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Research article

Preventing aircraft from wildlife strikes using trajectory planning based on the enhanced artificial potential field approach

  • Wildlife strikes refer to collisions between animals and aircraft during flight or taxiing. While such collisions can occur at any phase of a flight, the majority occur during takeoff and landing, particularly at lower altitudes. Given that most reported collisions involve birds, our focus was primarily on bird strikes, in line with statistical data. In the aviation industry, aircraft safety takes precedence, and attention must also be paid to optimizing route distances to minimize operational costs, posing a multi-objective optimization challenge. However, wildlife strikes can occur suddenly, even when aircraft strictly adhere to their trajectories. The aircraft may then need to deviate from their planned paths to avoid these collisions, necessitating the adoption of alternative routes. In this article, we proposed a method that combines artificial potential energy (APE) and morphological smoothing to not only reduce the risk of collisions but also maintain the aircraft's trajectory as closely as possible. The concept of APE was applied to flight trajectory planning (TP), where the aircraft's surroundings were conceptualized as an abstract artificial gravitational field. This field exerts a "gravitational force" towards the destination, while bird obstacles exert a "repulsive force" on the aircraft. Through simulation studies, our proposed method helps smooth the trajectory and enhance its security.

    Citation: Wenchao Cai, Yadong Zhou. Preventing aircraft from wildlife strikes using trajectory planning based on the enhanced artificial potential field approach[J]. Metascience in Aerospace, 2024, 1(2): 219-245. doi: 10.3934/mina.2024010

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  • Wildlife strikes refer to collisions between animals and aircraft during flight or taxiing. While such collisions can occur at any phase of a flight, the majority occur during takeoff and landing, particularly at lower altitudes. Given that most reported collisions involve birds, our focus was primarily on bird strikes, in line with statistical data. In the aviation industry, aircraft safety takes precedence, and attention must also be paid to optimizing route distances to minimize operational costs, posing a multi-objective optimization challenge. However, wildlife strikes can occur suddenly, even when aircraft strictly adhere to their trajectories. The aircraft may then need to deviate from their planned paths to avoid these collisions, necessitating the adoption of alternative routes. In this article, we proposed a method that combines artificial potential energy (APE) and morphological smoothing to not only reduce the risk of collisions but also maintain the aircraft's trajectory as closely as possible. The concept of APE was applied to flight trajectory planning (TP), where the aircraft's surroundings were conceptualized as an abstract artificial gravitational field. This field exerts a "gravitational force" towards the destination, while bird obstacles exert a "repulsive force" on the aircraft. Through simulation studies, our proposed method helps smooth the trajectory and enhance its security.


    Symmetries linked to Lie groups appear naturally in many control systems problems [1,2,3,4,5,6,7,8,9,10,11,12,13]. Numerical integrators evolving on Lie groups are commonly employed to improve its accuracy, as well as to avoid singularities by working with coordinate-free expressions in the Lie algebra associated to the Lie group, on which the system evolves. Typical tasks include trajectory tracking and estimation algorithms for the pose of mechanical systems.

    Optimization problems on Lie groups [14] have applications in control engineering. Indeed, there are many examples of robotic systems that possess invariant quantities steaming from existing symmetries. Invariant quantities can be used as leverage to reduce the complexity of the equations, by projecting them into lower dimensional spaces. Techniques taking advantage of symmetries have been studied in [15,16,17,18,19,20], among many others, mainly for applications in robotic and aerospace engineering and, in particular, for spacecraft attitude control and underwater vehicles [21]. Most of the applications provided in the literature focus on the single-agent situation and only a few works explore symmetry reduction in a multi-agent scenario (see for instance [22,23,24,25]). In this work, we employ symmetry reduction to study optimal control problems with broken symmetry for multi-agent systems on Lie groups, while agents avoid collisions and obstacles in the configuration space.

    In [26], the authors studied symmetry reduction in optimal control problems with broken symmetry for single-agent systems. In this work, we advance on the results of [26] by considering a multi-agent scenario. Hence, a generalization of the previous variational principle and reduction by symmetries performance are needed. The results in this paper are the Lagrangian/variational counterpart of those in [22]; we also develop a discrete-time version of the results. At the continuous-time side, we obtain the Euler–Poincaré equations from a constrained variational principle while after discretizing the variational principle in time, we obtain the discrete-time Lie–Poisson equations.

    The approach in this work to obtain the reduced optimality conditions is a generalization of the one followed in [26], and is as follows. First, we consider a collection of free agents evolving on a Lie group G and an artificial potential V0, used to prevent collisions with a fixed obstacle, which is not symmetry invariant. At the same time, we consider a representation of G on a dual vector space W and a graph G to describe interactions between neighboring agents. Though the artificial potential is not symmetry invariant, the interaction between neighbouring agents is, which is the standard situation when the interaction is done through a potential dependent on relative positions. Coupling the artificial potential a parameter depending on vectors in W, we obtain a symmetry invariant potential function under the action of G. Loosely speaking, the agents are coupled with vectors in W that are acted by the representation. The associated action considered at this stage restores the full Lie group symmetry in the cost function from our optimal control problem and allows us to apply the semi-direct product reduction theory [3,27,28,29,30,31,32], to obtain the corresponding Euler-Poincaré system on the semi-direct product Lie algebra gW at each node. This gives rise to a new system that finds no analogs in classical reduced-order models in optimal control of mechanical systems.

    The paper is organized as follows, in Section 2, we introduce some preliminaries about geometric mechanics on Lie groups and Lie group actions. In Section 3.1, we present the problem together with a motivating example. In Section 4, we study the Euler–Poincaré reduction of optimal control problems for left-invariant multi-agent control systems on Lie groups with partial symmetry breaking cost functions. Furthermore, we consider the discrete-time framework and obtain the discrete-time Lie–Poisson equations in Section 5. In Section 6, an example is considered to illustrate the theory. Finally, some concluding remarks are given in Section 7.

