Control of a Space Robot for Capturing a Tumbling Object - ESA

CONTROL OF A SPACE ROBOT FOR CAPTURING A TUMBLING OBJECT Ou Ma (1), Angel Flores-Abad (1), Khanh Pham (2) (1)

Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA Email: [email protected], [email protected] (2) U.S. Air Force Research Laboratory, Space Vehicle Directorate, Kirtland Air Force Base, NM 87117-5776, USA Email: [email protected]

ABSTRACT This paper presents an optimal control strategy for a space robot to capture a free-tumbling object under the condition of having minimal impact to the base satellite during the capturing operation. The idea is to first predict a target time, location and speed of the tumbling object for the robot to intercept with such that, when the robot hand physically touches the object, it will transfer a minimal angular impulse to the base satellite. Then, an optimal motion trajectory and the corresponding joint torques will be generated to control the robot to reach the object at the targeted time and location. Joint rate and torque limits will be taken into account in the optimal control solution. Since the control acts before a physical contact happens, it will not affect any existing compliance control capability of the space robot regarding both implementation and operation. Therefore, the proposed method can co-exist with an existing compliance control method in the robot control system. A numerical simulation example is presented to demonstrate the effectiveness of the proposed method. A numerical simulation example is presented to demonstrate the effectiveness of the proposed method. 1.

GENERAL SPECIFICATIONS

Space manipulators have been successfully used for many applications such as maneuvering astronauts, berthing and deploying large space structures, constructing and maintaining the International Space Station (ISS), exploring and sample-collecting, satellite on-orbit servicing (technology demos only), etc. All of these manipulation activities dealt with cooperative payloads or target objects and thus, the existing robotics technologies can handle them quite well, though many improvements such as the operational efficiency and dexterity may still be done. However, if a manipulator is expected to perform more challenging and riskier tasks, such as to capture an unknown object, like a piece of space debris, or a non-cooperative object, such as a tumbling satellite, the currently available space robotics technologies are still far from being ready. To make these challenging tasks practical, many enabling technologies have to be further advanced. This research develops enabling technology for a space manipulator to capture a tumbling satellite.

The operation of capturing a tumbling satellite after completion of the rendezvous process may be divided into four phases. The first phase is the observing and planning phase, which is to acquire motion (mainly rotational motion) information of the target satellite and then determine when and where to grasp the target satellite. The second phase is the final approaching phase, in which the robot is controlled to move its endeffector to the planned grasping location for grasping the target satellite at the planned time. The third phase is the capturing (or interception) phase, in which the robot captures the target satellite. This is the phase where physical contact happens and thus it is also the most risky phase. The last is the post-capture stabilization phase in which the tumbling target satellite is detumbled and stabilized by the robot and servicing satellite. The work reported in this paper is concerned only with the planning part of the first phase and the whole second phase. Our focus on the first phase is to determine an optimal time and location based on the tumbling motion of the target satellite for the robot’s end-effector to intercept with the target satellite. The focus on the second phase is to control the robot to reach the optimal location with a minimal disturbance to the attitude of the servicing satellite for safe capture of the tumbling target satellite. Research on the observation part of the first phase has been done by many researchers in the fields of computer vision and sensing technologies. The third and fourth phases are more risky and challenging because of the involvement of physical contact. Some research work has been done but much more future work is absolutely required in order to have guaranteed safe and successful future missions. We will not discuss these here because they are out of the scope of the paper. The impact minimization problem for capturing a target object has been studied by a few researchers from different perspectives. Yoshida et al [1] modeled the collision dynamics during the capturing process using the extended generalized tensor. They focused on the moments just before and after the impact using velocity relations. Yoshida and Nenchev [2] introduced the concept of reaction null-space to analyze the impact and post-impact moments of the capturing process. They found that choosing configurations within the reaction

