Optimal Trajectory Planning with the Dynamic Load Carrying Capacity of a Flexible Cable-Suspended Manipulator

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1 ransaction B: Mechanical Engineering Vol. 1, No., pp. 1{ c Sharif University of echnology, August 1 Research Note Optimal rajectory Planning with the Dynamic Load Carrying Capacity of a Flexible Cable-Suspended Manipulator Abstract. M.H. Korayem 1;, E. Davarzani 1 and M. Bamdad 1 his paper presents an indirect method for computing optimal trajectory, subject to robot dynamics, exibilities and actuator constraints. One key-issue that arises from mechanism exibility is nding the Dynamic Load Carrying Capacity (DLCC). he motion planning problem is rst formulated as an optimization problem, and then solved using Pontryagin's minimum principle. he basic problem is converted to the wo-point Boundary Value Problem (PBVP), which includes joint exibility. Some examples are employed to compare three models, dynamic, exible joint, and rigid. he results illustrate the eectiveness of this indirect method. Keywords: Path planning; Payload; Flexible cable-suspended manipulator; Optimal control; wo point boundary value problem. INRODUCION Several advantages distinguish cable-based systems from common manipulators. First, the mechanical architecture is rather simple and cost-eective even in the case of multiple DOF spatial systems. As a further consequence, these robots often present very high payload-to-weight ratios. RoboCrane, as one of the early works on cable-actuated manipulators, was developed based on the Stewart platform parallel link manipulator, providing a precise, six degrees of freedom control of loads. he Dynamic Load Carrying Capacity (DLCC) of a conventional serial manipulator is usually dened as the maximum load that a manipulator can lift and carry in a fully extended conguration. If the endeector trajectory is predened, DLCC would be the maximum value of load that the manipulator is able to carry [1,]. 1. Robotic Research Laboratory, College of Mechanical Engineering, Iran University of Science and echnology, P.O. Box 1, ehran, Iran. *. Corresponding author. hkorayemiust.ac.ir Received 1 September 9; received in revised form March 1; accepted May 1 he other denition of DLCC will be obtained by nding the maximum payload that a manipulator can carry between the given initial and nal position of the end-eector. In this case, nding the maximum payload and corresponding optimal path is formulated as a trajectory optimization problem []. In practice, the methods of solving deterministic optimal control problems are divided into two categories: direct and indirect methods. Direct methods are based on a discretization of dynamic variables (states, controls), leading to a parameter optimization problem, and it is solved by using one of the nonlinear programming codes. On the other hand, indirect methods are based on the minimum principle of Pontryagin. In this method, necessary conditions for optimality are found and the obtained equations establish a wo Point Boundary Value Problem (PBVP) that is solved by numerical techniques []. here have been a number of researchers who have developed trajectory planning and dynamic modeling for cable-actuated robots. Hiller et al. studied the workspace, forward kinematics and trajectory planning of one type of planer cable robot. In his paper, the optimal path is found by using a Powell algorithm and

