# Workspace Tracking Control of Two-Flexible-Link Manipulator Using Distributed Control Strategy

Authors: Fareh Raouf; Saad Mohamad; Saad Maarouf
Journal: Journal of Control Science and Engineering
 (2013)

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## Abstract

A distributed nonlinear control strategy for two-flexible-link manipulators is presented to track a desired trajectory in the robot’s workspace. The inverse dynamics problem is solved by transforming the desired trajectory from the workspace to the joint space using an intermediate space, called virtual space, and then using the quasi-static approach. To solve the nonminimum phase problem, an output redefinition technique is used. This output consists of the motor’s angle augmented with a weighted value of the link’s extremity. The distributed control strategy consists in controlling the last link by assuming that the first link is stable and follows its desired trajectories. The control law is developed to stabilize the error dynamics and to guarantee bounded internal dynamics such that the new output is as close as possible to the tip. The weighted parameter defining the noncollocated output is then selected. The same procedure is applied to control and stabilize the first link. The asymptotical stability is proved using Lyapunov theory. This algorithm is applied to a two-flexible-link manipulator in the horizontal plane, and simulations showed a good tracking of the desired trajectory in the workspace.


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## Body

## 1. Introduction

Many control strategies for manipulators have been the focus of several studies in recent years. The robot manipulators consist of a sequence of links and joints in various combinations. In industrial applications, most of the existing manipulators use rigid links and joints and are known as rigid manipulators. Rigid manipulators are generally slow, extremely rigid, and massive, and the useful load is very low compared to their weight. To improve the performance of the robot manipulators, their links must be lighter, and therefore they become more flexible. Flexible manipulators present more advantages when compared to rigid manipulators: they are faster and less massive and consume less energy. Some flexible manipulators are used in different areas, for example, the aerospace applications [1] and medical applications [2].For flexible manipulators, the problem of workspace tracking trajectory is less covered so far than that of joint space tracking. There are few solutions to the workspace tracking problem, particularly for manipulators with many flexible links. The workspace tracking trajectory is very important since most of the tasks are defined in the operational space, such as painting, welding, and assembly. The flexible link manipulators are a nonminimum phase system when controlling the position of the end effector [3]. Unlike rigid manipulators, the inverse kinematics of flexible manipulators is not sufficient to transform the desired trajectory from workspace to joint space because they are linked by kinematics and dynamics relationships. To solve this problem, many studies focused on the output redefinition technique [4–7]. This technique consists in selecting a new output as close as possible to the tip such that internal dynamics become bounded. In [5], this approach was used for a class of manipulators where the last link is flexible. An intermediate space between the joint space and the workspace called virtual space is used to transform the desired trajectory from the workspace to the joint space. The virtual space is linked with the workspace by a simple kinematics relation as in rigid manipulators. In [4], the output redefinition technique was used for one flexible link manipulator. This output consists of the motor’s angle augmented with a weighted value of the angle of links extremity.The nonlinear dynamics of flexible link manipulators combined with their under-actuated nature (the deflection variables are not actuated) present a challenging control problem. Multi-flexible-link manipulators can be controlled as one MIMO system so a single controller is used for all joints and links or as a set of interconnected subsystems so each pair of joint and link is controlled by its own controller. For the first case, many control schemes were used. Several studies used linearization around a nominal configuration of flexible manipulators model [8, 9]. Some nonlinear effects such as the variation of inertia matrix around an operating point were taken into account in the control design methodology. In [10], a control method based on the input-output linearization was developed to track a desired trajectory in the workspace for a class of flexible link manipulators when the last link is flexible. To select an output near the tip that guarantees the stability of the zero dynamics, authors used the output redefinition technique. A robust adaptive controller was developed in [11] for a class of flexible link manipulators where the last link is flexible. This controller is based on feedback linearization and uses the virtual joint space that is kinematically related to workspace. For trajectory tracking control problems, many other techniques were covered in the literature such as singular perturbation technique [12], sliding mode control [13], inversion-based nonlinear control [14], and so forth. All the above-mentioned control methods use a single controller for all joints and links as one MIMO system. Unfortunately, due to the complexity of the control structures, the real-time implementation in industrial applications is not easy [15]. A solution to this problem can be followed by considering the dynamics of the robot manipulators as interconnected subsystems (joint and link). Many control schemes consider this configuration. In [16], a decentralized control method for flexible link manipulators was used. The authors used a simple proportional derivative (PD) controller for the joints and a linear quadratic regulator (LQR) with output feedback for each link. In [17], a PD controller was used for each joint and the measurement of linear velocity of the tip position was used for controlling each link. A distributed control strategy was introduced in [18, 19] for rigid link manipulators. The distributed control strategy consists in controlling one joint at a time starting with the last joint and going backward until the first one. Lyapunov theory was used to prove the global stability of the error dynamics, and this controller was successfully applied on a 7 DOF manipulator. This control strategy was modified in [20] to take into account links’ flexibility. A good tracking performance in the workspace of a two-flexible-link manipulator was obtained. However, the control law, based on the feedback linearization approach, ensures only local stability. In this paper, a nonlinear distributed control strategy is presented for a two-flexible-link manipulator that ensures global stability. This strategy consists of controlling the second joint and link by assuming that the remaining joint and link are stable and follow their desired trajectories. Then, going backward, the first joint and link are controlled by following the same strategy. For the inversion dynamics problem, the virtual space and the quasi-static approach are used. The output redefinition technique is used to obtain the nearest point to the tip that ensures bounded internal dynamics. The Lyapunov approach is used to analyze stability of the tracking errors.The paper is organized as follows. Section2 presents the modeling of the two-flexible-link manipulator and presents its main properties that will be used in the control law design. Section 3 presents the distributed control strategy. The stability analysis is given in Section 4. The control method is applied on a two-flexible-link manipulator, and the simulations are given in Section 5. Finally, a conclusion is given in Section 6.


## 2. Modeling

### 2.1. System Description

Figure1 shows a two-flexible-link manipulator. This system moves in the horizontal plane and consists of two motors that generate two torques, two flexible links with mass mi, length Li, linear density ρi, and rigidity EIi(i=1,2), and a payload that 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. The flexible links are supposed uniform and are modeled as Euler-Bernoulli beams, and the deformations are assumed to be small.Figure 1
Two-flexible-link manipulator.Using Lagrange equations, the dynamical model of ann DOF flexible manipulator is given by [21]
(1)M(q)q¨+H(q,q˙)+Dq˙+Kq=τ,
where M is the inertia and mass matrix, H(q,q˙) is the Coriolis and centrifugal forces vector, D is the friction matrix, and K is the rigidity matrix. q represents the vector of the generalized coordinates and τ is the vector of the applied torques. For the n rigid coordinates and n flexible links, the deformation of the ith flexible link is given by the following equation:
(2)vi(x,t)=∑j=1zi∅ij(x)qfij(t),i=1,…,n,
where qfij is the jth generalized flexible coordinate, ∅ij(x) is its jth shape function, and zi is the number of the retained flexible modes of the ith flexible link. Note that the total number of the flexible modes is z=∑i=1nzi and the number of the rigid modes is n. The dynamical model (1) can be written as follows:
(3)[MrrMrfMrfTMff][qr¨qf¨]+[Hr(q,q˙)Hf(q,q˙)]+[000Dff][qr˙qf˙]+[000Kff][qrqf]=[I0]τ,
where Mrr∈ℜn×n and Mff∈ℜz×z are mass and inertia matrices for the rigid and flexible part, respectively. Mrf∈ℜn×z is a coupled element. Kff is the stifness diagonal matrix, and Dff is the damping diagonal matrix of the flexible part. The subscripts r and f denote the rigid and flexible modes.


