Pre-impact Configuration Planning for Capture Objectof Space Manipulator

CONG Pei-chao,SUN Zhao-weiResearch Center of Satellite Technology

Harbin Institute of TechnologyHarbin 150080, China

Abstract-A concept "Straight-Arm Capture" and a method ofconfiguration planning are proposed, which are based on themomentum conservation of space manipulator. Theconfiguration satisfies the proposed method can reduce thecoupling angular momentum, joint angular velocity and theburden of compound control, then avoids the limit of jointvelocity and torque when control the compound. Finally thesimulation results show that the effectiveness of the method.

Keywords-Space manipulator, straight-arm capture,planning, coupling angular momentum

I. INTRODUCTION

In order to reduce risks and improve efficiency in spaceenvironment, space manipulators are applied to replace orhelp astronauts to perform space missions, thus to. Spacemanipulator systems have been applied in space shuttles andstations, the first real sense ofspace robot system ofthe worldwas launched with the Japanese Experimental Satellite VII in

1997[1] .Object capture is one of important applications of space

manipulator, which contains the following phases: TrackingPhase, Approaching Phase, Capturing Phase, andPost-capture Phase.

The first and the second phases need to acquire the relativeposition and attitude parameters between the trackingspacecraft and the object, and the inertial character of object,meantime the approaching trajectory and configuration ofspace manipulator are planned to reduce the impact force andcoupling angular momentum generated in the capturing. Thethird and the fourth phases which involve the selection of asuitable gripper to execute capturing and the grasping point,the modeling ofthe collision between manipulator and object,the impact force, then post-capture motion control. Thechange of system's momentum caused by impact, particularlythe angular momentum has obvious effect on the attitude ofthe entire system. Therefore this paper focuses on the impactof capture.

There are many solutions regarding to the impact issuesduring the capturing operation. Most of them are proposed in

the sense that reduce impact force. L-B. Wee et al [2]adopted a "gradient projection algorithm" to optimize themotion track of manipulator to minimize the impact force.

Wee and Walker [3] put forward the controllable momentum

wheels ("speech leech") to absorb the angular momentum ofthe target, thus to reduce the impact force.

The other solutions are from the perspective of control

1-4244-2386-6/08/$20.00 ©2008 IEEE

algorithms. The utilization of impedance control to capture a

non-cooperative target was discussed by Yoshikawa [4] , and

a condition which guarantee the target is not pushed away was

given. Nenchev and Yoshida [5] - [6] applied "reaction null

space control algorithm" to decouple the base andmanipulator's dynamics, then apply it to the post-capturemotion control. Through it we can achieve that some jointvelocities have no effects on the momentum distribution.These solutions also subject to the limitations of joint and

torque. Besides, there are many other methods [7]-[9] .The significance of aircraft attitude requirement as

compared to its translation motion, it's crucial to control theeffect of impact on angular momentum. Before the impulsiveforce generated by the collision, to choose an appropriatemanipulator configuration that the direction of impulsiveforce in a line with or near the centroid of system(manipulator and base), so that to reduce the effect of impact,then to facilitate the base attitude control in the post-capturephase. For some objects that have relatively small angularmomentum, we adopt "Straight-Arm Capture" (SAC)strategy, of which the joint angles are zeros.

This paper at first formulates the dynamical model andmomentum conservation equation of space manipulator,analyzes the impact phenomenon of space manipulatorcapture object, presents a unique pre-captureconfiguration-"SAC" concept, then investigates themanipulator's pre-capture configuration that the centroids ofbase, system and end-effector are colinear, and obtains thecorresponding angular relationship. The manipulatorconfiguration satisfies above relationship can reduce theimpact effect on the system angular momentum, then avoidthe limitations of joint and torque when control thecompound. Finally the validity of the proposed method isverified by the simulation.

II. SYSTEM KINEMATICS AND DYNAMICS

The model consists ofa base (spacecraft) and a manipulatorwith n revolute joints. The coordinate systems used in this

article are the inertial coordinate LI of which the original

point is a point I on the orbit, the base coordinate LB of

which the original point is the center of mass of spacecraft,

the local coordinates fixed on each joint L i which rotate

along their z-axes and the x-axes direct to the i+1th joint.Based on the kinematic relation expressions derived by Y.

