Rotorcraft Dynamic Platform Landings Using Robotic Landing Gear

Throughout their history, rotorcraft have proven to be a vital tool for accessing landscapes that traditional fixed-wing aircraft are unable to reach due to their smaller landing zone footprint. Pilots of rotorcraft flight systems often face the complexity of landing on uneven, rugged, or moving surfaces in order to complete their mission. These complexities lead to operational limits for the landing conditions; pilots or flight control systems are allowed to operate in. For example, rotorcraft acceptable ground slope limits range from 6 deg to 15 deg, depending on their design (critical roll angle for dynamic rollover) and environmental conditions [1]. Additionally, pilot workload and safety risks increase for landings on mobile surfaces, such as maritime ship decks, because of the periodic, coupled landing surface dynamics [2–5]. Rotorcraft are critical to civil and military missions in maritime environments [6–8]. As such, there is a broad area of technological developments focused on improving mission safety.

The first set of technologies rely on modification to the maritime vessel itself. Deck lock mechanisms exist that lock the rotorcraft to a point on a landing deck in rough seas, see Fig. 1. The purpose of these tools is to ensure the rotorcraft does not bounce or slide during a landing. A drawback of these solutions is that they prevent the natural sliding of the rotorcraft and can cause damage to rotor mechanical components because of high lateral loads. Motion compensation ship decks have been conceptualized, but not commercially realized. These decks measure the maritime vessels' dynamic motion and adjust the landing deck's position and orientation accordingly [10]. Given the size and cost of this system, it has not been commercially viable to date.

Improved training simulations, sensors suites, and flight controllers have all been explored to reduce workload and maximize safety margins associated with shipboard rotorcraft landings as well. Reber and Bernard developed a game-based learning, high-fidelity training simulation that removed the need for highly skilled instructor oversight [11]. Their research showed that trainees of their program operated with improved safety margins during normal and extreme ship deck operations when compared to a control group, showing the importance of human factors and training. Xu et al. [12] and Saripalli et al. [13] approached the problem through a combination of sensor suite and control algorithm development to implement computer vision driven landing guidance. Using a reference object of known size, shape, and orientation on the landing surface, they showed preliminary results that IR and color spectrum imagery can be used to land on a moving platform. This body of research did not consider or attempt to verify operational capability during high sea state (SS) conditions, limiting its current use in the most challenging conditions. Numerous researchers have approached the problem through unique deep reinforcement learning [14], nature inspired controllers [15], or optimal control techniques to path plan and land on a moving platform [16–18]. These research topics showed increases in safety margins or reduced workload during approach, but none of them focus directly on dynamic rollover prevention or impact attenuation during the landing event. Robotic landing gear (RLG) for rotorcraft were developed as a solution to these challenges.

Robotic legs allow a given system to maneuver through irregular and rugged terrains, and they have been applied to a number of ground applications for quadrupedal and bipedal locomotion [19–22]. Modern research and development into RLG for rotorcraft was reintroduced by Manivannan et al. [23], but the concepts and idea of RLG for rotorcraft have existed since the 1950s [24,25]. Manivannan's concept and theoretical design provided a replacement to skid gear with three or four articulating legs that enabled rotorcraft landings on sloped or rugged static surfaces. This RLG design concept was shown by Kiefer et al. [26] to reduce peak loads on rotorcraft occupants through active control of the legs so they act as adaptive shock absorbers. An experimental flight vehicle based on Manivannan's work demonstrated, in a limited fashion, the viability and capability of RLG for rotorcraft on a 120 kg unmanned helicopter using a four-legged design [27]. A next generation crashworthy RLG designed by Di Leo et al. [28,29] implements a cable-driven, four-bar linkage mechanism. This mechanism provides single degree-of-freedom motion while housing all actuation electronics inside the fuselage and minimizes weight through the use of composite materials. León et al. presented an integration and flight tests of this next generation design on an S-100 Camcopter with a maximum takeoff weight of 200 kg, see Fig. 2 [30]. Similar concepts designed and experimentally verified by Stolz et al. [31] using a 78 kg helicopter and Huang et al. [32] on a 20 kg octocopter are additional examples of articulating landing gear concepts. The works by Manivannan et al., Stolz et al., and León et al. only used force feedback to control leg orientation during the landing, while Huang's work used infrared sensors on the bottom of each landing leg. The use of force feedback or foot-to-ground distance measurements is not sufficient for shipboard rotorcraft landings because the feedback saturates and is no longer meaningful once each leg makes contact and the rotorcraft drops collective.

For any system where the landing platform moves before and after all legs make contact, there is a need for closed-loop feedback channel(s) in addition to force/foot–ground distance information. Kim and Costello [33] developed one solution with a theoretical study on virtual model control for four-legged RLG landings on naval ship decks. This controller uses feedback from the rotorcraft orientation states to optimally compute the minimum actuator torques necessary to maintain a level fuselage. Their simulations found that such a controller in conjunction with four-legged RLG grasping mechanisms on each leg will absorb impact and conform to ship decks in rough seas with mean wave heights from 4 to 6 m and modal wave periods of 9–17 s, which is also referred to as sea state 6. Kim and Costello's work is promising but it did not provide experimental validation, and it may suffer from complications due to noise and bias in measurements or actuator response of the physical system [34]. Even with this broad scope of research, there is not at present an experimentally validated, robust RLG system for ship deck landings.

This work presents the design and experimental ground tests of roll and force feedback controlled cable-driven, four-bar link mechanism RLG for dynamic platform landings with an emphasis on the ship deck landing problem. Integration of the controller for this work is on the cable-driven, four-bar link RLG system previously used on the S-100 Camcopter during flight tests as presented by León et al. [30]. Section 2 is an overview of the system level design, including RLG kinematics, state estimation, and control for the S-100 platform. Section 3 details the multibody dynamics simulation tool used to guide the design and performance characterization of the S-100 RLG platform in a variety of static and mobile landing deck scenarios. Finally, experimental setup and results are presented and compared to simulations in Sec. 4. Section 5 concludes the paper.

This section details the subsystems pertinent to the cable-driven, four-bar RLG system as integrated with the S-100 Camcopter. These subsystems include the structural mechanism, sensing and state estimation, and multisensor feedback controller.

