Flight Dynamics Conceptual Design Exploration of Multirotor eVTOL

Flight Dynamics Conceptual Design Exploration of Multirotor eVTOL

Carlos Malpica Peter Suh Christopher Silva Aerospace Engineer Aerospace Engineer Aerospace Engineer Aeromechanics Branch Flight Controls & Dynamics Branch Aeromechanics Branch NASA Ames Research Center NASA Armstrong Flight Research Center NASA Ames Research Center Moffett Field, CA, USA Edwards, CA, USA Moffett Field, CA, USA ABSTRACT The advent of electric propulsion is revolutionizing the paradigm of rotorcraft design, leading to new electric Vertical Take-Off and Landing (eVTOL) aircraft. Direct drive topologies are common within these new designs, and some designers have chosen to utilize this mechanism for Primary Flight Control (PFC), effectively utilizing the aircraft engines as PFC actuators to control the speed of the rotors. This decision integrates the propulsion and flight control systems, and intrinsically couples the aircraft sizing and control. Four separate tools were exercised throughout this study to conduct a conceptual design exploration of eVTOL aircraft handling qualities. The main tasks for these tools were: 1) aircraft sizing and performance analysis, including the calculation of trim; 2) flight dynamics modeling and analysis; 3) handling qualities-centric control law optimization; and 4) electric motor sizing. Sizing of an RPM- controlled Hexacopter concept explored the dependency of aircraft size to fundamental design parameters: 1) disk loading and 2) blade loading coefficient. Increasing the design disk loading resulted in designs with high agility, but at the cost of significant growth in the design gross weight. Finally, Categories II and III pilot-induced-oscillation (PIO) are known potentially-catastrophic handling qualities deficiencies of fly-by-wire flight control systems such as those expected of eVTOL aircraft. Consideration of PIO predictive metrics in conceptual design control synthesis led to increased PIO robustness.

M Motor modulation ratio NOTATION M , M Pitch damping and torque control derivatives q Q i Symbols n Number of aircraft components c A Total rotor disk (reference) area (ft ) n Number of motor pole pairs ref p A Linear state-space model stability matrix n Number of rotors r B Linear state-space model control matrix N , N Yaw damping and torque control derivatives r Q i c Control allocation mixing coefficient p Body x -axis angular rate (rad / s) i , input c Motor viscous damping coefficient P Power available per engine group (hp or kW) μ av C / σ Design blade loading, W / ρ A V σ P Motor power heat losses (hp or kW) W D ref H tip C ( s ) MIMO command model matrix P Motor active, or input, electrical power (hp or kW) I f Fraction of power available for sizing P Motor load power (hp or kW) P L G Inverter voltage output gain inv P Max. power available per engine group (hp or kW) eng G Powertrain transmission gear ratio r P Power required per engine group (hp or kW) req G ( s ) Aircraft transfer matrix with rotor speed control q Body y -axis angular rate (rad / s) G ( s ) Bare-airframe aircraft transfer matrix a Q Rotor aerodynamic torque (ft · lb) H ( s ) Inner-loop MIMO feedback regulator matrix Q Rotor trim torque (ft · lb) H ( s ) Rotor speed controller feedback matrix Ω Q Rotor shaft torque margin (ft · lb) M I , I Motor stator D- and Q-axis currents (A) d q Q Mechanical torque limit of motor (ft · lb or N m) peak I , I , I Aircraft principal moments of inertia (slug · ft ) Q Rotor shaft torque (ft · lb) xx yy zz S ˆ I Rotor moment of inertia (slug · ft ) Q Input-aligned torque command (non-dimensional) R input k , k , k Normalized aircraft radii of gyration r Body z -axis angular rate (rad / s) x y z L ( s ) Low-order transfer function approximation of R Rotor radius (ft) G ( s ) R Motor stator resistance ( Ω ) s 25 ◦ L , L Motor rotor D- and Q-axis self-inductances (mH) d q R Motor stator resistance measured at 25 C ( Ω ) s L , L Roll damping and torque control derivatives p Q S ( s ) Feedback sensor transfer functions i m Aircraft mass (slug) T Rotor thrust (lb) T NDARC drivetrain control allocation matrix DN Presented at the Vertical Flight Society’s 80th Annual Forum & u Aircraft inputs (i.e., rotor speed commands) Technology Display, Montr´ eal, Qu´ ebec, Canada, May 7–9, 2024. a This is a work of the U.S. Government and is not subject to copy- u u u Linear state-space model input vector right protection in the U.S.

u u u Input vector from i -th aircraft component MRP Maximum Rated Power c i V DC-voltage measured at inverter terminal (V) MTOW Maximum Take-Off Weight b V Speed for best endurance (knot) NFW No-Flux Weakening be V Speed for best range (knot) OLOP Open-Loop Onset Point br V , V Motor steady state D- and Q-axis stator voltages PFC Primary Flight Control d q (V) PMSM Permanent Magnet Synchronous Motor V Motor battery voltage (V) m RCDH Rate Command-Direction Hold V Maximum speed (knot) max RCHH Rate Command-Height Hold V Motor stator voltage (V) s RSC Rotor Speed Controller V Rotor hover tip speed, given by Ω R (ft / s) tip w Body z -axis velocity (ft / s) W Design gross weight (lb) D x x x Linear state-space model state vector INTRODUCTION y Aircraft response vector Z , Z Heave damping and torque control derivatives The advent of electric propulsion is revolutionizing the w Q i α Motor voltage angle (rad) paradigm of rotorcraft design. New, emerging, electric Ver- ◦ tical Take-Off and Landing (eVTOL) aircraft offer a poten- α Permanent magnet thermal coefficient (1 / C) pm ◦ tially revolutionary new form of transportation, if certain tech- α Copper thermal coefficient (1 / C) c nical shortcomings are to be resolved (Refs. 1, 2). Industry, β Rotor blade flapping angle (deg or rad) while advancing quite rapidly, is not yet free from the expen- β , β , β Rotor coning and tilt degrees of freedom (deg or 0 1 c 1 s sive cycle of technology prototyping. Joby Aviation, to pick rad) δ Motor current angle (rad) one, went through the Monarch and the S2, before arriving δ δ δ Vector of pilot inputs at the S4 (of which there have been various prototype vari- η Motor power conversion efficiency m ants). Wisk Aero has gone through at least five generations (as λ Motor flux linkage (Wb) Zee Aero and Kitty Hawk) before arriving at their Cora pro- λ , λ , λ Dynamic inflow states 0 1 c 1 s totype; a Generation 6 has now been announced. Archer Avi- 20 ◦ λ Motor flux linkage measured at 20 C (Wb) ation built the 80%-scale technology demonstrator Maker be- σ Rotor thrust-weighted solidity fore rollout of their Midnight production prototype, although τ Command model equivalent time delay (s) e these efforts were almost in parallel. Vertical Aerospace, in τ Motor hysteresis friction loss (ft · lb or N m) h the United Kingdom, built the VA-X1 and VA-X2 prototypes τ Motor viscous torque losses (ft · lb or N m) l before pivoting aircraft type and developing the VX4 produc- τ Motor load torque (ft · lb or N m) L tion model.

θ Average temperature of motor permanent magnets r This is potentially indicative of either the inadequacy of early ◦ ( C) design tools, or of tools not being adopted. To address the ◦ θ Average temperature of motor winding ( C) w former, NASA developed the NASA Design and Analysis of Φ Motor power factor Rotorcraft (NDARC) software (Ref. 3), which allows the siz- ω Motor speed (rad / s) m ing and analysis of rotorcraft configurations based on mis- Ω Rotor speed (rad / s) sion performance requirements and technology assumptions.

Ω Trim rotor speed (rad / s) Flight dynamics, but more specifically the handling qualities Ω Rotor speed for i -th rotor (rad / s) i of aircraft, have historically not been considered at the con- Ω Rotor speed margin (rpm or rad / s) M ceptual design level (Refs. 4, 5). The framework of Ref. 5 was proposed as a means of addressing this gap, as a partner tool Superscripts to NDARC. Recently, some work in the area has recognized b Motor base condition this need as well (Refs. 6–8).

r Motor rated condition Commonly, many of these new configurations tend to have p Motor peak value large numbers of lifting rotors (typically between four and twelve) compared to conventional rotorcraft, with each rotor Key Acronyms usually driven by its own independent propulsion system, in a ACAH Attitude Command-Attitude Hold topology often referred to as Distributed Electric Propulsion BLDC Brushless Direct Current (DEP). Although not true to every configuration, some design- DGW Design Gross Weight ers have chosen to utilize the DEP mechanism for Primary DRB Disturbance Rejection Bandwidth Flight Control (PFC), effectively utilizing the aircraft engines DRP Disturbance Rejection Peak as PFC actuators to control the speed of the rotors. Crucially, eVTOL electric Vertical Take-Off and Landing this decision integrates the propulsion and flight control sys- EMC Electric Motor Controller tems, and intrinsically couples the aircraft sizing and control.

FW Flux Weakening MCP Maximum Continuous Power The objectives of this study are to investigate connections be- MCT Maximum Continuous Torque tween conceptual design sizing parameters and the flight dy- namics of eVTOL aircraft. Specifically, the paper will explore that Eq. 1 denotes a fundamental physical relationship, and how design parameters affect the control and maneuverability that proper bookkeeping of units is needed, especially when of DEP aircraft in low speed, rotor-borne flight. A sensitivity dealing with non-SI units.

analysis is conducted at the conceptual design level, utilizing The fraction of power available f is a means of allocating a P NDARC to size the aircraft. A flight dynamics analysis is then certain amount of power to the sizing task, effectively build- conducted for every design iteration to assess variations in the ing in power margins into the engine size. When sizing, power bare-airframe dynamics, firstly, and the effect of actuator size available P is fundamentally set by the power required P , av req when an optimal flight control system is synthesized to satisfy i.e., P = P , from the different sizing conditions. From av req explicit handling qualities requirements, secondly.

Eq. 1 it then follows, e.g., that METHODOLOGY P = P (2) eng req f P Vehicle Sizing such that P > P for f < 1.

eng req P Vehicle sizing is one of the defining tasks of conceptual de- It is noted that NDARC does not limit the motor speed ω m sign, where fundamental aspects such as fuselage shape, wing in any fundamental way, but the ratio of the maximum power and rotor configuration and location, engine size and type that to the torque limit for each motor rating, and thus the base satisfactorily meet all specified requirements look to be deter- speed, is an input parameter into NDARC. So is the ratio of mined. This is a multidisciplinary process that must consider maximum power to continuous power ratings (MRP/MCP).

several oft-competing factors in aerodynamics, propulsion, flight performance, structures and control systems (Refs. 4,9).

