Effect of Wing Flexibility and Motor Dynamics on
Split-Cycle Control of Flapping Wing Micro Air Vehicles
Stephen M. Nogar∗, Andrea Serrani†, Abhijit Gogulapati‡, Jack J. McNamara§ The Ohio State University, Columbus, OH, 43210
Michael W. Oppenheimer¶and David B. Domank
U.S. Air Force Research Laboratory, Wright Patterson Air Force Base, Ohio 45433-7531
The effect of unmodeled unsteady aerodynamics, wing flexibility, wing inertia and mo-tor dynamics are evaluated on a closed-loop feedback controller for a flapping wing micro air vehicle that employs split-cycle and wing biasing mechanisms to shape the wing-beat waveform. The aeroelastic model combines a geometrically nonlinear reduced order struc-tural model based on implicit condensation with an unsteady aerodynamic model. The model-based controller tracks 4 degrees of freedom using proportional-derivative control and motor dynamics are controlled using a servocompensator that linearizes the actuator dynamics of the rigid model. Results indicate that dynamics neglected at the control de-sign level de-significantly degrade the ability of the controller to track a reference trajectory, and in some cases destabilize the closed-loop system. This demonstrates the importance of considering coupled aeroelastic and actuator dynamics in the design of flight controllers for flapping-wing micro air vehicles.
∗
Ph.D. Candidate, Mechanical and Aerospace Engineering, Student Member AIAA †
Professor of Electrical and Computer Engineering, Senior Member AIAA ‡
Post Doctoral Researcher, Mechanical and Aerospace Engineering, Member AIAA §
Associate Professor of Mechanical and Aerospace Engineering, Senior Member AIAA ¶
Senior Electronics Engineer, Associate Fellow, AIAA k
Nomenclature
Fnaeroˆ External aerodynamic forces acting on the vehicle, Fˆnaero∈ R3
Fninertˆ External inertial forces acting on the vehicle, Fninertˆ ∈ R3
Jveh Vehicle rotational inertia, Jveh ∈ R3×3
Mˆnaero External aerodynamic moments acting on the vehicle, Mnaeroˆ ∈ R3
Mˆninert External inertial moments acting on the vehicle, Mˆninert∈ R3
P Vehicle position [x, y, z] ¨
rnwcgˆ Wing acceleration vector relative to the vehicle center of gravity. V Vehicle velocity [u, v, w]
ω Vehicle rotational velocities [p, q, r] Φ Vehicle Euler angles [φ, θ, ψ] A Wing stroke flapping amplitude kL, kD Rigid lift and drag coefficients
L, D Rigid lift and drag used to determine the control derivatives
mveh Vehicle mass
Qm Motor generalized force
um Motor applied voltage
v Motor linearizing feedback term
x, y, z Vehicle Vertical, lateral, and forward/backward translation directions, respec-tively
φ, θ, ψ Vehicle yaw, pitch and roll directions, respectively
φR, ˙φR, ¨φR Reference wing position, velocity and acceleration from the inverse kinematics
θR, ˙θR, ¨θR Reference motor position, velocity and acceleration from the inverse
kinemat-ics
1. Introduction and Problem Statement
Flapping wing micro air vehicles (FWMAVs) hold significant promise due to the potential for improved aerodynamic efficiency, enhanced maneuverability and hover capability compared to fixed and rotary configurations [1–4]. The integrated nature of the vehicle and inherent actuation
limitations create significant technical challenges in vehicle implementation, including unsteady aerodynamics, geometrically nonlinear wing deformations, actuator dynamics and interactions with control systems [5–8]. Experimental analysis and high fidelity modeling has suggested that aeroelastic effects can change the effective kinematics of the wing, potentially impacting stability margins for the controlled vehicle dynamics [9, 10]. However, many control studies for flapping wing vehicles do not consider these effects and instead validate a proposed control strategy under simple modeling assumptions including rigid wings, quasi-steady aerodynamics and no consider-ation of actuator dynamics [11–15]. The goal of this study is to incorporate motor dynamics, wing flexibility, wing inertia and unsteady aerodynamics into an existing feedback control design for a representative flapping-wing MAV and assess the impact of these effects on vehicle performance and stability.
Previous studies indicate that the highly integrated nature of FWMAV dynamics affects the net forces generated by the wings. High fidelity analyses of rigid and flexible hover-capable flapping wing vehicles, while intractable for control design and analysis, indicate that unsteady aerody-namic effects and geometrically nonlinear structural behavior significantly alter the aerodyaerody-namic forces generated by the wing and the resulting net forces on the vehicle [9, 16–22]. Furthermore, unwanted interactions between the motor and aeroelastic wing dynamics can reduce the forces produced by the wings [23, 24]. These aspects of the problem could create a significant challenge for development of control schemes, since FWMAVs depend on achieving a desired behavior for specific kinematic parameters (e.g., split-cycle parameters [15], wing biasing [25–27] and wing pitch angle [28]) of the flapping wings. However, the inclusion of motor dynamics and aeroelastic effects can change the effective kinematics of the wing [29, 30], potentially reducing the effective-ness of control strategies that manipulate these parameters.
Due to the high computational cost, studies that consider wing flexibiilty often are limited to using open-loop techniques, which requires the computation of only a limited number of cycles. These studies have determined that the combined effects of unsteady aerodynamics and wing deformations exacerbate the inherent unstable characteristics of the vehicle dynamics, requiring feedback control to maintain stability [31–34]. Additionally, while most studies neglect wing mass, the contribution of wing inertia can also induce increased instability [35]. Su and Cesnick coupled a nonlinear beam formulation with a quasi-steady aerodynamic approximation and found that
aeroelastic effects contribute to instability of the vehicle [36]. Thus, the presence of unmodeled aeroelastic effects gives rise to potentially serious challenges for the design of robust flight con-trollers.
Model-based control methodologies play an important role in FWMAVs, due to the complex dynamics governing the motion of this class of vehicles. Model based techniques depend on the adoption of a suitable control design model (CDM), which is used to determine the control inputs needed to produce a desired force or moment by the wings. The CDM is often taken to be linear, and in the case of FWMAVs, cycle averaged to obtain a time-invariant representation out of inher-ently time-varying dynamics [6,11,15,32,33]. While these models allow for direct implementation of well-established control methodologies for linear time-invariant systems, their limitations in providing adequate support for control design for complex systems can not be underestimated. The vehicle dynamics must be linearized about specific flight conditions, which in the case of FW-MAVs, is typically hover. The performance of the closed-loop system may degrade significantly away from the linearization point. CDMs are often limited to employing simple, quasi-steady aerodynamics [11, 15]. Additionally, wing inertia is often neglected, contributing to model uncer-tainty [7]. Last, linear CDMs do not consider nonlinear interactions within the vehicle subsystems, such as aeroelastic effects or nonlinearities in the actuator dynamics, that prevent the vehicle from achieving the target wing kinematics. Due to significant discrepancies between the CDM and the physics of the vehicle, it is important to validate a given model-based controller design using a more accurate control evaluation model (CEM) that captures these unsteady, nonlinear and cou-pled effects.
