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对气动人工肌肉驱动使用rnapm-SCARA机器人控制仿真外文文献翻译、中英文翻译、外文翻译
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附录一：对气动人工肌肉驱动使用rnapmSCARA机器人控制仿真 埃斯科瓦尔 美国韦恩 古铁雷斯 莱德妮娃 费尔南德斯 罗德里格斯 雷木思 群系统工程和机器人公司技术学院托卢卡Metepec、墨西哥 墨西哥的Open Access供资伙伴行为和SaludUnder为依据创用cc授权使用摘要：本课题介绍了运动控制的模拟一个自由度的机械臂，SCARA由一对McKibben气动人工肌肉驱动构成。气动人工肌肉致动器和模拟生物肌肉的行为,由于其非线性行为,还有一个需要制定机械手臂控制系统使用这种类型的驱动器。研究始于代表的传递函数,在数学语言,机械臂的关节运动；这允许使用PID控制器的传递函数和生成数据训练多层感知器神经网络(RNAPM)。到目前为止,PID控制系统能够控制机器人手臂的运动,但根据实验测试,事实证明,RNAPM优于PID控制的响应时间2.95秒,最大限度地降低1.3角误差,避免由于其持续的振荡问题,持续的行为。关键字: 控制系统；人工神经网络；多层感知器；气动人工肌肉1.引言: 进化的机器人手臂气动人工肌肉驱动，它的起源在1950年代，物理学家约瑟夫lMcKibben1的第一个原型设计和开发谁气动控制的人造肌肉的矫正法的扩张和收缩像一个真正的人类的手臂,这个人造设备是模拟生物肌肉2。 气动人工肌肉是几种类型的致动器,用于机器人手臂;其他则由液压、电子、气动执行机构(气缸)。然而,气动人工肌肉有好处就更加经典的致动，具有伟大的初始力,高加速能力,稳定运动;它是轻量级和坚固的,它可能是定位在不同的角度,而不丢失属性,这使得它适合于危险环境3,它可以直接驱动或与开环控制,滞后的理论数学行为接近实验物理测试期间观察到的行为1。 严格控制的气动人工muscle-actuated机器人手臂,由几个连接关节,关节之间的高度非线性和强耦合动力学4。机器人手臂由Tondu et al。5的机器人实验室国家应用科学研究所(INSA)在图卢兹,法国是McKibben气动人工肌肉表现最像人类的武器。Tondu指出的缺点之一是人工肌肉致动器的非线性是由于滞后,原因是编织壳内的纤维之间的摩擦的影响3，非线性气动人工肌肉致动器的结果的数学模型没有考虑到部分内胎不是一个气缸在肌肉的目的,他们在编织shell-inherent扭曲导致非线性收缩重复。气动人工muscle-actuated机器人手臂已经开发出来,像Situm et al。6,建立了臂段使用一对费斯托气动对手的肌肉,或者崔et al。7,机器人手臂由两对气动人工肌肉驱动的。这两款设计的目的是减少风险对人类有害的工业环境中,两个科学小组同意,将气动人工肌肉致动器的主要问题是很难开发控制系统由于其非线性。这显示控制系统的最大障碍之一:气动人工肌肉的非线性行为。PID控制技术提出了解决非线性问题5和可以接受的结果,因此,学习型的替代品也正在考虑,如人工神经网络,已将非线性效应的主要优势在训练阶段8,允许建设的网络拓扑,像这种行为来开发一个智能控制系统,提高了响应时间和PID控制稳定。 描述了一个自由度的控制仿真,RNAPM-actuated SCARA机器人(选择性合规组装机械臂)使用PID control-obtained数据。这PID控制器用于控制seven-degree-of-freedom机械臂建立国立应用科学学院在图卢兹,法国5。使用RNAPM,响应时间提高了2.95秒和1.3度角错误进行了优化。本文结构如下:第二部分描述的物理结构气动人工肌肉和传递函数用于控制仿真;第三节提供了一个可视化的仿真方法由这些流程:1)数据收集,2)RNAPM控制,3)训练,4)非线性近似。第四节提供了仿真测试和结果和第五节的重点是结论和未来的项目。2.气动人工肌肉 如今,McKibben气动人工肌肉已经成为最常用的执行机构在研究人员构建机器人手臂由于低成本,建设舒适和优于经典的致动器。研究人员喜欢Tondu1,骑手3,Situm6,崔7和斯莱姆9,Lezama11的10和迪亚兹在那些使用这种类型的传动装置,由一个内部充满压缩空气胶管,纤维编织成网覆盖着。末端有一个严格的,封闭的连接器之一,那里的肌肉施加一种外在的力量,而另一端一个连接器,允许压缩空气的入口;这样,内心的橡皮管合同或延伸,很像生物肌肉3。迪亚兹11和Tondu2认为,人造肌肉的行为紧密模拟人类的肌肉。图1描述了气动人工肌肉致动器的结构。气动人工肌肉致动器的结构。气动人工肌肉用来模拟RNAPM Lezama提出的控制是一个10,描述物理特性如气动人工肌肉的物理特性。气动人工肌肉初长23厘米，初始直径1.7厘米，网12角2.1.传递函数 在控制理论中,传输函数是用来表示组件或系统的输入-输出关系通过微分方程描述12;也就是说,传递函数的数学表达真实的生活行为,如操纵机械臂的运动。系统控制手段或施加力量申请工作根据特定的指令,也就是说,力量决定运动,速度和加速度13,因此,机械臂的仿真需要知道控制系统传递函数和表演。Tondu等2提出的传递函数联合由一对对手气动人工肌肉代表肘部动作的行为,如图2所示,与角指令响应范围在0到55和数学表示如下:方程(1)查看MathML sourceGs =U = KWn2s2 + 2 zwns + Wn2eTsSCARA-type机器人手臂的肘关节。U代表压力,在哪里位置(度),K是静态增益,Wn代表无阻尼角频率,Z是阻尼系数和变量T代表时间。根据实验测试,Lezama10也提出了特定的值变量组成,Z和T前面提到的在方程(1),然后代表传递函数如下:方程,查看MathML sourceGs =U = 502 s2 + 11.51 + 24.18 e0.0103.解决方法3.1.数据收集 如前所述,控制模拟一定需要知道所使用的传递函数的系统控制。本文使用方程(2)中所述的传递函数,编程与Matlab Simulink仿真工具和PID控制器,导致一个闭环控制系统,它提供了输入和输出数据训练RNAPM。Lezama10还提议使用PID控制器值控制系统图,P = 0.06,= 0.12和D = 0.0075;所需的位置变量提供0和55之间的角位置,将定义肘部的的活动范围。最后,收集数据通过测量输入变量的向量存储所有调整值的角误差和输出变量的向量,它负责存储提出的角位置值PID控制器达到指定位置的编译时间6秒。基于收集的数据,进行测试在555角位置每隔5,只允许可视化PID控制器行为进化。图4描述了PID行为进化在5角位置,用来收集数据来训练RNAPM。PID行为进化在5角位置。3.2.RNAPM神经控制 根据学习范式中,人工神经网络可以分为监督和非监督学习范例。依赖的神经网络的区别是为了识别和分类模式14。RNAPM用于本研究提出了一种多层感知器网络拓扑通过监督学习,其结构由实验测试,由一个节点在输入层,两个节点在隐藏层和输出层的一个节点。在多层神经网络,重要的是要确定激活函数最适合这个问题。有不同类型的功能:一步、乙状结肠、高斯和双曲正切。RNAPM的核心是基于双曲正切的激活函数。双曲正切是最常见的一种激活函数用于MLP拓扑由于其输出时间间隔(1 - 1),允许一个对称函数,允许非线性轨迹,提高速度在RNAPM训练阶段。3.3.培训 本文由梯度下降训练使用反向传播算法,从而优化神经元之间的连接根据误差的神经网络误差被网络输出和期望输出值之间的差异,最大限度地减少函数计算神经网络误差,称为成本函数。RNAPM培训中使用的成本函数是基于最小二乘的方法,它允许获得均方误差(MSE),数学中方程(3):方程(3)查看MathML sourceMES = 1 nnei i = 1其中N是总数的训练模式和“e”代表了错误。训练RNAPM 489模式。是批量使用的学习方法,它定义了数量的时代代表学习的迭代次数;然后,根据训练RNAPM已知的模式,获得的均方误差,允许调整网络权重,直到到达最后一个时代之前定义。显示了RNAPM培训和比较学习时代的价值在1000时代和均方误差值。表2显示了生成的训练价值,如最小和最终值的均方误差。最小MSE 0.004497979，最后MSE 0.0044979793.4.非线性近似 RNAPM用于提出网络拓扑构成一个理想的近似元素非线性行为,如操纵气动人工muscle-actuated机器人手臂,由于其泛化能力,也就是说,其学习能力基于输入和输出数据,定义为培训的证据,以及它们之间的非线性关系。