2.1 Weighted sum

The ideal solution for all presented methods is a minimal $U$ . In the case of the weighted sum, this scalar is calculated as per the following equation:

(3) \begin{equation}U = \mathop \sum \nolimits_{i = 1}^n {w_i}{o_i}(\boldsymbol{x})\end{equation}

where ${w_i}$ are the weightings of the objective functions ${o_i}(\boldsymbol{x})$ .

Weighted sums are the most common and intuitive strategy to transform a multi-objective optimisation problem into a single-objective optimisation problem in engineering [Reference Marler and Arora32]. Weights assign different importance to the different objectives of the multi-objective problem [Reference San Cristóbal Mateo33]. In the objective space, a given set of weights corresponds to an iso-cost line. In the case of a bi-objective problem, the weights define the slope of a set of parallel, straight lines in the objective function space. This method finds Pareto optimal points in non-convex regions of a Pareto front by alternating the weights of the weighted sum. However, it fails to find all non-dominated points in a non-convex Pareto front, as illustrated by Fig. 2. Minimal weight changes might lead to fundamentally different Pareto optimal points – one of the most significant weaknesses of the weighted sum approach.

Figure 2. Objective space with a non-convex region in the Pareto front. w ₁ and w ₂ belong to two minimally different sets of weights for the weighted sum.

A priori articulation of preferences identifies a single solution that presumably reflects the desired conditions. However, the sensitivity to the choice of weighting parameters, as indicated in Fig. 2, shows the difficulty of this task. Additionally, the magnitude of ${o_i}$ also defines the influence of various objectives [Reference Marler and Arora32]. If the magnitude of ${o_i}$ varies greatly, normalisation is required. The type of normalisation, however, defines the influences on the overall result and is arbitrary to a certain degree [Reference Tofallis34]. Various works investigate methods to set ideal weights, e.g. Ref. (Reference Marler and Arora9), or the broader influence of weights on the results, e.g. Ref. (Reference Marler and Arora32). However, none fully overcomes the disadvantages. The most common and intuitive engineering approach is associating individual objectives with importance. The weights ${w_i}$ accordingly represent the importance of the respective objectives. Therefore, the weighted sum focuses on optimising the more important objectives. However, the shape of the attainable set defines how sensitive the result is to the choice of weight. If the shape is non-convex, even minimal changes in weighting factors might lead to completely different solutions (see Fig. 2). Frequently, local path planning algorithms use path objectives (smoothness, obstacle vicinity, goal direction) to identify the optimal path. However, relevant mission objectives (success, distance, time, energy) may change during the flight (e.g. as a function of weather conditions) and are only ultimately known after the flight. Therefore, the weights of the path objectives are chosen such that they hopefully lead to the desired mission outcome. In practice, evidence suggests that the weighted sum might lead to a difficult-to-tune local path planner with unreliable performance because of the dependency of the result on the shape of the attainable set, the magnitude of the objectives and the chosen weights.

2.2 Weighted product

The weighted product is analogue to the weighted sum, but the individual objective functions ${o_i}$ are multiplied and weighted by an exponential ${w_i}$ as in:

(4) \begin{equation}U = \mathop \prod \nolimits_{i = 1}^n {o_i}{(\boldsymbol{x})^{{w_i}}}\end{equation}

Weighted products require ${o_i}(\boldsymbol{x}) \gt 0,\;\forall i$ . Contrary to weighted sums, weighted products do not require normalisation. However, weighted products also overvalue extremes: if one of the objective functions ${o_i}$ gives an exceptionally high or low value, the influence on the overall result is strong [Reference San Cristóbal Mateo33]. Moreover, they may incur saturation of computational precision of the calculator or library used, particularly in the presence of several objectives. Similar to weighted sum, the weighted product focus on optimising the most important objectives.

2.3 Weighted Chebyshev distance

The weighted Chebyshev distance optimises the worst of the objective functions:

(5) \begin{equation}U = {\rm{max}}\!\left\{ {{w_i}\!\left[ {{o_i} - o_i^0} \right]} \right\}\end{equation}

Here, $o_i^0$ denotes the utopian point with the best result for every individual objective function; $o_i^O = {\rm{min}}\{ {o_i}(\boldsymbol{x})|\boldsymbol{x} \in \boldsymbol{X}\} $ (Reference Cantrell35). The weighted Chebyshev distance aims to find the lowest worst objective relative to the utopian point possible.