A significant challenge in the development of control evaluation models is tractability, due to the requirement of computing time histories over large intervals in order to assess closed-loop performance. Many aeroelastic based solvers for flapping wings couple CFD to nonlinear finite element analysis (FEA) [9, 10, 37, 38]. However, these tools are too computationally expensive for long-term analysis and/or rapid prototyping of control schemes. Even approximate methods such as wake models [39–42] or panel methods [43] are still intractable for control evaluation pur-poses. The reduced-order models developed in [44–46] for hover-capable vehicles do not consider nonlinear wing deformation and unsteady aerodynamics.
mo-Fx Mx Fy My Fz Mz RW RW LW LW
(a) General diagram of the vehicle configuration
Motor
Gear-train Four-bar
linkage Flexible wing
(b) Schematic of the flapping wing mecha-nism.
Figure 1. Overview of the vehicle
tor dynamics, wing inertia and wing flexibility on vehicle control and stability has received only limited attention and remains an open area of research. The primary objective of this work study is to evaluate an existing control scheme in the context of these unmodeled effects using a re-cently developed comprehensive model that incorporates nonlinear wing deformations, unsteady aerodynamics, and actuator dynamics [23]. A second objective is to verify the suitability of the comprehensive model for use in control. The achievement of these objectives represents an im-portant step towards untethered control of flapping wing vehicles, and improved understanding on the importance of motor dynamics and wing flexibility on vehicle control and stability.
2. Overview of the Model
The representative hover-capable vehicle, depicted in Figure 1a, has a tailless configuration and is equipped with two wings that are actuated in a horizontal stroke plane. The goal of the wing actuation mechanism is to achieve the flapping waveform specified by the flight controller. The resultant aerodynamic loads are applied to the vehicle. This section provides an overview of the control evaluation model developed in [23] used to simulate the wing deformation and aerodynamic forces, and then describe the six degree of freedom (DOF) vehicle dynamics along with the external forces applied to the vehicle.
2.1. Motor-Wing Dynamics
The wing mechanism, shown in Figure 1b, is comprised of a motor, a gear-train, a flapping mech-anism and a flexible wing. A four-bar linkage is used to transform the rotational motion of the motor into a flapping motion. The equations of motion are derived using Lagrange’s equation, where each component is modeled individually, contributing to the kinetic and potential energy or a generalized force.
The equations of motion for the wing mechanism are written as
(1) Ja+ Jg+ Jl(θ) + Jwθθ (θ) Jθwξ(θ) Jξwθ(θ) Jξwξ ¨ θ ¨ ξ + Cm+ Cl κw(θ, ξ, ˙ξ) κw(θ, ξ, ˙ξ) Cξ ˙ θ ˙ξ + 0 0 0 K θ ξ + σl(θ) + σθw(θ, ξ) σξw(θ, ξ, ˙ξ) ˙ θ2= Qm+ Qθa(θ, ˙θ, ξ, ˙ξ) −Fnl+ Qξa( ˙θ, ξ, ˙ξ)
The degrees of freedom are the motor angle, θ, and the generalized coordinates of the wing, ξ ∈ RN, where N is the number of mode shapes. The terms J
a, Jg, Jl and Jw ∈ RN ×N are
the inertias of the motor armature, gear-train, linkage and wing, respectively. The term Cm
cap-tures the back-EMF force from the motor, Cl is the friction of the linkage and κw ∈ RN is the
centripetal dependent wing terms. The linear stiffness is represented by K ∈ RN ×N. The terms σl and σw ∈ RN captures the nonlinear kinetic energy of the linkage and wing, respectively. The
generalized forces applied to the system are represented by the motor torque, Qm, the nonlinear
stiffness, Fnl ∈ RN, and the generalized aerodynamic forces applied to the motor DOF (Qθa) and
the wing deformation (Qξa∈ RN). Coupling between the wing and motor dynamics is present in
the inertia terms Jw, σwand κwalong with the generalized aerodynamic force Qa.
Equation (1) yields N + 1 second-order ordinary differential equations, with N equations aris-ing due to the number of modes retained in the structural solution and the additional equation for the motor DOF. The total number of DOFs in the system is N + 1 + 2b after including the 2b states of the unsteady aerodynamics, where b is the number of elements contained in the aerodynamic mesh. These equations are time marched using an iterative Newmark-Beta integration scheme.
2.2. Vehicle Dynamics
The equations of motion for the vehicle are [47]
˙
P = RIˆn(Φ)V
mvehV = −m˙ vehω × V + Fnˆ− mvehgRnIˆeˆ1
˙
Φ = E(Φ)ω
Jvehω = −ω × J˙ vehω + Mnˆ
(2)
where the state variables are given by
P = x y z , V = u v w , Φ = φ θ ψ , ω = p q r (3)
The vehicle state is represented by the position of the center of mass for the vehicle in the body frame, P ∈ R3, the velocity in the body fixed frame, V ∈ R3, Euler angles, Φ ∈ R3, and the
rota-tional velocity in the in the body frame, ω ∈ R3. The rotation matrix RInˆ(Φ) ∈ R3×3transforms velocity into the body fixed frame representation, whereas the inverse of this relationship is de-noted by RnIˆ ∈ R3×3. The vehicle mass and inertia tensor are represented by mvehand Jveh∈ R3×3,
respectively. The external forces and moments acting on the vehicle are Fnˆ ∈ R3and Mnˆ ∈ R3,
re-spectively. Finally, E(Φ) ∈ R3×3is the Jacobian of the representation of the rotational kinematics.
In Eq. (2), the contribution of the wing to the vehicle mass and inertia tensor is neglected. How-ever, the inertial force generated by the acceleration of the wing is considered, and is described in the following section.
