Workspace trajectory tracking control for two-flexible-link manipulator through output redefinition

Int. J. Modelling, Identification and Control, Vol. 18, No. 2, 2013 119

Copyright © 2013 Inderscience Enterprises Ltd.

Workspace trajectory tracking control for two-flexible-link manipulator through output redefinition

Fareh Raouf* Department of Electrical Engineering, École de technologie supérieure, 1100, rue Notre-Dame ouest, Montréal, Québec, H3C 1K3, Canada E-mail: [email protected] *Corresponding author

Saad Mohamad College of Engineering, Université du Québec en Abitibi-Témiscamingue, 445, boul. de l’Université Rouyn-Noranda, Québec, J9X 5E4, Canada E-mail: [email protected]

Saad Maarouf Department of Electrical Engineering, École de technologie supérieure, 1100, rue Notre-Dame ouest, Montréal, Québec, H3C 1K3, Canada E-mail: [email protected]

Abstract: In this paper, a control strategy and stability analysis for two-flexible-link manipulator are presented to track a desired trajectory in the workspace. The inverse dynamics problem is solved using virtual space and the quasi-static approach. Flexible manipulators are non-minimum phase systems when the controlled output is the tip position. To overcome this problem, an output redefinition technique is used. Two steps are presented to control the manipulator’s tip position. First, assuming that the first link is stable, we develop the control law for the last link to stabilise the error dynamics using the feedback linearisation approach. The weighted parameter defining the non-collocated output is selected to guarantee bounded internal dynamics such that the output is as close as possible to the tip. Second, the same strategy is followed for the first link. Simulation results are presented to show good tracking of desired trajectory in the workspace.

Keywords: flexible manipulators; dynamic inversion; output redefinition; feedback linearisation; passivity.

Reference to this paper should be made as follows: Raouf, F., Mohamad, S. and Maarouf, S. (2013) ‘Workspace trajectory tracking control for two-flexible-link manipulator through output redefinition’, Int. J. Modelling, Identification and Control, Vol. 18, No. 2, pp.119–135.

Biographical notes: Fareh Raouf received his BE in Electrical Engineering from the National Engineering College of Gabes, Tunisia. He received his Masters in Automation from the National Engineering College of Monastir, Tunisia and DESS from École de Technologie Supérieure, University of Quebec, Montreal, Canada. He is a PhD student in Electrical Department at École de Technologie Supérieure, University of Quebec, Montreal, Canada. His main activities include robotics control.

Saad Mohamad received his Diplome d’ingénieur in 1989 in Electrical Engineering from the Lebaneese University, College of Engineering, and MScA in Industrial Electronics from Université du Québec à Trois-Rivières, QC, Canada, and PhD in Electrical Engineering from Ecole Polytechnique de Montréal, Montreal, QC, Canada in 1992 and 2001, respectively. From 2001 to 2005, he worked at CAE Inc as a Flight Control Specialist. In 2006, he joined Université du Québec en Abitibi-Témiscamingue, where he is currently teaching control theory and robotics courses. His research is mainly non-linear control applied to robotics.

120 F. Raouf et al.

Saad Maarouf received his BS and MS in Electrical Engineering from Ecole Polytechnique of Montreal, Montreal, QC, Canada in 1982 and 1984, respectively, and PhD in Electrical Engineering from McGill University, Montreal in 1988. In 1987, he joined Ecole de Technologie Supérieure, Montreal, where he is currently teaching control theory and robotics courses. His research is mainly in non-linear control and optimisation applied to robotics and flight control system.

1 Introduction

Control of flexible manipulators has been the focus of much research in recent years. This is due to the many advantages of flexible manipulators as compared to rigid manipulators. For instance, flexible manipulators are faster, less massive and consume less energy. Since the tasks are defined in the workspace, many studies were interested in controlling the tip of such manipulators (De Luca et al., 1998; Matsuno and Hatayama, 1999; Wang and Yu, 2010; Wang et al., 2009; Zhao-Hui, 2002, 2008). For flexible manipulators, the workspace and the joint space are linked by kinematic and dynamic relationships. The inverse kinematic is then not sufficient to transform the desired trajectory, defined in the workspace, to the joint space. To solve this problem, an intermediate space, called virtual space (Bigras et al., 2003; De Luca and Siciliano, 1989; Lucibello and Di Benedetto, 1993) is defined. The virtual space is linked to the workspace by a simple kinematic relation and has the same number of degrees-of-freedom (DOF) as the joint space. There are two steps to transform the desired trajectory from the workspace to the joint space. The first one represents the transformation from the workspace to the virtual space by using an inverse kinematic. In De Luca and Siciliano (1989), the virtual space coordinates for a one flexible link manipulator is used and in Bigras et al. (2003), it is used for a class of manipulators where the last link is flexible. In the second step, the transformation from the virtual space to the joint space is achieved by solving non-linear equations characterising the flexible part. In Pfeiffer (1989), the quasi-static approach is used to adjust the desired trajectory of the joints by taking into account link deformations. In Kwon and Book (1994), an iterative solution using a causal-anticausal integration method is proposed to solve the inverse dynamic problem. When controlling the position of the end-effector, flexible link manipulators are non-minimum phase systems (Cannon and Schmitz, 1984). To overcome this problem, the output redefinition technique is used in Lucibello and Di Benedetto (1993), Moallem et al. (2001) and Saad et al. (2000) to guarantee stable internal dynamics. The redefined output is usually chosen as the motor’s angle augmented by a weighted value of the tip angle. The weighting parameter is chosen such that the selected output is the closest to the extremity with stable zero-dynamics.

Multi-link flexible manipulators can be seen either as a one MIMO system or as interconnected SISO subsystems. In the first case, many control schemes use a single controller for all joints and links. The feedback linearisation technique is used in Atashzar et al. (2010), Bigras et al.

(2003), Modi et al. (1993) and Woosoon (1994) to control flexible manipulators in the joint space. In this case, the input torques and the joints’ position outputs are collocated; the internal dynamics are then bounded (Woosoon, 1994). The non-minimum phase characteristic limits the application of the feedback linearisation approach to control the end-effector position. This problem can be solved by the output redefinition technique. Adaptive control is also used for flexible link control in Bai (1998), Hoseini (2010) and Yang and Hua (1997). A sliding mode approach is used in Cheung (1995) and Elkaranshway (2011) for flexible link manipulators.

