1. Introduction

1.1. Broad context and why this study is important

Despite recent demonstrations of operations in space of capabilities at the system level of several autonomous functions [5,6] including robotics, contemporary space operations assessment and action planning rely upon pre-scripted sequences of commands from ground operations personnel. [7] Considering challenging and distant robotics missions (e.g., to Mars) ground operators are unlikely to predict all the likely encounters and physical interactions occurring in parts of space seldomly experienced before.[8] Limited knowledge in complex situations demands autonomy, perceivably establishing a new frontier for space exploration. Another obvious example is utilization of very small spacecraft in cislunar orbits to refuel, repair, and replenish earth-orbiting spacecraft at lower altitudes necessitating grappling potentially unknown masses of uncooperative spacecraft. Extreme fuel efficiency seems mandatory, especially due to the low mass and volume of the underactuated robotic repair spacecraft [8] like those depicted in Figure 2, leading naturally to control minimization as a primary figure of merit. [9]

1.1.1. Cislunar space

A recent primer on cislunar space published by the U.S. Air Force [13] intended to aid the development of expertise, capabilities, plans, and operational concepts. The importance of the orbits manifested in the December 2019 creation of a new branch of the military charged with the duties to defend and protect American space interests. Especially since highly perturbed orbits depart predictable locations; intercept, rendezvous, and proximity operations (depicted in Figure 3) become quite complicated and potentially unpredictable. Cislunar orbits are generally no longer planar and no longer elliptical (certainly not circular), spacecraft positions are no longer easy to articulate geometrically.

Figure 3. Operations in cislunar orbits [14] (a) Schematic defining features of cislunar space. Image credit NASA. [15] (b) Cislunar satellite inspector. Images: credit Air Force Research Laboratory and National Aeronautics and Space Administration [2,4].

Figure 3. Operations in cislunar orbits [14] (a) Schematic defining features of cislunar space. Image credit NASA. [15] (b) Cislunar satellite inspector. Images: credit Air Force Research Laboratory and National Aeronautics and Space Administration [2,4].

1.1.2. Cislunar robotic operations

Actuators for space robots were recently reviewed in [16] highlighting time–delays as a key limiting issue for successful operations, some of which are displayed in Figure 4. A disparate review [17] assembled (over the last three years), subdivided the trends in research development achievements. The reviews focus on two different treatments of modeling uncertainties in hopes of not needing to increase design margins: either increasing the accuracy of parameter discrimination or alternatively developing methods with inherent robustness. The later review [17] emphasized the fact that task performance execution success correlated with onboard computing power.

In the broadest sense, the space robot (like those depicted in Figure 1, Figure 3, and Figure 5) may be considered generically as a cylindrical main body with a robotic appendage attached, leading to both rigid–body and flexible–body treatments. Figure 5 displays two variations on laboratory hardware replicating space robots. Subfigure (b) is the Navy space robot laboratory hardware that serves as the baseline system analyzed in subsequent sections of this manuscript.

Table 1. Table of proximal variables and nomenclature ^1.

Table 1. Table of proximal variables and nomenclature ^1.

Variable/acronym	Definition	Variable/acronym	Definition
$\widehat{y}$	Centerline unit vector	$k$	Appendage stiffness
$T_C$	Control torque	$m_i,I_i\forall i=2\dots 5$	Flexible masses and inertias
$J_1$	Main body inertia mass moment	$J_2$	Flexible inertia mass moment
${\theta }_1$	Main body rotation angle	${\theta }_i,W_i\forall i=2\dots 5$	Translation & rotation angles

¹ Such tables are offered throughout the manuscript to aid readability.

The literature reveals many mathematical approaches [13] for modeling and also for control, while the readership might be bereft and confused about which method or combinations of methods should be considered for their particular application. This manuscript compares a recently proposed bio–inspired approach to many of the contemporary alternatives.

2. Materials and Methods

2.1. Space robot modeling

The highly flexible space robot laboratory hardware depicted in Figure 6, subfigure (a) is decomposed into a rigid-body base (simulating the near-rigid spacecraft) depicted in subfigure (b) with highly flexible, lightweight robotic appendages modeled using the finite element method in subfigure (c). The first node of the finite element representation is attached to the spacecraft, while flexible elements (mass and stiffnesses) are lumped at subsequent nodes, where Table 2 conveniently defines variables and nomenclature.

