The analysis of capture strategy for redundant space manipulator when performing snare capture

Abstract

Snare capture is a capture mode, which mainly relies on three flexible cables to enclose and then latch the target. Regarding the pre-impact phase of snare capture, this article presents a capture strategy for redundant space manipulator with the aim of improving reliability. It contains two steps: the attitude regulation of snare-type end-effector and the optimized control of space manipulator. In the first step, the contact model between grapple shaft and flexible cable is established based on the discrete flexible cable model. And then, with the purpose of reducing the peak value of contact force, a “safe contact region” is designed and the grapple shaft can get into it by regulating the end-effector’s attitude. In the second step, a description of the inertial properties perceived at the end-effector of space manipulator is provided first, and on this basis, the objective functions which aim to reduce the disturbance of the end-effector are established. Thereafter, the configuration of space manipulator is optimized by the null space method for its advantage of not affecting the regulated end-effector attitude. Finally, a simulation is carried out on a 7-degree-of-freedom space manipulator with snare-type end-effector and a target equipped with grapple shaft to prove the effectiveness of the proposed capture strategy.

Introduction

In the past decades, the importance of on-orbit capture operations by a manipulator has been increasing because of the key role it played in space servicing missions, such as payload/satellite deployment, servicing and retrieval, orbital replaceable units (ORUs) manipulation, as well as on-orbit construction and assembly.^1,2 End-effector, as the capture mechanism, will directly contact with the target and its performance will have significant impact on capture mission.

Traditional rigid end-effector, like the grippers which have been used in ROTEX ³ and ETS-VII, ⁴ has advantages of simple structure, convenient control, high stiffness connection, and so on. It is always used for cooperative target capture (the dynamic and configuration parameters of cooperative targets are often known or available). However, it faces two major problems when performing a capture task. First, the capture pose accuracy needs to be very high, which will increase the control difficulty of the whole system. Second, in the process of catching a large-scale space target, the collision impact could be huge, which will lead to a serious damage to the manipulator system and thus reduces its lifetime. ⁵ To deal with these problems, some new types of end-effectors begin to appear. Flexible tether-net system is proposed for in-cooperative satellite capture, like debris and malfunctioning satellites. The net capture is a kind of surface-to-point capture, and the in-cooperative capture can be easily performed without considering the target configuration and dynamic status. In addition, owing to its large capture distance, the collision between manipulator and target can be avoided effectively. ⁶ However, due to its complexity of dynamics and control, this type of capture system is still in concept design. Snare capture is another type of capture system, which mainly relies on three flexible cables to achieve flexible capture and rigid latch mechanism to realize high stiffness connection. This capture mode has been commonly used on orbit for capture missions;^7–9 however, its secure operation depends greatly on the skill of the operator. In microgravity environment, the capture operation becomes quite complicated, and it can be subject to high risk if improper contact happens.

It is generally known that a whole capture operation can be divided into three specific phases, namely, target chasing control phase, impact phase between the target and robotic hand, and stabilization control phase of tumbling motion. ¹⁰ The first phase, also called pre-impact phase, is very important for any capture mission. The first contact between space manipulator and the target will happen at the end of this phase, and if improper capture strategy is implemented, the target may be pushed away. What is worse, both the target and manipulator could suffer damage due to large impact force. For this reason, the minimization of impact force becomes a major challenge during pre-impact phase, and many researchers have made their own contributions.^10–13 However, in these literatures, the contact between manipulator and target is simply regarded as a force impulse and the type of end-effector is not considered.

With regard to snare capture, Tan et al. ¹⁴ proposed a discrete model of flexible cable based on torsion spring and then established the force and torque static equilibrium equations. The motion of flexible cable can be predicted through the equation, but the contact force has to be known previously. Li et al. ¹⁵ regarded the snare capture mechanism as a retractable rigid ring and built the dynamic model by Newton–Euler and hybrid coordinate methods. The physical properties of flexible cable cannot be reflected by this model, and the contact force is also treated as a force impulse. Uyama et al. ¹⁶ divided the flexible cable into a number of beam elements, and adjacent beam elements were connected by three tensile springs and three torsion springs. The contact force was solved based on the classical Hertz contact theory. However, this contact force model is insufficient to express the contact condition between flexible cable and the grapple shaft because the cable would not be a perfectly rigid body during penetration due to geometrical deformation. In Sawada et al., ¹⁷ Abiko et al., ¹⁸ and Yoshinitsu et al., ¹⁹ lots of ground experiments were conducted and the data of force and its corresponding bending displacement of flexible cable were collected by force/torque sensor. On this basis, the expression of contact force was approximated by a quadratic function or other empirical equations. Meanwhile, the disturbance to the manipulator was evaluated by a dynamic simulator. There are few literatures researching on the strategy of snare capture. Takahashi ²⁰ designed an impedance control method for a flexible joint manipulator with snare-type end-effector. It can make the flexible cable keep contact with the floating target during the whole capture process and thus reduce the possibility of pushing the target away. Pan ²¹ proposed a FA-SC-VT (fast-approach slow-contact velocity track) capture strategy, which mainly relies on slowing down the relative velocity to reduce the contact force, and a similar capture strategy based on Loop/Contact model can also been in Tan et al. ²²

In this article, regarding the pre-impact phase, a two-step capture strategy for redundant space manipulator when performing snare capture is proposed. The attitude of the snare-type end-effector is regulated in the first step, which aims to reduce the peak value of contact force between the grapple shaft and flexible cable. And in the second step, the configuration of space manipulator is optimized to reduce the end-effector pose disturbance from impact. Through the null space method, the self-adjustment of manipulator configuration in the second step will not affect the regulated end-effector attitude in the first step. The reliability of space snare capture can be improved by the proposed method.

