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Modelling and Simulation in Engineering
Volume 2012 (2012), Article ID 271705, 7 pages
Research Article

Robust Adaptive Tracking Control of a Class of Robot Manipulators with Model Uncertainties

1Departamento de Electrónica, CUCEI Universidad de Guadalajara, Avenida Revolución No. 1500, 44430 Guadalajara, JAL, Mexico
2Facultad de Ingeniería, Universidad Autónoma de Yucatán, Avenida Industrias no Contaminantes, Apdo. Postal 150 Cordemex, Mérida, Yucatán, Mexico

Received 17 April 2012; Accepted 13 June 2012

Academic Editor: Ahmed Rachid

Copyright © 2012 G. Solís-Perales and R. Peón-Escalante. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A robust tracking controller for robot manipulators measuring only the angular positions and considering model uncertainties is presented. It is considered that the model is uncertain; that is, the system parameters, nonlinear terms, external perturbations, and the friction effects in each robot joint are considered unknown. The controller is composed by two parts, a linearizing-like control feedback and a high-gain estimator. The main idea is to lump the uncertain terms into a new state which represents the dynamics of the uncertainties. This new state is then estimated in order to be compensated. In this way the resulting controller is robust. A numerical example for a RR robot manipulator is provided, in order to corroborate the results.

1. Introduction

Control of manipulators is a classical control problem [14], due to the innumerable applications, for instance, in manufacturing processes, biomedical engineering, and aeronautical. In the tracking problem, it is known that the tracking error should be maintained as small as possible, even in the presence of parameter variations, external perturbations, modeling errors, and even more the presence of faults in the system. These aspects produce, a source of uncertainties and then the controller should be robust against these phenomena.

Tracking and control of manipulators have been studied using several approaches. For instance, in [5] authors presented an approach for controlling a robot using only joint position measurements. In such approach an observer is proposed in order to estimate the joint velocities, but uncertainties in the model are not considered. A fuzzy scheme was presented in [6]; however, the methodology requires the knowledge of a nominal values for the inertia matrix. An alternative approach was presented in [7], where the problem of estimation of time-varying parameters was obtained via the multiestimation, which consists in the integration of an adaptive controller to a multi-model-based technique. An application of control of a dual cooperative manipulator based on singularly perturbed formulation and a position/force controller is designed [8], which does not require the velocity measures, as an observer to estimate the velocity and reduce the number of sensors.

The present contribution consists in considering uncertain the parameters of the manipulator and the effects of the friction; moreover, it is not necessary to estimate or to adapt to any parameter. In this sense, the controller is considered robust against parameter variations, nonmodeled dynamics (friction dynamics), and external disturbances. We illustrate that the robust control can reject external perturbations, attenuate parameter variation effects, and compensate nonmodeled dynamics. It is known that the friction is a physical phenomenon and many models exist for its study. Thus, the unknown of the friction model is a challenging problem and, from the control view point, compensating this phenomenon is determinant in the performance of the system. Thus in the present contribution a controller which compensate friction and modeling errors is described. In this sense, the controller is capable to compensate modeling errors, friction, parameter variations, and external perturbations using only the measure of the angular position of the robots.

The controller is constructed departing from the nonlinear control theory. First, a linearizing control law is determined, then all the uncertain terms are lumped into a new state, which is unknown. The linearizing-like controller requires the knowledge of such an unknown state. To tackle this problem, we use a high-gain state estimator to obtain an estimated value for this uncertain term [9]. This value is used by the controller which now can compensate all the uncertainties.

The paper is organized as follows: in Section 2, the dynamic model for a robot manipulator is presented, Section 3 contains the main contribution on nonlinear robust control, Section 4 presents the simulation results, and finally, in Section 5, some conclusions are given.

