Space Detumbling Robot Arm Deployment Path Planning Based on BiFMT* Algorithm
Abstract
:1. Introduction
2. Space Detumbling Robot Modeling
2.1. Kinematics
 (1)
 O_{i+1}X_{i+1} is perpendicular to O_{i}Z_{i};
 (2)
 O_{i+1}X_{i+1} intersects with O_{i}Z_{i}.
2.2. Dynamics
3. Space Detumbling Robot Arm Path Planning Based on BiFMT* Algorithm
3.1. BiFMT* Algorithm
3.2. Problem Definition
Given:  Θ_{initial}, t_{0}, Θ_{goal}, Θ_{free} 
Cost function: 
$$J\left(\Theta \right(t\left)\right)=tr\left(J{J}^{T}\right)$$

Constraints: 
$$\begin{array}{c}\Theta \left({t}_{0}\right)={\Theta}_{\mathit{initial}}\hfill \\ \Theta \left({t}_{f}\right)={\Theta}_{\mathit{goal}}\hfill \\ {t}_{0}{t}_{f}\hfill \\ g\left(\Theta \right(t),\tau (t),t)\le 0\hfill \\ h\left(\Theta \right(t),\tau (t),t)=0\hfill \end{array}$$

3.3. Constraint Analysis
3.3.1. Time Constraints
3.3.2. Stationary Constraints
3.3.3. Dynamic Characteristic Constraints
3.4. Path Planning Algorithm Description
4. Simulation
4.1. Robot Model
4.2. Path Planning
5. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Category  Schematic Diagram  Brief Description 

Injection [4,5,6,7,8]  The service satellite injects substances such as a gas, ion beam or laser into the target, changing the quality characteristics of the target, including mass and inertia. Thus, it is known from the angular momentum conservation law that the target will be detumbling. On the other hand, the injection could hinder the movement of the target, thereby achieving the purpose of eliminating the target rotation. Using this method, it is necessary to carry an additional end effector and substances for the purpose of detumbling, except for gas injection which can be injected through its own engine but needs more fuel.  
Auxiliary Device [9,10,11]  Attaching an auxiliary device to the target through the service satellite and using the auxiliary device to eliminate the target rotation. The service satellite can avoid contact with the target, and the detumbling mode can flexibly adopt various means according to the target situation. However, similar to the above, the detumbling mode needs to be an additional device that is dedicated to the service satellite and has certain maneuverability and controllability which increases the system complexity.  
Electrostatic [12,13,14,15]  Electrons are continuously emitted to the target through an electronemitting device carried on the service satellite, charging the target. By doing this, the target rotation is detumbled by Coulomb electrostatic force generated by the voltage difference between the service satellite and the target. This also avoids contact between the service satellite and target. However, in this method, it is necessary to continuously charge and discharge to change the potential of the service satellite and target. In addition, this method needs to be further studied in the identification of target charge and discharge characteristics, formation maintenance and charge and discharge control algorithms.  
Electromagnetic [16,17,18,19,20]  Space debris mostly contains conductive materials such as aluminum alloys and titanium alloys. Therefore, when the target is in an external magnetic field, eddy currents are internally induced to hinder the relative motion. By using a superconducting coil to construct an external magnetic field, the target can be detumbled. The electromagnetic damping effect passively eliminates the component angular rate perpendicular to the component of the external magnetic field but does not affect the angular rate component parallel to the magnetic field direction. Thus, the relative position between the magnetic field source and the target is to be changed. In addition, the use of superconducting coils to construct a wide range of electromagnetic fields requires a corresponding power supply and cooling system. How to superimpose the superconducting magnetic field source system with service satellites requires further study.  
Robotic Contact [21,22,23]  In this method, the service satellite touches the target intermittently by using the elastic deceleration device attached to the end of the arm. The target rotation is detumbled by the friction. Robotic contact detumbling can actively adjust the direction, size and time of the control force and provide a higher braking efficiency with an accurate torque control model. However, this type of detumbling mode needs a service satellite to perform a complex orbit maneuver before implementation, located at a position very close to the target, and the collision risk is also increased. In addition, it is suitable for a target with a lower speed considering the onorbit identification efficiency and the manipulator control precision.  
Net or Tether [24,25,26,27,28]  When the net or tether catches the target, the target rotational speed is reduced by the tension and damping force of the tether. This method is only used for debris. In addition, how to avoid failure in catching and preventing the entanglement of the rope also needs further research. 
Joint  a_{i}  α_{i}  d_{i}  θ_{i} 

