Nomenclature

B₀,free-floating base;

B_i,link of manipulator (i = 1,2);

B₃,target satellite;

O₀,centre of free-floating base;

O_i,joint centre (i = 1,2);

O-XY,inertial coordinate;

O_i-X_iY_i,local coordinate of B_i (i = 0,1,2);

C_m,centre of target satellite;

P,end-effector of space manipulator;

P′,handle on satellite surface which impact with end-effector of manipulator;

l₀,distance between l₀ and l₁;

l_i,length of link B_i(i = 1,2);

l₃,distance between C_m,and P′;

m_i,mass of B_i (i = 0,1,2,3);

J_i,center inertial momentum ofB_i(i=0,1,2,3). 1 Introduction

As the astronautic technology developing,the space manipulator will be employed to manage more assignments for helping the astronaut in the aerospace. It is attracting the attention of researchers^{[1, 2, 3, 4, 5]}. The space manipulator was employed in the space station firstly,then equipped in the space shuttle. Now,the free-floating space manipulator is becoming a new research area^[6]. The free-floating space manipulator can be employed in a wider work space on orbit to finish the repair,retrieve,and fuel refill of satellite services. In finishing the above services,the endeffector of space manipulator will impact with the satellite inevitably,the impact will cause the unstable motion or even cause damage of space manipulator,the dynamics and control of space manipulator capturing the satellite are more complex than the non-capture operation^{[7, 8]}.

In this paper,the impact dynamics and control of free-floating space manipulator capturing a satellite are analyzed. Firstly,based on the kinematics analysis,the dynamics equation of free-floating space manipulator system is derived by the second Lagrangian equation,and then according to the momentum conservation principle,the impact dynamics and effect of the capture process are analyzed by the momentum impulse. Considering the unstable motion of space manipulator which is caused by the impact of satellite,a robust adaptive compound control algorithm is designed to suppress the unstable motion of the space manipulator,it can eliminate the effect of uncertain and unknown parameters of space manipulator and satellite for the control,and the stability of the control algorithm is proved by the Lyapunov theorem. Finally,the simulation is proposed to reveal the impact effect and verify the validity of the control algorithm. 2 Kinematics analysis

The process of planar free-floating space manipulator capturing a target satellite on orbit is shown in Fig. 1. The capture process can be divided into three stages: the pre-impact stage between the manipulator end-effector and the target satellite handle,the impact stage,and the post-impact stage in which the space manipulator and satellite get combination.

The kinematic equation of the space manipulator which represents the relationship between the velocity of end-effector P and the generalized velocity is

where VP = ^T ∈ R^3×1 is the velocity of end-effector P in the inertial frame,q = (x₀ y₀ α θ₁ θ₂)^T ∈ R^5×1 is the generalized coordinates vector of space manipulator which represents the free-floating base position and attitude,and the manipulator joint angle,and J ∈ R^3×5 is the corresponding Jacobian matrix.

Similarly,the kinematic equation of the satellite can be expressed as follows:

where V_P′ = ^T ∈ R^3×1 is the velocity of the satellite handle P′ in the inertial frame,φ = (x_cm y_cm β)^T ∈ R^3×1 is the generalized coordinate vector of satellite which represents the position and attitude,and J_P′ ∈ R^3×3 is the corresponding Jacobian matrix. 3 Dynamics analysis

3.1 Dynamic equation of free-floating space manipulator of pre-impact stage

The free-floating base position of space manipulator is uncontrolled for saving the jet fuel as shown in Fig. 1,the dynamic equation can be derived by the second Lagrangian equation as follows:

where D ∈ L^5×5 is the symmetric positive definite inertial matrix, ∈ R^5×1 is the vector of the Coriolis and centrifugal force,τ = (0 0 τ₀ τ₁ τ₂)^T ∈ R^5×1 is the control input,the base position is uncontrolled for saving the jet fuel,and F_P = (F_Px F_Py M_P )^T ∈ R^3×1 is the external force which acts on the end-effector. 3.2 Impact dynamics and effect of impact stage

Assume the satellite is free motion on orbit which will be captured by the space manipulator, and the dynamic equation of satellite can be expressed as

where D_t = diag(m₃,m₃,J₃) ∈ R^3×3 is the inertial matrix,C_t ∈ R^3×1 is the vector of the Coriolis and centrifugal force,and F_P′ is the impact force which acts on the satellite handle during the impact stage.

