ( ) is bounded by (t)max In general, a is available Suppose that a,

it follows (t)z2z2 /1 The maximum value max /4 occurs at z2 /2

Eq (36) reveals that fort, it follows

k8kt2exp1k20Vkt2exp

dt

k2exp0Vkt2exp)(V

max 4

t

0 4 4

dy

dx

dy

Coordinate Transformation Based Contour Following Control for Robotic Systems 195

1 2

2

1 z

z    (27.b)

2 n

n 3

2 1

2

z    Λ   (27.c) Suppose that the parameter uncertainties and external disturbances satisfy the inequality

 3 n

max Λ , where  is an unknown positive constant, then the final control object is

to develop a controller that provides system robustness against 

Design a control law in the following form

n 12z1k3z22sgn(z2) (28) where  denotes the switching gain Select a Lyapunov candidate as

2/

z2

/z

2/

w

2 2

1 2

3   (29) From (28) and (29), one can obtain

 

2 3

2 2

2 2

3 2

2 2

1

2 n

2 2

1 2

1 2

2 1

1 1

3

kV2

zkV

2

zz

sgnz

zk

zk

wk

zz

zz

zw

where k1k2k3k is applied Therefore, system (27) is exponentially stable by using the

control law (28) when the selected  satisfies  Since the upper bound of  may not

be efficiently determined, the following well known adaptation law (Yoo & Chung, 1992),

which dedicates to estimate an adequate constant value a, is applied

),

z(

satz

ˆ

2 2

  (31) where ˆa is denoted as an estimated switching gain and

0

,

(32)

stands for an adaptation gain, where the use of dead-zone is needed due to the face that the

ideal sliding does not occur in practical applications For chattering avoidance, the

discontinuous controller (28) is replaced by

n 12z1k3z22ˆasat(z2,) (33) The saturation function is described as follows

2

z/zzsgn:zsat , , (34)

where  is relative to the thickness of the boundary layer

Let the estimative error be ~a, i.e., ~aaˆa and then select a Lyapunov function as

/zzkV2

z,zsatzkV2

,zsatz

~z,zsatˆzzkzkwk

/

~

~zzzzwwV

4

2 a 2 4

2 2

a 2 4

2 2 a 2 2

a 2 2 3 2 2 2 1

a a 2 2 1 1 4

where ( ) is bounded by (t)max In general, a is available Suppose that a,

it follows (t)z2z2/1 The maximum value max/4 occurs at z2 /2

Eq (36) reveals that fort, it follows

k8kt2exp1k20Vkt2exp

dt

k2exp0Vkt2exp)(V

max 4

t

0 4 4

dy

dx

dy

By (37), the system is exponentially stable with a guaranteed performance associated with

the size of control parameters  and k The overall contouring control architecture is

illustrated in Fig 6 It is similar with the standard feedback control loop, where the main

control components are highlighted in the dotted blocks

Remark 1 For illustration purpose, a PD controller is applied to the tangential dynamics

such that tangential tracking error cannot be eliminated completely Of course one can also

apply a robust controller to pursue its performance if necessary However, the following

simulations are going to show that even in the presence of tracking errors, the contouring

performance will not be degraded by using the proposed contouring control framework

Remark 2 The action of the adaptive law activates when z2  It means that for a given

small gain value, ˆa will be renewed in real time until the criterion 

2

z is achieved

5 Numerical Simulations

In this section, a robot system in consideration of nonlinear friction effects is taken as an

example The friction model used in numerical simulations is given by

2 si i ci

si ci i

F         (38)

where Fci is the Coulomb friction and Fsi is the static friction force  denotes an angular si

velocity relative to the Stribeck effect and si denotes the viscous coefficient The suffix

2

,

1

i  indicates the number of robot joint

The parameters used in friction model are:

025.0

Fc 1 , Fc 20.02,Fs 10.04, Fs 20.035

001.0

2 s 1

s  

  , s10.005 and s20.004 According to the foregoing analysis, an adequate switching gain is suggested to be

determined in advance for confirming system robustness Thus, estimations are performed

previously for two contouring control tasks, i.e., circular and elliptical contours Fig 7(a)

and (c) show the responses of sliding surface and Fig 7(b) and (d) depict the response of ˆa

during update

From Fig 7, it implies that the sliding surfaces were suppressed to the prescribed boundary

layer by integrating with the adaptation law The initial guess-value ˆa 0 12 and the

adaptation gain 100 are applied in (31) According to (32), an adequate value of ˆa was

determined when z2 0.0025 is achieved From the adaptation results shown in Fig

7(b) and 7(d), the conservative switching gains ˆa18 and 16 are adopted to handle 5

circular and elliptical profiles, respectively

k  , KVt9, KPt20, mˆ17.8, mˆ20.37Exact system parameters of two-arm robot are

344.8

Coordinate Transformation Based Contour Following Control for Robotic Systems 197

