AIAA Guidance, Navigation, and Control Conference and Exhibit 15-18 August 25, San Francisco, California AIAA 25-62 Design and modelling of an airship station holding controller for low cost satellite operations Nimrod Rooz and Eric N. Johnson Georgia Institute of Technology, Atlanta, GA, 3332 Current research has been directed in the use of autonomous airships as less expensive alternatives to satellites. A key component in these applications is the control and guidance systems. In this paper an existing airship has been considered, and the control and guidance laws have been tailored to it. Furthermore, the addressed airship is limited in its control authority. This presents a challenge in these designs. A realistic model of the airship was developed to test the performance of a guidance and control scheme. Simulation results show the effectiveness of the approach. Nomenclature Note: all quantities that do not contain a superscript, are expressed in the fixed body frame r Position vector of the location of the C.G with respect to the body frame. ω Angular velocity of the airship. ψ c Commanded heading ψ w Wind heading θ Engine Actuator position I Current inertia matrix of the airship. K d Proportional gain K p Proportional gain M Moment vector about the body frame origin. m a Current total mass of the airship. m i Mass of the i th element. p, q, r Angular velocities R Position vector of the body frame with respect to the inertial frame. r i Position vector of the i th element. T Engine Thrust u, v, w Linear velocities u i Velocity of the i th element. y, x Position coordinate Subscripts a Airship coordinate g Guidance reference coordinate i moving element within the system L, R, B Engine reference: Left, Right, Back Graduate Research Assistant, nimrod rooz@ae.gatech.edu Lockheed Martin Assistant Professor of Avionics Integration, eric.johnson@ae.gatech.edu 1 of 7 Copyright 25 by Georgia Institue of Technology. Published by the, Inc., with permission.
Conventions ξ 3 ξ 2 For any vector ξ R 3, ξ = ξ 3 ξ 1 ξ 2 ξ 1 Superscripts L Expressed in Local NED axes W Expressed in wind axes I. Introduction The airship is one of the oldest vehicles for aerial operations. After the appearance of the airplane, and a series of disasters, the airship lost its appeal and its use in the modern aviation era has been limited. This may be changing. The airship s ability to hover and lift heavy loads over long periods of time with very low fuel consumption has renewed the interest in these vehicles as possible aerial platforms. 1, 2 Recent research has considered using long endurance, high altitude airships as less expensive substitutes for communication satellite operations, wherein the design was done in conjunction with the design of the airframe. 3 In this paper an existing airframe was considered and the control and guidance algorithms were tailored to meet certain design features. The airship in Figure 1 was developed by 21 st Century Airships Inc., and currently holds the record for altitude 2,452 ft. The main challenge in designing the control law for this vehicle is its limited control authority. This paper is structured as follows. In section II the model developed for the simulation of the airship and its components are described in detail. In section III the design and implementation of the controller and guidance laws are described. The controller considered is based on a simple linear controller. Due to limited control authority, the controller incorporates a non-linear component that translates the desired accelerations to actuator commands. In section IV the simulation used is described and results are presented. Finally, in section V conclusions are presented. a b Figure 1: Airship modelled II. System description and modelling The vehicle considered is a high altitude ball shaped airship. Unlike conventional airships, the hull does not contain the lifting gas, rather the lifting gas is contained in a separate ballonet within the hull and is free to expand as the airship gains altitude. In order to maintain the shape of the airship hull, the presure within the hull is maintained above the external pressure. The propulsion of the airship consists of two engines mounted on the sides of the airship. These engines can only gimbal simultaneously ±4 deg in the 2 of 7