The NASA Design and Analysis of Rotorcraft (NDARC) soft- Handling Qualities ware helps manage many of these competing requirements and was utilized to size the aircraft and to conduct parame- A framework for including handling qualities in conceptual ter trade studies of interest. design of rotorcraft was first proposed in Ref. 5. The pro- cesses applicable to the assessment of RPM-control han- Following a component-based approach, NDARC relies on dling qualities have been progressively matured through anal- low-fidelity models typically appropriate for the conceptual ysis (Refs. 10–12) and experimental validation (Ref. 13).

design environment to represent the power and energy trans- Key processes are now being implemented into the new ro- fer between components, the aerodynamic (or other) forces torcraft Flight dynamics and control modeling and analysis and moments produced by components, component weight, tool for COnceptual DEsign (FlightCODE). FlightCODE en- etc. to characterize the aircraft design as a whole. NDARC it- visions improvements in terms of computational efficiency eratively solves for the vehicle dimensions, power and weight and capabilities over the framework of Ref. 5. New features given a prescribed set of design conditions and/or missions.

(Refs. 14,15) and uses (Ref. 16) are currently in development.

For this study, a set of mission requirements representative of The features directly relevant to this study are summarized in Urban Air Mobility (UAM) operations from Ref. 2 was uti- the following sections.

lized for the aircraft sizing (see Table 1 and Figure 1). From these requirements NDARC can determine the total engine power and the rotor radius for each propulsion group, as well Linear models The core capability of FlightCODE lies in as the design gross weight, maximum takeoff weight, drive the rapid generation of linear perturbation stability and con- system torque limit, and fuel tank capacity.

trol derivative models of the NDARC-sized aircraft designs.

Models generated are in the general state-space form Previous studies (Refs. 10–12) progressively explored some of the limitations of RPM control. Ref. 13, in particular, pro-       u u u c ˙ x x x x x x 1 vided powerful insight into the torque margins required from b b   u u u     c ˙ x x x x x x 2 the engine group to ensure acceptable handling qualities. The   β β     = A + B (3)  .      .

˙ x x x x x x sizing of the engine power for eVTOL, and specifically for   λ λ .

˙ x x x x x x eVTOL using rotor speed as the primary means of control, r r u u u c nc was thus of critical interest in this study.

where state vectors x x x , x x x , x x x and x x x include the aircraft rigid b β λ r The motor size in NDARC is defined by the maximum power body translational and rotational degrees of freedom, rotor available P and a peak torque limit Q representing the eng peak flapping degrees of freedom for each rotor, 3-state dynamic mechanical limit of the motor, such that power available is inflow states for each wake, and rotational degrees of free- given by dom of each rotor, respectively, and the input vectors u u u cor- c i { b respond to the control effector inputs associated with the i -th f ω Q , ω < ω P m peak m m P = (1) av b aircraft component. The specific definition of the control in- f P , ω ≥ ω P eng m m puts, as well as a default control allocation matrix mapping b where ω is the base speed of the motor, i.e., the speed at the four main aircraft controls to the component controls, are m which both power available and torque limits coincide. Note inherited from NDARC.

Table 1. Sizing mission segments and their associated properties.

Segments 1&10 2&11 3&12 4&13 5&14 6&15 7&16 8&17 9&18 19 Initial Alt. (MSL ISA) 6,000 6,000 6,050 6,050 10,000 6,050 6,050 6,050 6,000 10,000 Final Alt. (MSL ISA) 6,000 6,050 6,050 10,000 10,000 6,050 6,050 6,000 6,000 10,000 Time (s) 15 30 10 t t 10 30 30 15 1,200 climb cruise Distance (nmi) – 0 0 D 37 . 5 − D 0 0 0 0 – climb climb Speed – – 0 V V 0 0 – – V x br be ROC (ft/min) – 100 0 ≥ 900 0 0 0 − 100 – 0 Percent of Max Power 10% 100% 100% P P 100% 100% 100% 10% P climb cruise cruise Cruise at 4000 ft AGL Cruise at 4000 ft AGL 20 min cruise No credit No credit reserve Climb at no less than descent Climb at no less than descent Transition & Transition & 900 ft/min 900 ft/min Transition 30 s hover Transition 30 s hover Vertical climb to 50 ft Vertical descent to 50 ft Vertical climb to 50 ft Vertical descent to 50 ft Taxi Taxi Taxi Taxi Taxi Unload AGL at 100 ft/min at 100 ft/min Taxi AGL at 100 ft/min at 100 ft/min + Load 37.5 nmi 37.5 nmi Figure 1. Sizing mission profile.

For conventional rotorcraft, component controls typically Developments in flight control technology suggest that fly- consist of the rotor collective and cyclic inputs, and in some by-wire systems with increased stability augmentation and instances, airframe aerodynamic control surfaces. For aircraft advanced control modes to reduce pilot workload will be in- using rotor speed as the primary means of control, it was nec- creasingly more prevalent. It can in fact safely be assumed that essary to model the rotational dynamics of the rotor. Inputs new eVTOL aircraft will by default be equipped with fly-by- were defined by setting wire systems and highly augmented controls. These systems tend to shift the handling qualities burden away from the bare- u u u = Q for i = 1 , 2 , · · · n c S r i i airframe design and onto the flight control system design, pro- vided the actuators possess sufficient control authority. Fly- where Q are the torques to the rotor shafts and n is the num- S r i by-wire systems present new types of handling qualities de- ber of rotors.

ficiencies and certification challenges, however. In particular, The rotor blades are assumed rigid, with a hinge offset and real concerns exist about the potential for severe Category II spring stiffness implemented to match the fundamental flap- or III pilot induced oscillations (PIO) that can develop with ping frequency from NDARC. The second-order linear pertur- fly-by-wire controls (Ref. 18). Causes vary, but a key trigger bation equations for the flap motion in the non-rotating frame of these types of severe PIO is linked to the onset of nonlin- are symbolically generated and coded for computational expe- earities in the aircraft dynamics, such as those associated with diency, with the coning β and tip-path-plane tilt β and β 0 1 s 1 c actuator rate limiting.

degrees of freedom modeled for each rotor. The coupled na- These issues raise questions about the best approach for as- ture of the rotor-airframe dynamics is formally retained, with sessing the controllability and maneuverability or the han- shaft motion being accounted for in the flap equations of mo- dling qualities at the conceptual stage of design. The an- tion. The dynamic inflow formulation in the tool follows from swer lies possibly in the implementation of a flexible analysis the perturbation version of the generalized Peters-HaQuang framework that enables both classical stability and control as- model (Ref. 17). The finite-state wake model considers only sessments, and a more modern handling qualities-centric ana- the uniform, lateral and longitudinal variations in rotor inflow, lytical approach. It is observed that the linear models of Eq. 3 denoted by λ , λ and λ , respectively.

0 1 s 1 c can support both types of analysis and that trimmability as- sessment can be conducted using NDARC. The classical ap- Flight dynamics assessment Historical approaches to the proach would be informed by the open-loop bare-airframe sta- treatment of stability and control during conceptual design are bility and control characteristics determined from Eq. 3. When limited to basic sizing and placement of wings and aerody- considering the handling qualities of highly-augmented sys- namic control surfaces to guarantee adequate stability (static tems it is arguably unavoidable to consider the effect of feed- and dynamic) and trim, while handling qualities are typically back stabilization and control response mode shaping. The ul- deferred to later stages of design (Ref. 4). Rotorcraft introduce timate goal, however, should be to quantify the required sys- added complication in that these types of aircraft are often in- tem control authority so as to provide actionable information herently unstable. Also, the methods for estimating the rotor that can be used to adequately size the actuators.

dynamics effects are somewhat more involved than those em- ployed for fixed-wing aircraft. An explicit model-following control system architecture (Fig- − 1 L ( 𝑠 ) Table 2. Feedback optimization constraints.

Specification Requirement Loop y + + 𝑐𝑚𝑑 − 𝜏 𝑠 Hard Constraints e G ( 𝑠 ) H ( 𝑠 ) C ( 𝑠 ) 𝛿 y 𝛿 𝑒 u y – + a des Eigenvalue stability Re ( λ ) < 0 All y Steady-state error 0.5 dB RSC meas S ( 𝑠 ) Gain Margin 6.0 dB All Phase Margin 45 deg All (a) Nichols Margins Special PFC cmd Q Q S 1 1 Response damping 0.9 RSC . .

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Min. crossover frequency 0.5 rad/s Heave Min. crossover frequency 2.5 rad/s Roll ure 2) was used to enable the handling qualities assessment Min. crossover frequency 2.0 rad/s Pitch of conceptual designs. Figure 2a illustrates the key elements Min. crossover frequency 0.5 rad/s Yaw of the flight control system as implemented in Ref. 5. Addi- Min. crossover frequency 4.0 rad/s RSC tional feedback loops in Figure 2b were added for rotor speed Summed Objectives control. This choice of architecture offered a tractable method Max. crossover frequency 10 rad/s All b for independent feedback stabilization and command shaping Actuator RMS 1.5 All for the heave, roll, pitch and yaw axis, but is by no means the a Not applied to yaw axis b only choice.

Applied to engine group motors if used for RPM control A command model C ( s ) interprets pilot inputs δ δ δ , shaping motor control, the EMCs were assumed to act as saturation them into a desired vehicle response y . L ( s ) is a low-order des elements.

equivalent system (LOES) approximation of the open-loop aircraft rate dynamics G ( s ) , over a desired frequency range Requirements To adopt handling qualities into the con- of interest (1–10 rad/s, in this instance). This feed-forward ceptual design, it was necessary for the design framework component provides lead compensation through the inversion to accommodate appropriate performance requirements. The of the LOES model, estimating input u to achieve the de- a CONtrol Designer’s Unified InTerface (CONDUIT) software sired response. The role of the regulator H ( s ) is, firstly, to (Ref. 19) was used, for this purpose, to synthesize the feed- guarantee stability of the aircraft, and secondly, to ensure ac- back and command model control system gains based on re- curate tracking of the command by minimizing the error be- quirements from Tables 2 and 3.

tween commanded and measured output (via sensors S ( s ) ).

The basic elements of the rotor speed control loops were: a Feedback design requirements included stability margin and pre-defined control allocation matrix, T ; a rotor speed error closed-loop disturbance rejection performance requirements DN feedback compensator, H ( s ) ; the engine and engine speed from aeronautical standards (Refs. 20, 21) and others that Ω controller; and the bare-airframe aircraft aerodynamic model, encompass good general feedback design practices. Tuning x G ( s ) (a reduced-order model of Eq. 3 retaining only x x and of the command model gains responded to requirements a b x x x ). Here, rotor speed signals were fed back and compared for small- and large-amplitude attitude changes for attitude r against commanded speeds from the allocation matrix T command-attitude hold (ACAH) response types in pitch and DN to determine rotor speed error signals. The rotor speed con- roll, rate command-direction hold (RCDH) in yaw, and rate trol compensator H ( s ) specifies torque commands required command-height hold (RCHH) in yaw from the Aeronautical Ω to regulate rotor speeds with minimal error. The Electric Mo- Design Standard-33 (ADS-33E-PRF), now MIL-DTL-32742 tor Controller (EMC) units are torque feedback control sys- (Ref. 21), for military rotorcraft. Both limited and moderate tems that produce the commanded engine torque output. EMC agility sets of requirements were considered for the large- units are high-bandwidth electric components that respond at amplitude roll and pitch attitude change requirements. Open- frequencies much higher than those required for flight con- Loop Onset Point (OLOP, Ref. 22) criteria were not set as trol. Perfect response of the EMC to the commanded torque constraints, but were checked to investigate the integrated ef- was assumed unless a physical power or torque limit of the fect of torque limits on the inter-connection between response motor was reached. Neglecting the electrical complexities of type handling qualities specifications (bandwidth and agility) Torque Table 3. Command model optimization constraints.

