## I Introduction

Decentralized control strategies for multiple vehicles have gained increased attention in the last decades in the control community [4], [14], [16]. In particular, when the configuration space of the agents is on a Lie group the main applications involved the coordination and synchronization of spacecraft motions modeled by kinematic systems [18], [19]. Recently, researchers have shown an inceinterest in employing decentralized motion planning algorithms for multi-agent systems based on second-order dynamical models [8], [9]. The main motivation lies in that acceleration controls are more implementable in vehicle systems than velocity controls.

In this work we consider a set of agents evolving on a Lie group subject to collision avoidance constraints. We determine whether there are non-trivial trajectories in the collision avoidance problem of all agents that maintain the constraints. The results are applied to build a collision avoidance motion planning controller for the coordinated motion of the agents. We assume that the constraints for the distributed edge set should be non-conflicting, and the overall constraint for all the edges should be realizable in the full Lie group. The proposed mathematical framework for multi-agent systems on Lie groups was recently used in [6] for optimal control problems. We also build on the works [21], [20] by studying the problem of motion feasibility when agents evolves on Lie group manifolds.

The motion feasibility problem is studied in two different scenarios: when agents are described by kinematic left-invariant fully actuated control systems and when the agents are described by dynamical fully actuated control systems. While the kinematic approach has been studied more in the literature for the motion feasibility problem (single integrator dynamics), the main motivation for the second approach lies in the fact that acceleration control (double integrator dynamics) is more suitable under the real world requirements of sensors for multiple vehicles, than velocity controls. It also provides a first step towards the construction of distance-based numerical estimators via Lie groups variational integrators

[7]. The solution in the second approach is given by using techniques of calculus of variations on manifolds and the Lagrange multiplier theorem, while for the first one, we use techniques of differential calculus on manifolds.The main contribution of this work consists on providing a set of necessary conditions for non-trivial collision-free motions in multi-agent control systems where agents evolves on a Lie group manifold. The main results of this work are given in Theorem V.1 and Theorem VI.1. Theorem V.1 describes differential-algebraic conditions for the feasible motion, when the agents are given by kinematic left-invariant control systems, by finding the set of admissible velocities leaving the constraints invariant at a given point on a Lie group and describing it as a linear system of algebraic equations with the control inputs as unknown variables. Theorem VI.1, provides first-order necessary conditions for feasible motion when the dynamics of each agent is described by a Lagrangian function on through the constrained Euler-Lagrange equations, with being the Lie algebra associated with . Such a condition is given by a set of first order differential equations on .

The paper is structured as follows. Section II provides the nomenclature. Section III introduces Lie groups actions, constrained Euler-Lagrange equations and trivializations of the tangent bundle of a Lie group. Section IV describes left-invariant kinematic multi-agent control systems, dynamical multi-agent control systems and the formulation for the motion feasibility problem. In Section V we consider a differential-algebraic approach for the motion feasibility problem of kinematic left-invariant multi-agent systems. In Section VI we derive first-order necessary conditions for feasible motion through constrained Euler-Lagrange equations arising from a variational point of view. Section VII studies the applicability of the conditions found in Sections V and VI for the collision avoidance problem of three rigid bodies on modelling fully-actuated underwater vehicles.

## Ii Nomenclature

We begin by establishing the nomenclature used throughout this paper. The basic notation and methodology is fairly standard within the differential geometry literature and we have attempted to use traditional symbols and definitions wherever feasible. Table I provides the symbols will be used frequently along the paper.

Symbol | Description |
---|---|

Differentiable manifold | |

Tangent bundle of | |

Cotangent bundle of | |

Lie group | |

Lie algebra of | |

Dual of the Lie algebra | |

Dimension of | |

Number of agents | |

Number of edges in the communication graph | |

Transpose of a Matrix | |

Quantity of collision avoidance constraints | |

Lagrange multiplier | |

Collision avoidance constraints | |

Left-trivialization of | |

Left-trivialization of | |

Elements on | |

Element on | |

Element of | |

Identity element of | |

Left-translation | |

tangent map of at | |

the cotangent map of at | |

Adjoint action | |

co-Adjoint action | |

Adjoint operator | |

co-Adjoint operator |

## Iii Preliminaries

### Iii-a Differential calculus on manifolds

Let be a differentiable manifold with . Given a tangent vector , , and , the set of real valued smooth functions on , denotes how tangent vectors acts on functions on . denotes the differential of the function defined as

Just as a vector field is a “field” for tangent vectors, a differential -form is a “field” of cotangent vectors, one for every base point. A cotangent vector based at , is a linear map from to , and the set of all maps is the cotangent space , which is the dual to the tangent space . A -form on is a map such that for every . Differential one forms, can be added together and multiplied by scalar fields as , and .