### 2.2. Proprieties and Problem Formulation

The dynamical model of the flexible link manipulators has the following properties that will be used in the control law development:(P1)
M,Mrr,Mff,Dff,  andKff are symmetric positive definite matrices [21].(P2)
From (P1), we can deduce that the diagonal elements ofM are positive [22]:
(4)Mii(q)>0;fori=1,…,n.(P3)
There exists a matrixC(q,q˙) such that
(5)H(q,q˙)=C(q,q˙)q˙.(P4)
The inertia-mass matrixM(q) and the Coriolis matrix C(q,q˙) satisfy the following skew-symmetric property:
(6)XT(M˙(q,q˙)-2C(q,q˙))X=0,∀X∈ℛn+z.(P5)
The propriety (P4) is preserved for the diagonal elements ofM(q) and C(q,q˙). Then, we can write
(7)M˙ii(q)-2Cii(q,q˙)=0,fori=1,…,n.The new noncollocated output of theith link defined by the motor’s angle augmented with a weighted value αi of the endpoint angular position is given as follows:
(8)yi=qri+αiβiTqfi,0≤αi<1,
where βiT=(1/Li)[∅i1(Li),…,∅izi(Li)], with Li being the length of the ith flexible link, and ∅ij is the shape function of link i and mode j. In this paper, we consider only the first flexible mode of each link (i.e., zi=1). Thus, we have  z=n.In the distributed control strategy, when controlling theith link, we assume that the other links follow their desired trajectories. For the ith link, a new generalized coordinate Qi is defined such that the ith coordinate is the controlled one and the other links follow their desired trajectories such that(9)Qi=[qrd1⋯qrd(i-1)qriqrd(i+1)⋯qrdnqfd1⋯qfd(i-1)qfiqfd(i+1)⋯qfdn]T,where (qri,qfi) are the rigid and flexible modes associated with the ith motor and link, respectively. The objective of this work is to track a desired trajectory defined in the workspace of a two-flexible-link manipulator using distributed control. Three steps are followed to achieve this objective.Step 1.
Transform the desired trajectory from the workspace to the joint space using inverse kinematics and quasi-static approach.Step 2.
Develop the control law for the second and first links to stabilize the error dynamics and to guarantee bounded internal dynamics such that the output is as close as possible to the tip.Step 3.
Study the global stability.


## 2.1. System Description

Figure1 shows a two-flexible-link manipulator. This system moves in the horizontal plane and consists of two motors that generate two torques, two flexible links with mass mi, length Li, linear density ρi, and rigidity EIi(i=1,2), and a payload that 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. The flexible links are supposed uniform and are modeled as Euler-Bernoulli beams, and the deformations are assumed to be small.Figure 1
Two-flexible-link manipulator.Using Lagrange equations, the dynamical model of ann DOF flexible manipulator is given by [21]
(1)M(q)q¨+H(q,q˙)+Dq˙+Kq=τ,
where M is the inertia and mass matrix, H(q,q˙) is the Coriolis and centrifugal forces vector, D is the friction matrix, and K is the rigidity matrix. q represents the vector of the generalized coordinates and τ is the vector of the applied torques. For the n rigid coordinates and n flexible links, the deformation of the ith flexible link is given by the following equation:
(2)vi(x,t)=∑j=1zi∅ij(x)qfij(t),i=1,…,n,
where qfij is the jth generalized flexible coordinate, ∅ij(x) is its jth shape function, and zi is the number of the retained flexible modes of the ith flexible link. Note that the total number of the flexible modes is z=∑i=1nzi and the number of the rigid modes is n. The dynamical model (1) can be written as follows:
(3)[MrrMrfMrfTMff][qr¨qf¨]+[Hr(q,q˙)Hf(q,q˙)]+[000Dff][qr˙qf˙]+[000Kff][qrqf]=[I0]τ,
where Mrr∈ℜn×n and Mff∈ℜz×z are mass and inertia matrices for the rigid and flexible part, respectively. Mrf∈ℜn×z is a coupled element. Kff is the stifness diagonal matrix, and Dff is the damping diagonal matrix of the flexible part. The subscripts r and f denote the rigid and flexible modes.


## 2.2. Proprieties and Problem Formulation

The dynamical model of the flexible link manipulators has the following properties that will be used in the control law development:(P1)
M,Mrr,Mff,Dff,  andKff are symmetric positive definite matrices [21].(P2)
From (P1), we can deduce that the diagonal elements ofM are positive [22]:
(4)Mii(q)>0;fori=1,…,n.(P3)
There exists a matrixC(q,q˙) such that
(5)H(q,q˙)=C(q,q˙)q˙.(P4)
The inertia-mass matrixM(q) and the Coriolis matrix C(q,q˙) satisfy the following skew-symmetric property:
(6)XT(M˙(q,q˙)-2C(q,q˙))X=0,∀X∈ℛn+z.(P5)
The propriety (P4) is preserved for the diagonal elements ofM(q) and C(q,q˙). Then, we can write
(7)M˙ii(q)-2Cii(q,q˙)=0,fori=1,…,n.The new noncollocated output of theith link defined by the motor’s angle augmented with a weighted value αi of the endpoint angular position is given as follows:
(8)yi=qri+αiβiTqfi,0≤αi<1,
where βiT=(1/Li)[∅i1(Li),…,∅izi(Li)], with Li being the length of the ith flexible link, and ∅ij is the shape function of link i and mode j. In this paper, we consider only the first flexible mode of each link (i.e., zi=1). Thus, we have  z=n.In the distributed control strategy, when controlling theith link, we assume that the other links follow their desired trajectories. For the ith link, a new generalized coordinate Qi is defined such that the ith coordinate is the controlled one and the other links follow their desired trajectories such that(9)Qi=[qrd1⋯qrd(i-1)qriqrd(i+1)⋯qrdnqfd1⋯qfd(i-1)qfiqfd(i+1)⋯qfdn]T,where (qri,qfi) are the rigid and flexible modes associated with the ith motor and link, respectively. The objective of this work is to track a desired trajectory defined in the workspace of a two-flexible-link manipulator using distributed control. Three steps are followed to achieve this objective.Step 1.
Transform the desired trajectory from the workspace to the joint space using inverse kinematics and quasi-static approach.Step 2.
Develop the control law for the second and first links to stabilize the error dynamics and to guarantee bounded internal dynamics such that the output is as close as possible to the tip.Step 3.
Study the global stability.


## 3. Distributed Control Strategy

### 3.1. Inverse Dynamics

To achieve the objective of workspace tracking trajectories, we need to transform the desired trajectories from the workspace to the joint space. The flexible manipulator is a nonminimum phase system when the end effector is used as the output. In this case, kinematic and dynamic relationships link the workspace and the joint space. To overcome this problem, an intermediate space called virtual space can be used. Then the desired workspace trajectory is transformed to the virtual space using an inverse kinematics relation as in rigid manipulators. To transform the desired trajectories from the virtual space to the joint pace, the quasi-static approach can be used to solve a nonlinear equation for the flexible part.Using inverse kinematics as in rigid manipulators, the generalized coordinates in virtual space can be easily found. The deformation is assumed to be small. According to Figure2, we can write
(10)qvi-qrdi=arctan(vLiLi)≈vLiLi.
In the virtual space, the generalized coordinate is given as follows:
(11)qvi=qrdi+βiqfdi,
where βi=(∅i(Li)/Li) and ∅i is the shape function of link i.Figure 2
Virtual space.Using the Jacobian matrix as in rigid manipulators, the velocity and acceleration in the virtual space can be deduced. Then, for a two-DOF manipulator, the inverse kinematics is given by the following equation:(12)qv=[qv1qv2]=[arctan2(y,x)-arctan2(L2s2,L1+L2c2)arctan2(s2,c2)],q˙v=[q˙v1q˙v2]=J-1(qv)[x˙y˙],q¨v=[q¨v1q¨v2]=J-1(qv)[x¨y¨]-J-1(qv)J˙(qv)q˙v,
where x and y are the workspace desired positions; c2=(x2+y2-L12-L22)/2L1L2; s2=±1-c22 and J(qv)=[-L1s1-L2s12-L2s12L1c1+L2c12L2c12] is the Jacobian matrix. s1=sin(qv1); c1=cos(qv1); s12=sin(qv1+qv2); c12=cos(qv1+qv2); L1 is length of link 1 and L2 is length of link 2. The rigid and flexible coordinates are derived via quasi-static approach and using the generalized coordinates in the virtual space.The dynamical model (3) can be written as a function of the desired coordinates as follows:
(13)Mrr(qd)q¨rd+Mrf(qd)q¨fd+Hr(qd,q˙d)=τ,(14)MrfT(qd)q¨rd+Mff(qd)q¨fd+Hf(qd,q˙d)+Dffq˙fd+Kffqfd=0.
The generalized flexible and rigid coordinates in the joint space are deduced using the quasi-static approach. In the first step, the nonlinear equation of the flexible part (14) is solved to find the generalized flexible coordinates. The quasi-static approach neglects the desired velocity and acceleration of the flexible coordinates (q˙fd=q¨fd=0). Then, the equation of the flexible part (14) can be written as follows:
(15)MrfT(qd)q¨rd+Hf(qd,q˙d)+Kffqfd=0,
where
(16)qrdi=qvi-βiTqfdi;q˙rdi=q˙vi;q¨rdi=q¨vi.
Using (16), the generalized flexible coordinates qfd are found by solving (15). Then, the generalized rigid coordinates are given by (16).


