Keywords

1 Introduction

Satellites in orbit have greatly increased in recent years with the unceasing space development. With the limited space resources, space debris removal should be taken into action to make better use of the space.

1.1 Typical Target Capture Method

The debris removal method includes active debris removal and passive debris removal method. Passive debris removal usually puts a damper on the debris and drag it to the atmospheric layer by earth magnetic field or pneumatic forces. Active debris removal needs a space craft to perform the capture of the debris and then remove it with the spacecraft’s energy. And this removal procedure often equips with manipulator.

1.2 Target Manipulation with Manipulator

Robot manipulator, also named robotic arm, has been used in on-orbit refueling, module replacement and assisted space docking. These robot arms have different D-H parameters and joint numbers [1,2,3]. And researches on robot arms have extended to multi-robot and dedicated operation from single robot. Some research work focused on the eddy current brake have got wonderful results [4,5,6,7].

Chaser with a manipulator de-tumbles a large tumbling debris with an eddy current brake is shown in Fig. 1, and this paper is focused on the challenge.

Fig. 1.
figure 1

De-tumbling of large target with Eddy Current Brake in Hand

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1.3 Problems of Debris with High Angular Momentum

Former debris target to be captured by robot arm often has such characters: a, the target’s attitude is limited, it has a relatively constant pose to the spacecraft. b, the target is small and little in weight, it is easy to be handled by the robot arm’s gripper. c, the target’s angular velocity is negligible, the spacecraft can keep still to the target by orbital maneuver.

However, debris generated in recent years are mainly satellites and abandoned rocket’s upper stage which are large and weight. And these debris also have angular velocity which lead to large angular momentum.

Traditional method using rigid gripper to directly contact with the debris has following problems: a, the gripper may break at this moment when the arm first contact with the debris. b, the enormous torque transferred from the debris after the manipulator has linked to the debris may cause the joint motor to break down or even break the transmission gear. c, the sudden impact may cause the spacecraft out of control. d, the impact may drive the debris away and out of sight.

In this paper, a contactless debris removal strategy is proposed to accomplish the de-tumbling procedure of debris with large angular momentum. And avoid the problem of rigid contact by using eddy current brake. The kinematic model of the debris in spin state is established to analyze the de-tumbling procedure. Then an optimized de-tumbling method is proposed by calculation referring time consuming. At last, a simulation is designed to illustrate the efficiency of optimized strategy.

2 Target Recognition and De-Tumbling

The capture system consists of two parts: the target to be captured, meaning the space debris in orbit with large mass; the performer of the capture mission, meaning the space craft with manipulator on board Fig. 2.

The chaser keeps the target in the robot arm’s manipulation space by attitude adjustment after it reaches the target’s zone by orbital maneuver.

The eddy current brake in the hand of the manipulator starts to work after the manipulator has moved to the target coordinate by path planning. It will then gradually decrease the target’s spin movement. At the same time, the controller system maintains the chaser-target’s relative position and the manipulator’s action pose by real-time calculation.

2.1 Target Approach and Coordinates Building

The target’s orbit is assessed by observation, then the chaser can fly to the target’s zone through orbital maneuver Fig. 2.

Typical capturing method as described here is called active debris removal strategy. Debris in the orbit is the target of the chaser-target system while the capture vehicle acting as the chaser. First, the orbit of the target debris is observed.

Fig. 2.
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Capturing of large tumbling target

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Fig. 3.
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Coordinates of target and chaser

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When the chaser reaches the same orbit with the target. It slowly changes its relative pose to the target while maintaining their position until the end effector of the onboard manipulator is in flexible manipulation space.

The spinning motion of the target rotating around the coordinate axes can be reconstructed by vision camera. Two different frames are built here to explain the target’s angular velocity and angular momentum. The body frame’s origin of coordinate is the center of mass of the debris. And set the minimum inertia spindle as the \( Z_{TB} \) axis. And \( X_{TB} \), \( Y_{TB} \) axes are mutually perpendicular in the transection of the cylinder. Then the complex spin kinematic of the target can be described as rotation movement around three perpendicular axes \( {\text{O}}X_{TB} \) \( {\text{O}}Y_{TB} \) \( {\text{O}}Z_{TB} \) with the angular velocity \( \upomega_{X} \) \( \upomega_{Y} \) \( \upomega_{Z} \).

The origin of the inertia frame is also the target’s center of mass. Set \( \upomega \) as the resultant vector of angular velocity’s component \( \upomega_{X} \) \( \upomega_{Y} \) \( \upomega_{Z} \). Then the inertia frame’s \( Z_{IB} \) axis is parallel to \( \upomega \).

