LEAPING IN LOW GRAVITYModeling MASCOT’s hopping locomotion on asteroid Ryugu

R. Lichtenheldt*, J. Reill**

*German Aerospace Center, Institute of System Dynamics and Control, DLR, Germanye-mail: Roy.Lichtenheldt@dlr.de

**German Aerospace Center, Institute of Robotics and Mechatronics, DLR, Germany

Abstract

DLR’s lander MASCOT is an innovative system to ex-plore and traverse asteroid surfaces. Launched piggybackon JAXA’s Hayabusa II spacecraft in December 2014,MASCOT is already on its four years cruise phase tothe asteroid Ryugu-1999JU3. Even though MASCOTfeatures a 1-DOF mobility actuator only, it has to dealwith the complex interaction of the cuboid lander withthe terrain. Thus, a certain orientation of MASCOT can-not be achieved directly. Hence, the mobility unit de-veloped by DLR’s Robotics and Mechatronics Center en-ables MASCOT to up-right to the measurement positionand to relocate by hopping motion. In this article, theoptimization-based technique used to identify suitable androbust trajectories of the mobility unit is explained and ex-emplified for hopping and up-righting.

1 INTRODUCTION

The earlier exploration of our solar system mostly fo-cused on our moon and neighboring planets. In contrastasteroids, comets and small planetary bodies in generalare not yet well investigated. In the recent time explo-ration of these bodies has become further demanded andpopular (e.g. [1]). MASCOT is DLR’s lander [2] onboard of JAXA’s Hayabusa II mission to Ryugu-1999JU3,launched in December 2014. It is a cuboid system ofroughly 10 kg, whereby moret han 35% of this mass is sci-entific payload. Achieving an exceptionally high payloadrate was one of MASCOT’s goals right from the begin-ning. In order to allow the instruments to work, a certainlander orientation needs to be achieved. However, due tolow gravity and the resulting bouncing after landing, thisposture cannot be achieved directly after landing. ThusMASCOT features a novel locomotion system for hoppingand up-righting to measurement position [3]. The loco-motion system called mobility unit is an internal control-lable rotor with an excentrical rotor mass. As the depen-dence of the desired hopping trajectory on the motion pro-file of the mobility unit arm is complex, a suitable trajec-tory cannot be determined analytically or by experimentin advance. Even in parabolic flight campaigns the low-

gravity phases are not sufficiently long for trajectory tun-ing. As it is not possible to define the trajectories based onreal prototyping, a multibody model in conjunction with amathematical description of the asteroid’s terrain geome-try and gravity field is developed. Throughout the articlethe modeling approaches for MASCOT itself as well asthe asteroid’s model will be explained. The model fea-tures a description of MASCOT’s kinematics, dynamicsand contact mechanics to the asteroid surface in order torate a-priori created trajectories. Applying the modelingapproach to multi-objective optimization allows for thesystematic search for suitable trajectories in an automatedprocess. Using the optimization framework MOPS [4] de-veloped by DLR-SR a toolkit for trajectory optimizationon low gravity bodies is developed. Throughout the ar-ticle, the toolkit, the objectives as well as the stepwiseoptimization approach [5] will be explained. For hop-ping the optimization based technique is exemplified byfinding a trajectory for a maximum hopping distance. Forup-righting the most important objectives are short bounc-ing time and the binary goal, defined by the equilibriumorientation of MASCOT. Nevertheless up-righting needsto be as robust as possible against parameter deviationsexerted either by e.g. inhomogeneous regolith or controlinaccuracies due to the harsh environmental conditions.Non-robust solutions might cause MASCOT to land onthe wrong face and require repeated up-righting and thusshorter periods for the scientific measurements.