2 1 M.H. Korayem, E. Davarzani and M. Bamdad the method of simulated annealing []. One typical spatial cable robot was investigated and one direct method was used for stiness optimization and optimal trajectory planning []. Recently, Korayem et al. have introduced a procedure for nding the optimal path of maximum load for a cable suspended robot and a cable planar robot [,]. hey used an indirect method and, as mentioned before, the optimization problem is converted to a two-point boundary value problem as by solving that, we can have a precise solution of the problem. his method could be used for any kind of system wherein the state space form of equations is achievable. his method is used as a competent tool for analyzing nonlinear systems and the path planning of dierent types of system. On the other hand, considering exibility in modeling, the exibility of joints exists in all robots, so achieving better precision in the modeling and control of a robot must be considered. his kind of exibility could be the consequence of dierent determinatives, such as looseness of gears and forced transmission systems (like belts and shafts), as they are determinatives of error generation between theoretical and practical results. Flexibility of joints could generate a low resonance frequency in the structure and unwanted vibrations in the robot. owards this end, Korayem and Nikoobin devoted themselves to nding the optimal path of a two link manipulator by considering the exibility of joints. his paper also used the optimal control method and the minimum principle of Pontryagin [9]. Flexible manipulators vibrate not only during tracking, but also after reaching the goal point. Korayem et al. investigated this residual vibration, which continues with a specic amplitude and frequency after reaching the goal point and the eect of this on the carried payload [1]. he stiness and stability of cable-suspended manipulators with application to load optimal determination are studied. Korayem and Bamdad addressed computation of the maximum load carrying capacity of large cable-suspended parallel manipulators [11]. Among numerous papers in the eld of parallel manipulator path planning, exibility in the joint has rarely been considered. Moreover, in no previous work in the area of cable-suspended parallel robots, has joint exibility been considered in the dynamic equation. In this paper, the problem of the optimal path and maximum dynamic load carrying capacity of a sample spatial cable robot is studied. he open loop optimal control approach is applied, and using the indirect method and Pontryagins' minimum principle, the original problem is converted to a two-point boundary value problem. A number of simulations for a cable-suspended manipulator with exible joints are carried out to investigate the eciency of the presented method. he paper is organized as follows: First, the dynamic modeling of a system, considering the exibility of joints, for one type of six cable spatial robot, is introduced. hen, the optimal control problem and necessary conditions for optimality are dealt with. hen, based on the solution of PBVP, an algorithm is developed for nding the optimal path for a specic payload and another is then given for determining maximum payload and corresponding optimal path. Finally, simulation results are presented and discussed including verication. PROBLEM FORMULAION In cable-suspended manipulators, the weight of the end-eector provides tension in the cables. he type of cable robot is as an Incompletely Restrained Parallel Manipulator, IRPM, because the kinematics of the robot are not sucient to completely restrain the end-eector and, thus keep all cables in tension. Since the end-eector has six degrees of freedom, the minimum number of cables or actuators is six. First, the dynamic model of a typical cablesuspended robot will be presented and, then, the eects of exible joints are exerted in an ideal model. Finally, the full dynamic model is derived. he paper will be continued with the formulation of necessary conditions for optimality. Dynamic Modeling of Spatial Cable Robot A spatial cable robot has a triangular shaped endeector, as shown in Figure 1, which is suspended through cables and has degrees of freedom. If we show the direction and position of the endeector relative to the reference coordinate by six variables of X = [x; y; z; ; ; ], cable length by l = [q 1 ; q ; q ; q ; q ; q ] and cable tension by = [ 1 ; ; ; ; ; ], the dynamic model of the sys- Figure 1. Six cable spatial robot with exible joints [].

3 rajectory Planning of Flexible Cable Manipulator 1 tem will be as follows [1,]: D(X) X + C(X; _X) + G(X) = J = q 1 x q x q x q x q x q x q 1 y q y q y q y q y q y q 1 z q z q z q z q z q z q 1 q q q q q q 1 q q q q q J (X); q 1 q q q q q ; (1) in which J(X) is the conventional matrix of the robot's Jacobian, D(X) is the inertia matrix of the robot, C(X; _X) is the vector of Coriolis and centrifugal forces and G(X) is the vector of gravity forces. he cables must be capable of exerting a positive force and torque on the end eector. Dynamic Modeling of Cable Robot with Flexible Joints A torsional spring is considered between the motor and pulley.for the dynamics of the exible joint. In this case, the rotation of motors and pulleys can be dierent, and the following relations will be from the vibration modeling of the system: j + K( ) = r; () in which K (N.m /rad) is the torsional spring coecient, (rad) is the motor rotation angle and (rad) is the pulley rotation angle. Also, we have: j K( ) = ; () in which j (kg.m ) is motor inertia and (rad/s ) is angular acceleration of motors. he following equation is for the velocity and acceleration of pulleys: _ = X _ X; = d dt X X = 1 r J; _X + X X: () We need to rewrite the dynamic equations into a state space equation form for solving the optimization problem. With regard to exibility in the joints, we have 1 degrees of freedom. herefore, there are state variables dened as: X 1 = X = x 1 x x x x x x 1 x 1 x 1 x 1 x 1 x 1 = = x y z 1 ; X = ; X = x x x 9 x 1 x 11 x 1 x 19 x x 1 x x x = = _x _y _z _ ; _ 1 _ : () Now concerning these variables from Equation 1, we have: _X = D 1 ( J C G): () By the placement of Equations and in the above equation and the elimination of, we have: _X = hi + D 1 J jj i 1 h r D 1n J K( ) r C Goi ) _X = F 1 (X 1 ; X ; X ): () It is important to note that in the placement of the second derivative of the pulley rotation angle, the non-linear term is relinquished because, rstly, in previous studies conducted, the linear term is eective and, secondly, by regarding the nonlinear term, the size of the MALAB le becomes MB, which no software can solve. On the other hand from Equation we have: _X = = [ + K( )] j ) _X = F (X 1 ; X ; U): () Finally, state equations of the system will be as follows: _X 1 _X _X _X = X 1 F 1 (X 1 ; X ; X ) X : (9) F (X 1 ; X ; U) Problems of Optimal Control and Necessary Condition of Optimality In this optimal control problem, we want to determine the state function, X(t), and the control function, U(t). he general method is, rst, for a specied load; the optimal path of the robot for moving between the primary and nal point is found. hen, the maximum load is found through an iterative algorithm.

