Grégoire Lenoble
email: gregoire.lenoble@siemens.com

Marie Olivier

David Donjat
email: david.donjat@onera.fr

Anton Steblinkin
email: anton.steblinkin@tsagi.ru

Jimenez
email: ajimenezo@cesa.aero

Donjat David

Olazabal Jimenez

Andrés

Elevator actuator housing bay flight mission thermal integrated analysis

Keywords: Elevator, actuator, EHA, EMA, MBSE, LMS Amesim, CFD, Fluent, flight mission, thermal integration

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Context, Need for thermal analysis

The reduction of fuel burn has become a priority for most aircraft operators due to environmental and cost implications. Such interest has encouraged the aerospace industry to look at ways of minimizing fuel consumption, concluding that the approach towards a More Electrical Aircraft (MEA), i.e. an aircraft that uses electricity to operate its consumers, could provide some benefits. This approach tends to remove hydraulic actuators and the hydraulic systems that feed them, as they require a significant amount of maintenance, are heavier and consume significant amounts of energy. Additionally, electrical components allow for a higher integration of sensors that monitor actuators performance and structural integrity (from a safety point of view); and help implement a higher level of "intelligence", improving the actuation system integration within the aircraft controlling system as a result.

Despite the potential benefits indicated above, such as weight reduction, increased safety and reliability, lower consumption, reduced maintenance costs, etc.… electrical actuators are not fully embraced by the industry due the limitations they present in their current development stage, which prevent them from being used massively or in critical areas, such as aircraft primary flight control surfaces. These limitations are: high actuator weight, safety concerns due to the jamming probability of Electro-Mechanical Actuators (EMAs), and Electro Magnetic Compatibility (EMC) issues. Systems thermal behavior and integration is also another very challenging issue. Indeed, it is well known that the reliability of EMAs and Electro-Hydrostatic Actuators (EHAs) for flight controls depends on their operating temperature. The A380 flight control EHA is a great example [Bossche], being used as a backup actuator. Thus, EHAs and EMAs that are constantly working in an active regime need to see their thermal management improved: removing the traditional hydraulic network imposes the need to dissipate heat generated by the actuation system (motor, ECU, and harness) differently. The management of this thermal environment needs both new cooling technologies (see [Lawson]) and design optimization methods. The latter is a key enabler for designers to address thermal integration earlier in the design process.

For the above reasons, and the fact that the RESEARCH program is a wide objective oriented cooperative project, the decisions was taken to analyze the thermal integration of the actuation system under development by the consortium partners.

Cavity configuration

Based on the results of weight and safety analyses of the different possible FCAS architectures performed by the RESEARCH consortium in the first project phase, the elevator control system of a regional passenger aircraft (100seater) was chosen as the target application to be developed. The weight information available for the above mentioned hardware is summarized with Table 1. The actuator housing cavity is shown with Figure 4.

The flight mission

The flight scenario is based on a real flight record that contains the data arrays of Indicated Air Speed (IAS) and aircraft altitude (both shown with Figure 5), as well as external forces under hinge moments and other flight parameters. This flight record thus fully determines the actuation system heat losses and heat dissipation processes. Thermal analysis

MBSE cavity model

The methodology of the actuation system thermal behavior analysis described implies the wide use of simulation, which as a first step was represented by the simulation of the cavity system thermal behavior in the LMS Imagine.Lab Amesim software, over the flight mission described above.

1) Cavity boundary conditions

The elevator actuator operates inside a given thermal context, which will interact with the actuator bay, and even influence its thermal behavior. The following interfaces to the actuator are considered: outside air, Sun radiation, cavity structure.

Outside airflow provides a large cooling effect to the cavity outside shell. This is captured both on the top and bottom panels (see Figure 6 and Figure 7), assuming airflow over a horizontal flat plate configuration. Depending on the flight regime and air properties, forced and free convections are calculated with laminar or turbulent equations [Incropera].

In order to ensure a smooth mathematical model over the entire range of convection regimes (purely free, mixed, purely forced), a unique Nusselt number combines the free and forced ones, so that a unique convection Heat Transfer Coefficient (HTC) can be calculated as per (1). In this outside convection case, the characteristic dimension L is the total stabilizer outside area (top and bottom) divided by its perimeter. * / (1)

Note that the air viscosity, thermal conductivity, specific heat, density and speed are dynamically updated during the aircraft mission simulation. This Sun radiation is complemented with the radiative exchange the surface has with outside ambient air, as per (7), and using ε gw as the outside air and cavity panel emission factor.

⋅ ⋅ ⋅ (3)
Note that equation ( 3) is a particular case of the general radiation equation that assumes an infinite "radiating air surface" and thus a view factor of 1.

As shown with Figure 7, ground radiation is accounted for in a very similar way, only with no exposure angle factor, a different ε and the ability for it to be turned off when the aircraft has left ground.

Figure 7. Cavity bottom panel

The thermal behavior of the cavity that houses the stabilizer actuator has been simplified to that of a few panels: a vertical, and two horizontal ones: top and bottom. Top and bottom panels are the aggregate of an external layer (exposed to the Sun / ground and outside air), and an internal layer (in contact with internal air), separated by an insulation composite material, as shown with Figure 6 and Figure 7.

Finally, thermal conduction occurs from the top to the bottom panels, through the vertical one. This is captured with two instances of a "thermal conduction" component, that derive the heat flux between two solid material pieces 1 and 2; depending on is their contact surface, and contact thermal resistance. It predicts T 1 and T 2 temperatures, both taken at a distance d 1 and d 2 inside each material piece.

2) Cavity internal heat loads

The end objective of this analysis being the thermal integration of all heat sources and exchanges inside this cavity, each heat source has then been modeled individually.

The heat transfer modes involved in all of the following heat loads are all based on standard heat transfer equations: free and forced convection over a flat plate (or a cylinder for the EMA shaft and body, plus the EHA piston), radiation in an enclosure, and linear conduction between two pieces of different material.

First, EHAs are modeled with a focus put on their thermal interaction with the cavity. The EHA heat flux generated as a function of its duty cycle is obtained from the EHA Simulink-model developed by MAI, and split over the EHA parts: body, motor, pump, piston and reverse valve. The EHA parts are in obviously contact, and bolted to the cavity bottom (lower panel). They dissipate EHA heat losses through conduction (to themselves, and the lower panel), forced oil convection (driven by duty cycle oil consumption), as well as air convection (forced and free) and radiation, as will be described in the next section.

Figure 8. EHA model focus

Similarly, EMAs have identical thermal interactions with the cavity and its constituents. The EMA thermal capacitance behavior is captured with a body, motor and shaft, which obviously are not in contact with any oil. The EMA heat load is piloted by the EMA Simulink-model developed by TsAGI.

Figure 9. EMA model focus

Given the early nature of this analysis, the exact combination of EHAs / EMAs that would power the actuator has not been decided upon. Indeed, the output from this work was designed to support this decision! As a result, and leveraging the Systems Engineering approach, the actuator configuration implemented into the LMS Amesim software model is parametric, meaning that the number of EHAs and EMAs is only a model parameter, which is then used as a hiding / scaling factor of the relevant heat fluxes.

The ECU adds an additional third heat load into the cavity, through its thermal losses. This is captured with a single thermal capacitance, which also includes the behavior of the System ECU (SECU) when positioned in the cavity itself, as shown with Figure 10. Its heat load is calculated based on the power it provides to the EMA / EHA motor, and its own efficiency η. The ECU is in thermal contact with the lower panel, and is cooled by air convection and radiation.

Figure 10. ECU model focus

3) Cavity thermal integration

Thermal integration of the components described in the above sections is mostly piloted by the cavity internal convection and radiation.

First, internal convection thermally connects all cavity elements: EMA / EHA, ECU and cavity panels. As seen with Figure 7, Figure 8 and Figure 9, convection is modeled over "horizontal" and "vertical" faces, so as to capture all heat exchanges, distinguishing between the EMA cylindrical body and its flat end faces for example. Natural and forced convections are handled as per the equations given in section 1), with the additional subtlety that convection is piloted from the airspeed point of view. LMS Amesim software being a Systems Engineering tool, 2D or 3D airspeed maps cannot be predicted. Rather, this internal air speed estimation is based on a correlation derived by ONERA for similar actuator cavity configurations; that is driven by the ingested airflow. In the case of a sealed cavity, it is artificially set to a minimum of 0.2 m/s, as recommended in [Saury et al.].

Internal radiation completes the list of cavity internal heat exchanges, and has been allowed between all bodies (EHA, EMA, and ECU) and both cavity horizontal panels. Radiation between bodies, and from a panel to the other have been neglected given temperature levels and relative influence with regards to convection or conduction.

