Nicola De Carli
email: nicola.de-carli@irisa.fr

Paolo Salaris
email: paolo.salaris@unipi.it

Paolo Robuffo Giordano

P Robuffo

Multi-Robot Active Sensing for Bearing Formations

This paper proposes a novel distributed active sensing control strategy for formations of drones measuring relative bearings. To be able to localize their relative positions from bearing measurements, the drone formation must satisfy specific Persistency of Excitation (PE) conditions. We propose a solution that can meet these PE conditions by maximizing the information collected from onboard cameras via a distributed gradient-based algorithm. Additionally, we also consider presence of a (concurrent) position-based formation control task using Quadratic Program-based control with Control Lyapunov Functions (CLFs). The results show that the inclusion of active sensing in the formation control law enhances the localization accuracy and, as a consequence, the precision of reaching the desired formation. The improvement is especially important when the underlying graphs are not Infinitesimally Bearing Rigid (IBR), as it can be expected.

I. INTRODUCTION

Cooperative localization from relative sensing is a relevant topic in the multi-robot community [START_REF] Krick | Stabilisation of infinitesimally rigid formations of multi-robot networks[END_REF]- [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF], it is especially relevant in GPS-denied environments, such as indoors. One specific area of interest is cooperative localization from bearing measurements [START_REF] Schiano | A rigiditybased decentralized bearing formation controller for groups of quadrotor uavs[END_REF]- [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF], primarily due to the ease of retrieving relative bearings using onboard cameras. Much of the previous work in this domain has been built around the notion of Infinitesimal Bearing Rigidity (IBR) [START_REF] Schiano | A rigiditybased decentralized bearing formation controller for groups of quadrotor uavs[END_REF], [START_REF] Zhao | Bearing-only formation tracking control of multiagent systems[END_REF]. IBR characterizes the conditions, primarily related to graph topology, under which a group of drones measuring constant relative bearings can localize themselves up to a global translation and scaling factor. In case of constant bearings, to resolve the scale ambiguity, a distance measurement among a pair of robots is necessary.

Maintaining formation rigidity can be, however, quite restrictive in terms of motion flexibility, especially when the presence of edges among robots depends on sensing constraints such as limited range, occlusions, and limited field of view (FoV). Recently, in [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF], [START_REF] Tang | Localization and tracking control of autonomous vehicles in time-varying bearing formation[END_REF] the notion of Bearing Persistently Exciting (BPE) formations has been introduced, where at least a subset of the relative bearings is time-varying. By leveraging the concept of PE, this allows resolving the scale ambiguity and to relax the IBR assumption. A distributed observer is also proposed with Uniform Exponential (UE) convergence under the condition of the formation being BPE.

However, there remains the problem of how to ensure the BPE of the formation. Although a coordinated rotation was proposed as a reference trajectory satisfying the BPE condition [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF], this strategy may not be desirable in practical applications where more generic motions are needed. An alternative approach proposed in this work is to instead make use of an active sensing strategy that enables the optimization of robot motion for gathering informative measurements about the system state [START_REF] Krener | Measures of unobservability[END_REF]- [START_REF] Napolitano | Informationaware lyapunov-based mpc in a feedback-feedforward control strategy for autonomous robots[END_REF]. This can be exploited for producing a group motion that can (i) actively satisfy the necessary BPE conditions and (ii) be generic enough for fulfilling at the same time other tasks of interest related to the particular mission.

In the past, active sensing has been applied to the cooperative localization problem, e.g., in [START_REF] Ny | Localizability-constrained deployment of mobile robotic networks with noisy range measurements[END_REF]- [START_REF] Cano | Ranging-based localizability-constrained deployment of mobile robotic networks[END_REF]. where distance measurements together with rigidity and the presence of multiple anchors (i.e. robots knowing their poses) was assumed. An active sensing strategy was then used to steer the formation towards the best geometry from the localization point of view, without considering any PE condition. In [START_REF] Riz | Perception-aware trajectory planning for a pair of unicycle-like robots with absolute and relative ranging measurements[END_REF], the authors considered the active information gathering for ensuring observability of a multi-robot system with two anchors and range measurements. In our previous work [START_REF] Carli | Online decentralized perception-aware path planning for multi-robot systems[END_REF], we considered a formation with relative distance measurements, and we maximized the observability of the system over a future prediction horizon. The proposed method could be made distributed but at the cost of being suboptimal w.r.t. the centralized one and it was computationally heavy. In [START_REF] Spica | Active decentralized scale estimation for bearing-based localization[END_REF], an active sensing strategy to estimate the scale of an IBR formation was proposed. Most of the previous works mainly consider relative distance measurements instead of bearings, which are instead quite interesting from an application point of view. Some of these previous works also solely focus on optimizing the geometric arrangement of the group instead of the robots trajectories. In addition, the assumption of rigid formations is also often exploited (which, as explained, can be quite restrictive in terms of group mobility when considering sensing constraints).

In this paper, we propose a gradient-based control law that can address the above shortcomings. We consider maximization of the group observability by optimizing a global information measure: the derivative of the minimum nonzero eigenvalue of the weighted Observability Gramian (OG). This approach is simpler and computationally cheaper than receding horizon strategies and it allows maximizing a global quantity in a distributed way. Additionally, we address the coupling of the active cooperative localization strategy with presence of an independent primary task (formation control) whose performance is however enhanced by the active sensing action. This is obtained by leveraging distributed Quadratic Programming (QP)-based Control Lyapunov Functions (CLFs) [START_REF] Ames | Control barrier functions: Theory and applications[END_REF]. The combination of active sensing and CLFs has previously been successfully applied to single robot scenarios employing a receding horizon strategy [START_REF] Napolitano | Informationaware lyapunov-based mpc in a feedback-feedforward control strategy for autonomous robots[END_REF]. We believe that the approach we propose is a promising direction as it allows to achieve cooperative tasks while optimizing the trajectories to enhance the observability of the formation.

The rest of the paper is organized as follows: Section II introduces the main modeling assumptions and the employed observer. Section III presents a gradient-based active sensing controller. Section IV outlines a strategy to integrate active sensing into a position-based formation control task, accommodating not necessarily bearing rigid formations. Section V presents statistical results on various graph topologies, and finally, Sect. VI concludes the paper.