Table 2. Table of proximal variables and nomenclature ¹.

Table 2. Table of proximal variables and nomenclature ¹.

Variable/acronym	Definition	Variable/acronym	Definition
$\widehat{y}$	Centerline unit vector	$k$	Appendage stiffness
$T_C$	Control torque	$m_i,I_i\forall i=2\dots 5$	Flexible masses and inertias
$J_1$	Main body inertia mass moment	$J_2$	Flexible inertia mass moment
${\theta }_1$	Main body rotation angle	${\theta }_i,W_i\forall i=2\dots 5$	Translation & rotation angles

¹ Such tables are offered throughout the manuscript to aid readability.

Newton’s Law for translational motion is expressed in equation (1) where displacements are expressed in coordinates of an inertial reference frame, while similarly Euler’s moment equations are displayed in equation (2). Expressing motion in coordinates of a non-inertial reference frame modifies equations (1) and (2) to equations (3) and (4) respectively. A representative two-node system of equations including internal flexible “spring forces” that resist motion is displayed in equations (5) and (6) respectively for each node, and then assembled into matrix–vector notation in equation (7), where Table 2 conveniently defines variables and nomenclature.

$${\left.F\right|}_{inertial}={\left.ma\right|}_{inertial}={\left.m\dot{v}\right|}_{inertial}={\left.m\ddot{x}\right|}_{inertial}$$

(1)

$${\left.T\right|}_{inertial}={\left.J\alpha \right|}_{inertial}={\left.m\dot{\omega }\right|}_{inertial}$$

(2)

$$F=m\ddot{x}+m\frac{d\omega }{dt}\times r\text{'}+2m\omega \times v\text{'}+m\omega \times \left(\omega \times r\text{'}\right)$$

(3)

$$T=J\dot{\omega }+\omega \times J\omega$$

(4)

$$m_1{\ddot{x}}_1=-k_1{x}_1+k_2\left(x_2-x_1\right)$$

(5)

$$m_2{\ddot{x}}_2=-k_2\left(x_2-x_1\right)+F$$

(6)

$$\underset{\left[M\right]}{\underbrace{\left[\begin{array}{cc}{m}_1& 0\\ 0& m_2\end{array}\right]}}\underset{\left\{\ddot{x}\right\}}{\underbrace{\left\{\begin{array}{c}{\ddot{x}}_1\\ {\ddot{x}}_2\end{array}\right\}}}+\underset{\left[K\right]}{\underbrace{\left[\begin{array}{cc}{k}_1+k_2& -k_2\\ -k_2& k_2\end{array}\right]}}\underset{\left\{x\right\}}{\underbrace{\left\{\begin{array}{c}{x}_1\\ x_2\end{array}\right\}}}=\underset{\left\{F\right\}}{\underbrace{\left\{\begin{array}{c}0\\ 1\end{array}\right\}F}}$$

(7)

Table 2. Table of proximal variables and nomenclature ¹

Table 2. Table of proximal variables and nomenclature ¹

Variable/acronym	Definition
${\left.F\right|}_{inertial}$, ${\left.T\right|}_{inertial}$	Externally applied force & torque expressed in inertial coordinates
$F$, $T$	Externally applied force & torque expressed in non-inertial coordinates
$m$, $J$	Body’s mass & mass moment of inertia
${\left.a\right|}_{inertial}={\left.\dot{v}\right|}_{inertial}={\left.\ddot{x}\right|}_{inertial}$	Resulting accelerations expressed in inertial coordinates
$\omega$, $\dot{\omega }$	Angular velocity and acceleration vectors
$x_1,x_2$; ${\ddot{x}}_1,{\ddot{x}}_2$	Translational velocity and acceleration vectors
$k_1,k_2$	Flexible member stiffnesses
$\left[M\right],\left[K\right]$	Assembled matrices of masses and stiffnesses

¹ Such tables are offered throughout the manuscript to aid readability.

