# Mixed-Integer-Based Path and Morphing Planning for a Tensegrity Drone
## 1 Introduction  
Drones operating in large open areas have found a significant amount of practical applications, most notably ariel photography and cinematography [1]. Operation in smaller spaces, indoors, and in cluttered environments poses difficulties for control design and presents risks of unexpected collisions and crash-landings. This limits the practical use of drones in these environments. The potential uses of drones in such environments are numerous: in-door photography, building mapping, monitoring of industrial sites, construction, infrastructure, disaster response, etc. We can point to two complementary directions in drone operation in cluttered environments: lowering risks of unexpected collisions and landings occurring (e.g., via better trajectory planning and control design) and mitigating the negative consequences of the aforementioned collisions and crash-landings. The second direction often involves improved or more specialized mechanical designs for the drone body; an example is the use of protective cages [2, 3]. Here, we pursue this direction as well; however, rather than using additional structural elements to protect the drone, we propose to replace its rigid frame with a tensegrity structure. The following section will discuss the difference between this approach and the use of drone cages in more detail. Tensegrity structures are characterized by a design principle requiring that all members of the structure experience either purely tensile or purely compressive forces [4]. Thus, these members can be designed to be lightweight and damage-resilient. Moreover, the structure is pre-stressed, meaning that its elements are deformed and store potential energy [5]. The structure as a whole exhibits apparent elasticity, deforming under external forces andrestoring its original shape as the forces are lifted. Tensegrity structures are designed to withstand impacts without incurring mechanical damage [6], making them a favorable candidate for drone body design. The other important characteristic is their favorable mass-to-stiffness ratio [7]. Finally, tensegrity structures can be folded, simplifying their transportation [8; 9]. In this work, the first tensegrity drone prototype presented, based on a six-bar structure [10; 11] actuated with four rotors (see Figure 1). In this work, one additional advantage of using tensegrity structures for drone body design is explored: the possibility to control their shape by actuating one of the structural elements (one of the cables or struts). Through this, the drone can change its shape mid-flight in order to fit through narrow openings, lower the risks of collisions, or change the apparent stiffness of the structure. This concept has been discussed in a previous work [12], where the deformation of a drone was planned as a series of semidefinite programs (SDPs); in particular, a sequence of deformations of a bounding volume was planned and then mapped to actual configurations using a dataset with precomputed bounding volumes and their Eigendecompositions. The proposed approach required an assumption that the change in the robot's shape as it goes through a deformation can be accurately described by a positive-semidefinite matrix. This excludes non-linear deformations. Here, we propose a mixed-integer planning method that allows the simultaneously planning of the path of the robot and the selection of a configuration for each path point; configuration is a statically stable shape of the tensegrity structure, given the current state of the actuated structural elements. The resulting problem is a mixed-integer quadratic program (MIQP), which can be solved with a branch-and-bound method. This representation is of special interest in the cases when only a limited discrete set of configurations is expected to occur; for instance, if only two rods are actuated in order to control one of the dimensions of the robot and the range of deformations is divided into \(N\) discrete configurations.