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Üç serbestlik dereceli silindirlik bir manipülatörün tasarımı, simülasyonu ve kontrolu

Üç serbestlik dereceli silindirlik bir manipülatörün tasarımı, simülasyonu ve kontrolu

##### Dosyalar

##### Tarih

1990

##### Yazarlar

İçli, S. Haydar

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Tez konusunu oluşturan manipülatör, cam endüstrisinde belli bir is istasyonunda kullanılmak üzere programlanmış tır. Yapılacak iş, üretim hattına bağlı konveyörden gelen sıcak cam mamullerin tavlama fırınına giden ikinci konveyör üzerindeki tur net olarak isimlendirilen tutuculara yeri eştirilmesidir. Ancak transfer işleminin yapılacağı konveyörler senkronize çalışmadıkları için olay bir izleme yakalama problemine dönüşmektedir. Senkronizasyonun sağlanması için turnetierin bulunduğu gidiş konveyörü üzerine detektörler yerleştirilerek bunlardan gerekli bilgilerin alınması düşünülmüştür. Manipülatör boyutları ve özellikleri ortaya çıktıktan sonra kinematik, statik ve dinamik analizler yapılmış, robot dinamiğini veren ifade Lagrange Euler bağıntısından kapalı formda bulunmuştur. Manipülatörün kontrolü için dört değişik kontrol algoritması incelenmiştir. Bunlar, 1 -Tek girişli tek çıkışlı sistem yaklaşımıyla hız üzerin den PI kontrol algoritması, 2-Hesaplanmış moment yöntemi ile PD kontrol algoritması, 3-Kayan ufuklu hesaplanmış moment yöntemi algoritması, 4-Çok girişli çok çıkışlı sistem yaklaşımı ile il eri beslemeli PD kontrol algoritmasıdır. Simülasyon programı PASCAL dilinde yazılmış ve integrasyon metodu olarak lineer olmayan sistemlerin çözümünde tatmin edici sonuçlar veren RUNGE-KUTTA IV metodu kullanılmıştır. İncelenen kontrol algoritmalarının performanslarını karşılaştırmak için eksponansiyel ve kübik ifadelerden oluşan yörünge senaryoları kullanılmıştır. Simülasyon sonuçlarından en uygun kontrol algoritmasının "Çok girişli çok çıkışlı sistem yaklaşımı ile PD kontrol algoritması olduğu görülmüştür. Cam mamul transferi sırasında izlenecek yörüngenin planlanmasında kübik polinomlar kullanılmıştır. Transfer işleminin simülasyonu için iki değişik senaryo uygulanmıştır. Bunlardan birincisinde normal çalışma şartlarına göre simülasyon yapılırken ikincisinde tur neti er in abartılı ölçüde düzensiz hareket ettikleri varsayılmıştır. Elde edilen sonuçlardan geliştirilen algoritmanın düzensiz çalışma koşullarında bile işleyeceği görülmüştür.