Problem description

Working mechanism

Snare capture mechanism mainly consists of snare, rigid, and latch subsystem. The function of snare subsystem is to capture the grapple shaft and restrict its movement in space. It is mounted on the supporting shell and comprises a stationary ring, a rotary ring, and three flexible cables each having one end secured to the stationary ring and the other end secured to the rotary ring. When the grapple shaft gets into the region formed by three flexible cables, the rotary ring begins to rotate with respect to the stationary ring to enclose and then latch the target. The specific process is as follows:^8,20

1. The snare-type end-effector approaches the grapple shaft to make it into the enclosed area formed by three flexible cables.

2. After the grapple shaft is inserted, the rotary ring rotates to narrow the enclosed region until the flexible cables are fully tightened. And during the capture process, the grapple shaft may contact several times with three flexible cables.

3. The grapple shaft is then dragged into the cylinder of the end-effector by the cables, and it is latched when the end-effector and the grapple shaft fully contact. The whole process of snare capture is shown in Figure 1.

Figure 1.

The process of snare capture.

Capture strategy

In microgravity environment, snare capture operation would face two major risks. Risk 1: if the contact force is too large, the target may be pushed away and both the target and manipulator could suffer damage. Risk 2: if the pose of the end-effector is disturbed too much from impact, the capture process would become very unstable, and the target may be out of the enclosed region after the first contact. In order to improve the reliability of snare capture, the way of first contact between grapple shaft and flexible cable should be optimized and a two-step capture strategy is proposed as shown in Figure 2:

• Step 1. The attitude of the end-effector is regulated to make the plane formed by three flexible cables perpendicular to the grapple shaft, which will ensure the contact force lie in the plane as much as possible. Thereafter, the end-effector is rotated to adjust the contact point between grapple shaft and flexible cable to reduce the peak value of contact force.

• Step 2. The configuration of space manipulator is optimized to reduce the disturbance of the end-effector from impact. It is worth noting that the optimized control in this step should not affect the regulated attitude of the end-effector in the first step.

Figure 2.

Two-step capture strategy in the pre-impact phase of snare capture.

Contact model between grapple shaft and flexible cable

The snare capture mainly relies on three flexible cables to contact and enclose the grapple shaft; therefore, the contact problem during snare capture is actually the contact modeling problem between grapple shaft and flexible cable.

Discrete model of flexible cable

The flexible cable is assumed to be a series of point masses $(B_1-B_N)$ connected to each other by N−1 massless tension springs and there is a torsion spring installed on every revolute joint of the node (Figure 3). Based on this model, both tension and bending resistance properties of flexible cable can be reflected.

Figure 3.

The discrete model of flexible cable.

Contact assumption

The contact between grapple shaft and flexible cable is a very complex process, and the exact contact model is very hard to be established. Here, we make some assumptions below:

1. The contact force only acts on the node; actually, if the number of nodes is large enough, this could be realized.

2. The relative velocity between grapple shaft and flexible cable is low enough. And based on this, we assume that the velocity-dependent term does not make any contribution to the force equilibrium equation.

3. Considering the small deformation caused by contact, the contact force direction keeps the same during the contact process and only single point contact happens.

Assume that the contact happens on node n of flexible cable, and the force condition is shown in Figure 4.

Figure 4.

Force condition of node n.

The static force equilibrium equations for node $2,3,4\dots (N-1)$ can be established

$$\{\begin{array}{c}{\mathit{F}}_{\mathrm{c}}-{\mathit{T}}_{i-1}+{\mathit{T}}_i+{\mathit{F}}_{i,i}+{\mathit{F}}_{i-1,i}+{\mathit{F}}_{i+1,i}=0\mathrm{if}i=n\hfill \\ \multicolumn{1}{c}{-{\mathit{T}}_{i-1}+{\mathit{T}}_i+{\mathit{F}}_{i,i}+{\mathit{F}}_{i-1,i}+{\mathit{F}}_{i+1,i}=0\mathrm{else}}\end{array}$$

(1)

where ${\mathit{F}}_{\mathrm{c}}$ is the external force applied on node n, here we mean the contact force; ${\mathit{T}}_i$ is the tensile force of segment i, which can be calculated by ${\mathit{T}}_i=k_{\mathrm{s}}(l_i-{\hat{l}}_i)\mathit{j}$ , with $\mathit{j}$ the unit direction vector along segment i; $k_{\mathrm{s}}$ is the stiffness coefficient of tension spring; and $l_i\mathrm{and}{\hat{l}}_i$ are the instantaneous length and unformed length of segment i, respectively. ${\mathit{F}}_{i-1,i},{\mathit{F}}_{i,i},\mathrm{and}{\mathit{F}}_{i+1,i}$ are the equivalent structural forces acting on node i produced by bending moments ${\mathit{M}}_{i-1},{\mathit{M}}_i,\mathrm{and}{\mathit{M}}_{i+1}$ , respectively.

The magnitude of moment ${\mathit{M}}_i$ can be calculated by

$$M_i=k_{\mathrm{t}}{\beta }_i$$

(2)

where $k_{\mathrm{t}}$ is the stiffness coefficient of torsion spring, ${\beta }_i=({\phi }_{i-1}-{\phi }_i)$ is the angle between ${\mathit{j}}_{i-1}$ and ${\mathit{j}}_i$ , and ${\phi }_i$ is the acute angle from ${\mathit{j}}_i$ to X-axis, whose positive or negative sign conforms to anticlockwise or clockwise. The equivalent structural forces are required to balance the moment and at the same time must be self-equilibrated.^23,24 Under these conditions, the equivalent structural forces are given in Figure 5.

Figure 5.

The equivalent structural forces of ${\mathit{M}}_i$ .