2 MASCOT MOBILITY

The mobility subsystem consists of an actuator unit withan excentric arm (MobUnit), a controller and power elec-tronics PCB (MobCon) and an additional hall sensor PCBto detect the reference position of the excentric arm. Fig.1shows the main components before integration into theMASCOT electronics box. Since most of the scientific in-struments operations are dependent on MASCOT’s orien-tation on asteroid surface, the mobility subsystem is veryimportant for mission success. Therefore, every compo-nent was realized as redundant as possible. Nevertheless,due to space and weight limitations the up-righting and

Figure 1. : Flight model photo image that shows mobilitymotor connected to the controller and power electronicsboard

hopping maneuvers need to be performed with a single,non-redundant motor. Because of that a brushless DC mo-tor was chosen as it is a reliable and lightweight solutionthat offers also offers a high peak torque output. This maybe helpful to overcome friction and cold welding effectsin mechanics as well as to put high jerk to the system ifneeded. Together with a Harmonic Drive gearing the de-veloped actuator is very compact. Compared to DC mo-tors the brushless DC motor needs less mechanical partsand has no brushes at all. The commutation is realized bypower electronics and hall sensor information. The mobil-

Figure 2. : Flight model photo image that shows mo-bility motor and the excentric arm (l.), Orientations ofMASCOT (r.)

ity electronics is set up completely cold redundant and puton a single PCB. Each redundancy path is able to drivethe motor unit even if there might occur a failure in theother path. A special coupling network was developed toconnect two power electronics circuits to one single mo-tor [3]. Depending on housekeeping data and error mes-sages the OBC decides which redundancy path is to beused and powered. The communication to OBC, sensordata collection and interpretation, control of motor trajec-tory, computation of absolute position and safety issuesis all handled by a radiation hardened Microsemi FPGA.This FPGA needs to drive the motor power MOSFETs byuse of high and low side gate drivers. As the radiation tol-erant gate drivers did not fit on the restricted electronicsboard size an industrial BLDC motor controller was con-sidered. Of course the motor controller has successfullyundergone several radiation tests (see [6]) before.

3 MODELING OF THE LOCOMOTIONSYSTEM

In order to determine the trajectories for MASCOT’s mo-bility unit, the model previously used to support designdecisions described in [7] is enhanced. Therefor not onlythe lander’s system model, but also the description of theasteroid is adapted taking latest findings into account.

3.1 Multibody Model

In order to cover the kinematics and dynamics,MASCOT’s mechanical system is modeled using multi-body dynamics techniques, implemented in the commer-cial software SIMPACK. Dependent on the application ei-ther pure rigid body models or flexible multibody dynam-ics based on modal reduction are used. The flexible partis thereby limited to MASCOT’s main structural frame,including the electronics box and its interior, but treatingthe payload and the mobility as pure point masses. Thesubjacent FEM model [8] is provided by DLR Institute ofComposite Structures and Adaptive Systems.

3.2 Contact Dynamics Model

For MASCOT’s locomotion, based on hopping and re-peated low-energy impact on the asteroid’s surface, under-standing the corresponding contact dynamics is crucial.Yet, the knowledge on Ryugu and asteroids in general islimited. Thus without detailed knowledge, complex mod-els do not add beneficial detail to the simulation. For thatreason contact dynamics between the lander and the as-teroid has been modeled simplified as a visco-elastic sur-face model. The nominal parameters, i.e. Young’s Mod-ulus E and Poisson’s ratio ν, have been determined in [7]and improved during the mission development phase. Thecontact dynamics are based on SIMPACK’s PolygonalContact Model (PCM), enhanced by special approachesfor parameter estimation. Thereby the latest improve-ment bases on usage of the correlation between coefficientof restitution εr and critical damping kkrit to identify thedamping of an impacting body by [9] and the extensionby [10]:

k =2 · |ln(εr)|√ln(εr)2 + π2

·

√m · rc · E3(1 − ν2)

√|u0| (1)

where m is the mass of the lander, rc the equivalent contactradius for non-spherical bodies and u0 the allowed relativeoverlap. As the underlying assumptions by Lichtenheldthave been originally made for particle systems and stiffcontacts [11], the approach is slightly adapted and coeffi-cients of restitution based on experimental data are usedfor the contact between soil and the lander. Concluding arange of εr ∈ [0.2; 0.3] is found for granular soils.

