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Esnek Mekanizmaların Tasarlanması Ve Kontrolü

Esnek Mekanizmaların Tasarlanması Ve Kontrolü

##### Dosyalar

##### Tarih

2013-08-29

##### Yazarlar

Tekeş, Ayşe

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu tezde literatürde analizi yapılmamış olan iki farklı esnek mekanizma tasarımı ve kontrolü çalışması yapılmıştır. Mekanizmalardan ilki bekleme hareketi yapması istenen esnek çift kol mekanizmasıdır. Esnek çift kol mekanizmasıyla rijit krank biyel mekanizması hareket ettirilmektedir. Esnek çift kol mekanizması mikro ve makro seviyede üretilebilir. Mekanizmanın kinematik analizi Elastika teorisi ile elde edilmiştir. Büyük şekil değiştirmelerine maruz esnek kirişlerin yük-esneme eğrileri ankastre kiriş, tek kol ve tüm çift kol için hesaplanmış ve yük-esneme eğrisi bir polinomla ifade edilmiştir. Mekanizmanın dinamik cevabı doğrusal olmayan hareket denklemlerinin Runge Kutta metodu kullanılarak elde edilmiştir. Mekanizma lineer olmayan yay karakteristiğine sahip yay içerdiğinden hareket denklemi tüm durumlar için durum geri beslemesi ile doğrusallaştırılmıştır. Durum geri beslemesi ile doğrusallaştırılmış sisteme PD kontrolcüsü uygulanmıştır. İstenen referans hareketin gerçekleştirilebilmesi için kontrol katsayıları uygun şekilde ayarlanmıştır. İkinci mekanizma tamamen esnek 5 koldan oluşan bir mekanizmasıdır. İki serbestlik dereceli esnek beş kol mekanizmasının uç noktası istenen farklı hareketleri çizecek şekilde tasarlanmış ve kontrol edilmiştir. Mekanizma beş rijit kol ve çok esneyen beş esnek mafsaldan oluşmaktadır. Mekanizma taban kollarına uygulanan torklarla hareket ettirilmektedir. Mekanizmanın matematik modeli sistemin konum değişimini yöneten vektörel denklemler ve rijit kollara ait hareket denkelmleri kullanılarak elde edilmiştir. Taban kollara uygulanan torklarla hareket ettirilen mekanizmanın dinamik cevabı Runge Kutta metodu ile bulunmuştur. Mekanizmanın rijit eşlenik gövde modelinde, yüksek esneme özelliğine sahip kısa esnek mafsallar yerine aynı karakteristiğe sahip burulma yayı kullanılmıştır. Mekanizmanın istenen herhangi bir yörüngeyi takibi için PD kontrol kuralı kullanılmış ve en uygun katsayılar seçilmiştir.