By (37), the system is exponentially stable with a guaranteed performance associated with

the size of control parameters  and k The overall contouring control architecture is

illustrated in Fig 6 It is similar with the standard feedback control loop, where the main

control components are highlighted in the dotted blocks

Remark 1 For illustration purpose, a PD controller is applied to the tangential dynamics

such that tangential tracking error cannot be eliminated completely Of course one can also

apply a robust controller to pursue its performance if necessary However, the following

simulations are going to show that even in the presence of tracking errors, the contouring

performance will not be degraded by using the proposed contouring control framework

Remark 2 The action of the adaptive law activates when z2  It means that for a given

small gain value, ˆa will be renewed in real time until the criterion 

2

z is achieved

5 Numerical Simulations

In this section, a robot system in consideration of nonlinear friction effects is taken as an

example The friction model used in numerical simulations is given by

2 si

i ci

si ci

i

F        (38)

where Fci is the Coulomb friction and Fsi is the static friction force  denotes an angular si

velocity relative to the Stribeck effect and si denotes the viscous coefficient The suffix

2

,

1

i  indicates the number of robot joint

The parameters used in friction model are:

025

0

Fc 1 , Fc 20.02,Fs 10.04, Fs 20.035

001

0

2 s

1

s  

  , s10.005 and s20.004 According to the foregoing analysis, an adequate switching gain is suggested to be

determined in advance for confirming system robustness Thus, estimations are performed

previously for two contouring control tasks, i.e., circular and elliptical contours Fig 7(a)

and (c) show the responses of sliding surface and Fig 7(b) and (d) depict the response of ˆa

during update

From Fig 7, it implies that the sliding surfaces were suppressed to the prescribed boundary

layer by integrating with the adaptation law The initial guess-value ˆa 0 12 and the

adaptation gain 100 are applied in (31) According to (32), an adequate value of ˆa was

determined when z2 0.0025 is achieved From the adaptation results shown in Fig

7(b) and 7(d), the conservative switching gains ˆa18 and 16 are adopted to handle 5

circular and elliptical profiles, respectively

k  , KVt9, KPt20, mˆ17.8, mˆ2 0.37Exact system parameters of two-arm robot are

344.8

0 1 1

2 1

2 2 2 2 1 2

cosllsinltanx

ytan)0(

ll2llyxcos)0(

Referring to the simulations, Fig 8 obviously illustrates the contour following behavior It

shows that even though the real time command position (i.e., the moving ring) goes ahead

the end-effector, the end-effector still follows to the desired contour without significant

deviation The tracking responses are shown in Fig 9(a) and (b), where the tracking errors

exist significantly, but the contouring performance, evaluated by the contour index Ind,

remains in a good level The corresponding control efforts of each robot joint are drawn in

Fig.10(a)-(b) Examining Fig 8(a) and (b) again, it can be seen that the time instants where

the relative large tracking errors occur are also the time instants the relative large CIs are

induced The reason can refer to the dynamics of CI given in (16) Due to the face that (16b)

is perturbed by the coupling uncertain terms 3 when t 0, the control performance will

be (relatively) degraded when large tangential tracking errors occur

Fig 8 Behavior of path following by the proposed method

Fig 9 Performance of tracking and equivalent contouring errors

(a) (b) Fig 10 Applied control torque

end-The result is consistent with the behavior illustrated in Fig 2, i.e., the end-effector at A approaches to the real time command D through the desired path without causing short

cutting phenomenon Moreover, it has been demonstrated that the CI approach is also

capable of avoiding over-cutting phenomenon, which is induced by the T-N coordinate

transformation approach, in the presence of large tracking errors (Peng & Chen, 2007a) The simulation results confirm again that good contouring control performance does not necessarily rely on the good tracking level The corresponding continuous control efforts of joint-1 and -2 are shown in Fig 12(a) and (b), respectively

(a) (b) Fig 11 Partial behavior of path following by the proposed method

1 2

2 1

0 0

1 1

2

2 2

2 2

1 2

cosl

lsin

ltan

x

ytan

)0

(

ll

2l

ly

xcos

)0

Referring to the simulations, Fig 8 obviously illustrates the contour following behavior It

shows that even though the real time command position (i.e., the moving ring) goes ahead

the end-effector, the end-effector still follows to the desired contour without significant

deviation The tracking responses are shown in Fig 9(a) and (b), where the tracking errors

exist significantly, but the contouring performance, evaluated by the contour index Ind,

remains in a good level The corresponding control efforts of each robot joint are drawn in

