A Design Optimization Technique for Multi-Robot ...

AIAA 2017-0690 AIAA SciTech Forum 9 - 13 January 2017, Grapevine, Texas 55th AIAA Aerospace Sciences Meeting

A Design Optimization Technique for Multi-Robot Systems

Downloaded by GEORGIA INST OF TECHNOLOGY on January 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2017-0690

Jean-Guillaume Durand1, Frédéric Burgaud1, K. D. Cooksey2, Dimitri N. Mavris3 Georgia Institute of Technology, Atlanta, Georgia, 30332-0150, United States

Despite being more flexible, robust, and scalable than single robot solutions, multi-robot systems suffer from a lack of democratization. Most applications are found in academia or rare avant-gardist military projects with very few, if none, commercial implementations. This is due in part to the complex system of systems design process requiring the optimization of not only the group architecture, but also the individual agents, for the group operation. Such an intricate procedure, as well as the lack of link between the microscopic level and the macroscopic level of the system, leads to suboptimal designs. Focused on micro unmanned aerial vehicles, this paper proposes a novel bi-lebel optimization technique which differs from the classical sequential approach presently utilized by the community. The results of this global optimization scheme applied to a simple example show that performance improvements up to thirty percent can be achieved on designs. Moreover, the optimization step is shown to be not more computationally expensive than the current paradigm.

I. Introduction Largely democratized to the general public with the Parrot Drone, microdrones are experiencing a second wave of popularization thanks to a multitude of publicly available models. Besides, Unmanned Air Vehicles (UAVs) have been proving, not without some controversy, their efficiency in military operations since 2001. Fueled by the desire of automation, the development of this robotics field expands the spectrum of possible applications for the use of UAVs and promotes the proliferation of new architectures (Figure 1). However, this diversity in the existing drone fleet is not put to use and individual unmanned vehicles suffer from limitations, especially in terms of endurance and computational power for small UAVs.

(a) Multirotor [1]

(b) Twin boom design [2]

(c) Helicopter design [3]

(d) Hybrid design [4]

(e) Tilting wing hybrid design [5]

(f) Non-planar hexarotor [6]

1

Senior Graduate Research Assistant, School of Aerospace Engineering, 270 Ferst Dr, Mail Stop 0150, AIAA Student Member. 2 Research Engineer, Aerospace Systems Design Laboratory, 275 Ferst Drive #3, Atlanta, GA 30332 3 Regents Professor, Boeing Professor of Advanced Aerospace Analysis, Director of the Aerospace Systems Design Laboratory, School of Aerospace Engineering, 270 Ferst Dr, Mail Stop 0150, AIAA Fellow. 1 American Institute of Aeronautics and Astronautics

Copyright © 2017 by Jean-Guillaume Durand, Frédéric Burgaud, K. D. Cooksey, Dimitri N. Mavris. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

(g) Wall roller [7]

(h) Balloon [8]

(i) Ornithopter [9]


Figure 1: Examples of classical and unconventional designs Some of these shortcomings can be tackled by multi-robotics where cooperative teams of robots have agents work together in a coordinated fashion to accomplish a unique goal. This approach is partially inspired from the selforganized behavior exhibited in nature by social animals such as fish, birds, ants, or bees. For instance, the main idea of swarm robotics is to obtain a complex synergistic behavior through simple individual behavior rules and local interactions. Since several robots are used, this provides robustness, flexibility and scalability to the group. This collective approach of robotics also represents an interesting way to capitalize on the heterogeneity of the current fleet of robots. Robot groups can be used in real world applications for exploration, surveillance, search and rescue, humanitarian demining, intrusion tracking, cleaning, inspection, and transportation of large objects [10] (Figure 2).