And they satisfy the following conditions

$${\mathit{F}}_{i,i}=-({\mathit{F}}_{i,i-1}+{\mathit{F}}_{i,i+1})$$

(3)

$${\mathit{F}}_{i,i-1}=\frac{M_i}{l_{i-1}}{\mathit{e}}_{i,i-1},{\mathit{F}}_{i,i+1}=\frac{M_i}{l_i}{\mathit{e}}_{i,i+1}$$

(4)

where $i=1,2,3,\dots N$ . If $i=1$ , ${\mathit{F}}_{i,i-1}=0$ , and if $i=N$ , ${\mathit{F}}_{i,i+1}=0$ . ${\mathit{e}}_{i,i-1},{\mathit{e}}_{i,i+1}$ are the unit direction vectors of ${\mathit{F}}_{i,i-1},{\mathit{F}}_{i,i+1}$ , which can be calculated by

$$\{\begin{array}{c}{\mathit{e}}_{i,i-1}={\mathit{j}}_{i-1}\times ({\mathit{j}}_{i-1}\times {\mathit{j}}_i)\hfill \\ \multicolumn{1}{c}{{\mathit{e}}_{i,i+1}={\mathit{j}}_i\times ({\mathit{j}}_{i-1}\times {\mathit{j}}_i)}\end{array}$$

(5)

As equation (1) can be decomposed into two sets of scalar equations in x and y directions, $2(N-2)$ static force equilibrium equations can be obtained. And from Figure 4, the flexible cable in equilibrium state also conforms to the following geometric constraints

$$\{\begin{array}{c}{l}_1\mathrm{c}{\phi }_1+l_2\mathrm{c}{\phi }_2+\cdots +l_{N-1}\mathrm{c}{\phi }_{N-1}=d_{\mathrm{se}}\hfill \\ \multicolumn{1}{c}{l_1\mathrm{s}{\phi }_1+l_2\mathrm{s}{\phi }_2+\cdots +l_{n-1}\mathrm{s}{\phi }_{n-1}+l_n\mathrm{s}{\phi }_n}\\ \multicolumn{1}{c}{+l_{n+1}\mathrm{s}{\phi }_{n+1}+\cdots +l_{N-1}\mathrm{s}{\phi }_{N-1}=0}\end{array}$$

(6)

where $d_{\mathrm{se}}$ represents the start-end distance of flexible cable, and c and s stand for cos and sin, respectively. So far, $2(N-1)$ equilibrium equations have been set up, and if the external force ${\mathit{F}}_{\mathrm{c}}$ and the start-end distance $d_{\mathrm{se}}$ are known, the number of basic unknown variables $(l_i,{\phi }_i)$ will also be $2(N-1)$ , therefore, the equations are solvable.

As shown in Figure 6, considering the geometric constraints and initial state, we determine a reasonable range for $d_{\mathrm{se}}$ . When the flexible cable is in its initial position $\mathrm{AA}\prime$ , $d_{\mathrm{se}}$ has the minimum value, which can be calculated using $2R_{\mathrm{r}}\sin \gamma$ , where $R_{\mathrm{r}}$ is the radius of the ring and $\gamma$ is the half of the central angle corresponding to the arc $\widehat{\mathrm{AA}\prime }$ . And obviously, $2R_{\mathrm{r}}$ is the maximum value of $d_{\mathrm{se}}$ when flexible cable is in position $\mathrm{AA}″$ . Therefore, the reasonable range is $d_{\mathrm{se}}\in [2R_{\mathrm{r}}\sin \gamma ,2R_{\mathrm{r}}]$ .

Figure 6.

The determination of start-end distance of flexible cable.

Define the penetration $\delta$ as the displacement from initial contact position along normal contact direction (Figure 7), and the relationship between normal contact force $F_{\mathrm{n}}$ , start-end distance $d_{\mathrm{se}}$ , and the penetration $\delta$ is studied in the following:

1. The initial ranges for $d_{\mathrm{se}}\in [d_{min},d_{max}]$ and $F_{\mathrm{n}}\in [F_{\min },F_{\max }]$ are set first. The range of $d_{\mathrm{se}}$ has been obtained above; however, the range of $F_{\mathrm{n}}$ is set through a tentative process. The minimum value is of course 0 and the maximum value is set as $F_{\mathrm{A}}$ first.

2. A set of start-end distances $[2R_{\mathrm{r}}\sin \gamma ,d_{\mathrm{s}1},d_{\mathrm{s}2},\dots ,2R_{\mathrm{r}}]\in {\Re }^{1\times m}$ , with $2R_{\mathrm{r}}\sin \gamma <d_{\mathrm{s}1}<d_{\mathrm{s}2}<\cdots <2R_{\mathrm{r}}$ and normal contact forces $[0,F_{\mathrm{n}1},F_{\mathrm{n}2},\dots ,F_{\mathrm{A}}]\in {\Re }^{1\times n}$ , with $0<F_{\mathrm{n}1}<F_{\mathrm{n}2}<\cdots <F_{\mathrm{A}}$ are given. Setting $d_{\mathrm{se}}=2R_{\mathrm{r}}\sin \gamma ,d_{\mathrm{s}1},d_{\mathrm{s}2},\dots ,2R_{\mathrm{r}}$ in sequence and substituting each normal force into the equilibrium equations (equations (1) and (6)), the corresponding penetration $\mathbf{\delta }\mathbf{\in }{\mathbf{\Re }}^{\mathbf{m}\mathbf{\times }\mathbf{n}}$ can be obtained. Approximating the results with polynomial functions by curved surface fitting, the relationship between $F_{\mathrm{n}}$ , $\delta$ , and $d_{\mathrm{se}}$ can be obtained

   $$F_{\mathrm{n}}=f\left(\delta \right)+g\left(d_{\mathrm{se}}\right)+h(\delta ,d_{\mathrm{se}})$$

   (7)

   where $f(\delta )$ and $g(d_{\mathrm{se}})$ are the polynomial functions of $\delta$ and $d_{\mathrm{se}}$ , respectively, and $h(\delta ,d_{\mathrm{se}})$ represents the coupling function with variables $\delta$ and $d_{\mathrm{se}}$ . It is worth noting that if the precision of curved surface fitting is not high enough, the method of piecewise fitting can be used.