Finally, all heat loads, heat storage, and heat exchanges are resolved in the Cavity submodel shown in Figure 11.

CFD cavity model

Calculations are performed with FLUENT14's unsteady incompressible solver. Based on first estimations of flow conditions, the laminar approach is selected. For thermal considerations, both free convective and radiative behaviors are resolved. For radiative considerations, the S2S method is preferred for this 3D enclosed configuration. View factors are evaluated based on the geometry definition of the cavity and 3D localizations of the bodies. For each surface, the model assumes the identical values of emissivity and absorption characteristics. Solar radiation however, is not currently accounted for (although it is in the MBSE model), given initial model estimates predicting it to account for only 5% of the thermal heat exchanges at stake in the cavity.

Boundary conditions are extracted from the LMS Amesim software simulation in term of heat loads (i.e. heat power released by electric items) and external airflow (i.e. Static Air Temperature -SAT), altitude and Mach number.

Analysis focus

Given the early design phase of the EHA and EMA actuators envisaged for the elevator actuation, and the criticality of their thermal heat rejection, conservative assumptions have been applied, with a constant 60W heat rejection throughout the flight mission. This number is reduced to 6W (10%) in the "taxi out" and "taxi in" phases. This will replace the heat rejection obtained from the EHA and EMA Simulink models, until further confidence is gained in these models. Any other thermal analyses that does not need conservative assumptions, but rather a dynamic model of their heat rejection can use these Simulink models.

For the EHA heat rejection, the heat losses split between the pump and motor recommended by the Simulink model has been kept, with 31.4W generated by the pump and 28.6W by the motor.

Similarly, the ECU heat rejection has been assumed constant at 20W. Assuming that the ECU is powered throughout the entire aircraft mission, this value is not decreased during taxi.

The analysis results described in this article focus on a case which combines two EHAs with an ECU, with no EMA in the cavity. Out of the two EHAs, only one is active during the flight mission (nominal case). However the thermal inertia provided by the second one is accounted for.

Similarly, conservative assumptions have suggested running a HOT day analysis (ISA+35) in addition to the standard ISA day mission described in "The flight mission" section.

Simulation results

1) MBSE model results

The evolution of the temperature of the active EHA over the whole flight mission is shown with Figure 12. As expected, all temperatures clearly increase during both ground phases as a result of medium to high SAT and low aircraft speeds.

On the opposite, low outside temperatures and high aircraft velocities during cruise directly help reduce EHA temperatures. The steeper temperature rise during approach and landing than during takeoff is a direct consequence of the quick decrease of external heat convection following landing.

The temperature profiles observed for the EHA pump and motor are both higher than that of other EHA parts, but yet exhibit different time variations. This is explained by the fact that they both reject the same thermal power (approx. 30W), but have different weights (as summarized with Table 1), which results in different heat per unit mass values. The watchful reader may observe that the flight mission shown with Figure 5 has been extended by 160mins in the taxi-in phase. Although highly unrealistic, this was done so as to verify long-term temperature stabilizations, and made possible by the very low CPU run times of the LMS Amesim software model (100 CPU s for 16800s simulated). Despite this mission extension, all temperatures keep rising after landing and taxi-in phases, even with moderate heat release levels (6W for the EHA, and 20W for the ECU). This can be explained with the conservative assumption of a sealed cavity, combined with a low wind speed assumption (1 m/s) that do not provide sufficient cooling to the cavity and its actuators. This behavior does not hold when cavity ventilation is introduced, as described in section "EMA design support".

The results shown above have confirmed that, as expected, the flight condition that is most critical and demanding from a thermal point of view is the landing / taxiing phase. Therefore, it has been decided to further analyze the cavity thermal behavior straight after landing (t=95min) with the CFD model described previously. The end of cruise condition (t=76,7min) has also been analyzed, given that it is the operating point where actuators are designed to operate for the longest part of their life.

2) CFD model results

The temperatures of the cavity bodies obtained from CFD analysis in an EHA-EHA configuration, on a HOT day, in steady state conditions, are given below as body averages to allow for a comparison with MBSE results (extracted from Figure 12). End of cruise conditions (t=76,7min) and landing conditions (t=95min) are given and compared with MBSE results in Table 2 and Table 3 respectively.

Temperature (°C) CFD MBSE ∆

Mean In this result post-processing summary percentages, the CFD cavity temperature at End of Cruise is not included, as it clearly highlights a limitation to the current model. This is believed to be linked to the inaccurate accounting of thermal conduction along the cavity walls, currently modelled with the "shell-conduction" approach that uses an artificial meshing based on the real panel thicknesses. Improvements to this approach are currently being investigated.

Also left out of the post-processing summary percentages is the cavity upper panel external temperature since the end of cruise difference is 1degC only, but also since solar radiation

is not yet accounted for in the CFD model. This will be added in future versions of the model, given its importance when compared with other heat fluxes into the cavity roof, as understood from the MBSE model (250 to 350W, compared to 200W for external convection).

Keeping in mind the added value of the CFD model for an accurate prediction of radiation view factors, an additional run has be performed with the CFD model, increasing all emissivity values to 0.9 (starting from 0.5 for the glass fiber and aluminum, and 0.8 for the carbon fiber).

Temperature (°C) CFD MBSE ∆

Mean This additional CFD run provides further understanding of the cavity thermal behavior, and especially of the importance of internal radiation, which increases all internal temperatures by 2-3 degC. Just like the cavity air temperature discrepancy, this variation as a result of additional radiation can be explained with the "shell conduction" model that needs improving. Indeed, in the previous ( 0.5) run, inaccurate thermal conduction between the EHA constituents prevented realistic spread of the heat dissipated by the motor and pump. The rise in radiation ( 0.9) reduces this isolation of heat in one EHA constituent only (mostly the pump), and allows for heat propagation to the rest of the cavity and hardware.

Apart from conduction and solar radiation to be refined in the CFD model, this results comparison still validates the discretization assumptions made in the MBSE analysis, with the split of EHA actuators into a few bodies only. This is of great importance, as it gives confidence in further use of the MBSE model, whose flexibility and fast simulation capability are instrumental at this early stage of actuator design.

Finally, further understanding of the flow physics at stake in the cavity is provided with Figure 13, which shows the temperature-colored profile of some cavity air flow.

Figure 13. Cavity airflow profile colored by air temperature, HOT day, sealed cavity, taxi-in conditions

This illustration is key to the understanding of the natural convection happening within the cavity. A specific posttreatment will be designed during further work to postprocess this information for the update (improvement) of HTC estimations used in LMS Amesim software model, basing them on local air velocities as influenced by the cavity 3D geometry rather than lumped estimates.

EMA design support 1) Need for a thermal analysis

When designing the EMA, its endurance requirement is based on the assumption of a continuous torque requirement over the whole flight, resulting in continuous thermal dissipation during flight. This makes thermal dissipation of its motor windings (Joule effect) one of the most important challenges of EMA design, because it has a huge impact on the EMA internal temperature. Indeed, any increase in EMA internal temperature reduces its performance due to variations in motor electrical characteristics and magnets magnetic flux. This dissipation is directly driven by the current levels required to operate EMA loads, which is highest for direct-drive configurations (often preferred for increased reliability).

The sizing of EMA motors for flight control applications traditionally requires EMA motors to be oversized so that they can better dissipate heat, and therefore work continuously despite high cavity temperatures. This oversizing for thermal dissipation is always in terms of EMA volume and weight, so as to make up for the low EMA thermal dissipation efficiency.

2) Addressing EMA overdesign with cavity ventilation impact analysis

One way to address this EMA overdesign is to relax the conservative assumption of a sealed cavity, and introduce internal forced ventilation as a result of air ingestion. This is in line with the requirement that the cavity shall be certified for sand and dust ingestion, meaning that it is not sealed.

Of course, forced EMA ventilation can only beneficially support EMA thermal conditions, however it should not be strongly relied upon. Indeed, varying flight and ambient conditions may adversely affect it, and even reverse it at times. That's in addition to conditions where the aircraft is static on the runway, waiting for departure, with no relative speed or ambient pressure delta to drive air ingestion. For these reasons, it's rather the sensitivity of EMA thermal conditions to forced ventilation that has been tested, than its sensitivity to cavity ingestion of a target air flow, especially as no air scooping is envisaged.

3) MBSE sensitivity analysis output

The impact of air flow ingestion into the cavity on the EMA temperatures has been analyzed for air flows varying between 0 ("run 1") and 20g/s ("run 5") with steps of 5g/s, and is shown with Figure 14.

