Control of an automotive shape memory alloy mirror actuator 控制汽车形状记忆合金后视镜致动器
Eric A. Williams , Gordon Shaw , Mohammad Elahinia Eric A. Williams , Gordon Shaw , Mohammad Elahinia Robotics and Mechanisms Laboratory (RoMeLa), Department of Mechanical Engineering, Virginia Polytechnic Institute, Blacksburg, VA 24060, United States 弗吉尼亚理工学院机械工程系机器人与机械实验室（RoMeLa），弗吉尼亚州布莱克斯堡，24060，美国 Dynamic and Smart Systems Laboratory, Mechanical, Industrial and Manufacturing Engineering Department, University of Toledo, Toledo, OH 43606, United States 美国俄亥俄州托莱多 43606 托莱多大学机械、工业和制造工程系动态和智能系统实验室
A R T I C L E I N F O
Article history: 文章历史：
Received 15 February 2009 2009 年 2 月 15 日收到
Accepted 9 April 2010 2010 年 4 月 9 日接受
Keywords: 关键词：
Shape memory alloys 形状记忆合金
Automotive actuator 汽车制动器
Control 控制
Mirror actuator 后视镜致动器
Abstract 摘要
A B S T R A C T Automobile designers are continuously trying to improve the safety and comfort of new car models. In many cases this translates to the need for less expensive and compact actuators. There are many features that are commonly found in cars that utilize electromechanical type actuators. Power mirrors are one such application that has been becoming more and more standard on cars today. A cost effective mirror actuator was designed and built that utilizes shape memory alloy (SMA) wires to position the external side mirrors. This paper present a second generation SMA-actuated mirror. A robust control algorithm is developed for the mirror to provide stable and accurate positioning. This variable structure controller provides accurate positioning without adding unnecessary complexity to the computational requirements for the mirror. Experimental results are compiled to show the tracking response of the SMA mirror actuator A B S T R A C T 汽车设计师不断努力提高新车型的安全性和舒适性。在许多情况下，这意味着需要更便宜、更紧凑的执行器。汽车中常见的许多功能都使用机电式致动器。电动后视镜就是这样一种应用，如今已成为越来越多汽车的标准配置。我们设计并制造了一种具有成本效益的后视镜致动器，它利用形状记忆合金（SMA）线来定位外侧后视镜。本文介绍了第二代 SMA 驱动后视镜。为该反射镜开发了一种稳健的控制算法，以提供稳定而精确的定位。这种可变结构控制器在提供精确定位的同时，不会对反射镜的计算要求增加不必要的复杂性。实验结果汇编显示了 SMA 驱动镜的跟踪响应。
Smart materials offer better performance and improved safety for automobiles. Shock absorbers based on magnetorheological (MR) fluids offer variable damping and a better ride and handling [1]. MR fluids have also been used for developing engine and transmission mounts [2]. These mounts offer the possibility of having wider notch frequencies and can also be used as effective vibration isolators for sensitive equipment . Torsional MR dampers have been investigated to reduce the torsional vibration of the engines [5]. Another group of smart materials, piezoelectric foils and fibers, can be incorporated into the interior materials of cars for reducing noise [6]. 智能材料为汽车提供了更好的性能和更高的安全性。基于磁流变（MR）流体的减震器可提供可变阻尼和更好的驾驶和操控性[1]。磁流变液还被用于开发发动机和变速器悬置[2]。这些悬置件可提供更宽的缺口频率，也可用作敏感设备的有效隔振器 。扭转 MR 阻尼器已被研究用于减少发动机的扭转振动 [5]。另一类智能材料，即压电箔和纤维，可用于汽车内饰材料，以降低噪音[6]。
Shape memory alloys (SMAs) are a group of smart materials that are suitable for developing linear and rotary actuators. These alloys undergo a phase transformation that allows them to recover from a deformed state to a memorized shape. Through the phase transformation, SMA actuators can deliver large forces and large displacements [7]. SMA based actuators have been developed for aerospace, biomedical, and robot applications [8-14]. 形状记忆合金（SMA）是一类适用于开发线性和旋转致动器的智能材料。这些合金会发生相变，使其能够从变形状态恢复到记忆形状。通过相变，SMA 执行器可以产生较大的力和位移 [7]。基于 SMA 的致动器已被开发用于航空航天、生物医学和机器人应用领域 [8-14]。
SMA actuators are a class of lightweight actuators that offer advantages over traditional technology and allow for simple and compact solutions to the increasing demand for electrical actuation. Many researchers and companies have investigated and SMA 执行器是一类轻型执行器，与传统技术相比具有优势，可提供简单、紧凑的解决方案，满足日益增长的电动执行需求。许多研究人员和公司已经研究并
released products that employ SMA actuators because of their strong actuation force and compact nature. Toy companies employ SMAs to generate small motions on robotic toys without the use of DC motors or solenoids, which are typically heavier and more expensive. The aerospace applications for SMAs also require lightweight actuators with an excellent power to weight ratio to drive rudder and aileron controls [15]. 由于 SMA 执行器具有强大的执行力和紧凑的特性，许多玩具公司都推出了采用 SMA 执行器的产品。玩具公司使用 SMA 在机器人玩具上产生微小的运动，而无需使用直流电机或螺线管，因为直流电机或螺线管通常更重、更贵。SMA 在航空航天领域的应用也需要重量轻、功率重量比出色的致动器来驱动方向舵和副翼控制装置 [15]。
While SMAs offer superior characteristics for developing compact and reliable actuators, the automotive application of SMAs has been limited mostly to passive use of these materials for engine temperature control and active disassembly of automotive parts [16]. In the passive applications, either the superior thermomechanical properties of the materials are used or the SMA element is actuated only once. As an example of one of the very few automotive actuation application of SMAs, recently Coli et al. developed a SMA tumble flap actuator to regulate engine cylinder air intake [17]. The tumble flap actuator utilizes SMA springs to reduce pollution and to improve engine performance. When the flaps are in the lowered position, the manifold is opened and free intake occurs. When the flaps are in the raised position, the manifold is partially closed and tumble flow takes place, which improves the combustion efficiency and results in reduced pollution and increased performance. Laboratory experimental evaluation has shown that the SMA spring actuator is capable of rotating a shaft with an array of flaps to two limit positions (raised or lowered) inside the intake manifold of the engine [17]. 虽然 SMA 具有开发紧凑型可靠致动器的优越特性，但 SMA 在汽车上的应用主要限于被动使用这些材料控制发动机温度和主动拆卸汽车零件 [16]。在被动应用中，要么使用材料的优异热机械特性，要么只对 SMA 元件进行一次致动。最近，Coli 等人开发了一种 SMA 翻转襟翼致动器，用于调节发动机气缸的进气量[17]。该翻转襟翼致动器利用 SMA 弹簧减少污染并提高发动机性能。当襟翼处于下降位置时，歧管打开，自由进气。当襟翼处于升起位置时，歧管部分关闭，气流翻滚，从而提高燃烧效率，减少污染，提高性能。实验室实验评估表明，SMA 弹簧致动器能够将带有襟翼阵列的轴旋转到发动机进气歧管内的两个极限位置（升高或降低）[17]。
This paper introduces a second generation SMA actuator for automotive side mirrors [18]. The design of the new mirror has been updated to overcome some of the limitations of the previous design such as limited range of motion and the trade off between 本文介绍了用于汽车侧后视镜的第二代 SMA 执行器 [18]。新后视镜的设计经过了更新，克服了之前设计的一些局限性，例如运动范围有限以及在以下两个方面之间的权衡
the position stability of the mirror and its controllability. The newly designed mirror passively mitigates disturbances such as aerodynamic forces and variation in environment temperature. 镜的位置稳定性及其可控性。新设计的反射镜能被动地减轻空气动力和环境温度变化等干扰。
