Thomas Lagier

Piotr Dworakowski

Laurent Chédot

François Wallart

Bruno Lefebvre

Jose Maneiro

Juan Páez

Philippe Ladoux

Cyril Buttay
email: cyril.buttay@insa-lyon.frwww.ampere-lyon.fr

How Good are the Design Tools in Power Electronics?

Keywords: Virtual prototyping, Silicon Carbide (SiC), Transformer, Converter control, ZVS converters

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Introduction

Many research articles have been dedicated to the optimal design of converters [START_REF] Kolar | Performance trends and limitations of power electronic systems[END_REF][START_REF] Bortis | ηρ-Pareto Optimization and Comparative Evaluation of Inverter Concepts considered for the GOOGLE Little Box Challenge[END_REF][START_REF] Marxgut | Multiobjective optimization of ultraflat magnetic components with pcb-integrated core[END_REF]. It may appear that nowadays, building a new converter is only a matter of writing down some specifications, and then applying a design procedure. However, as we designed and built a new converter (i.e starting from a "blank page" instead of improving upon an existing design) in a limited time (12 months), we found that we had to rely heavily on assumptions and arbitrary decisions. This paper presents an overview of the design procedure which led to a working converter, with the objective of identifying the strengths and weaknesses of the process.

As a power converter is a complex system which requires multi-physics simulations (electrical, electromagnetic, thermal, mechanical. . . ), several modelling approaches have been proposed in the literature: [START_REF] Solomalala | Multi-domain simulation platform for virtual prototyping of integrated power systems[END_REF] describes a set of design tools mainly based on finite-elements (FE) modelling (associated with a circuit simulator), to take advantage of its accuracy. On the contrary, [START_REF] Biela | Towards Virtual Prototyping and Comprehensive Multi-Objective Optimisation in Power Electronics[END_REF] introduces a method to generate analytical models, which are much faster to solve than finite-elements. In some cases, a single, accurate FE simulation can be used to identify the parameters of fast compact models, allowing faster subsequent runs [START_REF] Evans | Automated fast extraction of compact thermal models for power electronic modules[END_REF].

System simulation of a power converter can be performed using dedicated system modelling language such as VHDL-AMS [START_REF] Pêcheux | Vhdl-ams and verilog-ams as alternative hardware description languages for efficient modeling of multidiscipline systems[END_REF] or Modellica, or using circuit simulators (Saber, SPICE, Simplorer, PSIM. . . ), with non-electrical quantities (such as temperature) calculated using equivalent circuit models. In any case, as the system becomes more complex, simulation time increases drastically. This is particularly true for time domain simulations of power electronic converters, which require a subnanosecond resolution (for switching transients) over milliseconds periods (to capture the fundamental frequency of signals), and ideally even minutes (to reach thermal equilibrium).

In this paper, we describe the design of a complete converter from the bottom up, including custom power electronic modules rather than off-the-shelf components. This is mainly justified by the lack of suitable components at the time, but also by the objective of evaluating the performance of the commercial design tools. An isolated dc-dc converter, the central part of a solid-state transformer (SST, [START_REF] Huber | Solid-state transformers -on the origins and evolution of key concepts[END_REF]), was chosen as the application case. Such converters have applications in traction [START_REF] Zhao | Power Electronic Traction Transformer-Medium Voltage Prototype[END_REF], wind turbines [START_REF] Lagier | A comparison of insulated DC-DC converters for HVDC off-shore wind farms[END_REF], etc. Here, we focused on a 100 kW, 1200 V/600 V bidirectional, isolated dc-dc converter.

Silicon IGBTs are commonly used as the power switches in this power/voltage range [START_REF] Bernet | Recent developments of high power converters for industry and traction applications[END_REF]. As recent improvements in SiC technology [START_REF] Millán | A Survey of Wide Bandgap Power Semiconductor Devices[END_REF] have resulted in the commercial availability of 1200 V and 1700 V MOSFETs and Schottky Barrier Diodes (SBDs), we focused on SiC power switches in this paper. Indeed, replacing Si IGBTs and diodes with SiC MOSFETs and SBDs in a comparable 100 kW dc-dc converter was shown in [START_REF] Akagi | Power-loss breakdown of a 750v 100-kw 20-khz bidirectional isolated dc -dc converter using sic-mosfet/sbd dual modules[END_REF] to result in a clear improvement in efficiency: 96.9 % at 60 kW max. output power for Si vs 97.9 % at 100 kW max. output power for SiC, while at the same time allowing a much higher switching frequency (20 kHz for SiC and 4 kHz for Si).

The next section briefly introduces the specifications of the converter, and the preliminary design stage which resulted in the selection of the converter topology (as well as some other general design parameters). Section 3 then details the design process all the way up to the physical implementation. The manufacturing and test of the converter are described in Section 4. The design process and its outcome are then discussed in section 5 and a conclusion is given in section 6

Pre-design

The specifications of the converter described in this paper are given in Fig. 1a. Some of them are purely functional (input/output voltage, nominal power, target efficiency. . . ), and some are arbitrary design choices (switches technology, switching frequency). These specifications are then used to assess the many dc-dc converter topologies available [START_REF] Lagier | A comparison of insulated DC-DC converters for HVDC off-shore wind farms[END_REF] (Dual Active Bridge -DAB -with or without resonance, resonant converter. . . ). This process of evaluating the topologies is conducted with theoretical analysis and circuit simulations (using ideal components). It is described in more details in [START_REF] Lagier | A comparison of insulated DC-DC converters for HVDC off-shore wind farms[END_REF], and is beyond the scope of this paper. The selection of the most suitable topology is based on the expected efficiency (transformer and power electronics), technical complexity and cost. Regarding the transformer, since its technology and geometry are not yet selected at this stage, the level of each current harmonic is used as a criterion for the losses estimation [START_REF] Perreira | Conception de transformateurs moyennes fréquences : application aux convertisseurs dc-dc haute tension et forte puissance[END_REF]. For the power electronic switches, conduction and switching losses are estimated based on the datasheets of the SiC dies [START_REF] Lagier | Potential of silicon carbide MOSFETs in the DC/DC converters for future HVDC offshore wind farms[END_REF] (on-state resistance and switching energy). The list of possible SiC dies is limited to only two references from Wolfspeed (one 1700 V and one 1200 V-rated MOSFET, along with the corresponding SBDs), because they are the only devices readily available at the time of the pre-design (2014). At this stage of the design, the number of dies to be used per switch position is simply based on their recommended RMS current (as quoted in their datasheet), without any thermal consideration.

As a result of this pre-design stage, a non-resonant DAB topology [START_REF] Dedoncker | Power conversion apparatus for dc/dc conversion using dual active bridges[END_REF] is selected (Fig. 1b). The leakage inductance L of the Medium Frequency Transformer (MFT) is a key element for the performance of the DAB converter, as it dictates the phase shift between the primary and secondary inverters: if too large, this phase shift would limit the transmitted power; if too low, it would make the control too complex [START_REF] Akagi | Power-loss breakdown of a 750v 100-kw 20-khz bidirectional isolated dc -dc converter using sic-mosfet/sbd dual modules[END_REF]. Considering a maximum phase shift range of 10 -25°(so the power factor is higher than 0.9 in nominal conditions) yields a leakage inductance in the 10 -25 µH range. Without more information (in particular regarding the control system timing constraints), it is decided to aim for the smallest value; an additional inductor would be connected in series to reach a higher leakage inductance value if needed. This allows to retain some freedom in the design process. Tab. I presents the main results of the pre-design stage.

Detailed Design and Implementation

Once the topology and is main parameters have been selected, the detailed design of the converter can begin. As presented in Fig. 2, the design is broken into three main tasks (transformer, power electronics and control), each managed by a different team using the specifications in Fig. 1a and Tab I.

