Showing papers in "International Journal of Aerospace Engineering in 2012"

Recent Experimental Efforts on High-Pressure Supercritical Injection for Liquid Rockets and Their Implications

Abstract: Pressure and temperature of the liquid rocket thrust chambers into which propellants are injected have been in an ascending trajectory to gain higher specific impulse. It is quite possible then that the thermodynamic condition into which liquid propellants are injected reaches or surpasses the critical point of one or more of the injected fluids. For example, in cryogenic hydrogen/oxygen liquid rocket engines, such as Space Shuttle Main Engine (SSME) or Vulcain (Ariane 5), the injected liquid oxygen finds itself in a supercritical condition. Very little detailed information was available on the behavior of liquid jets under such a harsh environment nearly two decades ago. The author had the opportunity to be intimately involved in the evolutionary understanding of injection processes at the Air Force Research Laboratory (AFRL), spanning sub- to supercritical conditions during this period. The information included here attempts to present a coherent summary of experimental achievements pertinent to liquid rockets, focusing only on the injection of nonreacting cryogenic liquids into a high-pressure environment surpassing the critical point of at least one of the propellants. Moreover, some implications of the results acquired under such an environment are offered in the context of the liquid rocket combustion instability problem.

Approaches to Low Fuel Regression Rate in Hybrid Rocket Engines

Abstract: Hybrid rocket engines are promising propulsion systems which present appealing features such as safety, low cost, and environmental friendliness. On the other hand, certain issues hamper the development hoped for. The present paper discusses approaches addressing improvements to one of the most important among these issues: low fuel regression rate. To highlight the consequence of such an issue and to better understand the concepts proposed, fundamentals are summarized. Two approaches are presented (multiport grain and high mixture ratio) which aim at reducing negative effects without enhancing regression rate. Furthermore, fuel material changes and nonconventional geometries of grain and/or injector are presented as methods to increase fuel regression rate. Although most of these approaches are still at the laboratory or concept scale, many of them are promising.

High-Velocity Impact Behaviour of Prestressed Composite Plates under Bird Strike Loading

Abstract: An experimental and numerical analysis of the response of laminated composite plates under high-velocity impact loads of soft body gelatine projectiles (artificial birds) is presented. The plates are exposed to tensile and compressive preloads before impact in order to cover realistic loading conditions of representative aeronautic structures under foreign object impact. The modelling methodology for the composite material, delamination interfaces, impact projectile, and preload using the commercial finite element code Abaqus are presented in detail. Finally, the influence of prestress and of different delamination modelling approaches on the impact response is discussed and a comparison to experimental test data is given. Tensile and compressive preloading was found to have an influence on the damage pattern. Although this general behaviour could be predicted well by the simulations, further numerical challenges for improved bird strike simulation accuracy are highlighted.

Preliminary Design of Debris Removal Missions by Means of Simplified Models for Low-Thrust, Many-Revolution Transfers

Abstract: This paper presents a novel approach for the preliminary design of Low-Thrust, many-revolution transfers. The main feature of the novel approach is a considerable reduction in the control parameters and a consequent gain in computational speed. Each spiral is built by using a predefined pattern for thrust direction and switching structure. The pattern is then optimised to minimise propellant consumption and transfer time. The variation of the orbital elements due to the thrust is computed analytically from a first-order solution of the perturbed Keplerian motion. The proposed approach allows for a realistic estimation of ΔV and time of flight required to transfer a spacecraft between two arbitrary orbits. Eccentricity and plane changes are both accounted for. The novel approach is applied here to the design of missions for the removal of space debris by means of an Ion Beam Shepherd Spacecraft. In particular, two slightly different variants of the proposed low-thrust control model are used for the different phases of the mission. Thanks to their low computational cost they can be included in a multiobjective optimisation problem in which the sequence and timing of the removal of five pieces of debris are optimised to minimise propellant consumption and mission duration.