Given a differentiable function with a smooth manifold, the pushforward of at is the linear map satisfying for all and . The pullback of at is the dual map satisfying

(1) |

for all and , where denotes how tangent covectors acts on tangent vectors.

###### Definition III.1

Let and be differentiable manifolds and be a differentiable map between them. The map is a submersion at a point if its differential is a surjective map.

###### Definition III.2

Let be an open set, be smooth. If is a submersion at all points in then for all , is a submanifold of . The value is said to be a regular value of .

###### Theorem III.1

[[13], Section , pp. 219] Let be a differentiable manifold and a regular value of . Given a function , by defining the function as

for some inner product on , the Lagrange multiplier theorem states that is an extrema of if and only if is an extrema of .

### Iii-B Lie group actions

###### Definition III.3

Let be a Lie group with the identity element . A left-action of on a manifold is a smooth mapping such that for all , for all and for every , the map defined by is a diffeomorphism.

Let be a finite dimensional Lie group. The tangent bundle at a point is denoted as and the cotangent bundle at a point is denoted as . will denote the Lie algebra associated to defined as , the tangent space at the identity . Given that the Lie algebra is a vector space, one may consider its dual space. Such dual of the Lie algebra is denoted by .

Let be the left translation of the element given by for . is a diffeomorphism on and a left-action from to . Their tangent map (i.e, the linearization or tangent lift of left translations) is denoted by . Similarly, the cotangent map (cotangent lift of left translations) is denoted by . It is known that the tangent and cotangent lift are actions (see [10], Chapter ).

Consider the vector bundles isomorphisms and defined as

(2) |

and are called left-trivializations of and respectively. Therefore, the left-trivialization permits to identify tangent bundle with , and by , the cotangent bundle can be identified with .

###### Definition III.4 ([10], Section pp.)

The natural pairing between vectors and co-vectors is defined by for and where is understood as a row vector and a column vector. For matrix Lie algebras .

Using the pairing between vectors and co-vectors, one can write the relation between the tangent and cotangent lifts as

(3) |

for , and .

Let for any a left action on ; a function is said to be invariant under the action , if , for any . The Adjoint action, denoted is defined by where . Note that this action represents a change of basis on

The co-adjoint operator , is defined by for all with the adjoint operator given by where denotes the Lie bracket of vector fields on .

The co-Adjoint action is given by with , . Note that Ad and are actions on Lie groups, while ad and are operators on the Lie algebra and its dual, respectively.

###### Example III.5

The co-Adjoint action of on , is given by (see [10], Ch. , pp. 224) , and identifying with , it is given by with and . .

Let be a Lagrangian function describing the dynamics of a mechanical system. After a left-trivialization of we may consider the trivialized Lagrangian given by .

The left-trivialized Euler–Lagrange equations on (see, e.g., [10], Ch. ), are given by the system of first order ode’s

(4) |

together with the kinematic equation , i.e., .

The left-trivialized Euler-Lagrange equations together with the equation are equivalent to the Euler–Lagrange equations for . Note that for a matrix Lie group, the previous equations means .

If does not depend on (for instance, as the Lagrangian for Euler’s equations on the Lie group ), equations (4) reduce to the Euler-Poincaré equations

(5) |

together with the kinematic equation , i.e., .

## Iv Multi-agent control system on Lie groups

In this section we introduce multi-agent control systems where the configuration space of each agent is a Lie group. It is described by an undirected static and connected graph. First, we introduce the motion feasibility problem of agents where each node of the graph is given by a dynamical control system (i.e., by the controlled trivialized Euler-Lagrange equations) governed by a Lagrangian function, and next, we consider that each node is given by a left-invariant control system.

