This paper considers systems of agents continuously evolving on , where . The interactions between the agents are changing as a function of time. For such systems we are analyzing a large class of distributed synchronization/consensus control laws. The analysis tool is a lifting method, where an equivalent consensus protocol is analyzed in the ambient space that embeds the sphere. In comparison to projection methods that have been used in this context—e.g., the gnomonic projection—the proposed method is not locally but globally defined on the unit sphere. The control action is performed in the tangent plane. Only relative information between neighboring agents is used in the control laws. Under the assumption that the time-varying graph is uniformly quasi-strongly connected, we show that the consensus manifold is globally uniformly asymptotically stable relative to any closed ball on the sphere contained in an open hemisphere.
Synchronization on the circle, i.e., , is closely related to synchronization of oscillators (Dörfler & Bullo, 2014) and it is equivalent to synchronization on , where several applications exist such as flocking in nature and alignment of multi-robot systems. Also for the two-dimensional sphere, i.e.,
, there are several applications such as formation flying and flocking of birds; consider for example a multi-robot system in 3D, where the relative directions between the robots are available and the goal is to align those. For higher dimensional spheres there are currently related problems such as distributed eigenvector computation, but concrete applications might arise in the future.
The control laws at hand—and slight variations or restrictions on the graph topologies, switchings of the graphs, dimensions of the sphere, and the nonlinear weights in the control laws etc.—have been studied from various perspectives (Scardovi et al., 2007; Sarlette, 2009; Olfati-Saber, 2006; Li & Spong, 2014; Li, 2015). There has recently been new developments (Pereira & Dimarogonas, 2015, 2016; Markdahl & Goncalves, 2016; Markdahl et al., 2016). In Markdahl et al. (2017), almost global consensus is shown by characterization of all equilibrium points when the graph is symmetric and constant (time-invariant). It is shown that the equlibria not in the consensus manifold are unstable and the equilibra in the consensus manifold are stable. A similar technique is used in Tron et al. (2012) to show that a consensus protocol on is almost globally asymptotically stable. Now, the above-mentioned results about almost global convergence come at a price. Static undirected graph topologies are assumed as well as more restrictive classes of weights in the control protocols. Furthermore, compared to Markdahl et al. (2017), the right-hand sides of the system dynamics is not necessarily an intrinsic gradient and the linearization matrices at equilibriums are not necessarily symmetric. Hence, we cannot use the result due to Lojasiewicz (1982) about point convergence for gradient flows. This inspired us to take a closer look at methods that transform the consensus problem on the unit sphere (or a subset thereof) to an equivalent consensus problem in . Before we address the method—referred to as a lifting method—we briefly make some connections to the related problem of consensus on .
The problem of consensus on has been extensively studied (Sarlette et al., 2009; Ren, 2010; Sarlette et al., 2010; Tron et al., 2013; Tron & Vidal, 2014; Deng et al., 2016; Thunberg et al., 2016). There is a connection between that problem and the problem of consensus on when the unit quaternions are used to represent the rotations. For those, the gnomonic projection can be used to show consensus on the unit-quaternion sphere (Thunberg, Song, Hong & Hu, 2014; Thunberg, Song, Montijano, Hong & Hu, 2014)
. In another line of research, several methods have been introduced where control laws based on only relative information have been augmented with additional auxiliary (or estimation) variables, which are communicated between neighboring agents. By doing so, results about almost global convergence to the consensus manifold are achieved(Sarlette & Sepulchre, 2009; Thunberg, Markdahl & Goncalves, 2017). The latter of these two publications provides a control protocol for Stiefel manifolds, with the unit sphere and as extreme cases. A similar technique had previously been used for the sphere (Scardovi et al., 2007). The idea of introducing auxiliary variables also extends to the related distributed optimization problem in Thunberg, Bernard & Goncalves (2017). In contrast to the mentioned works, in this paper we are not assuming additional communication between the agents by means of auxiliary variables. Instead only relative information is used in the protocols. In a practical setting (considering the case ), such information can be measured by for example a vision sensor and requires no explicit communication between the agents.
In the proposed lifting method, we lift the states from the -dimensional sphere into . The non-negative weights in the consensus protocol for the states in the lifting space are nonlinear functions. Each agent moves in a direction that is a weighted combination of the directions to the neighbors. The weights contain rational functions of the norms of the states of the agents. Since these rational functions are not well-defined at the origin, fundamental questions arise about existence, uniqueness, and invariance of sets. Those questions are answered with positive answers. The hope is that this lifting method will serve as a stepping-stone to future analysis on (almost) global convergence to the consensus manifold on the unit sphere. Compared to the approach in Markdahl et al. (2017) where all the “bad” equlibria on were characterized, we only need to characterize one point, which is the origin in the “lifted space”. If we were to show that this point has a region of attraction that is of measure zero, we would have equivalently shown the desired result about almost global convergence on the unit sphere (assuming ). However, the non-differentiability of this point remains an additional challenge.