null-space of the servicing manipulator system can result in an operation with minimum impact to the attitude of the servicing satellite. Papadopoulos and Paraskevas [3] proposed a methodology based on the percussion point of bodies to minimize the forces instead of the momentum transmitted to the base of the manipulator when grasping an object. In all the past studies the contact force was just an assumed impulsive force exerted on the tip of the manipulator without taking into account of the geometry of the contacting bodies and the tumbling motion of the target satellite. In this research, we move a step forward to consider the tumbling motion of the target satellite in the robot control strategy for achieving minimal attitude impact to the servicing satellite. The problem of optimal trajectory planning for a space manipulator was addressed earlier by Duvowsky and Torres [4]. They introduced an enhanced disturbance map, which can aid in selecting a path that reduces the disturbances of the base spacecraft by identifying the direction of each joint movement which results in minimum or maximum disturbances. Agrawal and Xu [5] proposed a global optimum path planning for redundant space manipulators using a variational approach to minimize the objective functional with constraints in the linear and angular momentum. Lampariello et al [6] proposed an optimal motion planning method using criteria in the joint space. Huang et al [7] proposed an optimal approach trajectory planning method for minimizing the impact on the base satellite, the optimal trajectory is found based on a genetic algorithm and in the dynamic coupling factor. Aghili in [8] designed an optimal controller to capture a tumbling satellite using an objective function minimizing the operation time and relative velocity between the robot tip and the target. T. Oki et al [9] also proposed an optimal control method to capture a tumbling satellite but they focused mainly on minimizing the operational time for fast capture. The main difference between our approach and those optimal control approaches is that we focus on the minimization of the reaction torque on the servicing satellite for safe capture operation. In this paper, the terms “servicing satellite” and “base satellite” are exchangeable, so are the terms “target object” and “target satellite” and the terms “manipulator” and “robot”. 2.

DYNAMICS MODELLING

2.1. Basic assumptions The development of the methodology described in this paper is based on following basic assumptions: (a) Both the servicing satellite and the target object are assumed to be rigid bodies. The manipulator also consists of rigid links.

(b) The mass properties and motion state of both the base satellite and the target object are assumed known. (c) The maneuvers are in close proximity range and thus the effect of orbital mechanics is neglected. (d) The attitude of the base satellite is fully controlled unless otherwise stated. Assumption (a) is a very usual assumption in the robotics field, especially for development and practical implementation of control methodologies because a rigid-body dynamical system is much easier to model and analyze. In many applications such assumption is also practically sufficient. This assumption may be too off reality for a long space manipulator to capture a fast tumbling object. Assumption (b) is to focus the research on the robot control problem and avoid dealing with the inertia identification and motion state estimation problems, which are two research areas having been well studied and are continuously being studied by many other researchers. Assumption (c) is to focus the research to the scope of proximity rendezvous and capture, where the forces/moments related to orbital mechanics are negligible compared to the inertia forces caused by the robot motion and the contact forces caused by the physical interception. Assumption (d) has been a common approach for all the practical capturing operations in space because uncontrolled attitude can significantly increase the possibility of mission failure. We are well aware that these assumptions may not be realistic in many application cases. They are imposed to facilitate our early development of the technology. We will be relaxing these assumptions in the future research. 2.2. Dynamics Modelling of the Servicing System The multibody system of the servicing satellite and the manipulator consists of n  1 rigid bodies connected by n joints, as shown in Fig. 1. Body 0 is the satellite which is also the base of the robot and body i (i  1, 2, , n) is the i-th link of the manipulator. Joint 0 has 6 degrees of freedom which connects the inertia frame to the servicing satellite and Joint i ( j  1, 2, , n) has only one degree of freedom which articulates links i-1 and i. The symbols appearing in Fig. 1 are defined as follows: θ  Rn : generalized joint coodinates τ  Rn : generalized joint torques ri  R3 : position vector of the CM of Body i

rc  R3 : position vector of the mass center of the entire servicing system re  R : position vector of the manipulator end-effector 3

ai  R3 : intrabody vector of link i expressedin Fi frame

Moreover, H is the generalized inertia matrix of the manipulator when it is attached to a free-floating base. The other variables of the system are defined as follows:

ci  R3 : position vector of the CM of Link i measured from Joint i zi  R : rotational axis of the i th joint 3

H    JT i Ii Ji   R nn : generalized inertia matrix of n

vi  R3 : linear velocity of the mass center of Link i

i 1

ωi  R3 : angular velocity of the i th link v0  R3 : linear velocity of the servicing satellite

the manipulator as it is attached to a fixed base.

