Pre-impact trajectory planning for minimizing base attitude disturbance in space manipulator systems for a capture task

Chinese Journal of Aeronautics, (2015), 28(4): 1199–1208

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

Pre-impact trajectory planning for minimizing base attitude disturbance in space manipulator systems for a capture task Zhang Long *, Jia Qingxuan, Chen Gang, Sun Hanxu School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China Received 17 November 2014; revised 26 January 2015; accepted 31 March 2015 Available online 22 June 2015

KEYWORDS Capture task; Minimum disturbance; Null space; Space manipulator; Trajectory planning

Abstract Aimed at capture task for a free-floating space manipulator, a scheme of pre-impact trajectory planning for minimizing base attitude disturbance caused by impact is proposed in this paper. Firstly, base attitude disturbance is established as a function of joint angles, collision direction and relative velocity between robotic hand and the target. Secondly, on the premise of keeping correct capture pose, a novel optimization factor in null space is designed to minimize base attitude disturbance and ensure that the joint angles do not exceed their limits simultaneously. After reaching the balance state, a desired configuration is achieved at the contact point. Thereafter, particle swarm optimization (PSO) algorithm is employed to solve the pre-impact trajectory planning from its initial configuration to the desired configuration to achieve the minimized base attitude disturbance caused by impact and the correct capture pose simultaneously. Finally, the proposed method is applied to a 7-dof free-floating space manipulator and the simulation results verify the effectiveness. ª 2015 The Authors. Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction The importance of capturing operations by a space manipulator has been increasing in recent years. A whole capture mission contains three specific phases: target chasing control * Corresponding author. Tel.: +86 10 61198259. E-mail addresses: [email protected] (L. Zhang), spacerobot@163. com (Q. Jia). Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

phase, impact phase between the target and robotic hand and stabilization control phase of tumbling motion.1 During the first phase, which is also called pre-impact phase, trajectory tracing or optimization sometimes is needed. And during the impact phase, due to control and sensing errors, there remain certain amounts of relative velocities between the robotic hand and the contact point of the target. Thus a force impulse is generated, which may damage the manipulator or the target if its magnitude is too large and disturb the base due to dynamic coupling. Therefore, the minimization of impact force impulse and the minimization of base attitude disturbance caused by the impact become two major problems for a capture task. In some cases, particularly when either (or both) manipulator and target are fragile or expensive, it is desired

http://dx.doi.org/10.1016/j.cja.2015.06.004 1000-9361 ª 2015 The Authors. Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1200 that the manipulation be as gentle as possible. Special attention needs to be paid to the minimization of impact force impulse. Regarding this issue, many scholars2–6 proposed their own solutions, and also our previous work.7,8 In other cases, especially when the manipulator moment of inertia is not negligible in comparison to the base, obvious base attitude disturbance will affect the communication with the ground and power supply for space manipulator, and compensating the disturbance using the attitude control system will consume large fuel which is limited in space. Therefore, proposing a method for minimizing base attitude disturbance caused by impact is the aim of the present study. So far, there have been some studies on the minimization of base attitude disturbance. The base attitude disturbance can be compensated by utilizing moment compensation system, for example, base-mounted reaction wheels, which apply moments to the base to cancel moments induced by the impact.9–11 Dimitrov and Yoshida12,13 proposed the idea of preloading bias momentum in the chaser’s manipulator, which is with equal magnitude and opposite direction to the one in the target. In this case, the momentum of the entire system at the end of the pre-impact phase is equal to zero, and after contact, the target momentum ‘‘entering’’ the chaser could be canceled out with the one preloaded in manipulator. Therefore, the momentum of the whole capture system can be stored in reaction wheels. And this idea can also be seen in Refs.14,15. Using a balance arm to compensate the base attitude is another way. Based on linear and angular momentum conservation law, the dynamics coupling between the base and its multi-arm manipulators are analyzed. On this basis, the base attitude disturbance can be minimized by controlling the balance arm.16,17 However, these kinds of compensation system may have several drawbacks. Firstly, they add significant mass to a system. Secondly, their capacity to compensate base disturbance is limited. Thirdly, they increase the complexity of the system, which is not preferred in control scheme. Therefore, methods to plan manipulator motion to minimize the base attitude disturbance are more interesting. Gattupalli et al.18 used holonomic distribution to reach closer to the target and task-level constraints to finally get to the capture point; during the point-to-point maneuvre, no reaction moment gets transferred to the base. Kaigom et al.19 also achieved minimized base disturbance based on the reaction null space and the constrained particle swarm optimization (PSO). However, the disturbance caused by impact is not considered in both methods. Nguyen-Huynh and Sharf20 presented an adaptive algorithm to generate reactionless motion for a space manipulator when capturing the target. Focused on the unstable motion of space manipulator due to the impact effect, Dong and Chen21 designed a robust adaptive compound control algorithm to suppress the unstable motion. They focused on the control strategy at the post-impact stage, which may affect the existing compliance control capability of space robot regarding both implementation and operation. Nenchev and Yoshida22 presented impact dynamic analysis of a freefloating space robot subject to a force impulse at the hand, especially focused on the study of the joint reaction and the base reaction, and the change of the respective partial momenta of the space robot. They showed that preferable directions of the impulsive force exist, such that impact momentum transfer toward the base can be minimized. Based on the idea, Cong and Sun23 introduced ‘‘straight arm