Transformer design

The details of the transformer design are given in [START_REF] Pereira | Comparison Between Numerical and Analytical Methods of AC Resistance Evaluation for Medium Frequency Transformers: Validation on a Prototype[END_REF]. Because of the short development time, the magnetic material is chosen a priori: among the possible candidates [START_REF] Hilzinger | Magnetic Materials[END_REF], ferrite is selected because of its availability, despite its low maximum induction level (≈ 0.2 T). Amorphous materials are discarded because of their strong magnetostrictive coefficients (risk of acoustic noise); nanocrystalline cores are only available in a limited range of sizes, with long lead times. In particular, 3C90 ferrite material is used because of its suitable performances.

Analytical design

The analytical design allows to quickly compare many configurations, especially regarding copper and core losses. The copper losses in the winding are estimated using Dowell's equations [START_REF] Dowell | Effects of eddy currents in transformer windings[END_REF]. These equations consider the current harmonics and estimate the AC resistance of the winding taking into account proximity and skin effects in the conductors [START_REF] Pereira | Comparison Between Numerical and Analytical Methods of AC Resistance Evaluation for Medium Frequency Transformers: Validation on a Prototype[END_REF]. Foil winding is preferred because of its lower cost, lower AC resistance compared to wire, and because it is well suited to the small number of turns being considered here (2 windings with 14 turns per column). The leakage inductance is calculated using [START_REF] Cougo | Integration of leakage inductance in tape wound core transformers for dual active bridge converters[END_REF] at 3.38 µH, a value compatible with the specifications (note that this value does not include the inductance of the connecting cables).

For the core losses, the improved General Steinmetz Equations (iGSE) are used [START_REF] Venkatachalam | Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only steinmetz parameters[END_REF]. IGSE requires detailed data regarding the Ferrite material. These data are extracted from the material's datasheet [START_REF] Ferroxcube | Datasheet -3C90 Material Specifications[END_REF], as we do not have experimental characterization data at this stage.

At the end of the analytical design, it is decided to have the transformer made, because of the long lead time (more than 8 weeks for manufacturing only). The main parameters (number of turns and dimensions of the foils, magnetic material reference) are given to the transformer manufacturer. The mechanical structure (required to hold together the various ferrite parts, the winding and the transformer terminals) is directly proposed by the manufacturer (it is reused from one of their former projects). Many parameters (in particular the location of the terminals of the transformer) are left to the discretion of the manufacturer.

Finite element simulations

While the transformer is being built, Finite Elements (FE) simulations are run. They allow to get a more accurate estimation of the losses (especially for the copper losses). They also allow to perform thermal simulations (temperature distribution over the transformer), electrostatic simulations (to calculate parasitic capacitance and check dielectric strength). All calculations are performed using Ansys software: Maxwell for the magnetic and electrostatic simulations, Ansys Mechanical for heat conduction, ICEpak for simulations which mix heat conduction and convection.

The magnetic simulations confirm the leakage inductance value which was calculated analytically (3.38 µH vs. 3.39 µH for FE). They are performed in 3D to represent effects such as current crowding around the power connections. Indeed, these connections are found to represent a large share of the transformer's ac resistance (up to 60 % [START_REF] Pereira | Comparison Between Numerical and Analytical Methods of AC Resistance Evaluation for Medium Frequency Transformers: Validation on a Prototype[END_REF]). 3D FE offers a much more accurate model of the current distribution in the copper conductors, which unfortunately results in a much greater ac resistance compared to the analytical model (Fig. 6a). 3D FE predicts that the ac resistance at 20 kHz is more than twice the dc resistance. At the time of these findings, the transformer is being manufactured, so it is decided to manage the increased losses through additional cooling fans, relying on ICEpak simulations for airflow calculations.

Power Electronics design

This task refers to the design and implementation of the two inverters of the DAB topology. It also addresses the mechanical structure of the whole converter.

Detailed power electronics circuit

At the time the design is initiated, 1700 V-SiC MOSFETs and Diodes are available on the market as bare dies or discrete, but not in module package [START_REF] Cree | Cree introduces industry's first 1.7-kv, all-sic power module[END_REF]. Given the power rating of the converter, a discrete-based circuit is not considered desirable, so we decide to develop a custom power module. Some simple thermal calculations are performed. First, the switching losses are estimated using the turn-off energy data given in the datasheet of the devices (as we assume Zero-Voltage Switching -ZVS -, turn-on losses are considered negligible). For the conduction losses, the evolution of R DSon with the temperature is non-negligible (it doubles between 25 and 150 °C for 1700 V MOSFETs), so it is considered in our calculations, assuming an (arbitrary) baseplate temperature of 70 °C and a module thermal resistance identical to that of the EconoDUAL package from Infineon (which is chosen as the outline for our module). These data confirm that for power modules with 5 MOSFET dies per switch position, the junction temperature remains below the maximum allowed value. On the secondary (600 V side), 1200 V MOSFETs (CPM2-1200-0025B, 25 mΩ R DSon ) and diodes (CPW5-1200-Z050B) are used instead of 1700 V devices. This way, the primary and secondary inverters remain geometrically equivalent, and their losses are comparable [START_REF] Lagier | A 100 kW 1.2 kV 20 kHz, DC-DC converter prototype based on the Dual Active Bridge topology[END_REF]. An existing gate driver circuit is used here. It was designed for IGBT modules, and is based on discrete components.

Finally, the last elements to be selected are the input/output capacitors. In the absence of any specification regarding input/output ripple voltage, we set it to an arbitrary value of 5 %, and we consider that the input and output capacitors manage the full ac currents of the converter. In the worst case (maximum voltage -1200 V -on the primary and the minimum conversion ratio -0.45) the current values is expected to reach 211 A. Because of the high switching frequency, the required capacitance remains relatively low (22 µF). The closest commercially available capacitors are found to be 40 µF capacitors which can sustain 57 A (AVX FFVS6U0406K). 4 such capacitors are then connected in parallel to provide sufficient current capability, and much higher capacitance value than initially required. For modularity and cost aspects, the same capacitors are used for the primary and secondary sides.

At this stage, by combining the estimation of losses from the power semiconductor devices and from the transformer (as calculated using the analytical model), we reach a first milestone where the efficiency of the converter ("first estimate" line, in Fig. 8b) can be compared to the target efficiency in the specifications (98 %, see Fig. 1a)

Physical implementation of the power electronics

At this point, the detailed circuit diagram of the converter has been drawn, and the exact references of all components are known. The most specific components (SiC MOSFETs and diodes, capacitors) are ordered, as they have relatively long lead times (more than 8 weeks).

As mentioned above, a custom power module is designed with the "EconoDUAL" outline (Fig. 3a). Although the SiC MOSFETs have an internal body diode, we are not sure we can use it reliably. As a consequence, we decide to use antiparallel SiC SBDs. To satisfy the current ratings of the diodes, 5 chips are used per switch position (i.e our module contains as many diodes as MOSFETs). This is probably far from optimal, but reflects the lacks of reliability data at the time the decision is taken (end 2014). Regarding the internal routing of the power module, various configurations are compared with regard to their parasitic inductances. The models are first designed with a 3D mechanical CAD software (CREO Parametric, Fig. 3b), then the parasitic inductances are calculated using Ansys Q3D Extractor. The gate and source connections of each die are routed independently, to keep some flexibility in case issues are encountered in the parallel operation of the MOSFETs: all transistors can be connected in parallel, or they can be driven through individual gate resistors, depending on the layout of the gate driver board. In our case, 5 Ω resistors are connected between the output stage of the gate driver and each gate terminal of the power module .

