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Robot kollarda optimal yörünge planlaması

Robot kollarda optimal yörünge planlaması

##### Dosyalar

##### Tarih

1994

##### Yazarlar

Özkıpçak, S. Hakan

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Robot kollarda, optimal yörünge planlaması genel olarak iki ana gruba ayrılmaktadır. Bunlardan birincisi, noktadan noktaya optimal yörünge planlamasıdır ve bu gruba giren çalışmalarda, tutucunun verilen iki nokta arasında izleyeceği optimal yörüngenin seçimi esastır. Diğeri ise optimal hız dağılımı planlamasıdır. Bu gruba giren çalışmalarda, robot kolun, önceden verilmiş bir yörünge üzerindeki optimal hız dağılımı incelenmektedir. Bu çalışmada, iki serbestlik dereceli bir robot kolun, verilmiş bir yörünge üzerindeki optimal hız dağılımı problemi ele alınmıştır. Zaman minimizasyonu kriteri, optimize edilecek olan performans indeksi olarak esas alınmıştır. Yörünge tanımlanmış olduğundan, problem yörünge boyunca robot kolun hareketli bir linkinin optimal hız dağılımının incelenmesi problemine indirgenmiştir. Yörünge üzerinde optimal hız dağılımını elde etmek üzere, esas olarak Dinamik Programlama metodu kullanılmıştır. Dinamik programlama metodu, çok adımlı karar problemlerinin optimizasyonuna çok iyi uyum sağladığı için tercih edilmiştir. Bu metod, ters kinematik eşitlikleri çözülebildiği sürece, herhangi ser bestlik derecesine sahip robot kollara uygulanabilmektedir. Çalışmanın birinci bölümünde, bu konuda yapılmış olan çalışmalara kısaca değinilmiştir. îkinci bölümde, robot kollar için temel kinematik esaslara yer verilmiş ve problemin uygulandığı robot kolun kinematik analizi yapılmıştır. Üçüncü bölümde, yine temel bazı dinamik esaslara değinildikten sonra, Lagrange-Euler yaklaşımı ile dinamik analiz yapılmıştır. Dördüncü bölümde, genel olarak dinamik programlama metoduna değinilmiştir. Beşinci bölümde ise, dinamik programlama metodu, tezin ana konusunu oluş turan probleme uygulanmış ve bu amaçla Fortran dilinde kullanılan bilgisayar programı tanıtılmıştır. Altıncı bölüm de ise, dinamik programlama metodu ile elde edilen sonuçları karşılaştırmak üzere Nonlineer Programlama metodu, aynı probleme uygulanmış ve yine Fortran dilinde kullanılan bilgisayar programı tanıtılmıştır. Son bölümde ise, bu çalışma ile elde edilen sonuçlar incelenmiştir.