In the same way, the chaser’s body frame can be built based on the center of its load’s setting surface. \( Y_{C} \) axis is perpendicular to the manipulator’s fixing panel Fig. 3.

First, when the captor reaches the target’s spin zone, maintaining the relative position between the spacecraft and the target.

$$ P_{{O_{C} }}^{{O_{TI} }} = const $$
(1)

Then, keeping the three axes of the captor’s body frame parallel to that of the target’s inertia frame by adjusting the spacecraft’s spatial attitude.

$$ R_{{O_{C} XYZ}}^{{O_{TI} XYZ}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$
(2)
$$ T_{{O_{C} XYZ}}^{{O_{TI} XYZ}} = \left[ {\begin{array}{*{20}c} {R_{{O_{C} XYZ}}^{{O_{TI} XYZ}} } & {P_{{O_{C} }}^{{O_{TI} }} } \\ 0 & 1 \\ \end{array} } \right] = const $$
(3)

By this way, the position and orientation matrix between the target’s inertia frame and the captor’s body frame is established. The manipulator onboard can calculate the target’s position through coordinate translation.

Fig. 4.
figure 4

The model of rocket upstage

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From [8,9,10,11], the dynamics motion of target with external force from action point between hand and target can be described as:

$$ H_{t} \ddot{r}_{t} + C_{t} = J_{t}^{T} f_{hd} $$
(4)

Where, \( \ddot{r}_{t} = {\text{col}}\left( {p_{t} , \varphi_{\text{t}} } \right) \in {\Re }^{6} \) is the generalized velocity vector consisting of the position of the center of mass \( p_{t} \in {\Re }^{3} \) and the attitude \( \varphi_{\text{t}} \in {\Re }^{3} \), respectively. Figure 4 \( C_{t} \in {\Re }^{6} \) represents the centrifugal and Coriolis terms. The vector \( f_{hd} \in {\Re }^{6} \) is the force and moment exerted by the arms. \( H_{t} \in {\Re }^{6 \times 6} \) is the generalized mass matrix which can be described as:

$$ H_{t} = \left[ {\begin{array}{*{20}c} {m1} & 0 \\ 0 & {I_{c} } \\ \end{array} } \right] $$
(5)

Where, \( m \) and \( I_{c} \) denote the mass and the inertia tensor of the target, respectively.

$$ c_{t} = \left[ {\begin{array}{*{20}c} {m\tilde{\omega }_{t} v_{0} } \\ {\tilde{\omega }_{t} I_{c} \omega_{0} } \\ \end{array} } \right] $$
(6)

Generalized Jacobin matrix \( A^{k} \) can be described following with \( \rho \in {\Re }^{3} \) being the position vector of the \( k^{th} \) action point with respect to its center of mass which detailed in 8. The operator ~ hat a vector denotes the cross product with the following vector.

$$ J_{t}^{k} = \left[ {\begin{array}{*{20}c} {1_{3} } & {\widetilde{\rho }_{k} } \\ 0 & {1_{3} } \\ \end{array} } \right] $$
(7)

At this point, the dynamic model of the tumbling target has been established, and the follow-up will carry out the study of the trajectory optimization of de-tumbling planning with eddy current brake based on the motion characteristics of the tumbling target.

2.2 Contactless De-Tumbling

The relative pose between the target and the manipulator’s end effector can be determined when the pose between the end and the robot’s base frame, the pose between the robot’s base frame and the captor’s body frame, the pose between the target’s inertia frame and the captor’s body frame. And the target’s pose in its inertia frame are determined.

Fig. 5.
figure 5

Brake force by eddy current

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A constant magnetic field will be generated when the eddy current brake at the robot’s end is powered on. During the de-tumbling procedure, make sure the brake is close to the external surface of the target’s spin state with the magnetic field’s direction parallel to the external surface’s normal direction. Current is produced when the target moves cutting the magnetic induction lines. And the direction of the force \( {\text{F}}_{B} \) produced by the brake is opposed to the target’s movement Fig. 5.

By keeping the magnetic induction lines’ direction perpendicular to the direction of the target’s resultant vector \( \upomega \) and the brake close to the target, an electromagnetic torque will continually decrease the target’s angular momentum. And eventually makes the target captured by the captor.

3 Optimization of De-Tumbling Strategy

As the brake force’s direction is always opposed to the target’s movement and the action time is relative to the brake’s shape size and the relative speed between the target and the manipulator. The major factor that affects the de-tumbling torque’s performance is the brake force’s point of action. In this section, the de-tumbling procedure of the target by cutting down its angular momentum through eddy current brake is analyzed. And an optimized de-tumbling strategy is proposed referring to time consuming.