RNAPM近似技术包括训练网络的输入和输出值获得PID控制系统描述,基于相同的向量的输入值用于训练,RNAPM有能力提出输出值的向量表示的非线性近似的行为学习训练阶段。显示了行为的非线性近似RNAPM提出的基于使用PID控制系统行为的训练。近似图RNAPM PID的行为。PID的演变行为定义的角误差和角位置的价值相当于的进化的行为提出RNAPM拓扑,证明这个网络能逼近非线性行为像操纵McKibben气动人工muscle-actuated机器人手臂。4.测试和结果 RNAPM获得的结果的非线性逼近阶段提供确凿的证据的效率是一个等价的行为,结果显示在图6图描述;然而,在这个阶段是不可能确定RNAPM提出值优于PID控制的行为。确定如果RNAPM控制器比PID在模拟环境中,控制系统中的network-proposed价值观取代,如图7所示。测试旨在比较对RNAPM PID控制系统的效率进行了555范围内,每隔5。下面的图8显示了结果的PID和RNAPM控制在5角位置,网络更有效率,导致稳定以3.8秒的成绩,没有振荡与PID控制相比,达到稳定在5.8秒和0.001的振荡的最高水平。在15角位置,另一个结果是,如图9所示,与RNAPM再次被证明是更有效的稳定时间为3.25秒,没有振荡,PID控制相比,达到稳定在4.73秒和0.027的振荡的最高水平。图10显示了结果在30角位置,RNAPM控制系统被证明是更有效的,稳定时间为3.1秒,没有振荡,PID控制相比,达到稳定在4.5秒和0.17的振荡的最高水平。行为图的PID和RNAPM控制在30角位置。图11显示了结果在45角位置,在减少RNAPM稳定时间的观察与PID控制相比,这反过来需要更多的时间来稳定的角位置定义。在该测试中,网络达到稳定以3.02秒的成绩,没有振荡,而PID控制在5秒内达到稳定振荡0.3的最高水平。最后,图12显示测试结果55角位置,在RNAPM稳定时间进一步减少相比,PID控制。在该测试中,网络达到稳定以2.95秒的成绩,没有振荡,而PID控制达到稳定在5.5秒内振荡1.3的最高水平。5.结论 基于测试的结果进行一个模拟环境在一个角位置5之间55,RNAPM被证明比PID控制系统。稳定2秒的时间和减少0.001角位置最小化在5的位置,观察这些数字与每个测试的改善进行定义的范围,直到到达最后的55角位置,导致一个稳定时间2.95秒和2.95错误最小化。此外,其连续的,持续的行为,RNAPM控制系统没有振荡。因此建议提出的神经网络拓扑结构能够学习非线性行为的气动人工muscle-actuated机器人手臂。未来的研究工作将包括RNAPM控制系统的物理测试自由度SCARA机器人手臂系统,建立与一个对手对气动人工肌肉,为了验证如RNAPM显示相同的效率在实际环境中。应答。 我们要感谢PROMEP(教员)的改善项目的支持和赞助开展这项研究,这是项目的一部分”的研究和表征气动人工Mini-Muscle原型在静态和动态条件下”,代码ittol - ptc - 002,从呼吁建议“支持承认新的全职教授,呼吁建议2009”。我们也愿意承认dg支持项目的批准“原型一双McKibben对手一个虚拟系统的气动人工肌肉mini-actuators”,代码托尔- dciet - 2010 - 102,一直继续这个项目研究的基础。文献：1B. Tondu, V. Boitier, P. Lpez.Naturally Compliant Robot-Arms Actuated By Mckibben Artificial Muscles, IEEE International Conference (1994), pp. 26352637 October2B. Tondu, P. Lpez.Modeling and Control of Mckkiben Artificial Muscle Robot Actuators, IEEE Control Systems Magazine (2000), pp. 2229 April View Record in Scopus | Citing articles (1)3Caballero Adolfo Hilario, Corts. Pablo Carbonell Prototipo Experimental para la Identificacin y Control de Actuadores por Msculo Neumtico, Elsevier -Revista Iberoamericana de Automtica e Informtica Industrial, Spain (2003), pp. 13 7-84Vivas. Oscar Predictive Control of a Scara Robot, Ingeniare - Revista Chilena de Ingeniera, Chile (2006), p. 1355B. Tondu, S. Ippolito, J. Guiochet, A. Daidie A Seven Degrees of Freedom Robot Arm Driven by Pneumatic Artificial Muscles for Humanoid Robots The International Journal of Robotics Research (2005), pp. 268272 April View Record in Scopus | Citing articles (1)6Situm Zeliko, Herceg. Srecko Design and Control of a Manipulator Arm Driven by Pneumatic Muscle Actuators, Mediterranean Conference on Control and Automation, Centre, Ajaccio, France (2008), pp. 927929 June7Choi Tae-Yong, Seok Joon-Hong, Lee. Ju-Jang Safe Robot with Artificial Pneumatic Muscle (2009), pp. 14341437 IEEE International Symposium on Industrial Electronics, July8Trujillano Javier, March Jaume, Sorribas. Albert Aproximacin Metodolgica al uso de Redes Neuronales Artificiales para la Prediccin de Resultados en MedicinaElsevier - Departament de Cincies Mdiques Bsiques, Universitat de Lleida, Spain (2004), pp. 59639Eskiizmirliler Selim, B. Tondu, C. Darlot Motor Control of a Limb Segment Actuated by Artificial Muscles, Proceedings 23rd Annual Conference IEEE, Turkey (2001), pp. 1210Morales. Ruth Lezama Modlisation et programmation dun robot anthropomorphe 7 degrs de libert actionn par muscles artificiels pneumatiques, Institut National des Sciences appliques, Touluse, France (2008), pp. 1012 PhD thesis, 43,63,68-7111Zagal. Sergio Daz Conception et Dveloppement d un Mini-Actionneur Muscle Artificial. Application la Robotique Mdicale, Institut National des Sciences appliques, Touluse, France (2007), pp. 4457 PhD thesis12Ogata. Katsuhiko Ingeniera de Control Moderna (2008), p. 60 Prentice Hall 4ta Ed.13Saha. Subir Kumar Introduccin a la Robtica (2010), p. 