Umetani and K.Yoshida[10] , and modeled with Lagrangian equation, the dynamics equation[11] is as the following,

hTm

TBB

m

Bb

mBm

BmB FJJF

ccx

HHHH

+=+τθ

(1)

Where

=ωHrM

rMMEH

g

Tg

B0

0~

~ )( 66×∈ R

=ωθ

ω

HJ

H TBm )( 6 nR ×∈

( )=

++=n

ii

Tiii IrrmIH

1000

~~ω )( 33×∈ R

( )=

+=n

iTiiiRii JrmJIH

10

~ωθ )( 3 nR ×∈

( )=

+=n

iTi

TTiiRii

TRim JJmJIJH

1

)( nnR ×∈

=

=n

iTiiT MJmJ

1/ω )( 3 nR ×∈

[ ]0,,0),(,),(),( 2211 iiiiiTi prkprkprkJ −×−×−×=)( 3 nR ×∈

[ ]0,0,,,, 21 iRi kkkJ = )( 3 nR ×∈

00 rrr gg −= )( 3R∈

00 rrr ii −= )( 3R∈

−−

−=

00

0~

xyxz

yzr

Where, im : mass of the i-th link ; M: total mass of

system; ir : position vector of the centroid of link i; ip :

position vector of the joint i ; ik : unit vector along axis i; 0r :

position vector of the centroid of base; gr : position vector of

the centroid of system; Bc : nonlinear forces acting at the

base; mc : nonlinear forces acting at the manipulator; BF :the

external forces acting at the base; hF :the external forces acting at the end-effector;τ : joint torques by joint motors.

III. IMPACT ANALYSIS

A. Conservation of MomentumThe key procedure of a capturing operation is during the

end-effector impacts the object. Due to the difference conditions between in the outer space and on the ground, the main point of

difference lies in that the manipulator configuration and the base attitude and position, the momentum of system will have notable change due to the mechanical collision in space. The main object of impact analysis is how to avoid, reduce and control this action.

The effect of impact force on system represents the momentum change of the base and the manipulator. Utilizing the conservation of momentum makes it consist of the above parts. The specific expressions are as the following,

=

=n

iiivmP

0

(2)

=

×+=n

iiiiii vmrIL

0)( ω 3

combing (2) and (3), one arrives the following equation,

θω θ

+=IJv

IrMrMME

LP Tg

Mg

g

0

003~

~ (4)

and then

×++=

PrH

vH

LP

cB00

0 0θ

ω (5)

Where

=Mg

Tg

B HrMrMME

H0

03~

~ )( 66×∈ R

=ωθH

JH Tg

c )( )6 nR ×∈

In practice, the significance of base attitude motion as compared to translational motion, so eliminates the 0v from the (5),