Fig.10(a)-(b) Examining Fig 8(a) and (b) again, it can be seen that the time instants where

the relative large tracking errors occur are also the time instants the relative large CIs are

induced The reason can refer to the dynamics of CI given in (16) Due to the face that (16b)

is perturbed by the coupling uncertain terms 3 when t 0, the control performance will

be (relatively) degraded when large tangential tracking errors occur

Fig 8 Behavior of path following by the proposed method

Fig 9 Performance of tracking and equivalent contouring errors

(a) (b) Fig 10 Applied control torque

end-The result is consistent with the behavior illustrated in Fig 2, i.e., the end-effector at A approaches to the real time command D through the desired path without causing short

cutting phenomenon Moreover, it has been demonstrated that the CI approach is also

capable of avoiding over-cutting phenomenon, which is induced by the T-N coordinate

transformation approach, in the presence of large tracking errors (Peng & Chen, 2007a) The simulation results confirm again that good contouring control performance does not necessarily rely on the good tracking level The corresponding continuous control efforts of joint-1 and -2 are shown in Fig 12(a) and (b), respectively

(a) (b) Fig 11 Partial behavior of path following by the proposed method

(a) (b)

Fig 12 Performance of tracking and contouring

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

In the robotic motion control field, positioning and tracking are considered as the main

control tasks In this Chapter, we have addressed a specific motion control topic, termed as

contouring control The core concept of the contouring control is different from the main

object of the tracking control according to its goal

For tracking control, the desired goal is to track the real time reference command as precise

as possible On the other hand, the main object is to achieve precise motion along prescribed

contours for contouring control Under this circumstance, tracking error is no longer a

necessary performance index requiring to be minimized To enhance resulting contour

precision without relying on tracking performance, a contour following control strategy for

robot manipulators is presented

Different from the conventional manipulator motion control, a contour error dynamics is

derived via coordinate transformation and an equivalent error called CI is introduced in

VCS to evaluate contouring control performance The contouring control task in the VCS

turns into a stabilizing problem in tangential dynamics and a regulation problem in

modified normal dynamics The main advantage of the control scheme is that the final contouring accuracy will not be degraded even if the tracking performance of the robot manipulator is not good enough; that is, the existence of tracking errors will not make harm

to the final contouring quality This advantage has been apparently clarified through numerical study

7 References

Chen, C L & Lin, K C (2008) Observer-Based Contouring Controller Design of a Biaxial

State System Subject to Friction, IEEE Transactions on Control Systems Technology,

Vol 16, No 2, 322-329

Chen, C L & Xu, R L (1999) Tracking control of robot manipulator using sliding mode

controller with performance robustness, Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, Vol 121, No 1, 64-70

Chen, S L.; Liu, H L & Ting, S C (2002) Contouring control of biaxial systems based on

polar coordinates, IEEE-ASME Transactions on Mechatronics, Vol 7, No 3, 329-345

Chin, J H & Lin, T C (1997) Cross-coupled precompensation method for the contouring

accuracy of computer numerically controlled machine tools, International Journal of

Machine Tools and Manufacture, Vol 37, No 7, 947–967

Chiu, G.T.-C & Tomizuka, M (2001) Contouring control of machine tool feed drive

systems: a task coordinate frame approach, IEEE Transactions on Control Systems Technology, Vol 9, No 1, 130-139

Dong, W & Kuhnert, K D (2005) Robust adaptive control of nonholonomic mobile robot

with parameter and nonparameter uncertainties, IEEE Transactions on Robotics, Vol

21, No 2, 261-266

Fang, R W & Chen, J S (2002) A cross-coupling controller using an H-infinity scheme and

its application to a two-axis direct-drive robot, Journal of Robotic Systems, Vol 19,

No 10, 483-497

Feng, G & Palaniswami, M (1993) Adaptive control of robot manipulators in task

space, IEEE Transactions on Automatic Control, Vol 38, No 1, 100-104

Ho, H C.; Yen, J Y & Lu, S S (1998) A decoupled path-following control algorithm based

upon the decomposed trajectory error, International Journal of Machine Tools and Manufacture, Vol 39, No 10, 1619-1630

Hsieh, C.; Lin, K C & Chen, C L (2006) Contour Controller Design for Two-dimensional

Stage System with Friction, Material Science Forum, Vol 505-507, 1267-1272

Koren, Y (1980) Cross-Coupled Biaxial Computer Control for Manufacturing Systems,

Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, Vol

102, No 4, 265-272

Lee, J H.; Dixon, W E.; Ziegert, J C & Makkar, C (2005) Adaptive nonlinear contour

coupling control for a machine tool system, IEEE/ASME International Conference on Advanced Intelligent Mechatronics Proceedings, 1629 – 1634

