ShArc: A Geometric Technique for Multi-Bend/Shape Sensing CCS Concepts Hardware Emerging interfaces; Sensors and actua- tors ShArc：一种用于多弯曲/形状感知的几何技术 CCS 概念 硬件 新兴界面；传感器和执行器
Conference Paper May 2020 会议论文 2020 年 5 月
DOI: DOI：
READS 读取
16
2 authors: 2 位作者:
Paul Dietz 保罗·迪茨
University of Toronto 多伦多大学
70 PUBLICATIONS 3,399 CITATIONS 70 篇出版物 3,399 次引用
SEE PROFILE 查看个人资料
ShArc:
A Geometric Technique for Multi-Bend/Shape Sensing 用于多弯曲/形状传感的几何技术
Fereshteh Shahmiri 费雷什特·沙赫米利Georgia Institute of Technology - Tactual Labs 佐治亚理工学院 - 触觉实验室Atlanta, USA 美国亚特兰大sshahmiri3@gatech.edu
Paul H. Dietz 保罗·迪茨Tactual Labs 触觉实验室Redmond, USA 美国雷德蒙德paul.dietz tactuallabs.com 保罗.迪茨 tactuallabs.com
Figure 1: a. A capacitive ShArc sensor uses a series of flexible strips that are joined together at one end. The outer electrode strips are held a constant distance apart via an elastic sleeve which compresses them against a series of spacers. A circuit measures the relative shift between the electrode layers as the strips are formed into the curves. b. When the strips are bent, the ends no longer align due to the varying radii of curvature. and . Interacting with the prototype and real-time reconstruction of dynamic bends. 图 1：a. 电容 ShArc 传感器使用一系列灵活的条带，在一端连接在一起。外电极条通过弹性套管保持恒定间距，压缩它们靠近一系列间隔块。电路测量电极层之间的相对位移，当条带形成曲线时。b. 当条带弯曲时，由于曲率半径不同，端部不再对齐。 和 。与原型交互并实时重建动态弯曲。
Abstract 摘要
We present ShArc, a precision, geometric measurement technique for building multi-bend/shape sensors. ShArc sensors are made from flexible strips that can be dynamically formed into complex curves in a plane. They measure local curvature by noting the relative shift between the inner and outer layers of the sensor at many points and model shape as a series of connected arcs. Unlike jointed systems where angular errors sum with each joint measured, ShArc sensors do not accumulate angular error as more measurement points are added. This allows for inexpensive, robust sensors that can accurately model curves with multiple bends. To demonstrate the efficacy of this technique, we developed a capacitive ShArc sensor and evaluated its performance. We conclude with examples of how ShArc sensors can be employed in applications like gesture input devices, user interface controllers, human motion tracking and angular measurement of free-form objects. 我们提出了 ShArc，一种用于构建多弯曲/形状传感器的精密几何测量技术。ShArc 传感器由可动态形成平面复杂曲线的柔性条带制成。它们通过注意传感器内外层之间在许多点的相对位移来测量局部曲率，并将形状建模为一系列连接的弧线。与关节系统不同，关节测量时角度误差会随着每个关节的测量而累积，ShArc 传感器在增加更多测量点时不会累积角度误差。这使得可以准确建模具有多个弯曲的曲线的传感器变得廉价且稳健。为了展示这一技术的有效性，我们开发了一种电容 ShArc 传感器并评估了其性能。最后，我们提供了 ShArc 传感器如何应用于手势输入设备、用户界面控制器、人体运动跟踪和自由形态物体的角度测量等应用示例。
-Hardware Emerging interfaces; Sensors and actuators; 硬件 新兴接口；传感器和执行器；
INTRODUCTION 介绍
Recent decades have seen tremendous progress in the development of high accuracy sensors and their low cost, mass production. Much of this has been driven by smartphones which include an impressive array of sensors. Despite these advancements, there are still many things about the physical world that have proven surprisingly difficult to sense with an inexpensive, precision device. We consider the challenging problem of sensing the shape of a dynamically deforming object. 近几十年来，高精度传感器的发展取得了巨大进步，而它们的低成本、大规模生产也得到了提升。其中很大一部分是由智能手机推动的，智能手机包含了令人印象深刻的传感器阵列。尽管取得了这些进展，但仍然有许多关于物理世界的事物，用廉价、精密设备感知起来令人意外地困难。我们考虑了感知动态变形物体形状的具有挑战性的问题。
The desire to understand shape arises in many applications. In robotics, rotary joints are frequently cascaded to allow dexterous, multi-axis motion that must be monitored to be actively controlled. Launching a large rocket has been compared to "pushing on a string", and it requires a detailed understanding of dynamic flexure. Bridges, storage tanks, planes, and many other structures are subject to repeated load cycling, and understanding deformation in these systems can help prevent catastrophe. More germane to the Human-Computer Interaction (HCI) community, our bodies are quite flexible. In medicine and sports performance, it is often important to understand the range and type of motion. Motion capture is critically important to both the gaming and movie industries. In virtual and augmented reality, a real-time understanding of 许多应用程序中都存在理解形状的愿望。在机器人技术中，旋转关节经常级联以实现灵巧的多轴运动，必须监测并主动控制。发射大型火箭被比作“拉绳子”，这需要对动态弯曲有详细的了解。桥梁、储罐、飞机和许多其他结构都会受到重复载荷循环的影响，了解这些系统中的变形可以帮助防止灾难发生。与人机交互（HCI）社区更相关的是，我们的身体非常灵活。在医学和运动表现中，理解运动的范围和类型通常很重要。运动捕捉对游戏和电影行业都至关重要。在虚拟和增强现实中，实时理解
detailed hand pose allows compelling interactions. For performance, musicians and other artists can manipulate shape intuitively to provide expressive control of key systems. 详细的手势可以实现引人入胜的互动。对于表演者来说，音乐家和其他艺术家可以直观地操纵形状，以提供对关键系统的表现控制。
In this work, we present ShArc - a new technique for low cost, precision, dynamic sensing of bends and the detailed shape of curves. ShArc - a portmanteau of "Shift Arc" - employs a stack of flexible strips that can be formed into complex curves in a plane. The sensor measures curvature by noting the relative shift between inner and outer layers of the sensor at many points. For example, as illustrated in Figure 2, if the sensor is wrapped around a cylinder, the inner most strip will be formed into a radius similar to the cylinder, while the outer strip will effectively conform to a slightly larger radius dependent on the spacing between the layers. The outer strip will need more length to cover the same angular extent. If equal length strips are conjoined on one end and bent around a cylinder, the other ends will separate, similar to what happens when the pages of a book are bent. This relative shifting provides a measure of curvature. We measure this relative shift at many locations along the strips, giving an accurate measure of curvature at many points. 在这项工作中，我们提出了 ShArc - 一种用于低成本、精密、动态感应弯曲和曲线详细形状的新技术。ShArc - “Shift Arc”的合成词 - 使用一堆可在平面内形成复杂曲线的柔性条带。传感器通过记录传感器内外层之间的相对位移来测量曲率。例如，如图 2 所示，如果传感器缠绕在圆柱周围，最内层条带将形成类似于圆柱的半径，而外层条带将有效地适应于略大半径，取决于层之间的间距。外层条带需要更多长度来覆盖相同的角度范围。如果等长的条带在一端连接并围绕圆柱弯曲，另一端将分开，类似于书页弯曲时发生的情况。这种相对位移提供了曲率的度量。我们在条带的许多位置测量这种相对位移，从而在许多点准确测量曲率。
ShArc sensors allow for a unique combination of true multibend/shape sensing, robustness to measurement errors, stability, and low cost. In this paper, we present the theory behind ShArc and some of its unique characteristics. We then examine a prototype implementation which uses capacitive sensing and characterize its performance. Finally, we consider some example applications in the HCI domain. ShArc 传感器允许独特的真正多弯曲/形状感知、对测量误差的稳健性、稳定性和低成本的组合。在本文中，我们介绍了 ShArc 背后的理论以及它的一些独特特性。然后，我们检查了一个使用电容感应的原型实现，并对其性能进行了表征。最后，我们考虑了在人机交互领域的一些示例应用。
RELATED WORK 相关工作
In this section, we consider prior work in flexible, selfcontained, bend/shape sensors. We first consider how bend is physically detected and then discuss specific implementations. 在这一部分，我们考虑了在柔性、自包含、弯曲/形状传感器方面的先前工作。我们首先考虑弯曲是如何被物理检测的，然后讨论具体的实现。
Strain Sensing 应变传感
The most common way of detecting flexure is by measuring the changing properties of a material under strain [36]. Strain is a problematic proxy for flexure. Stretching, environmental conditions, and other factors can induce strain that can not be easily distinguished from that due to bending . Continual strain cycles can also cause material fatigue . Most resistive strain sensors have high-latency and are unable to measure the absolute angles of bend. The hysteresis properties of conductive materials produce varying conductivity during cyclic loading [4, 22, 23, 24, 42, 44]. Most resistive and FBG (Fiber Bragg Grating) sensors are non-linear in response to large strains . 检测弯曲最常见的方法是通过测量材料在应变下的变化特性[36]。应变是弯曲的一个问题代理。拉伸、环境条件和其他因素可能引起应变，这些应变很难与由弯曲引起的应变区分开来。持续的应变循环也会导致材料疲劳。大多数电阻应变传感器具有高延迟，无法测量弯曲的绝对角度。导电材料的磁滞特性在循环加载过程中产生不同的导电性。大多数电阻和 FBG（光纤布喇格光栅）传感器对大应变的响应是非线性的。
Geometric Sensing 几何感知
An alternative to strain sensing is what we call geometric sensing. These sensors much more directly measure curvature by sensing geometric changes that are a result of bending. Examples include , which similar to ShArc, measure relative displacements of different sensor layers. ShArc improves on these earlier systems by supporting the direct detection of multiple bends, which we will show to have superior error performance. 一种替代应变传感的方法是我们所称的几何传感。这些传感器通过感知由弯曲引起的几何变化，更直接地测量曲率。例如 ，类似于 ShArc，测量不同传感器层的相对位移。ShArc 通过支持直接检测多次弯曲来改进这些早期系统，我们将展示其具有更优越的误差性能。
While [29] also measures shift via a change in capacitance, it uses an extended electrode pattern. If a sharp curve is introduced in the middle of the pattern, only the later half of the electrode pattern will be shifted. If the same sharp curve is introduced before the pattern, the shift will happen over the entire length, yielding a very different result. The issue with this extended electrode design is that even with a single bend the degree of bending and the placement of the bend are conflated. Our differential electrode design is much more compact and has superior noise and shielding performance. It provides a linear response with shift and is largely immune to misalignment in the direction orthogonal to shift. 虽然[29]也通过电容变化来测量位移，但它使用了扩展电极图案。如果在图案中间引入一个急转弯，那么只有电极图案的后半部分会发生位移。如果在图案之前引入相同的急转弯，位移将发生在整个长度上，产生非常不同的结果。这种扩展电极设计的问题在于，即使只有一个弯曲，弯曲的程度和弯曲的位置也会混淆。我们的差分电极设计更加紧凑，具有更优异的噪音和屏蔽性能。它提供了与位移相关的线性响应，并且在与位移正交方向的错位方面基本上不受影响。