pitch direction. In addition there is a fixed third engine mounted on the rear part of the airship. The fuel tanks and the ballast tanks are located in the lower part of the airship which ensures static stability of the airframe. Taking all these factors into account in addition to the ballonet constantly changing its volume, and the fuel being burnt, the airship s mass properties are constantly changing. Therefore the body frame is fixed to the airships center of volume C.V., rather than the center of gravity C.G.. The equations of motion for the airship derived with respect to the fixed body frame are: 4 M = m a r R + I ω + ω I ω + I ω + ω r i m i [ r i ] + r i m i [ r i ] r i m i u i r i m i ω r i 1 F = m a [ R + ω ω r + ω r + 2ω r + r] m i u i We assume that the rate of change in mass is small enough such that the rate at which the C.G. moves r, r is negligible. Therefore, the rate at which the moments of inertia change I is also negligible. Furthermore, we assume that all the mass components onboard the airship that are free to move ballast, fuel etc., do so at much smaller rates than the airship dynamics. Finally we arrive at the simplified version of Eqs. 1: M = m a r R + I ω + ω I ω F = m a R + ω r + ω ω r As seen from these equations, since the body frame origin does not coincide with the center of gravity, the linear and the rotational dynamics are coupled. We rewrite the equations using matrix notation in a way that will allow us to integrate both equations simultaneously. [ m a m a r m a R m a r ω = F m a ω ω r m a r R + [ ] [ R m = a ω m a r I ω = M ωiω ] [ ] [ ] m a r R F m = a ω ω r I ω M ωiω ] 1 [ ] m a r F m a ω ω r I M ωiω For vehicles the size of an airship, we must also bring into account the force required to accelerate the air mass around the body of the airship. This force is proportional to the acceleration of the airship, and therefore appears as an additional mass term. In Eq. 3 the airship mass has already been adjusted to include the added mass term. For a ball shaped hull the added mass term only appears in the linear motion, and is given by: M added = 2 3 πr3 hull ρ air 4 The forces and moments acting on the airship are modelled as following: Gravitational and Buoyancy - The forces and moments generated by the weight of different components of the airship. Since the C.G. does not coincide with the body frame origin, the weight of the airship hull also contributes to the sum of moments. Since the ballonet is free to move within the airship hull, its line of action always passes through the body frame origin, and therefore does not contribute to the moments generated. Aerodynamic - Since the airship hull is spherical, the aerodynamic forces are modelled as the forces acting on a sphere translating within a fluid. The main factor contributing to the aerodynamic moments on the airship hull are the engine struts on the sides and the rear of the airship. These struts are modelled as boxes with different cross-sectional areas in the different axes. The power plant model used for the initial stage of simulation was a model of a two stroke piston engine, This model was used during the development of the control and guidance systems. This model was later changed to a DC servo motor model, since using internal combustion engines for long endurance vehicles is not feasible. 2 3 3 of 7
III. Controller and Guidance design The main problems in the development of the control law for this airship is the lack of control authority and very slow dynamic response of the actuators it takes the engines 1 sec from full tilt up to full tilt down. The lack of control authority manifests itself in two ways: 1. The inability to achieve negative thrust commands 2. In most cases, inability to simultaneously track the commands on all axes. In order to overcome these difficulties, two features guided the design of the control and guidance laws. First, since drag is a substantial force acting on the airship hull the guidance makes use of the wind and lack of thrust, to achieve an overall negative acceleration. Secondly, when mapping the accelerations to actuator commands, we must give certain priority to commands on certain axes. To map the acceleration commands from the controller to the actual actuator commands, a simplified model of the airship was considered. The equations for the influence of the actuators are given by: ṙ = 1 I zz T L cos θ e T R cos θ e ẇ = 1 m a T L sin θ e + T R sin θ e u = 1 m a T L cos θ e + T R cos θ e + T B ṗ = 1 I xx T L sin θ e T R sin θ e 5 6 7 8 Since the airship is statically stable, the pitch and roll modes are naturally damped by gravity and the buoyancy forces. Therefore Eq. 8 can be excluded from the controller design. Since the mission objective is station holding, the main priority of the control law should be to keep the airship at the desired location. However, since the engine tilt actuators are considerably slow, the vertical axis is handled first. The commanded engine tilt angle is calculated to be proportional to the achievable vertical accelerations. Next, the engine throttle commands can be evaluated by solving Eq. 5 and Eq. 7, with the current engine angle position. As will be seen later, in the design of the guidance law, lateral correction is achieved by using a yawing motion. Therefore, Eq. 5 is first solved to obtain the desired yaw rate if at all achievable, and then these commands are adjusted to meet the required longitudinal command. The lack of control authority gave rise to another problem in the design of the guidance law. The need to compensate for errors in the lateral direction necessitates changing the airship s attitude. The complete guidance scheme is a two mode guidance law. The