Mekanik manipülatörlerde optimal yörünge planlaması
Optimum path planning for mechanical manipulators
- Tez No: 14430
- Danışmanlar: DOÇ.DR. CAN ÖZSOY
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1991
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 86
Özet
ÖZET Mekanik manipülatörlerde uç algılayıcının, verilen bir görevi çalışma sahası içindeki nesnelere çarpmadan başarılı bir şekilde tamamlamasını temin amacıyla, ön ceden planlanmış bir yörünge üzerinde hareket etmesi ge reklidir. Verimliliğin artırılması bakımından önemli olan nokta ise manipülatörün bu görevi en kısa zamanda yapmasıdır. Yörünge izleme problemi, yörünge üzerinde çok sayı da nokta belirlenerek manipülatör uç algılayıcısının bu noktalar üzerinden hareketinin kontrol edilmesiyle çözül müştür. Problemin f ormülasyonunda robot kinematiğine dayalı bir yaklaşım temel alınarak önceden belirlenmiş bir yörünge üzerinde hareket eden manipülatörün uç algı layıcısının, doğrusal ve açısal olarak hız ve ivme kısıt lamaları altında yörünge takibini minimum zamanda gerçek leştirmesine yönelik bir metod araştırılmıştır. Kullanılan kinematik yaklaşıma uygun olarak yörünge doğru parçalarına bölünmüş ve bu doğru parçaları yumuşak eğrilerle birleştirilerek hız ve ivme değerlerine uç al gılayıcının kartezyenyörüngesi içinde kısıtlamalar geti rilmiş, ayrıca planlanan yörünge ile hakiki yörünge ara sında meydana gelebilecek sapmalar gözönüne alınarak, manipülatör uç algılayıcısının yörünge üzerindeki hare ketinde konum ve durumuyla ilgili olarak yörüngedeki sapmaya kısıtlamalar getirilerek yörünge hareket zamanı nı minimize etmeyi amaçlayan bu problem bir nonlineer programlama problemi olarak formüle edilmiştir. Problemin çözümü için nonlineer programlama teknik lerinden FTM( Esnek Tolerans Metodu ) kullanılarak bu metodun elverişliliği araştırılmıştır.. Metodun ir delenmesi ve problemde elde edilen sonuçlara bağlı ola rak, yöntemin optimal yörünge planlaması problemini çözmede başarılı olduğu, buna ilaveten ön hazırlık za manının azlığı, kullanım kolaylığı gibi avantajları or taya çıkmıştır.
Özet (Çeviri)
OPTIMUM PATH PLANNING FDR MECHANICAL MANIPULATORS SUMMARY A mechanical manipulator can be modeled as an open- Iodp articulated chain uith several rigid bodies (links connected in series by either revolute or prismatic joints driven by actuators). One end of the chain is attached to a supporting base while other end is free and attached with a tool (usually called as the end- effector ar gripper) to manipulate objects or perform assembly tasks. The relative motion of the joints results in the motion of the links that positions the hand in a desired orientation. Although the advancement of technology makes it possible for a computer to perform on-line control of mechanical manipulators, in order to assure a successful completion of an assigned task without interruption, such as collisions with potential or existing obstacles in the work space, the hand of a mechanical manipulator often travels along a preplanned trajectory. For an increased level of productivity, it is very important that the end-effector of a robot manipulator moves from an initial location to the final position in the minimum traveling time subject to the constraints resulting from kinematlcal and physical characteristics of the robot mechanism itself and the actuators. A path planning problem is often solved by prescribing a large number of points along the path and controlling the end-effector to move through these points. In order to achieve a smooth motion, not only the position but the velocity must be prescribed at every intermediate point along the path. Planning the velocity and the associated force/torque at the intermediate points requires an optimization process if the path is to be followed closely and a performance index is to be optimized. VIApplication of optimum path planning for robotic manipulators leads to substantial practical difficul ties, so that significant simplifications are usually performed, either in model complexity or by meglecting some of the existing constraints. In this study, it is investigated an optimum path planning problem, which would take into account the manipulator kinematics so as to obtain a time schedule of velocities and accelerations along the path that the manipulator may adopt to obtain minimum traveling time. For the sake of simplicity and facilitating the controlled interaction with objects on a moving conveyor, the preplanned path is composed of straight line segments in cartesian coordinates connected by smooth arcs. It is also admitted that the manipulator rests its initial location before it moves, and stops at its final location (position and orientation). The real, physical constraints Dn the manipulators are the applied torques/forces at the joints. Because a serial link manipulatoris a highly nonlinear system and all its joints are coupled, therefore it is very difficult to convert the bounds on torques and forces into one Dn accelerations and velocities. Dne may suggest that the Cartesian path could be transformed into joint coordinates and then the const raints on joint velocities and accelerations could be imposed since Jacobien transformations are well developed, however such transformations are valid on a point-to- point basis. No transformations of functions of time between two coordinates are known. Based on the reasons indicated above, the velocities and accelerations are constrained in Cartesian path of the hand in this study. The position and orientation errors on the arcs of actual path fronthe corner of every two adjacent straight-line segments must be kept sufficiently small so that the actual path does not deviate significantly from the