Load being transported by four Partition-based surveillance Multiple observations of a map UAVs [114] strategy [115] landmark Figure 2: Increased capabilities of multi-robot solutions The focus in robotics has long been on intelligence and software architecture rather than on physical design. Ground-based robots have been the norm in the field of robotics until a few years ago, limiting the interest in design optimization since they are not subject to stringent weight restrictions and other realities that aerial vehicles experience. On the other hand, aerial robots design has to include optimization steps. Additionally, the field of collaborative robotics remains mostly confined to academic work such as the systems developed in [11], [12], and [13], and is not yet established in commercial applications. It is still at a preliminary stage in the research community and shy applications start to trigger interest only for military purposes with no known successful deployment [14]. Although these applications are outside the academic environment, they remain confined to pure research and are not used in operations. Reasons are varied and [15] mentions in particular: the need for a good laboratory infrastructure, the difficulty in building non-linear and stochastic models of the robot groups, and the fact that currently, no general method exists to go from the individuals to the group behavior. [15] mentions this latter reason as a key obstacle in the elaboration of design algorithms for swarming systems. As a matter of fact, most of the research community focuses on the behavior design of multi-robot systems, leaving aside their physical design [16]. These elements motivate the formulation of a key conjecture: a standard physical design process for multi-robot systems is needed to foster their democratization. The increase in capability offered by multi-robotics comes at the cost of an amplified complexity as a complete System of Systems (SoS) has to be designed instead of a single robot only: the properties of the whole system cannot be determined from the simple sum of its parts. Hence, a group of robots is hard to design as it is larger in scope and is subject to a more complex integration of its elements. The group may be comprised of robots which are not designed to fit the whole system and which are integrated after their design is finished [17]. The group (macroscopic level) has to be designed as well as the individual agents (microscopic level).

2 American Institute of Aeronautics and Astronautics


II. Motivation A. Sequential vs. Global Optimization The design space being considered is multi-architecture since different types of robots are to be integrated in the swarm, and multi-level since design choices have to be made at the macroscopic level and at the microscopic level. Such a design space present challenges for the optimization of generated alternatives against multiple criteria:  The architectures considered to constitute the swarm are not always defined by the same design variables so that a single conventional optimization algorithm may not be used.  There are highly non-linear relationships between the design variables of single agents and the group behavior, a multitude of design variables, and architectures. This prevents the use of surrogate models which could have sped up the design space exploration.  The variables describing the alternatives maybe either discrete (number of motors), continuous (geometric features), or even categorical (wing type). This motivates the use of methods other than the relatively fast gradient-based optimization algorithms which are not best suited for such problems.  It is expected that many local optima exist since a multitude of robots combinations could lead to pseudooptimal performance.  Multiple objectives have to be optimized so that multi-objective optimization techniques have to be used.  One slight modification at the vehicle level may completely disrupt the group level and vice-versa.  The design space is dynamic since the size of the vehicles level depends on how many vehicles were chosen at the group level These challenges call for the development of an appropriate optimization process and the following research question: how can swarm architectures be efficiently optimized in a multi-architecture multi-level design space? Due to the lack of link between the microscopic and macroscopic level of a group of robots, the research community tends to optimize the design of multi-robot systems through sequential optimization. Sequential optimization involves optimizing the different constituents of a system independently and sequentially, with their own set of constraints and cost function. For a robotic swarm for instance, it means that the agents of the swarm are first optimized with respect to individual mission requirements, the whole swarm is then optimized using this set of preoptimized agents, or vice-versa (Figure 3(a)). In particular, this makes it harder to consider the true advantages of heterogeneity. On the other hand global optimization tends to consider all constituents at once with all their interactions, in order to derive an optimal complete system (Figure 3(b)). Mission requirements

Mission Requirements Swarm Minimize mission cost w.r.t. high level performance of agents

Agent 1

Agent 2

Agent 3

Optimum swarm

Requirements agent 1



Swarm Agent 1

Agent 2

Agent 3

Optimum agent 1

Optimum agent 2

Optimum agent 3

Optimum swarm

(a) Sequential swarm design optimization (b) Example of global swarm design optimization Figure 3: Sequential vs. Global optimization


By benchmarking different Multi-Disciplinary Optimization (MDO) algorithms, [18] clearly states that strategies such as sequential optimization are often not able to find the true optimum of a system. In particular, it is underlined that the interactions between the components of the system must be properly accounted for. This statement is corroborated by [19] on an example of aerospace design: the partial (sequential) optimization approach does not lead to the optimal design of the complete system, except in special cases. On Figure 4, a wing is to be designed in order to achieve a minimum value of an aggregate function of weight and drag. The system optimizer acts on the span which is given to the aerodynamics group in charge of minimizing the drag by determining an optimal twist distribution. The aerodynamics loads acting on the wing are then forwarded to the structures group, in charge of minimizing the weight of the wing with respect to the skin thickness.