L. Zhang et al. capture’’ concept. In the methods, there will be no attitude disturbance neither during the impact nor after it, when all link centroid and the base centroid are aligned, and the force impulse direction is along that line. This is the most favorable condition, but it is a little hard to be obtained for plane robot, not to say space robot. Huang et al.24 suggested to find an optimal path for a space robot in joint space to minimize the base disturbance forces and momenta transmissed from the end-effector to the base. Cocuzza et al.25 presented an angular-acceleration-level solution based on constrained least-squares approach for the inverse kinematics of redundant space manipulators, which is aimed at locally minimizing the dynamic disturbances transferred to the base during trajectory tracking. And for capturing a free-tumbling object, FloresAbad et al.26 presented an optimal control strategy for a space robot under the conditions of having minimal impact on the base satellite during the capturing operation. In this paper, a scheme of pre-impact trajectory planning in space manipulator systems is proposed for a capture task. It is a combination of null space and PSO algorithm, where null space is used to search for the best configuration for capture and PSO algorithm is responsible for achieving the trajectory from initial state to the desired state. By this scheme, a pre-impact trajectory of space manipulator can be obtained to achieve the minimized base attitude disturbance caused by impact and the correct capture pose at the same time. 2. Base attitude disturbance caused by impact 2.1. Collision assumption The following assumptions are made in establishing the base attitude disturbance function.8 (1) The duration of contact is so short that the interaction forces act instantaneously. (2) The changes in position and orientation during impact are negligible and effects of other forces except the impact force can be disregarded. (3) From above assumptions (1) and (2), the inertia term of dynamic equation is dominant and other terms are less important. (4) Impulsive forces as well as moments are induced on an act-react principle at the single-point of contact.

2.2. Base attitude disturbance function Fig. 1 shows a general model of a free-floating space manipulator which is composed of n + 1 parts, and they are connected with revolute joints, where RI is the inertial frame, RE the end-effector frame, Ri the ith joint frame ði ¼ 1; 2; . . . ; nÞ, n the joint number, ro the vector from origin of RI to base center, rg the vector from origin of RI to total mass center of the system, rog the vector from base center to total mass center of the system, and roi the vector from base center to the ith link center. The equations governing the motion of a free-floating space manipulator as a multibody system are in general expressed in the following form:27

Pre-impact trajectory planning for minimizing base attitude disturbance in space manipulator systems for a capture task

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Recall the note we made previously on the significance of base attitude motion as compared to base translational motion and Eq. (2) is reformulated with respect to base attitude only by eliminating the base velocity acceleration term v_ b : " # " # ex e x/  x_ b   ~cx   ~sb  ~ H H JTbx þ ¼ þ ð3Þ Fe € eT e/ ~T ~cm ~sm h H H J x/

where

m

e x ¼ Hx þ H



H/ ¼ H/  HTv/ Hv/ =M,  s b ¼ s b  r og fb ,  HT T JTm ¼ Jm  Mv/ JTbv .

Fig. 1

A general model of free-floating space manipulator.

     " T# cb Fb x€b J ¼ þ Tb Fe Hs € þ cm sm h Jm 

ð1Þ

where Hs is the inertia matrix of the system; xb represents the €b ¼ ½_vb ; x_ b  describes the base acceleration, base pose and x with vb and xb the linear and angular velocities of the base, v_ b and x_ b the linear and angular accelerations of the base; € h ¼ ½€h1 ; €h2 ; . . . ; €hn , with hi and €hi ði ¼ 1; 2; . . . ; nÞ the ith joint angle and its acceleration; cb and cm are the velocity dependent non-linear terms for the base and the manipulator respectively; Fb ¼ ½ fb ; sb  is the external applied generalized force on the base, with fb and sb the external applied force and torque on base; Fe ¼ ½ fe ; se  is the external applied generalized force on robotic hand, with fe and se the external applied force and torque on robotic hand; sm ¼ ½s1 ; s2 ; . . . ; sn , with si ði ¼ 1; 2; . . . ; nÞ the ith joint torque; Jb ¼ ½JTbv ; JTbx  ½JTmv ; JTmx 

T

and

T

Jm ¼ are Jacobian matrices of the base and the manipulator, respectively, with Jbv and Jbx the linear and angular velocity Jacobian matrices of the base, Jmv and Jmx the linear and angular velocity Jacobian matrices of the manipulator. Rewrite Eq. (1) in a more specific one: 2

ME 6 Mr 4 og HTv/

MrT og Hx HTx/

32 3 2 3 2 3 2 T 3 Hv/ cv fb v_ b Jbv 7 6 7 6 7 6 7 6 Hx/ 7 54 x_ b 5 þ 4 cx 5 ¼ 4 sb 5 þ 4 JTbx 5Fe €h cm sm H/ JT m