### 3.2. Control Strategy

For two-flexible-link manipulator, the distributed control strategy consists of controlling and stabilizing the last joint and flexible link by assuming that the first joint and flexible link are stable and follow their desired trajectories. Then, we move backward and apply the same procedure to the first joint and link. For each step, a new noncollocated output and a control law are developed. The weighting parameter characterizing the noncollocated output is calculated such that the tracking error is asymptotically stable. Thus, the system becomes a minimum phase with the selected new weighted outputs.In this paper, the two-flexible-link model given in [21, 23] is modified by considering only the first flexible mode of each link [20]. Thus, we have n=z=2   and  z1=z2=1.Using (P3), the dynamical model of the two-flexible-link manipulator can be written as(17)M(q)q¨+C(q,q˙)q˙+Dq˙+Kq=Lτ,
where
(18)M=[M11M12M21M22M13M14M23M24M31M32M41M42M33M34M43M44];C=[C11C12C21C22C13C14C23C24C31C32C41C42C33C34C43C44];D=[000000000000Df100Df2];K=[0000000000Kf10000Kf2];q=[qr1qr2qf1qf2];L=[10010000];τ=[τ1τ2].In rigid and flexible part decomposition, the dynamical model (17) can be written as
(19)[MrMrfMrfTMf][q¨rq¨f]+[CrCrfCrfTCf][q˙rq˙f]+[000Df][q˙rq˙f]+[000Kf][qrqf]=[10]τ,
where
(20)Mr=[M11M12M21M22];Mrf=[M13M14M23M24];Mf=[M33M34M43M44];Crr=[C11C12C21C22];Crf=[C13C14C23C24];Cff=[C33C34C43C44];Df=[Df100Df2];Kf=[Kf100Kf2].To develop a control law, the dynamical model (17) is written as two interconnected subsystems. Each subsystem has a pair of joint and link. There exists a nonsingular matrix of transformation Tr such as
(21)q-=Trq,
where q=[qrqf]T=[qr1qr2qf1qf2]T is the original generalized coordinate, q-=[q1q2]T=[qr1qr2qf1qf2]T is the transformed one, and the matrix of transformation Tr is given by
(22)Tr=[1000001001000001].
Using the previous transformation, the dynamical model (17) can be written as follows:
(23)M(q)Tr-1q-¨+C(q,q˙)Tr-1q-˙+DTr-1q-˙+KTr-1q-=Lτ,
where q-¨=Trq¨ and q-˙=Trq˙.Equation (23) is equivalent to the following expression:
(24)M-(q)q-¨+C-(q,q˙)q-˙+D-q-˙+K-q-=L-τ,
where M-=TrM(q)Tr-1; C-=TrC(q,q˙)Tr-1; D-=TrDTr-1; K-=TrKTr-1; q-=[q1q2]T=[qr1qf1qr2qf2]T; L-=TrL=[10000010]T and τ=[τ1τ2].The modified dynamical model (24) can be written as follows:
(25)[M-11M-12M-21M-22][q¨1q¨2]+[C-11C-12C-21C-22][q˙1q˙2]+[D-11D-12D-21D-22][q˙1q˙2]+[K-11K-12K-21K-22][q1q2]=[L-1τL-2τ],
where
(26)M-11=[M11M13M31M33];M-12=[M12M14M32M34];C-11=[C11C13C31C33];C-12=[C12C14C32C34];D-11=[000Df1];D-12=[0000];K-11=[000Kf1];K-12=[0000];M-21=[M21M23M41M43];M-22=[M22M24M42M44];C-21=[C21C23C41C43];C-22=[C22C24C42C44];D-21=[0000];D-22=[000Df2];K-21=[0000];K-22=[000Kf2];q1=[qr1qf1];q2=[qr2qf2];L-1=[1000];L-2=[0100].Equation (25) can be written in the following form that will be used for control law:
(27)[M-1TM-2T]q-¨+[C-1TC-2T]q-˙+[D-1TD-2T]q-˙+[K-1TK-2T]q-=[L-1L-2]τ,
where M-1T=[M-11M-12]; C-1T=[C-11C-12];  D-1T=[D-11D-12]; K-1T=[K-11K-12]; M-2T=[M-21M-22]; C-2T=[C-21C-22];  D-2T=[D-21D-22]; K-2T=[K-21K-22].The first and the second subsystems can be characterized by two mass and inertia symmetric positive definite matrices:(28)M-11=[M11M13M31M33],M-22=[M22M24M42M44].The new generalized coordinate, used for the second joint and link while the first joint and link are assumed stable, is given as follows:(29)Q-2=[qd1q2]T=[qrd1qfd1qr2qf2]T.
Note that the coordinates of the first subsystem are the desired ones, and the coordinates of the second subsystem are the controlled ones. Using (27), the equation of motion of the second joint and link can be written as
(30)M-2T(Q-2)Q-¨2+C-2T(Q-2,Q-˙2)Q-˙2+D-2Q-˙2+K-2Q-2=L-2τ2.
The velocity Q-˙2 is the time derivative of Q-2 and the acceleration Q-¨2 is the time derivative of Q-˙2.According to (8), the new noncollocated output is given as follows:
(31)y2=qr2+α2β2Tqf2,0≤α2<1,
where β2T=β2=∅21(L2)/L2 when considering just one mode. α2 is the weighted parameter characterizing the second noncollocated output.Using (31), the equation of motion of the second subsystem is given as follows:
(32)M21q¨rd1+M22y¨2+M23q¨fd1+(M24-α2β2M22)q¨f2+C21q˙rd1+C22y˙2+C23q˙fd1+ℂ24q˙f2=τ2,(33)M41q¨rd1+M42y¨2+M43q¨fd1+(M44-α2β2M42)q¨f2+C41q˙rd1+C42y˙2+C43q˙fd1+ℂ44q˙f2+Df2q˙f2+Kf2qf2=0,
where ℂ24=C24-α2β2C22 and ℂ44=C44-α2β2C42.The internal dynamics of the second link is deduced from (33) as follows:
(34)q¨f2=-ℳ41q¨rd1-ℳ43q¨fd1-ℳ42y¨2-𝒞41q˙rd1-𝒞42y˙2-𝒞43q˙fd1-𝒞44q˙f2-𝒦f2qf2-𝒟f2q˙f2,
where ℳ41=(M44-α2β2M42)-1M41; ℳ42=(M44-α2β2M42)-1M42; ℳ43=(M44-α2β2M42)-1M43; 𝒞41=(M44-α2β2M42)-1C41; 𝒞42=(M44-α2β2M42)-1C42; 𝒞44=(M44-α2β2M42)-1ℂ44; 𝒦f2=(M44-α2β2M42)-1Kf2; 𝒟f2=(M44-α2β2M42)-1Df2, and α2 is chosen such as α2≠(M44/β2M42).Inserting (34) in (32), we get
(35)M21*q¨rd1+M22*y¨2+M23*q¨fd1+C21*q˙rd1+C22*y˙2+C23*q˙fd1+C24*q˙f2+Kf2*qf2+Df2*q˙f2=τ2,
where M21*=M21-(M24-α2β2M22)ℳ41; M22*=M22-(M24-α2β2M22)ℳ42; M23*=M23-(M24-α2β2M22)ℳ43; C21*=C21-(M24-α2β2M22)𝒞41; C22*=C22-(M24-α2β2M22)𝒞42; C23*=C23-(M24-α2β2M22)𝒞43; C24*=C-24-(M24-α2β2M22)𝒞44; Kf2*=-(M24-α2β2M22)𝒦f2 and Df2*=-(M24-α2β2M22)𝒟f2.To control the second link, we propose the following control law:(36)τ2=τr2+τf2+δτ2,
where τr2 is used for the rigid part, τf2 is for the flexible part, and δτ2 is the setting term:
(37)τr2=Kp2y~2+Kd2y~˙2+M22*y¨2*+M21*q¨rd1+C21*q˙rd1+C22*y˙d2,τf2=C23*q˙fd1+C24*q˙f2+α2β2(M21*q¨fd1+C21*q˙fd1)+Kf2*qf2+Df2*q˙f2,δτ2=δM22*y¨2*+δM21*y¨d1+δC21*y˙d1+δC22*y˙d2+δC23*q˙fd1+δC24*q˙f2,
where y~2=yd2-y2;    y~˙2=y˙d2-y˙2 and y¨2*=y¨d2+λ2y~˙2;  λ2>0.Now going backward to the first subsystem and assuming that the second subsystem is stable, the new generalized coordinate associated to the second subsystem becomes(38)Q-1=[q1qd2]T=[qr1qf1qrd2qfd2]T.
Using (27), the equation of motion of the first subsystem (i.e., first link and joint) is given as follows:
(39)M-1T(Q-1)Q-¨1+C-1T(Q-1,Q-˙1)Q-˙1+D-1Q-˙1+K-1Q-1=L-1τ1.
The new noncollocated output, defined as the angle of the first motor augmented by a weighted value of the link’s extremity angle, is given as follows:
(40)y1=qr1+α1β1Tqf1,0≤α1<1,
where β1T=β1=∅11(L1)/L1 for one mode.Using (39) and (40), the equation of motion of the first joint and link is given as follows:
(41)M11y¨1+M12q¨rd2+(M13-α1β1M11)q¨f1+M14q¨fd2+C11y˙1+C12q˙rd2+ℂ13q˙f1+C14q˙fd2=τ1,(42)M31y¨1+M32q¨rd2+(M33-α1β1M31)q¨f1+M34q¨fd2+C31y˙1+C32q˙rd2+ℂ33q˙f1+C34q˙fd2+Df1q˙f1+Kf1qf1=0.
Let us define ℂ13=C13-α1β1C11 and ℂ33=C33-α1β1C31.To find the internal dynamics of the first link,q¨f1 is given from (42) as follows:
(43)q¨f1=-ℳ31y¨1-ℳ32q¨rd2-ℳ34q¨fd2-𝒞31y˙1-𝒞32q˙rd2-𝒞33q˙f1-𝒞34q˙fd2-𝒦f1qf1-𝒟f1q˙f1,
where ℳ31=(M33-α1β1M31)-1M31; ℳ32=(M33-α1β1M31)-1M32; ℳ34=(M33-α1β1M31)-1M34; 𝒞31=(M33-α1β1M31)-1C31; 𝒞33=(M33-α1β1M31)-1ℂ33; 𝒞34=(M33-α1β1M31)-1C34; 𝒦f1=(M33-α1β1M31)-1Kf1; 𝒟f1=(M33-α1β1M31)-1Df1, and α1 is selected such as α1≠M33/β1M31. Using the same strategy, the control law of the first subsystem is given as follows:
(44)τ1=τr1+τf1+δτ1,
where
(45)τr1=Kp1y~1+Kd1y~˙1+M11*y¨1*+M12*q¨rd2+C11*y˙d1+C12*q˙rd2;τf1=C13*q˙f1+C14*q˙fd2+α1β1(M12*q¨fd2+C12*q˙fd2)+Kf1*qf1+Df1*q˙f1;δτ1=δM11*y¨1*+δM12*y¨d2+δC11*y˙d1+δC12*y˙d2+δC13*q˙f1+δC14*q˙fd2;
where y~1=yd1-y1;    y¨2*=y¨d2+λ2y~˙2 and y~˙2=y˙d2-y˙2; λ2>0. M1j*=M1j-(M13-α1β1M11)ℳ3j, j=1,2,4; C1j*=C1j-(M13-α1β1M11)𝒞3j,  j=1,2,3,4; Kf1*=-(M13-α1β1M11)𝒦f1; Df1*=-(M13-α1β1M11)𝒟f1. The distributed control strategy can be represented by Figure 3Figure 3
Distributed control strategy.