PrHHL gcB ×++= θω ~~0 6

Where TggB rrHH 00

~~~ −= ω Tggc JrHH 0~~ −= ωθ

In [8] the term θcH is called the coupling momentum and

in [9] the term θcH~ is referred to as the coupling angular momentum. These two concepts are crucial for the analysis of impact, the impact results in the change of above terms.

Cancelling the bx from the (1), to obtains,

hT FJcH θθ τθ ˆˆˆˆ +=+ (7)

Where

BmBTBm HHHHH 1ˆ −−= θθ

BBTBmm cHHcc 1ˆ −−=

BBTBm FHH 1ˆ −−= ττ

TBB

TBm

Tm

Tm JHHJJ 1ˆ −−=

B. The Pre-impact Configuration

In order to reduce the effect of impact force on system angular momentum, it will be reasonable to plan the manipulator pre-impact configuration. First, we can adopt a specific configuration of capturing operation, that is to set all the joints’ angle to 0 degree which is the so-called "Straight-Arm Capture”(SAC). Then the centroids of the base, the manipulator, the whole system and the end-effector are on the same line L. If the impact force coincides with L or within its vicinity, theoretically, there will be no or minor angular momentum disturbance either during the impact or post-impact, which is the 2C situation in [6] . Under some conditions that the impulsive force is bearable by manipulator joints, and the space is enough, we can use "SAC” to reduce the effect of impact during capturing operation. However, the conditions are not always satisfied, which means either the impulsive force generated by impact goes beyond the capacity of joints, or there is limitation of operation space, ”SAC” could not be applied. Therefore, we can plan the pre-impact configuration of manipulator so as to the impact angular momentum is accommodated by all parts of system, and the coupling momentum will be a minor quantity, that is the situation 1C mentioned but not detailed derived in [6] .In this paper, based on a two-link planar space manipulator, we derive the relationship between manipulator configuration and the vector of impulsive force, and the relationship of impact force and the centroids of the base and the whole system.

Fig.1 depicts the configuration of space manipulator, the coordinate system takes the centroid of base oC as origin, x- and y-axis parallels to two sides of the base respectively. C , oC , 1C , 2C , sC , E denote the positions of the centroids of system, the base, the first to the 2-th joint, the manipulator, and the end-effector, respectively in base coordinate system B . 1θ , 2θ , denote the 1st, the 2nd angles, β and γare the angles of vector of impulsive force w.r.t the end-link and the end-effector.

The coordinate of the centroid of Link 1, 1C is

1 1/ 2 cos( )X a l θ= + ×

1 1/ 2 sin( )Y l θ= × (8)

The coordinate of the centroid of Link 2, 2C is

2 1 1 2cos( ) / 2 cos( )X a l lθ θ θ= + × + × +

2 1 1 2sin( ) / 2 sin( )Y l lθ θ θ= × + × + (9)

Figure .1 The Configuration Of Two-Link Planar Space Manipulator

The coordinate of the centroid of manipulator sC is

1 1 2 2

1 2

1 1 2 1 1 2

1 2

[ /2 cos( )] [ cos( ) /2 cos( )]

sm X m XX

m mm a l m a l l

m mθ θ θ θ

× + ×=+

× + × + × + × + × +=+

1 1 1 2[ / 2 cos( )] [ cos( ) / 2 cos( )]2

a l a l lθ θ θ θ+ × + + × + × +=

1 1 2 2

1 2

1 1 2 1 1 2

1 2

/ 2 sin( ) [ sin( ) / 2 sin( )]

sm Y m YY

m mm l m l l

m mθ θ θ θ

× + ×=+

× × + × × + × +=+

1 1 1 2/ 2 sin( ) [ sin( ) / 2 sin( )]2

l l lθ θ θ θ× + × + × += (10)

The coordinate of the centroid of system C is

1 2

1 2

1 1 1 2

1 2

( ) 0( )

[2 3 / 2 cos( ) / 2 cos( )]( )

s oc

o

o

m m X mXm m m

m a l lm m m

θ θ θ

+ × + ×=+ +

× + × + × +=+ +

1 2

1 2

1 1 1 2

1 2

( ) 0( )

[3 / 2 sin( ) / 2 sin( )]( )

s oc

o

o

m m Y mYm m m

m l lm m m

θ θ θ

+ × + ×=+ +

× × + × +=+ +

(11)The coordinate of End-effector, E is

1 1 2cos( ) cos( )EX a l lθ θ θ= + × + × +

1 1 2sin( ) sin( )EY l lθ θ θ= × + × + (12)

oC as the original point, so oC (x,y)=(0,0).

When E, sC and oC are collinear in L, to obtains,

0

0

s E s

s E s

Y Y Y XX X X X

− −=− −

Then, 1 1 2

1 1 2

1 1 2

1 1 2

3 / 2 s in ( ) / 2 s in ( )2 3 / 2 c o s ( ) / 2 c o s ( )

s i n ( ) s i n ( )c o s ( ) c o s ( )

l la l l

l la l l

θ θ θθ θ θ

θ θ θθ θ θ

× + × ++ × + × +

× + × +=+ × + × +

If a=l, then, 1 1 2

1 1 2

1 1 2

1 1 2

3 s i n ( ) s i n ( )4 3 c o s ( ) c o s ( )

s i n ( ) s i n ( )1 c o s ( ) c o s ( )

θ θ θθ θ θ

θ θ θθ θ θ

× + ++ × + +

+ +=+ + +

(13)

If 1θ , 2θ satisfies above equation, E, sC and oC are

aligned in L, whiles the point C must be in L. α is the angle of line L with axes X, so

atan( )α λ= (14) Then

1 2[ ( )]β α θ θ= − + (15) γ is the vector of impulse force, we want to make this

Direction in line with L, therefore,

1 2[ ( )]γ π β

π α θ θ= +

= + − + (16)

