Inertia parameters identification for cellular space robot through interaction

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Inertia parameters identification for cellular space robot through interaction

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a, b

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Haitao Chang , Panfeng Huang Zhengxiong Liu a,b

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, Zhenyu Lu

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, Yizhai Zhang

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, Zhongjie Meng

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i n f o

a b s t r a c t

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Article history: Received 3 December 2016 Received in revised form 19 March 2017 Accepted 27 September 2017 Available online xxxx

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Research Center of Intelligent Robotics, School of Astronautics, Northwestern Polytechnical University, Xi’an, China b National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi’an, China

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Keywords: Cellular space robot Takeover control Inertia parameter identification Interactive Distributed

Most of the technologies are in high-speed evolution nowadays. But the spacecraft, however, is still high-priced and takes years to construct. Besides that, it is hardly to service since the conventional spacecraft are not serviceable designed. Facing those challenges, the concept of cellular space robot is presented in this paper for both spacecraft system construction and on-orbit service. As a typical onorbit service task, the non-cooperative target takeover control is considered in this paper. Specifically, the /colorreviseinertia/colorblack parameters identification for takeover control is studied in this paper. Because the cells in the cellular space robot are interconnected and networked, an /colorreviseinteractive/ colorrevise parameter identification algorithm is presented to solve the parameter identification problem by cells interaction. The algorithm is distributed and both synchronous and asynchronous interaction are supported. The algorithm is validated and analyzed by numerical simulations. © 2017 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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Spacecraft, especially the satellites are widely used in various fields and greatly facilitate our everyday life in different ways. With the developing of space technologies, the annual launch number of spacecraft grows enormously to meet the increasing demands. However, unlike the other technologies whose cost falls sharply after its /colorrevise inception /colorblack, the cost of the satellite has not been reduced for decades despite the development of the fundamental technologies. One of the major reasons is that a lot of testing is required during the spacecraft development in the conventional way. The heavy testing is not only costly but also time-consuming. Conventional construction of a big satellite takes 3 to 5 years, while nearly half of the time is consumed for testing [1]. Thanks to the development of small satellites like CubeSat, 94% of them built by commercial and military entities

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*

Corresponding author at: National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi’an, China. E-mail address: [email protected] (P. Huang). 1 Panfeng Huang received B.S. and M.S. degree from Northwestern Polytechnical University in 1998, 2001, respectively, and Ph.D. from the Chinese University of Hong Kong in the area of Automation and Robotics in 2005. He is currently a Professor of School of Astronautics and Vice Director of Research Center for Intelligent Robotics at the Northwestern Polytechnical University. His research interests include tethered space robotics, intelligent control, machine vision, space teleoperation. https://doi.org/10.1016/j.ast.2017.09.044 1270-9638/© 2017 Elsevier Masson SAS. All rights reserved.

will only take 24 months [2,3]. Even so, it is not enough for emergency response. Besides that, since the emergent of the satellite, its morphology has not changed [4]. The spacecraft nowadays are monolithic and as a consequence, the spacecraft with higher performance mostly have a greater mass. For a space mission, mass is a proxy of the cost, therefore, the monolithic morphology hinders the search of a lower cost solution. On the other hand, on-orbit service and debris removing are attractive fields nowadays [5], but the monolithic design also makes the spacecraft unrealizable for function adjustment which is considerably necessary for emergency response. Meanwhile, the conventional spacecraft without service-friendly design can hardly provide facilities for on-orbit service. Although the space manipulator is considered the most promising technique for on-orbit service [6–8] and maintenance, it is impractical to design a universal space manipulator to fit all diverse structures of heterogeneous spacecraft considering the variety of the spacecraft. On the other way, it is too costly and time-consuming to construct a specialized space manipulator for each mission. Therefore, it is difficult to reduce the costs or shorten the development cycle of a satellite without changing the conventional design pattern [9]. And also, trying to enable the flexible function adjustment without changing the morphology is a huge challenge. To overcome the above-mentioned problems, a new design pattern of the spacecraft by breaking down the monolithic morphology is proposed. That comes to the concept of cellularied design.

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Fig. 3. Concept of DLR’s iBOSS [18,19]. Fig. 1. Prototype of the CellSat [10].

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Fig. 2. Concept of DARPA’s Phoenix [12].

