Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June - 212 73 Dynamic Loa Carrying Capacity of Spatial Cable Suspene Robot: Sliing Moe Control Approach M. H. Korayem Department of Mechanical an Aerospace Engineering, Science an Research Branch, Islamic Aza University, ehran, Iran E-mail: hkorayem@iust.ac.ir *Corresponing author M. Jalali Department of Mechanical Engineering, Science an Research Branch, Islamic Aza University, ehran, Iran E-mail: mahsa_jalali_mec87@yahoo.com H. ourajizaeh Robotic Research Laboratory, School of Mechanical Engineering, Iran University of Science an echnology, ehran, Iran E-mail:hami1363@yahoo.com Receive: 27 November 211, Revise: 9 May 212, Accepte: 2 May 212 Abstract: his paper proposes a control strategy for a cable-suspene robot base on sliing moe approach (SMC) which is face to external isturbances an parametric uncertainties. his control algorithm is base on Lyapunov technique which is able to provie the stability of the en-effecter uring tracking a esire path with an acceptable precision. he main contribution of the paper is to calculate the Dynamic Loa Carrying Capacity (DLCC) of a spatial cable robot while tracking a esire trajectory base on SMC algorithm. In finale, the efficiency of the propose metho is illustrate by performing some simulation stuies on the ICaSbot (IUS Cable Suspene Robot) which supports 6 DOFs using six cables. Simulation an experimental results confirm the valiity of the authors claim corresponing to the accurate tracking capability of the propose control, its robustness an its capability towar DLCC calculation. Keywors: Cable Suspene Robot, Dynamic Loa Carrying Capacity (DLCC), Sliing Moe Control Reference: Korayem, M. H., Jalali, M. an ourajizaeh, H., Dynamic Loa Carrying Capacity of spatial Cable Suspene Robot: Sliing Moe Control Approach, Int J of Avance Design an Manufacturing echnology, Vol. 5/No. 3, 212, pp. 73 81. Biographical notes: M. Habibneja Korayem receive his BSc. (Hon) an MSc in mechanical engineering from the Amirkabir university of technology in 1985 an 1987, respectively. He has obtaine his PhD egree in mechanical engineering from the university of Wollongong, Australia, in 1994. M. Jalali receive her BSc. in robotic engineering from Shahroo university of technology in 28. She has obtaine her MSc. in mechatronic engineering from Islamic Aza university, science & research branch in 211. H. ourajizaeh receive his BSc. in mechanical engineering from KN university of technology in 26. He has obtaine his MSc. from Iran university of science an technology in 28 in the fiel of applie mechanical esign. He is now PhD caniate of IUS at the same fiel, branch of control an vibration. 212 IAU, Majlesi Branch
74 Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June 212 1 INRODUCION Cable transporter systems are wiely use in inustry such as high-rise elevators, cranes, conveyer belts, an tethere satellite systems, etc. A cable-suspene robot typically consists of a moving platform that is connecte to a fixe base by several cables. A cablesuspene robot can precisely manoeuvre large loas an is resistant to environmental perturbations. he main avantages of cable suspene robots over conventional robots are: 1) larger workspace for the same overall imension of the robot; 2) lightweight cables resulting in very safe an transportable system; an. he spatial sample is one particular type of cable robot with six cables. Alp an Agrawal [1] aresse the kinematic an ynamic moels, workspace, trajectory planning, an feeback controllers for parallel cable manipulators, an emonstrate through simulation an experiments on a six-dof cable suspene robot. However, cables have the unique property. hey cannot provie compression force on an en-effecter. his constraint leas to performance eterioration an even instability, if not properly accounte for in the esign proceure. Several techniques have been suggeste to guarantee positive tension in the cables while the en-effecter is moving [1], [2]. In aition, using the null space