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Pneumatic Artificial Muscle Optimal Control with Simulated Annealing ⁎
10th IFAC Symposium on Biological and Medical Systems 10th IFAC on and 10th IFAC Symposium Symposium on Biological Biological and Medical Medical Systems Systems São Paulo, Brazil, September 3-5, 2018 10th IFAC Symposium on Biological and Medical Systems 10th IFAC Symposium on Biological and Medical Systems São Paulo, Paulo, Brazil, Brazil, September September 3-5, 3-5, 2018 2018 Available online at www.sciencedirect.com São 10th IFAC Symposium on Biological and Medical Systems São Paulo, Brazil, September 3-5, 2018 São Paulo, Brazil, September 3-5, 2018 São Paulo, Brazil, September 3-5, 2018
ScienceDirect
IFAC PapersOnLine 51-27 (2018) 333–338
Pneumatic Artificial Muscle Optimal Pneumatic Pneumatic Artificial Artificial Muscle Muscle Optimal Optimal Control with Simulated Annealing Pneumatic Artificial Muscle Optimal Control with Simulated Annealing Control with Simulated Annealing Control with Simulated Annealing ∗,2 ∗,1 Oswaldo Horikawa ∗,2 William Scaff ∗,1
∗,1 Oswaldo Horikawa ∗,2 William Scaff William Scaff Horikawa∗,3∗,2 ∗,1 ∗,1 Oswaldo ∗,2 William Scaff Oswaldo ∗,3 Marcos de Sales Guerra Tsuzuki William Scaff Horikawa Marcos de Sales Guerra Horikawa Tsuzuki ∗,3 ∗,1 Oswaldo ∗,2 Marcos de Sales Guerra Tsuzuki ∗,3 William Scaff Oswaldo Horikawa ∗,3 Marcos de Sales Guerra Tsuzuki Marcos de Sales Guerra Tsuzuki ∗,3 Marcos de Sales Guerra Tsuzuki ∗ ∗ Mechatronics and Mechanical Systems Engineering Department, ∗ Mechatronics and and Mechanical Mechanical Systems Systems Engineering Engineering Department, Department, ∗ Mechatronics ∗ Mechatronics and Mechanical Engineering Department, Escola Polit´ e cnica da UniversitySystems of S˜ S˜ ao o Paulo, Paulo, S˜ ao o Paulo, Paulo, Brazil andda Mechanical Systems Engineering Department, Escola Polit´ eecnica University of a S˜ a Brazil ∗ Mechatronics Escola Polit´ cnica da University of S˜ a o Paulo, S˜ a o Paulo, Brazil Mechatronics and Mechanical Systems Engineering Department, Escola Polit´ Polit´eecnica cnica da da University University of of S˜ S˜ ao o Paulo, Paulo, S˜ S˜ ao o Paulo, Paulo, Brazil Escola a a Brazil Escola Polit´ecnica da University of S˜ ao Paulo, S˜ ao Paulo, Brazil Abstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal musAbstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal musAbstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal musAbstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal muscles. They are attractive for many applications where safety, human interaction or biomimetic Abstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal muscles. They are attractive for many applications where safety, human interaction or biomimetic cles. They are attractive for many applications where safety, human interaction or biomimetic Abstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal muscles. They are attractive for many applications where safety, human interaction or biomimetic behavior is required. However, its use is not yet widespread because of the difficulties of working cles. They are attractive for many applications where safety, human interaction or biomimetic behavior is required. However, its use is not yet widespread because of the difficulties of working behavior is required. However, its use is not yet widespread because of the difficulties of working cles. They are attractive for many applications where safety, human interaction or biomimetic behavior is required. However, its use is not yet widespread because of the difficulties of working with them, such as high nonlinearities, difficult to model and control accurately. This work behavior is required. However, its use is not yet widespread because of theaccurately. difficulties of working with them, such as high nonlinearities, difficult to model and control This work with them, such as high nonlinearities, difficult to model and control accurately. This work behavior is required. However, its use is not yet widespread because of the difficulties of working with them, such as high nonlinearities, difficult to model and control accurately. This work proposes a simple method for controlling the pneumatic muscle, which is using aa PID controller with them, such as high nonlinearities, difficult to model and control accurately. This work proposes a simple method for controlling the pneumatic muscle, which is using PID controller proposes a simple method for controlling the pneumatic muscle, which is using a PID controller with them, such as high nonlinearities, difficult to model and control accurately. This work proposes a simple method for controlling the pneumatic muscle, which is using a PID controller tuned with a simulated annealing optimization algorithm. proposes a simple method for controlling the pneumatic muscle, which is using a PID controller tuned with aa simulated annealing optimization algorithm. tuned with simulated annealing optimization algorithm. proposes a simple method for controlling the pneumatic muscle, which is using a PID controller tuned with a simulated annealing optimization algorithm. tuned with a simulated annealing optimization algorithm. © 2018,with IFACa (International Federationoptimization of Automatic algorithm. Control) Hosting by Elsevier Ltd. All rights reserved. tuned simulated annealing Keywords: Actuators, PID control, optimization, numerical simulation. Keywords: Actuators, PID control, optimization, Keywords: Actuators, Actuators, PID control, control, optimization, numerical numerical simulation. simulation. Keywords: Keywords: Actuators, PID PID control, optimization, optimization, numerical numerical simulation. simulation. Keywords: Actuators, PID control, optimization, numerical simulation. 1. INTRODUCTION There are many researches related to PAMs control. Clas1. INTRODUCTION INTRODUCTION There are many researches related to PAMs control. Clas1. There are many researches related to PAMs control. Clas1. There are many researches related to PAMs control. Classical linear control techniques such as PI and PID control 1. INTRODUCTION INTRODUCTION There are many researches related to PAMs control. Classical linear control techniques such as PI and PID control sical linear control techniques such as PI and PID control 1. INTRODUCTION There are many researches related to PAMs control. Classical linear control techniques such as PI and PID control have been proposed, because they are simple to design sical linear control techniques such as PI and PID control Pneumatic Artificial Muscles (PAMs) are interesting achave been proposed, proposed, because such theyasare are simple to control design been because they simple to design Pneumatic Artificial Artificial Muscles Muscles (PAMs) (PAMs) are are interesting interesting acac- have sical linear control techniques PI and PID Pneumatic have been proposed, because they are simple to design and have high adaptability (Sakthivelu and Chong, 2015). have beenhigh proposed, because they areand simple to design Pneumatic Muscles (PAMs) are actuators that have similar static and dynamic characterhave adaptability (Sakthivelu Chong, 2015). Pneumatic Artificial Muscles (PAMs) are interesting interesting ac- and and have adaptability (Sakthivelu Chong, 2015). tuators thatArtificial have similar similar static and dynamic dynamic characterhave beenhigh proposed, because they areand simple toby design tuators that have static and characterhave high adaptability (Sakthivelu and Chong, 2015). Pneumatic Artificial Muscles (PAMs) are therefore, interesting ac- and However, many of the PID controllers are tuned trial and have high adaptability (Sakthivelu and Chong, 2015). tuators that have similar static and dynamic characteristics to the human skeletal muscle and, have However, many of the PID controllers are tuned by trial tuators that have similar static and dynamic characterHowever, many of the PID controllers are tuned by trial istics to the human skeletal muscle and, therefore, have and have high adaptability (Sakthivelu and Chong, 2015). istics to the human skeletal muscle and, therefore, have However, many of the PID controllers are tuned by trial tuators that have similar static and dynamic characterand error or tuned based on the Ziegler-Nichols method, However, many of the PID controllers are tuned by trial istics to the human skeletal muscle and, therefore, have many applications in orthetics and biomimetic systems and error or tuned based on the Ziegler-Nichols method, istics to the human skeletal muscle and, therefore, have and error or tuned based on the Ziegler-Nichols method, many applications in orthetics and biomimetic systems However, many of the PID controllers are tuned by trial many applications in orthetics and biomimetic systems and error or tuned based on the Ziegler-Nichols method, istics to the human skeletal muscle and, therefore, have which is known to produce poor results in many cases and error or tuned based on the Ziegler-Nichols method, many applications in orthetics and biomimetic systems (Daerden and Lefeber, 2002; Tondu et al., 2009). They which is known known to based produce poor results in in many many cases many applications in orthetics and biomimetic systems which is to produce poor results cases (Daerden and Lefeber, 2002; Tondu et al., 2009). They and error or tuned on the Ziegler-Nichols method, (Daerden and Lefeber, 2002; Tondu et al., 2009). They ˚ which is known to produce poor results in many cases many applications intoorthetics and biomimetic systems (Sakthivelu and Chong, 2015; A str¨ oom and H¨ aagglund, which is known to produce poor results in many 2001; cases (Daerden and 2002; et 2009). They ˚ are lightweight, easy fabricate, low cost, and and Chong, 2015; A str¨ and H¨ (Daerden and Lefeber, Lefeber, 2002; Tondu Tondu et al., al.,compliant 2009). They ˚ (Sakthivelu and Chong, 2015; A str¨ oom m and H¨ aagglund, gglund, 2001; are lightweight, easy to to fabricate, fabricate, low cost, cost, compliant and (Sakthivelu ˚ which is et known to produce poor results in many 2001; cases are lightweight, easy low and (Sakthivelu and Chong, 2015; A str¨ m and H¨ gglund, 2001; ˚ (Daerden and Lefeber, 2002; Tondu et al.,compliant 2009). They Schr¨ ooder al., 2003). (Sakthivelu and Chong, 2015; A str¨ o m and H¨ a gglund, 2001; are lightweight, easy to fabricate, low cost, compliant and inherits the advantages of pneumatic systems. Schr¨ der et al., 2003). are lightweight, easy to fabricate, low cost, compliant and Schr¨ ooder et al., 2003). inherits the advantages advantages of pneumatic pneumatic systems. (Sakthivelu and Chong, 2015; ˚ Astr¨om and H¨agglund, 2001; inherits the of systems. Schr¨ der et al., 2003). are lightweight, easy to fabricate, low cost, compliant and Schr¨ o der et al., 2003). inherits the advantages of pneumatic systems. inherits the advantages of pneumatic systems. PID controllers are one one of of the the most most common common controllers controllers Schr¨ ocontrollers der et al., 2003). The invention of the PAM can be traced back to 1930, by are inherits the advantages of pneumatic systems. PID controllers are one of the most common controllers The invention of the the PAM PAM can be be traced traced back to to 1930, 1930, by by PID The invention of can back PID controllers are one of the most common controllers used today and may have similar performance to very comPID controllers are one of the most common controllers The invention of the PAM can be traced back to 1930, by S. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., used today and may have similar performance to very comThe invention of the PAM can be traced back to 1930, by used today and may have similar performance to very comS. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., PID controllers are one of the most common controllers S. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., used today and may have similar performance to very comThe invention of the PAM can be traced back to 1930, by plex control techniques with adequate gains. A method for used today and may have similar performance to very comS. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., 2016). Later, it was used by McKibben in the 1950’s to plex control techniques with adequate gains. A method for S. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., plex control techniques with adequate gains. A method for 2016). Later, it was used by McKibben in the 1950’s to used today and may have similar performance to very com2016). Later, it was used by McKibben in the 1950’s to plex control techniques with adequate gains. A method for S. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., choosing good PID gains can make the implementation of plex control techniques with adequate gains. A method for 2016). Later, it was used by McKibben in the 1950’s to power an orthetic upper limb (Tondu and Lopez, 1997; choosing good PID gains can make the implementation of 2016). Later, it was used by McKibben in the 1950’s to choosing good PID gains can make the implementation of power an orthetic upper limb (Tondu and Lopez, 1997; plex control techniques with adequate gains. A method for power an orthetic upper limb (Tondu and Lopez, 1997; choosing good PID gains can make the implementation of 2016). Later, it was used by McKibben in the 1950’s to PAMs simpler and promote it’s use in industry and other choosing good PID gains can make the implementation of power an orthetic upper limb (Tondu and Lopez, 1997; Daerden and Lefeber, 2002). However, it was not given PAMs simpler and promote it’s use in industry and other power an orthetic upper limb (Tondu and Lopez, 1997; PAMs simpler and promote it’s use in industry and other Daerden and Lefeber, 2002). However, it was not given choosing good PID gains can make the implementation of Daerden and Lefeber, 2002). However, it was not given PAMs simpler and promote it’s use in industry and other power an orthetic upper limb (Tondu and Lopez, 1997; areas. This method, however, should be flexible to address PAMs simpler and promote it’s use in industry and other Daerden and Lefeber, 2002). However, it was not given much attention because of the difficulty of working with areas. This method, however, should be flexible to address Daerden and Lefeber, 2002). However, it was not given This method, however, should be flexible to address much attention because2002). of the theHowever, difficulty itof ofwas working with areas. PAMs simpler and promote it’s use in industry and other much attention because of difficulty working with areas. This method, however, should be flexible to address Daerden and Lefeber, not given specific applications which may need different performance areas. This method, however, should be flexible to address much attention because of the difficulty of working with these muscles with the technology at that time (Daerden specific applications which mayshould need different different performance much muscles attentionwith because of the difficulty working with specific applications which may need performance these muscles with the technology technology at that thatof time (Daerden areas. This method, however, bebut flexible to address these the at (Daerden applications which may need performance much attention because of the difficulty oftime working with specific characteristics, such as slow response accurate posispecific applications which may need different different performance these muscles with the technology at that time (Daerden and Lefeber, 2002). With the progress on valve technology characteristics, such as slow response but accurate posithese muscles with the technology at that time (Daerden characteristics, such as slow response but accurate posiand Lefeber, 2002). With the progress on valve technology specific applications which may need different performance and Lefeber, 2002). With the progress on valve technology characteristics, such as slow response but accurate posithese muscles with the technology at that time (Daerden tioning or the fastest time response as possible. An easy characteristics, such as slow response but accurate posiand Lefeber, 2002). With the progress on valve technology control techniques, the interest on pneumatic muscles, tioning or the fastest time response as possible. An easy and Lefeber, 2002). With the progress on valve technology tioning or the fastest time response as possible. An easy and control techniques, the interest on pneumatic muscles, characteristics, such as slow response but accurate posiand control techniques, the interest on pneumatic muscles, tioning or the fastest time response as possible. An easy and Lefeber, 2002). With the progress on valve technology strategy for designing a controller to meet the application’s tioning or the fastest time response as possible. An easy control techniques, the interest on pneumatic muscles, more specifically the McKibben type PAM, is now back. strategy for designing a controller to meet the application’s and control techniques, the interest on pneumatic muscles, strategy for designing a controller to meet the application’s more specifically the McKibben type PAM, is now back. tioning or the fastest time response as possible. An easy more specifically the McKibben type PAM, is now back. strategy for designing a controller to meet the application’s and control techniques, the interest on pneumatic muscles, requirements is still missing. strategy for designing a controller to meet the application’s more specifically the McKibben type PAM, is now back. requirements is still still missing. missing. more McKibben specifically the McKibben type PAM, isPAM now back. requirements is strategy for designing a controller to meet the application’s The PAM is the most used today requirements is still missing. more McKibben specifically the McKibben now back. iscontrol still missing. The McKibben PAM is the the type most PAM, used isPAM PAM today requirements The PAM is most used today To address the problem, this paper proposes a PID The PAM the most PAM requirements iscontrol still missing. (Kothera et al., 2012). One interesting characteristic of To address the problem, this paper proposes a PID The McKibben McKibben PAM is is theinteresting most used used PAM today today To address the control problem, this paper proposes a PID (Kothera et al., 2012). One interesting characteristic of (Kothera et al., 2012). One characteristic of To address the control problem, this paper proposes a PID The McKibben PAM is the most used PAM today controller which is tuned by the Simulated Annealing algoTo address the control problem, this paper proposes a PID (Kothera et al., 2012). One interesting characteristic of the McKibben PAM is that it can be build without fercontroller which is tuned by the Simulated Annealing algo(Kothera et al., 2012). One interesting characteristic of controller which is tuned by the Simulated Annealing algothe McKibben PAM is that it can be build without ferTo address the control problem, this paper proposes a PID the McKibben PAM is that it can be build without fercontroller which is tuned by the Simulated Annealing algo(Kothera et al., 2012). One interesting characteristic of rithm. The method is tested in a mathematical nonlinear controller which is tuned by the Simulated Annealing algothe McKibben PAM is that it can be build without ferromagnetic and electric conductive materials (Scaff et al., rithm. Thewhich method is tested tested in Simulated mathematical nonlinear the McKibben is conductive that it can materials be build (Scaff without The method is in aaa mathematical nonlinear romagnetic andPAM electric conductive materials (Scaff et feral., rithm. controller is tuned by the Annealing algoromagnetic and electric et al., rithm. The method is tested in mathematical nonlinear the McKibben PAM is that it can be build without fermodel of a 1 DoF positioning system powered by a McKrithm. The is tested insystem a mathematical nonlinear romagnetic electric materials (Scaff et 2014). This characteristic makes them suitable to power of aa 11method DoF positioning powered by a McKromagnetic and electric conductive conductive materials (Scaff et al., al., model model of DoF positioning powered by a McK2014). This and characteristic makes them them suitable to power power The is tested insystem a mathematical nonlinear 2014). This characteristic makes suitable to model of aa 11method DoF positioning system powered by a McKromagnetic and electric conductive materials (Scaff etsuch al., rithm. ibben pneumatic artificial muscle. The parameters chosen model of DoF positioning system powered by a McK2014). This characteristic makes them suitable to power devices in intense electromagnetic field environments, ibben pneumatic artificial muscle. The parameters chosen 2014). This characteristic makes them suitable to power ibben pneumatic artificial muscle. The parameters chosen devices in intense electromagnetic field environments, such model of a 1 DoF positioning system powered by a McKdevices in intense electromagnetic field environments, such ibben pneumatic artificial muscle. The parameters chosen 2014). This characteristic makes them suitable to power by the algorithm is analyzed. This paper will address ibben pneumatic artificial muscle. The parameters chosen devices in intense electromagnetic field environments, such as in magnetic resonance imaging (MRI) guided surgery by thepneumatic algorithmartificial is analyzed. analyzed. This paper will address address devices in intense electromagnetic field environments, such by the algorithm is This paper will as in magnetic resonance imaging (MRI) guided surgery ibben muscle. The parameters chosen as in magnetic resonance imaging (MRI) guided surgery by the algorithm is analyzed. This paper will address devices in intense electromagnetic field environments, such the system model 2), the simulated by the algorithm is(Section analyzed. This paper willannealing address as magnetic resonance (MRI) guided surgery or functional MRI guided rehabilitation robotic devices, the system model 2), the simulated annealing as in in magnetic resonance imaging (MRI) guided devices, surgery the system model (Section 2), the simulated annealing or functional MRI guided imaging rehabilitation robotic devices, by the algorithm is(Section analyzed. This paper will address or functional MRI guided rehabilitation robotic the system model (Section 2), the simulated annealing as in magnetic resonance imaging (MRI) guided surgery algorithm (Section 3), the objective function (Section 4) the system model (Section 2), the simulated annealing or functional MRI guided rehabilitation robotic devices, unlike any other conventional actuator. It is still not widely algorithm (Section 3), the objective function (Section 4) or functional MRI guided rehabilitation robotic devices, (Section 3), the objective function (Section 4) unlike any other other conventional actuator. It It is isrobotic still not notdevices, widely algorithm the system model (Section 2), the simulated annealing unlike any conventional actuator. still widely algorithm (Section 3), the objective function (Section 4) or functional MRI guided rehabilitation and the results of the proposed method (Section 5). Section algorithm (Section 3), the objective function (Section 4) unlike any other conventional actuator. It is still not widely used in industry or outside academia because it is highly and the results of the proposed method (Section 5). Section unlike any other conventional actuator. It is still not widely and the results of the proposed method (Section 5). Section used in industry or outside academia because it is highly algorithm (Section 3), the objective function (Section 4) used in industry or outside academia because it is highly and the results of the proposed method (Section 5). Section unlike any other conventional actuator. It is still not widely 6 concludes the paper. and the results of the proposed method (Section 5). Section used in industry or outside academia because it is highly nonlinear and difficult to model and accurately control concludes theofpaper. paper. used in industry or outside academia because it iscontrol highly 66and concludes the nonlinear and difficult to model and accurately control the results the proposed method (Section 5). Section nonlinear and difficult to model and accurately 6 concludes the paper. used in industry or outside academia because it is control highly 6 concludes the paper. nonlinear and difficult to and (Thanh and Ahn, 2006; Repperger et al., 1999; Chan et al., nonlinear and difficult to model model and accurately (Thanh and Ahn, 2006; Repperger Repperger et al.,accurately 1999; Chan Chancontrol et al., al., 6 concludes the paper. (Thanh and Ahn, 2006; al., 1999; et nonlinear and difficult to model et and accurately control (Thanh and Ahn, 2006; Repperger et al., 1999; Chan et al., 2003; Anh, 2010; Medrano-Cerda al., 1995; Wu et (Thanh and Ahn, 2006; Repperger et al., 1999; Chan et al., 2003; Anh, 2010; Medrano-Cerda et al., al.,1999; 1995;Chan Wu et et al., 2003; Anh, Medrano-Cerda et al., 1995; Wu (Thanh and 2010; Ahn, 2006; Repperger et et al., 2. POSITIONING SYSTEM MODEL 2003; Anh, 2010; Medrano-Cerda al., 1995; Wu al., 2009; Situm and Herceg, 2008; Tang et al., 2016). 