ω0  R : angular velocity of the servicing satellite

Hb  JTm0

3

ve  R3 : linear velocity of the end-effector

 m1 Hb    mR 0c

fe  R6 : external force and moment exerted on the end-effector

f0  R6 : external force and moment exerted on the servicing satellite fr  R : reaction forces at the the root of the manipulator τ r  R : reaction torque at the the root of the manipulator 3

Link n-1

On-1

z 2 2

cn-1

Link n

Fe

an

Cn-1

cn

Fn

Cn

vn

re

C2

c2

v2

F2

ωe  ωn ve

rn a2

fe

inertia matrix of the

Ii  R3 : inertia matrix of link i with respect to its mass center

Ji   z1

z 2 ... zi

0 ...  R3n : Jacobian

matrix

J vi   z1  ρc1

z 2  ρc 2

zi  ρci

0

  R3n

Jacobian matrix for the linear velocity of the i-th body n

J m0   mi J vi  R3n i 1

rc

c

mRT0c  66 R : H 

for the angular velocity of the i-th body. n

an-1

Fn-1

ω2 Link 2

matrix

servicing satellite.

3

ωn1

inertia

between the servicing satellite and the manipulator.

ωe  R3 : angular velocity of the end-effector

zn

T

HT   R6n : coupling

Link 1

ω1

a1

1 F1

τr

H    I i Ji  mi Zi J vi   R3n n

v1 roc

c1

z1

fr

C1

r0

a0

ω0

f0

v0 C0

F0

i 1

m : total mass of the servicing system Service Satellite

mi : mass of the i-th body

Body 0 (B0 )

1  R33 : identity matrix

Fig. 1 Multibody dynamic system of a servicing satellite and a space manipulator In the final planning and approaching phases there are no external forces acting on the systems and thus, the momentum of the servicing system will be conserved, from which we can derive the dynamics equation of the space robot in terms of its joint variables θ as follows [10]: Hθ  Cθ  τ (1) where  1  Cθ    θT Hθ   R n (2) θ  2  which includes the nonlinear Coriolis and centrifugal forces acting on the system and H  H  HTb Hb 1Hb  Rnn

(3)

 zi (3) zi (2)   0  Zi   zi (3) 0  zi (1)   R33 : matrix form to   zi (2) zi (1) 0  represent the cross product operation r  for any vector.

 n  H    I i  mi RT0i R 0i   I 0  R33  i 1  ρcj  R3 : position vector from the jth joint to the mass center of the ith body. 2.3. Dynamics Modelling of the Target Satellite Since the target satellite is assumed to be a single rotating rigid body, its dynamics equation is rather simple, (4) It ωt + ωt ×It ωt = τ t where

I t  R33 : inertia matrix of the target satellite. ωt  R3 : angular velocity of the target satellite. ωt  R3 : angular acceleration of the target satellite. τ t  R3 : external torque applied to the target satellite. Based on the assumption of ignoring the orbital mechanics, the external torque τ t is zero before a capture operation and is the contact torque during capturing. 3.

DETERMINATION OF THE OPTIMAL CAPTURE TIME

As we have stated early, the overall goal for the control design is to capture the tumbling satellite with minimal impact on the attitude motion of the servicing satellite. The first step for achieving such a goal is to determine a best time for the robot to grasp. It is understandable that if the resultant contact force exerted at the robot tip (resulting from a capture action) passes the mass center of the servicing system, the contact force will not cause any attitude disturbance to the servicing satellite, as shown in Fig. 2. However, the direction of contact force depends on the relative velocity, contacting spots and contact geometry, which make it very difficult to predict in advance. Although such a prediction may not be impossible if we have an accurate contact dynamics model, this will require more extensive research work in the future. For this work, we approximate it by assuming that the contact force is along the direction of the relative velocity between the robot tip and the grasping handle of the target satellite. Therefore, no attitude disturbance to the servicing satellite can be achieved by either of the following conditions: 1) The relative velocity between the robot tip and the grasping handle of the target satellite is zero. 2) The relative velocity is nonzero but its direction passes through the mass center of the servicing system. The design of control strategies to meet first condition is a common approach such as the work described in [8]. It requires that the robot tip must move as fast as the grasping handle of the tumbling satellite. This is very difficult or impossible when the target satellite has a fast tumbling motion because the tip speed of a manipulator is always limited not only by the joint rate limits but also by the attitude tolerance of the servicing satellite. In such a case, a strategy using the second condition as its control goal becomes more attractive because it does not requires zero relative velocity, but such an approach has not been studied in the past Therefore, we will focus our study on achieving the second condition. As shown in Fig. 3, the second condition means