In the previous non-linear control strategies, a single controller is used for all joints and links of flexible manipulators which are regarded as one MIMO system. Unfortunately, due to the complexity of the control structures, it is not easy to use this configuration on the real-time implementation in industrial control systems (Fu et al., 1987). For this reason, many industrial controls are viewed as interconnected subsystems (joints and links). Many advantages exist in this case such as reduction of computational effort, simplicity of implementation, etc. For this configuration, many control strategies are used. The decentralised control approach is used in Bona and Li (1992) and Feiyue et al. (1993) for flexible link manipulators. An independent joint control method is also used in Hillsley (1993). A hierarchical control strategy is used in (Fareh et al., 2012 ) to solve the tracking control problem in the workspace for a 7-DOF hyper redundant articulated nimble adaptable trunks (ANAT) rigid manipulator. The hierarchical control strategy consists in controlling the last joint while assuming that the remaining joints are stable and follow their desired trajectories. Then, going backward, the same strategy is applied to control the (n – 1)th joint while assuming the remaining joints stable and follow their desired trajectories, and so on until the first joint. The feedback linearisation approach was used to develop the control law and asymptotical stability is proved using Lyapunov theory. This algorithm was experimented on the 7-DOF ANAT manipulator and gave effective results and good tracking of a desired trajectory defined in the workspace.

In this paper, a non-linear control strategy ensuring trajectory tracking in the workspace is proposed for two-flexible-link manipulators. To solve the non-minimum phase problem, the output redefinition method is used to select a non-collocated output ensuring stability of internal dynamics. First, to solve the inverse dynamic problem, a trajectory transformation from the workspace to the joint space through a virtual space is used. Secondly, a non-linear

Workspace trajectory tracking control for two-flexible-link manipulator through output redefinition 121

control strategy for the flexible manipulator is presented. This strategy is valid for n flexible links and is based on a hierarchical form. In fact, we begin by stabilising the last link by considering the remaining links stable. Thereafter, we go backward until the first link. The stabilisation of an ith link is ensured using the feedback linearisation approach. The non-collocated output is parameterised with a real parameter that is selected to guarantee bounded internal dynamics. In this paper, we use passivity and the circle criterion to analyse stability of the tracking error dynamics and to determine the new weighted outputs for two-flexible-link manipulators. The simulations show the effectiveness of the proposed method.

The paper is organised as follows: Section 2 presents a description and modelling of the two-flexible-link manipulator. Section 3 proposes a transformation of the desired trajectory from the workspace to the joint space. The non-linear control strategy is given in Section 4. Section 5 presents the application and simulation of the proposed control strategy to the system at hand. Finally, conclusions are given in Section 6.

2 Modelling

The system considered in this work is shown in Figure 1. It is a two-link flexible manipulator moving in the horizontal plane and connected by rigid revolute joints. It consists of two motors, two flexible links, and a payload. Two torques generated by the motors are acting on the system. The angle of the ith motor is θi(t), its inertia coefficient is Imi(i = 1, 2). The ith flexible arm, supposed uniform, has a mass mi and length Li, linear density ρi, and rigidity EIi. The payload has a mass mp. The first link is attached to the first motor and the second link is clamped to the rotor of the second motor.

0 0ˆ ˆ( , )X Y is the fixed reference frame, while (X1, Y1), and

(X2, Y2) are the links neutral axes and are moving with the first and the second link, respectively. 1 1

ˆ ˆ( , )X Y is linked to the second base. The flexible links are modelled as Euler-Bernoulli beams and the deformations are assumed to be small.

Using the Lagrangian formulation, the equation of motion of an n DOF manipulator can be written as:

( ) ( , )M q q C q q q Dq Kq τ+ + + = (1)

where M is the mass matrix, D is the friction matrix, K is the rigidity matrix, and ( , )C q q q is the Coriolis and centrifugal forces vector. q represents the vector of the generalised coordinates, and τ is the vector of the applied torques. Assume that there are a total of n rigid coordinates and n flexible links, the deformation of the ith flexible link can be expressed as follows:

1

( , ) ( ) ( ) 1, ,im

i ij fijj

v x t x q t i nφ=

= =∑ … (2)

where qfij is the jth generalised flexible coordinate of the ith flexible link, φij(x) is its jth shape function and mi is the number of the retained flexible modes. The total number of

the flexible modes is 1

.n

iim m

==∑ In Saad et al. (2006), a

detailed comparison between assumed-based models and finite element ones has been carried on a single flexible link system for one to eight flexible modes. This comparison shows that clamped beam shape functions for the AMM and cubic spline finite elements for the FEM describe best the shape functions of flexible link manipulators.

Figure 1 Two-flexible-link manipulator

In our case, we have modified the model of the two flexible links presented in De Luca and Siciliano (1990) and De Luca and Siciliano (1991) by only considering the first flexible mode of each link (see Appendix B). Then, we have n = 2, and m1 = m2 = 1. Equation (1) can be written as:

( , ) 0 00 ( , )

0 0

0 0

rr rf r r r

fff f ffr ff

r

ff f

M M q C q q qDq C q q qM M

q Iτ

K q

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤+ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎡ ⎤⎡ ⎤ ⎡ ⎤

+ =⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

(3)

where

1 1 1 2

2 1 2 2

1 1 1 2

2 1 2 2

1 1 1 2

2 1 2 2

;

;

;

r r r rrr

r r r r

r f r frf

r f r f

f f f fTfr rf ff

f f f f

M MM

M M

M MM

M M

M MM M M

M M

⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤= = ⎢ ⎥

⎢ ⎥⎣ ⎦

The subscripts ri and fi denote the rigid and flexible modes for the ith link. qr ∈ ℜn are the generalised coordinates associated with the movement of the rigid part, and qf ∈ ℜm are associated with the flexible part. The dynamical equation of motion of the flexible manipulators has the following properties:

122 F. Raouf et al.

P1 M, Mrr, Mff, Dff and Kff are symmetric positive definite matrices.

P2 The first and second links are characterised by two sub-matrices:

1 1 1 1 2 2 2 21 2

1 1 1 1 2 2 2 2

;

r r r f r r r f

f r f f f r f f

M M M MM M

M M M M⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

that are symmetric positive definite.

P3 There exists a matrix ( , )H q q such that

( , ) ( , )C q q H q q q= and , ( 2 ) 0.n m Tx x M H x+∀ ∈ℜ − =

3 Trajectory transformation

In this section, we transform in the desired trajectory from the workspace to the manipulator’s joint space. The flexible manipulator is a non-minimum phase system when the end-effector is considered as the output (Cannon and Schmitz, 1984). In the linear case, transfer functions have one or more zeros in the right half s-plane. For the non-linear system, the zero dynamics are unstable.

The workspace and the joint space are linked by kinematic and dynamic relationships. In the first step, we use the kinematic relation to transform the desired trajectory from the workspace to a virtual space. In the second step, we transform the desired trajectory from the virtual space to the generalised flexible coordinates space. The flexible coordinates are calculated by solving a non-linear equation for the flexible part using the quasi-static approach. The generalised rigid coordinates are deduced from the generalised flexible coordinates and the generalised coordinates defined in the virtual space. The transformation of the desired trajectories from the workspace to the joint space is shown in Figure 2.