first stage is used at the initial stage of flight while the airship is travelling towards the stationary point. In this phase the guidance is set to hold the airship s heading towards the stationary point with no compensation for measured disturbances. The guidance law is given by ψ c = tan 1 yg L ya L x L g x L 9 a Once the airship approaches the vicinity of the stationary point, the guidance switches to the second mode. In the second mode, the body x axis is pointed opposite to the wind direction. This allows the control system to compensate for errors in both the longitudinal and lateral directions. The guidance commands are now calculated in the wind axes. Lateral motion is achieved by generating an angle between the body x axis and the wind direction. This results in a thrust component normal to the wind direction, and is given by: ψ c = ψ w + π + K p y W g y W a + Kd v W a 1 4 of 7
IV. Simulation and Results The airship simulation was developed by using the described airship model along with the vehicle simulation tool developed by GeorgiaTech s UAV lab. 5, 6 Using the simulation, we investigate the performance of the designed controller. In Figure 2 we can see the response of the system, whilst the airship travels from the launch point to the stationary point. These simulations where run with a constant wind disturbance. From the trajectory of the airship we see that that although the guidance at the first stage is pointing directly towards the stationary point, due to the wind disturbance, the trajectory becomes more of an arc. 5 45 4 35 4 3 2 State Commanded y position [ft] 3 25 2 Angle [Deg] 1 1 15 1 5 denotes the station keeping point 2 3 5 5 1 15 2 25 3 35 4 45 x position [ft] a 1 4 1 2 3 4 5 6 7 8 9 Time [sec] Throttle command b Throttle position.8.6.4.2 1 2 3 4 5 6 7 8 9 5 Engine RPM 45 4 RPM 35 3 25 2 15 1 2 3 4 5 6 7 8 9 Time [sec] c Figure 2: a Trajectory of airship in NED coordinates; b Commanded and Actual engine tilt angle Vs. time; c Commanded and Actual engine throttle Vs. time In the second set of scenarios, we wish to determine the response of the controller to a sudden shift in the wind direction. The wind in the simulation changed direction at t = 1 sec, from a wind with components 1 ft/sec south and 7 ft/sec east, to 5 ft/sec south and 12 ft/sec west. The simulation results are shown in Figure 3. In figure 3a we see the point at which the wind shifted direction which is denoted by the symbol. Furthermore, we see that although after 6 sec there is still an error in the airship location, at t = 25 sec denoted by the symbol the error in location is 5.3 ft which in this application is negligible. Since the first stage of the guidance law does not compensate for disturbances. we see in Figure 4 the different trajectories of the airship under different wind regimes. 5 of 7
4 1 Throttle command y position [ft] 2 2 t = 6 t = 1 t = 25 Throttle position.8.6.4.2 5 1 2 3 4 5 6 Engine RPM 4 45 4 6 RPM 35 3 25 2 8 2 2 4 6 8 1 x position [ft] a 15 1 2 3 4 5 6 Time [sec] b Figure 3: a Trajectory of airship in NED coordinates; b Commanded throttle and Engine rpm Vs. time 5 4 3 y position [ft] 2 1 denotes the station keeping point 1 1 1 2 3 4 5 x position [ft] Figure 4: Comparison of trajectories under different wind regimes V. Conclusions and Future work In this paper we presented an airship model used at the GeorgiaTech UAV lab to simulate high altitude airships. Using this simulation environment, we have developed a control system for station holding of high altitude airships. From the simulation results, we see that the performance of the designed controller is relatively good. One of the main challenges for long endurance vehicles is energy management. This issue will be addressed by using adaptive control techniques to reduce the control effort. Also, a more intelligent guidance law would be employed at the first stage, to avoid vehicle drift. References 1 Kim, J., Keller, J., and Kumar, V., Design and Verification of controllers for Airships, Proceedings of the 23 IEEE/RSJ International Conference on Intelligent Robots and Systems IROS 23, IEEE, October 23, pp. 54 6. 2 Chang-Su Park, Hyunjae Lee, M.-J. T. and Bang, H., Airship control using neural network augmented model inversion, Proceedings of 23 IEEE Conference on Control Applications, Vol. 1, June 23, pp. 558 563. 6 of 7
3 J. Mueller, M. P. and Zhao, Y., Development of an Aerodynamic Model and Control Law Design for a High Altitude Airship, AIAA 3rd Unmanned Unlimited Technical Conference, Workshop and Exhibit, Chicago, Illinois, September 24. 4 Thomson, W. T., Introduction to Space Dynamics, chap. 7, Dover pulications, May 1986, pp. 23 235. 5 Johnson, E. N. and Mishra, S., Flight Simulation for the Development of an Experimental UAV, AIAA Modeling and Simulation Technology Conference, No. AIAA-22-4975, Monterey, CA, August 22. 6 Johnson, E. N. and Schrage, D. P., System Integration and Operation of a Research Unmanned Aerial Vehicle, Vol. 1, No. 1, Jan 24, pp. 5 18. 7 of 7