The impact between the manipulator end-effector and the target satellite is inevitably during the capture process,where F_P and F_FP′ are the interact forces. According to the impulse principle,it is known that the fact of the impact is the momentum transfer between the space manipulator and satellite. The impact affects the motion of space manipulator. By combining (3) and (4),and eliminating the interact force as follows,we can get

Assume the duration of the impact stage is △t. According to the momentum impulse,the integral of (5) for the duration △t can be expressed as follows:

where t₀ is the moment before impact. Since the duration of the impact stage is very short and the interact force is very large,the generalized coordinates of the space manipulator remain unchanged during the impact stage although the rate may change. Besides,there is not any control input during the impact stage to avoid joint damage by impact impulse,i.e.,τ = 0. It can be expressed in a mathematical form as Thus,the integral can be further written as follows: where the subscripts i and f of and represent before and after the impact,respectively. Since D,D_t,J,and J_P′ depend on the generalized coordinates only,they remain the same during the impact stage. D,D_t,J,and J_P′ are the values of D,D_t,J,and J_P′ at moment t₀. We know the left of (7) is O(1),the integrand of right is O(1),however the integration time △t = O(ε),the right of (7) is O(ε). (7) can be written as follows:

The derivation of (8) based on the momentum conservation can be used from perfect elastic impact to plastic impact situation. On the assumption that it is perfect plastic impact between the end-effector P and the satellite handle P′ during the capture process,i.e.,the manipulator captures the satellite successfully. Since the impact point plastic deformation is very small relative to the structure dimension,the end-effector P of space manipulator and handle P′ of satellite have the same velocity after impact. Thus,

By combining (8) and (9),at the moment t₀ + △t,the generalized velocity of space manipulator after impact can be expressed as

where

(10) represents the impact effect for the space manipulator motion during the impact stage,it will cause the unstable motion of space manipulator after capture the satellite. Therefore,the control for suppressing motion is necessary. 3.3 Impact dynamic equation of combination system of post-impact stage

After the space manipulator capturing the satellite successfully,they will become the combination system,the dynamic equation of combination system contains space manipulator subsystem and satellite subsystem. Thus,

The derivative of (11) can be rewritten as follows:

The interact force between the space manipulator and satellite becomes the internal force for the combination system during the post-impact stage,assuming there is not any other external force acting on the combination system of the capture process. Thus,(5) can be described as the dynamic equation of the combination system of the post-impact stage. By substituting (11) and (12) into (5) and with linear treating,we get

where

(13) is the dynamic equation of combination system of the post-impact stage. A relationship exists between Hz ∈ R^5×5 and a random vector z ∈ R^5×1[9],

4 Robust adaptive compound control algorithm design of space manipulator for suppressing unstable motion due to impact effect

The impact effect will cause the unstable motion of the combination system,such as the base movement,rotation,and joint irregular motion that will affect the normal operation of the solar panels and antenna,and damage the joint hinge due to over rotate. Therefore,in order to ensure the combination system normal operation,the applying of control for suppressing motion is necessary. Based on (13),a robust adaptive compound control algorithm of space manipulator is designed to suppress the unstable motion of the combination system,and it can be used to under the inertial parameters unknown or uncertain and the base position being uncontrolled.

Define the control output of the combination system as q_r = (α θ₁ θ₂)^T ∈ R^3×1,the virtual augmented output is q = (q_b^T q_r^T)^T ∈ R^5×1,and q_b = (x₀ y₀)^T ∈ R^2×1 is the position of the free-floating base. Assume the position qb,the velocity b,and the acceleration of the free-floating base can be measured or calculated,and the error between the desired augmented output qd = (q_b^T q_rd^T)^T ∈ R^5×1 and virtual augmented output is

Define the augmented error of system as

where λ> 0 is the constant,and s₁ = _r + λe_r. Define the reference output joint velocity and reference output joint acceleration as Therefore,according to (16) and (17),we can get This is the combination system error equation,and τ_r =(τ₀ τ₁ τ2)^T.

Based on (19),the robust adaptive compound control algorithm is proposed as follows:

where _z and _z are the estimation of D_z and H_z,the vector δ∈ R^2×1 ensures the upper two rows of the right of (19) to be zero,K₁ ∈ R^3×3 is a random symmetric positive definite matrix,and Q = diag( Q₁,Q₂,Q₃)∈ R^3×3 is a undetermined matrix.

Substituting (20) into (19),we can get

where WΦ = (Dsub>z-_z) + (H_z −Hcz) ,W is the function of q,, ,and that does not contain undetermined parameters,Φ = ξ- is the vector of estimation error of uncertain or unknown parameters,and ξ and are the true value and estimated value of the parameters,respectively.