Following the introductory material on the SMA materials and control of SMA actuators, the paper presents a simple yet effective controller logic for the mirror actuator. The selected controller does not require a model of the constitutive behavior that leads to the SMA actuation. This feature makes the control algorithm applicable for onboard implementation. Electronic design of the controller is discussed and closed-loop experimental results are presented. 在介绍了 SMA 材料和 SMA 执行器控制之后，本文介绍了一种简单而有效的镜面执行器控制器逻辑。所选控制器无需建立导致 SMA 驱动的构成行为模型。这一特点使得控制算法适用于机载实施。文中讨论了控制器的电子设计，并给出了闭环实验结果。
2. Shape memory alloys 2.形状记忆合金
Shape memory alloys (SMAs) are a group of smart materials that can effectively change their shape and provide actuation by restoring their memorized geometry. The reversible mechanism behind shape memory alloy actuation is a solid-state phase transformation that takes place in response to variation of temperature and stress. The distinct thermomechanical behavior of SMAs is the result of a transformation from the austenite (parent) phase to martensite (product) phase and vice versa [19]. These alloys have very high energy density; therefore, actuators that implement these alloys are compact and lightweight alternatives to other types of actuators such as DC motors and solenoids. SMA actuators are an effective way to reduce weight and to minimize the complexity of various systems. As actuators, these materials have been used in a wide range of applications from thin film actuators for MEMS [20] to helicopter rotor actuators [15,21]. 形状记忆合金（SMA）是一类智能材料，可有效改变形状，并通过恢复记忆中的几何形状来提供驱动力。形状记忆合金驱动背后的可逆机制是一种固态相变，这种相变会随着温度和应力的变化而发生。SMA 的独特热机械行为是奥氏体（母体）相向马氏体（产物）相转变的结果，反之亦然 [19]。这些合金具有极高的能量密度；因此，采用这些合金的致动器结构紧凑、重量轻，可替代直流电机和螺线管等其他类型的致动器。SMA 执行器是减轻重量和最大限度降低各种系统复杂性的有效方法。作为致动器，这些材料已被广泛应用于从微机电系统薄膜致动器 [20] 到直升机转子致动器 [15,21]。
Most of the SMA actuators utilize these alloys in the wire form. SMA actuators are designed to use the shape memory effect in generating motion and force. When a SMA wire is heated, as shown in 大多数 SMA 执行器都采用线状合金。SMA 执行器是利用形状记忆效应来产生运动和力的。当加热 SMA 金属丝时，如图所示
Fig. 1, the material applies a large amount of force while returning to a memorized length. Joule heating is an effective and simple way for actuating SMA components. Existing applications of SMA actuators include devices such as active endoscopes, rifle stabilizer systems and commercial linear actuators [22-24]. 如图 1 所示，材料在施加巨大力量的同时，会返回到记忆长度。焦耳加热是致动 SMA 元件的一种有效而简单的方法。SMA 执行器的现有应用包括主动内窥镜、步枪稳定器系统和商用线性执行器等设备 [22-24]。
Among the different models developed for the thermomechanical behavior of SMAs, phenomenological models have been most widely used for actuator design and control. A phenomenological model is comprised of two main parts: a constitutive model that defines the thermomechanical behavior of the material by expressing the stress as a function of temperature, strain, and martensite fraction; and a phase transformation kinetics model describes the martensite fraction as a function of temperature and stress of the material. More details on the thermomechanical modeling of SMAs can be found in [25-27]. 在针对 SMA 热机械行为开发的各种模型中，现象学模型在致动器设计和控制中的应用最为广泛。现象学模型由两个主要部分组成：一个是构成模型，通过将应力表示为温度、应变和马氏体分数的函数来定义材料的热机械行为；另一个是相变动力学模型，将马氏体分数描述为材料温度和应力的函数。有关 SMA 热机械模型的更多详情，请参阅 [25-27]。
Several methods of controlling SMA actuators have been investigated; most of these control systems, however, are non-modelbased with limited possibility for stability and robustness analysis. Different variations of linear Proportional Integral Derivative (PID) controls have been explored by several researchers [28-31] while many others have used Pulse Width Modulation [32-35]. Several nonlinear control schemes such as fuzzy logic, neural networks, feedback linearization, sliding mode control, and variable structure control have also been explored by researchers [36-38,14,39-44] Song et al. [45] have applied sliding mode feedback control in conjunction with a feedforward neural network to an SMA actuator with linear motion. Furthermore, other controllers have also been designed that include an additional open-loop part for improving the performance [46,47]. Elahinia et al. developed a model-based backstepping controller for SMA actuators [48]. The same group developed a stress-based and temperature-based controller which demonstrated enhanced tracking performance for rotary SMA actuators . 目前已研究出几种控制 SMA 执行器的方法；不过，这些控制系统大多不基于模型，稳定性和鲁棒性分析的可能性有限。一些研究人员探索了线性比例积分微分（PID）控制的不同变体 [28-31]，还有许多人使用了脉冲宽度调制 [32-35]。研究人员还探索了多种非线性控制方案，如模糊逻辑、神经网络、反馈线性化、滑动模式控制和可变结构控制 [36-38,14,39-44]。Song 等人 [45] 将滑动模式反馈控制与前馈神经网络结合应用于具有线性运动的 SMA 执行器。此外，还设计了其他控制器，其中包括额外的开环部分，以提高性能 [46,47]。Elahinia 等人为 SMA 执行器开发了基于模型的反步进控制器 [48]。同一研究小组还开发了一种基于应力和温度的控制器，该控制器提高了旋转 SMA 执行器的跟踪性能 。
(a)
(b)
Fig. 1. Shape memory alloys have two distinct extreme thermomechanical behaviors (a) shape memory effect, which is used for actuation in this work, and (b) superelasticity. 图 1.形状记忆合金具有两种截然不同的极端热机械行为（a）形状记忆效应，在本研究中用于驱动；以及（b）超弹性。
3. Second generation SMA mirror actuator 3.第二代 SMA 镜致动器
In an automotive external side mirror, the actuator typically consists of two DC motors and a gear assembly, as shown in Fig. 2. A SMA actuator for mirror positioning, on the other hand, can simply be comprised of joints and fixtures that can drastically reduce the number of necessary components and simplify the required assembly. The main objectives in developing the SMA mirror actuator are to reduce the number of moving parts and reduce the manufacturing cost of the mirror. The first generation SMA mirror actuator is shown in Fig. 3 [18]. While the actuator achieved desired range of motion, the main limiting factor was the trade-off between the stability of the mirror and range of motion. Since the joint was passive, a portion of the SMA actuation force was needed to overcome the friction. This in turn reduced the actuation range of the SMA wires. To overcome this limitation a second-generation mirror actuator is presented in this paper. 在汽车外侧后视镜中，致动器通常由两个直流电机和一个齿轮组件组成，如图 2 所示。而用于后视镜定位的 SMA 执行器则可以简单地由接头和夹具组成，从而大幅减少必要组件的数量并简化所需的装配。开发 SMA 镜面致动器的主要目的是减少运动部件的数量，降低镜面的制造成本。第一代 SMA 镜致动器如图 3 所示 [18]。虽然致动器达到了预期的运动范围，但主要的限制因素是反射镜的稳定性和运动范围之间的权衡。由于关节是被动的，因此需要使用部分 SMA 驱动力来克服摩擦。这反过来又缩小了 SMA 线的致动范围。为了克服这一限制，本文介绍了第二代反射镜致动器。