ShArc sensors use multiple, thin spacers providing a more consistent electrode spacing while maintaining high flexibility. Because shift is distributed among many layers, the relative movement between layers is reduced, increasing durability. ShArc 传感器使用多个薄间隔器，提供更一致的电极间距，同时保持高度的灵活性。由于位移分布在许多层之间，层之间的相对运动减少，增加了耐用性。
Multibend Sensors and Error Propagation 多弯曲传感器和误差传播
Most of the prior work uses sensors that give a single measure of bend. In order to sense complex curves, one can employ a series of single bend sensors [11, 21, 38, 40, 43], building a model of connected joints. This works best when the underlying thing to be sensed is well modelled as a series of linkages. However, placement of the sensors requires a priori understanding of the joint locations. For example, when modelling human joints such as a finger, there is significant variation in location from person to person, precluding a general solution . 大多数先前的工作使用传感器提供弯曲的单一测量。为了感知复杂的曲线，可以使用一系列单一弯曲传感器[11, 21, 38, 40, 43]，构建连接关节的模型。当被感知的基础事物被很好地建模为一系列连接时，这种方法效果最好。然而，传感器的放置需要对关节位置有先验的了解。例如，当对人类关节（如手指）进行建模时，人与人之间的位置存在显著差异，这使得一般解决方案不可行。
Complex curves may require a large number of single bend sensors to provide an adequate understanding of shape. Unfortunately, each additional bend sensor contributes measurement error which accumulates to progressively degrade the overall accuracy of the system [41]. This severely limits the maximum number of single bend sensors that can reasonably be employed. As we will show, ShArc sensors overcome this limitation. 复杂曲线可能需要大量的单弯曲传感器来提供对形状的充分理解。不幸的是，每个额外的弯曲传感器都会导致测量误差的累积，逐渐降低系统的整体准确性[41]。这严重限制了可以合理使用的单弯曲传感器的最大数量。正如我们将展示的那样，ShArc 传感器克服了这一限制。
Recently, machine learning approaches have been applied to understanding the output of systems with many single bend sensors . While these systems have the potential to combine the readings from large numbers of single bend sensors such that error does not accumulate in such a direct fashion, they require extensive training. It is also unclear if any reduction in accumulated error comes from imposing constraints that make the system less general. 最近，机器学习方法已被应用于理解具有许多单弯曲传感器的系统的输出 。虽然这些系统有潜力将大量单弯曲传感器的读数结合起来，使误差不会以如此直接的方式累积，但它们需要大量的训练。目前还不清楚是否通过施加使系统变得不太通用的约束来减少累积误差。
Specific Implementations 具体实施
The most common strain-based bend sensors are resistive , optical , 45] including Fiber Bragg Grating (FBG) sensors [23, 44], piezoelectric or capacitive . We consider each of these, and discuss their operation. 基于应变的弯曲传感器中最常见的类型包括电阻式、光学式、包括光纤布拉格光栅（FBG）传感器、压电式或电容式。我们将逐一讨论每种传感器的工作原理。
Resistive bend sensors are similar to resistive strain gauges, but are optimized for much larger bends. A layer of resistive material is placed on a flexible substrate and undergoes strain as the sensor is bent [4]. A bend away from the side with the resistive material causes tensile strain, increasing the resistance. 电阻弯曲传感器类似于电阻应变片，但针对更大的弯曲进行了优化。电阻材料层放置在柔性基底上，并在传感器弯曲时发生应变。远离带有电阻材料的一侧的弯曲会导致拉伸应变，从而增加电阻。
Resistive sensors suffer from significant drift due to fatigue, aging of materials and environmental conditions, and require constant re-calibration to achieve even modest accuracy . Because they provide only a single measure of bend, they can not distinguish shape for complex curves. Although resistive bend sensors have many limitations, they are quite inexpensive and easy to interface to, allowing use in many applications [6, . The best known of these is the Mattel's PowerGlove [39], an early consumer hand pose interface device used for gaming on Nintendo systems. [21, have embedded commercial flex sensors into both soft and rigid materials to create different control interactions like switches or sliders. have used ink jet printing on customized shapes to create game controllers and toys in two and three dimensions. 电阻式传感器由于疲劳、材料老化和环境条件等原因存在显著漂移，需要不断重新校准才能达到即使是适度的准确性。由于它们只提供弯曲的单一测量，无法区分复杂曲线的形状。尽管电阻弯曲传感器有许多局限性，但它们价格相当便宜且易于接口，可用于许多应用。其中最著名的是 Mattel 的 PowerGlove，这是一种早期的消费者手部姿势界面设备，用于在任天堂系统上进行游戏。已经将商用弯曲传感器嵌入软硬材料中，以创建不同的控制交互，如开关或滑块。已经使用喷墨打印在定制形状上创建了二维和三维的游戏控制器和玩具。
Fiber Optic Shape Sensors (FOSS) can be constructed from a flexible tube with reflective interior walls and a light transmitter and receiver. are examples of optical bend sensors used as input devices in the HCI domain. FOSS recover the bend shape by measuring changes in intensity, phase, polarization or wavelength of the light while the flexible tube is bent [3, 23, 24, 44, 42]. Fiber Bragg Grating sensors employ an optical fiber that has been processed to create a grating that interacts with light of a specific wavelength. As the fiber is bent, the grating is mechanically expanded or compressed, which shifts the wavelength of interest. A tunable laser is used to scan for the new wavelength of the deformed grating. Different wavelength grating patterns can be placed at different locations along the fiber, allowing the degree of bending to be independently measured at each location [23, 24]. 光纤形状传感器（FOSS）可以由具有反射内壁和光发射器和接收器的柔性管构建。这些是在人机交互领域中用作输入设备的光学弯曲传感器的示例。FOSS 通过测量柔性管弯曲时光线的强度、相位、偏振或波长的变化来恢复弯曲形状。光纤 Bragg 光栅传感器采用经过加工以与特定波长的光相互作用的光纤。当光纤弯曲时，光栅会被机械地扩展或压缩，从而改变感兴趣的波长。可调谐激光用于扫描变形光栅的新波长。不同波长的光栅图案可以放置在光纤沿不同位置，从而可以在每个位置独立测量弯曲程度。
FOSS can be extremely thin and light weight with little restriction on the length of the sensor [24]. They are relatively precise and immune to electromagnetic inference. While these sensors provide impressive performance, it comes at a high price. A tunable laser interrogator may cost as much as USD - a price that severely limits practical applications. While the fiber can be quite thin, the interrogators tend to be large and power hungry. They require sophisticated signal processing , complex fabrication processes and calibration. They have a restricted range of measurement for curvatures and fall into non-linearity very quickly .These properties limit their use cases to very specific applications like high-end medical devices [23, 24, 28, 31]. FOSS 可以非常薄且轻巧，对传感器长度几乎没有限制[24]。它们相对精确且不受电磁干扰影响。虽然这些传感器提供了令人印象深刻的性能，但代价高昂。一个可调谐激光询问器的成本可能高达 美元 - 这严重限制了实际应用。虽然光纤可以非常薄，但询问器往往体积庞大且耗电量大。它们需要复杂的信号处理 、复杂的制造工艺和校准。它们对曲率的测量范围受限，并且很快进入非线性 。这些特性限制了它们的用途范围，仅适用于高端医疗设备等非常特定的应用[23, 24, 28, 31]。
Piezoelectric bend sensors are based on deformation and strain in piezo materials. Such deformations change the surface charge density of the material and cause charge transfer between the electrodes. The amplitude and frequency of the signal is directly proportional to the applied mechanical stress [25]. Piezoelectric sensors, similar to triboelectric sensors, suffer from drift and only provide signal while in motion. This limits their application to dynamic bending only and not static or low-frequency deformations . 压电弯曲传感器基于压电材料中的变形和应变。这种变形会改变材料的表面电荷密度，并导致电极之间的电荷转移。信号的幅度和频率与施加的机械应力成正比[25]。压电传感器与摩擦电传感器类似，会出现漂移，并且只在运动时提供信号。这限制了它们仅适用于动态弯曲，而不适用于静态或低频变形。
Capacitive bend sensors work either by material strain [1, 2, 13, 32] or displacement between sensor layers [29]. Either way the geometric changes vary the effective overlapping surface areas for capacitive coupling and/or the spacing between conductors as a function of the bending angle. Capacitive sensors can be more linear than other techniques . They are inexpensive to produce and more stable than resistive sensors. Given these desirable properties, we chose to implement a capacitive version of the ShArc measurement technique. 电容弯曲传感器通过材料应变[1, 2, 13, 32]或传感器层之间的位移[29]工作。无论哪种方式，几何变化都会改变电容耦合的有效重叠表面积和/或导体之间的间距，这取决于弯曲角度。电容传感器可能比其他技术更线性。它们生产成本低廉，比电阻传感器更稳定。鉴于这些优点，我们选择实现 ShArc 测量技术的电容版本。
SHARC THEORY OF OPERATION SHARC 操作理论
The basic operation of a ShArc sensor can be understood by imagining a pair of measuring tapes of length , separated by a spacer of thickness as shown in Figure 2. On one end, the strips are joined together, much like the binding of a book. In the flat orientation, it is easy to see that the markings on the two tape measures should align. However, if the pair is formed around a cylinder of radius , the inner tape measure will be formed into a circular arc of radius , while the outer tape measure will be formed into a circular arc of radius . Because they are conjoined on one end, the zero markings of the two tape measures will still align, but the other markings will get progressively misaligned. This is because it takes more tape to subtend the same angle on a larger radius. Assuming that the sensor is formed around a circular arc, we can calculate the radius knowing only the spacing and the relative shift between the tape measures. Relative shift can similarly be measured at many points along the sensor, each allowing us to measure the curvature of successive segments. In this way, we can measure complex curves that are well modelled as a series of circular arcs. ShArc 传感器的基本操作可以通过想象一对长度为 的测量带，之间由厚度为 的间隔器分隔，如图 2 所示来理解。在一端，带子被连接在一起，就像书的装订一样。在平面方向上，很容易看出两个卷尺上的标记应该对齐。然而，如果这对卷尺围绕半径 的圆柱形成，内部卷尺将形成半径 的圆弧，而外部卷尺将形成半径 的圆弧。因为它们在一端连接在一起，两个卷尺的零标记仍然会对齐，但其他标记会逐渐错位。这是因为在较大半径上需要更多的卷尺来张开相同的角度。假设传感器围绕一个圆弧形成，我们可以仅通过间隔和卷尺之间的相对移位来计算半径。相对移位可以在传感器的许多点上类似地测量，每个点都允许我们测量连续段的曲率。 通过这种方式，我们可以测量被很好地建模为一系列圆弧的复杂曲线。