239 Mc Graw Hill 1era Ed.14Ahn Kyoung Kwan, Chau Nguyen. Huynh Thai Intelligent switching control of a pneumatic muscle robot arm using learning vector quantization neural network (2007), pp. 257259 Elsevier - Mechatronics附录二：Simulation of Control of a SCARA Robot Actuated by Pneumatic Artificial Muscles Using RNAPMF. Escobar , S. Daz, C. Gutirrez, Y. Ledeneva, C. Hernndez, D. Rodrguez, R. LemusGroup of Systems Engineering and Robotics Instituto Tecnolgico de Toluca Metepec, Estado de Mxico, MxicoOpen Access funded by Asociacin Mexicana del Comportamiento y SaludUnder a Creative Commons licenseAbstract： This paper describes the simulation of movement control of a one-degree-of-freedom articulated robot arm SCARA actuated by a pair of McKibben pneumatic artificial muscles. The pneumatic artificial muscle is the actuator and emulates the behavior of biological muscles; due to its nonlinear behavior, there is also a need to develop control systems for robot arms using this type of actuator. Research begins with the transfer function that represents, in mathematical language, the movement of the robot arms joints; this allows using a PID controller on the transfer function and generating data to train the Multilayer Perceptron Artificial Neural Network (RNAPM). So far, the PID control system has been able to control the movement of robot arms but, based on experimental tests, the RNAPM has proved to outperform the PID controls response time by up to 2.95 seconds, minimizing the angular error by 1.3 and avoiding the oscillation problem due to its continuous, constant behavior.Key words Control Systems; Artificial Neural Networks; Multilayer Perceptron; Pneumatic Artificial Muscle1. IntroductionThe evolution of robot arms actuated by pneumatic artificial muscles has its origin in the 1950s with physicist Joseph L. McKibben 1, who designed and developed the first prototype of an artificial muscle for the pneumatic control of an orthosis that would expand and contract like an actual human arm; this artificial device is a close emulation of biological muscles 2.The pneumatic artificial muscle is amongst several types of actuators that have been used in robotic arms; others are the hydraulic, electronic and pneumatic (pneumatic cylinder) actuators. However, the pneumatic artificial muscle has advantages with respect to more classic actuators: great initial force, high acceleration capacity, steady movement; it is both lightweight and sturdy, it may be positioned in different angles without losing properties, which makes it appropriate for hazardous environments 3, and it can be actuated directly or controlled with open loop, as hysteresis in the theoretical-mathematical behavior is close to the behavior observed during experimental physical tests 1.Strictly, the control of a pneumatic artificial muscle-actuated robot arm, composed by several connected joints, has highly nonlinear dynamics with strong couplings between joints 4. The robot arm developed by Tondu et al. 5 at the Robotics Laboratory of the National Institute of Applied Sciences (INSA) in Toulouse, France, is one of the McKibben pneumatic artificial muscles that behaves the most like human arms.Tondu states that one of the drawbacks of this actuator is the nonlinearity of the artificial muscle due to hysteresis, the reason being the effect of friction between fibers inside the woven shell 2.Caballero et al. 3 consider that the nonlinearity of the pneumatic artificial muscle results from the actuators mathematical models not considering that the section of the inner tube is not a cylinder at the muscles ends; they agree with Tondu in that the woven shell-inherent distortions result in