The design concept for the actuator is depicted in Figs. 4 and 5. This redesigned mirror solves the main limitations of the first generation SMA mirror actuator [18]. This second-generation mirror, 图 4 和图 5 展示了致动器的设计理念。这种重新设计的反射镜解决了第一代 SMA 反射镜致动器的主要局限性 [18]。第二代反射镜
Fig. 2. A conventional DC motor powered automotive side mirror actuator consists of two motors and a gear assembly. 图 2.传统的直流电机驱动汽车侧后视镜致动器由两个电机和一个齿轮组件组成。
Fig. 3. First-generation spherical joint designed to provide the passive braking force for stabilizing the SMA actuated side mirror when the wires are not activated. 图 3.第一代球形接头的设计目的是提供被动制动力，以便在导线未激活时稳定 SMA 驱动的侧后视镜。
Fig. 4. Second-generation prototype SMA mirror actuator. 图 4.第二代 SMA 镜致动器原型。
Fig. 5. Second-generation prototype SMA mirror actuator includes and active spherical joint to stabilize the mirror and to provide friction-free maximum range of motion. 图 5.第二代 SMA 镜面致动器原型包括一个主动球形关节，用于稳定镜面并提供无摩擦的最大运动范围。
as shown in Fig. 5, is primarily made of an innovative spherical joint and four SMA wires. The spring-loaded joint provides mirror stability between actuation. At the time of actuation, the force of the SMA wires disengage the joint to provide friction-free motion. This joint passively provides a larger range of motion with smaller actuation force. The actuation force, unlike in the first generation mirror, is not used to overcome friction. This feature of the joint, additionally, mitigates the disturbances such as aerodynamic forces and variation in environment temperature. 如图 5 所示，它主要由一个创新的球形接头和四根 SMA 线组成。弹簧加载的关节可在两次启动之间提供镜面稳定性。启动时，SMA 线的作用力会使关节脱离，从而实现无摩擦运动。这种关节能以较小的驱动力提供较大的运动范围。与第一代反射镜不同的是，驱动力不是用来克服摩擦的。此外，接头的这一特点还能减轻空气动力和环境温度变化等干扰。
The proposed actuator rotates the mirror about its two axes. As shown in Fig. 4, the SMA actuated mirror is much simpler than the traditional actuator. The two DC motors of the conventional actuator are replaced by two pairs of SMA wires. When one of these wires is heated through Joule heating, the wire contracts due to phase transformation, as shown in Fig. 1. By adjusting the electric current, it is possible to achieve a controlled temperature in the SMA wire. By controlling the temperature, the level of actuation is controlled. With the proper selection of a control algorithm, therefore, it is possible to position the mirror to a desired orientation. It is worth noting that the orientation of the mirror is defined by its rotation about two perpendicular axes. 拟议的致动器可使反射镜绕其两个轴旋转。如图 4 所示，SMA 驱动镜比传统致动器简单得多。传统致动器的两个直流电机由两对 SMA 线代替。如图 1 所示，当其中一根导线通过焦耳加热时，导线会因相变而收缩。通过调节电流，可以控制 SMA 金属丝的温度。通过控制温度，可以控制致动水平。因此，只要选择适当的控制算法，就可以将反射镜定位到所需的方向。值得注意的是，反射镜的方向是通过围绕两个垂直轴的旋转来确定的。
In order to identify the design objectives of the mirror, a standard powered side mirror was analyzed. This side mirror had a range of for each of the two angles of rotation. One of the design objectives for the SMA mirror was that this range should be set as the minimum goal for the SMA mirror. The mirror should also have the capability to be positioned quickly, maintain stability throughout operation and rest similarly to that of current mirrors with minimal sensitivity to variation in ambient temperature. 为了确定后视镜的设计目标，我们分析了一个标准的电动侧视镜。该侧反射镜的两个旋转角度范围分别为 。SMA 反射镜的设计目标之一是将这一范围设定为 SMA 反射镜的最低目标。该反射镜还应具备快速定位的能力，在整个操作过程中保持稳定，并与当前反射镜的静止状态相似，对环境温度变化的敏感性最小。
SMA wires provide the actuation force and displacement for the mirror. In order to properly select the SMA wire parameters, the required force and displacements of the wire must be determined. The amount of force is proportional to the diameter of the wire; larger diameter wires can be implemented to generate larger forces as needed. To increase the actuation life of the wire it is essential to limit the strain of the SMA actuator to . This strain limit and the required displacement of the actuator, define the required length of the wire. To determine the strain in each wire we can build a relationship: SMA 线为反射镜提供驱动力和位移。为了正确选择 SMA 线参数，必须确定所需的力和线的位移。力的大小与导线的直径成正比；根据需要，可以使用较大直径的导线来产生较大的力。为了延长金属丝的执行寿命，必须将 SMA 执行器的应变限制在 。这一应变限制和致动器所需的位移决定了所需的导线长度。为了确定每根导线的应变，我们可以建立一种关系：
where is the wire strain, is the length of the wire as a function of the orientation angles, referenced from a coordinate system collocated at the center of the joint and is the minimum length of the wire when it is in the austenite phase. The length of each wire changes very little due to the change in because the rotation about the mirror axis is very small and is considered negligible. It should also be noted that the strain of each wire at the reference position ( is nonzero. The effective torque of the SMA actuator can be written as: 其中 是线材应变， 是线材长度与方向角的函数关系，以位于接头中心的坐标系为参考， 是线材处于奥氏体阶段时的最小长度。由于围绕镜像轴的旋转非常小，可以忽略不计，因此 的变化导致每根金属丝的长度变化非常小。还应注意的是，每根金属丝在参考位置 ( ) 的应变都不为零。SMA 传动器的有效扭矩可写成
The proposed mirror is of the rotary type, meaning that the input to generate rotation is the wire torque. A relationship for the torque delivered by the wire about the joint can be found in (2), where is the moment arm of the wire and is the wire force. 拟议的反射镜是旋转型的，这意味着产生旋转的输入是导线扭矩。导线绕接头产生的扭矩关系见 (2)，其中 是导线的力矩臂， 是导线力。
In order to design the mirror such that it is powerful enough to position the mirror, one must first calculate the necessary torque and moment arm. 为了设计出足够强大的定位镜，必须首先计算出必要的扭矩和力矩臂。
4. Control of the mirror actuator 4.后视镜致动器的控制
In order to achieve accurate and robust positioning for the SMA actuated mirror, a variable structure controller (VSC), as schemati- cally shown in Fig. 8, is implemented. The controller is a simple yet effective method of control that does not require the computation of the entire system dynamics of the mirror. Details on the development of a VSC for SMA systems can be found in . Similar to our previous work, as shown in Fig. 6, the model of the SMA actuator can be represented by a set of nonlinear state equations as: 为了实现 SMA 驱动反射镜的精确和稳健定位，实施了一个可变结构控制器 (VSC)，如图 8 所示。该控制器是一种简单而有效的控制方法，无需计算反射镜的整个系统动态。有关为 SMA 系统开发 VSC 的详情，请参阅 。与我们之前的工作类似，如图 6 所示，SMA 执行器的模型可以用一组非线性状态方程来表示：
where the state vector and the control input are given as 其中，状态矢量 和控制输入 分别为
and 和
The variable structure control is introduced using the switching function as [52]: 如 [52] 所述，利用开关函数 引入了可变结构控制：
