# Mixed-Integer-Based Path and Morphing Planning for a Tensegrity Drone
## 8 Discussion  
In this paper, a new type of drone design based on tensegrity structures was discussed. This design allows the drones to withstand collisions, as was demonstrated in experimental studies. Actuating one or more of the structural elements (in particular, making the rest lengths of the elastic elements change) makes it possible to reshape the structure, as has been demonstrated before in tensegrity planetary lander designs, for instance. A mixed-integer method for simultaneously planning a path and producing a sequence of deformations was proposed, making it possible for the robot to fit through narrow gaps, corridors, and shafts when navigating through a cluttered environment. The proposed algorithm scales with the number of steps, however a proper cut-off value for the algorithm termination (such as Gurobi gap parameter) needs to be chosen to avoid exponential growth in the computational time at higher numbers of steps. In the example with five obstacle-free areas (three rooms connected via two tilted shafts), it was found that paths up to 15 steps can be found under 32 s, while 5-step paths require less than 3 s of CPU time. The main advantage of the proposed method compared with the previous one is qualitative rather than quantitative: casting the problem as a mixed-integer quadratic instead of mixed-integer semidefinite program makes it possible to use a much wider range of solvers, effectively relaxing the requirements on the software needed to support the motion planning.  
Two prototypes of tensegrity drones were demonstrated and tested: a six-bar pure tensegrity drone and a seven-bar higher pre-stress design. Both were tested for collision resilience, and motion capture and on-board logging were used to study the trajectory and frame vibrations of the seven-bar drone design in comparison with a rigid-body QAV-R 5\({}^{\prime\prime}\) Racer. Both drones performed, but with different distributions of frame vibration frequencies, with more pronounced and lower dominant frequencies for the tensegrity drone.  
Future work on tensegrity drones can take two directions: (1) optimal mechanical design of the drones and (2) state estimation and control design, taking into account the dynamic behavior of tensegrity frames.  
Conceptualization, S.S. and A.K.; methodology, S.S. and R.F.; software, S.S. and A.A.B.; validation, A.A.B., D.D., and R.F.; formal analysis, S.S. and A.A.B.; investigation, S.S., A.A.B., D.D., and R.F.; resources, S.S., R.F., and A.K.; data curation, S.S., D.D., and R.F.; writing--original draft preparation, S.S., A.A.B., and A.K.; writing--review and editing, S.S. and A.K.; visualization, S.S. and A.A.B.; supervision, S.S. and A.K.; project administration, S.S., R.F. and A.K.; funding acquisition, S.S. and R.F. All authors have read and agreed to the published version of the manuscript.  
This research is supported by a grant from the Russian Science Foundation (project No:19-79-10246).  
**Institutional Review Board Statement:** Not applicable.  
Figure 11: Spectrum for the second-generation tensegrity drone collected via PX4 log collector and plotted using PX4 Flight Review.  
Informed Consent Statement: Not applicable.  
Acknowledgments:We are thankful to [PERSON] for his advice during the design and early experimentation with tensegrity drone prototypes. We are also thankful to [PERSON] and [PERSON] for their involvement in the assembly and tests of the original tensegrity drone prototype, as well as to [PERSON] for his help with the experiments.