Specification Requirement Axis Peak Soft Constraints Torque Intermittent Heave mode 0.6 rad/s Heave Q Operation M Assumed Bandwidth 2 rad/s Roll G Flux r <latexit sha1_base64="NVhnN+wOltfRaVXNiLO0PI1ahIc=">AAACDnicbZDLSgMxGIUzXmu9dFTc6CZYBFdlRpG6LLjQjdCCvUA7DJk004YmM0OSEcswPoN7F2504Qu4E7e+gm+gb2Gm7cK2/hA4nJOTy+dFjEplWV/GwuLS8spqbi2/vrG5VTC3dxoyjAUmdRyyULQ8JAmjAakrqhhpRYIg7jHS9AYXWd68JULSMLhRw4g4HPUC6lOMlLZcs9DxBcJJzb1Ok0tXpK5ZtErWaOC8sCeiWNn/2Ss/vd5XXfO70w1xzEmgMENStm0rUk6ChKKYkTTfiSWJEB6gHmlrGSBOpJOMHp7CI+10oR8KvQIFR+7fRsKR6keK302dkyAu5ZB7up/lcjbLzP+ydqz8cyehQRQrEuDx9X7MoAphxgZ2qSBYsaEWCAuqfwBxH2k+ShPMazT2LIh50Tgp2aels5pmBMF4cuAAHIJjYIMyqIArUAV1gEEMHsEzeDEejDfj3fgYb10wJp1dMDXG5y/91J/i</latexit> Bandwidth 2 rad/s Pitch Weakening Rated Torque Region Bandwidth 0.5 rad/s Yaw No-Load Heave time delay 0.2 s Heave Speed G W r M <latexit sha1_base64="AxCjwIj2eqYRzENLWHBmQRWAUxs=">AAACCHicbVDNSgMxGMz6W+tf1ZtegkXwVHYV0ZsFBb2IFewPbJclm6ZtaJJdkqxYlr6Ad6/6Ct5EvPkQgm/gXb2bbXuwrQOBYSaTfN8EEaNK2/aHNTU9Mzs3n1nILi4tr6zm1tYrKowlJmUcslDWAqQIo4KUNdWM1CJJEA8YqQadk9Sv3hCpaCiudTciHkctQZsUI20k98yX9UtOWsi/8HN5u2D3ASeJMyT546/3n9PXze+Sn/usN0IccyI0Zkgp17Ej7SVIaooZ6WXrsSIRwh3UIq6hAnGivKQ/cg/uGKUBm6E0R2jYV/8mEo50O9L8duSdBHGlujww+dRX414q/ue5sW4eeQkVUayJwIPvmzGDOoRpK7BBJcGadQ1BWFKzAcRtJBHWprusqcYZL2KSVPYKzn7h4MrOFyEYIAO2wDbYBQ44BEVwDkqgDDAIwT14AI/WnfVkPVsvg6tT1jCzAUZgvf0CLNyfSg==</latexit> Continuous Phase delay 0.9 s All Operation Achievable rate 160 ft/min Heave Achievable attitude ± 15 deg Roll (limited agility) Base Rated Speed Speed Speed Achievable attitude ± 15 deg Pitch (limited agility) Achievable attitude ± 60 deg Roll Figure 3. Notional PMSM torque-speed curve.

(moderate agility) Achievable attitude − 30 , + 20 deg Pitch EXPLORATION OF THE DESIGN SPACE (moderate agility) Achievable rate ± 9.5 deg/s Yaw Concepts Check Only a Two vehicle architectures were explored in this study: 1) a OLOP Special fixed rotor speed, variable blade pitch quadrotor (Figure 4a), Attitude quickness Special All and 2) a variable rotor speed, fixed blade pitch hexacopter Summed Objectives b (Figure 4b). Both are battery powered. The drive system for Actuator RMS 1.5 All the quadrotor (Ref. 2) linked four engine groups with electric a Not applied to yaw axis motors to a single propulsion group that provided power to b Applied to engine group motors if used for RPM control all four rotors. The hexacopter distributed the propulsion over and PIO susceptibility (from engine torque limit saturation).

six independent groups, each with one engine group linked directly to a single rotor through a gear box. Both configu- rations had articulated rotor designs with matching flap fre- Motor Limits quency (1.03/rev), and baseline designs were sized to the same design disk loading (3 lb/ft ) and blade loading coefficient The handling qualities impact of motor torque saturation ( C / σ = 0.09). A summary of the main characteristics of the W nonlinearities in the flight dynamics of eVTOL aircraft was two designs is shown in Table 4.

demonstrated experimentally in Ref. 13. Torque-speed curves of the motor technology likely used with eVTOL aircraft, Effect of Disk and Blade Loading on Size especially with RPM-controlled eVTOL, were needed to account for these effects during the conceptual design. A Disk and blade loading were varied for the hexacopter physics-based derivation of the electric propulsion compo- from 3–9 lb / ft and 0 . 09–0 . 12, respectively. Consistent with nents is described in the Appendix. The model was capable Ref. 2, the lower design disk loading values arrived at more of producing the torque-speed curve of both Pulse Magnet lightweight designs (Figure 5). By contrast, the blade load- Synchronous Motors (PMSMs) and Brush-Less Direct Cur- ing coefficient had a much smaller but still appreciable effect rent (BLDC) motors, and was used to calculate the torque and (approximately 10–19% reduction for C / σ of 0 . 12 relative W speed margins at a given steady-state operating point of the the baseline 0 . 09). This reduction was most significant at high motor. A notional curve illustrating the definition of the torque disk loading.

and speed margins is shown in Figure 3.

It is worthwhile pointing out that aircraft principal axis radii The motor curves in Figure 3 were anchored by the peak and of gyration were assumed to scale with rotor diameter during rated torques calculated from the sizing task at the base and the sizing task, so effect of aircraft mass on moments of inertia rated (or specification) speeds. The curve was then projected is not direct.

out to the no-load condition. There is a region beyond the base motor curve where the motor can operate with specialized I = m ( k R ) (4) xx x flux-weakening control technology, up to the no-load speed.

I = m ( k R ) (5) It was also assumed that battery voltage could be slightly in- yy y creased beyond the motor’s rated voltage to allow torque to I = m ( k R ) (6) zz z remain constant between the base and rated speed. The no- load speed computed at the motor’s rated DC voltage marks where k = 0 . 6, k = 0 . 9 and k = 1 . 0 were estimated from the x y z the maximum speed limit of the motor. mass properties of the baseline design.

(a) Rotor 5 Rotor 3 Figure 5. Hexacopter Design Gross Weight.

Rotor 1 Rotor 6 Rotor 4 Rotor 2 (b) Figure 4. Renderings of the concept designs: (a) Quadro- tor, and (b) Hexacopter.

Table 4. Concept design aircraft characteristics.

Figure 6. Hexacopter rotor radius.

Characteristic Quad Hex Design Gross Weight (lb) 6,427 6,510 Rotor diameter was not constrained during sizing, resulting in Payload (lb) 1,200 1,200 diameters spanning the 15–20 ft range over the design space Empty Weight (lb) 5,216 5,299 (Figure 6). This approach differed from that in Ref. 2 for the Capacity (Pax + Crew) 6 6 Lift+Cruise, where a hard constraint on diameter of 10 ft was Number of Rotors 4 6 imposed in consideration for the possibility of the rotor inertia Design Disk Loading (lb/ft ) 3.0 3.0 increasing beyond the capability of RPM control to provide Number of Blades 3 3 enough responsiveness for maneuvering. The wisdom of this Blade Pitch@75% (deg) – 10.0 hypothesis is revised here.

Rotor Radius (ft) 13.1 10.7 Solidity, thrust-weighted 0.056 0.056 Caution should be employed, for example, when assuming Design Tip Speed (ft/s) 550 550 that a smaller radius will automatically translate to a smaller Design Rotor Speed (rad/s) 42 51.3 moment of inertia. Compared to the rotational moment of Design Rotor Speed (rpm) 401 490 inertia of the turbo-electric Lift-Cruise design from Ref. 2, Flapping Frequency (1/rev) 1.03 1.03 which was estimated at 14–27 slug · ft , rotor inertias for these Lock Number 5.16 4.61 Hexacopter designs were notably larger, ranging from 30 to Propulsion Group Central Direct over 70 slug · ft (Figure 7). While rotor diameter for the Hex- Number of Motors 4 6 acopter was minimum at the highest disk loading of 9 lb / ft , Engines rotor inertia was not. Finally, notable reductions in the rotor – MCP per Motor (hp) 111.5 87.0 inertia were achieved for higher blade loading because of re- – MRP per Motor (hp) 167.2 130.4 ductions in rotor solidity.

– Specification Speed (rpm) 8,000 8,000 While weight, rotor size and inertia parameters can affect ma- – Shaft Power Limit (hp) 319.9 211.7 neuverability in one way or another, it might be unwise to Drive Torque Limit (ft · lb) 8,320 1,136 extract strong conclusions about the overall maneuverability Battery Capacity (MJ) 1,316 1,449 from individual parameters because there are complex com- peting interdependencies at play. Referring back to the design Figure 7. Hexacopter rotor moment of inertia. Figure 8. Hexacopter mean relative torque margin.

The simplest dynamic system that captures these key factors assumptions for the Lift-Cruise from Ref. 2, for instance, it is a first-order ordinary differential equation such as Eq. 8: does take larger torques to impart the same acceleration on a rotor with larger rotational inertia, and presumably, radius.

˙ x ( t ) = ax ( t ) + bu ( t ) (8) The effectiveness of rotors for RPM control increases with size (Ref. 12), however, because or b X ( s ) = U ( s ) (9) ∂ T 2 T s − a ≈ R (7) ∂ Ω V tip in the Laplace domain. Here stability is measured by the a co- efficient, and the control effectiveness is determined by the For a constant thrust (set by aircraft design gross weight) and control derivative b . It is helpful to normalize the control design tip speed (usually set by acoustics or performance re- derivative from Eq. 8 such that the control input limits are quirements), Eq. 7 implies that smaller rotors need to be ac- ± 1.

celerated to higher speeds to get same thrust response.

If so, then the maximum theoretical steady state response is With smaller rotors and more weight, the capability of RPM given by the unit step input response of this system. From the control to provide enough responsiveness for maneuvering final value theorem applied to Eq. 9 this is: would seem to be put into question for higher disk loading.

