Dynamics of a Rotor-Pendulum With a Small, Stiff Propeller in Wind
As small rotorcraft grow in capability, the possibilities for their application increase dramatically. Many of these new applications require stable outdoor flight, necessitating a closer look at the aerodynamic response of the aircraft in windy environments. This paper develops the equations of motion for a single-propeller test stand by analyzing the blade-flapping response of a small-stiff propeller in wind. The system dynamics are simulated to show behavior under various wind conditions, and stable system equilibria are identified. Experiments with a rotor-pendulum validate the simulations, including system equilibria and gust response. INTRODUCTION Small unmanned aerial systems (UAS) are transforming from hobbyist entertainment into utilitarian machines. UAS have been tasked with objectives such as surveying farmland and aiding in natural disasters [1] that require multi-rotor aircraft to fly outdoors in potentially adverse weather. High winds pose a great challenge to small UAS [2–4], and developing an understanding of how they respond to wind and the mechanics behind that response is key to compensating for them. This paper uses a combination of theoretical and experimental work to describe the forces and moments experienced by a single quadrotor-type propeller in a uniform wind. Experimental results are collected with a test stand in wind using a Gemfan 5030 propeller commonly used on quadrotor helicopters. These results promise to ∗Address all correspondence to this author. improve guidance and control algorithms for small, multi-rotor helicopters. Although an important part of full-sized helicopter dynamics, blade-flapping is often assumed to be negligible in small quadrotor vehicles [5, 6]. For indoor flight with relatively small advance ratios, this assumption has proved valid even for highly aerobatic flight [7, 8]. However, for outdoor flight in wind, the effect of blade flapping and other aerodynamic phenomena must be re-evaluated [4]. When a helicopter rotor moves forward in air, the advancing side of the rotor produces more lift than the retreating side, which causes a roll moment on the blades [9]. Many studies [10–13] indicate that this moment causes the rotor blades to react with a maximum deflection at 90◦ phase delay, i.e., above the helicopter’s nose, due either to a gyroscopic effect or the blade frequency response. Others provide the equations for flapping without explicitly indicating the expected phase delay [14, 15]. This paper provides a more comprehensive look at the blade-flap dynamics of a small, stiff propeller commonly used in small UAS. Hoffmann et al. [12] and Yeo et al. [16] each measured a quadrotor propeller response to wind. Hoffmann et al. [12] tested a single propeller in wind to identify the flap angle, and showed that the hub experiences a force in the direction of the wind. Yeo et al. [16] tested a single-degree-of-freedom pitch stand with two propellers, as well as a fixed, rigid propeller in an edgewise flow, and found that forces and moments scale with free-stream velocity as suggested in [12]. This paper investigates the source of the forces and moments on a single propeller in wind, and describes blade-flapping dy1 Copyright c © 2016 by ASME namics based on first-principle analyses. A simplified set of analytically tractable equations predict the phase delay and flap amplitude of a small, stiff propeller, the results of which are compared to experimental data. The experimental testbed consists of a two-degree-of-freedom rotor-pendulum, which is a spherical pendulum affixed with a spinning propeller. The long arm of the spherical pendulum increases the effect of the hub forces, demonstrating the propeller’s response to wind. The contributions of this paper are (1) a detailed analysis of the blade-flapping response of a small, stiff propeller in uniform wind, yielding the derivation and solution of the equations of motion for blade flapping and the rotor-pendulum system from first-principles; (2) comparison of the first-principles model to existing experimental measurements of forces and moments at the hub of a propeller fixed in a uniform wind; and (3) the design, fabrication, and testing of a rotor-pendulum test stand that demonstrates the effect of wind on a single propeller. This work increases the theoretical and physical understanding of a small, stiff propeller’s response to wind, which has the potential to yield improved flight stability for small multi-rotor helicopters in adverse weather conditions by virtue of an improved feedback response using flow sensing and control [16]. The outline of the paper is as follows. The