II. MULTI-ROBOT SYSTEM MODEL

A. Graph Theory

We consider a group of N connected drones. The drone sensing and communication interactions are modeled by a constant connected and undirected graph G := (V, E), where V = {1, ..., N } is the set of vertices and E ⊆ V × V is the set of edges. Two drones i and j are called neighbors if (i, j) ∈ E. The set of neighbors of drone i is denoted by N i := {j ∈ V|(i, j) ∈ E} and its cardinality is |N i |. With a slight abuse of notation, we may refer to variables related to the edge e k := (i, j) with a subscript k or ij. The incidence matrix for an arbitrary orientation of the graph is denoted by E, with

[E] ik = 1 if vertex i is the head of the edge k and [E] ik = -1 if it is the tail, [E] ik =
0 otherwise. The Laplacian matrix associated to the graph G can be computed as L = EE T , while a weighted Laplacian is given by L W = EWE T , with W ⪰ 0 (i.e. positive semidefinite). The Laplacian matrix is positive semidefinite with L1 N = 0, i.e. it has zero row-sum, and it has positive diagonal and non-positive off-diagonal entries. Its eigenvalues are 0 = µ 1 ≤ µ 2 ≤ ... ≤ µ N with eigenvectors {1 N / √ N , v 2 , ..., v N } and µ 2 > 0 if and only if the graph is connected.

B. Formation Model

Each drone state is represented by its position p i ∈ R 3 and we refer to the stack of the drones position as

p = [p T 1 . . . p T N ]
T ∈ R 3N . The positions and linear velocities of the drones are expressed in a common inertial frame and the velocity of drone i is indicated as u i ∈ R 3 while the stack of the drones velocities as

u = [u T 1 . . . u T N ] T ∈ R 3N .
The dynamics of each drone is modeled as a simple single integrator:

ṗi = u i . ( 1
)
Each drone is able to measure the relative bearing with respect to its neighbors, i.e.:

β ij := p ij d ij ∈ S 2 (2)
where p ij = p j -p i and d ij = ∥p ij ∥ 2 . We also define the stack of bearing measurements

β = β T 1 ... β T |E| T ∈ R 3|E|
. The inclusion of sensing limitations is left as a future work.

C. Localization of Bearing Formations

The matrix-valued Laplacian [START_REF] Pan | Consensus on matrixweighted time-varying networks[END_REF] associated to the classical position-based formation control is given as L 3 := E 3 E T 3 [START_REF] Cortés | Coordinated control of multi-robot systems: A survey[END_REF], with E 3 = E ⊗ I 3 where ⊗ is the Kronecker product and I 3 the identity matrix. If the graph is connected, then rank(L 3 ) = 3N -3 and the null-space is N (L 3 ) = U, where U = 1 N ⊗ I 3 represents a basis for a common 3D translation of the formation. For bearing formations the bearing Laplacian matrix is defined as

L β := E 3 diag(Π β k )E T 3 , with diag(Π β k ) being the block diagonal matrix with blocks Π β k := I 3 -β k β T k ⪰ 0, which is an orthogonal projector, i.e. Π β k Π β k = Π β k and Π β k = Π T β k . A bearing forma- tion in R 3 is defined infinitesimally bearing rigid (IBR) if rank(L β ) = 3N -4.
IBR is a necessary and sufficient condition to localize the relative positions of the robots up to a scale factor from constant bearing measurements. As it was shown in [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF], this condition can be relaxed if the bearings are time-varying, implying the IBR condition is not anymore necessary and the scale ambiguity can be resolved. We give some necessary definitions before proceeding.

Definition II.1. Given a system with state x ∈ R d , state sensitivity matrix Φ(t, t 0 ) = ∂x(t) ∂x(t 0 ) , output y(t), and output Jacobian H(t) = ∂y(t) ∂x(t) , the Observability Gramian (OG) is defined as follows [START_REF] Krener | Measures of unobservability[END_REF]:

G(t) := G(t 0 ) + t t 0 Φ(τ, t 0 ) T H(τ ) T H(τ )Φ(τ, t 0 ) dτ (3)
Observability of the system linearized along the nominal trajectory is characterized by the invertibility of the OG [START_REF] Bernard | On the semi-global stability of an ek-like filter[END_REF], i.e. the state can be observed from the measured output y(t) if there exists some ϵ > 0 such that:

t t 0 Φ(τ, t 0 ) T H(τ ) T H(τ )Φ(τ, t 0 ) dτ ⪰ ϵI d . ( 4
)
Definition II.2.

[5] A formation is defined bearing persistently exciting (BPE) if the graph G is connected and given T > 0 and γ > 0 the following PE condition holds:

1 T t+T t L β (τ ) dτ ⪰ γL 3 (5)
It should be noted that this condition is less stringent than the classical PE condition since dim(N (L 3 )) = 3.

We point out that the previous integral in ( 5) represents a weighted Observability Gramian (OG) of the system. In fact, the state sensitivity matrix for a single integrator is the identity matrix and the output Jacobian is the so-called bearing rigidity matrix, i.e.:

B = ∂β ∂p = diag Π β k d k E T 3 . (6)
From ( 4) and ( 6), using the properties of the orthogonal projector, it follows that the OG of the system can be expressed as:

G(t) = G(t 0 ) + t t 0 E 3 diag Π β k (τ ) d k (τ ) 2 E T 3 dτ (7)
where we see the similarity to [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF], in which the information associated to each edge is weighted by the squared distance.

Since the scaling by the positive distances does not affect the rank of the matrix, it follows that the formation being BPE is a local observability condition of the subspace orthogonal to the common translation. We now introduce the observer presented in [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF], which we use for localizing the drones in a bearing formation. In [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF], it is shown that, if the formation is BPE, the bearings are always well-defined (i.e. no collisions occur) and the velocities are bounded, then the following distributed observer achieves UE convergence to the true state up to a common translation:

ṗ(t) = u(t) -k e L β (p(t))p(t) (8)
where p is the estimated position vector and k e > 0.

The convergence rate of this observer depends on the BPE parameter γ [START_REF] Tang | Relaxed bearing rigidity and bearing formation control under persistence of excitation[END_REF]. The aim of the following section is to then provide a control law that aims at indirectly increasing the value of γ for ensuring a satisfactory observer performance even for non-rigid bearing formations.

