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Inertial $ \mathcal N $ , perifocal $ \mathcal P $ , and body $ \mathcal B $ reference frames
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March
2022, 14(1): 29-55.
doi: 10.3934/jgm.2022002
Control and maintenance of fully-constrained and underconstrained rigid body motion on Lie groups and their tangent bundles
Aerospace Engineering, Embry-Riddle Aeronautical University, 1 Aerospace Boulevard, Daytona Beach, FL 32114, USA |
Presented herein are a class of methodologies for conducting constrained motion analysis of rigid bodies within the Udwadia-Kalaba (U-K) formulation. The U-K formulation, primarily devised for systems of particles, is advanced to rigid body dynamics in the geometric mechanics framework and a novel development of U-K formulation for use on nonlinear manifolds, namely the special Euclidean group $ {\mathsf{SE}(3)}$ and its second order tangent bundle ${\mathsf{T}^2\mathsf{SE}(3)} $, is proposed in addition to the formulation development on Euclidean spaces. Then, a Morse-Lyapunov based tracking controller using backstepping is applied to capture disturbed initial conditions that the U-K formulation cannot account for. This theoretical development is then applied to fully-constrained and underconstrained scenarios of rigid-body spacecraft motion in a lunar orbit, and the translational and rotational motions of the spacecraft and the control inputs obtained using the proposed methodologies to achieve and maintain those constrained motions are studied.
Keywords:
Constrained motion,
geometric mechanic Udwadia-Kalaba formalism,
Morse-Lyapunov function,
spacecraft pose control,
rigid body tracking problem.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
Citation:
Brennan McCann, Morad Nazari. Control and maintenance of fully-constrained and underconstrained rigid body motion on Lie groups and their tangent bundles. Journal of Geometric Mechanics,
2022, 14
(1)
: 29-55.
doi: 10.3934/jgm.2022002
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References:
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P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, New Jersey, 2008.
doi: 10.1515/9781400830244.
|
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J. Baumgarte,
Stabilization of constraints and integrals of motion in dynamical systems, Computer Methods in Applied Mechanics and Engineering, 1 (1972), 1-16.
doi: 10.1016/0045-7825(72)90018-7. |
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S. P. Bhat and D. S. Bernstein,
A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon, System Control Letters, 39 (2000), 63-70.
doi: 10.1016/S0167-6911(99)00090-0. |
| [5] |
H. Cho and F. E. Udwadia,
Explicit solution to the full nonlinear problem for satellite formation-keeping, Acta Astronautica, 67 (2010), 369-387.
doi: 10.1016/j.actaastro.2010.02.010. |
| [6] |
C. T. J. Dodson and G. N. Galanis,
Second order tangent bundles of infinite dimensional manifolds, Journal of Geometry and Physics, 52 (2004), 127-136.
doi: 10.1016/j.geomphys.2004.02.005. |
| [7] |
H. K. Khalil, Nonlinear Systems, 3rd edition, Upper Saddle River, New Jersey, 2002. |
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T. Lam,
A new approach to mission design based on the fundamental equations of motion, Journal of Aerospace Engineering, 19 (2006), 59-67.
doi: 10.1061/(ASCE)0893-1321(2006)19:2(59). |
| [9] |
A. D. Lewis,
The geometry of the gibbs-appell equations and gauss's principle of least constraint, Reports on Mathematical Physics, 38 (1996), 11-28.
doi: 10.1016/0034-4877(96)87675-0. |
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G. M. Low, Apollo 11 mission report, 1969, https://www.nasa.gov/specials/apollo50th/pdf/A11_MissionReport.pdf. |
| [11] |
C. G. Mayhew, R. G. Sanfelice and A. R. Teel, On quaternion-based attitude control and the unwinding phenomenon, in Proceedings of the 2011 American Control Conference, 2011,299–304.
doi: 10.1109/ACC.2011.5991127. |
| [12] |
B. S. McCann and M. Nazari, Conjugate Gradient Algorithm for Constrained Optimization on the Special Euclidean Group, 2021. |
| [13] |
B. S. McCann, W. T. Stackhouse and M. Nazari, Advancement of the Udwadia-Kalaba approach for rigid body constrained motion analysis in geometric mechanics, AIAA Scitech 2021 Forum.