of the Jacobian, workspace was stuie that have positive cable tensions [3].he statics workspace is efine as the set of positions wherein any operational force can be exerte on the en-effecter with all positive cables tensions [4]. here have been a number of researchers who have applie varie controller an ynamic moelling for cable-actuate robots such as the feeback linearization control metho. An algorithm towar calculation of Dynamic Loa Carrying Capacity (DLCC) for rigi cable robot which is uner controlling a close loop controller was presente [5]. Also esigning a sliing moe controller as a stabilizing controller for the given uncertain system has been use [6], fining the range of system states in terms of set points, substituting states in inequalities of the input so that constraints are satisfie. In this paper, sliing moe approach is employe for stabilization an tracking a preefine trajectory. Also the DLCC of a cable robot is evaluate in a close loop way using SMC metho. Sliing moe is a nonlinear feeback control with variable structure with respect to the system states. he main avantage of sliing moe control is that the system is insensitive to extraneous isturbance an internal parameter variations while the trajectories are on the switching surface. here are two important factors that shoul be consiere while calculating the DLCC of a robot at this metho an they are the maximum torques that can be applie by motors an the maximum acceptable bouns of errors that en-effecter is permitte to move within [7]. he require constraints can be easily satisfie by the ai of propose iterative algorithm in this paper which is base on SMC approach. Unlike general control laws, sliing moe control is more robust an is easy to be implemente. Sliing Moe Control algorithm as a robust control metho can also be iscusse for ientification of maximum ynamic loa carrying capacity, in the presence of isturbance an system uncertainties [8]. he paper is organize as follow: first ynamic equations of spatial cable robot are erive. Next a Sliing Moe Control as a powerful metho of uncertain nonlinear systems in the conition of absence of isturbance an in the presence of isturbance is presente. Finally an algorithm is propose to compute maximum allowable loa by consiering the limiting factors. In the last section some simulation results are erive an are compare with experimental tests in orer to show the valiation of the propose theories. 2 DYNAMICAL MODELLING For the spatial case, assume a triangular shape eneffecter like Fig. 1, which is suspene through 6 cables an has 6 egrees of freeom as x, y, ψ, Θ, φ. It is provable that the ynamical equation may be shown as [1]: Fig. 1 Spatial moel of cable robot DX ( ) X&& + CX (, X& ) X& + gx ( ) = S ( qx ( )) ; (1) where: 212 IAU, Majlesi Branch
Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June - 212 75 mi3 D= q i S = x j 3 ; C = ; g = ; P IP P { IP& & ο + ( P& ο) I( Pο)} mg 3 1 sinθ Ψ& ; P = cosψ sinψ cosθ & ; ο = Θ & j sinψ cosψ cosθ & i φ (2) is the vector of cables tension, X is the vector of DOFs of the system, m is the mass of the en-effecter, I is the moment of inertia of the en-effecter an q is the length of the cables. Furthermore, the ynamics of the motor is as follow: = 1/ r( τ J( / ( β/ X) X& + X&& ( β/ X)) C( β / X) X& ) (3) where J is the matrix of rotary inertia of the motors, c is the viscous friction matrix of the motors, β& is the vector of angular velocity of the motors an τ is the vector of motors torque. By coupling these two ynamics, we have: = 1/ r( τ J( / ( β/ X) X& + X&& ( β/ X)) C( β/ X) X& ) DX ( ) X&& + CX (, X& ) X& + gx ( ) = S ( qx ( )) ; 3 CONROL SCHEME he following equation is consiere: n 1 X f ( X ) b( X ) u (4) = + (5) In which u eq is Equivalent Control Law (as efine with Fillipov law). It is noticeable that by applying this control law we actually apply ss& < ε ( ε<) conition. In particular, the Lyapunov sliing conition forces system states to reach a hyperplane an keeps them sliing on this hyperplane. Essentially, a