Exfoliation Corrosion and Pitting Corrosion and Their Role in Fatigue Predictive Modeling: State-of-the-Art Review

Abstract: Intergranular attack (IG) and exfoliation corrosion (EC) have a detrimental impact on the structural integrity of aircraft structures of all types. Understanding the mechanisms and methods for dealing with these processes and with corrosion in general has been and is critical to the safety of critical components of aircraft. Discussion of cases where IG attack and exfoliation caused issues in structural integrity in aircraft in operational fleets is presented herein along with a much more detailed presentation of the issues involved in dealing with corrosion of aircraft. Issues of corrosion and fatigue related to the structural integrity of aging aircraft are introduced herein. Mechanisms of pitting nucleation are discussed which include adsorption-induced, ion migration-penetration, and chemicomechanical film breakdown theories. In addition, pitting corrosion (PC) fatigue models are presented as well as a critical assessment of their application to aircraft structures and materials. Finally environmental effects on short crack behavior of materials are discussed, and a compilation of definitions related to corrosion and fatigue are presented.

Feasibility Study and Demonstration of an Aluminum and Ice Solid Propellant

Abstract: Aluminum-water reactions have been proposed and studied for several decades for underwater propulsion systems and applications requiring hydrogen generation. Aluminum and water have also been proposed as a frozen propellant, and there have been proposals for other refrigerated propellants that could be mixed, frozen in situ, and used as solid propellants. However, little work has been done to determine the feasibility of these concepts. With the recent availability of nanoscale aluminum, a simple binary formulation with water is now feasible. Nanosized aluminum has a lower ignition temperature than micron-sized aluminum particles, partly due to its high surface area, and burning times are much faster than micron aluminum. Frozen nanoscale aluminum and water mixtures are stable, as well as insensitive to electrostatic discharge, impact, and shock. Here we report a study of the feasibility of an nAl-ice propellant in small-scale rocket experiments. The focus here is not to develop an optimized propellant; however improved formulations are possible. Several static motor experiments have been conducted, including using a flight-weight casing. The flight weight casing was used in the first sounding rocket test of an aluminum-ice propellant, establishing a proof of concept for simple propellant mixtures making use of nanoscale particles.

Flight Performance Analysis of an Image Processing Algorithm for Integrated Sense-and-Avoid Systems

Abstract: This paper is focused on the development and the flight performance analysis of an image-processing technique aimed at detecting flying obstacles in airborne panchromatic images. It was developed within the framework of a research project which aims at realizing a prototypical obstacle detection and identification System, characterized by a hierarchical multisensor configuration. This configuration comprises a radar, that is, the main sensor, and four electro-optical cameras. Cameras are used as auxiliary sensors to the radar, in order to increase intruder aircraft position measurement, in terms of accuracy and data rate. The paper thoroughly describes the selection and customization of the developed image-processing techniques in order to guarantee the best results in terms of detection range, missed detection rate, and false-alarm rate. Performance is evaluated on the basis of a large amount of images gathered during flight tests with an intruder aircraft. The improvement in terms of accuracy and data rate, compared with radar-only tracking, is quantitatively demonstrated.

Parachute-Payload System Flight Dynamics and Trajectory Simulation

Abstract: The work traces a general procedure for the design of a flight simulation tool still representative of the major flight physics of a parachute-payload system along decelerated trajectories. An example of limited complexity simulation models for a payload decelerated by one or more parachutes is given, including details and implementation features usually omitted as the focus of the research in this field is typically on the investigation of mission design issues, rather than addressing general implementation guidelines for the development of a reconfigurable simulation tool. The dynamics of the system are modeled through a simple multibody model that represents the expected behavior of an entry vehicle during the terminal deceleration phase. The simulators are designed according to a comprehensive vision that enforces the simplification of the coupling mechanism between the payload and the parachute, with an adequate level of physical insight still available. The results presented for a realistic case study define the sensitivity of the simulation outputs to the functional complexity of the mathematical model. Far from being an absolute address for the software designer, this paper tries to contribute to the area of interest with some technical considerations and clarifications.

Experimental Investigation into the Control and Load Alleviation Capabilities of Articulated Winglets

Abstract: An experimental investigation into the real-time flow and control characteristics of a flying wing with articulated winglets is described in this paper. The philosophy of the concept centres around the use of active, in-flight adjustment of each wing's winglet dihedral angle, both as a primary means of aircraft roll control (single winglet actuation) and though smaller equal and simultaneous winglet deflections, tailor and alleviate main wing load. Results presented in this paper do provide good evidence of the concept's ability to adequately perform both tasks, although for the current chosen wing/winglet configuration, roll control authority was unable to achieve, per unit of control surface deflection, the same level of performance set by modern aileron-based roll control methodologies.