2003; Situm Anh, 2010; Medrano-Cerda et et al.,al., 1995; Wu et et al., 2. POSITIONING SYSTEM MODEL 2009; Situm and Herceg, 2008; Tang et al., 2016). 2. POSITIONING SYSTEM MODEL 2009; and Herceg, 2008; Tang 2016). 2003; Anh, 2010; Medrano-Cerda et al., 1995; Wu et al., 2. POSITIONING SYSTEM MODEL 2009; Situm and Herceg, 2008; Tang et al., 2016). 2. POSITIONING SYSTEM MODEL 2009; Situm and Herceg, 2008; Tang et al., 2016). 2. POSITIONING SYSTEM MODEL 2009; Situm and Herceg, 2008; Tang et al., 2016). The The positioning positioning system system is is aaa muscle-mass-spring muscle-mass-spring system system The positioning system is muscle-mass-spring system This This work work is is supported supported by by the the National National Council Council for for Scientific Scientific and and The positioning system is aaThe muscle-mass-spring system work is supported by the National Council for Scientific and based on aa real test bench. muscle is fixed vertically The positioning system is muscle-mass-spring system This based on real test bench. The muscle is fixed vertically Technological Development (CNPq). This work is supported by the National Council for Scientific and based on a real test bench. The muscle is fixed vertically This work is supported by the National Council for Scientific and Technological Development (CNPq). The positioning system is a muscle-mass-spring system Technological Development (CNPq). based on a real test bench. The muscle is fixed vertically 1 at the top on one end and the other end is attached to a based on a real test bench. The muscle is fixed vertically This work is supported by the National Council for Scientific and e-mail: [email protected]. Technological Development (CNPq). 1 at the top on one end and the other end is attached to a Technological Development (CNPq). 1 e-mail: at the top on one end and the other end is attached to a e-mail: [email protected]. based on a real test bench. The muscle is fixed vertically [email protected]. 2 1 e-mail: [email protected]. at the top on one end and the other end is attached to a mass, which is connected with a return spring fixed at the 1 at the top on one end and the other end is attached to a Technological Development (CNPq). e-mail: [email protected]. 2 mass, which is connected with a return spring fixed at the [email protected]. 2 e-mail: e-mail: [email protected]. mass, which is connected with a return spring fixed at the e-mail: e-mail: [email protected]. 1 3 at the top on one end and the other end is attached to a 2 mass, which is connected with a return spring fixed at the 2 e-mail: e-mail: [email protected]. bottom in aa dynamometer dynamometer (seeaFig. Fig. 1). spring fixed at the [email protected]. e-mail: [email protected]. 3 mass, which is connected with return [email protected]. 3 bottom in (see 1). e-mail: [email protected]. bottom in aa dynamometer (see Fig. 1). e-mail: [email protected]. 2 3 mass, which is connected with a return spring fixed at the 3 e-mail: e-mail: [email protected]. bottom in dynamometer (see Fig. 1). [email protected]. bottom in a dynamometer (see Fig. 1). e-mail: [email protected]. 3 e-mail: [email protected]. bottombyinElsevier a dynamometer Fig. 1). 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting Ltd. All rights(see reserved.
Copyright © 2018 IFAC 333 Copyright ©under 2018 responsibility IFAC 333Control. Peer review© of International Federation of Automatic Copyright 2018 IFAC 333 Copyright © 333 Copyright © 2018 2018 IFAC IFAC 333 10.1016/j.ifacol.2018.11.618 Copyright © 2018 IFAC 333
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Fig. 3. Simulation workflow. 0, if valve off Pm (t)−Pt m ˙ = Pt −Patm s+f mmax 1 − e , if valve on
(3)
where Pm , Pt and Patm are muscle, tank and atmospheric pressures, respectively. Parameters mmax , s and f are determined experimentally for each PWM signal.
Fig. 1. Positioning system setup. Two fast acting solenoid valves (FESTO MHJ10-S-2.5QS-4-LF) are responsible for pressurization and depressurization of the muscle. When the muscle is pressurized it contracts, pulling the mass up, and when the muscle is depressurized the spring pulls the mass down. The controller receives the dynamometer signal and the PID output is sent as a PWM signal to the valves.
Because the resulting valve’s mass flow rate does not vary proportionally with the duty cycle, a correction factor is used to avoid distorting the controller’s output signal. Because this dynamic model nonlinear, it is solved numerically in the time domain using the Euler’s method (also referred as Runge-Kutta 1 or RK1). The total system simulation workflow is illustrated in Fig. 3. The controller’s objective is to position the mass at the setpoint by opening and closing the valves. The problem of finding specific PID gains to achieve the desired performance will be solved using an optimization algorithm called Simulated Annealing. The algorithm will choose the proportional, integrative and derivative gains that minimizes a cost function, also called objective function, which measures the performance of the simulated system with the chosen PID gains according to the desired criterion. 3. SIMULATED ANNEALING ALGORITHM
Fig. 2. Muscle-mass-spring dynamic model representation. This system is simulated according to the dynamic model represented in Fig. 2. The muscle is considered as a spring with variable elastic constant km , that is a function of the pressure p and the muscle length x, in parallel with a dashpot with damping factor c determined experimentally. The muscle elastic constant km is derived from the Gaylord force model p(L3 − 3x2 ) Fm = − (1) 4πn2 where p is the differential pressure, L the braid string length, x the muscle length and n the number of turns the braid string runs around the membrane. The dynamic model of the system is given by −Fm − cz˙ − mg − kz = m¨ z (2) where k is the spring elastic constant, m is the mass of the object to be positioned, g is the gravitational acceleration The air is considered to be an ideal gas and an empirical model is adopted to describe the mass flow rate of the valves. As mentioned before, the flux of air is regulated by on/off valves according to PWM pulses. Therefore, the simulation considers the mass flow rate, given by the empirical model, while the valve is opened, and zero when closed. The mass flow rate empirical model is 334
The Simulated Annealing (SA) is a probabilistic metaheuristic with the capacity of “escaping” from a local minima. It is based on the thermodynamic process of metals called annealing. In the annealing process, the atoms can easily migrate to form new states of lower energy as it gradually cools to a lower energy state, but might form a higher energy state sometimes. Similarly, the SA allows a bad solution to be accepted with a certain probability, which decreases over the cooling process (Tsuzuki et al., 2006; Martins and Tsuzuki, 2008; Tavares et al., 2011; Martins et al., 2011; Martins and Tsuzuki, 2013). The SA has two main phases: at higher temperatures, the SA performs exploration of the domain space (global search); and, at lower temperatures, the SA performs the refinement stage (exploitation, local search). The SA has two main parameters: the cooling schedule α, and the next candidate determination. The used SA with crystallization heuristic is shown in Algorithm 1. It has two main loops: the internal loop which iterates for a specific temperature until the thermal equilibrium is reached; and, the external loop which iterates until the process is frozen. Rejected solutions do not contribute to the progress of the optimization process. Therefore, the crystallization factor ck is adapted in order to increase the number of accepted solutions. Fig. 4 shows that if a candidate is
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Generate new solution
No
Accept solution?
Rejected: reduce exploration
Yes
Accepted: reduce refinement
Fig. 4. Crystallization factor feedback. The decision in the present iteration will influence the generation of the next candidate. rejected then it is necessary to perform more exploitation in the modified dimension and the crystallization factor is increased. On the other hand, if a candidate is accepted, then it is necessary to perform more exploration in the modified dimension and the crystallization factor is decreased. It is an instantaneous reaction to every possible situation, independent of any other parameter such as the temperature. Algorithm 1 SA with Crystallization Heuristic 1: x ←< random inital solution > 2: T0 ←< inital temperature > 3: while < Global condition not satisfied> do 4: T i ← Ti ∗ α 5: i←i+1 6: while < Local condition not satisfied> do 7: k ←< select parameter to modify > c 8: x∗ ← x + c1k 1k random(− 12 , 12 ) · ∆ri · ek 9: ∆E = F (x∗ ) − F (x) 10: if ∆E < 0 then 11: x ← x∗ 12: ck ← 1 Positive Feedback 13: else 14: if random(0, 1) < e− ∆E/kB Ti then 15: x ← x∗ 16: ck ← 1 Positive Feedback 17: else 18: c k ← ck + 1 Negative Feedback 19: end if 20: end if 21: end while 22: end while
Fig. 5. System metrics used by the penalization one and two objective functions. Each line in the plot represent a different penalization situation that is considered in the output of the objective function. Error is an example of a good penalization method, but that may have local minima. The design of the objective function will depend on the desired performance of the system. These objective functions target the fastest raise time without overshoot. The metrics used in the objective functions are illustrated in Fig. 5. 4.1 Penalization one The simplest way of designing objective functions is to directly penalize undesired behavior, such as overshoot, high raise time or high steady state error. One simple example would be to divide in three penalization situations: 0 to 98% raise time, under 98% of the setpoint P 1under and overshoot P 1over . The proposed penalization for the raising time situation is the raise time itself. The fastest raise time will have the lower penalization, which is compatible with the minimization procedure of the SA for finding a better solution. If the mass does not hit 98% of the setpoint sp, then the penalization score can depend on the distance d to the setpoint sp and the simulation time ts , described by P 1under = ts + d = ts + sp − z(ts ).
4. OBJECTIVE FUNCTION DESIGN The objective function design is very important because it is responsible for translating the system’s desired behavior into a scoring function, that will be used in the SA to choose the gains that achieve the desired performance, which corresponds to the coordinate of the minimum value of the objective function. Many objective functions can be designed to tune the controller to the desired performance, but some can take longer to converge to a good solution. For the sake of clarity, three objective functions will be analyzed: Penalization one, Penalization two and Integral of the Absolute Error. Penalization one is an example of a plausible but bad objective function. Penalization two fixes a problem of the Penalization one method and the Integral of the Absolute 335
(4)
This is because the maximum raise time is the total simulation time ts , and the distance d to the setpoint measures how bad are the PID gains. For the overshoot situation, the penalization can be a constant high value. If the overshoot is considered worse than the under 98% of the setpoint situation, P 1over can be set as P 1over = ts + P 1under max = ts + sp
(5)
where sp is a constant value that accounts for a bad solution. The score of this objective function depends on which situation the system’s performance fits, and the output value is calculated using the criterion of the specific situation.