Two external forces act on the vehicle: the aerodynamic forces generated by the wings and the inertial forces due to the acceleration of the wing mass. These forces are expressed in the body-fixed frame to be incorporated in the vehicle equations of motion. The aerodynamic force and moment for a given wing is
Fnaeroˆ = Z A TndˆˆdFdaeroˆ dA Mnaeroˆ = Z A rnwcgˆ × Tnˆˆ ddF ˆ d aerodA (4)
where Tndˆˆ∈ R6p×6pis the transformation from the deformed wing frame to the body-fixed frame,
rnwcgˆ ∈ R6p is the vector of the wing in the body-fixed frame relative to the center of gravity
rnwcgˆ = ∆rncgˆ + rnwˆ. The inertial forces and moments due to the wing are defined respectively as
Fninertˆ = Z A ρ¨rnwcgˆ dA Mninertˆ = Z A rnwcgˆ × ρ¨rnwcgˆ dA (5)
where ¨rnwcgˆ , the wing acceleration vector, is defined as
¨
rnwcgˆ = Tˆsnˆr¨swˆ + 2 ˙Tsˆnˆr˙ˆsw+ ¨Tsˆˆnrswˆ (6)
and ρ ∈ R6p is the spatially-varying wing density. The external forces and moments are then summed for a single wing to yield
Fˆnw= Fnaeroˆ + Fninertˆ Mˆnw= Mnaeroˆ + Mninertˆ
(7)
The forces and moments for each wing are calculated independently, then modified to account for the position of the left and right wings. The vehicle is assumed to be symmetric about the X-Z plane. Each wing system is actuated independently, and the contribution of the forces and moments from the left wing are mirrored about the X-Z plane.
3. Control Implementation
An effective control methodology for flapping wing MAVs that aims at shaping the wingbeat waveform, known as split-cycle, was developed in [11]. This method describes the wingbeat using three parameters, the flapping frequency, a shift in the waveform that increases the flapping velocity in either the fore or aft portion of the stroke, and wing biasing. Using two independent actuators (the left and right wings), yields six independent control inputs for which the overall rigid-body model is controllable in the first approximation and the evolution of four DOFs can be assigned by feedback.
Table 1. Vehicle force conventions
Direction Force/Moment Description
x Fx Vertical translation/thrust y Fy Lateral translation z Fz Forward/backwards translation φ Mx Yaw θ My Pitch ψ Mz Roll
α. The pitching angle is prescribed in the rigid wing model, whereas for the flexible wing the pitching angle is determined by the deformation of the wing structure. The body-fixed coordinate frame is oriented such that the x axis in the body frame is oriented in the direction of thrust. The conventions for the forces and moments in the body fixed frame are listed in Table 1.
3.1. Split-Cycle Overview
The underlying mechanism providing controllability in flapping-wing vehicles is the modifica-tion of the wing-beat momodifica-tion by means of manipulamodifica-tion of the wing kinematics, in particular of the wing-beat angle, φ. Forces and moments are generated in the stroke plane using an asymmet-ric waveform, i.e., shortening the forestroke and lengthening the aftstroke, and vice versa. The asymmetric shift is determined by the parameter δ, shown in Figure 2a. Wing biasing introduces a deviation in the center of pressure forward or backward in the stroke plane, generating a pitching moment [11]. The bias is described using η, shown in Figure 2b and the waveform has been mod-ified to be continuous by adjusting the second half of the aftstroke such that the wing traverses from the current wing bias η0 to the bias of the next stroke η1. Lastly, the flapping frequency is
denoted by ω.
Following [11], the fore and aft strokes are described in 3 segments, the forestroke, and then the first and second halves of the aftstroke. The separation of the aftstroke accounts for the difference in wing bias between the current stroke and the next stroke. The resulting wing beat function is as follows
φF(t) = A cos [(ω − δ)t] + η, 0 ≤ t < ω−δπ
φA1(t) = A cos [(ω + σ)t + ξ] + η, ω−δπ ≤ t < ω−δπ +
φA2(t) = A(1 + ∆A) cos [(ω + σ)t + ξ] + η, ω−δπ + ≤ t < 2πω
Cycle
0 0.2 0.4 0.6 0.8 1
Flapping Angle (Degrees)
-60 -40 -20 0 20 40 60 ForestrokeAftstroke Split-Cycle Shift (δ > 0)
(a) Asymmetric split-cycle shift
Cycle 0 0.5 1 1.5 2 2.5 3 Angle (rad) -1 -0.5 0 0.5 1 1.5 0 1 Flapping Angle Previous Stroke Wing Bias Next Stroke Wing Bias (b) Wing bias Figure 2. Diagrams of (a) the asymmetric split-cycle shift and (b) the wing bias
where σ = δω ω − 2δ ξ = −2πδ ω − 2δ = 2π ω − π ω−δ 2 ∆A = η1− η0 (9)
The flapping angle A defines the flapping motion amplitude, determined by the linkage kinemat-ics, σ and ξ are used to maintain a continuous waveform, describes the period of the first half of the aftstroke, and ∆Ais the difference between the wing bias of the current stroke and the next
stroke. To maintain continuity, the wing bias for the stroke n + 1 is computed at the beginning of stroke n, such that the difference in bias can be accounted for at the end of stroke n. This effectively delays the wing bias application by one stroke. These waveforms are defined independently for the left and right wings, depending on the specified wingbeat parameters.
3.2. Control Design Model
A control-oriented model of the vehicle that is averaged and linearized is constructed based on the vehicle configuration where a rigid wing, quasi-steady aeroydnamics, no wing inertia, and no motor dynamics are assumed. The result of the control design model is a control effectiveness matrix, which relates control inputs to the forces and moments generated by the wings. This
model was originally developed in [11] and is summarized here. The control effectiveness matrix is constructed using the following steps:
1. Defining the instantaneous aerodynamic forces
2. Deriving the cycle averaged force for the 6 total forces and moments of the vehicle 3. Taking partial derivatives of the forces and moments with respect to each control input. First, the time accurate aerodynamic force is defined as
L = kLφ˙2 D = kDφ˙2 (10) where kL= 1 2ρCL(α)IAA 2 kD = 1 2ρCD(α)IAA 2 (11)
and ρ is the air density, CLand CD are the aerodynamic coefficients for lift and drag and IAis the
area moment of inertia of the wing. The next step in the is to compute the cycle averaged forces ¯
G, for each of the total 6 forces and moments. The average generalized force is computed as
¯ G = ω 2π Z 2π ω G(φ(t))dt (12)
where G is the force or moment to be averaged. The control effectiveness matrix, B, is defined by taking the partial derivatives of ¯Gabout the linearization point, selected as 0 for δ and ω0, the
hovering frequency, for ω.