The concept of compliant mechanisms started with the challenge to exploit the flexibility of linkages for beneficial purposes. Instead of avoiding linkage flexibility problems of mechanisms, employing the flexibility itself for beneficial purposes can lead to some advantages over rigid body mechanisms. Compliant mechanisms can exploit the flexibility of its components for the creation of motion, which can be very difficult and expensive to mimic by rigid body mechanisms. Compliant mechanisms gain their mobility from the flexibility of its segments. Compliant mechanisms consist of at least one flexible member along with the conventional rigid links. The mobility of the compliant mechanisms is achieved not only due to relative movements of the joints, but also due to relative deflection of the flexible links. Using traditional rigid links and elastic joints, a compliant mechanism design might be achieved. A compliant mechanism is called fully or partially compliant depending on the existing of traditional links and joints. There are certain advantages to use the compliant mechanisms. Among these, the low production cost and the output precision can be considered the most significant ones. Compliant joints have certain disadvantages such as a decreased fatigue life and increased geometric stress concentration factor. The use of flexible members can make a mechanism comparatively lighter, thus enhancing its use in applications requiring low weight. The design of a compliant mechanism is more complicated than the design of a rigid mechanism as it requires large deflection analysis, consideration of the stored strain energy and the solution of transcendental loop closure equations. It is common in compliant mechanism design to explore the flexible member deflection to its outmost limit, thus requiring a large deflection analysis. The large deflection analysis of compliant mechanisms might be achieved using one of the three methods of; 1) using a Pseudo Rigid Body Model, 2) using a nonlinear FEM software package, and 3) using the Elastica theory. These methods are summarized as follows: A compliant mechanism might be modeled with a Pseudo Rigid Body Model (PRBM),which is a rigid body equivalent of the flexible linkage having equivalent torsional springs at its joints. PRBM makes the compliant mechanism analysis relatively easier. Compliant mechanisms are usually designed for simple motion requirements and the trial and the error method are commonly used to design novel compliant mechanisms. Once a desired designed is found, the structural properties of a compliant mechanism might be checked and revised according to the required load and flexible beam maximum bending stress. Another efficient analysis tool of compliant mechanisms is to use nonlinear FEM. Numerical techniques for large deflection analysis of structures are available in the literature. These techniques address geometric and material nonlinearities of structures. Most of these techniques use Finite Element Methods (FEMs) of different kinds. There are mainly two kinds of incremental nonlinear FEMs considering geometric nonlinearities, the total Lagrangian and the updated Lagrangian methods. Material nonlinearities have also been considered in compliant mechanism design. Elastica theory is another commonly used compliant mechanism analysis tool. Elastica theory provides a closed form of the solution. In other words, the functional solution relation between the applied load and the deflection cannot be expressed explicitly in the form of y=y(x). The mathematical exact equations need to be solved first to express such explicit y=y(x) relationship, then the exact solution might be represented by an accurate polynomial. The only disadvantage of the Elastica theory is that its application is restricted to simple geometries and simple loading conditions. Therefore compliant mechanism designs that can be investigated using the Elastica theory consist of simple flexible members and simple loading conditions. The design and control of compliant double arm mechanism and compliant five bar mechanism are investigated in this thesis. A compliant folded beam (double arm) compliant mechanism design is investigated both kinematically and dynamically. This mechanism consists of eight flexible arms, two rigid side arms, two rigid cranks, and a rigid coupler (a shuttle), actuated by a comb drive. First, the load deflection behavior of the parallel arms is obtained by using the cantilever beam’s large deflection results. The nonlinear spring rate of this member (a fixed-free beam) is found using the inextensible Elastica theory. Then, the spring rates of the arm beam and the complete mechanism are calculated. Finally, the calculated nonlinear spring rate of the complete mechanism is used to obtain the equation of motion of the whole system. The dynamic simulation of the compliant parallel arm mechanism is studied using numerical simulation procedures considering the desired force actuation coming from a comb drive unit. The simulation results of the compliant double arm mechanism have been obtained by including the translational inertia of the rigid coupler (the shuttle). The relation between the compliant double arm mechanism and the crank angle is derived by its geometry alone since the crank is assumed to be massless (its mass is quite small comparing to the shuttle). Load deflection characteristics of the flexible beams are represented by polynomial curve fits (obtained from nonlinear inextensible exact beam theory) that are then used as the nonlinear stiffness characteristics in the lumped parameter model. The compliant folded beam mechanism might be used as an indexing or a dwell mechanism if its rigid shuttle can be positioned at the desired locations at specified times and kept there for the specified durations. In this investigation, only the desired motion control is studied. It should be noted that the dwell motion is one of the many possible function generations that can be achieved with the trajectory control of this mechanism. This is also demonstrated by choosing several different functions at the non-dwell part of the motion. The magnetically actuated parallel arm mechanism with a control law is well suited for use as a function generator. If a desired velocity is achieved at a specified output location, the rigid coupler might be used as a punch mechanism. Even though local PID controllers are well suited for position control, they might cause an overshoot of the response. Overshoot is an undesired behavior in some applications. Simultaneous local control loops result in overall delays. Trajectory tracking is required in high-speed applications. In practice, the system dynamics performance depends on several different variables. If a variable under question is measurable, it can be used to characterize the system response or to tune the controller parameters . The mechanical system model concerning the equation of motion is nonlinear due to the nonlinear behavior of the elastic members modeled by higher order polynomials. In order to design linear controllers, a linear model of the system has to be obtained. A linear model is obtained by linearizing the nonlinear governing equations of motion about the operating point. This method will work best when the generated functions have the same amplitude as that of the operating point. An alternative method is to linearize the actual system behavior by feedback cancellation of the nonlinearities in feedback linearization. The feedback linearization method has superiority to the operating point linearization of the model, because it linearizes the system for the whole range of input variables instead of working with a linear model of a nonlinear system that is only accurate in the vicinity of the operating point. The feedback linearization system is found suitable for the compliant mechanism control due to its nonlinear stiffness function. Several examples, designs are presented and introduced where the proposed method may find application areas in both macro and micro dimensions. The second mechanism is the compliant five bar mechanism. In this part of the thesis, a compliant five bar mechanism consists of large deflecting compliant joints is investigated kinematically by using vector loop closures and dynamically by applying torques at the base links. The particular design and the control problem is new to the compliant mechanism literature. The particular compliant five bar mechanism consists of flexible links and two motors to actuate the base links. Vector closure loop techniques are used to derive mathematical model of the system. First, equivalent Pseudo Rigid Body Model (PRBM) of the compliant five bar mechanism is obtained. PRBM is composed of the rigid links which have the same load deflection characteristics with the fully compliant mechanism and the torsional springs attached between the links. The significant point of finding the equivalent PRB model is equivalent spring constant of the mechanism is supposed to have the same characteristic with the compliant mechanism. Then the kinematic and, the dynamic analysis are investigated by using vector loop closures. The first and the second derivative of the loop closure equations results in the velocity and the acceleration relations of the mechanism. The control problem for this system is achieved by inverting the relationship between the input and the output to find what an action is required to have the desired output . So how much effort required for the actuators of the mechanism to follow the desired trajectory by changing the input-output relation is investigated. Mechanism Mathematical Model is generated using MATLAB Simulink software. Two types of simulation performed: The first one running the motor with constant speed following either a straight line or a closed path (a circle). The second one is following a complicated path with a variable speed. A signature is chosen as a complicated path for the compliant five-bar mechanism end effector to follow. The desired trajectory is obtained by actuating the base links with constant angular velocity either running the motors with different directions so the mid joint could follow a straight path or running the motors with the same direction. After that both motors required torks are estimated. Finally the mechanism mid-elastic joint, the tip control is investigated to follow a desired trajectory. A line, a closed circular path and a signature path are defined as the tip trajectory of the compliant five bar mechanism. These trajectories are achieved using a PD control law. The PD control the coefficients are tuned until the desired motion is obtained.