Figure 2: ShArc sensors can be understood by considering two measuring tapes, bound together at one end, and held a fixed distance apart. When flat, the markings align. When bent, the markings become progressively more misaligned. 图 2：通过考虑两条测量带，在一端绑在一起并保持固定距离，可以理解 ShArc 传感器。当平放时，标记对齐。当弯曲时，标记逐渐变得不对齐。
Constructing Arcs 构建弧线
A ShArc sensor, shown in Figure 3, consists of two strips - a reference strip, and a sliding strip - conjoined at one end and held apart by a spacer of thickness . The goal is to acquire the shape of the reference strip. At known intervals along the reference strip, , it can measure the corresponding shifted position, , along the sliding strip. By corresponding, we mean that if you construct a normal to the curve of the reference strip at the measurement point, you measure where it intersects the sliding strip. ShArc 传感器如图 3 所示，由两条带子组成 - 一条参考带和一条滑动带 - 在一端连接在一起，并由 厚度的间隔器分开。目标是获取参考带的形状。在已知的间隔点 沿着参考带，它可以测量对应的滑动带上的移位位置 。所谓对应，是指如果您在测量点处构造参考带曲线的法线，您测量它与滑动带相交的位置。
Figure 3: A ShArc sensor consists of a reference and a sliding strip separated by a constant distance and joined on one end. Measurement points divide the unit into segments. 图 3：ShArc 传感器由一个参考和一个滑动条组成，它们之间由一个恒定距离 分隔，并在一端连接。测量点将该单元分成段。
Single Arc 单弧
We first consider the simple case of a single circular arc segment. As shown in Figure 4, this segment (segment ) is shaped into a circular arc of radius in a counter-clockwise direction. Thus, the reference strip has radius while the sliding strip is inside with a smaller radius of . We define the starting angle , which is the tangent at the beginning of the arc. The ending angle is the tangent to the arc at its end, . 我们首先考虑单个圆弧段的简单情况。如图 4 所示，该段（段 ）呈逆时针方向成圆弧形状，半径为 。因此，参考条的半径为 ，而滑动条在内部，半径较小为 。我们定义起始角 ，即圆弧开始处的切线。结束角是圆弧结束时的切线， 。
Figure 4: Definitions for segment 图 4： 段的定义
We define the length of the reference strip in segment as: 我们将参考条的长度在段 中定义为：
and the length of the corresponding sliding strip as: 对应滑动条的长度为：
Similarly, we define the total subtended angle of this segment as: 同样，我们将这一线段的总夹角定义为：
For the case of positive curvature ( and ), the sliding strip is shaped into a tighter curve than the reference strip. Thus, , even though they subtend the same angle, . 对于正曲率的情况（ 和 ），滑动条的形状比参考条更紧密。因此， ，即使它们所夹角度相同， 。
Working in radians, the length of the reference segment is: 在弧度制中，参考线段的长度是：
and the length of the corresponding sliding segment is: 对应滑动段的长度为：
Given the two lengths and the spacer thickness, we can solve for the radius of curvature of this segment: 给定两个长度和间隔厚度，我们可以解出这段的曲率半径：
This same equation applies when the curve proceeds clockwise, giving a more negative ending angle and negative radius of curvature. 当曲线顺时针前进时，同样的方程也适用，会得到一个更负的终止角和负的曲率半径。
We can also solve for the subtended angle of the arc: 我们也可以解出弧所对的角度：
Multiple Arcs 多个弧
Applying equations 6 and 7 provides a series of circular arcs of known length, angular extent, and radius of curvature. These must be pieced together to model the complete reference strip curve. Unlike jointed systems, ShArc sensors have continuous flexure along their length. They are inherently continuous in their first derivative. To maintain a continuous first derivative from segment to segment, we require that the tangents of adjoining segments match. To put this another way, the ending angle of each segment matches the starting angle of the next segment. 应用方程 6 和 7 提供了一系列已知长度、角度范围和曲率半径的圆弧。这些必须拼接在一起以建模完整的参考条带曲线。与接头系统不同，ShArc 传感器沿其长度具有连续的弯曲。它们在一阶导数上是连续的。为了保持从段到段的连续一阶导数，我们要求相邻段的切线匹配。换句话说，每个段的结束角度与下一个段的起始角度匹配。
Consider a single arc as shown in Figure 5. The arc begins at a known starting point, , and at an initial known angle of and proceeds to an unknown ending point, , at an unknown ending angle of . 考虑如图 5 所示的单弧。弧始于已知起点 ，初始已知角度为 ，并延伸至未知终点 ，终止角度为 。
Figure 5: Definitions for calculating the position of a segment. 图 5：计算部分位置的定义。
The change in angle from starting point to the ending point is just the turning of the segment angle, . 从起点到终点的角度变化只是线段角度的转动， 。
To find the translation, we find the increment in and over the arc and add this to the previous point. For convenience, we imagine that the center of the radius of curvature of the arc is at the origin and calculate the change in endpoint positions. This difference is then applied to the known starting point. 要找到 的翻译，我们找到弧线上 和 的增量，并将其添加到前一个点。为了方便起见，我们假设弧线的曲率半径的中心在原点，并计算端点位置的变化。然后将这种差异应用于已知的起点。
For this calculation, we need to know the angles from the center that form the arc. A normal angle to is . For an arc of positive radius of curvature, this gives the angle pointing out from the center of the radius of curvature. If the radius of curvature is negative, it points the opposite direction. This results in a sign flip that is corrected by using the signed radius of curvature. The endpoints can then be found iteratively via equations 8 and 9 : 对于这个计算，我们需要知道从中心形成弧的角度。到 的正常角度是 。对于正曲率半径的弧，这给出了从曲率半径中心指向外部的角度。如果曲率半径为负，则指向相反方向。这导致了一个符号翻转，通过使用带符号的曲率半径进行校正。然后可以通过方程 8 和 9 迭代地找到端点：
These equations can be slightly simplified using trig identities. 这些方程可以通过三角恒等式稍微简化。
These equations describe the series of circular arcs that model the bend. A circular arc is typically described by its center , its radius of curvature , a starting angle , and an angular extent . The center of an arc segment can be found by starting at (x[n],y[n]), and following the radius to the arc center . The starting angle is found from the normal at the point , which is . The center is then: 这些方程描述了模拟弯曲的圆弧系列。一个圆弧通常由其圆心 、曲率半径 、起始角 和角度范围 描述。圆弧段的中心可以通过从（x[n]，y[n]）开始，并沿半径到圆弧中心 找到。起始角是从点 处的法线找到的，该点为 。然后中心是：
Note that the use of the signed radius of curvature ensures that we are following the normal to the center. 请注意，使用有符号曲率半径可确保我们沿着指向中心的法线前进。
The starting angle is: 起始角度是：
The sign is needed to flip the angle if the arc proceeds clockwise. The extent of the arc is , which is also a signed value. 如果弧线顺时针前进，则需要翻转角度。弧线的范围是 ，也是一个有符号值。
Error Propagation Properties 错误传播特性
On multi-axis robot arms, position is usually determined via a series of encoders - one on each joint. High precision encoders are typically required because any errors in each joint measurement accumulate. For a planar arm, the angular error at the end of the arm is simply the sum of all the measurement errors in each joint. The location error of the endpoint is also wildly impacted by all of the joint errors - particularly the ones at the beginning of the arm. 在多轴机器人手臂上，位置通常是通过一系列编码器确定的 - 每个关节上都有一个。通常需要高精度编码器，因为每个关节测量中的任何错误都会累积。对于平面手臂，末端的角度误差简单地是每个关节中所有测量误差的总和。末端的位置误差也受到所有关节误差的影响 - 特别是在手臂开始部分的误差。
In stark contrast, ShArc sensors have far more benign error propagation properties. These arise from the fact that measurement errors for each arc are NOT independent. 与之形成鲜明对比的是，ShArc 传感器具有更为良性的误差传播特性。这是因为每个弧的测量误差并非独立产生的。
Consider the case of a ShArc sensor with two measurement points. To find the curvature of the first segment, we determine the relative shift at the first measurement point. Let us presume that this measurement is corrupted by noise, and our reading of shift at this point is incorrectly low. Next we measure the relative shift in the second segment. This measurement is made by taking the total shift at the second measurement point, and subtracting off the shift from the first measurement point. The error at the first point will now cause a corresponding error in the second segment that is opposite in sign from the error in the first segment. Thus, the two segments will end up with curvature errors that tend to cancel each other out. In fact, we will show that the error in final angle is completely unimpacted by the error at the first measurement point. 考虑一个具有两个测量点的 ShArc 传感器的情况。为了找到第一段的曲率，我们确定第一个测量点的相对位移。假设这个测量受到噪声的干扰，我们在这一点的位移读数过低。接下来我们测量第二段的相对位移。这个测量是通过在第二个测量点处的总位移减去第一个测量点的位移来进行的。第一个点的误差现在将导致第二段中的相应误差，其符号与第一段中的误差相反。因此，这两段最终会出现曲率误差，这些误差往往会互相抵消。事实上，我们将展示最终角度的误差完全不受第一个测量点的误差影响。
We define the starting point of the curve as: 我们将曲线的起点定义为：
By definition, and . We can now calculate the ending angle of segment 0 : 根据定义， 和 。我们现在可以计算段 0 的结束角度：
Next, we find the ending angle of segment 1 . 接下来，我们找到线段 1 的结束角。