Then, it is easy to find the virtual generalised coordinates using an inverse kinematic as done for rigid manipulators. For an ith link, the virtual coordinate Qi is the link’s angular position relative to 1

ˆiX − as shown in

Figure 3. The deformation is assumed to be small. According to

Figure 3, we can write:

arctan Li Lii ri

i i

v vQ q

L L⎛ ⎞

− = ≈⎜ ⎟⎝ ⎠

(4)

Then, the generalised coordinate in the virtual space is: T

i ri i fiQ q qβ= + (5)

where 11 [ ( ), , ( )],

i

Ti i i im i

iL L

Lβ φ φ= … with Li being the length

of the ith flexible link, and φij is the shape function of link i and mode j. The velocity and the acceleration in the virtual space can be found using the Jacobian matrix.

Figure 2 Transformation of the desired trajectories

Quasi-static

approach

Work space Kinematic relation Virtual space

Joint space

Figure 3 Virtual space

The quasi-static approach (Pfeiffer, 1989) can be used to compute the generalised flexible and rigid coordinates in the joint space. This approach neglects the velocity and acceleration of the flexible coordinates ( 0)f fq q= = when solving the inverse dynamics for the desired flexible trajectories. From (3), we get:

( ), 0fr r f f f f ff f ff fM q M q C q q D q K q+ + + + = (6)

Neglecting fq and fq and using (5), qfd is found by solving the following non-linear equation:

( ) ( ), 0fr d rd f d d ff fdM q q C q q K q+ + = (7)

where

; ; Trdi di i fdi rdi di rdi diq Q q q Q q Qβ= − = =

In the final step, the generalised rigid coordinates are given by (7).

4 Control strategy

4.1 Hierarchical control strategy for n flexible links

The control strategy presented in Fareh et al. (2012) was applied and experimented on a 7-DOF rigid manipulator. This strategy is modified to take into account link flexibility. In fact, the control strategy consists in stabilising the last joint and flexible link by assuming that the remaining joints and flexible links are stable. Then we go backward to the (n – 1)th joint and flexible link assuming the previous joints and links are stable, and so on until the first


joint and flexible link. In each step we define a new non-collocated output and we use the feedback linearisation approach to develop the control law. The weighted parameter characterising the non-collocated output, that ensures bounded internal dynamics, is found by studying the asymptotical stability of the resulting tracking error dynamics. The control strategy is shown in Figure 4.

For each link, we follow the following steps:

Step 1 Set i = n; assume that the first (i – 1) joints and links are stable, i.e., the generalised coordinates follow their desired trajectories.

Step 2 Stabilise the ith joint and flexible link by applying the following procedure: • Define a new non-collocated output as the

angle of the ith motor augmented by a weighted value of the ith link’s extremity angle:

, 0 1Tinew ri i i fi iy q qα β α= + ≤ < (8)

• Redefine the dynamical model with the new generalised coordinates:

Ti inew fiq y q⎡ ⎤= ⎣ ⎦ (9)

The new model is obtained by replacing in (3) the ith rigid generalised coordinate qri by the new non-collocated output yinew given in (8).

• Develop the control law τi by applying the feedback linearisation approach: first, separate the ith acceleration of the flexible generalised coordinate fiq from the flexible part of the new model. Second, insert this term in the rigid part to obtain the following form of the control law:

( ), ,i i i iτ f u q q= (10)

where ui is a control law that ensures tracking of the new output. Finally, the linearised dynamics and the internal dynamics take the following form:

( ), ,inew i

fi i i i

y uq g u q q

=⎧⎪⎨ =⎪⎩

(11)

• Find the critical value of the weighted parameter αi such that the output is the nearest to the ith link’s extremity while ensuring bounded internal dynamics. Then, study the asymptotical stability of the errors dynamics of the flexible part and choose the critical value which corresponds to its limit of stability. The passivity is used to analyse the stability.

Step 3 Set i = n – 1 and go back to Step 1.

Figure 4 Control strategy

i = n

New model

Control law (10)

Critical value of αi

i = i – 1

No

Yes

New non collocated output (8)

i = 0

End

4.2 Application to two-flexible-link manipulator

We now apply the previous strategy to a two-flexible-link manipulator as shown in Figure 1. From (3), the dynamical model is given by:

11 1 12 2 13 1 14 2 1 1r r f fM q M q M q M q C τ+ + + + = (12)

21 1 22 2 23 1 24 2 2 2r r f fM q M q M q M q C τ+ + + + = (13)

31 1 32 2 33 1 34 2 3

1 1 1 1 0r r f f

f f f f

M q M q M q M q C

D q K q

+ + + +

+ + = (14)

41 1 42 2 43 1 44 2 4

2 2 2 2 0r r f f

f f f f

M q M q M q M q C

D q K q

+ + + +

+ + = (15)

In the first step, we transform the desired trajectories in triangular form (Figure 5) from workspace to joint space. By following the same strategy given in Section 3, the virtual space is defined as if it was a 2-DOF rigid manipulator as follows:

( ) ( )( )

1 1 1 1

2 1

2 2 2 2 2 2

2 , 2 ,2 ,

d rd fd

d d

d rd fd

Q q qatan y x atan k k

Q q q atan s c

β

β

⎧ = +⎪

= −⎨⎪ = + =⎩

(16)

124 F. Raouf et al.

where

( )

( )( )

2 2 2 21 2

21 2

22 2

1 1 2 2

2 2 2

, 1, 2;

= ;2

= 1 ; and

;sin ; and

cos .

i ii

i

i di

i di

Li

L

x y L Lc

L L

s ck L L ck L ss Q

c Q

φβ = =

+ − −

± −

= +=

=

=

Using the Jacobian, the velocity and acceleration of the virtual coordinates are given by:

( )

( ) ( )

1 11

2 2

11

2

; d d dd

dd d

d dd d

d d

Q x QJ Q

yQ Q

x QJ Q j Q

y Q

−

−

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦⎧ ⎫⎡ ⎤⎡ ⎤⎪ ⎪= − ⎢ ⎥⎨ ⎬⎢ ⎥

⎢ ⎥⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭

(17)

where

( ) 1 1 2 12 2 12

1 1 2 12 2 12.d

l s l s l sJ Q

l c l c l c− − −⎡ ⎤

= ⎢ ⎥+⎣ ⎦

Using (16), the rigid coordinates and their time derivatives are replaced in (12) to (15) by the following expressions:

; ;

; 1,2rdi di i fdi rdi di

rdi di

q Q q q Q

q Q i

β= − =

= = (18)

The generalised flexible coordinates are calculated using the following non-linear equations:

( )31 1 32 2 3 1 1, , 0d d d d fd f fdM Q M Q C Q Q q K q+ + + = (19)

( )41 1 42 2 4 2 2, , 0d d d d fd f fdM Q M Q C Q Q q K q+ + + = (20)

The generalised rigid coordinates are then given by (18).