SMC esign is compose of two phases: hyperplane esign an controller esign. However, in this paper a metho propose by Slotine is use [9]. In this metho, the sliing surface is efine as: s = ( + λ)( x x ) = ( x& x& ) + λ( x x ) t (1) o etermine the control law, the erivative of the sliing surface must be etermine: s& = (&& x && x ) + λ( x& x& ) (11) Since sliing conition is efine by: s& ksing. ( s) (12) So, in orer to satisfy the sliing conition, Eq. (11) must be written as: (&& x && x ) + λ( x& x& ) = k. Sing( s) (13) By substituting X && from Eq. (1), Eq. (13) becomes: { λ } D( X) X&& ( X& X& ) k. Sing( s) + C( X, X& ) X& + g( X) = S ( q( X)) (14) As well as the following sliing surface: s t 1 = ( + λ) n x% where: (6) Combining Eq. (3), (14) they can be expresse base on motor torque: { ( ) λ( & X& ) && & } τ = SJ rd ksign s + rd X rdx rcx rg + J && β+ c & β (15) ( n 1) X = [ x%, x%, x%,..., x% ] (7) x% = x x (8) As it is shown in Ref. [8], control law for tracking ( x% ) has the following form: U = u k. Sign( s) (9) eq 4 SIMULAION OF CONROL PROCEDURE o investigate the propose controller, some simulation stuies are presente for spatial robot. In these stuies, a circular trajectory is assume that en-effecter with controller follow the path perfectly. he simulation results are shown in Figures. he parameters use in simulation are given in ables 1 an 2. 212 IAU, Majlesi Branch
76 Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June 212 able 1 Reference input for spatial simulation t <= 4, x =.5*cos(4*pi*(t^2)/64), y =.5*sin(4*pi*(t^2)/64), t > 4 x =.5*cos(4*pi*((-t + 8)^2)/64) y = -.5*sin(4*pi*((-t + 8)^2)/64) z =. 45 ; ψ = Θ = ; ϕ =.9.8.7.6.5.4.3.2 MOOR1 MOOR2 MOOR3 MOOR4 MOOR5 MOOR6 able 2 Characteristics of spatial system Name Symbol Value Unit Momentum of 2 I Ixx = Iyy = 1847779.15*1 kg.m inertia of the eneffecter Izz = 2* Ixx Control λ iag [2] Coefficient Control gain K iag [2] Raius of the r iag [.15] m motor Damping c iag [.1] N. m/ ra Coefficient ( 9) Momentum of J 2 iag[ 339.21*1 ] kg.m inertia of the pulley Mass of the eneffecter m 1 kg Motor input-output path, torque, cable tension an error profile will be as bellow: Cable ension(n).1 1 2 3 4 5 6 7 8 3.5 3.4 3.3 3.2 3.1 3 2.9 2.8 2.7 cable1 cable2 cable3 cable4 cable5 cable6 2.6 1 2 3 4 5 6 7 8.5 1 x 1-7.5.4 input circle output circle y(m).3.2.1 Error(m) -.5-1 -1.5 x Error y Error z Error eta Error Fi Error Si Error -.1 -.2 -.3 -.4 -.5 -.5 -.4 -.3 -.2 -.1.1.2.3.4.5 x(m) Fig. 2 Input-output path of spatial simulation in absence of isturbance It is observable that all of the cable tensions are positives as was expecte. Practical robotic systems have inherent system perturbations such as parametric uncertainties an external isturbances such as static friction, noise in control signals, etc. -2 1 2 3 4 5 6 7 8 Fig. 3 orques, tensions, an error profiles of spatial simulation in absence of isturbance In this paper, the behaviour of the cable robot in the presence of external isturbances is consiere. Let us enote the isturbances as =.5 Sin(5 t). he real ynamic equation of the cable robot with the isturbance term is represente as follows: { [ ( )] λ & ( & ) τ = S 1 rd ksign s + rd X X rdx&& rcx& rg + S J && β + S C & β} + (16) 212 IAU, Majlesi Branch
Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June - 212 77 Dynamic response of the system in presence of isturbance is shown in Figs. 4 an 5..5.4 input circle output circle.3.2.1 x Error y Error z Error eta Error Fi Error Si Error.3.2.1 Error(m) -.1 y(m) -.1 -.2 -.2 -.3 -.4 Fig. 4 -.5 -.6 -.4 -.2.2.4.6 x(m) Cable ension(n).9.8.7.6.5.4.3.2.1 Input-output path of spatial simulation in presence of isturbances MOOR1 MOOR2 MOOR3 MOOR4 MOOR5 MOOR6 1 2 3 4 5 6 7 8 4.2 4 3.8 3.6 3.4 3.2 3 2.8 2.6 cable1 cable2 cable3 cable4 cable5 cable6 1 2 3 4 5 6 7 8 -.3 1 2 3 4 5 6 7 8 Fig. 5 orques, tensions, an error profiles of spatial simulation in presence of isturbances It can be seen that the estructive external isturbance is successfully filtere by the ai of propose SMC controller since the fluctuations are neutralize by automatically switching of actuators torque which eventually provies a smooth tracking with an acceptable error. 