3.3 Asteroid ModelIn order to describe MASCOT’s locomotion on the as-teroid, a model of the latter is needed. This model in-cludes the surface description, as well as the correspon-dent gravity field and assumptions regarding the asteroidbody. Such are assumptions on the density and homogene-ity of the planetary body and its motion behavior. For thefinal trajectories, which will be used in the mission opera-tions, a farther detailed asteroid model including rotation,shape, gravity field and surface is needed. Therefore, thelanding site has to be known, in order to find suitable tra-jectories for the respective region, as different regions willfeature different gravity and environment. Additionally,with increasing model complexity, comparability and in-terpretability of the optimization results decrease. Thisphenomenon is due to effects exerted by gravity potentialand others. Thus for the evaluation of the optimization-based trajectory identification strategy’s applicability, asimplified model is used in this article. Simplificationsare also applied due to the lack of knowledge on Ryugu’ssurface at the present time. Anyway, further detailed mod-els for the mission are object to ongoing work.The simplified model features a spherical shape and ho-mogeneous density. Thus the gravity field is also spheri-cal and its effect on MASCOT is dependent on the heightof the center of gravity. The parameters of the sphericalbody are compiled from the current knowledge on the as-teroid, i.e. its assumed size and mass. The asteroid issimplified as a non-rotating body for comparability rea-sons. As the latter alters the effective escape velocity,MASCOT’s velocities are checked throughout the opti-mization process. Furthermore the asteroid is assumed tohave a smooth, homogeneous surface. The surface rough-ness and texture itself is covered by Coulomb friction. AsHayabusa II approaches Ryugu, all knowledge gathers un-til landing will be fed into the asteroid model in order toenhance the accuracy of the model. For further detail,the model is already prepared to cover rotating asteroidsof arbitrary shape (surface mesh) with elliptical gravityfields [12] solving Legendre’s elliptic integral of the sec-ond kind.

3.4 Experimental Model VerificationIn order to make sure, that the identified trajectories aresuitable for MASCOT’s mission on Ryugu, the simula-tion models need to be checked using measurements. Dueto the micro-gravity environment on the asteroid, verifica-tion on Earth is a demanding task. Thus in order to checkthe motion itself, parabolic flight campaigns using scaledexcentric arm masses have been used. These campaignsshowed a good agreement between measured and simu-lated motion behaviour. Nevertheless, due to the shortperiod of micro-gravity, a full maneuver demonstrationof MASCOT was not possible. Hence, to further check

Figure 3. : High-precision measurement of ground reac-tion forces on four force sensors in terrestrial conditions

the model of MASCOT itself, precision force measure-ments were taken in terrestrial environmental conditions.The configuration of the test is shown in Fig.3. Once the

0.1 0.2 0.3 0.4 0.5 0.6 0.7

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

t [s]

F [

N]

measurementsimulation

Figure 4. : Comparison of measured and simulated reac-tion force

MASCOT model is able to correctly simulate the contactforces at various operations of the mobility unit, it is as-sumed that the model is also able to reproduce the motionbehavior in the relevant environment. Fig.4 exemplarilyshows the congruence of measured and simulated forcesignals in the low frequency range of [3; 25] Hz, relevantfor hopping. Thereby the model computation and storageof results is 10 to 20 times faster than the real motion se-quence, hence it results in a real time factor of [0.05,0.1].

4 Optimization-based Approach forIdentification of Trajectories

Due to the complex interaction of the lander with the as-teroid surface, trajectory determination by model inver-sion or similar techniques used to implement feed-forwardcontrol of the mobility unit are not possible. As manualevaluation of trajectories would end up in long trial & er-ror campaigns, optimization is used to automatically tunethe trajectory parameters to the lander’s optimal motionbehaviour. A trajectory T consists of five parameters:

• start angle of the mobility arm θi

• angular acceleration of the motor aaarm

• maximum rotational velocity ωmax

• angular deceleration of the motor adarm

• end angle of the mobility arm θe

T =

{θi, θe, amax, ad

arm, ωmax

}v

(2)

These parameters are then compiled into the class C1 po-sition function.