4.3 Control and stability of the second link

We assume in this section that the first link is stable. Then its generalised coordinates, velocities, and accelerations follow their desired trajectories.

1 1 1 1 1 1 1 ; 0 ; 0 T T T

rd fd rd rdq q q q q q q⎡ ⎤= = =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎣ ⎦

The dynamical equations of the second link are given by (13) and (15).

The objective is to stabilise the second link and to track the desired trajectory in the workspace by selecting a non-collocated output close to the tip. The new output is given by:

2 2 2 2 2 2, 0 1r fy q qα α β α= + ≤ < (21)

where ( )2 2

22

.L

Lφ

β =

The generalised rigid coordinates of the second link and their derivatives can be expressed as follows:

2 2 2 2 2 2 2 2 2 2

2 2 2 2 2

; ;

r f r f

r f

q y q q y q

q y qα α

α

α β α β

α β

= − = −

= − (22)

By substituting (22) in (13) and (15), the new dynamical model is given by:

( )21 1 22 2 24 2 2 22 2 2 2rd fM q M y M M q C τα α β+ + − + = (23)

( )41 1 42 2 44 2 2 42 2 4

2 2 2 2 0rd f

f f f f

M q M y M M q C

K q D qα α β+ + − +

+ + = (24)

By isolating 2fq in (24), the internal dynamics of the second link is given by:

2 41 1 42 2 4

2 2 2 2 f rd

f f f f

q M q M y C

K q D qα= − − −

− − (25)

where 144 2 2 42( ) .X M M Xα β −= − and α2 is chosen such

that 442

2 42.

MM

αβ

≠

Inserting (25) in (23), the control law is chosen as follows:

* * * * *2 21 1 2 2 2 2 2 22 2rd f f f fτ M q C K q D q M uα= + + + + (26)

where

( )( )

( )( )( )

*21 21 24 2 2 22 41

*22 22 24 2 2 22 42

*2 2 24 2 2 22 4

*2 24 2 2 22 2

*2 24 2 2 22 2

;

;

;

;

.f f

f f

M M M M M

M M M M M

C C M M C

K M M K

D M M D

α β

α β

α β

α β

α β

= − −

= − −

= − −

= − −

= − −

Proposition 1: By considering the transformation matrix 2 21

0 1T

α β⎡ ⎤= ⎢ ⎥⎣ ⎦

between the new generalised coordinates

2 2 2[ ]Tfq y qα= and the former generalised coordinates

*2 2 2 22[ ] , T

r fq q q M= is then nonzero if 442

2 42.

MM

αβ

≠

The proof of this proposition is given in Appendix A. To ensure the stability of the output tracking error, we

choose the new control law as follows:

2 2 2 2 2 2d d pu y K y K yα α α α α α= + + (27)

where 2 2 2 2 2d rd fdy q qα α β= + is the desired acceleration of the new weighted output, and 2 2 2dy y yα α α= − is its tracking error. Kpα2, Kdα2 are positive constants.


The tracking error dynamics for the new output and internal dynamics are:

2 2 2 2dy y u δuα α α= − = (28)

( )( )( )

2 2 2

41 41 1

42 2 42 2

4 4

f fd f

d rd

d d

d

q q q

M M q

M y M u

C C

α α

= −

= − −

− −

− −

(29)

( ) ( )2 2 2 2 2 2 2 2fd fd f f fd fd f fK q K q D q D q− − − −

In state space form, (28) and (29) can be written as:

1 2

2 2 2 2 2 2

3 4

4 2 2

d d

fd f

x x

x y y y u δu

x x

x q q

α α α α

⎧ =⎪

= − = − =⎪⎨

=⎪⎪ = −⎩

(30)

Let

[ ]1 2 2 2

3 4 2 2

2 2 2

;

;

; .

TTr

TTf f f

Tr f f fd f

x x x y y

x x x q q

x x x q q q

α α⎡ ⎤= = ⎣ ⎦

⎡ ⎤= =⎡ ⎤⎣ ⎦ ⎣ ⎦

⎡ ⎤= = −⎣ ⎦

The origin of (30) is asymptotically stable if the system is output passive and zero-state observable (Khalil, 1996).

Proposition 2: The system (30) is output passive and

2 2, 0y yα α → as t → ∞ if 2yα is chosen as the output and the input δv is given by:

2 2 2p dδv δu K y K yα α= + + (31)

Proof: see Appendix A.

It remains now to verify the second condition (zero-state observability) by cancelling the input and the output in the error dynamics of the flexible part. After that we study the asymptotical stability of the origin and find the critical value of α2, if it exists.

Proposition 3: The error dynamics of the flexible part given in (29) can be represented as a feedback connection of a linear system and a non-linear memoryless element:

:

( , )

f f f fTf f

x A x B vG

y C x

v t y

⎧ = +⎪⎨

=⎪⎩= −ψ

(32)

where

( ) 10 1 44 2 2 42

00 1; ;

( ) ( )f fA Bt t M Mα α α β −

⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥− − −⎢ ⎥⎣ ⎦ ⎣ ⎦

2 2[ 0]TfC α β= − and ψ(t, y) is a non-linear element

(see Appendix A for the demonstration).

Proposition 4: If the non-linearity element ψ(t, y) is passive and the linear system is input strictly passive, then the origin

0fx = of (32) is asymptotically stable (Khalil, 1996). There exists α2 ≤ α2c that ensures the asymptotical stability of the origin.

The proof of Proposition 4 is given in Appendix A. The critical value of α2 is selected such that Af is

Hurwitz and the Nyquist diagram of Af lies in the right half-plan (Khalil, 1996).

4.4 Control and stability of the second link

After the stabilisation and control of the second link, we follow the same strategy to control and stabilise the first link. The dynamical equations of the first link are given in (12) and (14).

We define the new weighted output, velocity and acceleration as follows:

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

;

;

.

r f

r f

r f

y q q

y q q

y q q

α

α

α

α β

α β

α β

= +

= +

= +

Then,

1 1 1 1 1r fq y qα − α β= (33)

Substituting (33) and its derivatives into (14), the internal dynamics are:

1 31 32 2 3

1 1 1 1 f rd

f f f f

q M y M q C

K q D qα1= − − −

− − (34)

where 133 1 1 31( ) X M M Xα β −= − and α1 is chosen such that

331

1 31.

