Adaptive Robust Control Of Fully-constrained Cable Driven .
Mechatronics 25 (2015) 27–36Contents lists available at ScienceDirectMechatronicsjournal homepage: www.elsevier.com/locate/mechatronicsAdaptive robust control of fully-constrained cable driven parallel robotsReza Babaghasabha , Mohammad A. Khosravi, Hamid D. TaghiradAdvanced Robotics and Automated Systems (ARAS), Industrial Control Center of Excellence (ICCE), Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Irana r t i c l ei n f oArticle history:Received 24 June 2014Accepted 16 November 2014Available online 11 December 2014Keywords:Cable driven parallel robotsAdaptive robust controlStructured and parametric uncertaintiesInternal forceStability analysisExperimental verificationa b s t r a c tIn this paper, adaptive robust control (ARC) of fully-constrained cable driven parallel robots is studied indetail. Since kinematic and dynamic models of the robot are partly structurally unknown in practice, inthis paper an adaptive robust sliding mode controller is proposed based on the adaptation of the upperbound of the uncertainties. This approach does not require pre-knowledge of the uncertainties upperbounds and linear regression form of kinematic and dynamic models. Moreover, to ensure that all cablesremain in tension, proposed control algorithm benefit the internal force concept in its structure. The proposed controller not only keeps all cables under tension for the whole workspace of the robot, it is chattering-free, computationally simple and it does not require measurement of the end-effector acceleration.The stability of the closed-loop system with proposed control algorithm is analyzed through Lyapunovsecond method and it is shown that the tracking error will remain uniformly ultimately bounded(UUB). Finally, the effectiveness of the proposed control algorithm is examined through some experiments on a planar cable driven parallel robot and it is shown that the proposed controller is able to provide suitable tracking performance in practice.Ó 2014 Elsevier Ltd. All rights reserved.1. IntroductionCable driven parallel robots (CDPRs) remedy some shortcomings of the conventional serial and parallel robots. Using cablesinstead of rigid links in the robot structure has some positive features such as large workspace [1,2], high speed manipulation [3],high payload to robot weight ratio [4], transportability and easeof assembly/disassembly. CDPRs can be classified into two types,namely, fully-constrained and under-constrained robots. In theunder-constrained type, a passive force such as gravity is used tokeep all the cables under tension. While in the fully-constrainedtype, the number of actuators are be at least one more than thenumber of robot degree of freedoms [5]. The cable driven parallelrobots which are under study in this paper are restricted to thefully-constrained type and it is assumed that the motion controlis performed in the wrench-closure workspace.Using actuation redundancy in the structure of the parallelrobots improves some of kinematic and dynamic properties ofparallel robots such as stiffness and singularity avoidance [6,7].Moreover, actuation redundancy is an important requirement infully-constrained CDPRs, since cables are able to apply only tensileforces. Hence, the well-known control theories of the parallel Corresponding author.E-mail addresses: Reza.bgha@mail.kntu.ac.ir (R. Babaghasabha), Makh@ee.kntu.ac.ir (M.A. Khosravi), Taghirad@kntu.ac.ir (H.D. .2014.11.0050957-4158/Ó 2014 Elsevier Ltd. All rights reserved.robots should be modified such that the cables remain in tensionfor the whole workspace of CDPRs. For this reason, control of thecable driven parallel robots is more challenging than that of theconventional robots.Motion control algorithms of CDPRs may be classified based onthe coordinates used in the design procedure into two categories,namely the cable length coordinates and the task space coordinates. In the cable length coordinates, decentralized controllersare designed on each of the actuated cables [3,8,9], and the lengthof the cables which are simply measured by the encoders is used inthe feedback structure. However, due to the inherent flexibility ofthe cables, using cable length in