However, additional characteristics or parameters need to be 1 b lim x ( t ) = lim sX ( s ) = − (10) considered. For instance, because of the scaling of Eqs. 4– t → ∞ s → 0 s a 6, perhaps a drawback of the approach, the differences in the Applying these ideas to the Hexacopter model, to first-order moments of inertia tended to be small. The achievable maneu- approximation, the linear heave, roll, pitch and yaw dynamics verability also depends on the motor torque margins available can be simplified to for maneuvering (Figure 8), as well as the aircraft stability characteristics. A formal analysis of the vehicle dynamics that n r ˆ weighs all of these factors together follows in the next few ˙ w = Z w + Z c · Q (11)

w ∑ Q i , col col

i i = 1 sections.

n r ˆ ˙ p = L p + L c · Q (12) p Q i , lat lat

∑ i

Effect of Disk and Blade Loading on Maneuverability i = 1 n r ˆ Maneuverability of an aircraft is the quality it has of being ˙ q = M q + M c · Q (13) q Q i , lon lon

∑

i i = 1 easy to move or direct while in motion. There are three fun- n r damental characteristics of a dynamic system that govern its ˆ ˙ r = N r + N c · Q (14)

r ∑ Q i , dir dir

i maneuverability: 1) its stability, 2) the effectiveness of the i = 1 control mechanisms, and 3) the maximum control input that ˆ ˆ ˆ ˆ can be applied. It is an interesting aspect of dynamic systems where Q , Q , Q and Q represent axis-aligned shaft col lat lon dir that the more stable they are, the harder they are to maneuver. torque control inputs, and c are the control input allo- i , input High maneuverability therefore requires low stability. Effec- cation coefficients to the i -th rotor shaft. Stability and control tiveness of the controls measures how much rate of change derivatives are obtained from a 6-dof reduced-order equivalent can be generated per unit of control input. And of course, how model of Eq. 3. This simplification clearly neglects the rotor large of a control input can be applied to the system will deter- speed dynamics, but is appropriate for the theoretical estima- mine the overall maximum rate of change that can be imparted tion of the steady state responses in Figures 9–12, calculated on it. as per Eq. 10.

Figure 9. Maximum steady state heave response. Figure 11. Maximum steady state pitch rate response.

Figure 10. Maximum steady state roll rate response. Figure 12. Maximum steady state yaw rate response.

Control allocation coefficients c were defined such that for multirotor designs with fixed-pitch, variable-speed rotors i , input maximum torque was applied to each individual rotor for a versus variable-pitch, fixed-speed rotors. A Hexacopter de- unit step input of the designated axis-aligned torque input sign is first resized to ensure motor operating speed is within ˆ Q . With increased design gross weight, aircraft engines, the limits of the motor, and to build in additional torque mar- input which tended to be sized by the 900 ft / min rate of climb re- gins for maneuvering control. The resulting motor curves are quirement, were larger for the higher disk loading and offered then presented. Motor curves for the quadrotor design are then additional margins for maneuverability at the hover off-design shown for comparison.

flight condition. Results suggested a significant improvement in maneuverability is likely for higher disk and blade load- Resized hexacopter The baseline hexacopter configuration, ing. This was attributable to various factors, but mainly to: 2 with W / A = 3 lb / ft and C / σ = 0 . 09, was selected for D ref W 1) lower damping coefficients Z , L , M and N , but most w p q r redesign. Arguably, one of the first design parameters to be importantly, 2) larger powertrain torque margins.

varied should be the blade loading coefficient, as this could confer slight improvements in maneuverability and reductions in design gross weight. The design disk and blade loading Motor Sizing were kept constant, regardless. This configuration was still A key limitation not properly addressed in the preliminary siz- lightweight, which made it attractive from a design (and as- ing results shown above was the motor operating speed. In- sociated cost) perspective. It was in fact the potential maneu- spection of the vehicle state for each sizing flight condition verability deficiencies of this design that made it an interesting and mission segment, revealed that the no-load speed of the case study to exercise the broader capabilities of the toolset, motor was exceeded under certain conditions. Most critically, including a more detailed handling qualities analysis. From a the 900 ft / min climb to cruise altitude mission segment drove technical point-of-view it was of interest to: 1) assess han- the motors for the two rear rotors to operate over 10 , 000 rpm. dling qualities of the design, and 2) investigate whether han- Results presented in this section illustrate the motor torque- dling qualities could be improved through redesign of key pa- speed curve sizing process, and how motor operation differs rameters.

be negligible. The motor power factor was 0.942. The motor Table 5. Resized Hexacopter design characteristics.

power factor depends very much on the distance between the Characteristic Baseline Redesign rated speed and base speed, as well as on the ratio of MRP Design Gross Weight (lb) 6,510 6,758 to MCP. Increasing the distance between the base speed and Payload (lb) 1,200 1,200 rated speed would likely result in a lower power factor, which Empty Weight (lb) 5,299 5,548 may increase the no-load speed.

Max. Take-Off Weight (lb) 8,557 9,404 Number of Rotors 6 6 Table 6. Resized Hexacopter motor specifications Design Disk Loading (lb/ft ) 3.0 3.0 Motor Specifications Redesign Design Blade Loading ( C / σ ) 0.09 0.09 W Maximum Rated Power (hp) 177.4 Number of Blades 3 3 Peak Power @ 8,000 rpm (hp) 196.3 Blade Pitch@75% (deg) 10.0 12.5 Peak Torque (ft · lb) 128.9 Rotor Radius (ft) 10.7 10.9 Base Speed (rpm) 7,228 Solidity, thrust-weighted 0.056 0.056 Maximum Continuous Power (hp) 118.1 Design Tip Speed (ft/s) 550 550 Maximum Continuous Torque (ft · lb) 76.6 Design Rotor Speed (rad/s) 51.3 50.3 No-load speed (rpm) 8,621 Design Rotor Speed (rpm) 490 480 Rated Speed (rpm) 7,988 Flapping Frequency (1/rev) 1.03 1.03 Power Factor 0.942 Lock Number 4.61 4.67 Number of pole pairs 6 Engines Flux Linkage (Wb) 0.0692 – MCP per Motor (hp) 87.0 118.3 Back-EMF Constant (Vs / rad) 0.4154 – MRP per Motor (hp) 130.4 177.5 Stator Resistance ( Ω ) 0.0393 – Specification Speed (rpm) 8,000 8,000 Q-axis Self-Inductance (mH) 0.1448 – Shaft Power Limit (hp) 211.7 244.7 D-axis Self-Inductance (mH) 0.1448 – Transmission Gear Ratio 16.3 15.0 Rated Motor Voltage (V) 650 Drive Torque Limit (ft · lb) 1,136 1,337 Battery Capacity (MJ) 1,449 1,461 As seen in Table 5, the differences in size between the baseline Table 7. Hexacopter rotor torque margins ( ft · lb ) and redesigned hexacopter model were minimal. Again, the No. Flight Condition R1&2 R3&4 R5&6 first aspect to address was the motor speed, which was found 1 Hover MTOW, 6k/ISA 593 651 707 to exceed the no-load speed of the motor when contrasted 2 V , 500 ft/min ROC, 1,124 995 866 br against the associated motor torque-speed curves. Increas- 10k/ISA ing the rotor blade pitch was found to alleviate this slightly, 4 Miss. Segments 2 & 11 952 994 1,036 enabling rotors to trim to lower speeds. Adjusting the gear 5 Miss. Segments 3 & 12 967 1009 1,050 box ratio between the engine groups and rotors presented an- 6 Miss. Segments 4 & 13 948 827 662 other means of reducing the motor speed, while maintaining 7 Miss. Segments 5 & 14 1,358 1,218 1,079 the rotor speeds relatively constant over the sizing conditions.

8 Miss. Segments 6 & 15 967 1,009 1,050 Both of these design variations helped ensure feasible oper- 9 Miss. Segments 7 & 16 967 1,009 1,050 ating speeds for the motor, but neither provided better torque 10 Miss. Segments 8 & 17 981 1,023 1,064 margins.

21 Miss. Segment 19 1,480 1,371 1,267 The clearest way to improve the torque margin during the 22 V , 6k/ISA 1,355 1,217 1,082 br NDARC sizing task was to reduce the fraction of power avail- 23 V , 6k/ISA 1,479 1,370 1,267 be able f from Eq. 1. Fraction f for the sizing mission seg- P P 24 V , 6k/ISA 1,145 1,014 870 max ments was reduced to 0 . 75 from the baseline 0 . 95 value. Sub- 25 Hover, SL/ISA 968 1,010 1,052 sequent results in this section are solely for this resized design.

26 Hover, 6k/ISA 967 1,009 1,050 27 Hover, 10k/ISA 966 1,008 1,049 The hexacopter’s six motors were identical. Motor coeffi- cients, as derived by the sizing methodology described in the Appendix, are depicted in Table 6. It was observed that the The torque and speed margins of rotors determine the range motor’s base speed of 7,228 rpm and rated speed of 7,988 rpm that is left over for dynamic maneuvering after trimming the were close in value, but not exactly equal to, NDARC’s pre- aircraft. Rotor torque margins from sizing, mission and off- scribed base and specification speeds (7,200 and 8,000 rpm, design conditions are presented in Table 7. Mission segments respectively). Differences in the motor speeds were explained 1 and 8 were omitted as both corresponded to an idle condition by the convergence criteria chosen to solve for the flux link- with low torque and speed requirements. Condition 1, at max- age in Eq. A-18. The motor speed difference could be re- imum takeoff weight (MTOW), possessed the lowest torque st nd duced by tightening up convergence criteria and reducing step margin on the 1 and 2 rotors (R1 and R2 in Table 7), re- size of power factor iteration, but the difference was found to ported as 593 ft · lb. Note that MTOW here was calculated for f = 0 . 8. If computed at 100% of installed power there would When operating points cross NFW, the motor depends on P have been no margins for control available. Mission segments a combination of higher battery voltage than the rated mo- 4 and 13 (900 ft/min climb conditions) had the lowest torque tor voltage and flux-weakening control laws to operate effec- th th margin on the 5 and 6 rotors (662 ft · lb). Most rotor torque tively. The Flux-Weakening (FW) line depicted in Figure 13 margins were near 1,000 ft · lb. shows the predicted extensions of peak torque and MCT with speed. For typical motors operating at the rated motor volt- Rotor speed margins are presented in Table 8. The 900 ft / min age, torque starts to gradually roll off near the base speed, and climb condition of segments 4 and 13 (condition 6) was also power may or may not be constant immediately after the base observed to have the lowest speed margin of 31 rpm, for rotors speed depending on the flux-weakening algorithm employed.

5 and 6. This speed margin could potentially be improved, with a further reduction in gear ratio. Further reductions in gear ratio could mean an increase in torque and motor weight, however. Rotors 1 and 2 from condition 1 had a speed margin of 68 rpm. Rotors 5 and 6 from condition 2 had a speed margin of 51 rpm.