We begin this section with some set-definitions. The -dimensional unit sphere is
The special orthogonal group in dimension is
The set of skew symmetric matrices in dimensionis
The set is an open hemisphere if there is such that .
We consider a multi-agent system with agents. Each agent has a corresponding state for . The initial state of each agent at time is . Another way to represent the states of the agents is to use rotation matrices. Let satisfy for all and , where is the north pole; we also define as the south pole. Let for all , where is the initial -matrix at time . The -matrices can be interpreted as transformations from body coordinate frames—denoted by ’s—of the agents to a world coordinate frame
. They are transforming the unit vectorin the body frames to the corresponding unit vector (or point on the unit sphere) in the world coordinate frame. The ’s and their dynamics are not uniquely defined, but this is not of importance for the analysis. We choose to define the dynamics of the ’s according to (2) below.
The dynamics of the -vectors are given by
where for all . The -vectors are the controllers for the agents and those are defined in the body coordinate frames, i.e., the ’s. For the -matrices the dynamics is
The matrix on the right-hand side of in (2) is an element of . The control is performed in the tangent space of the sphere, which means that there are degrees of freedom for the control. This is the reason why the -vectors are -dimensional. Before we proceed, we provide some additional explanation for the expression in the right-hand side of (2). According to its definition, the first column of is equal to and by multiplying by from the right we obtain—due to (1)—the following expression
This means that
where the -parts are left to be chosen. We know that the matrix in the right-hand side above needs to be skew symmetric, since is a rotation matrix. We also know that the first column of it must be equal to . The matrix of minimum Euclidean norm that fulfills these two requirements is equal to
i.e., the one we chose in the right-hand side of (2).
We will study a class of distributed synchronization/consensus control laws on the unit sphere, where the agents are moving in directions comprising conical combinations of directions to neighbors. In this protocol only local and relative information is used. Before we provide these control laws we introduce directed graphs and time-varying directed graphs.
A directed graph is a pair , where is the node-set and is the edge-set. Each node in the node set corresponds to a unique agent. The set is the neighbor set or neighborhood of agent , where if and only if . We continue with the following definitions addressing connectivity of directed graphs.
In a directed graph , a directed path is a sequence of distinct nodes, such that any consecutive pair of nodes in the sequence comprises an edge in the graph. We say that is connected to if there is a directed path from to . We say that the graph is quasi-strongly connected if there is at least one node that is a center or a root node in the sense that all the other nodes are connected to it. We say that the graph is strongly connected if for all it holds that is connected to .
Now we define time-varying graphs. We define those by first defining time-varying neighborhoods. The time-varying neighborhood of agent is a piece-wise constant right-continuous set-valued function that maps from to . We assume that there is such that for all , where is the set of time points of discontinuity of . The constant is as a lower bound on the dwell-time between any two consecutive switches of the topology. We define the time-varying graph as
Furthermore, the union graph of during the time interval is defined by
where . We say that the graph is uniformly (quasi-) strongly connected if there exists a constant such that the union graph is (quasi-) strongly connected for all .
Now we provide the synchronization protocol to be studied. For each agent , the controller is is defined by
where , which is represented in the frame . The ’s are what we refer to as relative information and the control law (3) is constructed by only such information. For each , it holds that . The -functions are assumed to be Lipschitz and attain positive values for positive arguments. The ’s are neighborhoods of a time-varying directed graph , whose connectivity is at least uniformly quasi-strong. These control laws will be analyzed in the paper.
The expressions in (3) are more easily understood if they are expressed in the world frame . We define
for all , which is expressed in the frame . The vector is the sum of the positively weighted directions to the neighbors of agent , projected onto the tangent space at the point . Also for analysis purposes, (4) is easier to work with than (3). The closed loop system is
for all .
Let and . We define the set
which is the synchronization/consensus set. Throughout the paper we assume that the closed-loop dynamics of the system is given by (5). We study the convergence of to the consensus set . When we talk about convergence we refer to the concepts below.