The optimal trajectory search for robotic manipla- tors concerns generally two main performance criterions; minimal motion time criterion and minimal energy consump tion criterion. Minimal motion time criterion causes the increase of productivity, but causes also increase of energy cost. Minimal energy consumption criterion causes the decrease of energy cost, but causes also increase of motion time, which can be told a disadvantage for produc tivity. As a reference criterion, mixed minimal time and energy criterion which incorporates the two main criteri ons may be a good solution for search of the optimal tra jectory problem. Studies on the optimal trajectory search of robo tic maniplators can be seperated into two main groups: 1. Point to point optimal trajectory optimization search: The studies belong to this optimization method, try to find the optimal trajectory of robotic maniplator between two points. The solution of this problem may be the optimal time stories of the coordinates of the mani plator or the optimal time stories of the driving forces/ torques. 2. Optimization of velocity distribution along a predescribed path: This is main subject of this study. It is assumed the path which will be followed by the robotic maniplator is given. So, the problem is reduced to search of velocity distribution along the path. In many robot applications, the end effector is required to follow a specified path which will avoide the collisions with the obstacles. The solution of a path fol lowing problem is generally obtained by dividing the path into some number of points and then controlling the moti on of the end effector between those points. For getting a smooth motion, the velocity of the end effector had to be planned beside the position. But for planning the velo city and related driving force/torque along a path, opti mization of a performance index is necessary. Some of trajectory planning problems for robotic maniplators, considered kinematics or dynamics or optimi zation of a performance index. Paul [l] suggested a met hod which the maniplator passes from one straight line segment to another with the continuous motion in joint =? vi - displacements, velocities and accelerations. Taylor [2] proofed that, in cartesian space for obtaining minimum difference or error between the given and the planned path, the path had to be divided in a sufficient number of points. Kahn and Roth [3], studied on a time minimization problem. In their study, the path which will be followed by the maniplator was not given. But they showed that minimum time optimization results a bang-bang control. Luh and Lin [4], studied on a velocity distribu tion along a predescribed path, but they used cartesian velocities and acclerations as constraints. Shin and Mc Kay [" 6], applied dynamic programming method to optimal trajectory planning problem. But they defined the path by a scalar parameter, then they have written the dynamic equations by using that parameter. For using their method, the path must be parameterized. Also Vukobratovic and Kircanski [7 ] applied dyna mic programming to optimal trajectory problem. They cons trained the driving force/torques and they specified the motion time. They used the minimal energy consumption criterion. In this study, the velocity distribution along a given path has been searced. Joint moments and joint ve locities have been assumed as constraints. The path has been divided into segments. Constant acceleration has been supposed at every segment of the path. The minimal time criterion has been used for optimization of velocity distribution. As the path which would be followed by the maniplator, has been given, the problem has been reduced the search of optimal velocities of one moving link. At first, dynamic programming method has been used. Then nonlineer programming method has been also used to solve the problem. As preparations for solving the problem, the kine matic and dynamic analysis of two degree of freedom mani plator have been done. It is assumed that, maniplator has two mossless links but both of them have 1.0 kg of moss attached at the end of the links. Also both links assumed to move in the horizontal plane. The results of the inverse kinematic solution: 2. 2 "2 02 x +y -l1-^2 s <%2 = arc cos ( ) 2£1£2 - vii - q1 = arc tg (^) -arc tg {j-^ £2sinq2. i £2COsq~' q, and q2 are the joint displacements of link 1 and link £*. The Lagrange-Euler method has been used for obtain ing dynamic equations. The Lagrange (L) is defined as the difference of the kinetic energy (K) and the potential energy (P) of the system. L = K-P The dynamic equations in terms of the system can be defined as T. 3L i dt dq± dq± (i=l,...,n) After the necessary computations, the dynamic equa tions for the two link maniplator have been obtained; (m1fm2H1-hn2A2+2ni2£1£2cosq2 ir^-ta^Jl £2cosq2 m2_£ 2"1-in2 £, 5."cosq~ m2*2 r 9l ^2 +.2 -m2£1£2sinq2q2-2nuJi1£2sinq2q1q2.2 m2£1£2sinq2q. Decisions have to be made sequentially in the most of practical problems. The problems in which the decisi ons are to be made sequentially are called sequential - vxn - decision problems. Since those decisions are to be made at a number of stages, they are also called as multistage decision problems. The method of Dynamic Programming which was developed by Richard Bellman, is a mathematical method which is very suitable for the optimization of multistage decision problems. The dynamic programming method divides a multista ge decision problem as a sequence of single stage deci sion problems. The decomposition to N subproblems have to be done in a way that, the optimal solution of the original N-variable problem can be obtained from the opti mal solutions of N one dimensional problems. At stage n, there are Sn input state variables, Sn_i output state variables, xn decision variables and R" return functions. For all the multistage problem, the output from stage (ntl) is equal to input to stage n. The algorithm of dynamic programming as follows: 1. At stage 1, the interval from the high limit to low limit of input state variable is divided into equal steps. 2. Beginning at the lower limit of Si, the optimum return is searced in the range of xi decision variables f^S^ - opt [R1(S1, xx)] xl The state value is incremented sequentially and the opti mum is found for that value of Sj. This process is conti nued until maximum value of S^ is reached. 3. Starting from n=2, the process moves backwards. The optimum returns are found at every step. When the stage N is reached, the optimum return is found for all values of S^. 4. If the problem is a initial value or two point boundary problem, the value of SN and the resulting f'N and xN are found. The transformation equation is used to obtain Sn-1 and its corresponding values. This forward path process is followed until Sj, f^ and x^ are found. For the initial value problem, the optimum S^ is found for a given fin>?.l value. For applying the dynamic programming method, the given path which maniplator follows has been divided into N segments (N=10). k has been defined as the discrete points on the path. The joint displacements qj_(k) have been found by solving inverse kinematic equations. Mani plator has been assumed that, travelling from point k to (kfl). Link 1 has been chosen for the search of velocity distribution along the path. For a possible velocity - IX - of joint 1 at point k and an admissible velocity at (kfl) the acceleration of joint 1 was, Vk) " 2[q1(k+l)-q1(k)] and the motion time between point k and (k+1), 2[q1(k+l)-q1(k)] At(k) [q1(k-H)^q1(k)] Link 2 was assumed to take its own distance in the same motion time, so 2[cu(k+-l)-q:,(k)] ^k> - " At(k) Vk+1> For every possible q-,(k) and admissible q, (k-t-1) velocities, q- (k) has been computed. When <32(k) aid not satisfy its velocity constraint, the selected qi(k) has been rejected. When no violation case, the corresponding T^(k) and T2(k) moments have been computed. When T]_(k) or T2(k) did not satisfy its constraint, the selected qi(k) has been rejected. For the minimization the performance index was, j=£At(k). $(q^(k),k) gived the incremental performance index between point k and (ki-1), J°(qi(k),k) gived the minimum performance index to arrive to final state from the point k. Then Bellman's optimality principle has been applied, J°(qi(k),k)= min (*(q;L(k),k)+j0(q;L(kfl),kH)) q-^k+l) So, for every admissible qi(k) velocity, one opti mal q]_(k+l) velocity has been found. This process has been continued until initial state was reached. For per forming all the computations described above, the compu ter called Continuous Dynam Algoritm has been modified for the requirements of velocity distribution problem and then used. x - Also, Nonlineer Programming Method has been appli ed to the velocity distribution problem of robot manipula tors along a predescribed path. The Box (Complex) Algo- rith has been used. The results have showed that, Dynamic Programming Algorithm give more accurate results from Nonlineer Programming Algoritm.