System min 𝐽 = 𝐷 + 𝑘. 𝑊 w.r.t. span

Aerodynamics min 𝐷 for given lift, span w.r.t. twist distribution

Structures min 𝑊 for given lift, span, loading w.r.t. twist distribution

Figure 4: Wing design by sequential disciplinary optimization This process is iterated for different values of the wing span until an “optimum” design is reached. It is shown that this type of sequential procedure yields 11% more drag than an optimal solution derived with a global optimization technique [19]. This type of optimization procedure is notably used in [20] where a bi-level algorithm is utilized: one level dedicated to the design characteristics and high level variables (material, curvature, global thickness, struts…), and a second level reserved for design proportions (skin thickness, cross-sections…). [21] also constated the possible improvements achievable through simultaneous optimization by applying it to suborbital vehicles design programs. It was shown that programs that do not properly account for the interactions between their different business divisions (hence using sequential optimization schemes) proposed very different programs than the optimal ones found by simultaneous optimization. An improvement of up to 14.1% was announced as a result of using a global optimization scheme as opposed to a sequential one. The difference between sequential and global optimization is also well illustrated in the field of industrial engineering, and particularly for supply chain management problematics. In such problems, a system of suppliers, manufacturers, transportation, distributors, and vendors operates to transform raw materials to final products and supply those products to customers. Optimization is applied so that each component of the chain orders its necessary input in the right quantities, in order to be able to deliver the expected output at the right time. The goal of supply chain management is then to minimize the cost of the complete system despite conflicting objectives from the different facilities at stake [22]. The most evident of these tradeoffs is that the customer desires a short time to delivery at low prices while the warehouses focus on having low inventory to reduce their operating costs. In this context, [23] compares sequential optimization to global optimization on a simplified supply chain. In the first case, the agents of the chain derive their respective optimal solution independently of the others whereas a complete integrated supply chain system is considered for the second case (Figure 5). System

