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NASA NTRS1994

Simulation of Shuttle launch G forces and acoustic loads using the NASA Ames Research Center 20G centrifuge

The high cost and long times required to develop research packages for space flight can often be offset by using ground test techniques. This paper describes a space shuttle launch and reentry simulating using the NASA Ames Research Center's 20G centrifuge facility. The combined G-forces and acoustic environment during shuttle launch and landing were simulated to evaluate the effect on a payload of laboratory rates. The launch G force and acoustic profiles are matched to actual shuttle launch data to produce the required G-forces and acoustic spectrum in the centrifuge test cab where the rats were caged on a free-swinging platform. For reentry, only G force is simulated as the aero-acoustic noise is insignificant compared to that during launch. The shuttle G-force profiles of launch and landing are achieved by programming the centrifuge drive computer to continuously adjust centrifuge rotational speed to obtain the correct launch and landing G forces. The shuttle launch acoustic environment is simulated using a high-power, low-frequency audio system. Accelerometer data from STS-56 and microphone data from STS-1 through STS-5 are used as baselines for the simulations. This paper provides a description of the test setup and the results of the simulation with recommendations for follow-on simulations.

Shaw, T. L.↗

NASA NTRS2010

Interplanetary Transit Simulations Using the International Space Station

It has been suggested that the International Space Station (ISS) be utilized to simulate the transit portion of long-duration missions to Mars and near-Earth asteroids (NEA). The ISS offers a unique environment for such simulations, providing researchers with a high-fidelity platform to study, enhance, and validate technologies and countermeasures for these long-duration missions. From a space life sciences perspective, two major categories of human research activities have been identified that will harness the various capabilities of the ISS during the proposed simulations. The first category includes studies that require the use of the ISS, typically because of the need for prolonged weightlessness. The ISS is currently the only available platform capable of providing researchers with access to a weightless environment over an extended duration. In addition, the ISS offers high fidelity for other fundamental space environmental factors, such as isolation, distance, and accessibility. The second category includes studies that do not require use of the ISS in the strictest sense, but can exploit its use to maximize their scientific return more efficiently and productively than in ground-based simulations. In addition to conducting Mars and NEA simulations on the ISS, increasing the current increment duration on the ISS from 6 months to a longer duration will provide opportunities for enhanced and focused research relevant to long-duration Mars and NEA missions. Although it is currently believed that increasing the ISS crew increment duration to 9 or even 12 months will pose little additional risk to crewmembers, additional medical monitoring capabilities may be required beyond those currently used for the ISS operations. The use of the ISS to simulate aspects of Mars and NEA missions seems practical, and it is recommended that planning begin soon, in close consultation with all international partners.

Charles, J. B.↗

NASA NTRS2011

Time-Accurate Computational Fluid Dynamics Simulation of a Pair of Moving Solid Rocket Boosters

Since the Columbia accident, the threat to the Shuttle launch vehicle from debris during the liftoff timeframe has been assessed by the Liftoff Debris Team at NASA/MSFC. In addition to engineering methods of analysis, CFD-generated flow fields during the liftoff timeframe have been used in conjunction with 3-DOF debris transport methods to predict the motion of liftoff debris. Early models made use of a quasi-steady flow field approximation with the vehicle positioned at a fixed location relative to the ground; however, a moving overset mesh capability has recently been developed for the Loci/CHEM CFD software which enables higher-fidelity simulation of the Shuttle transient plume startup and liftoff environment. The present work details the simulation of the launch pad and mobile launch platform (MLP) with truncated solid rocket boosters (SRBs) moving in a prescribed liftoff trajectory derived from Shuttle flight measurements. Using Loci/CHEM, time-accurate RANS and hybrid RANS/LES simulations were performed for the timeframe T0+0 to T0+3.5 seconds, which consists of SRB startup to a vehicle altitude of approximately 90 feet above the MLP. Analysis of the transient flowfield focuses on the evolution of the SRB plumes in the MLP plume holes and the flame trench, impingement on the flame deflector, and especially impingment on the MLP deck resulting in upward flow which is a transport mechanism for debris. The results show excellent qualitative agreement with the visual record from past Shuttle flights, and comparisons to pressure measurements in the flame trench and on the MLP provide confidence in these simulation capabilities.

