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

* Corresponding author: Brennan McCann

Received  May 2021 Revised  September 2021 Published  March 2022 Early access  January 2022

Fund Project: The first author is supported by FIRST and GAANN

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.

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
References:
[1]

Geometry of the double tangent bundles of banach manifolds, 74.

[2] P.-A. AbsilR. 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. NielsenO. A. EidsvikM. 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. SunH. ZhaoS. ZhenK. HuangF. ZhaoX. 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. UdwadiaT. 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. YinY.-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. AbsilR. 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. NielsenO. A. EidsvikM. 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. SunH. ZhaoS. ZhenK. HuangF. ZhaoX. 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. UdwadiaT. 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. YinY.-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 1.  Inertial $ \mathcal N $, perifocal $ \mathcal P $, and body $ \mathcal B $ reference frames
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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