The robot constituting the subject of the thesis has been designed to be used at a particular work station in glass industry. The description of the work is, briefly, placing the products which are coming from the production line conveyor onto special holders on a second conveyor going to the furnace for heat treatment. However, since the conveyors are not synchronized, the application becomes a follow-and-catch problem rather than a point to paint transfer. If the manipulator would take the glass product from the first conveyor and came ta a fixed point on the axis of the second conveyor, it couldn't catch the holder because of the unsynchranized motion. Thus, the manipulator should know the position of the holder before or during its motion and plan the trajectory according to this information. Placing detectors on the second conveyor to receive necessary information seems a suitable solution for this problem. A cyl indi rical manipulator design has been chosen due to the description of the work. DC servo-motors have been preferred far driving the manipulator and the rotational motion has been transformed into transl ati anal motion by the use of ball -screws. Harmonic drive which provides high reduction ratios has been used at the base joint and timing belt has been used at the horizontal joint. Thin section bearings which has very rigid construction have been used for the bearing of the base joint. Kinematic, static and dynamic analyses have been done after determining the dimensions and specifications of the robot. Direct and inverse kinematic solutions giving the relation between the base and the end effector coordinate systems have been found, the Jacobian matrix which gives the relation between joint speeds and robot end effector speed has been obtained and static calculations have been done for the worst case. The equations for robot dynamics have been found in the closed form using Lagrange-Euler formulation. Motor dynamics was also included in the calculations to find the general dynamic equation. The VI Solutions of the analyses are indicated below. Direct Kinematic Solution: The position and orientation of the robot end effector far specific values of the joint variables have been formulated by Denavit-Hartenberg parameters as follows. T= n n n where t D tangential vector of the end effector, normal vector of the end effector, approach vector of the end effector perpendicular to the normal vector, P : position vector of the gripper, C =cosö, S =sin© 1 1 Ö : first joint variable, d : second joint variable, 2 d : third joint variable. Inverse Kinematics Solution: Inverse kinematics problem is to find the joint variables to provide the end effector to be at the desired position and orientation. By inverse transformation; C13> C23 C33 C40 VII As one can see from the definition, the Jacobian matrix gives the relation between the joint variables and small displacements of the robot end effector. After analyses, Jacobian matrix has been found to be; J» asin9-rcos9 O -sin9 -acosS-rsine O cos9 0 10 C50 Statical Analysis: The relation between the joint torques and the output torque or force acted by the robot end effector is as follows * = JT P From CcO ceo asin9-rcos9 -acos9-rsin9 0 -si n9 cos© 0 and t =F CasinQ-d cos93 -F Cacos9+d sin93 C8.a3 IX 3 y 3 T =F C8.W Z Z T =-F sin9 + F cos9 C8. cD 3 x y Since the only external force is the weight, it is found that F =F = O consequently t =t =0, t =F =mg. xy ^ J l 3 z y ^ Dynamical Analysis: Lagrange-Euler method which is an easy and systematic method has been used in the dynamic analysis of the cylindrical manipulator. Due to this method, dynamic equations for each of the links are as follows; Link 1 : CB1+J39 + CA2+2m Cr-1 Dr38 = All t 3 C3 1 Link 2 : Li nk 3 : B2z + A4z + AS ? A3I B3r + A7r + m Cr-1 38 = A6I 3 C3 3 C9.aD C9. W C9. c3 VTII Where Bl =NZJ, B2=m +m +NZJ, B3=m +NZJ, Al-N K 1 ml z 3 z mz 3 3 m3 1 ml A3=NZB, A3=N K, A4=N §, A5=Cm tm )g, 1 ml Z MZ Z mZ Z 3 A6=N K, A7=NZB, J =J +J +m lZ+m Cr -1 3Z 3 m3 3 m3 t 1 Z 3 3 3 C3 9,z and r are joint variables. A special attention is paid to the control problem because a trajectory control is necessary during the transfer application and it is desired that the tolerances should be within narrow limits. Thus, four different control algorithms have been investigated. PI Control Using Velocity Feedback By The Approach Of SI SOC Single Input Single Output? System: In this approach, the control law was found by modeling each of the joints as decoupled systems. Also, the non-linear effects have not been considered and the control parameters have been chosen by the approach of a second order linear system. Generally, the robot dynamics can be expressed as follows: MCqDq + NCq.qD + GCqD = T = K I C103 m for each link; mC qD q. + nC q, qD + gC qD = k. I. i=l.. 3 CUD i mi t and the control law: t. I = K CCq. -q3+ 1/T / Cq. - q 3 J C1ED ı p troft to ıraf t Substituting equationC123> into equationC113, the control parameters K and t, are found by any one of the pole P t placement methods. PD Control Algorithm with Computed Torgue Technigue; Salving Eqn.ClOD for joint accelerations one obtains: q = M_1CqDCT-k] C13D where k = NCq,q3 + GCqD The essential of computed torque technique is to take the joint accelerations as commands. Thus, equationC133 turns out to be; IX q = u q. = u. 1=1.. 3 CI 43 The choice of u, the input vector, according to PD control algorithm has been proposed: u.