A variety of micro air vehicle (MAV) technologies are now available which provide the fundamental flight capabilities required for basic survey and transport tasks. These aircraft exhibit highly coupled rotational and lateral dynamics which must be taken into account in the control design and when specifying aggressive required trajectories. A popular example is the planar quadrotor, whose flight state exists in six dimensions over position and body orientation but which is equipped with only four actuators. These aircraft only have control over their attitude moment vector and the magnitude of net thrust downward in the body frame, and so they must maneuver spatially by constantly changing their orientation. As a direct consequence of this underactuation they are incapable of independently regulating both position and orientation. Furthermore, even smooth spatial trajectories can be infeasible unless they are c^{3}c3, which excludes such common techniques as both minimum jerk and trapezoidal velocity multi-segment trajectories.
现在有各种微型飞行器 （MAV） 技术可用，它们提供了基本测量和运输任务所需的基本飞行能力。这些飞机表现出高度耦合的旋转和横向动力学，在控制设计和指定激进的所需轨迹时必须考虑到这一点。一个流行的例子是平面四旋翼，其飞行状态在位置和身体方向上以六个维度存在，但仅配备四个致动器。这些飞机只能控制其姿态矩矢量和机身框架中向下的净推力大小，因此它们必须通过不断改变方向来进行空间机动。作为这种驱动不足的直接后果，它们无法独立调节位置和方向。此外，即使是平滑的空间轨迹也可能是不可行的，除非它们是 c^{3}c3 ，这排除了最小加加速度和梯形速度多段轨迹等常见技术。

Fully actuated aircraft with independent control over body forces and moments could support a multitude of new capabilities. Such aircraft would be able to apply arbitrary wrenches on the environment, making them useful for construction or object manipulation. In flight they could independently point cameras, sensors, or high gain antennas independent of motion trajectories or the wind environment. In indoor environments with humans they would able to gesture with the aircraft posture to make their motion intentions more legible to bystanders, visually indicate objects or directions as a guide, or provide visual cues to aid in human-robot task coordination. These possibilities have inspired diverse efforts to realize new types of fully actuated MAV.
独立控制身体力和力矩的完全驱动飞机可以支持多种新功能。这种飞机将能够对环境使用任意扳手，使其可用于构造或对象操作。在飞行中，它们可以独立于运动轨迹或风环境独立指向相机、传感器或高增益天线。在有人类的室内环境中，他们能够用飞机的姿势做手势，使旁观者更容易理解他们的运动意图，以视觉方式指示物体或方向作为指导，或提供视觉提示以帮助人机任务协调。这些可能性激发了各种努力，以实现新型的完全驱动 MAV。

Many previous embodiments of fully actuated, holonomic, or omnidirectional MAV are conceptually inspired by the quadrotor and proceed by adding additional actuators. By configuring six conventional rigid rotors with their orientations canted out of plane it is possible to obtain independent control over forces and moments in proximity to hover, but the inability to reverse independent rotor thrust directions limit feasible forces and therefore feasible stable orientations [1], [2]. With seven unidirectional rotors it becomes in principle possible to hover in all orientations, even upside down [3]. Incorporating eight variable-direction rotors allows practical flight in all orientations and would potentially permit control strategies which avoid driving motors at low speeds or with rapid direction changes [4]. Similar capabilities in six-rotor configurations become possible with high performance reversing motor drivers [5].
许多先前的完全驱动、全息或全向 MAV 的实施例在概念上都受到四旋翼的启发，并通过添加额外的致动器进行。通过配置六个传统的刚性旋翼，使其方向倾斜出平面，可以独立控制靠近悬停的力和力矩，但无法反转独立的旋翼推力方向限制了可行的力，因此限制了可行的稳定方向 [1] 。 [2] 使用七个单向转子，原则上可以在所有方向上悬停，甚至 倒置 [3] .包含八个可变向旋翼允许在所有方向上进行实际飞行，并可能允许使用控制策略，避免以低速或快速改变 [4] 方向驱动电机。高性能换向电机驱动器 [5] 可以在六转子配置中实现类似的功能。