4.1 ApproachThe basic approach of the optimization based identifica-tion of suitable trajectories is to systematically evaluateand rate trajectories by their fitness. This fitness is therebycalculated from several objective functions, based on thesimulation results. These objectives will be explainedin the correspondent results section for hopping and up-righting. As the knowledge on the shape of the response-surface for the objectives is unknown, genetic algorithmsbased on evolutionary strategies are applied as the algo-rithm of choice. While the current knowledge on Ryuguis yet limited, robust solutions, working for a variety ofenvironmental parameters are needed. This robustness ofthe solution is assured by performance of several parallelsimulations per individual of the genetic algorithm. Thesesimulations will be called sub-individuals in this article, astheir entity forms one individual. Thereby, the variation ofenvironmental parameters is performed over the discretesub-individuals. To find robust solutions the mean gradi-ent of the objective Jk over the sub-individuals ς

dJk

dς= ||∇Jk ||1 (3)

is performed: Hence low gradients refer to low sensitiv-ity on the parameters and thus a robust solution, whereasa high gradient show a solution sensitive on the variedsub-individual parameters. For the up-righting optimiza-tions, the sub-individuals are divided in ten cases, listedin table 1. Thereby mainly soil parameters are varied inwide ranges, but also deviations in the trajectory itself areused. It shall be mentioned, that these variations are a firstset to evaluate the suitability of the approach using worstcase assumptions, which might even be worse than the ac-tual variation range on the asteroid. The total optimizationstatement yields:

T ′ ∈ arg minT

[J, ||∇Jk ||1

]⇔ T ∧ J(T ′) ≤ J(T ) (4)

∧ ||∇Jk(T ′)||1 ≤ ||∇Jk(T )||1; ∀ T ′ ∈ DT ∧ JC ≡ 1;

4.2 Framework - TOMATOThe approach described above is implemented in the Toolfor MASCOT’s Arm Trajectory Optimization (TOMATO,see Fig.5) which is based on the optimization frame-work MOPS [4], both developed by DLR Institute of Sys-tem Dynamics and Control. TOMATO mainly connects

SIMPACK to MOPS and takes over the job schedulingfor parallel processing of the sub-individuals includingunique naming and management of the results. TOMATOalso provides the infrastructure for determination of theobjectives based on the time dependent data in a fail safeway. Hence the main focus is to ensure, that long opti-mization runs will not be aborted by errors in single in-dividuals or sub-individuals in order to allow for full au-tomatization of the process. In order to decrease the timeneeded to finish optimizations TOMATO allows for multi-threading in terms of processing sub-individuals up to thenumber of cores or licenses available.

5 APPLICATION & RESULTS

In this section results used for verification of the optimiza-tion based approach are pointed out. Therefore a sce-nario for hopping as well as up-righting from every non-nominal face is shown. These results are not final resultsin sense of the mission yet, as asteroid modeling and pa-rameter setting are still work in progress and will be en-hanced while approaching Ryugu till 2018.

5.1 HoppingFor hopping the main objective has been to achieve an asfar as possible relocation distance to cover larger areas onRyugu. This distance is therefore divided into two objec-tives: The jumping distance up to the first impact δ onthe soil and the final distance δmax. Thereby the first im-pact distance is weighted higher, as it is the more reliableresult. This is due to the excessive amount of bouncing,occuring due to low gravity and rotational energy storedin the lander itself, which is then transferred into trans-lational energy at every impact. Thus the final distanceis mainly used to evaluate the radius of a circle in whichMASCOT will most likely come to rest, but not to eval-uate a final position as such, because every hit decreasesthe accuracy of the prediction.As MASCOT’s survival time is limited by the batterycharge another constraint, compiled into an objective, isthe time t(δz

max) needed to reach an equilibrated state. Thistime is minimized in order to enable the highest time forscientific output and to lower the amount of locomotiontime due to bouncing. Furthermore the maximum jump-ing height δmax, as well as the maximum velocity in ~ez areminimized and another constraint objective ensures, thatvelocity ~vz in ~ez is always safely lower than 0.5 times theescape velocity vesc.Thus the objective and constraints are:

J =

{δ−1, δ−1

max, t(δmax), δzmax, |~vz|

}v

(5)

JC =

{|~vz| < 0.5 · vesc

}v

(6)