= K.Cq. -q.3 + K.Cq. -q. 3 + q., \. pt, iref v vx. vr©ft iref CI S3 Rearranging equationC153 using q = u, e. = q. -q. V t "öl *# e. - q., -q. e. = q. -q.,one obtains the following equation in simple form: t + K e. + K e, = O i v t pi cie3 2 2 From the resemblance of equationC153 and s +2Çw s+w =0 o o K = w. pi 91 and K = 2Ç w VI \. oi CI 73 The control parameters might be calculated with relation to the desired values of w and Ç. o Receding Horizon Computed Torque Technique: Consider the decoupled linear model CI 43 and assume that the desired trajectory for joint i is given by q. fct3 q. fct3 q,Ct3 =rCt3. Defining the model state variables related to joint i as x = q., -q. > l ^Iref V 1 eads to x = q.,-q. 2 vref v dt O C183 The suboptimal solution of the minimum energy terminal control defined by C183 and the performance criterion J=l/2/ u.dt o t C193 C203 gives the control law, u = 6VTZCq. -q.3 + 4/TCq.,-q. 3 + Y. where the values of the time varying optimal feedback gains are selected and kept constant as a suboptimal solution. The generalized torque vector is obtained as T = MCqD C8/T Cq.,-q D + 4/TCq q 3 + y 3 + k iref ı % r©f % ' i, C213 From u. = K x + K x + y C22D v p\. i VI 2 I, one can obtain the following; wZ = K =S/T2.... w = Vfe/T op o 2Cw = K = 4/T.... T = 2/V6 = 0. 82 o v ^ PD Control with MI MO System Approach: Linearized Model of Robot Dynamics: In equatianClOD expanding N and G about the operating /N /?% I*. /\ point P=CT,q,q3 using Taylor series and assuming MCq+6cp = MCq3 for small joint position perturbation oq Eq. 10 can be linearized as A6qCt3+BaqCt3+CöqCtD=TCt3 C233 where the constant nxn matrices A, B and C are defined by A=CM] ; B=C - - ] ; C=[ ] P p p dq dq Eq. 23 gives a set of coupled linear time-invariant differential equations which describe the incremental behavior of the robot dynamics for perturbations in the neighborhood of the nominal operating point P. Multi variable Controller Design: The total control law is given by TCt3=T+K C6q -oqD +K C6q -<5qD +C6q +B6q +A<5q C243 The first component is the value of the joint torque vector at the nominal operating point P, namely T q, q and 6a,6a,6a are the reference perturbation vectors for r r r joint position, velocity and acceleration. By choosing the feedback gain matrices according to the equation below, the dynamics of tracking errors are decoupled and the response of each error is shaped as desired. K = AD -B ; K - AD -C C25I) VI' p z where D = diag<2C w> and D = diag-Cw > are the constant nxn matrices. The damping ratio and the undamped natural frequency of each tracking error e =£q. Ct3-<5q Ct3 are specified by the designer. XI While feedback controller determines the stability and the transient response of the tracking error, the feedforward controller determines the steady-state response of the tracking error. The operation during the pick and place of the glass product consist of four main steps. 1 -Picking the product from first conveyor and transferring it to a fixed point between the two conveyors. The intermediate point is selected for obstacle avoidance and information update. 2-Esti mating the target point from the information received from the detectors placed on the second conveyor. 3-Fol lowing and catching the holder and placing the product. 4-Going to the initial position for repeating, Since the conveyors are unsynchronized, first a fixed point is given as the reference point. During the motion of the manipulator to this point, the controller receives information about the position of the holder, calculates and updates the new reference point on the axis of the second conveyor. The trajectory followed until this time is composed of two cubic polynomials at the joint space whose coefficients are found from the initial and boundary conditions for position and velocity continuity. During the third stage, position and speed control is performed according to the signals coming from the detectors, and at the end of a predetermined time, the glass product is placed on the holder. Then the end effector returns to its initial position following a trajectory which consist of a cubic polynomial whose coefficients are determined by the same way indicated above. The simulation program had been written in FORTRAN before, however later PASCAL language has been preffered since it was modular, fast, easily understandable and more suitable for writing control algorithms in the case of constructing the system. RUNGE-KUTTA IV method which gives sufficient accuracy for the solution of non-linear systems, has been used for integration. The step length has been chosen as h=0. 004 sec. Trajectory scenarios consisting of exponential and cubic terms have been used for the comparison of investigated control algoritms. From the simulation results, it has been seen that the most suitable control algorithm is the PD control with MI MO system approach. XXI For the simulation of the real system two different scenarios have been tested. In the first scenario, the distances between the holder axes are considered to be equal and the conveyor speed fluctuates within some limits. From the results of this simulation, it has been observed that the maximum misalignment between the holder axis and the product axis was about 2mm. In the second scenario which was produced to measure the performance of the developed control system, it is assumed that the holders are placed with an exaggerated disorder. The results showed that the misalignment wouldn't exceed 3 mm in the worst case. Since the dimensions of the holders are quite larger than those of products, the misalignments do not significantly affect the performance of the operation. The measurements used in the simulation have been taken from a work station which is under operation at a factory. The simulation results show that the developed synchronization and control algorithms would operate with sufficient performance at the work station where the environment conditions are not suitable for human beings to work efficiently.