Mechanical design of converter

Based on the thermal calculations performed previously, plus some safety margins (we consider each module dissipates 500 W, has a baseplate temperature of 70 °C and shall operate at a maximum junction temperature of 120 °C so as not to degrade its R DSon too much), we select an aluminium heatsink with embedded copper heatpipes for better heat spreading (Mersen, 205×300×120 mm 3 ), with three 92 mm axial fans (Papst). This heatsink is used as the support to mount all the components of each inverter. A 3D implementation of the inverters is done using mechanical CAD software, with a custom laminated busbar used to connect the power modules. Inverters and transformer are then secured to an aluminium frame (Fig. 4a).

Circuit models

Electromagnetic simulations are performed using the Ansys Electromagnetic suite [START_REF]ANSYS Electronics Desktop[END_REF]: parasitic inductances, capacitances and resistances are calculated (using Q3D Extractor) from the 3D geometry of the inverters. An equivalent circuit model is automatically generated, and included in a time-domain simulation (Simplorer). As no models are available for the MOSFETs, we use the simplified macro model proposed in [START_REF] Fabre | Characterization and implementation of dual-sic mosfet modules for future use in traction converters[END_REF]. However, even this simplified model is found to be too complex for the converter simulation to run in a reasonable time (we have 20 dies per power module, and 4 power modules total). Finally, we end up using ideal switch and diode models with capacitors in parallel. These capacitors are important as, together with the inductors in the circuit, they set the switching transients. Fig. 4b presents a simulation result during turn-off.

From the simulated voltage and current waveforms (Fig. 4b), the global parasitic inductance of the commutation loop (capacitor, busbar and power module.) is evaluated at 68 nH [START_REF] Wang | Characterization and experimental assessment of the effects of parasitic elements on the mosfet switching performance[END_REF]. Finally, more detailed electromagnetic simulations are performed using Ansys Maxwell to analyze the current density distribution in the interconnects (wirebonds, busbar. . . ). However, these cannot be conducted using the complex current waveforms produced by the circuit simulation, and must be limited to dc. This limits their relevance (skin and proximity effects are not taken into account in the resulting current distribution As no obvious issue is identified through the simulation, the design stage is considered finished for the power electronics, and manufacturing can start. Conf. 1 is used for high level simulations. Conf. 2 simulates the operation of the FPGA, so it is complex and only short periods of time can be simulated (100 ms max.). Conf. 3 runs on the RCP platform connected to converter hardware in real-time (the indicated times correspond to the recurrence period of each task).

Control design

The control hardware solutions can be divided into three families: completely custom boards, development boards, and rapid control prototyping (RCP) platforms. RCPs provide very advanced code generation and debugging capabilities, but come at a cost of several tens of thousands of euros (for the hardware and the software licences). This is considered the most suitable solution in our case, because of the short development time. Many RCPs are available on the market. We select a system from Speedgoat [START_REF] Dworakowski | Supergrid institute -an efficient and compact power converter to enable the supergrids of the future[END_REF] because of its modularity, good integration with the Matlab/Simulink coding platform, and cost. This system includes a x86 Intel processor for all the calculation-intensive, low frequency tasks and a Spartan 6 Xilinx FPGA for the high frequency tasks such as the generation of the firing signals, the management of the error signals from the converter, and the acquisition of the measurements.

The control system architecture is depicted in Fig. 5. It is designed using Simulink, with three configurations. The first two are used for simulation (the code is run on a workstation, completely off-line), while the third is the Power Hardware-In-the-Loop (PHIL) configuration, where the code is implemented in the real-time hardware and actually controls the converter. This ensures that the same code is used for the simulation and the actual control [START_REF] Lagier | A 100 kW 1.2 kV 20 kHz, DC-DC converter prototype based on the Dual Active Bridge topology[END_REF].

In parallel, a "small scale mock-up" converter is assembled. It simply consists in a low voltage (50 V), low power (100 W) DAB converter built on a "breadboard"-type prototyping board. This smallscale mock-up is connected to the Speedgoat system. The mock-up has a low fidelity to the actual converter (in particular, its transformer, being an off-the-shelf item, has very different leakage and magnetizing inductances), so it cannot be used to fully validate the control algorithm. It is useful, however, to detect issues associated with the real-time operation. Indeed, a very detailed simulation of the control system including the internal operation of the FPGA (configuration 2 in Fig. 5) is possible but very slow. Practically, it is not possible to simulate more than a 100 ms of operation. Therefore, any issue that would occur over longer periods (such as a timer overflow, for example), would go unnoticed in simulation. This is where the PHIL operation, using the actual control hardware and the mock-up converter becomes useful: it runs in real-time, over much longer periods (minutes or hours), and in the worst case, a dramatic failure only involves a moderate amount of energy (compared to the full scale DAB).

Manufacturing and experimental Validation

This section describes the "Manufacturing" part of Fig. 2 (the dashed box at the bottom), as well as the evaluation tests performed on individual elements, and finally on the converter.

Transformer testing

Once the transformer built, it is characterized using an inverter available at Laboratoire Ampère: short-circuit test (which allows the measurements of the copper losses in the winding) and open circuit test (for the measurement of the core losses). All measurements are performed at the nominal frequency (20 kHz). The resistance increase with frequency is measured using an impedance analyzer (Keysight E4990A).

Fig. 6a confirms that the simplified equations [START_REF] Dowell | Effects of eddy currents in transformer windings[END_REF][START_REF] Venkatachalam | Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only steinmetz parameters[END_REF] tend to strongly underestimate the copper losses, especially as the frequency increases. It also confirms that the 3DFE modelling offers as satisfying level of precision. An analysis of these differences is available in [START_REF] Pereira | Comparison Between Numerical and Analytical Methods of AC Resistance Evaluation for Medium Frequency Transformers: Validation on a Prototype[END_REF].

In order to check the interactions between the transformer and the other components (switches, busbar. . . ), an equivalent model of the transformer is required. The circuit diagram of this equivalent model is depicted in Fig. 6b, after identification with the measurements.

Power electronics dynamic testing

The power modules are manufactured first (DeepConcept, Tarbes, France). They are characterized using a double pulse test circuit (Fig. 7a, principle described in [START_REF] Fabre | Characterization and implementation of dual-sic mosfet modules for future use in traction converters[END_REF]), which allows the measurements of the hard-switching losses. An example of measured waveforms is given in Fig. 7b. Thanks to such measurements, we can adjust the switching energy losses (turn-off losses only, as turn-on losses occur under zero voltage switching conditions) used in our estimation. As it can be seen in Fig. 8a, the use of the coefficients calculated from the datasheet of the dies causes an underestimation of the turn-off energy loss of up to 15-20 %. Together with the experimental characterization of the transformer, these results are used to produce the "second estimate" line in Fig. 8b. The double-pulse test circuit is also used to adjust the output impedance of the gate driver (a small capacitor is added to increase the gate-to-source capacitance of the MOSFETs). In parallel, the complete converter is assembled according to the CAD plans. The result is visible in Fig. 9a.

Full converter testing

Once the test bench has been set-up (a time-consuming task for a converter this size), the actual testing of the power converter is relatively straightforward: the control system is connected to the converter, and power is gradually increased. A waveform measured at the output of the primary inverter during operation is shown in Fig. 9b. Efficiency measurements are presented in Fig. 8b. The final efficiency is slightly below that predicted by the design, even after the second round of estimation. Note, however, that this efficiency measurement is performed with the voltage/current sensors used for the control of the converter, and therefore has a limited accuracy.

While the converter works, with an efficiency reaching almost 98% (the objective in Fig. 1a) over a large part of the power range, some issues must be highlighted (and will be discussed in the next section):

• EMC issues are encountered, causing a malfunction of the control system (the internal FPGA is reset by EMI above 50 kW). This is solved by re-routing the cables of the test bench (many were overlapping), by adding ferrite beads on auxiliary power supplies and driving signal cables, and by providing earth connection of all metal screen on one point only. The control software is also made more robust to EMI transients by filtering outliers from the I/Os.