As can be seen, the ending angle calculation has NO dependence on any earlier measurements. This means that any errors in earlier measurements do not contribute error in the ending angle of each segment. 正如所见，结束角度的计算不依赖于任何早期测量。这意味着早期测量中的任何错误都不会对每个部分的结束角度产生误差。
This property of ShArc sensors provides an important advantage over traditional solutions which string together a series of angular encoders. To accurately model a complex curve, many segments will be required. But the more angular encoders one adds to a system, the more angular error there will be. This places a practical limit on the number of encoders that can be strung together. In sharp contrast, ShArc sensors do not incur additional angular error when adding more measurement points. This makes ShArc sensors an ideal fit for measuring highly complex curves that require many segments for accurate modelling. ShArc 传感器的这一特性相对于传统解决方案具有重要优势，传统解决方案是将一系列角度编码器串联在一起。为了准确建模复杂曲线，将需要许多段。但是，系统中添加的角度编码器越多，角度误差就会越大。这对可以串联的编码器数量设置了实际限制。相比之下，ShArc 传感器在增加更多测量点时不会产生额外的角度误差。这使得 ShArc 传感器非常适合测量需要许多段进行准确建模的高度复杂曲线。
It should also be noted that these error propagation characteristics also make ShArc sensors highly robust in the face of local measurement errors. We have found that even relatively noisy measurement data tends to produce overall curves that match reality surprisingly well. 还应该注意到，这些误差传播特性也使得 ShArc 传感器在面对局部测量误差时具有很高的鲁棒性。我们发现，即使是相对嘈杂的测量数据也往往会产生与现实相当匹配的整体曲线。
THE SHARC SENSOR PROTOTYPE 鲨鱼传感器原型
In order to validate the ShArc sensing technique, we constructed the prototype device shown in Figure 1. ShArc sensors measure the relative shift between two flexible layers that are held a fixed distance apart. While there are many ways to craft such a system, we sought a design that would be easy to implement, provide reasonable precision, and allow for continuous flexure over the length of the device. 为了验证 ShArc 感应技术，我们制造了如图 1 所示的原型设备。ShArc 传感器测量两个柔性层之间的相对位移，这两个层保持固定距离。虽然有许多制作这样系统的方法，但我们寻求一种易于实现、提供合理精度并允许设备长度上连续弯曲的设计。
Top View: 顶视图:
Sliding Strip 滑动条
Reference Strip 参考条
Spacer Strip 间隔条
Side View: 侧视图:
Figure 6: Detailed layout of the sensor strips - units are in millimeters. 图 6：传感器条的详细布局 - 单位为毫米。
Our design was inspired by digital calipers that employ capacitive sensing to determine position. Baxter describes the widely adopted technique wherein a pattern of transmit electrodes moves along a corresponding pattern of receive electrodes. Position is determined by examining the change in coupling capacitance between the transmit and receive electrodes. 我们的设计灵感来自使用电容传感来确定位置的数字卡尺。 Baxter 描述了被广泛采用的技术，其中一组传输电极的图案沿着一组接收电极的相应图案移动。通过检查传输电极和接收电极之间耦合电容的变化来确定位置。
Our prototype device leverages standard flexible printed circuit board technology to create transmit and receive strips with precise electrode patterns. These are shown in Figure 6. The transmit strip has 8 equally spaced electrodes which align with 8 differential electrode pairs on the receive strip. When the strips are flat, each transmit electrode will be centered over a receive pair such that the differential capacitance is zero. As the two strips shift relative to each other, the transmit pads will move out of alignment with the receive pads, unbalancing the differential capacitance. The electrodes have been designed to have significant overlap to minimize the impact of skew and fringing fields, giving a linear change in differential capacitance with respect to shift. 我们的原型设备利用标准的柔性印刷电路板技术，创建具有精确电极图案的传输和接收条。这些显示在图 6 中。传输条具有 8 个等间距电极，与接收条上的 8 个差分电极对齐。当条是平的时候，每个传输电极将位于接收对的正中央，使差分电容为零。当两条相对移动时，传输垫将与接收垫失去对齐，使差分电容失衡。电极经过设计，具有显著的重叠，以最小化偏移和边缘场的影响，从而使差分电容随着移动而线性变化。
To keep the transmit and receive pads at a fixed spacing, we interpose a series of polyimide strips. The amount of shift is proportional to the thickness of the spacing, so we choose a reasonable value of . While this spacing could have been achieved with a single spacer, it would result in a fairly stiff sensor. More importantly, as the bending radius starts to approach the thickness of the spacer, one would expect significant stress deformations, creating uneven spacing. To solve this issue, we use 5 layers of spacers, which yields a device that is quite pliable while maintaining accurate spacing. Another advantage of using many thin spacers is that the shifting is spread among the layers. The relative shift between any two layers is extremely small, minimizing surface wear. 为了保持传输和接收垫片之间的固定间距，我们插入了一系列聚酰亚胺条。偏移量与间距的厚度成正比，因此我们选择了一个合理的值 。虽然这种间距可以通过单个 间隔器实现，但会导致传感器相当僵硬。更重要的是，当弯曲半径开始接近间隔器的厚度时，人们会预期出现显著的应力变形，导致间距不均匀。为了解决这个问题，我们使用了 5 层 间隔器，这样可以得到一个非常柔软且保持准确间距的装置。使用许多薄间隔器的另一个优点是，移位分布在各层之间。任意两层之间的相对移位非常小，最大程度地减少了表面磨损。
The strips are held pressed together via a Spandex sleeve, while still allowing them to shift against each other along the length. A clamp passes through alignment holes on the strips to constrain motion on that end. Gold finger contacts allow the strips to be insert into connectors on opposite sides of the controller board. 这些条带通过弹性袖套紧密压在一起，同时允许它们沿着长度相互移动。夹子穿过条带上的对准孔，以限制末端的运动。金手指接触点允许将条带插入控制器板的相对侧连接器中。
Figure 7 shows how the electrodes shift as the sensor is flexed. The transmit pads are shown in green, and the differential receive pads are shown in blue and red respectively. When flat, the transmit pads are centered under the receive pads. If the sensor is formed into a circular arc, the pads become increasingly misaligned. 图 7 显示了传感器弯曲时电极的移位情况。发射垫片显示为绿色，差分接收垫片分别显示为蓝色和红色。当传感器平放时，发射垫片位于接收垫片下方中心。如果传感器形成圆弧形，垫片会逐渐错位。
We use a single channel, 24-bit differential capacitance to digital converter (Analog Devices AD7745/AD7746) to perform the capacitance measurements. Using a series of ultra-low capacitance multiplexers (Texas Instruments TMUX1511), we can successively measure shift at 8 points along our strips. The capacitances we are measuring are sub-pico farad, and there are substantial parasitic capacitances due to the proximity of various traces. When the sensor is laid flat, we measure the static impact of these parasitic capacitances and subtract this value off of later readings to find the differential capacitance due to the electrodes. 我们使用单通道、24 位差分电容数字转换器（模拟设备 AD7745/AD7746）来进行电容测量。通过使用一系列超低电容多路复用器（德州仪器 TMUX1511），我们可以在我们的条带上的 8 个点上连续测量位移。我们正在测量的电容是亚皮可法拉德，由于各种线路的接近，存在大量寄生电容。当传感器放平时，我们测量这些寄生电容的静态影响，并从后续读数中减去这个值，以找到由电极引起的差分电容。
The current circuit has not been optimized for speed or power. It can do a full sweep of the sensor about 10 times per second, while drawing less than . Both of these specifications could be substantially improved with modest effort. 当前电路并未针对速度或功耗进行优化。它可以每秒大约进行 10 次传感器的完整扫描，同时功耗低于 。这两个规格都可以通过适度的努力大幅改进。
EVALUATION AND RESULT 评估和结果
In this section, we consider both the theoretical and actual performance of the sensor. 在本节中，我们考虑传感器的理论性能和实际性能。
We used a simple parallel plate model to calculate the theoretical change in differential capacitance as a function of shift. Assuming a dielectric constant of 3.5 for polyimide, our geometry should yield a sensitivity of of shift. 我们使用简单的平行板模型来计算不同位移下的理论电容变化。假设聚酰亚胺的介电常数为 3.5，我们的几何结构应该产生一个位移的灵敏度。
As shown in Figure 7, the transmit pads are designed to lay centered on the corresponding receive pad pairs when the sensor is laid flat. As the sensor is bent, the receive and transmit pads misalign. As shown in Figure 7c, the shift can only go before the transmit pad extends beyond the corresponding receive pads. This limits the maximum bend 如图 7 所示，当传感器放平时，传输垫设计为位于相应的接收垫对中心。当传感器弯曲时，接收和传输垫会错位。如图 7c 所示，在传输垫超出相应的接收垫之前，偏移只能达到 。这限制了最大弯曲。
Figure 7: The relationship between the relative shift and differential capacitance. a. Transmit electrodes are centered between two receive electrodes when sensor is laying flat, giving zero differential capacitance. b. When sensor is bent, the transmit electrodes overlap one receive electrode more than the other, creating a non-zero differential capacitance. The change in capacitance indicates the degree of shift. C. A top and side view of the overlap between transmit and receive electrodes when the differential capacitance is minimum and maximum. 图 7：相对位移与差分电容之间的关系。a. 当传感器平放时，传输电极位于两个接收电极之间，产生零差分电容。b. 当传感器弯曲时，传输电极与一个接收电极的重叠程度大于另一个，产生非零差分电容。电容的变化表示位移的程度。c. 传输和接收电极之间重叠的顶视图和侧视图，差分电容最小和最大时。
that the sensor can measure. As we have shown, the shift at any point is only a function of the ending angle. For any segment : 传感器可以测量的。正如我们所展示的，任何点的偏移量仅取决于结束角度。对于任何段 ：
With a shift of and a thickness of , our maximum ending angle is 6 radians, or about . The ending angle at any measurement point should not exceed this, and to ensure good linearity, needs to be somewhat less. If more range is required, the spacer layer can be decreased, creating less shift for a given curvature. The trade off is that this creates a corresponding loss of resolution. Alternatively, the electrodes can also be designed to accommodate more shift. 通过 的偏移和 的厚度，我们的最大结束角度为 6 弧度，约为 。任何测量点的结束角度不应超过此值，并且为了确保良好的线性度，需要略微减少。如果需要更大范围，可以减小间隔层，从而减少给定曲率的偏移量。这样做的代价是相应地降低分辨率。另外，电极也可以设计成能容纳更大的偏移量。