Raw material

Supplier

Manufacturer

Transportation

Distributor

(a) Sequential optimization 4 American Institute of Aeronautics and Astronautics

Vendor

Customers

System

Raw material

Supplier

Manufacturer

Transportation

Distributor

Vendor

Customers


Supply contracts, collaboration, information systems, decision support system

(b) Global optimization Figure 5: Optimization in supply chain management Hence in sequential management, each party optimizes its own profit with almost no regard to how its decisions affect the other members of the supply chain. On the other hand, a global optimization scheme tries to find what is best for the entire supply chain. Although the global optimization scheme is harder to put in place for supply chain management since all parties have to agree to share their inventory information, it results in better performance of the overall system than in the sequential optimization case. A total joint profit increase of 2.31% is demonstrated in the example of [23] and the result shows that the global optimization is better than the sequential optimization. B. Related Work Bearing in mind the motivations highlighted in the previous sections, it appears that a global optimization technique has to be utilized to yield better results in the design optimization of multi-robot systems. The scope of this literature review assumes that an a priori optimization scheme is used in conjunction with a metaheuristic approach. Indeed, a priori techniques are faster than a posteriori ones, which facilitates a quick exploration of the large multiarchitecture and multi-level design space. Moreover, metaheuristic optimization approaches present main advantages for the considered application: stochastic algorithms facilitate finding a global optimum, and no gradient or Hessian information is required. Indeed, considering the highly non-linear interactions between microscopic and macroscopic levels, it would be either too hard or too computationally expensive to compute such quantities. The techniques of multidisciplinary optimization are generally categorized based on their number of optimization levels: single-level, bi-level, and multi-level. The review proposed here below is based on [24]. Single-level techniques: single-level multidisciplinary optimization approaches only use one optimizer at the system level. The analysis may be distributed to the different partitioned subsystems but the optimization is kept centralized at the system-level. In the All-At-Once (AAO) method, all the variables of all disciplines are considered as optimization variables and the equations of each discipline are used as constraints. Thus, solutions are only consistent at convergence of the algorithm and there is no guarantee that at any iteration, the design will be feasible for all disciplines. If the algorithm experiences convergence issues and fails to reach a relative or absolute extremum, it will yield a design which is not only sub-optimal, but also not valid across the disciplines. However, this method has the advantage of not necessitating a complex analysis process. The Multi-Disciplinary Feasible (MDF) approach includes an analyzer which, at every iteration, solves the disciplinary equations using the design variables until the coupling variables converge. This ensures that the solution is consistent across all disciplines at each step of the optimization process. However this solution might be infeasible with respect to the constraints. Another limitation of this method is that it requires a complex system solver which coordinates all the subsystems in order to return a consistent solution to the optimization algorithm. This is not only hard to implement but implies a significant execution time. Finally, the Individual Disciplinary Feasible (IDF) method focuses on discipline feasibility at each iteration rather than multidisciplinary feasibility. The latter is only achieved at convergence thanks to constraints added for each of the coupling variables. The IDF method has improved convergence properties when compared to MDF but moves the complexity of the analysis to the optimization step which still requires consequent computational resources. Moreover, if the optimizer fails to converge, the produced solution might be inconsistent. In general, IDF preforms better than MDF when the coupling between the subsystems is more important. Multi-level techniques: as opposed to single-level techniques, the multi-level optimization methods use multiple optimizers at the subsystem level in addition to the traditional system-level optimizer. This type of approach is 5 American Institute of Aeronautics and Astronautics


preferred when the scale of the problem is too large for a single optimizer to handle [24]. In these techniques, analysis and design are distributed amongst the different subsystems. The Concurrent Sub-Space Optimization (CSSO) approach decouples the disciplines by letting each subspace carry out a separate optimization based uniquely on the design variables of that discipline. The coordination of all disciplines is handled by global sensitivity equations and a sensitivity analysis determining the non-local variables. This analysis can be carried out by equations or by response surface approximations in order to reduce the computation burden [25]. This method is useful for the industry as it is compatible with organizational features and the decoupling generally observed. However, the consistency of the final solution is generally mediocre due to difficulties in coordinating the subspaces optimization processes. This also makes it hard to guarantee a robust convergence of the whole optimization process [26], [27]. The Bi-Level Integrated System Synthesis (BLISS) approach divides the optimization problem into an upper level and a lower level. The subsystems of the lower level optimize on their design variables while the common variables are considered as constants. On the other hand, the upper level uses the common variables for optimization while the local variables of the lower level are regarded as constants [28]. A gradient-based approach is then used to reach an optimum. Hence, the computational cost of the BLISS method is quite important and limits its scalability. One approach by [29] to tackle this issue is to use response surface methodology. However, this is not applicable to multirobot systems due to the high non-linearity between microscopic variables and macroscopic responses. The Collaborative Optimization (CO) method is especially focused on early design phases when all disciplines are usually considered on the same level. The system-level optimizer establishes targets to be met by the partitioned subsystems and tries to minimize the system-level objective function. A set of equality constraints ensures that the design is driven towards consistency. At the subsystem level, the goal of the optimizers is to meet the targets and satisfy the constraints of the respective subsystems. With CO, each subsystems benefits from having its own optimizer, allowing for greater autonomy of the disciplines. However, the ability to handle coupling is limited since the interactions between the disciplines are handled by the main optimizer. The Analytical Target Cascading (ATC) technique uses a cascade of optimizers to propagate the design targets from the top level to the lower levels of the hierarchy. These lower levels are optimized to meet the targets and the resulting responses are forwarded to the higher levels in order to achieve consistency for the whole system. This iterative process is repeated until consistency is achieved globally for the targets and the responses. This approach is truly multilevel and considers analyzers and optimizers almost at the same level. On the other hand, it requires many executions and is computationally expensive. In the light of this review of existing work, no method seems to be able to directly handle the dynamic nature of the multi-architecture and multi-level design space. Moreover, no method considers a possible congregation of the different levels without altering the original algorithm. From this apparent gap, an adapted optimization method has to be designed.

