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Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft [chapter]

Yuichi Ikeda
<span title="2019-07-30">2019</span> <i title="IntechOpen"> Gyroscopes - Principles and Applications [Working Title] </i> &nbsp;
This chapter considers discrete-time nonlinear attitude tracking control problem of spacecraft. To this end, a Euler approximation system with respect to tracking error is first derived.  ...  Considerable research has been done about the nonlinear attitude tracking control of spacecraft, and these methods involve a continuous-time control framework.  ...  Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191 Gyroscopes -Principles and Applications Discrete-Time Nonlinear Attitude Tracking  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5772/intechopen.87191">doi:10.5772/intechopen.87191</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vvbpbswiqrg4zchngslvz64spa">fatcat:vvbpbswiqrg4zchngslvz64spa</a> </span>
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A Mixed Variable Speed Reaching Law of Sliding Mode Control for Spacecraft Tracking System

Yao Zhang, Yu-Xin Zhao
<span title="">2015</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wpareqynwbgqdfodcyhh36aqaq" style="color: black;">Mathematical Problems in Engineering</a> </i> &nbsp;
In this paper, spacecraft tracking system is considered as a discrete-time system, a mixed variable speed reaching law of SMC, and a controller for spacecraft tracking system has been investigated.  ...  For spacecraft tracking control system, the reaching law election and controller design are two crucial and important problems.  ...  Acknowledgments This work was supported by the National Natural Science Foundation of China (no. 51379049) and the Fundamental Research Funds for the Central Universities of China (nos.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2015/367146">doi:10.1155/2015/367146</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pnfc65twvbcs7p47nbleyh6evm">fatcat:pnfc65twvbcs7p47nbleyh6evm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190223152458/http://pdfs.semanticscholar.org/48fd/f5161ded2ce953aea1b7000df5c7914f116b.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/48/fd/48fdf5161ded2ce953aea1b7000df5c7914f116b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2015/367146"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> hindawi.com </button> </a>

Subject Index 1986

<span title="">1986</span> <i title="American Institute of Aeronautics and Astronautics (AIAA)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yekwqy6srncl5bew6tynp7vjsq" style="color: black;">Journal of Guidance Control and Dynamics</a> </i> &nbsp;
Design Using Improved Time-Domain Stability Robustness Bounds G86-117 Nonlinear Attitude Motion of a Dual-Spin Spacecraft Containing Spherical Dampers G86-113 An Asymptotic Perturbation Method for Nonlinear  ...  Its Retrieval G86-111 Nonlinear Feedback Control for Remote Orbital Capture G86-110 729 Modeling and Simulation of Spacecraft Solar Array Deployment G86-100 On-Off Attitude Control of Flexible Satellites  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2514/3.56425">doi:10.2514/3.56425</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5mei2cg6t5bs3eihjlq2cf74di">fatcat:5mei2cg6t5bs3eihjlq2cf74di</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/details/sim_journal-of-guidance-control-and-dynamics_november-december-1986_9_6/page/727" title="read fulltext microfilm" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Archive [Microfilm] <div class="menu fulltext-thumbnail"> <img src="https://archive.org/serve/sim_journal-of-guidance-control-and-dynamics_november-december-1986_9_6/__ia_thumb.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2514/3.56425"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

New Developments in Sliding Mode Control and Its Applications

Ligang Wu, Rongni Yang, Guanghui Sun, Xudong Zhao, Peng Shi
<span title="">2014</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wpareqynwbgqdfodcyhh36aqaq" style="color: black;">Mathematical Problems in Engineering</a> </i> &nbsp;
"A chattering free discrete-time global sliding mode controller for optoelectronic tracking system" by Y.  ...  "Adaptive backstepping finite time attitude control of reentry RLV with input constraint" by F.  ...  "A chattering free discrete-time global sliding mode controller for optoelectronic tracking system" by Y.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2014/481419">doi:10.1155/2014/481419</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6imzei5avzd5vkun5sqll7hry4">fatcat:6imzei5avzd5vkun5sqll7hry4</a> </span>
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Variable Structure Control and Applications

Yuqiang Wu, Xinghuo Yu, Lijun Zhang, Yu Kang, Ningsu Luo
<span title="">2013</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wpareqynwbgqdfodcyhh36aqaq" style="color: black;">Mathematical Problems in Engineering</a> </i> &nbsp;
Pukdeboon considered the attitude tracking control problem of a spacecraft nonlinear system with external disturbances and inertia uncertainties based on finite-time second-order sliding mode control schemes  ...  "Finitetime second-order sliding mode controllers for spacecraft attitude tracking" by C.  ...  Acknowledgments We would like to take this chance to express our deep gratitude to the authors for their excellent contributions and to the all of reviewers and our colleagues who have contributed directly  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2013/589759">doi:10.1155/2013/589759</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jalnwohw6ncmjocsvwwl4otcb4">fatcat:jalnwohw6ncmjocsvwwl4otcb4</a> </span>
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Model Predictive Control of three dimensional spacecraft relative motion