Coaxial helicopters offer a different point of departure for developing fully actuated aircraft. One technique is to vector the thrust of top and bottom rotors by reorienting the entire motor and rotor assembly using gimbals driven by additional pitch and roll servomotors. In static bench testing, [6] demonstrated that the resulting six-actuator system can obtain authority over net forces and moments. Alternatively, a pair of conventional swashplates and teetering rotors can be driven by four roll and pitch servos to tilt the rotor tip path plane and achieve a similar effect. Conventional flight capabilities using this technique were obtained by [7], but novel maneuvers unique to fully actuated MAV were not deeply explored. Both types of coaxial aircraft have an efficiency advantage over the aforementioned multirotors in that all of the rotor thrust can be directed downwards when in hover. However, each still require a minimum of six actuators for operation.
同轴直升机为开发全驱动飞机提供了不同的出发点。一种技术是通过使用由额外的变桨和滚转伺服电机驱动的万向节重新定向整个电机和转子组件来矢量化顶部和底部转子的推力。在静态台架测试中， [6] 证明了所得到的六作动缸系统可以获得对净力和力矩的权威。或者，一对传统的斜盘和摇摇欲坠的转子可以由四个滚动和俯仰伺服系统驱动，以倾斜转子尖端路径平面并实现类似的效果。使用这种技术的常规飞行能力是通过 [7] 获得的，但尚未深入探索完全驱动 MAV 独有的新型机动。与上述多旋翼飞机相比，这两种类型的同轴飞机都具有效率优势，因为在悬停时，所有旋翼推力都可以向下引导。但是，每个驱动器仍然需要至少 6 个执行器才能运行。

This paper introduces a new coaxial helicopter which emulates fully actuated aircraft using only two actuators. We do this by taking advantage of recent methods for controlling a flapping rotor's tip path plane by exciting a dynamic response to modulated shaft torques from the primary drive motor [8], [9]. Section II describes the idealized vehicle dynamics in terms of vectored thrusts derived from tilting top and bottom rotor tip path planes. Our method for controlling the tip path plane response without auxiliary actuators is described in Section III along with measurements of the individual rotor capabilities. The vehicle hardware design is summarized in Section IV, and the control architecture is described in Section V. Flight results in Section VI demonstrate decoupled lateral and rotational dynamics, confirming that this two-actuator MAV emulates the primary capabilities of a six-actuator, fully actuated MAV. This includes sustaining a stationary hover while pitching the aircraft up to 8°, and tracking trajectories with discontinuous accelerations up to 1 m/s² without pitching or rolling. We conclude by identifying areas for improvement and future work.
本文介绍了一种新型同轴直升机，它仅使用两个致动器即可模拟完全驱动的飞机。我们利用最新的方法，通过激发对主驱动电机的调制轴扭矩的动态响应来控制扑动转子的尖端路径平面 [8]，[9]。第二节描述了理想化的飞行器动力学，即从倾斜的顶部和底部转子尖端路径平面得出的矢量推力。第 III 节描述了我们在没有辅助促动器的情况下控制尖端路径平面响应的方法，以及各个转子能力的测量。第四节总结了车辆硬件设计，第五节描述了控制架构。第六节的飞行结果展示了解耦的横向和旋转动力学，证实了这款双致动器 MAV 模拟了六致动器、完全致动 MAV 的主要功能。这包括在将飞机俯仰至 8° 时保持静止悬停，以及以高达 1 m/s² 的不连续加速度跟踪轨迹，而不俯仰或滚动。最后，我们确定了需要改进的领域和未来的工作。

Fig. 1.   图 1.