3. Based on the contact assumptions, the velocity-dependent term is not considered. The dynamic contact process can be simplified as a static one in each step. And thus, the normal contact force can be calculated using equation (7) during snare capture when the penetration $\delta$ and the start-end distance $d_{\mathrm{se}}$ are known. However, it has a reasonable application range for the contact force expression, namely, $d_{\mathrm{se}}\in [d_{min},d_{max}]\mathrm{and}{F}_{\mathrm{n}}\in [F_{\min },F_{\max }]$ . The fitting precision can be ensured only in the fitting area. If it is found that the maximum normal contact force $F{\prime }_{\mathrm{A}}$ during the contact process is larger than the upper limit value $F_{\mathrm{A}}$ , then we need to reset the upper limit value as $F_{\mathrm{B}}$ to make sure that $F_{\mathrm{B}}>F{\prime }_{\mathrm{A}}$ and repeat this process. Of course, in the beginning, we can set the upper limit value as large as possible to avoid the repetitive work, but much more time may be needed to ensure the fitting precision.

So far, the normal contact force model has been established, and the friction force can be calculated by the Coulomb friction model $F_{\mathrm{f}}=\mu F_{\mathrm{n}}$ , where $F_{\mathrm{f}}$ is the friction force and $\mu$ is the friction coefficient.

Figure 7.

The definition of penetration.

Strategy for space snare capture

The regulation of end-effector attitude

First, the attitude of the end-effector is regulated to make the plane formed by three flexible cables perpendicular to the grapple shaft, which will ensure the contact force lie in the plane as much as possible. It is obvious that this step will greatly reduce the possibility of pushing the target out of the enclosed region after the first contact. Second, the end-effector is rotated to adjust the contact point between grapple shaft and flexible cable to reduce the peak value of contact force. This step is worth analyzing, which is detailed in the following.

In order to express more clearly, the contact characteristics between grapple shaft and flexible cable are analyzed by the simulation below. Without loss of generality, set the target mass $m_{\mathrm{t}}=20\mathrm{kg}$ . The target’s attitude is under control and the relative velocity between grapple shaft and the contact point of flexible cable along the normal contact direction is $v_{\mathrm{t}}=0.05\mathrm{m}/\mathrm{s}$ . The length of flexible cable is $l_{\mathrm{t}}=1\mathrm{m}$ , and it is divided into 50 segments, namely, N = 51. The stiffness coefficients of springs are set to be $k_{\mathrm{s}}=50,000\mathrm{N}/\mathrm{m}\mathrm{and}{k}_{\mathrm{t}}=1000\mathrm{N}\mathrm{m}/\mathrm{rad}$ . Let the grapple shaft contact from node 2 to node 50 be in sequence. And we can obtain the peak value of contact force corresponding to each node, as shown in Figure 8. The maximum value is 30.32 N corresponding to nodes 2 and 50. The minimum value is 5.21 N corresponding to nodes 14 and 38. The maximum value is almost six times of the minimum value, which means the contact point actually has a great influence. And from Figure 8, we can clearly see that the peak value has a sudden rise when the grapple shaft approaches the start and the end points. While from node 7 to node 45, there is no obvious change. On this basis, we define the “safe contact region,” which consists of the grapple shaft, node 7, and node 45 as shown in Figure 9.

Figure 8.

The peak value of contact force corresponding to the nodes.

Figure 9.

The grapple shaft in “safe contact region.”

It is worth noting that stiffness coefficients of flexible cable are affected by many cable parameters, like the structure, the length, and the material, and even if for the same flexible cable, its stiffness coefficients will also change with some external factors. ²⁵ However, the method proposed is applicable to any flexible cable with tension and bending resistance properties. For the cables with different stiffness coefficients, they all have the similar properties as shown in Figure 8. The only difference is the range of “safe contact region” according to different task requirements.

The optimized control of space manipulator

The inertial properties perceived at the end-effector

In order to improve the reliability of capture, the pose of the end-effector should not be disturbed too much during the contact process. Therefore, the inertial properties perceived at the end-effector of space manipulator are studied first. The dynamic equation of a free-floating space manipulator is generally expressed in the following form ¹³

$$\mathit{H}\left[\begin{array}{c}{\stackrel{··}{\mathit{x}}}_{\mathrm{b}}\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\mathbf{\theta }}}\end{array}\right]+\left[\begin{array}{c}{\mathit{c}}_{\mathrm{b}}\hfill \\ \multicolumn{1}{c}{{\mathit{c}}_{\varphi }}\end{array}\right]=\left[\begin{array}{c}{\mathit{F}}_{\mathrm{b}}\hfill \\ \multicolumn{1}{c}{{\mathbf{\tau }}_{\varphi }}\end{array}\right]+\left[\begin{array}{c}{\mathit{J}}_{\mathrm{b}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{\mathit{J}}_{\varphi }^{\mathrm{T}}}\end{array}\right]{\mathit{F}}_{\mathrm{e}}$$

(8)

where $\mathit{H}\mathrm{=}\left[\begin{array}{cc}{\mathit{H}}_{\mathrm{b}}& {\mathit{H}}_{\mathrm{b}\varphi }\\ {\mathit{H}}_{\mathrm{b}\varphi }^{\mathrm{T}}& {\mathit{H}}_{\varphi }\end{array}\right]$ is the inertia matrix of the system; ${\stackrel{··}{\mathit{x}}}_{\mathrm{b}}=[{\stackrel{·}{\mathit{\upsilon }}}_{\mathrm{b}},{\stackrel{·}{\mathit{\omega }}}_{\mathrm{b}}]$ describes the linear/angular acceleration of the base; $\stackrel{··}{\mathbf{\theta }}$ represents the joint angle acceleration; ${\mathit{c}}_{\mathrm{b}}\mathrm{and}{\mathit{c}}_{\varphi }$ are the velocity-dependent terms for the base and the manipulator, respectively; ${\mathit{F}}_{\mathrm{b}}$ and ${\mathit{F}}_{\mathrm{e}}$ are the external applied force/torque on base and end-effector, respectively; ${\mathbf{\tau }}_{\varphi }$ represents joint torque; and ${\mathit{J}}_{\mathrm{b}}$ and ${\mathit{J}}_{\varphi }$ are Jacobian matrices of the base and the manipulator, respectively.