a nonlinear contraction repetitiveness.Pneumatic artificial muscle-actuated robot arms have been developed, like the one by Situm et al. 6, who built the arm segment using a Festo Pneumatics antagonist pair of muscles, or that of Choi et al. 7, a robot arm driven by two pairs of pneumatic artificial muscles. The purpose of both designs is to diminish the risks to humans in hazardous industrial environments; both scientific teams agree that the main problem of incorporating pneumatic artificial muscle actuators is the difficulty to develop control systems due to their nonlinearity. This points to one of the biggest obstacles of control systems: the nonlinear behavior of pneumatic artificial muscles. PID control techniques have been proposed to solve the nonlinearity issue 5 with acceptable results and, therefore, learning-oriented alternatives are also being considered, like Artificial Neural Networks, which have the main advantage of incorporating nonlinear effects during the training stages 8, allowing for the construction of network topologies that resemble this behavior to develop an intelligent control system that improves response times and PID control stabilization. This paper describes the control simulation of a one-degree-of-freedom, RNAPM-actuated SCARA (Selective Compliance Assembly Robot Arm) robot using PID control-obtained data. This PID controller has been used to control a seven-degree-of-freedom robot arm built at the National institute of Applied Sciences in Toulouse, France 5. Using the RNAPM, response times were improved by 2.95 seconds and angular errors were optimized by 1.3 degrees. This paper is structured as follows: Section 2 describes the physical structure of the pneumatic artificial muscle and the transfer function used in control simulation; Section 3 offers a visualization of the simulation methodology constituted by these processes: 1) data collection, 2) RNAPM control, 3) training, 4) nonlinear approximation. Section 4 offers simulation tests and results and Section 5 focuses on conclusions and future projects.2. Pneumatic artificial muscle Nowadays, the McKibben pneumatic artificial muscle has become one of the most used actuators amongst researchers to build robot arms due to lower costs, construction easiness and advantages over classic actuators. Researchers like Tondu 1, Caballero 3, Situm 6, Choi 7 and Selim 9, Lezama 10 and Daz 11 are amongst those that have used this type of actuator, which consists of an inner rubber tube filled with pressurized air, covered with fibers woven to form a mesh. One of the ends has a rigid, closed connector, where the muscle exerts an outward force, and the other end has a connector that allows the entrance of pressurized air; this way, the inner rubber tube contracts or extends, very much like biological muscles 3. Daz 11 and Tondu 2 agree that the artificial muscles behavior closely emulates that of human muscles. Figure 1 depicts the structure of a pneumatic artificial muscle actuator.Full-size image (14 K)Figure 1. Structure of a pneumatic artificial muscle actuator.Figure options The pneumatic artificial muscle used to simulate the RNAPM control is the one proposed by Lezama 10, who describes the physical features shown on Table 1.Table 1. Physical features of the pneumatic artificial muscle.Pneumatic Artificial Muscle Initial length 23 cm Initial diameter 1.7 cm Mesh angle 12 Table options2.1. Transfer function In control theory, transfer functions are used to represent the input-output relation of components or systems that are described through differential equations 12; that is, transfer functions are mathematical representations of real life behaviors, like