doi: 10.2514/6.2021-0976. |
| [14] |
M. W. Memon, M. Nazari, D. Seo and E. A. Butcher, Fuel efficiency of fully- and under-constrained {C}oulomb formations in slightly elliptic reference orbits, IEEE Transactions on Aerospace and Electronic Systems. |
| [15] |
M. Nazari, M. Maadani, E. A. Butcher and T. Yucelen, Morse-Lyapunov-based control of rigid body motion on TSE(3) via backstepping, in SciTech, 2018.
doi: 10.2514/6.2018-0602. |
| [16] |
M. C. Nielsen, O. A. Eidsvik, M. Blanke and I. Schjølberg,
Constrained multi-body dynamics for modular underwater robots: Theory and experiments, Ocean Engineering, 149 (2018), 358-372.
doi: 10.1016/j.oceaneng.2017.12.007. |
| [17] |
C. M. Pappalardo and D. Guida,
On the Lagrange multipliers of the intrinsic constraint equations of rigid multibody mechanical systems, Archive of Applied Mechanics, 88 (2018), 419-451.
doi: 10.1007/s00419-017-1317-y. |
| [18] |
H. Patel, T. A. Henderson and M. Nazari, Application of Udwadia-Kalaba formulation to three-body problem, AAS/AIAA Astrodynamics Specialist Conference, AAS 19–706. |
| [19] |
A. A. Pothen and S. Ulrich, Close-range rendezvous with a moving target spacecraft using Udwadia- Kalaba equation, in 2019 American Control Conference (ACC), (2019), 3267–3272.
doi: 10.23919/ACC.2019.8815115. |
| [20] |
S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Springer, New York, NY, 1999.
doi: 10.1007/978-1-4757-3108-8. |
| [21] |
H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, 4th edition, American Institute of Aeronautics and Astronautics, 2018.
doi: 10.2514/4.861550. |
| [22] |
W. T. Stackhouse, M. Nazari, T. Henderson and R. J. Prazenica, Adaptive control design using the Udwadia-Kalaba formulation for hovering over an asteroid with unknown gravitational parameters, AIAA Scitech 2020 Forum.
doi: 10.2514/6.2020-0843. |
| [23] |
H. Sun, H. Zhao, S. Zhen, K. Huang, F. Zhao, X. Chen and Y.-H. Chen,
Application of the Udwadia-Kalaba approach to tracking control of mobile robots, Nonlinear Dynamics, 83 (2016), 389-400.
doi: 10.1007/s11071-015-2335-3. |
| [24] |
A. Suri, Higher order tangent bundles, Mediterranean Journal of Mathematics, 14 (2017), Paper No. 5, 17 pp.
doi: 10.1007/s00009-016-0812-7. |
| [25] |
F. E. Udwadia and R. E. Kalaba,
A new perspective on constrained motion, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439 (1992), 407-410.
doi: 10.1098/rspa.1992.0158. |
| [26] |
F. E. Udwadia and R. E. Kalaba, Analytical Dynamics: A New Approach, Cambridge University Press, New York, 2008.
doi: 10.1017/CBO9780511665479.
|
| [27] |
F. E. Udwadia and A. D. Schutte,
A unified approach to rigid body rotational dynamics and control, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468 (2011), 395-414.
doi: 10.1098/rspa.2011.0233. |
| [28] |
F. E. Udwadia, T. Wanichanon and H. Cho,
Methodology for satellite formation-keeping in the presence of system uncertainties, Journal of Guidance, Control, and Dynamics, 37 (2014), 1611-1624.
doi: 10.2514/1.G000317. |
| [29] |
D. A. Vallado, Fundamentals of Astrodynamics and Applications, 3rd edition, Microcosm
Press, 2007. |
| [30] |
A. Valverde and P. Tsiotras, Dual quaternion framework for modeling of spacecraft-mounted multibody robotic systems, Frontiers in Robotics and AI, 5.
doi: 10.3389/frobt.2018.00128. |
| [31] |
M. Wittal, G. Mangiacapra, A. Appakonam, M. Nazari and E. Capello, Stochastic spacecraft navigation and control in Lie group SE(3) around small irregular bodies, AAS/AIAA Astrodynamics Specialist Conference, AAS 20–690. |
| [32] |
H. Yin, Y.-H. Chen and D. Yu,
Vehicle motion control under equality and inequality constraints: A diffeomorphism approach, Nonlinear Dynamics, 95 (2019), 175-194.
doi: 10.1007/s11071-018-4558-6. |
| [33] |
G. A. Zupp, An analysis and a historical review of the apollo program lunar module touchdown dynamics, NASA/SP-2013-605. |
show all references
References:
| [1] |
Geometry of the double tangent bundles of banach manifolds, 74. |
| [2] |
P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, New Jersey, 2008.
