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A review on kinematic analysis and dynamic stable control of space flexible manipulators

  • Zhongliang JingEmail author
  • Qimin XuEmail author
  • Jianzhe Huang
Review
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Abstract

A review on state of the art of kinematic analysis and dynamic stable control of space flexible manipulators (SFMs) is presented. Specially, SFM as a significant assembled part of autonomous space robotics (ASRs) play an important role in precision-positioning and accurateness-controlling for space engineering application since this lightweight structure possesses a high-efficient payload-to-arm weight ratio. Further, the existing studies of kinematic analysis and dynamic stable control of SFMs are critically examined to ascertain the trends of research and to identify unsolved problems through comparing with different methods. Motivated by the current research results of the two aspects, some suggestions for future research are given concisely in our published literature: (1) a fast eliminate solution algorithm of forward kinematics is presented. (2) Two observer-based control methods are suggested after dynamic modeling of SFMs. (3) How to choose a suitable closed-loop strategy to describe system dynamic features is discussed in a comparison study of the two proposed observer-based control methods.

Keywords

Space flexible manipulators Variable geometry truss Kinematic analysis Dynamic stable control 

1 Introduction

With the sustaining exploration of extravehicular activities and the deepening research of space systems, more and more space missions need to be finished, such as spacecraft capturing and servicing in orbit, repairing disabled satellites, removing and reusing of space debris, and so on. Taking harshness and complexity of space environment into account, space flexible manipulators (SFMs) have important significance in space industrial applications [22, 39, 49, 52, 53, 54, 64, 66, 75, 76].

Generally, different changes of configurations have different flexibilities of the SFMs; space flexible manipulation and control is of special interest in space robotic systems and vehicles [39, 66, 75, 76]. Therefore, such a class of ASRs plays a significant role for long-distance operation in on-orbit servicing (OOS), especially in some special space tasks with unstructured environments [54, 64] since flexible characteristic deformation of SFMs in three-dimensional coordinate system can be freely reached. Therefore, SFM as an indispensable significant tool of autonomous OOS technologies becomes a more challenging [49, 53] and promising project, including the space station remote manipulator system (SSRMS) in Canada by Sallaberger et al. [64] and orbital express by defense advanced research projects agency (DARPA) by Friend [22]. Jing et al. [39] reported three main specific attributes of soft robots to account for SFMs had excellent prospects in practical space applications.

On the other hand, conventional rigidity manipulators are often established to be heavy and bulky for high structural stiffness [66, 71]. Some inherent drawbacks of the structures are difficult to overcome, such as actuators with high capacity, high power consumption, and low payload ratio, even if they can be easily controlled. Therefore, some new flexible manipulators with a high-efficient payload-to-arm weight ratio are preferred to be selected [39, 49, 53, 54, 66, 75, 76]. However, some drawbacks and new problems of flexible structures may be yielded. Wang et al. [71] pointed out that some new problems from flexibility of the structures were found such as a high degree of elastic vibration especially during the high-velocity maneuver of the manipulators, and joint friction resulted in a very complicated dynamics especially during low-velocity operation of the manipulators. Further, the dynamic equations of motion are nonlinear and of large dimensions. Ultimately, these problems will increase the difficulty of manipulation and control of SFMs.

To the best of the authors’ knowledge, there have been few papers to study kinematic analysis and dynamic stable control problems of space flexible manipulators, which are still open. Motivated by the works of kinematic analysis of SFMs (Stewart platform manipulator by Dasgupta and Mruthyunjaya [17], planar snake-like robot mechanism by Prada et al. [58]), and dynamic stable control of SFMs (control and sensor system by Kiang et al. [42], Cartesian control by Aghili [1]), the current research is an extensive review on kinematic analysis and dynamic stable control for SFMs is investigated.

2 Kinematic analysis of space flexible manipulators

As a typical kind of space structures, space flexible manipulators considered as a class of robotic arms play an important role in many industrial areas. Since most of the robotic arms were inspired by biological systems and exhibit optimal flexibility, the biological snake’s movement can be used for activities such as dimensionally confined space [56, 67], people after earthquakes [41], and pipe systems [40, 80] as shown in Fig. 1.
Fig. 1

Snake-like robots by Prada et al. [58]

(@ Copyright Science and Education Publishing)

On the framework of screw theory, a state-of-the-art review of the literature on the Stewart platform by Dasgupta and Mruthyunjaya [17] was investigated, and the authors reported that one of the salient features of theory of screws was the enunciation of the duality and reciprocity between instantaneous kinematics and statics. Thus, it means that different Stewart platforms of SFMs may be maintained with the same equivalent kinematic mechanism by Hesselroth and Hennessey [28]. For example, a cable-driven robot by Carricato [10] was considered as one class of the flexible robotics with a special structure of Stewart platform. Similarly, screw-drive mechanism by Fukushima et al. [24] was applied to one kind of snake-like robot by Prada et al. [58] as shown in Fig. 2.
Fig. 2

Snake-like robot using screw-drive mechanism by Fukushima et al. [24]

(@ Copyright 2012 IEEE)

Thus, the selection of biological systems is always preferred to some flexible structures in the requirements of a certain task function for OOS. With the help of SFMs, a demonstration of positioning and checking mission for the OOS is shown in Fig. 3.
Fig. 3

Positioning and checking mission of SFMs for the OOS by Debus and Dougherty [18]

(@ Copyright 2009 AIAA)

2.1 Different kinematic analysis methods

Whatsoever, kinematic analysis of flexible manipulators (FMs) is the first step in studying system performance. In recent years, many scholars devoted to deriving the kinematic model by the use of direct topology geometry relationships of flexible manipulators, such as space orthogonal property by Jafari and McInroy [37] in different planes, or an equivalent isomorphic kinematic model [48, 83, 84]. So many kinematic analysis approaches have been attracted, such as direct kinematic analysis (DKA) [13, 35, 36], algebraic geometry analysis (AGA) [87, 88], and general natural coordinated analysis (GNCA) [19, 20].