first section describes the rotor-pendulum system and develops the equations of motion for a static rotor for comparison to pre-existing experimental data. The second section investigates aerodynamic forces acting on the propeller and derives the equations of motion for the full rotor-pendulum system. The third section provides new experimental results for the rotor-pendulum, with a comparison to model predictions. The final section summarizes the paper and ongoing work. ROTOR DYNAMICS This paper utilizes a rotor-pendulum to investigate the effect of wind on a small, stiff propeller. The rotor-pendulum is a variation of the gyro-pendulum, which is a spherical pendulum with a rapidly spinning mass on the mobile end that causes the system to precess and nutate. Figure 1 shows the rotorpendulum system: a gyro-pendulum with the spinning mass replaced by a propeller. Consider inertial frame I , (O,e1,e2,e3) and intermediate frame A , (O,a1,a2,a3), where a3 = e3 and a1 · e1 = cosθ . Spherical frame B , (O,b1,b2,b3) satisfies b2 = a2 and b1 ·a1 = cosφ . The hub frame is C , (O,c1,c2,c3), where c3 = b3 and c1 ·b1 = cosψ . Let Nb represent the number of propeller blades and the superscript (n), where n = 1,2, or 3, denote the blade index, so that frame D(n) , (H(n),d 1 ,d (n) 2 ,d (n) 3 ) has origin at the blade hinge, and rotates about c2 by the flapangle β . (The blade index (n) is included only where needed for clarity.) Let r denote the displacement along the length of the blade of a point P with respect to O′, and dr be the differential position. The differential forces, moments, and mass are denoted Fdr , dF , Mdr , dM, and mdr , dm, where the quantities F , b 2 = a 2
Geometric Control of Quadrotor Attitude in Wind With Flow Sensing and Thrust Constraints
Quadrotor vehicles show great potential over a range of tasks, but effective control in windy environments continues to be a challenge. This paper develops a thrust-saturated controller on the Lie group SO(3) that uses flow sensing in order to reduce the effect of gusts on the vehicle. Designing the controller on SO(3) establishes almost-global exponential stability, and avoids the pitfalls of representing rigid-body kinematics using Euler angles. We prove that exponential stability is retained in the presence of thrust saturation. Aerodynamics are incorporated into the dynamics and control through a model of the blade-flapping phenomena experienced by rotorcraft. Numerical examples show that the system control remains effective despite thrust saturation, and that flow sensing improves both the initial response and steady-state error of the system in wind. INTRODUCTION Quadrotor helicopters have become popular in myriad tasks, ranging from entertainment to highly utilitarian work. Their ability to hover and maneuver with great agility provides advantages over fixed-wing aircraft, and their simplicity makes them preferable over traditional helicopters. A variety of control methods have been employed in order to stabilize quadrotors, including PID [1, 2], adaptive [3, 4], robust [5, 6], feedback linearization [7], and optimal [8] control. This paper employs a feedback linearization controller on the ∗Address all correspondence to this author. geometric Lie group SO(3) following [7], with the addition of saturation of the vehicle’s thrust inputs. Cao and Lynch [1], and Roza and Maggiore [9] approach thrust saturation using the nested saturation method from Teel [10], which is designed to address saturation in the case of a chain of integrators. Cao and Lynch [1] bound the roll and pitch angles of the system as well as the thrust by placing limits on system inputs, whereas Roza and Maggiore [9] place the bound on thrust only. Cutler and How [2] address saturation by choosing a trajectory that keeps the system states within the bounds required in order to avoid thrust saturation. This paper uses the method of Pappas et al. [11] to bound the thrust on the system in order to guarantee stability when the cost of feedback linearization does not saturate the thrust. Quadrotors are able to accomplish an array of tasks such as surveying farmland and aiding in natural disasters [12] that require multi-rotor aircraft to fly outdoors in potentially adverse weather. High winds pose a challenge to small UAS [13–15], and developing an understanding of how they respond to wind and the mechanics behind that response is key to compensation. This paper builds on previous work [16] in order to incorporate flow sensing in the attitude controller of a three-degree-of-freedom (DOF) quadrotor test stand. In [16], the blade-flapping response of a quadrotor propeller is analyzed in order to identify the aerodynamic moment acting