0  0  0  d_{0}  0 
1  0  90°  0  θ_{1} * 
2  L_{1}  0  0  θ_{2} * 
3  L_{2}  0  0  θ_{3} * 
4  0  90°  0  θ_{4} * 
5  0  90°  0  θ_{5} * 
6  0  0  d_{6}  θ_{6} * 
7  0  0  d_{7} *  0 
8  a_{8} *  0  0  0 
1:  given initial state Θ_{initial}, Θ_{goal}, task time T_{plan}, sampling point number N, maximum angular velocity ω_{max} and maximum angular acceleration ${\dot{\omega}}_{\mathrm{max}}$ 
2:  Calculating single step response time Δt by using (23) 
3:  The state space is sampled by means of Halton sampling method, and the set of sampling points Θ_{S} is obtained. 
4:  The path trees {S_{tree}, S_{check}, S_{cut}} and {S’_{tree}, S’_{check}, S’_{cut}} which are based on Θ_{initial} and Θ_{goal} is generated 
  While Do 
5:  Finding the intersection S_{m}_{eet} of S_{tree} and S’_{tree} 
6:  S_{meet} is not empty 
7:  Calculate the path cost J of each point 
8:  Find the sampling point θ_{meet} with the smallest J 
9:  By connecting S_{tree} and S’_{tree} with θ_{meet} as the connection point, the path between Θ_{initial} and Θ_{goal} is obtained 
10:  S_{meet} is empty 
11:  performing FMT*algorithms on {S_{tree}, S_{check}, S_{cut}} and {S’_{tree}, S’_{check}, S’_{cut}}, respectively, and updating S_{tree} and S’_{tree} 
  While Done 
12:  The local cubic polynomial interpolation is used to generate trajectory between sampling points in path 
Type  Subtype  Value 

Satellite platform  Central body  $\begin{array}{c}m=1000\mathrm{kg}\hfill \\ I=diag\left\{400,400,400\right\}\mathrm{kg}\cdot {\mathrm{m}}^{2}\hfill \\ V=1.6\mathrm{m}\times 1.6\mathrm{m}\times 1.6\mathrm{m}\hfill \end{array}$ 
Solar panel  $\begin{array}{c}m=20\mathrm{kg}\hfill \\ I=diag\left\{20,1,20\right\}\mathrm{kg}\cdot {\mathrm{m}}^{2}\hfill \\ V=2\mathrm{m}\times 1.6\mathrm{m}\times 0.015\mathrm{m}\hfill \end{array}$  
Robot arm  Joint1, Joint4~6  $\begin{array}{c}m=10\mathrm{kg}\hfill \\ I=diag\left\{1,1,1\right\}\mathrm{kg}\cdot {\mathrm{m}}^{2}\hfill \\ V=0.3\mathrm{m}\times \left(\pi \times 0.15\mathrm{m}\times 0.15\mathrm{m}\right)\hfill \end{array}$ 
Joint2~3  $\begin{array}{c}m=10\mathrm{kg}\hfill \\ I=diag\left\{1,1,1\right\}\mathrm{kg}\cdot {\mathrm{m}}^{2}\hfill \\ V=0.3\mathrm{m}\times \left(\pi \times 0.15\mathrm{m}\times 0.15\mathrm{m}\right)\hfill \\ L=1\mathrm{m}\hfill \end{array}$  
Flexible brush  Flexible brush  $L=0.75\mathrm{m}$ 
Simulation parameters  Initial joint state  $\left[\begin{array}{cccccc}{\theta}_{1}& {\theta}_{2}& {\theta}_{3}& {\theta}_{4}& {\theta}_{5}& {\theta}_{6}\end{array}\right]=\left[\begin{array}{cccccc}{0}^{\circ}& {0}^{\circ}& {0}^{\circ}& {90}^{\circ}& {90}^{\circ}& {0}^{\circ}\end{array}\right]$ 
Joint torque  $\left[\begin{array}{cccccc}{\tau}_{1}& {\tau}_{2}& {\tau}_{3}& {\tau}_{4}& {\tau}_{5}& {\tau}_{6}\end{array}\right]=\left[\begin{array}{cccccc}0.1& 0.1& 0.1& 0.1& 0.1& 0.1\end{array}\right]N\cdot m$ 
Joint  a_{i}  α_{i}  d_{i}  θ_{i} 

0  0  0  0.95  0 
1  0  90°  0  90° 
2  1  0  0  0 
3  1  0  0  0 
4  0  90°  0  90° 
5  0  90°  0  90° 
6  0  0  0  0 
7  0  0  0.75  0 
8  0  0  0  0 
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Chen, N.; Zhang, Y.; Cheng, W. Space Detumbling Robot Arm Deployment Path Planning Based on BiFMT* Algorithm. Micromachines 2021, 12, 1231. https://doi.org/10.3390/mi12101231
Chen N, Zhang Y, Cheng W. Space Detumbling Robot Arm Deployment Path Planning Based on BiFMT* Algorithm. Micromachines. 2021; 12(10):1231. https://doi.org/10.3390/mi12101231
Chicago/Turabian StyleChen, Ning, Yasheng Zhang, and Wenhua Cheng. 2021. "Space Detumbling Robot Arm Deployment Path Planning Based on BiFMT* Algorithm" Micromachines 12, no. 10: 1231. https://doi.org/10.3390/mi12101231