Abstract— In this paper we propose a new method todetumble a malfunctioning satellite. Large space debris suchas malfunctioning satellites generally rotates with nutationalmotion. Thus several researches have proposed the methodsto use a space robot for capturing and deorbiting thesedebris. The most of the past studies considered the method todetumble an uncontrollable satellite and then capture a singlespinning satellite. However these methods require physicalcontact with malfunctioning satellites, which has a risk ofaccident. Therefore, we propose a method with an eddy currentbrake [1]. The eddy current brake system can produce brakingforce to the target without any physical contact. Thus, we canreduce the risk of critical collision between the space robotand the target object. This paper firstly reviews dynamicsof a tumbling satellite and proposes a detumbling strategywith the eddy current brake. We carry out a fundamentalexperiment to evaluate the performance of the braking force ofthe developed eddy current brake system, and then we simulatedetumbling operation by using the experimental data and showan effectiveness of the proposed detumbling method.
I. INTRODUCTION
Recently a lot of space debris exist in the earth orbit.These debris have been considered to become the obstaclesof current and future space activities. The debris have beengenerated due to the past and current space missions and con-sist of, for example, rocket stages, malfunctioning satellites,and so on. Among them, large debris such as malfunctioningsatellites have high risk of colliding with ongoing operationalspacecrafts or any other debris. As a result of collisionwith large space debris, a big amount of small debris isscattered in the orbit. In fact, such an accidental collisionwas happened in 2009 and tons of debris have been dispersedaround the earth [2]. Hence, many researchers have studiedthe ways for space robots to deorbit large debris such as themalfunctioning satellites safely.
To achieve deorbiting the malfunctioning satellites bythe space robots, there are three elemental technologies;rendezvous, capturing, and deorbiting. In these elementaltechnologies, rendezvous and deorbiting technologies havebeen verified or are in the process of verification in orbit [3]–[6]. On the other hand, although many methods have beenproposed for capturing the target satellite, there is no absolutereliable solution since it involves risk to have a contingentcollision. Therefore the past proposed methods have beenverified by only numerical simulation or ground facilities.
F. Sugai is with Department of Mechanical Systems and Design,Graduate School of Engineering, Tohoku University, 6-6-01 Aramakiaza-Aoba, Aoba-ku, Sendai 980-8579, Japan. {sugai, abiko, tsujita,jiangxin, uchiyama}@space.mech.tohoku.ac.jp
In general, the malfunctioning satellites are considered tobe out of control due to the loss of the attitude control andthey rotate with nutational motion. In fact, the motion of sev-eral uncontrolled satellites have been observed from the earth[7]. Key technologies to capture such uncontrollable satellitesare how to deal with relative rotational motion between thetarget and the space robot and how to capture it safelywithout critical hard contact. In several related researches[8], [9], they assume that the space robot can follow suitabletrajectory which achieves zero relative velocity between thecapturing point of the target and the robot. However it isnot easy to achieve zero relative velocity between the spacerobot and the target due to complex motion of the target andthe limited movable area of the space robot. If there existsnon-zero relative velocity, contingent contact would happenand lead to the destruction of the robot arm or the target, orthe target satellite would be bounced away. In [10], [11], theyproposed the method that the space robot firstly detumbles(making nutational motion to single spin motion) a nutationalmotion of the target satellite, and then captures the target.This approach is more practical to capture the target sinceit is simpler to track certain point to grasp even when thetarget does not have a dedicated grasping point in the singlespin motion. However this method still has a drawback ofphysical contact in the process of the detumbling.
To reduce a risk of the hazardous contact between thetarget and the space robot, we propose a new methodfor detumbling an uncontrolled target without any physicalcontact by using an eddy current brake. Up to now, wedeveloped an eddy current brake system and evaluated theperformance of the first prototype brake system [1]. In thispaper first briefly reviews dynamics of a tumbling objectand proposes a detumbling strategy by using the developededdy current brake system. Then, we carry out a fundamentalexperiment to evaluate the performance of the braking forceof the developed eddy current brake system. Finally wesimulate a detumbling operation by using the experimentaldata and verify an effectiveness of the proposed detumblingmethod.
II. DYNAMICS OF A TUMBLING SATELLITE ANDA DETUMBLING STRATEGY
This section firstly explains a tumbling motion of thetarget satellite. Then we propose a strategy of detumblingan uncontrollable satellite by using the eddy current brake.
