4.2.1. Case 1

In this case, the MTTP for FFSR1 is studied. Figure 13 represents the variation trends of fitness values using IGA, showing that the minimum maneuver time of FFSR1 is 5.196 s. Figure 14 represents the movements of FFSR1, where the red points in each subfigure denote the corresponding values at the waypoints. The following explanation is useful to understand the subfigures: Figure 14a,b represents the time histories of the position and attitude of the end effector, respectively; Figure 14c,d represents the time histories of the seven joint angles; Figure 14e,f represents the time histories of the seven joint angular velocities. Figure 14a,b shows that the optimal waypoint sequence is 1 → 4 → 7 → 9 → 2 → 3 → 10 → 8 → 5 → 6, and all waypoints are accurately visited. Figure 14c,d shows that the joints move smoothly, while Figure 14e,f shows that the joint angular velocities at each waypoint are zero. Moreover, Figure 14c–f shows that the joint angles and the joint angular velocities are within the predefined region. In Figure 14, each joint angle *qi*(*i* = 1, ..., 7) denotes one component of the joint configuration **q** = [*q*1, ..., *q*7] *<sup>T</sup>*, and each joint angular velocity *dqi* denotes the 1st derivative of *qi*.

**Figure 13.** Variation trends of fitness values in the case of MTTP for FFSR1.

**Figure 14.** Case 1: FFSR1 movements. (**a**,**b**) Time histories of position and attitude of the end effector, respectively; (**c**,**d**) time histories of joint angles; (**e**,**f**) time histories of joint angular velocities.

#### 4.2.2. Case 2

In this case, the MTTP for FFSR2 is studied. Figure 15 represents the variation trends of the fitness values using IGA, where the minimum maneuver time of FFSR2 is 6.056 s, and the module value of the base attitude in Equation (8) is 0.22 deg. FFSR2 movements are represented in Figure 16, where the red parts in each subfigure denote the values at the corresponding waypoints. The legend of Figure 16 is explained as follows: Figure 16a,b represents the time histories of the position and the attitude of the end effector, respectively; Figure 16c represents the time histories of the base attitude; Figure 16d,e represents the time histories of the seven joint angles; Figure 16f,g represents the time histories of the seven joint angular velocities. Figure 16a,b shows that the optimal waypoint sequence

is 1 → 2 → 3 → 7 → 8 → 10 → 6 → 5 → 9 → 4, and all waypoints are accurately visited. It can be seen that the optimal waypoint sequence in this case is different from that obtained in Case 1, since one more objective function was considered in this case. Moreover, the randomness of IGA is a main factor. In Figure 16c, the variation amplitude of each Euler angle is less than 0.3 deg during the whole movement of FFSR2. Figure 16d,e shows that the joints move smoothly, while Figure 16f,g shows that the joint angular velocities at each waypoint are zero. Furthermore, Figure 16d–g shows that the joint angles and joint angular velocities are within the predefined region.

**Figure 15.** Variation trends of fitness values in the case of MTTP for FFSR2.

**Figure 16.** *Cont*.

(**g**)

**Figure 16.** Case 2: FFSR2 movements. (**a**,**b**) Time histories of position and attitude of the end effector, respectively; (**c**) time histories of attitude of the free-floating base; (**d**,**e**) time histories of joint angles; (**f**,**g**) time histories of joint angular velocities.

#### **5. Conclusions**

This work studied the MTTP for space robotics, including the free-flying space robot (FFSR1) and the free-floating space robot (FFSR2). The trajectory-planning problem was converted into an optimization problem by depicting the joint movements with piecewise- and continuous-sine functions. An IGA was proposed to solve the optimization problem. The main contributions are expressed as follows:


For future research, the MTTP for space robotics will be further studied in the case that one or more obstacles appear, especially the case when obstacles move. Furthermore, the corresponding trajectory-tracking control problem will be studied, which is of great significance for practical applications.

**Author Contributions:** conceptualization, S.Z. and Z.Z.; methodology, S.Z.; writing—original-draft preparation, S.Z.; writing—review and editing, S.Z. and Z.Z.; supervision, Z.Z. and J.L.; funding acquisition, Z.Z.

**Funding:** This research was funded by the National Natural Science Foundation of China (Grant No. 11472213).

**Acknowledgments:** The authors sincerely acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 11472213). The authors would also thank the members of the National Key Laboratory of Aerospace Flight Dynamics for their useful and valuable suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.

*Appl. Sci.* **2019**, *9*, 2226