the feedback control loop is notreliable in applications with high accuracy.In the task space coordinates, the pose of the end-effector ismeasured directly and it is used as the feedback to the controller[10,11]. Implementation of such measurement is more challengingthan that of the cable length measurement. Moreover, it mayrequire expensive instrumentation system such as laser rangingequipment [10,12], or differential GPS (Global Positioning System)augmented by accelerometers and rate gyros [11], or multiplecamera with high resolution and high frame rate [13,14]. For thisreason, only few researches focus on implementation of the taskspace controllers in practice, which is considered in this paper indetail.Motion control algorithms may be classified based on thecontroller designing technique, as well. Within this classification,
28R. Babaghasabha et al. / Mechatronics 25 (2015) 27–36classic controllers such as PID [15] are computationally simple andthey do not require complete dynamic knowledge of the CDPRs.However, lack of consideration of dynamic effects in the structureof the controller may limit the tracking performance. Nonlinearcontrollers such as Lyapunov based methods [16,17] and inversedynamics control (IDC) [18,19] may improve the tracking performance. However, they require complete kinematic and dynamicmodels of CDPRs with detailed information of the true parameters.It shall be noted that in practice the kinematic and dynamic modelsof CDPRs possess structured and parametric uncertainties andprecise knowledge of the models is unavailable. These issues significantly limit the performance of the nonlinear controllers intracking objectives. The shortcomings of the parametric uncertainties may be remedied by using an adaptive controller in task spacecoordinates in which the kinematic and dynamic parameters aresimultaneously adapted, which is proposed in this paper.As a representative of adaptive methods, in [20] robust PD controller with an adaptive compensation term are designed to identify near true kinematic parameters. However, a large number ofkinematic parameters has been selected for adaptation and therefore, the controller is computationally expensive. In order toimprove performance of such adaptive methods, in [21,22] twoadaptive control scheme is proposed by considering uncertaintiesin dynamic and kinematic parameters. However, these approachesrequire the linear regression form of kinematic and dynamicmodels. Moreover, structured uncertainties of the kinematic anddynamic models and external disturbances, directly affect theupdated parameters and degrade the performance of thecontroller.In order to reduce the effect of both structured and parametricuncertainties, in [23] sliding mode controller is designed based onknowledge of the uncertainties upper bound of the kinematic anddynamic models. However, from a practical point of view, determination of the uncertainties upper bound is a prohibitive task, andtherefore, it is often over-estimated which yields to excessive controller gains. This fact amplify the main drawback of the slidingmode control, namely the chattering phenomenon, and significantly degrade the performance of the controller.The goal of this paper is to design a controller to improve theperformance of the fully-constrained cable driven parallel robotsin presence of structured and parametric uncertainties. Todevelop the idea, an adaptive robust sliding mode controller isdesigned based on the adaptation of the uncertainties upperbound. It is assumed that all terms in the dynamic and kinematicmodel of the robot are uncertain and the precise pre-knowledgeabout their upper bounds is also unavailable in practice. Theproposed controller will significantly remove the effect of excessive gain and it is chattering-free. Moreover, this approach doesnot require the linear regression form of kinematic and dynamicmodels. In addition, the internal force concept is used in theproposed controller to provide positive tension of the cables.The stability of the closed-loop system with proposed controlalgorithm is analyzed through Lyapunov second method. Finally,the effectiveness of the proposed controller is examined throughsome experiments on a planar cable driven