1. Introduction

2. Materials and Methods

2.1. Rigid Body Dynamics

Rigid body is the starting point to consider for space robotics modeling by using the Newton’s second law of motion [11] to get the 6 DOF [12]. The following part in this subsection introduces some basic modeling information as well as the relationship between Hamiltonian and Newton’s law.

Equation (1) represents the basic Newton’s law of motion that provide the relationships between force, mass, and acceleration. And Equation (2) shows the relationship between torque and angular acceleration. Both are describing the non–rotating situations. Equation (3) and (4) are expressed in rotational conditions and describe the replenishment and assembly manipulation for the space robot working on orbit. Based on Equation (1) to (4), it can be combined to get 6 degrees of freedom.

$${F|}_{inertial}=m{a|}_{inertial}=\dot{mv}{|}_{inertial}=m\ddot{x}{|}_{inertial}$$

(1)

$${\tau }_{inertial}=J\alpha {|}_{inertial}=m\dot{\omega }{|}_{inertial}$$

(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)

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

(4)

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

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

Variable / acronym	Definition	Variable / acronym	Definition
F	Vector sum of external forces	$\tau$	Vector sum of external torque
J $v\text{'}$	Moments of inertia Velocity relative to reference frame	$r\text{'}$ $\omega$	Position relative to the reference frame Angular velocity

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

Figure 3. (a) U.S. Naval Postgraduate School simulate a very lightweight, low–stiffness, free–floating robot on a planar air–bearing table in autonomous robotic laboratory. [3] U.S. Department of Defense photographs and imagery, unless otherwise noted, are in the public domain [4]. Images courtesy U.S. Department of Defense [5]; (b) modeling and analysis of the flexible robot depicted in (a) discretized into four nodes depicted in (c) of the mass moment of inertias $I_2$ through $I_5$ appended to the rigid body depicted in (b) as $J_1$, where $J_2$ in (b) represents the sum of constituent inertias at nodes 2–5 in (c).

Figure 3. (a) U.S. Naval Postgraduate School simulate a very lightweight, low–stiffness, free–floating robot on a planar air–bearing table in autonomous robotic laboratory. [3] U.S. Department of Defense photographs and imagery, unless otherwise noted, are in the public domain [4]. Images courtesy U.S. Department of Defense [5]; (b) modeling and analysis of the flexible robot depicted in (a) discretized into four nodes depicted in (c) of the mass moment of inertias $I_2$ through $I_5$ appended to the rigid body depicted in (b) as $J_1$, where $J_2$ in (b) represents the sum of constituent inertias at nodes 2–5 in (c).

Equation (5) states the expression for the Lagrange method, where the potential energy is related to the height displacement and kinetic energy is related to the motion velocity. The left side of the Equation (6) is the representation for the Hamiltonian and the first term of the right side is Lagrangian. By doing the differentiation and minimization, it can produce the Euler–Lagrange equation shown in Equation (7). With certain steps of algebraic arrangement, Equation (7) can be modified to Newton’s Law shown in Equation (8).

$$L=Kinetics energy-potential energy=\frac{1}{2}m{V}^2+mgh=\frac{1}{2}m\dot{x^2+mgh}$$

(5)

$$H\left(\lambda ,x,u\right)\equiv L\left(x,u\right)+\lambda u$$

(6)

$$\frac{d}{dt}\left(\frac{\partial L\left(x,u\right)}{\partial u}\right)-\frac{\partial L\left(x,u\right)}{\partial x}=0$$

(7)

$$F=mA$$

(8)

Table 2. Table of proximal variables and nomenclature ^1.

Table 2. Table of proximal variables and nomenclature ^1.

Variable/acronym	Definition	Variable/acronym	Definition
$H(\lambda ,x,u)$	Hamiltonian	$L(x,u)$	Lagrangian

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

2.3. Modal System Identification

This section introduces the how the nodal degrees of freedom can be set up. It states the equation modification to get the torque relationship, which can be applied to the control topology calculation shown in Figure 4.

When modeling the Flexible robot, every node in the system should follow Newton’s second law of motion to get equation with the damping in accordance with equation (10).

$$\left[M\right] \{x ̈ \}+\left[C\right] \{x ̇ \}+\left[K\right] \left\{x\right\}=\left\{F\right\}$$

(10)

where the [M] is the global mass matrix, [C] is the global stiffness matrix, and [F] is the force matrix.

Here, when considering the nodal degrees of freedom, it should be constrained to be zero for both the flexible and rigid bodies. In such situation, only the corner node for the flexible body is considered, so the degrees of freedom for the horizontal and vertical direction are replaced by the angular rotation. Thus, the Newton’s second law can be modified as the combination of rotational form and the rigid–elastic coupling method as shown in the Equation (11), where $\sum \tau$ is the sum of the disturbance torques.