III. Proposed Approach A. Algorithm Considering the above-mentioned challenges, metaheuristic approaches are first preferred to establish an optimization method. In particular, genetic algorithms enable dealing with discrete variables and continuous variables at the same time. However they have to be adapted since several levels are to be handled at once and the architectures might have different design variables. First, the optimization technique should contain at least two layers. Indeed, a single layer would not be able to handle the dynamic aspect of the design space imposed by the macroscopic level design choices. However, it is important to notice that the layer segregation need not be done between macroscopic and microscopic levels. This would limit the congregation of the two levels and most probably yield optimization results similar to those obtained by the sequential optimization techniques. Additionally, the outer layer of the algorithm needs to only take care of the dynamics aspects of the design space which need to be instantiated at the microscopic level. As such, only the types of architectures considered and their number in the swarm are susceptible to affect the size of the design space and the microscopic level implementations. Other design variables, even macroscopic (spacing between agents, communication protocol, etc.), may be included in the inner loop optimizer (Figure 6).


Legend Design space Chromosome, design vector

Outer loop chromosome

Optimum Optimizer

×𝟏

` Downloaded by GEORGIA INST OF TECHNOLOGY on January 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2017-0690

×𝟑

×𝟒

Inner loop

Outer loop

Dynamically adjust size Inner loop chromosome

×𝟏

×𝟒

×𝟑

Figure 6: Dynamic size allocation for inner loop chromosomes In order to accelerate the process, it is possible to retain the good architectures, and possibly designs, so that they are used as an initial inner loop population for the next iteration of the outer loop. Indeed, since the mission is fixed during the optimization process, it is probable that an architecture performing well on the mission in a given swarm might show good performance in another swarm configuration. While improvements from a swarm configuration to another are not guaranteed due to heterogeneity and highly unpredictable macroscopic level effects, this assumption is susceptible to speed up the overall optimization process if the initial swarm populations are initialized with good designs. To summarize the optimization scheme, an outer loop optimizes the types of architectures to include in the swarm as well as their number. This configuration is then fed to an inner loop which optimizes the multi-robot system based on the remaining design variables. Figure 6 and Figure 7 illustrate both optimization loops with an implementation based on a genetic algorithm. One chromosome (or design vector) or the outer loop population can hence be written 𝑋𝑜𝑢𝑡 = [𝑁𝐴1 , 𝑁𝐴2 , … , 𝑁𝐴𝑁 ] with 𝑁𝐴𝑖 the number of agents for architecture 𝐴𝑖 (or 𝑖), and 𝑁 the total number of architectures considered. Now that the number for each architecture has been fixed in the outer loop, a chromosome of the inner loop can be written as shown in Equation 1. Equation 1: Inner loop design vector 𝑋𝑚𝑎𝑐𝑟𝑜 , 𝑁𝐴

𝑉𝐴11 , 𝑉𝐴21 , … , 𝑉𝐴1 1 , 𝑁

𝑋𝑖𝑛 = 𝑉𝐴12 , 𝑉𝐴22 , … , 𝑉𝐴 𝐴2 , 2 ⋮, 𝑁𝐴

𝑁 1 2 [𝑉𝐴𝑁 , 𝑉𝐴𝑁 , … , 𝑉𝐴𝑁 ]


Where 𝑋𝑚𝑎𝑐𝑟𝑜 is the remainder of the macroscopic variables not handled in the outer loop, and each row represents one architecture. For instance, the third row contains contains 𝑁𝐴2 smaller design vectors, each one of them representing one vehicle of the architecture type 2.