Teetering rotors allow independent control of force and moments.
摇晃的转子允许独立控制力和力矩。

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The vehicle dynamics can be approximated by considering a coaxial helicopter capable of tilting the direction of thrust from each rotor away from vertical. This thrust vectoring effect could conventionally be obtained from teetering rotors equipped with cyclic blade pitch control actuators. Our unique method for controlling the blade response using only the main drive torque will be examined in Section III, but first we address a generic thrust vectoring idealization. Figure 1 depicts one rotor mounted a distance r1 above the center of mass and a second counter rotating rotor mounted a distance r2 below the center of mass. The figure conceptually illustrates that the force vectors f1 and f2 can be directed counter to each other in order to produce a net pitching moment about the vehicle's center of mass while maintaining zero net lateral force. Alternatively, the force vectors can be pointed in similar directions, yielding a net lateral force on the aircraft while maintaining zero net moment.
可以通过考虑一架同轴直升机来近似计算，该直升机能够将每个旋翼的推力方向从垂直方向倾斜。这种推力矢量效应通常可以通过配备循环叶片变桨控制致动器的摇摇晃晃的旋翼获得。第三节将介绍我们仅使用主驱动扭矩控制叶片响应的独特方法，但首先我们讨论通用的推力矢量理想化。图 1 描述了一个安装在质心上方一段距离 r1 处的转子，另一个反向旋转的转子安装在质心下方一段距离 r2 处。该图从概念上说明了力矢量 f1 和 f2 可以相互相反的方向，以便在保持零净横向力的同时，围绕车辆质心产生净俯仰力矩。或者，力矢量可以指向类似的方向，在飞机上产生净侧向力，同时保持零净力矩。

Equation 1 develops the net force F and moment M vectors about the aircraft center of mass as a linear function of the individual rotor force vectors f1 and f2. In addition to these rotor forces, we model a corresponding reaction torque about the z axis for each rotor which is proportional to its thrust along the z axis by a constant coefficient kQ. This is a reasonable approximation for the small angular deflections in the forces considered here. Vectors F,M,f1, and f2 are written in component form in Eq. 1 with respect to body fixed x,y, and z axes.

View Source\begin{equation*} \begin{bmatrix} F_{x}\\ F_{y}\\ F_{z}\\ M_{x}\\ M_{y}\\ M_{z} \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 0 & 1\\ 0 & -r_{1} & 0 & 0 & r_{2} & 0\\ r_{1} & 0 & 0 & -r_{2} & 0 & 0\\ 0 & 0 & -k_{Q} & 0 & 0 & k_{Q} \end{bmatrix}\begin{bmatrix} f_{1x}\\ f_{1y}\\ f_{1z}\\ f_{2x}\\ f_{2y}\\ f_{2z} \end{bmatrix} \tag{1} \end{equation*}

方程 1 将围绕飞机质心的合力 F 和力矩 M 矢量发展为各个旋翼力矢量 f1 和 f2 的线性函数。除了这些转子力之外，我们还为每个转子模拟了绕 z 轴的相应反作用扭矩，该反作用矩与其沿 z 轴的推力成正比，系数为常数 kQ 。对于此处考虑的力中的小角度偏转，这是一个合理的近似值。向量 F,M,f1 ，以 f2 分量形式写在方程中 1 ，相对于 body fixed x,y 和 z axes。

⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢FxFyFzMxMyMz⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢1000r10010−r10000100−kQ1000−r20010r20000100kQ⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢f1xf1yf1zf2xf2yf2z⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥(1)

View Source\begin{equation*} \begin{bmatrix} F_{x}\\ F_{y}\\ F_{z}\\ M_{x}\\ M_{y}\\ M_{z} \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 0 & 1\\ 0 & -r_{1} & 0 & 0 & r_{2} & 0\\ r_{1} & 0 & 0 & -r_{2} & 0 & 0\\ 0 & 0 & -k_{Q} & 0 & 0 & k_{Q} \end{bmatrix}\begin{bmatrix} f_{1x}\\ f_{1y}\\ f_{1z}\\ f_{2x}\\ f_{2y}\\ f_{2z} \end{bmatrix} \tag{1} \end{equation*}

Fig. 2.   图 2.

