Hovering Stability of Helicopters with Elastic Constraints

Aerial vehicles are difficult to stabilize, especially when acted upon by external forces. A hovering vehicle in contact with objects and surfaces must maintain flight stability while subject to forces imparted to the airframe through the point of contact. These forces couple with the motion of the aircraft to produce distinctly different dynamics from free flight. While external contact is generally avoided, extending aerial robot functionality to include contact with the environment during flight opens up new and useful areas such as perching, object grasping and manipulation. In this paper, we present a general elastic contact constraint model and analyze helicopter stability in the presence of those contacts. As an example, we evaluate the stability of a proof-of-concept helicopter system for manipulating objects using a compliant gripper that can be modeled as an elastic linkage with angular reaction forces. An off-the-shelf PID flight controller is used to stabilize the helicopter in free flight, as well as during the aerial manipulation task. We show that the planar dynamics of the object-helicopter system in vertical, horizontal and pitch motion around equilibrium are shown to remain stable, within a range of contact stiffnesses, under unmodified PID control. NOMENCLATURE an, bn, etc. Routh-Hurwitz array elements. a Blade lift slope. cRD Rotor damping constant. C Flight controller transfer function. d Center of gravity-linkage offset. G Translational dynamics poles. Fx, Fz Elastic linkage reaction forces. FGE Ground effect force. ∗Address all correspondence to this author. FRD Vertical rotor damping force. g Acceleration due to gravity. H,H ′ Open-loop plant pitch-input transfer functions. Hcl Closed-loop plant pitch-input transfer functions. h Rotor height above center of gravity. I Helicopter pitch inertia. k PID flight controller system gain. kd PID flight controller derivative gain. ki PID flight controller integral gain. kGE Ground effect spring constant. kx, k′ x Linkage horizontal spring constants. kz, k′ z Linkage vertical spring constants. kθ, k′ θ Linkage pitch spring constants. l Gripper finger rotor axis offset. m, m0 Helicopter and object masses. q1 Translational rotor flapping coefficient. q2 Pitch rotor flapping coefficient. R Rotor radius. TFA, TGE Rotor thrust in free-air and ground effect. u Rotor cyclic blade pitch control input. x Helicopter longitudinal position. z Helicopter height above hover equilibrium, z0. β First harmonic longitudinal rotor flapping angle. θ Helicopter pitch angle. ρ Air density. σ Rotor solidity ratio. τ Elastic linkage torsional moment. ω Rotor angular velocity. INTRODUCTION Unmanned Aerial Vehicles (UAVs) are becoming increasingly important in a variety of fields [1]. However, UAVs ‘look, but don’t touch’, with minimal interaction with their environment, and significant effort has gone into preventing UAVs from Figure 1. AERIAL MANIPULATION TESTBED CARRYING A TUBE. contacting objects around them. It is desirable to extend the scope of UAV functionality to include contacting, picking-up and manipulating objects. The flying ability of UAVs promises capabilities not available to any wheeled or legged terrestrial robot: rapid traversal of impassible terrain, movement around 3D environments and an unlimited vertical workspace. Hovering UAVs such as helicopters and quadrotors are especially appealing platforms for object interaction: they can come to a stop over an object in order to interact with it, whereas a fixed-wing aircraft must maintain a minimum velocity. During contact with the environment, stability of the aircraft in flight must be guaranteed [2]. In this paper, we present a general contact model and examine the dynamic stability of planar helicopters with elastic constraints. Conventional external loads applied to helicopters, such as tether loads and landing gear oleo, apply unidirectional forces — either vertical compression or tension. However, generalized contact forces are applied bi-directionally — these loading conditions create structurally different dynamic behavior. Aircraft interacting with objects and surfaces require novel frameworks for analyzing flight stability; this model is applicable to extended cases of conventional aircraft loading, as well as to new cases such as aircraft perching and object grasping. Of specific interest is the possibility of manipulating objects while hovering, without the need to land. This minimizes the time and energy needed to perform a manipulation task and permits the robot to acquire objects from terrain not suitable for landing, such as at the tops of