It should be noted that this constraint does not limit the number of bends. For example, if the sensor was formed into a sinusoid, the shift would cyclically rise and fall, returning to zero at the end of each cycle. (If the amplitude was high enough, our maximum ending angle could be exceeded at some points, but this is not dependent on the number of bends.) However, if the sensor is formed into a single circular arc as in Figure 7a, the shift continues to linearly accumulate. If the sensor is wrapped too tightly, it will exceed the allowable range. 应注意，这种约束并不限制弯曲的次数。例如，如果传感器形成正弦曲线，偏移量会周期性地上升和下降，在每个周期结束时返回零。 （如果振幅足够大，我们的最大结束角度可能会在某些点超过，但这并不取决于弯曲的次数。）然而，如果传感器形成单个圆弧，如图 7a 所示，偏移量将继续线性累积。 如果传感器卷绕得太紧，它将超出允许的范围。
If the maximum angle is grossly exceeded, the transmit pad may completely move out of range of the receive pads, giving a differential capacitance of zero, which will report as flat. Interestingly, if the max angle is exceeded, but a compensating bend leaves the end of the strip less than the max angle, those later angles in the end piece will report correctly. 如果最大角度明显超过，传输垫可能会完全移出接收垫的范围，导致差分电容为零，这将报告为平坦。有趣的是，如果超过最大角度，但是补偿弯曲使条带的末端小于最大角度，那么末端的后续角度将会正确报告。
There is a numerical issue with using bend radius to describe essentially flat curves, where the bend radius goes to infinity. The sensor works quite well in reporting a flat curve, and there is no upper limit on radius of curvature. Because any sufficiently large radius is essentially flat, it makes little sense to consider the absolute radius error in these cases. This is why we chose to limit our measurement to a radius of . 使用弯曲半径来描述基本平坦曲线时存在一个数字问题，其中弯曲半径趋近于无穷大。传感器在报告平坦曲线时表现得相当好，并且曲率半径没有上限。因为任何足够大的半径基本上都是平坦的，所以在这些情况下考虑绝对半径误差几乎没有意义。这就是为什么我们选择将我们的测量限制在半径为 的原因。
If a curve can not be adequately described by the number of arcs available (e.g. when the number of bends exceeds the number of arcs), the failure mode of the sensor is relatively benign. So long as the sensor is still physically intact (i.e. that the strips are still held at a constant distance apart), the shift at any point will still reflect the total angle at that point, and this will be read correctly at each measurement point. This means that while some detail is lost, the understanding of overall low-spatial frequency shape will typically be pretty good, but could have confusing artifacts. It is fairly analogous to what happens when a waveform is sampled at less than the Nyquist rate. 如果一条曲线无法通过可用的弧数充分描述（例如，当弯曲次数超过弧数时），传感器的失效模式相对温和。只要传感器仍然在物理上完好（即带仍然保持在恒定距离之间），任何点的偏移仍将反映该点的总角度，并且这将在每个测量点正确读取。这意味着虽然会丢失一些细节，但对整体低空间频率形状的理解通常会相当不错，但可能会产生混淆的人为现象。这在某种程度上类似于以低于奈奎斯特率对波形进行采样时发生的情况。
To test the performance of our device, we created a number of circular arc test forms of known radii (Figure 9-a) onto which the sensor could be placed. We chose to test a range of radii from to , which given the length of our sensor, covers a reasonable range. 为了测试我们设备的性能，我们创建了一些已知半径的圆弧测试形式（图 9-a），传感器可以放置在上面。我们选择测试从 到 的一系列半径，考虑到我们传感器的长度，这涵盖了一个合理的范围。
We repeatedly placed the sensor onto the test forms (Figure 9b), collected data, and calculated the radius for each segment. The results are shown in Table 1 and Figure 10. There are a number of things to note about this data. First, the sensor did an excellent job of estimating the radii for values below , staying within of the true value. While the error appears to increase for very large radii, it should be understood 我们反复将传感器放置在测试表格上（图 9b），收集数据，并计算每个部分的半径。结果显示在表 1 和图 10 中。关于这些数据有几点需要注意。首先，传感器在估计小于 的半径时表现出色，保持在真实值的 范围内。虽然对于非常大的半径，误差似乎会增加，但应该理解
that these curves are almost flat, with very slight variations in curvature (normally defined as ). So these seemingly larger errors actually have little impact on the accuracy of the curve reconstruction. 这些曲线几乎是平坦的，曲率变化非常微小（通常定义为 ）。因此，这些看似较大的误差实际上对曲线重建的准确性影响不大。
Figure 8 shows example curve reconstructions when the sensor is placed on forms with radii of curvature ranging from to along with their ideal curves. Despite the presence of measurement errors, the curves closely track the ideal values. This is largely due to the error propagation properties of ShArc sensors. Looking closely at the graphs you can see that when a measured curve starts to drift off of the ideal, later segments help pull it back on. Again, this is in sharp contrast to systems that independently encode a series of joints. 图 8 显示了传感器放置在曲率半径范围从 到 的形式上的示例曲线重建，以及它们的理想曲线。尽管存在测量误差，曲线与理想值非常接近。这在很大程度上归因于 ShArc 传感器的误差传播特性。仔细观察图表，您会发现当测量曲线开始偏离理想曲线时，后续部分会帮助将其拉回。再次强调，这与独立编码一系列关节的系统形成鲜明对比。
Figure 8: Example sensor data taken on forms ranging in radii from 9 to . The dashed lines represent the ideal curves. 图 8：以半径从 9 到 变化的形式采集的传感器数据示例。虚线代表理想曲线。
Sources of Error 错误来源
In order to improve the sensor, it is important to understand what the major sources of error are in the system. 为了改进传感器，重要的是了解系统中的主要误差来源是什么。
Systematic sources of error: 系统误差来源:
As noted previously, even when the transmit and receive pads are aligned in the flat orientation, there are parasitic capacitances due to the asymmetric layout that cause a non-zero differential capacitance. Since these are static, they can be measured in the flat position, and then subtracted off of all future measurements. 正如前面所指出的，即使在传输和接收垫在平面方向上对齐时，由于不对称布局而导致寄生电容，会产生非零差分电容。由于这些是静态的，可以在平面位置上进行测量，然后从所有未来的测量中减去。
Our understanding of permativity and spacer thickness are relatively imprecise. Sheet material is typically specified with a thickness tolerance. There may also be small errors in the sizing of the electrodes and the gain of the converter. All of these factors result in a change in the sensitivity, and can be compensated for with a single constant. In practice, we find this constant by adjusting it until there is a good fit to a known curve. 我们对介电常数和间隔厚度的理解相对不够精确。板材通常规定有 的厚度公差。电极尺寸和转换器增益可能也存在小误差。所有这些因素导致了灵敏度的变化，可以通过一个恒定值进行补偿。在实践中，我们通过调整这个常数直到与已知曲线拟合良好来确定这个常数。
While we ignored fringing fields in our analysis, they are a significant source of non-linearity, particularly when approaching the shift limit. It would be straightforward to characterize this non-linearity and compensate for it. 在我们的分析中忽略了边缘场，但它们是非线性的重要来源，特别是在接近移位极限时。很容易对这种非线性进行表征并进行补偿。
Random sources of error: 随机误差来源:
We observed two major types of random errors in system. 我们观察到系统中存在两种主要类型的随机错误。
参考半径（毫米）
Reference
Radii(mm)
构造半径
Constructed
Radii
Error (%) 错误 (%)
STD
CV
40
40.23
-0.58
0.22
-0.39
50
49.85
0.29
0.10
0.35
60
60.82
-1.37
0.15
-0.12
70
70.76
-1.07
0.03
-0.03
80
79.01
1.25
0.13
0.11
90
89.90
0.15
0.30
1.97
100
99.08
1.11
0.55
0.49
125
126.16
-0.92
0.27
-0.23
150
147.43
1.80
0.42
0.24
175
175.35
-0.17
0.18
-1.03
200
201.58
-0.60
0.35
-0.59
300
285.03
4.98
0.73
0.15
400
380.83
4.79
0.56
0.12
500
404.60
19.07
1.04
0.05
600
453.69
24.38
0.46
0.02
Table 1: Measurement Results 表 1：测量结果
Electrical noise limits the practical resolution of the capacitance to digital converter. More significantly, we saw issues with mechanical repeatability. 电气噪声限制了电容到数字转换器的实际分辨率。更重要的是，我们发现了机械重复性问题。
To investigate electrical noise, we conducted a simple experiment in which we laid the sensor flat and read the output for 500 counts. The maximum observed deviation for all 8 channels was . We then lifted the sensor, and returned it to the flat position five times. Under these conditions we observed a maximum change of - about an order of magnitude worse than the electrical noise floor. We are investigating the cause of this lack of mechanical repeatability, and suspect that it is largely due to insufficient restoring force being provided by the elastic sleeve to keep the layers tightly pressed together. 为了调查电噪声，我们进行了一个简单的实验，将传感器放平并读取 500 个计数的输出。所有 8 个通道的最大观察偏差为 。然后我们将传感器抬起，并将其放回平放位置五次。在这些条件下，我们观察到最大变化为 ，大约比电噪声底线差一个数量级。我们正在调查这种机械重复性不足的原因，并怀疑这主要是由于弹性套筒提供的恢复力不足，无法使层紧密压在一起。
Calibration 校准
The relationship between curvature and differential capacitance is determined by geometry and the dielectric constant of the spacer. Given the accuracy of the electrode patterns, most of the variation from device to device is caused by the small changes in spacer material thickness. We have found that this variation is small enough as to be negligible for most applications, and that a single constant can be used for all devices of the same design. The parasitic capacitances are more impacted by the precise placement of components on the PC board, and since these are currently assembled by hand, it makes sense to subtract off a unique baseline for each device. So no per device calibration is required - just the baseline subtraction. 曲率与微分电容之间的关系由几何形状和间隔层的介电常数确定。鉴于电极图案的准确性，从一个设备到另一个设备的大部分变化是由于间隔层材料厚度的微小变化引起的。我们发现这种变化足够小，以至于对于大多数应用来说可以忽略不计，并且可以为所有相同设计的设备使用单个常数。寄生电容更受 PC 板上组件的精确放置影响，由于这些组件目前是手工组装的，因此逐个设备减去独特的基线是合理的。因此，不需要每个设备的校准 - 只需进行基线减法。