Teetering rotor with skewed lag-pitch hinges.
摇摇欲坠的转子，带有倾斜的滞后螺距铰链。

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The constant coefficient matrix has a determinant of 2kQ(r1+r2)2 and so will be full rank and invertible so long as the two rotors are not co-located. As a result the relationship can be inverted, and Eq. 2 provides a unique solution for allocating individual rotor controls f1 and f2 given a desired net vehicle force and moment. If the six components of rotor forces f1 and f2 are available as independent inputs, the aircraft will be fully actuated in all six operational degrees over orientation and position.

View Source\begin{equation*} \begin{bmatrix} f_{1x}\\ f_{1y}\\ f_{1z}\\ f_{2x}\\ f_{2y}\\ f_{2z} \end{bmatrix}=\begin{bmatrix} \frac{r_{2}}{r_{1}+r_{2}} & 0 & 0 & 0 & \frac{1}{r_{1}+r_{2}} & 0\\ 0 & \frac{r_{2}}{r_{1}+r_{2}} & 0 & \frac{-1}{r_{1}+r_{2}} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0 & 0 & -\frac{1}{2k_{Q}}\\ \frac{r_{1}}{r_{1}+r_{2}} & 0 & 0 & 0 & \frac{-1}{r_{1}+r_{2}} & 0\\ 0 & \frac{r_{1}}{r_{1}+r_{2}} & 0 & \frac{1}{r_{1}+r_{2}} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0 & 0 & \frac{1}{2k_{Q}} \end{bmatrix} \begin{bmatrix} F_{x}\\ F_{y}\\ F_{z}\\ M_{x}\\ M_{y}\\ M_{z} \end{bmatrix} \tag{2} \end{equation*}

常数系数矩阵的行列式为 2kQ(r1+r2)2 ，因此只要两个转子不位于同一位置，它将是全秩和可逆的。因此，这种关系可以反转，方程 2 提供了一种独特的解决方案，用于分配单个转子控制 f1 ，并 f2 给出所需的净车辆力和力矩。如果旋翼力 f1 f2 的六个分量都可以作为独立输入，则飞机将在方向和位置的所有六个操作角度上完全驱动。

⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢f1xf1yf1zf2xf2yf2z⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢r2r1+r200r1r1+r2000r2r1+r200r1r1+r20001200120−1r1+r2001r1+r201r1+r200−1r1+r20000−12kQ0012kQ⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢FxFyFzMxMyMz⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥(2)

View Source\begin{equation*} \begin{bmatrix} f_{1x}\\ f_{1y}\\ f_{1z}\\ f_{2x}\\ f_{2y}\\ f_{2z} \end{bmatrix}=\begin{bmatrix} \frac{r_{2}}{r_{1}+r_{2}} & 0 & 0 & 0 & \frac{1}{r_{1}+r_{2}} & 0\\ 0 & \frac{r_{2}}{r_{1}+r_{2}} & 0 & \frac{-1}{r_{1}+r_{2}} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0 & 0 & -\frac{1}{2k_{Q}}\\ \frac{r_{1}}{r_{1}+r_{2}} & 0 & 0 & 0 & \frac{-1}{r_{1}+r_{2}} & 0\\ 0 & \frac{r_{1}}{r_{1}+r_{2}} & 0 & \frac{1}{r_{1}+r_{2}} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0 & 0 & \frac{1}{2k_{Q}} \end{bmatrix} \begin{bmatrix} F_{x}\\ F_{y}\\ F_{z}\\ M_{x}\\ M_{y}\\ M_{z} \end{bmatrix} \tag{2} \end{equation*}