Euler’s moment equations in equation (2) are elaborated for the flexible space robot in equation (8) and resembled in equation (9) to more closely resemble the basic expression of Newton’s Law, where variables and nomenclature are conveniently defined in Table 3.

$$I_{zz}\ddot{\theta }+\sum _{i=1}^n{D}_i{\ddot{q}}_i+{I_w\ddot{\theta }}_w=T_D$$

(8)

$$I_{zz}\ddot{\theta }+\sum _{i=1}^n{D}_i{\ddot{q}}_i=\sum \mathit{T}$$

(9)

Isolating the first term of equation (9) leads to equation (10), and slight arithmetic leads to equation (11).

$$\ddot{\theta }+\frac{\sum _{i=1}^n{D}_i}{I_{zz}}{\ddot{q}}_i=\frac{\sum \mathit{T}}{I_{zz}}$$

(10)

$$\ddot{\theta }=\frac{\sum \mathit{T}}{I_{zz}}-\frac{\sum _{i=1}^n{D}_i}{I_{zz}}{\ddot{q}}_i$$

(11)

Detailed implementation on the flexible space robot depicted in Figure 6 is included in the appendix to aid repeatability. Mode shapes and (constant) natural vibrational frequencies are obtained by spectral decomposition (i.e. the eigenvalue problem). Meanwhile, mass and moments of inertia (locations of mass) vary leading to time–invariant frequenices and shapes, thus motivating the proposals in this manuscript with the anticipation of future research into deterministic artificial intelligence.

3. Results

3.3. Comparing commanded trajectories with filtered feedback

Having discerned the advantages of single-sinusoidally shaped, commanded trajectories and time-delay input-shaped feedforward control, this section repeats the comparison in Section 3.1 (which included only unfiltered feedback), but this time iterates the options when structural filters are use. Figure 13 reveals the lowest control effort (i.e., fuel usage) with nominal tracking performance using unshaped step commands, no feedforward controls with structurally filtered feedback; however, substantially improved target tracking performance is achievable at using higher control efforts by commanding optimal (fuel-minimizing) trajectories that are constrained by rigid-body dynamics equations.

Figure 13. Selectable commanded trajectory comparative simulation experiments performed in Simulink®. Solid blue line indicates step trajectory, no feedforward, filtered feedback; solid black line indicates rigid body optimal trajectory, no feedforward, filtered feedback; red dotted line indicates whiplash trajectory, no feedforward, filtered feedback; and green dashed line indicates single sine trajectory, no feedforward, filtered feedback (a) control; (b) tracking error, (c) rotation angle. Qualitative results correspond to quantitative figures of merit in Table 8.

Figure 13. Selectable commanded trajectory comparative simulation experiments performed in Simulink®. Solid blue line indicates step trajectory, no feedforward, filtered feedback; solid black line indicates rigid body optimal trajectory, no feedforward, filtered feedback; red dotted line indicates whiplash trajectory, no feedforward, filtered feedback; and green dashed line indicates single sine trajectory, no feedforward, filtered feedback (a) control; (b) tracking error, (c) rotation angle. Qualitative results correspond to quantitative figures of merit in Table 8.

  Interim summary. When comparing commanded trajectories with filtered feedback and no feedforward, step trajectory commands surprisingly led to the least control effort, while rigid body optimal trajectories achieved an order of magnitude higher accuracy with 2345% more control effort.

3.4. Comparing mode 1 filtering with single-sinusoidal trajectories and no feedforward

The final sections of the manuscript present experiments with iterated structural filtering: mode 1, mode 2, mode 3, mode 4, and then all of modes 1–4. Single-sinusoidally commanded trajectories are carried over from Section 3.1, Section 3.2 and Section 3.3, identified as a burgeoning best practice. This section iterates compensation of the lowest (first) flexible mode by sequentially compensating for the anti-resonance, the resonance, and then both modal features. The prequel literature predominantly designs structural filters for the sake of stability, often leading to emphasizing notch filtering the first resonant peak, while the results here (designed for target tracking performance rather than stability) indicate superior performance is obtainable by only compensating for the first anti-resonance frequency. This assertion is buttressed by the qualitative results in Figure 13 whose quantitative companion is displayed in Table 9’s display of meaningful performance figures of merit.

