3.1. Calculation Results Analysis
To demonstrate the effectiveness of the proposed method, an example is presented. The structural parameters of two identical manipulators are shown in
Table 1. Each link of the manipulator is assumed to have a mass of 0.05 kg/mm, while the trunk has a mass of 45 kg. The cross-section of the manipulator link is cylindrical with a radius of 300 mm, and that of the trunk link is rectangular with dimensions of 0.6 m × 0.8 m.
Suppose the space station model is a cuboid with dimensions of 32.5 m × 23.7 m × 18.6 m, and the fixed coordinate system of the robot during its first cycle as a reference. As depicted in
Figure 4a, there are multiple obstacles within the region defined by points (−2 m, −3 m, 0 m), (2.80 m, −3 m, 0 m), (2.80 m, 3.47 m, 0 m), and (−2 m, 3.47 m, 0 m) on the lower wall plane, with a height range of (0.65 m to 1 m). A rectangular boss is present on the lower wall plane, with vertices at (−2.50 m, 0 m, 0 m), (−2.50 m, 11.30 m, 0 m), (30 m, 11.30 m, 0 m), and (30 m, 0 m, 0 m), respectively. This boss has a height of 0.3 m. On the front wall plane, there is a prohibited contact area where precision electronic components are located, situated within a rectangular region defined by points (16.73 m, −3 m, −1.55 m), (16.73 m, −3 m, 4.03 m), (17.98 m, −3 m, 4.04 m), and (17.98 m, −3 m, 1.55 m). The proposed algorithm possesses general applicability and is not dependent on the shape of the space station or the positioning of obstacles and prohibited contact areas. The coordinates for point
O1 during the four cycles are (−0.50 m, 0.15 m, 2.70 m), (7.38 m, 10.77 m, 13.61 m), (16.74 m, −0.72 m, 3.02 m) and (28.47 m, 14.48 m, 9.34 m), respectively. The given parameters are shown in
Table 2. Here,
v1i,
v2i, and
v3i are the velocities of the trunk at points
O1k,
O2k, and
O3k, respectively, while
and
are the posture of the trunk at points
O1k and
O2k, respectively. Moreover,
ω1k and
ω2k are the angular velocity of the trunk at points
O1k and
O2k. For the given constraints, Δ
d in Equation (19) is 0.15 m, while
rq =
rs = 0.30 m.
SAkx and
SAkz in Equation (20) are 1.25 m and 2.48 m, respectively, and
rA is 200 mm. The rotation angle range for the four joints at
Aj,
Bj,
Cj, and
Fj is [−90°, 90°]. The rotation angle range for the two joints at
D1 and
E2 and the rotation angle range for the two joints at
D2 and
E1 are [−180°, 0°] and [0°, 180°], respectively. Moreover, the range of posture angles of the trunk around both the
X0-axis and
Y0-axis in a fixed coordinate system is [−25°, 25°], while the range around the
Z0-axis is [−45°, 45°]. The range for the mean of the total inertia moment is [−55 Nm, 55 Nm]. For the OA-ABC, the number of honey sources, as well as the leading bees and following bees, is 100, with the maximum number of iterations being 100. The expansion coefficient for the honey source search range is 1. The threshold value for the leading bee to become an investigating bee is set at 55. The range of the two contact points are {[−200 mm, 500 mm], [−2000 mm, −500 mm]} and {[200 mm, 500 mm], [500 mm, 2000 mm]}, respectively. The above range is divided into a 71 × 151 grid, with each unit being 10 mm. The machine spec used is as follows: the CPU model is EPYC 7742, with 64 physical cores and 128 GB of RAM (Random Access Memory). Based on the method proposed in this paper, the optimization results of motion parameters are shown in
Table 3. The optimization results of driving torques are shown in
Table A1 in
Appendix A.