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This is a simple objective function that runs very fast, because it is not necessary to simulate until ts if an overshoot happens. 4.2 Penalization two This method is similar to Penalization one but without the simplification in the case an overshoot happens. It is slower but it is possible to determine the magnitude of the overshoot and penalize it proportionally.
Table 1. Tests results for the penalization 2 method. Test 1 2 3 4
P 778.1981 817.6187 808.1250 770.8543
I 33.0148 251.7287 113.9967 136.3697
D 16.2071 18.8097 17.6768 16.4484
Score 0.1501 0.1492 0.1512 0.1505
This method has four different situations: 0 to 98% raise time, under 98% of the setpoint sp, raise only P 2raise and overshoot P 2over . The first two situations remains the same of Penalization one. The other two situations have a proportional penalty, with the objective of faster convergence to the first two situations. The raise only situation is the worst of all, meaning that the PID gains are too high, in which the mass only raises without going back to the setpoint during ts . For this situation the penalization is proportional to the raise above the setpoint, given by P 2raise = ts + 600(z(ts ) − sp) (6) For the overshoot case, the penalization will depend on the magnitude of the first overshoot and the time of the last overshoot tlast , given by P 2over = ts + 350os + tlast (7)
Fig. 6. Test one of the penalization 2 method acceptance ratio.
Constants are used to weight each penalization criteria and are determined by the designer. The most undesired behaviors should produce higher scores. Similarly to penalization one, the score of this objective function depends on which situation the system’s performance fits, and the output value is calculated using the criterion of the specific situation. 4.3 Integral of the Absolute Error The score of this method is based on the integral of the absolute error e(t), which is the difference of the current position z(t) to the setpoint sp. Therefore, the score is given by ts Pint = |e(t)|dt (8) 0
This method provides a more continuous score and accounts for all deviations to the setpoint. 5. RESULTS
All three methods used the same initial settings. The simulation time is ts = 2s. Randomly chosen initial PID gains are P = 1015; I = 130; D = 3.17. Average simulation execution time is 1 second per test. Four processes are exR ecuted simultaneously on an Intel CoreTM i3-2120 CPU @ 3.30Ghz 64-bit GNU/Linux machine using Python3.5. 5.1 Penalization one The algorithm virtually runs forever with this method. All processes had to be manually terminated. There was no convergence and, thus, the final solutions are meaningless. 336
Fig. 7. Penalization 2 system responses with PID gains for each test. Legend shows the raise time of each test. 5.2 Penalization two All of the tests converged to a good result (low raise time score). Fig. 6 shows the acceptance ratio. The temperature where no solution is accepted is the frozen temperature (global condition was reached). Tests results are given in Table 1. Fig. 7 compares the system response with the PID gains of each solution. 5.3 Integral of the Absolute Error Fig. 8 illustrates the variation of the acceptance ratio as the temperature decreases. All tests results are in Table 2. System responses with the calculated PID gains are in Fig. 9.
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does not find a good solution soon, it will probably get stuck because the temperature is getting colder, meaning that the area of exploration is getting shorter.
Table 2. Tests results for the integral of the absolute error method. Test 1 2 3 4
P 737.3473 1690.7083 384.6905 2049.9084
I 371.0319 1.9676 264.2752 147.7134
D 16.3310 14.4064 2.1301 19.9065
Score 16.3775 18.2092 16.1116 18.0224
337
Duration 12.29h 7.44h 7.60h 5.51h
One way of detecting a bad objective function or bad SA parameters, e.g. initial temperature, is by measuring the acceptance ratio per temperature. Objective functions with plain regions tend to have acceptance ratios near 100% almost all of the time, which means that the search is almost completely random. Thus, penalization methods like this should be avoided. 6.2 Penalization two Differently from the penalization one, the acceptance ratio per temperature decreases gradually over time for all tests. This method is the most stable in terms of convergence. The solutions of the algorithm are consistent with the penalization criterion used, which corresponds to low raise time without overshoot. The solutions are close to each other in the hypersurface of the objective function and the simulation of the system responses with the obtained solutions are very similar. 6.3 Integral of the Absolute Error
Fig. 8. Test one of the integral of the absolute error method acceptance ratio.
The execution time of the algorithm varies among tests, but the mean duration is about 8h. Solutions also varies significantly between tests, meaning that they are more widespread in the objective function hypersurface. This indicates that there might be many local minima and multiple executions might be necessary for confirming a good solution. Similar to penalization two, acceptance drops over time and crystallization factors raise over time, indicating a smooth convergence over temperature. The solutions corresponding to the tests two and four produces overshoot and finish with higher scores than the other tests. The good solutions, however, are the best over all other objective functions, with the lowest raise times and without overshoot.
Fig. 9. Integral of the absolute error system responses with PID gains for each test. Legend shows the raise time of each test. 6. DISCUSSION 6.1 Penalization one This penalization method is very difficult for the algorithm because many of the possible parameters gives the same output value, which is a consequence of the overshoot criterion of a constant penalty. The algorithm gets stuck on a plain region or in a local minimum very easily and many executions are needed before an actual good solution appears. This is because the SA compares two points to check whether this is a candidate solution or it should discard it. If the values are equal, then it accepts the solution as a possible candidate and continues trying to find a better one. The problem is that, if the algorithm 337
Possible improvements of this method are penalizing the area above the setpoint more heavily than the bellow area, this should make the tests two and four harder to be selected as solutions, requiring less tests to determine the final solution. 6.4 General considerations Because the SA is a probabilist metaheuristic method, starting with the same initial PID gains leads to different solutions, as was shown in the previous section. This characteristic differs from other deterministic methods such as Gradient Descent or Newton-Raphson, where the initial condition is critical for producing different solutions and for the solution quality. Because of that, the initial PID gain in SA can be chosen by randomly selecting a point in the exploration range of the hypersurface, without affecting the algorithm performance. There are several advantages to this method for tuning the controller parameters over classical tuning. It is simpler because no model linearization is necessary and, additionally,
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it is not affected by linearization errors. Does not need trial and error for performance optimization. Does not require highly qualified personnel for tuning the controller. More flexible because the objective function can be designed to meet the application’s specific performance criterion. Can easily be used for multiobjective systems, with multiple objective functions combined with user defined weights, or by taking the union of the solutions. Producing better solutions can be as easy as just rerunning the algorithm. And, although it was used to tune the PID controller, this method can also be used to tune any controller, with any number of parameters. 7. CONCLUSIONS It has been demonstrated that using the simulated annealing algorithm for generating PID gains is a viable method, even for a nonlinear system. Algorithm execution time depends heavily in the simulation time and difficulty of the objective function and, thus, it may not be viable for systems that take too long to simulated and for some penalization or scoring methods. This method for PID tuning is very flexible because it is possible to design objective functions that meets the desired criterion, the quality of the PID parameters for the muscle-mass-spring system are compatible with the established requirements and the execution time of the algorithm was reasonable. Simulated annealing can also act in the real system, avoiding modeling and producing more relevant solutions. ACKNOWLEDGEMENTS This research was developed with the resources of High Performance Computing of the STI of USP. O. Horikawa and M. S. G. Tsuzuki were partially supported by CNPq (procs. 301547/2017-3 and 305959/2016–6). W. Scaff was supported by CNPq. REFERENCES Anh, H.P.H. (2010). Online tuning gain scheduling mimo neural PID control of the 2-axes pneumatic artificial muscle (PAM) robot arm. Expert Syst Appl, 37, 6547– 6560. ˚ Astr¨om, K.J. and H¨agglund, T. (2001). The future of PID control. Control Eng Pract, 9, 1163–1175. Chan, S., Lilly, J.H., Repperger, D.W., and Berlin, J.E. (2003). Fuzzy PD+ I learning control for a pneumatic muscle. In The 12th IEEE Int Conf Fuzzy Systems, volume 1, 278–283. St Louis, USA. Daerden, F. and Lefeber, D. (2002). Pneumatic artificial muscles: actuators for robotics and automation. Eur J Mech Env Eng, 47, 11–21. Kothera, C.S., Philen, M., and Tondu, B. (2012). Modelling of the McKibben artificial muscle: A review. J Intel Mat Syst Str, 23, 225–253. Martins, T.C., Camargo, E.D.L.B., Lima, R.G., Amato, M.B.P., and Tsuzuki, M.S.G. (2011). Electrical impedance tomography reconstruction through simulated annealing with incomplete evaluation of the objective function. In Proc 33rd A Int Conf IEEE EMBS, 7033–7036. Boston, USA. Martins, T.C. and Tsuzuki, M.S.G. (2008). Rotational placement of irregular polygons over containers with 338
fixed dimensions using simulated annealing and no-fit polygons. J Braz Soc Mech Sci, 30, 196–203. Martins, T.C. and Tsuzuki, M.S.G. (2013). Electrical impedance tomography reconstruction through simulated annealing with multi-stage partially evaluated objective functions. In Proc 35th IEEE EMBS, 6425–6428. Osaka, Japan. Medrano-Cerda, G.A., Bowler, C.J., and Caldwell, D.G. (1995). Adaptive position control of antagonistic pneumatic muscle actuators. In Proc IEEE/RSJ Int Conf Intelligent Robots and Systems, volume 1, 378–383. Pittsburgh, USA. Repperger, D., Johnson, K., and Philips, C. (1999). Nonlinear feedback controller design of a pneumatic muscle actuator system. In Proc American Control Conf, volume 3, 1525–1529. San Diego, USA. Sakthivelu, V. and Chong, S.H. (2015). Motion control of a 1-DOF pneumatic muscle actuator positioning system. In Proc 10th Asian Control Conf, 1983–1987. Kota Kinabalu, Malaysia. Scaff, W., Hirakawa, A.R., and Horikawa, O. (2014). Nonconducting manipulator for live-line maintenance: Use of pneumatic artificial muscles. In 3rd Int Conf Applied Robotics Power Industry, 1–6. Foz de Igua¸cu, Brazil. Schr¨oder, J., Kawamura, K., Gockel, T., and Dillmann, R. (2003). Improved control of a humanoid arm driven by pneumatic actuators. In Proc Int Conf Humanoid Robots, 29–30. Munich, Germany. Situm, Z. and Herceg, S. (2008). Design and control of a manipulator arm driven by pneumatic muscle actuators. In 16th Mediterranean Conf Control Automation, 926– 931. Ajaccio Corsica, France. Takosoglu, J.E., Laski, P.A., Blasiak, S., Bracha, G., and Pietrala, D. (2016). Determining the static characteristics of pneumatic muscles. Meas Control, 49, 62–71. Tang, T., Chong, S., Chan, C., and Sakthivelu, V. (2016). Point-to-point positioning control of a pneumatic muscle actuated system using improved-PID control. In IEEE Int Conf Automatic Control Intelligent Systems, 45–50. Shah Alam, Malaysia. Tavares, R.S., Martins, T.C., and Tsuzuki, M.S.G. (2011). Simulated annealing with adaptive neighborhood: A case study in off-line robot path planning. Expert Syst Appl, 38, 2951–2965. Thanh, T.D.C. and Ahn, K.K. (2006). Nonlinear PID control to improve the control performance of 2 axes pneumatic artificial muscle manipulator using neural network. Mechatronics, 16, 577–587. Tondu, B., Emirkhanian, R., Math´e, S., and Ricard, A. (2009). A pH-activated artificial muscle using the McKibben-type braided structure. Sensor Actuat APhys, 150, 124–130. Tondu, B. and Lopez, P. (1997). The McKibben muscle and its use in actuating robot-arms showing similarities with human arm behavior. Ind Robot, 24, 432–439. Tsuzuki, M.S.G., Martins, T.C., and Takase, F.K. (2006). Robot path planning using simulated annealing. In Proc IFAC INCOM, 173–178. Saint Etienne, France. Wu, J., Huang, J., Wang, Y., Xing, K., and Xu, Q. (2009). Fuzzy PID control of a wearable rehabilitation robotic hand driven by pneumatic muscles. In Int Symp Micro-NanoMechatronics and Human Science, 408–413. Nagoya, Japan.