4 1 M.H. Korayem, E. Davarzani and M. Bamdad Deriving Equations for Flexible Joints Case According to [,], the overall form of the objective function is as follows: J (U) = 1 ke p 1 (t f )k W p ke p (t f )k W p where: + 1 ke v 1 (t f )k W v ke v (t f )k W v Z tf + L(X; U)dt; (1) t e p1 (t f ) = X 1 (t f ) X 1f ; e p (t f ) = X (t f ) X f ; e v1 (t f ) = X (t f ) X f ; e v (t f ) = X (t f ) X f ; L(X; U) = 1 (kx 1k W 1 + kx k W + kx k W + kx k W + kuk R); (11) where t and t f are known as the initial and nal times, and the integrand, L, is a smooth dierentiable function in the argument. kxk K = X KX is the generalized squared norm, W p and W v are symmetric positive semi-denite (n n) weighting matrices, W 1 ; ; W and R are symmetric positive denite (n n) matrices. X 1f, X f, X f and X f are the desired values of the position and velocity of the cable at the nal time. he performance criteria dened by Equation 1 are minimized on the total range of motion. On the other hand, the boundary conditions at the beginning and end of the path are: X 1 () = X 1 ; X () = X ; X () = X ; X () = X ; X 1 (t f ) = X 1f ; X (t f ) = X f ; X (t f ) = X f ; X (t f ) = X f ; (1) which represent the position and velocity of the end-eector at the initial and nal time. Each of the pulleys works on an under specic curve and in a particular range; U = fu i U i U + i g; (1) in which the upper and lower limits of the torques according to the current-torque characteristics of DC motors are dened as [1]: U + = K 1 K X ; U = K 1 K X ; (1) where: K 1 = s1 s sn ; K = dig s1! 11 sn! mn : s is the still torque and! mi is the maximum no load speed of the motor. It is noticeable that since the cables are only capable of being in tension, the upper limit of the torque for our robot will be zero. As mentioned, for solving the optimality problem, we use the indirect method, thus rst the Hamiltonian function is dened as follows: H = L + where: W 1 = W = R = = 1 X + w w F 1 + X + ; W = F ; (1) w..... w 1 w 1 w ; W =..... w 1 w R R x 1 x x x x x 9 >= >; ; = ; 1 = x x x 9 x x 1 x 9 >= >; x x x x x 9 x 9 >= >; ; ; = x x x x x x 9 >= >; ; ; ; (1) Finally, by using the Pontryagin minimum principle, we have the following relations: 1. _X = H! _X 1 = X _X = F 1 (X 1 ; X ; X ) _X = X _X = F (X 1 ; X ; U)