The system of equations in Eq. (2) is linearized at the hover condition. The equilibrium condi-tion at hover entails V = [0 0 0], ω = [0 0 0], Φ = [0 0 0], whereas the posicondi-tion P at hover is left unspecified (but constant). Without loss of generality, the position at hover is selected at the origin of the inertial frame, that is, P = [0 0 0]. As far as the control input is concerned, the equi-librium condition entails ωRW = ωLW = ω0, δRW = δLW = 0and ηRW = ηLW = 0. Carrying out
the standard procedure for Jacobian linearization, one obtains the expression for the linearized dynamics, where the state and input variables denote variations from the equilibrium at hover. With a mild abuse of notation, the same variables as in Eq. (2) are used to denote the states and inputs of the linearized model, with the exception of the flapping frequencies, which are replaced
by the corresponding deviations from trim, ∆ωRW = ωRW − ω0and ∆ωLW = ωLW− ω0.
The linearized dynamics assumes the following form
˙ x = u ˙ u = B11∆ωRW+ B11∆ωLW ˙ y = v ˙v = gψ ˙ z = w ˙ w = −gθ + B31δRW + B31δLW ˙ φ = p ˙ p = B41δRW − B41δLW ˙ θ = q ˙ q = B51δRW + B53δLW + B55ηRW + B55ηLW ˙ ψ = r ˙r = B62∆ωRW− B62∆ωLW (13) where B11= kLA2ω0 m B31= 1 mkDAJ1(A)ω0 B41= − 1 2Jxx
kDAω0(AycpW P + wJ1(A))
B51=
1 Jyy
AJ1(A)ω0kLxW Pcp cos α + kD(xW Pcp sin α + ∆xBR)
B53=
1 Jyy
AJ1(A)ω0kLxW Pcp cos α + kD(xW Pcp sin α + ∆xBR)
B55= 1 Jyy Aω20ycpW PkLJ1(A) B62= 1 Jzz 2kLω0 AyW Pcp J1(1) + wA2 4 (14)
Letting u =
δRW ∆ωRW δLW ∆ωLW ηRW ηLW
one obtains the control effectiveness ma-trix B = 0 B11 0 B11 0 0 0 0 0 0 0 0 B31 0 B31 0 0 0 B41 0 −B41 0 0 0 B51 0 B53 0 B55 B55 0 B62 0 −B62 0 0 (15)
The hovering frequency, ω0, is a function of the vehicle weight and aerodynamic thrust. It is
important to note that no direct actuation for ∆ ˙v is possible, but translation in the y direction can be achieved by rolling the vehicle and then translating. The sparsity of the matrix leads to im-proved control by a strategy that aims at minimizing coupling in the system. However, coupling is still present in ∆ ˙wand ∆ ˙q, which must be accounted for in the control design.
3.3. Controller Synthesis
A general block diagram depicting the control evaluation strategy is shown in Figure 3. It is assumed that a flight controller is available to generate appropriate wing beat commands. A reference trajectory is specified in the earth-fixed frame of reference for the flight controller to track. The flight controller computes the difference between the current state and the target state, and commands the target wingbeat for the right (φRW) and left (φLW) wings needed to stabilize
the vehicle and track the target trajectory. The control evaluation model computes the forces and moments generated by each wing. The vehicle dynamics block in Figure 3 represents the vehicle dynamics in the earth-fixed frame, providing the state of the rigid body dynamics.
The controller adopted in this study is defined in [11] and uses a split-cycle wingbeat with wing bias. The control inputs that determine the wingbeat are the split-cycle parameter δ, change in flapping frequency from hover ∆ω, and wing bias η. Each of these are defined independently for each wing, resulting in 6 control inputs. The desired forces and moments are used as inputs to the control design model, which yields the control inputs, as shown in Figure 4. Each controlled DOF is modeled as a damped oscillator and tuning is determined by adjusting stiffness and damping coefficients. The stiffness (ωx), determines the proportional gain, while damping (ξxωx), is the
Control Evaluation Model Left Wing Control Evaluation Model
Right Wing Vehicle
Dynamics Flight Controller Generate LW(t) Control Logic Control Design Model Generate RW(t) RW Parameters ParametersLW
-+ Reference Trajectory E P, ,V, FRW, MRW LW(t) RW(t) + + FLW, MLWFigure 3. General block diagram of the closed loop control system for the flapping wing MAV.
Table 2. Tuning parameters for each degree of freedom.
Degree of freedom Value
ωF x 10 ξF x 0.5 ωF z 1 ξF z 1 ωM x 10 ξM x 1 ωM y 20 ξM y 0.5 ωM z 10 ξM z 1.5
derivative gain. The gains and damping chosen for all DOFs are listed in Table 2. A second order filter is used on each tracked DOF to provide improved representation of the cycle averaged dynamics.
This controller is capable of controlling directly 5 DOFs of the vehicle. Specifically, the position (x, z) in the longitudinal plane in the inertial coordinate system and the vehicle heading ψ are directly controlled to track given reference trajectories. Tracking of the position y in the inertial frame is achieved indirectly through rolling and translating. While it theoretically possible to actuate z and θ independently, due to the limited ability of the split-cycle mechanism to produce a force in F z, these DOFs are effectively coupled. Actuation of coupled DOFs are achieved by specifying a target rotation angle based on the error in the translational direction, i.e. error in z is
Figure 4. Control design implementation [11].
transformed into an error in pitch, such that a pitching command is given to correct the z error. The block diagram of the control law yielding a desired translation command in the z direction is shown in Figure 5. While this method accounts for coupling within the vehicle, it neglects the body force produced by the split-cycle parameters, hence the coupling between pitching moment and translational dynamics.
3.4. Inner-Loop
Due to the inclusion of the motor dynamics, an inner-loop controller is required to track a desired waveform for the wingbeat angles, φRW and φLW, by specifying the motor input voltage. A block
diagram of the motor controller is shown in Figure 6. The controller design is accomplished in two steps. First, the kinematics of the linkage is inverted so that the target waveform can be converted to a target motor angular position. Second, the controller tracks the target position
+ z zdes des qdes q w wdes z 2 2z z 1 g + 2 2 Iyy Mydes + + +
Figure 5. Block diagram used to determine the desired pitching moment.
Wing Dynamics Motor Controller MAV Dynamics um ,
F, M
Invert Kinematics R R-
+
Linearization by Feedback vFigure 6. Inner-loop block diagram that includes the inverse kinematics and motor controller.
using a linearization by feedback approach.
3.4.1. Inverse Kinematics
The inverse kinematics are derived in a similar fashion as the forward kinematics, but solving for the reference crank angle, θRl, instead of φR. However, the solution for θRl is not unique, so
measures must be taken to track the correct solution, depending on the motor state in the previous timestep. The parameters are the same as those defined in [23], and a description of the terms are listed in Table 3.