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

(11)

After rearranging the modified equation, the angular acceleration of the system rotation angle can be represented as the Equation (12) states.

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

(12)

Table 3. Table of proximal variables and nomenclature ^1.

Table 3. Table of proximal variables and nomenclature ^1.

Variable/ acronym	Definition	Variable/ acronym	Definition
$\left[M\right]$	Mass matrix	$\left[K\right]$	Stiffness matrix
$\ddot{q}$	Acceleration in generalized displacement coordinates	$I_{zz}$	Principal moment of inertia with respect to z-axis
$\left\{F\right\}$	Force vector	$D$	Coupling term for rigid-elastic

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

2.4. Classical Second–Order Structural Filtering

In this section, it uses an example equation to introduce a PID type of control system with the Laplace transformation. In Figure 4, the SIMULINK model used to get the result is presented with the filter function provided. The commonly used controller in feedback control is the PID control, which uses proportional, integral, and Derivative components to make up the control algorithm. Equation (13) states the close–loop equation example for the proportional and velocity components controller. By doing the Laplace transformation, Equation (12) can be converted into Equation (14). It can not only guide on the performance specification, e.g., stability, overshoot, settling time, etc., but also help to get the structural filtering to adapt the compensation with the natural frequency of the system.

$$I\ddot{\theta }+K_V\dot{\theta }+K_P\theta =K_P{\theta }_d$$

(13)

$$\frac{\theta \left(s\right)}{{\theta }_d\left(s\right)}=\frac{K_p}{I{s}^2+K_{v}s+K_p}\to C.E. : s^2+K_{v}s+K_p{|}_{I=1}=s^2+s\xi {\omega }_{n}s+{{\omega }_n}^2$$

(14)

Equation (15) provides the transfer function for the structural filters used to augment the system compensation. It defines the frequencies and the damping ratio for the zeros and poles. According to the equation, the steady–state gain, maximum gain, and the phase can be measured numerically. Based on the parameters: ${\omega }_z , {\omega }_p , {\xi }_z$,and ${\xi }_p$ , the point that occur maximum lead and lag can also be found.

$$\frac{Output\left(s\right)}{Input\left(s\right)}=\frac{\frac{s^2}{{\omega }_z^2}+\frac{2{\xi }_z}{{\omega }_z^2}s+1}{\frac{s^2}{{\omega }_p^2}+\frac{2{\xi }_p}{{\omega }_p^2}s+1}$$

(15)

Table 4. Table of proximal variables and nomenclature ^1.

Table 4. Table of proximal variables and nomenclature ^1.

Variable/acronym	Definition	Variable/acronym	Definition
$K_V\dot{\theta }$	Active damping	$K_P\theta$	Active stiffness
$K_P{\theta }_d$	Feedforward

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

2.5. Simulation Parameters

The simulation (depicted in Figure 4) and the Appendix are created by MATLAB/SIMULINK with version r2022b. The results are shown in the Section 3.

Figure 4. Control topology used in this article with proportional, integral, and derivative control combined with the second–order structural filter, which are bandpass (BP) and notch. It is designed by using the Equation (14) with the parameters of the filter.

Figure 4. Control topology used in this article with proportional, integral, and derivative control combined with the second–order structural filter, which are bandpass (BP) and notch. It is designed by using the Equation (14) with the parameters of the filter.