A. Weiss, I. Kolmanovsky, M. Baldwin, R. S. Erwin
<span title="">2012</span> <i title="IEEE"> 2012 American Control Conference (ACC) </i> &nbsp;
Bandwidth constraints of the spacecraft attitude control system and exhaust plume direction constraints are also addressed.  ...  The MPC controller is validated on the full nonlinear model of spacecraft orbital motion and augmented with an Extended Kalman Filter (EKF) to estimate spacecraft states based on relative angles and relative  ...  Discrete-time prediction model We convert the model (2) to discrete-time to be used as a prediction model in MPC.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/acc.2012.6314862">doi:10.1109/acc.2012.6314862</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bnjtch7tyvevjepw35cvlhuvgm">fatcat:bnjtch7tyvevjepw35cvlhuvgm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200507164200/http://www-personal.umich.edu/%7Eavishai/papers/mpcrdv.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/bc/b8/bcb8134d2e05914962ecee6b242adb64d39f21cd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/acc.2012.6314862"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Adaptive Sliding Mode Attitude-Tracking Control of Spacecraft with Prescribed Time Performance

Runze Chen, Zhenling Wang, Weiwei Che
<span title="2022-01-27">2022</span> <i title="MDPI AG"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ye33srllvnanjouxn4tmrfgjsq" style="color: black;">Mathematics</a> </i> &nbsp;
In this article, a novel finite-time attitude-tracking control scheme is proposed by using the prescribed performance control (PPC) method for the spacecraft system under the external disturbance and an  ...  Unlike the existing attitude-tracking control results, the proposed PPC scheme guarantees the performance of spacecraft system under the static and transient conditions.  ...  Reference [27] studied a fuzzy SMC scheme to realize the attitude-tracking control of the spacecraft with the unknown nonlinear term.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/math10030401">doi:10.3390/math10030401</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v5nihrgrq5f5tncpulug3kgpkq">fatcat:v5nihrgrq5f5tncpulug3kgpkq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220501164842/https://mdpi-res.com/d_attachment/mathematics/mathematics-10-00401/article_deploy/mathematics-10-00401.pdf?version=1643292050" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/87/cd/87cd657a84ad86cae337df0f27119f81724268b6.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/math10030401"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> mdpi.com </button> </a>

Subject Index 1993

<span title="">1993</span> <i title="American Institute of Aeronautics and Astronautics (AIAA)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yekwqy6srncl5bew6tynp7vjsq" style="color: black;">Journal of Guidance Control and Dynamics</a> </i> &nbsp;
Discretization Formulas for Real-Time Simulation G93-085 Control System Design Optimal Control System Design with Prescribed Damping and Stability Characteristics G93-185 Inversion-Based Nonlinear Control  ...  Open Multistep Discretization Formulas for Real-Time Simulation G93-085 Saturating and Time-Optimal Feedback Con- trols G93-083 Optimal Discrete-Time Dynamic Output-Feed- back Design: A w-Domain Approach  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2514/3.56621">doi:10.2514/3.56621</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/76b2r2m4bndvvmomgfwrajx7gm">fatcat:76b2r2m4bndvvmomgfwrajx7gm</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/details/sim_journal-of-guidance-control-and-dynamics_november-december-1993_16_6/page/1201" title="read fulltext microfilm" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Archive [Microfilm] <div class="menu fulltext-thumbnail"> <img src="https://archive.org/serve/sim_journal-of-guidance-control-and-dynamics_november-december-1993_16_6/__ia_thumb.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2514/3.56621"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

2012 Subject Index

<span title="">2012</span> <i title="American Institute of Aeronautics and Astronautics (AIAA)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yekwqy6srncl5bew6tynp7vjsq" style="color: black;">Journal of Guidance Control and Dynamics</a> </i> &nbsp;
G12-123 Dynamic Neural Units for Adaptive Magnetic Attitude Control of Spacecraft G12-074 Finite-Time Attitude Stabilization for a Spacecraft Using Homogeneous Method G12-111 Tradeoff Analysis of Attitude-Control  ...  Attitude-Control Algorithms for an Underactuated Satellite Spacecraft Attitude Determination G12-078 Nonlinear Control Analysis of a Double- Gimbal Variable-Speed Control Moment Gyro- scope G12-076 Numerical  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2514/1.60510">doi:10.2514/1.60510</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pmdtengfbzhu5njb4ux3mls5sm">fatcat:pmdtengfbzhu5njb4ux3mls5sm</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/details/sim_journal-of-guidance-control-and-dynamics_november-december-2012_35_6/page/1928" title="read fulltext microfilm" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Archive [Microfilm] <div class="menu fulltext-thumbnail"> <img src="https://archive.org/serve/sim_journal-of-guidance-control-and-dynamics_november-december-2012_35_6/__ia_thumb.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2514/1.60510"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Attitude Trajectory Optimization for Agile Satellites in Autonomous Remote Sensing Constellation [article]