A malfunctioning satellite generally loses its attitude con-trol and rotates freely. Firstly let us consider, for simplicity,motion of a spinning satellite whose moment of inertia is
2013 IEEE/RSJ International Conference onIntelligent Robots and Systems (IROS)November 3-7, 2013. Tokyo, Japan
Fig. 1. Nutational motion. Fig. 2. Principle of an eddy current brake. Fig. 3. Precession by an eddy current brake.
I = diag[Ix, Iy, Iz ] = diag[IT , IT , Is]. Assuming thatangular momentum and angular velocity of the satellite areH and ω respectively, the equation of motion of the satelliteis expressed by the following Euler equation of a rigid body.
dH
dt+ ω ×H = M . (1)
where M is torque. and z axis is considered as the spin axisof the satellite. When the satellite is not subjected to externaltorque, M becomes zero. Substituting a moment of inertiaI = diag[IT , IT , Is] and ω = [ωx ωy ωz]
T to the aboveequation, the following equation is derived.
IT ωx + (Is − IT )ωyωz = 0IT ωy + (IT − Is)ωzωx = 0Isωz = 0
⎫⎬⎭ . (2)
From the third equation, the spin rate ωz is constant. Thefirst and second equations are expressed as follows:
ωx + λωy = 0ωy − λωx = 0
}, (3)
λ =Is − IT
ITωz. (4)
From these equations, angular velocity ωT = ωxxB+ωyyB
rotates around spin-axis with angular velocity λ in a directionperpendicular to spin axis. xB and yB denote unit vectoralong each axis.
In this dynamic motion, ω, H , and zB (unit vector alongzB axis) lie in the same plane. When a spin satellite is notapplied external force, an angular momentum H is constant.Therefore spin axis zB rotates around H with maintaininga certain amount of angle between zB and H (Fig. 1). Thisrotational motion is called nutational motion and that angleis called nutational angle θ.
In order to detumble a nutational motion of the targetsatellite, we have to add an external torque so as to makingnutational angle decreasing. The external torque vector has tobe a vector which lies in the same plane of angular momen-tum vector and spin-axis. Direction of the external torquevector has to be a direction from the angular momentumvector to the spin-axis.
In this detumbling strategy, we use an eddy current brakein order to exert a force to the target. In this paper, we assumethat the eddy current brake system is attached on the tip ofa robotic arm on a space robot. Fig. 2 shows a principle
of the eddy current brake. When a conductive object hasa relative velocity to a magnetic field, an eddy current isgenerated so as to disturb changing a magnetic field. Thus,the rotating object is exerted by an interaction force betweenthe eddy current and magnetic field so as to brake a rotatingconductive object. The structure of most of satellites is madeby nonmagnetic material such as aluminum alloy. Thereforethe eddy current brake is feasible as to exert a braking forceto the target satellites.
As shown in Fig. 3 a coil of the eddy current brake isplaced near by the target. The coil applied a force f tobrake a spinning motion. By applying a force f angularmomentum H is precessed (changing an angular momentumwith external torque) and the nutational angle θ changed withδθ as the following equation.
δθ = cos−1 H ·H ′
|H||H ′| , (5)
H ′ = H + nΔt, (6)
n = r × f . (7)
where r is a vector from the center of mass of the targetto the position where an external force is applied. n isthe torque exerted to the target satellite and Δt is a timeduration when the eddy current brake applies force to thetarget satellite. The eddy current brake generates brakingforce at the time when the surface of the target passes throughthe efficient area of a coil of the eddy current brake whilekeeping no contact between them. As shown in Fig. 3, thevector of the angular momentum, H becomes closer to thespin-axis, zB and the nutational angle, θ decreases to zero.
Fig. 4 shows numerical simulation when one eddy currentbrake applies the braking force to the tumbling target. Thefigure shows trajectories of the tip of the spin axis and thevector of the angular momentum. In the simulation, initialangular momentum is along z axis and nutational angle is30 [◦]. A coil of the eddy current brake is placed near thetarget satellite at initial condition. As shown in the figure,the angular momentum is precessed by the external torqueand the trajectory of the spin axis moves to +x directionand circular path becomes smaller. However this strategy hasa problem that a space robot gets a reaction force when aspace robot has only one robot arm on which a coil of theeddy current brake is attached. Therefore in order to keep arelative position between a space robot and a target satellite,we use two coils of the eddy current brake and two coils
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−]
Trajectory of Spin AxisTrajectory of Angular Momentum
Start
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Fig. 4. Trajectory of spin axis and angular momentum (using one coil).
are placed opposite each other. These coils generate brakingforce to opposite direction and reaction forces are canceledout each other. Then, the space robot does not affect anyreaction force and it gets only reaction torque.