MOPSMOPS-setup.m

MOPS-script.m

-define MOPS-script

-define tuners

-interaction with GUI

-modify tuners

-modify paramters

-define add. Cases

-control optimization

MOPS-GUI

-saved setup data

MOPS-setup.mat

-define cases

-define tuners

-define parameters

-define criteria

-define results

-define plots

-define interface

-postprocessing

-simulation

Define interface-functions to system-shell-calls

(bash)

defineDemOpt.m

Prepare solver calls and file-IO

exchange parameters & results with DEMETRIA

File-IO and interface-functions to simpack-

solver and model

shell layer (bash, powershell)

DEMETRIA.writeVarfile()

DEMETRIA.runSim()Compute criteria out of

time dependent results

from DEMETRIA

computeCriteria.m

DEMETRIA.getSimRes() results to struct

MOPS optimization algorithm

Matlab

MOPSMOPS-setup

MOPS-script.m

MOPS-GUI

-saved setup data

MOPS-setup.mat

- compute objectives and their derivatives

- define and run sub-individuals

Define interface-functions to system-shell-calls

(BaSh)

defineSimp()

Prepare solver calls and file-IO

exchange parameters and results with Simpack

File-IO and interface-functions to simpack-

solver and model and parallel computing

shell layer (bash, powershell)

Sim.writeParameters()

Sim.runSimulation() Compute criteria out of

time dependent results

from simpack

computeObjective()

Sim.getSimRes() result files to struct

Parameters &

tuners

MOPS optimization algorithm

TOMATO(Tool for MASCOT Arm

Trajectory Optimization)

Sim.parallelJobControl()

Compute objective

derivative over sub-

individuals to measure

sensitivity

SensitivityObjective()

Figure 5. : Overview of the framework TOMATO

The optimization has been carried out for face A and B(nominal and opposite face, largest faces of the box, seeFig.2) as starting faces. In order to improve the conver-gence of the optimization, it is carried out stepwise, firstindentifying the two angles. This approach is based on theassumption, that certain parameter’s choice is dependenton the optimal choice of other parameters. As a secondstep, the angles are held constant and suitable accelera-tions are to be found. Last the maximum rotational ve-locity is determined. For details on the approach and itsverification refer to [5].Fig.6 shows the results of the penultimate step for the fourdimensional parameter space of the first two steps, illus-trating all the individuals of the optimization run. As theresult the jumping distance δ up to the first hitpoint isused. Two maxima concerning the angles are clearly vis-ible, whereas one of them is a global maximum for theset conditions. Fig.7 shows the same points as in Fig.6,

0

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6

8

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−100

1020

3040

0

5

10

15

θe [rad]

Optimum

θi [rad]

δ [m

]

aar

md

[ra

d/s2 ]

5

10

15

20

25

30

35

Figure 6. : Results on hopping distance up to the firstimpact in four dimensional parameter space for startingon face A cf. [13]; Marker size denotes maximum armacceleration

but with the focus on the accelerations instead of the an-gles. Thereby it is shown, that for certain ranges of thearm acceleration, there is low sensitivity on changes ofthe latter, but a clear maximum exists at 33.55 rad

s2 . For thedeceleration, the results are showing a maximum plateauwhich is compliant to the basic assumption, that higher de-celerations would result in higher forces and thus higherdistances. Fig.8 shows the influence of the last step - tun-

010

2030

40

010

2030

400

5

10

15

aaarm

[rad/s2]

Optimum

adarm

[rad/s2]

δ [m

]

θe [

rad]

−20

−10

0

10

20

30

Figure 7. : Results on hopping distance up to the firstimpact in four dimensional parameter space for startingon face A cf. [13]; Marker size denotes initial angle of thearm

ing of the maximum velocity. As already shown for thedeceleration, the expected result is also shown by the op-timization: increasing rotational velocity increases the en-ergy stored in the system and thus results in larger jump-ing distances. As an additional result, the equilibrated to-tal jumping distance is shown (the range of the y-axis isshrinked) to illustrate that an increase in input energy alsoincreases the range of possible end positions and thus de-creases the accuracy of the prediction. In the figures 9 and10 the parabola of the first jump (blue) and the subsequent