such that (switching surface) can be reached in finite time. The sliding mode takes place on the switching surface, following the desired system dynamics to ensure that the overall VSC system is globally asymptotically stable. A boundary layer around the switching surface can be introduced to solve the chattering problem which is a common phenomenon in many VSC designs. The magnitude of this boundary layer is determined experimentally by starting at a value less than the required resolution and increasing it until the system reaches the sliding surface. 使 （切换面）能在有限时间内到达。滑动模式在开关面上进行，遵循所需的系统动态，以确保整个 VSC 系统在全局上渐近稳定。开关面周围可引入一个边界层，以解决许多 VSC 设计中常见的颤振问题。该边界层的大小可通过实验确定，即从小于所需分辨率的值开始，不断增大直到系统达到滑动面。
To position the mirror, a two-stage VSC is designed that consists of a simple switch between a high voltage current for quick heating (i.e. mirror rotation), a low voltage current for cooling and reverse motion and a proportional gain for the boundary layer. The switching is defined based on the position error, . A boundary layer is introduced around the switching surface , to suppress chattering. The layer thickness is defined as , and the control input for each of the four wires is defined as: 为了定位反射镜，设计了一个两级 VSC，其中包括一个用于快速加热（即反射镜旋转）的高压电流、一个用于冷却和反向运动的低压电流之间的简单开关，以及一个用于边界层的比例增益。开关的定义基于位置误差 。在开关表面周围引入边界层 ，以抑制颤振。层厚度定义为 ，四根导线中每根导线的控制输入定义为：
where is a proportional gain. Generally, increasing the boundary layer thickness, reduces chattering but increases steady-state error. To reduce the overshoot possibility, the switching condition (surface) can be further modified using a combination of position and velocity errors, 其中 为比例增益。一般来说，增加边界层厚度可减少颤振，但会增加稳态误差。为了降低过冲的可能性，可以结合位置和速度误差进一步修改开关条件（表面）、
In this system, is slope of the sliding surface in the phase plane. Although the angular velocity is not measured experimentally, a 在该系统中， 是滑动面在相位平面上的斜率。虽然角速度没有通过实验测量，但 a
Fig. 6. Block diagram of SMA mirror model. 图 6.SMA 镜模型框图。
numerical derivative approximation is obtained by filtering out the noise. 数值导数近似值是通过滤除噪声得到的。
5. Experimental setup 5.实验装置
A prototype mirror is developed, as shown in Fig. 7, to evaluate the controller performance. It is worth noting that the side mirror is controlled by the driver using a joystick. The variable structure controller designed in this paper provides the robustness needed to overcome the environmental variations and to correct unwanted motions of the mirror. The prototype mirror is simply comprised of a spherical joint and four SMA wires. The Flexinol SMA wires have a diameter of and were supplied by Dynalloy Inc. To verify the performance of the control system, three resolution laser displacement sensors are used in order to obtain the three points needed to detect the mirror plane angle. It is worth noting that these sensors will not be needed when the driver uses a joystick adjusts the mirror. Each of the three measurements has its particular and coordinates determined by the distance between each laser beam. The value determined by the sensor is then taken as the coordinate and a unit vector is defined by the three points coinciding with each laser measurement. A dSPACE 1104 controller is used along with an Agilent 6543 A power supply for conducting the experiments (see Fig. 8). 为评估控制器的性能，开发了一个后视镜原型，如图 7 所示。值得注意的是，侧后视镜是由驾驶员使用操纵杆控制的。本文设计的可变结构控制器具有克服环境变化和纠正后视镜意外运动所需的鲁棒性。原型后视镜由一个球形接头和四根 SMA 线组成。Flexinol SMA 线直径为 ，由 Dynalloy 公司提供。为了验证控制系统的性能，使用了三个 分辨率激光位移传感器，以获得检测镜面角度所需的三个点。值得注意的是，当驾驶员使用操纵杆调节后视镜时，将不需要这些传感器。三个测量点中的每一个都有其特定的 和 坐标，由每束激光之间的距离决定。然后将传感器确定的值作为 坐标，并由与每个激光测量重合的三个点定义一个单位向量。使用 dSPACE 1104 控制器和 Agilent 6543 A 电源进行实验（见图 8）。
The VSC controller is implemented in Simulink/dSPACE. The electronics for the mirror system are designed around having one voltage controlled power supply. The controller receive a joystick signal and routes an electrical current to the proper SMA wire to reduce the error in the angular position about two axes. To this VSC 控制器是在 Simulink/dSPACE 中实现的。镜像系统的电子设备是围绕一个电压控制电源设计的。控制器接收操纵杆信号，并将电流输送到适当的 SMA 线，以减少两个轴的角度位置误差。为此
Fig. 7. Prototype mirror actuator used in experimental study. 图 7.实验研究中使用的镜面致动器原型。
Fig. 8. Implementation of the variable structure control algorithm for the antagonistic SMA actuator. end the controller compares the two angular errors in the controller to determine which axis error is of higher magnitude. This means that the two degrees of freedom are controlled one at a time. As can be seen from the experimental results, the controller quickly switches between priorities and maintains fast and robust control in spite of this discontinuity in the implementation. 图 8.拮抗式 SMA 执行器的可变结构控制算法的实施情况。控制器末端会比较控制器中的两个角度误差，以确定哪个轴的误差更大。这意味着两个自由度一次控制一个。从实验结果中可以看出，尽管实施过程中存在这种不连续性，但控制器仍能在优先级之间快速切换，并保持快速、稳健的控制。
The control voltage is routed to the proper SMA wire through a Boolean switching circuit based on switching transistors that is controlled by a Boolean control signal (on or off). The controlled analog voltage is then routed to the necessary wire one at a time. 控制电压通过基于开关晶体管的布尔开关电路传输到适当的 SMA 导线，该电路由布尔控制信号（开或关）控制。然后，受控模拟电压被逐次分配到所需的导线上。
6. Experimental results 6.实验结果
Results of modeling, model verification and functionality evaluation of the first generation mirror were previously reported [18]. This section includes the results related to experimental evaluation of the control system. 第一代反射镜的建模、模型验证和功能评估结果已在之前报道过 [18]。本节包括与控制系统实验评估相关的结果。
Using the setup shown in a series of experiments were performed to evaluate the performance of the SMA actuated mirror and the controller. The parameters for the controller are listed in Table 1. 利用如图所示的装置进行了一系列实验，以评估 SMA 驱动反射镜和控制器的性能。表 1 列出了控制器的参数。
The first of group tests was aimed at determining the tracking response of the system to a series of step changes in desired orientation. The full range of motion of the mirror was tested by in these tests ( in both directions). The results are presented in Fig. 9 . 第一组测试的目的是确定系统对预期方向的一系列阶跃变化的跟踪反应。在这些测试中，对镜子的整个运动范围进行了测试（ ）。测试结果如图 9 所示。
Table 1 表 1
Controller parameters used for experiments. 实验使用的控制器参数
I-High I 高
I-Low
Fig. 9. Tracking results of the mirror to step changes in desired orientation, solid line is the desired horizontal axis and the dashed line is the desired vertical axis. 图 9.镜面对所需方向阶跃变化的跟踪结果，实线为所需水平轴，虚线为所需垂直轴。
Fig. 10. Tracking results to step changes in desired position showing that a range of is controllable, solid line is the desired horizontal axis and the dashed line is the desired vertical axis. 图 10.对预期位置阶跃变化的跟踪结果显示， ，实线为预期水平轴，虚线为预期垂直轴。