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Cellular Space Robot (CSR) is a cellularized design that provides a new approach for both spacecraft construction and space operation. By disaggregating the architecture of the spacecraft according to the subsystem function (e.g. sensing, communication, attitude control, etc.), we can get different kinds of architectural units. Each architectural unit is physical independent, functional independent and equipped with standardized structure and standardized interface. Aggregating the units together brings out a full functional spacecraft system. The independence and standardization of the cells conduce to the functional flexibility and configuration flexibility of the aggregation system. The aggregation system has a morphology architecture similar to the biological organization of the multicellular species. Each architectural unit can be regarded as a cell of the aggregation system, and hence comes the name of cellular space robot. The concept of cellular space robot proposed in this paper is on the foundation of other researchers’ significant work. The CellSat as shown in Fig. 1 is a cellularized satellite proposed by Hideyuki Tanaka and Noritaka Yamamoto [10,11]. The CellSat includes the cells and connector pin, and the system is maintained by a specialized space manipulator. Hideyuki Tanaka and Noritaka Yamamoto also built a prototype and studied the accurate assembly by the manipulator [10]. In consideration of the advantage in rapid response of the cellularization. Several organizations have proposed their research program on it. DARPA presented the Phoenix program [12–15] to harvest the antenna of the retired satellites and aggregate the cells named “satlets” [15–17] with the antenna to build a new satellite. The concept of the Phoenix program is shown in Fig. 2. With the help of the space manipulator FREND, the new satellite might be built in situ rather than launched from the ground. The iBOSS [18,19] (Intelligent Building Blocks for On-orbitSatellite Servicing) is another cellularized spacecraft to facilitate the on-orbit service and maintenance, as shown in Fig. 3. The standardized “building blocks” can be easily replaced on orbit and that releases the need of the manipulation skills. As a conclusion, a specialized space robot is needed in the projects mentioned above, for example the FREND in the Phoenix program. Meanwhile, the cellularized design system can be easily manipulated because of its standardization. Owing to that, a simple

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mechanism named manipulation cell can substitute for the manipulator to manipulate the cells. The system can manipulate the cells and execute various manipulations with help of the manipulation cells and end effectors. Hence, the cellular space robot is capable of self-maintenance and servicing the other spacecraft. Such a system can be implemented as both conventional spacecraft and space robot. That is the connotation of the Cellular Space Robot. As reported, a lot of the high valued spacecraft reach their end of life unexpectedly due to the propellant depletion or control system malfunctions [20]. However, the valuable payloads onboard are still functional. Therefore, there is an urgent need for on-orbit service to restore the control capability or replenish consumables. To accomplish the on-orbit service, the target should be captured first. Besides the conventional space manipulators, a lot of capturing methods including tethered space robot [21–23] and maneuverable tethered space net [24,25]. After capturing the target, many servicing tasks can be implemented. Among them the takeover control [26–28] offers a feasible solution to extend the working life of the spacecraft with propellant depletion or control system malfunctions. The OLEV (including CX-OLEV [29] and SMART-OLEV [30]), SUMO [31,32] and Orbital ATK MEV [33] are representative projects focus on spacecraft life extension. By attaching on the target spacecraft the service spacecraft provide a fully functional substitute for the original control system, allowing the remaining payloads to perform normally. While attaching to the target spacecraft to service, the cellular space robot can take over the attitude control system of the target and restore the control capability. The cells attached on the target constitute the proxy control system. Moreover, thanks to the flexibility of the cellular space robot, it suits more target spacecraft by adjusting amount and configuration of the cells. Since the target spacecraft on orbit nowadays are not servicefriendly designed, i.e. the target spacecraft are non-cooperative. The parameters, especially the inertia parameters which are crucial to takeover control are unavailable. Consequently, the dynamic parameters identification is the primary problem for the takeover control. Moreover, the inertia parameters identification is also essential for self-reconfiguration of the cellular space robot since the switch of the system configuration might lead to change of the parameters. The topics of the inertia parameter identification for space manipulator [34], tethered space robot [35] and conventional satellite are widely researched, there are summarily two types of methods: the method based on Newton–Euler method [36–38] and the method based on the law of conservation of momentum [34, 39]. The takeover control scenario of the cellular space robot indicates that multiple cells are needed because the ability of a single cell is not sufficient to drive a target spacecraft with much greater mass. The cells attached on the target spacecraft have to work in coordination. The cells can interact with each other via the data interface and hence the configuration of the aggregated system is networked and each cell can be regarded as a node in the network. As a consequence, the model identification for the cellular

