Finite-time attitude synchronization with distributed discontinuous protocols

Finite-time attitude synchronization
with distributed discontinuous protocols

Jieqiang Wei, Silun Zhang, Antonio Adaldo, Johan Thunberg, Xiaoming Hu, Karl H. Johansson *This work is supported by Knut and Alice Wallenberg Foundation, Swedish Research Council, and Swedish Foundation for Strategic Research.J. Wei, A. Adaldo and K.H. Johansson are with the ACCESS Linnaeus Centre, School of Electrical Engineering. S. Zhang and X. Hu are with School of engineering sciences. KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden. {jieqiang, silunz, adaldo, hu, kallej}@kth.se. Johan Thunberg is with Luxembourg Centre for Systems Biomedicine, Université du Luxembourg,6, avenue du Swing, L-4367 Belvaux. {johan.thunberg}@uni.lu
Abstract

The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed controllers using the vectorized signum function are proposed, which guarantee almost global and local convergence, respectively. Filippov solutions and non-smooth analysis techniques are adopted to handle the discontinuities. Sufficient conditions are provided to guarantee finite-time convergence and boundedness of the solutions. Simulation examples are provided to verify the performances of the control protocols designed in this paper.

Agents and autonomous systems, Finite-time attitude synchronization, Network Analysis and Control, Nonlinear systems

I Introduction

Originally motivated by aerospace developments in the middle of the last century [5, 15], the rigid body attitude control problem has continued to attract attention with many applications such as aircraft attitude control [2, 32], spacial grabbing technology of manipulators [21], target surveillance by unmanned vehicles [24], and camera calibration in computer vision [20]. Furthermore, the configuration space of rigid-body attitudes is the compact non-Euclidean manifold , which poses theoretical challenges for attitude control [3].

Here we review some related existing work. As attitude systems evolves on —a compact manifold without a boundary—there exists no continuous control law that achieves global asymptotic stability [6]. Hence one has to resort to some hybrid or discontinuous approaches. In [17], exponential stability is guaranteed for the tracking problem for a single attitude. The coordination of multiple attitudes is of high interest both in academic and industrial research, e.g., [11, 26, 29]. In [18] the authors considered the synchronization problem of attitudes under a leader-follower architecture. In [23], the authors provided a local result on attitude synchronization. Based on a passivity approach, [25] proposed a consensus control protocol for multiple rigid bodies with attitudes represented by modified Rodrigues parameters. In [31], the authors provided a control protocol in discrete time that achieves almost global synchronization, but it requires global knowledge of the graph topology. Although there exists no continuous control law that achieves global asymptotic stability, a methodology based on the axis-angle representation obtains almost global stability for attitude synchronization under directed and switching interconnection topologies is proposed in [30]. These control laws were later generalized to include various types of vector representations including the Rodrigues Parameters and Unit Quaternions [29]. Besides these agreement results, [16, 28, 34] provided distributed schemes for more general formation control of attitude in space .

Among all the properties of attitude synchronization schemes, the finite-time convergence is an important one, because in practice it is desired that the system reaches the target configuration within a certain time-interval; consider, for instance, satellites in space that shall face a certain direction as they move in their orbits, or cameras that shall reach a certain formation to quickly follow an object. So far, finite-time attitude control problems are studied in different settings, e.g., [12, 35]. In [12], finite-time attitude synchronization was investigated in a leader-follower architecture, namely all the followers tracking the attitude of the leader. In [35], quaternion representation was employed for finite-time attitude synchronization. Both works used continuous control protocols with high-gain.

In this paper, we shall focus on the finite-time attitude synchronization problem, based on the axis-angle representations of the rotations without a leader-follower architecture, using discontinuous control laws. Two intuitive control schemes are proposed. The first scheme employs a direction-preserving sign function to guarantee finite-time synchronization almost globally, namely, the convergence holds for almost all the initial conditions. The other scheme, motivated by binary controllers for scalar multi-agent systems, e.g., [7, 19, 9, 14], uses the component-wise sign function. Compared to the first scheme, the second one is more coarse, in the sense that only finite number of control outputs are employed, and guarantees finite-time convergence locally. Since these control schemes are discontinuous, nonsmooth analysis is employed throughout the paper.

The structure of the paper is as follows. In Section II, we review some results for the axis-angle representation for rotations in and introduce some terminologies and notations in the context of graph theory and discontinuous dynamical systems. Section III presents the problem formulation of the finite-time attitude synchronization problem. The main results of the stability analysis of the finite-time convergence are presented in Section IV, where an almost global and local stability are given in Subsection IV-A and IV-B respectively. Then, in Section V, the paper is concluded.