For the system (5), we say that the set is attractive relative to a forward invariant set if
where . Furthermore, we say that the set is globally uniformly asymptotically stable relative to a forward invariant compact set if
for every there is such that
for every there is such that
for all ).
The equivalent definitions to the above will also be used (after changing the sets and ) for other systems evolving in or linear subspaces thereof. Forward invariance, or simply invariance, of a set means that if the initial state is contained in the set, then the state is contained in the set for all future times.
The two concepts of global convergence respective almost global convergence relative to a forward invariant set refer to, respectively, the situations where convergence occur for all initial points in and convergence occur for all initial points in a set where has measure zero.
3 Projection methods
Before we continue to present the lifting method, we show how projection based methods can be used to analyze consensus on hemispheres. In particular we consider two such methods. The two methods are such that the -vectors are projected down onto a -dimensional linear subspace of . The symbol is used to denote the projection variable for in both methods.
3.1 Equatorial plane projection
The equatorial plane projection simply projects all the states onto a
-dimensional hyperplane (that contains the origin). This plane separates the sphere into two hemispheres. If all the agents are positioned on one of those hemispheres, one can easily show that they reach consensus provided that the graph has strong connectivity. This projection method is appealing because the projections are simple and the convergence proof is straightforward. It is interesting that results from the literature about convergence on hemispheres (and slightly more general ones where the graph is assumed to be time-varying) can easily be shown with this simple projection.
Now, formally, the -states are projected onto the equatorial plane whose normal is equal to in the world coordinate frame .
The projected state is defined by
This is illustrated in Fig. 1 for the dimension . Points on the northern hemisphere, i.e., the ’s satisfying , are projected down onto the equatorial plane. For each point there is a blue dotted line between the point and its projection.
On the northern hemisphere is a diffeomorphism. The mapping is defined by
where and are the ’th elements of and , respectively.
By using this projection one obtains a local convergence result for hemispheres.
Proof: We will use Theorem 1 in Thunberg, Hu & Goncalves (2017). Under the condition that the graph is uniformly strongly connected, if we can show that any closed disc (or ball) in the equatorial place with radius less than is forward invariant for the ’s and we can find a function such that is 1) positive definite, 2) is decreasing as a function of , and 3) is strictly negative if and there is such that . Then the ’s converge to a consensus formation. This in turn implies, since is a diffeomorphism, that the ’s converge to a consensus formation, i.e., the set is attractive.
Let , i.e., , which obviously satisfy condition 1). For it holds that
Now, at time , assume and assume . The following observations imply that conditions 2) and 3) hold. It holds that and the inequality is strict if . It holds that and the inequality is strict if . It holds that
We also see that any closed disc with radius less than is forward invariant for the ’s.
By a change of coordinates, we obtain the following generalization. Suppose controller (3) is used for each agent and suppose is uniformly strongly connected. If all the ’s are contained in an open hemisphere it holds that is attractive for the closed loop system (5).
A main problem, with the equatorial plane projection is that the convex hull of the projected variables is not necessarily forward invariant. This means that the projected variables are not following a consensus protocol. This is also the reason why we settle for the strong connectivity assumption about the graph, i.e., that it is uniformly strongly connected. However, the projected variables under the gnomonic projection—introduced in the subsequent section—do follow a consensus protocol, which, in turn, allows for more general convergence results.
3.2 The gnomonic projection
The gnomonic projection projects an open hemisphere onto a tangent plane at a point on the sphere. We will use the convention of projecting the points on the southern hemisphere defined as onto the tangent plane at the south-pole, i.e., at the point . The projection of is the intersection between the tangent plane and the line that passes through the origin and . This projection is illustrated in Fig. 2, where several points are projected.
The gnomonic projection has the property that segments of great circles on the sphere (geodesics) correspond to straight line segments in the projection plane. One can show that a consensus algorithm on the open hemisphere corresponds to a consensus protocol for the projected states. It should be emphasized that the gnonomic projection method is not new. It is claimed to have been invented by the Greek philosopher Thales of Miletus somewhere around 624–546 BCE (Alsina & Nelsen, 2015)
. Its first appearance in a subject related to the one addressed in this paper, was probably inHartley & Dai (2010) and subsequently in Hartley et al. (2013) in the context of rotation averaging. Later the gnomonic projection was used as a tool to show consensus on the open hemisphere (Thunberg, Song, Hong & Hu, 2014; Thunberg et al., 2016). In those latter works, the three-dimensional (unit-quaternion) sphere was considered in the context of attitude synchronization. Recently, the gnomonic projection was also considered for arbitrary dimensions (Lageman & Sun, 2016). It should be emphasized that the graph was not time-varying in that context.