Outer loop optimizer (Macroscopic)

Inner loop optimizer (Macroscopic, microscopic)

Initial architectures


Architectures selection

Crossover

Initial fitness evaluation

Selection

Crossover

Mutation

Initialize population

Convergence?

Replacement

Fitness evaluation

Mutation

Save optimal configurations

Convergence?

Replacement

Optimum

Figure 7: Proposed optimization scheme A key particularity lies in the fact that the outer loop handles only macroscopic variables but the inner loop deals with both microscopic and macroscopic design variables. The advantages of this method are multifold:  Microscopic and macroscopic levels optimizations are now combined, an approach different than the usual sequential optimization scheme.  It provides augmented capabilities since it enables multi-architecture and multi-level optimization.  The retention of optimal microscopic configurations accelerates the convergence process and the design space exploration.  There is no need to implement a new optimization algorithm for each architecture of robot, only two generic optimization algorithms are to be implemented. B. Experiment The experiment designed to test this hypothesis aims at assessing the quality of the optimization scheme. It first requires the implementation of both inner and outer loops, and a modular genetic algorithm. Then, the optimization algorithm is evaluated on the number of iterations, number of objective function calls, and accuracy in terms of optimal swarm performance. Again, additional metrics used to assess optimization algorithms can be used, such as the ones reviewed in [30]. To assess this precision, it is possible to compare the obtained optimum against a “ground truth” optimum obtained by randomized sampling, full factorial or by taking the best known solution [31]. The parameters to be varied during this experiment include the initial swarm population constitution, the typical genetic algorithm parameters and the retention scheme for the optimized microscopic architectures. The validation criteria of hypothesis are based on statistical hypothesis evaluation:  On average, the global optimization scheme is able to find a better solution than the sequential optimized solution, with respect to the main mission performance metric.  The optimization scheme is fast enough for the design space exploration of multi-robot systems.




The time increase between the proposed scheme and sequential optimization is less than an order of magnitude

Mapping Start

End

Base 𝑙𝑦 Downloaded by GEORGIA INST OF TECHNOLOGY on January 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2017-0690

𝑑0

𝒜1

𝒜2

𝒜3

𝑙𝑥 Figure 8: Example of mapping configuration for 3 agents The proposed algorithm is first validated with a dynamic analytical test function which optimum location and value depend on the constitution of the swarm and is theoretically known. This also enabled to test the performance of the algorithm for different levels of elitism. The results are presented in the next section. Additionally, the algorithm is applied to a canonical mission which is representative of typical group robotics operations: a mapping and exploration mission (see Figure 8). The mission is constrained by a cost for the whole system with a fixed cost for the swarm and for each vehicle, as well as variable costs depending on the number of agents and the technology (velocity).

IV. Results This section presents the validation and characterization campaign carried out on the proposed optimization algorithm by varying its main parameters. It also proposes an application to an actual group robotics mission to demonstrate the possible improvements achievable with the method. A. Characterization The algorithm is first validated through a unit test campaign which tests separately the inner loop, the outer loop, and the complete algorithm with their corresponding test function. The tests generate random number of architectures, random swarm compositions and run the optimizers to obtain optimal designs which are compared to the theoric values. These random tests are repeated for 1,000 times to make sure the validation is robust. Then, the characterization is performed to assess the effect of the elite retention scheme on the performance of the algorithm. For this experiment, the elitism rate is fixed at 50% so that half the population of the inner loop is initialized at an elite population for every iteration of the outer loop. The data recorded consists of the number of generations required for each inner loop chromosome to reach convergence, for each generation of the outer loop. These results are averaged and replicated one thousand times to ensure the robustness of the conclusions. The stopping criteria is fixed at 20 stalled generations. The results are presented on Figure 9.