5 rajectory Planning of Flexible Cable Manipulator 19. _ = H x!. H _ 1 = ([w 1 x 1 ; w x ; w x ; w x F1 w x ; w x ] + f g+ X 1 F1 f g) X 1 _ = ([w x ; w x ; w 9 x 9 ; w 1 x 1 w 11 x 11 ; w 1 x 1 ] + f 1 g+ F1 f g) X _ = ([w 1 x 1 ; w 1 x 1 ; w 1 x 1 ; w 1 x 1 F1 w 1 x 1 ; w 1 x 1 ] + f g+ X F f g) X _ = ([w 19 x 19 ; w x ; w 1 x 1 ; w x w x ; w x ] + f g u =! 1 = R 1 F U u i = U i + i U i U i <! i > U + i i < U + i i < U i (1) We have totally dierential equations, which with the last control law form all the required equations. In solving process, rst of all control Equation would be substituted in two other equations for a known load (Equations 1). Consequently, the resultant equations establish a set of n ordinary dierential equations, while Equation 1 describes a n boundary value condition n of which are dened as t = t and the other n as t = t f. he algorithm iterates on the initial values of the costate until the nal error converges to the desired accuracy, ". o put it another way, the following relation must be satised by PBVP solving: 1 kx 1(t f ) X 1f k W p1 + 1 kx (t f ) X f k W p + 1 kx (t f ) X f k W kx (t f ) X f k W ": (1) ALGORIHM OF FINDING OPIMAL PAH AND MAXIMUM PAYLOAD Finding Optimal Path for Specic Payload 1. Select " and penalty matrices.. Select two points that indicate the initial and nal positions of the manipulator in the workspace of the robot, as they use in the boundary condition.. We go to the next step, if the workspace in two selected points does not include singularity, otherwise, if Jacobian values at the initial or nal conguration are equal to zero, the algorithm jumps back to the second step and two new points should be selected (Condition I).. Select a primary path as the initial guess for solving the problem.. Solve the two point boundary value problem using the given boundary conditions and the BVPC command in MALAB software.. Calculate the amount of torque (and if necessary the other problems, such as tension of cable, length of cable etc.).. If the desired objectives and purposes and the required accuracy are satised, the obtained path is optimal and the running should be stopped, otherwise the algorithm jumps back to step (Condition II). Calculation of Maximum Payload he algorithm of the maximum payload has also been shown in Figure. he rst to third step of this algorithm is similar to the previous algorithm. In this algorithm, by selecting penalty matrices, maximum load is calculated per specic objective function. m p is the value of the primary load and determines the increasing value of the load at each stage. he solving method is based on increasing the minimum amount of load until the maximum is obtained. SIMULAION In this section, simulation is done for dierent exibilities and under rigid conditions. It is noticeable that the rigid optimal path is used as the initial guess in each exible case. Simulation is done for a specic load of 1 (kg), and moving between the point (-.1, -.1, 1.) and point (.1,.1, 1.) for two dierent exible and rigid cases to guarantee the results. Applied boundary conditions in the simulation are according to able 1, and other specications are listed in able. In Figure, comparisons between motor torques for the rst and sixth ones are shown. It is clear, by considering joint exibility, that the torques will have vibrations around the rigid torque and that, by increasing the stiness of the modeled spring, these vibrations decrease. hen values of the maximum load and also the required optimal path are obtained for two dierent exible and rigid cases. In this simulation, also the boundary conditions are the same as in the last simulation and able. According to conducted simulations, the maximum

6 M.H. Korayem, E. Davarzani and M. Bamdad Figure. (a) Finding optimal path for specic payload. (b) Calculation of maximum payload. able 1. Boundary conditions of exible joints case. X 1() = [ :1; :1; +1:; ; ; ] X 1(t f = 1) = [+:1; +:1; +1:; ; ; ] X () = [; ; ; ; ; ] X (t f = 1) = [; ; ; ; ; ] X () = [; ; ; ; ; ] For K = : X (t f = 1) = [ :9; :9; :1; :1; :; :119] For K = : X (t f = 1) = [ :99; :9; :9; :9; :; :1] X () = [; ; ; ; ; ] X (t f = 1) = [; ; ; ; ; ] payload for a rigid case is obtained as.9 (kg), and by adding joint exibility this value is reduced; for spring stiness of (N.m/rad), it is decreased to 19 (kg) and for spring stiness of (N.m/rad), it is decreased to 1. (kg). Figure shows the optimal path of each case that is listed above. It should be mentioned that in each of the simulation cases, before the rst motor saturation, the optimal paths of dierent loads are the same and, then, if the load increases, the plan attempts to change the path to still keeping the torques within the allowed range. Also, increasing the load is possible until the needed accuracy in the problem is provided. In Figure, for example, in spring stiness (N.m/rad), the motor torques achieve maximum payload. It is noticeable that in the case of spring stiness equal to (N.m/rad), we