Formally we define , the projection of , by
which is a diffeomorphism from the open southern hemisphere to the tangent plane of the south pole. Suppose controller (3) is used, i.e., the closed loop dynamics is given by (5). If for all , i.e., all the ’s are located on the southern hemisphere, it holds that the dynamics of the ’s is on the form
where , and it can be shown that the ’s are locally Lipschitz and globally Lipschitz on any bounded set. This can be used (as an alternative to using the lifting method) to prove the result in Proposition 4 in next Section, which is stronger than that in Corollary 3.1.
4 The lifting method
In this section we propose a method where the ’s are not projected onto a ()-dimensional plane, but rather relaxed to be elements in . Those elements, we call them ’s, are then projected down onto the sphere to create the ’s (which in this case are equivalent to the ’s). The method can thus be seen as the inverse procedure to the two in the previous section. Provided , the projection is given by
This projection as well as the lifting is illustrated in Figure 3. Points in are projected down onto the sphere in the sense of minimizing the least squares distance.
We let be governed by the following dynamical system
for all . Let the initial state of the system be . Equation (13) describes a consensus protocol with nonlinear weights that contain rational functions of the norms of the states. The question is how this dynamical system is related to (5). The following proposition provides the answer.
Suppose that all the ’s are not equal to zero. On the time interval the dynamics for the ’s is given by
i.e., it is the same as (5).
Proof: Proposition 4 below provides the result that the solution to (13) is well-defined on the interval and is forward invariant. Given that result, the ’s and their derivatives are well defined. Now,
Suppose the dynamics for the ’s is governed by (13). Suppose there is no such that . Let be the convex hull of the ’s during when the initial condition is . Let be such that the solution exists during . Then the solution to (13) exists and is unique for all times , is forward invariant, and the set is forward invariant.
Proof: We first address the claim that is forward invariant. It suffices to verify that for each , the right-hand side of (13) is either inward-pointing relative to , or equal to . Now, due to the structure of (13), this is true.
Now we address the invariance of , and, by doing that, obtain the existence and uniqueness result for the solution during for free, since the right-hand side of (13) is locally Lipschitz on . Now, suppose there is and a finite time such that and there is no and such that . This means that there is a first finite time for which at least one state, that is, attains the value . The assumption is equivalent to assuming that is not forward invariant.
For and it holds that
where and . For , it also holds that
We define for all and write the equation above as
Let be an upper bound for the ’s, which is equivalent to an upper bound for the ’s. Such a bound must exist (since the set is compact and the function is continuous on , where is the function that returns the Euclidean distance between two points in ).
Let . On it holds that
where is the upper Dini-derivative. By using the Comparison Lemma for (15), we can conclude that is bounded from above by on
Now, for and it holds that
By using the Comparison Lemma, we can conclude that . But this, in turn, means that , which is a contradiction.
In the following proposition we make use of , which was defined in Proposition 4.
Suppose the dynamics for the ’s are governed by (13) and is uniformly quasi-strongly connected. Suppose is not contained in the convex hull for . Then the consensus set —defined as the set where all the ’s are equal in —is globally uniformly asymptotically stable relative to . Furthermore, there is a point that all the ’s converge to.
Proof: Invariance of is an indirect consequence of the fact that (13) is a consensus protocol. On this set the right-hand side of (13) is Lipschitz continuous in and piece-wise continuous in . Now the procedure in the rest of the proof is analogous to the one in Proposition 4. Since the right-hand side of (13) is Lipschitz continuous in and piece-wise continuous in , we can use Theorem 2 in Thunberg, Hu & Goncalves (2017) to find a continuously differentiable function such that 1) is decreasing as a function of ; and 2) is strictly negative if and there is such that or there is such that . The existence of such a function guarantees that the consensus set is globally uniformly asymptotically stable relative to . It holds that the function is such a -function. Convergence to a point for all the ’s can be shown by using the facts that is forward invariant for all and converges to .
Suppose the graph is uniformly quasi-strongly connected. Then for any closed ball contained in the hemisphere, the consensus set is globally uniformly asymptotically stable relative to under (5).
Proof: Forward invariance holds for due to the structure of the right-hand side of (5). Let . Due to Proposition 4 we know that the consensus set is globally uniformly asymptotically stable relative to and that there is a point that all the ’s converge to. We also know that the projected -variables follow the protocol (14), which is the same as (5). The norms of the ’s are uniformly bounded on and is forward invariant for all . Thus the desired result readily follows.