III. ACTIVE SENSING CONTROL

In this section, we propose a gradient-based control law for actively collecting information about the drone relative positions. To quantify the amount of information, we adopt the minimum nonzero eigenvalue, specifically the fourth eigenvalue denoted as λ 4 , of the weighted OG (5) but with an additional forgetting factor, that is, an OG with dynamics

Ġβ (t) = -ρG β (t) + L β (t). (9)
The forgetting factor ρ > 0 ensures that the OG remains bounded, since otherwise the term L β ⪰ 0(t), ∀t ≥ 0 would make the OG grow unbounded over time. Notice that if G β (t 0 ) is a matrix-weighted Laplacian [START_REF] Pan | Consensus on matrixweighted time-varying networks[END_REF] associated to the graph G, then this also holds for G β (t), ∀t ≥ t 0 , that is, (i) the required definiteness pattern of the matrix weights (negative semidefinite off-diagonal terms) is preserved:

[G β (t)] (ij) = e -ρ(t-t 0 ) [G β (t 0 )] (ij) + t t 0 e -ρ(t-τ ) [L β (τ )] (ij) dτ = e -ρ(t-t 0 ) [G β (t 0 )] (ij) - t t 0 e -ρ(t-τ ) Π β ij (τ ) dτ ⪯ 0 (10)
where

[G β (t 0 )] (ij) ⪯ 0 is the ij-th block of G β (t 0 ) and Π β ij ⪰ 0; (ii) U ⊆ N (G β (t)): G β (t)U = e -ρ(t-t 0 ) G β (t 0 )U+ t t 0 e -ρ(t-τ ) L β (τ )U dτ = 0 3N ×3N (11)
and;

(iii) if ij / ∈ E then [G β (t)] (ij) = 0 ∀t ≥ 0.
These properties are used in the next derivations. In particular, the preservation of the sparsity pattern allows for a distributed implementation of the proposed control law.

Consider the state of the full system ζ = p T vec G β T T , where vec(•) is the vectorization operator. Then, the system dynamics can be written as:

ζ(t) = f (ζ) + g u ( 12
)
where the system has a cascade structure with

f (ζ) = 0 T 3N vec Ġβ T T
and g = I 3N 0 3N ×(3N ) 2 T . Note that the first derivative of λ 4 (our 'information metric') depends on the positions of the drones, so that λ 4 has relative degree 2 w.r.t. the drone velocities (the available control inputs). This poses a challenge for direct control since the drone positions are solely involved in the dynamics of λ 4 within the orthogonal projector Π β . However, the derivative of λ 4 can be more easily controlled. This has expression

λ4 = L f λ 4 = (v 4 ⊗ v 4 ) T vec( Ġβ ) = - (i,j)∈E (v 4i -v 4j ) T [ Ġβ (t)] (ij) (v 4i -v 4j ) (13)
where

L f λ 4 = ∂λ4 ∂ζ f (ζ)
is the Lie derivative of λ 4 along f (analogously we also define L gi ) and

v 4 = v 41 . . . v 4N
T is the eigenvector associated to λ 4 . For convenience, let us define

v 4ij := v 4i -v 4j .
The second order derivative of λ 4 is then

λ4 = N i=1 Lg i L f λ 4 u i + L 2 f λ 4 = - 1 2 N i=1   j∈N i [v 4ij ⊗ v 4ij ] T ∂ vec([ Ġβ ] ij ) ∂p i   u i + vec( Ġβ ) T H λ vec( Ġβ ) -ρ(v 4 ⊗ v 4 ) T vec( Ġβ ) (14)
where

H λ := ∂ 2 λ4 ∂ vec(G β ) vec(G β )
T is the hessian of λ 4 with respect to the matrix entries. The important point to note here is that ( 14) has a direct (affine) dependence on the control inputs u i , which can then be exploited for controlling the quantity λ4 , for instance by applying the gradient-based control law

ui := k (Lg i L f λ4) T . (15)
The control action [START_REF] Spica | Active decentralized scale estimation for bearing-based localization[END_REF] implements the sought active sensing since it aims at maximizing λ4 that, indirectly, maximizes the eigenvalue λ 4 itself (which is the metric we care about, but which is less directly controllable by acting on the control inputs u i ).

We now discuss some properties of the control law [START_REF] Spica | Active decentralized scale estimation for bearing-based localization[END_REF].

Proposition III.1. From an information perspective, it is intuitive that the gradient of λ4 associated to an edge is orthogonal to the corresponding bearing, i.e.:

[v 4ij ⊗ v 4ij ] T ∂ vec([ Ġβ ] ij ) ∂p i β ij = 0 (16)
Proof. First of all, notice that

[ Ġβ ] (ij) = -ρ[G β ] (ij) -Π β ij
and the first term does not depend on p i . Furthermore, using the properties of the Kronecker product, we have

vec(β ij β T ij ) = β ij ⊗ β ij . It follows that: ∂ vec(Π β ij ) ∂p i = - ∂(β ij ⊗ β ij ) ∂p i = 1 d ij Π β ij ⊗ β ij + β ij ⊗ Π β ij (17)
One can then use the Kronecker product properties and show that:

Π β ij ⊗ β ij + β ij ⊗ Π β ij β ij = Π β ij ⊗ β ij (β ij ⊗ 1) + β ij ⊗ Π β ij (1 ⊗ β ij ) = Π β ij β ij ⊗ β ij + β ij ⊗ Π β ij β ij = 0. ( 18
)
Proposition III.2. Assuming that no collisions happen i.e. d ij ≥ D min > 0, ∀i, j ∈ {1, 2, ..., N }, then the resulting input is always bounded.

Proof. w.l.o.g. considering k = 2:

u T i = j∈N i [v 4ij ⊗ v 4ij ] T 1 d ij Π β ij ⊗ β ij + β ij ⊗ Π β ij (19)
and

∥u i ∥ 2 ≤ j∈N i ∥v 4ij ⊗ v 4ij ]∥ 2 2 D min Π β ij ⊗ β ij 2 ≤ 8 D min |N i | (20)
where we used the subadditivity property of norms, the facts that

∥v 4i ∥ 2 ≤ ∥v 4 ∥ 2 = 1, ∥X ⊗ Y∥ 2 = ∥X∥ 2 • ∥Y∥ 2 , ∥Π β ∥ 2 = 1 and β ij 2 = 1.
Finally, the active sensing gradient control (15) requires each robot i to only know quantities which are locally available or communicated by neighboring robots, i.e.,

• v 4i : the components of the eigenvector corresponding to the i-th robot itself, which can be estimated in a distributed way by suitably modifying the distributed power iteration method (see e.g. [START_REF] Yang | Decentralized estimation and control of graph connectivity for mobile sensor networks[END_REF], [START_REF] Malli | Robust distributed estimation of the algebraic connectivity for networked multi-robot systems[END_REF]) as done e.g. in [START_REF] Zelazo | Decentralized rigidity maintenance control with range measurements for multi-robot systems[END_REF]; • v 4j ∀j ∈ N i : which can be communicated by neighboring robots; • d ij : for which an estimate dij can be computed from the estimated positions; pi , locally available, and pj which can be communicated by the neighbors • β ij which is measured. In the next section, we show how to embed the active sensing action (15) within a primary task of formation control.

