This paper considers multi-agent systems continuously evolving on , i.e., the set of rotation matrices. The agents interact locally with each other and the neighborhood structure is determined by an interaction graph that is quasi-strongly connected. For such systems, we address the following synchronization problem. How to design control laws in the body fixed coordinate frames of the agents such that specific columns of the rotation matrices asymptotically synchronize (converge to the set where they are the same and equal to the columns of a constant matrix) as time goes to infinity. The problem is, in general, a synchronization problem on a Stiefel manifold. The control laws shall be designed by using the corresponding columns of the relative rotations between the agents (and not the other columns). Such control laws can be used to solve the synchronization problem on the unit sphere; consider for example the case where satellites in space only monitor one axis of each of its neighbors. But it can also be used in problems where various degrees of reduced attitudes are available, or the problem where complete rotations are available. To solve the problem we introduce auxiliary variables and use a QR-factorization approach. The benefit of this approach is that the dynamics of the columns considered can be decoupled from the dynamics of the remaining ones.
Two important special cases of the problem considered are synchronization of whole rotation matrices, i.e., synchronization on
, and synchronization of one specific column vector, i.e., synchronization on the-sphere. In these cases, for obvious reasons of applicability, the dimensions and have been mostly considered. The distributed synchronization problem on the unit sphere has been studied from various aspects Sarlette (2009); Olfati-Saber (2006); Li & Spong (2014); Li (2015). Recently there have been some new developments Markdahl et al. (2016); Markdahl & Goncalves (2015). Markdahl et al. (2017, 2016); Markdahl & Goncalves (2016); Pereira & Dimarogonas (2015); Lageman & Sun (2016). In Markdahl et al. (2017) the classical geodesic control law is studied for undirected graph topologies. Each agent moves in the tangent space in a weighted average of the directions to its neighbors. Almost global synchronization is de facto shown by a characterization of all the equilibria; the equlibria that are not in the synchronization set are shown to be unstable and the equilibra in the synchronization set are shown to be stable. The analysis can be seen to parallel the one in Tron et al. (2012) (also for undirected topologies) for the case of synchronization on , where intrinsic control laws are designed for almost global synchronization. For the case an almost global synchronization approach has been presented for directed topologies and the -sphere Scardovi et al. (2007). That approach is a special case of the one in Sarlette & Sepulchre (2009).
The problem of synchronization on has been extensively studied Ren (2010); Sarlette et al. (2010); Tron et al. (2013); Tron & Vidal (2014); Thunberg et al. (2014); Deng et al. (2016). Often the control algorithms are of gradient descent types and assume undirected topologies Thunberg et al. (2011); Sarlette et al. (2009). Local convergence results are often obtained Thunberg et al. (2014, 2016). If a global reference frame is used, one can show almost global convergence Thunberg et al. (2014)—this is not allowed in the design of our control laws, only relative information is to be used. As mentioned above, Tron et al. (2012) provides a control algorithm for almost global convergence. The idea is to use so-called shaping functions where a gain constant can be chosen large enough to guarantee almost global consenus. The algorithm is defined in discrete time.
By introducing auxiliary state variables based on the QR-factorization of matrices, this work provides a dynamic feedback control algorithm for synchronization of the
first columns of the rotation matrices of the agents. The dynamics of the auxiliary variables follow a standard consensus protocol. The idea of using auxiliary or estimation variables with such dynamics is not new. Early works includeScardovi et al. (2007) and Sarlette & Sepulchre (2009), where the former addresses the -sphere and the latter addresses manifolds whose elements have constant norms and satisfy a certain optimality condition. Such manifolds are and the Grassmann manifold . If the approach in Sarlette & Sepulchre (2009) is used to synchronize columns of the rotation matrix where , then the entire relative rotations are used in the control design, which is not in general allowed in the problem considered here. In our proposed QR-factorization approach, only the corresponding columns of the relative rotations are used in the controllers. Under the control scheme, the closed loop dynamics achieves almost global convergence to the synchronization set for quasi-strong interaction topologies.
We start this section with some set-definitions. We define the special orthogonal group
and set of skew symmetric matrices
The -dimensional unit sphere is
The set of invertible matrices in is
We will make use of directed graphs, which have node set and edge sets . Such a directed graph is quasi-strongly connected if it contains a rooted spanning tree or a center, i.e., there is one node to which there is a directed path from any other node in the graph. A directed path is a sequence of (not more than ) nodes such that any two consecutive nodes in the path comprises an edge in the graph. For we define for all .