## 3.1. Inverse Dynamics

To achieve the objective of workspace tracking trajectories, we need to transform the desired trajectories from the workspace to the joint space. The flexible manipulator is a nonminimum phase system when the end effector is used as the output. In this case, kinematic and dynamic relationships link the workspace and the joint space. To overcome this problem, an intermediate space called virtual space can be used. Then the desired workspace trajectory is transformed to the virtual space using an inverse kinematics relation as in rigid manipulators. To transform the desired trajectories from the virtual space to the joint pace, the quasi-static approach can be used to solve a nonlinear equation for the flexible part.Using inverse kinematics as in rigid manipulators, the generalized coordinates in virtual space can be easily found. The deformation is assumed to be small. According to Figure2, we can write
(10)qvi-qrdi=arctan(vLiLi)≈vLiLi.
In the virtual space, the generalized coordinate is given as follows:
(11)qvi=qrdi+βiqfdi,
where βi=(∅i(Li)/Li) and ∅i is the shape function of link i.Figure 2
Virtual space.Using the Jacobian matrix as in rigid manipulators, the velocity and acceleration in the virtual space can be deduced. Then, for a two-DOF manipulator, the inverse kinematics is given by the following equation:(12)qv=[qv1qv2]=[arctan2(y,x)-arctan2(L2s2,L1+L2c2)arctan2(s2,c2)],q˙v=[q˙v1q˙v2]=J-1(qv)[x˙y˙],q¨v=[q¨v1q¨v2]=J-1(qv)[x¨y¨]-J-1(qv)J˙(qv)q˙v,
where x and y are the workspace desired positions; c2=(x2+y2-L12-L22)/2L1L2; s2=±1-c22 and J(qv)=[-L1s1-L2s12-L2s12L1c1+L2c12L2c12] is the Jacobian matrix. s1=sin(qv1); c1=cos(qv1); s12=sin(qv1+qv2); c12=cos(qv1+qv2); L1 is length of link 1 and L2 is length of link 2. The rigid and flexible coordinates are derived via quasi-static approach and using the generalized coordinates in the virtual space.The dynamical model (3) can be written as a function of the desired coordinates as follows:
(13)Mrr(qd)q¨rd+Mrf(qd)q¨fd+Hr(qd,q˙d)=τ,(14)MrfT(qd)q¨rd+Mff(qd)q¨fd+Hf(qd,q˙d)+Dffq˙fd+Kffqfd=0.
The generalized flexible and rigid coordinates in the joint space are deduced using the quasi-static approach. In the first step, the nonlinear equation of the flexible part (14) is solved to find the generalized flexible coordinates. The quasi-static approach neglects the desired velocity and acceleration of the flexible coordinates (q˙fd=q¨fd=0). Then, the equation of the flexible part (14) can be written as follows:
(15)MrfT(qd)q¨rd+Hf(qd,q˙d)+Kffqfd=0,
where
(16)qrdi=qvi-βiTqfdi;q˙rdi=q˙vi;q¨rdi=q¨vi.
Using (16), the generalized flexible coordinates qfd are found by solving (15). Then, the generalized rigid coordinates are given by (16).