the manipulation of a robot arms movements. System control means applying or exerting forces for it to work according to specific instructions, that is, the force determines terms like movement, velocity and acceleration 13; therefore, the simulation of a robotic arm requires knowing the transfer function and a control system acting on it. Tondu et al 2 propose the transfer function of a joint composed by an antagonist pair of pneumatic artificial muscles to represent the behavior of elbow movements, as depicted in Figure 2, with angle instructions responding at a range between 0 to up to 55 and the following mathematical representation: equation(1),View the MathML sourceGs=U=KWn2s2+2ZWns+Wn2eTsFull-size image (26 K)Figure 2. Elbow joint of a SCARA-type robot arm.Figure options Where U represents pressure (bars), is the position (degrees), K is the static gain, Wn represents the undamped angular frequency, Z is the damping coefficient and variable T represents time. Based on experimental tests, Lezama 10 also proposes specific values for variables Wn, Z and T mentioned previously in equation (1); the transfer function is then represented as follows:equation(2)View the MathML sourceGs=U=502s2+11.51s+24.18e0.010s3. Solution methodology3.1. Data collection As mentioned earlier, control simulation necessarily requires knowing the transfer function used by the system control. This paper uses the transfer function described in equation (2), programmed with the Matlab Simulink simulation tool and the PID controller, resulting in a closed loop control system that provides input and output data to train the RNAPM. Figure 3 shows the PID control system diagram.Full-size image (15 K)Figure 3. PID control system diagram.Figure options Lezama 10 also proposes the PID controller values used in the control system diagram of Figure 3, like P=0.06, I=0.12 and D=0.0075; the Required Position variable provides the angular position between 0 and 55 that will define the elbows range of movement. Finally, data is collected by measuring the vector of the Input variable storing all correction values of the angular error and the vector of the Output variable, which is in charge of storing the angular position values proposed by the PID controller to reach the user-specified position at a compilation time of 6 seconds. Based on the data collected, tests were carried out at a 5 to 55 angular position at five-degree intervals, which allow visualizing only the PID controller behavior evolution. Figure 4 depicts the PID behavior evolution at a 5 angular position, which was used to collect data to train the RNAPM.Full-size image (14 K)Figure 4. PID behavior evolution at a 5 angular position.Figure options3.2. RNAPM neural control Depending on the learning paradigm, artificial neural networks can be divided into supervised and unsupervised learning paradigms. The difference relies on the way in which neural networks are designed to recognized and classify patterns 14.The RNAPM used in this research paper proposes a Multilayer Perceptron network topology by means of supervised learning, its structure derived from experimental testing, formed by one node at the input layer, two nodes at the hidden layer and one node at the output layer. Within multilayer neural networks, it is important to identify the activation function most adequate for this problem. There are different types of functions: Step, Sigmoid, Gaussian and Hyperbolic Tangent. The core of the RNAPM is based on the Hyperbolic Tangent as activation function. Hyperbolic Tangent is one of the most common activation functions used in MLP topology due to its output interval (-1 to 1), allowing a symmetrical function