An Effective Three-Dimensional Layout of Actuation Body Force for Separation Control

Abstract: We conducted large eddy simulations of the control of separated flow over an airfoil using body forces and discuss the role of a three-dimensional vortex structure in separation control. Two types of cases are examined: (1) the body force is distributed in a spanwise uniform layout and (2) the body force is distributed in a spanwise intermittent layout, with three-dimensional vortices being expected to be generated in the latter cases. The flow fields in the latter cases have a shorter separation bubble than those in the former cases although the total momentum of the body force in the latter cases is the same as or half of the former cases. In the flow fields of the latter type, the three-dimensional vortices, which are not observed in the former cases, are generated by the body force downstream of the body force distributed. Thus, three-dimensional vortices are considered to be effective in controlling the separated flow.

Indirect Optimization of Satellite Deployment into a Highly Elliptic Orbit

Abstract: The analysis of the optimal strategies for the deployment of a spacecraft into a highly elliptic orbit is carried out by means of an indirect optimization procedure, which is based on the theory of optimal control. The orbit peculiarities require that several perturbations are taken into account: an 8 × 8 model of the Earth potential is adopted and gravitational perturbations from Moon and Sun together with solar radiation pressure are considered. A procedure to guarantee convergence and define the optimal switching structure is outlined. Results concerning missions with up to 4.5 revolutions around the Earth are given, and significant features of this kind of deployment are highlighted.

Burning Characteristics of Ammonium-Nitrate-Based Composite Propellants with a Hydroxyl-Terminated Polybutadiene/Polytetrahydrofuran Blend Binder

Abstract: Ammonium-nitrate-(AN-) based composite propellants prepared with a hydroxyl-terminated polybutadiene (HTPB)/polytetrahydrofuran (PTHF) blend binder have unique thermal decomposition characteristics. In this study, the burning characteristics of AN/HTPB/PTHF propellants are investigated. The specific impulse and adiabatic flame temperature of an AN-based propellant theoretically increases with an increase in the proportion of PTHF in the HTPB/PTHF blend. With an AN/HTPB propellant, a solid residue is left on the burning surface of the propellant, and the shape of this residue is similar to that of the propellant. On the other hand, an AN/HTPB/PTHF propellant does not leave a solid residue. The burning rates of the AN/HTPB/PTHF propellant are not markedly different from those of the AN/HTPB propellant because some of the liquefied HTPB/PTHF binder cover the burning surface and impede decomposition and combustion. The burning rates of an AN/HTPB/PTHF propellant with a burning catalyst are higher than those of an AN/HTPB propellant supplemented with a catalyst. The beneficial effect of the blend binder on the burning characteristics is clarified upon the addition of a catalyst. The catalyst suppresses the negative influence of the liquefied binder that covers the burning surface. Thus, HTPB/PTHF blend binders are useful in improving the performance of AN-based propellants.

Experimental and Numerical Failure Analysis of Adhesive Composite Joints

Abstract: In the first section of this work, a suitable data reduction scheme is developed to measure the adhesive joints strain energy release rate under pure mode-I loading, and in the second section, three types of adhesive hybrid lap-joints, that is, Aluminum-GFRP (Glass Fiber Reinforced Plastic), GFRP-GFRP, and Steel-GFRP were employed in the determination of adhesive hybrid joints strengths and failures that occur at these assemblies under tension loading. To achieve the aims, Double Cantilever Beam (DCB) was used to evaluate the fracture state under the mode-I loading (opening mode) and also hybrid lap-joint was employed to investigate the failure load and strength of bonded joints. The finite-element study was carried out to understand the stress intensity factors in DCB test to account fracture toughness using J-integral method as a useful tool for predicting crack failures. In the case of hybrid lap-joint tests, a numerical modeling was also performed to determine the adhesive stress distribution and stress concentrations in the side of lap-joint. Results are discussed in terms of their relationship with adhesively bonded joints and thus can be used to develop appropriate approaches aimed at using adhesive bonding and extending the lives of adhesively bonded repairs for aerospace structures.