Figure 13. Selectable mode 1 feedback filtering comparative simulation experiments performed in Simulink®. Solid black line indicates single sine trajectory, no feedforward, mode 1 bandpass filtered; solid blue line indicates single sine trajectory, no feedforward, mode 1 notch filtered; red dotted line indicates single sine trajectory, no feedforward, mode 1 bandpass and notch filtered. (a) control; (b) tracking error, (c) rotation angle. Qualitative results correspond to quantitative figures of merit in Table 9.

Figure 13. Selectable mode 1 feedback filtering comparative simulation experiments performed in Simulink®. Solid black line indicates single sine trajectory, no feedforward, mode 1 bandpass filtered; solid blue line indicates single sine trajectory, no feedforward, mode 1 notch filtered; red dotted line indicates single sine trajectory, no feedforward, mode 1 bandpass and notch filtered. (a) control; (b) tracking error, (c) rotation angle. Qualitative results correspond to quantitative figures of merit in Table 9.

Interim summary. When comparing mode 1 filtering options bandpass only not-surprisingly led to the least control effort, but surprisingly also produced the most accurate tracking.

4. Discussion

References

1. AF, Navy baseball teams square off for 2018 Freedom Classic. Available online: https://www.nellis.af.mil/News/Article/1452330/af-navy-baseball-teams-square-off-for-2018-freedom-classic/ (accessed 12 December 2023).
2. Department of Defense Policy. “U.S. Department of Defense photographs and imagery, unless otherwise noted, are in the public domain.” Available online: https://www.defense.gov/Contact/Help-Center/Article/article/2762906/use-of-department-of-defense-imagery (accessed 12 December 2023).
3. Nile, R. First Humanoid Robot In Space Receives NASA Government Invention of the Year. Jun 17, 2015. Available online: https://www.nasa.gov/mission_pages/station/research/news/invention_of_the_year (accessed 17 February 2023).
4. NASA Image Use Policy. “NASA content (images, videos, audio, etc.) are generally not copyrighted and may be used for educational or informational purposes without needing explicit permissions.” Available online: https://gpm.nasa.gov/image-use-policy (accessed October 8, 2023).
5. Nesnas,I.; Fesq, L.; Volpe,R. Autonomy for Space Robots: Past, Present, and Future. Current Robotics Reports 2021, 2, 251–263. [CrossRef]
6. Sagan C, Reddy R. Machine intelligence and robotics: report of the NASA Study Group FINAL REPORT. 715-32. 1980. Carnegie Mellon University. Retrieved 2011 7 September. http://www.rr.cs.cmu.edu/NASA%20Sagan%20Report.pdf.
7. NASA Autonomous Systems – Systems Capability Leadership Team. Autonomous systems taxonomy. NASA Technical Reports Server. Document ID 20180003082. 2018 14 May. https://ntrs.nasa.gov/citations/20180003082.
8. Marshall Spaceflight Center, Advanced Space Transportation Program: Paving the Highway to Space. Available online: https://www.nasa.gov/centers/marshall/news/background/facts/astp.html (accessed on 15 January 2022).
9. Curiosity Bot, Advantages and disadvantages of using Robots instead of Astronauts. Available online: https://mycuriositybot.wordpress.com/2019/02/18/advantages–and–disadvantages–of–using–robots–instead–of–astronauts/ (accessed on 18 February 2019).
10. NASA is Building a Mission That Will Refuel and Repair Satellites in Orbit. Available online: https://www.universetoday.com/155863/nasa-is-building-a-mission-that-will-refuel-and-repair-satellites-in-orbit/ (accessed October 8, 2023).