Suppose the dynamics for the ’s is governed by (13) and suppose the graph is quasi-strongly connected. If the ’s converge to a point that is not equal to zero, then the ’s converge to a point . Furthermore, if the convex hull of the ’s does not contain the point zero, then the ’s converge to a point .
Variations of Proposition 4 has appeared in the literature before. The idea of using the gnomonic projection to show consensus on the hemisphere was used in Thunberg, Song, Hong & Hu (2014); Thunberg et al. (2016) where restricted versions of Proposition 4 were given for the dimension in the context of attitude synchronization. Recently the attractivity of relative to open hemispheres was established under quasi-strong graph connectivity (Lageman & Sun, 2016) using the gnomonic projection. The graph was not time-varying in that context.
. In this example there are five agents with a uniformly quasi-strongly connected interaction graph. The agents were initially uniformly distributed on a hemisphere and the-functions were chosen to be constant; either equal to or . In the figure, the red discs denote the initial positions and the yellow disc denote the final consensus point. We have also denoted two points on the trajectories where the graph switches.
Now, the case when is contained in the convex hull of the ’s is more intriguing. We provide the following result.
Suppose the dynamics for the ’s is governed by (13) and is uniformly strongly connected. Suppose is contained in the convex hull of the ’s, i.e., in , and there is no such that . Furthermore, suppose that the ’s are contained in a compact set where the ’s are bounded from below by a positive constant . Then the set —defined as the set where all the ’s are equal in the convex hull of the ’s—is attractive. Furthermore, there is a fixed point that all the ’s converge to.
Proof: We need to prove that the ’s converge to .
In light of Proposition 4, the only case left to consider is when the point is contained in the convex hull of the ’s for all , i.e., it is contained in for all . We will thus only consider this case in the following where we need to prove that all the ’s converge to . We partition this case into two sub-cases:
1) the omega-limit set, denoted by , does not contain a point for which a .
2) the omega-limit set contains at least one point where at least one of the ’s is equal to zero.
We begin by considering 1). There must be a ball around the origin such that there is no time for which a is contained in the ball. This is proven in the following way. Proposition 4 guarantees that no can reach the origin in finite time. Thus, at any finite time there exists a largest open ball with radius such that no is contained in the ball during . Assume that . This implies that there is a point that is in the closure of for which one of the ’s is equal to zero. But is compact, hence we know that such a point also will be contained in . This is a contradiction to the statement that such points are not contained in .
Now, in the set we replace the weights in the right-hand side of (13) by functions such that the total weights, consisting of outside and inside , are globally Lipschitz on a set containing in the interior. Furthermore, is chosen to be positive when .
Now let us study the solution starting at at time of this modified system with the replaced weights in . At any finite time the solution is the same as the original system. However, we can use the results in Thunberg, Hu & Goncalves (2017) to show that the solution to the modified system converges to and in particular all the ’s converge to a fixed point that is nonzero. This means that after some
finite time , the point is not contained in for the modified as well as for the original system. But this is a contradiction to our assumption that the point is contained in the convex hull of the ’s for all .
Now we consider 2). Let us first introduce . is, besides continuous, monotonically decreasing. We assume that . This means that for any , it holds that the set is nonempty. We will show that this assumption leads to a contradiction in the end of the proof. This, in turn, means that all the ’s converge to .
Since the continuous ’s are defined on a compact set, there is such that the ’s are bounded from above by . Furthermore, there is also an such that for all and . Take for example .
We continue by formulating a series of claims, each of which is followed by a proof. After these claims have been introduced, they are used as building blocks in the final part of the proof. Roughly, the claims can be understood as follows. The first claim says that if a is close to the origin, it will remain so for a specified time interval. The second claim says that if a has a neighbor that is close to the origin, it will will be “dragged” to the origin by this neighbor. The third claim simply says that there must be a close to the origin at some time. Then we show that that the that is close to the origin will drag all the other states close to the origin; so close that their distances to the origin is smaller than , which, in turn, is a contradiction.
Claim 1: There is satisfying and such that for time
there is such that has smaller norm than , then for , where is the lower bound on the dwell time and is the length of the time interval such that the union graph is guaranteed to be strongly connected, see Section 2.
It is assumed that and . Suppose there is and such that . Let us consider the dynamics for . It is
Now we can use the Comparison Lemma to deduce that for . Now, if is sufficiently small, the expression will be smaller than during .
Claim 2: There is satisfying , such that for any time , if an agent has a neighbor during for which it holds that , then there is a time during such that