Thrust vectoring for control through tilting of the tip path plane can be implemented without adding any additional actuators beyond the top and bottom drive motors themselves. In previous work it has been shown that a single motor can control both the mean operating speed and cyclic blade pitch variation of a rotor by modulating the applied drive torque [8]. Two blades are attached to a hub with skewed lag-pitch hinges, as shown in Fig. 2. Modulating the motor torque sinusoidally at one-per-rev excites a synchronous lead-lag motion in each blade within the plane of rotation. The skewed lag-pitch hinge couples this lag oscillation into a blade pitch oscillation. The two blades are mounted on asymmetric hinges so that one has a positive lag-pitch coupling and the other a negative lag-pitch coupling. As a result a one-per-rev sinusoidal modulation in motor torque causes the blades to pitch 180° out of phase with each other, phase locked with the rotor rotation. By controlling the amplitude and phase of the motor torque the amplitude and azimuthal phase of the blade pitch can be controlled. The aircraft in [8] is capable of attitude control like a standard quadrotor or helicopter and maneuvers by taking advantage net direct hub moments arising when, for example, both blades cyclically pass across the aircraft nose at minimum pitch and lift but pass across the tail at maximum pitch and lift.
通过倾斜尖端路径平面进行控制，无需在顶部和底部驱动电机本身之外添加任何其他致动器即可实现推力矢量。在以前的工作中已经表明，单个电机可以通过调制施加的驱动扭矩 [8] 来控制转子的平均运行速度和循环叶片螺距变化。两个叶片通过倾斜的 Lag-Pitch 铰链连接到轮毂上，如 所示。 Fig. 2 以每转 1 的速度正弦调制电机扭矩，会激发旋转平面内每个叶片的同步超前滞后运动。倾斜的滞后-螺距铰链将这种滞后振荡耦合成叶片螺距振荡。两个叶片安装在不对称的铰链上，因此一个叶片具有正滞后-螺距耦合，另一个具有负滞后-螺距耦合。因此，电机扭矩的每转 1 正弦调制导致叶片彼此异相 180°，与转子旋转锁相。通过控制电机扭矩的幅度和相位，可以控制叶片俯仰的幅度和方位相位。该 [8] 飞机能够像标准四旋翼或直升机一样进行姿态控制，并利用产生的净直接轮毂力矩进行机动，例如，两个叶片以最小俯仰和升力周期性地穿过飞机机头，但在最大俯仰和升力下穿过尾部。

Independent offset flap hinges were added in [9] to allow each blade to individually flap up and down during each revolution in response to changing blade pitches and the resulting blade lift. In addition to direct moments on the hub, this causes an apparent tilting of the tip path plane and redirection of the thrust vector.
添加了独立的偏置襟翼铰链 [9] ，以允许每个叶片在每次旋转期间单独上下摆动，以响应不断变化的叶片间距和由此产生的叶片升程。除了轮毂上的直接力矩外，这会导致尖端路径平面的明显倾斜和推力矢量的重定向。

The operational principle depicted in Fig. 1 benefits from large flapping angles and a pure thrust vectoring effect with no direct moments applied to the hub, properties which neither of the rotor designs in [8] or [9] achieve. This is now obtained by incorporating a single, central teetering hinge as seen in Figs. 2 and 3. For each degree of cyclic blade pitch authority a teetering rotor enjoys one degree of flap and tip path plane inclination, and the thrust force may be thought of as remaining perpendicular to this tip path plane. At the same time, no direct torques can be transfered to the hub through the teetering hinge.
其中 Fig. 1 描述的工作原理受益于大的扑动角和纯粹的推力矢量效应，没有直接的力矩施加到轮毂上，这是两个旋翼都没有设计 [8] 或 [9] 实现的特性。现在，这是通过合并一个中央摇摇欲坠的铰链来实现的，如 Figs. 2 和 3 所示。对于每度循环叶片俯仰授权，摇摇欲坠的旋翼享有一度的襟翼和尖端路径平面倾斜，推力可以被认为是保持垂直于该尖端路径平面。同时，没有直接扭矩可以通过摇晃铰链传递到轮毂。

The change in blade flap angle β as a function of azimuthal angle ψ is conventionally described as