ð2Þ 

 ME MrT og ,Hb is the base inertia matrix, with E where Mr Hx og an identity matrix, M the total mass of the system, Hx the angular velocity inertia matrix of the base and if 2 3 0 z y  rog ¼ ½x; y; z, then rog ¼ 4 z 0 x 5; H/ is the manipuy x 0 T

lator inertia matrix; ½HTv/ ; HTx/  ,Hb/ is called the coupling inertia matrix, with Hv/ and Hx/ the linear and angular coupling inertia matrix; cv and cx are the linear and angular velocity-dependent nonlinear terms of the base.

e x/ ¼ Hx/  r Hv/ , H og

 Mr og rog ,

~cx ¼ cx  r og cv ,



s m ¼ sm 

HT v/ M

~cm ¼ cm  T

fb ,

Jbx ¼

JTbx



HT v/ M

cv ,

T r og Jbv ,

Integrate Eq. (3) over an infinitesimally small time period dt from an arbitrary time ta , cancel velocity-dependent terms and internal forces and replace all accelerations with respective finite changes of velocity. The change of any velocity will be denoted as dðÞ, then Eq. (4) can be obtained. " # " # ex e x/  dxb  H H JeTbx  Fe ¼ ð4Þ eT e/ dh_ H H JeTm x/  R where Fe ¼ ta ta þdt Fe dt is the force impulse; dxb and dh_ are the changes of base angular velocity and joint angle velocity, respectively. In order to calculate the base attitude disturbance caused by impact, we eliminate dh_ from Eq. (4): 1

T  ex  H e x/ H e 1 H e T Þ ðe e x/ H e 1 e J Tbx  H dxb ¼ ð H / x/ / J m ÞFe

ð5Þ

As derived in our previous work,8 force impulse between the target and robotic hand can be expressed as Eq. (6), which is related to relative velocity vr , collision direction N and the restitute coefficient e besides the kinematic and dynamic parameters. vr  e ¼ ð1 þ eÞ ð6Þ F T N ðDm þ Dt ÞN where Dm and Dt are called the Jacobian inertia of manipula e in Eq. (5) with tor and the target, respectively. Replace F Eq. (6), we can establish the base attitude disturbance function as dxb ¼ fðh; N; vr Þ ð7Þ 3. Capture configuration optimization for minimizing base attitude disturbance Assume that the initial momentum and angular momentum are zero, the kinematics relationship of space manipulator in velocity-level is x_ e ¼ Jf h_ ð8Þ where xe represents the robotic hand pose, and x_ e ¼ ½ve ; xe  describes the robotic hand velocity, with ve and xe the linear and angular velocity of robotic hand; Jf is the Jacobian matrix of a free-floating space manipulator. Non-minimum-norm solutions to Eq. (8) based on Jacobian pseudoinverse can be written in the general form:28 ð9Þ h_ ¼ Jy x_ e þ ðE  Jy Jf Þkc u f

f

1

where Jyf ¼ JTf ðJf JTf Þ is the pseudoinverse of Jacobian matrix Jf ; E  Jyf Jf is called null space; kc is a gain coefficient, which

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can be a constant or a function; u is an arbitrary vector. By observing that null space velocities produce a change in the configuration of the manipulator without affecting its velocity at the end-effector and the characteristics can be exploited to achieve additional goals, in this paper, we mean base attitude disturbance minimization and joint limits avoidance. For minimizing the base attitude disturbance caused by impact, we define the optimization function as g ¼ kdxb k

ð10Þ

where k  k is 2-norm of an arbitrary column vector. Therefore, the aim turns to minimize the value of g. We can see that the base attitude disturbance can be minimized by optimizing manipulator’s configuration, when N and vr are determined. It is worth noting the fact that during optimization, the joint angles may exceed their limits, and this is additional but very significant problem we should consider. Zghal et al.29 designed the joint limits avoidance function as Eq. (11): HðhÞ ¼ q

n X i¼1

ðhi max  hi min Þ2 ðhi max  hi Þðhi  hi min Þ

ð11Þ

where hi max and hi min are the top and bottom limitations of the ith joint angle; q is a regulation coefficient. When the current joint angle approaches the middle value ðhi max þ hi min Þ=2, the value of HðhÞ tends to be 4nq, which is the minimum value. On the contrary, when the current joint angle goes to hi max or hi min , the value of HðhÞ will be infinity. Its performance criterion is shown in Fig. 2. During optimization, when joint angles approach their limits, some measures need to be taken to make the roaring value of HðhÞ drop. Given these factors, the optimization factor in null space is designed as u ¼ q1 u1 þ q2 u2

ð12Þ 



@g @g @g is used to realize the ; ;...; @h1 @h2 @hn minimization of base attitude disturbance and   @H @H @H is used to avoid the joint ; ;...; u2 ¼ rH ¼ @h1 @h2 @hn limits; q1 and q2 are the weight coefficients which need to be regulated automatically to achieve the goal: during normal optimization, u1 is dominant, and if the joint angles are close to their limits, u2 plays the leading role. In light of these requirements, q1 and q2 are designed as where u1 ¼ rg ¼