Figure 3(a) illustrates a wireframe diagram of the S-100 cable-driven, four-bar RLG in the neutral configuration. A continuous cable (marked in red), gas spring, and actuation/spool are the three primary components that enable motion. The gas spring maintains tension in the cable by applying an upward force (i.e., retracts the legs upward). The cable acts as the opposing force to move legs downward, and it is routed through a spool such that the spool's rotation changes the effective ground angle absorbed by the legs, see Fig. 3(b). This cable doubles back around the crank-leg pin joint and terminates within the fuselage on a shock absorber. The total cable length determines the neutral stance and the maximum achievable ground angle, as studied by Di Leo et al. [29].

The authors chose to implement single degree-of-freedom differential motion based on (1) the normal flight operations of rotorcraft on sloped surfaces, (2) the dynamics of a rolling and pitching ship deck, and (3) the specific configuration of the S-100 landing gear. Normal sloped surface operation suggested by the Federal Aviation Administration (FAA) to prevent dynamic rollover [1] recommends landing on a slope aligned with the roll axis of the rotorcraft.

In addition to this recommendation by the FAA, representative ship deck motion up through sea state 6 does not require rotorcraft pitch control which would require a symmetric retraction/extension of the legs. Illustrated in Fig. 4(a) is a representative maritime vessel, the DDG-54 Arleigh-Burke class destroyer with a length of 154 m, a width of 18.8 m, and a draft of 7 m. These physical properties were used to generate the sea state 6 roll and pitch data in Fig. 4(b) for a period of 1 min. This simulation, described further in Sec. 3.2, used a modal wave period of 17 s and a significant wave height of 6 m with the ship moving at 10 kn. This simulation produced roll on the order of ±12 deg on the landing deck and a pitch of approximately ±5 deg. Even for passive landing gear, ±5 deg of pitch is well within most rotorcraft operational limits, but the roll is above ±10 deg repeatedly. Between the FAA recommendation, and ship deck dynamics, any rotorcraft operating on sloped or mobile surfaces would see a large expansion in mission capabilities with a single degree-of-freedom aligned with the aircraft's roll axis. In addition to these general observations, there are two S-100 specific factors that limit the need for pitch-axis control: (1) the S-100 has a shock-absorber tail and (2) the forward landing leg contact points are lower than the rear leg of the S-100 (i.e., the forward legs will make contact first for small pitch angles of the landing surface).

Each of the provided considerations shows that there are clear advantages to having RLG for control along the rotorcraft's roll axis. Therefore, the authors chose to focus on actuation along the roll axis and left actuation along the pitch axis to future work, since pitch control may be valuable for other rotorcraft. A more detailed analysis of the design constraints and crashworthiness (up to 5 m/s) of the S-100 design of a cable-driven, four-bar RLG (see Fig. 2) was initially presented by Di Leo et al. [28,29] and expanded upon by León et al. [30]. The same RLG flight tested by León et al. is the system used throughout the remainder of this work.

Equation (5) now relates the rotation rate of the legs relative to the fuselage and the actuator's rotation speed.

These relationships give useful insight when input into the relationship for ground angle absorption rate, $γ˙$⁠, where B is the horizontal distance between the legs, and $Δh˙$ is the descent velocity that the legs are able to track. B is presumed constant. This assumption is acceptable given that B changes only by a small amount, $≈$3%, over the gear's entire range of motion. Equations (5)–(9) and actuator performance (⁠$β˙$⁠) are combined to yield kinematic limits for the S-100 RLG such as maximum achievable ground angle γ_max and its associated rate $γ˙max$⁠. Table 1 lists the necessary supporting specifications to compute the maximum ground roll angle and ground roll angle absorption rate. The S-100 RLG's unique structure constrains the motion such that the maximum roll angle absorption rate, $γ˙max$⁠, changes as a function of ground roll angle, $γmax$⁠.

Figure 5 shows this limit boundary up to the maximum achievable ground roll angle. The RLG configuration are able to conform up to $≈$17 deg with a maximum leg velocity of 0.69 m/s. Maximum leg velocity, which stems from $β˙max$⁠, is fundamentally limited by the S-100's onboard available power and the original design constraints from Refs. [28–30]. Actuator performance was experimentally measured, so it inherently incorporates cable friction and gas damper force and viscous damping. This maximum leg velocity does not change substantially across the system's range of motion, but the geometry causes a change in the maximum ground angle absorption rate, see Fig. 5. Angular roll rates consistent with the kinematic equations and computed results were achieved by León et al. [30] during their ground and flight test operations with this system. The x's in the plot show a sample of the performance achieved by the S-100 RLG during previous ground and flight tests. León et al. also showed that landings at 0.5 m/s were possible with minimal perturbations on roll angle on slopes up to 15 deg (3× the S-100's safe operational roll angle), but landings faster than 0.5 m/s inevitably have some notable roll perturbations because the legs are physically limited in acceleration and maximum speed.

Overlaid on Fig. 5 is a region of roll angles and angular rates from a sea state 6 simulation with a significant margin between the peak performance limit of the RLG compared to expected landing deck conditions. Based on the kinematic limits alone, the design of the S-100 RLG from León et al. appears have adequate performance for dynamic platform landing on the deck of the DDG-54, a modern maritime vessel. Note that this overlay only considers the rolling rate of the deck. It does not incorporate the combined ranges of aircraft descent rate and deck heave velocity. The following section detail the control algorithm used to conduct dynamic platform landings.

Previous research on experimental testing of RLG systems implemented force feedback control using ground contact sensors with no fuselage orientation feedback during or after the landing [23,27]. The primary assumptions of those systems are (1) that the closed-loop control will react faster than the aircraft descends and (2) the aircraft holds a steady descent rate throughout the landing. Both are idealizations of the problem. The first assumption means that the legs will not cause the aircraft to roll or pitch throughout the landing. This assumption is acceptable for low descent rates while landing on static surfaces with ideal force sensors and high update rate feedback control because the closed-loop system reacts faster than the aircraft descends. This is not a valid assumption during rapid landings or on helicopters with unique landing gear configurations (see Ref. [30]). The second assumption is true for static landing surfaces, but it cannot be true for landings on dynamic platforms. The orientation and position of the landing surface results in variable velocity between the aircraft and landing surface. For this reason, the second assumption used on force feedback controlled landing gear is not valid for landing conditions considered in this work. The lack of closed-loop roll and/or pitch feedback is a limitation of contact feedback RLG controllers.