The flight mission is as per the previous one, except that the elevator is now powered by two EMAs, with only one active EMA. Thermal conditions being EMA design drivers, focus is put on a HOT day configuration. Based on the above observation that an increase of the radiation emission factors increases EHA temperatures by 2-3degC and EMA temperatures by 4-5 degC (both in the CFD and MBSE models), a conservative assumption of a uniform 0.9 is made for the analysis of the forced ventilation impact. However what is interesting to note is that once some initial ventilation is introduced, its impact is fairly limited, hardly bringing final EMA temperatures below 59 to 60degC at the end of the simulation (160mins after aircraft mission end). This means that EMA temperature transients between "run 2" (5g/s, green curve) and "run 3" (10g/s, blue curve) can reasonably be assumed when designing the EMA and its constituents. Indeed, it is very likely that the cavity will not behave as 100% sealed throughout the entire flight, meaning that even under a reduced (conservative) ventilation assumption, EMA operating temperatures are expected to reach a 55degC maximum (for 10-15min peak approximately, during the Top Of Climb flight phase), and remain below the 50degC mark until the end of the aircraft mission (i.e. t=110min).

4) CFD analysis output

Further confidence in the MBSE model predictions can be obtained from additional runs of the CFD model. For thermal design consideration purposes, this model has been run with the conservative assumption of a HOT day, with increased radiation emission factors (all set to 0.9) in the steady state conditions after aircraft landing (t=95min). The EHA FCAS configuration is chosen based due to CFD model availability, and has been compared to the outputs of an EHA FCAS configuration MBSE model run, to validate the accurate prediction of the impact of ventilation in the MBSE model.

Air is ingested from the three forward-facing holes and a leakage on the inboard face, assuming ideal (identical) mass flow repartition between all inlets. A Pressure boundary condition is used for the air exit from the cavity.

Average temperatures obtained from post-processing of two CFD runs are shown with Table 5 and Table 6. A number of conclusions may be drawn from the results given with Table 5 and Table 6.

First, although discrepancy between MBSE and CFD results is higher than for sealed cavity scenarios, it stays below 10 degC difference for the cavity mean temperature, and below 20degC difference for the EHA mean temperature. Such discrepancy values have to be kept in mind, but are believed to be acceptable, especially when recalling the number of simplifications made in the MBSE model: lumped mass discretization, local air velocities, HTC, air flow split. Local air velocity patterns in particular are complex, and cannot be determined a-priori, as shown with a visualization of the cavity internal flow patterns shown with Figure 15. The simple fact that EHA temperatures exhibit more discrepancy than the cavity average temperatures highlights the fact that local air velocities have a greater impact on the temperature of local, 3D particular shapes than on volume averaged ones, as expected of course.

Second, one can observe a slight tendency for temperature discrepancies to increase from the first (10g/s) run to the second (20g/s) one, which would tend to align with the observation made on the importance of 3D flow patterns.

Finally, particular attention should be paid to the ECU temperatures provided by the two software packages. As shown with Figure 15, it is indeed located right by an airflow entry in the CFD model, but only receives "side-licking" ventilation rather than an impinging one. This makes a large difference in the calculation of local HTC values, especially when the MBSE model assumes no difference between the calculation of ECU HTC values and EHA HTC ones.

5) Cavity ventilation benefits to the EMA

Following LMS Amesim software analysis of EMA sensitivity to cavity air ingestion, additional EMA design calculations performed during the project estimated that continuous EMA forced ventilation could allow for a reduction of a maximum of 30% to 40% in EMA motor weight. This calculation is based on the assumption that flight controls are not going to work and dissipate heat in a continuous manner while at 70 degC as initially assumed (worst case scenario assumption). Indeed, the EMA operating temperature could rather be in the 40-50 degC range for most of its operating life (i.e. except for the initial short peak to 55 degC), see Figure 14), in the conservative assumption of a HOT day (radiation emission factors increased to 0.9).

Further work 1) Mission definition

In order to obtain the duty cycle of the actuators during some pessimistic design scenarios such as highly loaded flights, a «virtual» aircraft flight-record will also developed. It will include the influence of predefined turbulence, as well as some special maneuvers during flight (e.g. sudden altitude drop during cruise, s-maneuver just before landing and large turbulence levels) so as to put the actuation system into the most critical regime and make sure it will not generate overheat conditions in the cavity.

Finally, more diverse missions may be analyzed, with varied configurations, such as {EHA & EMA} for example. This will support MEA investigations in particular. More complexity can also be brought in with flight missions that include the failure of an actuator, or maximum loads applied to the surface as a result of a tear-down for example.

2) MBSE model

When enough CFD runs have been run, HTC variations may be characterized with dedicated Nu correlations that would be specific to the elevator cavity configuration and materials, and thus bring the 3D perspective and added value of CFD into the MBSE model. A simpler approach would consist in running smaller series of CFD runs (e.g. 4 runs) per flight phase (i.e. taxi-out / climb / cruise / descent / taxi-in) and capture HTC correlations specific to each aircraft phase only.

Similarly, it is planned to update the MBSE model with the radiation view factors generated by the 3D thermal model.

3) CFD model

The modeling of the thermal conduction within the cavity layers (currently based on the shell-conduction" approach) will be investigated for improvements. Particular attention will be paid to the impact it has on the cavity temperature.

The introduction of solid models of EHA and ECU into the CFD model will allow for the analysis of the impact of the aerothermal flow inside the cavity during a transient part of the flight mission, such as landing for example.

4) Testing for validation

In order to validate the MBSE and CFD models generated during this work, experimental testing of the "actuator in cavity" system is planned for execution by TsAGI. This test will rely on a full-scale composite stabilizer and the real EHA / EMA actuators installed inside the cavity.

Conclusions

This collaboration between project partners allowed for the development of models that can reliably predict the influence of actuators design & characteristics (mass, dimensions) on the thermal behavior of the cavity they live in. This provides a better understanding of the thermal conditions they should be designed to operate in, even at early design stages.

As a result, these models and the verification methodology proposed allow actuation design engineers to account for system thermal considerations at early design stage, which is instrumental in supporting MEA applications development, and is expected to eventually lead to aircraft lifecycle costs reduction through faster and better systems development.

Last but not least, the methodology developed gives the ability to all the project partners to merge and leverage the outputs of their work in the common LMS Amesim software platform. This makes it possible for them to focus on their core expertise (e.g. component development), while being confident that they can then analyze their design in a realistic context, with interfaces to other components being taken care of in the MBSE model. Of course, this model can be complemented with additional insight gained from CFD analysis, supporting flow pattern, local HTC, and radiation view factors identifications in particular. Once this additional insight is implemented into the MBSE model however, the MBSE model can be used as a standalone one, leveraging its flexibility and simulation power for further pre-design analyses.

  The control system should consist of an EMA (under CESA's development, Spain) and EHA (under Moscow Aviation Institute (MAI) development, Russia) operating in parallel (active/active configuration). The preliminary actuators characteristics for such configuration were the following:  Stroke ±30mm  Maximum rod velocity 55 mm/s  Maximum rod velocity under 25 mm/s  Stall load 20 kN EMA and EHA designs identified as candidates for this analysis are shown with Figure 1, Figure 2 and the ECU envisaged for their control is shown with Figure 3.

Figure 1 .

 1 Figure 1. EMA isometric view (CESA)

Figure 4 .

 4 Figure 4. Stabilizer cavity with actuating devices: EHA/EHA/ECU (ONERA)

Figure 5 .

 5 Figure 5. Aircraft IAS and altitude from the flight-record

Figure 6 .

 6 Figure 6. Cavity top panel As shown with Figure 6, Sun radiation onto the stabilizer cavity is accounted for with (2), where θ is the angle between the cavity surface (normal vector) and the Sun direction (set to 0° so as to consider maximum Sun radiation), and abscoeff stands for the surface absorption factor. * cos * * 2

Figure 11 .

 11 Figure 11. Cavity model focusThis submodel computes the cavity temperature and Pressure based on mass and energy conservation, respectively owing from airflows (ingested and leaked), and the heat exchanges detailed above. Note that in the scope of this project, relative humidity dynamics have been neglected.

Figure 12 .

 12 Figure 12. Time profiles of the active EHA lumped temperatures on HOT (top) and ISA (bottom) days, for a sealed cavity (no air ingestion)

Figure 14 .

 14 Figure 14. EMA temperatures as influenced by cavity air ingestion, HOT day, all emission factors set to 0.9 Forced ventilation has a large impact on EMA motor, body and shaft temperatures, as illustrated with Figure 14.However what is interesting to note is that once some initial ventilation is introduced, its impact is fairly limited, hardly bringing final EMA temperatures below 59 to 60degC at the end of the simulation (160mins after aircraft mission end). This means that EMA temperature transients between "run 2" (5g/s, green curve) and "run 3" (10g/s, blue curve) can reasonably be assumed when designing the EMA and its constituents. Indeed, it is very likely that the cavity will not behave as 100% sealed throughout the entire flight, meaning that even under a reduced (conservative) ventilation assumption, EMA operating temperatures are expected to reach a 55degC maximum (for 10-15min peak

Figure 15 .