The robot constituting the subject of the thesis has been designed to be used at a particular work station in glass industry. The description of the work is, briefly, placing the products which are coming from the production line conveyor onto special holders on a second conveyor going to the furnace for heat treatment. However, since the conveyors are not synchronized, the application becomes a follow-and-catch problem rather than a point to paint transfer. If the manipulator would take the glass product from the first conveyor and came ta a fixed point on the axis of the second conveyor, it couldn't catch the holder because of the unsynchranized motion. Thus, the manipulator should know the position of the holder before or during its motion and plan the trajectory according to this information. Placing detectors on the second conveyor to receive necessary information seems a suitable solution for this problem. A cyl indi rical manipulator design has been chosen due to the description of the work. DC servo-motors have been preferred far driving the manipulator and the rotational motion has been transformed into transl ati anal motion by the use of ball -screws. Harmonic drive which provides high reduction ratios has been used at the base joint and timing belt has been used at the horizontal joint. Thin section bearings which has very rigid construction have been used for the bearing of the base joint. Kinematic, static and dynamic analyses have been done after determining the dimensions and specifications of the robot. Direct and inverse kinematic solutions giving the relation between the base and the end effector coordinate systems have been found, the Jacobian matrix which gives the relation between joint speeds and robot end effector speed has been obtained and static calculations have been done for the worst case. The equations for robot dynamics have been found in the closed form using Lagrange-Euler formulation. Motor dynamics was also included in the calculations to find the general dynamic equation. The VI Solutions of the analyses are indicated below. Direct Kinematic Solution: The position and orientation of the robot end effector far specific values of the joint variables have been formulated by Denavit-Hartenberg parameters as follows. T= n n n where t D tangential vector of the end effector, normal vector of the end effector, approach vector of the end effector perpendicular to the normal vector, P : position vector of the gripper, C =cosö, S =sin© 1 1 Ö : first joint variable, d : second joint variable, 2 d : third joint variable. Inverse Kinematics Solution: Inverse kinematics problem is to find the joint variables to provide the end effector to be at the desired position and orientation. By inverse transformation; C13> C23 C33 C40 VII As one can see from the definition, the Jacobian matrix gives the relation between the joint variables and small displacements of the robot end effector. After analyses, Jacobian matrix has been found to be; J» asin9-rcos9 O -sin9 -acosS-rsine O cos9 0 10 C50 Statical Analysis: The relation between the joint torques and the output torque or force acted by the robot end effector is as follows * = JT P From CcO ceo asin9-rcos9 -acos9-rsin9 0 -si n9 cos© 0 and t =F CasinQ-d cos93 -F Cacos9+d sin93 C8.a3 IX 3 y 3 T =F C8.W Z Z T =-F sin9 + F cos9 C8. cD 3 x y Since the only external force is the weight, it is found that F =F = O consequently t =t =0, t =F =mg. xy ^ J l 3 z y ^ Dynamical Analysis: Lagrange-Euler method which is an easy and systematic method has been used in the dynamic analysis of the cylindrical manipulator. Due to this method, dynamic equations for each of the links are as follows; Link 1 : CB1+J39 + CA2+2m Cr-1 Dr38 = All t 3 C3 1 Link 2 : Li nk 3 : B2z + A4z + AS ? A3I B3r + A7r + m Cr-1 38 = A6I 3 C3 3 C9.aD C9. W C9. c3 VTII Where Bl =NZJ, B2=m +m +NZJ, B3=m +NZJ, Al-N K 1 ml z 3 z mz 3 3 m3 1 ml A3=NZB, A3=N K, A4=N §, A5=Cm tm )g, 1 ml Z MZ Z mZ Z 3 A6=N K, A7=NZB, J =J +J +m lZ+m Cr -1 3Z 3 m3 3 m3 t 1 Z 3 3 3 C3 9,z and r are joint variables. A special attention is paid to the control problem because a trajectory control is necessary during the transfer application and it is desired that the tolerances should be within narrow limits. Thus, four different control algorithms have been investigated. PI Control Using Velocity Feedback By The Approach Of SI SOC Single Input Single Output? System: In this approach, the control law was found by modeling each of the joints as decoupled systems. Also, the non-linear effects have not been considered and the control parameters have been chosen by the approach of a second order linear system. Generally, the robot dynamics can be expressed as follows: MCqDq + NCq.qD + GCqD = T = K I C103 m for each link; mC qD q. + nC q, qD + gC qD = k. I. i=l.. 3 CUD i mi t and the control law: t. I = K CCq. -q3+ 1/T / Cq. - q 3 J C1ED ı p troft to ıraf t Substituting equationC123> into equationC113, the control parameters K and t, are found by any one of the pole P t placement methods. PD Control Algorithm with Computed Torgue Technigue; Salving Eqn.ClOD for joint accelerations one obtains: q = M_1CqDCT-k] C13D where k = NCq,q3 + GCqD The essential of computed torque technique is to take the joint accelerations as commands. Thus, equationC133 turns out to be; IX q = u q. = u. 1=1.. 3 CI 43 The choice of u, the input vector, according to PD control algorithm has been proposed: u.= K.Cq. -q.3 + K.Cq. -q. 3 + q., \. pt, iref v vx. vr©ft iref CI S3 Rearranging equationC153 using q = u, e. = q. -q. V t "öl *# e. - q., -q. e. = q. -q.,one obtains the following equation in simple form: t + K e. + K e, = O i v t pi cie3 2 2 From the resemblance of equationC153 and s +2Çw s+w =0 o o K = w. pi 91 and K = 2Ç w VI \. oi CI 73 The control parameters might be calculated with relation to the desired values of w and Ç. o Receding Horizon Computed Torque Technique: Consider the decoupled linear model CI 43 and assume that the desired trajectory for joint i is given by q. fct3 q. fct3 q,Ct3 =rCt3. Defining the model state variables related to joint i as x = q., -q. > l ^Iref V 1 eads to x = q.,-q. 2 vref v dt O C183 The suboptimal solution of the minimum energy terminal control defined by C183 and the performance criterion J=l/2/ u.dt o t C193 C203 gives the control law, u = 6VTZCq. -q.3 + 4/TCq.,-q. 3 + Y. where the values of the time varying optimal feedback gains are selected and kept constant as a suboptimal solution. The generalized torque vector is obtained as T = MCqD C8/T Cq.,-q D + 4/TCq q 3 + y 3 + k iref ı % r©f % ' i, C213 From u. = K x + K x + y C22D v p\. i VI 2 I, one can obtain the following; wZ = K =S/T2.... w = Vfe/T op o 2Cw = K = 4/T.... T = 2/V6 = 0. 82 o v ^ PD Control with MI MO System Approach: Linearized Model of Robot Dynamics: In equatianClOD expanding N and G about the operating /N /?% I*. /\ point P=CT,q,q3 using Taylor series and assuming MCq+6cp = MCq3 for small joint position perturbation oq Eq. 10 can be linearized as A6qCt3+BaqCt3+CöqCtD=TCt3 C233 where the constant nxn matrices A, B and C are defined by A=CM] ; B=C - - ] ; C=[ ] P p p dq dq Eq. 23 gives a set of coupled linear time-invariant differential equations which describe the incremental behavior of the robot dynamics for perturbations in the neighborhood of the nominal operating point P. Multi variable Controller Design: The total control law is given by TCt3=T+K C6q -oqD +K C6q -<5qD +C6q +B6q +A<5q C243 The first component is the value of the joint torque vector at the nominal operating point P, namely T q, q and 6a,6a,6a are the reference perturbation vectors for r r r joint position, velocity and acceleration. By choosing the feedback gain matrices according to