• At 1200 V, two power modules are found to fail, because of arcing between the internal busbars (the black and red metal sheets in Fig. 3b). This is attributed to an insufficient clearance between the parts, and possibly to a bubble in the silicone encapsulant. The other modules do not seem to be affected.

A 100 kW converter was entirely designed and built over the course of one year. Its final performance, although not fully in line with the specifications (the efficiency, in particular, is slightly below 98%), are remarkably close. The converter can operate continuously, indicating that the thermal management system is able to absorb the higher losses level.

As explained in the introduction, the objective of this article is to reflect upon the design process. As the converter was essentially "built from scratch", the design was not based on incremental improvements of an existing converter. This means that in many cases, questions were asked in absolute terms (will it work?) rather than in relative terms (will it work better?). This is an essential difference, in particular for computer simulation, as in the "absolute" case, no reference data are available. Therefore, the accuracy of the modelling is unknown at the design stage.

When looking back on the design process, summarized in Fig. 2, one striking fact emerges: despite the effort which was put in the modelling and simulation, almost none of the simulation results were actually fed back in the process: in Fig. 2, simulation results are outputs (red boxes), mere "milestones" used to check that the converter meets its specifications. In no occasion did the simulation results led to changes in the design, which was finally mainly based on simple analytical approaches and on the datasheets of the components. However, this observation masks fundamental differences between models, which we will discuss now.

For the transformer design, the numerical modeling is found to be remarkably accurate, capable to identify issues such as an increase in ac resistance caused by the geometry of the terminals. 3D FE simulations are not simply a refinement of the analytical calculations: they offer a very different picture, which may have resulted in changes in the design of the transformer, had we not sent it to manufacturing beforehand. The main issue here is that simulations take time: one week for the electromagnetic calculations, not including the preparation of the model, the analysis of the results, and the subsequent thermal simulations. It is, however, a worthwhile effort, and the simulation time could be reduced by using more powerful computing hardware.

Regarding the design of the control system, the RCP provides a very efficient way to (almost) seamlessly transfer from a block-diagram description to real-time code. An intermediary test using a smallscale mock-up remains valuable, as it helps capturing issues which might not be detected in simulation (typically the overflow of a register which occurs over a long period). This is especially important, as the debugging capabilities are limited for the FPGA circuit. All designs we currently work on include such small-scale mock-up stage, with three major improvements: the mock-ups are now made on PCBs (as opposed to breadboard, which was found not to be reliable enough), they now use exactly the same interface as the final converter, and are designed to have the same dynamics, so the transfer to the full-scale converter is even more straightforward.

The power electronics simulations present a completely different picture: over the course of the project, we have never been able to perform a complete circuit simulation of the converter (i.e including accurate models of the switches, their gate drive circuits, as well as the circuit parasitics and the transformer model). This level of complexity is too demanding for the circuit simulator. Ideal switches (on/off behaviour) with a linear capacitor in parallel were used as a model for the SiC MOSFETs instead. A second (and related issue) is that SiC MOSFETs models are not available for all simulation platforms (Wolfspeed provides SPICE models, and we use Ansys Simplorer). This required the adjustment of a compatible model using a dedicated tool (in Simplorer) and the datasheet of the MOSFET. Generating a model this way is convenient, but does not give any information about its level of accuracy, or about its domain of validity.

Even if we had been able to simulate the power circuit of the converter, this would probably have been insufficient to predict the EMC issues we encountered during the tests (spurious resets of the control systems above a certain power level). This would have required to take into consideration the entire converter, including models of the gate drivers, the auxiliary power supplies, the low power wiring, the grounding configuration, etc. Not only would such approach require tremendous computing power, but it would also require models for these elements. Another approach could be to perform model reduction, to adapt the models to the analyses (i.e. to use a different model to predict EMI or power losses). This, however, requires considerable expertise, time, and data. Some of these data being not available, it requires experimental characterization, and therefore advance purchase of some elements [START_REF] Foissac | System simulation for emc network analysis[END_REF]. Finally, for subsequent designs, we use a more protective approach regarding EMC, for example using fiber optics as a link between the control system and the converter (for driving signals as well as for measurements). While it provides an ideal barrier to EMI, it comes at a certain cost.

As for the other issue encountered (arcing inside one power module), preventing it through simulation, if possible at all, would have required a totally different modelling. It is not clear yet if the issue was related to thermo-mechanical effects (deformation of the internal leadframe as a consequence of heating), manufacturing issues (inaccuracy in the forming of the metal parts, in their alignment, bubble in the silicone encapsulant), design issue (insufficient clearance between the parts), or (more probably) a combination of the three. The multiplicity of the possible causes makes modelling quite complicated, and one can consider that hardware prototyping is unavoidable here. This is, however, a specific case: custom modules were required as no 1700 V SiC power modules were available at the time. With "off-the-shelf" modules, any such design issue would have been addressed beforehand by the manufacturer.

Conclusion

A 100 kW converter was designed "from scratch", over a relatively short time-frame (1 year from beginning of design to test), with the objective of assessing the design methodology. The design process was separated between three main elements: transformer, power electronics, and control system.

Thanks to RCP techniques, which rely on seamless transition from computer model to firmware generation, the control was developed independently from the transformer and the converter. This is not so true for hardware elements such as the transformer and the power electronic circuits, for which computer simulation was found to be less straightforward.

For the transformer, 3D-FE simulation was found to be accurate, but requires a lot of effort and time to set the model up, run the simulation, and analyze the results. However, it is worth the effort, and new transformer designs we have made since have relied heavily on 3D-FE simulation.

For the power electronics, on the contrary, no complete simulation could be run, because of the complexity of the model. Simplifications were required which, together with un-verified models for the semiconductor devices, produced simulation results of unknown validity. As a consequence, the power electronics design was mainly based on datasheet information, safety margin, or designer expertise.

More importantly, the lack of complete circuit simulation did not allow us to identify EMC issues, which required fixing the converter on the test bench. Here, the solutions rely more on EMI-safe approaches (such as fiber optics data transfer) than on computer simulation.

This does not mean that simulation is useless. For incremental changes (e.g. improving a prototype before moving to production), a reference is available to check the validity of the models. In this case, simulation gives an invaluable analysis tool. Simulation is also quite useful for optimization or to evaluate the consequences of changing a design parameter. But in the case of a relatively large converter such as the one presented here, electrical simulation does not currently allow the generation of a complete "virtual prototype" which could replace a hardware prototype.

Figure 1 :

 1 Figure 1: (a): List of specifications and arbitrary design choices used as an input to the design process. (b): Circuit diagram of the chosen converter topology: a Dual Active Bridge (DAB). The transformer has an input/output ratio of 2. The input and output capacitors are not depicted here.

Figure 2 :

 2 Figure 2: Flowchart of the detailed design procedure (the corresponding timeline is discussed in the article).

Figure 3 :

 3 Figure 3: (a) Photograph of the custom power module with the output stage of the gate driver mounted. Length/width of the power module: 152/62 mm. (b) CAD view of the internal interconnects of the power module (top) and layout of the ceramic substrates (where the MOSFETs are located in close proximity with the diode of the opposite switch).

Figure 4 :

 4 Figure 4: (a): CAD view of the complete converter (without cables). (b): Time-domain simulation result for the turn-off (V = 1000 V ; I = 300 A ; di/dt = -10 kA/us), including the modelling of the interconnects (especially the inductive behaviour), in a double-pulse configuration (hard switching).

Figure 5 :

 5 Figure5: Architecture of the control. Conf. 1 is used for high level simulations. Conf. 2 simulates the operation of the FPGA, so it is complex and only short periods of time can be simulated (100 ms max.). Conf. 3 runs on the RCP platform connected to converter hardware in real-time (the indicated times correspond to the recurrence period of each task).