Strutzenberg, Louise L.↗

NASA NTRS2017

Implementation of Real-Time Hardware in the Loop Simulation for WAVE Instrument Avionics

The Regolith and Environment Science and Oxygen and Lunar Volatile Extraction (RESOLVE) payload will lead the Resource Prospector rover to hydrogen-rich locations on the moon supporting NASA's in-situ resource utilization (ISRU) mission. The Water Analysis and Volatile Extraction (WAVE) system will be responsible for heating up regolith samples and analyzing their volatiles in a vaporized state. Given the space environment, testing flight hardware and software using the scientific instruments can be costly and time consuming, which can hold back progress involving the instruments. A hardware-in-the-loop (HITL) simulation will test the Avionics Data Acquisition as well as the Instrument Interface Unit, through simulating sensors and actuators involved in supporting the WAVE instruments. HITL is a platform for testing and developing WAVE's avionics and software, where the simulation plant will imitate the LAVA and OVEN instruments, thus allowing for an accessible, efficient, and replicable testing environment.

Al Qaraghuli, Ali↗

NASA NTRS2011

Implementation of Motion Simulation Software and Visual-Auditory Electronics for Use in a Low Gravity Robotic Testbed

The Jet Propulsion Laboratory (JPL) is developing the All-Terrain Hex-Limbed Extra-Terrestrial Explorer (ATHLETE) to assist in manned space missions. One of the proposed targets for this robotic vehicle is a near-Earth asteroid (NEA), which typically exhibit a surface gravity of only a few micro-g. In order to properly test ATHLETE in such an environment, the development team has constructed an inverted Stewart platform testbed that acts as a robotic motion simulator. This project focused on creating physical simulation software that is able to predict how ATHLETE will function on and around a NEA. The corresponding platform configurations are calculated and then passed to the testbed to control ATHLETE's motion. In addition, imitation attitude, imitation attitude control thrusters were designed and fabricated for use on ATHLETE. These utilize a combination of high power LEDs and audio amplifiers to provide visual and auditory cues that correspond to the physics simulation.

gravity↗

NASA NTRS1983

Large space structures controls research and development at Marshall Space Flight Center: Status and future plans

Work performed in Large Space Structures Controls research and development program at Marshall Space Flight Center is described. Studies to develop a multilevel control approach which supports a modular or building block approach to the buildup of space platforms are discussed. A concept has been developed and tested in three-axis computer simulation utilizing a five-body model of a basic space platform module. Analytical efforts have continued to focus on extension of the basic theory and subsequent application. Consideration is also given to specifications to evaluate several algorithms for controlling the shape of Large Space Structures.

Buchanan, H. J.↗

NASA NTRS1997

Efficient Parallel Kernel Solvers for Computational Fluid Dynamics Applications

Distributed-memory parallel computers dominate today's parallel computing arena. These machines, such as Intel Paragon, IBM SP2, and Cray Origin2OO, have successfully delivered high performance computing power for solving some of the so-called "grand-challenge" problems. Despite initial success, parallel machines have not been widely accepted in production engineering environments due to the complexity of parallel programming. On a parallel computing system, a task has to be partitioned and distributed appropriately among processors to reduce communication cost and to attain load balance. More importantly, even with careful partitioning and mapping, the performance of an algorithm may still be unsatisfactory, since conventional sequential algorithms may be serial in nature and may not be implemented efficiently on parallel machines. In many cases, new algorithms have to be introduced to increase parallel performance. In order to achieve optimal performance, in addition to partitioning and mapping, a careful performance study should be conducted for a given application to find a good algorithm-machine combination. This process, however, is usually painful and elusive. The goal of this project is to design and develop efficient parallel algorithms for highly accurate Computational Fluid Dynamics (CFD) simulations and other engineering applications. The work plan is 1) developing highly accurate parallel numerical algorithms, 2) conduct preliminary testing to verify the effectiveness and potential of these algorithms, 3) incorporate newly developed algorithms into actual simulation packages. The work plan has well achieved. Two highly accurate, efficient Poisson solvers have been developed and tested based on two different approaches: (1) Adopting a mathematical geometry which has a better capacity to describe the fluid, (2) Using compact scheme to gain high order accuracy in numerical discretization. The previously developed Parallel Diagonal Dominant (PDD) algorithm and Reduced Parallel Diagonal Dominant (RPDD) algorithm have been carefully studied on different parallel platforms for different applications, and a NASA simulation code developed by Man M. Rai and his colleagues has been parallelized and implemented based on data dependency analysis. These achievements are addressed in detail in the paper.