2.1.1 Direct kinematic analysis

In different hybrid kinematic mechanisms, the research of the position kinematic problem is still challenged. Innocenti and Parenti-Catelli reported direct position analysis (DPA) for the Stewart platform mechanism by Innocenti and Parenti-Catelli [35], 6–4 fully parallel mechanisms by Innocenti [36]. With the help of algebraic geometry, Zsombor-Murray and Hyder [87] analyzed the mobility of a double equilateral tetrahedral mechanism, and they suggested a unified kinematic approach based on point transformation by Zsombor-Murray and Gfrerrer [88]. Such a unified approach involves some advantages such as reducing the complexity of solution and keeping good dynamic performance. More details are presented as follows. If p and q are homogeneous point coordinate vectors with respect to a point P and its image Q under \( \beta \), the general Euclidean displacement \( \beta \) in 3-space can be described:
$$ \varvec{q} = {\mathbf{Mp}}, $$
(1)
where M is a 4 \( \times \) 4 matrix:
$$ \varvec{M} = \left[ {\begin{array}{*{20}c} {t_{0} } & 0 & 0 & 0 \\ {t_{1} } & {a_{1} } & {a_{2} } & {a_{3} } \\ {t_{2} } & {b_{1} } & {b_{2} } & {b_{3} } \\ {t_{3} } & {c_{1} } & {c_{2} } & {c_{3} } \\ \end{array} } \right], $$
(2)
where ai, bi, ci, ti are study parameters for i = 1,2,3 seen in Zsombor-Murray and Gfrerrer [88]. Another interesting result of a unified kinematic approach was given by Ding et al. [20], they attempted to construct a unified topological representation model by Ding and Huang [19] based on the characteristics of topological graphs of planar kinematic chains.

2.1.2 Geometry algebra analysis

It is not easy to solve inverse kinematics [59, 60] of flexible manipulators when mathematical statements of forward kinematic equations are complicated, highly nonlinear, coupled and multiple solutions. However, the problem of inverse kinematics can be dealt with well since several extensive related investigations of kinematics were discussed using several geometry algebra (GA) approaches [87, 88]. For example, Fu et al. [23] applied theory of GA to kinematic modeling of 6R robot manipulators, so a group of closed-form kinematic equations was generated to reformulate the generalized eigenvalue problem [4]. The merit of GA approach is to have a universal meaningfulness on geometric intuition, computation and real time for all serial robot manipulators. Some details of kinematic transformation matrix are presented, using the Euler representation for a rotation \( \theta \):
$$ R = \exp \left( { - \frac{\theta }{2}\pi } \right) = \cos \left( {\frac{\theta }{2}} \right) - \pi \sin \left( {\frac{\theta }{2}} \right). $$
(3)
Baroon and Ravani [5] used line geometry to develop a three-dimensional generalized approach [51] based on the classical Reuleaux method by Eberharter and Ravani [21]. In this theory, direction moment of the screw can be determined by utilizing the common perpendicular relation of intersect lines, so moment of the screw is transformed into a Lagrangian conditional minimum problem:
$$ \mathop {\hbox{min} }\limits_{{\bar{s}}} E_{{\bar{s}}} = \mathop {\hbox{min} }\limits_{{\bar{s}}} \left\{ {\left( {A\bar{s}^{T} + Bs^{T} } \right) \cdot \left( {A\bar{s}^{T} + Bs^{T} } \right) - \lambda^{0} \left( {\bar{s}s^{T} } \right)} \right\}, $$
(4)
where \( \bar{s} \) is the moment vector, \( \lambda^{0} \) is a Lagrangian multiplier, A is a series of direction vectors with respect to screw axis s, B is an adjoint matrix with respect to A in Pl\( {\ddot{\text{u}}} \)cker coordinates S = (s,\( \bar{s} \)).

With the mapping mechanism of distance geometry, Rojas and Thomas [63] used point geometry (PG) theory to trace coupler curves of pin-jointed linkages well. Further, a general GA approach for geometric error modeling of lower mobility parallel manipulators (LMPMs) was investigated by Lin et al. [52]. The main advantage of this approach is it is not only that some desired poses can be separated by dealing with the error of different signal sources but also several prior conditions can be obtained by providing for suitable measurements.

2.1.3 General natural coordinated analysis

Through making full use of modularized decoupling relations of feedback controllers, the manipulation and control capability at the end-effector of the structure was presented by switching different sub-models of kinematics in Cartesian coordinates Kim and Kim [44]. The merit of this method is to minimize structural errors and to weaken negative effect of measurement errors by means of generalized inverse of kinematics.

However, the above methods of kinematic analysis are often referred to coordinate transformation, so the occurrence of transcendental functions in velocity and acceleration analyses is not easy to be avoided. Motivated by the works of Waldron and Hunt [70], Zhao et al. [85] utilized tetrahedron coordinate method to discuss kinematic mechanism of spatial parallel manipulators with four non-coplanar points’ Cartesian coordinates. The rotational matrix is given as
$$ \varvec{R} = \left[ {\begin{array}{*{20}c} {c\psi c\phi - c\theta s\phi s\psi } & { - s\psi c\phi - \text{c} \theta s\phi c\psi } & {s\theta s\phi } \\ {c\psi s\phi + c\theta c\phi s\psi } & { - s\psi s\phi + c\theta c\phi c\psi } & { - s\theta c\phi } \\ {s\psi s\theta } & {c\psi s\theta } & {c\theta } \\ \end{array} } \right], $$
(5)
where s and c denote sine and cosine, respectively, for the Euler angles (\( \varphi ,\theta ,\psi \)) that describe the orientation of the manipulator. The merit of this approach is that: (1) extra new variables are not introduced as the increasing of investigated points; (2) the inherent intrinsic characteristics of natural coordinate (NC) method are not lost.