The inertial parameters of space manipulator are always uncertain,and they vary within a range as a result of fuel consuming. The inertial parameters of the target satellite are always unknown as an external body of space manipulator. Therefore,WΦ can be decomposed as follows:

where Φ_R = ξ_<>R-_R represents the vector of estimation error of uncertain inertial parameters of space manipulator,and Φ_S = ξ_S-_S represents the vector of estimate error of unknown inertial parameters of target satellite.

Assume that the inertial vector of space manipulator varies in a definite range as follows:

The above control algorithm does not need to self-adjust the uncertain inertial parameters,and will keep robust if the estimated parameters do not exceed the definite range,which will increase the computational efficiency of the control system. The following theorem exists for the control algorithm.

Lemma 1 If under the appropriative determination of the undetermined matrix Q,the following adaptive law equation (24) is used to adjust the unknown inertial parameters for the control algorithm,the control algorithm will guarantee = 0.

where is the positive constant.

Remark 1 Assuming s = 0 and Φ= 0 represent no interference motion. (21) and (24) are interference motion equations,and the Lyapunov function is defined as follows:

By combining (14),(21),and (24),we can get

where K₂ = diag(I_2×2,K₁). It is obviously that-γs^TK₂s 6 0,and the first term of (26) can be written as where n is the number of elements of the matrix Φ_R,Φ_Rj represents the corresponding element of Φ_R,W_R3j ,W_R4j ,and W_R5j represent the elements of the matrix W_R,and s₁₁,s₁₂,and s₁₃ represent the elements of the vector s₁. If the undetermined matrix Q is determined as where μ_i is the positive constant,then (27) satisfies the following relationship:

By substituting (29) into (26),we can get

By analyzing (30),we known that is negative definite,and = 0 only exists when s = 0. Therefore,= 0 is satisfied. 5 Simulations

To reveal the impact effect during the capture operation and verify the validity of the control algorithm,a simulation of free-floating space manipulator capturing a satellite shown in Fig. 1 is carried out. The parameters are taken as follows:

The uncertain and unknown parameters are taken as

The initial moment of the simulation is 0 s,and the duration of the simulation is 10 s. At 0 s,it is assumed that the space manipulator is adjusted to a desired position and configuration,and waited for capturing the satellite. At t₀ = 1 s,the free motion satellite (velocity: = −0.3 m/s, = 0.3 m/s, = 0.1 rad/s) is flying toward the end-effector of space manipulator,and then satellite impacts with end-hand,the duration of impact stage is △t (assume △t = 0.2 s here). The impact effect will cause the unstable motion of combination system. Therefore,it is necessary to employ the active control for suppressing motion of the combination system.

The parameters of control algorithm are taken as

The estimations of uncertain inertial parameters of the space manipulator are taken as The estimation of unknown parameters of the target satellite are taken as Two different simulations are carried out for comparing.

Simulation 1 There is not any control algorithm used for suppressing the unstable motion of combination system after the impact between the space manipulator and satellite,the space manipulator is under uncontrolled state. Figure 2 shows the time history of the base attitude and joint angle under uncontrolled after impact. Figure 3 shows the time history of base rotation velocity and joint angle velocity under uncontrolled after impact. The above simulation results reveal the impact effect for the motion during the capture operation.

Simulation 2 A robust adaptive compound control algorithm is used for suppressing the unstable motion of combination system after the impact between the space manipulator and satellite,the space manipulator is under active controlled state. Figure 4 shows the time history of base attitude and joint angle under active controlled after impact. Figure 5 shows the time history of the base rotation velocity and joint angle velocity under active controlled after impact. By comparing the results of Simulation 1 and Simulation 2,the validity of the active control for suppressing motion is obvious.

6 Conclusions

In this paper,the dynamics of free-floating space manipulator capturing a target satellite on orbit are analyzed,and the active control for suppressing the unstable motion of combination system during the capture operation is proposed. By the analyzing,the following conclusions can be drawn.

Firstly,by the dynamics analysis and simulations,we can find that the impact effect for the stability of space manipulator system is obvious. Therefore,the active control for suppressing the unstable motion of space manipulator during capturing the satellite is necessary.

Secondly,some difficulties need to be considered for the active control in advance: the uncertain or unknown parameters effect for the control algorithm,the uncontrolled of freefloating base during the active control applied. In this paper,the robust adaptive compound control can be applied under the above situation.

Finally,the process of space manipulator capturing a satellite is complex,there are many detail and considerate problems need to be considered. This will also be the in-depth research work in future.