3.1 Dynamics of Target De-Tumbling

The relationship between the target’s angular velocity vector and angular momentum is shown as below when it is in spin state Fig. 6.

Fig. 6.
figure 6

Rotation model

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Fig. 7.
figure 7

Angular momentum of target

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The target’s Euler equations of spin movement in orbit are:

$$ \begin{aligned} & J_{x} \dot{\omega }_{x} + \omega_{x} \times J_{x} \omega_{x} = \sum T_{x} \\ & J_{y} \dot{\omega }_{y} + \omega_{y} \times J_{y} \omega_{y} = \sum T_{y} \\ & J_{z} \dot{\omega }_{z} + \omega_{z} \times J_{z} \omega_{z} = \sum T_{z} \\ \end{aligned} $$
(8)

The angular velocity around X Y Z axes remain unchanged when the target’s resultant momentum is zero, and can be expressed as below:

$$ \begin{aligned} & H_{x} = J_{x} \omega_{x} \\ & H_{y} = J_{y} \omega_{y} \\ & H_{z} = J_{z} \omega_{z} \\ \end{aligned} $$
(9)

Usually, the target’s angular velocity vector is not parallel to its angular momentum vector. The eddy current brake generates the angular momentum \( {\text{H}}_{F} \) by applying the brake force \( {\text{F}}_{B} \) on the target to decrease the target’s angular momentum H Fig. 7.

The eddy-current-brake angular momentum applies once when the target circles one time around its angular velocity vector. And the target’s angular momentum is decreased from H to \( {\text{H}}^{'} \). Which also means a decrease in the target’s angular velocity. Its angular velocity turns out to be zero by continuous de-tumbling method.

3.2 Calculation of Optimized De-Tumbling Strategy

The eddy-current-brake angular momentum vector’s value and direction depends on the brake force’s point of action.

The total time to finish the de-tumbling procedure is set as the principle to judge the strategy’s efficiency. The less the remained angular momentum of the target is after one revolution, the better the de-tumbling strategy is.

So, the optimized de-tumbling strategy is to find out the minimum value of \( H^{{\prime }} \).

$$ {\text{Min}}\left\{ {H^{{\prime }} } \right\} $$
(10)

Assuming the value of \( {\text{F}}_{B} \) is constant, \( {\text{L}}_{C} \) is the width of the eddy current brake’s magnetic field.

  • \( \theta^{'} \) is the angle between the target’s angular velocity vector and the arm of \( F_{B} \).

  • θ is the angle between the target’s angular velocity vector and the Z axis of the target’s body frame.

  • δ is the angle between the target’s angular velocity vector and its angular momentum, calculated as the angle between H and the target’s Z axis minus θ (Fig. 8).

    Fig. 8.
    figure 8

    Calculation of \( {\text{H}}^{'} \)

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The eddy-current-brake angular momentum is expressed as:

$$ \begin{aligned} H_{F} & = F_{B} \cdot l_{f} \cdot t = F_{B} \cdot l_{f} \cdot \left( {\frac{{L_{C} }}{{\omega \cdot l_{f} \cdot \,\sin \,\theta^{{\prime }} }}} \right) \\ & = \frac{{F_{B} \cdot L_{C} }}{{\omega \cdot \,\sin \,\theta^{{\prime }} }} = \frac{{K_{F} }}{\omega } \cdot \frac{1}{{\sin \,\theta^{{\prime }} }} \\ \end{aligned} $$
(11)

Where, \( K_{F} = F_{B} \cdot L_{C} \). Apply the law of cosines in the triangle built by \( H_{F} \), \( H^{{\prime }} \), \( H \). The detailed process is