Peng, C C & Chen, C L (2007a) Biaxial contouring control with friction dynamics using a

contour index approach, International Journal of Machine Tools & Manufacture, Vol

2007, No 10, 1542-1555

Coordinate Transformation Based Contour Following Control for Robotic Systems 201

(a) (b)

Fig 12 Performance of tracking and contouring

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

In the robotic motion control field, positioning and tracking are considered as the main

control tasks In this Chapter, we have addressed a specific motion control topic, termed as

contouring control The core concept of the contouring control is different from the main

object of the tracking control according to its goal

For tracking control, the desired goal is to track the real time reference command as precise

as possible On the other hand, the main object is to achieve precise motion along prescribed

contours for contouring control Under this circumstance, tracking error is no longer a

necessary performance index requiring to be minimized To enhance resulting contour

precision without relying on tracking performance, a contour following control strategy for

robot manipulators is presented

Different from the conventional manipulator motion control, a contour error dynamics is

derived via coordinate transformation and an equivalent error called CI is introduced in

VCS to evaluate contouring control performance The contouring control task in the VCS

turns into a stabilizing problem in tangential dynamics and a regulation problem in

modified normal dynamics The main advantage of the control scheme is that the final contouring accuracy will not be degraded even if the tracking performance of the robot manipulator is not good enough; that is, the existence of tracking errors will not make harm

to the final contouring quality This advantage has been apparently clarified through numerical study

7 References

Chen, C L & Lin, K C (2008) Observer-Based Contouring Controller Design of a Biaxial

State System Subject to Friction, IEEE Transactions on Control Systems Technology,

Vol 16, No 2, 322-329

Chen, C L & Xu, R L (1999) Tracking control of robot manipulator using sliding mode

controller with performance robustness, Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, Vol 121, No 1, 64-70

Chen, S L.; Liu, H L & Ting, S C (2002) Contouring control of biaxial systems based on

polar coordinates, IEEE-ASME Transactions on Mechatronics, Vol 7, No 3, 329-345

Chin, J H & Lin, T C (1997) Cross-coupled precompensation method for the contouring

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with parameter and nonparameter uncertainties, IEEE Transactions on Robotics, Vol

21, No 2, 261-266

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its application to a two-axis direct-drive robot, Journal of Robotic Systems, Vol 19,

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space, IEEE Transactions on Automatic Control, Vol 38, No 1, 100-104

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coupling control for a machine tool system, IEEE/ASME International Conference on Advanced Intelligent Mechatronics Proceedings, 1629 – 1634

Peng, C C & Chen, C L (2007a) Biaxial contouring control with friction dynamics using a

contour index approach, International Journal of Machine Tools & Manufacture, Vol

2007, No 10, 1542-1555

Peng, C C & Chen, C L (2007b) A 3-dimensional contour following strategy via

coordinate transformation for manufacturing applications, International Conference

on Advanced Manufacture, Tainan, Taiwan, November, Paper No B4-95

Ramesh, R.; Mannan, M A & Poo, A N (2005) Tracking and contour error control in CNC

servo systems, International Journal of Machine Tools and Manufacture, Vol 45, No 3

301-326

Sarachik, P & Ragazzini, J R (1957) A Two Dimensional Feedback Control System,

Transactions AIEE, Vol 76, 55–61

Shih, Y T.; Chen, C S & Lee, A C (2002) A novel cross-coupling control design for Bi-axis

motion, International Journal of Machine Tools and Manufacture, Vol 42, No 14,

1539-1548

Slotine, J J E & Li, W P (1988) Adaptive manipulator control: a case study, IEEE

Transactions on Automatic Control, Vol 33, No 11, 995-1003

Spong, M W & Vidyasagar, M (1989) Robot dynamics and control, John Wiley & Sons, Inc

Sun, M.; Ge, S S & Mareels, I M Y (2006) Adaptive repetitive learning control of robotic

manipulators without the requirement for initial repositioning, IEEE Transactions on

Robotics, Vol 22, No 3, 563-568

Wang, L S & Zhang, J (2004) The research of static de-coupled contour control technique of

the ultra-precision machine tool, Proceedings of the 5 th World Congress on Intelligent

Control and Automation, June 15-19, Hangzhou, China

Yeh, S S & Hsu, P L (1999) Analysis and design of the integrated controller for precise

motion systems, IEEE Transactions on Control Systems Technology, Vol 7, No 6,

706-717

Zhong, Q.; Shi, Y.; Mo, J & Huang, S (2002) A linear cross-coupled control system for

high-speed machining, The International Journal of Advanced Manufacturing Technology,

Vol 19, No 8, 558-563

Zhu, W H.; Chen, H T & Zhang, Z J (1992) A variable structure robot control algorithm

with an observer, IEEE Transactions on Robotics and Automation, Vol 8, No 4, 1992,