2.1.4 Finite element modular analysis

Recently, Aguirrebeitia et al. [2] proposed eigensensitivity analysis methodologies by comparing with similar research [7, 8] for VGT redundant multi-body systems. The reason is that a complete finite element (FE) model of a VGT manipulator can be substituted by groups of fewer equivalent parametric macroelements (EPMs):
$$ \left\{ {U\left( {\left[ \delta \right]} \right)} \right\} = \left\{ {V\left( {\left[ \delta \right],\left\{ A \right\}} \right)} \right\}, $$
(6)
where all the submodel elastic energies are stored in the vector {U}, all the EPM elastic energies are stored in the vector {V}, and the parameters are stored in vector {A}. Then the related EPM parameters in an equivalent mechanism can be optimized using nonlinear least square principles. More discussion of the parameter estimation is given in Bai et al. [3], and Chen and Jackson [14].

2.1.5 Structural geometry analysis

  1. 1.

    Symmetrical geometry analysis

     
As a matter of fact, the topology structure for fundamental units of many SFMs in design is symmetrical since some symmetry (or mirror) technologies can be straightly applied to get space vector representation of kinematics. Based on Denavit–Hartenberg transformation (DHT) method, Williams II [72] reported the kinematic solution process of an adaptive structure with the structural symmetry using the constraints between flexible-link length (Li, i = 1,2,3) and separating points (BQi, i = 1,2,3) of the structure:
$$ \left\{ {\begin{array}{*{20}c} {L_{1}^{2} = \overrightarrow {{Q_{1} Q_{2} }} = \left\| {{}^{B}Q_{2} - {}^{B}Q_{1} } \right\|^{2} } \\ {L_{2}^{2} = \overrightarrow {{Q_{2} Q_{3} }} = \left\| {{}^{B}Q_{3} - {}^{B}Q_{2} } \right\|^{2} } \\ {L_{3}^{2} = \overrightarrow {{Q_{3} Q_{1} }} = \left\| {{}^{B}Q_{1} - {}^{B}Q_{3} } \right\|^{2} } \\ \end{array} } \right.. $$
(7)
As for one class of symmetrical 6–6 Stewart platforms, a concise algebraic elimination algorithm of the closed-form forward kinematics was presented by solving a univariate polynomial equation with a relatively small size by Huang et al. [33]. If the position vector P between the two origin points O1 and O2, and the transformation matrix R in two different coordinate systems (Ai, Bi, for i = 1,…,6) are given, the leg vectors (Li, i = 1,…,6) satisfied
$$ L_{i} = RB_{i} + P - A_{i} . $$
(8)
But with the increase of rotational degrees of freedom, the kinematic analysis of flexible manipulators is not feasible since the corresponding Jacobian matrix cannot be obtained. To overcome this drawback, some reasonable suggestions were proposed by Zhao et al. [85]. Further, motion control of free-floating system comprised by multiple VGTs was studied thoroughly by taking into account two aspects of kinematic mechanisms: forward kinematics (FKs) and inverse kinematics (IKs) as in Huang et al. [31, 32]. Based on conservation of momentum without external force, the geometrical definition of the center of mass of flexible structure is given by
$$ M_{G} p_{G} = \sum\limits_{i = 1}^{N} {\left( {M_{p}^{i} p_{i} + M_{a}^{i} p_{i,a} } \right)} + M_{p}^{0} p_{0} , $$
(9)
where pi,a is the center-of-mass vector of the active plane in the ith bay, and the masses of a passive plane and an active plane are M p i and M a i for i = 1,…, N, respectively.
  1. 2.

    Asymmetrical geometry analysis

     

Compared with a similar symmetrical structure, an asymmetrical topology structure of space flexible manipulators possesses larger workspace [79], and more changeable configurations, so many kinematic analysis approaches of asymmetrical structures have been studied extensively [26, 38, 45, 46, 47, 50, 57, 77]. A lot of contributions to forward displacement analysis (FDA) of different asymmetrical parallel manipulators were made, such as a quadratic 3T1R PM by Kong and Gosselin [46], third-class analytic 3-RPR PM by Kong and Gosselin [45], and a special 2 degrees of freedom (2-DOF) 5R spherical PM by Kong and Gosselin [47].

In more recent times, Gallardo-Alvarado et al. [26] reported kinematics of one class of asymmetrical manipulators by means of screw theory; if a set of generalized coordinates are given, the expression of kinematic equations is represented in echelon form:
$$ \left\{ {\begin{array}{*{20}l} {\left( {S_{3} - S_{1} } \right) \cdot \left( {S_{3} - S_{1} } \right) - S_{13}^{2} = 0} \hfill \\ {\left( {S_{3} - S_{2} } \right) \cdot \left( {S_{3} - S_{2} } \right) - S_{23}^{2} = 0} \hfill \\ {\left( {S_{3} - B_{3} } \right) \cdot \left( {S_{3} - B_{3} } \right) - q_{3}^{2} = 0} \hfill \\ \end{array} } \right., $$
(10)
where qi are generalized coordinates, Si are coordinates of the points for i = 1,2,3. Then a good solution of the FDA is found. B3 is the coordinate of the points attached at the moving platform.

2.2 The existing problems and some suggestions on kinematic analysis

2.2.1 The existing problems on kinematic analysis

Throughout the mentioned methods of kinematic analysis, comparison studies among different kinematic analysis methods are shown in Table 1.
Table 1

Comparison analysis among different kinematic analysis methods

Analysis methods

Advantages

Disadvantages

Applications

DKA

Equivalent closed-form solution

Analysis difficultly, transcendental equations

Velocity and acceleration analyses

GAA

Did not induce new variables

Exaggerated computation consumptions

Kinematics and dynamics analyses

GNCA

Unified description

Limited on GA theory

Suitable for any system in theory

FEA

Eigensensitivity analysis

Decreased precision of modules

Space bionic structures

SGA

Symmetry, simplified solution

Applied DH transformation

Space adaptive system

Asymmetry, larger than workspace

Limited on PG theory and GA theory

Suitable for dimensionally confined space

It is seen from Table 1 that kinematic mechanisms of different SFMs are able to be found in different kinematic analysis methods. For instance, Lee and Bejczy [50] reported an inverse solution method using a group of joint parameter equations, but this method cannot be directly applied to a limited joint angle case. Usually, kinematic analysis was focused on some characteristics, such as position, direction, velocity, acceleration, and the corresponding high-order derivative form, so some potential problems of kinematics are still open.
  1. 1.