Fig. 5 shows a simulation result of detumbling by twocoils of the eddy current brake. By using two coils, angularmomentum changes around initial direction in x axis, andcenter of the nutational motion does not move. Thereforethe distance between the coils and the target gets longer. Inorder to keep the distance between the coils and the target,two coils have to move.
Fig. 6 shows a strategy of detumbling and capturingan uncontrolled satellite. The free-flying space robot hastwo arms, on which two coils of the eddy current brakeare attached. Firstly the space robot moves two arms andapproximates the coils to the target. Then the eddy currentbrake applies a braking force to the target in order to precessthe motion of the target. At the same time the space robotcontrol arms and keeps a distance between the coils andthe target when nutation angle of the target gets smaller.Secondarily the eddy current brake reduces a spinning rateafter the target motion becomes single spin. Finally the spacerobot captures the target satellite by two arms.
III. DEVELOPED EDDY CURRENT BRAKE &FUNDAMENTAL EXPERIMENT
We clarified the required specifications of the eddy currentbrake system and developed a linear induction motor typededdy current brake system as shown in Fig. 7 and conductedfundamental experiment to measure a braking force of thedeveloped eddy current brake [1]. In [1], we measured anoutput braking force to an aluminum flat plate as a target.The aluminum plate is fixed and the relative velocity betweenthe coil of the eddy current brake and the plate is zero. Inthe actual detumbling mission, the target object is not a flatplate. In our assumption, the target has a curved surface andthe relative velocity between the coil and the target object isnot zero because the target object is tumbling.
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Coil
Coil
Start End
Target
Coil
Spin Axis
Fig. 5. Trajectory of spin axis and angular momentum (using two coils).
1, Detumbling 2, Reducing spin rate 3, Capturing
Fig. 6. Strategy of detumbling and capturing a malfunctioning satellite.
In this section, first we measure braking force of thedeveloped eddy current brake to an object that rotates arounda single axis with constant angular velocity. Then we carryout experimental verification of the braking performance toa one-axis free rotating object.
A. Experiment 1: measuring braking force
1) Experimental setup: Fig. 8 shows an experimentalsetup. In the experiment, we use an acrylic cylinder as thetarget object, whose diameter was 400 [mm] and height was90 [mm]. The target is placed on one axis spinning motiontable. An aluminum plate, whose thickness was 2 [mm], wasattached on outer surface of the acrylic cylinder. A coil ofthe eddy current brake is placed on the side of the cylindricaltarget with 5 [mm] gap. The coil is mounted on a force sensorand a braking force of the eddy current brake is measuredby the force sensor.
We control a linear induction motor typed eddy currentbrake with a field oriented control(FOC). Fig. 9 shows ablock diagram of FOC. We can control an output forcedirectly by setting a reference of currents Idref and Iqref .The magnitude of these currents correspond to the magnitudeof magnetic field and torque, respectively. The FOC requiresthe relative velocity between the coil and the target. Inactual operation, we need to measure a spinning rate of thetarget satellite. However in this experiment we predetermineconstant angular velocity of the target and provide it to thecontroller in advance.
Furthermore, in this study, the plug braking approach isused, in which the direction of movement of the magnetic
785
(a) Coil.
(b) Controller.
Aluminum Plate
Acrylic Cylinder
Coil
Force Sensor
Spining Motion Table
Park Transform Clarke Transform
SVM3-phaseInverter
CurrentSensors
d, q
α, β
α, β
a, b, c
PI
PI
+-
+
+
+
-
Id
Idref
Iqref
Ia
Ib
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Iα
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Ta
Tb
Tc
U V W
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TargetObject
Coil
ωs
ωs
ωm
1s
Fig. 7. Overview of the eddycurrent brake system.
Fig. 8. Experimental setup. Fig. 9. Block diagram of field oriented control.
field using the linear induction motor is inverse direction ofthe movement of the target. This method can provide largebraking force when the relative velocity between the coiland the target is slow compared to the regenerating brakingsystem.
In the experimental condition, the target object ro-tates four patterns of the angular velocity, namely{5π/6, 5π/3, 5π/2, 10π/3} [rad/s]. We set two patterns ofreference current: Idref = Iqref = {5, 10}[A]. Then wemeasure braking force by the force sensor in each case.
2) Experimental result and remarks: Here firstly let usconsider the relationship between the angular velocity of coilcurrent ω1, slip angular frequency ωs and electric angularfrequency ωm which is obtained from the relative velocityas shown in (9).