We will consider a multi-agent system with agents. There are coordinate systems , each of which corresponding to a unique agent in the system. There is also a world (or global) coordinate system . At each time , each coordinate system is related to the global coordinate system via a rotation . This means that transforms vectors in to vectors in .
For all , let be the “tall matrix” consisting of the first columns of . Thus, and is the first column of . All the columns of are obviously mutually orthogonal and each one an element of the -sphere. Let and for all . These matrices comprise the relative transformations between the coordinate frames and and the first columns thereof, respectively.
The matrix , or shorthand , is an element of for all , . The matrix is the upper left block matrix of the matrix . These ’s are communicated between the agents. For invertible we define as . Observe the difference in terms of the matrix inverse between and , i.e., , whereas . Let and .
The functions and are defined for matrices in for all . The function returns a matrix of the same dimension as the input, a matrix in that is, where each -element of the matrix is equal to that of the input matrix if and equal to if . The function returns a matrix in ; each -element of the matrix is equal to that of the input matrix if and equal to if .
We continue by introducing two assumptions that will be used in the problem formulation in the next section.
Assumption 1 (Connectivity).
It holds that is quasi-strongly connected.
Assumption 2 (Dynamics).
The time evolution of the state of each agent is given by
where and . In particular it holds that
The ’s are the controllers we are to design. An important thing to note in (1) is that , or rather the columns thereof, are defined in the -frames. If those would have been defined in the world frame , the agents would have needed to know their own rotations to that frame, i.e., the -matrices. Those matrices are not assumed to be available for the agents.
We let be the following subset of ,
We let be the following subset of ,
3 Problem formulation
The goal is to design (and in particular ) as a dynamic feedback control law such that the -matrices asymptotically aggregate or converge to the synchronization set. In this problem, formally presented below, a key assumption in the control design is that the information available for the agents is both relative and local. This means that no knowledge of a global coordinate system is assumed and that only relative rotations and vectors between local neighboring agents are used.
Let Assumption 1 and Assumption 2 hold. Let and . Design the -controllers as continuous functions of and the elements in the collection such that the following is fulfilled: there is a unique continuous solution for the ’s and there is such that
In Problem 3 we made the assumption that . This restriction of the ’s to those smaller than or equal to can be made without loss of generality. If all the ’s are equal, so are all the ’s. This means that synchronization of the ’s is equivalent to synchronization of the ’s.
For , our problem was studied in e.g., Markdahl et al. (2016); Markdahl & Goncalves (2016); Pereira & Dimarogonas (2015) and for , in Scardovi et al. (2007). For the case , the problem has been studied in e.g., Thunberg et al. (2014); Deng et al. (2016); Ren (2010); Sarlette et al. (2010); Tron et al. (2013); Tron & Vidal (2014). In general, for , Problem 3 is related to synchronization on the Grassmann manifold , which was studied in Sarlette & Sepulchre (2009). It is de facto synchronization on a compact Stiefel manifold.
4 The proposed control algorithm
In this section we propose a control algorithm as a candidate solution to Problem 3.
Before we introduce the control algorithm, Algorithm 1 below, we define the set
The set comprise the upper triangular matrices in , whose diagonal elements are positive. The algorithm below is restricted to those , ’s contained in .
Initialization at time : for all , choose and , let be upper triangular with positive elements on the diagonal, and let .
Inputs to agent at time : and .
Controllers at time when :
let, for all ,
We will show in the next section that the restriction of the , ’s to those contained in does not comprise a limitation from a practical point of view since the set is invariant under the proposed control scheme for all but a set of measure zero of the initial points.
It is important to note in Algorithm 1 (see next page) that (and not ) is the controller that is used by agent . The latter is however the part of the controller that affects according to (2). We remind the reader that comprises the first columns of , i.e., it is a restriction of . At the initialization step, the matrix is chosen (or constructed) by each agent , whereas the matrix is not. The latter is not known by the agent under our “relative information only”-assumption in the control design.
) in Algorithm 1 seem, at a first glance, a bit non-intuitive. As it turns out, the closed-loop system under Algorithm 1 is, almost everywhere, equivalent to the QR-decompositions of the-matrix variables in Dynamical System 1 below, which is a standard linear continuous time consensus protocol. In the next section we will, through a series of technical results, prove the equivalence between Algorithm 1 and Dynamical System 1 (almost everywhere).