View Source\begin{equation*} \beta(\psi)=\beta_{c}\cos(\psi)+\beta_{s}\sin(\psi) \tag{3} \end{equation*}

f1x=kTΩ21βcf2x=kTΩ32βcf1y=kTΩ21βsf2y=−kTΩ22βsf1z=kTΩ21f2z=kTΩ22(4)

View Source\begin{align*}& f_{1x}=k_{T}\Omega_{1}^{2}\beta_{c}\qquad f_{2x}=k_{T}\Omega_{2}^{3}\beta_{c}\\ & f_{1y}=k_{T}\Omega_{1}^{2}\beta_{s}\qquad f_{2y}=-k_{T}\Omega_{2}^{2}\beta_{s}\\ & f_{1z}=k_{T}\Omega_{1}^{2}\qquad \quad f_{2z}=k_{T}\Omega_{2}^{2} \tag{4} \end{align*}

叶片襟翼角 β 随方位角 ψ 的变化通常被描述为

β(ψ)=βccos(ψ)+βssin(ψ)(3)

View Source\begin{equation*} \beta(\psi)=\beta_{c}\cos(\psi)+\beta_{s}\sin(\psi) \tag{3} \end{equation*} 在船尾方向上的位置 ψ=0 ， ψ 在旋转方向上增加的位置。因此，对于逆时针顶部转子， βc 描述了尖端路径平面向前的纵向倾斜，并 βs 描述了向后退叶片侧面的横向倾斜。推力可以表示为转子速度 Ω1 的函数，推力系数 kf1 可以表示为 kTΩ21 。在 β 转子力矢量 f1 中采用小角度近似和类似结构 f2 的构造用方程表示， 4 其中符号的差异是由于它们的旋转方向相反。

f1x=kTΩ21βcf2x=kTΩ32βcf1y=kTΩ21βsf2y=−kTΩ22βsf1z=kTΩ21f2z=kTΩ22(4)

View Source\begin{align*}& f_{1x}=k_{T}\Omega_{1}^{2}\beta_{c}\qquad f_{2x}=k_{T}\Omega_{2}^{3}\beta_{c}\\ & f_{1y}=k_{T}\Omega_{1}^{2}\beta_{s}\qquad f_{2y}=-k_{T}\Omega_{2}^{2}\beta_{s}\\ & f_{1z}=k_{T}\Omega_{1}^{2}\qquad \quad f_{2z}=k_{T}\Omega_{2}^{2} \tag{4} \end{align*}

The motor torques driving the gross propeller rotation as well as the cyclic blade pitch and flapping response are a result of modulating the applied motor voltage. The applied voltage V is the sum of two parts: a proportional-integral control on error between the observed rotor speed ψ˙ and desired speed Ω with gains kF and kI, and an additional voltage modulation V~.

View Source\begin{equation*} V=-k_{P}(\dot{\psi}-\Omega)-k_{I}\int(\dot{\psi}-\Omega)dt+\tilde{V} \tag{5} \end{equation*}

驱动螺旋桨总旋转以及循环叶片俯仰和扑动响应的电机扭矩是调制施加的电机电压的结果。施加的电压 V 是两部分之和：对观察到的转子速度 ψ˙ 与带增益 kF kI 和的期望速度 Ω 之间的误差的比例积分控制，以及附加的电压调制 V~ 。

V=−kP(ψ˙−Ω)−kI∫(ψ˙−Ω)dt+V~(5)

View Source\begin{equation*} V=-k_{P}(\dot{\psi}-\Omega)-k_{I}\int(\dot{\psi}-\Omega)dt+\tilde{V} \tag{5} \end{equation*}

Previous dynamical modeling and experimental validation of similar rotors in [9] motivates a useful approximation for the flap response in terms of the applied voltage modulation. The flap response in β lags the voltage modulation V~ by an angle ϕβ. The flap amplitude is proportional to the voltage amplitude V~ in excess of a minimum threshold V~min by a linear constant kβ. Parameters ϕβ,V~min, and kβ are functions of the rotor physical properties, electromechanical motor properties, and software speed control gains. They are valid near a trim thrust condition, and are readily determined with a bench test. The final expression for V~ is then given by Eq. 6, where it is convenient to write the desired flapping in terms of polar amplitude a and phase ϕ.