For increased accuracy, one could do a more detailed measurement of the change in capacitance with curvature, and model the small non-linearity due to fringing fields. We did not do this because the device was surprisingly accurate without this step. But we could imagine that demanding applications might benefit from a model of this non-ideality. 为了提高准确性，可以对电容变化进行更详细的测量，并对由于边缘场引起的小非线性进行建模。我们没有这样做，因为该设备在没有这一步骤的情况下表现出令人惊讶的准确性。但我们可以想象，苛刻的应用可能会受益于对这种非理想性的建模。
APPLICATIONS AND INTERACTION TECHNIQUES 应用和交互技术
Gesture Input Devices 手势输入设备
A common application for resistive bend sensors has been finger tracking on glove input devices. Having only a measurement of gross flexure, these provided a crude estimate of finger position. Even that requires a model of finger structure 电阻性弯曲传感器的常见应用是手套输入设备上的手指跟踪。只有粗略弯曲的测量，这些传感器提供了手指位置的粗略估计。即使如此，也需要手指结构的模型。
Figure 9: a. Calibration frames with reference radii from 4.0 to . Placing the sensor on to the test forms to measure the known radii. 图 9：a. 校准框架，参考半径从 4.0 到 。将传感器放在测试形式上，测量已知半径。
Figure 10: Top: Constructed vs reference radii in green line and the ideal characteristic curve in blue dashed line. The red dots show the variation of measured data, arises from random noise. Bottom: Measurement errors 图 10：顶部：绿线中的构建与参考半径 以及蓝色虚线中的理想特征曲线。红点显示由随机噪声引起的测量数据变化。底部：测量误差
Figure 11: ShArc sensor as a gesture input device, allows tracking joints in the wrist and index finger. and specific joint locations which vary dramatically among different users. ShArc sensors can overcome these limitations. 图 11：ShArc 传感器作为手势输入设备，允许跟踪手腕和食指的关节，以及在不同用户之间差异巨大的特定关节位置。ShArc 传感器可以克服这些限制。
To demonstrate this, we attached the prototype sensor to a user's wrist and index finger as shown in Figure 11. We then used the resulting data to drive a Unity model, updating joint positions in real-time. The sensor reports an 8 segment shape model over its length, which is adequate resolution for mapping the deformation data onto the wrist and finger joint positions. This provides accurate and continuous motion tracking, enabling in-air gesture control of user interfaces for navigation, selection, hover, pressing, scrolling, etc. 为了证明这一点，我们将原型传感器连接到用户的手腕和食指上，如图 11 所示。然后，我们使用得到的数据驱动一个 Unity 模型，实时更新关节位置。传感器在其长度上报告一个 8 段形状模型，这对于将变形数据映射到手腕和手指关节位置是足够的分辨率。这提供了准确和连续的运动跟踪，使用户可以通过手势控制界面进行导航、选择、悬停、按压、滚动等操作。
Posture Monitoring 姿势监测
ShArc sensors are ideal for many health applications where a detailed understanding of body motion is desired. Because they are precise, low-cost, low power, light weight and slim in form factor, they can easily be incorporated into patient wearable systems. ShArc 传感器非常适用于许多健康应用场景，需要对身体运动进行详细了解。由于它们精确、低成本、低功耗、重量轻、体积小，因此可以轻松地集成到患者可穿戴系统中。
As a simple demonstration, we constructed a long ShArc sensor, shown in Figure 12, about half a meter long, wide and about thick. The electrode patterns are the same as used in the other prototype, but spread out to cover the longer distance. We attached the sensor to a compression garment using Velcro strips so that it tracks the motion of the spine. Again, we implemented a simple posture monitoring application in Unity (Figure 13) to visualize the range of motion of the spine. 作为一个简单的演示，我们制作了一个长的 ShArc 传感器，如图 12 所示，大约半米长， 宽，厚约 。电极图案与其他原型中使用的相同，但展开以覆盖更长的距离。我们使用尼龙贴条将传感器固定在压缩服上，以便跟踪脊柱的运动。同样，我们在 Unity 中实现了一个简单的姿势监测应用程序（图 13），以可视化脊柱的运动范围。
Figure 12: ShArc sensor built in two different size 图 12：ShArc 传感器内置两种不同尺寸
We discovered several issues in using this second prototype for skeletal tracking. First, with only 8 segments in our model there is not enough resolution to track all of the vertebrae independently. Second, our Unity application did not have a detailed spine model limiting the accuracy of our model. Third, better integration of the sensor into a wearable form is required. A smaller, wireless version would be vastly preferred than the needlessly bulky, tethered prototype. 我们在使用这个第二个原型进行骨骼跟踪时发现了几个问题。首先，在我们的模型中只有 8 个部分，分辨率不足以独立跟踪所有椎骨。其次，我们的 Unity 应用程序没有详细的脊柱模型，限制了我们模型的准确性。第三，需要更好地将传感器集成到可穿戴形式中。一个更小、无线的版本比那种不必要笨重的、有线的原型更受欢迎。
Figure 13: Using a ShArc sensor to track spine curvature 图 13：使用 ShArc 传感器跟踪脊柱曲度
Angular Ruler 角尺
While it is easy to measure rectilinear objects, many aesthetically designed things feature free-form curves which are difficult to characterize. ShArc can act as a simple angular ruler which reports the precise shape of such geometries. As Figure 14 shows, we have designed an interactive user interface so that a user can click on any point along the curve and collect detailed information about the radii of the curvatures and the bend degree at that specific point. 尽管测量直线物体很容易，但许多审美设计的物品具有自由曲线，这些曲线很难描述。ShArc 可以作为一个简单的角度标尺，报告这些几何形状的精确形状。如图 14 所示，我们设计了一个交互式用户界面，用户可以单击曲线上的任意点并收集关于该特定点的曲率半径和弯曲度的详细信息。
Figure 14: A ShArc sensor used as an angular ruler. An interactive user interface allows real time collection of curve data. 图 14：将 ShArc 传感器用作角度尺。交互式用户界面允许实时收集曲线数据。
DISCUSSION 讨论
When we first conceived the idea of measuring relative shift between two strips to measure complex curves, we recognized that measurement error on one segment would cause a compensating error on the other direction on the next. As we have shown, both theoretically and in practice, this allows for excellent performance, even in the face of significant measurement errors. This is a startling result and it is why ShArc techniques should be preferred over joint encoding for curves with significant complexity. 当我们最初构想测量两条带之间的相对移位来测量复杂曲线的想法时，我们意识到一个段的测量误差会导致下一个方向上的补偿误差。正如我们已经理论上和实践中展示的那样，这使得即使在面对显著的测量误差时，也能实现出色的性能。这是一个惊人的结果，这就是为什么在处理具有显著复杂性的曲线时，应该优先选择 ShArc 技术而不是联合编码。
Our capacitive prototype demonstrates that sensors with good performance can be easily produced. However, this is clearly a first generation implementation, with much room for improvement in the future. 我们的电容原型证明了 传感器可以轻松生产且性能良好。然而，这显然是第一代实现，在未来有很大的改进空间。
Mechanical Issues: 机械问题:
Mechanical repeatability was seen to be a major cause of error. When replaced on the same form, we would often see change in segment measurements from the prior measurement (even if the overall curve fit was good). We note that the elastic sleeve mostly provides force on the edges of the strips rather than the faces. Very small changes in layer separation can cause significant errors, effectively scaling the differential capacitance. An improved mechanical design should address this issue. This can also be addressed electrically by switching from a differential capacitance measurement to a ratiometric one. 机械重复性被视为错误的主要原因。当在相同形式上进行更换时，我们经常会看到段测量值与先前测量值有所变化（即使整体曲线拟合良好）。我们注意到弹性套筒主要对条带的边缘施加力，而不是对面。层间距的微小变化可能导致显著误差，有效地扩大差分电容。改进的机械设计应该解决这个问题。这也可以通过从差分电容测量切换到比值测量来在电气上解决。
No special effort was made to use materials with high flatness. Thin sheets are often specified with thickness variation. We made no effort to calibrate for thickness variations along the spacer strips. 没有特别努力使用平整度高的材料。薄板通常规定 厚度变化。我们没有努力校准间隔条上的厚度变化。
Modelling Issues: 建模问题:
As presented, ShArc sensors describe curves using a series of circular arcs. However, not all curves are well modelled by a small number of arcs. Higher order models may provide significant advantages for some geometries. 如所示，ShArc 传感器使用一系列圆弧描述曲线。然而，并非所有曲线都能用少量圆弧很好地建模。对于某些几何形状，更高阶模型可能提供显著优势。
Our implementation uses time division multiplexing, making each shift measurement serially in time. Thus, a single curve is constructed from data taken at different times. This causes a dynamic distortion. In future versions, we could time align the data in software, or preferably, use a circuit that does true simultaneous measurement at all points. 我们的实现使用时分复用，在时间上依次进行每次移位测量。因此，从不同时间获取的数据构建了单个曲线。这会导致动态失真。在未来版本中，我们可以在软件中对齐数据的时间，或者更好地使用一个可以在所有点上进行真正同时测量的电路。
Shielding: 屏蔽:
Proper shielding of the electrodes will improve tolerance to handling and the proximity of nearby conductors. While electrical noise has not limited current performance, it may in the future. Shielding will help. 适当屏蔽电极将提高对操作和附近导体的容忍度。虽然电气噪音目前并未限制电流性能，但将来可能会。屏蔽将有所帮助。
Extension to multiple dimensions: 扩展到多个维度：
The current ShArc sensor measures the bend in one plane. However, the underlying technique of measuring relative shifts can be extended to three dimensions. 当前的 ShArc 传感器测量一个平面上的弯曲。然而，测量相对位移的基本技术可以扩展到三维。
CONCLUSION 结论
In this paper, we introduced , a precision, geometric measurement technique to sense the shape of a dynamically deforming object. ShArc devices are able to sense complex curves which are modelled as a series of connected, circular arcs. We outlined the operating principle of measuring relative shift between inner and outer layers of the sensor at many points and showed the theoretical tolerance to measurement errors. A practical capacitive implementation was described, and its performance characterized. Compared to traditional bend sensors, ShArc sensors are inexpensive, precise and do not suffer from drift of strain characteristics. This makes them ideal for a number of applications. 在本文中，我们介绍了 ，一种精密的几何测量技术，用于感知动态变形物体的形状。ShArc 设备能够感知被建模为一系列相连的圆弧的复杂曲线。我们概述了测量传感器内外层之间相对位移的操作原理，并展示了对测量误差的理论容忍度。描述了一种实用的电容实现，并对其性能进行了表征。与传统的弯曲传感器相比，ShArc 传感器价格低廉、精确，并且不会受到应变特性漂移的影响。这使它们非常适用于许多应用场景。