 15 Figure 15. Air flow lines in the cavity, colored by Temperature (K), 10g/s air ingestion

Table 1 :

 1 Hardware weight summary (partners)

  Interpretation of the MBSE and CFD results provided in Table2and Table3is of great interest, providing average differences around 6 to 7% for the values that should be compared, 5% as a minimum discrepancy, and 11% as a maximum one. The main contributors to these discrepancies are believed to be differences in local HTC values, and radiation view factors. Of course both predictions are expected to be more reliable in the CFD model, and will fed back to the MBSE model in due time.

	EHA	51.9	54.3	5%
	ECU	47	52.4	11%
	Cavity	15.1	45	198%
	Upper panel -internal	--	35.6	--
	Upper panel -external	-3	-4	33%
	Table 2 : CFD averaged temperatures, steady state end of
	cruise conditions, sealed cavity, HOT day
	Temperature (°C) CFD MBSE	∆
	Mean EHA	50.7	54.9	8%
	ECU	59.6	55.8	6%
	Cavity	50.5	53.6	6%
	Upper panel -internal	--	53.2	--
	Upper panel -external	50	50	0%
	Table 3 : CFD averaged temperatures, steady state at
	landing conditions, sealed cavity, HOT day

Table 5

 5

	: EHA temperatures as influenced by forced
	ventilation, analysis at landing, HOT day, 10g/s
	Temperature (°C) CFD MBSE	∆
	Mean EHA	50.8	40,9	19%
	ECU	57.6	44,2	23%
	Cavity	50.65	47,4	6%
	Upper panel -internal	--	47,3	--
	Upper panel -external	50	50	0%
	Table 6 : EHA temperatures as influenced by forced
	ventilation, analysis at landing, HOT day, 20 g/s

ACKNOWLEDGEMENTS

The authors wish to thank all the RESEARCH project partners for their participation and support.

The authors would also like to express their gratitude to the European Commission and Russian Ministry of Trade and Industry for their support to the RESEARCH project.

NOTATIONS


Vilhena Bruno

Adorno
email: adorno@ufmg.br

Philippe Fraisse

The cross-motion invariant group and its application to kinematics

Keywords: kinematics, unit dual quaternion, rigid motion, cross-motion invariance

) and the differences between this group and the standard group Spin(3) R 3 of unit dual quaternions, as well as the kinematic equations under a sequence of CMI(3) operations. Two numerical examples are presented in order to illustrate the main characteristics of CMI(3).

Introduction

The study of rigid motions is a classic topic in robot kinematics and is covered by virtually all modern robotics textbooks (for example, see [START_REF] Murray | A Mathematical Introduction to Robotic Manipulation[END_REF][START_REF] Craig | Introduction To Robotics, Mechanics and Control[END_REF][START_REF] Spong | Robot Modeling and Control[END_REF][START_REF] Siciliano | Springer Handbook of Robotics[END_REF][START_REF] Siciliano | Robotics: Modelling, Planning and Control. Advanced Textbooks in Control and Signal Processing[END_REF]. Its importance relies on the fact that rigid motions (or rigid transformations) play an important role in robot modelling and control [START_REF] Spong | Robot Modeling and Control[END_REF], as well as in motion planning and high-level task definitions [START_REF] Lana | A new algebraic approach for the description of robotic manipulation Tasks[END_REF], as most robots are composed of rigid components serially or parallelly attached to each other, resulting in complex kinematic chains. The group that represents rigid motions in three dimensions is the Special Euclidean Group SE(3) and it is well known that this group is basically the semi-direct product of SO(3) and R 3 [START_REF] Selig | Geometric Fundamentals of Robotic[END_REF]; that is, given R ∈ SO(3) and p ∈ R 3 such that X = (R, p) ∈ SE(3), then

X 1 X 2 = R 1 , p 1 R 2 , p 2 = R 1 R 2 , p 1 + R 1 p 2 .
(1)

Elements of SE(3) are usually written in the form of homogeneous transformation matrices (HTM); that is,

X = R p 0 1×3 1 ,
where 0 m×n is the zero matrix with m rows and n columns, such that

R 1 p 1 0 1×3 1 R 2 p 2 0 1×3 1 = R 1 R 2 p 1 + R 1 p 2 0 1×3 1 . 2 B.
V. ADORNO AND P. FRAISSE An alternative to the group SE(3) in the representation of rigid transformations is the group Spin(3) R 3 of unit dual quaternions, which double covers SE(3) [START_REF] Mccarthy | Introduction to Theoretical Kinematics[END_REF][START_REF] Selig | Geometric Fundamentals of Robotic[END_REF]. The advantage of using unit dual quaternions is that they are more compact than HTM, as the former has eight elements, whereas the latter has sixteen. In addition, dual quaternions have strong algebraic properties and can be used to represent rigid motions, twists, wrenches and several geometrical primitives-e.g. Plücker lines and planes-in a very straightforward way [START_REF] Daniilidis | Hand-eye calibration using dual quaternions[END_REF][START_REF] Radavelli | Points, lines, screws and planes in dual quaternions kinematics[END_REF]. Moreover, it is easy to extract geometric parameters from a given unit dual quaternion (translation, axis of rotation and angle of rotation) and dual quaternions multiplications are less expensive than HTM multiplications (Adorno, 2011, p. 42). Also, unit dual quaternions do not have representational singularities (although this feature is also present in HTM) and are easily mapped into a vector structure, which can be particularly convenient when controlling a robot, as there is no need to extract parameters from the dual quaternion to perform such task [START_REF] Adorno | Dual position control strategies using the cooperative dual taskspace framework[END_REF]. Thanks to the aforementioned advantages, unit dual quaternions will be used throughout the article in order to represent rigid motions.

Let us introduce some facts about quaternions and dual quaternions. Let î, ĵ , k be the three imaginary units such that î2 = ĵ 2 = k2 = î ĵ k = -1. The set of quaternions naturally extends the set of complex numbers and is defined as

H a + bî + c ĵ + d k : a, b, c, d ∈ R ,
whereas the set of pure quaternions is composed of all quaternions with real component equal to zero,

H p aî + b ĵ + c k : a, b, c ∈ R .
It is important to note that H p is isomorphic to R 3 under the addition operation; that is, R 3 , + ∼ = H p , + . The conjugate of a quaternion is analogous to the complex conjugate: given h = a + bî + c ĵ + d k, its conjugate is h * abî + c ĵ + d k . In addition, the quaternion norm is defined as h √ hh * . A unit quaternion (i.e. a quaternion with unit norm) r = cos(φ/2)+sin(φ/2)n represents the rotation angle φ around the rotation axis n = n x î + n y ĵ + n z k and form the group Spin(3) [START_REF] Kuipers | Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality[END_REF][START_REF] Selig | Geometric Fundamentals of Robotic[END_REF]. The identity element of Spin(3) is the real number 1, since 1r = r1 = r and the group inverse is the conjugate r * cos(φ/2)sin(φ/2)n, such that rr * = r * r = 1. Thanks to the isomorphism between R 3 and H p , pure quaternions represent translations and points in three dimensions, hence p = xî +y ĵ +z k represents the point with coordinates (x, y, z).

Elements of the group Spin(3) R 3 behave analogously to elements of SE(3), that is, given

x = (r, p) ∈ Spin(3) R 3 , then x 1 x 2 = r 1 , p 1 r 2 , p 2 = r 1 r 2 , p 1 + Ad (r 1 ) p 2 , ( 2
)
where Ad (r) p rpr * represents the point p rotated by r (it is important to recall that (2) is equivalent to (1)). However, instead of being represented by the pair (r, p), elements of Spin(3) R 3 are usually represented in a compact way by unit dual quaternions, which provides convenient ways of exploiting algebraic properties. The set of dual quaternions is defined as where ε is the nilpotent Clifford unit, also known as dual unit [START_REF] Selig | Geometric Fundamentals of Robotic[END_REF]. The dual quaternion conjugate of h = g + εh is h * g * + εh * and the dual quaternion norm is defined as h hh * . Usually g and h are called primary part and dual part of h, respectively, and we define the operators

H g + εh : g, h ∈ H, ε 2 = 0, ε = 0 ,

P h g,

D h h

(3) to retrieve those components from a dual quaternion.