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IFAC PapersOnLine 51-27 (2018) 333–338
Pneumatic Artificial Muscle Optimal Pneumatic Pneumatic Artificial Artificial Muscle Muscle Optimal Optimal Control with Simulated Annealing Pneumatic Artificial Muscle Optimal Control with Simulated Annealing Control with Simulated Annealing Control with Simulated Annealing ∗,2 ∗,1 Oswaldo Horikawa ∗,2 William Scaff ∗,1
∗,1 Oswaldo Horikawa ∗,2 William Scaff William Scaff Horikawa∗,3∗,2 ∗,1 ∗,1 Oswaldo ∗,2 William Scaff Oswaldo ∗,3 Marcos de Sales Guerra Tsuzuki William Scaff Horikawa Marcos de Sales Guerra Horikawa Tsuzuki ∗,3 ∗,1 Oswaldo ∗,2 Marcos de Sales Guerra Tsuzuki ∗,3 William Scaff Oswaldo Horikawa ∗,3 Marcos de Sales Guerra Tsuzuki Marcos de Sales Guerra Tsuzuki ∗,3 Marcos de Sales Guerra Tsuzuki ∗ ∗ Mechatronics and Mechanical Systems Engineering Department, ∗ Mechatronics and and Mechanical Mechanical Systems Systems Engineering Engineering Department, Department, ∗ Mechatronics ∗ Mechatronics and Mechanical Engineering Department, Escola Polit´ e cnica da UniversitySystems of S˜ S˜ ao o Paulo, Paulo, S˜ ao o Paulo, Paulo, Brazil andda Mechanical Systems Engineering Department, Escola Polit´ eecnica University of a S˜ a Brazil ∗ Mechatronics Escola Polit´ cnica da University of S˜ a o Paulo, S˜ a o Paulo, Brazil Mechatronics and Mechanical Systems Engineering Department, Escola Polit´ Polit´eecnica cnica da da University University of of S˜ S˜ ao o Paulo, Paulo, S˜ S˜ ao o Paulo, Paulo, Brazil Escola a a Brazil Escola Polit´ecnica da University of S˜ ao Paulo, S˜ ao Paulo, Brazil Abstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal musAbstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal musAbstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal musAbstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal muscles. They are attractive for many applications where safety, human interaction or biomimetic Abstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal muscles. They are attractive for many applications where safety, human interaction or biomimetic cles. They are attractive for many applications where safety, human interaction or biomimetic Abstract: Pneumatic artificial muscles are soft actuators which resemble human skeletal muscles. They are attractive for many applications where safety, human interaction or biomimetic behavior is required. However, its use is not yet widespread because of the difficulties of working cles. They are attractive for many applications where safety, human interaction or biomimetic behavior is required. However, its use is not yet widespread because of the difficulties of working behavior is required. However, its use is not yet widespread because of the difficulties of working cles. They are attractive for many applications where safety, human interaction or biomimetic behavior is required. However, its use is not yet widespread because of the difficulties of working with them, such as high nonlinearities, difficult to model and control accurately. This work behavior is required. However, its use is not yet widespread because of theaccurately. difficulties of working with them, such as high nonlinearities, difficult to model and control This work with them, such as high nonlinearities, difficult to model and control accurately. This work behavior is required. However, its use is not yet widespread because of the difficulties of working with them, such as high nonlinearities, difficult to model and control accurately. This work proposes a simple method for controlling the pneumatic muscle, which is using aa PID controller with them, such as high nonlinearities, difficult to model and control accurately. This work proposes a simple method for controlling the pneumatic muscle, which is using PID controller proposes a simple method for controlling the pneumatic muscle, which is using a PID controller with them, such as high nonlinearities, difficult to model and control accurately. This work proposes a simple method for controlling the pneumatic muscle, which is using a PID controller tuned with a simulated annealing optimization algorithm. proposes a simple method for controlling the pneumatic muscle, which is using a PID controller tuned with aa simulated annealing optimization algorithm. tuned with simulated annealing optimization algorithm. proposes a simple method for controlling the pneumatic muscle, which is using a PID controller tuned with a simulated annealing optimization algorithm. tuned with a simulated annealing optimization algorithm. © 2018,with IFACa (International Federationoptimization of Automatic algorithm. Control) Hosting by Elsevier Ltd. All rights reserved. tuned simulated annealing Keywords: Actuators, PID control, optimization, numerical simulation. Keywords: Actuators, PID control, optimization, Keywords: Actuators, Actuators, PID control, control, optimization, numerical numerical simulation. simulation. Keywords: Keywords: Actuators, PID PID control, optimization, optimization, numerical numerical simulation. simulation. Keywords: Actuators, PID control, optimization, numerical simulation. 1. INTRODUCTION There are many researches related to PAMs control. Clas1. INTRODUCTION INTRODUCTION There are many researches related to PAMs control. Clas1. There are many researches related to PAMs control. Clas1. There are many researches related to PAMs control. Classical linear control techniques such as PI and PID control 1. INTRODUCTION INTRODUCTION There are many researches related to PAMs control. Classical linear control techniques such as PI and PID control sical linear control techniques such as PI and PID control 1. INTRODUCTION There are many researches related to PAMs control. Classical linear control techniques such as PI and PID control have been proposed, because they are simple to design sical linear control techniques such as PI and PID control Pneumatic Artificial Muscles (PAMs) are interesting achave been proposed, proposed, because such theyasare are simple to control design been because they simple to design Pneumatic Artificial Artificial Muscles Muscles (PAMs) (PAMs) are are interesting interesting acac- have sical linear control techniques PI and PID Pneumatic have been proposed, because they are simple to design and have high adaptability (Sakthivelu and Chong, 2015). have beenhigh proposed, because they areand simple to design Pneumatic Muscles (PAMs) are actuators that have similar static and dynamic characterhave adaptability (Sakthivelu Chong, 2015). Pneumatic Artificial Muscles (PAMs) are interesting interesting ac- and and have adaptability (Sakthivelu Chong, 2015). tuators thatArtificial have similar similar static and dynamic dynamic characterhave beenhigh proposed, because they areand simple toby design tuators that have static and characterhave high adaptability (Sakthivelu and Chong, 2015). Pneumatic Artificial Muscles (PAMs) are therefore, interesting ac- and However, many of the PID controllers are tuned trial and have high adaptability (Sakthivelu and Chong, 2015). tuators that have similar static and dynamic characteristics to the human skeletal muscle and, have However, many of the PID controllers are tuned by trial tuators that have similar static and dynamic characterHowever, many of the PID controllers are tuned by trial istics to the human skeletal muscle and, therefore, have and have high adaptability (Sakthivelu and Chong, 2015). istics to the human skeletal muscle and, therefore, have However, many of the PID controllers are tuned by trial tuators that have similar static and dynamic characterand error or tuned based on the Ziegler-Nichols method, However, many of the PID controllers are tuned by trial istics to the human skeletal muscle and, therefore, have many applications in orthetics and biomimetic systems and error or tuned based on the Ziegler-Nichols method, istics to the human skeletal muscle and, therefore, have and error or tuned based on the Ziegler-Nichols method, many applications in orthetics and biomimetic systems However, many of the PID controllers are tuned by trial many applications in orthetics and biomimetic systems and error or tuned based on the Ziegler-Nichols method, istics to the human skeletal muscle and, therefore, have which is known to produce poor results in many cases and error or tuned based on the Ziegler-Nichols method, many applications in orthetics and biomimetic systems (Daerden and Lefeber, 2002; Tondu et al., 2009). They which is known known to based produce poor results in in many many cases many applications in orthetics and biomimetic systems which is to produce poor results cases (Daerden and Lefeber, 2002; Tondu et al., 2009). They and error or tuned on the Ziegler-Nichols method, (Daerden and Lefeber, 2002; Tondu et al., 2009). They ˚ which is known to produce poor results in many cases many applications intoorthetics and biomimetic systems (Sakthivelu and Chong, 2015; A str¨ oom and H¨ aagglund, which is known to produce poor results in many 2001; cases (Daerden and 2002; et 2009). They ˚ are lightweight, easy fabricate, low cost, and and Chong, 2015; A str¨ and H¨ (Daerden and Lefeber, Lefeber, 2002; Tondu Tondu et al., al.,compliant 2009). They ˚ (Sakthivelu and Chong, 2015; A str¨ oom m and H¨ aagglund, gglund, 2001; are lightweight, easy to to fabricate, fabricate, low cost, cost, compliant and (Sakthivelu ˚ which is et known to produce poor results in many 2001; cases are lightweight, easy low and (Sakthivelu and Chong, 2015; A str¨ m and H¨ gglund, 2001; ˚ (Daerden and Lefeber, 2002; Tondu et al.,compliant 2009). They Schr¨ ooder al., 2003). (Sakthivelu and Chong, 2015; A str¨ o m and H¨ a gglund, 2001; are lightweight, easy to fabricate, low cost, compliant and inherits the advantages of pneumatic systems. Schr¨ der et al., 2003). are lightweight, easy to fabricate, low cost, compliant and Schr¨ ooder et al., 2003). inherits the advantages advantages of pneumatic pneumatic systems. (Sakthivelu and Chong, 2015; ˚ Astr¨om and H¨agglund, 2001; inherits the of systems. Schr¨ der et al., 2003). are lightweight, easy to fabricate, low cost, compliant and Schr¨ o der et al., 2003). inherits the advantages of pneumatic systems. inherits the advantages of pneumatic systems. PID controllers are one one of of the the most most common common controllers controllers Schr¨ ocontrollers der et al., 2003). The invention of the PAM can be traced back to 1930, by are inherits the advantages of pneumatic systems. PID controllers are one of the most common controllers The invention of the the PAM PAM can be be traced traced back to to 1930, 1930, by by PID The invention of can back PID controllers are one of the most common controllers used today and may have similar performance to very comPID controllers are one of the most common controllers The invention of the PAM can be traced back to 1930, by S. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., used today and may have similar performance to very comThe invention of the PAM can be traced back to 1930, by used today and may have similar performance to very comS. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., PID controllers are one of the most common controllers S. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., used today and may have similar performance to very comThe invention of the PAM can be traced back to 1930, by plex control techniques with adequate gains. A method for used today and may have similar performance to very comS. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., 2016). Later, it was used by McKibben in the 1950’s to plex control techniques with adequate gains. A method for S. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., plex control techniques with adequate gains. A method for 2016). Later, it was used by McKibben in the 1950’s to used today and may have similar performance to very com2016). Later, it was used by McKibben in the 1950’s to plex control techniques with adequate gains. A method for S. Garasiev (Daerden and Lefeber, 2002; Takosoglu et al., choosing good PID gains can make the implementation of plex control techniques with adequate gains. A method for 2016). Later, it was used by McKibben in the 1950’s to power an orthetic upper limb (Tondu and Lopez, 1997; choosing good PID gains can make the implementation of 2016). Later, it was used by McKibben in the 1950’s to choosing good PID gains can make the implementation of power an orthetic upper limb (Tondu and Lopez, 1997; plex control techniques with adequate gains. A method for power an orthetic upper limb (Tondu and Lopez, 1997; choosing good PID gains can make the implementation of 2016). Later, it was used by McKibben in the 1950’s to PAMs simpler and promote it’s use in industry and other choosing good PID gains can make the implementation of power an orthetic upper limb (Tondu and Lopez, 1997; Daerden and Lefeber, 2002). However, it was not given PAMs simpler and promote it’s use in industry and other power an orthetic upper limb (Tondu and Lopez, 1997; PAMs simpler and promote it’s use in industry and other Daerden and Lefeber, 2002). However, it was not given choosing good PID gains can make the implementation of Daerden and Lefeber, 2002). However, it was not given PAMs simpler and promote it’s use in industry and other power an orthetic upper limb (Tondu and Lopez, 1997; areas. This method, however, should be flexible to address PAMs simpler and promote it’s use in industry and other Daerden and Lefeber, 2002). However, it was not given much attention because of the difficulty of working with areas. This method, however, should be flexible to address Daerden and Lefeber, 2002). However, it was not given This method, however, should be flexible to address much attention because2002). of the theHowever, difficulty itof ofwas working with areas. PAMs simpler and promote it’s use in industry and other much attention because of difficulty working with areas. This method, however, should be flexible to address Daerden and Lefeber, not given specific applications which may need different performance areas. This method, however, should be flexible to address much attention because of the difficulty of working with these muscles with the technology at that time (Daerden specific applications which mayshould need different different performance much muscles attentionwith because of the difficulty working with specific applications which may need performance these muscles with the technology technology at that thatof time (Daerden areas. This method, however, bebut flexible to address these the at (Daerden applications which may need performance much attention because of the difficulty oftime working with specific characteristics, such as slow response accurate posispecific applications which may need different different performance these muscles with the technology at that time (Daerden and Lefeber, 2002). With the progress on valve technology characteristics, such as slow response but accurate posithese muscles with the technology at that time (Daerden characteristics, such as slow response but accurate posiand Lefeber, 2002). With the progress on valve technology specific applications which may need different performance and Lefeber, 2002). With the progress on valve technology characteristics, such as slow response but accurate posithese muscles with the technology at that time (Daerden tioning or the fastest time response as possible. An easy characteristics, such as slow response but accurate posiand Lefeber, 2002). With the progress on valve technology control techniques, the interest on pneumatic muscles, tioning or the fastest time response as possible. An easy and Lefeber, 2002). With the progress on valve technology tioning or the fastest time response as possible. An easy and control techniques, the interest on pneumatic muscles, characteristics, such as slow response but accurate posiand control techniques, the interest on pneumatic muscles, tioning or the fastest time response as possible. An easy and Lefeber, 2002). With the progress on valve technology strategy for designing a controller to meet the application’s tioning or the fastest time response as possible. An easy control techniques, the interest on pneumatic muscles, more specifically the McKibben type PAM, is now back. strategy for designing a controller to meet the application’s and control techniques, the interest on pneumatic muscles, strategy for designing a controller to meet the application’s more specifically the McKibben type PAM, is now back. tioning or the fastest time response as possible. An easy more specifically the McKibben type PAM, is now back. strategy for designing a controller to meet the application’s and control techniques, the interest on pneumatic muscles, requirements is still missing. strategy for designing a controller to meet the application’s more specifically the McKibben type PAM, is now back. requirements is still still missing. missing. more McKibben specifically the McKibben type PAM, isPAM now back. requirements is strategy for designing a controller to meet the application’s The PAM is the most used today requirements is still missing. more McKibben specifically the McKibben now back. iscontrol still missing. The McKibben PAM is the the type most PAM, used isPAM PAM today requirements The PAM is most used today To address the problem, this paper proposes a PID The PAM the most PAM requirements iscontrol still missing. (Kothera et al., 2012). One interesting characteristic of To address the problem, this paper proposes a PID The McKibben McKibben PAM is is theinteresting most used used PAM today today To address the control problem, this paper proposes a PID (Kothera et al., 2012). One interesting characteristic of (Kothera et al., 2012). One characteristic of To address the control problem, this paper proposes a PID The McKibben PAM is the most used PAM today controller which is tuned by the Simulated Annealing algoTo address the control problem, this paper proposes a PID (Kothera et al., 2012). One interesting characteristic of the McKibben PAM is that it can be build without fercontroller which is tuned by the Simulated Annealing algo(Kothera et al., 2012). One interesting characteristic of controller which is tuned by the Simulated Annealing algothe McKibben PAM is that it can be build without ferTo address the control problem, this paper proposes a PID the McKibben PAM is that it can be build without fercontroller which is tuned by the Simulated Annealing algo(Kothera et al., 2012). One interesting characteristic of rithm. The method is tested in a mathematical nonlinear controller which is tuned by the Simulated Annealing algothe McKibben PAM is that it can be build without ferromagnetic and electric conductive materials (Scaff et al., rithm. Thewhich method is tested tested in Simulated mathematical nonlinear the McKibben is conductive that it can materials be build (Scaff without The method is in aaa mathematical nonlinear romagnetic andPAM electric conductive materials (Scaff et feral., rithm. controller is tuned by the Annealing algoromagnetic and electric et al., rithm. The method is tested in mathematical nonlinear the McKibben PAM is that it can be build without fermodel of a 1 DoF positioning system powered by a McKrithm. The is tested insystem a mathematical nonlinear romagnetic electric materials (Scaff et 2014). This characteristic makes them suitable to power of aa 11method DoF positioning powered by a McKromagnetic and electric conductive conductive materials (Scaff et al., al., model model of DoF positioning powered by a McK2014). This and characteristic makes them them suitable to power power The is tested insystem a mathematical nonlinear 2014). This characteristic makes suitable to model of aa 11method DoF positioning system powered by a McKromagnetic and electric conductive materials (Scaff etsuch al., rithm. ibben pneumatic artificial muscle. The parameters chosen model of DoF positioning system powered by a McK2014). This characteristic makes them suitable to power devices in intense electromagnetic field environments, ibben pneumatic artificial muscle. The parameters chosen 2014). This characteristic makes them suitable to power ibben pneumatic artificial muscle. The parameters chosen devices in intense electromagnetic field environments, such model of a 1 DoF positioning system powered by a McKdevices in intense electromagnetic field environments, such ibben pneumatic artificial muscle. The parameters chosen 2014). This characteristic makes them suitable to power by the algorithm is analyzed. This paper will address ibben pneumatic artificial muscle. The parameters chosen devices in intense electromagnetic field environments, such as in magnetic resonance imaging (MRI) guided surgery by thepneumatic algorithmartificial is analyzed. analyzed. This paper will address address devices in intense electromagnetic field environments, such by the algorithm is This paper will as in magnetic resonance imaging (MRI) guided surgery ibben muscle. The parameters chosen as in magnetic resonance imaging (MRI) guided surgery by the algorithm is analyzed. This paper will address devices in intense electromagnetic field environments, such the system model 2), the simulated by the algorithm is(Section analyzed. This paper willannealing address as magnetic resonance (MRI) guided surgery or functional MRI guided rehabilitation robotic devices, the system model 2), the simulated annealing as in in magnetic resonance imaging (MRI) guided devices, surgery the system model (Section 2), the simulated annealing or functional MRI guided imaging rehabilitation robotic devices, by the algorithm is(Section analyzed. This paper will address or functional MRI guided rehabilitation robotic the system model (Section 2), the simulated annealing as in magnetic resonance imaging (MRI) guided surgery algorithm (Section 3), the objective function (Section 4) the system model (Section 2), the simulated annealing or functional MRI guided rehabilitation robotic devices, unlike any other conventional actuator. It is still not widely algorithm (Section 3), the objective function (Section 4) or functional MRI guided rehabilitation robotic devices, (Section 3), the objective function (Section 4) unlike any other other conventional actuator. It It is isrobotic still not notdevices, widely algorithm the system model (Section 2), the simulated annealing unlike any conventional actuator. still widely algorithm (Section 3), the objective function (Section 4) or functional MRI guided rehabilitation and the results of the proposed method (Section 5). Section algorithm (Section 3), the objective function (Section 4) unlike any other conventional actuator. It is still not widely used in industry or outside academia because it is highly and the results of the proposed method (Section 5). Section unlike any other conventional actuator. It is still not widely and the results of the proposed method (Section 5). Section used in industry or outside academia because it is highly algorithm (Section 3), the objective function (Section 4) used in industry or outside academia because it is highly and the results of the proposed method (Section 5). Section unlike any other conventional actuator. It is still not widely 6 concludes the paper. and the results of the proposed method (Section 5). Section used in industry or outside academia because it is highly nonlinear and difficult to model and accurately control concludes theofpaper. paper. used in industry or outside academia because it iscontrol highly 66and concludes the nonlinear and difficult to model and accurately control the results the proposed method (Section 5). Section nonlinear and difficult to model and accurately 6 concludes the paper. used in industry or outside academia because it is control highly 6 concludes the paper. nonlinear and difficult to and (Thanh and Ahn, 2006; Repperger et al., 1999; Chan et al., nonlinear and difficult to model model and accurately (Thanh and Ahn, 2006; Repperger Repperger et al.,accurately 1999; Chan Chancontrol et al., al., 6 concludes the paper. (Thanh and Ahn, 2006; al., 1999; et nonlinear and difficult to model et and accurately control (Thanh and Ahn, 2006; Repperger et al., 1999; Chan et al., 2003; Anh, 2010; Medrano-Cerda al., 1995; Wu et (Thanh and Ahn, 2006; Repperger et al., 1999; Chan et al., 2003; Anh, 2010; Medrano-Cerda et al., al.,1999; 1995;Chan Wu et et al., 2003; Anh, Medrano-Cerda et al., 1995; Wu (Thanh and 2010; Ahn, 2006; Repperger et et al., 2. POSITIONING SYSTEM MODEL 2003; Anh, 2010; Medrano-Cerda al., 1995; Wu al., 2009; Situm and Herceg, 2008; Tang et al., 2016). 