For a free-floating space manipulator, ${\mathit{F}}_{\mathrm{b}}=[0,0,0,0,0,0{]}^{\mathrm{T}}$ . Separating equation (8), we can obtain the following equation

$$\begin{array}{c}{\mathit{H}}_{\mathrm{b}}{\stackrel{··}{\mathit{x}}}_{\mathrm{b}}+{\mathit{H}}_{\mathrm{b}\varphi }\stackrel{··}{\mathbf{\theta }}+{\mathit{c}}_{\mathrm{b}}={\mathit{J}}_{\mathrm{b}}^{\mathrm{T}}{\mathit{F}}_{\mathrm{e}}\hfill \\ {\mathit{H}}_{\mathrm{b}\varphi }^{\mathrm{T}}{\stackrel{··}{\mathit{x}}}_{\mathrm{b}}+{\mathit{H}}_{\varphi }\stackrel{··}{\mathbf{\theta }}+{\mathit{c}}_{\varphi }={\mathbf{\tau }}_{\varphi }+{\mathit{J}}_{\varphi }^{\mathrm{T}}{\mathit{F}}_{\mathrm{e}}\hfill \end{array}$$

(9)

Multiply both sides of the first equation of equation (9) by the inverse of ${\mathit{H}}_{\mathrm{b}}$ , we can obtain the expression of ${\stackrel{··}{\mathit{x}}}_{\mathrm{b}}$ as

$${\stackrel{··}{\mathit{x}}}_{\mathrm{b}}={\mathit{H}}_{\mathrm{b}}^{-1}{\mathit{J}}_{\mathrm{b}}^{\mathrm{T}}{\mathit{F}}_{\mathrm{e}}-{\mathit{H}}_{\mathrm{b}}^{-1}{\mathit{H}}_{\mathrm{b}\varphi }\stackrel{··}{\mathbf{\theta }}-{\mathit{H}}_{\mathrm{b}}^{-1}{\mathit{c}}_{\mathrm{b}}$$

(10)

Substituting equation (10) into the second equation of equation (9), the base acceleration ${\stackrel{··}{\mathit{x}}}_{\mathrm{b}}$ can be eliminated and the dynamic equation of a free-floating space manipulator in joint space can be obtained

$${\mathit{H}}_{\mathrm{f}}\stackrel{··}{\mathbf{\theta }}+{\mathit{c}}_{\mathrm{f}}={\mathbf{\tau }}_{\varphi }+{\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}{\mathit{F}}_{\mathrm{e}}$$

(11)

where ${\mathit{H}}_{\mathrm{f}}={\mathit{H}}_{\varphi }-{\mathit{H}}_{\mathrm{b}\varphi }^{\mathrm{T}}{\mathit{H}}_{\mathrm{b}}^{-1}{\mathit{H}}_{\mathrm{b}\varphi }$ , ${\mathit{c}}_{\mathrm{f}}={\mathit{c}}_{\varphi }-{\mathit{H}}_{\mathrm{b}\varphi }^{\mathrm{T}}{\mathit{H}}_{\mathrm{b}}^{-1}{\mathit{c}}_{\mathrm{b}}$ , and ${\mathit{J}}_{\mathrm{f}}={\mathit{J}}_{\varphi }-{\mathit{J}}_{\mathrm{b}}{\mathit{H}}_{\mathrm{b}}^{-1}{\mathit{H}}_{\mathrm{b}\varphi }$ .

Assume that the initial momentum and angular momentum are zero, the kinematic relationship of space manipulator in velocity level is

$${\stackrel{·}{\mathit{x}}}_{\mathrm{e}}={\mathit{J}}_{\mathrm{f}}\stackrel{·}{\mathit{\theta }}$$

(12)

where ${\stackrel{·}{\mathit{x}}}_{\mathrm{e}}$ describes the velocity of end-effector, and $\stackrel{·}{\mathit{\theta }}$ is the joint angular velocity. Taking the derivative of equation (12) with respect to time, we can obtain

$${\stackrel{··}{\mathit{x}}}_{\mathrm{e}}={\mathit{J}}_{\mathrm{f}}\stackrel{··}{\mathbf{\theta }}+{\stackrel{·}{\mathit{J}}}_{\mathrm{f}}\stackrel{·}{\mathit{\theta }}$$

(13)

Meanwhile, the relationship between joint torques and operational forces for redundant manipulator is

$${\mathbf{\tau }}_{\varphi }={\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}{\mathit{F}}_{\varphi }+\mathit{\aleph }\left({{\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}}^{\mathbf{†}}\right){\mathbf{\tau }}_0$$

(14)

where $\mathit{\aleph }({\mathit{J}}_{\mathrm{f}}^{\mathrm{T}\mathbf{†}})=(\mathit{E}-{\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}{\mathit{J}}_{\mathrm{f}}^{\mathrm{T}†})$ is the null space of ${\mathit{J}}_{\mathrm{f}}^{\mathrm{T}\mathbf{†}}$ , with $\mathit{E}$ the identity matrix and ${\mathit{J}}_{\mathrm{f}}^{\mathrm{T}\mathbf{†}}$ the generalized inverse of ${\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}$ . ${\mathit{F}}_{\varphi }$ is the operational force at the end-effector and ${\mathit{\tau }}_0$ is an arbitrary joint torque vector which will be projected into the null space. Considering the dynamic consistency,^26,27 the unique expression of ${\mathit{J}}_{\mathrm{f}}^{\mathbf{†}}$ can be obtained as

$${\mathit{J}}_{\mathrm{f}}^{\mathbf{†}}={\mathit{H}}_{\mathrm{f}}^{-1}{\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}{\left({\mathit{J}}_{\mathrm{f}}{\mathit{H}}_{\mathrm{f}}^{-1}{\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}\right)}^{-1}$$