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space robot is more complicated than the conventional spacecraft faced. In the traditional ways, the measurement data can be easily gathered to the data handling subsystem and processed there. It is much easier because the conventional spacecraft is a centralized system. But applying the centralized algorithm to a decentralized system like cellular space robot could bring several issues. For example, one of the cells must be the central node to process all the measurement data, and that will bring great computational load to the central node. The need of the central node also leads to heavy traffic load for the communication bus of the central node. Moreover, both the computational load and the traffic load will raise as the amount of the cells increases. For the parameter identification problem of the multi-node system, Liu and Cao [40] present an algorithm but the algorithm can only be used in the singlehop networked system. Ding Feng [41] also provide a valid method named coupled-least-squares (C-LS) to identify the common parameters for the distributed system. But the transmission of the covariance matrix is required and that might lead to communication pressure. Moreover, the C-LS can only be applied to the system with ring topology and the data must be transmitted in the ring topology in order. Siavash Dorvash proposed the interactive model identification (IMID) [42,43] for structural health monitoring, and it is based on the method of maximum likelihood. This paper presents an interactive parameter identification method that combines the C-LS and IMID. In this method, the need of the transmission of the covariance matrix is released. Moreover, the method suits more topologies besides the ring. The rest of this paper is organized as follows: section 2 describes the concept of the cellular space robot, including the component and the design principles of the cellular space robot and categories of the cells. Section 3 presents dynamic model of the cellular space robot and formulation of the parameter identification. Section 4 proposes the interactive parameter identification algorithm for the cellular space robot. Section 5 is the simulation and section 6 is the analysis and discussion of the algorithm. At last, section 7 concludes this paper.

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The cellular space robot is a disaggregation of the conventional monolithic spacecraft. It breaks down the monolithic architecture of the spacecraft into standardized cells. Inside each cell, the applicable subsystem is accommodated. Each cell has standardized outside structure and standardized interfaces for physical docking and resource sharing. Although a single cell is structural and functional simplified, once assembled, they can provide enhanced function. The organization of the cellular space robot is similar to the biological organization of the tissues/organisms of the multicellular creature. The concept of the cellular space robot is illustrated in Fig. 4. The cellular space robot is equipped with manipulation cells which can manipulate its own cells. The manipulation cell, as a component of the cellular space robot system, can move the other cells from an original position to a target position i.e. the cellular space robot can fulfill the self-reconfiguration process. The common features of the cellular space robot and multicellular creature are listed as follows:

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• Limited types of the components: the types of the fundamen-

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tal cells in both the biological system and cellular space robot are limited; • Large amount of components: the amount of the cells contained in the living creature or aggregative cellular space robot is extremely large;

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Fig. 4. Concept of the Cellular Space Robot.

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• Simple components: both the structure and function of a single cell are simple in both system; • Interconnected aggregated system: the cells in the biological organism are physically interconnected and share resources/information, e.g. the neurotransmitters are transferred between the neurons in the nervous tissue for synaptic communication. In the cellular space robot system, the cells are physical docked and share information through the data interface; • Enhanced aggregative system: although the components are simple, the aggregated system can accomplish complex functions by coordinately, a great example is the human brain, i.e. the aggregated system are enhanced.

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2.1. Concept and description

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2.2. Disaggregation scheme

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2. The cellular space robot

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An intuitive disaggregation scheme is a division by spacecraft subsystems. Different subsystems accommodate in the standardized cells make different types of cells. Meanwhile, some equipment relies on external information or jet of propellant to work properly, for example the star sensor, camera, antenna, thruster, etc. Consequently, the cells with those equipment must dock on the outer side of the aggregative system due to the positional constraint. Base on the above considerations, the cells can be categorized according to the function and positional constraint. Typical cells are listed as follows and shown in Fig. 5.

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• Brain Cell: the brain cell provides the basic computation and

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data handling ability for the aggregative system and contains computational devices, data handling devices and memory; • Actuator Cell: the actuator cell provides the actuation ability for attitude/orbit control and maneuver and contains actuators like thrusters, reaction wheels, magnetorquers, etc.; • Sensor Cell: sensor cell provides fundamental measurement function for attitude determination, it contains a suite of sensors like gyroscopes, star trackers, earth sensors, accelerometers and so on; • Communication Cell: communication cell provides fundamental function for communication with ground station or other spacecraft. it contains antenna, modulator and other communication management devices;

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Fig. 6. General model of cellular space robot system.

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Fig. 5. Typical cells.

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• Power Management Cell: power management cells provide the power generation, storage and distribution function for all the other cells, and contains solar panel, battery and power distribution hardwire; • Payload Cell: Contains the payload hardwire designing for the mission; • Operation Cell: operation cell provides the manipulation function for self-reconfiguration and other on-orbit operations and contains a rotational joint and end effector adapter. 2.3. Design principles

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Owing to the cellularization, low-cost and rapid response spacecraft development is possible through cells assembly. During design phase, the functional adjustment can be achieved by adjusting constitution and configuration of the cells rather than start over. Through on-orbit self-reconfiguration, the cellular space robot can adjust to the temporary or permanent change of the requirement. To ensure the flexibility of the cellular space robot system, several principles [18,19] should be followed in the cellular space robot design, especially for the interface design:

• Structure standardization: it will boost the designing, reduce • • •

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•

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(1)

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T

and T = τ1 τ2 · · · τnt . J f and J t are the control effectiveness matrices determined by the configuration of force cells and torque cells respectively and can be expressed as follows.



k1 Jf = a× 1 k1



Jt =

0 d1

··· kn f · · · an×f kn f  0 ∈ R6×nt

k2 a× 2 k2

··· ···

0 d2



∈ R6×n f

(2)