IV. FORMATION CONTROL

We establish a hierarchical framework that prioritizes position-based formation control over active sensing by using distributed QP-based Control Lyapunov Functions [START_REF] Ames | Control barrier functions: Theory and applications[END_REF]. We now formally introduce the definition of CLFs which will be instrumental for the next developments.

Definition IV.1. A function α : R ≥0 → R ≥0 is of class K if it is strictly increasing and α(0) = 0 Definition IV.2. [23] A smooth positive definite function V : R d → R is a Control Lyapunov Functions (CLF) for a given system ẋ = f (x) + g(x)u if it satisfies: inf u∈R m {L f V (x) + L g V (x)u} ≤ -α(V (x)) (21)
with α of class K.

The position-based formation control task considered in this work is defined based on the error associated with each edge, denoted as e ij = p j -p i -p d ij . Here, p d ij represents the desired relative position between robot i and j. We also define the desired position vector as

p d = [(p d 1 ) T ... (p d N ) T ]
T , which relates to the relative desired position vector

p d E = stack(p d ij ) through p d E = E T 3 p d .
It is worth noting that p d is defined up to a common translation. Then, the position error

is e = p -p d = [e T 1 ... e T N ]
T with dynamics ė = uṗd .

We consider the following potential function:

V (e) = 1 2 (i,j)∈E ∥e i -e j ∥ 2 2 . ( 22
)
We emphasize that, while this function is only positive semidefinite i.e. V (w) = 0 for any vector w ∈ span(U) representing a common translation, it is positive definite with respect to the desired equilibrium set {e 1 = e 2 = ... = e N }. Let us de-

fine c i (ζ) := (Lg i L f λ 4 ) T / Lg i L f λ 4 2 if Lg i L f λ 4 2 ̸ = 0 and c i (ζ) = 0 otherwise (vector c i (ζ)
is thus the unit-norm direction of the active sensing control ( 15)). The centralized QP, including the active sensing task, is formulated as:

min u i ,i=1,...,N . 1 2 N i=1 c i (ζ) T (u i -kc i (ζ) 2 + η ∥Πc i u i ∥ 2 2 s.t. N i=1   j∈N i (e i -e j )   T u i + 1 2 (i,j)∈E α i (∥e i -e j ∥ 2 2 ) ≤ 0 ( 23
)
where Π ci = I 3 -c i c T i is the orthogonal projector onto the plane perpendicular to c i , η > 0 and α i (•) is an extended class K.

The first term in the cost function aims at achieving the same information gain as the one obtained by u i = kc i (ζ) and, therefore, it represents the active sensing task. The second (regularization) term in the cost function is meant to address two issues, namely (i) avoiding excessive inputs in the direction orthogonal to the active sensing task due to constraint satisfaction, and (ii) obtaining a strongly convex cost function. Indeed, the Hessian of the QP is given by

H QP := diag(H QP,i ) = diag(c i c T i + ηΠc i ) ≻ 0.
Our objective now is to transform this problem into one that is suitable for distributed implementation. Inspired by [START_REF] Tan | Distributed control barrier functionbased control scheme for multi-agent systems under a collective constraint[END_REF], we compute the analytical expression for the solution of the QP. First, we define a = [a T

i := 1 4 j∈Ni α(∥e i -e j ∥ 2
2 ), so that the constraint can be written as

N i=1 (a T i u i + b i ) ≤ 0.
We point out that, in absence of input limits, the constraint is always feasible, i.e. a = 0 implies b = 0, in which case the constraint is trivially satisfied. The analytical solution to the QP can be obtained by using the Karush-Kuhn-Tucker (KKT) optimality conditions. Let us define the Lagrangian of the problem

L := 1 2 N i=1 c i (ζ) T (u i -kc i (ζ) 2 + η ∥Πc i u i ∥ 2 2 + λ L (a T u + b), (24)
with λ L being the Lagrange multiplier. The resulting KKT conditions, using * to indicate the optimal solution, are:

         ∂L ∂u i = (u * i -kc i ) T c i c T i + η(u * i ) T Πc i + λ * L a T i = 0 ∀i N i=1 (a T i u * i + b i ) ≤ 0 λ * L ≥ 0 λ * L = 0 if N i=1 (a T i u * i + b i ) < 0 . (25)
From the first equation one obtains:

u * i = H -1 QP,i (kc i -λ * L a i ) (26)
from which two cases are possible: 1) the constraint is not active at the unconstrained solution kH -1 QP,i c i , hence from the last condition one has λ * L = 0 and u * i = kH -1 QP,i c i ; 2) the constraint is active, hence substituting (26) into the constraint equation with equality yields

λ * L = N i=1 (a T i H -1 QP,i kc i + b i ) N i=1 a T i H -1 QP,i a i . (27)
Substituting back in [START_REF] Freeman | Stability and convergence properties of dynamic average consensus estimators[END_REF], one obtains the complete solution for the two cases as:

u i = kH -1 QP,i c i -max 0, N i=1 (ka T i H -1 QP,i c i + b i ) N i=1 a T i H -1 QP,i a i H -1 QP,i a i . (28)
Since H QP,i ≻ 0, a zero denominator implies a i = 0 ∀i, that is, accomplishment of the formation control task. Hence, in this case the input can be set to zero. Otherwise, the first term represents the solution to the unconstrained problem. The term in the numerator

N i=1 (ka T i H -1 QP,i c i + b i
) is negative when the constraint is satisfied by the solution to the unconstrained problem. Since the term at denominator is always greater than zero, then the second term is different from zero only when the constraint is active and it acts to correct the input in order to satisfy the constraint. Also, notice that the proposed controller is piecewise (because of the eigenvalues re-ordering) locally Lipschitz continuous [START_REF] Xu | Robustness of control barrier functions for safety critical control[END_REF].

The two terms

N i=1 (ka T i H -1 QP,i c i + b i ) and N i=1 a T i H -1
QP,i a i are not locally available but one can estimate their average through dynamic consensus, e.g. [START_REF] Freeman | Stability and convergence properties of dynamic average consensus estimators[END_REF]. Finally notice that the terms a i and b i depend on the estimated positions of robot i and its neighbors, which must then be estimated as accurately as possible to correctly achieve the formation task. This is achieved by the gradient-based active sensing strategy embedded in the cost function of [START_REF] Khalil | Nonlinear control[END_REF].