Fig. 11. Tracking results of a circular pattern, solid line is the desired horizontal axis and the dashed line is the desired vertical axis. 图 11.圆形图案的跟踪结果，实线为所需的水平轴，虚线为所需的垂直轴。
Fig. 12. Tracking results of the mirror to step changes in desired orientation with a fan cooling the wires (solid line is the desired horizontal axis and the dashed line is the desired vertical axis). 图 12.在风扇冷却导线的情况下，镜子对所需方向阶跃变化的跟踪结果（实线为所需水平轴，虚线为所需垂直轴）。
Fig. 13. Tracking results of a circular pattern with a fan cooling the wires (solid line is the desired horizontal axis and the dashed line is the desired vertical axis). 图 13.使用风扇冷却导线的圆形图案的跟踪结果（实线为所需的水平轴，虚线为所需的垂直轴）。
At the beginning, the two angles quickly converge to zero. Each desired angular position changes repeatedly at different times. Each angle of rotation is actuated independently. The tracking performance of the mirror is acceptable except for the far endpoints of the range of motion. This is due to the combined stiffness of the two passive wires which are opposing the motion. 开始时，两个角度迅速趋于零。每个所需的角度位置在不同时间反复变化。每个旋转角度都是独立驱动的。除了运动范围的远端点外，反射镜的跟踪性能是可以接受的。这是由于与运动相对的两根无源导线的综合刚度造成的。
In the second test, variable desired angles about the two axes are also tracked. In these tests, the overall range of motion for both horizontal and vertical axes is limited to . The test shows acceptable and repeatable tracking performance. Fig. 10 shows this these results. 在第二项测试中，还对两个轴的可变预期角度进行跟踪。在这些测试中，水平轴和垂直轴的总体运动范围限制在 。测试结果表明，跟踪性能可以接受且可重复。图 10 显示了这些结果。
In another test, the mirror desired path was to follow a circular motion. This test was performed to show the mirror's performance in following a smooth change in desired coordinates. As shown in Fig. 11 this test requires that the two desired angles of rotation change according to sinusoidal signals with similar frequencies. The figure makes it clear that for smooth transitions, the mirror controller is capable of providing the desired tracking performance. 在另一项测试中，镜子的预期路径是跟随圆周运动。进行这项测试是为了显示反射镜在跟随预期坐标平滑变化时的性能。如图 11 所示，该测试要求两个预期旋转角度根据频率相似的正弦信号变化。从图中可以清楚地看出，对于平滑过渡，反射镜控制器能够提供所需的跟踪性能。
To test the robustness of the controller, several tests were performed. The robustness was tested by adding additional mass to the mirror or by inducing sudden change in the actuating wire temperatures. Additional mass represents unexpected forces applied to the mirror during the actuation. These forces include aerodynamic and road roughness related forces. The controller should ideally mitigate the adverse effect of these forces. It is worth noting that the effect of these disturbances are mitigated by the spherical joint when the mirror is not being actuated. In rare occasions if these forces are large enough to move the wire surface in the passive stage of the mirror, the controller can become active to automatically readjust the mirror position. The second disturbance, in the form of temperature variation, takes place due to environmen- 为了测试控制器的稳健性，我们进行了多项测试。测试稳健性的方法是在反射镜上增加额外的质量，或使驱动线温度发生突然变化。附加质量表示在致动过程中施加到后视镜上的意外力。这些力包括与空气动力和路面粗糙度相关的力。控制器最好能减轻这些力的不利影响。值得注意的是，在不启动后视镜时，球形接头可减轻这些干扰的影响。在极少数情况下，如果这些力大到足以使反射镜被动阶段的导线表面发生移动，控制器就会主动重新调整反射镜的位置。第二种干扰以温度变化的形式出现，是由于环境因素造成的。
Fig. 14. Tracking results of the mirror to step changes in desired orientation with a weight attached to the mirror surface (solid line is the desired horizontal axis and the dashed line is the desired vertical axis). 图 14.在镜面上附有 重物的情况下，镜面对所需方向阶跃变化的跟踪结果（实线为所需水平轴，虚线为所需垂直轴）。
Fig. 15. Tracking results of a circular pattern with a weight attached to the mirror surface (solid line is the desired horizontal axis and the dashed line is the desired vertical axis). 图 15. 重物附着在镜面上的圆形图案的跟踪结果（实线为所需的水平轴，虚线为所需的垂直轴）。
tal variation. Again the controller is expected to compensate for ambient temperature variations. 温度变化。同样，控制器也要对环境温度变化进行补偿。
The first of the robustness tests was to track the desired angle of rotation about two axes in full range. This test was performed while a fan was continuously cooling the wires. The results of the experiment are shown in Fig. 12. The figure shows that the alternating actuation of the two separate degrees of freedom causes the performance to suffer more acutely when the controller is not tuned for the particular disturbance. The second disturbance rejection test was to track a circular motion with a fan blowing on the wires (Fig. 13). 第一项稳健性测试是全范围跟踪两个轴的预期旋转角度。这项测试是在风扇持续冷却导线的情况下进行的。实验结果如图 12 所示。从图中可以看出，当控制器没有针对特定干扰进行调整时，两个独立自由度的交替驱动会导致性能受到更严重的影响。第二个干扰抑制测试是跟踪电线上吹着风扇的圆周运动（图 13）。
Another group of the robustness tests involved adding a mass on the surface of the mirror. The added mass is thought to be a valid perturbation to test the robustness of the mirror controller and the results show that there is not significant difference in the full scale range test show in Fig. 14 and the circular tracking test shown in Fig. 15. 另一组鲁棒性测试包括在反射镜表面添加 质量。添加的质量被认为是测试反射镜控制器鲁棒性的有效扰动，结果显示，图 14 所示的全量程测试和图 15 所示的圆形跟踪测试没有明显差异。
7. Conclusions 7.结论
In this paper a second-generation SMA-actuated mirror was presented and control of its rotation about two axes was investigated. To this end, a prototype SMA actuator was developed. A variable structure control was designed to position the mirror in response to the driver's command. Simplicity and robustness of this control algorithm are essential features for automotive applications. The controller successfully positioned the mirror in the desired working range of the automotive mirror. The robustness of the controller to environmental and disturbance force variation was investigated. The controller provides acceptable motion and disturbance rejection performance. 本文介绍了第二代 SMA 驱动镜，并研究了其双轴旋转控制。为此，开发了一个 SMA 执行器原型。设计了一个可变结构控制装置，用于根据驾驶员的指令对后视镜进行定位。这种控制算法的简单性和鲁棒性是汽车应用的基本特征。控制器成功地将后视镜定位在汽车后视镜所需的工作范围内。研究了控制器对环境和干扰力变化的鲁棒性。该控制器具有可接受的运动和干扰抑制性能。
Acknowledgment 鸣谢
The authors would like to thank the Ohio Board of Regents for providing partial support for this research through the UT Challenge Grant. 作者感谢俄亥俄州执政委员会通过 UT 挑战补助金为本研究提供部分支持。
References 参考资料
[1] Lee Hwan-Soo, Choi Seung-Bok. Control and response characteristics of a magneto-rheological fluid damper for passenger vehicles. J Intell Mater Syst Struct 2000;11(January):80-7. [1] Lee Hwan-Soo，Choi Seung-Bok。用于乘用车的磁流变流体减振器的控制和响应特性。J Intell Mater Systruct 2000；11（1 月）：80-7.