Fig. 2

Performance criterion of HðhÞ.

q1 ¼

k1 ; q2 ¼ k2 HðhÞ HðhÞ

ð13Þ

where k1 and k2 are constants. And the designed coefficients have the following characteristics: (1) If hi 2 ½hi min þ hs ; hi max  hs , where hs means a safe threshold, then q1 > q2 . Thus, u1 is dominant during normal optimization. (2) If hi ! hi max or hi min , q1 ! 0 and q2 ! þ1, then u2 plays the leading role. (3) Both q1 and q2 can be regulated automatically according to the current space manipulator configuration. Before optimization, the fact that the correct capture pose of robotic hand should be kept during the optimization needs to be emphasized. Generally speaking, for a capture task, its capture pose is always determined. Therefore, for successful capture, making the end-effector keep the correct capture pose has higher priority over any other task. Set x_ e ¼ 0 in Eq. (10) to keep the correct capture pose when adjusting its configuration and replace u with Eq. (12):   k1 ð14Þ u1 þ k2 HðhÞu2 h_ ¼ ðE  Jyf Jf Þkc HðhÞ It is hoped that the value of optimization function, either g or HðhÞ, as small as possible, therefore kc needs to be negative. Besides, to make the initial joint angular velocities zero and run stably, we design the transition function kc as 8 0 6 t 6 T0 > <0 p kc ¼ sinðs  2Þ  1 T0 < t 6 Tf ð15Þ > : 2 T0 < t 0Þ where s ¼  pðtT , with T0 and Tf the start and the end time of Tf T0 the transition curve as shown in Fig. 3. The manipulator configuration will be optimized until reaching a balanced point. Through the method above, we can obtain the best configuration within joint limits, which can satisfy the base attitude disturbance minimization and the capture pose of robotic hand correction.

4. Pre-impact trajectory planning for minimizing base attitude disturbance On the basis of successful capture, the best configuration for minimizing base attitude disturbance caused by impact has been obtained in Section 3, and the following problem is how to plan the trajectory from its initial state to the desired state. For a free-floating space manipulator, the planning and control face more additional problems than those on earth

Fig. 3

Curve of transition function kc.

Pre-impact trajectory planning for minimizing base attitude disturbance in space manipulator systems for a capture task because of the strong dynamic coupling between space manipulator and its base. PSO algorithm is a choice to complete the pre-impact trajectory planning. 4.1. A brief review of PSO PSO, a stochastic optimization method based on the simulation of the social behavior of bird flocks, was originally developed by Kennedy and Edberhart.30,31 In this algorithm, the system is initialized with a population of random solutions, and each potential solution is also assigned a randomized velocity; the potential solutions, called particles, are then ‘‘flown’’ through hyperspace. And each particle will change the velocity toward its best previous value pbest and global best value gbest locations according to its own flying experience and its companion’s flying experience. In a d-dimension space, the velocity and location of particle k are updated by Eqs. (16) and (17): vdk ðs þ 1Þ ¼ wvdk ðsÞ þ c1 nðpdk  xdk ðsÞÞ þ c2 nðpdg  xdk ðsÞÞ

ð16Þ

xdk ðs þ 1Þ ¼ xdk ðsÞ þ vdk ðsÞ

ð17Þ

where s is the iteration step of PSO; xdk ðsÞ and vdk ðsÞ are the position and velocity of the kth particle at s step; n is a random number uniformly distributing in the range [0, 1]; c1 and c2 represent the weighting of the stochastic acceleration terms that pull each particle toward its best previous value and overall best value positions; the inertia weight w plays a role of balancing the global and local search, it can be a positive constant or a positive linear or nonlinear function of time; pdk is the best previous position of the kth particle and pdg is the swarm’s best position. The steps for implementing PSO algorithm are simply shown in Fig. 4.

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4.2. Particles of PSO As joint variables can be used directly to adjust the manipulator configuration, they are parameterized to be the particles, and in actual applications, the requirements of Eqs. (18) and (19) have to be met. _ 0Þ ¼ 0 hðt0 Þ ¼ H0 ; hðtf Þ ¼ Hd ; hðt _ f Þ ¼ 0; € € hðtf Þ ¼ 0 hðt0 Þ ¼ 0; hðt hi min 6 hi ðtÞ 6 hi max

ð18Þ ð19Þ

where t0 and tf are the initial and the final time of the preimpact trajectory planning task; H0 and Hd are the initial and the desired final angles, respectively. The polynomial functions are usually used to obtain smooth joint motion, and considering the joint angle limits, we parameterize the joint trajectory by a sinusoidal function, whose argument is a seven-order polynomial. hi ðtÞ ¼ vi1 sinðai7 t7 þ ai6 t6 þ ai5 t5 þ ai4 t4 þ ai3 t3 þ ai2 t2 þ ai1 t þ ai0 Þ þ vi2