This work fuses roll feedback with previously tested foot-force feedback loop to eliminate the assumptions of previous works. The roll and force feedback control designed here follows three operational stages illustrated in Fig. 6. The first stage has the aircraft descend with no contact on either foot. Regardless of the roll angle during the descent phase, the legs do not actuate. Once one leg makes contact with the landing surface, the second phase initiates. The RLG controller incorporates roll angle and force feedback to minimize dispersion in inertial roll angle until both legs make contact. The force sensor measurement range on both legs limits the effectiveness of force data after landing; thus, only roll angle feedback is used to reach a safe roll angle in the final stage. Nominally, this last phase ends when the aircraft reaches a minimal roll angle within an acceptable deadband, $ϕDB$⁠, on a static surface or when the frame is at a minimal roll angle and the legs are in their neutral position. The included roll feedback allows the swift detection of whether or not the landing surface is in motion.

Figure 7 shows the high-level block diagram of the combined roll and force feedback controller used in this work. There are two control loops using three feedback signals. The outer loop control combines foot-force data and fuselage roll data to command a desired leg angle, then the inner loop control monitors and changes the motor's speed, $β˙$⁠, to achieve the desired leg angle. The leg-angle estimator and foot-force feedback estimators were previously presented by León et al. [30], and the roll estimate used for loop closure is provided by an attitude and heading reference system sensor (AHRS). The RLG control starts with a conditional proportional–derivative (PD) controller that uses desired and estimated foot forces in addition to the desired and estimated roll angles as inputs. A lower-level outline of the conditional PD control operates and calculations are outlined in Algorithm 1.

Algorithm 1 generally mimics the procedures of the flight software, but it does not include specific logic of health monitor checks, control value clamping, and additional sanity checks on control outputs that exist in the flight software. In this algorithm, $eFk, eθk$⁠, and $eϕk$ are the errors in foot force, leg angle, and fuselage roll angle, respectively. The desired force for the legs to keep in ground contact is F_d, and the desired fuselage roll angle is $ϕd$⁠. The desired roll angle, $ϕd$⁠, is always $0 deg$⁠, so the controller tries to maintain a level fuselage.

The initial conditional PD algorithm is interrupt driven upon initial contact on either foot, so asynchronous inputs do not impede performance. The forces measured by the left or right foot that initiates control are F_L or F_R, respectively. The loop runs continuously after a force sensor crosses the predetermined desired force F_d. Force sensors designed and characterized in Ref. [36] were configured in an array on the bottom of each leg. The force threshold and how it was chosen for the S-100 implementation were discussed by León et al. [30] in more detail, but is suffices for this work to say the threshold is 5 N. Line 22 of the algorithm denotes that the primary leg measurement used to track leg-angle control is the left leg. Therefore, the force error force computations (lines 4 and 6) are different depending on whether the left or right food makes contact first, so that the leg-angle control computations drive the legs in the proper direction. Note that the system measures the left and right leg angles, and it can dynamically switch between the left and right leg as the primary leg-angle measurement for feedback control based on sensor health or loss of communication.

The derivative control equations implement a first-order, discrete low-pass filter on the derivative control to limit noise from the incoming signals [37]. Each channel of control (roll and foot force) has their own unique low-pass filter parameter, α, which is chosen based on the frequency characteristics of each input channel. There are more elegant solutions for the filtering of this data, but the authors found the first-order, low-pass filter to be effective for this application.

Notable from this algorithm is that the controller uses force feedback until the aircraft exceeds the allowable deadband along the roll axis. For this reason, it is advantageous to minimize the deadband of the system. Methods to minimize deadband include high dynamic accuracy roll measurements and/or actuation methods that can run continuously with little or no regard for power consumption. For this system, a VectorNAV VN-100 was used for roll and pitch feedback. The company lists the sensor's static roll accuracy to 0.5 deg and dynamic roll accuracy to 1.0 deg, so it is suitable for testbed and experimental use but a higher accuracy sensor would be ideal for a robust, fielded system. This dynamic roll accuracy was used as the roll deadband, $ϕDB$⁠, value for the experiments in work.

and the use of this alternate computation depends on the quality of roll rate feedback from an AHRS.

This section elucidates the simulation tools and their integration as a performance prediction model for the S-100 RLG. First, a multibody dynamics tool is described that has a proven track record for RLG development. Then, a ship deck motion simulation tool and standardized sea state conditions that are commonly used are described.

Proper selection of the damping ratio, ζ, and the natural frequency, ω_n, provide a stable constraint stabilization controller. The optimal values of ζ and ω_n depend on the simulation time-step and dynamics or vibration frequencies of the system, but a wide range of values maintain constraints and system stability.

This tool has a number of subsystem models that can be included or neglected in the force and moment calculations of any body in a modular manner. Figure 8 illustrates how the subsystem models are defined, then incorporated into the computation of $F¯$⁠. This figure also calls out the three models used for this work, which includes gravity, ground contact, and a first-order rotor thrust model.

for each leg that is in contact. A coefficient of friction, static and kinetic, is applied to the normal force computation for tangential sliding force. While not a perfect assumption of contact, it is a suitable and fast model of contact for RLG contact and feedback control. In this work, we are interested in dynamic landing surfaces as well. A simple extension to the SLS model is used for dynamic ground contact where δ and $δ˙$ are computed from the relative velocity of a foot and the ground at any given point. The nominal values of the spring constants and damping coefficients are provided in Table 2. These ground contact parameter values were tuned by experimentation with drop tests of the experimental landing gear with an S-100 airframe surrogate in Refs. [28] and [29].

where T is the rotor thrust, η_long is the thrust angle along the rotorcraft longitudinal plane, and η_lat is the thrust angle along the rotorcraft lateral plane. The angles η_long and η_lat stem from cyclic input to the rotor, but are not actively controlled within this work's simulations. The parameters τ_T and $τη$ are the time constants for the first-order responses of the rotor thrust and thrust angles. Finally, $Tc, ηlongc, and ηlatc$ are the desired rotor trust, desired thrust angle along the rotorcraft longitudinal plane, and the desired thrust angle along the rotorcraft lateral plane, respectively.

There are 15 rigid bodies used for dynamic simulation and constraint stabilization using the aforementioned framework, see Fig. 10. While most bodies are direct counterparts to their physical counterparts, such as the fuselage, BD1, and rotor, BD15, other parts are decomposed into multiple rigid bodies in order to introduce elastic joints. These are necessary for understanding all coupled dynamic interactions such as pitch rate to roll inertial coupling of the rotor to the system [42–44]. For example, the cable on the left leg assembly (right side of the image) is represented by two bodies, BD10 and BD11. These bodies have variable length to mimic changes in rope length resultant of spool rotations. More important than the ability to vary geometry, these bodies are connected by an elastic joint, denoted by a black dot in Fig. 10. This allows us to capture some of the elastic behavior of the real bodies. Similarly, the carbon fiber reinforced legs are represented by BD4 and BD13 as well as BD7 and BD14 with elastic joints in between. The spring stiffness and damping of each elastic joint (marked by black dots) were experimentally derived from drop tests of the full-scale S-100 RLG [28,29]. The rear leg of the aircraft, BD12, also connects to the fuselage via an elastic joint, though not shown in the figure. Joint elastic parameters for the rear leg were directly provided by the aircraft supplier, but were not experimentally verified. Finally, BD15 rotates with the S-100's rotor nominal frequency, and all thrust model outputs are applied to this body's center of gravity.