ACKNOWLEDGEMENT 确认
We would like to thank all the reviewers for their detailed feedback. We also thank Steven Sanders and our colleagues at Tactual Labs for their support of this project. Special thanks go to Cathy Dietz, who created the sensor sleeves and Alex Dietz for lending his voice to the accompanying video. 我们要感谢所有审阅者提供的详细反馈。我们还要感谢 Steven Sanders 和我们在 Tactual Labs 的同事们对这个项目的支持。特别感谢 Cathy Dietz 制作传感器袖子，以及 Alex Dietz 为配套视频提供配音。
REFERENCES 参考资料
[1] J Aebersold, K Walsh, M Crain, M Voor, M Martin, W Hnat, J Lin, D Jackson, and J Naber. 2006. Design, modeling, fabrication and testing of a MEMS capacitive bending strain sensor. In Journal of Physics: Conference Series, Vol. 34. IOP Publishing, 124. J Aebersold, K Walsh, M Crain, M Voor, M Martin, W Hnat, J Lin, D Jackson, 和 J Naber。2006 年。MEMS 电容弯曲应变传感器的设计、建模、制造和测试。在《物理学杂志：会议系列》中，第 34 卷。IOP 出版社，124。
[2] Julia W Aebersold, Kevin Walsh, Mark Crain, and Michael Voor. 2010. MEMS capacitive bending and axial strain sensor. (Dec. 21 2010). US Patent 7,854,174. 朱莉娅·W·艾伯索尔德（Julia W Aebersold）、凯文·沃尔什（Kevin Walsh）、马克·克雷恩（Mark Crain）和迈克尔·沃尔。2010 年。MEMS 电容弯曲和轴向应变传感器。 （2010 年 12 月 21 日）。美国专利 7,854,174。
[3] Moe Amanzadeh, Saiied M Aminossadati, Mehmet S Kizil, and Aleksandar D Rakić. 2018. Recent developments in fibre optic shape sensing. Measurement 128 (2018), 119-137. Moe Amanzadeh，Saiied M Aminossadati，Mehmet S Kizil 和 Aleksandar D Rakić。2018。光纤形状传感的最新发展。《测量》2018 年第 128 期，119-137。
[4] Morteza Amjadi, Ki-Uk Kyung, Inkyu Park, and Metin Sitti. 2016. Stretchable, skin-mountable, and wearable strain sensors and their potential applications: a review. Advanced Functional Materials 26, 11 (2016), . Morteza Amjadi，Ki-Uk Kyung，Inkyu Park 和 Metin Sitti。2016。可伸缩、可贴合皮肤和可穿戴应变传感器及其潜在应用：综述。《先进功能材料》26，11（2016 年）， 。
[5] Asli Atalay, Vanessa Sanchez, Ozgur Atalay, Daniel M Vogt, Florian Haufe, Robert J Wood, and Conor J Walsh. 2017. Batch fabrication of customizable silicone-textile composite capacitive strain sensors for human motion tracking. Advanced Materials Technologies 2, 9 (2017), 1700136 阿斯利·阿塔拉伊（Asli Atalay），瓦内萨·桑切斯（Vanessa Sanchez），奥兹古尔·阿塔拉伊（Ozgur Atalay），丹尼尔 M·沃格特（Daniel M Vogt），弗洛里安·豪夫（Florian Haufe），罗伯特 J·伍德（Robert J Wood）和康纳 J·沃尔什（Conor J Walsh）。2017 年。用于人体运动跟踪的可定制硅-纺织复合电容应变传感器的批量制造。《先进材料技术》2，9（2017），1700136
[6] Moritz Bächer, Benjamin Hepp, Fabrizio Pece, Paul G Kry, Bernd Bickel, Bernhard Thomaszewski, and Otmar Hilliges. 2016. Defsense: Computational design of customized deformable input devices. In Proceedings of the 2016 CHI Conference on Human Factors in Computing Systems. ACM, 3806-3816. [6] Moritz Bächer, Benjamin Hepp, Fabrizio Pece, Paul G Kry, Bernd Bickel, Bernhard Thomaszewski, and Otmar Hilliges. 2016. Defsense: 计算设计定制可变形输入设备。在 2016 年人机交互计算系统 CHI 会议论文集中的论文。ACM，3806-3816。
[7] Ravin Balakrishnan, George Fitzmaurice, Gordon Kurtenbach, and Karan Singh. 1999. Exploring interactive curve and surface manipulation using a bend and twist sensitive input strip. SI3D 99 (1999), 111-118. Ravin Balakrishnan、George Fitzmaurice、Gordon Kurtenbach 和 Karan Singh。1999 年。使用弯曲和扭曲敏感输入条探索交互式曲线和曲面操作。SI3D 99（1999），111-118。
[8] Larry K Baxter. 1997. Capacitive sensors. Design and Applications (1997). [8] Larry K Baxter. 1997. 电容传感器。设计与应用（1997）。
[9] Larry K Baxter and Robert J Buehler. 1986. Capacitive displacement measuring instrument. (May 6 1986). US Patent 4,586,260. Larry K Baxter 和 Robert J Buehler。1986 年。电容位移测量仪器。 （1986 年 5 月 6 日）。美国专利 4,586,260。
[10] Shaohua Chen, Zhengrong Tong, Qida Zhao, Zhiguo Liu, and Xiaoyi Dong. 2004. A smart bending sensor with a novel temperature-and strain-insensitive long-period grating. Sensors and actuators A: Physical 116,1 (2004), 103-106. 陈少华，童正荣，赵启达，刘志国和董晓义。2004 年。一种具有新型温度和应变不敏感的长周期光栅的智能弯曲传感器。传感器和执行器 A：物理 116,1（2004），103-106。
[11] Chin-yu Chien, Rong-Hao Liang, Long-Fei Lin, Liwei Chan, and Bing-Yu Chen. 2015. Flexibend: Enabling interactivity of multi-part, deformable fabrications using single shape-sensing strip. In Proceedings of the 28th Annual ACM Symposium on User Interface Software & Technology. Citeseer, 659-663. [11] 钱金玉，梁荣浩，林龙飞，陈立伟和陈炳宇。2015。Flexibend：使用单一形状感应条实现多部分可变形制品的互动性。在第 28 届 ACM 用户界面软件与技术年会论文集中。Citeseer，659-663。
[12] Sung Yun Chung, Hwa-Jin Lee, Duck-Jae You, Sanghun Cho, Buil Nam, Tae Il Lee, and Youn Sang Kim. 2016. Piezoelectric Bending Motion Sensor for Simultaneous Detection of Bending Curvature and Speed. ECS Transactions 75, 16 (2016), 277-279. 宋润钟，李华津，尤德杰，曹尚勋，南必尔，李泰一，金润相。2016 年。压电弯曲运动传感器用于同时检测弯曲曲率和速度。ECS Transactions 75，16（2016），277-279。
[13] Darryl Cotton. 2018. Apparatus having touch and bending sensing functions. (Nov. 6 2018). US Patent App. 10/120,483. 达里尔·科顿。2018 年。具有触摸和弯曲感应功能的装置。 （2018 年 11 月 6 日）。美国专利申请。10/120,483。
[14] Yanyan Dong, Yanfang Lu, Changyu Shen, Jinlei Chu, Xin Zou, Chuan Zhong, and Xinyong Dong. 2013. Novel bending sensor based on a meniscus shaped beam with LPG. Optik 124, 24 (2013), 6737-6739. Yanyan Dong，Yanfang Lu，Changyu Shen，Jinlei Chu，Xin Zou，Chuan Zhong 和 Xinyong Dong。2013。基于 LPG 的新型弯曲传感器，其基于一个弯曲的弯曲形梁。Optik 124，24（2013），6737-6739。
[15] Khaled Elgeneidy, Gerhard Neumann, Michael Jackson, and Niels Lohse. 2018. Directly printable flexible strain sensors for bending and contact feedback of soft actuators. Frontiers in Robotics and AI 5 (2018), 2. Khaled Elgeneidy，Gerhard Neumann，Michael Jackson 和 Niels Lohse。2018。用于软执行器弯曲和接触反馈的可直接打印的柔性应变传感器。《机器人与人工智能前沿》2018 年第 5 期，2。
[16] Abu Ilius Faisal, Sumit Majumder, Tapas Mondal, David Cowan, Sasan Naseh, and M Jamal Deen. 2019. Monitoring Methods of Human Body Joints: State-of-the-Art and Research Challenges. Sensors 19, 11 (2019), 2629. [16] Abu Ilius Faisal，Sumit Majumder，Tapas Mondal，David Cowan，Sasan Naseh 和 M Jamal Deen。2019。人体关节监测方法：现状和研究挑战。传感器 19，11（2019），2629。
[17] Nicholas Fellion, Thomas Pietrzak, and Audrey Girouard. 2017. FlexStylus: Leveraging bend input for pen interaction. In Proceedings of the 30th Annual ACM Symposium on User Interface Software and Technology. ACM, 375-385. [17] 尼古拉斯·费利昂（Nicholas Fellion），托马斯·皮耶特扎克（Thomas Pietrzak）和奥德丽·吉鲁瓦尔（Audrey Girouard）。2017 年。FlexStylus：利用弯曲输入进行笔交互。在第 30 届 ACM 用户界面软件和技术研讨会论文集中。ACM，375-385。
[18] Kevin C Galloway, Yue Chen, Emily Templeton, Brian Rife, Isuru S Godage, and Eric J Barth. 2019. Fiber Optic Shape Sensing for Soft Robotics. Soft robotics (2019). 凯文 C.加洛韦（Kevin C Galloway）、陈悦（Yue Chen）、艾米丽·坦普尔顿（Emily Templeton）、布莱恩·赖夫（Brian Rife）、伊苏鲁·S·戈达格（Isuru S Godage）和埃里克·J·巴斯（Eric J Barth）。2019 年。软机器人的光纤形状感知。软机器人（2019）。
[19] Giada Gerboni, Alessandro Diodato, Gastone Ciuti, Matteo Cianchetti, and Arianna Menciassi. 2017. Feedback control of soft robot actuators via commercial flex bend sensors. IEEE/ASME Transactions on Mechatronics 22, 4 (2017), 1881-1888. Giada Gerboni、Alessandro Diodato、Gastone Ciuti、Matteo Cianchetti 和 Arianna Menciassi。2017 年。通过商用柔性弯曲传感器对软机器人执行器进行反馈控制。《IEEE/ASME 机电学刊》22, 4 (2017), 1881-1888。
[20] Oliver Glauser, Shihao Wu, Daniele Panozzo, Otmar Hilliges, and Olga Sorkine-Hornung. 2019. Interactive hand pose estimation using a stretch-sensing soft glove. ACM Transactions on Graphics (TOG) 38, 4 (2019), 41. Oliver Glauser, Shihao Wu, Daniele Panozzo, Otmar Hilliges 和 Olga Sorkine-Hornung。2019。使用拉伸传感软手套进行交互式手部姿势估计。ACM Transactions on Graphics (TOG) 38, 4 (2019), 41。
[21] Antonio Gomes, Andrea Nesbitt, and Roel Vertegaal. 2013. MorePhone: a study of actuated shape deformations for flexible thin-film smartphone notifications. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. ACM, 583-592. 安东尼奥·戈麦斯（Antonio Gomes），安德烈亚·内斯比特（Andrea Nesbitt）和罗尔·维尔特加尔（Roel Vertegaal）。2013 年。MorePhone：一项关于用于柔性薄膜智能手机通知的驱动形状变形的研究。在人机交互计算系统 SIGCHI 会议论文集中。ACM，583-592。
[22] RL Hollingshead, L Henry-Etesse, E Tankere, D Kamper, and T Tan. 2017. Characterization of hysteresis in resistive bend sensors. In 2017 International Symposium on Wearable Robotics and Rehabilitation (WeRob). IEEE, 1-2. [22] RL Hollingshead, L Henry-Etesse, E Tankere, D Kamper, and T Tan. 2017. Characterization of hysteresis in resistive bend sensors. In 2017 International Symposium on Wearable Robotics and Rehabilitation (WeRob). IEEE, 1-2.
[22] RL Hollingshead, L Henry-Etesse, E Tankere, D Kamper, 和 T Tan. 2017. 电阻式弯曲传感器中滞后特性的表征。在 2017 年可穿戴机器人和康复国际研讨会（WeRob）中。IEEE，1-2。
[23] Minsu Jang, Jun Sik Kim, Soong Ho Um, Sungwook Yang, and Jinseok Kim. 2019. Ultra-high curvature sensors for multi-bend structures using fiber Bragg gratings. Optics express 27, 3 (2019), 2074-2084. [23] Minsu Jang, Jun Sik Kim, Soong Ho Um, Sungwook Yang, and Jinseok Kim. 2019. Ultra-high curvature sensors for multi-bend structures using fiber Bragg gratings. Optics express 27, 3 (2019), 2074-2084.