### Iv-a Left-invariant dynamical multi-agent control systems

Consider a set consisting of free agents evolving each one on a Lie group with dimension . Along this work we assume that the configuration space of each agent has the same Lie group structure. Same configuration does not mean the same agent. For instance, each agent can have different masses and inertia values, and therefore agents are heterogeneous. We denote by the configuration (positions) of an agent and describes the evolution of agent at time . The element denotes the stacked vector of positions where represents the cartesian product of copies of . We also consider the Lie algebra associated with the Lie group for the agent where is the neutral element of with the neutral element of the -Lie group which determines .

The neighbor relationships are described by an undirected static and connected graph where the set denotes the set of ordered edges for the graph. The set of neighbors for agent is defined by .

The dynamics of each agent is determined by a Lagrangian function together with collision avoidance (holonomic) constraints. Each tangent space can be left-trivialized and therefore, instead of working with we shall consider . Note also that the left-trivialization is not an extra assumption, we can always identify with by using the isomorphism (2).

Each agent is assumed to be a fully-actuated dynamical Lagrangian control system associated with the Lagrangian , that is,

(6) |

where for each , the -tuple of control inputs take values in and where with , and where is an arbitrary interval of .

We also assume that each agent occupies a disk of radius on . The quantity is chosen to be small enough so that it is possible to pack disks of radius on . We say that agents and avoid mutual collision if where is the canonical projection from over its -factor and is an appropriated distance function on the Lie group .

Consider the set given by the (holonomic or position-based) constraints indexed by the edges set with , the cardinality of the set of edges. Each for is a set of constraints for the edge , that is, being the number of constraints on the edge . Let be the total number of constraints in the set , that is,

(7) |

For each edge , is a function on defining an inter-agent collision avoidance constraint between agents and for all . The constraint is enforced if and only if .

The constraints on edge , induce the constraints as . If the map is a submersion at any point of its domain, then is an -dimensional submanifold of . Its -dimensional tangent bundle is given by

(8) |

where denotes the Jacobian matrix of the contraints.

Denote , where the Lie algebra structure of is given by with and

We also denote and the corresponding canonical projections over its -factors. Note that, after a left-trivialization, can be seen as a submanifold of given by

where .

The control policy for the motion feasibility problem for multi-agent systems can be determined by solving the corresponding dynamics (trivialized Euler-Lagrange equations (4)) for each subject to the constraint for each edge, as a unique system of differential equations, by lifting the dynamics of each vertex to , the constraints to , and to study the dynamics for the formation problem as a holonomically constrained Lagrangian system on .

### Iv-B Left-invariant kinematic multi-agent control systems

Let be a vector field on . The set denotes the set of all vector fields on . The tangent map shifts vectors based at to vectors based at . By doing this operation for every we define a vector field as for .

###### Definition IV.1

A vector field is called left-invariant if .

In particular for , Definition IV.1 means that a vector field is left-invariant if for . Note that if is a left invariant vector field, then .

Consider an undirected static and connected graph , describing the kinematics of each agent given by left invariant kinematic control systems each one on , together with the constraints defining the set . As before, denotes the set of vertices of the graph, but now, each is a fully-actuated left invariant kinematic control system, that is, the kinematics of each agent is determined by

(9) |

and the set denotes, as before, the set of edges for the graph, where , is fixed and is a curve on the Lie algebra of . Alternatively, the left-invariant control system (9) can be written as . Each curve on the Lie algebra determines a control input , where for each , the -tuple of control inputs takes values in .

If for each agent, , then satisfies and therefore (9) is given by the drift-free kinematic left invariant control system

(10) |

Note that we have not made any reference to coordinates on . We only require a basis for . This is all that is necessary to study left-invariant kinematic systems.

## V A differential-algebraic approach to characterize motion feasibility of LIMS

In this section, inspired by [21], we consider a differential-algebraic approach for the motion feasibility problem for formation control of kinematic left-invariant multi-agent systems (LIMS’s) introduced in Section IV-B.

Given the collision avoidance constraint , consider the new constraint and consider the corresponding projection into the denoted by .

###### Definition V.1

The constraint is said to be left invariant if , that is, the pullback to the identity of the constraint corresponds to the constraint at the identity.

Note that and evaluated at a point , i.e., , is a one form on .