486-492

Yoo, D S & Chung, M J (1992) A variable structure control with simple adaptation laws

for upper bounds on the norm of the uncertainties, IEEE Transactions on Automatic

Control, Vol 37, No 6, 860-865

Acknowledgements

Acknowledgments: Part of this work was supported by the National Science Council,

Taiwan, under the Grant No NSC96-2221-E006-052

M M M M

lm

sinllmsin

llm2

2 1 2 c 1 2

2 2 2 c 1 2

2 2 c 1 2

2 1 2 c 1 1 2 1 1 c 1

cosglm

coslcoslgmcosglm

i ci

2 2 2 c 2 22

2 2 2 c 2 2 c 1 2 21

2 2 2 c 2 2 c 1 2 12

2 1 2 2 c 1 2 2 c 2 1 2 2 1 c 1 11

lm

121I,l2

1l

Ilm

Ilcosllm

Ilcoslm

IIcosll2llmlm

2 1 2 1 1

sinlsinl

coslcoslt

l)sin(

l)sin(

l

2 2

1 2 2 1 2 1 1

1 2 2 1 2 1 1

sincos

Coordinate Transformation Based Contour Following Control for Robotic Systems 203

Peng, C C & Chen, C L (2007b) A 3-dimensional contour following strategy via

coordinate transformation for manufacturing applications, International Conference

on Advanced Manufacture, Tainan, Taiwan, November, Paper No B4-95

Ramesh, R.; Mannan, M A & Poo, A N (2005) Tracking and contour error control in CNC

servo systems, International Journal of Machine Tools and Manufacture, Vol 45, No 3

301-326

Sarachik, P & Ragazzini, J R (1957) A Two Dimensional Feedback Control System,

Transactions AIEE, Vol 76, 55–61

Shih, Y T.; Chen, C S & Lee, A C (2002) A novel cross-coupling control design for Bi-axis

motion, International Journal of Machine Tools and Manufacture, Vol 42, No 14,

1539-1548

Slotine, J J E & Li, W P (1988) Adaptive manipulator control: a case study, IEEE

Transactions on Automatic Control, Vol 33, No 11, 995-1003

Spong, M W & Vidyasagar, M (1989) Robot dynamics and control, John Wiley & Sons, Inc

Sun, M.; Ge, S S & Mareels, I M Y (2006) Adaptive repetitive learning control of robotic

manipulators without the requirement for initial repositioning, IEEE Transactions on

Robotics, Vol 22, No 3, 563-568

Wang, L S & Zhang, J (2004) The research of static de-coupled contour control technique of

the ultra-precision machine tool, Proceedings of the 5 th World Congress on Intelligent

Control and Automation, June 15-19, Hangzhou, China

Yeh, S S & Hsu, P L (1999) Analysis and design of the integrated controller for precise

motion systems, IEEE Transactions on Control Systems Technology, Vol 7, No 6,

706-717

Zhong, Q.; Shi, Y.; Mo, J & Huang, S (2002) A linear cross-coupled control system for

high-speed machining, The International Journal of Advanced Manufacturing Technology,

Vol 19, No 8, 558-563

Zhu, W H.; Chen, H T & Zhang, Z J (1992) A variable structure robot control algorithm

with an observer, IEEE Transactions on Robotics and Automation, Vol 8, No 4, 1992,

486-492

Yoo, D S & Chung, M J (1992) A variable structure control with simple adaptation laws

for upper bounds on the norm of the uncertainties, IEEE Transactions on Automatic

Control, Vol 37, No 6, 860-865

Acknowledgements

Acknowledgments: Part of this work was supported by the National Science Council,

Taiwan, under the Grant No NSC96-2221-E006-052

12 11

M M

M M

lm

sinl

lm

sinl

lm

2

2 1

2 c

1 2

2 2

2 c

1 2

2 2

2 c

2 c

2

2 1

2 c

1 1

2 1

1 c

1

cosgl

m

cosl

cosl

gm

cosgl

i ci

2 2 2 c 2 22

2 2 2 c 2 2 c 1 2 21

2 2 2 c 2 2 c 1 2 12

2 1 2 2 c 1 2 2 c 2 1 2 2 1 c 1 11

lm

121I,l2

1l

Ilm

Ilcoslm

Ilcoslm

IIcosll2llmlm

2 1 2 1 1

sinlsinl

coslcoslt

l)sin(

l)sin(

l

2 2

1 2 2 1 2 1 1

1 2 2 1 2 1 1

sincos

Design of Adaptive Controllers based on Christoffel Symbols of First Kind 205

Design of Adaptive Controllers based on Christoffel Symbols of First Kind

Juan Ignacio Mulero-Martínez

X

Design of Adaptive Controllers based on

Christoffel Symbols of First Kind

Juan Ignacio Mulero-Martínez

Technical Universty of Cartagena

Spain

1 Introduction

The present chapter is aimed at systematically exposing the reader to certain modern trends