Dual quaternions with unit norm are called unit dual quaternions and belong to the set

S h ∈ H : h = 1 .
Elements of S equipped with the multiplication operation represent elements of Spin(3) R 3 , the group of rigid motions that double covers SE(3). Let r ∈ Spin(3) and p ∈ H p , the unit dual quaternion corresponding to the translation p followed by the rotation r is given by x = r + ε(1/2)pr and the inverse motion is given by the dual quaternion conjugate x * , which corresponds to the group inverse such that x * x = xx * = 1 [START_REF] Selig | Geometric Fundamentals of Robotic[END_REF]. (Consequently, 1 is the group identity of Spin(3) R 3 .) Remarkably, the composition of rigid transformations is given by a sequence of dual quaternion multiplications. More specifically,

x 3 r 1 + ε 1 2 p 1 r 1 r 2 + ε 1 2 p 2 r 2 = r 1 r 2 + ε 1 2 r 1 p 2 r 2 + p 1 r 1 r 2 . ( 4 )
In order to see the equivalence between ( 4) and ( 2) first we start with the rotation, which is given by the primary part of the unit dual quaternion, P x 3 = r 1 r 2 (here we see that the P (•) operator, when applied to a unit dual quaternion, return elements of the subgroup of pure rotations). This is the same rotation of (2). To see that the translation of both equations is the same, we define the operator T (x) for x ∈ Spin(3) R 3 :

T (x) x P x * . ( 5
)
The T operator, when applied to the unit dual quaternion x, returns its translation in dual quaternion form. Thus, this operator returns elements of the subgroup of pure translations. For example, given

x = r + ε(1/2)pr, T (x) = x P x * = r + ε 1 2 pr r * = 1 + ε 1 2 p.
This way, applying the T operator to (4) we obtain

T x 3 = r 1 r 2 + ε 1 2 r 1 p 2 r 2 + p 1 r 1 r 2 (r 1 r 2 ) * = 1 + ε 1 2 Ad (r 1 ) p 2 + p 1 ,
which corresponds to the translation of (2). From (5) we see that x = T x P x , which means that if x represents the rigid transformation from a inertial frame F to frame F b then x can be interpreted as a sequence of two transformations: first a transformation T x that translates the origin of F to the origin of frame F b and then a transformation P x that rotates the translated frame in order to align it with F b , as shown in Fig. 1. Any subsequent rigid transformation, when performed by a right multiplication, uses F b as the reference frame for the transformation. However, in case a subsequent rigid transformation is performed by a left multiplication, the inertial frame F is used as reference frame for the transformation.

Motivation

Because of the construction of Spin(3) R 3 , the final translation after a sequence of rigid motions always depends on the intermediate rotations (note that this is true also for SE( 3)). This dependence, which is intrinsic to Spin(3) R 3 , can be called cross-motion variance, and has important consequences when describing rigid motions. For instance, some specific motions cannot be directly specified, without intermediate transformations, with respect to arbitrary frames when only Spin(3) R 3 operations are used.

In order to see this fact, consider a manipulator robot and let x 0 ∈ Spin(3) R 3 be the initial pose of the frame F E attached to the end effector with respect to the base frame F b (Fig. 2). A rigid motion m E ∈ Spin(3) R 3 can be specified with respect to the end-effector frame F E by performing a right multiplication on x 0 , that is x 1 = x 0 m E , where x 1 is the pose of the end effector after the motion m E . Alternatively, an equivalent motion m B can be specified with respect to the base frame F B to generate the same final end-effector pose; that is,

x 1 = m B x 0 . (6) Since x 0 m E = x 1 = m B x 0 , then m B = x 0 m E x * 0 . ( 7
)
Let us consider a motion m E r E that consists of just a rotation of the end effector using F E as the reference frame. The final end-effector pose is

x 1 = x 0 r E (8)
and T x 1 = T x 0 , which implies that the end-effector position does not change. However, if the motion is to be described with respect to the base frame but resulting in the same final end-effector pose, the solution given by ( 7) is Fig. 2. Common frames in manipulator robots: F B is the base frame and F E is the end-effector frame. The unit dual quaternion x 0 corresponds to the initial pose of the robot end effector with respect to the base frame.

x 1 = x 0 r E x * 0 x 0 , (9)
which yields exactly (8); that is, using the group operation of Spin(3) R 3 , a movement of rotation without translation must be essentially specified with respect to the end-effector frame. Otherwise, if the movement is specified directly with respect to the base frame, one must also take into account the coupling between rotations and translations and then specify a suitable translation in order to maintain the end-effector position constant, which can be quite counterintuitive.

To better clarify this point, consider that the desired motion is a pure rotation m B = r B of the end effector but defined with respect to F B . The result of m B x 0 is not the desired one, since T m B x 0 = 1 + ε (1/2) Ad (r B ) p 0 , which is different from T x 0 = 1 + ε (1/2) p 0 (although at least the final orientation is the desired one). Actually, the desired value for m B such that

T m B x 0 = T x 0 is m B = r B + ε (1/2) p 0 -Ad (r B ) p 0 r B .
Not only this transformation is counterintuitive but it also has a fundamental problem since it depends on the initial position of the end effector and this position may be uncertain. This way, because of the uncertainty of p 0 the desired value of m B is also uncertain, which does not happen when the task is defined at the end-effector frame.

It is important to highlight that those issues arise not only in the context of manipulator robots but in any type of kinematic chain. It suffices to have different coordinate systems in which some of them are inertial and others are moving and motions must be specified in different frames, which is very typical in cooperative robot systems. In this context, it would be much more convenient if there existed a more intuitive operation, for example ⊗, acting on Spin(3) R 3 operands such that rigid motions could be defined more intuitively, independently of the choice of coordinate system; for instance, it would be convenient to describe the motion of rotation without translation, defined at the base frame, as

x 1 = r B ⊗ x 0 = 1 + ε (1/2) p 0 r B r 0 . More generally, given x 1 = r 1 + ε (1/2) p 1 r 1 and x 2 = r 2 + ε (1/2) p 2 r 2 , we seek an operation ⊗ such that x 1 ⊗ x 2 = r 1 r 2 + ε 1 2 p 1 + p 2 r 1 r 2 . 6 B. V. ADORNO AND P. FRAISSE
In terms of groups, this is achievable simply by taking the direct product of Spin(3) and R 3 , that is Spin(3) × R 3 . However, more than just a convenient notation such operation should provide useful features to describe motions that cannot be easily described using Spin(3) R 3 operations. Moreover, this new operation should be complementary to the standard one in order to enrich the algebra of rigid motions.

In this article, we propose a new group, the Cross Motion Invariant group CMI( 3), which has the following characteristics: (i) rotation operations do not affect translations and vice-versa (cross-motion invariance), (ii) motions are invariant with respect to the position of the reference frame, (iii) translations under CMI(3) operations commute with arbitrary rigid motions and (iv) the direction of trajectories is preserved under CMI(3) operations. Furthermore CMI(3) operations can be naturally mixed with Spin(3) R 3 operations, leading to a richer algebra and an abstraction level that can be quite useful when describing rigid motions. Also, we derive the kinematic equation of a sequence of rigid motions under the CMI(3) group operation.

This article is organized as follows: the next section presents the group CMI( 3) and its underlying operation, the decompositional multiplication, along with the corresponding geometrical interpretation. Section 3 presents the kinematic equation of a sequence of rigid motions under CMI(3) operations, whereas Section 4 presents two numerical examples to illustrate the behaviour of rigid motions under CMI(3) operations. Lastly, Section 5 concludes the article.

The group CMI(3) and decompositional multiplications

Before formally introducing the group CMI( 3), let us first recall that a group is a manifold G equipped with an identity element e and two operations

mult : G × G → G, inv : G → G,
such that mult, also known as the group operation, must be associative and obey the closure property and inv must be a bijection [START_REF] Selig | Geometric Fundamentals of Robotic[END_REF]. So, in order to define the group CMI(3) first we must establish the underlying manifold of unit dual quaternions and then we must define the group operation, the group identity and the group inverse.

The underlying manifold of unit dual quaternions is S 3 × R 3 , where S 3 is the unit 3-sphere in fourdimensional space and thus the set S of unit dual quaternions 'is a six-dimensional algebraic submanifold of R 8 ' [START_REF] Mccarthy | Introduction to Theoretical Kinematics[END_REF]. Since CMI( 3) is a group acting on unit dual quaternions, thus G S 3 × R 3 . The group operation of CMI( 3) is denoted by ⊗ to distinguish it from the standard multiplication of Spin(3) R 3 and is defined in the following.

Definition 1 Given x a , x t ∈ S, the decompositional multiplication, represented by ⊗, is defined as

x a ⊗ x t T x a T x t P x a P x t . ( 10
)
Lemma 1 (Closure) The set S is closed under the decompositional multiplication.