## 3.2. Control Strategy

For two-flexible-link manipulator, the distributed control strategy consists of controlling and stabilizing the last joint and flexible link by assuming that the first joint and flexible link are stable and follow their desired trajectories. Then, we move backward and apply the same procedure to the first joint and link. For each step, a new noncollocated output and a control law are developed. The weighting parameter characterizing the noncollocated output is calculated such that the tracking error is asymptotically stable. Thus, the system becomes a minimum phase with the selected new weighted outputs.In this paper, the two-flexible-link model given in [21, 23] is modified by considering only the first flexible mode of each link [20]. Thus, we have n=z=2   and  z1=z2=1.Using (P3), the dynamical model of the two-flexible-link manipulator can be written as(17)M(q)q¨+C(q,q˙)q˙+Dq˙+Kq=Lτ,
where
(18)M=[M11M12M21M22M13M14M23M24M31M32M41M42M33M34M43M44];C=[C11C12C21C22C13C14C23C24C31C32C41C42C33C34C43C44];D=[000000000000Df100Df2];K=[0000000000Kf10000Kf2];q=[qr1qr2qf1qf2];L=[10010000];τ=[τ1τ2].In rigid and flexible part decomposition, the dynamical model (17) can be written as
(19)[MrMrfMrfTMf][q¨rq¨f]+[CrCrfCrfTCf][q˙rq˙f]+[000Df][q˙rq˙f]+[000Kf][qrqf]=[10]τ,
where
(20)Mr=[M11M12M21M22];Mrf=[M13M14M23M24];Mf=[M33M34M43M44];Crr=[C11C12C21C22];Crf=[C13C14C23C24];Cff=[C33C34C43C44];Df=[Df100Df2];Kf=[Kf100Kf2].To develop a control law, the dynamical model (17) is written as two interconnected subsystems. Each subsystem has a pair of joint and link. There exists a nonsingular matrix of transformation Tr such as
(21)q-=Trq,
where q=[qrqf]T=[qr1qr2qf1qf2]T is the original generalized coordinate, q-=[q1q2]T=[qr1qr2qf1qf2]T is the transformed one, and the matrix of transformation Tr is given by
(22)Tr=[1000001001000001].
Using the previous transformation, the dynamical model (17) can be written as follows:
(23)M(q)Tr-1q-¨+C(q,q˙)Tr-1q-˙+DTr-1q-˙+KTr-1q-=Lτ,
where q-¨=Trq¨ and q-˙=Trq˙.Equation (23) is equivalent to the following expression:
(24)M-(q)q-¨+C-(q,q˙)q-˙+D-q-˙+K-q-=L-τ,
where M-=TrM(q)Tr-1; C-=TrC(q,q˙)Tr-1; D-=TrDTr-1; K-=TrKTr-1; q-=[q1q2]T=[qr1qf1qr2qf2]T; L-=TrL=[10000010]T and τ=[τ1τ2].The modified dynamical model (24) can be written as follows:
(25)[M-11M-12M-21M-22][q¨1q¨2]+[C-11C-12C-21C-22][q˙1q˙2]+[D-11D-12D-21D-22][q˙1q˙2]+[K-11K-12K-21K-22][q1q2]=[L-1τL-2τ],
where
(26)M-11=[M11M13M31M33];M-12=[M12M14M32M34];C-11=[C11C13C31C33];C-12=[C12C14C32C34];D-11=[000Df1];D-12=[0000];K-11=[000Kf1];K-12=[0000];M-21=[M21M23M41M43];M-22=[M22M24M42M44];C-21=[C21C23C41C43];C-22=[C22C24C42C44];D-21=[0000];D-22=[000Df2];K-21=[0000];K-22=[000Kf2];q1=[qr1qf1];q2=[qr2qf2];L-1=[1000];L-2=[0100].Equation (25) can be written in the following form that will be used for control law:
(27)[M-1TM-2T]q-¨+[C-1TC-2T]q-˙+[D-1TD-2T]q-˙+[K-1TK-2T]q-=[L-1L-2]τ,
where M-1T=[M-11M-12]; C-1T=[C-11C-12];  D-1T=[D-11D-12]; K-1T=[K-11K-12]; M-2T=[M-21M-22]; C-2T=[C-21C-22];  D-2T=[D-21D-22]; K-2T=[K-21K-22].The first and the second subsystems can be characterized by two mass and inertia symmetric positive definite matrices:(28)M-11=[M11M13M31M33],M-22=[M22M24M42M44].The new generalized coordinate, used for the second joint and link while the first joint and link are assumed stable, is given as follows:(29)Q-2=[qd1q2]T=[qrd1qfd1qr2qf2]T.
Note that the coordinates of the first subsystem are the desired ones, and the coordinates of the second subsystem are the controlled ones. Using (27), the equation of motion of the second joint and link can be written as
(30)M-2T(Q-2)Q-¨2+C-2T(Q-2,Q-˙2)Q-˙2+D-2Q-˙2+K-2Q-2=L-2τ2.
The velocity Q-˙2 is the time derivative of Q-2 and the acceleration Q-¨2 is the time derivative of Q-˙2.According to (8), the new noncollocated output is given as follows:
(31)y2=qr2+α2β2Tqf2,0≤α2<1,
where β2T=β2=∅21(L2)/L2 when considering just one mode. α2 is the weighted parameter characterizing the second noncollocated output.Using (31), the equation of motion of the second subsystem is given as follows:
(32)M21q¨rd1+M22y¨2+M23q¨fd1+(M24-α2β2M22)q¨f2+C21q˙rd1+C22y˙2+C23q˙fd1+ℂ24q˙f2=τ2,(33)M41q¨rd1+M42y¨2+M43q¨fd1+(M44-α2β2M42)q¨f2+C41q˙rd1+C42y˙2+C43q˙fd1+ℂ44q˙f2+Df2q˙f2+Kf2qf2=0,
where ℂ24=C24-α2β2C22 and ℂ44=C44-α2β2C42.The internal dynamics of the second link is deduced from (33) as follows:
(34)q¨f2=-ℳ41q¨rd1-ℳ43q¨fd1-ℳ42y¨2-𝒞41q˙rd1-𝒞42y˙2-𝒞43q˙fd1-𝒞44q˙f2-𝒦f2qf2-𝒟f2q˙f2,
where ℳ41=(M44-α2β2M42)-1M41; ℳ42=(M44-α2β2M42)-1M42; ℳ43=(M44-α2β2M42)-1M43; 𝒞41=(M44-α2β2M42)-1C41; 𝒞42=(M44-α2β2M42)-1C42; 𝒞44=(M44-α2β2M42)-1ℂ44; 𝒦f2=(M44-α2β2M42)-1Kf2; 𝒟f2=(M44-α2β2M42)-1Df2, and α2 is chosen such as α2≠(M44/β2M42).Inserting (34) in (32), we get
(35)M21*q¨rd1+M22*y¨2+M23*q¨fd1+C21*q˙rd1+C22*y˙2+C23*q˙fd1+C24*q˙f2+Kf2*qf2+Df2*q˙f2=τ2,
where M21*=M21-(M24-α2β2M22)ℳ41; M22*=M22-(M24-α2β2M22)ℳ42; M23*=M23-(M24-α2β2M22)ℳ43; C21*=C21-(M24-α2β2M22)𝒞41; C22*=C22-(M24-α2β2M22)𝒞42; C23*=C23-(M24-α2β2M22)𝒞43; C24*=C-24-(M24-α2β2M22)𝒞44; Kf2*=-(M24-α2β2M22)𝒦f2 and Df2*=-(M24-α2β2M22)𝒟f2.To control the second link, we propose the following control law:(36)τ2=τr2+τf2+δτ2,
where τr2 is used for the rigid part, τf2 is for the flexible part, and δτ2 is the setting term:
(37)τr2=Kp2y~2+Kd2y~˙2+M22*y¨2*+M21*q¨rd1+C21*q˙rd1+C22*y˙d2,τf2=C23*q˙fd1+C24*q˙f2+α2β2(M21*q¨fd1+C21*q˙fd1)+Kf2*qf2+Df2*q˙f2,δτ2=δM22*y¨2*+δM21*y¨d1+δC21*y˙d1+δC22*y˙d2+δC23*q˙fd1+δC24*q˙f2,
where y~2=yd2-y2;    y~˙2=y˙d2-y˙2 and y¨2*=y¨d2+λ2y~˙2;  λ2>0.Now going backward to the first subsystem and assuming that the second subsystem is stable, the new generalized coordinate associated to the second subsystem becomes(38)Q-1=[q1qd2]T=[qr1qf1qrd2qfd2]T.
Using (27), the equation of motion of the first subsystem (i.e., first link and joint) is given as follows:
(39)M-1T(Q-1)Q-¨1+C-1T(Q-1,Q-˙1)Q-˙1+D-1Q-˙1+K-1Q-1=L-1τ1.
The new noncollocated output, defined as the angle of the first motor augmented by a weighted value of the link’s extremity angle, is given as follows:
(40)y1=qr1+α1β1Tqf1,0≤α1<1,
where β1T=β1=∅11(L1)/L1 for one mode.Using (39) and (40), the equation of motion of the first joint and link is given as follows:
(41)M11y¨1+M12q¨rd2+(M13-α1β1M11)q¨f1+M14q¨fd2+C11y˙1+C12q˙rd2+ℂ13q˙f1+C14q˙fd2=τ1,(42)M31y¨1+M32q¨rd2+(M33-α1β1M31)q¨f1+M34q¨fd2+C31y˙1+C32q˙rd2+ℂ33q˙f1+C34q˙fd2+Df1q˙f1+Kf1qf1=0.
Let us define ℂ13=C13-α1β1C11 and ℂ33=C33-α1β1C31.To find the internal dynamics of the first link,q¨f1 is given from (42) as follows:
(43)q¨f1=-ℳ31y¨1-ℳ32q¨rd2-ℳ34q¨fd2-𝒞31y˙1-𝒞32q˙rd2-𝒞33q˙f1-𝒞34q˙fd2-𝒦f1qf1-𝒟f1q˙f1,
where ℳ31=(M33-α1β1M31)-1M31; ℳ32=(M33-α1β1M31)-1M32; ℳ34=(M33-α1β1M31)-1M34; 𝒞31=(M33-α1β1M31)-1C31; 𝒞33=(M33-α1β1M31)-1ℂ33; 𝒞34=(M33-α1β1M31)-1C34; 𝒦f1=(M33-α1β1M31)-1Kf1; 𝒟f1=(M33-α1β1M31)-1Df1, and α1 is selected such as α1≠M33/β1M31. Using the same strategy, the control law of the first subsystem is given as follows:
(44)τ1=τr1+τf1+δτ1,
where
(45)τr1=Kp1y~1+Kd1y~˙1+M11*y¨1*+M12*q¨rd2+C11*y˙d1+C12*q˙rd2;τf1=C13*q˙f1+C14*q˙fd2+α1β1(M12*q¨fd2+C12*q˙fd2)+Kf1*qf1+Df1*q˙f1;δτ1=δM11*y¨1*+δM12*y¨d2+δC11*y˙d1+δC12*y˙d2+δC13*q˙f1+δC14*q˙fd2;
where y~1=yd1-y1;    y¨2*=y¨d2+λ2y~˙2 and y~˙2=y˙d2-y˙2; λ2>0. M1j*=M1j-(M13-α1β1M11)ℳ3j, j=1,2,4; C1j*=C1j-(M13-α1β1M11)𝒞3j,  j=1,2,3,4; Kf1*=-(M13-α1β1M11)𝒦f1; Df1*=-(M13-α1β1M11)𝒟f1. The distributed control strategy can be represented by Figure 3Figure 3
Distributed control strategy.