500 1000 1500 2000 2500 3000 3500 4000 4500 50000

2

4

6

8

10

12

14

16

vmax

[rpm]

δ, δ

tot [

m]

Figure 8. : Results on hopping distance up to the first im-pact and final position dependent on maximum rotationalvelocity of the mobility arm for starting on face A cf. [13]

hit points (red) of the simulation are shown. The greencircle illustrates the circle in which MASCOT’s end po-sition is most likely situated. Thereby Fig.9 shows thebest manually tuned trajectory, which has been chosenusing common engineering assumptions, whereas Fig.10denotes the optimized trajectory using the proposed ap-proach. By comparing the two plots a factor of ≈ 6 canbe evaluated for the increase in distance up to the firsthit point and an even larger factor for the final equilib-rium position. Nevertheless the longer equilibrated jump-ing distance also means larger uncertainty of MASCOT’sfinal position. As a second case to verify the abilities

0

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−4

−2

0

2

4

0.51

1.5

x [m]

y [m]

z [

m]

Figure 9. : Best manually tuned trajectory, blue paraboladenotes first jump, red dots are subsequent hitpoints dur-ing bouncing for starting on face A cf. [13]

−200−100

0100

200

−200−100

0100

200

015

x [m] y [m]

z [

m]

Figure 10. : Optimized hopping trajectory, blue paraboladenotes first jump, red dots are subsequent hitpoints dur-ing bouncing for starting on face A cf. [13]

of the algorithm, jumping from face B has been used, asthe case is not perfectly symmetric to case A. Therewithit has been checked if on one hand similar dependenciesare found and if on the other hand parameters like the armangles are sufficiently different from the face A solution.Fig.11 shows that the response surface, as well as the way

through parameter space for the acceleration values is ingeneral agreement with the ones of face A. Additionally itcan be seen, that a more clearly visible maximum exists.In Fig.12 similar qualitative behaviour like in Fig.7 is vis-ible, but with significantly different quantitative results onthe optimal angles.

0 1 2 3 4 5 6 7 −40

−20

0

20

400

5

10

15

θe [rad]

Optimum

δ [m

]

a arm

d [

rad/

s2 ]

5

10

15

20

25

30

35

θi [rad]

Figure 11. : Results on hopping distance up to the firstimpact in four dimensional parameter space for starting onface B; Marker size denotes maximum arm acceleration

010

2030

40

010

2030

400

5

10

15

aaarm

[rad/s2]

Optimum

adarm

[rad/s2]

δ [m

]

θe [

rad]

−20

−10

0

10

20

30

Figure 12. : Results on hopping distance up to the firstimpact in four dimensional parameter space for startingon face B; Marker size denotes initial angle of the arm

5.2 Up-rightingSo called up-righting is the motion bringing MASCOTfrom any non-nominal face to the nominal measurementface A, as most of the scientific instruments are only ableto measure in this position. Dependent on which faceMASCOT lands on, manual solutions can be found moreor less easily: e.g. for the faces E and F the rotation axisof the mobility unit points in normal direction of the soil,which is the most problematic condition. Due to soil fric-tion and its anisotropic behaviour it is possible to up-rightMASCOT from face E and F nonetheless, however withlower probability to equilibrate on face A. Fig.13 dividesthe solutions which landed on face A (ΘA=1) from thenon-successful cases. By the scattered pattern of success-ful up-rights it can be seen, that finding suitable solutionsis not a trivial task, moreover if robust solutions are de-sired.The main objective for up-righting is to minimize the face-membership-function ΞF, which rates each equilibrium

Table 1. : Variation ranges for the environmental and mobility parameters of sub-individuals

Case µh εr E [MPa] ωmax adarm

(1-3) [0.7; 1.2] - - - -(4-6) - [0.1; 0.6] - - -(7-8) - - [0.1; 1] - -(9-10) - - - [0.9; 1] · ωdes

max [1; 1.1] · ad,desarm

orientation based on its severity for the mission. Therebyface A results in the lowest values, followed by equalvalues for face C and D. Face B features slightly highervalues, as up-righting from this face requires higher in-put energy and thus takes longer than from faces C andD. Worst cases are faces E and F as mentioned above.Additionally equilibrating time t(δmax), jumping distance