2. Dynamic Model of the Manipulator

Let us consider the well-known standard equation describing the dynamics of a 𝑛-DOF rigid joint robot manipulator with friction 𝐌(𝑞)̈𝑞+𝐕(𝑞,̇𝑞)+𝐆(𝑞)+𝐁(̇𝑞)=𝐓(𝑡)𝐓𝑑(𝑡),(1) where 𝐌(𝑞)𝑛×𝑛 is a positive definite inertia matrix, 𝐕(𝑞,̇𝑞)𝑛×1 the Coriolis and centrifugal forces, 𝐆(𝑞)𝑛×1 is the vector for the gravitational forces and 𝐁(̇𝑞) is a vector field of the friction terms, 𝐓,𝐓𝑑𝑛×1 stand for the control and disturbance torques, respectively, and 𝑞(𝑡)𝑛 is the state vector for the joint angular positions of the manipulator. In this contribution we consider the friction terms as follows: 𝐵𝑖̇𝑞𝑖=𝐵𝜈,𝑖̇𝑞𝑖+𝐵𝑓1,𝑖211+𝑒2𝜔1̇𝑞𝑖+𝐵𝑓2,𝑖211+𝑒2𝜔2̇𝑞𝑖,(2) where 𝐵𝜈,𝑖 stands for the viscous friction; the second and third terms model the Coulomb and Stribeck friction effects in the 𝑖th robot link; for details see [10, 11] and references therein.

3. Nonlinear Control of Robot Manipulators

Let us consider (1) and defining 𝑥1=𝑞=[𝑞1,𝑞2,,𝑞𝑛]𝑇 as the 𝑛-vector of the joint positions and 𝑥2=̇𝑞=̇𝑥1=[̇𝑞1,̇𝑞2,,̇𝑞𝑛]𝑇 as the 𝑛-vector of angular velocities; thus one can write the system as follows: ̇𝑥1=𝑥2,̇𝑥2𝑥=𝐌11𝐓𝐓𝑑𝑥𝐕1,𝑥2𝑥𝐆1𝑥𝐁2,𝑦=𝑥1,(3) where the output 𝑦 is the vector of angular positions, without losing generality, for a manipulator with 𝑛 joints, the affine system can be written as follows: ̇𝜒=Ψ(𝜒)+𝑛𝑗=1Γ𝑗(𝜒)𝜏𝑗𝑦𝑖=𝑖(𝜒),(4) where 𝜒2𝑛 is the state vector of positions and velocities of the manipulator given by 𝜒=[𝑞1,𝑞2,,𝑞𝑛,̇𝑞1,,̇𝑞𝑛]𝑇,Ψ(𝜒):2𝑛2𝑛 is a sufficiently smooth vector field describing the robot dynamics, Γ𝑗(𝜒)2𝑛 is the input vector given by Γ𝑗(𝜒)=[0𝑛×1,𝔪1,𝑗(𝜒),,𝔪𝑛,𝑗(𝜒)]𝑇 where 𝔪𝑖,𝑗 are the entries of the inverse inertia matrix, and 𝜏𝑗 is the control torque in the 𝑗th joint. It is important to stress that the friction terms are included into the dynamics Ψ(𝜒). System (4) consists of a system with 𝑛 inputs and 𝑛 outputs; therefore the multiple-input and multiple-output nonlinear control tools are used [12]. The underlying idea of the nonlinear feedback control is to find functions 𝜏𝑗, such that the desired dynamical behavior is induced for any initial condition 𝜒0=𝜒(0) in an attraction basin 𝑈02𝑛. To begin with, we depart from the relative degree condition for a MIMO system [12].