11. DARPA teams with Northrop Grumman to build robotic service satellites. Available online: https://newatlas.com/space/robotic-service-satellite-darpa-northrop-grumman/ (accessed October 8, 2023).
12. Usage Policy. “Images posted on the DARPA website may be used for educational or informational purposes.” Available online: https://www.darpa.mil/policy/usage-policy (accessed October 8, 2023).
13. Holzinger, M.; Chow, C.; Garretson, P. A Primer on Cislunar Space, technical report, Air Force Research Labs, May 2021, 7, https://www.afrl.af.mil/Portals/90/Documents/RV/A%20Primer%20on%20Cislunar%20Space_Dist%20A_PA2021-1271.pdf?ver=vs6e0sE4PuJ51QC-15DEfg%3d%3d (accessed October 9, 2023).
14. Available online: https://www.ssc.spaceforce.mil/Portals/3/SDA%20Briefings/13.%20Bates_Tyler_SDA%20Industry%20Day_Cislunar%20Space%20SDA_V4.pdf?ver=fhTzhW7RFhI1MDcP90KAuQ%3D%3D (accessed 14 December 2023).
15. Image created by NASA used here: https://en.wikipedia.org/wiki/Halo_orbit#/media/File:Lagrange_points2.svg.
16. Ellery, A. Tutorial Review on Space Manipulators for Space Debris Mitigation. Robotics 2019, 8(2), 34. [CrossRef]
17. Zhang, W.; Li, F.; Li, J.; Cheng, Q. Review of On-Orbit Robotic Arm Active Debris Capture Removal Methods. Aerospace 2023, 10(1), 13. [CrossRef]
18. Ai, H.; Zhu, A.; Wang, J.; Yu, X.; Chen, L. Buffer Compliance Control of Space Robots Capturing a Non-Cooperative Spacecraft Based on Reinforcement Learning. Appl. Sci. 2021, 11(13), 5783. [CrossRef]
19. Armanini, C; Boyer, F; Mathew, A; Duriez, C; Renda, F. Soft robots modeling: A structured overview. IEEE Transactions on Robotics, Oct. 2022, pp. 1–21. [CrossRef]
20. Encyclopaedia Britannica, Britannica, Euclidean space. Available online: https://www.britannica.com/science/Euclidean–space (accessed on 10 October 2022).
21. Newton, I. Principia, Jussu Societatis Regiæ ac Typis Joseph Streater; Cambridge University Library: London, UK, 1687.
22. Mazalov, V; Parilina, E. The Euler–equation approach in average–oriented opinion dynamics, Mathematics, vol. 8, no. 3, p. 355, 2020. [CrossRef]
23. Sands, T. Comparison and Interpretation Methods for Predictive Control of Mechanics. Algorithms 2019, 12(11), 232. [CrossRef]
24. A.C.M. Operations, What–are–the–6–degrees–of–freedom, industrial Inspection & Analysis (IIA). Available online: https://industrial–ia.com/what–are–the–6–degrees–of–freedom–6dof–explained/ (accessed on 21 March 2023).
25. Doyle, J; Francis, A; Tannenbaum, A. Feedback control theory. Mineola, NY: Dover, 2009.
26. Johansson, R; Robertsson, A; Nilsson, K; Verhaegen, M. State–Space System Identification of Robot Manipulator Dynamics. Mechatronics, vol. 10, no. 3, pp 403–418, 2000. [CrossRef]
27. Corke, P, “High–performance visual closed–loop robot control,” thesis.
28. Li, Y; Ang, K; Chong, G.C.Y., “PID Control System Analysis and design”, IEEE Control Systems, vol. 26, no. 1, pp.32–41, 2006. [CrossRef]
29. Bemporad, A; Morari, M; Dua, V; Pistikopoulos, E, “The explicit linear quadratic regulator for constrained systems,” Automatica, vol. 38, no. 1, pp. 3–20, 2002. [CrossRef]
30. 30. Tao, K; Kosut, R; Aral, G, “Learning feedforward control,” Proceedings of 1994 American Control Conference –ACC ’94.
31. Chang, T; Seufert, C; Eminaga, O; Shkolyar, E, Hu, J; Liao, J, “Current trends in artificial intelligence application for endourology and robotic surgery,” Urologic Clinics of North America, vol 48, no. 1, pp. 151–160. [CrossRef]
32. Sands, T. Inducing Performance of Commercial Surgical Robots in Space. Sensors 2023, 23(3), 1510. [CrossRef]
33. Fu, S; Bhavsar, P, “Robotic Arm Control based on internet of things,” 2019 IEEE Long Island Systems, Applications and Technology Conference (LISAT), 2019.