in designing advanced robot controllers More specifically, it focuses on a new and

improved method for building suitable adaptive controllers guaranteeing asymptotic

stability It covers the complete design cycle, while providing detailed insight into most

critical design issues of the different building blocks In this sense, it takes a more global

design perspective in jointly examining the design space at control level as well as at the

architectural level

The primary purpose is to provide insight and intuition into adaptive controllers based on

Christoffel symbols of first kind for a serial-link robot arm, (Mulero-Martínez, 2007a) These

controllers are referred to as static since the positional dependence of the nonlinear

functions In this context, the preferred method of nonlinear compensation is the method of

building emulators Often, however, the full power of the method is overlooked, and very

few works deal with these techniques at the level of detail that the subject deserves As a

result, the chapter fills that gap and includes the type of information required to help control

engineers to apply the method to robot manipulators Developed in this chapter are several

deep connections between dynamics analysis and implementation emphasizing the

powerful adaptive methods that emerge when separate techniques from each area are

properly assembled in a larger context

After beginning with a comprehensive presentation of the fundamentals of these techniques,

the chapter addresses the problem of factorization of the Coriolis/centripetal matrix,

(Mulero-Martinez, 2009) This aspect is crucial when designing non-linear compensators by

emulation At this point, it is provided a concise and didactically structured description of

the design of emulators as matters stand, (Mulero-Martinez, 2006) Specifically, emulators

are split up into sub-emulators to improve and simplify the design of controllers while

making faster the updating of parameters From a practical point of view, the

implementation is developed by resorting to parametric structures This means to obtain a

set of system's own function as regression functions

Most of the adaptive schemes start from the notable property of linearity in the parameters,

which lead naturally to equivalent structures when designing emulators for the nonlinear

terms When the linearity in the parameters (LIP) is considered as a first assumption in the

10

development of adaptive schemes, it is clear that there exists a strong connection to the LIP

emulators formulated in terms of a regression matrix and a vector of parameters The main

difference between standard adaptive schemes and the proposed approach stems from the

idea of developing efficient controllers The present work is aimed by attempts to mitigate

the "curse of dimensionality" by exploiting the representation properties associated with the

matrix of Coriolis/centripetal effects By recalling the connection between LIP

representation of robot manipulators and LIP adaptive emulators, it can be asserted that

standard scheme matches perfectly with a dynamic emulator Thus, the regression matrix,

depend not only on the position joint variables but also on the velocity and acceleration

variables

As regards to the control, a novel theorem guarantees the stability for the whole system and

is based on the Lyapunov energy The proof is generalized to cope with a realistic case

where both a functional reconstruction error and an external disturbance are present It

should be observed that the functional reconstruction error is caused by not using a number

of regression functions appropiately distributed in the space As a result, these

considerations lead to a quite different approach, since it is required to analyze the initial

conditions of the errors to guarantee the validity of the approximation The specification of a

range of validity causes that the stability holds only inside a compact set As a consequence,

the proof guarantees semi-global stability as opposed to the standard schemes where the

stability is attained in the whole state space, in a global sense Apart from these

considerations, a number of remarks have been made to address some special aspects such

as the boundedness of the parameters, the ultimately uniformly boundedness of all the

signals and the stability in the ideal case

The main benefit of the proposed controller is that it allows to derive tuning laws only for

inertia, gravitational and frictional parameters The Coriolis parameters are not necessary to

be used because of the approximation based on Christoffel symbols This is very useful to

implement adaptive controllers since the number of nodes diminishes and the

computational performance improves Previously, an extensive analysis of the mechanical

properties for a robot has been discussed The regression functions for the adaptive

controller depend on the non-linear functions associated with the inertia matrix, and

therefore, a discretization of positions could be done for the inertia matrix This is a very

useful aspect because the position space for a revolute robot is compact and in consequence,

the number of nodes is limited to approximate a non-linear function

The plan of the chapter is as follows In section 2 the representation properties for the

Coriolis/centripetal matrix are analysed An interpretation for the Coriolis/centripetal

matrix is presented and the description by means of the Christoffel symbols of first kind and

fundamental matrices are provided In section 3, emulators are used to approximate the

non-linearities of a robot using the properties presented in the previous section and the