Proof. Let x a , x t ∈ S and consider x f x a ⊗ x t = T x a T x t P x a P x t , then

x f x * f = T x a T
x T x * = 1, hence x f x * f = 1, which implies x f = 1. Thus x f ∈ S. Lemma 2 Let x 1 , x 2 ∈ S, then T x 1 ⊗ x 2 = T x 1 T x 2 and P x 1 ⊗ x 2 = P x 1 P x 2 .
Proof. Considering that a unit dual quaternion x i can always be written as

x i = r i + ε (1/2) p i r i , then x 1 ⊗ x 2 = 1 + ε 1 2 p 1 1 + ε 1 2 p 2 r 1 r 2 = r 1 r 2 + ε 1 2 p 1 + p 2 r 1 r 2 .
Thus,

T x 1 ⊗ x 2 = 1 + ε 1 2 p 1 + p 2 = T x 1 T x 2 and P x 1 ⊗ x 2 = r 1 r 2 = P x 1 P x 2 .
Lemma 3 (Associativity) The decompositional multiplication is associative.

Proof. Considering x 1 , x 2 , x 3 ∈ S and using Lemma 2,

x 1 ⊗ x 2 ⊗ x 3 = T x 1 ⊗ x 2 T x 3 P x 1 ⊗ x 2 P x 3 = T x 1 T x 2 T x 3 P x 1 P x 2 P x 3 = T x 1 T x 2 ⊗ x 3 P x 1 P x 2 ⊗ x 3 = x 1 ⊗ x 2 ⊗ x 3 .
Lemma 4 (Group identity) The identity element under the decompositional multiplication is the real number 1.

Proof. First, it is clear that 1 ∈ S, since R ⊂ C ⊂ H ⊂ H and 1 = 1. Also, P (1) = 1 hence T (1) = 1. Thus, considering any x ∈ S we have that x ⊗ 1 = 1 ⊗ x = T x P x = x.
Lemma 5 (Group inverse) Given x ∈ S, the inverse operation under decompositional multiplication is a bijective mapping given by

x † = T x * P x * . ( 11
)
Proof. Since T x † = T x * and P x † = P x * , then we verify by direct calculation that

x † ⊗ x = x ⊗ x † = 1. 8 B.
V. ADORNO AND P. FRAISSE

In addition, we see that

x † † = T x † * P x † * = T x P x = x. ( 12
)
In order to show that the inverse operation is injective, consider x 1 , x 2 ∈ S such that

x 1 = x 2 but x † 1 = x † 2 .
Inverting both sides of the equality gives x † 1 † = x † 2 † , which implies from (12) that x 1 = x 2 , which is a contradiction. Hence the inverse operation under decompositional multiplication is injective.

In order to show that the inverse operation is surjective, let us assume that there exists a unit dual quaternion a in the codomain of the inverse operation but not in its image. Because ( 11) is defined for all unit dual quaternions, there is a unit dual quaternion a in the domain such that a † = b. Since the domain also contains b, then b † = a † † = a; that is, a must be in the image of the inverse operation, contradicting the initial claim that a is in the codomain of the inverse operation, but not in its image, thus implying surjectivity.

Theorem 1 The set S of unit dual quaternions under the decompositional multiplication form a group.

Proof. First, S is a manifold (S 3 × R 3 , see [START_REF] Mccarthy | Introduction to Theoretical Kinematics[END_REF] and the decompositional multiplication is closed and associative (Lemmas 1 and 3). In addition, there is an identity element under the decompositional multiplication (Lemma 4) and the inverse operation is a bijection (Lemma 5), consequently unit dual quaternions under decompositional multiplication form a group, which is named CMI(3).

Geometrical interpretation of CMI(3) operations

Note that both operators T (•) and P (•) return unit dual quaternions, the former corresponding to a pure translation and the latter corresponding to a pure rotation. This way, the decompositional multiplication corresponds to a rigid motion that behaves differently from the rigid motion performed by the standard multiplication of Spin(3) R 3 , which is given by

x a x t = T (x a ) P x a T (x t ) P x t . (13)
In order to illustrate this difference, let x t and x a be unit dual quaternions such that x t represents a target frame F t (which could be attached to any relevant rigid body, for example an object to be manipulated or the robot end effector) with respect to the inertial frame F and x a represents a transformation that will be applied on x t . In addition, let T (x t ) = 1 + ε ĵ d /2 and P x t = cos (π/4) + k sin (π/4); that is, considering the reference frame F, x t is the pose of the target frame obtained after a translation d along the y axis followed by a rotation of π/2 around the z axis. Also, consider T (x a ) = 1 + ε ĵ d /2 and P x a = cos (π/4) + î sin (π/4); that is, the transformation x a corresponds to a translation d along the y axis and a rotation of π/2 around the x axis.

The reference frame for the transformation x a defines if it corresponds to a left multiplication or a right multiplication: if the target frame is used as reference, right multiplication is used (i.e. x t x a in case of the standard multiplication and x t ⊗ x a in case of the decompositional multiplication); conversely, if the inertial frame is the reference for the transformation, left multiplication is used (i.e. x a x t in case of the standard multiplication and x a ⊗ x t in case of the decompositional multiplication).

Assuming that the desired transformation is given with respect to the inertial frame F, the resulting pose under the standard Spin(3) R 3 multiplication, given by ( 13), can be interpreted as follows. Given x t (see step 1 in Fig. 3a), first we rotate it by applying P x a T x t P x t resulting in the intermediate frame F i (step 2 in Fig. 3a). Since this is a left multiplication, the x-axis of the inertial frame F is used for the transformation (see bent contour arrow in Fig. 3a). Then we translate F i by applying the last transformation, T x a P x a T (x t ) P x t , which results in the final frame F f (step 3 in Fig. 3a). Again, since this is a left multiplication, the y axis of the inertial frame F is used for the transformation. On the other hand, the resulting pose under the CMI(3) multiplication given by ( 10) can be interpreted as follows. Given x t = T x t P x t (see step 1 in Fig. 3b), first we rotate it by applying T x t P x a P x t resulting in the intermediate frame F i (see step 2 in Fig. 3b). Differently from the standard multiplication, the transformation P x a acts locally, as the reference for this transformation is a frame aligned with F but with the same origin of F t . (Note that T x t 'brings' the origin of the reference frame to the origin of F t without changing the alignment, as T (•) corresponds to a pure translation.) The result is that the orientation of the resulting frame F i is different from the orientation of F t but the position is the same. Finally, F i is translated by applying the last transformation T x a T x t P x a P x t resulting in the final frame F f (see step 3 in Fig. 3b). Again, since this is a left multiplication, the y axis of the inertial frame F is used for the transformation.

Remark 1 Note that in Fig 3a the position of the final frame clearly depends on the rotation P x a . Even if the action of x a on x t were applied by the right side (i.e. x f = x t x a ), the final position of F f would depend on the initial orientation of F t . The influence of rotation on translation can be called cross-motion variance. This is a property intrinsic to the group Spin(3) R 3 (and also of SE( 3)). On the other hand, decompositional multiplication is cross motion invariant, meaning that the translation will never be affected by the rotation part of the operation and vice versa, as shown in Fig. 3b. If the action were applied by the right (i.e. x f = x t ⊗ x a ), the initial orientation of F t would not affect the final position of F f . This is a property that cannot be achieved by using the standard group operation of Spin(3) R 3 (and also of SE( 3)).

Remark 2 For any transformation under the CMI(3) multiplication, there is a corresponding sequence of transformations given by standard Spin(3) R 3 multiplications. For instance, consider a target frame F t with the corresponding unit dual quaternion x t and a transformation y ∈ CMI(3), such that the final transformation is given by x f = y⊗x t . Now, consider that we want to find a corresponding transformation z ∈ Spin(3) R 3 that results in the same final modified frame, that is, x f = zx t . Consequently,

zx t = y ⊗ x t =⇒ z = y ⊗ x t x * t = T y T x t P y P x t P x t * T x t * = T y T x t P y T x t * . ( 14
)
This way, both operations can be used together because the result is always a unit dual quaternion. The choice of a particular operation depends mostly on the desired final transformation and which one provides a more convenient geometrical interpretation for a specific application. This will be better illustrated in Section 4.

Example 1 In Section 1, we showed that a motion of rotation without translation can be defined directly with respect to the end-effector frame (that is, x 1 = x 0 r E ), but when this motion is specified with respect to the base frame and Spin(3) R 3 operations are used, then a transformation must be performed involving terms of the target frame (i.e. x 1 = x 0 r E x * 0 x 0 ). On the other hand, if CMI(3) operations are used, the specification of this motion is much more straightforward. For instance, let us define a pure rotation with respect to the base frame as r B . When using Spin(3) R 3 operations the final position will be different from the initial position, because for x 0 = r 0 + ε(1/2)p 0 r 0 the new position will be

T r B x 0 = 1 + ε 1 2 r B p 0 r * B ,
which is clearly different from the initial position given by T x 0 = 1 + ε(1/2)p 0 . On the other hand, if the group operation of CMI( 3) is used, the motion of rotation without translation can be easily defined directly with respect to the base frame (left action):

x 1 = r B ⊗ x 0 = T (r B ) T x 0 P (r B ) P x 0 = T x 0 r B r 0 .
In this case, the rotation r B is specified with respect to the base frame but applied locally, thus the position does not change; that is, T x 1 = T x 0 .