2.4 Factorial achievement scalarising function

Achievement scalarising functions are based on natural decision-making and using desirable reference objectives rather than weighting factors [Reference Wierzbicki36].

(6) \begin{equation}U = \mathop {\max }\limits_{{I^q}\underline C {N_n}\,:\,\left| {{I^q}} \right| = q} \left\{ {\mathop \sum \nolimits_{i \in {I^q}} {\rm{max}}\!\left[{w_i} \cdot {\left({o_i}(\boldsymbol{x}) - o_i^u\right),0} \right]} \right\}\end{equation}

In parameterised achievement scalarising functions, as in Equation (6), an integer parameter $q\; \in {N_n}$ controls the degree of flexibility from a linear ${L_1}$ metric to a Chebyshev ${L_\infty }$ . ${I^q}$ denotes a subset of ${N_n}$ of cardinality $q$ [Reference Nikulin, Miettinen and Mäkelä37]. Achievement scalarising functions aim to find the lowest worst combination of objectives relative to the utopian point depending on the degree of flexibility.

2.5 Epsilon-constraint method

(7) \begin{equation}U = \min \!\left\{ {{F_p}(\boldsymbol{x})} \right\}\end{equation}

(8) \begin{equation}{\rm{Subject\ to}}\,\,{F_i}(\boldsymbol{x}) \le {\epsilon _i},{\rm{\;\;}}i = 1, \ldots ,n,{\rm{\;\;}}i \ne p\end{equation}

The Epsilon-constraint method converts all objectives except one, denoted as ${F_p}$ , into bounds – leading to a single optimisation problem [Reference Mavrotas38]. By systematically varying bounds on the constraints, the Pareto front is generated. Therefore, the epsilon-constraint method does not lead to an aggregated objective function. This approach’s advantage is that the parts of a non-convex Pareto front are also identified. However, it is also difficult to choose an appropriate value for epsilon, shifting the problem of defining adequate weights to a problem of defining good epsilon values. The epsilon constraint method is also computationally heavy because it has to solve several optimisation problems at once; therefore, this method is not further investigated.

2.6 Experimental setup

We use the 3DVFH^* as a testing algorithm. The 3DVFH^* combines a 3D vector field histogram with the A^* algorithm. It generates several candidate paths, usually consisting of five waypoints 2m apart, based on the search behaviour of A^*. It then evaluates all candidate paths with a cost function. The standard 3DVFH^* has a weighted sum as a cost function, as in Equation (9). The 3DVFH^* was introduced in Ref. (Reference Baumann39).

(9) \begin{equation}U = {w_{yaw}} \cdot {o_{yaw}} + {w_{pitch}} \cdot {o_{pitch}} + {w_{smooth}} \cdot {o_{smooth}} + {o_{obstacle}} + {o_{heuristics}}\end{equation}

All ${o_i}$ in Equation (9) depend on $\boldsymbol{x}$ giving ${o_i}(\boldsymbol{x})$ ; however, we omitted the $(\boldsymbol{x})$ in Equation (9) for abbreviation. The total cost $U$ consists of the cost for horizontal deviation from the direct goal direction ${o_{yaw}}$ , weighted by ${w_{yaw}}$ , the cost for vertical deviation from the direct goal direction ${o_{pitch}}$ , weighted by ${w_{pitch}}$ , the cost for deviation from the previous flight direction ${o_{smooth}}$ weighted by ${w_{smooth}}$ , the cost of proximity to obstacles ${o_{obstacle}}$ and heuristic cost ${o_{heuristic}}$ , that describe the remaining distance to the goal. The algorithm estimates the cost of alternative paths and chooses the cheapest path.

We define two main categories of test scenarios: simple obstacle avoidance and urban environments. We defined 900 different short-distance flight scenarios in which the UAV has to navigate past one to 40 flat obstacles, standing or floating in the air. We also defined 1,408 challenging urban environments to evaluate the performance of the different cost functions in different, realistic situations. The urban environments consist of various cuboidal obstacles in shapes and sizes typical for various buildings. The obstacles stand in rows with different distances between them. Figure 3 shows sample images of both types of scenarios. The urban environments represent anything from a small rural village to a large American-type city centre.