At this time, aerodynamics and ship wake are not modeled in these simulations. The simulations considered in this work assume the rotorcraft approach is complete, and the simulations initiate just prior to first contact of the gear during landing. Other works have studied aircraft (, namely, rotor) feedback [45,46] and path planning [17] through the ship's wake field. This work focuses on the landing problem when a rotorcraft may enter a dangerous dynamic rollover scenario caused by aggressive roll and roll rate of a ship deck.

The authors used the simulation tool to characterize roll dynamics of S-100 RLG in a rigid, locked configuration. This configuration is the comparison baseline for improvements by the active gear. Table 3 lists the variables for the Monte Carlo. The variable μ_f is the friction with the landing surface, u is the aircraft's body forward velocity, w is the aircraft's body descent velocity, and θ_AC is the aircraft's pitch angle. Each range of values considered is within the aircraft's expected landing envelope, and values simulated were generated using a uniform distribution.

The simulations all initiate with one foot in contact with the ground surface, and the commanded thrust changes to the minimum value. This is designed to mimic how pilots land the aircraft, where they initiate contact then lower collective to the minimum value. Figure 11 shows cumulative distribution functions (CDFs) of the peak fuselage roll angle absolute value of eight landing surfaces, $γ=±5 deg,7 deg,10 deg, and 15 deg$⁠, from this study. The x-axis is limited to 20 deg to show the fine detail for each slope, any portion of the CDF beyond 20 deg represents dynamic rollover. The results of the study show that the S-100 RLG configuration with the gear inactive, or locked, shows no risk with landings at $γ=5 deg$⁠. The risk for dynamic rollover first appears at $γ=7 deg$⁠. The simulations suggest that less than 5% of landings in the nominal envelope on a 7 deg slope result in dynamic rollover. By $γ=10 deg$⁠, the risk increases with more than 40% of landings in an otherwise nominal landing envelope result in dynamic rollover, while $γ=15 deg$ results in more than 85% of landings with dynamic rollover. These results show that the S-100 would benefit from active gear on static surfaces at low as 7 deg. The authors note that these CDF plots will shift downward on dynamic platforms with similar γ because of the added interface dynamics between the aircraft and landing surface.

Three types of landing surface dynamic motion programs were simulated and experimented with in this work. The first is a static surface. This is the most basic landing surface for RLG to interact with and has been extensively simulated and tested (see Fig. 11 and Refs. [29] and [30]). This section describes the sinusoidal and maritime ship deck landing platforms and how their motion is generated for simulation and experiments.

A sine wave is generated for the simulation using the resultant frequency, f, and time shift, t₀, and input into the multibody dynamic simulation tool.

A ship deck simulation tool provides realistic landing deck 6DOF motion, given the vessel's geometry, mass properties, and sea state parameters. This simulation tool is crucial to understanding the benefits of a roll-fused RLG controller because rotorcraft often must land on dynamic maritime ship decks. The ship motion simulation consists of two parts, the Ship Motion Program (SMP) [47] and the Simulation Time History (STH) [48]. Given a ship with specified dimensions and hull characteristics, the SMP creates 6DOF response transfer functions for regular waves. The STH uses the transfer functions from the SMP to predict the ship motion due to ocean waves. This ship motion simulation generates deck motion as a function of wave motion parameters.

The origin of the ship is located at the intersection of the ship's forward perpendicular, centerline, and baseline. This point is easily identifiable and does not vary with changes in ship draft or trim. The direction of the x, y, and z coordinates from the ship's origin is depicted in Fig. 12. Inputs to the simulation are the hull characteristics (ship length, draft, and trim). The ship configuration used for the simulation shown here is the same as defined in Sec. 2. The SMP generates the ship's 6DOF response transfer functions for regular waves. It also provides absolute and relative 6DOF information (surge, sway, heave, roll, pitch, and yaw) of the ship at the origin and other desired points on the ship. The STH provides realistic, random wave time histories of 6DOF ship responses using the regular wave response transfer functions obtained from the SMP. Random waves are simulated using a two parameter Bretschneider wave spectral model to define the frequency content of the random sea waves [49]. The two parameters used are the significant wave height and the modal (peak-to-peak) wave period. Strip theory is used to obtain the response for a ship advancing at constant forward speed with arbitrary heading in regular sinusoidal waves. The Access Time History Computer Program then uses ship origin responses from the STH to generate response time histories at other locations on the ship. The inputs to the STH are the transfer functions for a particular ship generated by the SMP, sea state (significant wave height and mean wave frequency) and the steaming condition (ship heading relative to predominant wave direction, ship speed). The outputs from the STH used in this work are 6DOF time histories for motion of the helicopter landing surface of the ship.

Nominal ranges of wave height and wave modal period are correlated to standard SS codes. Table 4 lists the sea state parameters from SS 3 through SS 6. These values come from the nominal conditions list by Bales [50] as measured in the North Pacific. Simulations in this work use the largest wave height and average wave modal period. An example of roll and pitch data using SS 6 wave conditions with STH was shown in Fig. 4(b). Sections 4.2.1–4.2.3 use a combination of static and dynamic landing surfaces to better understand the theoretical limits of the S-100 RLG.

Experiments of the integrated system were performed at Boeing's Mesa, AZ facility with a Sarnicola Simulation Systems hexad series 6DOF hydraulic motion table. This S-100 RLG system uses the same drivetrain as designed and flight tested by León et al. [30]. Figure 13 illustrates the integrated S-100 RLG on the motion table. Denoted by arrows are the shore power cable that was always connected during RLG operation on the aircraft and a crane connection point.