Definition 1. The multiple input and multiple output affine system (4) has relative degree (𝜌1,𝜌2,,𝜌𝑛) at the point 𝜒0 if(i)𝐿Γ𝑗𝐿𝑘Ψ𝑖(𝜒)=0, for all 1𝑗,𝑖𝑛, 𝑘<𝜌𝑖1, and for all 𝜒 in the neighborhood of 𝜒0,(ii)the 𝑛×𝑛 matrix 𝒜𝜒=𝐿Γ1𝐿𝜌1Ψ11(𝜒)𝐿Γ𝑛𝐿𝜌1Ψ11𝐿(𝜒)Γ1𝐿𝜌2Ψ12(𝜒)𝐿Γ𝑛𝐿𝜌2Ψ12(𝐿𝜒)Γ1𝐿𝜌𝑛Ψ1𝑛(𝜒)𝐿Γ𝑛𝐿𝜌𝑛Ψ1𝑛(𝜒)(5)is nonsingular at 𝜒=𝜒0.

In the fully actuated manipulator, the relative-degree matrix 𝒜𝜒 is square, and it is given by the positive definite matrix 𝑀(𝑥1)1. Thus, due to the invertibility of the matrix 𝒜𝜒, a diffeomorphic transformation can be determined based on the Lie derivatives, of the outputs along the vector fields. Without losing generality, one can propose the transformation 𝑧=Φ(𝜒)=[𝜒1,𝜒𝑛+1,𝜒2,𝜒𝑛+2,,𝜒𝑛,𝜒2𝑛]𝑇, Φ2𝑛2𝑛, for 𝑧Ω𝑈, such that the affine form (4) takes a linearizable canonical form. In terms of the Lie derivatives the transformation is given as 𝑧2𝑘1=𝑘(𝜒)=𝜒𝑘 and 𝑧2𝑘=𝐿Ψ𝑘(𝜒)=𝜒𝑛+𝑘,𝑘=1,2,,𝑛. Therefore, the system in the new-state variables is described by 𝑛 set of equations as follows:̇𝑧2𝑘1=𝑧2𝑘,̇𝑧2𝑘=𝜁𝑘(𝑧)+𝑛𝑗=1𝜗𝑘,𝑗(𝑧)𝜏𝑗,𝑦𝑘=𝑧2𝑘1,𝑘=1,2,,𝑛.(6) This system corresponds to the 𝑘th joint, and it is fully linearizable via feedback, and 𝜁𝑘(𝑧)=𝐿2Ψ𝑘(Φ(𝑧)1), 𝜗𝑖,𝑗(𝑧)=𝐿Γ𝑗𝑘(Φ(𝑧)1). These functions represent the dynamics of the system and the way as the control is acting onto the system, respectively. Note that the friction term, the Coriolis, centrifugal forces the gravity effect, and all possible non modeled dynamics are in function 𝜁𝑘(𝑧) and the inertia terms are in function 𝜗𝑘,𝑗(𝑧). From system (6), the vector of control torques is calculated as follows: 𝑇=𝒜𝜒1(𝜒)(𝑧)+𝐪𝑟(2)+𝐊1𝐙1𝐪𝑟+𝐊2𝐙2𝐪𝑟(1),(7) where (𝑧) is the vector of functions 𝜁𝑘(𝑧)=𝐿2Ψ𝑘(Φ(𝑧)1), 𝐪𝑟 is the reference angular position vector, 𝐊1 and 𝐊2 are control gain matrices, and 𝐙1 and 𝐙2 are the vector of transformed angular positions and velocities, respectively. This controller is called the perfect controller since it requires the perfect knowledge of the dynamics (𝑧), the angular velocities 𝐙2, and the matrix 𝒜𝜒1(𝜒) in order to track the reference trajectory. In this sense such a controller is not realistic and cannot be implemented. Thus, we assume that only the angular positions are available for feedback and that ̂𝜗sign[𝑘,𝑗(𝑧)]=sign[𝜗𝑘,𝑗(𝑧)] at any 𝑧𝑈02𝑛 of 𝑧0 is known. Note that it is not necessary to know a nominal value of the inertia matrix.