ð20Þ

where ai7 ; ai6 ; . . . ; ai0 are the coefficients of the polynomial; hi max  hi min hi max þ hi min vi1 ¼ ; vi2 ¼ : 2 2 The joint velocity and joint acceleration can be obtained by the derivative and second-derivative of Eq. (20). Considering Eq. (18), the results of Eq. (21) are found.   8 i2 > ai0 ¼ arcsin hi0vv > > i1 > > > > ai1 ¼ ai2 ¼ 0 > > n h    io > < hi0 vi2 i2  arcsin =t3f ai3 ¼  3ai7 t7f þ ai6 t6f  10 arcsin hidvv vi1 i1 n h     i o > > > a ¼ 8a t7 þ 3a t6  15 arcsin hid vi2  arcsin hi0 vi2 > =t4f > i4 i7 f i6 f vi1 vi1 > > > n h     i o > > : ai5 ¼  6ai7 t7 þ 3ai6 t6  6 arcsin hid vi2  arcsin hi0 vi2 =t5f f f vi1 vi1 ð21Þ

where hi0 and hid are the initial and the desired angle of the ith joint. After parameterization, only two parameters ai6 and ai7 are included in each joint function. Thereby, let a ¼ fa16 ; a17 ; a26 ; a27 ; . . . ; an6 ; an7 g be the particles. 4.3. Objective function of PSO The optimized capture configuration for minimizing base attitude disturbance can be achieved by introducing the constraints in Eq. (18) and then the correct capture pose can be set as the objective function of PSO. The position of robotic hand is determined by velocity inteRt gral method, namely, Pe ðtÞ ¼ 0f Jmv h_ dt. The attitude of robotic hand is represented by quaternion, which is a popular nonsingular four-parameter representation.32 A unit quaternion Q is defined as a complex number Q ¼ g þ q1 i þ q2 j þ q3 k

Fig. 4

Flowchart of PSO.

ð22Þ

formed from four different units ð1; i; j; kÞ by means of the real parameters g; q1 ; q2 and q3 , where i; j; k are three orthogonal unit spatial vectors, and the real parameters are constrained by g2 þ q21 þ q22 þ q23 ¼ 1. Combining the kinematics knowledge, the variation rate of Q is given by33

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 T _ ¼ 1 q Jmx h_ Q 2 gE  q

ð23Þ

Table 1 Link No.

hi ð Þ

di ðmÞ

ai1 ðmÞ

ai1 ð Þ

1 2 3 4 5 6 7

0 90 0 0 0 90 0

0.6 0.5 0 0.5 1.0 0.5 0.6

0 0 0 5 5 0 0

0 90 90 0 0 90 90

T

where q ¼ ½q1 ; q2 ; q3  ; for two coordinate systems, their attitudes are represented by ga ; qa and gb ; qb , and the relative attitude is given by Dg and Dq:  Dg ¼ ga gb þ qTa qb ð24Þ Dq ¼ ga qb  gb qa  q a qb When the two frames coincide: Dg ¼ 1; Dq ¼ 0

ð25Þ

It’s worth noting that Dq ¼ 0 implies Dg ¼ 1. Suppose gef ; qef and ged ; qed are the actual and desired attitude of robotic hand, and Pef ; Ped are the actual and desired position. And the pose deviations are Dqe ¼ gef qed  ged qef  q ef qed DPe ¼ Ped  Pef ¼ Ped 

DH parameters of space manipulator.

Z

tf

ð26Þ Jmv h_ dt

ð27Þ

0

where Dqe is the deviation between actual and desired attitude in terms of quaternion; DPe is the deviation between actual and desired position. Thus, the objective function of PSO can be designed as f ¼ Wa

ð28Þ

where a ¼ ½kDqe k; kDPe kT ; W ¼ ½w1 ; w2  is the weight matrix, with w1 and w2 the weight coefficients of kDqe k and kDPe k, which can balance the convergence result by setting their values. 5. Simulation 5.1. Studied space manipulator system The studied space manipulator is composed of the base and a 7-dof manipulator. Its joint frames according to Denavit Hartenberg (DH) method are shown in Fig. 5, where Xi and Zi ði ¼ 0; 1; . . . ; 7Þ represent the vectors of X axis and Z axis of the ith frame and set a ¼ k ¼ 0:6 m, b ¼ l ¼ m ¼ n ¼ h ¼ 0:5 m, c ¼ d ¼ 5 m. The relative

Fig. 5

Table 2

Dynamic parameters of space manipulator.

Part

mi ðkgÞ

Pi ðmÞ

Ii ðkg  m2 Þ

42.5 42.5 70 70 42.5 42.5 42.5 10000

½0; 0:25; 0:6 ½0:25; 0; 0:5 ½2:5; 0; 0:5 ½2:5; 0; 0:5 ½0; 0; 0:25 ½0; 0; 0:25 ½0; 0; 0:3 ½0; 0; 0

diagð0:89; 0:05; 0:89Þ diagð0:05; 0:89; 0:89Þ diagð0:09; 145:83; 145:83Þ diagð0:09; 145:83; 145:83Þ diagð0:89; 0:89; 0:05Þ diagð0:89; 0:89; 0:05Þ diagð1:28; 1:28; 0:05Þ diagð2000; 2000; 2000Þ