7 rajectory Planning of Flexible Cable Manipulator 1 able. Simulation parameters in exible joints case. Parameter Value Unit Moment of inertia (I xx ) I zz = b t p + b t p kg.m Moment of inertia (I yy ) I zz = b t p + b t p kg.m Moment of inertia (I zz ) I zz = b p t kg.m a. m b. m Motor's max. no load speed Rad/s Motor stall torque. N.m Pulley radius. m Pulley rotational inertia 1 kg.m Rotational inertia for motor collection 1 kg.m Figure. Optimal path for maximum payload in dierent cases of joint exibility in XYZ space. Figure. Comparison of motor torques in rigid and exible joint cases for specic payload. have seen, on the maximum payload, that only the fth motor has reached the saturation limit; in the second case, the fth motor has reached the saturation limit and the second motor is also near the end of saturation. But, in rigid cases, the th, nd and th motors were saturated and the th motor also reposed near to saturation. In this simulation, the torques tried to reach their upper bound. In Figure, motor torques in maximum payload for rigid cases are shown. o evaluate the accuracy of the obtained curves, one of the methods is nding the values of the tensions that are used here. Concerning Figures and, the tension of whole cables, when carrying the maximum load in the obtained optimal path, is positive. As can be seen, obtained optimal paths in both exible cases are very close to rigid optimal paths. Inci-

8 M.H. Korayem, E. Davarzani and M. Bamdad Figure. Diagram of motor torques to reach maximum payload for K = (N.m/rad). dentally, by increasing spring stiness and, technically, decreasing the exibility of the system, the vibrations of the motor torques are reduced and the maximum load carrying capacity is also increased. VERIFICAION In this paper, the proposed algorithm has been demonstrated as a promising way to predict the payload capacity. It is more uncertain and challenging for eld implementation. herefore, it is urged that eld verication of the result should be carried out. In [1], direct and indirect kinematic and dynamic modeling of one cable spatial robot, considering the exibility of joints, is undertaken. In this reference, the author has used a Matlab simulink toolbox and, then, veried the results with Sim-Designer software. For verication of our result, rst, for a specic payload, the optimal path and motor torque are found from the optimal control program. hen, the obtained optimal path is given to the inverse dynamic program [1] and the motor torque is obtained. Finally, the motor torques obtained from the two methods are compared. It can be observed that the parameter of simulation is according to able, and the boundary conditions are as in able 1. A comparison of results for a. (kg) payload and a spring stiness of

9 rajectory Planning of Flexible Cable Manipulator Figure. Diagram of motor torque reaching maximum payload in a rigid case. Figure. Cable tension in maximum payload for K = (N.m/rad). Figure. Cables tension in maximum payload for rigid case.

10 M.H. Korayem, E. Davarzani and M. Bamdad Figure 9. Comparison of torques of optimal control and inverse dynamic in optimal path for. kg payload at K = (N.m/rad). (N.m/rad) is shown in Figure 9. hese gures show good agreement between results. CONCLUSION he problem of the optimal path and maximum carrying capacity of a DOF spatial cable robot was studied. Considering exibility of joints, the simulation was undertaken. his paper deals with the verication of results. he maximum payload without considering exibility at the given boundary condition was equal to.9 (kg), and by adding the exibility of joints, this value reduced, as for a spring stiness of (N.m/rad) to 19 (kg), and for a spring stiness of (N.m/rad) to 1. (kg). Actually, the vibration has grown in the exible systems. he actuators saturation bounds are xed for both exible and rigid models. Meanwhile, for controlling the vibration of the end-eector, the maximum value of the produced torque in a exible system versus a rigid