The crane used for experiments lowers the system at a constant 0.18 m/s for these experiments. The orientation of the aircraft during crane descent is approximately level, unlike the flight system that trims at $≈$−2 deg. The Sarnicola motion table can maintain angular rates in excess of 30 deg/s, but has hard limits at ±14 in. of heave (vertical) motion [51]. This limitation meant experiments could not fully mimic ship deck surge–sway–heave conditions, but experiments were conducted up to the acceptable limits of the table for translation motion. Operation of the configuration was limited to 10 deg/s in roll, pitch, and yaw rate, and SS 5 with the restriction to surge, sway, and heave to 10% of maximum values. The experimental setup as provided does not mimic the full range of landing speeds, and it does not include dynamic interactions from spinning rotor blades, so this experimental system will perform better than the flight system. Nevertheless, the experiments prove necessary to compare with the results of the simulation tools, so that the entire landing envelope may be further explored with the simulation tools.

Two categories of dynamic platform landings were tested as a part of this work; sinusoidal and sea state. Sinusoidal table motion required a wave frequency and amplitude, but provides a consistent and decoupled input for the RLG to track. These tests were completed to better understand the hardware's physical limitations before experimental tests on sea state platform landings.

Sinusoidal landing experiments with roll amplitude of ±10 deg and peak roll rates between 3 and 5 deg/s were conducted. These experiments were conducted to measure pure roll rate response of the RLG decoupled from pitch motion and translational velocities of the landing surface because the roll is axis where this RLG was designed to mitigate dispersions. These experiments began with the S-100 on the table, then the table activated. With the aircraft on the table, the force feedback portion of control is effectively turned off from the beginning. Numerous experiments were conducted prior to the examples presented in this section to converge on a smooth and fast response of the RLG. Figure 14 shows still images of the experiment (a), still images of the simulations (b), and experimental data (c) and (d), of a motion table commanded ±10 deg, 3 deg/s peak roll rate sinusoid as an example of these tests. The images in Figs. 14(a) and 14(b) illustrate the RLG and fuselage undergoing roll feedback control and maintaining the fuselage within or at the bounds of $ϕDB$ for the experiment and simulation, respectively. As expected from simulations and kinematic relations, the feedback controller and drivetrain are consistently able to absorb roll dynamics of the landing platform. Experiments were run for less than 30 s to prevent any motor overheating issues.

The plot in Fig. 14(c) shows fuselage, RLG, and estimated table angle data from this experiment. Roll controlled RLG maintains the fuselage roll angle (green data points) within or at the $ϕDB$ (green shaded region) boundary for the entirety of the table's dynamic motion (red dashed line). Figure 14(d) shows the simulated response of the S-100 subjected to the similar landing conditions. The simulated sinusoid was computed using the method described in Sec. 3.2.1. All sinusoid simulations for the remainder of this work were precomputed using the peak rate and amplitude of the associated experimental test. These example simulations may not directly match the dynamics of the table, since there is inertia and lag in the table's motion and the virtual table moves without lag or inertia. The motion table had a peak roll rate of 3.5 deg/s, while the simulation was ideal at 3.0 deg/s. The authors computed the peak angular rate of the RLG using measurements from all angular encoders and computing a filtered, moving average derivate of the measurements. During the experiment, the RLG achieved angular rates between 4.5 and 4.8 deg/s. Figure 14(c) shows the table reached peaks of ±11 deg based on the onboard computed estimate of ground angle, $γ̂$⁠, rather than the commanded ±10 deg. This was accounted for in the matching simulation. Regardless of the motion platform's performance, the roll feedback controller operated nominally for the entire sinusoid experiment. Furthermore, the simulation tool accurately represented the response of the experimental system.

The authors conducted additional experiments where the aircraft was lowered onto the platform to close the force and roll feedback loops. Figure 15(a) shows data from all RLG sensors during one of these experiments where the table was commanded a 5 deg/s peak roll rate sinusoid of ±10 deg with the aircraft lowered onto the table at 0.18 m/s. The table only achieves ±8–9 deg but it had peak angular rates of 10 deg/s for reference, which were both accounted for in the comparable simulation shown in Fig. 15. The table angle and rate appreciably lags when compared to the simulation at t = 1–2 s as the table adjusts its control gains to keep up with the added weight of the aircraft. Simulation results with the S-100 landed on a deck undergoing similar motion to the experiment are shown in Fig. 15(b). The differences between the experimental data and simulation data are due to the variable motion rates of the table that the simulation does not account for in this work. Visible in Figs. 15(a) and 15(b) are the bumps in $ϕ$ and γ_RLG when the landing gear first make contact and begin to conform on the surface. While the aircraft does not undergo landing in the simulation at the same time as the experiment, the motion dynamics from 3 s onward match within 0.5 deg.

These examples of roll sinusoid responsiveness of the S-100 and integrated RLG show the ability of the prototype system to operate with force feedback and roll feedback closed-loop control of the gear from initial contact through the full landing sequence. The RLG prototype conformed on the table with platform roll rates up to 10 deg/s on slopes up to 11 deg. Finally, the simulations most comparable to the experiments show comparable response of the aircraft and the gear.

The final set of experiments conducted on the S-100 were SS 3 and 5 platform landings. Experiments began in a similar fashion to the sinusoidal experiments, with the aircraft on the platform when the experiment began. The authors choose to show two of these experiments in this work, one SS 3 and one SS 5. Both of these experiments used the maximum allowable surge–sway–heave for the table, up to 10% of the maximum value output from STH and SMP as described in Sec. 3.2. Figures 16(a) and 16(b) depict the angle measurements, and (c) and (d) present the angular rates of the gear for these two experiments. The SS 3 data in (a) and (c) show mild waves where RLG is not required for safe aircraft operation due to the low roll angles associated with the platform. This is in stark contrast to the results of the SS 5 experiment. The particular SS 5 test chosen begins with mild waves, and within 10 s the waves begin to increase such that the ship deck peaks near $±10 deg$ and angular rates at and above 5 deg. The operational range of the gear is apparent from these two vastly different experiments. For both systems, the gear maintains the experimental platform within 0.5 deg of the $ϕDB$ bounds. Comparable simulations were executed to further validate the simulation design. Figures 16(e) and 16(f) illustrate the roll angle dynamics of the simulated systems undergoing the same motion table dynamics as the experimental system. The simulations closely match the response of the experimental system, with one minor difference. The SS 3 experimental system overcorrected the roll angle of the fuselage, resulting in a minor difference in roll angle response between the real and simulated systems (see Fig. 16(c) for the spike in $γ˙RLG$⁠). The set of experiments conducted with the aircraft initialized on the motion table showed enough success to conduct experiments with the S-100 descending onto the platform while in motion.