Figure 6 :Figure 7 :

 67 Figure 6: (a):Evolution of the ac resistance (relative to the dc resistance), as forecast by various analytical models, by FEM simulation, and as measured. (b): Equivalent model of the medium frequency transformer identified from measurements.

Figure 8 :Figure 9 :

 89 Figure 8: (a): Comparison between the switching losses estimated from the manufacturer's datasheet and from the double-pulse test, for 5 1700 V-SiC MOSFETs and a dc voltage of 1200 V. (b): comparison between the global efficiency as forecast initially, after refinement of the design and the models, and eventually measured.

Table I :

 I Outcome of the pre-design stage.

	Parameter	Value	Comment
	General
	Topology Minimum power transfer	Non-resonant DAB 1 kW	(arbitrary) 1 % of the nominal power
	Transformer
	Apparent power Primary nominal voltage range Maximal RMS current on primary Operating frequency range Turns ratio (N2/N1) Leakage Inductance	180 kVA 900-1200 V 150 A 17-23 kHz 0.5 < 10 µH	specifications from Tab. 1a calculated from the ideal waveforms arbitrary: ± 15 % frequency variation allowed for control specifications from Tab. 1a final objective 10 -25 µH, to be adjusted using a series inductor if needed
	SiC switches
	# of MOSFETs and diodes per switch position	5 (1700 V rating)	cf. datasheet for chips CPM2-1700-0040B and CPW5-1700-Z050B

Aknowledgements

This work was supported by a grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" Program (ANE-ITE-002-01).


Mahmoud Hamandi

Kapil Sawant

Marco Tognon

Antonio Franchi

Omni-Plus-Seven (O 7 + ): An Omnidirectional Aerial Prototype with a Minimal Number of Uni-directional Thrusters

The aim of this paper is to present the design of a novel omnidirectional Unmanned Aerial Vehicle (UAV) with seven uni-directional thrusters, called O 7

+ . The paper formally defines the O + design for a generic number of propellers and presents its necessary conditions; then it illustrates a method to optimize the placement and orientation of the platform's propellers to achieve a balanced O + design. The paper then details the choice of the parameters of the O 7 + UAV, and highlights the required mechanical and electrical components. The resultant platform is tested in simulation, before being implemented as a prototype. The prototype is firstly static-bench tested to match its nominal and physical models, followed by hovering tests in multiple orientations. The presented prototype shows the ability to fly horizontally, upside down and at a tilted angle.

I. INTRODUCTION

Unmanned Aerial Vehicles (UAVs) have been widely studied in the literature, with many applications that attempted to push their ability to the limits, such as aerial physical interaction [START_REF] Ollero | The AEROARMS project: Aerial robots with advanced manipulation capabilities for inspection and maintenance[END_REF], [START_REF] Ollero | Aerial robotics cooperative assembly system (arcas): First results[END_REF], surveying, photography, etc.

In its generic form, a UAV hovering (or flying) in the threedimensional world is a rigid body able to apply forces and moments in (at most) a six-dimensional space. The applied forces and moments usually have the task to counteract external forces such as gravity, wind, etc, in addition to internal and external gyroscopic effects. The applied control wrench is generally allocated with a group of motors placed around the Center of Mass (CoM), with fixed or actuated orientations. Their placement and aerodynamic properties define the shape of the platform's feasible wrench space.

The most common UAV in the literature is the quadrotor [START_REF] Mahony | Multirotor Aerial Vehicles: Modeling, Estimation, and Control of Quadrotor[END_REF], [START_REF] Faessler | Differential flatness of quadrotor dynamics subject to rotor drag for accurate tracking of high-speed trajectories[END_REF], which can apply uni-directional forces and threedimensional moments. Given its wrench space, such a vehicle has to modify its orientation to be able to move in the threedimensional world.

To decouple the platform's forces from its orientation, multiple designs in the literature presented platforms that exploit the full six-dimensional wrench space. [START_REF] Ryll | 6D interaction control with aerial robots: The flying end-effector paradigm[END_REF] and [START_REF] Romero | A new UAV configuration having eight rotors: dynamical model and real-time control[END_REF] use tilted uni-directional thrusters to apply 3-dimensional LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France, mahmoud.hamandi@laas.fr, kapil.sawant@laas.fr, antonio.franchi@laas.fr, Autonomous Systems Lab, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zürich, Switzerland, mtognon@ethz.ch Robotics and Mechatronics lab, Faculty of Electrical Engineering, Mathematics & Computer Science, University of Twente, Enschede, The Netherlands a.franchi@utwente.nl

This work was partially funded by the European Union's Horizon 2020 research and innovation programme under grant agreement ID: 871479 AERIAL-CORE Fig. 1: The built prototype showing the motor assembly, the electronics assembly and the motion capture tracking markers.

forces independently from the applied moment. However, these platforms cannot apply forces in any direction, and are usually limited to force directions in the upper hemi-sphere.

Conversely, [START_REF] Ryll | Modeling and control of FAST-Hex: a fully-actuated by synchronized-tilting hexarotor[END_REF]- [START_REF] Long | Omnicopter: A Novel Overactuated Micro Aerial Vehicle[END_REF] actively tilt the propellers to achieve omnidirectional flight: [START_REF] Ryll | Modeling and control of FAST-Hex: a fully-actuated by synchronized-tilting hexarotor[END_REF] synchronizes the tilt angle of the propellers of a hexarotor, [START_REF] Ryll | A novel overactuated quadrotor unmanned aerial vehicle: modeling, control, and experimental validation[END_REF] actively tilts the angles of a quadrotor independently about their radial axes, and [START_REF] Long | Omnicopter: A Novel Overactuated Micro Aerial Vehicle[END_REF] actively tilts the propellers of a trirotor, while adding a fixed central propeller to carry the weight of the vehicle. While popular in the literature, actuated propellers add weight to the platform due to the extra actuators. In addition, propeller tilting is achieved via servo motors, which cannot guarantee instantaneous force exertion because of the time required to re-orient the propellers.

On the other hand, [10]- [START_REF] Dyer | Energy optimal control allocation in a redundantly actuated omnidirectional uav[END_REF] achieve omnidirectional flight with 6 or 8 bidirectional thrusters. While their solutions are very interesting, as illustrated in [START_REF] Hamandi | Survey on Aerial Multirotor Design: a Taxonomy Based on Input Allocation[END_REF], bidirectional thrusters exhibit a singularity near the zero thrust region. Moreover, commercial hardware solutions for bidirectional thrusters are not satisfactory, where commercial ESCs allowing the control of bidirectional propellers are scarce. Additionally, bidirectional propellers provide less thrust than their uni-directional equivalent.

In our previous work [START_REF] Tognon | Omnidirectional aerial vehicles with unidirectional thrusters: Theory, optimal design, and control[END_REF], we investigated the required properties and conditions to achieve omnidirectional flight with fixed uni-directional thrusters, and proved the number of propellers n ≥ 7 to be a necessary condition. Furthermore, we presented an optimization method to find propeller tilts for any generic number of propellers (n ≥ 7), that can guarantee the omnidirectional property of the platform, while enforcing equal sharing of the desired forces between the propellers.

In this work we aim to design, manufacture and test an omnidirectional platform with the minimal number (n = 7) of uni-directional thrusters. Our design will rely on the optimization proposed in [START_REF] Tognon | Omnidirectional aerial vehicles with unidirectional thrusters: Theory, optimal design, and control[END_REF]. To the knowledge of the authors, this is the first manuscript in the literature showing such a working prototype with the properties mentioned above; the manufactured platform is shown in Fig. 1.

The rest of this paper is organized as follows. In section II we model a generic UAV platform. In sections III and IV we define the properties that guarantee the omnidirectionality of a UAV platform with uni-directional thrusters, and define the optimization problem with an emphasis on the assumptions made for the prototype. Section V presents the controller used to fly the platform, while section VI presents the final prototype. Sections VII-VIII show the numerical and real experiments that test the feasibility of the platform. Finally, section IX concludes the paper.