Thus, system (6) can be written in an extended form defining 𝛿𝑘(𝑧)=𝜗𝑘,𝑗̂𝜗(𝑧)𝑘,𝑗(𝑧), Θ𝑘(𝑧,𝑢)=𝜁𝑘(𝑧)+𝛿𝑘(𝑧)𝜏𝑗, and 𝜂𝑘=Θ𝑘(𝑧,𝜏𝑗). Thus system (6) takes the form ̇𝑧2𝑘1=𝑧2𝑘̇𝑧2𝑘=𝜂𝑘+𝑛𝑗=𝑖̂𝜗𝑘,𝑗(𝑧)𝜏𝑗,̇𝜂𝑘=Ξ𝑘𝑧,𝜂𝑘,𝜏𝑗,̇𝜏𝑗𝑦𝑘=𝑧2𝑘1,𝑘=1,2,,𝑛,(8) where the function Ξ𝑘()=Σ𝑛𝑘=1((𝜕Θ𝑘()/𝜕𝑧2𝑘1)𝑧2𝑘+(𝜕Θ𝑘()/𝜕𝑧2𝑘)(𝜂𝑘+Σ𝑛𝑗=1̂𝜗𝑘,𝑗(𝑧)𝜏𝑗)+𝛿𝑘(𝑧)̇𝜏𝑗). The new state 𝜂𝑘 provides the dynamics of the uncertain terms, such as the friction and the uncertain parameter values [9].

System (8) is dynamically equivalent to system (6) and comprises the lumping state; however, such a state is again unknown. Although the state 𝜂𝑘 is unknown, there are only two unknown variables, the new-state 𝜂𝑘 and the state 𝑧2𝑘 which corresponds to the angular velocity.

At this point the controller is given by 𝑇=𝒜𝜒1(𝜒)𝐇+𝐪𝑟(2)+𝐊1𝐙1𝐪𝑟+𝐊2𝐙2𝐪𝑟(1),(9) where 𝐇 is the vector with entries 𝜂𝑘 and it cannot be implemented since the states 𝜂𝑘 and 𝑧2𝑘 are not available for feedback. To tackle this problem we use a state estimator which provides an estimated value of 𝜂𝑘 and 𝑧2𝑘 [9]. The state estimator for the 𝑘th link is given by ̇̂𝑧2𝑘1=̂𝑧2𝑘+𝜆𝑘𝜅1,𝑘𝑧2𝑘1̂𝑧2𝑘1̇̂𝑧2𝑘=̂𝜂𝑘+𝑛𝑗=𝑖̂𝜗𝑘,𝑗(̂𝑧)𝜏𝑗+𝜆2𝑘𝜅2,𝑘𝑧2𝑘1̂𝑧2𝑘1,̇̂𝜂𝑘=𝜆3𝑘𝜅3,𝑘𝑧2𝑘𝑖̂𝑧2𝑘1,𝑦𝑒,𝑘=̂𝑧2𝑘1,𝑘=1,2,,𝑛,(10) With these estimates the control command can be given by 𝑇=𝒜𝑒1𝐇+𝐪𝑟(2)+𝐊1𝐙1𝐪𝑟+𝐊2𝐙2𝐪𝑟(1),(11) where 𝒜𝑒 is an estimated matrix where its elements satisfy the condition ̂𝜗sign[𝑘,𝑗(𝑧)]=sign[𝜗𝑘,𝑗(𝑧)]. This controller only requires the value of the angular position vector 𝐙1, the time derivatives of the reference signals 𝑦𝑟,𝑘, and the estimated values of the angular velocity. The estimation parameters 𝜆𝑘 and 𝜅𝑖,𝑘 are the high gain parameters and the state estimator gains, respectively, which are chosen in such way that the close-loop system be stable.