22.2

2.42.6

2.83

01

23

450

0.2

0.4

0.6

0.8

1

θi [rad]θ

e [rad]

ΘA

[ ]

Col

or:

a d [ra

d/s2 ];

Mar

kers

ize

aa [

rad/

s2 ]

2

4

6

8

10

Figure 13. : Results on up-righting from face C, ΘA is themembership function to evaluate if the final face is face A

δmax and height δzmax, as well as velocity in ~ez are min-

imized. Else the same escape velocity constraint as forhopping applies.

J =

{ΞF, δ, δmax, t(δmax), δz

max, |~vz|, ωmax

}v, ||∇Jk ||1 (7)

JC =

{|~vz| < 0.5 · vesc

}v

(8)

Furthermore for up-righting the sub-individual based ro-bust optimization applies. Table 1 shows the ranges for theten sub-individuals, which vary soil and actuation param-eters and thus model inaccuracy in the current knowledgeon the asteroid and unlikely system errors. The rangesare chosen such, that worst cases - most likely worse thanwhat is to expect on Ryugu - are used to verify the applica-bility of the algorithm. Based on the ten cases the deriva-tives of the objectives are evaluated in order to rate thesensitivity as well. For final mission value optimization,more than ten cases may be used. As post-optimizationcheck, a verification of the qualitative motion behaviourfor up-righting is carried out. As the main result of thecampaign, using optimized, robust trajectories MASCOTwas able to up-right from every non-nominal face to faceA in ten out of ten cases. Before optimization, up-righting

from face E and F was not possible in every case and noneof the trajectories were robust against bigger changes inenvironmental conditions. Additionally none of the solu-tions violated the escape velocity constraints and the set-tling time decreased from ≈ 800 − 900 s before optimiza-tion to ≈ 300 s after optimization. The only case that took≈ 500 s has been face E to face A, for which an even bet-ter trajectory will be found in prolonged optimization runsin future work. In order to illustrate the achieved goal ofup-righting and additionally to underline that reasonabletrajectories are found, Fig.14 shows a picture series of theup-righting process for starting face B, C and F. Henceall levels of difficulty are present in the visualized resultsand it can be seen, that no additional bouncing on non-nominal/intermediate faces is occuring.

6 CONCLUSION

In this article an optimization based technique to de-termine suitable arm trajectories for the asteroid landerMASCOT has been proposed. Using this approach tra-jectories for both, hopping and up-righting have been op-timized successfully. Thereby the performance of the re-sults goes far beyond the results of manual tuning, whilebeing a fully automated process. For hopping longerjumping distances were achieved in order to allow plan-etary science to cover larger areas of the asteroid. Espe-cially for up-righting robust solutions for a wide variety ofsoil parameters as well as unlikely actuator control inac-curacies were found. Thereby ten out of ten cases for up-righting from any non-nominal face are achieved, whileeven decreasing the settling time of the lander and thus en-hancing the time for planetary science. The results for theintermediate specimen of the evolutionary strategies mayalso be used to analyze influences of the tuner parametersas well as to be treated as training data for operations.Nevertheless the approach still has limitations: genetic al-gorithms need a sufficiently large number of specimen andgenerations to find suitable solutions, thus optimizationruns need at least one week on nowadays powerful work-stations, even though multi-threading using 20 cores is ap-plied. Even though this runtime is still sufficiently fast tobe able to update the trajectories once final information onRyugu is gathered before the last data upload.

Figure 14. : Optimized up-righting motion in nominal case for face B, C and F; motion is completed within ≈ 300 s

For further work, the asteroid model as well as the systemmodel will be farther enhanced as knowledge is gained ap-proaching Ryugu. Thereby further optimization runs willstep by step refine the trajectories aiming for the final setused during the mission after descent.

Acknowledgment

Special thanks are going to the main author’s sister PiaLichtenheldt who designed the logo for the frameworkTOMATO.

References

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