Kinematic equation and additional properties of CMI(3)

Given x ∈ Spin(3) R 3 , the kinematic equation is given by

ẋ = 1 2 ξ x, ( 15
)
where ẋ is the time derivative of x = r + ε (1/2) pr (recall that r ∈ Spin(3) and p ∈ H p ) and ξ = ω + ε (v + p × ω) is the twist expressed in the inertial frame [START_REF] Wu | Strapdown inertial navigation system algorithms based on dual quaternions[END_REF]. In addition, ω, v ∈ H p are pure quaternions that represent the angular and linear velocity, respectively, and p × ω is the cross product defined in quaternion form as

p × ω pω -ωp 2 . ( 16
)
Thanks to the isomorphism R 3 , + ∼ = H p , + , the cross-product ( 16) between pure quaternions yields the same result of the cross product between the corresponding vectors in R 3 . If x is the result of the composition of several rigid motions (e.g. the combination of several intermediate rigid motions of a serial kinematic chain), that is x = x 1 • • • x n , then the resultant twist is the sum of all individual twists expressed in the inertial frame [START_REF] Adorno | Two-arm Manipulation: From Manipulators to Enhanced Human-Robot Collaboration[END_REF]. It is important to recall that, due to the coupling between rotation and translation of Spin(3) R 3 , the resultant linear velocity is strongly influenced by the angular velocities of intermediate elements of the kinematic chain.

Hence, we seek the twist associated to a composition of rigid motions under the CMI(3) operation, which tends to be much simpler as motions are decoupled. More specifically, we want to find the twist ξ d that satisfies

ẋ = 1 2 ξ d x, ( 17
)
where x = x 1 ⊗ • • • ⊗ x n .
Before deriving the final result, first we introduce some auxiliary lemmas.

Lemma 6 A sequence of decompositional multiplications always result in a decomposed movement, that is

x 1 ⊗ x 2 ⊗ . . . ⊗ x n = n l=1 T x l n l=1 P x l .
Proof. From Lemmas 2 and 3 we obtain

x 1 ⊗ x 2 ⊗ x 3 ⊗ . . . ⊗ x n = x 1 ⊗ x 2 ⊗ x 3 ⊗ . . . ⊗ x n = T (x 1 ⊗ x 2 )T (x 3 ) P x 1 ⊗ x 2 P x 3 ⊗ . . . ⊗ x n . . . = T (x 1 ) . . . T (x n ) P x 1 . . . P x n .
Lemma 6 is important because a sequence of transformations under the decompositional multiplication preserves the cross-motion invariance, since the decomposition propagates through the whole transformation.

Lemma 7 Given x ∈ S and h ∈ H, then T x (εh) = (εh) T x = εh.

Proof. Since x has unit norm, then T x = 1 + ε(1/2)p and the result is obtained by direct calculation using the fact that ε 2 = 0.

Lemma 8 Given x 1 , . . . , x n ∈ S, d dt n l=1 P x l = 1 2 n m=1 ω m n l=1 P x l ,
where

ω m = Ad m-1 l=1 P x l ω m and ω m ∈ H p satisfies P ẋm = (1/2) ω m P x m .
Proof. From ( 15), we see that the kinematic equation of only the primary part is P x l = P ẋl = (1/2) ω l P x l , where P x l d dt P x l -this is commonly known as the quaternion propagation equation [START_REF] Kuipers | Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality[END_REF]. Hence Theorem 2 Given x ∈ CMI( 3)

such that x = x 1 ⊗ • • • ⊗ x n with x l = r l + ε (1/2) p l r l , the twist that satisfies the kinematic equation ẋ = 1 2 ξ d x is ξ d = ω d + ε v d + p d × ω d
, where the total linear velocity is v d = n l=1 v l , the total translation is p d = n l=1 p l and the total angular velocity is where

ω d = n m=1 ω m , where ω m = Ad m-1 l=1 P x l ω m . Proof. Since x = x 1 ⊗ • • • ⊗ x n ,
ω m = Ad m-1 l=1 P x l ω m . Since T x m T x l = T x l for any x m , x l , hence ẋ = n l=1 T x l n l=1 T x l + 1 2 n m=1 ω m n l=1 P x l = 1 2 n l=1 T x l n m=1 ω m + ε n l=1 v l n l=1 P x l = 1 2 n l=1 T x l [ω d + εv d ]
ξ d = Ad n l=1 T x l ξ d = n l=1 T x l (ω d + εv d ) n l=1 T x l * = 1 + ε n l=1 p l 2 (ω d + εv d ) 1 -ε n l=1 p l 2 = ω d + ε v d + p d × ω d .
It is important to note that the twist ξ d given in Theorem 2 is in fact simpler than the one of ( 15), since the translational components p d and v d are just the sum of the contributions of any intermediate translations and velocities, respectively, without any influence of the intermediate rotations and angular velocities. This is not the case for Spin(3) R 3 . The resultant angular velocity, however, is the same for both groups.

Additional properties of CMI(3)

Lemma 9 (Commutativity with translation)

T (x 1 ) ⊗ x 2 = x 2 ⊗ T (x 1 ) = T (x 1 )x 2 .
Proof. Since T x 1 T x 2 = T x 2 T x 1 and P T x 1 = 1, the result follows from direct calculation.

Proposition 1 The left action x a on the target frame x t , under the decompositional multiplication, is independent of the position of the reference frame.

Proof. Let us consider a left action x ref a , specified with respect to a reference frame F ref , which not necessarily is the inertial frame F. This action will be applied to a target frame F t , represented by x t , where x t is specified with respect to F. The final transformation, with respect to F ref , is given by

x ref f = x ref a ⊗ x ref t , (18)
where

x ref t = x ref x t and x ref is the transformation from F ref to F.
If the final transformation is to be specified with respect to F, then (18) becomes

x f = x ref * x ref a ⊗ x ref t = Ad P x ref* x ref a ⊗ x t , (19)
where left action x a on the target frame x t , under the decompositional multiplication, is independent of the position of the reference frame.

x
Proposition 2 The left action x a on the target frame x t , under the standard Spin(3) R 3 multiplication, is dependent of the position of the reference frame.

Proof. The proof is similar to the one of Proposition 1. Since

x ref f = x ref a x ref t , then x f = x ref * x ref a x ref t = x ref * x ref a x ref x t = Ad x ref * x ref a x t . (20)
In this case, since the transformation Ad x ref * x ref a takes into account both position and orientation information of x ref , we conclude that the left action x a on the target frame x t , under the standard Spin(3) R 3 multiplication, depends on the position of the reference frame.

The last two propositions show an interesting property of the decompositional multiplication when compared to the standard Spin(3) R 3 multiplication. When transforming a frame F t using Spin(3) R 3 operations and specifying a left action x ref a with respect to a reference frame different from the inertial frame F, uncertainties in the position of this reference frame will affect the final transformation. On the other hand, if this left action is applied using the decompositional multiplication, uncertainties in the position of the reference frame will not affect the final transformation; in fact, the position of the reference frame is completely irrelevant.

Number of elementary operations involved in the computation of CMI(3) operations

In this section we present the number of elementary operations (addition and multiplication) involved in the calculation of both the standard multiplication of Spin(3) R 3 and the decompositional multiplication of CMI(3).

The cost of multiplying two unit dual quaternions x 1 and x 2 under the standard multiplication is equivalent to the cost of the following operation

x 1 x 2 = P x 1 P x 2 + ε P x 1 D x 2 + D x 1 P x 2 ,
where P (•) and D (•) are given by (3). Hence, the resulting cost is the cost of three multiplications of quaternions (one multiplication for the primary part and two multiplications for the dual part) and one addition of quaternion for the dual part. According to [START_REF] Adorno | Two-arm Manipulation: From Manipulators to Enhanced Human-Robot Collaboration[END_REF], the multiplication of quaternions requires 16 multiplications of real numbers and twelve additions of real numbers, whereas the addition of quaternions requires four additions of real numbers. This way, the total cost of multiplying two dual quaternions under the standard multiplication of Spin(3) R 3 is 48 multiplications of real numbers and 40 additions of real numbers.