The space manipulator configuration which satisfiesequation (16) can reduce the effect of impact force on the coupling angular momentum, lower the joints’ angular velocities, and then circumvent the limitations of joints and torques of the compound control problems. Then, the vector of force is colinear with L, the impact generated angular momentums of manipulator and base are equal and in opposite direction, and the sum of the system momentum is zero. The impact effects on the system will be accommodated as translational momentum, thus the system attitude keeps stable. The next section gives the simulation verification.

IV. SIMULATION Table 1 Model Parameters

Parameters im kg il m iI ( 2kgm )

base 1000.0 4.0 1332

Link 1 100.0 2.0 33.3

Link 2 100.0 2.0 33.3

In order to guarantee the vector of impulsive force is colinear with L, the angle of force w.r.t. end-effectorγ must

satisfy γ π β= + 16

The two-link planar space manipulator system is simulated to verify the proposed method. Table 1 gives the parameters, impact happenes at t=0.2s, the impulsive force

10000NhF = , and 186oγ = . For (14) and (15), we set

1 30oθ = and 2 24oθ = − ( 1θ and 2θ which fulfill the equation (15) are not unique). And "Reaction Null Space" method[6]is adopted as control strategy . The units used in simulation results are attitude in rad , angular momentum in Nms , angular velocity in rad/s , time of map a in ms, and time of others in s. The simulation step is 0.01s.

0 50 100 150 200 250 300 3500

1

2

3

4

5

6

7x 10

-7 Base angle

Figure 2-a Base Attitude

0 0.5 1 1.5 2 2.5 3-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1angle momentum

Figure 2-b Angular momentum

0 0.5 1 1.5 2 2.5 3-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-4 angle 1 velocity

Figure 2-c Joint1 Angular Velocity

0 0.5 1 1.5 2 2.5 3-1.2

-1

-0.8

-0.6

-0.4

-0.2

0x 10

-4 angle 2 velocity

Figure 2-d Joint2 Angular Velocity

The same impulsive force, 1 30oθ = , 2 30oθ = − ,

0 50 100 150 200 250 300 3500

1

2

3

4

5

6

7

8x 10

-5 Base angle

Figure 3-a Base Attitude

Figure 3-b Angular momentum

0 0.5 1 1.5 2 2.5 3-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0angle 1 velocity

Figure 3-c Joint1 Angular Velocity

0 0.5 1 1.5 2 2.5 3-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0angle 2 velocity

Figure 3-d Joint2 Angular Velocity

The same impulsive force, 1 30oθ = , 2 18oθ = − ,

0 50 100 150 200 250 300 350-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-5 Base angle

Figure 4-a Base Attitude

0 0.5 1 1.5 2 2.5 3-10

0

10

20

30

40

50

60angle momentum

Figure 4-b Angular momentum

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035angle 1 velocity

Figure 4-c Joint1 Angular Velocity

0 0.5 1 1.5 2 2.5 3-60

-50

-40

-30

-20

-10

0

10angle momentum

0 0.5 1 1.5 2 2.5 30

0.002

0.004

0.006

0.008

0.01

0.012angle 2 velocity

Figure 4-d Joint2 Angular Velocity

The simulation results show that: 1. The direction of impulsive force which follows the

configuration derived in this paper, the attitude of base, the coupling angular momentum and the joint velocities are minor during capturing operation, accordingly the impact effect on the system behaves as translational motion, [as shown in Fig.2-a,b,c,d].

2. When “Reaction Null Space” is adopted as control strategy, through the planning of manipulator‘s pre-capture configuration, it can resolve the limit of joint and torque when control the post-impact compound [as shown in Fig.2-c,d].

3. Fig.3 and 4 show that when the direction of impulsive force deviates from the line L, the attitude of the base, the coupling angular momentum and the joint velocities are all in controllable ranges. Which means, when the impact direction couldn’t be precisely determined, it could also to reduce the burden of post-impact control just when makes the direction in a certain area. So, by selecting a appropriate space manipulator

pre-capture configuration that the centroids of base, manipulator, system and end-effector are in line with L, at this time, if the direction of impact force coincides or approaches the line L, then the change of angular momentum generated by impulsive force during impact is a minimal mount, the simulation verified the method’s availability.

CONCLUSION In this article, we discussed the impact phenomenon for

space manipulator capture object, presented “SAC” concept, got the corresponding angular relationship that the centroids of base, system and end-effector are in a line. Through the planning of space manipulator configuration can reduce the effect of impulsive force on the coupling angular momentum, then overcome the limit of post-capture compound’s control strategy.

As in the actual operation, it is difficult to determine the direction of impulsive force, mainly owing to the modeling of impact is still in the developing stagy, the existing theories aren’t very mature. So, future research should have following directions :

a) The choice of gripper of space manipulator is vital to control the direction of impulsive force, it should give more attention.

b) The modeling of impact during capture operation needs in-depth research.

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