parallel robot withfour actuated cable-driven limbs and it is shown that the proposed controller is able to provide suitable tracking performancein practice.The paper is organized as follows. In Section 2 some importantproperties of kinematic and dynamic models of the CDPRs aredenoted. Section 3 focuses on the controller design in which theproposed adaptive robust controller is introduced and the adaptation law is defined based on the adaptation of the uncertaintiesupper bound. Then, stability of the closed loop system is analyzedthrough Lyapunov second method. Finally, in Section 4 experimental results on a planar CDPR is discussed in detail.2. Robot kinematics and dynamicsCable driven parallel robot has a closed kinematic chain mechanism in which a number of actuated cables provide connectionbetween the base and the end-effector. Furthermore, the electricalactuators lead the end-effector toward a desired pose by changingthe length of the cables. In [15] kinematic and dynamic analysis offully-constrained CDPRs have been reported in detail. In this paper,we leave the details of the formulation and only denote someimportant properties of kinematic and dynamic formulations,which are used in the controller design. The dynamic model of afully-constrained CDPR without considering the flexibility of thecables may be written as: þ Cðx; xÞx þ GðxÞ þ NðxÞþ T d ¼ J T sMðxÞxð1Þin which,¼ F d x þ F s ðxÞNðxÞwhere x denotes the generalized coordinates vector for pose of thedenotes Coriend-effector, MðxÞ is mass matrix of the robot, Cðx; xÞolis and centrifugal terms and GðxÞ indicates the vector of gravityterms, F denotes the vector of Cartesian wrench, s is the vector ofcable forces and J denotes the Jacobian matrix of the robot. Furthermore, F d denotes the coefficient matrix of viscous friction, F s is theCoulomb friction term and T d denotes disturbance which may represent any inaccuracy in dynamic model. Although the dynamicmodel described by (1) is nonlinear and multi-input/multi-output(MIMO), it has some useful properties that are listed as follows [15]:Property 1. The mass matrix MðxÞ is symmetric, uniformly positivedefinite and bounded from above and below for all x by:m 6 kMðxÞk 6 mð2ÞProperty 2. Upper bound of the Coriolis and centrifugal matrix isindependent of x, and is a function of only x as:6 nc kxkkCðx; xÞkð3Þis skew-symmetric and 2Cðx; xÞProperty 3. The matrix MðxÞtherefore, for all Z: 2Cðx; xÞÞZZ T ðMðxÞ¼0ð4ÞProperty 4. Coulomb and viscous friction terms are dissipative andthey have an upper bound as:6 nf 0 kxkþ nf 1kF d x þ F s ðxÞkð5ÞProperty 5. Gravity vector and disturbance term have upper boundsof:kGðxÞk 6 ng ;kT d k 6 ntð6Þ3. Adaptive robust control of cable robotsIn this section, considering the structured and parametricuncertainties in kinematic and dynamic models of the robot, anadaptive robust control algorithm is proposed. This control algorithm consists of an adaptive robust sliding mode control whoseadaptation law is only based on the adaptation of only the uncertainties upper bound. To derive the control and adaptation laws,consider the following Lyapunov function candidate as:
29R. Babaghasabha et al. / Mechatronics 25 (2015) 27–36VðtÞ ¼1 T T Cq ÞðS MS þ q2ð7ÞfM¼cM M;in which, þ Kx S¼xð8ÞTwhere, S ¼ ½S1 ; . . . ; Sn is a sliding surface vector, C ¼ diagðC1 ; . . . ; Cn Þ and K ¼ diag ðK1 ; . . . ; Kn Þ are symmetric diagonal positive definite matrices, n denotes the degrees of freedom of the robot, ¼ x xd is the tracking error vector and q ¼q q denotes thexuncertainties upper bound estimation error vector. DifferentiateVðtÞ with respect to time:1 T CqVðtÞ¼ S T M S þ S T MSþq2 1 þ Kx Þ þ S T ðM T Cq 2CÞ þ C S þ q¼ S T Mðx2e ¼Nb NNkðJ T bJ T ÞQ k 6 nQkðI J T bJ y ÞFk 6 nJ kSkð18Þð19ÞAccording to the above inequalities and the properties of therobot dynamic model, which are denoted in Section 2, it may beconcluded that:6ð10ÞnXqi ðSi ; tÞð20Þi¼1asUsing this inequality one may rewrite VðtÞVðtÞ6nXjSi j½ kdi þ qi þ q i Ci q ið21Þi¼1Define the virtual reference trajectory asNow, assume that:t dtxe¼Gb G;Ge x r þ GeþNe Þ T d ðI J T bJ y ÞF ðJ T bJ T ÞQ k r þ CMxkð f þ CS þ q d þ MKx T CqVðtÞ¼ S T ½ J T s ðC x þ G þ N þ T d Þ M xZe¼Cb C;CThe uncertain Jacobian matrix bJ T , and the internal forces may beassumed to be bounded by the following relations:ð9Þ 2C and substitutionUsing skew-symmetricity property of Mfrom (1) yields to:xr ¼ xd Kin which,ð11Þ0kdi ¼ ki þ q ið22ÞDifferentiate xr twice with respect to time, and substitute intoEq. (10) yields to:in which, ki is a positive parameter, and substitute it into Eq. (21): r þ C x r þ G þ N þ T d Þ þ q T CqVðtÞ¼ S T ½ J T s ðM xVðtÞ6ð12ÞLet us propose the following adaptation law:ð13Þin which,8Si 0 1Si ¼ 0sgn ðSi Þ ¼ 0 : 1 Si 0VðtÞ6assumed that the attachment points are not precisely mounted inpractice. Therefore, Jacobian matrix of the robot is uncertain dueto the uncertainty in some of the kinematic parameters. The generalsolution of (13) for s isð14Þwhere s is the minimum norm solution of (13) which is derived byusing the pseudo-inverse of bJ T as: 1s ¼ bJ ðbJ T bJ Þ F ¼ bJ y Fð15ÞThe term Q may be physically interpreted as the internal forces thatspans the null space of bJ T , by whichbJ T Q ¼ 0ð16ÞInternal forces are used to keep all cables under tension withinthe whole workspace of the robot. Moreover, this term may beused to increase the robot stiffness [24]. Now, substitute the control law (14) into Eq. (12). This yields to:T by r þ C x r þ G þ N þ T d Þ þ q T CqVðtÞ¼ S ½ðJ J ÞF J Q ðM xTq i ¼ q i ¼ C 1i jSi jð24ÞBy using the above adaptation law, the resulting expression ofVðtÞis reduced to:b ; G;b Nb are available estimations of dynamic terms,where cM; CK D ¼ diag ðkd1 ; . . . ; kdn Þ is symmetric diagonal positive definitematrix and bJ T denotes estimated Jacobian matrix of the robot. It iss¼sþQð23Þi¼1Let us define the control law asb x r þ GbþNb K D sgn ðSÞ r þ CF ¼ bJ T s ¼ cMxnXjSi j½ ki q i þ q i Ci q inX ki jSi j 6 0ð25Þi¼1which is negative semi-definite. This result shows that the proposedcontroller can stabilize the system and the trajectories of the closedloop system will eventually converges to the sliding surface, þ Kx ¼0S¼xð26ÞTherefore, the proposed adaptive controller guarantees zerosteady-state tracking error. The main feature of this approach isthat the result is not in debt of finding a linear regression formfor kinematic and dynamic models, and furthermore, it does notrequire a priori knowledge on the uncertainties upper bound.As it can be seen in adaptation law (24), the value of q i is zeroonly when the value of Si is zero, otherwise the value of q i alwaysincreases. However, in practice, steady-state tracking error cannotremain on sliding surface due to the measurement noise. In thiscase, the control gain kdi is clearly over-estimated with respect touncertainties, which induces chattering. In order to avoid increasing the control gains, it is proposed to modify the control law (13)and the adaptation law (24) as following: b x r þ GbþNb K D tanh S r þ CMxF ¼ bJ T s ¼ c ð27ÞTand r þ C x r þ G þ N¼ S T ½ðJ T bJ y ÞF þ F F J T Q þ bJ T Q ðM x q i ¼ C 1i jSi j sgn ððjSi j i Þqi Þ T Cqþ T d Þ þ q rMx¼ S T ½ K D sgn ðSÞ ðI J T bJ y ÞF ðJ T bJ T ÞQ þ ð fe x r þ GeþNe Þ T d þ q T CqþCð17Þð28Þwhere ¼ diag ð 1 ; . . . ; n Þ is a threshold width on the S variableand it is chosen based on the measurement noise amplitude. Thismodification of control and adaptation law will significantlyremoves the effect of excessive gain and generates a chattering free
30R. Babaghasabha et al. / Mechatronics 25 (2015) 27–36output. This modification will induce the controller gains asfollows: If jSi j P i , then the sign of adaptation law (28) is positive andthe gain kdi is increasing up to a large enough value to counteract the bounded uncertainty, and as a result, the tracking errorapproaches the sliding surface. As long as the trajectories reaches within the boundary layer inwhich, the jSi j i , then the sign of adaptation law (28)becomes negative and the gain kdi reduces to a value which isproportional to the current uncertainties. With this adaptationlaw, the minimum value of kdi is set to ki . Now, if an additional load or an external disturbance is appliedto the robot and it makes jSi j P i , according to abovealgorithm, the control gain kdi is adjusted with respect to thecurrent uncertainties.Although such modification improve the performance of theproposed controller in practice, the stability