Figure 3. Examples of two test scenarios. (a) is an environment with flat rectangular obstacles and (b) represents an urban environment with various buildings of different sizes and dimensions.

We simulated flights with MATLAB and the UAV toolbox. We used our own implementation of the 3DVFH^* in Matlab, as previously discussed in Ref. (Reference Thomeßen, Thoma and Braun40). This version is a baseline with a weighted sum as a cost function. We replaced this cost function with a weighted product, two variants mixing weighted sum and weighted product, a weighted Chebyshev function, and two versions of factorial achievement scalarising functions with different levels of factorisation ( $q = 2$ and $q = 3$ for Equation (6)). The cost of proximity to obstacles plays a special role because it ensures safety. Therefore, we tested the Chebyshev distance and the factorial achievement scalarising function, with ${o_{obstacle}}$ as part of the cost function and ${o_{obstacle}}$ as additional cost.

We also investigated the influence of multiple definitions of the objective cost ${o_i}$ itself. Most optimisation-based path planning algorithms use two or three main types of cost: deviation from the goal direction/a global path, obstacle cost, and smoothness cost. While the first two are often formulated similarly across multiple algorithms, the latter is formulated in multiple ways. Therefore, we investigated the influence of multiple definitions of the smoothness cost ${o_{smooth}}$ itself. We tested the standard formulation of velocity cost of the 3DVFH^* as published in Ref. (Reference Baumann39), as well as four alternatives representing different approaches seen in the broader field of local path planning algorithms.

Changing the cost functions also requires the identification of suitable weights. This work also investigates the sensitivity of the solution on the chosen weight. Therefore, we chose a design of experiments (DoE)-inspired approach to get as much information as possible in the shortest time possible. We fully simulate every flight, which means that our tool needs to render the sensor data for every flight position, use the avoidance algorithm for these situations and simulate any physical and dynamic limitations. This approach is computationally heavy. Our limited computational resources do not allow the simulation of millions of flights in a short time. Therefore, we start with a full factorial surface design for every cost function. Sensitive weights are further refined with additional evaluation points in the DoE. The approach quickly identifies statistically significant weights and reduces the parameter room. This trade-off between high-fidelity simulations and optimised experimental design allowed over 600,000 realistic test flights in a reasonable amount of time.

2.7 Performance evaluation

The essential requirement of an obstacle avoidance algorithm is that it has to be safe. Therefore, a collision has to be avoided in any case. An obstacle avoidance algorithm’s second most important requirement is to fulfil its mission successfully – its reliability. This work evaluates reliability and safety combined with failure probability. The failure probability $f$ is defined as:

(10) \begin{equation}f = \frac{{{n_f}}}{{{n_f} + {n_s}}}\end{equation}

Where ${n_f}$ denotes the number of failed flights and ${n_s}$ denotes the number of successful flights. A failed flight is a flight that does not reach its goal. The reason for failure, e.g. crashing or poor path planning, is irrelevant to its reliability. However, we investigate its safety separately.

Besides the algorithm’s reliability in fulfilling its mission, its efficiency plays a crucial role, especially in small and nano UAV operations. We use estimated energy consumption as a measure of efficiency. Determining the energy consumption from the simulation is complex and inaccurate. Therefore, we use a simplified approach to only roughly estimate the energy consumption. This approach is simple but sufficient to compare the efficiency of the different cost function types.

The simulation provides accurate position data in $x$ , $y$ and $z$ and velocities in all three directions at 30 Hz. With this information, potential energy and kinetic energy are easy to determine. However, a quadcopter hovering in a fixed position has constant potential energy and no kinetic energy, i.e. no change in energy. In reality, the energy consumption in this situation is the highest. Therefore, a potential and kinetic energy approach is invalid for quadcopter UAVs. A more precise approach based on all motors’ power consumption is impossible because the simulation does not obtain this data. However, the required thrust can be estimated from movements. Therefore, the power consumption is estimated with a simple thrust estimation model. This procedure is inspired by the simplified effective hover mode endurance presented in Ref. (Reference Dorrington41).

The whole flight mission is discretised into time steps. The total energy ${E_{tot}}$ consumed for a mission is the sum of all energies consumed during the individual time steps ${E_i}$ .