The optimal trajectory search for robotic manipla- tors concerns generally two main performance criterions; minimal motion time criterion and minimal energy consump tion criterion. Minimal motion time criterion causes the increase of productivity, but causes also increase of energy cost. Minimal energy consumption criterion causes the decrease of energy cost, but causes also increase of motion time, which can be told a disadvantage for produc tivity. As a reference criterion, mixed minimal time and energy criterion which incorporates the two main criteri ons may be a good solution for search of the optimal tra jectory problem. Studies on the optimal trajectory search of robo tic maniplators can be seperated into two main groups: 1. Point to point optimal trajectory optimization search: The studies belong to this optimization method, try to find the optimal trajectory of robotic maniplator between two points. The solution of this problem may be the optimal time stories of the coordinates of the mani plator or the optimal time stories of the driving forces/ torques. 2. Optimization of velocity distribution along a predescribed path: This is main subject of this study. It is assumed the path which will be followed by the robotic maniplator is given. So, the problem is reduced to search of velocity distribution along the path. In many robot applications, the end effector is required to follow a specified path which will avoide the collisions with the obstacles. The solution of a path fol lowing problem is generally obtained by dividing the path into some number of points and then controlling the moti on of the end effector between those points. For getting a smooth motion, the velocity of the end effector had to be planned beside the position. But for planning the velo city and related driving force/torque along a path, opti mization of a performance index is necessary. Some of trajectory planning problems for robotic maniplators, considered kinematics or dynamics or optimi zation of a performance index. Paul [l] suggested a met hod which the maniplator passes from one straight line segment to another with the continuous motion in joint =? vi - displacements, velocities and accelerations. Taylor [2] proofed that, in cartesian space for obtaining minimum difference or error between the given and the planned path, the path had to be divided in a sufficient number of points. Kahn and Roth [3], studied on a time minimization problem. In their study, the path which will be followed by the maniplator was not given. But they showed that minimum time optimization results a bang-bang control. Luh and Lin [4], studied on a velocity distribu tion along a predescribed path, but they used cartesian velocities and acclerations as constraints. Shin and Mc Kay [" 6], applied dynamic programming method to optimal trajectory planning problem. But they defined the path by a scalar parameter, then they have written the dynamic equations by using that parameter. For using their method, the path must be parameterized. Also Vukobratovic and Kircanski [7 ] applied dyna mic programming to optimal trajectory problem. They cons trained the driving force/torques and they specified the motion time. They used the minimal energy consumption criterion. In this study, the velocity distribution along a given path has been searced. Joint moments and joint ve locities have been assumed as constraints. The path has been divided into segments. Constant acceleration has been supposed at every segment of the path. The minimal time criterion has been used for optimization of velocity distribution. As the path which would be followed by the maniplator, has been given, the problem has been reduced the search of optimal velocities of one moving link. At first, dynamic programming method has been used. Then nonlineer programming method has been also used to solve the problem. As preparations for solving the problem, the kine matic and dynamic analysis of two degree of freedom mani plator have been done. It is assumed that, maniplator has two mossless links but both of them have 1.0 kg of moss attached at the end of the links. Also both links assumed to move in the horizontal plane. The results of the inverse kinematic solution: 2. 