2003; Situm Anh, 2010; Medrano-Cerda et et al.,al., 1995; Wu et et al., 2. POSITIONING SYSTEM MODEL 2009; Situm and Herceg, 2008; Tang et al., 2016). 2. POSITIONING SYSTEM MODEL 2009; and Herceg, 2008; Tang 2016). 2003; Anh, 2010; Medrano-Cerda et al., 1995; Wu et al., 2. POSITIONING SYSTEM MODEL 2009; Situm and Herceg, 2008; Tang et al., 2016). 2. POSITIONING SYSTEM MODEL 2009; Situm and Herceg, 2008; Tang et al., 2016). 2. POSITIONING SYSTEM MODEL 2009; Situm and Herceg, 2008; Tang et al., 2016). The The positioning positioning system system is is aaa muscle-mass-spring muscle-mass-spring system system The positioning system is muscle-mass-spring system This This work work is is supported supported by by the the National National Council Council for for Scientific Scientific and and The positioning system is aaThe muscle-mass-spring system work is supported by the National Council for Scientific and based on aa real test bench. muscle is fixed vertically The positioning system is muscle-mass-spring system This based on real test bench. The muscle is fixed vertically Technological Development (CNPq). This work is supported by the National Council for Scientific and based on a real test bench. The muscle is fixed vertically This work is supported by the National Council for Scientific and Technological Development (CNPq). The positioning system is a muscle-mass-spring system Technological Development (CNPq). based on a real test bench. The muscle is fixed vertically 1 at the top on one end and the other end is attached to a based on a real test bench. The muscle is fixed vertically This work is supported by the National Council for Scientific and e-mail: [email protected]. Technological Development (CNPq). 1 at the top on one end and the other end is attached to a Technological Development (CNPq). 1 e-mail: at the top on one end and the other end is attached to a e-mail: [email protected]. based on a real test bench. The muscle is fixed vertically [email protected]. 2 1 e-mail: [email protected]. at the top on one end and the other end is attached to a mass, which is connected with a return spring fixed at the 1 at the top on one end and the other end is attached to a Technological Development (CNPq). e-mail: [email protected]. 2 mass, which is connected with a return spring fixed at the [email protected]. 2 e-mail: e-mail: [email protected]. mass, which is connected with a return spring fixed at the e-mail: e-mail: [email protected]. 1 3 at the top on one end and the other end is attached to a 2 mass, which is connected with a return spring fixed at the 2 e-mail: e-mail: [email protected]. bottom in aa dynamometer dynamometer (seeaFig. Fig. 1). spring fixed at the [email protected]. e-mail: [email protected]. 3 mass, which is connected with return [email protected]. 3 bottom in (see 1). e-mail: [email protected]. bottom in aa dynamometer (see Fig. 1). e-mail: [email protected]. 2 3 mass, which is connected with a return spring fixed at the 3 e-mail: e-mail: [email protected]. bottom in dynamometer (see Fig. 1). [email protected]. bottom in a dynamometer (see Fig. 1). e-mail: [email protected]. 3 e-mail: [email protected]. bottombyinElsevier a dynamometer Fig. 1). 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting Ltd. All rights(see reserved.
Copyright © 2018 IFAC 333 Copyright ©under 2018 responsibility IFAC 333Control. Peer review© of International Federation of Automatic Copyright 2018 IFAC 333 Copyright © 333 Copyright © 2018 2018 IFAC IFAC 333 10.1016/j.ifacol.2018.11.618 Copyright © 2018 IFAC 333
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Fig. 3. Simulation workflow. 0, if valve off Pm (t)−Pt m ˙ = Pt −Patm s+f mmax 1 − e , if valve on
(3)
where Pm , Pt and Patm are muscle, tank and atmospheric pressures, respectively. Parameters mmax , s and f are determined experimentally for each PWM signal.
Fig. 1. Positioning system setup. Two fast acting solenoid valves (FESTO MHJ10-S-2.5QS-4-LF) are responsible for pressurization and depressurization of the muscle. When the muscle is pressurized it contracts, pulling the mass up, and when the muscle is depressurized the spring pulls the mass down. The controller receives the dynamometer signal and the PID output is sent as a PWM signal to the valves.
Because the resulting valve’s mass flow rate does not vary proportionally with the duty cycle, a correction factor is used to avoid distorting the controller’s output signal. Because this dynamic model nonlinear, it is solved numerically in the time domain using the Euler’s method (also referred as Runge-Kutta 1 or RK1). The total system simulation workflow is illustrated in Fig. 3. The controller’s objective is to position the mass at the setpoint by opening and closing the valves. The problem of finding specific PID gains to achieve the desired performance will be solved using an optimization algorithm called Simulated Annealing. The algorithm will choose the proportional, integrative and derivative gains that minimizes a cost function, also called objective function, which measures the performance of the simulated system with the chosen PID gains according to the desired criterion. 3. SIMULATED ANNEALING ALGORITHM
Fig. 2. Muscle-mass-spring dynamic model representation. This system is simulated according to the dynamic model represented in Fig. 2. The muscle is considered as a spring with variable elastic constant km , that is a function of the pressure p and the muscle length x, in parallel with a dashpot with damping factor c determined experimentally. The muscle elastic constant km is derived from the Gaylord force model p(L3 − 3x2 ) Fm = − (1) 4πn2 where p is the differential pressure, L the braid string length, x the muscle length and n the number of turns the braid string runs around the membrane. The dynamic model of the system is given by −Fm − cz˙ − mg − kz = m¨ z (2) where k is the spring elastic constant, m is the mass of the object to be positioned, g is the gravitational acceleration The air is considered to be an ideal gas and an empirical model is adopted to describe the mass flow rate of the valves. As mentioned before, the flux of air is regulated by on/off valves according to PWM pulses. Therefore, the simulation considers the mass flow rate, given by the empirical model, while the valve is opened, and zero when closed. The mass flow rate empirical model is 334
The Simulated Annealing (SA) is a probabilistic metaheuristic with the capacity of “escaping” from a local minima. It is based on the thermodynamic process of metals called annealing. In the annealing process, the atoms can easily migrate to form new states of lower energy as it gradually cools to a lower energy state, but might form a higher energy state sometimes. Similarly, the SA allows a bad solution to be accepted with a certain probability, which decreases over the cooling process (Tsuzuki et al., 2006; Martins and Tsuzuki, 2008; Tavares et al., 2011; Martins et al., 2011; Martins and Tsuzuki, 2013). The SA has two main phases: at higher temperatures, the SA performs exploration of the domain space (global search); and, at lower temperatures, the SA performs the refinement stage (exploitation, local search). The SA has two main parameters: the cooling schedule α, and the next candidate determination. The used SA with crystallization heuristic is shown in Algorithm 1. It has two main loops: the internal loop which iterates for a specific temperature until the thermal equilibrium is reached; and, the external loop which iterates until the process is frozen. Rejected solutions do not contribute to the progress of the optimization process. Therefore, the crystallization factor ck is adapted in order to increase the number of accepted solutions. Fig. 4 shows that if a candidate is
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Generate new solution
No
Accept solution?
Rejected: reduce exploration
Yes
Accepted: reduce refinement
Fig. 4. Crystallization factor feedback. The decision in the present iteration will influence the generation of the next candidate. rejected then it is necessary to perform more exploitation in the modified dimension and the crystallization factor is increased. On the other hand, if a candidate is accepted, then it is necessary to perform more exploration in the modified dimension and the crystallization factor is decreased. It is an instantaneous reaction to every possible situation, independent of any other parameter such as the temperature. Algorithm 1 SA with Crystallization Heuristic 1: x ←< random inital solution > 2: T0 ←< inital temperature > 3: while < Global condition not satisfied> do 4: T i ← Ti ∗ α 5: i←i+1 6: while < Local condition not satisfied> do 7: k ←< select parameter to modify > c 8: x∗ ← x + c1k 1k random(− 12 , 12 ) · ∆ri · ek 9: ∆E = F (x∗ ) − F (x) 10: if ∆E < 0 then 11: x ← x∗ 12: ck ← 1 Positive Feedback 13: else 14: if random(0, 1) < e− ∆E/kB Ti then 15: x ← x∗ 16: ck ← 1 Positive Feedback 17: else 18: c k ← ck + 1 Negative Feedback 19: end if 20: end if 21: end while 22: end while
Fig. 5. System metrics used by the penalization one and two objective functions. Each line in the plot represent a different penalization situation that is considered in the output of the objective function. Error is an example of a good penalization method, but that may have local minima. The design of the objective function will depend on the desired performance of the system. These objective functions target the fastest raise time without overshoot. The metrics used in the objective functions are illustrated in Fig. 5. 4.1 Penalization one The simplest way of designing objective functions is to directly penalize undesired behavior, such as overshoot, high raise time or high steady state error. One simple example would be to divide in three penalization situations: 0 to 98% raise time, under 98% of the setpoint P 1under and overshoot P 1over . The proposed penalization for the raising time situation is the raise time itself. The fastest raise time will have the lower penalization, which is compatible with the minimization procedure of the SA for finding a better solution. If the mass does not hit 98% of the setpoint sp, then the penalization score can depend on the distance d to the setpoint sp and the simulation time ts , described by P 1under = ts + d = ts + sp − z(ts ).