Dynamical system 1.
Let Assumption 1 hold and let . Let for all , where the ’s (including the initial points at time ) are elements in for all . The time evolution of is governed by
5 Convergence results for Algorithm 1
A verbal interpretation of the set is the following one. It is the set of initial points in such that 1) all the ’s are invertible at all times larger than , and 2) the ’s and the ’s converge to matrices respective , where is in and is invertible.
Now, it would be good if we could prove that the set contains most of . The following proposition provides such a result; it is the main result of the paper. It is claiming almost global convergence to the synchronization set for Algorithm 1. For all but a set of measure zero of ’s and ’s, the matrices converge to the synchronization set. The rest of this section is dedicated to the proof of the claim in the proposition.
By inspection, we can verify that if all the ’s are equal and all the ’s are equal, then all the expressions in (3)-(6) are equal to zero, which means that the system is at equilibrium. Now, what we want to establish is the almost global convergence to such an equilibrium. However, the structure of the system seems at first hand complicated, which might make the convergence analysis cumbersome.
Now, instead of studying the dynamics of the closed loop system under (3)-(6), the main idea of the proof of Proposition 5 is to show that (as already mentioned in Section 4) when is contained in , there is a change of coordinates so that after this change of coordinates the dynamics of the system is described by the simple consensus protocol in Dynamical system 1 for the -matrices in , see (7) below. The idea is to, instead of studying the convergence of a seemingly complicated dynamical system, use well-known results for the simple consensus protocol. This type of approach is indeed not new, see for example Scardovi et al. (2007) and Sarlette & Sepulchre (2009).
A complication here is that in order to establish the convergence result in Proposition 5 we need to guarantee that in general the matrices evolving under the consensus protocol have full rank for all time points along the trajectories (as well as in the limit). This result is provided by Lemma 5 below. After the lemma (and its proof) has been provided, we give the proof of Proposition 5.
We begin by recalling the following known result.
[Dieci & Eirola (1999)] Any full column rank time-varying -matrix has a QR-decomposition.
and define the set as those ’s in for which it holds that for all , when is generated by Dynamical System 1 and there exists that has full column rank such that .
For Dynamical System 1 the following statements hold
there is a unique analytic solution for (7);
let denote the convex hull of the ’s. The set is forward invariant for any and there is (as a function of the initial state) such that goes to zero as goes to infinity;
for any time interval during which the ’s have full column rank, if (instead of being produced by Algorithm 1) the ’s and the ’s correspond to QR-decompositions of the ’s (i.e., for all ), then the ’s and the ’s can be chosen as smooth functions (of ), where , and for all ;
for all but a set of measure zero of the initial points , the matrix in (2) has full column rank;
has measure zero and is nowhere dense. The set is open in .
Proof: (1) The dynamics for can be written as
where is the weighted graph Laplacian matrix defined by: if and ; if and ; . The matrix is implicitly parameterized by the ’s. The unique analytic solution to (8) is
(3) Direct application of (1) and Proposition 5. We can without loss of generality assume that the ’s have positive elements on the diagonal. Suppose it was not the case, then we multiply and with a diagonal matrix from the right and the left, respectively. This diagonal matrix is constant and has ’s at -entries where is positive and ’s at the -entries where is negative. Since
, we can choose this diagonal matrix in such a way that the resulting orthogonal matrix after the multiplication (from the right) with the diagonal matrix is an element of.
(4) (8) is translation invariant. Translation invariance means that if we disturb the initial ’s by adding a matrix to all the ’s, the matrix is canceled out in the dynamics (8), i.e., it is invariant to this common translation of the initial states. As a consequence, the state trajectories for the disturbed initial conditions are equal to those without the disturbance up to the added . This also holds in the limit so the equivalent matrix to for such disturbed initial conditions is .
It holds that has measure zero. The rest of this proof is about showing that the subset of under consideration comprises all but a measure zero of . Now we define
In the above, and are linear subspaces of . The set is the synchronization set and the set is its orthogonal complement defined via the standard trace inner product. Each can be written as a sum of two matrices and , where is an element of and an element of . Furthermore it holds that where is given as a function of the initial condition. Now, for any fixed it holds that has full column rank for all but a set of measure zero of the ’s in .
(5) We begin by proving the zero measure of . The columns of are linearly independent for all and analytic in . Thus the matrix has a smooth QR-factorization , see Proposition 5.