The concept of compliant mechanisms started with the challenge to exploit the flexibility of linkages for beneficial purposes. Instead of avoiding linkage flexibility problems of mechanisms, employing the flexibility itself for beneficial purposes can lead to some advantages over rigid body mechanisms. Compliant mechanisms can exploit the flexibility of its components for the creation of motion, which can be very difficult and expensive to mimic by rigid body mechanisms. Compliant mechanisms gain their mobility from the flexibility of its segments. Compliant mechanisms consist of at least one flexible member along with the conventional rigid links. The mobility of the compliant mechanisms is achieved not only due to relative movements of the joints, but also due to relative deflection of the flexible links. Using traditional rigid links and elastic joints, a compliant mechanism design might be achieved. A compliant mechanism is called fully or partially compliant depending on the existing of traditional links and joints. There are certain advantages to use the compliant mechanisms. Among these, the low production cost and the output precision can be considered the most significant ones. Compliant joints have certain disadvantages such as a decreased fatigue life and increased geometric stress concentration factor. The use of flexible members can make a mechanism comparatively lighter, thus enhancing its use in applications requiring low weight. The design of a compliant mechanism is more complicated than the design of a rigid mechanism as it requires large deflection analysis, consideration of the stored strain energy and the solution of transcendental loop closure equations. It is common in compliant mechanism design to explore the flexible member deflection to its outmost limit, thus requiring a large deflection analysis. The large deflection analysis of compliant mechanisms might be achieved using one of the three methods of; 1) using a Pseudo Rigid Body Model, 2) using a nonlinear FEM software package, and 3) using the Elastica theory. These methods are summarized as follows: A compliant mechanism might be modeled with a Pseudo Rigid Body Model (PRBM),which is a rigid body equivalent of the flexible linkage having equivalent torsional springs at its joints. PRBM makes the compliant mechanism analysis relatively easier. Compliant mechanisms are usually designed for simple motion requirements and the trial and the error method are commonly used to design novel compliant mechanisms. Once a desired designed is found, the structural properties of a compliant mechanism might be checked and revised according to the required load and flexible beam maximum bending stress. Another efficient analysis tool of compliant mechanisms is to use nonlinear FEM. Numerical techniques for large deflection analysis of structures are available in the literature. These techniques address geometric and material nonlinearities of structures. Most of these techniques use Finite Element Methods (FEMs) of different kinds. There are mainly two kinds of incremental nonlinear FEMs considering geometric nonlinearities, the total Lagrangian and the updated Lagrangian methods. Material nonlinearities have also been considered in compliant mechanism design. Elastica theory is another commonly used compliant mechanism analysis tool. Elastica theory provides a closed form of the solution. In other words, the functional solution relation between the applied load and the deflection cannot be expressed explicitly in the form of y=y(x). The mathematical exact equations need to be solved first to express such explicit y=y(x) relationship, then the exact solution might be represented by an accurate polynomial. The only disadvantage of the Elastica theory is that its application is restricted to simple geometries and simple loading conditions. Therefore compliant mechanism designs that can be investigated using the Elastica theory consist of simple flexible members and simple loading conditions. The design and control of compliant double arm mechanism and compliant five bar mechanism are investigated in this thesis. A compliant folded beam (double arm) compliant mechanism design is investigated both kinematically and dynamically. This mechanism consists of eight flexible arms, two rigid side arms, two rigid cranks, and a rigid coupler (a shuttle), actuated by a comb drive. First, the load deflection behavior of the parallel arms is obtained by using the cantilever beam’s large deflection results. The nonlinear spring rate of this member (a fixed-free beam) is found using the inextensible Elastica theory. Then, the spring rates of the arm beam and the complete mechanism are calculated. Finally, the calculated nonlinear spring rate of the complete mechanism is used to obtain the equation of motion of the whole system. The dynamic simulation of the compliant parallel arm mechanism is studied using numerical simulation procedures considering the desired force actuation coming from a comb drive unit. The simulation results of the compliant double arm mechanism have been obtained by including the translational inertia of the rigid coupler (the shuttle). The relation between the compliant double arm mechanism and the crank angle is derived by its geometry alone since the crank is assumed to be massless (its mass is quite small comparing to the shuttle). Load deflection characteristics of the flexible beams are represented by polynomial curve fits (obtained from nonlinear inextensible exact beam theory) that are then used as the