V. SIMULATION RESULTS

In this section, we show the effectiveness of our approach through extensive numerical simulations. We compare the results obtained applying the proposed control law (28), which achieves the desired formation while performing active sensing against a control law which only implements formation control by satisfying the constraint in [START_REF] Khalil | Nonlinear control[END_REF] while minimizing the input norm. For convenience, we refer to the two methods, respectively, as AS and CLF-only. We consider a group of N = 7 drones, which can sense the neighboring robots without any sensing constraints and three different graph topologies with different levels of connectivity. We performed a set of 50 simulations starting from random initial positions for each of the following graph topologies (ordered in increasing connectivity level): line graph, cycle graph and 1-redundantly bearing rigid (RBR) graph [START_REF] Trinh | Minimal and redundant bearing rigidity: Conditions and applications[END_REF] (Fig. 1). Notice that the first two graphs are not bearing rigid. The drones initial estimated position is drawn from a gaussian distribution centered around the real position with standard deviation 0.8m 2 along each axis. As desired formation, we chose to have the drones equally spaced along a circle lying on a plane parallel to the x-y plane. The decrease of the CLF imposed in the constraint is the same for CLF-only and AS, hence, the converge speed of the formation is not of particular interest. We instead focus on the (more relevant) evaluation of the steady state formation error at convergence, and of the estimation error. In the simulations, we used the following parameters: the observer gain is k e = 0.1, the forgetting factor is ρ = 0.04, the OG is initialized as a matrix weighted Laplacian with matrix weights 0.5I 3 , the active sensing gain is k = 0.2, the regularization term is η = 0.01. We also implemented Control Barrier Functions (CBFs), which filters the inputs provided by the proposed controller to ensure collision-free trajectories [START_REF] Wang | Safety barrier certificates for collisions-free multirobot systems[END_REF].

We present the results for the line graph in Fig. 2, the cycle graph in Fig. 3, and the 1-redundant bearing rigid graph in The results clearly show that the use of active sensing leads to higher acquired information and, thus, a more accurate localization and, as a consequence, a better performance for the formation control. The use of active sensing provides benefits, particularly in scenarios where the connectivity is lower, as for the line and cycle graphs. In these cases, the active sensing task reduces both the estimation and formation errors. Conversely, in highly connected and rigid graphs like the 1-redundantly rigid graph, the improvement margin is quite small. Indeed, in this case, rigidity of the graph greatly simplifies the localization task since the robot group becomes "instantaneously localizable" (up to a scale factor). This is also evident from Fig. 4(c), where the information growth is considerably high even when active sensing is not used (CLF-only case).

VI. CONCLUSIONS AND FUTURE WORKS

In this paper, we have presented a novel distributed control strategy for bearing formations to maximize the information gathered for cooperative localization. Our active sensing strategy is based on the minimum eigenvalue of the weighted Observability Gramian with a forgetting factor as an information measure. This active sensing strategy can be combined with other additional tasks of interest (a position-based formation control task in our case) by making use of distributed QP-based control with Control Lyapunov Functions. This integration enables the execution of higherlevel tasks while concurrently enhancing the minimum level of "localizability" for the robot group, thereby improving the overall task execution.

Numerical simulations over graphs with different levels of connectivity demonstrated the benefits of the approach, in particular for non-rigid graphs. Future works will focus on including sensing constraints such as limited range and field of view, as well as accounting for the drones orientation. By addressing these factors, we can develop a more comprehensive and practical framework that considers the corresponding directed time-varying graph. Such advancements will enhance the applicability and robustness of our proposed strategy in real-world scenarios.

1 .

 1 a i := j∈Ni (e i -e j ), and b = N i=1 b i , with b

  (a) Line graph (b) Cycle graph (c) 1-RBR graph Fig. 1: Graphs corresponding to the sets of simulations.

Fig. 4 .

 4 In each case, the figures are arranged from left to right to display: (a) the trajectory of λ 4 (t), (b) the estimation error, (c) the formation error, (d) and (e) the violin plot respectively for the estimation and formation error at the end of the trajectory. As a reminder, λ 4 (t) is the minimum non-zero eigenvalue of the OG and it is a measure of the collected information which affects the worst case convergence rate of the employed localization observer. The plots (a), (b) and (c) show the average trajectories along with the standard deviation of the results across simulations.

  Fig. 2: Line graph

This work was supported by the ANR-20-CHIA-0017 project "MULTI-SHARED"


Kyohei Yoshida

Yutaka Kuwahara

Nanami Hano

Yumi Horie

Makoto Takafuji

Naoya Ryu

Shoji Nagaoka

Reiko Oda

Hirotaka Ihara
email: ihara@kumamoto-u.ac.jp

| Nanami Hano

| Shoji Nagaoka

Chiral H -aggregation-induced large stokes shift with CPL generation

Keywords: chiroptical, circularly polarized luminescence, enhanced chirality, induced circular dichroism, poly (amino acid)

| INTRODUCTION

Luminescent nanomaterials have been widely used in various applications, including optical signal processing, [START_REF] Mamun | Recent progress in nanomaterial enabled chemical sensors for wearable environmental monitoring applications[END_REF] fluorescence imaging, [START_REF] Ji | Organic dye assemblies with aggregation-induced photophysical changes and their bio-applications[END_REF] and environment-sensitive probes. [START_REF] Xue | Aggregation-induced emission nanofiber as dual sensor for aromatic amine and acid vapor[END_REF][START_REF] Caroleo | Advances in optical sensors for persistent organic pollutant environmental monitoring[END_REF] Doping fluorescent materials onto various carriers, such as colloidal silica, [START_REF] Li | Classification, synthesis, and application of luminescent silica nanoparticles: a review[END_REF][START_REF] Ihara | Luminescent supramolecular gel for light management technology[END_REF] magnetic particles, [START_REF] Fang | Multifunctional magneticfluorescent nanoparticle: fabrication, bioimaging, and potential antibacterial applications[END_REF] graphenes, [START_REF] Krishnan | A review on graphene-based nanocomposites for electrochemical and fluorescent biosensors[END_REF] selfassemblies, [START_REF] Zhang | Self-assembled fluorescent organic nanomaterials for biomedical imaging[END_REF][START_REF] Cao | Amplified polarization properties of electrospun nanofibers containing fluorescent dyes and helical polymer[END_REF] and polymer nanofibers, [START_REF] Kaerkitcha | A review on electrospun luminescent nanofibers: photoluminescence characteristics and potential applications[END_REF] is a common technique for obtaining them. Accordingly, extending the variants of these "luminophore-on-carrier" systems is an ongoing process to expand their potential applications.