Notation. With and we denote the sets of negative, positive, non-negative, non-positive real numbers, respectively. The rotation group The vector space of real by skew symmetric matrices is denoted as . denotes the -norm and the -norm is denoted simply as without a subscript.

Ii Preliminaries

In this section, we briefly review some essentials about rigid body attitudes [27], graph theory [4], and give some definitions for Filippov solutions [13].

Next lemma follows from Euler’s Rotation Theorem.

Lemma 1.

The exponential map

(1)

is surjective.

For any and given as

(2)

Rodrigues’ formula is the right-hand side of

(3)

where and is the rotation matrix through an angle anticlockwise about the axis . For where is not symmetric, we define the inverse of as

(4)

where . We define as the zero matrix in . Note that (4) is not defined for . The Riemannian metric for is defined as where is the Frobenius norm.

One important relation between and is that the open ball in with radius around the identity, which is almost the whole , is diffeomorphic to the open ball in via the logarithmic and the exponential map defined in (4) and (3).

An undirected graph consists of a finite set of nodes and a set of edges of unordered pairs of . To each edge , we associate a weight . The weighted adjacency matrix is defined by if and otherwise. Note that and that as no self-loops are allowed. For each node , its degree is defined as . The graph Laplacian is defined as where is a diagonal matrix such that . As a result, . We denote the set of neighbors of node as . If the edges are ordered pairs of , the graph is called a directed graph, or digraph for short. An edge of a digraph is denoted by (with ) representing the tail vertex and the head vertex of this edge. A digraph with unit weights is completely specified by its incidence matrix , where , with equal to if the th edge is towards vertex , and equal to if the th edge is originating from vertex , and otherwise. The incidence matrix for undirected graphs is defined by adding arbitrary orientations to the edges of the graph. Finally, we say that a graph is connected if, for any two nodes and , there exists a sequence of edges that connects them. In order to simplify the notation in the proofs, we set the weight to be one. All the results in this paper hold for the general case where the ’s are elements in .

In the remainder of this section, we discuss Filippov solutions. Let be a map from to and let denote the collection of all subsets of . The Filippov set-valued map of , denoted , is defined as

where is a subset of , denotes the Lebesgue measure, is the ball centered at with radius and denotes the convex closure of a set . If is continuous at , then contains only the point .

A Filippov solution of the differential equation on is an absolutely continuous function that satisfies the differential inclusion

(5)

for almost all . A Filippov solution is maximal if it cannot be extended forward in time, that is, if it is not the result of the truncation of another solution with a larger interval of definition. Next, we introduce invariant sets, which will play a key part further on. Since Filippov solutions are not necessarily unique, we need to specify two types of invariant sets. A set is called weakly invariant if, for each , at least one maximal solution of (5) with initial condition is contained in . Similarly, is called strongly invariant if, for each , every maximal solution of (5) with initial condition is contained in . For more details, see [10, 13]. We use the same definition of regular function as in [8] and recall that any convex function is regular.

For locally Lipschitz, the generalized gradient is defined by

(6)

where is the gradient operator, is the set of points where fails to be differentiable and is a set of measure zero that can be arbitrarily chosen to simplify the computation, since the resulting set is independent of the choice of [8].

Given a set-valued map , the set-valued Lie derivative of a locally Lipschitz function with respect to at is defined as

(7)

If is convex and compact , then is a compact interval in , possibly empty.

The th row of a matrix is denoted as . For any matrix , we denote as and as . A positive definite and semidefinite (symmetric) matrix is denoted as and , respectively. The vectors denote the canonical basis of . The vectors and represents a -dimensional column vector with each entry being and , respectively. In this paper, we define the direction-preserving signum as

(8)

for . The component-wise signum is denoted as

(9)

where . Notice that for scalars, these two signum functions coincide. Furthermore, component-wise signum is coarser than direction-preserving in the sense that there is only a finite number of elements in the range for a fixed dimension .

Iii Problem formulation

We consider a system of agents (rigid bodies). We denote the world frame as and the instantaneous body frame of agent as where . Let be the attitude of relative to at time , and, when , the corresponding axis-angle representation be given by

(10)

The kinematics of is given by [27]

(11)

where is the control signal corresponding to the instantaneous angular velocity of relative to expressed in the frame , and the transition matrix is defined as

(12)

where is defined as for all and , see [27]. Note that for , the function is concave and belongs to . Then the symmetric part of , denoted by , is positive semidefinite. More precisely, if . Moreover, is Lipschitz on for any (see [30]).