the equation below, the dynamics of tracking errors are decoupled and the response of each error is shaped as desired. K = AD -B ; K - AD -C C25I) VI' p z where D = diag<2C w> and D = diag-Cw > are the constant nxn matrices. The damping ratio and the undamped natural frequency of each tracking error e =£q. Ct3-<5q Ct3 are specified by the designer. XI While feedback controller determines the stability and the transient response of the tracking error, the feedforward controller determines the steady-state response of the tracking error. The operation during the pick and place of the glass product consist of four main steps. 1 -Picking the product from first conveyor and transferring it to a fixed point between the two conveyors. The intermediate point is selected for obstacle avoidance and information update. 2-Esti mating the target point from the information received from the detectors placed on the second conveyor. 3-Fol lowing and catching the holder and placing the product. 4-Going to the initial position for repeating, Since the conveyors are unsynchronized, first a fixed point is given as the reference point. During the motion of the manipulator to this point, the controller receives information about the position of the holder, calculates and updates the new reference point on the axis of the second conveyor. The trajectory followed until this time is composed of two cubic polynomials at the joint space whose coefficients are found from the initial and boundary conditions for position and velocity continuity. During the third stage, position and speed control is performed according to the signals coming from the detectors, and at the end of a predetermined time, the glass product is placed on the holder. Then the end effector returns to its initial position following a trajectory which consist of a cubic polynomial whose coefficients are determined by the same way indicated above. The simulation program had been written in FORTRAN before, however later PASCAL language has been preffered since it was modular, fast, easily understandable and more suitable for writing control algorithms in the case of constructing the system. RUNGE-KUTTA IV method which gives sufficient accuracy for the solution of non-linear systems, has been used for integration. The step length has been chosen as h=0. 004 sec. Trajectory scenarios consisting of exponential and cubic terms have been used for the comparison of investigated control algoritms. From the simulation results, it has been seen that the most suitable control algorithm is the PD control with MI MO system approach. XXI For the simulation of the real system two different scenarios have been tested. In the first scenario, the distances between the holder axes are considered to be equal and the conveyor speed fluctuates within some limits. From the results of this simulation, it has been observed that the maximum misalignment between the holder axis and the product axis was about 2mm. In the second scenario which was produced to measure the performance of the developed control system, it is assumed that the holders are placed with an exaggerated disorder. The results showed that the misalignment wouldn't exceed 3 mm in the worst case. Since the dimensions of the holders are quite larger than those of products, the misalignments do not significantly affect the performance of the operation. The measurements used in the simulation have been taken from a work station which is under operation at a factory. The simulation results show that the developed synchronization and control algorithms would operate with sufficient performance at the work station where the environment conditions are not suitable for human beings to work efficiently.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1990

##### Anahtar kelimeler

Benzetim,
Manipülatör,
Tasarım,
Simulation,
Manipulator,
Design