We will without loss of generality only consider the case and prove that will be of full rank for all for all but a set of measure zero of the ’s in . We can make this “wlog-assumption” since the procedure of the proof is equivalent for the other choices of after a permutation of the -matrices.
Let be the first column of , the second column and so on. Let , where
Now we define
The ’s (besides ) are implicitly parameterized by the ’s. It holds that for all . The columns of comprise a smooth orthogonal basis for . The ’s for are built in the proposed way to take into account that, for a singular , one of the columns of a -matrix is a linear combination of previous columns in the matrix. Formally, these matrices are used in the following condition: if there is a such that the rank of is smaller than , there must be a with a vector such that . We refer to this as Condition a).
Now, the reminder of the proof amounts to showing that for only for a set of measure zero of ’s there is such that Condition a) is fulfilled. Once this is showed we can conclude that has full rank for all for all but a set of measure zero of the initial conditions (i.e., ). If the latter would not have been true, there would have been a for which Condition a) is satisfied for a positive measure of ’s, but that would have been a contradiction.
For all we define
which is a smooth mapping from to (we allow for negative ’s here). It is a nonlinear (and smooth) function of and a linear function of . The elements of comprise weights in the sum of columns of , which is the returned vector of . We want to show that we cannot in general choose and such that this column sum is equal to , thereby showing that Condition a) is not fulfilled in general. Now, the rank of the Jacobian matrix of is at most for all . Hence all are critical points and, due to Sard’s theorem, has measure zero in . This means that Condition a) is only fulfilled for a set of measure zero of ’s.
We know that for all but a set of measure zero of the initial points , the matrix —the matrix that the states converge to—has full column rank. It holds that has measure zero. We know that Condition a) is only fulfilled for a set of measure zero of ’s, see above. Thus we can conclude, has measure zero.
Now, we prove that is nowhere dense in . We prove this by showing that for any neighborhood in , the set is not dense in . Note that is open in .
Consider an arbitrary neighborhood in . There must be a the point , since the set has measure zero. Now, since , the following holds (each statement in the following list continues on the previous ones): there is such that for initialized at the point at time , the ’s converge to ; there is a closed ball with radius centered at ; there is a finite time after which , where is the closed ball around with radius . The two balls above are defined with respect to the Frobenius norm.
Now, define , which is a continuous function on . This means that the composite function is continuous in on (we treat and the initial condition as parameters). Furthermore, the function is strictly positive on and attains a minimum there. Now choose such that . Due to the continuous dependency theorem on initial conditions, there is such that if , then for all . For any point it holds that if is initialized at , then the ’s are contained after time . The ball is forward invariant under (7) since the set is contained in and is forward invariant, see Lemma 5 (2). Thus, for all times , all the ’s have full rank (as well as for the limit matrix) when the initial condition for was chosen to .
What we have shown is that there is a ball around for which no point is contained in , i.e., it is an interior point of as well as . This means that is not dense in . But was arbitrarily chosen in . Thus the proof is complete. We also obtain the result that is open in .
In the proof of (5) when we prove that has measure zero, we did not use the property that the matrix in the right-hand side of (8) is . That partial result also holds for any other matrix in . Furthermore, a straightforward generalization is that only for a measure zero set of the initial matrices in , any matrix block of with more rows than columns loses rank at a finite time.
There is a diffemorphism between and . For every in , provides the unique in such that for all . To see that it is a diffeomorphism we note the following. Let . The -matrix is the upper triangular matrix obtained by Cholesky’s factorization of . Cholesky’s factorization is analytic on the set of positive definite matrices Lee (2014). The -matrix is given by , where is analytic on the set of invertible matrices. As for , it holds that , where the right-hand side in the last equation is analytic in the two matrices.
Figure 1 illustrates relations between the matrix sets.
It holds that has measure zero. It also holds that set has measure zero, see (4) and (5) in Lemma 5. The proof amounts to showing that .
We know that has measure zero and that a measure zero set is mapped to a measure zero set under a diffeomorphism. Hence, if we show that , then we know that and that the set on the right-hand side has measure zero. Furthermore, open sets are mapped to open sets and is open. Thus is open if . This latter fact in combination with the fact that has measure zero can be used to conclude that is nowhere dense.
We first show that 1) and then show that 2) .
1) We assume that is in and define
where is the unique QR-decomposition of with positive diagonal elements for . These QR-decompositions are smooth according to Proposition 5.