The motion trajectory of the center of mass of the robot’s trunk in the space station is shown in
Figure 4. Despite the complexity of the aforementioned environment, the robot can achieve continuous movement across various inner walls of the space station, with no intervening pauses between adjacent cycles. In the first cycle, the robot moves from the lower wall plane to the upper wall plane, followed by movement from the upper wall plane to the front wall plane during the second cycle, and from the front wall plane to the rear wall plane in the third cycle. During the fourth cycle, the robot stops when it reaches its closest point to the inner wall. The coordinates of the contact points between the manipulators and the inner wall during the four cycles are shown in
Table 3. The robot will encounter obstacles during the first cycle and prohibited contact areas during the third cycle.
The variations in the joint angles of the two manipulators are shown in
Figure 5a,b. Throughout the contact phases, the maximum angle range for the four joints at
Ai,
Bi,
Ci, and
Fi is [−37.74°, 73.83°], the maximum angle range for the two joints at
D1 and
E2 is [−122.71°, −47.01°], and the maximum angle range for the two joints at
D2 and
E1 is [26.37°, 122.58°]. The rotation angles remain within the permissible range. During the flight phase, as shown in
Figure 5a,b, the joint angles of the manipulators undergo significant changes, with manipulator 1 and manipulator 2 reaching maximum joint rotation angles of 31.10° and 39.48°, respectively. The movement of the manipulators’ joints during this flight phase enables the robot to achieve the desired posture at the onset of the next cycle, ensuring movement continuity and superior performance. In terms of obstacle avoidance, during the contact phase, the minimum distance between the simplified capsules of two manipulators is 1.64 m, while the minimum distance between the capsules and the obstacles is 0.20 m. There are no collisions between the manipulators and obstacles. Similarly, during the flight phase, the minimum distance between the simplified capsules of the two manipulators is 1.48 m, which adequately satisfies the requirements for obstacle avoidance.
The variations in the angle and angular velocity of the trunk are presented in
Figure 5c,d, respectively. As can be seen from
Figure 5c, the posture of the trunk is not constant and exhibits a polynomial variation law during the contact phase, which is obtained by optimization. Across the four cycles, the maximum rotation angles of the trunk around the
X0,
Y0, and
Z0 axes are 9.58°, 7.00°, and 22.26°, respectively. If the robot’s trunk posture remains unchanged, it results in a significant increase in the calculated mean and variance of
dZMP, the variance of the total inertia moment and the total inertia moment at the moment of detachment from the inner wall, and the energy consumption of the robot, thereby leading to a decrease in overall performance.
The change in
dZMP is shown in
Figure 6a. It can be seen from
Figure 6a that
dZMP fluctuates within a minimal range, reaching maximum values of 0.43, 0.42, 0.40, and 0 during the four cycles, respectively. Notably, the sudden variation in
dZMP during the transition from the
O1O2 phase to the
O2O3 phase can be attributed to the change in acceleration direction, but this does not compromise the robot’s dynamic stability. By controlling the trajectory and posture of the trunk,
dZMP remains at zero during the fourth cycle. The change in total inertia moment is shown in
Figure 6b, which exhibits a smooth variation. The robot possesses a relatively large total inertia moment at the beginning of the
O1O2 phase to the
O2O3 phase due to the high acceleration when the direction of movement changes. The mean values of the total inertia moment for the four cycles are 33.22 Nm, 33.97 Nm, 50.70 Nm, and 16.47 Nm, respectively, while the variances are 331.45 Nm, 407.66 Nm, 238.43 Nm, and 10.78 Nm, all within small ranges. At the moment of detachment from the inner wall, the total inertia moments are 12.29 Nm,13.84 Nm, and 28.40 Nm, which are close to zero. These results demonstrate that the robot possesses good dynamic stability. The changes in energy consumption are shown in
Figure 6c. Prior to optimization, the robot’s energy consumption during the four cycles, using the initial optimization values as the known values, is 3303.27 J, 3299.09 J, 3388.37 J, and 1348.79 J, respectively. Following optimization, the energy consumption of the robot during the four cycles is reduced by 51.24%, 46.48%, 43.57%, and 54.24%, respectively. These findings satisfy the requirement for low energy consumption by robots within the space station.