that the angle β (between the relative velocity and the position vector of the grasping handle of the target satellite) should be zero. In such a case, the major component of the impact force (assuming mainly along the relative-velocity direction) will pass through the mass center of the servicing system and thus, cause no angular moment to the servicing satellite. Of course, this is only an ideal case. In a general tumbling case, the direction of the relative velocity may never pass through the mass center of the servicing satellite. However, even if the β angle can never reach zero, it will always have a minimal value at a certain time. Hence, we will just focus on the problem to determine such a minimal angle. This can be formulated as: given a set of initial motion conditions of the target satellite, find a future time t1 such that the velocity of the grasping handle,

v(t1 ) , will have zero or minimum momentum about the mass center of the servicing system. Mathematically, this can be expressed as

max cos   max t

t

v(t )  r(t ) v(t ) r (t )

(5)

where

v(t )  vt  R(ωt  a) r(t )  rt  Ra

(6)

fc Mass center of the servicing system

Fig. 2 Interception for minimal impact to the base satellite based on the contact force direction. In the above problem definition, R  R33 is the rotation matrix defining orientation of the satellite frame Ft with respect to the orbital frame F0 . Position vector a points to the grasping handle from the mass center of the target satellite, expressed in the satellite’s body-fixed frame Ft . Note that both matrix R and vector ω t are nonlinear functions of time although the satellite is undergoing a torquefree rotation. R(t ) and ωt (t ) can be solved from the following dynamics equation and known initial conditions:

I t ωt + ωt × I t ωt = 0

been controlled to keep a fixed distance to the target satellite such that the target satellite is within the reach of the robotic arm.

(7)

r(0)=r0 , v(0)  v 0 , ωt (0)  ω0

To develop the optimal control, the robot’s dynamics equation (1) is rewritten into a state space form as follows

Solution of ωt (t ) for any given time t in the satellite local frame can be obtained in closed-form through the evaluation of Jacobian Elliptic functions [13]. However, the closed-form solution of ωt (t ) in global frame and the closed-form

x  f(x)  G(x)τ

solution of the orientation R(t ) are very difficult, as discussed recently in [14]. At any rate, numerical solution of the optimization problem (5) is always possible. When there are no external forces and moments applied on the target satellite, its tumbling motion should be periodical over the time. Thus, the abovediscussed optimal capture time will be repeated with the period of the satellite’s tumbling motion. This means that we can have enough time to prepare for a safe and optimal capture because the desired capture time and opportunity will come repeatedly over the time. Of course, in the reality, the target will unlikely be doing perfectly periodic rotation because there always exist some non-zero external forces and moments as well as damping factors in the system. Therefore, the tumbling motion may not be kept in the same period forever. However, such changing is likely be slow in time and thus, we will still have time to plan and perform an optimal capture task, as described in the next section. t0

t  t1

h(0)

ω (0)  ω 0

h(t1 )

ω (t1 )  ω 1  ?

R (0)  R 0

R (t1 )  R 1  ?

v (0)

zt

Grasping handle

zt a

v(t1 )

v t (0)

Ft

yt xt

yt



vt Ft

ω (t1 )

ω (0) r (t1 ) Mass center of target satellite

rt (0)

Mass center of servicing system

z0

rt (t1 )

y0

F0

 x0

Fig. 3 Interception for minimal impact to the base satellite based on the contact force direction 4.