View Source\begin{align*}& a=\sqrt{\beta_{c}^{2}+\beta_{s}^{2}}\\ & \phi= \text{a}\tan 2 (\beta_{s},\ \beta_{c})\\ & \tilde{V}=(\tilde{V}_{min}+k_{\beta}a)\cos(\psi-\phi-\phi_{\beta}) \tag{6} \end{align*}

先前对类似转子 [9] 的动力学建模和实验验证激发了在施加电压调制方面对襟翼响应的有用近似。的襟翼响应 中的 β 抖动与电压调制 V~ 滞后一个角度 ϕβ 。襟翼振幅与 超过最小阈值 V~min 的电压振幅 V~ 成线性常 kβ 数 的比例。Parameters ϕβ,V~min 和 kβ 是转子物理属性、机电电机属性和软件速度控制增益的函数。它们在配平推力条件下有效，并且很容易通过台架测试来确定。最后的表达式 V~ 由方程 给出 6 ，其中可以方便地用极振幅 a 和相位 ϕ 来写出所需的拍打。

a=β2c+β2s−−−−−−√ϕ=atan2(βs, βc)V~=(V~min+kβa)cos(ψ−ϕ−ϕβ)(6)

View Source\begin{align*}& a=\sqrt{\beta_{c}^{2}+\beta_{s}^{2}}\\ & \phi= \text{a}\tan 2 (\beta_{s},\ \beta_{c})\\ & \tilde{V}=(\tilde{V}_{min}+k_{\beta}a)\cos(\psi-\phi-\phi_{\beta}) \tag{6} \end{align*}

Fig. 3.   图 3.

Top rotor of coaxial helicopter.
同轴直升机的顶部旋翼。

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Fig. 4.   图 4.

Coaxial helicopter.  同轴直升机。

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The flight vehicle is shown in Fig. 4, incorporating two counter-rotating propeller systems which are depicted in Fig. 3. The rotors are 32 cm in diameter, and are driven to a trim hover speed of approximately 370 rad/s by two size 2212 BLDC motors. The rotor blades are commercial symmetric airfoils attached to custom 3D printed hub pieces which are joined by steel pin hinges with PTFE plastic washers added to reduce friction. The full aircraft mass is 380 g, with the center of mass approximately equidistant between the two rotors which are themselves 16 cm apart.
飞行 Fig. 4 器如 所示，包括两个反向旋转的螺旋桨系统，如 Fig. 3 所示。转子直径为 32 厘米，由两台 2212 型 BLDC 电机驱动至约 370 rad/s 的配平悬停速度。转子叶片是商业对称翼型，连接到定制的 3D 打印轮毂件上，这些轮毂件由钢销铰链连接，并添加了 PTFE 塑料垫圈以减少摩擦。飞机的全质量为 380 g，两个旋翼之间的质心大约相等，这两个旋翼本身相距 16 cm。

A commercial flight controller using the PX4 autopilot software [10] runs an attitude tracking control law to generate desired body moments M. The desired body attitude as well as additional body force commands F are passed in through a WiFi radio link. The flight controller calculates speed Ω and flap parameters βc,βs for each rotor based on linear combinations of F and M consistent with Eqs. 2 and 4 near trim. These parameters are passed to the motor controller as three PWM encoded values.
使用 PX4 自动驾驶仪软件 [10] 的商用飞行控制器运行姿态跟踪控制律以生成所需的身体力矩 M 。所需的身体姿势以及额外的身体力命令 F 通过 WiFi 无线电链路传递。飞行控制器根据方程 2 和 4 近配平的线性组合 F 计算 Ω 每个旋翼的速度和襟翼参数 βc,βs ，并与 M 方程 2 和 4 一致。这些参数作为三个 PWM 编码值传递给电机控制器。