Figure 9: Effect of elitism Figure 9 shows that when there is no elitism, the number of inner loop generations remains quite constant as the outer loop converges around 115 generations as the outer loop converges. Note that this number seems to slightly decrease between generations one and six, before increasing and settling again at 115: the genetic algorithm is still exploring the design space a lot and is trying out swarm configurations that converge a bit more easily. Moroever, variability of the results is more important in the first 10 generations of the outer loop by looking at the confidence interval. By activating elite retention for 50% of the population of the inner loops, the number of generations required for their convergence reduces drastically from the first generations of the outer loop. Indeed, the number of required inner generations decreases by 30 percent of its value from the second to the third generation and then goes on to settle at around 35 generations when the outer loop converges. This represents a speedup of around 70 percent for the algorithm. After a couple generations from the outer loop, enough swarm configurations have been considered so that the optimal microscopic configurations of the memory buffer are actually meaningful and untied to specific outer loop configurations. Giving efficient microscopic configurations as starting points for some of the chromosomes of the inner loops accelerates the convergence tremendously for these latter. Then, as the outer loop progresses and starts focusing on efficient swarm compositions, the memory buffer is refined little by little around optimal microscopic configurations which are particularly adapted to the already converging group composition. This helps in continuing to reduce progressively the number of required generations for the inner loops until this number comes very close to the convergence criterion of 20 stall generations. With the stopping criterion of the algorithms fixed at 20 stall generations, it is expected that the outer loop converges at values a bit above 20 generations. Few instances have the outer loop converge in more than 25 or 30 generations, which explains why the variability of the response increases after 25 generations. Indeed, there are less replications to average the results over. Moreover, it can be seen that with elitism, the lower bound of the confidence interval is at 22 generations for the inner loops (see lower portion of the 95 percent confidence interval). Although the average number of inner loop generations are clearly separated by around 80 generations with and without elitism, the confidence intervals exhibit some overlap. In particular, the lower portion of the 95 percent confidence interval without elitism is under the higher portion of the interval with 50 percent of elitism. This indicates that while elitism promises a faster convergence on average, some cases with elitism might still take equal or superior time to converge than without elitism. B. Application When applied to the canonical mission presented on Figure 8, the proposed algorithm is compared with the two other sequential optimization algorithms: a micro-macro optimizer, and a macro-micro optimizer. The first one 10 American Institute of Aeronautics and Astronautics

corresponds to what is mostly used by the community at the moment: optimization of agents individually before optimization of a group composition. The second one is more rarely used and consists in deriving an optimal number of agents for each architecture based on baseline vehicles, before optimizing the performance of these vehicles in the proposed group.


Figure 10 enables to understand how the microscopic level and macroscopic level variables affect the performance of the system, here represented by the mapping time 𝑇(𝑣, 𝑁) with 𝑣 the individual agent velocity, and 𝑁 the number of agents. As one can expect, increasing the number of agents in the swarm decreases the mapping time hence improving the mission performance. Enhancing the capabilities of each agent constituting the homogeneous swarm also results in a performance improvement. The constrained part of the design space where the cost exceeds the assigned budget is represented in red. As for the mission performance, design points vary between a mapping rate of 20 𝑚2 /𝑠 and 40 𝑚2 /𝑠 for the numbers given in this particular case.

Figure 10: Evolution of mapping rate with the design variables The whole set of results is presented in Table 1, with the relative difference with respect to the solution found by the simultaneous optimizer (in bold). Table 1: Results comparison Macro-micro Micro-macro Simultaneous 7 (+40%) 4 (-20%) 5 N 5.57 (-38%) 10 (11%) 9 v (m/s) 323 (+23%) 285 (+9%) 262 Mapping time (s) 70 (0%) 62 (-11%) 70 Cost (no units) Note that when deriving the optimal design with usual sequential optimization approaches, sub-optimal designs are obtained (in cyan and magenta). By being stuck at such local optima, a 9% to 23% performance difference is observed between the optimization methods. In this particular example, the optimization seems to favor individual performance over the numerality in the swarm. One may also notice that since the micro-macro optimizer was subjected to the group cost constraints (fixed at 70 cost units in the case of Figure 10), the first step of the optimization tried to derive an optimal performance for one individual agent with an assigned budget of 70 units. Hence, the optimizer naturally chose the maximum achievable velocity (10 m/s). This might not be representative of real-world applications where the design of single vehicles is 11 American Institute of Aeronautics and Astronautics