on a propeller in the presence of a wind gust. This moment is described analytically and may be included in the model directly, rather than using an uncertainty block characteristic of a robust-control scheme. Using traditional inertial sensing along with flow sensing to predict the aerodynamic mo1 Copyright c © 2017 by ASME ment yields improved results as compared to using inertial feedback alone. Flow sensing will be performed with two-port, foreand aft-facing probes that measure flow using differential pressure measurements [17]. Flight-dynamics applications often apply Euler angles [18] due to the intuitive nature of measuring each angle directly. However, Euler angles kinematics suffer singularities at gimbal lock and each angle is not necessarily constrained to the unit circle, which allows for non-physical error representations in the case of deviations over one rotation. We therefore use a geometric approach on the Lie group SO(3), which is a compact set representing the configuration space of the orientation of a rigid body. Additionally, a geometric description of the rotational kinematics in SO(3) does not encounter singularities, allowing for potentially global solutions. The contributions of this paper are (1) the addition of propeller aerodynamics to the quadrotor dynamics, yielding a more accurate description in the presence of wind disturbances; (2) a nonlinear, feedback-linearizing controller on SO(3) with saturated thrust inputs; and (3) an assessment of the relative merits of adding flow sensing to the vehicle controller versus using inertial feedback alone. We show improved stabilization through the use of flow sensing, which promises to allow for controlled flight in unfavorable weather and improved safety when flight is required in spite of weather concerns. The outline of the paper is as follows. The first section describes the dynamics of the quadrotor vehicle. The second section describes the controller and proves stability under bounded thrust constraints. The third section shows numerical results comparing the controller with and without thrust saturation, and with and without flow sensing. The final section summarizes the paper and discusses ongoing work. QUADROTOR DYNAMICS This paper investigates a quadrotor constrained to operate on an attitude stand, such that the quadrotor is mounted on a ball joint at its center of mass, allowing for full attitude motion while constraining the translational degrees of freedom. Let rotation matrix R ∈ SO(3) represent the orientation of the vehicle’s body frame with respect to the inertial frame and employ rigid-body kinematics and Euler’s second law to describe the rotational dynamics. We have Ṙ = RΩ̂ JΩ̇ =−Ω× JΩ+Mthrust +Maero, (1) where Ω= [p,q,r]T is the angular velocity of the quadrotor in the body frame, Mthrust is the moment acting on the system due to the thrust input, Maero is the aerodynamic moment on the system due to the interaction between the rotors and the wind, and J is the moment of inertia of the quadrotor. Here, J is a diagonal matrix due to the symmetry of the quadrotor; specifically [J1,J2,J3] = [m``/12+2mm`,m``/12+2mm`,m``/6+4mm`] where m` is the mass of one cross-beam of the quadrotor, ` is the length of one cross-beam, and mm is the mass of each motor. The wedge operator ∧ converts a vector in R3 to a 3×3 skew symmetric matrix in so(3), which can also be used to represent a cross product, such that for any vectors x and y in R3, x̂y = x×y. The inverse of the wedge operator is the vee operator ∨, which transforms a matrix in so(3) to a vector in R3.
Column Axial Load Effects on Performance of Panel Zone in Welded-Flange-Plate Connections
In this study the effects of different levels of column axial load on seismic performance of Welded Flange Plate (WFP) connections with different panel zone strength are presented. Finite element method was used to do this investigation and reliability of the modelling method was verified by comparison with experimental available records in documents. Fifteen sub-assemblages of WFP connections were generated by means of ABAQUS software. The aim was to enhance the knowledge about the consequence of column axial load level on cyclic behaviour and fracture potential of the connections with different panel zone strength for external joints in Steel Moment Resisting Frames (SMRFs). Results showed that cyclic behaviour including hysteresis loops and energy dissipation were not affected significantly by increasing column axial load regardless of the panel zone strength. It was also found that increase of the fracture potential due to increase in column axial load became less pronounced when the panel zone strength was decreased.