(15)

Substituting equation (15) into equation (14), we can get the dynamically consistent relationship between joint torques and the end-effector operational forces for redundant space manipulator. It provides a decomposition of joint torques into two dynamically decoupled control vectors: joint torques corresponding to forces acting at the end-effector and joint torques that only affect internal motions. Substituting equations (12), (13), and (14) into equation (11), we can get the dynamic equation expressed in operational space

$$\hat{\mathit{H}}{\stackrel{··}{\mathit{x}}}_{\mathrm{e}}+\hat{\mathit{c}}={\mathit{F}}_{\varphi }+{\mathit{F}}_{\mathrm{e}}$$

(16)

where $\hat{\mathit{H}}=({\mathit{J}}_{\mathrm{f}}{{\mathit{H}}_{\mathrm{f}}}^{-1}{\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}{)}^{-1}$ and $\hat{\mathit{c}}=({\mathit{J}}_{\mathrm{f}}^{\mathrm{T}}{)}^{†}{\mathit{c}}_{\mathrm{f}}-\hat{\mathit{H}}{\stackrel{·}{\mathit{J}}}_{\mathrm{f}}\stackrel{·}{\theta }$ .

So far, the dynamic behavior of the end-effector can be described by equation (16), and its inertial properties can be calculated by virtue of the positive definite matrix $\hat{\mathit{H}}$ . In the following, the inertial properties perceived at the end-effector are analyzed in terms of two types of tasks in operational space: the translational task and the rotational task.

For the translational task, the Jacobian matrix ${\mathit{J}}_{\mathrm{f}}$ in $\hat{\mathit{H}}$ needs to be replaced by the linear velocity Jacobian matrix ${\mathit{J}}_{\mathrm{f}\nu }$ , which consists of the first three rows of ${\mathit{J}}_{\mathrm{f}}$ . The effective mass perceived at the end-effector along unit direction vector $\mathit{u}$ can be represented as

$$m_{\mathrm{e}}=\frac{1}{{\mathit{u}}^{\mathrm{T}}{\hat{\mathit{H}}}_{\nu }^{-1}\mathit{u}}$$

(17)

where ${\hat{\mathit{H}}}_{\nu }=({\mathit{J}}_{\mathrm{f}\nu }{\mathit{H}}_{\mathrm{f}}^{-1}{\mathit{J}}_{\mathrm{f}\nu }^{\mathrm{T}}{)}^{-1}$ . And in the same way, the effective inertia perceived at the end-effector about unit direction vector $\mathit{z}$ can be represented as

$$I_{\mathrm{e}}=\frac{1}{{\mathit{z}}^{\mathrm{T}}{\hat{\mathit{H}}}_{\omega }^{-1}\mathit{z}}$$

(18)

where ${\hat{\mathit{H}}}_{\omega }=({\mathit{J}}_{\mathrm{f}\omega }{\mathit{H}}_{\mathrm{f}}^{-1}{\mathit{J}}_{\mathrm{f}\omega }^{\mathrm{T}}{)}^{-1}$ , with ${\mathit{J}}_{\mathrm{f}\omega }$ the angular velocity Jacobian matrix consisting of the last three rows of ${\mathit{J}}_{\mathrm{f}}$ .

Through the analysis of the inertial properties perceived at the end-effector, in some way, the space manipulator can be regarded as a single body from the perspective of the end-effector. Thus, the following equation can be established when contact happens

$$\{\begin{array}{c}{\mathit{F}}_{\mathrm{c}}=m_{\mathrm{e}}{\stackrel{·}{\mathit{v}}}_{\mathrm{e}}\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{\mathrm{c}}=I_{\mathrm{e}}{\stackrel{·}{\mathit{\omega }}}_{\mathrm{e}}}\end{array}\Rightarrow \{\begin{array}{c}\delta {\mathit{v}}_{\mathrm{e}}=\frac{\underset{t_0}{\overset{t_0+\delta t}{\int }}{\mathit{F}}_{\mathrm{c}}\mathrm{d}t}{m_{\mathrm{e}}}\hfill \\ \multicolumn{1}{c}{\delta {\mathbf{\omega }}_{\mathrm{e}}=\frac{\underset{t_0}{\overset{t_0+\delta t}{\int }}{\mathit{M}}_{\mathrm{c}}\mathrm{d}t}{I_{\mathrm{e}}}}\end{array}$$

(19)

where ${\mathit{F}}_{\mathrm{c}}$ and ${\mathit{M}}_{\mathrm{c}}$ represent the force and moment caused by contact. $m_{\mathrm{e}}$ is the effective mass along the contact force direction, and $I_{\mathrm{e}}$ is the effective inertia about the contact moment direction. $\delta {\mathit{v}}_{\mathrm{e}}$ and $\delta {\mathbf{\omega }}_{\mathrm{e}}$ represent the disturbance of linear velocity and angular velocity of the end-effector, respectively. $t_0$ is the contact time and $\delta t$ is the contact duration.

As the flexible cable is installed on the end-effector, the contact force produced at the flexible cable will transmit to the end-effector. Due to the short duration of contact, the end-effector’s pose could be assumed still during this period. Based on this assumption, the contact force impulse and the impulsive moment are only relevant to the flexible cable and independent of the end-effector’s inertial properties. Therefore, in order to reduce the disturbance of end-effector pose from impact, the mass/inertia property perceived at the end-effector should be as large as possible along/about the direction of contact force/moment. And we design the objective functions as follows

$$g_{\mathrm{em}}=\left({\mathit{u}}^{\mathrm{T}}{\hat{\mathit{H}}}_{\nu }^{-1}\mathit{u}\right),g_{\mathrm{ei}}=\left({\mathit{z}}^{\mathrm{T}}{\hat{\mathit{H}}}_{\omega }^{-1}\mathit{z}\right)$$

(20)

$g_{\mathrm{em}}$ and $g_{\mathrm{ei}}$ are the inverse of the effective mass and inertia, respectively. When they are minimized, the corresponding effective mass and inertia are maximized. And thus, the disturbance can be reduced.