MM

αβ

≠

The control law is chosen by inserting (34) in (12); * * * * *

1 11 1 12 1 1 1 1 1 1d f f f fτ M u M y C K q D qα α= + + + + (35)

where

( )( )

( )( )( )

*11 11 13 1 1 11 31

*12 12 13 1 1 11 32

*1 1 13 1 1 11 3

*1 13 1 1 11 1

*1 13 1 1 11 1

;

;

;

;

.f f

f f

M M M M M

M M M M M

C C M M C

K M M K

D M M D

α β

α β

α β

α β

α β

= − −

= − −

= − −

= − −

= − −

*11M is nonzero using the Schur complement as in the

previous section. To track the desired new output trajectory, we choose

the following control law:

126 F. Raouf et al.

1 1 1 1 1 1d d pu y K y K yα α α α α α= + + (36)

where 1dyα is the desired acceleration, 1 1 1dy y yα α α= − is the tracking error of the new output, Kdα1 and Kpα1 are positive constants.

The error dynamics of the new output and the flexible coordinates are given by the following state space model:

1 1 2

2 1 1 1

3 1 4

4 1 1 1

d

f

f fd f

x y x

x y u δu

x q x

x q q q

α

α α

⎧ = =⎪

= − =⎪⎨

= =⎪⎪ = = −⎩

(37)

Let:

[ ]1 2

3 4 1 1

;

;

.

TTr

TTf f f

Tr f

x x x y y

x x x q q

x x x

α1 α1⎡ ⎤= = ⎣ ⎦

⎡ ⎤= =⎡ ⎤⎣ ⎦ ⎣ ⎦

⎡ ⎤= ⎣ ⎦

As already shown, the system is output passive and 0rx → as t → ∞ if we consider the following strictly passive control (Ortega, 1998; Saad, 2004).

1 1 1p dδu K y K y δvα α= − − + (38)

The error dynamics of the flexible part (34) can be expressed as:

1( , , )f f ux F t x δ= (39)

where 21 1 3 2 4

1( , , ) , , ,

( , , )f

f u f ff u

xF t x δ x x x x

f t x δ⎡ ⎤

= = =⎢ ⎥⎣ ⎦

and

1( , , )f uf t x δ is given in Appendix A. It remains to check the zero-state observability by

setting the input and output to zero and studying the asymptotic stability of the origin of the error dynamics.

Proposition 5: The asymptotical stability of the origin of (39) is equivalent to the asymptotical stability of the following system:

1 1

0 1f f

f fx x

K D⎡ ⎤

= ⎢ ⎥− −⎣ ⎦ (40)

where 1fK and 1fD are given in (34). The proof of proposition 5 is given in Appendix A.

The critical value of α1 is given when the eigenvalues of (40) change their signs from negative to positive.

5 Simulation results

The parameters of the two-flexible-link manipulator are described in Table 1. The controller’s parameters are chosen by using a trial and error method as follow:

1 1 2 2100 and 250.p d p dK K K K= = = =

Table 1 System parameters

Parameter Value

Hub inertia (Jhi) 0.1 kg.m2 Link length (Li) 0.5 m

Link linear density (ρ) 1 kg/m

Link rigidity (EIi) 10 N.m2 Link mass (mi) 0.5 kg Payload mass (mp) 0.1 kg Payload inertia (Jp) 0.0005 kg.m2

The workspace desired trajectory in triangular form is shown by Figure 5.

For the system parameters given in Table 1, the critical values of the weighted parameters of the second link and the first link are, respectively, α2c = 0.92 and α1c = 0.85 (Figures 6 and 7). A Nyquist diagram is shown for Af0 given in (53). Simulation results were obtained with α2 = 0.9 < α2c and α1 = 0.8 < α1c. Note that these values respect the conditions given in equations (25) and (34).

The workspace tracking is given in Figure 8. The virtual and joint space tracking are given in Figures 9 and 10, respectively. The joints’ tracking errors are given in Figure 11.

Figure 5 Workspace desired trajectory: (a) Yd vs. Xd and (b) xd(t), yd(t) (see online version for colours)

0.3 0.35 0.4

0.3

0.32

0.34

0.36

0.38

0.4

Xd(m)

Yd(

m)

0 1 2 3 4

0.3

0.32

0.34

0.36

0.38

0.4

time (s)

X(t)

, Y(t)

X(t)Y(t)

(a) (b)


Figure 6 Stability of second link, (a) Nyquist diagram, and (b) eigenvalues evolution (see online version for colours)

-2 0 2 4 6 8

x 10-3

-1 .5

-1

-0.5

0

0.5

1

1 .5x 10

-3

(a)

yq g

Real Axis

Imag

inar

y Ax

is

0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

2000

2500

Alpha2

Eig

en v

alue

First eigen valuesecond eign value

(a) (b)

Figure 7 Stability of first link: eigenvalues vs. α1 (see online version for colours)

0 0.2 0.4 0.6 0.8 1-500

0

500

1000

1500

2000

Alpha 1

Eig

n V

alue

first eign valuesecond eign value

Alpha1=0.85

Figure 8 Tracking of X-Y trajectory (see online version for colours)

0.3 0.32 0.34 0.36 0.38 0.4

0.3

0.32

0.34

0.36

0.38

0.4

X

Y

XYdXY

128 F. Raouf et al.

Figure 9 Generalised coordinates in virtual space, (a)–(b) position of virtual joints, (c)–(d) velocity of virtual joints, (e)–(f) acceleration of virtual joints (see online version for colours)

0 1 2 3 4-0.5

-0.4

-0.3

-0.2

-0.1

time (s)

Q1(

rad)

0 1 2 3 42

2.1

2.2

2.3

time (s)

Q2

(rad)

(a) (b)

0 1 2 3 4-2

-1

0

1

2

time (s)( )

dQ1

(rad/

s)

0 1 2 3 4-2

-1

0

1

2

time (s)dQ

2 (ra

d/s)

(c) (d)

0 1 2 3 4-4

-2

0

2

time (s)

ddQ

1 (ra

d/s2 )

0 1 2 3 4-4

-2

0

2

4

time (s)(f)

ddQ

2 (ra

d/s2 )

(e) (f)

Figure 10 Generalised coordinates in joint space, (a)–(b) flexible coordinates (c)–(d) rigid coordinates (see online version for colours)

0 1 2 3 4-0.5

-0.4

-0.3

-0.2

-0.1

time (s)

qr1

0 1 2 3 42

2.1

2.2

2.3

time (s)

qr2

(a) (b)

(c) (d)


Figure 11 (a)–(b) Errors of joints 1 and 2, (c)–(d) errors of flexible coordinates 1, 2 (e)–(f) tracking errors of workspace trajectories x(t) and y(t) (see online version for colours)