[23] Minsu Jang，Jun Sik Kim，Soong Ho Um，Sungwook Yang 和 Jinseok Kim。2019。使用光纤 Bragg 光栅的多弯曲结构的超高曲率传感器。光学快报 27，3（2019），2074-2084。
[24] Useok Jeong and Kyu-Jin Cho. 2016. A novel low-cost, large curvature bend sensor based on a Bowden-cable. Sensors 16, 7 (2016), 961. [24] 郑宇奭和赵奎镇。2016 年。基于波登电缆的一种新型低成本、大曲率弯曲传感器。传感器，16(7)，961。
[25] Takashi Kato, Akio Yamamoto, and Toshiro Higuchi. 2003. Shape recognition using piezoelectric thin films. In IEEE International Conference on Industrial Technology, 2003, Vol. 1. IEEE, 112-116. [25] 加藤隆、山本昭夫和樋口敏郎。2003 年。利用压电薄膜进行形状识别。在 2003 年 IEEE 国际工业技术会议上，第 1 卷。IEEE，112-116。
[26] Sean Jason Keller, Tristan Thomas Trutna, David R Perek, and Bruce A Cleary III. 2018. 肖恩·杰森·凯勒，特里斯坦·托马斯·特鲁特纳，大卫·R·佩雷克和布鲁斯·A·克利里三世。2018。
[27] Johan Kildal, Susanna Paasovaara, and Viljakaisa Aaltonen. 2012. Kinetic device: designing interactions with a deformable mobile interface. In CHI'12 Extended Abstracts on Human Factors in Computing Systems. ACM, 1871-1876. [27] Johan Kildal, Susanna Paasovaara 和 Viljakaisa Aaltonen. 2012. 动力学设备：设计与可变形移动界面的交互。在 CHI'12 人机交互计算系统的扩展摘要中。ACM，1871-1876。
[28] Vandana Mishra, Nahar Singh, DV Rai, Umesh Tiwari, GC Poddar, SC Jain, SK Mondal, and Pawan Kapur. 2010. Fiber Bragg grating sensor for monitoring bone decalcification. Orthopaedics & Traumatology: Surgery Research 96, 6 (2010), 646-651. [28] Vandana Mishra，Nahar Singh，DV Rai，Umesh Tiwari，GC Poddar，SC Jain，SK Mondal 和 Pawan Kapur。2010。用于监测骨钙化的光纤 Bragg 光栅传感器。《骨科与创伤外科：手术 研究》96，6（2010），646-651。
[29] James S Neely and Phillip J Restle. 1997. Capacitive bend sensor. (March 11 1997). US Patent 5,610,528. [29] James S Neely 和 Phillip J Restle。1997 年。电容弯曲传感器。 （1997 年 3 月 11 日）。美国专利 5,610,528。
[30] Yanhui Qi, Lin Ma, Jiang Sun, Zexin Kang, Yunlong Bai, and Shuisheng Jian. 2015. Highly sensitive bending sensor based on multimode-multimode-coreoffset fiber structure. Optics & Laser Technology 75 (2015), 52-56. [30] 齐彦辉，马林，孙江，康泽鑫，白云龙和简水生。2015 年。基于多模-多模-核偏移光纤结构的高灵敏度弯曲传感器。光学与激光技术 75（2015），52-56。
[31] Ginu Rajan and B Gangadhara Prusty. 2016. Structural health monitoring of composite structures using fiber optic methods. CRC press. [31] Ginu Rajan 和 B Gangadhara Prusty。2016。使用光纤方法对复合结构进行结构健康监测。CRC 出版社。
[32] Shawn P Reese. 2015. Angular displacement sensor of compliant material. (Dec. 29 2015). US Patent . [32] 肖恩·P·里斯。2015。一种由顺应性材料制成的角位移传感器。 （2015 年 12 月 29 日）。美国专利 。
[33] Christian Rendl, David Kim, Sean Fanello, Patrick Parzer, Christoph Rhemann, Jonathan Taylor, Martin Zirkl, Gregor Scheipl, Thomas Rothländer, Michael Haller, and others. 2014. FlexSense: a transparent self-sensing deformable surface. In Proceedings of the 27th annual ACM symposium on User interface software and technology. ACM, 129-138.
[34] Giovanni Saggio and Giancarlo Orengo. 2018. Flex sensor characterization against shape and curvature changes. Sensors and Actuators A: Physical 273 (2018), 221-231. [34] Giovanni Saggio 和 Giancarlo Orengo。2018 年。Flex 传感器对形状和曲率变化的特性化。传感器和执行器 A：物理 273（2018），221-231。
[35] Giovanni Saggio, Antonio Pallotti, Laura Sbernini, Vito Errico, and Franco Di Paolo. 2016. Feasibility of Commercial Resistive Flex Sensors for Hand Tracking Applications. Sensors & Transducers 201, 6 (2016), 17. [35] Giovanni Saggio, Antonio Pallotti, Laura Sbernini, Vito Errico, and Franco Di Paolo. 2016. 商用电阻式弯曲传感器在手部跟踪应用中的可行性。传感器与变换器 201, 6 (2016), 17.
[36] Giovanni Saggio, Francesco Riillo, Laura Sbernini, and Lucia Rita Quitadamo. 2015. Resistive flex sensors: a survey. Smart Materials and Structures 25, 1 (2015), 013001. [36] Giovanni Saggio, Francesco Riillo, Laura Sbernini 和 Lucia Rita Quitadamo. 2015. 电阻式柔性传感器：一项调查。智能材料与结构 25, 1 (2015), 013001.
[37] Fereshteh Shahmiri, Chaoyu Chen, Anandghan Waghmare, Dingtian Zhang, Shivan Mittal, Steven L Zhang, Yi-Cheng Wang, Zhong Lin Wang, Thad E Starner, and Gregory D Abowd. 2019. Serpentine: A [37] Fereshteh Shahmiri，Chaoyu Chen，Anandghan Waghmare，Dingtian Zhang，Shivan Mittal，Steven L Zhang，Yi-Cheng Wang，Zhong Lin Wang，Thad E Starner 和 Gregory D Abowd。2019 年。Serpentine：A
Self-Powered Reversibly Deformable Cord Sensor for Human Input. In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems. ACM, 545. 自供电可逆变形绳传感器用于人体输入。在 2019 年人机交互计算系统 CHI 会议论文集中。ACM，545。
[38] Ronit Slyper, Ivan Poupyrev, and Jessica Hodgins. 2011. Sensing through structure: designing soft silicone sensors. In Proceedings of the fifth international conference on Tangible, embedded, and embodied interaction. ACM, 213-220. [38] Ronit Slyper，Ivan Poupyrev 和 Jessica Hodgins。2011。通过结构感知：设计软硅胶传感器。在第五届国际切实可行、嵌入式和具身交互会议论文集中。ACM，213-220。
[39] David J Sturman and David Zeltzer. 1994. A survey of glove-based input. IEEE Computer graphics and Applications 14, 1 (1994), 30-39. 大卫·J·斯图曼（David J Sturman）和大卫·泽尔策（David Zeltzer）。1994 年。基于手套的输入调查。IEEE 计算机图形与应用，14 卷，1 期（1994 年），30-39。
[40] Nirzaree Vadgama and Jürgen Steimle. 2017. Flexy: Shape-customizable, single-layer, inkjet printable patterns for 1d and 2d flex sensing. In Proceedings of the Eleventh International Conference on Tangible, Embedded, and Embodied Interaction. ACM, 153-162. [40] Nirzaree Vadgama 和 Jürgen Steimle。2017。Flexy：可定制形状、单层、喷墨打印的 1D 和 2D 柔性传感图案。在第十一届国际有形、嵌入和具象化交互会议论文集中。ACM，153-162。
[41] Sean P Walker, June Park, Neal A Tanner, and others. 2016. Reducing incremental measurement sensor error. (March 22 2016). US Patent 9,289,578. 肖恩·P·沃克（Sean P Walker）、June Park、Neal A Tanner 等人。2016 年。减小增量测量传感器误差。《2016 年 3 月 22 日》。美国专利 9,289,578。
[42] Lefan Wang, Turgut Meydan, and Paul Williams. 2017. A two-axis goniometric sensor for tracking finger motion. Sensors 17, 4 (2017), 770. [42] 王乐凡，图尔古特·迈丹和保罗·威廉姆斯。2017 年。用于跟踪手指运动的双轴测角传感器。传感器，17(4)，770。
[43] Kristen Warren, Jessica Lo, Vaibhav Vadgama, and Audrey Girouard. 2013. Bending the rules: bend gesture classification for flexible displays. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. ACM, 607-610. 克里斯汀·沃伦（Kristen Warren）、杰西卡·洛（Jessica Lo）、瓦伊巴夫·瓦德加马（Vaibhav Vadgama）和奥德丽·吉鲁瓦尔（Audrey Girouard）。2013 年。《弯曲规则：柔性显示器的弯曲手势分类》。收录于《人机交互计算系统 SIGCHI 会议论文集》。ACM，607-610。
[44] Ran Xu, Aaron Yurkewich, and Rajni V Patel. 2016. Curvature, torsion, and force sensing in continuum robots using helically wrapped FBG sensors. IEEE Robotics and Automation Letters 1, 2 (2016), 1052-1059. [44] Ran Xu, Aaron Yurkewich, and Rajni V Patel. 2016. Curvature, torsion, and force sensing in continuum robots using helically wrapped FBG sensors. IEEE Robotics and Automation Letters 1, 2 (2016), 1052-1059.
[44] 徐然，亚伦·尤克维奇和拉吉尼·V·帕特尔。2016 年。使用螺旋包覆 FBG 传感器在连续机器人中进行曲率、扭转和力传感。IEEE 机器人与自动化快报 1，2（2016），1052-1059。
[45] Thomas G Zimmerman. 1985. Optical flex sensor. (Sept. 17 1985). US Patent 4,542,291. [45] 托马斯 G.兹默曼。1985 年。光学弯曲传感器。(1985 年 9 月 17 日)。美国专利 4,542,291。
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions @ acm.org. 未经费用许可，允许制作本作品的全部或部分数字或硬拷贝，供个人或课堂使用，但不得为牟利或商业优势而制作或分发拷贝，并且拷贝必须带有本通知和第一页上的完整引用。对于 ACM 以外的其他所有者拥有的本作品组成部分的版权必须受到尊重。允许带有信用进行摘要。要复制其他内容，或重新发布，或发布到服务器上，或重新分发到列表上，需要事先获得特定许可和/或支付费用。请从 permissions@acm.org 请求权限。
CHI '20, April 25-30, 2020, Honolulu, HI, USA. CHI '20，2020 年 4 月 25 日至 30 日，美国夏威夷州檀香山。
⑳20 Copyright is held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 978-1-4503-6708-0/20/04 . 2020 版权归所有者/作者所有。出版权许可给 ACM。ACM ISBN 978-1-4503-6708-0/20/04 。