11 rajectory Planning of Flexible Cable Manipulator case is increased. It can be shown that this indirect method is suitable for high degree of freedom systems and, moreover, can optimize some objective function simultaneously. NOMENCLAURE D(X) G(X) C(X; _X) J(X) l a b K j j r ; inertia matrix of the manipulator vector of gravity forces vector of Coriolis and Centrifugal forces matrix of robot Jacobian vector of cable length half distance between points A & B on base of robot half distance between points D & F on end-eector diagonal joints stiness matrix diagonal pulleys inertia matrix diagonal motors inertia matrix vector of pulley rotation angle vector of motor rotation angle radius of pulleys vector of motors torque U(t) control function s stall torque of motor! m maximum no load speed of motor vector of cable tension J (U) objective function H Hamiltonian function U i + ; U I extrimal bound of motor torque U vector of admissible control torque X 1 ; X X ; X W 1 ; W ; W ; W ; W p ; W v ; R state vectors weighting matrices 1; ; ; costate vectors " desired accuracy in PBVP solution REFERENCES 1. Korayem, M.H. and Bamdad, M. \Dynamic load carrying capacity of cable-suspended parallel manipulators", Int. J. Adv. Manuf. echnol.,, pp. 9- (9).. Korayem, M.H. and Shokri, M. \Maximum dynamic load carrying capacity of a UPS-Stewart platform manipulator", Scientia Iranica, 1(1), pp. ().. Wang, L.. and Ravani, B. \Dynamic load carrying capacity of mechanical manipulators: II. Computational procedure and applications", J. Dyn. Syst. Meas. Control, 11, pp. -1 (19).. Korayem, M.H. and Nikoobin, A. \Formulation and numerical solution of robot manipulators in point-topoint motion with maximum load carrying capacity", Scientia Iranica, rans. B, 1(1), pp (9).. Hiller, M., Fang, S., Mielczarek, S., Verhoeven, R. and Franitza, D. \Design, analysis and realization of tendon-based parallel manipulators", J. of Mechanism and Machine heory, (), pp. 9- ().. Alp, A.B. \Cable-suspended parallel robots", MS hesis in Mechanical Engineering at the University of Delaware (1).. Korayem, M.H., Bamdad, M. and Bayat, S. \Optimal trajectory planning with dynamic load carrying capacity of cable-suspended manipulator", IEEE International Symposium Mechatronics and Its Applications, ISMA, pp. 1- (9).. Korayem, M.H., Bayat, S. and Bamdad, M. \Maximum payload of cable-based planar manipulator for two given end points of end-eector using optimal control approach", 1th. Annual International Conference on Mechanical Engineering ISME, University of ehran (9). 9. Korayem, M.H. and Nikoobin, A. \Maximum payload for exible joint manipulators in point-to-point task using optimal control approach", Int. J. Adv. Manuf. echnol., (9/1), pp. 1-1 (). 1. Korayem, M.H., Heidari, A. and Nikoobin, A. \Eect of payload variation on the residual vibration of exible manipulators at the end of the given path", Scientia Iranica, ransaction B: Mechanical Engineering, 1(), pp. - (9). 11. Korayem, M.H. and Bamdad, M. \Stiness modeling and stability analysis of cable-suspended manipulators with elastic cable for maximum load determination", Kuwait J. Sci. Eng.,, pp. 1-1 (1). 1. Iranpoor, M. \Modeling and simulation of kinematic and dynamic of cable actuated robot", MS hesis in Mechanical Engineering at the Iran University of Science and echnology (9). BIOGRAPHIES Moharam Habibnejad Korayem was born in ehran, Iran, in 191. He received his BS (Hon) and MS degrees in Mechanical Engineering from Amirkabir University of echnology, in Iran, in 19 and 19, respectively. He obtained his PhD degree in Mechanical Engineering from the University of Wollongong, Australia, in 199. He is Professor of Mechanical Engineering at the Iran University of Science and echnology, where he has been involved with teaching and research activities in the area of robotics for the last 1 years.

12 M.H. Korayem, E. Davarzani and M. Bamdad His research interests include: Dynamics of Elastic Mechanical Manipulators, rajectory Optimization, Symbolic Modeling, Robotic Multimedia Software, Mobile Robots, Industrial Robotics Standard, Robot Vision, Soccer Robot, and the Analysis of Mechanical Manipulators with Maximum Load Carrying Capacity. He has published more than papers in international journals and conference proceedings in the eld of robotics. Ehsan Davarzani was born in ehran, Iran, in 19. He received his BS degree in Mechanical Engineering from ehran University in and his MS degree from the Iran University of Science and echnology in 9. His research interests include: Robotic Systems, Optimization, Parallel Manipulators, and Hydraulic Systems. Mehdi Bamdad was born in Sabzevar, Iran, in 19. He received his BS in Mechanical Engineering from KN University of echnology, Iran, in, and his MS degree from Semnan University, Iran, in. He is now a PhD candidate in Mechanical Engineering at the Iran University of Science and echnology. His research interests include: Robotic Systems, Optimization, Parallel Manipulators, Industrial Automation and Mechatronic Systems.