5 DEERMINING MAXIMUM LOAD_CARRYING CAPACIY he ynamic loa-carrying capacity (DLCC) of a robot is efine as the loa that the cables can carry on a efine trajectory. he ynamics of cable robot can be use to exten its payloa capability while taking into account torque an tension as realistic constraints. By consiering the actuator torque an accuracy constraints an aopting a logical computing metho, the maximum loa-carrying capacity of a cable robot for a preefine trajectory can be compute. he actuator torque constraint is formulate on the basis of typical torque-spee characteristics of DC motors. τ = k k q& ; τ = k k q& (17) u 1 2 l 1 2 whereτ u an τ l are the upper boun an the lower boun of actuator constraint, respectively. he coefficients k i are efine as k = ; k = / w (18) 1 s 2 s nl where is the stall torque an s w nl is the maximum no-loa spee of the motor. he algorithm use for fining DLCC in close-loop case is shown in Fig. 6. 212 IAU, Majlesi Branch
78 Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June 212 SAR.4.3 Choose an initial value for m for a given p Compute control input with SMC controllerτ Eq. (15).2.1 -.1 -.2 MOOR1 -.3 Compute the actuators bouns base on Eq. (16), (17) -.4 1 2 3 4 5 6 7 8.4.3 If τ < τ < τ Check the accuracy of en-effecter tracking l u It s violate DLCC = m p.2.1 -.1 -.2 MOOR2 It s not violate -.3 Increase m p -.4 1 2 3 4 5 6 7 8 Fig. 6 Flowchart of computing ynamic loa carrying capacity he actuator torque at each point is compute, an using the computational proceure, the upper an lower bouns on torque available for ynamic loa an the accuracy of en effecter tracking is etermine. Using this metho we can fin DDLC in each time so it will be ientifie for whole trajectory. 6 SIMULAION OF HE DLCC ALGORIHM.4.3.2.1 -.1 -.2 -.3 MOOR3 -.4 1 2 3 4 5 6 7 8.4 Saturation motors torque are presente in Fig.7, an as it can be seen the first an fifth motors are saturate at the mile of the simulation. As it is shown in Fig. 8 that error constraint is saturate for the system. By applying the propose algorithm for close loop plant the maximum allowable loa compute as m 4.7kg p =..3.2.1 -.1 -.2 -.3 MOOR4 -.4 1 2 3 4 5 6 7 8 212 IAU, Majlesi Branch
Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June - 212 79.4.3.2.1 -.1 MOOR5 -.2 -.3 -.4 1 2 3 4 5 6 7 8.4.3 Fig. 9 Scheme of the esigne IUS cable robot.2.1 -.1 MOOR6 he geometrical properties of the cable suspene robot in IUS are liste in able 3. able 3 features of the esigne IUS cable robot -.2 -.3 Fig. 7 -.4 1 2 3 4 5 6 7 8.1 Saturation profile of motor torques of spatial simulation Boy Height Sie length of base triangle Weight En effecter Sie length of base triangle hickness weight 12cm 1-2 cm 35 kg 17 cm 8 cm 1,1 gr Error(m).8.6.4.2 -.2 he controller achieves the esire steay-state values with a small error, which may be ue to system uncertainties such as friction in the pulley. racking of the en-effecter in comparison to simulation result is shown in Fig. 1. -.4 -.6 -.8 -.1 1 2 3 4 5 6 7 8 Fig. 8 Error profiles of spatial simulation 6 EXPERIMENAL ESS In this step, these simulation results shoul be verifie by experiment. o provie an example of the possible uses of spatial cable suspene robot which is esigne an manufacture in IUS supporting six DOFs (Fig. 9), a simple curve experiment is performe. Fig. 1 racking of the en-effecter in comparison to simulation results Relate angular velocity of the motors an its comparison to simulation results are also shown in Fig. 11. 212 IAU, Majlesi Branch
8 Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June 212 Fig. 11 Angular velocity of the motors an its comparison to simulation results As it is shown in Fig. 11, the system response of the experiment shows a goo match with simulation results. Cable tensions are shown in Fig. 12. 212 IAU, Majlesi Branch