Table 3. Linkage parameters used in inverse kinematics derivation. Description Variable Linkage 0 length l0 Crank length l1 Coupler length l2 Linkage 3 length l3 Gear-ratio Γ Linkage 1-2 angle αl
Figure 7. Diagram of the drivetrain system [24]
The closure equations for the linkage shown in Figure 7 are as follows
l1cos θRl+ l2cos αl= l0+ l3cos φR
l1sin θRl+ l2sin αl= l3sin φR
(16)
where the crank angle is related to the motor angle by
θRl = ΓθR (17)
Solving for αlyields
l2cos αl = l0+ l3cos φR− l1cos θRl
l2sin αl = l3sin φR− l1sin θRl
(18)
in Eq. (18) and setting the terms containing αlequal to 1, yielding
H1(φR) = −2l1l3sin φR
H2(φR) = −2l1(l0+ l3cos φR)
H3(φR) = l20+ l21− l22+ l32+ 2l0l3cos φR
(19)
This sets up a quadratic equation of the form
(H3(φR) − H2(φR))τ2+ 2H1(φR)τ + (H3(φR) + H2(φR)) = 0 (20) where τ = tan(θRl/2) sin θRl = 2τ 1 + τ2 cos θRl = 1 − τ2 1 + τ2 (21)
Solving the quadric equation and substituting for θRlyields the reference motor position
θR(φR) = ΓθRl(φR) = 2Γ arctan 2 −H1(φR)s +pH1(φR) 2− H 3(φR)2+ H2(φR)2 H3(φR) − H2(φR) ! (22)
The reference flapping angle, φR, is not unique, as at any given angle it could either be in the fore
or aft portion of the stroke. Thus, θR is also not unique. To account for this non-uniqueness, a
parameter is introduced to keep track of the fore or aft stroke. Additionally, Eq. (22) is restricted to the domain between 0 and 2π, thus a counter is implemented to maintain continuity at these limits and is incremented once every rotation. The rotational velocity of the motor, ˙θR, is obtained
as ˙ θR= T2(φR) ˙φR= Γ l1sin(αl− φR) l2sin(αl− ΓθRl) (23) 3.4.2. Motor Control
Motor control is achieved using a linearization by feedback scheme. The equations of motion for the motor are solved for the motor voltage, um, which is regarded as the overall control input.
trajectory θR, computed in the previous section, which yields the desired reference for the wing
beat angle, φR. Since it is not possible to observe the wing deformation, the wing is assumed to be
rigid and flexibility introduces a disturbance load to be rejected by the controller. First, the equations describing the motor dynamics are recalled from Eq. (1)
˙ θ1 = θ2 ˙ θ2 = J (θ)−1 h C ˙θ + σ(θ) ˙θ2− Qm(u) − Qa(θ, ˙θ) i (24) where Qmis Qm = KT Ra um (25)
The generalized coordinates of the wing deformation are set to zero, eliminating the deformation dependent terms. The linearizing control is provided by
um = Ra KT h C ˙θ + σ(θ) ˙θ2− Qa(θ, ˙θ)i+J (θ)Ra KT v (26)
where v is a tracking control defined as
v = ¨θR+ a0(θ − θR) + a1( ˙θ − ˙θR) (27)
where a0and a1 are the negative gain parameters. The reference signals, θR, ˙θR, are determined
using Eqs. (22) and (23), respectively. The feed-forward term, ¨θR, is determined numerically by
high-pass filtering of ˙θR. This reduces the system equations to
˙ θ1= θ2
˙
θ2= ¨θR+ a0(θ − θR) + a1( ˙θ − ˙θR) + d(θ, ˙θ)
(28)
where the disturbance term d(θ, ˙θ) represents the effect of model approximation.
4. Results and Discussion
Two studies are presented in this section. First the suitability of the dynamics model developed in [23] for use in control evaluation is verified. Second, closed-loop simulations are presented, demonstrating the impact of motor dynamics and wing flexibility on vehicle performance and
Right Wing Dynamics Generate Generate RW(t) LW(t) Left Wing Dynamics RW RW RW LW LW LW + + F, M
Figure 8. Diagram of the open-loop system. Results are generated by prescribing control inputs and measuring cycle averaged forces.
Table 4. Control inputs and degrees of freedom used in generating open-loop results.
Left and Right Wing
DOF Control Input Symmetric/Differential Range ¯ Fx ∆ω Symmetric −0.3ω0 ≤ ∆ω ≤ 0.3ω0 ¯ Fy N/A N/A ¯ Fz δ Symmetric −0.3ω0≤ δ ≤ 0.3ω0 ¯ Mx δ Differential −0.3ω0≤ δ ≤ 0.3ω0 ¯ My η Symmetric −0.175 ≤ η ≤ 0.175 ¯ Mz ∆ω Differential −0.3ω0 ≤ ∆ω ≤ 0.3ω0 stability.
4.1. Open-loop Dynamics Verification
An important consideration is the accuracy of the control design model against the control eval-uation and high fidelity models. This serves the purpose of verifying the appropriateness of the model simplifications for the generation of the control forces and moments, and revealing the ex-tent of the inherent nonlinear effects that add to the model uncertainty. The "open-loop" results are generated by prescribing input control parameters and measuring the cycle-averaged forces and moments, as shown in Figure 8. The decoupled nature of the model in Eq. (15) allows a primary control input to be specified for each force or moment acting on the vehicle, as listed in Table 4. Note that it is not possible to generate forces in ¯Fy, thus no control input is provided. The
left and right wing symmetric/differential column indicates whether the specified control input is actuated with the same sign on both wings, or if opposite signs are used. All results in this section are flapped at a hovering frequency of 50 Hz.
Table 5. Summary of the dynamic effects included in the open-loop models for the rigid wing cases shown in Figure 9.
Case Aerodynamics Inertia Flexibility
Control Design Model Quasi-steady, Linearized No No
Simple Quasi-steady No No
Enhanced Unsteady ROM Yes No
DVM Quasi-steady + Unsteady Wake Yes No
4.1.1. Rigid wing with Unsteady Aerodynamics and Wing Inertia
Open-loop simulation results for a rigid wing configuration are shown in Figure 9. The primary control input for each force and moment coincides with the x axis, and the resulting averaged force or moment with the y axis. In the figure, the green line indicates the simple control eval-uation model, which only includes the quasi-steady aerodynamics and represents the time accu-rate version of the control design model, while the enhanced control evaluation model includes unsteady aerodynamics and wing inertia. The red line refers to results obtained with the high fidelity model employing the discrete vortex method [39]. These models are listed in Table 5.