Figures 17(a)–17(f) show two sea state 5 experimental landings and (g) and (h) depict simulation results with similar conditions as the experiments. Both of the landings descended onto the platform at 0.18 m/s, and operators initiated descent at random times during the platform's dynamic motion. Therefore, the dynamics seen in the experiments look significantly different. Additionally, the dynamic motion programs are limited to 5% of the maximum surge, sway, and heave outputs from SMP and STH simulations for operator and aircraft safety.

Data in the plots of Figs. 17(a), 17(c), and 17(e) present the landing response when the gear make first contact, while the table increases roll angle up to a peak near 10 deg. First contact is made, and the combine impact force of the aircraft and table's heave pushed the force sensor measurement beyond the minimum threshold of force feedback control. Roll and force fused control maintains the aircraft at the boundary of $ϕDB$ until the RLG achieves full contact. RLG tracked angular rates of $≈$9.5 deg/s to match the combined dynamics of the platform and aircraft. The gear makes full contact with both legs and matches the deck motion within 2 s of descent onto the platform. For the next 10 s, the RLG maintains the aircraft within 1.5 deg of $ϕDB$⁠. The larges deviation in roll occurs at 8 s, at which time the cable friction under the aircraft's weight causes significant differential torque on the drivetrain's spool. The torque on the motor was too large to keep up with the table's motion, so the angular rate trended toward 0 deg/s at this moment. Simulation of this landing showed similar behavior during contact with the table, where the fuselage moves beyond $ϕDB$ before rapid RLG motion starts (see Fig. 17(g)). For the majority of the test, the simulation and experiment match together except for the peak in roll on the experimental system at 8 s. While the simulation includes a motor thermal model for degradation of control, it assumes the motor is in free space with air at 25 °C. Since the experimental drivetrain is enclosed in the fuselage and operated for long periods of time before this experiment, these assumptions cannot mimic the torque (heating) limitation of motor speed at 8 s. The thermal model is accurate for short bursts of operation less than 4 s, as verified from the multiple experiments in this section.

The data in the plots of Figs. 17(b), 17(d), 17(f), and 17(h) present a mild portion of SS 5 conditions. The largest angle magnitude achieved by the motion table during this landing was −6 deg, and the RLG never moved at angular rates above 6 deg/s to track the deck's motion. A simulation chosen for comparison has the same contact timing, but different roll response of the simulated platform. Even with this minor difference, the simulation and experimental system have matching responses to contact and platform motion in the first 4–5 s. The gear never exceeded 0.5 deg outside of $ϕDB$ for either the simulation or experiments. These results further show the simulations as a useful characterization tool for the S-100 RLG system.

In both landing scenarios, the contact force varies depending on contact angle with the deck. The amount of force variation is an unexpected find of the experimental research, and it warrants study of force feedback inclusion after full contact in the future. Both landing scenarios presented in this work support that the experimental prototype has the sensor measurements and drivetrain response to land on a deck in sea state 5 conditions. These experimental results show at a limited level that the experimental prototype S-100 RLG meets the design performance expectations. Experiments and simulations of the same or similar landing conditions match well, so the authors explored additional landing envelope and dynamic platform conditions through simulation.

Additional simulations were conducted on static and dynamic landing platforms for three RLG operational cases: no feedback control (legs locked), force feedback controlled, and roll and force feedback control. No feedback control (i.e., rigid landing gear) was simulated as a part of this section to show a direct comparison between the locked and active landing gear configuration responses with respect to dynamic rollover. This does not represent the dynamic rollover scenario for the nominal static gear on the aircraft. Force feedback control scenarios were simulated to compare to a controller used for previous flight tests [30]. The roll and force feedback controller is not expected to enhance performance during the short time horizon from initial contact to contact on all legs because of the fundamental acceleration (power) limits of the drivetrain as noted in Sec. 2.1. Nevertheless, this roll and force fused controller is expected to prevent dynamic rollover by pushing the rotorcraft's roll angle toward the ideal deadband of $ϕDB=±$0.1 deg once full contact is achieved. This deadband was chosen to simulate high quality AHRS that are integrated into rotorcraft, not the VN-100 used on the experimental platform. The last consideration for these simulations is the landing velocity. Each simulation presented here considers the case where the aircraft starts at $≈$0 m/s descent, but collective is dropped at first contact, similar to how remote pilots fly the aircraft [30]. Typical aircraft descent rates once both legs have made contact range from 0.55 to 0.6 m/s for this scenario, at the upper end of the RLG's performance specification. Any additional vertical velocity from the platform will push the RLG up to its original design boundary and beyond. However, this method is similar to how pilots land the aircraft, and it accurately represents the flight-tested actuation. The landing simulations presented here represent the approximate performance boundary for the S-100 RLG, not a full range of performance estimates. Furthermore, they are meant to show general trends of landing performance outside the range of the experimental validation in Sec. 4 because of the numerous nonlinearities that may stem from faster, more aggressive landing situations.

The first landing cases simulated were on static surfaces with the aircraft descending with a roll angle of −2 deg, the nominal approach roll angle of the aircraft [30]. Figure 18(a) illustrates the roll angle results for a 15 deg even slope landing without any feedback control (green line), force feedback controlled RLG (blue line), and force and roll feedback controlled RLG (black dashed line). Four shaded regions are shown in (a) to distinguish risk of dynamic rollover for the given roll angle. These regions were analytically observed as low risk (orange, 5–10% chance) and high risk (red, > 40% chance) for dynamic rollover in Sec. 3.1. As previously noted, the authors have no knowledge of the aircraft's dynamic rollover risk when the original equipment manufacturer rigid landing gear is installed, so this data is a baseline comparison of the active RLG system to the RLG without any active control (i.e., locked in the neutral position). The maximum descent speed achieved during this landing for the force and roll feedback controlled RLG is 0.56 m/s, so it is expected for the locked RLG to diverge and experience dynamic rollover during the short landing period. Since this descent rate is within the performance specification of the gear's drivetrain, the system compensates and maintains a roll angle throughout the landing with no risk of dynamic rollover. Figures 18(b)–18(d) give a physical picture of the simulations performed and shown in Fig. 18(a). These images and the simulation outputs clearly show that any feedback control of RLG is beneficial to prevent dynamic rollover on extreme slopes. The benefit of including roll feedback is clearer for landings where the aircraft's inertial cross-coupling of the rotor impacts roll rate.