Table 8. Hexacopter rotor speed margins (rpm) No. Flight Condition R1&2 R3&4 R5&6 1 Hover MTOW, 6k/ISA 68 79 91 2 V , 500 ft/min ROC, 104 78 51 br 10k/ISA 4 Miss. Segments 2 & 11 143 152 162 5 Miss. Segments 3 & 12 145 155 164 6 Miss. Segments 4 & 13 81 56 31 7 Miss. Segments 5 & 14 146 120 94 8 Miss. Segments 6 & 15 145 155 164 9 Miss. Segments 7 & 16 145 155 164 Figure 13. Hexacopter torque, speed and efficiency map.

10 Miss. Segments 8 & 17 148 157 167 21 Miss. Segment 19 209 193 177 22 V , 6k/ISA 170 146 123 br 23 V , 6k/ISA 209 193 178 be 24 V , 6k/ISA 127 99 71 max 25 Hover, SL/ISA 182 191 200 26 Hover, 6k/ISA 145 155 165 27 Hover, 10k/ISA 118 128 138 The location of operating points with respect to maximum continuous torque (MCT) which was at 76.6 ft · lb is crucial.

Motors for most design and off-design flight conditions were under MCT as observed by tight groupings of the torque- speed-efficiency motor map produced within Figure 13. The th th 5 and 6 motors from condition 6 operated above MCT by th th 4.2 ft · lb, in the region of flux-weakening. The 5 and 6 motors from condition 6 also exceeded the rated speed by 168 rpm. All motors from condition 1, at MTOW, operated above MCT.

It was observed that almost all motors operated above 95.5% Figure 14. Hexacopter power, speed and efficiency map.

efficiency; 95% motor efficiency was a target efficiency to match the prescribed model in NDARC. The efficiency map in Figure 13 did not include losses from 3-phase inverter ef- The hexacopter design and off-design motor operating points ficiency, which is why 96% efficiency was originally targeted were placed on a speed-power-efficiency map as in Figure 14.

during coefficient sizing instead of 95% efficiency assumed in Here the power was allowed to increase after the base speed NDARC. The computed motor map efficiencies include wind- through a combination of flux-weakening control and higher ing and permanent magnet temperature variations, away from battery voltage than the motor rated voltage of 650 V, up un- the intersection of No-Flux Weakening (NFW) and MCT in til the rated speed of 7,988 rpm, where it was held constant.

Figure 13. Motor manufacturers do provide a range of voltage for their motors which exceeds the rated motor voltage often by several hundred volts.

A key point of Figure 14 is that power available could be made to exceed the MRP depending on specific architectural de- sign choices of the electric powertrain. Notably, NDARC was not allowed to utilize this additional power during sizing, but flux-weakening control could conceivably be used to increase powertrain margins for maneuvering. Thermal implications of continuous operation in this region will be discussed next.

th th The 5 and 6 motor exceeded the rated speed and MCT dur- ing the 900 ft / min climb to cruise altitude (condition 5); this called into question if operations in an intermittent region of the motor map, the motor winding temperature might exceed ◦ 140 C. The entire NDARC design mission was simulated to check the winding temperature of each motor over the mis- sion. Conditions 1 and 2 were not technically part of the suc- cessive design mission points, so they were excluded in the Figure 16. Quadrotor torque, speed and efficiency map.

analysis. So were the reserve mission segment 19 (condition 21) and off-design analysis conditions (conditions 22–27).

giving temperature time to converge. A rise in temperature was observed in all motor windings for mission segment 4, corresponding to the approximately 4.5 minute climb at 900 ft/min. The temperatures gradually reduced over the rest of the mission, until segment 13 when another 4.5 minute climb at 900 ft/min was initiated. It was observed that for the de- th sign mission points, the winding temperature of the 5 and th 6 motors, never exceeded the rated temperature. There are two main reasons for this. The first reason was that prior and following design mission segments end at significantly lower winding temperatures than the rated temperature. The second reason was that the time constant of the motor temperature model was approximately 10 minutes.

Quadrotor For comparison, a collective blade-pitch con- trolled quadrotor torque-speed-efficiency motor map was also generated (Figure 16). It was observed that the motors were placed at the specification speed and appear in a vertical line for all flight conditions. The most critical conditions were 1 and 6 which overlap almost on top of each other in Figure 16.

These same conditions were also observed to be crucial for the hexacopter. For the blade-pitch controlled vehicles, where the engine group does not double as primary flight control actuator, speed and torque margins are less critical. Specific tabulated margins are therefore not shown. However, it was observed that, like the hexacopter, quadrotor motor torque re- quirements occur above MCT of 73 ft · lb, by 26 ft · lb. There- fore a temperature analysis was required on the design mis- sion points.

The temperature analysis presented in Figure 17 indicated that operating condition did not adversely affect the motor wind- Figure 15. Top) Hexacopter motor winding temperature; ing temperature of the quadrotor during the design mission.

bottom) individual mission segment time.

Driven by the design mission, the temperatures remained be- ◦ low the rated condition of 140 C for the same reasons as the hexacopter. However, the winding temperature after climb The temperature simulation results shown in Figure 15) ini- segments at 900 ft/min, corresponding to conditions 6 and tially assumed that the motors were at idle for 300 minutes, 15, were higher in the case of the quadrotor, than it was for (a) (b) Figure 18. Baseline limited agility ACAH case: (a) small- amplitude response criteria, and (b) OLOP criteria.

Figure 17. Top) Quadrotor motor winding temperature; bottom) individual mission segment time.

back loops closed. Although not directly evident from the Fig- the hexacopter (see Figure 15). This result could indicate that ures 18–20, results also showed the handling qualities effects temperature is an important factor in sizing. If the design mis- of large-amplitude attitude change requirements on these two sion were to be extended and include another climb segmentt, designs. Figures 18 and 20 compare the handling qualities for the temperature progression in Figure 17 would suggest that a flight control system tuned to enable limited agility maneu- the winding temperature may climb above the rated tempera- verability as defined by achievable roll and pitch changes of ture, leading to reduced reliability of the powertrain. ± 15 deg. Figure 19 shows results for a control system tuned to provide moderate agility maneuvering ( ± 60 deg in roll and ± 30 deg in pitch).

Handling Qualities Across Figures 18a, 19a and 20a it is seen how through the Figures 18–20 compare the small-amplitude attitude response control synthesis it was possible to instill identical (small- and (bandwidth and phase delay) and OLOP (phase and ampli- large-amplitude response) handling qualities. Although these tude) margin characteristics for the baseline and redesigned two configurations had very similar bare-airframe dynamics, hexacopter models (both designed to W / A = 3 lb / ft and this would still hold true for aircraft with differing dynamics D ref C / σ = 0 . 09). Bandwidth and phase delay characterize the characteristics. The differences in handling qualities showed W short-term response for small attitude changes (pitch and roll up in Figures 18b, 19b and 20b when assessing the impact of in this case). These aircraft response qualities govern the pre- the motor torque limit on the OLOP criteria. The redesigned cision with which a pilot can make small attitude correc- hexacopter exhibited additional protection against PIO across tions. The OLOP criteria quantify the likelihood of the cou- all three control system sets due to the extra installed torque pled pilot-vehicle system becoming unstable if an actuator margins. It can be seen from Figure 18 that when asked for rate limit were to be activated. The OLOP criteria measure limited agility both aircraft designs were fairly robust to PIO.

the actuator broken-loop frequency-response phase and am- In contrast, command model gain sets delivering moderate plitude at the so-called onset frequency, i.e., the frequency agility were found to be significantly more PIO prone (Fig- at which the actuator rate limit is activated with all feed- ure 19). Similarly, Figure 20 showed there were limits to how (a) (a) (b) (b) Figure 19. Low-bandwidth moderate agility ACAH case: Figure 20. High-bandwidth limited agility ACAH case: (a) small-amplitude response criteria, and (b) OLOP cri- (a) small-amplitude response criteria, and (b) OLOP cri- teria. teria.

much bandwidth could be extracted from these two designs.

the torque-speed characteristic curves and the efficiency map of the motor from the available power and peak torque.

Despite longer moment arms, only four of the rotors were ac- tive for pitch control. This, coupled with a doubling of the The various tools are not currently tightly integrated, so a cer- pitch moment of inertia, would explain why pitch control tain amount of manual iteration was needed. For instance, left tended to be more prone to PIO: motors needed to work about unconstrained, NDARC produced motor speeds during the 65% harder in pitch than in roll. Roll control only exhibited sizing task that invalidated the no-load speed predicted by the a propensity for PIO with the moderate agility flight control physics based motor models. The motor sizing tool was used laws, but it is pointed out that large-amplitude roll attitude here as a means of verifying the feasibility of the design after change requirements were twice as high. Incidentally, this re- sizing was completed. Future work should investigate how to sult was consistent with Figures 10 and 11.

streamline the toolchain “data flow” and interaction.

Sizing of an RPM-controlled Hexacopter concept explored DISCUSSION the dependency of aircraft size to the fundamental design pa- rameters: 1) disk loading and 2) blade loading coefficient. De- Four separate tools were exercised throughout this study to sign tip speed was kept constant at 550 ft / s. The design gross conduct a conceptual design exploration of eVTOL aircraft weight was observed to almost double for a disk loading of handling qualities. Of primary concern was the investigation 2 2 9 lb / ft compared to the baseline 3 lb / ft value. Rotor diam- of the handling qualities of RPM-controlled eVTOL multiro- eter decreased by about 20%, commensurate with the weight tor aircraft. The tools utilized were: 1) NDARC for the aircraft and disk loading changes.

sizing and performance analysis, including the calculation of trim; 2) FlightCODE, a flight dynamics modeling and analysis Perhaps the more interesting results from this parametric in- tool capable of rapid generation of linear stability and control vestigation lay in the assessment of the flight dynamics: a derivative models and control analysis; 3) CONDUIT aided first-order approximation of the stability and control param- in the handling qualities-centric control law optimization; and eters revealed that the heavier aircraft designs, sized to the 4) an electric motor physics-based sizing tool used to model higher disk loading and blade loading coefficients values, were potentially and significantly more agile. Overall, agility CONCLUSIONS increased approximately three-fold compared to the baseline disk and blade loading design. This large difference in agility The following conclusions were established on the basis of was attributed to a number of factors including a reduction in the results and discussion presented in this paper: the damping coefficient, a slight increase in the control effec- tiveness, but also notable growth in the engine size conferring • A framework that automated the generation of linear these designs with higher control authority.

perturbation state space models from the size and per- formance characteristics of the aircraft designs provided The final discussion point pertains to the different approaches sufficient stability, control effectiveness and control mar- taken for the conceptual design assessment of flight dynam- gin to conduct a rapid assessment of the flight dynamics ics. The flight dynamics modeling tool enabled a very rapid suitable for conceptual design.

assessment of design maneuverability based purely on the lin- ear perturbation model of Eq. 3. For context, given the aircraft • Agility of the hexacopter designs improved significantly description and trim information from NDARC, the genera- when increasing the design disk loading and design blade tion of the data for Figures 5–12 was executed in a matter of 2 2 loading coefficient to 9 lb / ft and 0 . 12 (from 3 lb / ft and minutes (less than a minute per case for an ordinary 2.4 GHz 0 . 09), respectively.

consumer grade processor). This approach weighed the three fundamental flight dynamics governing parameters: 1) stabil- • Increasing the disk loading up to 9 from 3 lb / ft also ity (linear stability damping coefficient), 2) control effective- caused significant growth in the design gross weight, ness (linear control derivative), and 3) control margins (the from about 6 , 500 to nearly 12 , 000 lb.

physical motion, rate or load limits of the actuator, e.g., the torque limit of the engine for RPM control). It, however, ne- • Post-sizing, designs frequently exceeded electric motor glected transient, actuator or relevant modal dynamics. This and thermal model speed and torque limits, suggesting is not to say that further analysis of the models from Eq. 3 that motor model limits should be integrated into the siz- could not deliver this type of information, simply that it was ing task.

not presented here.