Alternatively, the magnitude of the Stokes shift is a crucial component in defining the characteristics of luminescent nanomaterials. Using a luminophore with a significant Stokes shift, for instance, the excitation-inducing light and fluorescence can be separated in fluorescence microscopy of living cells. [START_REF] Sednev | Fluorescent dyes with large Stokes shifts for super-resolution optical microscopy of biological objects: a review[END_REF][START_REF] Gao | A fluorescent dye with large Stokes shift and high stability: synthesis and application to live cell imaging[END_REF] The capacity to alter the Stokes shift in spectral converters facilitates the matching of the solar cell wavelength pattern and manufacturing devices that effectively harvest solar light, [START_REF] Albano | Synthesis of new bis[1-(thiophenyl)propynones] as potential organic dyes for colorless luminescent solar concentrators (LSCs)[END_REF][START_REF] Albano | Photophysical properties of new p-phenylene-and benzodithiophene-based fluorophores for luminescent solar concentrators (LSCs)[END_REF] which increases the conversion efficiency of solar cells. [START_REF] Jintoku | Molecular gel-mediated UV-to-visible spectral conversion for enhancement of power-conversion efficiency[END_REF][START_REF] Jintoku | Molecular gelation-induced functional phase separation in polymer film for energy transfer spectral conversion[END_REF][START_REF] Li | Large stokes shift and high efficiency luminescent solar concentrator incorporated with CuInS 2 /ZnS quantum dots[END_REF] Accordingly, several strategies to augment the Stokes [This article is part of the Special issue: Chiral Materials. See the first articles for this special issue previously published in Volumes 34:12, 35:2, 35:3, 35:4, 35:5 and 35:6. More special articles will be found in this issue as well as in those to come.] shift have been developed, including the construction of an excited-state molecule with intramolecular proton transfer. [START_REF] Zhang | Alkyl chain-dependent ESIPT luminescent switches of phenothazine derivatives in response to force[END_REF] Another approach involves the formation of a donor and acceptor unit within the same molecule via chemical modification to induce fluorescence resonance energy transfer or through-bond energy transfer. [START_REF] Xue | Through-bond energy transfer cassette with dual-Stokes shifts for "double checked" cell imaging[END_REF][START_REF] Liu | A FRET chemosensor for hypochlorite with large Stokes shifts and long-lifetime emissions[END_REF] However, these approaches yield molecules with complicated architectures, necessitating complex synthesis and purification processes.

In this paper, we demonstrate a facile approach for increasing the Stokes shift without carrying out any chemical modification of the dyes. In particular, we focused on the phenomena of dye aggregation by utilizing the more pronounced absorption advantages of a blue-shifting H-aggregation over a red-shifting J-aggregation. In addition, red-shifting emission is anticipated to occur during excimer formation, whereas Haggregation demonstrates fluorescence quenching. [START_REF] Kasha | The exciton model in molecular spectroscopy[END_REF][START_REF] Deng | H-and J-aggregation of fluorene-based chromophores[END_REF][START_REF] Basak | Tailor-made design of Jor H-aggregated naphthalenediimide-based gels and remarkable fluorescence turn on/off behaviour depending on solvents[END_REF] Thus, we designed the chirally twisted H-aggregation of fluorescent dyes, employing a chiral host template comprising poly (lysine) ( pLys) that may form a righthanded α-helical structure by modifying external conditions such as pH (Figure S1). Consequently, the side chain amino groups that serve as electrostatic dye binding sites are ordered and chirally organized along the helical backbone. [START_REF] Shibata | Unique property of cyanine dyes on charged poly(L-lysine)[END_REF] On the other hand, our approach can have fascinating effects in addition to the large Stokes shift because it is known that chiral perturbation to nonchiral dyes can induce circular dichroism (CD) and circularly polarized luminescence (CPL). [START_REF] Liu | Supramolecular chirality in selfassembled systems[END_REF][START_REF] Albano | Chiroptical properties in thin films of π-conjugated systems[END_REF][START_REF] Pescitelli | ECD exciton chirality method today: a modern tool for determining absolute configurations[END_REF] Particularly, CPL generation, which is useful for chiroptical application such as chemical or biological sensors, [START_REF] Imai | A smart sensing method for object identification using circularly polarized luminescence from coordination-driven self-assembly[END_REF][START_REF] Yuasa | Fingerprint signatures of lanthanide circularly polarized luminescence from proteins covalently labeled with a β-diketonate europium (iii) chelate[END_REF] light emitting devices, [START_REF] Zhang | Recent advances in circularly polarized electroluminescence based on organic light-emitting diodes[END_REF] security devices, [START_REF] Mackenzie | Circularly polarized lanthanide luminescence for advanced security inks[END_REF] has been actively studied in recent years. Therefore, this study also focused on CPL generation. Figure 1 schematically illustrates our approach for realizing a blue shift due to chiral Haggregation of the dye with α-helical pLys, a large Stokes shift through a red shift due to excimer emission, and CPL generation based on chiral orientation of dyes.

| MATERIALS AND METHODS

| Materials

Poly(L-or D-lysine) hydrobromides, pLys (Mw: 70,000-150,000) [the polymer host], as well as NK2751, NK2012, and NK77 [fluorescent dyes] were purchased from Sigma-Aldrich and Hayashibara Co., Ltd., respectively. The other dyes such as pyrene-1-sulfonic acid, methyl orange, methylene blue, and acridine orange were purchased from Nacalai Tesque Inc. The dyes used in this study are shown in Figure S1. Spectrophotometric grade methanol and millipore-deionized water were used throughout this study. Additionally, all the chemicals were used as obtained, without further purification.

| Sample preparation

A typical dye-on-pLys system was prepared as follows. First, pLys from L-enantiomer and the dye (NK2751, NK77, methyl orange, pyrene-1-sulfonic acid, etc.) were dissolved separately in methanol. Each solution was then adjusted to respectively yield a 1 unit-mM pLys solution and 1 mM of dye solution. Subsequently, 50 μL of dye solution and 1 mL of pLys solution were combined to achieve a dye concentration of 5 μM. Using a similar procedure, water-methanol mixed solutions of varying compositions were prepared for the NK2751-on-pLys system.

| Measurements

Fluorescence (FL) spectroscopy was performed on a JASCO FP-6500 spectrofluorometer. UV-visible absorption and CD spectroscopy were performed using a JASCO V-630 spectrophotometer and a JASCO J-725 spectrodichrometer, respectively. In addition, the CPL spectra were acquired using a JASCO CPL-300 CPL spectrophotometer. Moreover, a long-pass filter (R60, HOYA) was used to reduce the excitation light during the CPL measurement. Additionally, all spectroscopic measurements were carried out at 20 C, in a cell with a 10 mm optical path length.

F I G U R E 1 Strategic approach used in the current study to achieve a large Stokes shift with induction of circular dichroism (CD) and circularly polarized luminescence (CPL) via a "dye-oncarrier" approach using α-helical poly (lysine) as a chiral template.