[2] Stelzer GJ, Schulz MJ, Kim J, Allemang RJ. A magnetorheological semi-active isolator to reduce noise and vibration transmissibility in automobiles. J Intell Mater Syst Struct 2003;14(December):743-65. [2] Stelzer GJ, Schulz MJ, Kim J, Allemang RJ.减少汽车噪音和振动传播的磁流变半主动隔离器。J Intell Mater Systruct 2003；14（12 月）：743-65.
[3] Ahmadian M, Ahn Y. Performance analysis of magneto-rheological mounts. J Intell Mater Syst Struct 1999;10(March):248-56. [3] Ahmadian M, Ahn Y. 磁流变支架的性能分析.J Intell Mater Systruct 1999；10（3 月）：248-56.
[4] Ahn Y, Yang B, Ahmadian M, Morishita S. A small-sized variable-damping mount using magnetorheological fluid. J Intell Mater Syst Struct 2005;16(2):127-33. [4] Ahn Y, Yang B, Ahmadian M, Morishita S. 使用磁流变液的小型可变阻尼支架。J Intell Mater Systruct 2005;16(2):127-33.
[5] Ye S, Williams KA. MR fluid brake for control of torsional vibration. In: Proceedings of the SAE world congress detroit, Michigan, April 11-14, 2005. [5] Ye S, Williams KA.用于控制扭转振动的磁共振流体制动器。In：美国密歇根州底特律市 SAE 世界大会论文集，2005 年 4 月 11-14 日。
[6] Manz H, Breitbach EJ. Application of smart materials in automotive structures. In: Proceedings of SPIE smart structures and materials: industrial and commercial applications of smart structures technologies, vol. 4332; 2001. p. 197-204. [6] Manz H, Breitbach EJ.智能材料在汽车结构中的应用。In：Proceedings of SPIE smart structures and materials: industrial and commercial applications of smart structures technologies, vol. 4332; 2001. p. 197-204.
[7] Mavroidis C, Pfeiffer C, Mosley M. Conventional actuators, shape memory alloys, and electrorheological fluids. In: Bar-Cohen Y, editor. Invited chapter in automation, miniature robotics and sensors for non-destructive testing and evaluation; 1999 [7] Mavroidis C, Pfeiffer C, Mosley M. 传统致动器、形状记忆合金和电流变液。In：Bar-Cohen Y, editor.自动化、微型机器人和用于无损检测和评估的传感器特邀章节；1999 年
[8] Myose R, Horn W, Hwang Y, Herrero J, Huynh C, Boudraa T. Application of shape memory alloys for leading edge deicing. In: Proceedings of the SAE general, corporate and regional aviation meeting and exposition (GCRAM), Wichita, Kansas, April 20-22, 1999 [8] Myose R、Horn W、Hwang Y、Herrero J、Huynh C、Boudraa T.形状记忆合金在前缘除冰中的应用。In：1999年4月20-22日在堪萨斯州威奇托举行的美国航空工程师学会通用、企业和地区航空会议暨博览会（GCRAM）论文集
[9] Singh K, Sirohi J, Chopra I. An improved shape memory alloy actuator for rotor blade tracking. J Intell Mater Syst Struct 2003;14(December). [9] Singh K, Sirohi J, Chopra I. 用于转子叶片跟踪的改进型形状记忆合金致动器。J Intell Mater Systruct 2003;14(December).
[10] Cragg A, Lund G, Rysavy J, Castaneda F, Castaneda-Zuniga W, Amplatz K. Nonsurgical placement of arterial endoprostheses: a new technique using nitinol wire. Radiology 1983;147(April):261-3. [10] Cragg A, Lund G, Rysavy J, Castaneda F, Castaneda-Zuniga W, Amplatz K. 《非手术置入动脉内膜假体：使用镍钛诺丝的新技术》。Radiology 1983;147(April):261-3.
[11] Maeno T, Hino T. Miniature five-fingered robot hand driven by shape memory alloy actuators. In: Robotics and applications, IASTED international conference, Honalulu, Hawaii; August 2006. [11] Maeno T, Hino T. 由形状记忆合金致动器驱动的微型五指机械手。In：机器人与应用，IASTED 国际会议，夏威夷霍纳鲁鲁；2006 年 8 月。
[12] Song G, Ma N, Lee H, Arnold S. Design and control of a proof-of-concept variable area exhaust nozzle using shape-memory alloy actuators. Smart Mater Struct 2007;16. [12] Song G、Ma N、Lee H、Arnold S. 利用形状记忆合金致动器设计和控制概念验证型可变面积排气喷嘴。Smart Mater Struct 2007;16.
[13] Price A, Jnifene A, Naguib H. Design and control of a shape memory alloy based dexterous robot hand. Smart Mater Struct 2007;16. [13] Price A, Jnifene A, Naguib H.基于形状记忆合金的灵巧机器手的设计与控制。Smart Mater Struct 2007;16.
[14] Elahinia M, Ashrafiuon H. Nonlinear control of a shape memory alloy actuated manipulator. ASME J Vib Acoust 2002;124(October):566-75. [14] Elahinia M, Ashrafiuon H. 形状记忆合金致动器的非线性控制.ASME J Vib Acoust，2002；124（10 月）：566-75.