analysis performedon the closed loop system is not valid any more, and this changein the control and adaptation law may affect the asymptotic tracking performance. In order to analyze stability of the closed loopsystem, consider the same Lyapunov function given in (7), takethe derivative with respect to time and substitute the modifiedcontrol law (27) and the modified adaptation law (28). ForjSi j P i , the same stability analysis for the closed loop systemholds and the trajectory tracking eventually converges to theboundary of sliding surface. However, as long as the trajectoryreaches within the boundary layer, and jSi j i , then the derivativeof the Lyapunov function with respect to time, given in (21)changes to:VðtÞ6 nXjSi jþ qi þ q i Ci q ijSi j kdii¼1 ið29ÞUse the modified adaptation law (28) and simplify as follows:VðtÞ6 nXjSi jþ ðqi q i ÞjSi j kdii¼1 ii¼1 i nXjSi jþ ð2qi q i ÞjSi j kdi6 nXjSi jþ 2qijSi j ki6i¼1 ið30ÞIn this case, VðtÞis negative semi-definite only ifjSi j 2qi i ¼ dsikið31ÞTherefore, the proposed controller can stabilize the system andthe tracking error will remain uniformly ultimately bounded(UUB). However, as it is seen in Eq. (31), the radius of ultimatesteady state tracking error dsi is dependent on the uncertaintiesupper bound qi , the parameter ki and the threshold i . This impliesthat in order to decrease the steady state tracking error, we mayincrease the gain ki and therefore, increase the required controleffort.4. Experimental resultsIn order to verify the effectiveness of the proposed adaptiverobust controller, it is applied to KNTU planar cable driven parallelrobot. This manipulator is under investigation for high speed andwide workspace applications in Advanced Robotics and AutomatedSystems (ARAS) group of K.N. Toosi University of Technology.4.1. Experimental setupKNTU planar cable driven robot consists of four actuated cabledriven limbs with three degrees of freedom planar motion which isshown in Fig. 1. Actuators are located on the vertices of a rectanglewith dimension of 2:24 m 2:1 m to provide a relatively largeFig. 1. KNTU planar cable-driven parallel robot.
31R. Babaghasabha et al. / Mechatronics 25 (2015) 27–36workspace. Inertial parameters of the planar cable robot is given inTable 1. Moreover, the actual tensions of the cables are measuredby the load cells located near the end-effector attachment points.TLL500 product from Transducer Techniques is used as suitableload cell in the experiments due to their relatively large measurement range and low weight.The block diagram of control system setup is shown in Fig. 2.The host computer serves as the user interface and enables theuser to edit and modify control structure and parameters. The target computer is a real time proces
In this paper, adaptive robust control (ARC) of fully-constrained cable driven parallel robots is studied in detail. Since kinematic and dynamic models of the robot are partly structurally unknown in practice, in this paper an adaptive robust sliding mode controller is proposed based on the adaptation of the upper bound of the uncertainties.
a fully-actuated subsystem and an under-actuated subsys-tem [9]. Then, he controlled them with a PID controller and a sliding mode controller, respectively. As a result, . Section 2 so that the adaptive robust control for the alti-tude control of the helicopter can be designed in Section 3. Then, Section 3 will discuss an adaptive robust .
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Adaptive Control, Self Tuning Regulator, System Identification, Neural Network, Neuro Control 1. Introduction The purpose of adaptive controllers is to adapt control law parameters of control law to the changes of the controlled system. Many types of adaptive controllers are known. In [1] the adaptive self-tuning LQ controller is described.
H Control 12. Model and Controller Reduction 13. Robust Control by Convex Optimization 14. LMIs in Robust Control 15. Robust Pole Placement 16. Parametric uncertainty References: Feedback Control Theory by Doyle, Francis and Tannenbaum (on the website of the course) Essentials of Robust Control by Kemin Zhou with Doyle, Prentice-Hall .File Size: 1MB
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