All models perform qualitatively similar to the control design model, indicating that nonlinear effects are not prominent for the rigid wing configuration. Both the enhanced model and DVM exhibit increased thrust production in ¯Fx, indicating that the simple control evaluation model and
the control design model underestimate thrust. Small deviations from the control design are ob-served for negative control inputs in ¯Fz. This may be due to the split-cycle waveform description,
as prescribing a negative split-cycle shift does not shift the mid-stroke period as much as a posi-tive shift. The DVM model exhibits reduced moment production in yaw ( ¯Mx). This may be due to
unsteadiness or wing inertia, as this effect is partially captured by the enhanced control evaluation model, but is not exhibited by the simple control evaluation model.
4.1.2. Flexible Wing
Open-loop results are also generated for a flexible wing configuration with a natural frequency of 149 Hz. The open loop results for this configuration are shown in Figure 10 and the models used by each result are listed in Table 6. This configuration has reasonable qualitative agreement to the high fidelity FEA+DVM model. Small differences are observed in both ¯Fzand ¯Mx, which is
dependent on the split-cycle parameter δ. Additionally, the enhanced model under predicts thrust in ¯Fx. Results for the enhanced model suggest that the latter may be a reasonable representation
Control Input (-0.2 0 "!/0.2!0) 7 Fx (N ) -0.2 -0.1 0 0.1 0.2 0.3 0.4 Control Input -0.2 0 0.2 7 Fy (N ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Control Input (//!0) -0.2 0 0.2 7 F(Nz ) -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Control Input (//!0) -0.2 0 0.2 7 Mx (N m ) -0.015 -0.01 -0.005 0 0.005 0.01 0.015 Control Input (2) -0.2 -0.1 0 0.1 0.2 7 My (N m ) #10-3 -5 -4 -3 -2 -1 0 1 2 3 4 5 Control Input (-0.2 0 "!/!0.20) 7 Mz (N m ) -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02
Simple Enhanced DVM Control Design Model
Figure 9. Open-Loop results computing the cycle averaged force or moment for each DOF. The simple model includes only quasi-steady aerodynamics while the enhanced model includes unquasi-steady aerodynamics and wing inertia. The control input is specified on the x axis, and the cycle averaged force or moment is on the y axis.
of the high fidelity model for this configuration.
Results for a more flexible configuration with a natural frequency of 68 Hz are shown in Fig-ure 11. While the cycle averaged forces in ¯Fx are qualitatively similar, significant errors are
ob-served in ¯Fzand ¯Mx. Surprisingly, the FEA+DVM model predicts an opposite force in ¯Fzand
mo-ment ¯Mx compared to the CDM. This behavior is known as control inversion. This effect would
likely destabilize a controller based on the CDM, as the vehicle dynamics are generating the op-posite force or moment than predicted by the CDM. Both of these DOFs are dependent on the split-cycle parameter δ, suggesting that specific kinematics can be disproportionately affected by significant wing flexibility. This result also demonstrates the wide variety of dynamics that are exhibited by flapping wings by modifying the hovering frequency of the vehicle (e.g., by adding a payload thereby increasing the vehicle mass), and the importance of understanding its effect on
Control Input ("!/!0) -0.4 -0.2 0 0.2 0.4 7 Fx (N ) -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Control Input 0 0.5 1 7 F(Ny ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Control Input (//!0) -0.4 -0.2 0 0.2 0.4 7 Fz (N ) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Control Input (//!0) -0.4 -0.2 0 0.2 0.4 7 Mx (N m ) -0.015 -0.01 -0.005 0 0.005 0.01 0.015 Control Input (2) -0.2 -0.1 0 0.1 0.2 7 My (N m ) #10-3 -5 -4 -3 -2 -1 0 1 2 3 4 5 Control Input ("!/!0) -0.4 -0.2 0 0.2 0.4 7 Mz (N m ) -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02
Enhanced FEA+DVM Control Design Model
Figure 10. Open-Loop results computing the cycle averaged force or moment for each DOF for the flexible configuration.
vehicle controllability.
The enhanced model does not exhibit control reversal, and instead predicts significantly re-duced forces and moments, which would still introduce challenges in achieving vehicle controlla-bility. Additionally, this shows that use of the control evaluation model is restricted to configura-tions in which the flapping frequency does not approach the natural frequency of the wing. This is consistent with the validation of the control evaluation model in [23].
4.2. Closed-Loop Performance
Evaluation of the controller is performed by selectively enabling unsteady aerodynamics, wing inertia, flexibility and motor dynamics in the evaluation model. These additional dynamic effects, unaccounted for in the control design model, add uncertainty to the closed-loop system. Conse-quently, it is of importance to evaluate the degree of robustness provided by the controller against
Table 6. Summary of the dynamic effects included in the open-loop models for the flexible wing cases shown in Figures 10 and 11.
Case Aerodynamics Inertia Flexibility
Control Design Model Quasi-steady, Linearized No No
Enhanced Unsteady ROM Yes Nonlinear ROM
FEA+DVM Quasi-steady + Unsteady Wake Yes Nonlinear FEA
Control Input ("!/!0) -0.4 -0.2 0 0.2 0.4 7 F(Nx ) -0.2 -0.1 0 0.1 0.2 0.3 0.4 Control Input 0 0.5 1 7 F(Ny ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Control Input (//!0) -0.4 -0.2 0 0.2 0.4 7 F(Nz ) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Control Input (//!0) -0.4 -0.2 0 0.2 0.4 7 Mx (N m ) -0.015 -0.01 -0.005 0 0.005 0.01 0.015 Control Input (2) -0.2 -0.1 0 0.1 0.2 7 My (N m ) #10-3 -6 -4 -2 0 2 4 6 8 Control Input ("!/!0) -0.4 -0.2 0 0.2 0.4 7 Mz (N m ) -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025
Enhanced FEA+DVM Control Design Model
Figure 11. Open-Loop results computing the cycle averaged force or moment for each DOF for the flexible L3B1 configuration.
these. Listed in Table 7 are the parameters for the vehicle. The motor and linkage parameters are the same as those in [23].
The target trajectory, shown in Figure 12, was selected as a representative case study. The vehicle is commanded to start at the origin of the earth-fixed frame and to move forward by 1 m, followed by a rotation of 90◦about the x axis. The vehicle is then commanded to simultaneously move upward and forward by 1 m ending at the point (1, −1, 1). The reference trajectory for each segment of the commanded motion is prescribed as a filtered step function. The total time of the simulation is 20 s.
Table 7. Closed-loop vehicle parameters.