Link Link Link Link Link Link Link Base

1 2 3 4 5 6 7

parameters are listed in Tables 1 and 2, where hi ; di ; ai1 and ai1 are DH parameters; mi ; Pi and Ii represent the mass of the ith part, mass center vector of the ith part in the ith frame, and inertia tensor in the frame attached at the geometric center of the ith part. The inertia tensor is a diagonal matrix, and the diagonal elements are given. Initial joint angles and base attitude are set as H0 ¼ ½50; 170; 150; 60; 130; 170; 0 ð Þ c0 ¼ ½0; 0; 0 ð Þ. Final desired position and attitude of the robotic hand are ½7; 0; 3 m and ½1:0; 0:5; 2:0 rad. Assume the target is a rigid sphere, whose mass is mt ¼ 30 kg, and radius is R ¼ 0:3 m. Its inertia tensor is a diagonal matrix, whose elements are Ixx ¼ Iyy ¼ Izz ¼ 1:08 kg  m2 . Vector from target mass center to contact point is rtc ¼ ½0:07; 0:26; 0:13T , the contact direction is N ¼ ½0:30; 0:30; 0:91T , and the relative velocity is 0:05 m=s .

7-dof free-floating space manipulator.

Pre-impact trajectory planning for minimizing base attitude disturbance in space manipulator systems for a capture task 5.2. Configuration optimization for minimizing base attitude disturbance In order to obtain the best configuration for minimizing base attitude disturbance caused by impact, method in Section 3 is employed. Adopt any method to obtain a group of joint angles (within joint limits) that are suitable for capture pose. Take the suitable manipulator configuration H1 ¼ ½92:6; 192:1; 142:1; 91:0; 188:3; 187:2; 38:2 ð Þ and Euler angle of base c1 ¼ ½13:03; 8:00; 8:33 ð Þ as an example. Set the limits of the joint angles as  Hmax ¼ ½160; 200; 180; 170; 200; 200; 180 ð Þ Hmin ¼ ½160; 220; 180; 150; 180; 226; 80 ð Þ First, joint limits are not considered, namely u2 in Eq. (12) is neglected. The optimized results of objective function g and the change of joint angles are shown in Fig. 6. From Fig. 6(a), it can be seen that through optimization, the value of objective function g changes from 0.1317 to almost zero, which means that in the end we get the configuration that makes the base attitude disturbance caused by impact become zero. And from Fig. 6(b), we can see joint angles change smoothly from initial to the desired configuration. However, because the joint limits avoidance factor u2 is not introduced, joint 5 and joint 6 exceed their limits, which is not expected in practical application. Therefore, in order to prove the effectiveness of the composite factor we design, we use the same initial conditions and obtain the following results.

Fig. 6

Optimization results without joint limits.

Here, we define optimization capacity f as g  gfi f ¼ in  100% gin

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ð29Þ

where gin and gfi stand for the initial value and the final optimization result of g, respectively. Fig. 7(a) shows that due to joint limits, g value changes from 0.1317 to 0.0651, whose optimization capacity is 50.57%, lower than the one neglecting the joint limits, which is almost 100%. However, benefited from the joint limits avoidance factor’s effective work, all joint angles are limited to their joint operation range, as shown in Fig. 7(b). Take joint 5 as an example and make a comparison with the previous one in Fig. 8, we can see the effectiveness of joint limits’ avoidance factor. Fig. 9 shows the changes of joint angle velocity during optimization; at about 12 s, all joint velocities approach zero, which means optimization factor u1 and u2 have reached a balance point. The desired configuration is ½94:74; 171:80; 136:50; 89:82; 195:76; 198:96; 22:02 ð Þ which meets three requirements: (1) correct capture pose; (2) minimized base attitude disturbance caused by impact; (3) within joint limits. 5.3. Pre-impact trajectory planning for minimizing base attitude disturbance The desired configuration of space manipulator has been obtained, and the following job is to achieve the trajectory

Fig. 7

Optimization results within joint limits.

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Fig. 8

Comparison of joint 5 within/without joint limits.

Fig. 11 Fig. 9 Curves of joint angle velocity during optimization (within joint limits).

Curves from initial point to desired point.

And the value of objective function f is 0.0047, which means the convergence result is satisfactory. The convergence process is shown as Fig. 10. Substitute the optimal particles into Eq. (20) and we can obtain the changes of the joint angle and joint angle velocity in Fig. 11. In light of these figures, it can be seen that the joint angle velocity curves are smooth and steady, and all joint angles are within their physical limits, which means this method can be applied to actual operation. The final capture position and attitude of robotic hand are ½6:99; 0:00; 3:00 m and ½1:0; 0:5; 2:0 rad and the deviations are acceptable. As for the effect for the minimization of the base attitude disturbance caused by impact, its optimization capacity has been stated in Section 5.2. 6. Conclusions

Fig. 10

Variation of global best fitness evaluation.

from initial state to the desired state using the method in Section 4. Set relevant parameters as follows: c1 ¼ c2 ¼ 2:0, w1 ¼ w2 ¼ 50. The population size is 24. After 200 iterations we can get the optimal particles: ½0:63; 0:04; 0:33; 0:31; 0:59; 0:16; 0:43;0:17;1:00;0:44;0:55;0:32;0:11;0:36 108 :