In this study, we expected that pLys, which possesses an ordered arrangement of side chain amino groups on a rigid and cylindrical structure based on α-helix, plays a crucial role in inducing chirality when nonchiral fluorophores were mixed. Accordingly, we experimentally applied seven fluorescent dyes (Figure S1) in righthanded α-helical pLys from L-enantiomers, to carry out a preliminary study, due to the electrostatic interactions occurring on the side chain functional groups on pLys. Based on the outcome, a disulfonic NK2751 dye was selected as one of the best candidates In contrast, the structurally identical NK2012 and NK77 dyes were not used for further research because NK2012 exhibited similar results to NK2751 as shown in Figure S2 but was relatively weak for UV-irradiation, and the cationic NK77 dye had no meaningful response, perhaps due to its cationic nature. Similarly, other cationic dyes such as acridine orange and methylene blue did not exhibit no significant spectral change. In contrast, only minor spectral changes were observed with monosulfonic dyes such as methyl orange and pyrene-1-sulfonic acid, making them unsuitable for our strategy. Based on these preliminary results, it was determined that the structure in which the two sulfonic acid groups linked to the planar structure of the dyes are bound to pLys-like double anchors matches the objective of this study.

Figure 2 shows the absorption and CD spectra of NK2751 in the visible region. As observed from the black line in Figure 2A, the maximum absorption of NK2751 in the absence of pLys was observed at 571 nm, corresponding to a monomeric dispersion state. Due to its lack of chirality in NK2751, no CD signal was detected around the absorption band (shown in the black line of Figure 2B). In contrast, for the same system including pLys, the maximum absorption distinctly shifted to the shorter wavelength of 104 nm (shown in the bold red line of Figure 2A). In addition, a strong Davidoff splitting of the CD pattern was found around the red-shifted absorption band of NK2751 (shown in the bold red line of Figure 2B). These spectral behaviors can be described by the dimer formation depicted in the inset of Figure 2B, where the red-shift is attributed to a head-to-head orientation, such as H-aggregation, and the chirally bimolecular stacks near the center of the molecule produce a split CD pattern. Accordingly, pLys interacted with NK2751 to induce chirality in the dye.

Figure 2C illustrates the FL spectra of NK2751. Excited at 571 nm and lacking pLys, a black-line spectrum with a maximum emission at 605 nm was produced for the system. Accordingly, the Stokes shift was observed to be 34 nm. In contrast, the maximum emission was seen at 705 nm when pLys was present and excited at 475 nm, as depicted by the bold red line in Figure 2C. In this instance, the Stokes shift reached 230 nm, indicating several unique features. First, the maximum emission wavelength was 100 nm longer than the monomer emission wavelength. Second, the FL intensity with the pLys system was comparable with half of what it would have been without the system. Finally, the emission band was much broader with pLys than it was without pLys, with respective half-value widths of 50 and 108 nm. These characteristics match those of an excimeric emission. Therefore, a large Stokes shift can be achieved through excimer formation based on H-aggregation with pLys.

Additionally, we explored a second matter. Generally, H-aggregation prevents electron transitions, resulting in a minuscule emission intensity based on the quenching mechanism with the exciton theory. [START_REF] Kasha | The exciton model in molecular spectroscopy[END_REF][START_REF] Deng | H-and J-aggregation of fluorene-based chromophores[END_REF][START_REF] Basak | Tailor-made design of Jor H-aggregated naphthalenediimide-based gels and remarkable fluorescence turn on/off behaviour depending on solvents[END_REF] However, our system is unique in that it can achieve both Haggregation and fluorescence and induce secondary chirality that brings CPL as discussed later. Here, compared with the previously reported examples on H-aggregation, Rösch et al. reported the formation of fluorescent, twisted H-aggregates with merocyanine derivatives while chirality induction was not realized in this system. [START_REF] Rösch | Fluorescent Haggregates of merocyanine dyes[END_REF] On the other hand, we recently reported twisted H-aggregates exhibiting fluorescence and chirality by immobilizing nonchiral dyes onto chiral lipid aggregates but were unable to achieve CPL. [START_REF] Ryu | Fluorescence emission originated from the H-aggregated cyanine dye with chiral gemini surfactant assemblies having a narrow absorption band and a remarkably large stokes shift[END_REF] In addition, as for the large blue shift and sharpening of the absorption peak, the H*aggregation mechanism has been proposed. [START_REF] Belko | H*-and J-aggregates of an indotricarbocyanine dye: steady-state spectral properties and excitedstate dynamics[END_REF] In this regard, our system may also apply this mechanism; however, the previously reported ones were nonfluorescent and did not include the importance of chirality. [START_REF] Belko | H*-and J-aggregates of an indotricarbocyanine dye: steady-state spectral properties and excitedstate dynamics[END_REF][START_REF] Herz | Aggregation of sensitizing dyes in solution and their adsorption onto silver halides[END_REF] Therefore, we speculate that both twisted H-aggregation and chiral configuration are important in realizing our special phenomenon.

The influence of pLys helicity on the H-aggregation of dyes was also investigated. In general, ionic repulsion between side chain groups is known to influence the conformational variations of poly (amino acid)s. [START_REF] Grigsbya | Effect of secondary structure on the potential of mean force for poly-L-lysine in the α-helix and β-sheet conformations[END_REF][START_REF] Gupta | Helix-coil transition in poly-L-lysine and poly-L-glutamic acid[END_REF] Accordingly, the degree of helicity can be adjusted by altering the polarity and pH of the solvent. Figure S3 depicts the CD spectra of pLys in the mixed solvent system. When methanol was utilized as the sole solvent, a typical righthanded α-helical CD pattern was observed. Since the material pLys employed in this work is an HBr salt, essentially no electrostatic repulsion was seen in methanol. However, the addition of water to the mixed solvent triggered a fall in helicity. The concentration dependence of the α-helicity content of water is illustrated in Figure 3. When more than 40% water was added, the helicity almost disappeared and became a random coil.