Sun, Xian-He↗

NASA NTRS1986

Flight simulator requirements for airline transport pilot training - An evaluation of motion system design alternatives

The effect of motion platform systems on pilot behavior is considered with emphasis placed on civil aviation applications. A dynamic model for human spatial orientation based on the physiological structure and function of the human vestibular system is presented. Motion platform alternatives were evaluated on the basis of the following motion platform conditions: motion with six degrees-of-freedom required for Phase II simulators and two limited motion conditions. Consideration was given to engine flameout, airwork, and approach and landing scenarios.

Lee, A. T.↗

NASA NTRS2011

ARC+(Registered Trademark) and ARC PC Welding Simulators: Teach Welders with Virtual Interactive 3D Technologies

123 Certification Inc., a Montreal based company, has developed an innovative hands-on welding simulator solution to help build the welding workforce in the most simple way. The solution lies in virtual reality technology, which has been fully tested since the early 90's. President and founder of 123 Certification Inc., Mr. Claude Choquet Ing. Msc. IWE. acts as a bridge between the welding and the programming world. Working in these fields for more than 20 years. he has filed 12 patents world-wide for a gesture control platform with leading edge hardware related to simulation. In the summer of 2006. Mr Choquet was proud to be invited to the annual IIW International Weld ing Congress in Quebec City to launch the ARC+ welding simulator. A 100% virtual reality system and web based training center was developed to simulate multi process. multi-materiaL multi-position and multi pass welding. The simulator is intended to train welding students and apprentices in schools or industries. The welding simulator is composed of a real welding e[eetrode holder (SMAW-GTAW) and gun (GMAW-FCAW). a head mounted display (HMD), a 6 degrees of freedom tracking system for interaction between the user's hands and head. as well as external audio speakers. Both guns and HMD are interacting online and simultaneously. The welding simulation is based on the law of physics and empirical results from detailed analysis of a series of welding tests based on industrial applications tested over the last 20 years. The simulation runs in real-time, using a local logic network to determine the quality and shape of the created weld. These results are based on the orientation distance. and speed of the welding torch and depth of penetration. The welding process and resulting weld bc.1d are displayed in a virtual environment with screenplay interactive training modules. For review. weld quality and recorded process values can be displayed and diagnosed after welding. To help in the le.tming process, a learning curve for each student and each Virtual Welding Class'" can be plotted, for an instructor's review or a required third party evaluation.

Choquet, Claude↗

NASA NTRS2004

The Formation Control Testbed

This paper describes the design and construction of the Formation Control Testbed. The FCT consists of three robots navigating on an air bearing floor, propelled by cold gas thrusters. Each robot contains an attitude platform supported on a spherical air bearing which provides three rotational degrees of freedom. The sixth degree of freedom, vertical translation, will be provided by a powered vertical stage, actively controlled to provide a simulated zero-g environment for the attitude platform.

Singh, Gurkirpal↗

NASA NTRS2000

NPSS on NASA's IPG: Using CORBA and Globus to Coordinate Multidisciplinary Aeroscience Applications