II. MODELING

Let us define a world frame F W with origin O W following the East-North-Up (ENU) convention with axes {x W , y W , z W }. Let us define a body frame F R with center O R fixed to the geometric center of the robot (assumed to coincide with the robot's CoM), and with axes {x R , y R , z R }. We refer to p R ∈ R 3 as the position of O R in F W and to R R ∈ SO(3)1 as the orientation of F R with respect to (w.r.t.) F W . We parameterize the rotation matrix R R with the classical Euler angles roll, pitch and yaw (φ , θ , ψ), such that (s.t.) R R = R z (ψ)R y (θ )R x (φ ). We refer to v R ∈ R 3 as the translational velocity of O R in F W , and to ω R ∈ R 3 as the angular velocity of F R w.r.t. F W , expressed into F R .

Let m R ∈ R >0 and J R ∈ R 3×3 >0 define the mass and the positive definite inertia matrix of the robot w.r.t. F R . Then, following the Newton-Euler formalism, we can write the robot equations of motion as ṗR

= v R , ṘR = R R Ω R , and m R vR J R ωR = - g m R e 3 ω R × J R ω R + G w, (1)
where Ω R = S(ω R ) is the skew symmetric matrix relative to ω R , e 3 = [0 0 1] , g is the gravitational constant, and w ∈ R 6×1 is the total wrench applied on O R w.r.t. F R . In particular, w = [f m ] , where f ∈ R 3 and m ∈ R 3 are the corresponding force and moment components of w. Finally, G is the 6-by-6 matrix of the form

G = R R 0 3 0 3 I 3 . (2)
We denote by n the number of propellers, and by F ∈ R 6×n the full allocation matrix. F 1 ∈ R 3×n and F 2 ∈ R 3×n are the force and moment allocation matrices, s.t.

w = [F 1 F 2 ] [u 1 ... u n ] = F u, (3)
where u i is the control thrust of the corresponding i-th propeller, and u ∈ R n×1 their concatenation. It is noted that in this formalism it was assumed that the propellers are the only source of wrench being applied on the robot, and that any other sources are neglected as secondary disturbances.

Following this notation, we can write F 1 and F 2 as follows:

F 1 = [v 1 ... v n ], (4)
F 2 = [d 1 × v 1 ... d n × v n ] + [c 1 kv 1 ... c n kv n ], (5)
where v i ∈ R 3 and d i ∈ R 3 are the thrust direction and CoM position of the i-th propeller in F R , respectively. c i = -1 (c i = 1) if the i-th propeller angular velocity vector has the same direction of v i (-v i ) when u i > 0, i.e., the propeller spins counter-clockwise (clockwise); k ∈ R is the drag to lift ratio of each propeller, where we assumed all propellers to be identical.

III. OPTIMUM OMNIPLUS

From the previous section, we can define a fixed propeller aerial vehicle design as the tuple T = (n, m R , v, d, c, k, u min , u max ) representing the number of propellers n, the platform mass m R , the thrust direction and position of each propeller in F R -v and d, respectively -, the rotation direction of the corresponding propellers c, the aerodynamic drag to lift coefficient k, and minimum and maximum thrust of each propeller, u min and u max , where 0 ≤ u min < u max .

While u min and u max are not shown in the previous formalism, they are crucial for any platform design to guarantee the feasibility of a desired wrench w d ∈ W, where W is the set of desired wrenches necessary for platform flight. As such, we can write the following condition:

∀ w d ∈ W ∃ u d s.t. w d = F u d and u d ∈ U , ( 6
)
where U is the set of allowable control thrust, defined as

the n-dimensional hypercube s.t. U = × n i=1 [u min , u max ].
We denote with 1 the column vector with all ones. Its size is understood from the context. Given two vectors x and y, the notations x ≥ y, x > y are intended component-wise.

Definition 1. a design tuple T is said to be OmniPlus O + if one of the following equivalent conditions holds [START_REF] Tognon | Omnidirectional aerial vehicles with unidirectional thrusters: Theory, optimal design, and control[END_REF]

∀ w ∈ R 6 ∃ u ≥ u min 1 s.t. F u = w (7) ∀ w ∈ R 6 ∃ u ≥ 0 s.t. F u = w (8) rank(F ) = 6, ∃ b = [b 1 . . . b n ] > 0 s.t. F b = 0 (9)

A. Allocation Strategy

Given an O + design and a desired wrench, following condition (9), one may calculate the thrust u * s.t. u * = F † w d where F † is the Moore-Penrose pseudo inverse of F . As was proven in [START_REF] Tognon | Omnidirectional aerial vehicles with unidirectional thrusters: Theory, optimal design, and control[END_REF], u * always has at least a negative entry, and as such violates condition [START_REF] Ryll | Modeling and control of FAST-Hex: a fully-actuated by synchronized-tilting hexarotor[END_REF].

Let us consider an ellipsoid that represents the attainable wrench space S w = {w ∈ R 6 |w Σw ≤ 1} ⊂ R 6 , where Σ ∈ R 6×6 is a positive definite matrix. The set U w maps S w through the linear transformation F such that U w = {u ∈ R n |w = F u, ∀w ∈ S w }. U w maps S w one to one, however, as stated in the previous paragraph, not all solutions u * ∈ U w have all positive entries, and as such U w ⊂ U .

Let's define a vector b ∈ null(F

) ∩ R n + , then b ⊥ u * . Any solution such that u * * = u * + λ b with λ > 0 ∈ R satisfies w d = F u * * .
As such, the objective of the allocation strategy would be to find λ such that u * * ∈ U as follows:

λ = argmin u * * ∈U u * * (u * , b) = argmin u * * ∈U u * + λ b . ( 10
)
The control thrust found in [START_REF] Brescianini | Design, modeling and control of an omni-directional aerial vehicle[END_REF]

satisfies u * * ∈ U * w , where U * w = U w ∩ U . It is noted that U *
w also maps S w one to one, and as such, in what follows we refer to u * * (w) as the control thrust in U * w that allows the platform to apply wrench w.

Definition 2. an O + design is said to be optimal if its space U * w has minimum eccentricity, and if its propellers equally share the effort to keep u * * ∈ U calculated in [START_REF] Brescianini | Design, modeling and control of an omni-directional aerial vehicle[END_REF].

Minimizing the eccentricity of U * w allows the platform to apply lower maximum thrust for each desired wrench since the platform will be sharing the load equally among its propellers; this problem can be solved by minimizing the condition number of Σ -1 F . On the other hand, to satisfy the second condition of Definition 2, it is easy to be convinced that the best choice to have b = 1. For more details we refer the reader to [START_REF] Tognon | Omnidirectional aerial vehicles with unidirectional thrusters: Theory, optimal design, and control[END_REF].

IV. PARAMETER OPTIMIZATION

In this section we detail the choice of parameters that allow the design T to satisfy the conditions and requirements mentioned above. First, we make the following assumptions:

• platform dimensions are chosen separately and fixed throughout the optimization, • all motors and propellers used in the platform are identical, • motor and propeller choice is made separately from this optimization problem, • propeller rotation directions are chosen prior to the optimization, with these directions alternating between one propeller and the next, i.e. c i = (-1) i for i = 1 ... n. With these assumptions, we can clearly see that the eterovectoring part of T (n, m R , d, c, k, u min , u max ) is fixed, in addition to the norm of the vectoring part v i for i = 1 ... n, while the optimization problem should choose the direction of the vectoring part (v). It is noted that v i = 1 as the allocation matrix F is assumed to map wrench w to propeller thrust u.