OTIMAL CONTROL FOR THE ROBOT’S FINAL APPROACHING

Once an optimal time for capturing is determined, the corresponding motion state of the target satellite can also be calculated. This optimal time and motion state will be used as the final time and target pose of the endeffector for developing the control of the robot to perform the capture task. To focus on the robotics control, it is assumed that the servicing satellite has

(8)

where x  R is the state vector; f  R is the state function, G  R2nn is the control matrix; and τ  Rn is the joint control torques. They are defined as  x  θ  x   1     x 2  θ  1 0   x1  f(x)   (9)  x  1 0  H(x ) C(x)  1  2  0  G(x)   1   H(x1 )  2n

2n

Assuming that the servicing satellite is fully controlled, we can find the impact of the robot motion to the servicing satellite by deriving the reaction force f r and moment τ r (see Fig. 1) on the root of the robot (at the first robot joint), namely, n   fci     fr  i 1  τ    n   r     τ ci  (ri  a 0 )  fci    i 1  (10) n    m v  i i    f (x, x)  i 1  r  n     τ r (x, x)   I ω  ( r  a )  m v   i 0 i i    ci i  i 1  where f ci  R 3 and τ ci  R 3 are the inertia force and moment acting at the mass center of the ith body, respectively; f r  R 3 and τ r  R 3 are the total force and moment the robot applies at the its root, respectively. Therefore, the total reaction force and moment caused by the robot motion at the mass center of the servicing satellite are n   fci     f0  i 1  τ    n   0     τ ci  ri  fci    i 1  (11) n    mi v i    f (x, x)  i 1  0  n     τ 0 (x, x)   I ω  m R v    ci i i i i    i 1  Our control goal is then to find a time history of each joint’s control torque such that, when the manipulator’s tip is controlled by this set of joint torques to move from its initial pose to its final pose, it

will have minimal attitude disturbance to the servicing satellite. To find this set of optimal control torques, we can use the following objective function t1 θ  θ  J   τT0 τ 0 dt , x(0)  x0   0  , x(t1 )  x1   1  (12) 0 θ θ1   0 For this optimal control problem, the initial state x 0 is known and the final time t1 and final state x(t1 ) are determined by solving the constrained optimization problem defined in section 3. The Maximum principle [12] tells us that a necessary condition for an optimal control τ (t ) is to maximize the following Pontryagin Hamiltonian H  H (x, λ, τ)  0 τT0 τ0  λT (f  Gτ)

(13)

where λ  R is the vector of costate variables. Then, the necessary conditions for the solution can be written in the following form H H λ   , (14) x λ x 2n

with the initial and final conditions given in (12). For our problem, the final time and final motion state of the optimal control are known. In other words, it is a fixed time and fixed boundary problem. However, because of its complex nonlinear nature, it still has to be solved numerically. 5.

To show the application of the afore-mentioned optimal control strategy, we present an example using a 2-DOF planar manipulator in this section. The parameters of the satellites and robot are defined in Fig. 4 and Table 1.

v (t f )

c2

2

f r 0 0.5m

Fully Controlled Base Satellite

1

Mass center of the entire servicing satellite system

C

m r

i i

i 0

m

  0.2 0.41 0 m T

We also assume that :

a  0 0.5 0 m s2 T

1 0 0  R (0)  0 1 0  , 0 0 1  ωt =0.147 rad/sec,

r(0)  [2.117 1.41 0]T m, v(0)   0.735 0 0 m/s T

In the 2-D case, the rotational motion is simple and thus, we can easily find that the optimal time and orientation of the target satellite for the robot to capture with zero attitude impact are t f  4s

 0.8321 0.5547 0  R (t f )  -0.5547 0.8321 0   0 0 1 

tf

J   τTr τ r dt  J (x, τ) 0

subject to: t f  4sec

re (t0 )  [2 0 0 0]T ; x(t0 )  [0 0 0 0]T

re (t f )  [1.417 1.32 0]T ; x(t f )  [0.5 0.5 0 0]T

Table 1. Parameters of the 2-DOF space manipulator Body no.

n

dx3 dx1 dx 2 dx 4  x3 ;  x3 and  x4 ;  x4 , dt dt dt dt

2

Fig. 4 Example of 2-DOF space manipulator approaching a square target for capturing

Body

25 2.5 2.5 18

The system is assumed as shown in Fig. 4, the center of mass of the servicing system including the robot can be found to be:

Minimize

End-effector trajectory with minimun reaction torque

c1

1

250 25 25 180

5.1. Determination of the optimal time and state

ω  t f   ω  t0 

1

r

---0.5 0.5 ----

Following the method introduced in Section 5, the optimal control of the robot is defined as follows:

4

2

0.5 0.5 0.5 0.5

5.2. Optimal Controller for the Final Approaching

Grasping handle v (0)

0 1 2 4

v(t f )  R(ωt  a)  [0.612 0.0408 0]T m/s

SIMULATION EXAMPLE

Relative velocity pointing to the servicing satellite’s COM

Base satellite Robot link 1 Robot link 2 Target satellite

i (m)

ci (m)

mi (Kg)

Ii (Kg m2)

The optimal control problem was solved numerically using the Tomlab software. Figure 5 shows the manipulator’s joint displacements. Fig. 6 shows the applied joint torques driving the robot from its initial

4

0.5

0.2

0 0

0.5

1

1.5

2 Time (sec)

2.5

3

3.5

4

Fig. 8 Joint trajectories of the robot from the solution of a non-optimal control 0

Reaction moment (Nm)

2 1

-1 0

0.5

1

1.5

2 Time (sec)

2.5

3

3.5

4

-5

-10 rx ry

-15

rz

Fig. 5 Joint trajectories of the robot from the solution of the optimal control 30

-20 0

0.5

1

1.5

2

2.5

3

3.5

4

time (sec)

Fig. 9 Joint control torques from the solution of a non-optimal control

1 2

20 Joint torques (Nm)

0.3

0.1

0

10

6.

0 -10 -20 -30 -40 0

0.5

1

1.5

2 Time (sec)

2.5

3

3.5

4

Fig. 6 Joint control torques from the solution of the optimal control x 10

-6

0

Reaction moment (Nm)

1 2

0.4

1 2

3 Joint angles (rad)

control. As a result, such a robot maneuver will have a significant risk of destabilizing the base satellite.

Joint angles (rad)

configuration to the target configuration for capturing the target satellite. It can be noticed that both joints reach their desired final positions at the desired time. It is inter4esting to see that the second link moves a lot (forward first and then backward). This is necessary for the purpose of achieving the goal of having a minimum accumulated reaction moment on the base satellite during the arm maneuvering. Fig. 7 clearly indicates that he reaction torque at the reaction torque on the servicing satellite is very small (in the level of micro Newtons). Therefore, we can conclude that the desired optimal control goal has been achieved.



rx



ry

-5

rz

-10

-15

-20 0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

Fig. 7 Reaction torque on the base satellite In order to see the advantage of the proposed optimal control, we also implemented a usual robot controller for achieving the same tip motion target within the same time limit. The resulting joint trajectories of such a nonoptimal control are shown in Fig. 8. Obviously, the behavior of the joint motion, as shown in Fig. 8, seems better than that from the optimal control shown in Fig. 5. However, the reaction torque in this case, as depicted in Fig. 9, is much larger than that from the optimal

CONCLUSIONS

An optimal control strategy for a space manipulator to capture a tumbling satellite was presented. The goal of the control strategy is to minimize the impact on the attitude of the servicing satellite. This is done in two steps: 1) find an optimal time and the corresponding motion state of the tumbling satellite such that the physical interception from capturing operation will have zero or minimal attitude impact on the servicing satellite; 2) control the robot tip to reach the tumbling satellite at the optimal time and also cause minimal reaction moment on the servicing satellite during the motion of the robot. This approach is mainly aimed at safe operation for capturing a fast tumbling satellite, which is otherwise a very difficult and risky operation. A 2D example was presented to demonstrate the application of the proposed method. The example shows that, with the optimal control, the rotational disturbance to the base satellite is almost zero while the target can still be reached at the optimal time. Currently, we are studying the strategies and performance for the cases where the first step returns a result having random errors (i.e., the estimated optimal capture time and state have uncertainties). Further, the study of the performance of the proposed technology for general 6DOF manipulators including capturing contact dynamics is also in the planned future work of the this research project.

7.

KNOWLADGMENTS

This material is based on the research work sponsored by the US Air Force Research Laboratory (AFRL) under agreement number FA9453-11-1-0307. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory or the US Government. 8.

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