**Autonomous Multirotor Navigation with Obstacle Avoidance **

**Fuzzy systems are capable of robust modelling and control of complex systems like **
multi-rotors. Designing and fine tuning of fuzzy systems is a complex challenge.

**Gradient descent optimisations are capable of finding only the nearest extremum **
point, often a sub-optimal solution.

Optimisation methods based on thorough search are excessively computation expensive.

Various **stochastic search optimisation methods based on stochastic gradient descent **
variations like simulated annealing, taboo search; swarm intelligence methods like ant
colonies, bacterial search; and evolutionary algorithms like genetic algorithms have
been successfully used for fuzzy system optimisation [32], [16], [2], [9]. **Genetic **
**algorithms are efficient, well studied simple stochastic optimisation methods of proven **
convergence, capable of **global multi-objective search and optimisation of versatile **
complex systems.

Mathematical model design of complex real systems can readily take the so-called black-box common approach, which uses exclusively numerical system input-output data pairs for constructing the model. Without deeper understanding of the problem, these black box models can easily end up being clumsy and working only in some specific setups, without any guaranties for general precision or robustness.

In contrast to black-box there is white box (also called glass box or clear box) modelling, which uses extensive, state of the art physics and mathematics analysis, presuming to know all necessary information; still just to end up with only simplified models, as real complex nonlinear systems can in the end be only approximated.

**Grey-box modelling builds on both input-output data and also on essential expert **
knowledge; it efficiently incorporates them into the model structure used for system
identification. Fuzzy logic system (FLS) modelling can be conducted as black-box
modelling where all the system knowledge is mere input-output data, however when
**expert knowledge is readily available, we should take advantage of it – fuzzy **
grey-box modelling is a rational choice.

Multi-input single-output complete first order Takagi-Sugeno-Kang type FLSs are
having large number of interdependent nonlinear parameters, whose number is
proportional to the number of antecedent membership functions. The number of FLS
linear parameters is even larger, proportional to the product of the number of
membership functions over each input. **Singular value decomposition (SVD) based **
least squares optimization can determine the optimal value of these linear parameters
[s10], [s11].

To overcome the problem complexity of finding good values for the nonlinear
parameters of a flexible structured FLS, **global search and optimization methods as **
genetic algorithm (GA) can be used [88]. Genetic fuzzy system (GFS) optimisation
problems like system identification inherently require multi-objective approach as not
just the maximum absolute error and the mean square error of the identification has to
be simultaneously minimal, but also the system complexity as the number of
membership functions and number of rules should also be minimized for computation
efficiency [s4]. Based on the standard approximation theory these objectives are clearly
competing, thus the required multi-objective optimisation problem is of high
complexity.

**Angular orientations and induced torques of flying body systems are naturally **
**continuous and periodic. It is our [0, 2π) orientation representation that results in a **
discontinuity at full turn when returning to the origin. A proper dynamic model, be it

continuously changes between any two posture orientation attitude angles. One possible solution is to transform the intuitive 3D Euler angles to quaternions, and perform the entire maths in this transformed space. Quaternion solutions may be called elegant, by whoever likes them, but are surely not simple and intuitive. For a proper intuitive soft computing approach to flying body modelling new efficient tools have to be designed [s13].

From autonomous multi-rotors it is expected to precisely track the desired path and
**to avoid obstacles ** [75]. For quadrotor flight efficiency **minimizing the energy **
**consumption of the system is more beneficial than planning for a minimum time or a **
minimum distance trajectory; a proposal for designing minimum fuel trajectories is
elaborated in [14].

For quad rotors off the shelf, camera based products exist for implementing simple
visual point-to-point waypoint tracking, where **the biggest challenge is agility of the **
**quad rotor and precision of path tracking [29]. **

Minimum-snap polynomial trajectories are proven very effective as quadrotor
trajectories, “since the motor commands and attitude accelerations of the vehicle are
proportional to the snap, or forth derivative, of the path” – citation from [37]. The rotor
blade velocity is considered as an arbitrary control input. As 7^{th} order minimum-snap
polynomial trajectories are discontinuous in displacement crackle, the fifth time
derivative of displacement, my claim is that this is still a sub-optimal approach. My
research considers the **reality of the rotor blade velocity not being an arbitrary **
theoretical control signal, but a real, electro-mechanical physical system, subject to aero
dynamical load conditions, and as such having a specific transient behaviour. As I will
present the second time derivative of the actuator torque has to be continuous; for multi
rotor flight dynamics this is equivalent to having a continuous displacement pop, the
sixth time derivative of the body displacement. To achieve feasible energy efficient
trajectories, one must take into account both the base system and the control actuator
dynamics.

**To efficiently generate trajectories for agile quadrotor flight through maps of **
real-world environments is addressed in [59], where the straight-line route is translated into
a smooth dynamically feasible polynomial trajectory and iteratively refined by a time
allocation scheme that naturally performs a trade-off to minimize accelerations while
attempting to fly at a desired velocity [59].

Based on analysis above the ultimate goal of my research is to achieve improvements of autonomous multi-rotor navigation with obstacle avoidance. In forthcoming chapters the above highlighted essential system design tools are analysed and, possible improvements are suggested for the following subjects:

a.) multi objective search and optimisation

b.) robust function approximation by fuzzy logic systems using a.) c.) complex dynamic system identification by b.)

d.) optimal feasible trajectory design for c.) e.) reduction of d.) to training data set for c.)