0 1 2 3 4-5

0

5x 10-3

time (s)

er1

0 1 2 3 4-10

-5

0

5x 10-3

time (s)

er2

(a) (b)

(c) (d)

(e) (f)

According to the simulation results, we can conclude that the trajectory defined in the workspace was well transformed to the joint space by using the virtual space and the quasi-static approach. This accurate transformation is shown in Figure 9, which represents the position, velocity and acceleration in virtual space, and Figure 10, which represents the rigid and flexible coordinates in joint space. The control strategy was effective to ensure the tracking of the joints’ trajectories and reduce vibrations at the extremity. Good tracking in the joint space is shown in Figures 11(a) to 11(d), which represent tracking errors in the joint space. In the workspace, good tracking is shown in Figure 8 and the tracking errors are given in Figures 11(e) and 11(f). The new selected outputs are near links’ extremities. This explains the satisfactory results obtained using this approach. This control strategy has many advantages. Indeed, it is easier to study one arm at each step than to study the stability of the whole system. In addition, this strategy takes a hierarchical form.

6 Conclusions and future work

In this paper, non-linear control laws and stability analysis have been presented to stabilise the tracking errors of a two-flexible-link manipulator. A desired trajectory defined in the workspace has been transformed to the joint space using a virtual space transformation and the quasi-static approach. A desired trajectory (triangle) chosen in the workspace has been successfully transformed to the joint space using this transformation procedure. The control strategy used in this paper consists of stabilising the system starting with the last flexible link and proceeding backward until the first link. In each step, the output redefinition technique was used to select the nearest point to the extremity. This strategy shows good tracking trajectory in the workspace. Asymptotical stability of the tracking errors has been guaranteed using the passivity theory. In the present work, we were interested by the local stability of two-flexible-link manipulators. The global stability of the

130 F. Raouf et al.

same control strategy is being investigated and will be considered in future work.

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada under the Discovery Grants Programme.

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Appendix A

Proof of proposition 1: The new generalised coordinates

2 2 2[ ]Tfq y qα= are linked with the former generalised

coordinates 2 2 2[ ]Tr fq q q= by a non-singular

transformation matrix T:

2 2 2 22 2

2

r f

f

q qq Tq

qα β+⎡ ⎤

= =⎢ ⎥⎣ ⎦

(41)

where 2 21.

0 1T

α β⎡ ⎤= ⎢ ⎥⎣ ⎦

Then we can write:

( ) ( ) ( )* 1 1 12 2 2 2 2 2M q M q T M T q T− − −= = (42)

where ( ) 22 24 2 2 22*2 2

42 44 2 2 42

M M MM q

M M Mα βα β

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

and

( ) 22 242 2

42 44

M MM q

M M⎡ ⎤

= ⎢ ⎥⎣ ⎦

is represented in Section 2. Since

M2(q2) is a non-singular positive definite matrix and (M44 – α2β2M42) is a nonzero element, then the Schur complement *

22M of M44 – α2β2M42 is a nonzero element (Zhang, 2005).

Proof of proposition 2: Let δu2 be the input and 2yα the output. We consider the following energy function:

( ) 21

12 rV x x= (43)

The time derivative of V is:

( ) 1 1 2 2r r uV x x x δ yα= =

The system (30) is then passive. This passivity is preserved with the following control law (Ortega, 1998):

2 2u pδ K y δvα= − + (44)

where δv is a new control input. With δv as the input and

2yα as the output, the system is output strictly passive. The control law is given by:

2 2 2u p pδ K y K y δvα α= − − + (45)

where Kp and Kd > 0. Then we can conclude that

2 2, 0t

y yα α→∞→ (Ortega, 1998).

Proof of proposition 3: The error dynamics of the flexible part [(29) and (30)] is written as:

[ ]

1 2

2 2 1 2 2

144 2 2 42 1 ( , )

f f

f f f f f

x x

x K x D x

M M t yα β −

⎧ =⎪⎪ = − −⎨⎪

− −⎪⎩ ψ

(46)

where 2 2 1 2 2 2[ 0] , , Tf f f f f f fy C x x x q x qα β= = − = =

and 1 41 41 1 42 2 42 2

4 4

( , ) ( ) ( )( ).

d rd d d

d

t y M M q M y M uC C

α α= − − − −

− −

ψ

Using the system parameters (Appendix B), ψ1(t, y) can be written as follows:

( )( ) ( )

( ) ( )( ) ( )

21 501 1 153 11 1

2 2

152 2 2

504 11 1 1 2 2 2 2

( , )

sin sin

cos cos

cos cos

rd fd

T Td d

T Td d

fd rd rd rd r r

t y h q b t q

C q C q y

b C q C q y

h t q q q q q q

α α

α α

= +

⎡ ⎤− −⎣ ⎦⎡ ⎤+ − −⎣ ⎦

⎡ ⎤+ −⎣ ⎦

ψ

and hijk, bijk and tij are constants and are given in Appendix B.

Using Taylor series, the last element can be written as:

( ) ( )( )

( )

2 2 2 2

2 2 2 2 2

2 2 2 2

cos cos

sin

cos

rd rd r r

rd rd f

rd f

q q q q

q q q

q q

α β

α β

−

=

+

Then, the memoryless time-varying non-linearity can be expressed as follows:

( )( )

1 2 2 2 2 2

2 2 2 2

( , ) ( )sin

( ) cos ( , )rd rd f

rd f

t y q c t q q

c t q q t y

α β

α β

=

+ +

ψ

ψ (47)

where 504 11 1 1( ) fd dc t h t q Q= and the new non-linearity element is given by:

( ) ( )( ) ( )

( ) ( )

1 2 2

2 2

504 11 1 1 2 2 2 2

( , ) ( ) sin sin

( ) cos cos

cos cos

T Td d

T Td d

fd rd rd rd r r

t y a t C q C q y

b t C q C q y

h t q q q q q q

α α

α α

⎡ ⎤= − −⎣ ⎦⎡ ⎤+ − −⎣ ⎦

⎡ ⎤+ −⎣ ⎦

ψ

(48)

2501 1 153 11 1 152( ) ; ( ) .rd fda t h q b t q b t b= + =

132 F. Raouf et al.

Thus, we can rewrite the error dynamics of the flexible part (46) as follows:

( ) ( , )f f f fx A t x B t y= − ψ (49)

where

( )( )

( )( )( )

( )( )

0 1

144 2 2 42

10 44 2 2 42

2 2 2 2 2

11 44 2 2 42

2 2 2 2 2

0 1( ) ;

( ) ( )

0

( )

( )sin

( )