that permits nonlinear trajectories, improving velocity during the RNAPM training stage.3.3. Training This paper uses the Backpropagation by gradient descent training algorithm, which optimizes connections between neurons according to the error of the neural network - error being the difference between the network output and the desired output - and minimizes the function calculating the neural network error, known as cost function.The cost function used in RNAPM training is based on the method of least squares, which allows obtaining the mean-squared error (MSE), mathematically represented in equation (3):equation(3)View the MathML sourceMES=1Ni=1NeiWhere N is the total number of training patterns and “e” represents the error. To train the RNAPM, 489 patterns were used. The learning method used was Batch, which defines the number of epochs representing the number of learning iterations; then, based on the training patterns known by the RNAPM, the mean-squared error is obtained, allowing to adjust the network weights until reaching the last epoch previously defined. The graph in Figure 5 shows the RNAPM training and a comparison between the value of learning epochs in the range of up to 1000 epochs and the mean-squared error value. Table 2 shows the resulting training values, like the minimum and final values of the mean-squared error.Full-size image (14 K)Figure 5. Training graph of the RNAPM.Figure optionsTable 2. RNAPM training data.Best NetworkTrainingNumber of Epoch 1000 Minimum MSE 0.004497979 Final MSE 0.004497979 Table options3.4. Nonlinear approximation The RNAPM used in the proposed network topology constitutes an ideal approximation element to nonlinear behaviors, like the manipulation of a pneumatic artificial muscle-actuated robot arm, due to its generalization capacities, that is, its learning ability based on input and output data, defined as training evidence, and the nonlinear relation between them. The RNAPM approximation technique consists of training the network with input and output values obtained from the PID control system described in Figure 3. Based on the same vector of input values used for training, the RNAPM has the capacity to propose the vector of output values representing the nonlinear approximation to the behavior learned at the training stage. Figure 6 shows the behavior of the nonlinear approximation proposed by the RNAPM based on the training that used the PID control system behavior.Full-size image (18 K)Figure 6. Approximation graph of RNAPM to the PID behavior.Figure options The evolution of the PID behavior defined by the angular error and the value of the angular position is equivalent to the evolution of the behavior of the proposed RNAPM topology, proving that this network is capable of approximating a nonlinear behavior like manipulating a McKibben pneumatic artificial muscle-actuated robot arm.4. Tests and results The results obtained by the RNAPM up to the nonlinear approximation stage provide solid evidence of efficiency for being an equivalent behavior, according to the results depicted in the Figure 6 graph; however, at this stage it is not possible to determine if the RNAPM proposed values outperform the PID control behavior. To determine if the RNAPM is a better controller than the PID in a simulated environment, the network-proposed values were replaced within the control system, as shown in Figure 7.Full-size image (14 K)Figure 7. Diagram of the RNAPM control system.Figure options The tests aimed to compare the efficiency of the PID control system against the RNAPM were carried out in the 5 to 55 range, at five-degree intervals. Figure 8 below shows the result obtained from the PID and