To highlight the optimization problem, let us rewrite F 1 and F 2 as follows:

F 1 = [I 3 v 1 ... I 3 v n ] (11)
F 2 = [(S(d 1 ) + c 1 kI 3 )v 1 ... (S(d n ) + c n kI 3 )v n ]. (12)
Then we can rewrite the second part of (9) as:

I 3 b 1 ... I 3 b n (S(d 1 ) + c 1 kI 3 )b 1 ... (S(d n ) + c n kI 3 )b n A(n, d, c, k, b) v = 0, (13)
where I i is the i-by-i identity matrix .

Following this formalism, the O + parameter optimization can be written as follows:

min cond(Σ -1 F ) (14)
subject to the following constraints

v D 1 v = 1, ... , v D n v = 1 (15) rank(F (c, k, d, v)) = 6 (16) A(n, d, c, k, 1)v = 0 (17)
where

D i = diag(D i1 ... D in ) is a 3n-by-3n diagonal matrix, with D i j = 0 if j = i and D ii = I 3 otherwise.
We note that in the choice of the propeller placements, we chose all propellers to be coplanar, placed in a star shape with the first propeller arm along x R , i.e. d 1 = [d, 0, 0] , and As the aim of this paper is to build an omnidirectional platform, it is desired that the force and moment ellipsoids to resemble a sphere, where the platform will be invariant to its flight direction. As such we choose Σ of the form

d i = d 1 R z (2π(i -1)/n) for i = 2 ... n,
Σ = σ f I 3 0 3×3 0 3×3 σ m I 3 ( 18
)
where σ f , σ m ∈ R >0 . Finally, we analyze the maximum propeller thrust u d max while the platform is in hover, where we define Definition 3. hovering (or static hovering) as the ability of the platform to stabilize its position and orientation for some orientation R d ∈ SO(3) with zero linear and angular velocity, i.e. (

p d R , R d , v d R , ω d R ) = ( p d R , R d , 0, 0).
Hovering is of particular interest for the design as it is a base point for the platform to apply forces and moments in any direction. The analysis of u d max is required for the motor choice and it is an important feature to study the feasibility of the design.

The chosen Σ, and the ensuing minimization of cond(Σ -1 F ), enforces a uniformity in the platform's generated force (moment) about its corresponding directions, and as such guarantees invariance of the platform to R at hovering.

While hovering, the thrust of each propeller can be calculated as

u d (R d ) = u * * d m R g R d e 3 0 3 (19)
Due to the invariance of the platform to its hovering direction and to the chosen optimization constraints, it is straight forward to prove that

u d max = max u d (R d ) = max u d (I), (20)
where the right hand part of the equality is for hovering at an identity rotation matrix. While in theory maximum propeller thrust should be the same irrespective of the hovering orientation, in practice the condition number never reaches unity, and as such there is always a difference between the max u d (R d ) at different R d , and as such, u d max is found with a grid search algorithm over possible orientations.

V. CONTROLLER Given a desired position and orientation p d R (t) and R d (t), the control strategy is straight forward as the allocation matrix F is full rank. The desired wrench is calculated as the one that brings the platform to the desired position and orientation (along with their corresponding derivates) while compensating for gravity and gyroscopic moments. As such the desired wrench can be written as follows:

w d = G -1 m R (ge 3 + vd R ) + K P e p + K D ėp + K IP t 0 e p ω R × J R ω R + K R e R + K ω e ω + K IR t 0 e R (21
) where K P , K D , K IP , K R , K ω and K IR are diagonal positive definite matrices ∈ R 3×3 representing the controller tunable gains.

e p = p d R -p R , e ω = ω d R -ω R and e R = 1/2(R d R -R R d ) ∧
, where [.] ∧ is the inverse skew symmetric operator. Then for each desired wrench, a control thrust is calculated as described earlier in [START_REF] Brescianini | Design, modeling and control of an omni-directional aerial vehicle[END_REF].

A. State Estimation

The platform is endowed with an IMU that captures the platform's specific linear acceleration and angular velocity. Furthermore, its position and orientation are tracked with a motion capture system.

All measurements from the IMU are filtered with the regression-based filter introduced in [START_REF] Hamandi | Direct acceleration feedback control of quadrotor aerial vehicles[END_REF]. The filter is designed to reduce the noise caused by the propellers' vibration; however, as the motors are controlled in open-loop (i.e. propeller rotational velocities are not measured), the filter fit the second order polynomial to the IMU signal without separation between its signal and noise constituents.

Both filtered IMU measurements and motion capture measurements are fused using an Unscented Kalman Filter (UKF) [START_REF] Wan | The unscented kalman filter for nonlinear estimation[END_REF] to retrieve the full pose estimate of the platform.

VI. PROTOTYPE

For our prototype we chose to construct a platform with n = 7, the least number of uni-directional propellers necessary to achieve omnidirectional thrust. The platform is built to be the smallest possible to increase its stability by reducing any possible oscillations in the arms connecting the motors to the body; as such, we chose an arm length d = 0.16 [m]. We then chose 5" propellers, as it is the largest diameter that can be installed on the platform without any collision between adjacent propellers. The propellers we chose enforced a drag to lift coefficient k = 0.002 [m], and a lift coefficient k f = 0.5e -4[N/Hz 2 ]. Finally, the wrench ellipsoid was chosen such that Σ = diag([1, 1, 1, 0.5, 0.5, 0.5]). We estimated the platform mass before the platform construction to be around 1.1 [kg] following the component-wise weight estimation shown in Tab. I.

A. Numerical Optimization

The O + optimization algorithm calculated the vectoring part of the design, and reached a minimum condition number of cond(Σ -1 F ) = 2.052, with a vectoring part as follows: v =   0.36 -0.35 0.29 -0.81 -0.37 0.78 0.10 -0.90 0.44 0.76 -0.12 0.45 -0.57 -0.07 0.25 0.83 -0.58 -0.58 0.81 0.26 -0.99   .

(22) The above thrust directions are illustrated in Fig. 2.

With the current parameters, the maximum propeller thrust was found to be u d max = 11.18 [N], corresponding to a maximum rotational speed of W max = 472 [Hz] for the chosen propellers; as such we chose a motor that can provide a The chosen motor is controlled in PWM via an Electronic Speed Controller (ESC), i.e. , the motor is controlled in openloop, where the thrust generated at each PWM was identified with a force-torque sensor. Finally, the PWMs delivered to the ESCs are generated using an onboard microcontroller.

B. Platform Implementation

The platform body is constructed with 7 aluminum bars connecting the CoM of the platform to the CoM of the propellers. Each propeller is connected with a separate arm to ensure the stability of the platform. Aluminum bars are fastened together using 3D printed plates connected to one of their edges, while the second edge is connected to a 3D printed adapter that ensures the motors' connection at the calculated direction. Fig. 3 shows the CAD drawing of one of the adapters, while Fig. 4 shows the CAD drawing of the body frame assembly.

Finally, the necessary electronics and motion capture markers are placed on top of the platform, with the full prototype shown in Fig. 1.

The final weight of this setup without a battery is measured at m R = 0.835 [kg].

C. Design Drawbacks

The platform prototype as presented above exhibits the following drawbacks that can affect its performance: • Propeller open-loop control: as the propellers are controlled in open-loop, the propeller speed is not guaranteed, and correspondingly the individual motor thrust. Therefore, the applied wrench can differ from the desired one. • Propeller airflow cylinder intersection. We define the airflow cylinder as the cylinder containing the corresponding propeller, of radius equal to the propellers', and direction similar to the corresponding propeller. Since the propellers are placed close to each other, with each producing thrust in any direction, it is impossible for the airflow cylinders of adjacent propellers not to intersect as shown in Fig. 5. This intersection affects their aerodynamics, as propellers have to withdraw air from the inflow/outflow of their adjacent propellers instead of withdrawing air from the free stream assumed static. It should be noted that it is difficult to estimate the effect of this intersection, as the change in the thrust produced by each propeller will depend on the amount of thrust provided by the adjacent propeller.