Multidisciplinary Integrated Framework for the Optimal Design of a Jet Aircraft Wing

Abstract: The preliminary design of a jet aircraft wing, through the use of an integrated multidisciplinary design environment, is presented in this paper. A framework for parametric studies of wing structures has been developed on the basis of a multilevel distributed analysis architecture with a “hybrid strategy” process that is able to perform deterministic optimizations and tradeoff studies simultaneously. The particular feature of the proposed multilevel optimization architecture is that it can use different set of variables, defined expressly for each level, in a multi-level scheme using “low fidelity” and “high fidelity” models, as well as surrogate models. The prototype of the design environment has been developed using both commercial codes and in-house tools and it can be implemented in a geographically distributed and heterogeneous IT context.

Plasma Reactors and Plasma Thrusters Modeling by Ar Complete Global Models

Abstract: A complete global model for argon was developed and adapted to plasma reactor and plasma thruster modeling. It takes into consideration ground level and excited Ar and Ar

A Low-Cost Launch Assistance System for Orbital Launch Vehicles

Abstract: The author reviews the state of art of nonrocket launch assistance systems (LASs) for spaceflight focusing on air launch options. The author proposes an alternative technologically feasible LAS based on a combination of approaches: air launch, high-altitude balloon, and tethered LAS. Proposed LAS can be implemented with the existing off-the-shelf hardware delivering 7 kg to low-earth orbit for the 5200 USD per kg. Proposed design can deliver larger reduction in price and larger orbital payloads with the future advances in the aerostats, ropes, electrical motors, and terrestrial power networks.

Chemical rocket propulsion

Abstract: 1Department of Aerospace Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3 2Aerophysics Branch, Air Force Research Laboratory, Edwards AFB, CA 93524, USA 3Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, 10129 Torino, Italy 4Department of Aeronautical Engineering, Kumaraguru College of Technology, Coimbatore, Tamil Nadu 641006, India 5 School of Mechanical and Mining Engineering, The University of Queensland, Brisbane, QLD 4072, Australia

Corrosion Sensor Development for Condition-Based Maintenance of Aircraft

Abstract: Aircraft routinely operate in atmospheric environments that, over time, will impact their structural integrity. Material protection and selection schemes notwithstanding, recurrent exposure to chlorides, pollution, temperature gradients, and moisture provide the necessary electrochemical conditions for the development and profusion of corrosion in aircraft structures. For aircraft operators, this becomes an important safety matter as corrosion found in a given aircraft must be assumed to be present in all of that type of aircraft. This safety protocol and its associated unscheduled maintenance requirement drive up the operational costs of the fleet and limit the availability of the aircraft. Hence, there is an opportunity at present for developing novel sensing technologies and schemes to aid in shifting time-based maintenance schedules towards condition-based maintenance procedures. In this work, part of the ongoing development of a multiparameter integrated corrosion sensor is presented. It consists of carbon nanotube/polyaniline polymer sensors and commercial-off-the-shelf sensors. It is being developed primarily for monitoring environmental and material factors for the purpose of providing a means to more accurately assess the structural integrity of aerospace aluminium alloys through fusion of multiparameter sensor data. Preliminary experimental test results are presented for chloride ion concentration, hydrogen gas evolution, humidity variations, and material degradation.

Solving Coplanar Power-Limited Orbit Transfer Problem by Primer Vector Approximation Method

Abstract: The coplanar orbit transfer problem has been an important topic in astrodynamics since the beginning of the space era. Though many approximate solutions for power-limited orbit transfer problem have been developed, most of them rely on simplifications of the dynamics of the problem. This paper proposes a new approximation method called primer vector approximation method to solve the classic power-limited orbit transfer problem. This method makes no simplification on the dynamics, but instead approximates the optimal primer-vector function. With this method, this paper derives two approximate solutions for the power-limited orbit transfer problem. Numerical tests show the robustness and accuracy of the approximations.

Boron Particle Ignition in Secondary Chamber of Ducted Rocket

Abstract: In the secondary chamber of ducted rocket, there exists a relative speed between boron particles and air stream. Hence, the ignition laws under static conditions cannot be simply applied to represent the actual ignition process of boron particles, and it is required to study the effect of forced convective on the ignition of boron particles. Preheating of boron particles in gas generator makes it possible to utilize the velocity difference between gas and particles in secondary chamber for removal of the liquid oxide layer with the aid of Stoke's forces. An ignition model of boron particles is formulated for the oxide layer removal by considering that it results from a boundary layer stripping mechanism. The shearing action exerted by the high-speed flow causes a boundary layer to be formed in the surface of the liquid oxide layer, and the stripping away of this layer accounts for the accelerated ignition of boron particles. Compared with the King model, as the ignition model of boron particles is formulated for the oxide layer removal by considering that it results from a boundary layer stripping mechanism, the oxide layer thickness thins at all times during the particle ignition and lower the ignition time.