Description Variable Value Units
Vehicle mass mveh 25 g
Height h 10 cm
Width w 3 cm
Depth d 1 cm
Wing area moment of inertia IA 416 cm4
Wing radius R 7.5 cm
Right wing offset ∆rB
R [5 1.5 0] cm
Left wing offset ∆rLB [5 -1.5 0] cm Rigid wing pitch angle α0 25◦ deg
−0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 −0.5 0 0.5 1 1.5 y (m) z (m) x (m) Start End
Figure 12. Commanded trajectory in the earth fixed frame.
4.2.1. Unsteady Aerodynamics and Wing Inertia
The controller is first evaluated with the addition of unsteady aerodynamics and wing inertia with a rigid wing and no motor dynamics. Studying these effects first will establish the robustness of the controller against relatively benign model uncertainty, before more prominent nonlinear and unmodeled dynamic effects like wing flexibility and motor dynamics are taken into consideration. The simple CEM is the time accurate version of the CDM, with only quasi-steady translation, while the enhanced CEM considers unsteady aerodynamics and wing inertia effects. For both CEMs, a flapping frequency of 46.9 Hz is assumed. The abbreviations listed in the table are provided for reference and are used throughout the remainder of this work.
The closed-loop simulation results are shown in Figure 13, including the tracked vehicle state variables and the corresponding control inputs. The simulations exhibit steady state error for the y
Table 8. Results of simple vs enhanced CEM simulations.
Case Uns Inert Flex Motor Performance
Simple CEM × × × × Good
Enhanced CEM X X × × Fair
coordinate in the earth fixed frame. Due to the vehicle rotation, this is an offset in the forward and backward direction in the vehicle reference frame. Tracking the initial forward motion command appears to be a challenging task for the controller, possibly due to the dynamic coupling between translational and rotational motion. These effects are exacerbated by the enhanced CEM, due to added uncertainty of the forces and moments generated by the wings. Tracking errors are also present in the yaw (φ) response, possibly due to the contribution of wing inertia. Wing inertia could be a large source of uncertainty, since yaw tracking depends on the use of the split-cycle mechanism; as prescribing an asymmetric waveform generates a net moment due to inertia, in addition to the aerodynamic force, which is not accounted for in the CDM.
The results of each study will be henceforth qualitatively labeled as good, fair and unstable. For the case study under consideration, the evaluation with the simple CEM is labeled as "good," due to the fact that the closed-loop system remains stable, with a relatively small steady-state error exhibited by the regulated outputs. Results obtained with the enhanced CEM are labeled as "fair," due to the significant tracking error seen over large portions of the trajectories. The results are listed in Table 8. Note, that in Table 8 the labels for the dynamic effects have been abbreviated.
4.2.2. Wing Flexibility
The effect of wing flexibility across a range of flapping frequencies on the vehicle tracking per-formance is shown in Figure 14. The hovering frequencies are determined by varying the vehicle mass, as listed in Table 9. No motor dynamics are considered in these cases. The configurations with slower flapping frequencies exhibit the same steady state offset at the end of the simulation in the output y as the rigid wing case. This may be due to tracking error in forward translation, as significant errors are observed for the output z for the first 15 s of the simulation. The 64 Hz flapping frequency exhibits instability after performing the 90◦ yaw maneuver. Due to the build up of errors in multiple DOFs, the controller is unable to correct the errors simultaneously. This may be due to changes in the aerodynamic forces applied to the vehicle caused by flexibility. A
(a) Vehicle state in earth-fixed frame.
(b) Control inputs.
Figure 13. Closed-loop simulation of the controller and rigid wings comparing the simple and enhanced CEMs. The coordinates x, y, and z are in the earth fixed frame.
Table 9. Vehicle mass and hovering frequencies for the flexible wing simulations shown in Figure 14.
Vehicle Mass (g) Hovering Frequency (Hz)
25 44
34 51
52 64
Table 10. Summarized performance of the flexible wing configurations for the controller.
Flapping Frequency Uns Inert Flex Motor Performance
44 Hz X X X × Fair
51 Hz X X X × Fair
64 Hz X X X × Unstable
summary of the performance obtained in these cases is listed in Table 10.
Further analysis is provided by the open loop results for the 44 Hz and 64 Hz flexible config-urations, shown in Figure 15. Open-loop results help diagnose the source of instability from the analysis of the cycle averaged output corresponding to a given input trajectory. The dashed lines in Figure 15 represent the linear control design model, i.e., the predicted forces for a given control input. The solid lines are determined from the cycle averaged forces from the control evaluation model for each control input. It is important to note that the control design model changes depend-ing on the flappdepend-ing frequency, as increased flappdepend-ing frequencies leads to increased aerodynamic forces. This can be seen in Fx, as the flapping frequency increases, more thrust is produced due to
the coupling of increased flapping frequency resulting in an increased pitching angle caused by the aerodynamic loading deforming the wing. The DOFs that depend on the split-cycle parame-ter δ produce less force in Fz and yaw moment Mx for the 64 Hz flapping frequency case. This
is due to the aerodynamic loading deforming the wing, and reducing the forces and moments produced in these DOFs. The large errors between the CDM and CEM in the 64 Hz case produce uncertainty in relating the control inputs to cycle averaged forces and moments, destabilizing the vehicle. However, despite the reduction in forces, the vehicle still maintains controllability, as changes in the forces and moments are still produced in Fz and Mx, albeit with reduced
magni-tude compared to the CDM. Thus, the controller performs poorly when the forces and moments exhibit uncertainty due to flexibility.
(a) Vehicle state in earth-fixed frame.
(b) Control inputs.
Control Input (-0.2 0 "!/0.2!0) 7 Fx (N ) -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Control Input -0.2 0 0.2 7 F(Ny ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Control Input (//!0) -0.2 0 0.2 7 Fz (N ) -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Control Input (//!0) -0.2 0 0.2 7 Mx (N m ) -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 Control Input (2) -0.2 -0.1 0 0.1 0.2 7 My (N m ) #10-3 -8 -6 -4 -2 0 2 4 6 8 Control Input ("!/!0) -0.2 0 0.2 7 Mz (N m ) -0.03 -0.02 -0.01 0 0.01 0.02 0.03 44 Hz - CEM 44 Hz - CDM 64 Hz - CEM 64 Hz - CDM
Figure 15. Open-loop results of the 44 Hz and 64 Hz flexible configurations. The solid lines are the control evaluation model and the dashed lines are the control design model.