When the constraint is left-invariant, there exists a left-invariant distribution of feasible velocities for the formation given by the annhilator of the constraints at each point, and it determines a subgroup of . In other words, is a subgroup of and classical reduction by symmetries [10], [13] can be performed in the multi-agent system to obtain an unconstrained reduced problem on . Given that the collision avoidance constraints in general are defined by the distances among configurations of the agents, we are mainly interested in constraints depending explicitly on the variables on .

The co-Adjoint action on each Lie group induces a co-Adjoint action on , denoted as , and given by with , and the inverse element of .

To give a necessary condition for the existence of feasible motion in the formation problem for kinematic LIMS’s, we want to find the set of velocities satisfying the kinematics and the constraints, that is, the set of admissible velocities leaving the constraints invariant at a given point on .

###### Theorem V.1

The set of admissible velocities allowing feasible motion in the formation problem is given by the set of elements such that for a fixed ,

Proof: The interaction between agents in the formation, given by the formation constraints on induces the constraint on , . Motion feasibility requires that the constraints holds along the trajectories of the LIMS (10).

Differentiating the constraint on we get the constraint , that is, where with and is a one-form on , .

The one-forms can be translated back to using left translations as , where and are the Lie algebra evaluated constraints, and where we used that is equal to the identity map on and equation (3).

Note that our problem not only involves that solutions must satisfy the constraints, which means that for , . Solutions must also be left invariant vector fields, solving (10), that is, (see Definition IV.1). To solve the combined problem we proceed as in [2] (Section 4), to unify the solution in a unique algebraic condition. In order to find the left-invariant vector fields satisfying the constraints, we must study how much the vector fields changes from . As a transformation connecting and the identity in we use the Adjoint operator, this means that we have to find such that for a fixed

(11) |

The operator represents a change of basis on and equation (11) gives the subspace of annihilated by . Therefore the problem consists on finding such that for a fixed ,

(12) |

###### Remark V.2

As pointed out in [2], equation (12) gives a linear system of algebraic equations with as unknown variables and as the known coefficients. Thus, solutions of the latter equation give rise to linear combinations of the control inputs in the linear subspace of annihilating the constraints.This means that in order for the solution not leave the submanifold which defines the constraints, the set of velocities must satisfy equations (12)

###### Remark V.3

Note that the one-forms are not left-invariant and may change at any point , but by using the co-adjoint action, we can study vector fields at any point . Therefore, the problem of finding the orthogonal subspace of the constraints at points of given in [21], to characterize the physical allowable directions of motion, in the context of LIMS’s, is equivalent to finding the annihilator of the co-adjoint action for .

###### Remark V.4

Following [21] (Section III-B), when more than one solutions exist, the solution space can be exploited to find a new distribution which is called in [21] a group abstraction of the kinematic LIMS (10), that is, a new control system keeping the formation along solutions. This new control system is given by studying the kernel of the co-distribution defined by the union of a basis of and a basis for the co-distribution describing the kinematics of the agents.

For LIMS’s, the space of solutions is determined by equation (12), that is,

for a fixed . As in [21], one may use to find an abstraction for LIMS. The new control system is given by the Kernel of , that is,

giving rise to a group abstraction that describes the set of admissible velocities keeping the formation of the LIMS.

By considering a basis of , denoted , we can write such a group abstraction as the left-invariant control system (note that does not depend on and therefore its basis is given by left-invariant elements)

(13) |

where are the new control inputs that activate the elements of the base with . The abstracted control system (13) provides certain insights on different types of feasible motions for the agent according to different choices of for .

###### Example V.5

As an application we consider the motion feasibility problem for three agents moving in the plane. The configuration of each agent at any given time is determined by the element , given by where .

The kinematic equations for the multi-agent system are

(14) |

where

The Lie algebra of is determined by

(15) |

where , , with and we identify the Lie algebra with via the isomorphism .

Equations (14) gives rise to a left-invariant control system on with the form describing all directions of allowable motion, where the elements of the basis of are

which satisfy . Using the dual pairing, where , for any and , the elements of the basis of are given by

The communication topology is given by an equilateral triangle where each node communicates with its adjacent vertex.

The formation is completely specified by the (holonomic) constraints , (i.e., ) determined by a prescribed distance among the positions of all agent at any time. The constraint for the edge is given by where is the Frobenius norm,

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