The system (11) can be written in a compact form as

(13)

where

(14)

For the multi-agent system (13), we assume that the agents can communicate with each other about the state variables via an undirected connected graph . The aim is to design control protocols for such that the absolute rotations of all agents converge to a common rotation in the world frame in finite time, i.e.,

(15)

This is equivalent to that converges to the consensus space

(16)

in finite time. We shall propose two distributed controllers that achieve this goal.

Iv Main result

In this section, we shall first present a control law that guarantees that the rotations of all the rigid bodies converge to a common rotation for any initial condition for all . Note that this initial condition in is equivalent to under the axis-angle representation. In order to avoid the singularity of the logarithmic map (4), the control law makes sure that the constraint is met for all and for all time . We consider controllers of the following form

(17)

with maps and the elements in the set are the neighbors of agent . Now the closed-loop system can be written in a compact form as

(18)

where , and is the Laplacian of the graph. Our control design is based on the signum function. More precisely, we consider the case when some of the functions are or , while the others satisfy certain continuity assumptions to be defined in the following subsections. We propose two control protocols which guarantee almost global, in the sense of , and local convergence, respectively. As discontinuities are introduced into (18) by the signum functions, we shall understand the trajectories in the sense of Filippov, namely an absolutely continuous function satisfying the differential inclusion

(19)

for almost all time, where we used Theorem 1(5) in [22].

Iv-a Control law for global convergence

In this subsection, we shall design a controller such that finite-time synchronization is achieved for any initial condition by using the direction preserving defined in (8). It might seem natural to let for all . However, the following example shows that this simple controller does not guarantee for all Filippov solutions.

Example 1.

Consider the system

defined on a complete graph. We show that for such that , the trajectories can violate the constraints , for some .

First, by Theorem 1(1) in [22], we have for any , there exists an , independent of , such that the ball . Second, suppose for some . Then there exists such that the vector . Hence,

is a Filippov solution. Indeed,

where the second equality follows from

Here we used the fact that . Then for large enough , can be larger than . The solutions of the type with a non-constant function is called sliding consensus.

The previous example motivates us to consider the following assumption.

Assumption 1.

For some set , the function in (18) satisfies the following conditions:

  1. For , is locally Lipschitz continuous and satisfies and for all ;

  2. For , the function .

Note that Condition (i) in Assumption 1 corresponds to that is direction preserving.

Before showing the result for finite-time convergence, we formulate a condition for the controller (17) satisfying Assumption 1 such that the set is strongly invariant for the dynamics (18).

Lemma 2.

Consider the differential inclusion (19) satisfying Assumption 1. If one of the following two conditions is satisfied

  1. and ;

  2. and ,

then the set , where is a constant, is strongly invariant. This implies that is strongly invariant.

Proof.

We use a Lyapunov-like argument to prove that for any initial condition in , all the solutions of (19) will remain within the set.

Consider the Lyapunov function candidate . Notice that is convex, hence regular. Let

(20)

The generalized gradient of is given as

(21)

Next, let be defined as

(22)

Since is absolutely continuous (by definition of Filippov solutions) and is locally Lipschitz, by Lemma 1 in [1] it follows that for a set of measure zero and that

(23)

for all , so that the set is nonempty. For , we have that is empty, and hence by definition. Therefore, we only consider in the rest of the proof.

By using Theorem 1(4) and (5) in [22], the differential inclusion can be enlarged as follows

(24)

where the first equality follows from Assumption 1 and the fact that is continuous for with , thus we can use Theorem 1(5) in [22]. Moreover, we obtain that for all . For the rest of the proof, we shall show by considering two cases.

Case 1: For , i.e., , the following two subcases can be distinguished.

  1. and . There is such that is locally Lipschitz and direction preserving. Then, by using the definition of the Filippov set-valued map, one can show that for all (recall that ). As is nonempty (by considering ), there exists such that for all , see the definition (7). By choosing , it follows that , which implies that .

  2. and . In the following, we consider the Filippov solution of system (19), which can be written as

    (25)

    Then it can be shown that, for (i.e., ), any element in the Filippov set-valued map of (25) satisfies . Stated differently, the following implication holds for :

    (26)

    Next, by recalling that , it follows that

    (27)

    with . Now, following a similar reasoning as in item (i) on the basis of the definition of the set-valued Lie derivative in (7), it can be concluded that , so that for all .

Case 2: For , take an index such that . Note that such indeed exists. Namely, assume in order to establish a contradiction that for all . Then, it holds that

(28)

Since , it follows from (28) that for all . By repeating this argument and recalling that the interconnection topology is connected, it follows that for all , i.e., . This is a contradiction to .

For the index satisfying , it follows from Assumption 1 that there exists such that

(29)

i.e., for any it holds that . Note that this is a result of the direction-preserving property of either the vectorized signum function (for nonzero argument, then ) or the Lipschitz continuous function (by Assumption 1). Then, choosing as (recall that ), it follows that for any we have

(30)

where the second equality is based on .