    The solution process of kinematics is more complicated since kinematic analytic process of space flexible structures is difficult;

     
  2. 2.

    the analytic closed-loop solutions of transcendental equations are not easy to obtain since the computational redundant problem of kinematics is challenged.

     

2.2.2 Some suggestions on kinematic analysis

Here, the prototype of a variable geometry structure of SFMs is designed in Fig. 4. In fact, the kind of structure is equivalent to a class of double Stewart platforms by Hesselroth and Hennessey [28].
Fig. 4

SFMs comprised of asymmetrical variable geometry structures by Xu et al. [77] (@ Copyright 2017 SAGE) with a asymmetrical VGT platform and b geometry analysis of A-VGT platform

Inspired by these works of Williams II [72] and Gallardo-Alvarado et al. [26], it is seen from Fig. 4b that the structure geometry relationship for an asymmetrical VGT platform (A-VGTP) is analyzed by constructing some geometry auxiliaries. Also, using DH parameter transformation by Jain and Kramer [38] of coordinates in Cartesian space, the target position, position–direction and motion configurations can be formulated within a vector-parameter expression on a measurable vector space. See a similar report in Page et al. [57]; space coordinates of the target vector can be expressed as a group of equations:

$$ \left\{ {\begin{array}{*{20}l} {{\text{X}} = \frac{{A_{2} + B_{2} + C_{2} }}{3}} \hfill \\ {{\text{n}} = \frac{{\bar{n}}}{{\left\| {\bar{n}} \right\|}}} \hfill \\ \end{array} } \right., $$
(11)
where A2, B2, and C2 are three vertexes for the mobile platform. \( \bar{n} \) is the orientation vector. X and n are the position center vector and the orientation vector of SFMs.
As for the mentioned solution problem of kinematic analysis process, we find transcendental equations of A-VGTP are not easy to solve, but a fast eliminate solution process of forward kinematics of A-VGTP in our published paper by Xu et al.[77] is shown in Fig. 5.
Fig. 5

The fast eliminate solution analysis of forward kinematics

Therefore, the main contributions of our proposed approach are incarnated in two aspects:
  1. 1.

    The key transcendental equations are successfully formulated into compact polynomial ones using a group of parameter transformation.

     
  2. 2.

    Compared with the screw approach by Gallardo-Alvarado et al. [26] with the complexity of \( O\left( {11 \times n^{5} } \right) \), our proposed approach with the complexity is \( O\left( {8 \times n^{4} } \right) \), so the computational efficiency of our approach is increased by 16.27%.

     

2.3 The trends of research on kinematic analysis

Some potential trends of research on kinematic analysis are summarized as follows:
  1. 1.

    Adaptive characteristics of structure addressed by Miura and Furuya [55], which can make arbitrary configurations in three-dimensional plane. To describe structure flexible deformation by means of adaptive characteristics, some structure characteristics of SFMs are adopted such as co-existing in stiffness and flexibility, modularity and mobility. To solve structure redundant problem of a flexible multi-body system, FEA method is an effective strategy to find groups of fewer equivalent parametric macroelements.

     
  2. 2.

    To satisfy flexibility or stiffness of SFMs during executing space tasks for OOS, several new theories of kinematics will be developed, such as point or line geometry theory and algebra geometry theory. For example, screws of theory by Hong and Choi [29] or Grassmann–Cayley algebra (GCA) by Ben-Horin and Shoham [6] can be used to analyze kinematics of different equivalent Stewart platforms. Although kinematics of an n-DOF (n is sufficiently large) flexible structure is not easy to solve straightly, equivalent mechanism of kinematics can be derived by constructing combination motion of several of geometry groups by means of geometry algebra theory. In additional, some geometry methods based on coordinate transformation in Cartesian coordinated system are applied extensively.

     
  3. 3.

    To find more kinematic extensive performance, the application of kinematics of SFMs will be broadened, such as Jacobian analysis [34], kinematic singularity, optimal geometry planning. For examples, Gosselin and Angeles [27] used an input–output mapping of velocity transmission to study singularities of a Gough–Stewart Platform (GSP). Zlatanov et al. [86] extended this approach to arbitrary non-redundant mechanisms.

     
According to the mentioned trends of research, we will find kinematic analysis.
  1. 1.

    It lays the foundation of dynamic stable control.

     
  2. 2.

    It offers a powerful support of space assignment for the long-distance effect, even in hostile surroundings by Takayama and Hirose [67] or in dimensionally confined space by Kakogawa and Ma [40] for the OOS.

     

3 Dynamic stable control of space flexible manipulators

To the best of the authors’ knowledge, the main purpose of dynamic feature analysis of SFMs is to delineate the corresponding deformation of flexible systems or to design system stable controllers. Generally, two representative methods with description of deformation are as follows:
  1. 1.

    Characteristic representative: a series of tests (such as bending moment test or tension test) are implemented to detect dynamic characteristics of SFMs;

     
  2. 2.

    model-based representative: an equivalent rigid-body dynamic model, such as a descending dimension method by Zhang et al. [83] was considered to make the whole flexible system translated into a combination of slow time-varying sub-system and fast time-varying sub-system. But the application of this approach has a limitation of variable payloads of the end-effector.

     

Thus, if an equivalent stiffness of dynamics is found, the description of system dynamic features can be expressed analytically.