[15] Mabe James H, Ruggeri Robert T, Rosenzweig Ed, Mike Yu. NiTinol performance characterization and rotary actuator design. In: Proceedings of SPIE - the international society of optical engineering, vol. 5388. San Diego (CA); 2004. p. 95-109. [15] Mabe James H, Ruggeri Robert T, Rosenzweig Ed, Mike Yu.镍钛诺性能表征和旋转致动器设计。In：Proceedings of SPIE - the international society of optical engineering, vol. 5388.圣迭戈（加利福尼亚州）；2004 年。第 95-109 页。
[16] Jones N, Harrison D, Chiodo J, Billett E. Design for automotive glass removal using active disassembly. In: Proceedings of the 2002 SAE international body engineering conference and automotive & transportation technology conference Paris, France, July 9-11, 2002 [16] Jones N, Harrison D, Chiodo J, Billett E. 利用主动拆卸设计汽车玻璃。In：2002 SAE 国际车身工程会议暨汽车与运输技术会议论文集 法国巴黎，2002 年 7 月 9-11 日
[17] Colli M, Bellini A, Concari C, Toscani A, Franceschini G. Current-controlled shape memory alloy actuators for automotive tumble flap. In: Proceedings of 32nd annual conference on IEEE industrial electronics, IECON, Paris, France, November 7-10, 2006 [17] Colli M、Bellini A、Concari C、Toscani A、Franceschini G.用于汽车翻转襟翼的电流控制形状记忆合金致动器。In：第 32 届 IEEE 工业电子学年会论文集，IECON，法国巴黎，2006 年 11 月 7-10 日
[18] Williams E, Elahinia MH. An automotive SMA mirror actuator: modeling, design and experimental evaluation. J Intell Mater Syst Struct 2008;19(12):1425-34. [18] Williams E, Elahinia MH.汽车 SMA 后视镜致动器：建模、设计和实验评估。J Intell Mater Systruct 2008; 19(12):1425-34.
[19] Brinson LC. One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J Intell Mater Syst Struct 1993;4(April):229-42. [19] Brinson LC.形状记忆合金的一维构成行为：非恒定材料函数和重新定义的马氏体内部变量的热力学推导。J Intell Mater Systruct 1993;4(April):229-42.
[20] Roch I, Bidaud Ph, Collard D, Buchaillot L. Fabrication and characterization of an SU-8 gripper actuated by a shape memory alloy thin film. J Micromech Microeng 2003(February). [20] Roch I, Bidaud Ph, Collard D, Buchaillot L. 由形状记忆合金薄膜驱动的 SU-8 机械手的制造和特性分析。J Micromech Microeng，2003（2 月）.
[21] Chopra I. Review of state of art of smart structures and integrated systems. AIAA J 2002;40(11):2145-87 [21] Chopra I. 智能结构和集成系统的最新进展综述.AIAA J 2002;40(11):2145-87
[22] Ikuta K, Tsukamoto M, Hirose S. Shape memory alloy servo actuator system with electric resistance feedback and application for active endoscope. In: IEEE international conference on robotics and automation, Philadelphia (PA, USA); April 1998 [22] Ikuta K, Tsukamoto M, Hirose S. 带电阻反馈的形状记忆合金伺服致动器系统及其在主动内窥镜中的应用。In：电气和电子工程师学会机器人和自动化国际会议，费城（宾夕法尼亚州，美国）；1998 年 4 月
[23] Barnes B, Brei D, Luntz J, Lavigna C. Development of an antagonistic SMA actuator for INSTAR rifle stabilization system. In: ASME international mechanical engineering congress and exposition, Orlando (FL, USA) November 2005 [23] Barnes B, Brei D, Luntz J, Lavigna C..为 INSTAR 步枪稳定系统开发拮抗 SMA 执行器。In：美国机械工程师学会国际机械工程大会暨博览会，奥兰多（佛罗里达州，美国），2005 年 11 月
[24] McGregor R. Shape memory alloy actuators and control methods. USPTO, June 2003. US Patent 6574958 [24] McGregor R. Shape memory alloy actuators and control methods.美国专利商标局，2003 年 6 月。美国专利 6574958
[25] Brinson LC. One-dimensional constitutive behavior of shape memory alloys thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J Intell Mater Syst Struct 1993(April). [25] Brinson LC.采用非恒定材料函数和重新定义的马氏体内部变量的形状记忆合金热力学推导的一维构成行为。J Intell Mater Systruct 1993（4 月）.
[26] Liang C. The constitutive modeling of shape memory alloys. PhD thesis, Virginia Tech, Blacksburg, VA, 1990. [26] Liang C. The constitutive modeling of shape memory alloys.弗吉尼亚州布莱克斯堡弗吉尼亚理工大学博士论文，1990 年。
[27] Elahinia M. Effect of system dynamics on shape memory alloy behavior and control. PhD thesis, Virginia Tech, Blacksburg, VA, August 2004. [27] Elahinia M. 《系统动力学对形状记忆合金行为和控制的影响》。弗吉尼亚州布莱克斯堡弗吉尼亚理工大学博士论文，2004 年 8 月。
[28] Arai S, Aramaki K, Yanagisawa Y. Continuous system modeling of shape memory alloy (SMA) for control analysis. In: Proceedings of the 5th IEEE international symposium on micro machine and human science; 1994. p. 97-9 [28] Arai S, Aramaki K, Yanagisawa Y. 用于控制分析的形状记忆合金（SMA）连续系统建模。In：第 5 届 IEEE 国际微型机器与人体科学研讨会论文集》，1994 年。第 97-9 页
[29] Carpenter B, Head RJ, Gehling R. Shape memory-actuated gimbal. In Proceedings of SPIE - the international society of optical engineering, vol. 2447. San Diego (CA); 1995. p. 91-101. [29] Carpenter B, Head RJ, Gehling R. Shape memory-actuated gimbal.In Proceedings of SPIE - the international society of optical engineering, vol. 2447.圣迭戈（加利福尼亚州）；1995 年。第 91-101 页。
[30] Gorbet RB, Wang DWL. General stability criteria for a shape memory position control systems. In: Proceedings of IEEE international conference on robotics and automation, vol. 3. 1995. p. 2313-9. [30] Gorbet RB，Wang DWL.形状记忆位置控制系统的一般稳定性标准。In：Proceedings of IEEE international conference on robotics and automation, vol. 3. 1995. p. 2313-9.
[31] Ikuta K. Micro/miniature shape memory alloy actuator. In: IEEE robotics and automation society, vol. 3. Los Alamitos (CA): IEEE Computer Society Press; 1990. p. 2156-61. [31] Ikuta K. 微型形状记忆合金致动器。In：Los Alamitos (CA): IEEE Computer Society Press; 1990. p. 2156-61.
[33] Hashimoto M, Takeda M, Sagawa H, Chiba I. Application of shape memory alloy to robotic actuators. J Robotic Syst 1985;2(1):3-25. [33] Hashimoto M, Takeda M, Sagawa H, Chiba I. 形状记忆合金在机器人致动器中的应用.J Robotic Syst 1985;2(1):3-25.