⎡

dnt

M v˙

˙ + ω× I ω Iω



=

Fe Te

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According to Newton–Euler method, the dynamic equation of the system can be written as



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a2 −a1 ⎦ 0

0 a1

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103

⎤

−a3

0 a× = ⎣ a3 −a2

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(3)

In (2), the superscript ‘×’ represents the cross product operator, e.g. a× is the skew-symmetry matrix of a defined as follows:



+ ξe

(4)

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3.1. Dynamic model

where

The cellular space robot system studied in this paper is shown in Fig. 6. The target spacecraft and the cells attached on it constitute an aggregative system. Assuming that the cells are solid docked on the target spacecraft, the aggressive system can be considered as a rigid body. The rigid body is denoted as B, and O cm is its centroid. The centroid frame O cm X cm Y cm Z cm is fixed with rigid body B and its origin is at O cm . O I X I Y I Z I is the inertial frame. For the actuator cells, they can be categorized into

H (X) =

3×3





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H ( X ) X¨ + c X , X˙ = J f F + J t T + ξ e

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= J f F + JtT

where F ∈ Rn f ×1 denotes force cells’ output and F =  T f 1 f 2 · · · f n f . T ∈ Rnt ×1 denotes the torque cells’ output



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3. Modeling and formulation

•

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Fe Te



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In which, M is the mass of the system, I ∈ R is the inertial tensor, v ∈ R3×1 is the linear velocity of the centroid O cm , ω ∈ R3×1 is the angular velocity of O cm and ξ e ∈ R6×1 is the external disturbance. Equation (4) can be rewritten as a general form as

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testing and make mass-produce possible; Space operation friendly: it will release the need of manipulation skills; Thermal control feasibility: it is the basic requirement for spacecraft; Androgynous interface (including physical interface, power interface and data interface): it will lead to dock combination flexibility i.e. arbitrarily pair of the cells can be docked with each other; Axial symmetry interface: it will lead to dock direction flexibility i.e. the cells can be docked in multiple directions along the symmetry axis; Interface releasable: it will ease the self-reconfiguration of the cellular space robot.

torque cells and force cells according to the type of the actuators: the force cells contain thrusters and are denoted as C f i (i = 1, 2, ..., n f ); the torque cells contain momentum wheels and are denoted as C t j ( j = 1, 2, ..., nt ). The docked position of the C f i relative to the centroid is ai (i = 1, 2, ..., n f ), and the output orient and value are ki (i = 1, 2, ..., n f ) and f i (i = 1, 2, ..., n f ) respectively. The torque output orient and value are d j ( j = 1, 2, ..., nt ) and τ j ( j = 1, 2, ..., nt ) respectively. As a joint result of the force cells and torque cells, the external force F e and external torque T e acting on the rigid body B can be written as





M E3 0

c X , X˙ =





(5)

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0 ∈ R6×6 I

ω × Iω

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123



0

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∈R

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6×1

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6×1 In which H ( X ) is the inertial matrix of the system,  X ∈R

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is the non-

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denotes the position and orientation of O cm , c X , X˙ linear force and E 3 is an identity matrix.

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The latter is the parameter to be identified and the relationship between I and b I can be expressed as follows.

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b

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Fig. 7. Illustration of the sensor cells attached on the target.

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= Ab

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Last section presents the dynamics model of the spacecraft with actuator cells docked on. The actuator cells are necessary for the takeover control because they provide the force and torque for attitude/orbit maneuver. Besides that, the sensor cells are also indispensable because they provide the sensor data for takeover control and parameter identification. Assuming that there are ns sensor cells attached on the target spacecraft, and all the sensor cells are equipped with accelerators and gyroscopes. As shown in Fig. 7, the displacement from O cm to the attach point of the sensor C sk are r sk (k = 1, 2, ..., ns ) respectively. For convenience, we need to define a body-fixed frame O b X b Y b Z b as a reference to locate the center of mass. Therefore, its origin should be a geometric feature point of the system. Without loss of generality, let it be coincident with the fixed frame of the sensor cell C s1 . The position of the centroid can be described in the body frame by b r cm . Then the acceleration of the kth sensor cell is ×

˙ r sk + ω ask = ac + ω =

Fe M

×

×

ω r sk

×

(6)

×

+ ω˙ r sk + ω ω r sk

r sk = A b



b

r bk − b r cm



(7)

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where A b ∈ R3×3 is the direction cosine matrix of the body frame, b r sk is displacement of C sk relative to O b X b Y b Z b and the presuperscript “b” represents the reference frame. Assuming that the relative configuration of the cells is known, the r bk can be obtained by the configuration. Combining (6) and (7), we can get a regression model for mass and center of mass identification based on the C sk ’s measurement: T ymk = ϕ mk θ mk