Proposition 2. Let 𝑒𝑘3 be an estimation error vector whose components are defined as 𝑒1,𝑘=𝜆𝑘(𝑧2𝑘1̂𝑧2𝑘1), 𝑒2,𝑘=𝜆2𝑘(𝑧2k1̂𝑧2𝑘1), and 𝑒3,𝑘=𝜂𝑘̂𝜂𝑘. For sufficiently large 𝜆𝑘, the dynamics of the estimation error decays globally exponentially to zero if ̂𝜗sign(𝑘,𝑗(𝑧))=sign(𝜗𝑘,𝑗(𝑧)).

Proof. Combining systems (8) and (10), the dynamics of the estimation error can be written as follows: ̇𝑒𝑘=𝜆𝑘𝐃𝑒𝑘+Δ𝑧,𝜂𝑘,𝜏𝑒,𝑗,(12) where Δ(𝑧,𝜂𝑘,𝜏𝑒,𝑗)=[0,0,Ξ(𝑧,𝜂𝑘,𝜏𝑒,𝑗)]𝑇 and the matrix 𝐃 is 𝐃=𝜅1,𝑘10𝜅2,𝑘01𝜅3,𝑘00,(13) since ̂𝜗sign(𝑘,𝑗(𝑧))=sign(𝜗𝑘,𝑗(𝑧)), thus 𝐃 is obviously Hurwitz; thus there exists a positive definite and symmetric matrix 𝐏 such that 𝐏𝐃+𝐃𝑇𝐏=𝐐, with 𝐐 a positive definite and symmetric matrix. Choosing 𝑉(𝑒𝑘)=𝑒𝑇𝑘𝐏𝑒𝑘 as the Lyapunov-like function candidate, one has ̇𝑉𝑒𝑘=̇𝑒𝑇𝑘𝐏𝑒𝑘+𝑒𝑇𝑘𝐏̇𝑒𝑘=𝜆𝑘𝑒𝑇𝑘𝐏𝐃+𝐃𝑇𝐏𝑒𝑘+2𝑒𝑇𝑘𝐏Δ𝑧,𝜂𝑘,𝜏𝑒,𝑗̇𝑉𝑒𝑘𝜆𝑘𝜎min𝑒(𝐐)𝑘2+2𝜎max𝑒(𝐏)𝑘Δ𝑧,𝜂𝑘,𝜏𝑒,𝑗,̇𝑉𝑒𝑘𝜆𝑘𝜎min𝑒(𝐐)𝑘2+2𝜎max𝑒(𝐏)𝑘𝑟,(14) where 𝜎min(𝐐) and 𝜎max(𝐏) are the minimum and maximum eigenvalues of the matrices 𝐐 and 𝐏, respectively, and the function Δ(𝑧,𝜂𝑘,𝜏𝑒,𝑗) satisfies Δ(𝑧,𝜂𝑘,𝜏𝑒,𝑗)𝑟 for some 𝑟>0. Therefore, the error 𝑒𝑘 decays, and it can be seen that it is ultimately bounded. Also note that as 𝜆𝑘 increases, 𝑒𝑘 will decrease, which also decreases the exponential estimation error bound; therefore, 𝜆𝑘 should be made as large as possible, and this achieves the proof.

It is important to note that the proposition is valid for 𝑘=1,2,,𝑛, therefore it is possible to obtain an estimated value for the unknown states. The previous proposition states that there exists a Lyapunov function which depends on the parameter 𝜆𝑘 and provides the global exponential convergence of the estimation error dynamics.

Proposition 3. System (1) asymptotically tracks a prescribed sufficiently smooth reference under the feedback (11) via the stabilization of the extended system (8).