$$ \begin{aligned} \left| {H^{{\prime }} } \right|^{2} & = \left| H \right|^{2} + \left| {H_{F} } \right|^{2} - 2 \cdot \left| H \right| \cdot \left| {H_{F} } \right| \cdot \,\cos \,\gamma \\ & = \left| H \right|^{2} + \left| {H_{F} } \right|^{2} - 2 \cdot \left| H \right| \cdot \left| {H_{F} } \right| \cdot \,\sin \left( {\uptheta^{{\prime }} +\updelta} \right) \\ & = \left| H \right|^{2} + \left( {\frac{{K_{F} }}{\omega } \cdot \frac{1}{{\sin \theta^{{\prime }} }}} \right)^{2} - 2 \cdot \left| H \right| \cdot \left( {\frac{{K_{F} }}{\omega } \cdot \frac{1}{{\sin \theta^{{\prime }} }}} \right) \cdot \,\sin \left( {\uptheta^{{\prime }} +\updelta} \right) \\ & = \left| H \right|^{2} + \left( {\frac{{K_{F} }}{\omega }} \right)^{2} \cdot \frac{1}{{\sin^{2} \theta^{{\prime }} }} - 2 \cdot \left| H \right| \cdot \frac{{K_{F} }}{\omega } \cdot \frac{{\sin \,\uptheta^{{\prime }} \, \cdot \,\cos \,\delta + \cos \,\uptheta^{{\prime }} \cdot \,\sin \,\delta }}{{\sin \,\theta^{{\prime }} }} \\ & = \left| H \right|^{2} + \left( {\frac{{K_{F} }}{\omega }} \right)^{2} \cdot \frac{1}{{\sin^{2} \theta^{{\prime }} }} - 2 \cdot \left| H \right| \cdot \frac{{K_{F} }}{\omega } \cdot \frac{{\cos \,\uptheta^{{\prime }} \cdot \,\sin \,\delta }}{{\sin \,\theta^{{\prime }} }}2 \cdot \left| H \right| \cdot \frac{{K_{F} }}{\omega } \cdot \,\cos \,\delta \\ \end{aligned} $$
(12)

In the above equation, the change of the brake force’s point of action will lead to the change of θ’, which will affect the value of \( H^{{\prime }} \). Set \( \left| {H^{{\prime }} } \right|^{2} \) as the function with θ’ as the independent variable:\( f\left( {\theta '} \right) \).

Calculate the function’s minimum value by its first derivative:

$$ \begin{aligned} f^{{\prime }} \left( {\theta^{{\prime }} } \right) & = - 2 \cdot \left( {\frac{{K_{F} }}{\omega }} \right)^{2} \cdot \frac{{\cos \,\uptheta^{{\prime }} }}{{sin^{3}\uptheta^{{\prime }} }} - 2 \cdot \left| H \right| \cdot \frac{{K_{F} }}{\omega } \cdot \,\sin \,\delta \cdot \frac{{ - \sin \,\uptheta^{{\prime }} \cdot \,\sin \,\uptheta^{{\prime }} - \cos \,\uptheta^{{\prime }} \cdot \,\cos \,\uptheta^{{\prime }} }}{{sin^{2}\uptheta^{{\prime }} }} \\ & = 2 \cdot \left( {\frac{{K_{F} }}{\omega }} \right)^{2} \cdot \frac{{\cos \,\uptheta^{{\prime }} }}{{sin^{3}\uptheta^{{\prime }} }} + 2 \cdot \left| H \right| \cdot \frac{{K_{F} }}{\omega } \cdot \,\sin \,\delta \cdot \frac{1}{{sin^{2}\uptheta^{{\prime }} }} \\ & = 2 \cdot \frac{{K_{F} }}{\omega } \cdot \,\frac{1}{{sin^{2}\uptheta^{{\prime }} }}\left( {\left| H \right| \cdot \,\sin \,\delta - \frac{{K_{F} }}{\omega } \cdot \frac{{\cos \,\uptheta^{{\prime }} }}{{\sin \,\uptheta^{{\prime }} }}} \right) \\ \end{aligned} $$
(13)
$$ \theta^{{\prime }} { \in }\left( {\uptheta^{{\prime }}_{min} ,\frac{\pi }{2}} \right);\uptheta^{{\prime }}_{min} = \theta + \tan^{ - 1} \left( {\frac{2R}{h}} \right) $$
(14)

Set \( f^{{\prime }} \left( {\theta^{{\prime }} } \right) \) = 0, then:

$$ \uptheta^{{\prime }} = \tan^{ - 1} \left( {\frac{{K_{F} }}{\omega \left| H \right| \cdot \,\sin \,\delta }} \right) $$
(15)

It can be concluded that:

$$ \begin{aligned} & f^{{\prime }} \left( {\theta^{{\prime }} } \right) > 0,0 <\uptheta^{{\prime }} < \tan^{ - 1} \left( {\frac{{K_{F} }}{\omega \left| H \right| \cdot \,\sin \,\delta }} \right) \\ & f^{{\prime }} \left( {\theta^{{\prime }} } \right) < 0,\tan^{ - 1} \left( {\frac{{K_{F} }}{\omega \left| H \right| \cdot \,\sin \,\delta }} \right) <\uptheta^{{\prime }} < \frac{\pi }{2} \\ \end{aligned} $$
(16)

Function \( f\left( {\theta^{{\prime }} } \right) \) has its minimum value when \( \theta^{'} = \tan^{ - 1} \left( {\frac{{K_{F} }}{\omega \left| H \right| \cdot \,\sin \,\delta }} \right) \).