Summarizing the results of the two cases leads to

(31)

if for all . Since the trajectory is absolutely continuous, we have that if for all , all the trajectories remain within the set . ∎

Remark 1.

As indicated in Example 1, sliding consensus can happen when and . This will violate the strong invariance of the set with , which will introduce singularity for the axis-angle representation for rotations.

Before we prove the finite-time convergence, we provide a sufficient condition for that all Filippov solutions of (19) converge to consensus asymptotically.

Lemma 3.

Under the same assumptions of Lemma 2, all Filippov solutions of (19) asymptotically converge to static consensus.

Proof.

Similar to the proof of Lemma 2, we shall prove that the conclusion holds for the bigger inclusion given by (24). In this proof cases (i) and (ii) can be handled with the same arguments.

Consider the Lyapunov function candidate , which is convex, hence regular. The generalized gradient of is given as follows:

(32)

Next we shall calculate the Lie derivative of by considering two cases.

  1. If , the Lie derivative is given as

    (33)

    Here we have for . Indeed, it is because: (1), the conditions that the matrix for satisfying ; (2), Assumption 1 about direction preservation, and (3), the set is strongly invariant for . Moreover, if , the set . Hence .

  2. If , it can be seen that implies . Hence if , it has to be . In fact, by taking , we have that .

Next, by Theorem 3 in [9], it holds that all Filippov solutions of (24) converge to the set asymptotically. The remaining task is to characterize the set . So far we have shown that , which implies that . By the fact that is closed, we have . Moreover, when , where , which implies that remains constant. In conclusion, asymptotic convergence to static consensus is guaranteed.

Now we are ready for the main result of this section.

Theorem 4.

Assume that for all and that the graph is connected. Consider the multi-agent system (18) satisfying Assumption 1 and the corresponding differential inclusion (19). Then, all Filippov solutions converge to consensus in finite time if one of the following conditions holds:

  • and ;

  • and .

Proof.

The proof is separated into two parts, one for each of the conditions.

(i) Without loss of generality, we assume that satisfies the condition (i) in Assumption 1 while are . Similar to Lemma 2, instead of proving the conclusion for the differential inclusion (19), we shall show that it holds for the bigger inclusion given by (24).

Consider the Lyapunov function candidate . We shall show that there exists such that for any initial condition with .

In the proof of Lemma 3, we have shown that for , the Lie derivative is given by (33). By the fact that with and is direction preserving, we have

(34)

Furthermore, the Filippov set-valued map

(35)

which implies that

(36)

Note that, for any satisfying for all , there exists , which only depends on , such that for all . This implies that

(37)

So far we have shown that, for any , it holds that

(38)

Furthermore, by using that , which is based on the connectivity of the graph , we have

(39)

where the triangle inequality is used. Then, the use of (39) in (38) yields

(40)

By exploiting the observation that is a graph Laplacian, it holds that

(41)

where is a diagonal matrix with real-valued eigenvalues satisfying and for . The matrix collects the corresponding right-eigenvectors. From (41), it can be seen that

(42)

for any . Consequently, using , it follows that

(43)

After taking the square root (note that for all ) in (43) and substituting the result in (40), the result

(44)

follows, which proves finite-time convergence to consensus by Proposition 4 in [9] and Lemma 3.

(ii) By using a similar reasoning, we have that for any , it satisfies that where satisfying for all . This again implies finite time convergence.

Remark 2.

Theorem 4 provides sufficient conditions for finite-time convergence of the protocol (17) satisfying Assumption 1. However, we conjecture these sufficient conditions to be necessary as well. Namely, for the case , we expect that all the Filippov solutions of (19) converge to consensus in finite time if and only if ; and for the case , we expect that the finite-time synchronization is achieved if and only if . We show that the latter statement holds according to the following argument.

If and , then . In this case, we can only have asymptotic convergence if . Indeed, the right-hand side of (18) is Lipschitz; therefore, finite-time convergence to an equilibrium can not occur.

Unfortunately, for the case , we can not prove the necessity, which leaves it as an open question.

We close this subsection by demonstrating the result in Theorem 4 and conjecture in Remark 2 by an example.

Example 2.

Consider the three-agent system

(45)

defined on a line graph with. Notice that this system meets condition (i) in Theorem 4. A phase portrait and trajectory of this system are depicted in Fig. 1. There, we can see that finite-time consensus is achieved.

(a) Phase portraits of in .
(b) Evolution of the three coordinates of . Finite-time consensus is achieved.
Fig. 1: The simulation of Example 2.

Next, modify the system to