( ) cos

f

f

f rd rd

f rd rd

A ta t a t

BM M

a t M M

K q c t q

a t M M

D q c t q

α β

α β

α β

α β

α β

−

−

−

⎡ ⎤= ⎢ ⎥− −⎣ ⎦⎡ ⎤

= ⎢ ⎥−⎢ ⎥⎣ ⎦

= −

−

= −

−

The error dynamics can be represented as a feedback connection of a linear system and a non-linear memoryless system as follows:

:

( , )

f f f fTf f

x A x B vG

y C x

v t y

⎧ = +⎪⎨

=⎪⎩= −ψ

(50)

Proof of proposition 4: The origin of (32) is asymptotically stable if the non-linear element ψ(t, y) is passive and the linear system is input strictly passive (Khalil, 1996). For the non-linear element, we can write:

( ) ( )( )

( ) ( )( )

2 2

2

2 2

2

sin sin

2cos / 2 sin ( / 2)

cos cos

2sin / 2 sin ( / 2)

T Td d

Td

T Td d

Td

C q C q y

C q y y

C q C q y

C q y y

α α

α

α α

α

− −

= −

− −

= − −

Then, we can rewrite the memoryless non-linearity as follows:

( )( )

2

2

( , ) 2 ( )cos / 2 sin ( / 2)

2 ( )sin / 2 sin ( / 2)

Td

Td

t y a t C q y y

b t C q y y

α

α

= −

− −

ψ (51)

where a(t), b(t), and c(t) are given in (47) and (48) and are bounded. Then we can write: 152( ) ; ( )a a t a b t b≤ ≤ = is constant and the non-linear term as:

2 2( , )ky y t y ky− ≤ ≤ψ (52)

where k b a= + and .k b a= +

We can conclude that the non-linear element ψ(t, y) is passive (Khalil, 1996).

For the linear system, we can write:

0( ) Δ( )f fA t A t= + (53)

where

( )

( )

02 2

0 1

2 2 2 20

551 2 2 251

2 2 21

551 2 2 251

0 1

0 0Δ( )

( ) ( )

( )sin( )

( ) cos( )

ff f

rd rd

rd

AK D

tδ t δ t

q c t qδ t

b bc t q

δ tb b

α βα β

α βα β

⎡ ⎤= ⎢ ⎥− −⎣ ⎦⎡ ⎤

= ⎢ ⎥⎣ ⎦

=−

=−

If Af0 is asymptotically stable, then the uniform asymptotical stability is preserved for any Δ( ) 0

tt

→∞→ (Coppel, 1965).

However, c(t) → 0 as t → ∞. Then, Δ(t) → 0 as t → ∞. Finally, the passivity of the linear system depends only on Af0. This can be easily done by an appropriate choice of α2.

Proof of proposition 5: The error dynamic of the flexible part are represented by:

( ), ,f fx F t x u= (54)

where ( ) ( )2

1; , ,, ,

ff

f

xu δu F t x u

f t x u

⎡ ⎤= = ⎢ ⎥

⎢ ⎥⎣ ⎦

( ) 21 1 2 1 3 1 0, , ( ) ( ) ( ) ( )f f f ff t x u a t q a t q a t q c t= + + + (55)

11 33 1 1 31

21 234 2 31 2 319 2 2 31

( )

f rd rd

a t M M

K b s t q h q c t

α β −= − −⎡ ⎤⎣ ⎦⎡ ⎤+ +⎣ ⎦

( )

12 33 1 1 31

1 307 2 2 311 2 2

317 2 2 21 2

1 1 315 21 2 2 2 302 2 2

1 1 301 2 1

( )

2

f rd fd

rd fd

fd rd rd

rd

a t M M

D h q s h s q

h q c t q

h t q q c h s q

h s q

α β

α β

α β

−= − −⎡ ⎤⎣ ⎦

⎡ + +⎣+

− +

+ ⎤⎦

13 33 1 1 31 1 1 301 2 1( ) 2 rda t M M h s qα β α β−

= − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

1 20 33 1 1 31 301 2 1( ) rdc t M M h s qα β −

= − −⎡ ⎤⎣ ⎦

To facilitate the stability analysis, equation (54) can be expressed as (Khalil, 1996):

( ) ( ), ,f f fx F t x G t x= + (56)

where


( )

( )

2

1 1 2 2 0

23 2

, ;( ) ( ) ( )

0,

( )

ff

f f

ff

xF t x

a t x a t x c t

G t xa t x

⎡ ⎤= ⎢ ⎥+ +⎣ ⎦⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

It is clear that: G(t, 0) = 0 and || ( , ) || || ||f fG t x γ x≤ where γ = sup(a3(t)) a non-negative constant. Then, the asymptotical stability of the origin of (56) is equivalent to:

( ) ( ), ( )f f fx F t x A t x C t= = + (57)

where

1 2

0

0 1( ) ;

( ) ( )

0 0( ) as

( ) 0

A ta t a t

C t tc t

⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤

= → →∞⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

The asymptotical stability of the origin of (57) depends only on the subsystem:

( )f fx A t x= (58)

Now, we analyse the asymptotical stability of the previous subsystem to choose the critical value of α1. A(t) can be written as:

0( ) ( ) Δ( )A t A t t= + (59)

where

010 20

1 2

10 2 10 1

1 21 33 1 1 31 234 2 31 2 319 2 2 31

12 33 1 1 31 307 2 2 311 2 2

317 2 2 2

0 1( ) ;

0 0Δ( ) ;

( ) ( )

; ;

( )

( )

f f

rd rd

rd fd

rd

A ta a

tδ t δ t

a K a D

δ t M M b s t q h q c t

δ t M M h q s h s q

h q c t

α β

α β

−

−

⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥⎣ ⎦

= − = −

⎡ ⎤= − − +⎡ ⎤⎣ ⎦ ⎣ ⎦

⎡= − − +⎡ ⎤⎣ ⎦ ⎣+

( )1 2

1 1 315 21 2 2 2 302 2 2

1 1 301 2 1

2

fd

fd rd rd

rd

q

h t q q c h s q

h s q

α β

α β

− +

+ ⎤⎦

We know that 1 2 2 2 2, , , , 0rd rd rd fd fdq q q q q → as t → ∞, then δ1(t), δ2(t) → 0, (Δ(t) → 0).

Then the asymptotical stability of the origin of (59) is equivalent to the asymptotical stability of the origin of the following system:

1 1

0 1f f

f fx x

K D⎡ ⎤

= ⎢ ⎥− −⎣ ⎦ (60)

where 1fK and 1fD are given in (34). The critical value of α1 is given when the eigenvalues of (60) change their signs from negative to positive.

Appendix B

Dynamical model of two flexible links with one mode

The following dynamical model is deduced from De Luca and Siciliano (1990, 1991) by considering only one flexible mode to alleviate the computation.