Within NASA's High Performance Computing and Communication (HPCC) program, the NASA Glenn Research Center is developing an environment for the analysis/design of aircraft engines called the Numerical Propulsion System Simulation (NPSS). The vision for NPSS is to create a "numerical test cell" enabling full engine simulations overnight on cost-effective computing platforms. To this end, NPSS integrates multiple disciplines such as aerodynamics, structures, and heat transfer and supports "numerical zooming" between O-dimensional to 1-, 2-, and 3-dimensional component engine codes. In order to facilitate the timely and cost-effective capture of complex physical processes, NPSS uses object-oriented technologies such as C++ objects to encapsulate individual engine components and CORBA ORBs for object communication and deployment across heterogeneous computing platforms. Recently, the HPCC program has initiated a concept called the Information Power Grid (IPG), a virtual computing environment that integrates computers and other resources at different sites. IPG implements a range of Grid services such as resource discovery, scheduling, security, instrumentation, and data access, many of which are provided by the Globus toolkit. IPG facilities have the potential to benefit NPSS considerably. For example, NPSS should in principle be able to use Grid services to discover dynamically and then co-schedule the resources required for a particular engine simulation, rather than relying on manual placement of ORBs as at present. Grid services can also be used to initiate simulation components on parallel computers (MPPs) and to address inter-site security issues that currently hinder the coupling of components across multiple sites. These considerations led NASA Glenn and Globus project personnel to formulate a collaborative project designed to evaluate whether and how benefits such as those just listed can be achieved in practice. This project involves firstly development of the basic techniques required to achieve co-existence of commodity object technologies and Grid technologies; and secondly the evaluation of these techniques in the context of NPSS-oriented challenge problems. The work on basic techniques seeks to understand how "commodity" technologies (CORBA, DCOM, Excel, etc.) can be used in concert with specialized "Grid" technologies (for security, MPP scheduling, etc.). In principle, this coordinated use should be straightforward because of the Globus and IPG philosophy of providing low-level Grid mechanisms that can be used to implement a wide variety of application-level programming models. (Globus technologies have previously been used to implement Grid-enabled message-passing libraries, collaborative environments, and parameter study tools, among others.) Results obtained to date are encouraging: we have successfully demonstrated a CORBA to Globus resource manager gateway that allows the use of CORBA RPCs to control submission and execution of programs on workstations and MPPs; a gateway from the CORBA Trader service to the Grid information service; and a preliminary integration of CORBA and Grid security mechanisms. The two challenge problems that we consider are the following: 1) Desktop-controlled parameter study. Here, an Excel spreadsheet is used to define and control a CFD parameter study, via a CORBA interface to a high throughput broker that runs individual cases on different IPG resources. 2) Aviation safety. Here, about 100 near real time jobs running NPSS need to be submitted, run and data returned in near real time. Evaluation will address such issues as time to port, execution time, potential scalability of simulation, and reliability of resources. The full paper will present the following information: 1. A detailed analysis of the requirements that NPSS applications place on IPG. 2. A description of the techniques used to meet these requirements via the coordinated use of CORBA and Globus. 3. A description of results obtained to date in the first two challenge problems.

Lopez, Isaac↗

NASA NTRS2015

A Modular Framework for Modeling Hardware Elements in Distributed Engine Control Systems

Progress toward the implementation of distributed engine control in an aerospace application may be accelerated through the development of a hardware-in-the-loop (HIL) system for testing new control architectures and hardware outside of a physical test cell environment. One component required in an HIL simulation system is a high-fidelity model of the control platform: sensors, actuators, and the control law. The control system developed for the Commercial Modular Aero-Propulsion System Simulation 40k (C-MAPSS40k) provides a verifiable baseline for development of a model for simulating a distributed control architecture. This distributed controller model will contain enhanced hardware models, capturing the dynamics of the transducer and the effects of data processing, and a model of the controller network. A multilevel framework is presented that establishes three sets of interfaces in the control platform: communication with the engine (through sensors and actuators), communication between hardware and controller (over a network), and the physical connections within individual pieces of hardware. This introduces modularity at each level of the model, encouraging collaboration in the development and testing of various control schemes or hardware designs. At the hardware level, this modularity is leveraged through the creation of a SimulinkR library containing blocks for constructing smart transducer models complying with the IEEE 1451 specification. These hardware models were incorporated in a distributed version of the baseline C-MAPSS40k controller and simulations were run to compare the performance of the two models. The overall tracking ability differed only due to quantization effects in the feedback measurements in the distributed controller. Additionally, it was also found that the added complexity of the smart transducer models did not prevent real-time operation of the distributed controller model, a requirement of an HIL system.

propulsion simulation↗

NASA NTRS2014

A Modular Framework for Modeling Hardware Elements in Distributed Engine Control Systems