Figure 19(a) presents the roll angle data for a −10 deg roll angle sloped surface. In this scenario, the pitch–roll rate inertial cross-coupling exacerbates the roll rate increase during landing which the roll feedback controller is able to mitigate. The simulation with force feedback control leaves the aircraft above 5 deg of roll. This is consistent with the findings of León et al. [30] during flight test experiments of the force feedback controller. The force and roll feedback controller is not able to completely mitigate roll angle increase prior to full contact of the legs. This is due to the physical power and speed constraints of the RLG. The added feedback channel (roll) allows the gear to return the aircraft to a safe operating region rapidly after full contact is made. A custom, higher power actuator would be able to mitigate this issue, though. Nevertheless, the roll and force feedback fused control is especially important for limiting the dynamic rollover risk during subsequent takeoff when collective is increased and thrust begins to push the aircraft to roll about contact points. With the fused control system, the aircraft is able to takeoff from a near level configuration unlike the other configurations.

Simulations were conducted landing the S-100 on a moving platform with only sinusoidal roll motion. The sine waves used to generate the deck motion have motion from ±15 deg in the frequency ranges that correspond to waves with maximum angular rates of 3–8 deg/s, respectively. Sinusoid waves of these amplitudes and frequencies are within the sea state 6 simulated ship deck response as shown in Fig. 4(b). The primary benefit of simulations using only roll sinusoidal is the lack of surge, sway, and heave motion which adds to the dynamic interaction. These simulations are designed to be compared to sea state platform simulations to determine the impact of decoupled angular motion versus translation motion on the RLG's overall performance.

Figure 20(a) shows the roll angle data for the three operation modes landing on a platform undergoing sinusoidal motion. The platform's sinusoid had a peak roll rate of ±5 deg/s and was initialized at −10 deg. Figure 20(b) illustrates each operation mode's orientation at t = 1.0 s, when the locked RLG is recovering from a bounce. The aircraft for each simulation has a peak descent speed of 0.61 m/s, near the bound of design specification for the S-100 RLG and well above the experimental tests conducted in Sec. 4.1. Simulation with locked RLG does not roll over, but it has a significant and dangerous bounce. The authors note that the aircraft does not enter dynamic rollover because the platform was moving toward $γ=0 deg$ and the risk to enter dynamic is low at that roll angle. Both RLG controllers prevent this large roll angle bounce and keep the aircraft outside the low risk region. Again the force and roll feedback controlled system quickly (<0.5 s) returns the aircraft to a level position even as the deck continues to move. The force feedback controller does not have any sensor feedback to track the deck motion, so the aircraft is a ride along on the platform with a constant offset. This puts the aircraft in an unusual configuration for recovery or take off at a later time. All of these issues are dealt with by the force and roll feedback controlled gear, since it converges toward and maintains the allowable $ϕDB$⁠.

Similar behavior is observed from a landing where the platform's initial condition is 10 deg with positive angular rate. Figure 21(a) shows the roll feedback of the simulations for each operating mode. The authors note that the aircraft has a significant bounce, but does not enter dynamic rollover in these ideal conditions (no wind modeled). In the locked RLG case, there is a $≈$1 deg bias between the aircraft roll and the deck roll angle, which is due to semipermanent stretch of the cable on one side of the RLG assembly. This semipermanent asymmetry is caused by the large impact force on the right leg.

Some oscillations are visible on the force and force and roll feedback RLG simulations, which Fig. 21(b) further elucidates for the force and roll feedback RLG simulation. The oscillations are caused by momentary loss of contact. The force feedback control portion of both controllers is not perfectly tuned across a wide range of landing speeds; therefore, the gear can momentarily lose contact. Any momentary loss of contact results in a deceleration of the gear until contact is regained. This prevents the gear from impacting the surface with significant force, further destabilizing roll angle. In this case, the gear loses contact four times during the landing sequence (see Fig. 21(b)). Regardless of this contact bouncing, the RLG is fast enough to bound the roll angle within a safe operating region during the time before full contact is established around 0.4 s. There is then a transient period where the torque on the spool from both cables is too high to achieve significant $γ˙RLG$⁠. By 0.6 s, the torque stabilizes to a point where the drivetrain can drive the legs at a rate of $≈$7.7 deg/s until the roll is within $ϕDB$⁠. These sinusoid platform simulations show the force and roll feedback RLG to bound the roll angle to a safe operating region. The force and roll-fused control maintains the roll angle near the desired deadband even with significant pitch–roll inertial cross-coupling caused by the rotor blade and landing gear geometry.

The final set of simulations conducted were on dynamic platforms simulated SS 5 and 6 conditions. These two conditions were chosen since they represent a combined 37% probability of occurrence in the northern hemisphere [50], and the dynamics of the ship decks at these SS pose a risk to crew and aircraft.

Figure 22 depicts the fuselage roll angle output of a 4 s simulation where the aircraft initiates contact on the landing surface while it is near 10 deg. This particular case leads to dynamic rollover of the locked RLG system. There is rolling motion of the fuselage for the active controlled systems as well, but no dynamic rollover or dangerous bouncing behavior. The significant rolling of the fuselage of the actively controlled simulations is because the combined impact velocity of the helicopter decks is 1.3 m/s because of the deck's heave motion. This is almost two times the maximum speed of the active RLG, so they are unable to absorb all of the roll angle. The force and roll feedback RLG returns the fuselage to a safe operating condition within 1 s of initial contact even though the combine system dynamics are well outside of the nominal operating specification.

More detail of the drivetrain's response to this aggressive landing is illustrated in Fig. 23. It is clear the system is accelerating as quickly as possible to keep up with the combined deck and aircraft dynamics until contact is made on all legs and the torque is too high to continue accelerating at the same magnitude. Once the drivetrain slows and the torque stabilizes across the spool, the RLG system begins to drive toward $ϕDB$ at t = 0.7 s. Noise in the data between the full contact and return toward level is from response dynamics of the cable, ground, and elastic joints during the impact.

Similar responses result from sea state 6 simulated landings. Figure 24(a) depicts the roll response to a sea state 6 landing when contact begins with the deck near −7 deg and 1.28 m/s relative vertical velocity, while Fig. 24(b) shows the response to a sea state 6 landing with contact initiated at 10 deg and a relative velocity between the deck and aircraft being 0.98 m/s. In Fig. 24(a), the difference between the roll angles of the three systems is small because of the large relative velocity and inertial cross-coupling effect on roll rate. This represents an aggressive landing outside the nominal operation bounds of the experimental RLG. The active RLG systems have a minor bounce but do not enter dynamic rollover. The gear's ability to push against the slope prevents the aircraft from a dangerous bounce and rollover. The locked RLG does enter dynamic rollover in the low risk region because of the high landing velocity relative to the nominal value (see Table 3). This ability to combat dynamic rollover is more pronounced for the force and roll feedback system, which pushes the aircraft into the safe operating region well before the deck reaches a safe roll angle.