ω1 = ωs + ωm, (8)
ωm =2πvmp
, (9)
f =φ2
2
r2ωs. (10)
where vm represents relative velocity between the coil andthe target. Here we assume that the coil is fixed and the targetis tumbling. Therefore, vm can be expressed as follows:
vm =d
2ωz (11)
d and ωz are a diameter and a spin rate of the targetrespectively. p is a pole pitch of the coil. f is an outputforce of the eddy current brake. r2 [Ω] and φ2 [Wb] denoteresistance and interlinkage magnetic flux of the target object,respectively.
From the above equations, one can obtain the followingrelationship between the output force, f , and the angularvelocity of the coil current, ω1.
f =φ2
2
r2(ω1 − ωm)
=φ2
2
r2(ω1 − 2πvm
p). (12)
Equation (12) implies that in order to provide constantbraking force to the target, ω1 should be determined larger
as the relative velocity is getting smaller. Therefore, highervoltage is required as the relative velocity gets smallerbecause inductive reactance becomes large. However powersupply of the eddy current brake cannot supply requiredcurrent due to the limitation of rated voltage. As a result,the braking force becomes smaller as the relative velocitygets slower.
In general, FOC determines the current for the outputmagnetic field and torque, Idref and Iqref , as mentionedbefore, which indicates that the constant slip angular fre-quency, ωs, is controlled to be constant. This approach ishereafter called as “Constant Braking Force”. However, inthe plug braking approach with FOC, one can actively usethe effect of the relative velocity, ωm while holding theangular velocity of the coil, ω1, constant. In this case, theslip angular frequency, ωs, gradually increases as the relativevelocity gets larger which leads to the eddy current brakecan generate larger braking force. This approach is hereaftertermed as “Maximized Braking Force”.
We conduct an experiment in order to verify how largethe eddy current brake can provide larger braking force withthe “Constant Braking Force(CBF)” approach and with the“Maximized Braking Force(MBF)” approach.
Fig. 10 shows an experimental result. In the case ofIdref = Iqref = 5[A] (CBF), braking force is approximately0.06 [N] and maintained constant despite of the change of theangular velocity of the target. On the other hand, in the caseof Idref = Iqref = 10[A](CBF), the braking force graduallyincreases from 0.22 to 0.26 [N] when the angular velocitybecomes large from 5π/6 to 10π/3 [rad/s]. This happeneddue to the limitation of the rated voltage as mentioned inthe above. In Fig. 10, Idref = Iqref = 5[A] (MBF) andIdref = Iqref = 10[A] (MBF) show the experimentalresult of the “Maximized Braking Force” approach. Theexperimental result shows that in the case of the “MaximizedBraking Force” approach, braking force becomes larger thanthe case of the “Constant Braking Force” approach.
Consequently, it is shown that the developed eddy currentbrake system can generate constant braking force despite ofthe change of the relative velocity in certain range of thecurrent. But, more efficiently, the eddy current brake cangenerate larger force by using the relative velocity actively
Without Eddy Current BrakeMaximized Braking ForceConstant Braking Force
Fig. 10. Experimental result. Fig. 11. Overview of a rotating table. Fig. 12. Experimental result.
in plug braking.
B. Experiment 2: braking a free rotating object
From the result of experiment 1, it is shown that the eddycurrent brake system can work well for the constant relativevelocity. However the relative velocity between the targetand the coil generally varies due to the braking force. In thissubsection, we verify if the eddy current brake can generateproper braking force during the decreasing of the relativevelocity and finally the spinning motion of the object isstopped or not.
1) Experimental setup: A target object and a coil of theeddy current brake are same as in the experiment 1. Thetarget is mounted on a free rotating table as shown in Fig. 11which can rotate one-axis freely. A rotary encoder is attachedon a rotating shaft and it measures an angle of the targetobject.
The angular velocity of the target when we start applyingthe braking force is 10π/3 [rad/s]. We set a reference currentIdref = Iqref = 10[A]. We measure the angle of the targetand the braking force during the angular velocity is from10π/3 to 0 [rad/s].
We carried out two patterns of braking approaches asmentioned in experiment 1, namely one is “Constant BrakingForce” that feeds the relative velocity information back tothe controller for keeping a braking force constant. Anotherone is “Maximized Braking Force” that makes braking forcelarger by passively using relative velocity. The “ConstantBraking Force” approach sends the measurement data withthe rotary encoder to the controller every 1 [s] to keep aconstant braking force.