doi: 10.1515/9781400830244.
|
| [3] |
J. Baumgarte,
Stabilization of constraints and integrals of motion in dynamical systems, Computer Methods in Applied Mechanics and Engineering, 1 (1972), 1-16.
doi: 10.1016/0045-7825(72)90018-7. |
| [4] |
S. P. Bhat and D. S. Bernstein,
A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon, System Control Letters, 39 (2000), 63-70.
doi: 10.1016/S0167-6911(99)00090-0. |
| [5] |
H. Cho and F. E. Udwadia,
Explicit solution to the full nonlinear problem for satellite formation-keeping, Acta Astronautica, 67 (2010), 369-387.
doi: 10.1016/j.actaastro.2010.02.010. |
| [6] |
C. T. J. Dodson and G. N. Galanis,
Second order tangent bundles of infinite dimensional manifolds, Journal of Geometry and Physics, 52 (2004), 127-136.
doi: 10.1016/j.geomphys.2004.02.005. |
| [7] |
H. K. Khalil, Nonlinear Systems, 3rd edition, Upper Saddle River, New Jersey, 2002. |
| [8] |
T. Lam,
A new approach to mission design based on the fundamental equations of motion, Journal of Aerospace Engineering, 19 (2006), 59-67.
doi: 10.1061/(ASCE)0893-1321(2006)19:2(59). |
| [9] |
A. D. Lewis,
The geometry of the gibbs-appell equations and gauss's principle of least constraint, Reports on Mathematical Physics, 38 (1996), 11-28.
doi: 10.1016/0034-4877(96)87675-0. |
| [10] |
G. M. Low, Apollo 11 mission report, 1969, https://www.nasa.gov/specials/apollo50th/pdf/A11_MissionReport.pdf. |
| [11] |
C. G. Mayhew, R. G. Sanfelice and A. R. Teel, On quaternion-based attitude control and the unwinding phenomenon, in Proceedings of the 2011 American Control Conference, 2011,299–304.
doi: 10.1109/ACC.2011.5991127. |
| [12] |
B. S. McCann and M. Nazari, Conjugate Gradient Algorithm for Constrained Optimization on the Special Euclidean Group, 2021. |
| [13] |
B. S. McCann, W. T. Stackhouse and M. Nazari, Advancement of the Udwadia-Kalaba approach for rigid body constrained motion analysis in geometric mechanics, AIAA Scitech 2021 Forum.
doi: 10.2514/6.2021-0976. |
| [14] |
M. W. Memon, M. Nazari, D. Seo and E. A. Butcher, Fuel efficiency of fully- and under-constrained {C}oulomb formations in slightly elliptic reference orbits, IEEE Transactions on Aerospace and Electronic Systems. |
| [15] |
M. Nazari, M. Maadani, E. A. Butcher and T. Yucelen, Morse-Lyapunov-based control of rigid body motion on TSE(3) via backstepping, in SciTech, 2018.
doi: 10.2514/6.2018-0602. |
| [16] |
M. C. Nielsen, O. A. Eidsvik, M. Blanke and I. Schjølberg,
Constrained multi-body dynamics for modular underwater robots: Theory and experiments, Ocean Engineering, 149 (2018), 358-372.
doi: 10.1016/j.oceaneng.2017.12.007. |
| [17] |
C. M. Pappalardo and D. Guida,
On the Lagrange multipliers of the intrinsic constraint equations of rigid multibody mechanical systems, Archive of Applied Mechanics, 88 (2018), 419-451.
doi: 10.1007/s00419-017-1317-y. |
| [18] |
H. Patel, T. A. Henderson and M. Nazari, Application of Udwadia-Kalaba formulation to three-body problem, AAS/AIAA Astrodynamics Specialist Conference, AAS 19–706. |
| [19] |
A. A. Pothen and S. Ulrich, Close-range rendezvous with a moving target spacecraft using Udwadia- Kalaba equation, in 2019 American Control Conference (ACC), (2019), 3267–3272.
doi: 10.23919/ACC.2019.8815115. |
| [20] |
S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Springer, New York, NY, 1999.
doi: 10.1007/978-1-4757-3108-8. |
| [21] |
H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, 4th edition, American Institute of Aeronautics and Astronautics, 2018.