preplanned path. VllOtherwise it may have a risk of running into obstacles in the work space. All these requirements imposed on angular and linear velocities, position and orientation errors on the arcs Df line segment corners, angular/linear accelerations are expressed by a set of linear and nonlinear inequalities. As a result, minimum time traveling problem is formulated as a nonlinear programming problem. Such a nonlinear programming problem can be solved by two ways. The first method involves in linearizing the nonlinear inequality constraints and then reformu lating the problem as a linear programming problem. After iterative linearization and reformulation, conventional linear programming methods may be implemented. Tha here above mentioned solution was adopted by Luh and Lin [k\. However, such a method requires numeric deri vatives thereby as the number of nonlinear inequalities becomes larger, the time required for preperation would be even longer. Second disadvantage is that such a long preperation may easily result in human error since taking derivatives is the major source of human error. In addition to those explained above, the Method of Approximate Programming (MAP) applied by Luh and Lin requires the starting vector to be a feasible one, which is not considered as, preferrable point from the wiev of simplicity of use. These are the cons of MAP revealed at first glance. In this study, by considering the disadvantages of MAP indicated above, instead of applying linear programming methods Flexible Tolerance Method, Dne of nonliear programming methods, as an alternative method is inves tigated. Flexible Tolerance method is a direct search method. The algorithm improves the value of the objective function by using information provided by feasible points, as well as certain nonfeasible points termed nearfeasible points. The near-feasibility limits are gradually made more restrictive as the search proceeds toward the solution of the nonlinear programming problem until in the limit only feasible variable vectors are accepted. Dne advan tage of this basic strategy is that the extent of the Vlllviolation of the constraints is progressively decreased as the search moves tomards the solution. Because the equality and/or the inequality const raints are ^loosely satisfied in the early stages of the search, and more tightly satisfied only as the search approaches the solution, hence the overall computation effort required in optimization is considerably reduced. As the Flexible Tolerance method is direct search method, no linearization of nonlinear constraints is required. The advantage of this is the ease of pre- peration of problem to enter the computer, which is very important criterion in evaluating the methods. In contrast to Method of Approximate Programming, Flexible Tolerance Method doesnot need the starting vector to be a feasible one. This is an additional simplicity for preperation of problem and data set into computer. The Flexible Tolerance algorithm is written in FORTRAN, and an illustrative example is solved by this code to see the effectiveness of this method. According to the results a better value of optimal solution is attained. In conclusion, the Flexible Tolerance Method is found to be capable of solving such path planning problem and well matches the requirements such as ease of preperation, simplicity of use, ease of data input etc. Spending too much time on satisfying the inequality constraints is the only exception that overshadows the merit of Flexible Tolerance Algorithm. IXIn this problem, however, the physical constraints on velocities, accelerations and the error constraint constraints on the traveling path from a nonconvex constraint set. hence iterated solutions show that these different feasible solutions are merely different points on the boundary of the constraint set which are not optimum points. Consequently the computed optimum solution is false. The reason why the false solution may take place is that the iterated feasible solution is close to the curved section of the boundary of the nonconvex constraint set, along which the optimum solution lies. Thus a logical way to overcome the obstacle is to move the feasible solution of each iteration away from that section of boundary as it is performed in Direct App roximate Programming Algorithm (DAPA). Optimum value is obtained as 19.835 seconds by DAPA algorithm, and the execution time is 6.554 seconds for nine iteration. Flexible Tolerance Algorithm reaches the optimum value by ten iteration in 157 seconds. Although such execution time seems to be very long, it is somehow misleading. Because this execution time is not the net computing time that computer's CPU wastes and includes the delays in time sharing, entering data, displaying, transmission and the like. Second is also difference between computers on which FTFl and DAPA algorithms are run. It should be noted that Flexible Tolerance Algorithm suffers from the appropriate choices of the size of initial polyhedron (SIZE), ALFA, BETA, GAMA which are namely reflection, contraction and expansion factors, respectively. The choice of the size of size of initial poly hedron directly affects the execution time. Bigger values of SIZE parameter leads to bigger changes between iterated feasible points thus the computational process quickly reduces the value of performance index by consecutive iterations in somewhat ossillating way. In order to utilize this result relatively large values of SIZE parameters are used at the beginning of execu tion. As the process approaches to the optimum point GAMA factor needs to be decreased and BETA needs to be increased.
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