The importance of capturing operations by a space manipulator has been increasing in recent years. When a space manipulator is in its free-floating mode, due to dynamic coupling, the base attitude will be disturbed by impact, which will affect the communication with the ground and power supply. Therefore, a scheme for minimizing base attitude disturbance caused by impact is proposed in this study, and the simulation results verify the effectiveness. (1) The base attitude disturbance is established as a function of joint angles, collision direction and relative velocity

Pre-impact trajectory planning for minimizing base attitude disturbance in space manipulator systems for a capture task between robotic hand and the target. This function expresses that how various factors disturb the base attitude and provide a criterion which describes the degree of base attitude disturbance. (2) A novel optimization factor in null space is designed, which can minimize base attitude disturbance and ensure that the joint angles do not exceed their limits simultaneously. And the optimization factor can be regulated automatically according to different cases. During normal optimization, minimizing base attitude disturbance is dominant, and if the joint angles are close to their limits, avoiding joint limits plays the leading role. Therefore, the designed optimization factor can minimize the base attitude disturbance as much as possible within joint limits. (3) The significance of the work is that we propose a preimpact trajectory planning method for minimizing base attitude disturbance in space manipulator systems for a capture task, which can meets three requirements: (1) correct capture pose (2) minimized base attitude disturbance caused by impact (3) within joint limits. From the simulation results, it can be seen that this scheme can greatly reduce the amplitude of base attitude disturbance, however, because of some additional constraints, the impact still introduces some attitude disturbance to the base in this scheme, and the stabilization control after impact is considered for the future work.

Acknowledgments This study was supported by the National Basic Research Program of China (No. 2013CB733000), the National Natural Science Foundation of China (No. 61175080) and BUPT Excellent Ph.D. Students Foundation of China (No. CX201427). References 1. Yoshida K, Sashida N. Modeling of impact dynamics and impulse minimization for space robots. Proceedings of IEEE/RSJ international conference on intelligentrobots and systems; 1993 Jul 26–30; Yokohama, Japan. Piscataway, NJ: IEEE Press; 1993. p. 2064–9. 2. Walker ID. The use of kinematic redundancy in reducing impact and contact effects in manipulation. Proceedings of IEEE international conference on robotics and automation; 1990 May 13–18; Cincinnati, USA. Piscataway, NJ: IEEE Press; 1990. p. 434–9. 3. Wee LB, Walker MW. Space based robot manipulators: dynamics of contact and trajectory planning for impact minimization. Proceedings of American control conference; 1992 Jun 24–26; Chicago, USA. Piscataway, NJ: IEEE Press; 1992. p. 771–6. 4. Wee LB, Walker MW. On the dynamics of contact between space robots and configuration control for impact minimization. IEEE Trans Rob Autom 1993;9(5):581–91. 5. Shibli M, Aghili F, Su CY. Modeling of a free-flying space robot manipulator in contact with a target satellite. Proceedings of IEEE conference on control applications; 2005 Aug 28–31; Toronto, Canada. Piscataway, NJ: IEEE Press; 2005. p. 559–64. 6. Huang PF, Xu WF, Liang B, Xu YS. Configuration control of space robots for impact minimization. Proceedings of IEEE international conference on robotics and biomimetics; 2006 Dec 17–20; Kunming, China. Piscataway, NJ: IEEE Press; 2006. p. 357–62.