Proof. Substituting the robust feedback (9) into (8) and considering the estimation error system (12), the closed-loop dynamics is given by ̇𝑧2𝑘1=𝑧2𝑘,̇𝑧2𝑘=𝜂𝑘+𝑛𝑗=𝑖̂𝜗𝑘,𝑗(𝑧)𝜏𝑗,̇𝜂𝑘=Ξ𝑘𝑧,𝜂𝑘,𝜏𝑗,̇𝜏𝑗,̇𝑒𝑘=𝜆𝑘𝐃𝑒𝑘+Δ𝑧,𝜂𝑘,𝜏𝑗.(15) Since 𝜂𝑘=Θ𝑘(𝑧,𝜏𝑗) and 𝜏𝑗̂𝜗=(1/𝑗(𝑧))(̂𝜂𝑘+𝜈𝑘), it follows that 𝜂𝑘=𝑘(𝑧,𝜂𝑘,𝑒𝑘,𝜏𝑗,𝑡) (which can be computed from the first integral of ̇𝜂𝑘=Ξ𝑘(𝑧,𝜂𝑘,𝜏𝑗,̇𝜏𝑗); i.e, 𝜂𝑘=Ξ𝑘(𝑧,𝜂𝑘,𝑒𝑘,𝜏𝑗,𝜎)𝑑𝜎). Then, according to the contraction mapping theorem the state 𝜂𝑘 can be expressed globally and uniquely as a function of the coordinates (𝑧,𝑒𝑘). Now, note that since the matrix 𝐃 is Hurwitz by construction and the nonlinear function Δ(𝑧,𝜂𝑘,𝜏𝑗) is bounded, the estimation error system (12) is asymptotically stable. In this sense given a compact set of initial conditions Ω𝑘3 containing the origin, there exists an upper bound 𝜏𝑗,max such that 𝜏𝑗𝜏𝑗,max and a high-gain estimator parameter 𝜆𝑘 such that Ω𝑘 is contained into the attraction basin. Hence, the closed-loop system is semiglobally practically stable; that is, (𝑒𝑘,𝜂𝑘)(0,0), and this achieves the proof.

The previous proposition is a simple strategy for robust tracking of robot manipulators, despite the uncertain knowledge of the models.

4. Numerical Simulations

To corroborate the performance of the robust controller in the effects of parameter variations, modeling errors, and external perturbations, we propose to apply the robust controller to a RR manipulator (rotational-rotational, 𝑛=2,𝑞=[𝑞1,𝑞2]𝑇). We propose a smooth chaotic trajectory as a reference, which is obtained from the solutions of the Duffing chaotic system ̇𝑦1=𝑦2,̇𝑦2=𝑦1+0.25𝑦2+𝑦3,𝑌𝑅=𝒞𝑦.(16) This system provides irregular, unpredictable but bounded smooth trajectories in the cartesian coordinates of the manipulator; therefore using the inverse kinematics transformation the reference trajectories in the articular space are given by the functions 𝑞𝑟=[𝜑1(𝑌𝑅)𝜑2(𝑌𝑅)]𝑇. In this case we used these trajectories in order to illustrate that the controller makes the manipulator tracks an irregular trajectory. The reference trajectories are illustrated in Figure 1.

Figure 1: Reference irregular (chaotic) trajectory used to be followed by the RR manipulator.