3.1 Modeling of dynamics

To avoid computational efficiency problems of flexibility–rigidity-coupled systems in a finite element approach, Hu and Hong [30] borrowed the Hamilton principle to establish the key dynamic equations by giving a complete quantitative relation with respect to an equivalent stiffness of dynamics:
$$ \frac{\text{d}}{{{\text{d}}t}}\left( {\frac{{\partial \varvec{T}_{SM} }}{{\partial \dot{\varvec{q}}}}} \right) - \frac{{\partial \varvec{T}_{SM} }}{{\partial \varvec{q}}} + \frac{{\partial \varvec{U}_{SM} }}{{\partial \varvec{q}}} = \varvec{Q}_{SM} , $$
(12)
where TSM is the kinetic energy of the single flexible-link manipulator with a moving base (SFLMB), USM is the potential energy of the SFLMB, and QSM is the generalized force vector corresponding to a generalized coordinate q.
To derive the closed-form dynamic equations of a SFM with velocity constraints of the end-effector, the natural orthogonal basis of the Jacobian matrix in a linear homogeneous form is obtained using Lagrange approach in an independent generalized coordinate. Considering the physical meanings of nonlinear Euler–Bernoulli equation (Eq. 12), a dynamic equation based on a discretized spring-mass-damper model by Simo and Quoc [65] is established:
$$ \varvec{M}\left( \varvec{q} \right)\varvec{\ddot{q}} + \varvec{F}\left( {\varvec{q},\dot{\varvec{q}}} \right) + \varvec{Kq} = \varvec{B}\tau , $$
(13)
where M is the mass matrix, F(q,\( \bar{\varvec{q}} \)) represents the summation of the Coriolis, centrifugal and gravity forces, K is the stiffness matrix, B is a constant matrix, and \( \tau \) is a combination of the torques of the link’s actuator.
However, different types of flexible constraints may be with respect to different configurations; Vogtmann et al. [69] presented a characteristic representative of the elastic joints to increase control precision of dynamic error model of flexible systems. Briot and Khalil [9] used a symbol regression law to investigate a dynamic modeling approach satisfying Newton–Euler principle and then the global elastodynamic model (GEM) of the virtual structure is addressed in the following form:
$$ \left[ {\begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {\varvec{\tau}_{t} } \\ {0_{{n_{e} }} } \\ \end{array} } \right]} \\ {\varvec{f}_{p} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\sum\nolimits_{i,j} {\varvec{J}_{ij}^{T} \varvec{M}_{ij} \varvec{J}_{ij} } } & {\varvec{M}_{p} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{\ddot{q}}_{t} } \\ {\dot{\varvec{t}}_{p} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\varvec{c}_{ij}^{s} } \\ {\varvec{c}_{p} } \\ \end{array} } \right] = \varvec{M}_{t} \left[ {\begin{array}{*{20}c} {\varvec{\ddot{q}}_{t} } \\ {\dot{\varvec{t}}_{p} } \\ \end{array} } \right] + \varvec{c}_{t} , $$
(14)
where \( \varvec{\tau}_{\varvec{t}} \) is the vector of the tree-structure input efforts, fp is the platform reaction wrench, Jij is the Jacobian matrix, Mij is the global Jacobian matrix, Mp is the platform mass matrix, qt is an assembled vector of passive and elastic variables, tp is the platform velocity screw, and the other variables are defined in Briot and Khalil [9]. It implies that the distributed flexibility can be extended by taking effort to reduce the number of the symbol regression operator.
Model-based method in Rigatos [61] was robust for all unknown parameters of dynamics. Applying nonlinear Euler–Bernoulli equation (Eq. 12), a group of dynamic equations is expressed:
$$ \left\{ {\begin{array}{*{20}l} {\varvec{M}_{11} \ddot{\theta } + \varvec{M}_{12} \varvec{\ddot{v}} + \varvec{F}_{1} \left( {z,\dot{z}} \right) = \varvec{T}\left( t \right)} \hfill \\ {\varvec{M}_{21} \ddot{\theta } + \varvec{M}_{22} \varvec{\ddot{v}} + \varvec{F}_{2} \left( {z,\dot{z}} \right) + \varvec{D\dot{v}} + \varvec{Kv} = 0} \hfill \\ \end{array} } \right., $$
(15)
where unknown parameters defined in Eq. (15) are similar as that of Eq. (13). If energy-based control of flexible-link robots is considered, the torque of the ith motor (control output) can be given by choosing a suitable PD-type controller. But the whole system performance could not be evaluated accurately due to nonlinear multiplier effects. On the other hand, some negative effects of system performance in free-model-based method were avoided, but system inverse dynamics could be out of control due to inaccurate or non-equivalent description of flexibility.

3.2 Control strategies of dynamic systems

Due to additive effects of flexible systems assembled by multiple adaptive model cells, the precision of system control is difficult to be guaranteed. To gain a good robustness of a flexible system, Rodriguez et al. [62] adopted a linearization feedback control strategy to make flexible joints be equivalent to a linear rotating spring, but system control precision was still not guaranteed. To overcome these drawbacks, some main control approaches of flexible systems have been proposed, such as linear feedback control (LFC) [11, 25, 43], nonlinear control (NC) [12, 15, 74], and adaptive control (AC) [16, 68].

3.2.1 Linear feedback control method

To reduce vibration of a micro-damping flexible structure, a linear input feedback control (LIFC) method was reported by Kim et al. [43] in a comparison study of negative and positive position feedback (PPF) control. If the output-dependent integral sliding surface without any observers and extra-perturbations is
$$ \left\{ {\begin{array}{*{20}l} {\dot{x}\left( t \right) = \left( {A + DG\left( t \right)H} \right)x\left( t \right) + B\left( {u\left( t \right) + f\left( {x,u,t} \right)} \right) + Ed\left( t \right)} \hfill \\ {y\left( t \right) = Cx\left( t \right)} \hfill \\ \end{array} } \right., $$
(16)
where \( x\left( t \right) \) is the state, \( u\left( t \right) \) is the control input, \( y\left( t \right) \) is the output, \( f\left( {x\left( t \right)} \right) \) are nonlinear perturbations, d(t) is extra disturbances, system matrices (A, B, C, D, H, and E) are known with \( G(t)G(t) \le I \). To eliminate perturbations and nonlinear disturbances, Chang [11] used a dynamic output feedback sliding mode control (DOF-SMC) strategy by Gadewadikar et al. [25], taking advantage of SMC:
$$ s\left( t \right) = \left( {CB} \right)^{ + } y\left( t \right) - \int_{0}^{t} {v\left( \tau \right)d\tau } , $$
(17)
where (CB)+ is a generalized inverse of CB, and the dynamic output feedback controller is \( v\left( t \right) = - Fy\left( t \right). \)

3.2.2 Nonlinear control method

NC method includes several different types, such as shape control, variable structure control, and SMC. Different nonlinear control strategies may be adopted for different control conditions of nonlinear systems; SMC method, as one representative of NC methods, whose physical meaning is defined as a dynamic behavior that non-continuous system signals are forced to reach an equivalent stable sliding mode surface.