Each simulation presented shows that the force feedback controller is, at minimum, sufficient for reducing roll instability, but the danger with such a controller is clear in Fig. 24(b). The gear using force feedback absorbs significant ground angle (⁠$≈$8 deg) which in turn puts the aircraft at a significant inertial roll angle of 17 deg by the end of the simulation at t = 5 s. At this roll angle, the aircraft is in significant danger of rollover if any collective is applied to the rotor. This is an example of a case where force feedback RLG is worse than a rigid gear configuration, showing the need for continued feedback after landing. The force and roll feedback controlled RLG eliminates this issue by maintaining the aircraft roll near level.

The set of simulations presented here are representative of the conditions; the S-100 RLG would be subjected to in the real world, from static to dynamic simulated ship decks. A rigid landing gear configuration enters dynamic rollover repeatedly even with the ideal conditions for collective (thrust) drop and no simulated wind. Both active RLG configurations prevent roll instability under the same static and mobile platform landing conditions, but the system using only force feedback has a significant probability of the aircraft entering a dangerous condition when landing on a dynamic platform (see Fig. 24(b)). Even with the most aggressive sea state 5 and 6 conditions with platform-aircraft relative impact velocities at 2× the maximum rating of the experimental landing gear, the aircraft with force and roll feedback did not enter dynamic rollover. These simulations show that the force and roll feedback controller is theoretically effective to eliminate roll dynamic instability and ensures safe operating conditions for the S-100 in conditions up to the extremes of sea state 6.

This work presented a cable-driven, four-bar link RLG integrated onto the S-100 Camcopter as a method to aid landings on dynamic platforms—namely, maritime ship decks. The experimental S-100 RLG was proven to have ample dynamic response landing on moving platforms using a discrete, real-time roll and foot-force feedback control. Experiments on dynamic motion platform simulating sinusoidal and SS 5 conditions showed the feedback controller and drivetrain are able to maintain a user-defined deadband and maintain safe operating orientation for the S-100. Each rotor-off experiment verified simulation results at a basic level by testing the hardware on a real motion table. The simulation tools complement the experimental findings and further demonstrate that the drivetrain and feedback control chosen for the system are capable of preventing dynamic rollover in aggressive sea state 6 conditions. This work is the first necessary step toward autonomous landings with RLG on static or moving platforms, and commercial applications of this technology are far reaching for rotorcraft and the expanding unmanned vertical takeoff and landing aircraft space. There are a number of future work tasks that could improve the usability of such a four-bar RLG system. A few of these future work tasks are itemized below:

1. Incorporate aerodynamic and ship wake flow fields into the multibody dynamics simulation to better understand how the wind and gusts impact landing stability after the rotorcraft collective has been dropped. This will aid in the statistical understanding of how RLG performs as compared to locked (rigid) landing gear.

2. RLG sensing and control systems may be implemented with onboard flight controls and landing approach flight software in order to fully realize the commercial potential of the technology.

3. The RLG control may be integrated with vision or LIDAR-based landing guidance systems. This would give the landing gear a preemptive image of the landing surface to preconform or prime the actuators prior to landing.

4. Study the scalability of the four-bar RLG structure for full-size helicopters. The scalability for a cable-driven design is not immediately clear. However, the four-bar link mechanism has distinct advantages that will scale to full-size helicopters.

The authors would like to thank DARPA, and in particular, Ashish Bagai, Timothy Chung, Ryan Hofmeister, Natalie Van Osch, Matthew Tallent, and Michael Ol for providing valuable comments and guidance throughout the S-100 program. The authors would like to thank the Boeing-Mesa team, and in particular, Ray Lopez, James Roos, Steve Low, John Waxler, Roy Bergman, James Bonds, and Bill Brady for their assistance during ground and flight testing of the S-100 program. Finally, the authors thank Jacob Wachlin and Kyle Solomon for their guidance and assistance throughout the program. The views, opinions, and/or findings expressed are those of the author(s) and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

• Defense Advanced Research Projects Agency (DARPA) Tactical Technology Office (TTO) (Contract No. FA8650-12-C-7276; Funder ID: 10.13039/100000185).

B =

distance between the contact points of the left and right legs

c =

standard linear solid contact model parallel damper constant

C_F =

linear scaling coefficient between pressure on each foot and the force applied on that foot

F =

standard linear solid contact model force

$F¯$ =

vector containing six degrees-of-freedom dynamic equations of motion

$F̂d$ =

desired foot contact force during landing

$F̂L/R$ =

force estimate on the left or right foot

$[G]$ =

linear transform to map constraint forces and moments to multibody equations of motion

k =

standard linear solid contact model parallel spring constant

k_v =

standard linear solid contact model series spring constant

K_D =

derivative control gain

K_P =

proportional control gain

L₁ =

crank length

L₂ =

distance between the crank and follower

$P̃$ =

normalized pressure measured on a RLG foot

r_spool =

RLG drivetrain spool radius

$sL/R$ =

cable length of the left or right half of the RLG assembly

T =

rotor thrust

T_c =

commanded rotor thrust

u =

aircraft body relative forward velocity

$U¯$ =

vector containing multibody constraint forces and moments

w =

aircraft body relative downward velocity

$X¯$ =

state vector

α =

low-pass filter time constant

$β˙$ =

angular speed of the RLG drivetrain spool

γ =

ground angle

$γ̂$ =

onboard estimate of the ground angle based on roll angle and RLG leg angles

$γ˙RLG$ =

angular rate of the active landing gear

δ =

standard linear solid contact model displacement

$Δh$ =

vertical height between the left and right leg contact points

ζ =

damping ratio

η_lat =

thrust angle relative to the fuselage's lateral plane

η_long =

thrust angle relative to the fuselage's longitudinal plane

$ηlatc$ =

commanded thrust angle relative to the fuselage's lateral plane

$ηlongc$ =

commanded thrust angle relative to the fuselage's longitudinal plane

θ =

crank orientation angle relative to the fuselage body relative vertical axis

μ =

friction coefficient

τ_T =

first-order model time constant for rotor thrust

$τη$ =

first-order model time constant for the angle of rotor thrust

$ϕ$ =

roll angle

$ϕ̂$ =

roll angle estimate

$ϕd$ =

desired fuselage roll angle in feedback controller

$ϕDB$ =

deadband roll angle for roll feedback control

ω_n =

natural frequency