Progress toward the implementation of distributed engine control in an aerospace application may be accelerated through the development of a hardware-in-the-loop (HIL) system for testing new control architectures and hardware outside of a physical test cell environment. One component required in an HIL simulation system is a high-fidelity model of the control platform: sensors, actuators, and the control law. The control system developed for the Commercial Modular Aero-Propulsion System Simulation 40k (C-MAPSS40k) provides a verifiable baseline for development of a model for simulating a distributed control architecture. This distributed controller model will contain enhanced hardware models, capturing the dynamics of the transducer and the effects of data processing, and a model of the controller network. A multilevel framework is presented that establishes three sets of interfaces in the control platform: communication with the engine (through sensors and actuators), communication between hardware and controller (over a network), and the physical connections within individual pieces of hardware. This introduces modularity at each level of the model, encouraging collaboration in the development and testing of various control schemes or hardware designs. At the hardware level, this modularity is leveraged through the creation of a Simulink(R) library containing blocks for constructing smart transducer models complying with the IEEE 1451 specification. These hardware models were incorporated in a distributed version of the baseline C-MAPSS40k controller and simulations were run to compare the performance of the two models. The overall tracking ability differed only due to quantization effects in the feedback measurements in the distributed controller. Additionally, it was also found that the added complexity of the smart transducer models did not prevent real-time operation of the distributed controller model, a requirement of an HIL system.

numerical simulation↗

NASA NTRS2014

A Modular Framework for Modeling Hardware Elements in Distributed Engine Control Systems

Progress toward the implementation of distributed engine control in an aerospace application may be accelerated through the development of a hardware-in-the-loop (HIL) system for testing new control architectures and hardware outside of a physical test cell environment. One component required in an HIL simulation system is a high-fidelity model of the control platform: sensors, actuators, and the control law. The control system developed for the Commercial Modular Aero-Propulsion System Simulation 40k (40,000 pound force thrust) (C-MAPSS40k) provides a verifiable baseline for development of a model for simulating a distributed control architecture. This distributed controller model will contain enhanced hardware models, capturing the dynamics of the transducer and the effects of data processing, and a model of the controller network. A multilevel framework is presented that establishes three sets of interfaces in the control platform: communication with the engine (through sensors and actuators), communication between hardware and controller (over a network), and the physical connections within individual pieces of hardware. This introduces modularity at each level of the model, encouraging collaboration in the development and testing of various control schemes or hardware designs. At the hardware level, this modularity is leveraged through the creation of a Simulink (R) library containing blocks for constructing smart transducer models complying with the IEEE 1451 specification. These hardware models were incorporated in a distributed version of the baseline C-MAPSS40k controller and simulations were run to compare the performance of the two models. The overall tracking ability differed only due to quantization effects in the feedback measurements in the distributed controller. Additionally, it was also found that the added complexity of the smart transducer models did not prevent real-time operation of the distributed controller model, a requirement of an HIL system.

numerical simulation↗

NASA NTRS2022

Enabling Innovative Research on the International Space Station to Solve the Challenges of A Human Mission to Mars: Results of the ISS4Mars International Workshops 2020–2021

During the ISS4Mars workshops in 2020–2021, personnel from the International Space Station (ISS) partner agencies convened to reflect on scenarios for how the ISS could be used and its operations possibly modified to simulate aspects of a human mission to Mars. Scientific leaders, operations experts, crewmembers, managers, and flight surgeons discussed the five hazards of human spaceflight—gravity transitions, radiation, isolation and confinement, distance from Earth, and hostile closed environments—and considered how an ISS-based analog of Mars transit could benefit assessments and mitigations of these hazards. A focused writing team then discussed the advantages and disadvantages of each approach identified by the workshop participants before developing a set of eight use cases to consider the feasibility of implementing on the ISS. The writing team also identified the prerequisites needed, including ground analog studies simulating a mission to Mars required to verify measurements and procedures, before testing could begin on the ISS. Five of the use cases were considered feasible to assess in simulations using an ISS-based analog of Mars transit if some ground rules and assumptions were met. These five use cases were Earth-independent medical operations, Earth-independent integrated operations, life support and food for a one year duration, lower-body negative pressure as a countermeasure against the effects of exposure to microgravity, and fitness levels after landing. In addition, three more extensive interventions—extended Mars surface operations, a small-volume transit analog, and artificial gravity—were deemed unfeasible for testing on the ISS. Experience gained from the five use cases executed on the ISS may help answer some of the questions in the deferred scenarios, or it may be possible to complete them on another platform (e.g. commercial space station, lunar habitat). Simulating conditions during a Mars mission on the ISS will afford higher fidelity for assessing multiple integrated hazards of human spaceflight, however, ground analogs of Mars missions can be used to ensure effective measures and experimental design before testing begins on the ISS. The strategic concepts refined as part of these workshops were brought to a multilateral forum, Mulitlateral Human Research Planel for Exploration (MHRPE), where ISS partner agencies are now discussing implementation plans to provide new opportunities to use the ISS to prepare for deep space exploration over the coming decade. In this publication we present a summary of the international strategic plans for future research that will enable operations, software, and countermeasures to be developed that will reduce the risk to humans during future crewed missions to Mars.