Testing and Modeling Fuel Regression Rate in a Miniature Hybrid Burner

Abstract: Ballistic characterization of an extended group of innovative HTPB-based solid fuel formulations for hybrid rocket propulsion was performed in a lab-scale burner. An optical time-resolved technique was used to assess the quasisteady regression history of single perforation, cylindrical samples. The effects of metalized additives and radiant heat transfer on the regression rate of such formulations were assessed. Under the investigated operating conditions and based on phenomenological models from the literature, analyses of the collected experimental data show an appreciable influence of the radiant heat flux from burnt gases and soot for both unloaded and loaded fuel formulations. Pure HTPB regression rate data are satisfactorily reproduced, while the impressive initial regression rates of metalized formulations require further assessment.

On Singular Perturbations of Flexible and Variable-Speed Wind Turbines

Abstract: A model for the mechanical dynamics of a wind turbine is developed, which is the composition of three physical mechanisms: flexion, torsion, and rotational dynamics. A first contribution is the identification of the essential physical parameters that provide a time-scale separation of these three mechanisms. Under the assumption of singular perturbations the time-scale separation allows to work with a reduced model of order one. This reduction has been essential for the control of this system allowing to control designers to take into account only the reduced-order model. A second contribution consists in employing a measurement of the fore-aft nacelle acceleration with the reduced model, together with a Kalman filter to estimate the flexible DOFs of the system (tower and average blade deflection). The successful approach is tested on high-order nonlinear aeroelastic simulator (FAST).

Flow Measurements Using Particle Image Velocimetry in the Ultra Compact Combustor

Abstract: The potential for the ultracompact combustor (UCC) lie in future research to reduced fuel consumption and improved engine performance. Velocity measurements performed on the UCC test rig at the Air Force Institute of Technology revealed flow patterns and time-averaged turbulence statistics for data taken burning hydrogen fuel in a straight and a curved cavity vane configuration. Over an equivalence ratio from 0.7 to 1.5, the straight vane configuration showed spanwise velocity decreased linearly with distance from the cavity vane over the width of the main channel. Increasing the flow rates and holding the equivalence ratio and ratio of cavity to main airflow rates constant, flow velocities in the main channel showed an increase with the curved circumferential configuration but a decrease with the straight circumferential configuration. Turbulence intensity is expected to be a major contributing factor, specifically since measured at 15% and 21% in the main channel for the straight and curved configurations, respectively. The results also show how the radial vane cavity (RVC) created strong vorticity throughout the main flow supporting a recirculation zone for mixing. Peak vorticity occurred farthest from the cavity vane suggesting the angle of the radial vane cavity is effective in generating increasing flow rotation.

Radiation Analysis for Two Trajectory Points of the Fire II Entry

Abstract: Numerical rebuilding of two trajectory points (t=1634 s and t=1643 s) of the Fire II mission has been carried out to predict the radiative heat flux for nonequilibrium and close-to-equilibrium conditions. The simulations have been performed with eilmer3 in an uncoupled way and using a tangent slab method for the radiation transport. Different population models (QSS and Boltzmann) have been compared, and the influence of catalytic wall condition was taken into account. An analysis on the spectral range and on the spectral resolution has also been carried out.

Multisized Inert Particle Loading for Solid Rocket Axial Combustion Instability Suppression

Abstract: In the present investigation, various factors and trends, related to the usage of two or more sets of inert particles comprised of the same material (nominally aluminum) but at different diameters for the suppression of axial shock wave development, are numerically predicted for a composite-propellant cylindrical-grain solid rocket motor. The limit pressure wave magnitudes at a later reference time in a given pulsed firing simulation run are collected for a series of runs at different particle sizes and loading distributions and mapped onto corresponding attenuation trend charts. The inert particles’ presence in the central core flow is demonstrated to be an effective means of instability symptom suppression, in correlating with past experimental successes in the usage of particles. However, the predicted results of this study suggest that one needs to be careful when selecting more than one size of particle for a given motor application.