• A handling qualities trade-off analysis, capable of deliv- Further analysis was needed to generate actionable handling ering actionable concrete handling qualities information, qualities information. CONDUIT was used to synthesize an required casting the linear perturbation models and siz- optimal set of flight control laws, given an assumed control ing information into a control synthesis and optimization system architecture. When recast into this form the problem framework.

allowed for the trade-off analysis of concrete handling quali- ties specifications, e.g., pilot input response type characteris- • For the Hexacopter, reducing the fraction of power avail- tics, while assessing whether the design had sufficient control able to 75% during the sizing task conferred improved authority. This CONDUIT-enabled approach presented a pow- control authority over the baseline design, with only a erful capability, but was dramatically more computationally modest increase in aircraft gross weight (250 lb).

intensive. Only one of the designs was therefore picked to un- dergo a more detailed handling qualities trade-off study. There was no intent to produce an optimally redesigned aircraft, but Author contact: Carlos Malpica, carlos.a.malpica@nasa.gov; only to demonstrate tool utilization. Reducing the fraction of Peter Suh, peter.m.suh@nasa.gov; Christopher Silva, christo- power available for the sizing task produced slightly heav- pher.silva@nasa.gov.

ier RPM-controlled hexacopter designs with larger engines, higher control authority, and ultimately more robustness to APPENDIX PIO (and hence, improved handling qualities).

Motor Map Design In a sense, this approach was representative of the problems likely faced by fly-by-wire flight control system designs: ad- vanced flight control modes that simplify the task of the pi- This section includes two subsections which cover the motor lot and reduce workload can indeed confer Level 1 handling map design methodology. The motor map was considered an qualities, but differences in the bare-airframe dynamics may intuitive way of depicting available torque, power and speed require varying levels of authority from the control actuators. of the motor, which together with design points from NDARC Failure of the designer to ensure adequate actuator authority was used to determine torque and speed limits useable by can lead the aircraft exposed to potentially catastrophic han- FlightCODE. The first subsection covers motor coefficient dling qualities cliffs, i.e., aircraft states that when entered will sizing which was the process used to convert NDARC motor result in severe and sudden degradation of the handling quali- specifications into physics-based motor coefficients. The sec- ties. Crucially, the methods proposed here were demonstrated ond subsection covers the prediction of peak torque and speed to enable the conceptual design sizing of actuators to mini- values, which in turn were used to determine rotor speed and mize these handling qualities deficiencies. torque limits.

Motor coefficient sizing Sized NDARC vehicles report mo- Here R is stator resistance, I is motor stator current, λ is s q tor specification speed (set as input), MRP, MCP, peak torque, motor flux linkage, n is number of pole pairs, and ω is rota- p m and a power limit defined from peak torque at the specification tional mechanical speed of the motor. The back-EMF constant speed. This Appendix addresses how to convert these parame- was computed by λ n . Motor speed ω was assumed to oper- p m ters into a feasible motor map which would likely be provided ate at a multiple of the gear ratio of the gear box connected to by a motor manufacturer. The first step is to use NDARC mo- the aerodynamic rotor of the rotorcraft.

tor specifications to derive motor coefficients by way of motor The D-axis stator voltage for steady state operations was com- physics equations. Some decisions had to be made along the puted as in Eq. A-2 way for various constants used in the motor physics equations, and although not called out directly, values chosen were typi- ∼ V = − I L n ω (A-2) d q q p m cally reflected in relevant motor data sheets.

Where L is the rotor Q-axis self-inductance. The motor stator The Direct-Quadrature (D-Q) frame is traditionally used to q voltage in Figure 21 was computed as in Eq. A-3.

analyze operations of both Pulse Magnet Synchronous Mo- tors (PMSMs) and brushless direct current (BLDC) motors.

√ For direct current (DC) analysis of alternating current (AC) V = V + V (A-3) s q d motors, rotations are performed successively from the three- phase frame to the alpha-beta frame to the D-Q frame fixed to AC motors are voltage limited by the DC-link voltage on the the motor rotor. The steady-state vectorial schema of a circuit inverter. The voltage limitation was conveniently captured by equivalent plotted on the D-Q axis from Ref. 23 is depicted in the following constraint (Ref. 24) as provided in Eq. A-4 Figure 21.