Summing up the above, the optimized de-tumbling strategy performs best when \( \theta^{'} = \tan^{ - 1} \left( {\frac{{K_{F} }}{\omega \left| H \right| \cdot \,\sin \,\delta }} \right) \) in the target’s every spin cycle.

4 Simulation of Optimized De-Tumbling Method

4.1 Simulation

A simulation is performed to estimate the optimized method using MATLAB tools. The target’s model is based on the abandoned rocket’s upper stage with a launch weight of 2154 kg. The sizes are shown below, where R = 2.6 m, r = 2.5 m, h = 7.4 m Fig. 9.

Fig. 9.
figure 9

Shape of target

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Fig. 10.
figure 10

Iteration cycle

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Its moment of inertia around each axis are:

$$ \begin{aligned} J_{X} & = \frac{m}{12} \cdot \left[ {3 \cdot \left( {R^{2} + r^{2} } \right) + h^{2} } \right] = \frac{{16835\,{\text{m}}^{2} }}{\text{kg}} \\ J_{Y} & = \frac{m}{12} \cdot \left[ {3 \cdot \left( {R^{2} + r^{2} } \right) + h^{2} } \right] = \frac{{16835\,{\text{m}}^{2} }}{\text{kg}} \\ J_{Z} & = \frac{m}{2} \cdot \left( {R^{2} + r^{2} } \right) = \frac{{14012{\text{m}}^{2} }}{\text{kg}} \\ \end{aligned} $$
(17)

Also, set the target’s initial angular velocity ω = \( \frac{\pi }{18}\left( {10^{^\circ } } \right) \) and the angle between the angular velocity and the Z axis of the target’s body frame θ = \( \frac{\pi }{6}\left( {30^{^\circ } } \right) \).

Fig. 11.
figure 11

Angular velocity \( \omega_{x} \), \( \omega_{y} \), \( \omega_{z} \), angular momentum \( H \) relative to n (iteration cycle)

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The simulation is carried out as following process, the parameters’ value of every cycle is calculated by iteration Fig. 10. The angular parameters θ, \( \uptheta_{min}^{'} \), δ are calculated using angular velocity data first.

Then put the angular parameters in to the Equation to calculate \( \theta^{{\prime }} \). Then the remained angular momentum of target \( H^{{\prime }} \) of present cycle is computed, which can be used to determine the initial angular velocity of the target’s next spinning cycle.

Finish the iteration by using revolution numbers n as the iteration times. The de-tumbling process will end when \( H^{{\prime }} \) is close to zero.

Above figures show that the target’s angular velocity around each axis is gradually decreased cycle by cycle Fig. 11. But the de-tumbling rate becomes slow as the target’s angular velocity slows down.

Fig. 12.
figure 12

Effective time of force in each cycle, Total time relative to n

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Eventually, the process stops at n = 3740 when \( H^{{\prime }} \) is close to zero. The effective time of force in each cycle and the total time of the de-tumbling procedure can be calculated Fig. 12.

On the other hand, if the brake force’s point of action is not optimized, but constantly acting at the middle part of the target Fig. 13. The simulation results as below Fig. 14:

Fig. 13.
figure 13

Un-optimized method of de-tumbling

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Fig. 14.
figure 14

Total time of un-optimized method

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Put the angular parameters into iteration, the total time can be computed relative to cycle time. Comparing to the total time in optimized de-tumbling, it can be concluded that the time consuming in debris removal can be spared by optimized de-tumbling strategy based on the debris’ kinematic parameters of spin movement.

5 Conclusions and Future Work

In this paper, we presented our motivation for studying capturing of large tumbling target. In most cases, the space debris who loses power and control always floats freely with large residual angular momentum on-orbit. Especially when the collision occurs between debris and other objects, the residual angular momentum will increase, which will further increase the challenge of capture safety. In this paper, we propose a method of de-tumbling a large tumbling target with the eddy current brake. The eddy current brake is mounted on the hand of the manipulator to brake the tumbling target, and the residual angular momentum is reduced smaller. Thereby providing safer conditions for arm-based capture. In view of the characteristics of the motion of the large tumbling target, this paper presents an optimized method of braking, and obtains the optimized planning trajectory by simulation. Then, the validity of the above method is verified by simulation.

The effectiveness of large debris de-tumbling based on eddy current brake in hand of manipulator is verified by the study of the noncontact de-tumbling technology by the simplified model. The ground experiment based on the 6 DoF space manipulator [12] and air-bed experimental system [13] will be carried out in the future and the results should be applied to practice.