2 "2 02 x +y -l1-^2 s <%2 = arc cos ( ) 2£1£2 - vii - q1 = arc tg (^) -arc tg {j-^ £2sinq2. i £2COsq~' q, and q2 are the joint displacements of link 1 and link £*. The Lagrange-Euler method has been used for obtain ing dynamic equations. The Lagrange (L) is defined as the difference of the kinetic energy (K) and the potential energy (P) of the system. L = K-P The dynamic equations in terms of the system can be defined as T. 3L i dt dq± dq± (i=l,...,n) After the necessary computations, the dynamic equa tions for the two link maniplator have been obtained; (m1fm2H1-hn2A2+2ni2£1£2cosq2 ir^-ta^Jl £2cosq2 m2_£ 2"1-in2 £, 5."cosq~ m2*2 r 9l ^2 +.2 -m2£1£2sinq2q2-2nuJi1£2sinq2q1q2.2 m2£1£2sinq2q. Decisions have to be made sequentially in the most of practical problems. The problems in which the decisi ons are to be made sequentially are called sequential - vxn - decision problems. Since those decisions are to be made at a number of stages, they are also called as multistage decision problems. The method of Dynamic Programming which was developed by Richard Bellman, is a mathematical method which is very suitable for the optimization of multistage decision problems. The dynamic programming method divides a multista ge decision problem as a sequence of single stage deci sion problems. The decomposition to N subproblems have to be done in a way that, the optimal solution of the original N-variable problem can be obtained from the opti mal solutions of N one dimensional problems. At stage n, there are Sn input state variables, Sn_i output state variables, xn decision variables and R" return functions. For all the multistage problem, the output from stage (ntl) is equal to input to stage n. The algorithm of dynamic programming as follows: 1. At stage 1, the interval from the high limit to low limit of input state variable is divided into equal steps. 2. Beginning at the lower limit of Si, the optimum return is searced in the range of xi decision variables f^S^ - opt [R1(S1, xx)] xl The state value is incremented sequentially and the opti mum is found for that value of Sj. This process is conti nued until maximum value of S^ is reached. 3. Starting from n=2, the process moves backwards. The optimum returns are found at every step. When the stage N is reached, the optimum return is found for all values of S^. 4. If the problem is a initial value or two point boundary problem, the value of SN and the resulting f'N and xN are found. The transformation equation is used to obtain Sn-1 and its corresponding values. This forward path process is followed until Sj, f^ and x^ are found. For the initial value problem, the optimum S^ is found for a given fin>?.l value. For applying the dynamic programming method, the given path which maniplator follows has been divided into N segments (N=10). k has been defined as the discrete points on the path. The joint displacements qj_(k) have been found by solving inverse kinematic equations. Mani plator has been assumed that, travelling from point k to (kfl). Link 1 has been chosen for the search of velocity distribution along the path. For a possible velocity - IX - of joint 1 at point k and an admissible velocity at (kfl) the acceleration of joint 1 was, Vk) " 2[q1(k+l)-q1(k)] and the motion time between point k and (k+1), 2[q1(k+l)-q1(k)] At(k) [q1(k-H)^q1(k)] Link 2 was assumed to take its own distance in the same motion time, so 2[cu(k+-l)-q:,(k)] ^k> - " At(k) Vk+1> For every possible q-,(k) and admissible q, (k-t-1) velocities, q- (k) has been computed. When <32(k) aid not satisfy its velocity constraint, the selected qi(k) has been rejected. When no violation case, the corresponding T^(k) and T2(k) moments have been computed. When T]_(k) or T2(k) did not satisfy its constraint, the selected qi(k) has been rejected. For the minimization the performance index was, j=£At(k). $(q^(k),k) gived the incremental performance index between point k and (ki-1), J°(qi(k),k) gived the minimum performance index to arrive to final state from the point k. Then Bellman's optimality principle has been applied, J°(qi(k),k)= min (*(q;L(k),k)+j0(q;L(kfl),kH)) q-^k+l) So, for every admissible qi(k) velocity, one opti mal q]_(k+l) velocity has been found. This process has been continued until initial state was reached. For per forming all the computations described above, the compu ter called Continuous Dynam Algoritm has been modified for the requirements of velocity distribution problem and then used. x - Also, Nonlineer Programming Method has been appli ed to the velocity distribution problem of robot manipula tors along a predescribed path. The Box (Complex) Algo- rith has been used. The results have showed that, Dynamic Programming Algorithm give more accurate results from Nonlineer Programming Algoritm.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1994

##### Anahtar kelimeler

makine mühendisliği,
robot kolu,
yörünge planlaması,
mechanical engineering,
robot arm,
orbit planning