On the other hand, the cost of multiplying two unit dual quaternions x 1 and x 2 under the decompositional multiplication of CMI(3) corresponds to the cost of the following operation

x 1 ⊗ x 2 = T x 1 T x 2 P x 1 P x 2 = x 1 P x 1 * x 2 P x 2 * P x 1 P x 2
Table 1 Main differences between Spin(3) R 3 and CMI( 3)

Spin(3) R 3 CMI(3)
Cross-motion variant (rotation affects translation)

Cross-motion invariant (rotation does not affect translation) Translation does not commute with arbitrary rigid motions: T (x 1 )x 2 = x 2 T (x 1 )

Translation commutes with arbitrary rigid motions: T (x 1 ) ⊗ x 2 = x 2 ⊗ T (x 1 ) Transformation depends on the position of the reference frame

Transformation does not depend on the position of the reference frame Does not preserve the direction of transformed trajectories

Preserves the direction of transformed trajectories More complex kinematic equations for a sequence of rigid motions Less complex kinematic equations for a sequence of rigid motions Standard multiplication requires 48 multiplications of real numbers and 40 additions of real numbers Decompositional multiplication requires 67 multiplications of real numbers and 52 additions of real numbers

= P x 1 + ε D x 1 P x 1 * P x 2 + ε D x 2 P x 2 * P x 1 P x 2 = P x 1 P x 2 + ε D x 2 P x 2 * P x 1 + D x 1 P x 2 ,
which requires four multiplications of quaternions (one multiplication for the primary part and three multiplications for the dual part), one addition of quaternion and one conjugation of quaternion. Thus, the total cost of multiplying two unit quaternions under the decompositional multiplication of CMI( 3) is 67 multiplications of real numbers (the conjugation of a quaternion requires three multiplications of real numbers) and 52 additions of real numbers. These results are summarized in Table 1.

Numerical examples

In this first example, consider a trajectory expressed in the inertial frame F, given by h 4, whereas g(t) is represented by a solid-green trajectory and f (t) is represented by a dashedred curve. Because the group operation used to generate g(t) belongs to the cross motion invariant group, translations do not affect rotations and vice-versa. This way, both h(t) and g(t) go upward independently of the orientation and the decompositional multiplication preserves the direction of the trajectory. On 3) transformation (solid green) and modified trajectory f (t) using Spin(3) R 3 transformation (dashed red). F is the fixed frame and F ref is the reference frame for the transformation. Observe that CMI(3) transformation preserves the direction of the trajectory whereas Spin(3) R 3 transformation does not preserve the direction. the other hand, f (t) is transformed using Spin(3) R 3 operations and thus the dashed-red curve is rotated with respect to the original trajectory, hence the final trajectory has a different direction from the original one.

(t) = r(t) + ε(1/2)p(t)
In this second example, consider a 7-DOF robot manipulator and the associated coordinate systems as shown in Fig. 5, where F and F e are the inertial and end-effector frames, respectively. Considering the initial end-effector pose given by the unit dual quaternion x e , suppose that we want to define a desired rigid motion using the inertial frame F as the reference frame. This rigid motion consists of a displacement of 0.1 m in the positive direction of the x axis and 0.3 m in the negative direction of the y axis-that is, p = 0.1î -0.3 ĵ . In addition, we want the end effector to be locally rotated of -π/2 rad around the z axis-that is, r = cos (-π/4) + k sin (-π/4). It means that we want to rotate the end effector by using the z axis of a frame that is aligned with the inertial frame but whose origin coincides with the origin of the end-effector frame. The desired rigid motion is given by x a = r + ε (1/2) pr and, since we want the rotation to be performed locally (that is, decoupled from the translation), the decompositional multiplication is very convenient, as the final desired pose is given by

x d = x a ⊗ x e
and is represented as frame F d in Fig. 5. Note that, because rotation and translation are decoupled, we can define them separately without worrying about translations being affected by rotations. On the other hand, if we consider x a to perform the standard multiplication, the final pose is given by and is represented as frame F s in Fig. 5. Since the standard multiplication performs a coupled motion, we see that the final rotation is the same, but the final translation is not the intended one. Of course, it is possible to find an equivalent transformation y such that yx e = x a ⊗ x e as shown in Remark 2. However, the advantage of using the decompositional multiplication is that it is usually much more intuitive to think about a decoupled motion to define the desired pose for the end effector.

x s = x a x e ,

Conclusions

This article presented the group CMI(3) and its mathematical properties, as well as some examples in order to show its application to kinematics and to the study of rigid motions in general. Also a thorough and formal comparison with the group Spin(3) R 3 has been presented, as summarized in Table 1. Because both CMI(3) and Spin(3) R 3 operate on the same underlying structure-the unit dual quaternions-the idea is to use both in a complementary way, leading to a richer algebra of rigid motions. For instance, CMI(3) operations can be, in some cases, more convenient than Spin(3) R 3 operations. Because CMI(3) group operation guarantees cross-motion invariance such that rotations and translations do not affect each other, these motions can be specified separately. On the other hand, in some cases the goal is to really have the position affected by the rotation movement; in those cases, Spin(3) R 3 operations would be more appropriate.

The kinematic equation was derived for a sequence of rigid motions under the decompositional multiplication. resultant twist is in fact simpler for CMI(3) than for Spin(3) R 3 , since in CMI(3) the translational components are just the sum of the contributions of any intermediate translations and linear velocities, without any influence of the intermediate rotations and angular velocities, respectively. This is not the case for Spin(3) R 3 . The resultant angular velocity, however, is the same for both groups.

Lastly, although the goal of the article was to formally present the mathematical description of the CMI group and only few illustrative examples were shown, this group and its underlying group operation-the decompositional multiplication-can be quite useful in practice. For instance, our implementation of the decompositional multiplication, both in Matlab and C/C++ (see dqrobotics.sourceforge.net, last accessed on May 10, 2016), was already used in the context of humanrobot interaction and bimanual manipulation in order to simplify the definition of tasks [START_REF] Adorno | Towards a cooperative framework for interactive manipulation involving a human and a humanoid[END_REF].

Funding

Agence Nationale de la Recherche (ANR) (ANR 07-ROBO-01); Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) (APQ-00967-14) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (456826/2013-0).

Fig. 1 .

 1 Fig. 1. The unit dual quaternion x = T x P x represents the rigid transformation from the inertial frame F to F b .

Fig. 3 .

 3 Fig. 3. F is the inertial frame, F i is an intermediate frame and F t is the target frame. The transformation T x a corresponds to a translation d along the y axis and P x a corresponds to a rotation of π/2 around the x axis. The contour arrows are the ones used to perform the translation and rotation. Each numbered circle corresponds to one intermediate step of the complete transformation.

  the time derivative of T x l -denoted by T x l d dt T x l -is T x l = ε (v l /2), where v l ṗl so we use Lemmas 7 and 8 to obtain

  . ADORNO AND P. FRAISSE Let p d = n l=1 p l and recall (16) to obtain

Fig. 4 .

 4 Fig. 4. Original trajectory h(t)(black dotted), modified trajectory g(t) using CMI(3) transformation (solid green) and modified trajectory f (t) using Spin(3) R 3 transformation (dashed red). F is the fixed frame and F ref is the reference frame for the transformation. Observe that CMI(3) transformation preserves the direction of the trajectory whereas Spin(3) R 3 transformation does not preserve the direction.

Fig. 5 .

 5 Fig.5. of different motions when performing CMI(3) and Spin(3) R 3 operations. F and F e are the inertial and endeffector frames, respectively, whereas F d is the final frame obtained by using the CMI(3) decompositional multiplication and F s is the final frame obtained by using the Spin(3) R 3 standard multiplication.

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  using the frame F ref as the reference for the action, where p ref = 2.5î + ĵ +3.5 k and r ref = cos (π/2)+ k sin (π/2).In CMI(3), when the transformation is specified with respect to F ref , but the trajectory is expressed in the inertial frame F, the transformed trajectory is given by g(t) = Ad P h ref href ref provides the rigid transformation from F to F ref (see (19) of Proposition 1). When the transformation h ref a belongs to Spin(3) R 3 and is specified with respect to F ref , but the trajectory is expressed in F, the transformed trajectory is given by f (t) = Ad h ref h ref

	a	h(t)-see (20) of Proposition 2. Note that,
	although the action h ref a is specified with respect to F ref , both original and modified trajectories are
	specified with respect to the fixed frame. The original trajectory h(t) is represented by a black-dotted
	curve in Fig.

r(t), where p = î cos t + ĵ sin t + kt and r = cos (t/2) + î sin (t/2). The goal is to apply the transformation h ref a , where p ref a = î + ĵ and r ref a = cos (π/4) + ĵ sin (π/4) , to the trajectory h(t) a ⊗ h(t), where h

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Acknowledgements

The authors would like to thank Prof. B. Siciliano, Prof. M. Uchiyama and Prof. AP Murray for their helpful remarks regarding the decompositional multiplication.