However, a single SMC method has been used rarely due to the complexity of engineering applications. If a nonlinear control affine system with n-generalized coordinates and m input variables was considered,
$$ \ddot{q} = f\left( {q,\dot{q},t} \right) + B\left( {q,t} \right)u\left( t \right). $$
(18)
An adaptive SMC method was given by Zeinali and Notash [82],
$$ u\left( t \right) = \hat{B}^{ - 1} \left( {\ddot{q}_{r} - \hat{f} - K_{b} \text{sgn} \left( S \right)} \right), $$
(19)
to eliminate structure vibration of robot manipulators without prior knowledge, where q is the vector of generalized coordinates, the vector u(t) is the control input, and f(q,\( \dot{q} \),t) is a bounded nonlinear vector function that represents the nonlinear deriving terms. B(q,t) is an invertible and bounded positive definite nonlinear function. Kb is the bounds of uncertainty vector. Chiu [15] introduced a labeled symbol and decomposition technology to force system state trajectory to be tracked consistently to an equivalent stable TSMC surface during a finite time by implanting a differentiation–integration (DI) controller into a sign integral terminal sliding function:
$$ s\left( t \right) = e\left( t \right) + \alpha \int_{0}^{t} {{\text{sign}}\left( {e\left( \tau \right)} \right){\text{d}}\tau } ,s\left( 0 \right) = 0^{\prime\prime}, $$
(20)
where \( \alpha \) > 0, and e(t) is defined as the tracking error.

Thus, a conclusion is given that convergence velocity of DI-TSMC method is faster than traditional terminal sliding mode control (TSMC) method. Since system error was defined as a residual signal that can be mapped into a matrix null-space with linear measurement distribution, Wu and Shi [74] suggested a sub-optimal adaptive variable structure state estimation (AVS-SE) method to ensure all system errors maintain an ultimate terminal boundary region.

3.2.3 Adaptive control method

At special space surroundings, inertia of payloads of space flexible manipulators is often beyond inertia of themselves. It implies that manipulation and control of SFMs is very difficult since the control precision of position tracking of the end-effectors may be reduced, especially for position drifting or unknown disturbances from the changes of payloads of end-effectors. Damaren [16] introduced the concept of passivity to make dynamic systems of SFMs with the payload changes of the end-effectors be global stable by adjusting an adaptive control law. Ulrich et al. [68] gave a definition of almost strict passivity to describe decentralized modified simple adaptive control (DMSAC) techniques:

$$ u = K_{e} \left( t \right)e_{y} + K_{x} \left( t \right)x_{m} + K_{u} \left( t \right)u_{m} , $$
(21)
for dynamic equations:
$$ \left\{ {\begin{array}{*{20}l} {M\left( q \right)\ddot{q} + C\left( {q,\dot{q}} \right)\dot{q} - k\left( {q_{m} - q} \right) = 0} \hfill \\ {J_{m} \ddot{q}_{m} + k\left( {q_{m} - q} \right) = \tau } \hfill \\ \end{array} } \right., $$
(22)
where Ke is the time-varying stabilizing control gain matrix, Kx and Ku are time-varying feed-forward control gain matrices that contribute to maintaining the stability of the controlled system, and the other parameters defined in (22) are similar as Eq. (13).

However, adaptive control method is not suitable for all control systems. For instance, some overstrike problems of AC method may bring some serious outcomes for dynamic behaviors of flexible systems. In other words, velocity response is not decreased sharply, even if switch gain of system controller becomes large enough due to initial tracking error.

3.3 The existing problems and some suggestions on dynamic stable control

3.3.1 The existing problems on dynamic stable control

  1. 1.

    The existing problems on modeling of dynamics

    Here, if an equivalent mechanism of dynamic characteristics for a SFM is found, the description of system dynamic features can be expressed analytically. Thus, modeling problem of dynamics concentrates on the two following aspects:
    1. 1.

      The control variables of the manipulator are strictly less than the DOF of dynamic system.

       
    2. 2.

      Linear excitation of flexible parts cannot be separated from nonlinear excitation.

       
     
In practice, an ideal dynamic model with some dynamic feature constraints is difficult to gain since dynamic parameters of flexible systems are not easily determined.
  1. 2.

    The existing problems on dynamic stable control

    Throughout the mentioned control strategies of SFMs, comparison analysis of different control strategies is shown in Table 2.
    Table 2

    Comparison study of three control strategies

    Approaches

    Advantages

    Disadvantages

    LFC

    Non-conditional stability

    Excessive frequency, sensitive to noises

    NC

    SMC, conditional stability

    More DOF, large computation

    AC

    Adaptive

    Overstrike phenomenon

    It is seen from Table 2 that there is not an optimal strategy for all control systems with SFMs since each control strategy has certain advantages with unexpected disadvantages. In practice, some potential problems on dynamic stable control are incarnated on two aspects as follows:
    1. 1.

      Dynamic system of SFMs with coupling relation between dynamic characteristics and controller is a complicated and strongly coupled nonlinear one by Hu and Hong [30]. But the research of dynamic modeling approaches or dynamic stable control approaches for such a complicated system is relative rarely and not completed.

       
    2. 2.