## 4. Stability Analysis

This section presents the stability analysis of the tracking errors of the rigid and the flexible parts. First, we study the global stability of the rigid part by inserting the two control laws in the initial dynamical model. Second, the stability of the flexible part is studied to select the weighted parameters and guarantee bounded internal dynamics such that the new output is as close as possible to the tip.


### 4.1. Stability of Rigid Part

In compact form, the first (36) and second (44) control laws can be written as follows:
(46)τ=τr+τf+δτ,
where
(47)τr=Kpy~+Kdy~˙+Mrd*y¨*+Mrnd*q¨rd+Crd*y˙d+Crnd*q˙rd;τf=Crf*q˙f+αβ(Mrnd*q¨fd+Crnd*q˙fd)+Kf*qf+Df*q˙f;(48)δτ=δMrd*y¨*+δMrnd*y¨d+δCr*y˙d+δCrf*q˙f;(49)Kp=[Kp100Kp2];Kd=[Kd100Kd2];Mrd*=[M11*00M22*];Mrnd*=[0M12*M21*0];Crd*=[C11*00C22*];Crnd*=[0C12*C21*0];Crf*=[C13*C14*C23*C24*];δMrd*=[δM11*00δM22*];δMrnd*=[0δM12*δM21*0];δCr*=[δC11*δC12*δC21*δC22*];δCrf*=[δC13*δC14*δC23*δC24*];δMij* are given in (B.3).To study the global stability, we insert the control law given in (46) in the dynamical model given in (1).Proposition 1.
The equation of motion of the two-flexible-link manipulator (17) is equivalent to the following model:
(50)Mr*(qα)y¨+Cr*(qα,q˙α)y˙+Crf*(qα,q˙α)q˙f+Kf*qf+Df*q˙f=τ,
where qα=[y1y2qf1qf2]T; Mr*=[M11*M12*M21*M22*] is positive definite matrix; Cr*=[C11*C12*C21*C22*] and Crf*=[C13*C14*C23*C24*]; Mr* and Cr* satisfy the following skew-symmetric property:
(51)XT(M˙r*-2Cr*)X=0,∀X∈ℛn.Proof.
See AppendixA.M
r
* can be written as
(52)Mr*=Mrd*+Mrnd*,
where Mrd* is the diagonal elements and Mrnd*  is the non-diagonal elements of Mrd*. Then, the dynamical model (50) can be written as follows:
(53)Mrd*(qα)y¨+Mrnd*(qα)y¨+Cr*(qα,q˙α)y˙+Crf*(qα,q˙α)q˙f+Kf*qf+Df*q˙f=τ.
Using (52), the control law (46) is equivalent to
(54)τ=Kpy~+Kdy~˙+Mrd*(Q)y¨*+Mrnd*(Q)y¨d+Cr*(Q,Q˙)y˙d+Crf*(Q,Q˙)q˙f+Kf*qf+Df*q˙f+δτ.
Now, the global stability is studied by inserting the control law (54) in the dynamical model (53). Then, the error dynamics are given by the following equation:
(55)Kpy~+Kdy~˙+[Mrd*(Q)y¨*-Mrd*(qα)y¨]+[Mrnd*(Q)y¨d-Mrnd*(qα)y¨]+[Cr*(Q,Q˙)y˙d-Cr*(qα,q˙α)y˙]+[Crf*(Q,Q˙)q˙f-Crf*(qα,q˙α)q˙f]+δτ=0.Proposition 2.
When using the control law (54) and the dynamical model (53),  the error dynamics given by (55) are equivalent to
(56)Kpy~+Kdy~˙+Mr*(qα)y~¨+Cr*(qα,q˙α)y~˙+λMrd*(qα)y~˙=0.Proof.
See AppendixB.To prove the asymptotical stability of the error dynamics, we propose the following Lyapunov function:(57)V=12y~˙TMr*y~˙+12y~TKdy~.
The time derivative of V is
(58)V˙(t)=y~˙TMr*y~¨+12y~˙TM˙r*y~˙+y~TKpy~˙=y~˙T(-Cr*y~˙-λMrd*y~˙-Kpy~-Kdy~˙)+12y~˙TM˙r*y~˙+y~TKpy~˙.
Using Proposition 1, we can conclude that
(59)V˙(t)=-y~˙T(Kd+λMrd*)y~˙.Kd, λ,  and  Mrd* are positive definite diagonal matrices. Then, using LaSalle theorem [24], the error dynamics are globally asymptotically stable.