Human space exploration↗

NASA NTRS1997

Global Transport of Aerosol and CO: Initial 3-D Simulations of MAPS, TOMS, and AVHRR

Carbon monoxide concentrations and aerosol properties provide the tracers of global tropospheric perturbation by humankind that are most easily observed from satellite platforms. Aircraft field observations are additionally needed for us to simulate and understand the patterns observed. We report on our accumulating experience in making detailed, situation-specific, 3-D simulations of these tropospheric constituents as observed from the MAPS CO sensor, the Advanced Very High Resolution Radiometer (AVHRR) aerosol scattering data, and the Total Ozone Mapping Spectrometer (TOMS) absorbing-aerosol information. We have found a strong complimentarity of satellite and aircraft data in this effort, and recommend modelers use: (1) satellite data to pose initial questions; (2) aircraft data for modelers' first detailed simulations; and (3) a return to satellite data for global generalization. This work provides one example. Our main tools are the MM5 numerical model for meteorological assimilation and GRACES, our NASA Ames tracer-chemistry model, which incorporates emissions estimates. We report on several six-week simulations of global biomass burning effects as observed in the MAPS October 1994 dataset, and show the usefulness of two aircraft datasets, the TRACE-A (1992) and PEM-Tropics missions of NASA's Global Tropospheric Experiment. Additional information is contained in the original extended abstract.

Chatfield, Robert B.↗

NASA NTRS2005

Real-Time Simulation for Verification and Validation of Diagnostic and Prognostic Algorithms

To verify that a health management system (HMS) performs as expected, a virtual system simulation capability, including interaction with the associated platform or vehicle, very likely will need to be developed. The rationale for developing this capability is discussed and includes the limited capability to seed faults into the actual target system due to the risk of potential damage to high value hardware. The capability envisioned would accurately reproduce the propagation of a fault or failure as observed by sensors located at strategic locations on and around the target system and would also accurately reproduce the control system and vehicle response. In this way, HMS operation can be exercised over a broad range of conditions to verify that it meets requirements for accurate, timely response to actual faults with adequate margin against false and missed detections. An overview is also presented of a real-time rocket propulsion health management system laboratory which is available for future rocket engine programs. The health management elements and approaches of this lab are directly applicable for future space systems. In this paper the various components are discussed and the general fault detection, diagnosis, isolation and the response (FDIR) concept is presented. Additionally, the complexities of V&V (Verification and Validation) for advanced algorithms and the simulation capabilities required to meet the changing state-of-the-art in HMS are discussed.

Aguilar, Robet↗

NASA NTRS2005

Integration of Launch Vehicle Simulation/Analysis Tools and Lunar Cargo Lander Design

Simulation and analysis of vehicle performance is essential for design of a new launch vehicle system. It is more and more demand to have an integrated, highly efficient, robust simulation tool with graphical user interface (GUI) for vehicle performance and simulations. The objectives of this project are to integrate and develop launch vehicle simulation and analysis tools in MATLAB/Simulink under PC Platform, to develop a vehicle capable of being launched on a Delta-IV Heavy Launch Vehicle which can land on the moon with the goal of pre-implanting cargo for a new lunar mission, also with the capability of selecting other launch vehicles that are capable of inserting a payload into Trans-Lunar Injection (TLI). The vehicle flight simulation software, MAVERIC-II (Marshall Aerospace VEhicle Representation In 'C'), developed by Marshall Space Flight Center was selected as a starting point for integration of simulation/analysis tools. The goals are to convert MAVERIC-II from UNIX to PC platform and build input/output GUI s in the MATLAB environment, and then integrate them under MATLAB/Simulink with other modules. Currently, MAVERIC-II has been successfully converted from UNIX to PC using Microsoft Services for UNIX subsystem on PC. Input/Output GUI's have been done for some key input/output files. Calling MAVERIC-II from Simulink has been tested. Details regarding Lunar Cargo Lander Design are described in Part 2/2 of the paper on page X-1.

Shiue, Yeu-Sheng Paul↗