      Space flexible manipulator is a complicated structure with some properties such as high DOF and MIMO; therefore, dynamic model of SFMs has uncertainties, nonlinearities, or some strongly coupled features.

       
     

Moreover, we find that a single control method has been adopted rarely. But to discover further system dynamic characteristics or to strengthen system performance, one control method together with the other control methods to generate a new assembled method is still open.

3.3.2 Some suggestions on dynamic stable control

  1. 1.

    Some suggestions on modeling of dynamics

    Since the purpose of dynamic modeling is to choose the function of forces or torques to act on its kinematic equations when some inertia of kinematics are differentiable, in our published paper by Xu et al. [78], an equivalent description of inertia variables of flexible system was given, and then a Lagrangian derivation process was proposed to describe modeling of dynamics. If an independent variable is introduced by utilizing a generalized coordinate parameter and Hamilton principle, a state-space representative of dynamic equation is obtained:

    $$ \left\{ {\begin{array}{*{20}l} {\dot{x}\left( t \right) = A\left( x \right)x\left( t \right) + B\left( x \right)u\left( t \right) + E\left( x \right)f\left( {x\left( t \right)} \right)} \hfill \\ {y\left( t \right) = C\left( x \right)x\left( t \right)} \hfill \\ \end{array} } \right., $$
    (23)
    where \( x\left( t \right) \in \Re^{n} \) is the state, \( u\left( t \right) \in \Re^{n} \) is the control input, \( y\left( t \right) \in \Re^{n} \) is the output, and \( f\left( {x\left( t \right)} \right) \in \Re^{n} \) are nonlinear perturbations due to the gravity and its strain energy, and external disturbance of the flexible manipulator from the link’s flexibility. When the position of the key joint or the actuator is detected to be measurable, system matrices (A(x), B(x), C(x), and E(x)) can be dealt with in a local state-feedback linear parameterization approach by Lee and Bejczy [50] at operation point \( x\left( t \right) \); a linearized expression of nonlinear system (23) is considered:
    $$ \left\{ {\begin{array}{*{20}l} \begin{aligned} \dot{x}\left( t \right) = \left( {A + \Delta A} \right)x\left( t \right) + \left( {B + \Delta B} \right)u\left( t \right) \hfill \\ \begin{array}{*{20}l} {} \hfill \\ \end{array} \begin{array}{*{20}l} {} \hfill \\ \end{array} \begin{array}{*{20}l} {} \hfill \\ \end{array} \begin{array}{*{20}l} {} \hfill \\ \end{array} + \left( {E + \Delta E} \right)f\left( {x\left( t \right)} \right) + Dw\left( t \right) \hfill \\ \end{aligned} \hfill \\ {y\left( t \right) = \left( {C + \Delta C} \right)x\left( t \right)} \hfill \\ \end{array} } \right., $$
    (24)
    where \( w\left( t \right) \in \Re^{r} \) denotes a Gaussian white-noise stochastic process, and uncertainty matrices \( \Delta A,\Delta B,\Delta C,\Delta E \) are measurable approximately.
    1. 2.

      Some suggestions on dynamic stable control based on observer

       

    As is well known, dynamic system of SFMs is a class of non-minimum phased one with time variation and uncertainty, which may bring about destruction seriously for the designed controller by Wu et al. [73]. To keep high velocity and good payload capacity for the end-effector of SFMs, space tasks may be much heavy. It means that the precision of kinematic displacement and the accurateness of dynamic control need to reach a higher level. Therefore, it is necessary to study dynamic stable control problems of SFMs to strengthen system performance or to improve system control precision.

    Here, two observer-based control methods will be discussed by constructing suitable closed-loop mechanisms.
    1. 1.

      To overcome reduction of the precision of flexible system (24) due to system transmission errors.

       
    2. 2.

      To maintain system performance of stability or robustness against system nonlinearity or uncertainty.

       
    3. 1.

      Some suggestions on linear state observer-based control

       
     
In our proposed algorithm by Xu et al. [78], a linear state observer-based control (LS-OBC) with rank-full by Wang et al. [71] is considered:
$$ \left\{ {\begin{array}{*{20}l} {\dot{\hat{x}}\left( t \right) = G\hat{x}\left( t \right) + Ky} \hfill \\ {u\left( t \right) = L_{r} x\left( t \right)} \hfill \\ \end{array} } \right.. $$
(25)

With a linear state-observer gain

$$ \left\{ {\begin{array}{*{20}l} {K = P_{2}^{ - 1} \left( {\varTheta^{T} R^{ - 1} + {\text{SUR}}^{{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} } \right)} \hfill \\ {{\text{G}} = \hat{A} - K\hat{C}} \hfill \\ \end{array} } \right., $$
(26)
where the unknown parameters defined in the above equation can be seen in Xu et al. [78]. As a result, asymptotical stability (AS) of nonlinear system (24) is derived using Lyapunov stability theory.
  1. 2.

    Some suggestions on nonlinear state observer-based control

     
Since all dynamic behaviors can be regarded as a combination of motion assembled by all motions of each structure fundamental cells of SFMs, some error sources may be yielded due to additive effect of structure flexibility. Therefore, dynamic responses of flexible systems remain strong nonlinear characteristics in Yan [81], and then a new control strategy is chosen by taking the place of a linear state observer that is inevitable. Here, a nonlinear state observer-based control (NS-OBC) is worthy to study further in a suitable closed-loop mechanism:
$$ \left\{ {\begin{array}{*{20}l} \begin{aligned} \dot{\hat{x}}\left( t \right) = \left( {A + \Delta A} \right)\hat{x}\left( t \right) + \left( {B + \Delta B} \right)u\left( t \right) \hfill \\ \begin{array}{*{20}l} {} \\ \end{array} \begin{array}{*{20}l} {} \\ \end{array} \begin{array}{*{20}l} {} \\ \end{array} + \left( {E + \Delta E} \right)f\left( {\hat{x}\left( t \right)} \right) + G_{ * } \left( {y\left( t \right) - \hat{y}\left( t \right)} \right) \hfill \\ \end{aligned} \hfill \\ \begin{aligned} u\left( t \right) = K_{1} x\left( t \right) \hfill \\ \hat{y}\left( t \right) = \left( {C + \Delta C} \right)\hat{x}\left( t \right) \hfill \\ \end{aligned} \hfill \\ \end{array} } \right.. $$
(27)
  1. (a)