constrained by budgets that are much lower than for multi-robot solutions. To account for such particularities a set of experiments is designed with changing constraints, first by constraining the optimization of individual vehicles to adequate budget limitations, and then by optimizing a baseline swarm of 5 agent as opposed to optimizing a single vehicle. The set of experiments is the following:  Experiment 1: linear cost constraint at 70 units for both optimizers  Experiment 2: linear cost constraint at 70 units for the groups, 10 units for individual agent optimization  Experiment 3: quadratic cost constraint at 100 units for both optimizers  Experiment 4: quadratic cost constraint at 100 units for the groups, 10 units for individual agent optimization  Experiment 5: linear cost constraint at 70 units, micro-macro optimization based on baseline swarm  Experiment 6: quadratic cost constraint at 100 units, micro-macro optimization based on baseline swarm The results on the main performance metric are presented in Table 2. Table 2: Experimentation results Micro-macro Macro-micro optimizer optimizer Design Design Mapping Mapping Cost Cost [𝑵, 𝒗] time (s) [𝑵, 𝒗] time (s) [4, 10] 285 62 [7, 5.57] 323 70 1 [5, 7] 337 60 [7, 5.57] 323 70 2 [2, 10] 530 96 [5, 5.6] 422 100 3 [8, 3.44] 472 90 [5, 5.6] 422 100 4 [5, 9] 262 70 [7, 5.57] 323 70 5 [5, 5.6] 422 100 [5, 5.6] 422 100 6

Simultaneous optimizer Design Mapping [𝑵, 𝒗] time (s) [5, 9] 262 [5, 9] 262 [7, 4.3] 418 [6, 4.87] 417 [5, 9] 262 [6, 4.87] 417

Cost 70 70 100 100 70 100

This set of experiments focused on a canonical group robotics mission to prove the benefits of simultaneous optimization when applied to swarming systems. They show that the optimal designs derived by the sequential algorithms always exhibit worse performance than the ones derived by the simultaneous optimization algorithm, and for the same costs. This means that even though the sequential algorithms are able to get close to utilizing the full assigned budget, their proposed solutions are far from an optimal performance. On this particular mission example, the performance could be improved up to 29% (16% on average) by using the proposed optimizer. These results are most probably lower bounds with respect to the improvements that could be achieved in real-world situations. Indeed, in the proposed experiments, the two steps of the sequential algorithms are optimized with respect to identical mission requirements. However in reality, individual vehicles are optimized with respect to their own separate and uncorrelated requirements, before being put to work together on a group mission with its own and again different requirements.

V. Conclusion The growing market observed in robotics and micro UAVs in particular unleashes a multitude of unforeseen applications, hereby fostering the diverisifcation of the robotic fleet in terms of designs and capabilities. However, this broadening spectrum of architectures is not exploited. Additional limitations in endurance and cognitive capabilities can be addresses by multi-robotics: a field inspired by nature with groups of robots performing complex missions beyond the capabilities of their simple individual components. Heterogeneous groups of robots could capitalize on the diversity of the current robotics fleet. However, multi-robotics is a confined field at a preliminary stage and applications demonstrated by the military are avant-gardist and far from deployment in real world situations. This lack of democratization can be explained by the complex design process involved when dealing with a multiarchitecture and multi-level design space. This complexity causes the research community to use suboptimal designs obtained through a sequential optimization technique. This paper introduced a global optimization technique aimed at finding better optimal designs thanks to a bi-level algorithm mixing the microscopic and macroscopic levels of the robotics group instead of separating them. Results show that when applied to a canonical mission, up to thirty percent performance improvements can be achieved thanks to the simultaneous scheme. Moreover, an elite retention scheme helps in making the method not more computationally expensive than current optimization techniques.



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