3. Results

4. Discussion

References

1. Nile, R.; First Humanoid Robot In Space Receives NASA Government Invention of the Year. 17 June 2015. Available online: https://www.nasa.gov/mission_pages/station/research/news/invention_of_the_year (accessed on 17 February 2023).
2. NASA. NASA Image Use Policy. 2022. Available online: https://gpm.nasa.gov/image-use-policy (accessed on 14 April 2023).
3. Naval Postgraduate School, Flexible Spacecraft Simulator Laboratory. Available online: https://nps.edu/web/srdc/laboratories (accessed on 12 April 2023).
4. Use of Department of Defense Imagery. Available online: https://www.defense.gov/Help-Center/Article/Article/2762906/use-of-department-of-defense-imagery/#:~:text=Department%20of%20Defense%20photographs%20and,use%2C%20subject%20to%20specific%20guidelines (accessed on 14 April 2023).
5. Defense Imagery Management Operations Center. Public Use Notice of Limitations. Available online: https://www.dimoc.mil/resources/limitations/ (accessed on 14 April 2023).
6. 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).
7. 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).
8. Armanini, C.; Boyer, F.; Mathew, A.T.; Duriez, C.; Renda, F. Soft Robots Modeling: A Structured Overview. IEEE Trans. Robot. 2023, 39, 1728–1748. [Google Scholar] [CrossRef]
9. Encyclopaedia Britannica, Britannica, Euclidean Space. Available online: https://www.britannica.com/science/Euclidean–space (accessed on 10 October 2022).
10. Sands, T. Comparison and Interpretation Methods for Predictive Control of Mechanics. Algorithms 2019, 12, 232. [Google Scholar] [CrossRef]
11. Newton, I. Principia, Jussu Societatis Regiæ ac Typis Joseph Streater; Cambridge University Library: London, UK, 1687. [Google Scholar]
12. Mazalov, V.; Parilina, E. The Euler-Equation Approach in Average-Oriented Opinion Dynamics. Mathematics 2020, 8, 355. [Google Scholar] [CrossRef]
13. 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).
14. Doyle, J.; Francis, A.; Tannenbaum, A. Feedback Control Theory; Dover: Mineola, NY, USA, 2009. [Google Scholar]
15. Johansson, R.; Robertsson, A.; Nilsson, K.; Verhaegen, M. State-space system identification of robot manipulator dynamics. Mechatronics 2000, 10, 403–418. [Google Scholar] [CrossRef]
16. Corke, P, “High–performance visual closed–loop robot control,” thesis.
17. Li, Y.; Ang, K.H.; Chong, G. PID control system analysis and design. IEEE Control. Syst. 2006, 26, 32–41. [Google Scholar] [CrossRef]
18. Bemporad, A.; Morari, M.; Dua, V.; Pistikopoulos, E. The explicit linear quadratic regulator for constrained systems. Automatica 2002, 38, 3–20. [Google Scholar] [CrossRef]
19. Tao, K; Kosut, R; Aral, G, “Learning feedforward control,” Proceedings of 1994 American Control Conference –ACC ’94.
20. Chang, T.C.; Seufert, C.; Eminaga, O.; Shkolyar, E.; Hu, J.C.; Liao, J.C. Current Trends in Artificial Intelligence Application for Endourology and Robotic Surgery. Urol. Clin. North Am. 2020, 48, 151–160. [Google Scholar] [CrossRef]
21. Fu, S; Bhavsar, P, “Robotic Arm Control based on internet of things,” 2019 IEEE Long Island Systems, Applications and Technology Conference (LISAT), 2019.
22. Sands, T. Inducing Performance of Commercial Surgical Robots in Space. Sensors 2023, 23, 1510. [Google Scholar] [CrossRef]
23. Sikdar, S.; Rahman, M.H.; Siddaiah, A.; Menezes, P.L. Gecko–Inspired Adhesive Mechanisms and Adhesives for Robots—A Review. Robotics 2022, 11, 143. [Google Scholar] [CrossRef]
24. McDonnell, J.M.; Ahern, D.P.; Doinn, T.; Gibbons, D.; Rodrigues, K.N.; Birch, N.; Butler, J.S. Surgeon proficiency in robot-assisted spine surgery. Bone Jt. J. 2020, 102-B, 568–572. [Google Scholar] [CrossRef]
25. Gebremeskel, M.; Shafiq, B.; Uneri, A.; Sheth, N.; Simmerer, C.; Zbijewski, W.; Siewerdsen, J.H.; Cleary, K.; Li, G. Quantification of manipulation forces needed for robot-assisted reduction of the ankle syndesmosis: an initial cadaveric study. Int. J. Comput. Assist. Radiol. Surg. 2022, 17, 2263–2267. [Google Scholar] [CrossRef] [PubMed]
26. Rogowski, A. Scenario–Based Programming of Voice–Controlled Medical Robotic Systems. Sensors 2022, 22, 9520. [Google Scholar] [CrossRef] [PubMed]