(8)





θ mk =

b

1 M

r cm

ymk = ask −



∈ R4×1

ω˙ + ω A b

×

ω˙ A b + ω× ω× A b



b

b

 r sk

I

ω

(11)

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where,

⎡

ω˙ x

M ωd = ⎣ 0

0

ω˙ y

0

⎡

0 0

M ω = ⎣ ωx ω z −ω x ω y

ω˙ z ω˙ y

ω˙ z

0

0

0

−ω y ω z 0

ωx ω y

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⎤

ω˙ z ω˙ y ˙x⎦ 0 ω ω˙ x 0

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⎤ ω y ωz ω2y − ω2z ωx ω y −ω x ω z −ωx ωz −ωx ω y ω2z − ωx2 ω y ωz ⎦ 0 ωx ω z −ω y ωz ωx2 − ω2y

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and



Ma =

M a11 M a21

M a12 M a22

90



91

(12)

⎤

⎡

⎡

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2α12 α13

2α13 α11

2α11 α12

2α32 α33

2α33 α31

2α31 α32

100

⎤

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⎤

⎡

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⎤

α22 α33 + α23 α32 α21 α33 + α23 α31 α21 α32 + α22 α31 ⎣ Ma22 = α12 α33 + α13 α32 α11 α33 + α13 α31 α11 α32 + α12 α31 ⎦ α12 α23 + α13 α22 α11 α23 + α13 α21 α11 α22 + α12 α21 3×3

In (12), of A b and

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α21 α31 α22 α32 α23 α33 M a21 = ⎣ α31 α11 α32 α12 α33 α13 ⎦ ∈ R3×3 , α11 α21 α12 α22 α13 α23

∈R

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2 2 2 α11 α12 α13 ⎢ 2 2 2 ⎥ ⎥ ∈ R3×3 , α α22 α23 M a11 = ⎢ ⎦ ⎣ 21 2 2 2 α31 α32 α33

⎡

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In which M a11 , M a12 , M a21 , M a22 are blocks of the partitioned matrix M a :

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α11 , α12 , α13 , α21 , α22 , α23 , α31 , α32 , α33 are elements

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⎤

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y Ik = ϕ TIk θ Ik

(13)

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3.3. Formulation for inertial tensor identification

where y Ik = T e and

ϕ TIk = ( M wd + M w ) M a .

According to (4), the following equation holds while the external disturbance is negligible.

4. Interactive model identification method

˙ + ω× I ω = T e Iω

Last section deduces the regression models for inertia parameters (including mass, center of mass and inertia tensor) identification for cellular space robot. For regression model like y (t ) = ϕ T (t ) θ + δ (t ), where θ ∈ R p×1 is the parameter to identify, p is

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A bT

Then we can get the regression model for inertia tensor identification based on the C sk ’s measurement:

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×

α11 α12 α13 A b = ⎣ α21 α22 α23 ⎦ α31 α32 α33

T ϕ mk = F e ω˙ × A b + ω× ω× A b ∈ R3×4



I

A bT

⎡

where



b

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75



M a12 = ⎣ 2α22 α23 2α23 α21 2α21 α22 ⎦ ∈ R3×3 ,

where ac is the acceleration of O cm . r sck can be expressed as

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×



= ( M ωd + M ω ) M a θ Ik

3.2. Formulation for mass identification

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T

˙ + ω× I ω T e = Iω

10

15

(10)

Define θ Ik as θ Ik = I xx b I y y b I zz b I yz b I xz b I xy as parameter maintained in the kth sensor cell. Consider (10) we can rewrite (9) as:

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I = A b I A bT

b

5

13

5

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(9)

In which, I is the inertia tensor expressed in the inertial frame. Relatively, b I is the inertia tensor expressed in the body frame.

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the dimension of the θ ; y (t ) ∈ Rq×1 is the output vector of the regression model and q is the output dimension; ϕ T (t ) ∈ Rq× p is the regression information matrix and δ (t ) ∈ Rq×1 is the measurement noise. For (8), p = 4, q = 3, and for (13), p = 6, q = 3. To composite the measurement data of ns sensor cells, a commonly used method is to gather all the sensor cells’ measurement data and send to a central node. The measurement data of the ns sensor cells can form a larger scale regression model Y (t ) = T (t ) θ +  (t ) as follows

⎡

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⎡

⎤

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⎡

ϕ T1 (t ) ⎢ ϕ T (t ) ⎥ ⎥ ⎢ 2

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Fig. 8. Data transmission in the C-LS.