Optimized control based on null space

Non-minimum-norm solutions to equation (12) based on Jacobian pseudoinverse can be written in the general form

$$\stackrel{·}{\mathit{\theta }}={\mathit{J}}_{\mathrm{f}}^{†}{\stackrel{·}{\mathit{x}}}_{\mathrm{e}}+\mathit{\aleph }\left({\mathit{J}}_{\mathrm{f}}\right)\rho \mathbf{\xi }$$

(21)

where $\mathit{\aleph }({\mathit{J}}_{\mathrm{f}})=(\mathit{E}-{\mathit{J}}_{\mathrm{f}}^{†}{\mathit{J}}_{\mathrm{f}})$ is the null space of ${\mathit{J}}_{\mathrm{f}}$ ; $\rho$ is the gain coefficient, which can be a constant or a function; and $\mathit{\xi }$ is an arbitrary vector. By observing that null space velocities produce a change in the configuration of manipulator without affecting its velocity at the end-effector, the characteristics can be exploited to optimize the objective functions in equation (20). The optimized vector in null space is designed as

$$\mathit{\xi }={\lambda }_1{\mathit{\xi }}_1+{\lambda }_2{\mathit{\xi }}_2$$

(22)

where ${\lambda }_1\mathrm{and}{\lambda }_2$ are the weight coefficients which are needed to be regulated according to the requirements of different tasks. ${\mathit{\xi }}_1\mathrm{and}{\mathit{\xi }}_2$ are the gradient vectors of the objective functions, with ${\mathit{\xi }}_1=\nabla g_{\mathrm{em}}=[(\partial g_{\mathrm{em}}/\partial {\theta }_1),(\partial g_{\mathrm{em}}/\partial {\theta }_2),\dots ,(\partial g_{\mathrm{em}}/\partial {\theta }_n)]\mathrm{and}{\mathit{\xi }}_2=\nabla g_{\mathrm{ei}}=[(\partial g_{\mathrm{ei}}/\partial {\theta }_1),(\partial g_{\mathrm{ei}}/\partial {\theta }_2),\dots ,(\partial g_{\mathrm{ei}}/\partial {\theta }_n)]$ . To ensure the regulated pose of the end-effector is not affected, set ${\stackrel{·}{\mathit{x}}}_{\mathrm{e}}=0$ in equation (21), and then we get the following equation

$$\stackrel{·}{\mathit{\theta }}=\rho (\mathit{E}-{\mathit{J}}_{\mathrm{f}}^{†}{\mathit{J}}_{\mathrm{f}})({\lambda }_1{\mathit{\xi }}_1+{\lambda }_2{\mathit{\xi }}_2)$$

(23)

Through equation (23), the configuration of space manipulator will be optimized until the values of objective functions do not change any more. And during the whole optimized control process, the pose of the end-effector will not be disturbed.

Simulation

In this simulation, the proposed capture strategy is applied on a 7-degree-of-freedom (DOF) free-floating space manipulator with snare-type end-effector. Its joint frames according to Denavit–Hartenberg (DH) method are shown in Figure 10, where $X_i\mathrm{and}{Z}_i(i=0,1,\dots ,7)$ represent the vectors of X-axis and Z-axis of the ith frame, respectively. Set $a=k=1.2\mathrm{m}$ , $b=c=f=h=0.5\mathrm{m}$ , $d=e=5\mathrm{m}$ . Its relative parameters are listed in Tables 1 and 2, where ${\theta }_i,d_i,a_{i-1},\mathrm{and}{\alpha }_{i-1}$ are DH parameters; $m_i,{\mathit{P}}_i,\mathrm{and}{\mathit{I}}_i$ represent the mass of the ith part, mass center vector of the ith part with respect to the ith frame, and inertia tensor with respect to the frame attached at the geometric center of the ith part.

Figure 10.

A 7-DOF free-floating space manipulator with snare-type end-effector.

Table 1.

DH parameters of space manipulator.

Link $i$	${\theta }_i(°)$	$d_i(m)$	$a_{i-1}(m)$	${\alpha }_{i-1}(°)$
1	0	1.2	0	0
2	90	0.5	0	90
3	0	0	0	−90
4	0	0.5	5	0
5	0	1	5	0
6	−90	0.5	0	90
7	0	1.2	0	−90

Table 2.

Dynamic parameters of space manipulator.

Part	$m_i(\mathrm{kg})$	${\mathit{P}}_i(\mathrm{m})$	${\mathit{I}}_i(\mathrm{kg}{\mathrm{m}}^2)$
Link 1	42.5	$[0,-0.25,0]$	$\mathrm{diag}(0.89,1.33,0.89)$
Link 2	42.5	$[0,0.25,0]$	$\mathrm{diag}(0.89,1.33,0.89)$
Link 3	70	$[2.5,0,0.5]$	$\mathrm{diag}(2.19,145.83,145.83)$
Link 4	70	$[2.5,0,0.5]$	$\mathrm{diag}(2.19,145.83,145.83)$
Link 5	42.5	$[0,0,-0.25]$	$\mathrm{diag}(0.89,0.89,1.33)$
Link 6	42.5	$[0,0,-0.25]$	$\mathrm{diag}(0.89,0.89,1.33)$
Link 7	42.5	$[0,0,-0.6]$	$\mathrm{diag}(5.10,5.10,5.31)$
Base	10,000	$[0,0,0]$	$\mathrm{diag}(20,000,20,000,20,000)$

The relevant parameters of the snare-type end-effector are set the same as above. The target, whose mass is 20 kg and inertia matrix is $\mathrm{diag}([3.36,3.36,3.32])$ , is equipped with a grapple shaft. For better description, the velocity, the position, and the contact force are expressed with respect to the end-effector frame, whose origin is at the center of the ring. The initial position of the grapple shaft is $[0.2,0.2,0]\mathrm{m}$ , and its initial velocity is $[0.05,0,-0.02]\mathrm{m}/\mathrm{s}$ . Through regulating the end-effector’s attitude, the grapple shaft has moved into the “safe contact region” and will contact with node 24 of flexible cable. The contact force is approximated with the polynomial function as in equation (24), with its coefficients shown in Table 3

$$F_{\mathrm{n}}=(p_{00}+p_{10}\delta +p_{20}{\delta }^2+p_{30}{\delta }^3+p_{40}{\delta }^4)+\left(p_{01}{d}_{\mathrm{se}}\right)+(p_{11}\delta d_{\mathrm{se}}+p_{21}{\delta }^2{d}_{\mathrm{se}}+p_{31}{\delta }^3{d}_{\mathrm{se}})$$

(24)

Table 3.