nonlinear stiffness characteristics in the lumped parameter model. The compliant folded beam mechanism might be used as an indexing or a dwell mechanism if its rigid shuttle can be positioned at the desired locations at specified times and kept there for the specified durations. In this investigation, only the desired motion control is studied. It should be noted that the dwell motion is one of the many possible function generations that can be achieved with the trajectory control of this mechanism. This is also demonstrated by choosing several different functions at the non-dwell part of the motion. The magnetically actuated parallel arm mechanism with a control law is well suited for use as a function generator. If a desired velocity is achieved at a specified output location, the rigid coupler might be used as a punch mechanism. Even though local PID controllers are well suited for position control, they might cause an overshoot of the response. Overshoot is an undesired behavior in some applications. Simultaneous local control loops result in overall delays. Trajectory tracking is required in high-speed applications. In practice, the system dynamics performance depends on several different variables. If a variable under question is measurable, it can be used to characterize the system response or to tune the controller parameters . The mechanical system model concerning the equation of motion is nonlinear due to the nonlinear behavior of the elastic members modeled by higher order polynomials. In order to design linear controllers, a linear model of the system has to be obtained. A linear model is obtained by linearizing the nonlinear governing equations of motion about the operating point. This method will work best when the generated functions have the same amplitude as that of the operating point. An alternative method is to linearize the actual system behavior by feedback cancellation of the nonlinearities in feedback linearization. The feedback linearization method has superiority to the operating point linearization of the model, because it linearizes the system for the whole range of input variables instead of working with a linear model of a nonlinear system that is only accurate in the vicinity of the operating point. The feedback linearization system is found suitable for the compliant mechanism control due to its nonlinear stiffness function. Several examples, designs are presented and introduced where the proposed method may find application areas in both macro and micro dimensions. The second mechanism is the compliant five bar mechanism. In this part of the thesis, a compliant five bar mechanism consists of large deflecting compliant joints is investigated kinematically by using vector loop closures and dynamically by applying torques at the base links. The particular design and the control problem is new to the compliant mechanism literature. The particular compliant five bar mechanism consists of flexible links and two motors to actuate the base links. Vector closure loop techniques are used to derive mathematical model of the system. First, equivalent Pseudo Rigid Body Model (PRBM) of the compliant five bar mechanism is obtained. PRBM is composed of the rigid links which have the same load deflection characteristics with the fully compliant mechanism and the torsional springs attached between the links. The significant point of finding the equivalent PRB model is equivalent spring constant of the mechanism is supposed to have the same characteristic with the compliant mechanism. Then the kinematic and, the dynamic analysis are investigated by using vector loop closures. The first and the second derivative of the loop closure equations results in the velocity and the acceleration relations of the mechanism. The control problem for this system is achieved by inverting the relationship between the input and the output to find what an action is required to have the desired output . So how much effort required for the actuators of the mechanism to follow the desired trajectory by changing the input-output relation is investigated. Mechanism Mathematical Model is generated using MATLAB Simulink software. Two types of simulation performed: The first one running the motor with constant speed following either a straight line or a closed path (a circle). The second one is following a complicated path with a variable speed. A signature is chosen as a complicated path for the compliant five-bar mechanism end effector to follow. The desired trajectory is obtained by actuating the base links with constant angular velocity either running the motors with different directions so the mid joint could follow a straight path or running the motors with the same direction. After that both motors required torks are estimated. Finally the mechanism mid-elastic joint, the tip control is investigated to follow a desired trajectory. A line, a closed circular path and a signature path are defined as the tip trajectory of the compliant five bar mechanism. These trajectories are achieved using a PD control law. The PD control the coefficients are tuned until the desired motion is obtained.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2013

Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2013

Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2013

##### Anahtar kelimeler

esnek mekanizmalar,
durum geri beslemesi,
PID kontrolcüsü,
yörünge kontrolü,
beş kol mekanizması,
Compliant mechanims,
state feedback linearization,
PID controller,
trajectory controller,
five bar mechanism