RNAPM controls at a 5 angular position, where the network is more efficient, resulting in a stabilization time of 3.8 seconds and no oscillations when compared with the PID control, which reaches stabilization at 5.8 seconds and oscillations of up to 0.001 at its highest level.Full-size image (22 K)Figure 8. Behavior graph of PID and RNAPM control at 5 angular position.Figure options At a 15 angular position, another result was obtained, as seen in Figure 9, with the RNAPM again proving to be more efficient with a stabilization time of 3.25 seconds and no oscillations, compared to the PID control, which reaches stabilization at 4.73 seconds and oscillations of 0.027 at its highest level.Full-size image (24 K)Figure 9. Behavior graph of PID and RNAPM control at 15 angular position.Figure options Figure 10 shows the result at a 30 angular position, the RNAPM control system proving to be more efficient, with a stabilization time of 3.1 seconds and no oscillations, compared to the PID control, which reaches stabilization at 4.5 seconds and oscillations of 0.17 at its highest level.Full-size image (24 K)Figure 10. Behavior graph of PID and RNAPM control at 30 angular position.Figure options Figure 11 depicts the results at a 45 angular position, where a decrease of the RNAPM stabilization time is observed when compared to the PID control, which conversely requires more time to stabilize at the defined angular position.Behavior graph of PID and RNAPM control at 45 angular position.Figure options In this test, the network achieves a stabilization time of 3.02 seconds and no oscillations, while the PID control reaches stabilization at 5 seconds with oscillations of 0.3 at its highest level. Finally, Figure 12 shows the results of the test at 55 angular position, where the RNAPM stabilization time further decreases when compared to the PID control. In this test, the network achieves a stabilization time of 2.95 seconds and no oscillations, while the PID control reaches stabilization at 5.5 seconds with oscillations of 1.3 at its highest level.Full-size image (26 K)Figure 12. Behavior graph of PID and RNAPM control at 55 angular position.Figure options5. Conclusions Based on the results obtained from the tests carried out in a simulated environment at an angular position range between 5 to 55, the RNAPM proved to outperform the PID control system. A decrease of 2 seconds in the stabilization time and a 0.001 angular position minimization was observed at a 5 position, these numbers improving with each of the tests carried out at the defined range until reaching the final angular position of 55, which resulted in a stabilization time of 2.95 seconds and a 1.3 error minimization. Furthermore, with its continuous, constant behavior, the RNAPM control system shows no oscillations. It is therefore suggest that the neural network topology proposed is capable of learning a nonlinear behavior like that of pneumatic artificial muscle-actuated robot arms. Future research work will include physical tests of the RNAPM control system on a one-degree-of-freedom SCARA robotic arm, built with an antagonist pair of pneumatic artificial muscles, in order to validate if the RNAPM shows the same efficiency in a real environment.Acknowledgments We would like to thank PROMEP Program for the Improvement of the Faculty for its support and sponsorship to carry out this research, which is part of the project “Study and Characterization of a Pneumatic Artificial Mini-Muscle Prototype under Static and Dynamic Conditions”, code ITTOL-PTC-002, from the call for proposals “Support to Admit New Full-Time Professors, Call for Prop
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