34. Liu, X.-F.; Zhang, X.-Y.; Cai, G.-P.; Chen, W.-J. Capturing a Space Target Using a Flexible Space Robot. Appl. Sci. 2022, 12(3), 984. [CrossRef]
35. Li, J., Ma, K., Wu, Z. Tracking control via switching and learning for a class of uncertain flexible joint robots with variable stiffness actuators. Neurocomputing 2022, 469, 130–137. [CrossRef]
36. SEMINAL REFERENCE: INPUT SHAPING .
37. Yavuz H.; Mıstıkoğlu, S.; Kapucu, S. Hybrid input shaping to suppress residual vibration of flexible systems. Journal of Vibration and Control 2012, 18(1), 132-140. [CrossRef]
38. Sands, T. Flattening the Curve of Flexible Space Robotics. Appl. Sci. 2022, 12(6), 2992. [CrossRef]
39. Sands, T. Optimization Provenance of Whiplash Compensation for Flexible Space Robotics. Aerospace 2019, 6(9), 93. [CrossRef]
40. Carabis, D.; Wen, J. Trajectory Generation for Flexible-Joint Space Manipulators. Frontiers in Robotics and AI 2022, 9, 687595. [CrossRef]
41. Yang, T.; Xu, F.; Zeng, S.; Zhao, S.; Liu, Y.; Wang, Y. A Novel Constant Damping and High Stiffness Control Method for Flexible Space Manipulators Using Luenberger State Observer. Appl. Sci. 2023, 13(13), 7954. [CrossRef]
42. Hamad, R. Modelling and Feed-Forward Control of Robot Arms with Flexible Joints and Flexible Links. Master’s Thesis, Chalmers University of Technology, Gothenburg, Sweden, June 2016.
43. Li, X.; Yang, D.; Liu, H.. China’s space robotics for on-orbit servicing: the state of the art. National Science Review 2023, 10, 129. [CrossRef]
44. Papadopoulos E, Aghili F and Ma O et al. Front Robot AI 2021; 228: 686723.
45. Wie, B. Space Vehicle Dynamics and Control. 2nd, American Institute of Aeronautics and Astronautics, Reston, VA, 2008.
46. Singhose, W.; Seering, W.; Singer, M. Input Shaping for Vibration Reduction with Specified Insensitivity to Modeling Errors. In Proceedings of Japan-USA Symposium on Flexible Automation, 1. Boston, Massachusetts, USA. July 7-10, 1996.
47. Pao, L. Multi-input shaping design for vibration reduction, Automatica 1999 35, 1, 81-89 doi: 10.1016/S0005-1098(98)00124-1. [CrossRef]
48. Gorinevsky, D., Vukovich, G.; Nonlinear Input Shaping Control of Flexible Spacecraft Reorientation Maneuver. J. Guid. Con. Dyn 1998 21(2), 264-270. [CrossRef]
49. K., Zhengxian, Y. Combined feedback control and input shaping for vibration suppression of flexible spacecraft. In proceedings of the International Conference on Mechatronics and Automation, pp. 3257-3262, Changchun, China. 18 September 2009. https://10.1109/ICMA.2009.5246238.
50. Pontryagin, L.; Boltyanskii, V.; Gamkrelidze, R.; Mischenko, E. The Mathematical Theory of Optimal Processes; Neustadt, L.W., Ed.; Wiley: New York, NY, USA, 1962.
51. Banginwar, P.; Sands, T. Autonomous Vehicle Control Comparison. Vehicles 2022, 4, 1109–1121. [CrossRef]
52. Cooper, M.; Heidlauf, P.; Sands, T. Controlling Chaos—Forced van der Pol Equation. Mathematics 2017, 5(4), 70. [CrossRef]
53. Smeresky, B.; Rizzo, A.; Sands, T. Optimal Learning and Self-Awareness Versus PDI. Algorithms 2020, 13(1), 23. [CrossRef]
54. Osburn, P.; Whitaker, H.; Kezer, A. New Developments in the Design of Model Reference Adaptive Control Systems. Institute of the Aerospace Sciences papers 1961, 61(39).
55. MDPI Image Use Policy. “No special permission is required to reuse all or part of article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited.” Available online: https://www.mdpi.com/openaccess#:~:text=Permissions-,No%20special%20permission%20is%20required%20to%20reuse%20all%20or%20part,original%20article%20is%20clearly%20cited. (accessed on 14 February 2023).