2) Experimental result and discussion: Fig. 12 shows anexperimental result. By using the eddy current brake, thespin motion of the target converges to zero faster than thecase without using any braking system. Comparing the caseof “Constant Braking Force” with the case of “MaximizedBraking Force”, “Maximized Braking Force” can brakequicker than the other case. Fig. 13 shows a braking forcein the case of “Maximized Braking Force” and “ConstantBraking Force”. The braking force in the case of “MaximizedBraking Force” depends on the relative velocity. When therotational speed of the target object is fast, the braking forceis large and the braking force decreases as the rotational
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Braking Force (Maximized Braking Force)Braking Force (Constant Braking Force)Angular Velocity (Maximized Braking Force)Angular Velocity (Constant Braking Force)
Fig. 13. Comparison of “Maximized” and “Constant” braking force.
speed gets slow. However, the “Maximized Braking Force”approach still can keep larger braking force than the case of“Constant Braking Force”. On the other hand, in the caseof “Constant Braking Force” the eddy current brake outputsconstant braking force despite the rotational speed varies.But the magnitude of the output force is smaller than thatof the “Maximized Braking Force” approach as mentionedbefore. In each case, there is cyclic jerk on the braking force.This is because an aluminum plate on the outer surface ofthe target object is not a perfect circle and the gap betweenthe target object and the coil varies in a cycle.
As a result, the eddy current brake can apply braking forceto the rotating target properly. By using the relative velocitydata, it can generate constant braking force. Furthermore, itcan generate larger braking force by making effective use ofthe relative velocity.
IV. DETUMBLING SIMULATION
In this section, we simulate detumbling operation for morerealistic target with using the experimental result in SectionIII. As a target satellite, we assume a spinning satellite whichis out of control and rotates with 100 [rpm] = 10π/3 [rad/s]spin rate and 30 [◦] nutation angle. The size of the target isassumed to be that of the climate satellite Himawari 5 whosemass is 345 [kg], diameter is 2.15 [m], and height is 3.54 [m].
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Coil
Coil
Z [
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Fig. 14. Trajectory of spin axis and angular momentum.
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0
500
1000
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ular
Mom
entu
m [
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s]
Time [h]
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Fig. 15. Angular momentum of the target satellite.
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Fig. 16. Nutation angle of the target satellite.
However since we have no data about mass distribution, weset a simulation model as a homogenous cylindrical objectand height of the model so as to make inertia ration 1.05 (thisvalue is used as a design guide). Consequently, the height ofthe simulation model is 1.77 [m]. The moment of inertia isIS = 199 [kg ·m2], IT = 190 [kg ·m2] respectively. Initialattitude of the target satellite is set so as to align the directionof the angular momentum vector with z axis.
In the simulation, the coil of the eddy current brake isplaced on the side of the target with 5 [mm] gap. The coilmoves in order to keep the gap to the target constant when thetarget satellite is precessed. The eddy current brake appliesthe force at the moment when the target comes near the coil.The simulation will end if nutation angle will be less or equal0.01 [rad].
Figs. 14 to 16 shows a simulation result. Fig. 14 shows
a trajectory of the tip of the spin axis and the angularmomentum vector. The trajectory of the spin axis is plotting5 [s] per half hour. From Fig. 14 the trajectory of the spinaxis converges to a single spin motion, which means x and ybecomes zero. Fig. 15 shows the angular momentum of thetarget satellite. This figure shows that x and y components ofthe angular momentum show little change around zero and atthe same time z component of the angular momentum getssmaller. Fig. 16 shows nutation angle. From this figure, it isclearly shown that nutation angle becomes smaller while theeddy current brake system is applied to the target and finallythe nutational motion converges to a single spinning motion
V. CONCLUSIONS AND FUTURE WORKS
In this paper we proposed a detumbling method using aneddy current brake system. We discussed detumbling dynam-ics and proposed a strategy for detumbling an uncontrollablesatellite with two coils of the eddy current brake in order tocancel reaction force to the space robot. Then we carried outan experimental verification of the developed eddy currentbrake system to observe the generated braking force to thespinning target. Finally we simulated detumbling motionusing the data of experimental verification and showed theefficiency of the proposed method.
In the future works, we will carry out an experiment ofdetumbling a target which rotates in 3 degrees of freedom.Besides, we will simulate detumble mission including anattitude control of a space robot itself and control of robotarm mounted on the space robot. After fundamental researchis finished, we will study toward practical use of this detum-bling method.
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