For the RR configuration and from (3), we have 𝑃𝑀(𝑞)=1+𝑃2+2𝑃3cos𝑞2𝑃2+𝑃3cos𝑞2𝑃2+𝑃3cos𝑞2𝑃2,𝑉(𝑞,̇𝑞)=2𝑃3sin𝑞2̇𝑞1̇𝑞2𝑃3sin𝑞2̇𝑞22𝑃3sin𝑞2̇𝑞22,𝑔𝐺(𝑞)=1cos𝑞1+𝑔2𝑞cos1+𝑞2𝑔2𝑞cos1+𝑞2,(17) where 𝑃𝑖, 𝑔𝑗,𝑖=1,2,3,𝑗=1,2 are constant parameters given by 𝑃1=𝑚1𝑙𝑐𝑚21+𝑚2𝑙21+𝐼𝑧𝑧1, 𝑃2=𝑚2𝑙𝑐𝑚22+𝐼𝑧𝑧2, 𝑃3=𝑚2𝑙1𝑙𝑐𝑚2, 𝑔1=(𝑚1𝑙𝑐𝑚1+𝑚2𝑙1)𝑔, and 𝑔2=𝑚2𝑙𝑐𝑚2𝑔 and are considered unknown. Now, performing the change of variable 𝑥1(𝑡)=[𝑞1,𝑞2]𝑇=[𝜒1,𝜒2]𝑇 and 𝑥2(𝑡)=[̇𝑞1,̇𝑞2]𝑇=[𝜒3,𝜒4]𝑇, the system in affine form is written as follows: ̇𝜒1=𝜒3,̇𝜒2=𝜒4,̇𝜒3=𝑓1(𝜒)+𝑀11𝜏1+𝑀12𝜏2,̇𝜒4=𝑓2(𝜒)+𝑀21𝜏1+𝑀22𝜏2,𝑌=𝒞𝜒,(18) where 𝑓1(𝜒)=𝔪11(𝑉1(𝜒)+𝐺1(𝜒)+𝐵1(𝜒))𝔪1,2(𝑉2(𝜒)+𝐺2(𝜒)+𝐵2(𝜒)) and 𝑓2(𝜒)=𝔪21(𝑉1(𝜒)+𝐺1(𝜒)+𝐵1(𝜒))𝔪22(𝑉2(𝜒)+𝐺2(𝜒)+𝐵2(𝜒)) are the nonlinear terms (considered uncertain) of the robot manipulator and the output is given by the matrix 𝒞=[𝐼2×2,02×2]𝑇. Notice that, in this nonlinear function appear the friction terms (2), 𝔪𝑘,𝑗 stand for the elements of the inverse inertia matrix, 𝑉1(𝜒)=2𝑃3sin𝜒2𝜒3𝜒4𝑃3sin𝜒2𝜒24, 𝑉2(𝜒)=𝑃3sin𝜒2𝜒24, 𝐺1(𝜒)=𝑔1cos𝜒1+𝑔2cos(𝜒1+𝜒2), and 𝐺2(𝜒)=𝑔2cos(𝜒1+𝜒2). Therefore, calculating the relative degree matrix (see Definition 1) for this system we have 𝒜𝜒=𝐿Γ1𝐿Ψ1(𝜒)𝐿Γ2𝐿Ψ1𝐿(𝜒)Γ1𝐿Ψ2(𝜒)𝐿Γ2𝐿Ψ2(=𝔪𝜒)11𝔪12𝔪21𝔪22=𝑀(𝜒)1.(19) Note that the relative degree matrix is invertible due to the positive definiteness of the inertia matrix; however, it is considered uncertain and a nominal model is not required. The parameter values used for this example but considered unknown in the controller were for link 1: 𝑙1=0.35m, 𝑙𝑐𝑚1=0.175m, 𝐼𝑧𝑧1=0.0064Kgm2, 𝑚1=0.479Kg. For link 2: 𝑙2=0.30m, 𝑙𝑐𝑚2=0.145m, 𝐼𝑧𝑧2=0.004Kgm2, 𝑚2=0.341Kg. Thus for 𝑘=2 the transformed extended system is written as follows: ̇𝑧1=𝑧2,̇𝑧2=𝜂1+̂𝜗1,1(𝑧)𝜏1+̂𝜗1,2(𝑧)𝜏2,̇𝜂1=Ξ1𝑧,𝜂1,𝜏𝑘,̇𝜏𝑘,̇𝑧3=𝑧4,̇𝑧4=𝜂1+̂𝜗2,1(𝑧)𝜏1+̂𝜗2,2(𝑧)𝜏2,̇𝜂2=Ξ2𝑧,𝜂2,𝜏𝑘,̇𝜏𝑘,𝑦1=𝑧1,𝑦2=𝑧3;(20) since we assume that only the measured states are available for feedback and by Propositions 2 and 3, we can design the state estimator as follows: ̇̂𝑧1=̂𝑧2+𝜆1𝜅1,1𝑧1̂𝑧1,̇̂𝑧2=̂𝜂1+̂𝜗1,1(̂𝑧)𝜏1+̂𝜗1,2(̂𝑧)𝜏2+𝜆21𝜅2,1𝑧1̂𝑧1,̇̂𝜂1=𝜆31𝜅3,1𝑧1̂𝑧1,̇̂𝑧3=̂𝑧4+𝜆2𝜅1,2𝑧2̂𝑧2,̇̂𝑧4=̂𝜂2+̂𝜗2,1(̂𝑧)𝜏1+̂𝜗2,2(̂𝑧)𝜏2+𝜆22𝜅2,2𝑧2̂𝑧2,̇̂𝜂2=𝜆32𝜅3,2𝑧2̂𝑧2;(21) thus, the required robust control vector is given by 𝜏1𝜏2=𝐴𝑒1̂𝜂1+𝑞(2)𝑟,1+𝜈1̂𝜂2+𝑞(2)𝑟,2+𝜈2,(22) where 𝜈1 and 𝜈2 are the new control inputs and are given by 𝜈1𝜈2=𝐾1,1𝑧1𝑞𝑟,1+𝐾2,1̂𝑧2𝑞(1)𝑟,1𝐾1,2𝑧3𝑞𝑟,2+𝐾2,2̂𝑧4𝑞(1)𝑟,2.(23) Therefore, with these controllers the RR robot manipulator tracks the trajectory given by the Duffing equation. The tracking of the reference trajectory is illustrated in Figure 2, where 𝑌𝑟 and 𝑌 are the reference and the robot manipulator trajectories, respectively, and 𝑌 is obtained by the kinematics transformation. Note that the perturbation torques do not affect the performance. That is, the perturbations are rejected by the controller. The error in the robot workspace is illustrated in Figure 3, where it remains close to the origin even in the presence of the uncertainties.