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(14)

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where

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⎤

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⎤ δ 1 (t ) ⎢ y (t ) ⎥ ⎢ δ (t ) ⎥ ⎢ 2 ⎥ ⎥ ⎢ 2 ⎢ . ⎥ = ⎢ . ⎥θ + ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ δns (t ) yns (t ) ϕ nTs (t ) y 1 (t )

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Y (t ) =



···

T

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qns ×1

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∈R   T T (t ) = ϕ 1 (t ) ϕ 2 (t ) · · · ϕ ns (t ) ∈ Rqns × p  T  (t ) = δ T1 (t ) δ T2 (t ) · · · δnTs (t ) ∈ Rqns ×1 y T1 (t )

y T2 (t )

ynTs

(t )

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Fig. 9. Data transmission in the interactive parameter identification algorithm, N i is the set of C si ’s neighbors. ns 

g i j = 0. di is the degree of the cell C si and di =

  ⎧ T ⎪ ⎪ θˆ (t ) = θˆ (t − 1) + L (t ) Y (t ) −  (t ) θˆ (t − 1) ⎪ ⎪ ⎪ ⎪  −1 ⎨ L (t ) = P (t − 1)  (t ) E qns + T (t ) P (t − 1)  (t ) ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎩ P (t ) = E − L (t ) ϕ T (t ) P (t − 1)

teraction between cells in the interactive parameter identification algorithm can be demonstrated in Fig. 9. Unlike the C-LS, only the parameter estimation is needed to transmit to the neighbor cells. The procedures of the interactive parameter identification algorithm are listed as follows: 1) Initialize the parameter and set the initial estimation θˆ i (0) = 1 p ×1 / p 0 , P i (0) = p 0 E p for i = 1, 2, ..., q, and p 0 = 1 × 105 . 2) Each sensor cell collects the output data y i (t ) and information matrix ϕ Ti (t ) respectively.

(15)

where L (t ) ∈ R p ×qns is the gain matrix, P (t ) ∈ R p × p is the covariance matrix, and E qns is a qns − order identity matrix. For multi-node system like cellular space robot, the RLS method using (15) leads to several issues as follows. (1) A central cell is required. As all the data should be processed in the central node, reliability of the central cell must be high enough. (2) Heavy communication pressure. The central cell should gather all the measurement data, and that leads to heavy communication pressure. (3) Great calculation pressure. As we can  see, the matrix inverse  operation of E qns + T (t ) P (t − 1)  (t ) is required in (15), and it is a qns × qns matrix. The calculation pressure will rise severely to an unacceptable level while the amount of the sensor cells is big enough. The RLS algorithm expressed in (15) is a centralized algorithm, to solve the issues above, an alternative decentralized algorithm is required. The C-LS is a decentralized algorithm for parameter identification. But in C-LS, the data should be transmitted orderly from one node to the next node in the system with a ring-topology, as showed in Fig. 8. Moreover, the covariance matrix transmission is required. Combining the C-LS and IMID, we present a decentralized parameter identification algorithm based on interaction. The interactive parameter identification algorithm presented in this paper suits not only the ring topology but also many others. The communication topology of the cells is described by undirected graph in graph theory, and the cells can be regarded as vertices and data links can be regarded as the edges in the graph. Assume that the adjacency matrix of the graph is G ∈ Rns ×ns and the g i j = 1 is the element of the adjacency matrix. g i j describes the connectivity of cell C si and C sj , if they are connected, g i j = 1 and otherwise,

g i j . The data in-



⎧ ⎪ θˆ i (t ) = θ¯ai + L i (t ) y i (t ) − ϕ iT (t ) θ¯ai ⎪ ⎪ ⎪ ⎨   L i (t ) = P i (t − 1) ϕ iT (t ) / I + ϕ iT (t ) P i (t − 1) ϕ i (t ) ⎪ ⎪   ⎪ ⎪ ⎩ P (t ) = I − L (t ) ϕ T (t ) P (t − 1) i i i i

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3) Estimate the parameter θˆ i (t ) in sensor cell C si with the initial condition set in 1). 4) Update the parameter θˆ i (t ) of cell C si based on the mean value of the neighbors’ estimation θ¯ ai by (16).



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Normally, the recursive least square (RLS) is used to solve the regression model above using

j =1

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(16)

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where θ¯ ai is the mean value of the ith sensor cell’s neighbors’ estimation. It is based on the interaction strategy. For synchronous interaction,

θ¯ai =

n 1 

di

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g i j θˆ j (t − 1)

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and for asynchronous interaction,

⎡

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⎤

 1 ⎢  ⎥ θ¯ ai = ⎣ g i j θˆ j (t ) + g ik θˆk (t − 1)⎦ di

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j ∈ V fast i

k∈ V slow i

where V ifast and V islow are the set of C si ’ neighbors that update their estimates faster and slower than C si respectively. 5) Continue the procedures till terminate conditions. The consensus and convergence properties of the algorithm are proven in the previous work for the ring topology and the interactive parameter identification problem is equivalent to the multi-agent consensus problem. And all the all the nodes’ states