doi: 10.2514/4.861550. |
| [22] |
W. T. Stackhouse, M. Nazari, T. Henderson and R. J. Prazenica, Adaptive control design using the Udwadia-Kalaba formulation for hovering over an asteroid with unknown gravitational parameters, AIAA Scitech 2020 Forum.
doi: 10.2514/6.2020-0843. |
| [23] |
H. Sun, H. Zhao, S. Zhen, K. Huang, F. Zhao, X. Chen and Y.-H. Chen,
Application of the Udwadia-Kalaba approach to tracking control of mobile robots, Nonlinear Dynamics, 83 (2016), 389-400.
doi: 10.1007/s11071-015-2335-3. |
| [24] |
A. Suri, Higher order tangent bundles, Mediterranean Journal of Mathematics, 14 (2017), Paper No. 5, 17 pp.
doi: 10.1007/s00009-016-0812-7. |
| [25] |
F. E. Udwadia and R. E. Kalaba,
A new perspective on constrained motion, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439 (1992), 407-410.
doi: 10.1098/rspa.1992.0158. |
| [26] |
F. E. Udwadia and R. E. Kalaba, Analytical Dynamics: A New Approach, Cambridge University Press, New York, 2008.
doi: 10.1017/CBO9780511665479.
|
| [27] |
F. E. Udwadia and A. D. Schutte,
A unified approach to rigid body rotational dynamics and control, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468 (2011), 395-414.
doi: 10.1098/rspa.2011.0233. |
| [28] |
F. E. Udwadia, T. Wanichanon and H. Cho,
Methodology for satellite formation-keeping in the presence of system uncertainties, Journal of Guidance, Control, and Dynamics, 37 (2014), 1611-1624.
doi: 10.2514/1.G000317. |
| [29] |
D. A. Vallado, Fundamentals of Astrodynamics and Applications, 3rd edition, Microcosm
Press, 2007. |
| [30] |
A. Valverde and P. Tsiotras, Dual quaternion framework for modeling of spacecraft-mounted multibody robotic systems, Frontiers in Robotics and AI, 5.
doi: 10.3389/frobt.2018.00128. |
| [31] |
M. Wittal, G. Mangiacapra, A. Appakonam, M. Nazari and E. Capello, Stochastic spacecraft navigation and control in Lie group SE(3) around small irregular bodies, AAS/AIAA Astrodynamics Specialist Conference, AAS 20–690. |
| [32] |
H. Yin, Y.-H. Chen and D. Yu,
Vehicle motion control under equality and inequality constraints: A diffeomorphism approach, Nonlinear Dynamics, 95 (2019), 175-194.
doi: 10.1007/s11071-018-4558-6. |
| [33] |
G. A. Zupp, An analysis and a historical review of the apollo program lunar module touchdown dynamics, NASA/SP-2013-605. |
Figure 2.
Fully-constrained translational motion comparison between formulation on $ \mathbb{R}^{6} $ and $ {\mathsf{T}^2\mathsf{SE}(3)}$
Figure 3.
Fully-constrained rotational motion comparison between formulation on $ \mathbb{R}^{6} $ and $ {\mathsf{T}^2\mathsf{SE}(3)}$
Figure 4.
Fully-constrained control input comparison between formulations on $ \mathbb{R}^{6} $ and $ {\mathsf{T}^2\mathsf{SE}(3)}$
Figure 5.
Translational motion in underconstrained (UC) case versus that in the fully-constrained (FC) case
Figure 6.
Rotational motion in underconstrained (UC) case versus that in the fully-constrained (FC) case
Figure 7.
Control inputs in underconstrained (UC) case versus those in the fully-constrained (FC) case
Figure 8.
Underconstrained translational motion comparison between formulation on $ \mathbb{R}^{6} $ and $ {\mathsf{T}^2\mathsf{SE}(3)}$
Figure 9.
Underconstrained rotational motion comparison between formulation on $ \mathbb{R}^{6} $ and ${\mathsf{T}^2\mathsf{SE}(3)} $
Figure 10.
Underconstrained control inputs comparison between formulation on $ \mathbb{R}^{6} $ and ${\mathsf{T}^2\mathsf{SE}(3)} $
Figure 11.
Position response using U-K and M-L control with disturbed ICs
Figure 12.
Velocity response using U-K and M-L control with disturbed ICs
Figure 13.
Attitude response using U-K and M-L control with disturbed ICs
Figure 14.
Angular velocity response using U-K and M-L control with disturbed ICs
Figure 15.
Total control input using U-K and M-L control with disturbed ICs
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