Figure 2: Tracking of the reference trajectory, using the robust approach.
Figure 3: Tracking error in the robot workspace.

In Figure 4 the dynamics of the friction and perturbations in joints are illustrated. The friction was obtained with 𝐵𝜈=0.45, 𝐵𝑓1=0.1, 𝐵𝑓2=0.27, 𝜔1=0.005, and 𝜔2=0.0025 for the first link and 𝐵𝜈=0.45, 𝐵𝑓1=1, 𝐵𝑓2=2.7, 𝜔1=0.005, and 𝜔2=0.0025 for the second. The perturbation torques are given by a sinusoidal function of amplitude 8 Nm and 7 Nm for the first and second robot elements. Note that this friction parameters and the specific functional for the friction are completely unknown. Therefore, it can be considered as time varying the parameters and the friction parameter, without retuning the controller, that is, the tracking of the trajectory, is sustained.

Figure 4: Friction dynamics in each robot joint and external torque perturbations.

Finally, in Figure 5 the resulting control torques for the robot are illustrated. Note that the perturbation torques are compensated by the controllers; however, the controllers do not have any knowledge about them.

Figure 5: Control torques provided by the robust controller.

5. Conclusions

In this work we present a robust tracking control for the compensation of modeling errors, parameter variations, and external perturbation for a class of robot manipulators. The robust scheme is such that it only requires to measure the angular positions. In this sense, the controller can compensate the uncertainty in parameter values, uncertain knowledge of the model, and external perturbations via an augmented state. This approach does not require to estimate or adapt any parameter; it is only needed to know the sign of some parameters but not a nominal value. In this sense the control strategy is considered robust; in the case of the friction it is not required of the precise function to be compensated. The key feature is to lump the uncertain terms into a new state, which provides the dynamics of the uncertainties via a state estimator, such that instead of estimate every parameter it is only required to estimate the dynamic of the uncertainties. This robust control approach can be applied to other robot configurations provided that the relative degree matrix be invertible.


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