Emmanuel Sin, Sreeja Nag, Vinay Ravindra, Alan Li, Murat Arcak
<span title="2021-02-16">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The Minimum-Time Slew Optimal Control Problem determines the minimum time, required energy, and optimal trajectory to slew between any two orientations given nonlinear quaternion kinematics, gyrostat and  ...  minimizing attitude error and control effort.  ...  attitude control.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2102.07940v1">arXiv:2102.07940v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sr6c6e75rzarjg7bbekuvrvn2m">fatcat:sr6c6e75rzarjg7bbekuvrvn2m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210218021406/https://arxiv.org/pdf/2102.07940v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c0/03/c0030f47a296eb1ad38e465d956354d0ead1390d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2102.07940v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Model Predictive Control of an Underactuated Spacecraft with Two Reaction Wheels

Christopher D. Petersen, Frederick Leve, Ilya Kolmanovsky
<span title="">2017</span> <i title="American Institute of Aeronautics and Astronautics (AIAA)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yekwqy6srncl5bew6tynp7vjsq" style="color: black;">Journal of Guidance Control and Dynamics</a> </i> &nbsp;
Simulations on the full nonlinear model demonstrate that model predictive control can successfully control the attitude of an underactuated spacecraft when the attitude maneuvers are sufficiently small  ...  This paper presents an application of model predictive control to the attitude dynamics of an underactuated spacecraft with two reaction wheels and zero angular momentum.  ...  Its application to the spacecraft attitude control problem began in the mid-1990s [1, 2] . In [1] , attitude tracking was performed using a one-step-ahead prediction of the states.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2514/1.g000320">doi:10.2514/1.g000320</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y2nal5njh5cqjctwfoce7pb4zq">fatcat:y2nal5njh5cqjctwfoce7pb4zq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190504152031/https://deepblue.lib.umich.edu/bitstream/handle/2027.42/143105/1.G000320.pdf;jsessionid=D3DF9F95BB372C132F617FB5EF7860B0?sequence=1" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6d/82/6d82df44948231f7306b8b8a15e5fe81b27bd968.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2514/1.g000320"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

The prescribed time sliding mode control for attitude tracking of spacecraft [article]

Zhirun Chen, Xiaozhe Ju, Ziwei Wang, Qing Li
<span title="2020-04-05">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Finally, this paper presents a disturbance observer based prescribed time sliding mode control method for attitude tracking of spacecraft and the efficiency of this method has been verified through the  ...  More importantly, as time approaches the prescribed time, the singularity of control input can be avoided.  ...  Acknowledgments This work is sponsored by the National Natural Science Foundation of China No. 61771281, the "New generation artificial intelligence" major project of China No. 2018AAA0101605, the 2018  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2004.02092v1">arXiv:2004.02092v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mh67lqrn6fet5kt5usnv3s4qxy">fatcat:mh67lqrn6fet5kt5usnv3s4qxy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200408013727/https://arxiv.org/ftp/arxiv/papers/2004/2004.02092.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2004.02092v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Discrete-Time Attitude Tracking Synchronization for Swarms of Spacecraft Exploiting Interference

Peiran Li, Xin Wen, Mohong Zheng, Haiying Liu, Dizhi Long, Yuping Lu
<span title="2022-03-03">2022</span> <i title="MDPI AG"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnqw7sgwrve3dace7vnhziq654" style="color: black;">Aerospace (Basel)</a> </i> &nbsp;
The attitude tracking synchronization control of an orbit-predetermined leader–follower spacecraft swarm for the space moving target is discussed in this paper.  ...  Then the backstepping control algorithm is designed to let the spacecraft (β, A)-practically stably synchronize their states and track a time-varying trajectory in the presence of unknown fading channels  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/aerospace9030134">doi:10.3390/aerospace9030134</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rkqaloqoqbcvxd4e7do2ndbjzm">fatcat:rkqaloqoqbcvxd4e7do2ndbjzm</a> </span>
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Subject Index

<span title="">2011</span> <i title="American Institute of Aeronautics and Astronautics (AIAA)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yekwqy6srncl5bew6tynp7vjsq" style="color: black;">Journal of Guidance Control and Dynamics</a> </i> &nbsp;
Proportional-Inte- gral-Derivative Control Laws for Rigid-Body Attitude Tracking G11-085 Fault-Tolerant Attitude Control for Spacecraft Under Loss of Actuator Effectiveness G11-064 Optimal Nonlinear Feedback  ...  G11-011 Continuous-Time Kalman Filtering with Implicit Discrete Measurement Times G11-156 Solution Strategies for an Extension of Wahba’s Problem to a Spinning Spacecraft Spacecraft Dynamics G11-044 Spectral  ... 
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Integrated attitude determination and control system via magnetic measurements and actuation

Mohammad Abdelrahman, Sang-Young Park
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7ka2sksbjnckhhvewyc2zlrjey" style="color: black;">Acta Astronautica</a> </i> &nbsp;
A nonlinear control scheme using a Modified State-Dependent Riccati Equation MSDRE has been developed through a pseudo-linearization of spacecraft augmented dynamics and kinematics.  ...  The full-state knowledge, required for the control loop is provided through a generalized algorithm for spacecraft three-axis attitude and rate estimation based on the utilization of magnetometer measurements  ...  Acknowledgments This work is supported by the Korean Science and Engineering Foundation (KOSEF) through the National Research Laboratory Program funded by the Ministry of Science and Technology (No.  ... 
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