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Table 1 Dynamics parameters of the aggregative spacecraft system. Value

Cell No.

b

Mass (kg)

M

1700

Centroid position b r cm (m)

b

r cm (x) b r cm ( y ) b r cm ( z)

−0.5839

b

3505.1 3130.7 2264.2 −861.0 362.3 −124.7

1 2 3 4 5 6 7 8

[0.0, 0.0, 0.0] [−0.2912, −2.0742, 0.3128] [−0.2600, −2.0589, 0.3997] [−0.2399, 0.4044, −2.0284] [−0.1169, −2.0746, −1.7223] [0.0052, −0.0090, −1.9229] [1.8891, −2.0412, −1.6531] [0.0200, 0.3443, −1.7435]

8

Inertia Tensor b I (kg · m2 )

10 11 12 13

I xx b I yy b I zz b I yz b I xz b I xy

1.0942 1.0563

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will approach to the same constant, i.e. reach a consensus value. Moreover, for a system yns (k) = ϕ nTs θ + σns (k), the convergence property holds [44]. Combining the consensus property and the convergence property, we can draw a conclusion that all the nodes’ estimates will converge to the true value of the parameter. To sum up, the consensus and convergence of the algorithm are proven for the ring topology. According to the consensus theory in the field of multi-agent system, higher connectivity will accelerate the convergence speed. Hence, the application of the interactive parameter identification in the cellular space robot with a higher connectivity topology is effective.

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5. Simulation

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Table 2 Attaching location of the sensor cells.

Description

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Consider an aggregative spacecraft system with the inertia parameters listed in Table 1. Assume the amount of the sensor cells attached on the target ns = 16. And the attaching locations of the cells are listed in Table 2. Set the connection possibility of arbitrary pair of cells as 0.3 and generate the topology. The topology should be connected and can be described by an undirected graph. The topology generated is shown in Fig. 10. The synchronous interaction is used in this section.

39 40

r i (m)

b

9 10 11 12 13 14 15 16

[1.9509, 0.2132, −1.7693] [−0.0563, 0.0663, −1.5852] [−0.2176, −0.0850, 0.2749] [1.7172, −1.8939, −1.6851] [0.0835, −1.6848, −1.9253] [0.0241, −1.8276, −1.8067] [−0.1483, 0.1426, −2.0334] [1.8327, 0.2556, 0.2744]

r i (m)

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In order to valid the interactive model identification algorithm, simulations for both mass property and inertia tensor identification are conducted. The simulation time is 100 s and the sampling interval is 0.1 s. The external force acting on the target for mass identification is shown in Fig. 11a and the external torque acting on the target for inertia tensor identification is shown in Fig. 11b. The identification results in 10 s are shown in Fig. 12 and Fig. 13. As we can see the convergence of the interactive parameter identification algorithm is validated. In Fig. 12 and Fig. 13 all the nodes reach a consensus in a short time.

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6. Analysis and discussion

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Last section validates the convergence and consensus of the interactive parameter identification algorithm for both mass and inertia tensor identification. This section analyses the characteristics of the interactive parameter identification algorithm. Different factors are considered in this section, especially the system noise, topology switching, system topology and interaction strategy. Two key properties we care the most in parameter identification for cellular space robot are the convergence and consensus. The following indicators are selected to evaluate the convergence and consensus performance of the algorithm respectively: (1) Mean value of the estimated values of all the cells. (2) Standard deviation of the estimated values of all the cells. Without the loss of generality, the simulations about mass and center of mass identification are discussed. For simplification, the elements of b r cm in (8) can be denoted as

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R x , R y and R z respectively i.e. b r cm = R x R y R z . Moreover, the mass rather than its inverse is concerned. The mean values ¯ (t ), R¯ x (t ), R¯ y (t ) and R¯ z (t ) are denoted as M

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ns ns   ¯ (t ) = 1 ˆ i (t ), R¯ x (t ) = 1 M M Rˆ xi (t ),

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Cell No.

Fig. 10. Communication topology of the sensor cells.

R¯ y (t ) =

1 ns

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Rˆ zi (t ),

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Fig. 11. External excitation for identifications.

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  n

2  1  σ y (t ) =  Rˆ yi (t ) − R¯ y (t ) , n−1 i =1   n

2  1  σz (t ) =  Rˆ zi (t ) − R¯ z (t ) . n−1

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Fig. 12. Identification results of the mass M and center of mass.

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6.2. Link disconnection

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System noise is inevitable in practical system. To analyze the effect of the noises, diverse simulations with different degrees of noise are conducted. The signals with normal distributed noise here include the acceleration a sk , the angular velocity ω , the angu˙ and external force F e . Set the 3σ of the noise as lar acceleration ω 1%–60% of the maximum output of the measurement on the basis of original set in section 5. The mean values and standard deviations of the cells’ estimation are shown in Fig. 14 and Fig. 15. As we can see in Fig. 14 and Fig. 15, the algorithm acclimates for the measurement noises. The measurement noise slows the convergence and consensus of the cells’ estimation. Nevertheless, the convergence and consensus are still guaranteed.