Figure 4 depicts the effect of water content on the visible, CD, and FL spectra for the NK2751-pLys mixed system. All spectroscopic results suggested that NK2751 dimerization diminished as the water content in the solvent increased. In particular, H-aggregation was eliminated (Figure 4A), and chirality and excimer emission disappeared (Figure 4B,C, respectively). Many studies have described the changes in the chiral stacking states due to the solvent changes. [START_REF] Resta | Natural α-amino acidfunctionalized poly (phenyleneethynylene)s (PPEs): synthesis and chiroptical characterization of aggregate states[END_REF][START_REF] Korevaar | Pathway complexity in π-conjugated materials[END_REF][START_REF] Hassan | Impact of chirality on the aggregation modes of L-phenylalanine-and Dglucose-decorated phenylene-thiophene oligomers[END_REF] However, in our system, the addition of water to the solvent induced α-helix-torandom coil transition of pLys as a primary effect. Therefore, the main factor of the spectral change due to the change of solvent is considered to be the dimer-tomonomer transition of dyes on pLys instead of the change of solvation. Considering that no such effects were observed with monosulfonic acid dyes such as methyl orange and pyrene-1-sulfonic acid, it is estimated that the disulfonic acid structure of NK2751 plays a significant role, as it acts as a double anchor to attach to the two amino groups of the lysine moiety. As depicted in the inset of Figure 3, the chiral H-aggregation of NK2751 can only be produced by the α-helical conformation of pLys. In contrast, NK2751 in pure water exhibited a visibly distinct spectrum in the presence and absence of pLys (Figure S4), but no CD was detected (Figure S5). Therefore, NK2751 interacted with randomly coiled pLys in pure water without promoting dimerisation.

As our system generates both chirality and fluorescence, applications to circularly polarized light emission (CPL) need to be addressed. As a chiroptical light source, CPL represents one of the most recent periods of technological advancement in optical science. Various CPL generation compounds with precise molecular designs have been produced in the last decade. [START_REF] Morisaki | Planar chiral tetrasubstituted [2.2]paracyclophane: optical resolution and functionalization[END_REF][START_REF] S Anchez-Carnerero | Circularly polarized luminescence from simple organic molecules[END_REF][START_REF] Homberg | Combined reversible switching of ECD and quenching of CPL with chiral fluorescent macrocycles[END_REF][START_REF] Tanaka | Circularly polarized luminescence and circular dichroisms in small organic molecules: correlation between excitation and emission dissymmetry factors[END_REF][START_REF] Takaishi | Chiral exciplex dyes showing circularly polarized luminescence: extension of the excimer chirality rule[END_REF][START_REF] Baciu | Dithia[9]helicenes: molecular design, surface imaging, and circularly polarized luminescence with enhanced dissymmetry factors[END_REF][START_REF] Ouyang | Intramolecular energy and solvent-dependent chirality transfer within a BINOL-perylene hetero-cyclophane[END_REF][START_REF] Zullo | Tunable excimer circularly polarized luminescence in isohexide derivatives from renewable resources[END_REF] However, these approaches combine elements both of fluorescence and chirality into a single molecule, resulting in a lengthy and complex synthetic pathway and purifying process. To solve this issue, a new way of combining nonchiral fluorescent agents with a chiral template has been proposed and developed. [START_REF] Goto | Induction of strong and tunable circularly polarized luminescence of nonchiral, nonmetal, low-molecular-weight fluorophores using chiral nanotemplates[END_REF][START_REF] Jiang | Circularly polarized luminescence of achiral cyanine molecules assembled on DNA templates[END_REF][START_REF] Han | Full-color tunable circularly polarized luminescent nanoassemblies of achiral AIEgens in confined chiral nanotubes[END_REF][START_REF] Yang | Highly efficient full-color and white circularly polarized luminescent nanoassemblies and their performance in light emitting devices[END_REF][START_REF] Harada | Lanthanide ion-doped silica nanohelix: a helical inorganic network acts as a chiral source for metal ions[END_REF][START_REF] Zhang | Circularly polarized luminescence of agglomerate emitters[END_REF] The NK2751-on-pLys system is a strong candidate for CPL generation using this method. Figure 5 depicts an example of two CPL spectra in the presence of pLys from L-or D-enantiomers. Excitation at the absorption maximum of the H-aggregates produced a unique CPL with a mirror image between the L-and Denantiomers. The emission area of the resulting CPL is nearly identical to that of the NK2751 excimer. The g lum value, which refers to the optical purity of CPL, is roughly 2 Â 10 À3 , which is not particularly high. However, it is the first CPL generation achieved by assisting with α-helical poly (amino acid), and this achievement is noteworthy as a new step forward in CPL research.

| CONCLUSION

Using functional groups arranged chirally on the stiff and cylindrical structure of poly(a-amino acid) and the way of binding nonchiral fluorescent agents to it, a new method for achieving a significant Stokes shift was established in the current study. By selecting the appropriate dye, Haggregates were produced on the chiral template, which altered the excimer emission's absorption wavelength to shorter wavelengths and emission wavelength to longer wavelengths. In addition, we successfully demonstrated the efficient luminescence of H-aggregates, where the phenomenon of quenching is prevalent. In other words, the chiral twisted H-aggregation of the dyes allows for the necessary electron transfer for light emission. In addition, because our system is capable of both fluorescence and chirality, we have succeeded in developing CPL generation, which is likely to garner significant attention in optical research. It is, therefore, anticipated that additional research on this system will demonstrate the significant development potential of the suggested technology for CPL generation.

F

  I G U R E 2 UV-visible (A), circular dichroism (CD) (B), and fluorescence (FL) (C) spectra of NK2751 in methanol at 20 C. The bold red lines and the black lines show the results obtained with and without pLys, respectively. The concentrations of NK2751 and pLys were fixed to 5 μM and 0.1 mM, respectively. The FL spectra (C) were obtained by excitation at 475 and 575 nm with and without pLys, respectively.

F

  I G U R E 3 Concentration dependencies of water on the α-helicity of pLys (left vertical axis) and the dimer-monomer ratio of NK2751 (right vertical axis).

F

  I G U R E 4 Concentration dependencies of water on visible (A), circular dichroism (CD) (B), and fluorescence (FL) (C) spectra for the NK2751-pLys mixed system at 20 C. The concentrations of NK2751 and pLys were fixed to 5 μM and 0.1 mM, respectively. The excitation wavelength was performed at 475 nm for the FL spectra.

ACKNOWLEDGMENTS

This work was supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan and by the Laboratoire International Associé based on Chiral Nanoobjects for Photonic Application (LIA-CNPA) in Centre National de la Recherche Scientifique (CNRS) Universite de Bordeaux-Bordeaux INP-Kumamoto University-Kyoto University.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon request. F I G U R E 5 Circularly polarized luminescence (CPL) (A) and fluorescence (FL) (B) spectra of NK2751 in methanol solutions with pLys from L-or D-enantiomers. Temperature: 20 C, excitation wavelength: 475 nm, [NK2751] = 25 μM, [pLys] = 0.5 mM. Gray lines in (A) indicate their CPL spectra without smoothing. The CPL and FL spectra were measured using different spectrometers with JASCO CPL-300 and JASCO FP-6500, respectively.

SUPPORTING INFORMATION

Additional supporting information can be found online in the Supporting Information section at the end of this article.