The coefficients of contact force function.

$p_{00}$	1.00	$p_{30}$	$1.22\times {10}^6$
$p_{10}$	$5.29\times {10}^3$	$p_{21}$	$-1.31\times {10}^5$
$p_{01}$	−1.08	$p_{40}$	$-2.02\times {10}^6$
$p_{20}$	$1.22\times {10}^5$	$p_{31}$	$-9.75\times {10}^5$
$p_{11}$	$-5.12\times {10}^3$

Set $d_{\mathrm{se}}=0.91\mathrm{m}$ and $\mu =0.2$ , and the contact force applied on the target can be obtained as shown in Figure 11(a), it starts at 2.64 s and ends at about 3.18 s. The peak value of contact force is about −5.02N, −1.55N, and 1.03 N in x, y, and z directions, respectively. Figure 11(b) shows the first contact process; the red circle represents the grapple shaft and it moves along the arrow direction. The dotted line represents the contact direction.

Figure 11.

The first contact between flexible cable and grapple shaft: (a) the contact force applied on the target and (b) the contact process.

Assume that the regulated pose of the end-effector is $[7\mathrm{m},0\mathrm{m},3\mathrm{m},-1.0\mathrm{rad},-0.5\mathrm{rad},-2.0\mathrm{rad}]$ and the initial joint angles and base attitude are $[-1.61,-3.39,2.66,-1.73,3.29,3.27,0.69](\mathrm{rad})$ and $[0.22,-0.09,-0.16](\mathrm{rad})$ , respectively. The initial inertial properties perceived at the end-effector are $m_{\mathrm{e}}=15.79\mathrm{kg}$ and $I_{\mathrm{e}}=7.50\mathrm{kg}{\mathrm{m}}^2$ . Set ${\lambda }_1=30\mathrm{and}{\lambda }_2=10$ . The changes in joint angles during optimized control process are shown in Figure 12, and it can be seen that optimized control ends at about 20 s, and the final joint angles and base attitude are $[0.35,-4.22,4.70,-1.89,3.62,2.20,-1.90](\mathrm{rad})$ and $[0.23,-0.11,-0.24](\mathrm{rad})$ , respectively. As shown in Figure 13, the values of objective function $g_{\mathrm{em}}$ and $g_{\mathrm{ei}}$ reduce to 0.0355 and 0.1154 from 0.0633 and 0.1334, respectively, which means the corresponding effective mass and effective inertia are with an increase of 78.31% and 15.66%.

Figure 12.

The changes of joint angles.

Figure 13.

The values of objective function $g_{\mathrm{em}}$ and $g_{\mathrm{ei}}$ .

Equation (25) describes the variations of end-effector velocity due to the impact before and after optimized control, which are represented by $\Delta {\mathit{v}}_{\mathrm{e}},\Delta {\tilde{\mathit{v}}}_{\mathrm{e}}$ and $\Delta {\mathbf{\omega }}_{\mathrm{e}},\Delta {\tilde{\omega }}_{\mathrm{e}}$ , respectively. Obviously, the disturbance is reduced

$$\{\begin{array}{c}\Delta {\mathit{v}}_{\mathrm{e}}=[-2.81,10.72,-3.35]\times 10^{-2}\mathrm{m}/\mathrm{s}\hfill \\ \multicolumn{1}{c}{\Delta {\tilde{\mathit{v}}}_{\mathrm{e}}=[-1.58,6.01,-1.88]\times 10^{-2}\mathrm{m}/\mathrm{s}}\end{array},\{\begin{array}{c}\Delta {\mathit{\omega }}_{\mathrm{e}}=[-2.46,-0.93,-0.91]\times 10^{-2}\mathrm{rad}/\mathrm{s}\hfill \\ \multicolumn{1}{c}{\mathit{\Delta }{\tilde{\omega }}_{\mathrm{e}}=[-2.13,-0.81,-0.79]\times 10^{-2}\mathrm{rad}/\mathrm{s}}\end{array}$$

(25)

The whole optimized control process can be seen in Figure 14, the red configuration is the final optimized configuration, and we can see that the pose of the end-effector is not disturbed during the control process.

Figure 14.

The whole optimized control process.

Conclusion

Snare capture performed by a manipulator is becoming more and more popular for its large tolerance and high stiffness connection for space payload operations. In order to improve the reliability, a two-step capture strategy for redundant space manipulator is proposed regarding the pre-impact phase. From the simulation results, we can see that the peak value of contact force can be greatly reduced when the grapple shaft gets into the “safe contact region,” and with the purpose of reducing the disturbance to the end-effector, the inertial properties perceived at the end-effector are optimized by adjusting the manipulator configuration. The simulation results prove the effectiveness of the proposed snare capture strategy. The potential application of our work is to predict the contact process and reduce the risks during snare capture.

The contributions include that the determination of “safe contact region,” the calculation of contact force, and the reduction of end-effector pose disturbance are applicable to the target with any initial conditions as long as we know the dynamic parameters and the relative velocity between the target and the contact point of flexible cable. However, the capture strategy proposed is more applicable to the cooperative target, especially the one with attitude under control. Regarding the tumbling target, how to make the grapple shaft insert into the capture region is still a challenge.