Operational Readiness Simulator: Optimizing Operational Availability Using a Virtual Environment

Abstract: The maintenance and logistics systems that support aircraft fleets are complex and often very integrated. The complexity of these systems makes it difficult to assess the impact of events that affect operational capability, to identify the need for resources that can affect aircraft availability, or to assess the impact and potential benefits of the system and procedural changes. This problem is further complicated by the adoption of condition-based maintenance approaches resulting in dynamic maintenance planning as maintenance tasks are condition directed instead of scheduled or usage based. A proof of concept prototype for an aircraft operational readiness simulator (OR-SIM) has been developed for the Canadian Forces CH-146 Griffon helicopter. The simulator provides a synthetic environment to forecast and assess the ability of a fleet, squadron, or aircraft to achieve desired flying rates and the capability of the sustainment systems to respond to the resultant demands. The prototype was used to assess several typical scenarios including adjustment of preventative maintenance schedules including impact of condition-based maintenance, variation of the annual flying rate, and investigation of deployment options. This paper provides an overview of the OR-SIM concept, prototype model, and sample investigations and a discussion of the benefits of such an operational readiness simulator.

Solid Rocket Motor Design Using Hybrid Optimization

Abstract: A particle swarm/pattern search hybrid optimizer was used to drive a solid rocket motor modeling code to an optimal solution. The solid motor code models tapered motor geometries using analytical burn back methods by slicing the grain into thin sections along the axial direction. Grains with circular perforated stars, wagon wheels, and dog bones can be considered and multiple tapered sections can be constructed. The hybrid approach to optimization is capable of exploring large areas of the solution space through particle swarming, but is also able to climb “hills” of optimality through gradient based pattern searching. A preliminary method for designing tapered internal geometry as well as tapered outer mold-line geometry is presented. A total of four optimization cases were performed. The first two case studies examines designing motors to match a given regressiveprogressive-regressive burn profile. The third case study studies designing a neutrally burning right circular perforated grain (utilizing inner and external geometry tapering). The final case study studies designing a linearly regressive burning profile for right circular perforated (tapered) grains.

Implementation of Surface Radiation and Fluid-Structure Thermal Coupling in Atmospheric Reentry

Abstract: During atmospheric reentry, radiative heating is one of the most important component of the total heat flux. In this paper, we investigate how the thermal radiation coming from the postshock region interacts with the spacecraft structure. A model that takes into account the radiation reflected by the surface is developed and implemented in a solid solver. A partitioned algorithm performs the coupling between the fluid and the solid thermal fields. Numerical simulation of a hollow cone head and a deployed flap region shows the effects of the radiative cooling and the significance of the surface radiation.

A DES Procedure Applied to a Wall-Mounted Hump

Abstract: This paper describes a detached-eddy simulation(DES) for the flow over a wall mounted hump using MBFLO3 1,2 . The Reynolds number based on the hump chord is Rec = 9.36◊10 5 with an inlet Mach number of 0.1. Solutions of the three-dimensional Reynolds-averaged NavierStokes(RANS) procedure are obtained using the Wilcox k ! equations. The DES results are obtained using the model presented by Bush and Mani 3 and are compared with RANS solutions and experimental data from NASA’s 2004 Computational Fluid Dynamics Validation on Synthetic Jets and Turbulent Separation Control Workshop 4,5 . The DES procedure exhibited a three-dimensional flow structure in the wake, with a 13.65% shorter mean separation region compared to RANS and a mean reattachment length that is in good agreement with experimental measurements. DES predictions of the pressure coecient in the separation region also exhibit good agreement with experiment and are more accurate than RANS predictions.

Implementation of a Differential Geometric Filter for Spacecraft Formation Orbit Estimation

Abstract: The differential geometric filter is implemented to estimate the absolute and relative positions of the spacecraft in a formation. The extended Kalman Filter is also implemented as a benchmark for the differential geometric estimation. Only relative positions between the spacecraft are measured. Relative positions are measured using a wireless local positioning system (WLPS) installed in all spacecraft. Two different scenarios are studied: (1) the observations include WLPS measurements only and (2) the observations include WLPS measurements in addition to measurements for the absolute position of one spacecraft made by a radar that takes measurements from the earth’s surface. Results show that the differential geometric estimation has better stability performance and a faster convergence rate compared to the extended Kalman filter.