    For a non-delay time case, a new closed-loop augment system with NS-OBC (27) is proved to be AS in the square mean, if AS algorithm is designed with nonlinear state observer gain

     
$$ \left\{ {\begin{array}{*{20}l} {K_{2} = \frac{{\kappa^{{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} V_{{a^{ * } }}^{T} U_{{a^{ * } }}^{T} - \kappa I}}{{B_{\lambda }^{T} P_{2} + P_{1} B_{\lambda } }}} \hfill \\ \begin{aligned} K_{1} = \left( {R_{{b^{ * } }}^{T} } \right)^{{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} \left( {V_{{b^{ * } }}^{T} U_{{b^{ * } }}^{T} - \beta^{{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} \varTheta_{{b^{ * } }}^{T} } \right) \hfill \\ G_{ * } = P_{2}^{ - 1} \left( {\varTheta_{{c^{ * } }}^{T} - U_{{c^{ * } }} V_{{c^{ * } }} R_{{c^{ * } }}^{{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} } \right)R_{{c^{ * } }}^{ - 1} \hfill \\ \end{aligned} \hfill \\ \end{array} } \right., $$
(28)
where unknown parameters defined in the above equation are similar as (26) seen in Xu et al. [78].
  1. (b)

    For a delay time case, once a delay factor d is introduced to the designed nonlinear state observer (27), then the original system is transformed into a new singular augment system with time delay. Further, AS of the new singular system is also derived using LMI technology.

     
Throughout the design process of two nonlinear observers, a general framework of NS-OBC for SFMs is shown in Fig. 6.
Fig. 6

General framework of NS-OBC for SFMs

3.3.3 Performance comparison study of two state OBC methods

How to select a suitable closed-loop mechanism to describe system dynamic characteristics is? A comparison study of the two proposed observer-based control methods is discussed in the following aspects.
  1. 1.

    Comparison of system control effect with two kinds of state observers

     
To evaluate system dynamic performance of SFMs, two observer-based control gains are compared with the same physical conditions as described in Table 3.
Table 3

Comparison of two observer-based control gains

Maximum eigenvalues

LS-OBC gain

NS-OBC gain

\( \lambda_{\hbox{max} } \left( G \right) \)

6.7106

6.8594

\( \lambda_{\hbox{max} } \left( K \right) \)

− 4.9616

− 1.4563

Remark 1

It is seen from Table 3 that
  1. 1.

    from the standpoint of measurement gain, the change of measurement estimation is not large in different state OBC methods;

     
  2. 2.

    from the standpoint of state-feedback gain, the change of state-feedback estimation is large in different state OBC methods.

     
Therefore, the precision of system state estimation of NS-OBC will be superior to LS-OBC case. It means that the change of state observer from LS-OBC to NS-OBC is helpful for eliminating system uncertainties, nonlinearities, and perturbations since some potential system performances (such as stability and robustness) with NS-OBC of dynamic system (24) are preformed sufficiently.
  1. 2.

    Comparison of state trajectories and H robust performance in different state observers

     
To further illustrate the results of Remark 1, the change of system state trajectories and the change of H norm in different state observers are displayed in Fig. 7, respectively.
Fig. 7

Comparison analysis of system performance of different state observers on two aspects: a state trajectories and b comparisons of H norm

Remark 2

From Fig. 7a, the fastest convergence speed is NS-OBC for a transient case. Also, the second convergence speed is LS-OBC for a transient case. The slowest convergence speed is NS-OBC for a delay case since the delay effect of dynamic behavior is due to system flexibility. From Fig. 7b, some similar conclusions with respect to Fig. 7a can be obtained by analyzing H robustness of system (24). Thus, the delay time may destroy system stability and robustness.

3.4 The trends of research on dynamic stable control

Some potential trends of research on dynamic stable control are concluded.
  1. 1.

    An efficient and equivalent description of dynamic mechanism plays a significant role in modeling of dynamics. Therefore, dynamic parameters can be obtained using several neural networks or on-line learning machine. Then a relative ideal dynamic model of SFMs may be gained if some assumptions and flexible constraints are addressed in a concise and reasonable way.

     
  2. 2.

    Synthesis of multiple conventional control methods or development of a new unconventional control method may be a good strategy. Control system of SFMs is often attributed to a class of nonlinear, uncertain, and non-minimum phased systems, so an optimal assembled control strategy is a feasible path to solve such a class of complicated systems.

     
  3. 3.

    The selection of control strategies may be limited by dynamic environment and different requirements of space tasks. In aerospace applications, the manipulation and control of SFMs is usually complex and difficult to be finished. But to study a large-scale system of SFMs, a single control method is preferred to realize one-to-one matched control for each sub-system. Also, for a complex space-task assignment, a combination of multiple control methods is preferred to excavate much more dynamic characteristics.

     

4 Conclusion

In this article, a state-of-the-art review on kinematic analysis and dynamic stable control of space flexible manipulators (SFMs) is investigated. First, some open problems in the field of kinematic analysis for SFMs are enumerated, and then one new approach of kinematic analysis is suggested in our published literature. Further, different dynamic stable control methods of SFMs are concisely reviewed. Also, two observer-based control methods on dynamic stable control are proposed in our published literature. Several challenging problems on modeling of dynamics and dynamic stable control for SFMs are enumerated. At last, some trends of research on kinematic analysis and dynamic stable control of SFMs are pointed out.

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant nos. 61673262 and 61175028) and Shanghai key project of basic research (Grant no. 16JC1401100).

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© Shanghai Jiao Tong University 2019

Authors and Affiliations

  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiChina

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