27. Stauffer, T.P.; Kim, B.I.; Grant, C.; Adams, S.B.; Anastasio, A.T. Robotic Technology in Foot and Ankle Surgery: A Comprehensive Review. Sensors 2023, 23, 686. [Google Scholar] [CrossRef] [PubMed]
28. Sühn, T.; Esmaeili, N.; Mattepu, S.Y.; Spiller, M.; Boese, A.; Urrutia, R.; Poblete, V.; Hansen, C.; Lohmann, C.H.; Illanes, A.; Friebe, M. Vibro–Acoustic Sensing of Instrument Interactions as a Potential Source of Texture–Related Information in Robotic Palpation. Sensors 2023, 23, 3141. [Google Scholar] [CrossRef]
29. Ahn, B.; Kim, Y.; Oh, C.K.; Kim, J. Robotic palpation and mechanical property characterization for abnormal tissue localization. Med Biol. Eng. Comput. 2012, 50, 961–971. [Google Scholar] [CrossRef]
30. Zhang, H.; Liu, W.; Yu, M.; Hou, Y. Design, Fabrication, and Performance Test of a New Type of Soft–Robotic Gripper for Grasping. Sensors 2022, 22, 5221. [Google Scholar] [CrossRef]
31. Yu, M.; Liu, W.; Zhao, J.; Hou, Y.; Hong, X.; Zhang, H. Modeling and Analysis of a Composite Structure–Based Soft Pneumatic Actuators for Soft–Robotic Gripper. Sensors 2022, 22, 4851. [Google Scholar] [CrossRef]
32. Qin, Y.; Zhang, H.; Wang, X.; Han, J. Active Model–Based Hysteresis Compensation and Tracking Control of Pneumatic Artificial Muscle. Sensors 2022, 22, 364. [Google Scholar] [CrossRef]
33. Ma, Z.; Sameoto, D. A Review of Electrically Driven Soft Actuators for Soft Robotics. Micromachines 2022, 13, 1881. [Google Scholar] [CrossRef]
34. Palmieri, P.; Melchiorre, M.; Mauro, S. Design of a Lightweight and Deployable Soft Robotic Arm. Robotics 2022, 11, 88. [Google Scholar] [CrossRef]
35. Herron, C.W.; Fuge, Z.J.; Kogelis, M.; Tremaroli, N.J.; Kalita, B.; Leonessa, A. Design and Validation of a Low–Level Controller for Hierarchically Controlled Exoskeletons. Sensors 2023, 23, 1014. [Google Scholar] [CrossRef] [PubMed]
36. Chellal, A.A.; Lima, J.; Gonçalves, J.; Fernandes, F.P.; Pacheco, F.; Monteiro, F.; Brito, T.; Soares, S. Robot–Assisted Rehabilitation Architecture Supported by a Distributed Data Acquisition System. Sensors 2022, 22, 9532. [Google Scholar] [CrossRef] [PubMed]
37. Song, H. –S.; Yoon, H.–S.; Lee, S.; Hong, C.–K.; Yi, B.–J. Surgical Navigation System for Transsphenoidal Pituitary Surgery Applying U–Net–Based Automatic Segmentation and Bendable Devices. Appl. Sci. 2019, 9, 5540. [Google Scholar] [CrossRef]
38. Lu, Y.; Zhou, Z.; Kokubu, S.; Qin, R.; Tortós Vinocour, P.E.; Yu, W. Neural Network–Based Active Load–Sensing Scheme and Stiffness Adjustment for Pneumatic Soft Actuators for Minimally Invasive Surgery Support. Sensors 2023, 23, 833. [Google Scholar] [CrossRef] [PubMed]
39. Cunha, F.; Ribeiro, T.; Lopes, G.; Ribeiro, A.F. Large–Scale Tactile Detection System Based on Supervised Learning for Service Robots Human Interaction. Sensors 2023, 23, 825. [Google Scholar] [CrossRef] [PubMed]
40. Wei, H.; Li, K.; Xu, D.; Tan, W. Design of a Laparoscopic Robot System Based on Spherical Magnetic Field. Appl. Sci. 2019, 9, 2070. [Google Scholar] [CrossRef]
41. Wang, H.; Yang, Y.; Lin, Z.; Wang, T. Multi–Agent Reinforcement Learning with Optimal Equivalent Action of Neighborhood. Actuators 2022, 11, 99. [Google Scholar] [CrossRef]
42. Sands, T. Optimization Provenance of Whiplash Compensation for Flexible Space Robotics. Aerospace 2019, 6, 93. [Google Scholar] [CrossRef]
43. Sands, T. Virtual Sensoring of Motion Using Pontryagin’s Treatment of Hamiltonian Systems. Sensors 2021, 21, 4603. [Google Scholar] [CrossRef]
44. Zhang, J.; Dai, X.; Huang, Q.; Wu, Q. AILC for Rigid–Flexible Coupled Manipulator System in Three–Dimensional Space with Time–Varying Disturbances and Input Constraints. Actuators 2022, 11, 268. [Google Scholar] [CrossRef]
45. Yan, X.; Shao, G.; Yang, Q.; Yu, L.; Yao, Y.; Tu, S. Adaptive Robust Tracking Control for Near Space Vehicles with Multi–Source Disturbances and Input–Output Constraints. Actuators 2022, 11, 273. [Google Scholar] [CrossRef]
46. Sands, T. Flattening the Curve of Flexible Space Robotics. Appl. Sci. 2022, 12, 2992. [Google Scholar] [CrossRef]
47. Huang, B.; Sands, T. Novel Learning for Control of Nonlinear Spacecraft Dynamics. J. AppliedMath 2023, 1. [Google Scholar] [CrossRef]
48. Smeresky, B.; Rizzo, A.; Sands, T. Optimal Learning and Self-Awareness Versus PDI. Algorithms 2020, 13, 23. [Google Scholar] [CrossRef]