V s M = ≤ 1 (A-4) w n L I + L I m p q q qd d G V inv b <latexit sha1_base64="Cg/wbPNvEBMFNMeVUypRg8To/pI=">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</latexit> + Q <latexit sha1_base64="CZD9EGInYC2jsDGsx9UWbuyPym8=">AAAB/3icbVDLSgMxFL1TX7W+qi7dBIsgCGXGB7osuHHZin1AW0omzbShSWZIMmIZunDvVn/Bnbj1U/wDP8NMOwvbeiBwOCcnuff4EWfauO63k1tZXVvfyG8WtrZ3dveK+wcNHcaK0DoJeahaPtaUM0nrhhlOW5GiWPicNv3Rbeo3H6nSLJQPZhzRrsADyQJGsLHS/VmtVyy5ZXcKtEy8jJQgQ7VX/On0QxILKg3hWOu250amm2BlGOF0UujEmkaYjPCAti2VWFDdTaaTTtCJVfooCJU90qCp+jeRCGyGkRFPc+8kWGg9Fr7Np75e9FLxP68dm+CmmzAZxYZKMvs+iDkyIUrLQH2mKDF8bAkmitkNEBlihYmxlRVsNd5iEcukcV72LspXtctSBWUl5eEIjuEUPLiGCtxBFepAIIAXeIU359l5dz6cz9nVnJNlDmEOztcv5WOWXQ==</latexit> where V is the DC-voltage measured at the inverter terminal b R I w n L I + L I s s m p d d dq q <latexit sha1_base64="uv8dknPG5reuab1igP82S7/eiqY=">AAACA3icbVC7TsMwFHXKq5RXgZHFokJiqhIegrESC2wF0YfURpHjOq2pnUT2DaKKOrKzwi+wIVY+hD/gM3DaDLTlSJaOzvGx7z1+LLgG2/62CkvLK6trxfXSxubW9k55d6+po0RR1qCRiFTbJ5oJHrIGcBCsHStGpC9Yyx9eZX7rkSnNo/AeRjFzJemHPOCUgJGad56+8bRXrthVewK8SJycVFCOulf+6fYimkgWAhVE645jx+CmRAGngo1L3USzmNAh6bOOoSGRTLvpZNoxPjJKDweRMicEPFH/JlJJYBCDfJp5JyVS65H0TT7z9byXif95nQSCSzflYZwAC+n0+yARGCKcFYJ7XDEKYmQIoYqbDTAdEEUomNpKphpnvohF0jypOqfV89uzSg3nJRXRATpEx8hBF6iGrlEdNRBFD+gFvaI369l6tz6sz+nVgpVn9tEMrK9fUR6YSA==</latexit> <latexit sha1_base64="RWZW1hah8ytEEXBihNHO3R7+h5w=">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</latexit> and G is the inverter voltage output gain. During motor op- inv erations, the modulation ratio, M , when equal to 1 suggests inverter saturation, resulting in open loop behavior and limit- V d <latexit sha1_base64="kXuzYBmUKsRQG89Ptmv82FQtOv4=">AAACAHicbVC7TsMwFHXKq5RXgZHFokJiqhIegrESC2MR9CG1UeU4bmvVdiL7BlFFXdhZ4RfYECt/wh/wGThtBtpyJEtH5/jY954gFtyA6347hZXVtfWN4mZpa3tnd6+8f9A0UaIpa9BIRLodEMMEV6wBHARrx5oRGQjWCkY3md96ZNrwSD3AOGa+JAPF+5wSsNJ9sxf2yhW36k6Bl4mXkwrKUe+Vf7phRBPJFFBBjOl4bgx+SjRwKtik1E0MiwkdkQHrWKqIZMZPp6NO8IlVQtyPtD0K8FT9m0glgWEM8mnunZRIY8YysPnMN4teJv7ndRLoX/spV3ECTNHZ9/1EYIhw1gYOuWYUxNgSQjW3G2A6JJpQsJ2VbDXeYhHLpHlW9c6rl3cXlRrOSyqiI3SMTpGHrlAN3aI6aiCKBugFvaI359l5dz6cz9nVgpNnDtEcnK9fBs+XBA==</latexit> ing torque. The voltage constraint of the motor described by E = w n l m p <latexit sha1_base64="45Cf1fRgQyWVGIX8ubZJ8qL+eW0=">AAACFXicbVDLSgMxFM34rPU1KrhxEyyCqzLjA90IBRFcVrAP6AxDJpO2oUlmSDJiGfsd7t3qL7gTt679Az/DTDsL23ogcDgnJ7n3hAmjSjvOt7WwuLS8slpaK69vbG5t2zu7TRWnEpMGjlks2yFShFFBGppqRtqJJIiHjLTCwXXutx6IVDQW93qYEJ+jnqBdipE2UmDv31x5MSc9FHAoggR6zGQjFNgVp+qMAeeJW5AKKFAP7B8vinHKidCYIaU6rpNoP0NSU8zIqOyliiQID1CPdAwViBPlZ+P5R/DIKBHsxtIcoeFY/ZvIONL9RPPHqXcyxJUa8tDkc1/Nern4n9dJdffSz6hIUk0EnnzfTRnUMcwrghGVBGs2NARhSc0GEPeRRFibIsumGne2iHnSPKm6p9Xzu7NKDRYllcABOATHwAUXoAZuQR00AAZP4AW8gjfr2Xq3PqzPydUFq8jsgSlYX7+DAp7H</latexit> Eq. A-4 was affected by the type of inverter and gate switch- V s <latexit sha1_base64="F5IkGI9XVHQSVn1ATLL4gE4qmJU=">AAACAHicbVC7TsMwFL0pr1JeBUYWiwqJqUp4CMZKLIxF0IfURpXjOq1V24lsB1FFXdhZ4RfYECt/wh/wGThtBtpyJEtH5/jY954g5kwb1/12Ciura+sbxc3S1vbO7l55/6Cpo0QR2iARj1Q7wJpyJmnDMMNpO1YUi4DTVjC6yfzWI1WaRfLBjGPqCzyQLGQEGyvdN3u6V664VXcKtEy8nFQgR71X/un2I5IIKg3hWOuO58bGT7EyjHA6KXUTTWNMRnhAO5ZKLKj20+moE3RilT4KI2WPNGiq/k2kApthbMTT3DspFlqPRWDzma8XvUz8z+skJrz2UybjxFBJZt+HCUcmQlkbqM8UJYaPLcFEMbsBIkOsMDG2s5KtxlssYpk0z6reefXy7qJSQ3lJRTiCYzgFD66gBrdQhwYQGMALvMKb8+y8Ox/O5+xqwckzhzAH5+sXHtWXEw==</latexit> I I V q ing logic. Gate switching logic such as sinusoidal pulse width d <latexit sha1_base64="KlN4U7fHl6+k8f5RSFXcRiXdI4I=">AAACAHicbVC7TsMwFL3hWcqrwMhiUSExVQkPwViJBbYi6ENqo8px3Naq7US2g6iiLuys8AtsiJU/4Q/4DJw2A205kqWjc3zse08Qc6aN6347S8srq2vrhY3i5tb2zm5pb7+ho0QRWicRj1QrwJpyJmndMMNpK1YUi4DTZjC8zvzmI1WaRfLBjGLqC9yXrMcINla6v+2G3VLZrbgToEXi5aQMOWrd0k8njEgiqDSEY63bnhsbP8XKMMLpuNhJNI0xGeI+bVsqsaDaTyejjtGxVULUi5Q90qCJ+jeRCmwGsRFPM++kWGg9EoHNZ76e9zLxP6+dmN6VnzIZJ4ZKMv2+l3BkIpS1gUKmKDF8ZAkmitkNEBlghYmxnRVtNd58EYukcVrxzioXd+flKspLKsAhHMEJeHAJVbiBGtSBQB9e4BXenGfn3flwPqdXl5w8cwAzcL5+AfHUlvc=</latexit> s <latexit sha1_base64="YuVBFFGsXxp2j7inKlu/KTm9tBc=">AAACAHicbVC7TsMwFHV4lvIqMLJYVEhMVcJDMFZiga0I+pDaqHJcp7VqO5F9g6iiLuys8AtsiJU/4Q/4DJw2A205kqWjc3zse08QC27Adb+dpeWV1bX1wkZxc2t7Z7e0t98wUaIpq9NIRLoVEMMEV6wOHARrxZoRGQjWDIbXmd98ZNrwSD3AKGa+JH3FQ04JWOn+tmu6pbJbcSfAi8TLSRnlqHVLP51eRBPJFFBBjGl7bgx+SjRwKti42EkMiwkdkj5rW6qIZMZPJ6OO8bFVejiMtD0K8ET9m0glgUEM8mnmnZRIY0YysPnMN/NeJv7ntRMIr/yUqzgBpuj0+zARGCKctYF7XDMKYmQJoZrbDTAdEE0o2M6KthpvvohF0jiteGeVi7vzchXnJRXQITpCJ8hDl6iKblAN1RFFffSCXtGb8+y8Ox/O5/TqkpNnDtAMnK9fCemXBg==</latexit> <latexit sha1_base64="QXvxz3qhGooEE4XJaYnm/NnZZFg=">AAACAHicbVC7TsMwFL3hWcqrwMhiUSExVQkPwViJhbEI+pDaqHJcp7VqO8F2EFXUhZ0VfoENsfIn/AGfgdNmoC1HsnR0jo997wlizrRx3W9naXlldW29sFHc3Nre2S3t7Td0lChC6yTikWoFWFPOJK0bZjhtxYpiEXDaDIbXmd98pEqzSN6bUUx9gfuShYxgY6W7RvehWyq7FXcCtEi8nJQhR61b+un0IpIIKg3hWOu258bGT7EyjHA6LnYSTWNMhrhP25ZKLKj208moY3RslR4KI2WPNGii/k2kAptBbMTTzDspFlqPRGDzma/nvUz8z2snJrzyUybjxFBJpt+HCUcmQlkbqMcUJYaPLMFEMbsBIgOsMDG2s6KtxpsvYpE0TiveWeXi9rxcRXlJBTiEIzgBDy6hCjdQgzoQ6MMLvMKb8+y8Ox/O5/TqkpNnDmAGztcvG6GXEQ==</latexit> modulation (SPWM) or space vector pulse width modulation d <latexit sha1_base64="mErlhmH36tzlKEAKn+l9A+Z43nQ=">AAACA3icbVC7TsMwFHXKq5RXgZHFokJiqhIegrESC2OR6ENqo8px3NbUdiL7BlFFHdlZ4RfYECsfwh/wGThtBtpyJEtH5/jY954gFtyA6347hZXVtfWN4mZpa3tnd6+8f9A0UaIpa9BIRLodEMMEV6wBHARrx5oRGQjWCkY3md96ZNrwSN3DOGa+JAPF+5wSsFKzGzIBpFeuuFV3CrxMvJxUUI56r/zTDSOaSKaACmJMx3Nj8FOigVPBJqVuYlhM6IgMWMdSRSQzfjqddoJPrBLifqTtUYCn6t9EKgkMY5BPc++kRBozloHNZ75Z9DLxP6+TQP/aT7mKE2CKzr7vJwJDhLNCcMg1oyDGlhCqud0A0yHRhIKtrWSr8RaLWCbNs6p3Xr28u6jUcF5SER2hY3SKPHSFaugW1VEDUfSAXtArenOenXfnw/mcXS04eeYQzcH5+gWQC5hv</latexit> 1 I q <latexit sha1_base64="ncye0sCi16afCJrhbOHzXrEPUtQ=">AAACAHicbVDLTgIxFO3gC/GFunTTSExckRkf0SWJG91hlEcCE9IpHWhoO2N7x0gmbNy71V9wZ9z6J/6Bn2GBWQh4kiYn5/S0954gFtyA6347uaXlldW1/HphY3Nre6e4u1c3UaIpq9FIRLoZEMMEV6wGHARrxpoRGQjWCAZXY7/xyLThkbqHYcx8SXqKh5wSsNLdTeehUyy5ZXcCvEi8jJRQhmqn+NPuRjSRTAEVxJiW58bgp0QDp4KNCu3EsJjQAemxlqWKSGb8dDLqCB9ZpYvDSNujAE/Uv4lUEujHIJ9m3kmJNGYoA5sf+2beG4v/ea0Ewks/5SpOgCk6/T5MBIYIj9vAXa4ZBTG0hFDN7QaY9okmFGxnBVuNN1/EIqmflL3T8vntWamCs5Ly6AAdomPkoQtUQdeoimqIoh56Qa/ozXl23p0P53N6NedkmX00A+frFwa1lwQ=</latexit> (SVPWM) affects G to be equal to . In the context of inv D + D <latexit sha1_base64="n1rmVFhzdn9jt6A7akFDfcV0ZDE=">AAAB/3icbVDLTgIxFL2DL8QX6tJNIzFxI5nxEV2S6MIlGnkkQEindKCh7UzajpFMWLh3q7/gzrj1U/wDP8MOzELAkzQ5Oaenvff4EWfauO63k1taXlldy68XNja3tneKu3t1HcaK0BoJeaiaPtaUM0lrhhlOm5GiWPicNvzhdeo3HqnSLJQPZhTRjsB9yQJGsLHS/clNt1hyy+4EaJF4GSlBhmq3+NPuhSQWVBrCsdYtz41MJ8HKMMLpuNCONY0wGeI+bVkqsaC6k0wmHaMjq/RQECp7pEET9W8iEdgMIiOeZt5JsNB6JHybT30976Xif14rNsFVJ2Eyig2VZPp9EHNkQpSWgXpMUWL4yBJMFLMbIDLAChNjKyvYarz5IhZJ/bTsnZUv7s5LFZSVlIcDOIRj8OASKnALVagBgQBe4BXenGfn3flwPqdXc06W2YcZOF+/08eWUg==</latexit> <latexit sha1_base64="PQ2Wft6Uu+8GA5SV1jVGAvpgVe0=">AAAB/3icbVDLSgMxFL1TX7W+qi7dBIsgCGXGB7os6MJlFfuAtpRMmmlDk8yQZMQydOHerf6CO3Hrp/gHfoaZdha29UDgcE5Ocu/xI860cd1vJ7e0vLK6ll8vbGxube8Ud/fqOowVoTUS8lA1fawpZ5LWDDOcNiNFsfA5bfjD69RvPFKlWSgfzCiiHYH7kgWMYGOl+5ObbrHklt0J0CLxMlKCDNVu8afdC0ksqDSEY61bnhuZToKVYYTTcaEdaxphMsR92rJUYkF1J5lMOkZHVumhIFT2SIMm6t9EIrAZREY8zbyTYKH1SPg2n/p63kvF/7xWbIKrTsJkFBsqyfT7IObIhCgtA/WYosTwkSWYKGY3QGSAFSbGVlaw1XjzRSyS+mnZOytf3J2XKigrKQ8HcAjH4MElVOAWqlADAgG8wCu8Oc/Ou/PhfE6v5pwssw8zcL5+AdCRllA=</latexit> Eq. A-4, an SVPWM inverter was assumed and G was de- inv a <latexit sha1_base64="QmnKjQ3CU+1JGWxi08kYHeMs54U=">AAACA3icbVC7TsMwFL0pr1JeBUYWiwqJqUp4CMZKLIxFog+pjSrHdVpT24lsB1FFHdlZ4RfYECsfwh/wGThtBtpyJEtH5/jY954g5kwb1/12Ciura+sbxc3S1vbO7l55/6Cpo0QR2iARj1Q7wJpyJmnDMMNpO1YUi4DTVjC6yfzWI1WaRfLejGPqCzyQLGQEGys1u5jHQ9wrV9yqOwVaJl5OKpCj3iv/dPsRSQSVhnCsdcdzY+OnWBlGOJ2UuommMSYjPKAdSyUWVPvpdNoJOrFKH4WRskcaNFX/JlKBzTA24mnunRQLrccisPnM14teJv7ndRITXvspk3FiqCSz78OEIxOhrBDUZ4oSw8eWYKKY3QCRIVaYGFtbyVbjLRaxTJpnVe+8enl3UamhvKQiHMExnIIHV1CDW6hDAwg8wAu8wpvz7Lw7H87n7GrByTOHMAfn6xeJqJhr</latexit> √ termined through simulation to be .

The design of the motor coefficients took place at the rated Figure 21. Steady-state phasor diagram of the PMSM.

motor torque and speed, the condition defined as immediately before requiring flux-weakening control and the intersection of MCT. NDARC would associate the rated design point with Several assumptions were made for motor coefficient sizing.