power lines, radio masts or on the surface of water. The contact constraints imposed by this task are challenging, and may serve to destabilize the craft if not properly understood and accounted for. In addition to presenting a general contact model and stability analysis, we examine the specific case of grasping external objects from a helicopter platform with a compliant gripper mounted ventrally under the airframe. Aerial Manipulation The difficulties of aerial manipulation are numerous. The vehicle must maintain hover position of the helicopter over the target object accurately enough for capture with an end effector Figure 2. COMPLIANT GRIPPER MODULE. such as a gripper or hook, and reject disturbances and biases from aerodynamic effects. Earlier efforts to overcome the imprecision of hovering vehicles have relied heavily on structuring of the target object to simplify the task, for example using magnets [3, 4] and hoops [5]. However, this greatly limits the variety of objects the robot can grasp. The approach taken in this work is to use a compliant underactuated manipulator, based on the SDM Hand [6], mounted ventrally between the skids of a 4.3 kg, 1.5 m rotor, T-Rex 600 ESP radio control helicopter (Fig. 1). The gripper consists of four fingers with two elastic joints each, actuated by a parallel tendon mechanism that balances loads across each digit; it has a grasp span of 115 mm (Fig. 2). The helicopter is stabilized with a Helicommand flight controller, directed by a human pilot. The aircraft and gripper can carry loads above 1 kg. The special characteristics of the hand design — open-loop adaptive grasping, wide finger span, insensitivity to positional error [6] — closely match the challenges associated with the UAV manipulation task, allowing for a very simple, light-weight mechanism, without the need for imposed structural constraints on the load. To acquire an object, the helicopter approaches the target, descends vertically to hover over the target and then closes its gripper. Once a solid grasp is achieved, the helicopter ascends with the object. Flight Stability and External Contact Contacting and grasping objects while flying raises questions of continued aircraft flight stability [2]. When a robot helicopter with an elastic gripper has hold of an object, but has not yet applied enough thrust to ascend with it, contact forces will be transmitted through the gripper to the airframe. It is possible that these added dynamics will not be correctly compensated-for by the flight controller and thus destabilize the helicopter, leading to a crash. Throughout all modes of operation, the vehicle must guarantee flight stability to remain in the air. Automatic flight controllers for small-scale helicopters are now commercially available. Use of off-the-shelf avionics is beneficial as it keep costs and overhead down and reduces development time. However, these flight controllers may not be adaptable to deal with the additional forces transmitted through the compliant gripper. It is desirable to show that a standard control architecture, in this case Proportional Integral Derivative (PID) control, will remain stable during external contact for a given gripper and helicopter configuration. A compliant gripper may be approximated as a linkage (conceptualized as a compressible tether) connecting the immovable object and the helicopter. Research into tethered unmanned helicopter stability has been conducted since the 1960s. An early paper describes two fundamental flight modes of tethered helicopters [7]: stability of attitude due to the low connection point of the tether, and pure instability in position, the so-called ‘pendulum’ mode. These dynamics have been exploited to produce a stable unmanned rotor platform that flew at the end of its tether in a local equilibrium where the tether tension, weight and rotor thrust were balanced by automatic control [8]. More recently, efforts have focussed on autonomous landing of helicopters on ships in rough weather using a tether winch [9, 10]. All of these papers consider the helicopter to be flying far from the tether point, where the line tension and direction is approximately constant. In this case, where the elastic link is short, the aircraft will be operating exclusively around the equilibrium directly above the tether point, in low-velocity flight. The applied load cannot be treated as a constant either in magnitude or direction, nor always in tension, and consequently the mechanics are quite distinct from previous models. A different analytical approach must be taken which specifically includes the unique dynamics loadings transmitted to the airframe through the gripper.