4. OBJECTIVE FUNCTION DESIGN The objective function design is very important because it is responsible for translating the system’s desired behavior into a scoring function, that will be used in the SA to choose the gains that achieve the desired performance, which corresponds to the coordinate of the minimum value of the objective function. Many objective functions can be designed to tune the controller to the desired performance, but some can take longer to converge to a good solution. For the sake of clarity, three objective functions will be analyzed: Penalization one, Penalization two and Integral of the Absolute Error. Penalization one is an example of a plausible but bad objective function. Penalization two fixes a problem of the Penalization one method and the Integral of the Absolute 335
(4)
This is because the maximum raise time is the total simulation time ts , and the distance d to the setpoint measures how bad are the PID gains. For the overshoot situation, the penalization can be a constant high value. If the overshoot is considered worse than the under 98% of the setpoint situation, P 1over can be set as P 1over = ts + P 1under max = ts + sp
(5)
where sp is a constant value that accounts for a bad solution. The score of this objective function depends on which situation the system’s performance fits, and the output value is calculated using the criterion of the specific situation.
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This is a simple objective function that runs very fast, because it is not necessary to simulate until ts if an overshoot happens. 4.2 Penalization two This method is similar to Penalization one but without the simplification in the case an overshoot happens. It is slower but it is possible to determine the magnitude of the overshoot and penalize it proportionally.
Table 1. Tests results for the penalization 2 method. Test 1 2 3 4
P 778.1981 817.6187 808.1250 770.8543
I 33.0148 251.7287 113.9967 136.3697
D 16.2071 18.8097 17.6768 16.4484
Score 0.1501 0.1492 0.1512 0.1505
This method has four different situations: 0 to 98% raise time, under 98% of the setpoint sp, raise only P 2raise and overshoot P 2over . The first two situations remains the same of Penalization one. The other two situations have a proportional penalty, with the objective of faster convergence to the first two situations. The raise only situation is the worst of all, meaning that the PID gains are too high, in which the mass only raises without going back to the setpoint during ts . For this situation the penalization is proportional to the raise above the setpoint, given by P 2raise = ts + 600(z(ts ) − sp) (6) For the overshoot case, the penalization will depend on the magnitude of the first overshoot and the time of the last overshoot tlast , given by P 2over = ts + 350os + tlast (7)
Fig. 6. Test one of the penalization 2 method acceptance ratio.
Constants are used to weight each penalization criteria and are determined by the designer. The most undesired behaviors should produce higher scores. Similarly to penalization one, the score of this objective function depends on which situation the system’s performance fits, and the output value is calculated using the criterion of the specific situation. 4.3 Integral of the Absolute Error The score of this method is based on the integral of the absolute error e(t), which is the difference of the current position z(t) to the setpoint sp. Therefore, the score is given by ts Pint = |e(t)|dt (8) 0
This method provides a more continuous score and accounts for all deviations to the setpoint. 5. RESULTS
All three methods used the same initial settings. The simulation time is ts = 2s. Randomly chosen initial PID gains are P = 1015; I = 130; D = 3.17. Average simulation execution time is 1 second per test. Four processes are exR ecuted simultaneously on an Intel CoreTM i3-2120 CPU @ 3.30Ghz 64-bit GNU/Linux machine using Python3.5. 5.1 Penalization one The algorithm virtually runs forever with this method. All processes had to be manually terminated. There was no convergence and, thus, the final solutions are meaningless. 336
Fig. 7. Penalization 2 system responses with PID gains for each test. Legend shows the raise time of each test. 5.2 Penalization two All of the tests converged to a good result (low raise time score). Fig. 6 shows the acceptance ratio. The temperature where no solution is accepted is the frozen temperature (global condition was reached). Tests results are given in Table 1. Fig. 7 compares the system response with the PID gains of each solution. 5.3 Integral of the Absolute Error Fig. 8 illustrates the variation of the acceptance ratio as the temperature decreases. All tests results are in Table 2. System responses with the calculated PID gains are in Fig. 9.
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does not find a good solution soon, it will probably get stuck because the temperature is getting colder, meaning that the area of exploration is getting shorter.
Table 2. Tests results for the integral of the absolute error method. Test 1 2 3 4
P 737.3473 1690.7083 384.6905 2049.9084
I 371.0319 1.9676 264.2752 147.7134
D 16.3310 14.4064 2.1301 19.9065
Score 16.3775 18.2092 16.1116 18.0224
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Duration 12.29h 7.44h 7.60h 5.51h
One way of detecting a bad objective function or bad SA parameters, e.g. initial temperature, is by measuring the acceptance ratio per temperature. Objective functions with plain regions tend to have acceptance ratios near 100% almost all of the time, which means that the search is almost completely random. Thus, penalization methods like this should be avoided. 6.2 Penalization two Differently from the penalization one, the acceptance ratio per temperature decreases gradually over time for all tests. This method is the most stable in terms of convergence. The solutions of the algorithm are consistent with the penalization criterion used, which corresponds to low raise time without overshoot. The solutions are close to each other in the hypersurface of the objective function and the simulation of the system responses with the obtained solutions are very similar. 6.3 Integral of the Absolute Error
Fig. 8. Test one of the integral of the absolute error method acceptance ratio.
The execution time of the algorithm varies among tests, but the mean duration is about 8h. Solutions also varies significantly between tests, meaning that they are more widespread in the objective function hypersurface. This indicates that there might be many local minima and multiple executions might be necessary for confirming a good solution. Similar to penalization two, acceptance drops over time and crystallization factors raise over time, indicating a smooth convergence over temperature. The solutions corresponding to the tests two and four produces overshoot and finish with higher scores than the other tests. The good solutions, however, are the best over all other objective functions, with the lowest raise times and without overshoot.
Fig. 9. Integral of the absolute error system responses with PID gains for each test. Legend shows the raise time of each test. 6. DISCUSSION 6.1 Penalization one This penalization method is very difficult for the algorithm because many of the possible parameters gives the same output value, which is a consequence of the overshoot criterion of a constant penalty. The algorithm gets stuck on a plain region or in a local minimum very easily and many executions are needed before an actual good solution appears. This is because the SA compares two points to check whether this is a candidate solution or it should discard it. If the values are equal, then it accepts the solution as a possible candidate and continues trying to find a better one. The problem is that, if the algorithm 337
Possible improvements of this method are penalizing the area above the setpoint more heavily than the bellow area, this should make the tests two and four harder to be selected as solutions, requiring less tests to determine the final solution. 6.4 General considerations Because the SA is a probabilist metaheuristic method, starting with the same initial PID gains leads to different solutions, as was shown in the previous section. This characteristic differs from other deterministic methods such as Gradient Descent or Newton-Raphson, where the initial condition is critical for producing different solutions and for the solution quality. Because of that, the initial PID gain in SA can be chosen by randomly selecting a point in the exploration range of the hypersurface, without affecting the algorithm performance. There are several advantages to this method for tuning the controller parameters over classical tuning. It is simpler because no model linearization is necessary and, additionally,
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it is not affected by linearization errors. Does not need trial and error for performance optimization. Does not require highly qualified personnel for tuning the controller. More flexible because the objective function can be designed to meet the application’s specific performance criterion. Can easily be used for multiobjective systems, with multiple objective functions combined with user defined weights, or by taking the union of the solutions. Producing better solutions can be as easy as just rerunning the algorithm. And, although it was used to tune the PID controller, this method can also be used to tune any controller, with any number of parameters. 7. CONCLUSIONS It has been demonstrated that using the simulated annealing algorithm for generating PID gains is a viable method, even for a nonlinear system. Algorithm execution time depends heavily in the simulation time and difficulty of the objective function and, thus, it may not be viable for systems that take too long to simulated and for some penalization or scoring methods. This method for PID tuning is very flexible because it is possible to design objective functions that meets the desired criterion, the quality of the PID parameters for the muscle-mass-spring system are compatible with the established requirements and the execution time of the algorithm was reasonable. Simulated annealing can also act in the real system, avoiding modeling and producing more relevant solutions. ACKNOWLEDGEMENTS This research was developed with the resources of High Performance Computing of the STI of USP. O. Horikawa and M. S. G. Tsuzuki were partially supported by CNPq (procs. 301547/2017-3 and 305959/2016–6). W. Scaff was supported by CNPq. REFERENCES Anh, H.P.H. (2010). Online tuning gain scheduling mimo neural PID control of the 2-axes pneumatic artificial muscle (PAM) robot arm. Expert Syst Appl, 37, 6547– 6560. ˚ Astr¨om, K.J. and H¨agglund, T. (2001). The future of PID control. Control Eng Pract, 9, 1163–1175. Chan, S., Lilly, J.H., Repperger, D.W., and Berlin, J.E. (2003). Fuzzy PD+ I learning control for a pneumatic muscle. In The 12th IEEE Int Conf Fuzzy Systems, volume 1, 278–283. St Louis, USA. Daerden, F. and Lefeber, D. (2002). Pneumatic artificial muscles: actuators for robotics and automation. Eur J Mech Env Eng, 47, 11–21. Kothera, C.S., Philen, M., and Tondu, B. (2012). Modelling of the McKibben artificial muscle: A review. J Intel Mat Syst Str, 23, 225–253. Martins, T.C., Camargo, E.D.L.B., Lima, R.G., Amato, M.B.P., and Tsuzuki, M.S.G. (2011). Electrical impedance tomography reconstruction through simulated annealing with incomplete evaluation of the objective function. In Proc 33rd A Int Conf IEEE EMBS, 7033–7036. Boston, USA. Martins, T.C. and Tsuzuki, M.S.G. (2008). Rotational placement of irregular polygons over containers with 338
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