1207

7. Jia QX, Zhang L, Chen G, Sun HX. Pre-impact trajectory optimization of redundant space manipulator with multi-target fusion. J Astronaut 2014;35(6):639–47 Chinese. 8. Zhang L, Jia QX, Chen G, Sun HX. The precollision trajectory planning of redundant space manipulator for capture task. Adv Mech Eng 2014;2014:1–10. 9. Longman RW, Lindberg RE, Zedd MF. Satellite-mounted robot manipulators-new kinematics and reaction moment compensation. Int J Rob Res 1987;6(3):87–103. 10. Oki T, Nakanishi H, Yoshida K. Whole-body motion control for capturing a tumbling target by a free-floating space robot. Proceedings of IEEE/RSJ international conference on intelligent robots and systems; 2007 Qct 29–Nov 2; San Diego, USA. Piscataway, NJ: IEEE Press; 2007. p. 2256–61. 11. Pathak PM, Mukherjee A, Dasgupta A. Interaction torque control by impedance control of space robots. Simulation 2009;85(7):451–9. 12. Dimitrov DN, Yoshida K. Utilization of the bias momentum approach for capturing a tumbling satellite. Proceedings of IEEE/ RSJ international conference on intelligentrobots and systems; 2004 Sept 28–Qct 2; Sendai, Japan. Piscataway, NJ: IEEE Press; 2004. p. 3333–8. 13. Dimitrov DN, Yoshida K. Momentum distribution in a space manipulator for facilitating the post-impact control. Proceedings of IEEE/RSJ international conference on intelligentrobots and systems; 2004 Sept 28–Qct 2; Sendai, Japan. Piscataway, NJ: IEEE Press; 2004. p. 3345–50. 14. Yoshida K, Dimitrov D, Nakanishi H. On the capture of tumbling satellite by a space robot. Proceedings of IEEE/RSJ international conference on intelligentrobots and systems; 2006 Qct 9–15; Beijing, China. Piscataway, NJ: IEEE Press; 2006. p. 4127–32. 15. Shui HT, Li X, Peng SJ, Ma HX. Zero disturbance planning for space robots during target capture. Proceedings of IEEE international conference on control and automation; 2010 Jun 9–11; Xiamen, China. Piscataway, NJ: IEEE Press; 2010. p. 471–5. 16. Huang PF, Xu YS, Liang B. Dynamic balance control of multi-arm free-floating space robots. Int J Adv Rob Syst 2005;2(2):117–24. 17. Cong PC, Zhang X. Preimpact configuration analysis of a dualarm space manipulator with a prismatic joint for capturing an object. Robotica 2013;31(6):853–60. 18. Gattupalli A, Shah SV, Krishna KM, Misra AK. Control strategies for reactionless capture of an orbiting object using a satellite mounted robot. Proceedings of conference on advances in robotics; 2013 Jul 4–6; Pune, India. New York: ACM; 2013. p. 1–6. 19. Kaigom EG, Jung TJ, Rossmann J. Optimal motion planning of a space robot with base disturbance minimization. Proceedings of symposium on advanced space technologies in robotics and automation; 2011 Apr 12–15; Noordwijk, Netherlands. Noordwijk: ESTEC; 2011. p. 1–6. 20. Nguyen-Huynh TC, Sharf I. Adaptive reactionless motion for space manipulator when capturing an unknown tumbling target. Proceedings of IEEE international conference on robotics and automation; 2011 May 9–13; Shanghai, China. Piscataway, NJ: IEEE Press; 2011. p. 4202–7. 21. Dong Q, Chen L. Impact dynamics analysis of free-floating space manipulator capturing satellite on orbit and robust adaptive compound control algorithm design for suppressing motion. Appl Math Mech Engl Ed 2014;35(4):413–22. 22. Nenchev DN, Yoshida K. Impact analysis and post-impact motion control issues of a free-floating space robot subject to a force impulse. IEEE Trans Rob Autom 1999;15(3):548–57. 23. Cong PC, Sun ZW. Pre-impact configuration planning for capture object of space manipulator. Proceedings of international symposium on systems and control in a erospace and a stronautics; 2008 Dec 10–12; Shenzhen, China. Piscataway, NJ: IEEE Press; 2008. p. 1–6. 24. Huang PF, Chen K, Xu YS. Optimal path planning for minimizing disturbance of space robot. Proceedings of international

1208

25. 26.

27.

28.

29.

30.

conference on control, automation, robotics and vision; 2006 Dec 5–8; Singapore, Singapore. Piscataway, NJ: IEEE Press; 2006. p. 1–6. Cocuzza S, Pretto I, Debei S. Least-squares-based reaction control of space manipulators. J Guid Control Dyn 2012;35(3):976–86. Flores-Abad A, Wei Z, Ma O, Pham K. Optimal control of space robots for capturing a tumbling object with uncertainties. J Guid Control Dyn 2012;37(6):2014–7. Chen G, Jia QX, Sun HX, Hong L. Analysis on impact motion of space robot in the object capturing process. Robot 2010; 32(3):432–8 Chinese. Liegeois A. Automatic supervisory control of the configuration and behavior of multibody mechanisms. IEEE Trans Syst Man Cybern 1977;7(12):868–71. Zghal H, Dubey RV, Euler JA. Efficient gradient projection optimization for manipulators with multiple degrees of redundancy. Proceedings of IEEE international conference on robotics and automation; 1990 May 13–18; Cincinnati, USA. Piscataway, NJ: IEEE Press; 1990. p. 1006–11. Eberhart R, Kennedy J. A new optimizer using particle swarm theory. Proceedings of the international symposium on micro

L. Zhang et al. machine and human science; 1995 Qct 4–6; Nagoya, Japan. Piscataway, NJ: IEEE Press; 1995. p. 39–43. 31. Kennedy J, Eberhart R. Particle swarm optimization. Proceedings of IEEE international conference on neural networks; 1995 Nov 27– Dec 1; Perth, Australia. Piscataway, NJ: IEEE Press; 1995. p. 1942–8. 32. Chou JCK. Quaternion kinematic and dynamic differential equations. IEEE Trans Rob Autom 1992;1(8):53–64. 33. Hong JZ. Computational dynamics of multibody systems. Beijing: High Education Press; 2009 Chinese. Zhang Long is a Ph.D. student at School of Automation, Beijing University of Posts and Telecommunications, China. He received the B.S. degree in mechanical engineering from the same university in 2011. His area of research covers the kinematic and dynamic modeling, as well as motion control of space manipulator. Jia Qingxuan is a professor and Ph.D. supervisor at School of Automation, Beijing University of Posts and Telecommunications, China. His current research interests are space manipulator control and virtual reality technology.