Int J Avance Design an Manufacturing echnology, Vol. 5/ No. 3/ June - 212 81 cable systems, but also provies a goo close loop calculation of DLCC. Moreover, experimental results emonstrate the valiity of the propose controller. he results showe a goo match between the theory an the experiment. ACKNOWLEDGMENS his paper escribes a research one in Robotic Research Laboratory, College of Mechanical Engineering, Iran University of Science an echnology (IUS). Authors thank IUS University for supporting the program uring their research activity. In aition, the authors woul like to appreciate robotics team for their cooperation. Fig. 12 Cable tensions in comparison to simulation results Accoring to these results it can be seen that experimental results are consierably compatible with simulation results an thus a trajectory tracking can be easily one by the ai of propose close loop controller base on SMC approach. he efficiency of the propose metho is illustrate by simulations an laboratory experiments on a six egree-of freeom cable suspene robot. herefore, in this paper Sliing Moe Controller (SMC) as a robust control algorithm is use for controlling the stability of the system while tracking a esire trajectory. he avantage of the propose algorithm is calculation of the DLCC of the cable robot using close-loop computational technique base on SMC algorithm whereas in recent works, open-loop methos were use to calculate this parameter. 7 CONCLUSION his paper aresse the issue of controlling the six egrees-of-freeom spatial cable robot of IUS. A sliing moe control was implemente on the mentione uner constraine cable robot as a stabilizing controller which is face to external isturbances an parametric uncertainties. his paper presente an iterative approach for calculation of the Dynamic Loa Carrying Capacity of cable suspene robots in a close loop way an base on SMC approach. It was seen that the estructive effect of isturbances can be consierably neutralize by the ai of propose controller. he maximum payloa of the robot, consiering parametric uncertainties for the close-loop system was also calculate which was equal to 4.7 (kg). Simulation results have prove that not only SMC is applicable for accurate tracking of the REFERENCES [1] Alp, A. B. an Agrawal, S. K., Cable suspene robots: esign, planning an control, in proceeing of international conference of robotics an automation, Washington, DC, 22, pp. 4275 428. [2] Oh, S. an Agrawal, S. K., Cable-suspene planar robots with reunant cables: controllers with positive cable tensions, IEEE transaction on robotics, Vol. 21, No. 3, Jun. 25, pp. 457 464. [3] Williams II, R. L. an Gallina, P., Planar cableirect-riven robots, part I: kinematics an statics, in proceeing of ASME esign technical 27th esign. automation conference., Pittsburgh, DEC21/DAC-21 145, 21. [4] Gallina, P., Rossi, A. an Williams II., R. L., Planar cable-irect-riven robots, part ii: ynamics an control, in proceeing of the ASME esign engineering technical conference, Vol. 2, 21, pp. 1241 1247. [5] Korayem, M. H., ourajizaeh, H. an Bama, M., Dynamic loa carrying capacity of flexible cable suspene robot: robust feeback linearization control approach, Journal of Intelligent Robot an Systems, Vol. 6, 21, pp. 341-363. [6] Oh, S. R. an Agrawal, S. K., Sliing moe control an feasible workspace analysis for a cable suspene robot, in Proceeing amer. control conf., Boston, MA, 24, pp. 4631 4636. [7] Korayem, M. H., Firoozi, S. an Heiari, A., Dynamic loa carrying capacity of mobile-base flexible link manipulators: feeback linearization approach, IEEE international conference on robotic, 27. [8] Korayem, M. H. an Pilechian, A. Maximum ynamic loa carrying capacity in flexible joint robots using sliing moe control, international congress on manufacturing engineering (ICME25) ecember 12-15, ehran, Iran, 25. [9] Slotine, J. E. an Sastry, S. S., racking control of nonlinear systems using sliing surfaces, with application to robot manipulators, International.Journal of Control, Vol. 38, NO. 2, 1983, pp. 465 492. 212 IAU, Majlesi Branch