( )( )( )( )

111 12 13 14

221 22 23 24

131 32 33 34

41 42 43 44 2

1

2

1 1 1 13

2 2 2 24

1

2

, 0 0, 0 0

,

,

00

r

r

f

f

f f f f

f f f f

qM M M MqM M M MqM M M M

M M M M q

C q q

C q qK q D qC q qK q K qC q q

ττ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

where

( )11 111 112 2 113 1 114 2 2M b b c b t b t s= + + +

( )12 121 122 2 123 1 124 2 2M b b c b t b t s= + + +

13 131 132 2 133 2 2M b b c b t s= + +

15 151 152 2 153 1 2M b b c b t s= + +

22 221M b=

( )23 231 232 2 233 2 234 3 2M b b c b t b t s= + + +

25 251M b=

33 331 332 2 333 2 2M b b c b t s= + +

35 351 352 2 353 3 2M b b c b t s= + +

55 551M b=

1 11 1 2 21 1 3 31 1; ; f f ft t q t t q t t q= = =

The bijk coefficients are given as follow:

( )2

111 1 1 2 2 1

2 2 22 2 1 1 2

n o n n

o p p

b J J J m l

J m l J m l l

= + + +

+ + + + +

134 F. Raouf et al.

( )112 2 2 2 12 pb m d m l l= +

( )113 2 2 22 pb m d m l= +

114 12b l= −

2121 2 2 2n o p pb J J J m l= + + +

( )122 2 2 2 12 pb m d m l l= +

123 2 2 2pb m d m l= +

124 1b l= −

( )( )

2131 11 2 2 2 1,

2 2 1 1,

n o p p e

n p e

b w J J J m l

m m m l

φ

φ

′= + + + +

+ + +

( )( )132 2 2 2 1, 1 1,p e eb m d m l lφ φ′= + +

( )133 1, 1 1,e eb lφ φ′= − +

151 2 2, 2 2,p e p eb w J m lφ φ′= + +

( )152 2 2, 1p eb v m lφ= +

153 2 2,p eb v m φ= +

2221 2 2 2n o p pb J J J m l= + + +

( )2131 2 2 2 1,n o p p eb J J J m l φ′= + + +

( )232 2 2 2 1,p eb m d m l φ= +

233 1,eb φ= −

( )234 2 2 2 1,p eb m d m l φ= +

251 2 2, 2 2,p e p eb w J m lφ φ′= + +

331 1b m=

( )332 2 2 2 1, 1,2 p e eb m d m l φ φ′= +

333 1, 1,2 e eb φ φ′= −

( )351 2 2, 2 2, 1,p e p e eb w J m lφ φ φ′ ′= + +

( )352 2 2, 1,p e eb v m φ φ= +

( )353 2 2, 1,p e eb v m φ φ= − +

551 2b m=

11 1, 1 1,e et lφ − φ′=

21 2 2,p et v m φ= −

31 1,et φ′=

The Coriolis and centrifugal forces vector coefficients are:

( )( )

( )( )

1 101 2 102 1 104 2 1

106 2 107 1 109 2 1

111 2 1 2

115 1 116 2 117 2 1

119 1 120 2 121 1 2 2 2

f f

f f

f f

f

f

C h θ h q h q θ

h θ h q h q θ

h q q s

h θ h θ h q t

h θ h θ h q t θ c

⎡= + +⎣

+ + +

⎤+ ⎦⎡+ + +⎣

⎤+ + + ⎦

( )( ){( )

}

2 201 1 202 1 1 2

204 1 205 2 1

207 1 208 1 2 1

211 1 2 214 2 3 1 2

+

f

f

f

f f f

C h θ h q θ s

h θ h q t

h θ h q t θ

h q t h q t q c

= +

⎡+ +⎣

⎤+ + ⎦⎡ ⎤+⎣ ⎦

( )( )

( )( )

3 301 1 302 2 304 2 1

306 2 307 1 309 2 2

311 2 11 2

315 1 316 2 317 1 2

319 2 320 2 3 2 2

f

f f

f f

f

f

C h θ h θ h q θ

h θ h q h q θ

h q q s

h θ h θ h q t

h θ h q t θ c

⎡= + +⎣

+ + +

⎤+ ⎦⎡+ + +⎣

⎤+ + ⎦

( )4 501 1 502 1 1 2

504 1 1 505 1 3 2 2

f

f

C h θ h q θ c

h t θ h q t θ c

= +

⎡ ⎤+ +⎣ ⎦

The hijk coefficients are given as follow:

( )101 2 2 2 12 ph m d m l l= − +

( )( )102 2 2 2 1, 1 1,2 p e eh m d m l lφ − φ′= +

( )104 2 2, 12 p eh v m lφ= − +

( )106 2 2 2 1ph m d m l l= − +

( )( )107 2 2 2 1 1,p eh m d m l l φ′= − +

( )109 2 2, 12 p eh v m lφ= − +

( )111 2 2, 1 1,2 p e eh v m lφ φ′= − +

( )115 2 2 22 ph m d m l= +

116 2 2 2ph m d m l= +

( )117 21 2,p eh v m φ= − +


119 12h l= −

20 1h l= −

( )121 1, 1 1,e eh lφ − φ′= −

( )201 2 2 2 1ph m d m l l= +

( )202 2 2 2 1,2 p eh m d m l φ= +

( )204 2 2 2ph m d m l= − +

( )205 2 2,p eh v m φ= − +

207 1h l=

208 1, 1 1,e eh lφ φ′= +

212 1, 1,e eh φ φ′=

( )214 2 2, 1,p e eh v m φ φ= +

( )( )301 2 2 2 1, 1 1,p e eh m d m l lφ − φ′= − +

( )302 2 2 2 1,2 p eh m d m l φ= − +

( )304 2 2, 1,2 p e eh v m φ φ= − +

( )306 2 2 2 1,p eh m d m l φ= − +

( )307 2 2 2 1, 1,2 p e eh m d m l φ φ′= − +

( )309 2 2, 1,2 p e eh v m φ φ= − +

( )311 2 2, 1, 1,2 p e e eh v m φ φ φ′= − +

( )315 1, 1 1,e eh lφ − φ′= −

316 1,eh φ= −

317 1, 1,2 e eh φ φ′= −

( )319 2 2 2 1,p eh m d m l φ= − +

( )320 2 21, 1,2 p e eh v m φ φ= − +

( )501 2 2, 1p eh v m lφ= +

( )502 2 2, 1,2 p e eh v m φ φ= +

504 2 2,p eh v m φ= +

( )505 2 2, 1,p e eh v m φ φ= − +

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Workspace trajectory tracking control for two-flexible-link manipulator through output redefinition