Kronecker product The next section presents the design of the adaptive controller in terms

of a control law and a parameter updating law This section concludes with a theorem that

guarantees the stability for the whole system and is based on the Lyapunov energy Finally

an example of a 2-dof robot arm is used to illustrate the theorem

2 Representation of the Coriolis/Centripetal Matrix Fundamental Matrices

In this section some notions regarding the representation of the Coriolis/centripetal

matrices are introduced All the ideas presented here constitute an original contribution and have many interesting implications in the field of robotics To this end, fundamental matrices are introduced and described in terms of their structure Moreover, some emerging properties are analyzed, allowing one to build the Coriolis/centripetal matrix in a simple way Let start with the definition of the matrix MD which from now on will be called the inertia derivative matrix

2x M q x relative to the joint position can be written as 1 T( )

Design of Adaptive Controllers based on Christoffel Symbols of First Kind 207

development of adaptive schemes, it is clear that there exists a strong connection to the LIP

emulators formulated in terms of a regression matrix and a vector of parameters The main

difference between standard adaptive schemes and the proposed approach stems from the

idea of developing efficient controllers The present work is aimed by attempts to mitigate

the "curse of dimensionality" by exploiting the representation properties associated with the

matrix of Coriolis/centripetal effects By recalling the connection between LIP

representation of robot manipulators and LIP adaptive emulators, it can be asserted that

standard scheme matches perfectly with a dynamic emulator Thus, the regression matrix,

depend not only on the position joint variables but also on the velocity and acceleration

variables

As regards to the control, a novel theorem guarantees the stability for the whole system and

is based on the Lyapunov energy The proof is generalized to cope with a realistic case

where both a functional reconstruction error and an external disturbance are present It

should be observed that the functional reconstruction error is caused by not using a number

of regression functions appropiately distributed in the space As a result, these

considerations lead to a quite different approach, since it is required to analyze the initial

conditions of the errors to guarantee the validity of the approximation The specification of a

range of validity causes that the stability holds only inside a compact set As a consequence,

the proof guarantees semi-global stability as opposed to the standard schemes where the

stability is attained in the whole state space, in a global sense Apart from these

considerations, a number of remarks have been made to address some special aspects such

as the boundedness of the parameters, the ultimately uniformly boundedness of all the

signals and the stability in the ideal case

The main benefit of the proposed controller is that it allows to derive tuning laws only for

inertia, gravitational and frictional parameters The Coriolis parameters are not necessary to

be used because of the approximation based on Christoffel symbols This is very useful to

implement adaptive controllers since the number of nodes diminishes and the

computational performance improves Previously, an extensive analysis of the mechanical

properties for a robot has been discussed The regression functions for the adaptive

controller depend on the non-linear functions associated with the inertia matrix, and

therefore, a discretization of positions could be done for the inertia matrix This is a very

useful aspect because the position space for a revolute robot is compact and in consequence,

the number of nodes is limited to approximate a non-linear function

The plan of the chapter is as follows In section 2 the representation properties for the

Coriolis/centripetal matrix are analysed An interpretation for the Coriolis/centripetal

matrix is presented and the description by means of the Christoffel symbols of first kind and

fundamental matrices are provided In section 3, emulators are used to approximate the

non-linearities of a robot using the properties presented in the previous section and the

Kronecker product The next section presents the design of the adaptive controller in terms

of a control law and a parameter updating law This section concludes with a theorem that

guarantees the stability for the whole system and is based on the Lyapunov energy Finally

an example of a 2-dof robot arm is used to illustrate the theorem

2 Representation of the Coriolis/Centripetal Matrix Fundamental Matrices

In this section some notions regarding the representation of the Coriolis/centripetal

matrices are introduced All the ideas presented here constitute an original contribution and have many interesting implications in the field of robotics To this end, fundamental matrices are introduced and described in terms of their structure Moreover, some emerging properties are analyzed, allowing one to build the Coriolis/centripetal matrix in a simple way Let start with the definition of the matrix MD which from now on will be called the inertia derivative matrix

2x M q x relative to the joint position can be written as 1 T( )

The inertia velocity matrix M q,xv( ) receives its name from the fact that when x q=  , the

term M q,q will be the time differentiation of the generalized inertia matrix, i.e.v( ) M q( )

The following property provides an alternative way to write the matrix Mv

Property 2: The inertia velocity matrix can be also expressed as

2.1 Properties of the fundamental matrices

Subsequently, some properties related to the fundamental matrices are analyzed Following

a systematic methodology, the properties have been classified into two groups:

commutation properties and representation properties

2.1.1 Commutation properties

Commutation properties permit interchange of an external arbitrary vector y and a vector

x passed to a fundamental matrix as an argument The following property makes possible

the commutation while keeping the type of the fundamental matrices This means that the

transpose of the inertia derivative matrix can be transformed into the same structure by

simply interchanging the roles of x and y

M q,x y M q,y x=

The proof of the last property follows directly from the definition of MD The following

property allows to pass from a type of fundamental matrix to another commuting the