[34] Honma D, Miwa Y, Iguchi N. Micro robots and micro mechanisms using shape memory alloy. In: The third toyota conference, integrated micro motion systems, micro-machining, control and applications, Nissan, Aichi, Japan; 1984. [34] Honma D, Miwa Y, Iguchi N. 使用形状记忆合金的微型机器人和微型机构。In：第三届丰田会议，集成微型运动系统、微型加工、控制和应用，日本爱知县日产市；1984 年。
[35] Kuribayashi K. A new actuator of a joint mechanism using NiTi alloy wire. Int J Robotics Res 1986;4(4):103-8 [35] Kuribayashi K. 一种使用镍钛合金线的新型关节机构致动器。Int J Robotics Res 1986;4(4):103-8
[36] Arai S, Aramaki K, Yanagisawa Y. Feedback linearization of SMA (shape memory alloy). In: Proceedings of the 34th SICE annual conference; 1995. p . [36] Arai S, Aramaki K, Yanagisawa Y. SMA（形状记忆合金）的反馈线性化。In：第 34 届 SICE 年会论文集；1995 年。p .
[37] Choi S, Cheong CC. Vibration control of flexible beam using shape memory alloy actuators. J Guid Control Dyn 1996;19(5):1178-80. [37] Choi S, Cheong CC.使用形状记忆合金致动器的柔性梁振动控制。J Guid Control Dyn，1996；19（5）：1178-80.
[38] Dickinson CA, Wen JT. Feedback control using shape memory alloy actuators. J Intell Mater Syst Struct 1998;9(4):242-50. [38] Dickinson CA, Wen JT.使用形状记忆合金致动器的反馈控制。J Intell Mater Systruct 1998；9（4）：242-50.
[39] Grant D, Hayward V. Constrained force control of shape memory alloy actuators. In: Proceedings of the IEEE international conference on robotics and automation; 2000. p. 1314-20 [39] Grant D，Hayward V.形状记忆合金致动器的约束力控制。In：IEEE 机器人与自动化国际会议论文集》，2000 年，第 1314-20 页
[40] Kumagai P, Hozian A, Kirkland M. Neuro-fuzzy model based feedback controller for shape memory alloy actuators. In: Proceedings of SPIE - the international society for optical engineering, society of photo-optical instrumentation engineers, vol. 3984. Bellingham (WA, USA); 2000. p. 291-9. [40] Kumagai P, Hozian A, Kirkland M. 基于神经模糊模型的形状记忆合金致动器反馈控制器。In：Proceedings of SPIE - the international society for optical engineering, society of photo-optical instrumentation engineers, vol. 3984.Bellingham (WA, USA); 2000. p. 291-9.
[41] Nakazato T, Kato Y, Masuda T. Control of push-pull-type shape memory alloy actuators by fuzzy reasoning. Trans Jpn Soc Mech Eng, Part C 1993;59(568):226-32 [41] Nakazato T, Kato Y, Masuda T. 用模糊推理控制推拉型形状记忆合金致动器.Trans Jpn Soc Mech Eng, Part C 1993;59(568):226-32
[42] Rao R, Damle V, Lashlee R, Kern F. Identification and robust control of smart structures using artificial neural networks. Smart Mater Struct 1994;3: . [42] Rao R, Damle V, Lashlee R, Kern F. 利用人工神经网络对智能结构进行识别和鲁棒控制。Smart Mater Struct 1994; 3: .
[43] Elahinia M, Ashrafiuon H, Ahmadian A, Inman D. Application of the extended kalman filter to control of a shape memory alloy arm. In: ASME international mechanical engineering congress and R&D expo, Washington (DC), November . [43] Elahinia M, Ashrafiuon H, Ahmadian A, Inman D. 扩展卡尔曼滤波器在形状记忆合金臂控制中的应用。In：ASME 国际机械工程大会暨研发博览会，华盛顿特区，11 月 。
[44] Elahinia M, Ashrafiuon H, Ahmadian M. A temperature-based controller for a shape memory alloy actuator. ASME J Vib Acoust 2005;127(June):285-91. [44] Elahinia M, Ashrafiuon H, Ahmadian M. 基于温度的形状记忆合金致动器控制器。ASME J Vib Acoust，2005；127（6 月）：285-91.
[45] Song G, Chaudhry V, Batur C. Precision tracking control of shape memory alloy actuators using neural networks and a sliding-mode based robust controller Smart Mater Struct 2003;12:223-31. [45] Song G, Chaudhry V, Batur C. 使用神经网络和基于滑动模式的鲁棒控制器对形状记忆合金致动器进行精确跟踪控制 Smart Mater Struct 2003; 12:223-31.
[46] Reynaerts D, Van Brussel H. Development of a SMA high performance robotic actuator. In: Fifth international conference on advanced robotics, vol. 2. New York; 1991. p. 19-27. [46] Reynaerts D, Van Brussel H. SMA 高性能机器人致动器的开发。In：New York; 1991. p. 19-27.
[47] van der Wijst MWM, Schreurs PJG, Veldpaus FE. Application of computed phase transformation power to control shape memory alloy actuators. Smart Mater Struct 1997;6(2):190-8 [47] van der Wijst MWM, Schreurs PJG, Veldpaus FE.应用计算相变功率控制形状记忆合金致动器.智能材料结构，1997；6（2）：190-8
[48] Elahinia M, Koo J, Ahmadian M, Woolsey C. Backstepping control of an SMA actuated robotic arm. J Vib Control 2005;11(3):407-29. [48] Elahinia M, Koo J, Ahmadian M, Woolsey C. SMA 驱动机械臂的后步法控制。J Vib Control 2005; 11(3):407-29.
[49] Elahinia M, Seigler M, Leo D, Ahmadian M. Nonlinear stress-based control of rotary SMA-actuated manipulator. J Intell Mater Syst Struct 2004;15(6):495-508. [49] Elahinia M, Seigler M, Leo D, Ahmadian M. 旋转 SMA 驱动机械手的非线性应力控制.J Intell Mater Systruct 2004；15（6）：495-508.
[50] Elahinia M, Ahmadian M, Ashrafiuon H. Design of Kalman filter for rotary shape memory alloy actuators. Smart Mater Struct 2004;13(4):691-7. [50] Elahinia M, Ahmadian M, Ashrafiuon H. 旋转形状记忆合金致动器的卡尔曼滤波器设计.Smart Mater Struct 2004;13(4):691-7.
[51] Grant D. Accurate and rapid control of shape memory alloy actuators. PhD thesis. Canada: McGill University; 1999. [51] Grant D. 形状记忆合金致动器的精确和快速控制。博士论文。加拿大：加拿大：麦吉尔大学；1999 年。
[52] Stepanenko Y, Su CY. Variable structure control of robot manipulators with nonlinear sliding manifolds. Int J Control 1992;58(2):285-300. [52] Stepanenko Y, Su CY.具有非线性滑动流形的机器人机械手的可变结构控制.Int J Control 1992; 58(2):285-300.