MCP. A superscript r is appended to variables to indicate a It was assumed at steady state that change in current is zero, parameter evaluated at a motor rated condition.

cross-inductance terms were negligible and D-axis current was zero. The last of these three assumptions was made to- The motor power factor was computed from angles depicted gether with a decision to identify the NDARC motor spec- in Figure 21 as in Eq. A-5 ification speed as the motor’s rated speed. The NDARC re- ( ) π ported MRP, was determined to be the peak power of the mo- Φ = cos ( ϕ ) = cos + δ − α (A-5) tor. The motor’s rated speed was defined as the highest speed on the torque map before flux-weakening control laws were where the voltage angle is solved for as in Eq. A-6 required (i.e., D-axis current is non-zero) coinciding with the ( ) V d − 1 MCT operation of the motor. For simplicity, the motor type δ = − tan (A-6) V was assumed to be non-salient meaning equivalence in D-axis q and Q-axis self-inductance; although in the paper, the self- and the current angle was solved for as in Eq. A-7.

inductance terms were presented as separate quantities. With ( ) these assumptions in place, the steady state Q-axis stator volt- I q − 1 α = tan (A-7) age was approximated as the summation of voltage due to I d torque and back-EMF as in Eq. A-1.

The current angle, α , at the rated condition of the motor is π ∼ V = R I + λ n ω (A-1) equal to , due to the assumption that I = 0. Therefore at the q s q p m d rated condition the motor power factor can be computed as in It is assumed that load power, friction and hysteresis losses Eq. A-8. subtracted from input power is equal to heating losses. The winding losses were determined as in Eq. A-15.

( ) V d − 1 Φ = cos ( ϕ ) = cos − tan (A-8) V ( ) q 2 r r r r r r r P = P − P − τ ω , R I (A-15) H I L l m s q By substituting Eqs. A-1 and A-2 into Eq. A-8, the motor The rated stator resistance was computed by substituting rated inductance can be solved for as in Eq. A-9.

power terms from Eqs. A-11–A-13 into Eq. A-15. Solving for r rated stator resistance resulted in Eq. A-16.

− 1 tan ( cos ( Φ )) λ r r L = L = (A-9) d q r I q r 2 P r H R = (A-16) s r ( I ) q The three unknown terms in Eq. A-9 are the rated current, power factor and flux linkage. The rated current was deter- The rated flux linkage was computed by substituting Eqs. A-2 mined as in Eq. A-10 and A-3 into Eq. A-1. The result is substituted into Eq. A-4 which was used to form Eq. A-17, where the inequality con- r r 2 τ 2 τ + τ I r L l straint has been converted to an equality at the rated motor I = = (A-10) q r r 3 λ n 3 λ n p p condition.

r where τ is the sum of torque loss and load torque, τ is the I l √ r torque loss experienced at the rated motor condition and τ is L 2 r r 2 r r r r r r G V = ( R I + λ n ω ) + ( − L I n ω ) (A-17) inv p p m s q m q q m the rated load torque. The friction model was assumed to be as in Eq. A-11 Equation A-17 is crucial in this paper to sizing motor coeffi- r cients. The rated battery motor voltage, V , is defined as the m r r ∼ τ τ + c ω (A-11) = h μ l m DC-rating provided by motor manufacturers when providing a motor map. Like the number of pole pairs of the motor, an where τ is hysteresis friction loss and c is the viscous damp- μ h independent choice must be made on the rated motor voltage.

ing coefficient. Hysteresis torque loss was modeled to assist The map itself will not change with the choice, but the rea- in sizing a hysteresis coefficient for the motor temperature sonableness of the resulting motor coefficients are affected by model. For simplicity, the damping coefficient was calculated r r the choice of V . For this research, 650 V was chosen for V .

m m by assuming a percentage of torque loss with respect to the The pole pair count, n , was set equal to six in the motor co- p rated motor load torque. It was assumed that torque losses efficient sizing procedure. Other choices could be acceptable would amount to 2% of the rated load torque at the rated op- depending on desired speed range of the motor; a lower num- erating condition. It was assumed that hysteresis torque was ber of pole pairs is more ideal for high-speed applications. In- 10% of the viscous damping torque at the rated motor condi- ductance, resistance, and current computed respectively from tion. The motor load power was determined as in Eq. A-12.

Equations A-9, A-10 and A-15 were directly substituted into Eq. A-17. The rated flux linkage is solved for as in Eq. A-18.

r r r P = τ ω (A-12) L L m r r V G where ω is the rated motor speed. With the ratio of peak inv m r m √ λ = ( ) power to maximum continuous power known, the torque load, 2 r r P P 2 − 1 r 2 r H H r n [ 1 + tan ( cos Φ )] ω + 2 ω + p m m τ , is solved for at the rated condition. The rated input power τ τ I I L was set equal to the motor load power with an assumed power (A-18) conversion efficiency of 96% as in Eq. A-13 To solve Eq. A-18, the power factor and speed are iteratively changed until convergence in the rated speed is achieved equal r P r L to the NDARC rated speed. With power factor, flux link- P = (A-13) I r η age, and rated current determined, the rated inductance from m Eq. A-9 can be computed.

The rated input power is recast as motor active power which Rated motor coefficients change with temperature. For gen- is a function of the stator voltages and currents as in Eq. A-14.

eration of motor maps, a lumped mass temperature model from Appendix A in Ref. 25 was utilized to capture varia- 3 3 r r r r r P = V I + V I (A-14) tion of the motor coefficients with temperature. For simplicity I q q d d 2 2 the coolant temperature was assumed to be held constant at ◦ ◦ Casting input power as active power allows for changes to 60 C. The rated temperature was assumed to be 140 C, con- efficiency as a function of motor temperature to be directly sistent with material properties assumed in the assumed ther- computed. mal model. Because NDARC produces motor models with higher maximum continuous power than the experimentally The rated superscript was dropped from motor coefficients, to fit motor of 46 hp in Ref. 25, thermal parameters from an ex- account for possible changes of the flux linkage and induc- perimentally derived thermal model of a 82 hp motor (Ref. 26) tance due to temperature and current draw at different mo- was utilized. The area of contact contributing to all thermal tor conditions. However, when plotting the boundaries of the resistances was scaled up by a linear factor until the average motor maps and determining peak conditions, the rated coef- ◦ rated winding temperature of the motor converged to 140 C ficients were used for simplicity. The rated motor constants in time domain simulations. The temperature model allowed were substituted into Eq. A-21 for a reasonable peak speed insight into the trend of motor winding and rotor temperatures calculation. Since current is a function of speed, Eqs. A-21 p under specific motor loads. The thermal modeling efforts did and A-22 were easiest to solve by iteratively varying ω .

m not include a model of changes to self-inductance due to tem- It was assumed that peak torque in the flux-weakening region perature. The flux linkage change with temperature was mod- could be allowed to remain constant between the base speed eled as in Eq. A-19 and rated speed. This assumption may be supported with a combination of a higher battery voltage than the rated motor ( ) λ = λ 1 + α ( θ − 20 ) (A-19) pm r voltage and flux-weakening control. The assumption results in the peak power to continue climbing between the base speed 20 ◦ where λ is the flux linkage measured at 20 C, α is pm and rated speed. After the rated speed, the power was assumed the thermal coefficient for permanent magnet material set to to level off to a constant value with increasing motor speed.

− 0 . 001, and θ is the average temperature of the motor rotor r The assumption requires torque to fall to maintain a constant permanent magnets. The sized flux linkage coefficient is used power relationship after the rated speed. The researchers as- to solve for λ when the average winding temperature θ is w sumed this would be accomplished with flux-weakening con- ◦ equal to 140 C from a time domain simulation at the rated trol technology fully engaged at the rated speed. To model the condition and θ has moved to its steady state temperature r impact of flux-weakening control technology past the rated ◦ typically near 90 C.

speed, the peak speed was defined as in Eq. A-23 The stator resistance was modeled as in Eq. A-20 r b ω τ p m L nl ω , ≤ ω (A-23) m m R = R ( 1 + α ( θ − 25 )) (A-20) τ s c w s L 25 ◦ where R is the stator resistance measured at 25 C, α is the nl b c s where ω is the no-load speed and τ is the peak load torque m L − 3 25 copper coefficient set to 3 . 93 × 10 . The R term is solved s achievable at the base speed.

for by setting the left-hand side of Eq. A-20 to the sized sta- b The base speed of the motor, ω , is defined only at the motor’s m tor resistance and the average winding temperature is equal to ◦ rated DC voltage and peak power condition. The base speed 140 C from a time domain simulation of the thermal model.

solution, provided by NDARC, may be computed as in Eq. A- The changes in resistance and flux linkage with temperature have a significant impact on the efficiencies computed in the motor map.

p P b L ω = (A-24) m b Speed and Torque Limits Expressions for the motor speed τ L and torque limits are derived in this subsection. The limits p b where τ is the peak load torque at the base speed and P may be computed at each segment of the NDARC mission if L L is peak load power or what NDARC refers to as MRP. The the maximum available motor speed and torque can also be base speed may be iteratively computed by substituting peak computed. The definition of maximum available motor speed p current, I , defined as in Eq. A-25, as well as the motor’s rated and peak torque changes when including or excluding flux- q r DC voltage, V , into Eq. A-21.

weakening control technology. Solutions to peak motor speed m and torque were presented with and without flux-weakening [ ] p control. By rearranging Eq. A-17 and inserting battery volt- 2 P p b L I = + τ + c ω (A-25) h μ q m b age, the peak motor speed for any load torque up to the peak 3 λ n ω p m load torque can be computed for motors which do not use The no-load speed may also be computed by substituting no- flux-weakening control. The peak motor speed is defined in nl load current, I , defined as in Eq. A-26, as well as the motor’s Eq. A-21 q r rated DC voltage, V , into Eq. A-21 and repeating the same m √ speed iteration until convergence.

( ) 2 2 2 2 2 2 2 2 2 − R I λ + G V λ + L I − L I R I s q inv q q q q s q b p [ ] ω = ( ) (A-21) m 2 2 2 nl b n λ + L I p q q I = τ + c ω (A-26) h μ q m 3 λ n p where nl The no-load speed, ω , was crucial to compute and was used m ( ) p 2 τ + τ + c ω as a maximum theoretical motor speed. NDARC did not pos- L h μ m I = (A-22) q sess knowledge of the no-load speed and may report motor 3 λ n p designs which can exceed the no-load speed. Exceeding the and rated speed which would require a larger battery voltage no-load speed with a significant torque load – while theoreti- than the motor rated voltage. The second assumption was to cally possible with a larger voltage source and/or flux weak- allow the motor to operate past the rated speed with a con- ening control – may lead to mechanical damage of the motor. stant peak power relationship up to the no-load speed. The third assumption was that continuous torque was modeled to At every segment of the NDARC mission the aerodynamic fall assuming constant continuous power past the rated speed.

rotor speed margin as a function of discharge was computed The second and third assumptions are more consistent with as in Eq. A-27.

certain flux-weakening integrated motor manufacturer speci- p fication sheets. A combination of a larger battery voltage than ω m Ω = − Ω (A-27) the rated voltage as well as flux-weakening control may be M 0 G r used to achieve constant torque between the base speed and rated speed.

where G is the rotor transmission gear ratio, and Ω is the r 0 NDARC-derived aerodynamic rotor trim speed for any condi- tion. The speed margin is a useful quantity for RPM controlled REFERENCES vehicles which require aerodynamic rotor speed changes to control the vehicle.

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