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ˆ i (t ), Rˆ xi (t ), Rˆ yi (t ) and Rˆ zi (t ) are the estimates of M, R x , where M R y and R z respectively by cell i. The standard deviations denoted as σz (t ) are

  n

2  1  ˆ i (t ) − M ¯ (t ) , σM (t ) =  M n−1 i =1

  n

2  1  σx (t ) =  Rˆ xi (t ) − R¯ x (t ) , n−1 i =1

σM (t ), σx (t ), σ y (t ) and

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In practice, the data interface might disconnect due to all kinds of reasons. On the basis of the topology shown in Fig. 10, set the disconnection probabilities pd of the arbitrary pair of cells. The mean values of all the cells’ estimates are shown in Fig. 16 and the standard deviations of the cells’ estimates are shown in Fig. 17. As shown in Fig. 16 and Fig. 17, the algorithm acclimates for the topology switching. The higher the probability of the link disconnection, the slower the convergence speed and consensus speed but the influence is negligible. Constant connectivity is not prerequisite for the convergence and consensus.

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6.3. System topology

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This part focuses on the influence of the system topology on the convergence and consensus. All the parameters keep the same as

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Fig. 13. Identification results of the inertia tensor.

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in the section 5 except the system topology. As the huge number of the possible topologies, a few typical topologies are considered in this part, which are star topology (ST), chain topology (CT) and strong connected topology (SC), as a comparison, the random topology (RT) generated in section 5 is also considered. The results in 10 s are shown in Fig. 18 and Fig. 19, the cells’ estimation converges and consents at the fastest speed among all the topologies above-mentioned. And the chain topology is quite the contrary. In Fig. 18, the identification suffers conspicuous chattering respect to the star topology, this phenomenon is analyzed in the next part.

52 53

6.4. Interaction strategy

Fig. 17. Standard deviation of the cells’ estimates with different disconnect possibilities.

sequently, the estimations are mainly based on different origins for adjacent steps, and that leads to the chattering of the central node’s estimation. The other cells’ estimation chatters for the same reason. To verify the analysis above, we can plot the estimated values separately by the odd and even numbered steps as in Fig. 22. In this figure Line CO and line CE represent the estimated values of the central node in odd numbered steps and even numbered steps respectively. Line PO and line PE represent the estimated values of the peripheral nodes in odd numbered steps and even numbered steps respectively. We can tell that line CO coincides well with line PE and line CE coincides well with line PO. That phenomenon indicates that our analysis above is reasonable.

54 55 56 57 58 59 60 61 62 63 64 65 66

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

In section 4, we provide two interaction strategies for the interactive parameter identification, as expressed in (17) and (18). Keep the parameters set in section 5 unchanged except for the topology and interaction strategy. The synchronous strategy and asynchronous strategy are compared in this part. Set the system topology as the star topology, the simulation results in 10 s are compared in Fig. 20 and Fig. 21. It is interesting that the chattering disappear while using asynchronous interaction. Assuming that the central node of the topology is the sensor cell C ss , according to (16) and (17), the estimation θˆ s (t ) is a minor correction based on the mean value of estimation θˆ j (t − 1) , j = s, and θˆ j (t − 1) , j = s is a minor correction based on θˆ s (t − 2). Con-

7. Conclusion

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In this paper, the concept of the cellular space robot for both spacecraft construction and space operation is presented. And system constitution and the design principles of the cellular space robot is introduced. To study the typical application of the cellular space robot, the dynamic model of the cellular space robot in takeover control is deduced. After that, the regression models for mass identification and inertial tensor identification are also deduced respectively. And the interactive model identification method for the distributed system like cellular space robot is presented. To valid the interactive model identification method, simu-

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Fig. 20. Mean value of the cells’ estimates with different interactive strategies.

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lations for mass, centroid position and inertial tensor identification are conducted and the result shows that the method presented can guarantee the consensus and convergence of the identification. The analysis of the method shows that: (1) In the case of minor noise, asynchronous interaction, low probability of link disconnection, the identification results converge faster; (2) The algorithm can effectively endure to the noise and link disconnection, and can ensure the convergence and consensus of the identification; (3) In the star topology, synchronous interaction might cause chattering of the identification results, using asynchronous interaction can eliminate it. Besides that, the implementation difficulties of the asynchronous interaction strategy is lower because time synchronization and waiting are non-essential. Hence, asynchronous interaction strategy is more practical for cellular space robot.

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Fig. 22. Comparison of the estimates in odd and even numbered steps.

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Conflict of interest statement

2 3

None declared.

4 5

Acknowledgements

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