While these drawbacks can induce an error in the applied wrench, we assume it equivalent to an external disturbance that can be compensated by the feedback controller shown in section V.

VII. PRELIMINARY TESTS

A. Dynamic Simulation

To assess the performance and flyability of the prototype described in section VI, we simulated its dynamical system in Matlab/Simulink with the estimated mass. The simulation is made closer to reality with the addition of measurement noise and signal delays.

Figure 6 shows the performance of the platform's flight with z R circling the unit radius sphere. This figure shows that the platform is able to fly while in a variety of orientations. The simulation also shows that the platform can apply independent force and moments, and as such, orient F R independently of its translation, while the propeller rotational velocities are kept within the allowable range.

B. Wrench Tests

To assess the discrepancy between the ideal model and the built prototype, a force-torque sensor was used to measure the generated wrench in a static experiment as shown in Fig. 7. Tab. II shows the force and moment constituents of the desired nominal wrench w d and the corresponding error between the measured and nominal wrench w error = w measured -w d .

We can observe from the data in Tab. II an error between the measured and nominal values of the applied wrench; it can also be observed that the value of this error changes depending on the desired nominal wrench. While we could not identify clearly the cause of these errors, they were expected due to the motor speed control and aerodynamic interaction between adjacent propellers, in addition to manufacturing imperfections (see section VI-C). VIII. EXPERIMENTS

A. System Setup

As stated in section V-A, the platform is endowed with an IMU, which exports the raw specific linear acceleration and angular velocity measurements at 1 [kHz]; in addition, the platform position and orientation are tracked with a motion capture system at 100 [Hz]. Both measurements are fused by a UKF running at 1 [kHz], and providing an estimate of the platform state. The platform controller is implemented in Matlab/Simulink at 500 [Hz], while the onboard microcontroller delivers the desired PWM to the ESCs. Most software (excluding the controller), including those used for the communication between Matlab and the platform, are developed in C++ using Genom3 [START_REF] Mallet | Genom3: Building middleware-independent robotic components[END_REF], a code generator and formal software component description language that allows assembling middleware-independent components in a modular system. These software can be found here: https: //git.openrobots.org/projects/telekyb3

The platform is top-connected to a power supply cable, in addition to multiple data cables allowing the back and forth communication with the off-board controller PC.

B. Hovering

To preliminarily test the omnidirectional flight ability of the platform, we ask the vehicle to lift off from its hanged position and hover in place in multiple orientations as shown in Fig. 8. The performance of the platform in each of the desired orientations is shown in Figures 9 through 11. These figures show that the platform is able to hover horizontally, upside down, and at a tilted angle. Furthermore, for these orientations, the desired propeller rotational velocities are within the acceptable range. Remark 1. In all these experiments, the propeller rotational velocities are within the acceptable range. Moreover, the maximum propeller rotational velocity is around 380 [Hz], far from the maximum allowed rotational velocity of 550 [Hz]. Remark 2. While the platform is able to hover at these orientations, there is a noticeable tracking error in all experiments even if the controller gains have been tuned specifically for each experiment to reduce tracking error while assuring the platform's stability.

IX. CONCLUSIONS

In this paper we have presented a novel prototype of an O + UAV with 7 propellers. To the knowledge of the authors, the design is the first of its kind. We conducted an experimental campaign to assess the model and the prototype. Our experiments show that the platform is able to hover in many directions. From our tests, we have realized that there is a discrepancy between the nominal desired wrench and the one actually applied by the platform. This descrepancy is expected to be caused by : 1) the open-loop control of the propeller rotational velocities, 2) and the aerodynamic interactions between adjacent propellers due to the small size of the frame. While the aerodynamic interactions are caused by the proximity between the propellers, the openloop control of the propellers is necessary to ensure such high speed rotational velocity of the propellers, where off the shelf ESCs providing closed loop speed control at the required rotational speed are still rare.

In the future, we aim at ameliorating the design along the following lines

• Reduce the weight of the platform by replacing aluminum bars with lighter carbon fiber material. This will allow the platform to fly with lower propeller rotational velocities, and as such be able to achieve close-loop control of the propellers. • Increase the platform's arm length to reduce the aerodynamic interactions between propellers. • Implement an adaptive/learning based controller that can compensate for the unmodeled aerodynamic interactions.

  where d is the norm of the arm connecting O R to the CoM of any propeller, and R z is the transformation matrix corresponding to the rotation about z R .

  Representing the estimated weight of each of the platform components prior to its final design and construction. It should be noted that the weight of parts that are certainly used were reported as they are, while others are estimated.

Fig. 2 :

 2 Fig. 2: Optimized propeller direction of the O 7 + design, with n = 7, d = 0.16 [ m] and k = 0.002 [m] at cond(Σ -1 F ) = 2.052.

Fig. 3 :

 3 Fig. 3: CAD design of the motor adapter for motor # 1. The adapter is designed to fasten the motor to the body frame at position d 1 and orientation v 1 in F R .

Fig. 5 :

 5 Fig. 5: Example intersection between adjacent propeller airflow cylinders, showing the intersection between 7 th propeller airflow cylinder with the airflow cylinders of the 1 st and 6 th propeller.

Fig. 6 :Fig. 7 :

 67 Fig. 6: Tracking results of the simulated platform while following a desired position and orientation. The platform orientation is chosen such that z R circles the unit sphere multiple times, while the position is chosen to change smoothly and simultaneously on all axes. The desired and actual z R are superimposed in the top right plot. A step change in the z-position is required at time t = 0s, where the platform is required to lift from the height of 0 [m] to 4 [m]. The figure also shows the propeller rotational velocities w i for i = 1 ... 7, where the dashed line constitutes the limiting maximum velocity.

8 ]Fig. 8 :

 88 Fig. 8: Preliminary hovering tests of the prototype platform: a) platform hovering horizontally, b) platform hovering upside down, c) platform hovering at a tilted orientation such that φ d = 130 • .

Fig. 9 :Fig. 10 :

 910 Fig. 9: Performance of the platform while hovering horizontally: (top-left) shows the desired and estimated platform height, (topright) shows the angular errors, (bottom-left) shows the average xy error, and (bottom-right) shows the propeller desired rotational velocities.

Fig. 11 :

 11 Fig. 11: Performance of the platform while hovering at a tilted orientation such that φ d = 130 • , θ d = 0 • ψ d = 90 • : (top-left) shows the desired and estimated platform height, (top-right) shows the angular errors, (bottom-left) shows the average xy error, and (bottom-right) shows the propeller desired rotational velocities. The platform starts its maneuver while being oriented near upside down, i.e. φ (t = 0s) ≈ 180 • .

SO(3) = {R ∈ R 3×3 | R R = I 3 , det(R) = 1}, where I i ∈ R i×i is the identity matrix of dimension i.

Preprint version final version at http://ieeexplore.ieee.org/ 2 Accepted for 2020 Int. Conf. on Unmanned Aircraft Systems

Preprint version final version at http://ieeexplore.ieee.org/ 3 Accepted for 2020 Int. Conf. on Unmanned Aircraft Systems

Preprint version final version at http://ieeexplore.ieee.org/ 4 Accepted for 2020 Int. Conf. on Unmanned Aircraft Systems

Preprint version final version at http://ieeexplore.ieee.org/ 5 Accepted for 2020 Int. Conf. on Unmanned Aircraft Systems

Preprint version final version at http://ieeexplore.ieee.org/ 6 Accepted for 2020 Int. Conf. on Unmanned Aircraft Systems

Preprint version final version at http://ieeexplore.ieee.org/

Accepted for 2020 Int. Conf. on Unmanned Aircraft Systems

Preprint version final version at http://ieeexplore.ieee.org/ 8 Accepted for 2020 Int. Conf. on Unmanned Aircraft Systems