Figure 7 shows the prohibited contact areas and contact points between robots and inner walls in the fixed coordinate system during the third cycle. The length and width sides of these areas are parallel to the
X0-axis and
Z0-axis, respectively. The positions of the centers of the three rectangles in the moving coordinate system for the third cycle are (17.35 m, 1.79 m), (17.35 m, 2.79 m), and (17.35 m, 3.78 m). Without considering the prohibited contact areas, the optimized positions of the two contact points are (17.50 m, 1.43 m) and (17.35 m, 4.08 m), respectively. However, these contact points fall within the prohibited contact areas. Thus, the contact position is optimized twice, resulting in new contact points at (17.55 m,1.28 m) and (17.37 m, 4.35 m), respectively. Analysis shows that the modified contact point positions have not significantly impacted the overall performance of the robot.
The results from the above analysis demonstrate that the method proposed in this paper enables the robot to achieve continuous movement among the various complex inner walls of the space station while maintaining good comprehensive performance. This provides a theoretical basis for the control of robots within a space station.
3.2. Simulation Results Analysis
To further substantiate the feasibility of the method proposed in this paper, a simulation is conducted. Software Webots is employed to simulate the examples discussed in
Section 3.1. Webots is an open-source and multi-platform desktop application used to simulate various types of robots, including industrial manipulators and legged robots. We employ Webots purely for simulation purposes, making no modifications to the software itself, in compliance with the software license. Webots is capable of providing realistic dynamic simulation effects and can replicate complex environments, making it an ideal match for our research. During the simulation process, the structural parameters of the robot (including the size and weight) and environmental settings are aligned with the theoretical calculations. The friction between joints is disregarded. In addition to the known values shown in
Section 2.2, the positions of the contact points and the driving torques are also provided as known values.
The motion process of the robot within the space station is shown in
Figure 8. It can be intuitively seen from
Figure 8 that the motion trajectory and the change in the posture of the robot. The discrepancy between the theoretical calculation results and the simulation results for the motion law of the robot is shown in
Figure 9.
Figure 9a shows the difference in the motion trajectory of the robot. The cumulative errors result in an increasing trajectory difference. The difference in the motion trajectory along the three axes is 110.23 mm, 123.41 mm, and 110.51 mm at the end of the continuous movement, respectively. Compared with the total movement distance (60.32 m in four cycles), the error remains within a marginal range.
Figure 9b shows the variation in the trunk velocities, with maximum and minimum values of 5.72 × 10
−3 m/s and 0 m/s, respectively. Moreover, the ratio of the maximum difference to the maximum motion velocity of the robot is merely 0.01.
Figure 9c,d show the difference between the theoretical calculation results and simulation results regarding the trunk angle and angular velocity, where the difference in both angle and angular velocity fluctuates within small ranges. The variation range for the angle difference is [−6.91°, 7.10°], and for angular velocity difference, it is [−0.013 rad/s, 0.011 rad/s]. These ranges also remain within a reasonable range when compared with the actual rotation angles and angular velocities of the robot’s trunk. The disparity between the theoretical calculation results and the simulation results can primarily be attributed to the following key factor. The inability to completely immobilize the contact points, and the micro-slip of the end of the manipulator relative to the inner wall, leads to changes in the motion law of the robot. Hence, during the prototype production stage, high-friction materials or adsorption devices can be employed at the end of the manipulators to prevent contact point slips. The aforementioned simulation was conducted without considering joint friction. However, joint friction could potentially influence omnidirectional continuous movement [
44]. Assuming a friction coefficient of 0.15, the difference in the motion trajectory of the trunk, both with and without considering joint friction, is shown in
Figure 10. It can be seen from
Figure 10 that the trunk’s trajectory, after factoring in friction, deviates from the previous simulation trajectory. This deviation occurs because the presence of friction can alter the force distribution at the manipulator joints, thereby affecting the trunk’s trajectory. However, this deviation falls within a small range, with a maximum error of 2.7 mm. A larger friction coefficient may lead to a greater deviation in movement and increased energy consumption.
The aforementioned analysis indicates that the simulation results of the robot align closely with the theoretical calculation results, supporting the feasibility of the method proposed in this paper.