(11) \begin{equation}{E_{tot}} = \mathop \sum \nolimits_i {E_i}\end{equation}

The energy per timestep ${E_i}$ is the sum of the change in kinetic energy ${E_{kin}}$ , potential energy ${E_{pot}}$ , and the energy required to generate a certain thrust ${E_{thrust}}$ .

(12) \begin{equation}{E_i} = \Delta {E_{kin,i}} + \Delta {E_{pot,i}} + {E_{thrust,i}}\end{equation}

The velocity and attitude in each time step are assumed to be constant. The energy required for attitude control is neglected. The energy required to accelerate the UAV between two timesteps is estimated by the acceleration work, which is equal to the change in kinetic energy ${\rm{\Delta }}{E_{kin}}$ , given by:

(13) \begin{equation}{\rm{\Delta }}{E_{kin,i}} = \frac{1}{2} \cdot m \cdot {\rm{\Delta }}v_i^2\end{equation}

(14) \begin{equation}{\rm{\Delta }}{v_i} = \sqrt {{{\left( {{v_{x,i + 1}} - {v_{x,i}}} \right)}^2} + {{\left( {{v_{y,i + 1}} - {v_{y,i}}} \right)}^2} + {{\left( {{v_{z,i + 1}} - {v_{z,i}}} \right)}^2}} \end{equation}

With $\boldsymbol{v} = \left( {{v_x},\;{v_y},\;{v_z}} \right)$ denoting the flight velocity. The energy required for altitude changes is given by the change in potential energy, according to:

(15) \begin{equation}{\rm{\Delta }}{E_{pot,i}} = m \cdot g \cdot \left( {{z_{i + 1}} - {z_i}} \right)\end{equation}

Finally, the energy required to provide sufficient thrust is derived from the definition of the figure of merit [Reference Leishman42] and given by:

(16) \begin{equation}{E_{thrust,\;i}} = \frac{{T_i^{\frac{3}{2}}}}{{\sqrt {2 {\cdot} {S_{ref}} {\cdot} \rho \;} }} \cdot \frac{{{\rm{\Delta }}t}}{{FOM}}\end{equation}

${\rm{\Delta }}t$ is the length of the time step, ${S_{ref}}$ is the reference area, i.e. the disk area, $\rho $ denotes the air density, and $FOM$ is the figure of merit of the rotors. $FOM\; = 0.72$ is assumed for the simulation. The thrust of the UAV is denoted by $T$ and estimated via:

(17) \begin{equation}{T_i} \cong \sqrt {{{\left( {m \cdot g} \right)}^2} + {D^2}} \end{equation}

$m$ denotes the weight of the UAV, $g$ the gravitational acceleration, and $D$ the drag force acting on the UAV. $D$ is estimated from Ref. (Reference Russell, Jung, Willink and Glasner43) with assuming low pitch angles. In all UAV applications, energy is a valuable resource. The lower the energy consumption, the more versatile the UAV application. Energetically very efficient path planning increases the mission range and mission duration. On the contrary, too high energy consumption might limit the UAV in its applicability. Therefore, the influence on energy consumption requires monitoring as well.

The sensitivity of cost function weights is the last parameter to evaluate the different cost function types. If only a specific set of weights leads to good performance while most other parameters lead to bad performance, the cost function has low robustness against wrongly chosen weights. The higher the sensitivity, the lower the robustness of this approach.

2.8 Cost function parameter identification

Local path planning is a two-staged multi-optimisation problem. The local path planning algorithm identifies the best path by considering aspects like the deviation from the goal direction, vicinity to obstacles and path smoothness. However, failure probability, flight distance, flight time and energy consumption measure the algorithm’s performance. The cost function parameters are only loosely coupled with the actual performance parameters of the algorithm and do not show deterministic behaviour. Therefore, the ideal choice of cost function parameters is unknown. We conducted a DoE-inspired search for ideal weights. First, we identified a reasonable parameter room for the different cost function weights. Second, we conduct a full factorial surface Design and identify those parameters with a statistically significant influence. Third, those significant parameters are further investigated with a refined sampling around the optimum found by the DoE.

Figure 4. Comparison of the failure probability of different variants of a cost function for the 3DVFH^* for flights in all environments. The bars indicate the minimal failure probability. The error bars indicate the range of failure probabilities.