4.2.3. Limited Motor Torque
A fundamental consideration for flapping wing vehicles is the ability of the wing to attain the desired motion. If the motor is undersized, the ability of the inner-loop controller to reproduce the target waveform is reduced. The motor torque is limited by a saturation to the voltage prescribed to the motor via the motor controller, which limits the torque that can be produced. However, it should be noted that this method of limiting the motor torque is not physically representative of motors of varying size and torque, as motor properties such as armature inertia, back-EMF and torque constants are not varied. Additionally, the wing bias is modeled without consideration of the mechanism necessary to rotate the motor and linkage assembly, and is directly prescribed. This is a reasonable assumption, since this mechanism could be implemented using a simple servo motor, and has minimal coupling with the motor and aeroelastic dynamics.
Table 11. Summarized performance of the limited motor torque configurations.
Motor Saturation (V) Uns Inert Flex Motor Performance
5 X X × X Unstable
7 X X × X Unstable
12 X X × X Fair
None X X × × Fair
Configurations of ±5 V, ±7 V, ±12 V and no saturation are shown. These results assume a rigid wing at a 46.9 Hz flapping frequency. In both the 5 V and 7 V cases, the closed-loop systems are unable to maintain hover, and become unstable. The 12 V case performs similarly to the unsaturated case, indicating that the motor is adequately sized. A summary of these results are listed in Table 11. These results demonstrate the importance of considering motor limitations when selecting control inputs, as parameters that include modulating the wingbeat increase the torque required to generate the wingbeat and achieve closed-loop stability.
Open-loop results are presented in Figure 17. Each motor configuration can achieve the desired thrust in Fx, illustrating that an adequate thrust-to-weight ratio of the vehicle is easily achieved by
each motor configuration. However, the control input effectiveness in Fz and Mx, which depend
on the split-cycle parameter, are reduced. In the 5 V case, effectively no yaw moment is produced regardless of the control input. Interestingly, as the magnitude of δ/ω0 is increased, the yaw
mo-ment produced actually decreases, indicating that inner-loop controller dynamics, specifically the behavior of the controller once the input has been saturated, may be a factor. From these results, it is apparent that increased torque is required to achieve the split-cycle wingbeat, due to the quick actuation required to shorten either the fore or aft stroke. This result is intuitive, as increasing the velocity of the fore or aft stroke would increase the aerodynamic loading on the wing, increasing the required motor torque.
4.2.4. Limited Motor Torque with Wing Flexibility
Qualitative results for simulations with limited motor torque and wing flexibility are listed in Table 12. The flapping frequency of the wing is 51 Hz. The 12 V configuration is unstable, despite stability exhibited by both the 51 Hz flexible case and the 12 V saturation case. This demonstrates the coupling effects result in reduced closed-loop performance, due to uncertainty in the forces and moments generated by the wing. Additionally, increased control effort is necessary by the
(a) Vehicle state in earth-fixed frame.
(b) Control inputs.
Figure 16. Closed-loop simulations with varying motor torque. Available motor torque is determined by saturating the motor voltage.
Control Input (-0.2 0 "!/0.2!0) 7 Fx (N ) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Control Input -0.2 0 0.2 7 Fy (N ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Control Input (//!0) -0.2 0 0.2 7 F(Nz ) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Control Input (//!0) -0.2 0 0.2 7 Mx (N m ) -0.015 -0.01 -0.005 0 0.005 0.01 0.015 Control Input (2) -0.2 -0.1 0 0.1 0.2 7 My (N m ) #10-3 -5 -4 -3 -2 -1 0 1 2 3 4 5 Control Input (-0.2 0 "!/!0.20) 7 Mz (N m ) -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 5 V 7 V 12 V No Saturation CDM
Figure 17. Open-loop results of the configurations with limited motor torque.
motor controller to achieve the target kinematics due to the coupled interactions. The controller does not achieve "fair" performance until the saturation is increased to 25 V, approximately double the required voltage compared to a rigid wing. Thus, dynamic coupling within the vehicle is a challenge, and requires additional actuator capacity to account for unmodeled dynamics in the control evaluation model and maintain closed-loop stability.
4.2.5. Summary of Results
Ultimately, the added uncertainty of the enhanced control evaluation model compared to the con-trol design model yields poor performance and instability for the ensuing system. This is evident in the reduced tracking performance of the controller with the enhanced CEM compared to the simple CEM. In the flexible cases, increased flapping frequencies lead to increased wing deforma-tions, producing large errors between the CDM and CEM. The controller is unable to stabilize the
Table 12. Summarized performance of the limited motor torque with wing flexibility configurations.
Motor Saturation (V) Uns Inert Flex Motor Performance
12 X X X X Unstable
20 X X X X Unstable
25 X X X X Fair
vehicle in the presence of large uncertainty in the forces and moments. When the motor torque is limited, further sources of instability are introduced due to the inability of the motor to repro-duce the split-cycle waveform. In these cases, some degree of controllability of the cycle averaged forces is maintained, and it is reasonable to assume that a controller with improved robustness could achieve improved closed-loop performance, without modifying the control design model. However, in the combined limited motor torque and flexible wing cases, the additional uncer-tainty yields a significant reduction in performance, requiring a considerable increase in actuator authority to stabilize the vehicle.
5. Conclusions
The study investigates the performance of a comprehensive vehicle dynamics model in a con-troller based on a split-cycle approach. Geometrically nonlinear wing deformation, which arises due to a combination of the flapping motion and the unsteady aerodynamic loading, is modeled using an implicit condensation approach. Important conclusions from this study are as follows:
1. Investigations reveal that moderate-to-large wing deformation increases the overall force generated by the wings but reduces the component of the aerodynamic force in the stroke plane. This leads to vehicle instability due to reduced effectiveness in the yaw moment. 2. By design, the split-cycle requires greater torque from the motor compared to a symmetric
waveform. Thus undersized motors are found to produce vehicle instability due to reduced control effectiveness.
3. In some cases controllability is maintained despite a noticeable reduction in control effec-tiveness. This highlights the necessity of control techniques that maintain control while accounting for uncertainties.
The completion of this study represents an important step in the control of flapping wing flight, and highlights the importance of considering wing flexibility and motor dynamics in vehicle trol and design. The inclusion of dynamics in the control evaluation model unmodeled in the con-trol design model exacerbates the inherent vehicle instability. In particular, the work presented here motivates the development of robust control techniques that are suitable for vehicle control in the presence of unmodeled dynamics.
Acknowledgment
This research is funded by the AFRL/DAGSI Ohio Student-Faculty Fellowship along with the AFRL summer internship program. The authors would like to thank Mr. Gregory Knapik and the Biodynamics Laboratories at the Ohio State University for access to their computational resources. This work is also supported in part by an allocation of computing time from the Ohio Supercomputer Center.
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