### 4.2. Stability of the Flexible Part

In this section, asymptotical stability of the flexible part is studied using the internal dynamics. From the dynamical model given in (1) and (3), the equation of motion of the flexible part of the second link is given by
(60)M41q¨r1+M42q¨r2+M43q¨f1+M44q¨f2+H4+Df2q˙f2+Kf2qf2=0.
Using the new generalized coordinate (29) and the new noncollocated output (31), the equation (60) is equivalent to
(61)M41q¨rd1+M42y¨2+M43q¨fd1+(M44-α2β2M42)q¨f2+H4+Df2q˙f2+Kf2qf2=0.
The internal dynamics of the second link are given by subtracting  q¨f2  from (61) as follows:
(62)q¨f2=-ℳ41q¨rd1-ℳ42y¨2-ℳ43q¨fd1-ℋ4-𝒦f2qf2-𝒟f2q˙f2,
where ℳ41=(M44-α2β2M42)-1M41; ℳ42=(M44-α2β2M42)-1M42; ℳ43=(M44-α2β2M42)-1M43; ℋ4=(M44-α2β2M42)-1H4; 𝒟f2=(M44-α2β2M42)-1Df2; 𝒦f2=(M44-α2β2M42)-1Kf2, and α2 is chosen such as α2≠M44/β2M42.The quasi-static approach, using the inverse dynamics, neglectsq¨fd1. Then, the tracking errors of the rigid and flexible part of the second link are given by the following expressions:
(63)q~¨f2=q¨fd2-q¨f2=-(ℳd41-ℳ41)q¨rd1-ℳd42y~¨2-(ℋd4-ℋ4)-(𝒦fd2qfd2-𝒦f2qf2)-(𝒟fd2q˙fd2-𝒟f2q˙f2).
In the above section, the tracking error of the rigid part was proved to be asymptotically stable; then, y~2,y~˙2,y~¨2→0 as t→∞. Then, the tracking error of the flexible part can be written as
(64)q~¨f2=-(ℳd41-ℳ41)q¨rd1-(ℋd4-ℋ4)-𝒦f2q~f2-𝒟f2q~˙f2.
In state space form, the tracking error of the flexible part takes the same form as the one obtained in [20] and is given by the following expression:
(65)x~˙1=x~2,x~˙2=q¨fd2-q¨f2.
Let x~f=[x~1x~2]T=[q~f2q~˙f2]T; q~f2=qfd2-qf2.The critical value ofα2 is selected such that Af2 is Hurwitz and the Nyquist diagram of Af2 is in the right half plan [20], where Af2=[01-𝒦f2-𝒟f2]. 𝒦f2 and 𝒟f2 are given in (62).The same procedure is now applied, proceeding backward, to the first joint-link subsystem. Using (1) and (3), the equation of motion of the flexible part of the first link is given by
(66)M31q¨r1+M32q¨r2+M33q¨f1+M34q¨f2+H3+Df1q˙f1+Kf1qf1=0.
Using the new generalized coordinate (38) and the new noncollocated output (40), the equation of motion becomes
(67)M31y¨1+M32q¨rd2+(M33-α1β1M31)q¨f1+M34q¨fd2+H3+Df1q˙f1+Kf1qf1=0.
Subtracting q¨f1 from (67), the internal dynamics are given as follows:
(68)q¨f1=-ℳ31y¨1-ℳ32q¨rd2-ℳ34q¨fd2-ℋ3-𝒦f1qf1-𝒟f1q˙f1,
where ℳ31=(M33-α1β1M31)-1M31; ℳ32=(M33-α1β1M31)-1M32; ℳ34=(M33-α1β1M31)-1M34; ℋ3=(M33-α1β1M31)-1H3; 𝒦f1=(M33-α1β1M31)-1Kf1; 𝒟f1=(M33-α1β1M31)-1Df1, and α1 is chosen such as α1≠M33/β1M31.In the state space form, the tracking error of the flexible part of the first link can be written as follows:(69)x~˙f1=q~˙f1=x~f2,x~˙f2=q~¨f1=q¨fd1-q¨f1.
Let x~f=[x~f1x~f2]T=[q~f1q~˙f1]T.As already shown in the stability analysis of the rigid part,y~1,y~˙1,  and  y~¨1→0 as t→∞. The error dynamics of the flexible part (69) take the same form as the one obtained in [20].The critical value ofα1 is selected when the eigenvalues of Af1 change their signs from negative to positive; for example, the internal dynamics become instable [20], where Af1=[01-K¯f1-D¯f1].


## 4.1. Stability of Rigid Part

In compact form, the first (36) and second (44) control laws can be written as follows:
(46)τ=τr+τf+δτ,
where
(47)τr=Kpy~+Kdy~˙+Mrd*y¨*+Mrnd*q¨rd+Crd*y˙d+Crnd*q˙rd;τf=Crf*q˙f+αβ(Mrnd*q¨fd+Crnd*q˙fd)+Kf*qf+Df*q˙f;(48)δτ=δMrd*y¨*+δMrnd*y¨d+δCr*y˙d+δCrf*q˙f;(49)Kp=[Kp100Kp2];Kd=[Kd100Kd2];Mrd*=[M11*00M22*];Mrnd*=[0M12*M21*0];Crd*=[C11*00C22*];Crnd*=[0C12*C21*0];Crf*=[C13*C14*C23*C24*];δMrd*=[δM11*00δM22*];δMrnd*=[0δM12*δM21*0];δCr*=[δC11*δC12*δC21*δC22*];δCrf*=[δC13*δC14*δC23*δC24*];δMij* are given in (B.3).To study the global stability, we insert the control law given in (46) in the dynamical model given in (1).Proposition 1.
The equation of motion of the two-flexible-link manipulator (17) is equivalent to the following model:
(50)Mr*(qα)y¨+Cr*(qα,q˙α)y˙+Crf*(qα,q˙α)q˙f+Kf*qf+Df*q˙f=τ,
where qα=[y1y2qf1qf2]T; Mr*=[M11*M12*M21*M22*] is positive definite matrix; Cr*=[C11*C12*C21*C22*] and Crf*=[C13*C14*C23*C24*]; Mr* and Cr* satisfy the following skew-symmetric property:
(51)XT(M˙r*-2Cr*)X=0,∀X∈ℛn.Proof.
See AppendixA.M
r
* can be written as
(52)Mr*=Mrd*+Mrnd*,
where Mrd* is the diagonal elements and Mrnd*  is the non-diagonal elements of Mrd*. Then, the dynamical model (50) can be written as follows:
(53)Mrd*(qα)y¨+Mrnd*(qα)y¨+Cr*(qα,q˙α)y˙+Crf*(qα,q˙α)q˙f+Kf*qf+Df*q˙f=τ.
Using (52), the control law (46) is equivalent to
(54)τ=Kpy~+Kdy~˙+Mrd*(Q)y¨*+Mrnd*(Q)y¨d+Cr*(Q,Q˙)y˙d+Crf*(Q,Q˙)q˙f+Kf*qf+Df*q˙f+δτ.
Now, the global stability is studied by inserting the control law (54) in the dynamical model (53). Then, the error dynamics are given by the following equation:
(55)Kpy~+Kdy~˙+[Mrd*(Q)y¨*-Mrd*(qα)y¨]+[Mrnd*(Q)y¨d-Mrnd*(qα)y¨]+[Cr*(Q,Q˙)y˙d-Cr*(qα,q˙α)y˙]+[Crf*(Q,Q˙)q˙f-Crf*(qα,q˙α)q˙f]+δτ=0.Proposition 2.
When using the control law (54) and the dynamical model (53),  the error dynamics given by (55) are equivalent to
(56)Kpy~+Kdy~˙+Mr*(qα)y~¨+Cr*(qα,q˙α)y~˙+λMrd*(qα)y~˙=0.Proof.
See AppendixB.To prove the asymptotical stability of the error dynamics, we propose the following Lyapunov function:(57)V=12y~˙TMr*y~˙+12y~TKdy~.
The time derivative of V is
(58)V˙(t)=y~˙TMr*y~¨+12y~˙TM˙r*y~˙+y~TKpy~˙=y~˙T(-Cr*y~˙-λMrd*y~˙-Kpy~-Kdy~˙)+12y~˙TM˙r*y~˙+y~TKpy~˙.
Using Proposition 1, we can conclude that
(59)V˙(t)=-y~˙T(Kd+λMrd*)y~˙.Kd, λ,  and  Mrd* are positive definite diagonal matrices. Then, using LaSalle theorem [24], the error dynamics are globally asymptotically stable.