2.1.2 Properties of representation of the Coriolis/centripetal matrix

These properties are very important to describe the Coriolis/centripetal matrix from the

J q,q M q,q M q,q   is a skew symmetric matrix, i.e J q,q  J q,qT  

Proof: This is an immediate consequence of the representation of C q,q by means of the  property 1 and the fact that the inertia velocity matrix is M q,qv  M q 

In a general way, the following representation can be derived

D 2

C q,z z M q,z J q,z z An interesting property which is a direct implication of the property 4 is that, by setting x y in C q,x y  

Property 7: The Coriolis/centripetal force can be represented as

Design of Adaptive Controllers based on Christoffel Symbols of First Kind 209

The inertia velocity matrix M q,xv( ) receives its name from the fact that when x q=  , the

term M q,q will be the time differentiation of the generalized inertia matrix, i.e.v( ) M q( )

The following property provides an alternative way to write the matrix Mv

Property 2: The inertia velocity matrix can be also expressed as

2.1 Properties of the fundamental matrices

Subsequently, some properties related to the fundamental matrices are analyzed Following

a systematic methodology, the properties have been classified into two groups:

commutation properties and representation properties

2.1.1 Commutation properties

Commutation properties permit interchange of an external arbitrary vector y and a vector

x passed to a fundamental matrix as an argument The following property makes possible

the commutation while keeping the type of the fundamental matrices This means that the

transpose of the inertia derivative matrix can be transformed into the same structure by

simply interchanging the roles of x and y

M q,x y M q,y x=

The proof of the last property follows directly from the definition of MD The following

property allows to pass from a type of fundamental matrix to another commuting the

2.1.2 Properties of representation of the Coriolis/centripetal matrix

These properties are very important to describe the Coriolis/centripetal matrix from the

J q,q M q,q M q,q   is a skew symmetric matrix, i.e J q,q  J q,qT  

Proof: This is an immediate consequence of the representation of C q,q by means of the  property 1 and the fact that the inertia velocity matrix is M q,qv  M q 

In a general way, the following representation can be derived

D 2

C q,z z M q,z J q,z z An interesting property which is a direct implication of the property 4 is that, by setting x y in C q,x y  

Property 7: The Coriolis/centripetal force can be represented as

where x is an arbitrary vector of dimension n :

Property 8: The Coriolis/centripetal matrix commutes with external vectors

Proof: In order to see this point the representation of the Coriolis/centripetal matrix will be

used as a sum of the inertia velocity matrix, M q,x and the skew symmetric matrix v( )

3 Design of Emulators for Robot Manipulators

3.1 Functional and Linear Parameterization

The approach that follows is founded on the idea to find an emulator as a function close to

the non-linear terms involved in the dynamics equations of a robot manipulator In order to

get a model from a practical point of view, uncertainties in the nonlinear terms getting arise

from the partial information about the exact structure of the dynamics, must be taken into

account The inaccuracies of a model can be classified into two classes: structured and

unstructured uncertainties The first kind of uncertainties comes out from the inaccuracies of

the parameters whereas the unstructured uncertainties are related to unmodeled dynamics,

see (Slotine & Li,1991) Thus, the uncertainties can be adaptatively compensated by defining

each coefficient as a separate parameter so that the dynamics can be expressed in the linear

in the parameters (LIP) and this means that nonlinearities can be split up into an unknown

vector of physical parameters P and a known matrix of basis nonlinear functions

Ψ q,q,x,y comprising the elements of M q , ( ) C q,q , ( ) G q and ( ) F q,q , referred to as ( )

regression matrix Therefore, the nonlinear function f x can be written in this sense adding ( )

a term of error ε , see (Ge et al., 1998)

The linearity of the parameters is the major structural property of robot manipulators and has been analyzed in (Lewis et al., 2003) This linear factorization is always possible to be done for the rigid body dynamics of a fixed-based manipulator as long as the physical uncertainty is on the mass properties of the robot links Furthermore, linearity of the parameters is the first assumption in the most of adaptive controllers An alternative representation of the nonlinear component is as follows

where R q,q( )=(M q C q,q G q( ) ( ) ( ) ) n 2n 1 ´ ( + ) and v=(yT xT 1) 2n 1 + This factorization is always attainable whereas the linearity in the parameters (LIP) is only obtained under some circumstances In the literature, emulators based on regression matrices have been used to approximate the nonlinear dynamics as a whole, as follows

arguments so that each component can be uniformly approximated on any compact subset

of the state space by an appropriately designed emulator

From now on we assume that the number of parameters to approximate the column i of a matrix is li