## 4.2. Stability of the Flexible Part

In this section, asymptotical stability of the flexible part is studied using the internal dynamics. From the dynamical model given in (1) and (3), the equation of motion of the flexible part of the second link is given by
(60)M41q¨r1+M42q¨r2+M43q¨f1+M44q¨f2+H4+Df2q˙f2+Kf2qf2=0.
Using the new generalized coordinate (29) and the new noncollocated output (31), the equation (60) is equivalent to
(61)M41q¨rd1+M42y¨2+M43q¨fd1+(M44-α2β2M42)q¨f2+H4+Df2q˙f2+Kf2qf2=0.
The internal dynamics of the second link are given by subtracting  q¨f2  from (61) as follows:
(62)q¨f2=-ℳ41q¨rd1-ℳ42y¨2-ℳ43q¨fd1-ℋ4-𝒦f2qf2-𝒟f2q˙f2,
where ℳ41=(M44-α2β2M42)-1M41; ℳ42=(M44-α2β2M42)-1M42; ℳ43=(M44-α2β2M42)-1M43; ℋ4=(M44-α2β2M42)-1H4; 𝒟f2=(M44-α2β2M42)-1Df2; 𝒦f2=(M44-α2β2M42)-1Kf2, and α2 is chosen such as α2≠M44/β2M42.The quasi-static approach, using the inverse dynamics, neglectsq¨fd1. Then, the tracking errors of the rigid and flexible part of the second link are given by the following expressions:
(63)q~¨f2=q¨fd2-q¨f2=-(ℳd41-ℳ41)q¨rd1-ℳd42y~¨2-(ℋd4-ℋ4)-(𝒦fd2qfd2-𝒦f2qf2)-(𝒟fd2q˙fd2-𝒟f2q˙f2).
In the above section, the tracking error of the rigid part was proved to be asymptotically stable; then, y~2,y~˙2,y~¨2→0 as t→∞. Then, the tracking error of the flexible part can be written as
(64)q~¨f2=-(ℳd41-ℳ41)q¨rd1-(ℋd4-ℋ4)-𝒦f2q~f2-𝒟f2q~˙f2.
In state space form, the tracking error of the flexible part takes the same form as the one obtained in [20] and is given by the following expression:
(65)x~˙1=x~2,x~˙2=q¨fd2-q¨f2.
Let x~f=[x~1x~2]T=[q~f2q~˙f2]T; q~f2=qfd2-qf2.The critical value ofα2 is selected such that Af2 is Hurwitz and the Nyquist diagram of Af2 is in the right half plan [20], where Af2=[01-𝒦f2-𝒟f2]. 𝒦f2 and 𝒟f2 are given in (62).The same procedure is now applied, proceeding backward, to the first joint-link subsystem. Using (1) and (3), the equation of motion of the flexible part of the first link is given by
(66)M31q¨r1+M32q¨r2+M33q¨f1+M34q¨f2+H3+Df1q˙f1+Kf1qf1=0.
Using the new generalized coordinate (38) and the new noncollocated output (40), the equation of motion becomes
(67)M31y¨1+M32q¨rd2+(M33-α1β1M31)q¨f1+M34q¨fd2+H3+Df1q˙f1+Kf1qf1=0.
Subtracting q¨f1 from (67), the internal dynamics are given as follows:
(68)q¨f1=-ℳ31y¨1-ℳ32q¨rd2-ℳ34q¨fd2-ℋ3-𝒦f1qf1-𝒟f1q˙f1,
where ℳ31=(M33-α1β1M31)-1M31; ℳ32=(M33-α1β1M31)-1M32; ℳ34=(M33-α1β1M31)-1M34; ℋ3=(M33-α1β1M31)-1H3; 𝒦f1=(M33-α1β1M31)-1Kf1; 𝒟f1=(M33-α1β1M31)-1Df1, and α1 is chosen such as α1≠M33/β1M31.In the state space form, the tracking error of the flexible part of the first link can be written as follows:(69)x~˙f1=q~˙f1=x~f2,x~˙f2=q~¨f1=q¨fd1-q¨f1.
Let x~f=[x~f1x~f2]T=[q~f1q~˙f1]T.As already shown in the stability analysis of the rigid part,y~1,y~˙1,  and  y~¨1→0 as t→∞. The error dynamics of the flexible part (69) take the same form as the one obtained in [20].The critical value ofα1 is selected when the eigenvalues of Af1 change their signs from negative to positive; for example, the internal dynamics become instable [20], where Af1=[01-K¯f1-D¯f1].


## 5. Simulation Results

The two-flexible-link manipulator shown in Figure2 is used to test the control strategy performance. Using the trial and error method, the controllers’ gains are chosen as Kp1=12, Kp2=10, Kd1=15, and Kd2=18.In Sections3 and 4, the theoretical development of the control law and stability analysis are given in the general case, that is, for any higher order. In the simulation case, a finite order of Taylor series is fixed. In this section, for simplicity, the Taylor series is limited to the first order. Then, RMnj=RCnj=RGn=0.Table1 presents the parameters of the two-flexible-link manipulator.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·m2Simulation results are given Figures4–14.Figure 4
X
-
Y desired workspace trajectory.Desired workspace trajectory: (a)-(b) position trajectory, (c)-(d) velocity trajectories, and (e)-(f) acceleration trajectories.
(a)
(b)
(c)
(d)
(e)
(f)Desired virtual space trajectories: (a)-(b) position trajectories, (c)-(d) velocity trajectories, and (e)-(f) acceleration trajectories.
(a)
(b)
(c)
(d)
(e)
(f)Desired joint space trajectories: (a)-(b) position trajectories of rigid coordinates, (c)-(d) velocity trajectories of rigid coordinates, (e)-(f) acceleration trajectories of rigid coordinates, and (g)-(h) position trajectories of flexible coordinates.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Stability of second link. (a) Nyquist diagram and (b) eigenvalues evolution [20].
(a)
(b)Figure 9
Stability of first link: eigenvalues versusα1 [20].Tracking trajectories of noncollocated outputs.
(a)
(b)Tracking errors in joint space. (a)-(b) tracking errors of noncollocated outputs and (c)-(d) tracking errors of the flexible coordinates.
(a)
(b)
(c)
(d)Tracking in the workspace. (a)x-position tracking and (b) y-position tracking.
(a)
(b)Workspace tracking errors. (a)x-position tracking errors and (b) y-position tracking errors.
(a)
(b)Figure 14
X
-
Y workspace tracking.For two-flexible-link manipulators, the desired workspace trajectory is chosen as lozenge form. The results of the inverse kinematics problem are given in Figures4–7. Workspace trajectory is shown in Figures 4 and 5, virtual space trajectories are given in Figure 6, and joint space trajectories are shown in Figure 7. According to simulation results, the workspace desired trajectory was well transformed to the joint space using the virtual space and the quasi-static approach. For the second subsystem, the Nyquist diagram and the evolution of eigenvalues of Af2 are given in Figure 8. For the first subsystem, the eigenvalues evolution of Af1  is given in Figure 9. Using the system parameters given in Table 1, the critical value of the second and first links are α2c=0.92 and α1c=0.85, respectively. The simulation results were obtained with α2=0.9<α2c and α1=0.8<α1c. Note that these values satisfy the conditions given in (30) and (38).A good tracking performance of the new noncollocated output was obtained, as shown in Figure10. This tracking is confirmed by Figure 11, which shows the tracking error. In the workspace, a good tracking performance is shown in Figure 12, and the tracking error, shown in Figure 13, does not exceed 2 mm in x-position and y-position. The good tracking in the workspace is confirmed in Figure 14 that shows the tracking of the desired lozenge. Thus, we can conclude that the control strategy was effective to ensure the tracking of the noncollocated output and to reduce vibrations at the extremity; this explains the satisfactory results obtained using this approach.


## 6. Conclusion and Future Work

This paper presents a nonlinear distributed control for two-flexible-link manipulator. For the inverse dynamics, a virtual space, linked with the workspace by a simple kinematics relation as in rigid manipulators, and a quasi-static approach were used. Using this transformation procedure, a workspace desired trajectory (lozenge) has been successfully transformed to the joint space. The distributed control strategy presented in this paper uses the output redefinition technique and consists of stabilizing the flexible manipulators starting with the last joint and flexible link and going backward until the first joint and link. Lyapunov theory was used to prove the asymptotical stability. An adaptive version of this control strategy will be investigated in future work.



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*Source: 617465-2013-12-03.xml*