    Consider a manifold Q to be the configuration space of a mechanical system and let it be differentiable of dimension d which can be described locally by coordinates q=(q1,,qd). Denote by TQ the tangent bundle of Q, with local coordinates described by positions and velocities for the system, vq=(q1,,qd,˙q1,,˙qd)TQ with dim(TQ)=2d. Let TQ be its cotangent bundle, locally described by the positions and the momenta for the system, i.e., (q,p)TQ with dim(TQ)=2d. The tangent space at qQ has a vector space structure, and it is denoted as TqQ. The cotangent space at qQ is just the dual space of TqQ and is denoted as TqQ. The dynamics of a mechanical system is described by the equations of motion determined by a Lagrangian function L:TQR given by L(q,˙q)=K(q,˙q)V(q), where K:TQR denotes the kinetic energy and V:QR the potential energy of the system. The equations of motion are given by the Euler-Lagrange equations ddt(L˙qi)=Lqi,i=1,,d, which determine a system of second-order differential equations. In case, the configuration space of the system is a Lie group, Euler-Lagrange equations can be reduced to a system of first-order ordinary differential equations.

    Definition 2.1. Let G be a Lie group and Q a smooth manifold. A left-action of G on Q is a smooth map Φ:G×QQ, such that Φ(ˉe,g)=g and Φ(h,Φ(g,q))=Φ(hg,q) for all g,hG and qQ, where ˉe is the identity of the group G and the map Φg:QQ given by Φg(q)=Φ(g,q) is a diffeomorphism for all gG.

    Definition 2.2. A function f:QR is called left invariant under Φg if fΦg=f for any gG.

    For a finite dimensional Lie group G, its Lie algebra g is defined as the tangent space to G at the identity, g:=TˉeG. Let Lg:GG be the left translation of the element gG given by Lg(h)=gh, where hG. Lg is a diffeomorphism on G and a left-action of G on G [33]. Its tangent map (i.e, the linearization or tangent lift) is denoted by ThLg:ThGTghG. In a similar way, the cotangent map (cotangent lift), is denoted by (ThLg), and is defined as the dual map of the tangent lift denoted by ThLg:TghGThG, and determined by the relation (ThLg)(αgh),yh=αgh,(ThLg)yh, yhThG, αghTghG. As it is known, the tangent and cotangent lift of a Lie group action are Lie group actions as well. Here, ,:W×WR with W a finite-dimensional vector space denotes the usual natural pairing between vectors and co-vectors, and defined by y,x:=yx for yW and xW. For a matrix Lie algebra y,x=yTx (see [33], Section 2.3). Using the natural pairing between vectors and co-vectors, for g,hG, yg and xg, we write TgLg1(y),TˉeLg(x)=y,x.

    Denote by ad:g×gg, (ξ,μ)adξμ the co-adjoint operator, defined by adξμ,η=μ,adξη for all ηg, where the ad:g×gg denotes the adjoint operator on g given by the Lie-bracket, i.e., adξη=[ξ,η], ξ,ηg. We also define the adjoint action of G on g, denoted by Adg:gg and, in the case the Lie algebra is a matrix Lie group, it is given by Adgχ:=gχg1, where χg, and the co-adjoint action of G on g, denoted by Adg:gg, and given by Adgα,ξ=α,Adgξ with αg.

    For qQ, the isotropy (or stabilizer or symmetry) group of Φ at q is given by Gq:={gG|Φg(q)=q}G. Since the map Φ(,q)=Φq:GQ is a continuous, Gq=(Φq)1(q) is a closed subgroup, and thus, a Lie subgroup of G (see [34] Sec. 9.3 for instance).

    Example 1. Consider the special Euclidean group SE(2) of rotations and translations on the plane. Elements on SE(2) can be expressed by transformations of R2 of the form zRz+r, with rR2 and RSO(2). This transformation can be represented by g=(R,r), for R=(cosθsinθsinθcosθ) and r=[x,y]T. The composition law is (R,r)(S,s)=(RS,Rs+r) with identity element (I,0) and inverse (R,r)1=(R1,R1r). Under this composition rule, SE(2) has the structure of the semidirect product Lie group SO(2)R2. Here, as usual in the literature, we denote by the semidirect product of Lie groups.

    The Lie algebra se(2) of SE(2) is determined by se(2)={(Ab00)|Aso(2)R,bR2}. In the following, for simplicity, we write A=aJ, aR, where J=(0110). Therefore, we denote ξ=(a,b)se(2). The adjoint action of SE(2) on se(2) is given by Ad(R,r)(a,b)=(a,aJr+Rb) (see [35], pp. 153 for instance), so, Ad(R,r)1(a,b)=(a,RT(baJr)).

    Next, we provide the infinitesimal description of a Lie group action, that will be an important concept in the remainder of the paper.

    Definition 2.3. Given a Lie group action Φ:G×QQ, for ξg, the map Φexp(tξ):QQ is a flow on Q, where exp is the exponential map of G. The corresponding vector field on Q, given by ξQ(q):=ddtt=0Φexp(tξ)(q) is called the infinitesimal generator of the action corresponding to ξ.

    Definition 2.4. Let us denote the set of vector fields on a Lie group G by X(G). A left invariant vector field is an element X of X(G) such that ThLg(X(h))=X(Lg(h))=X(gh) g,hG.

    Especially, for h=ˉe, we have that a vector field X is left-invariant if X(g)=TˉeLgξ for ξ=X(ˉe)g. As a consequence, left invariant vector fields are identified with its corresponding element of g. Thus, for every gG and ξg, we define the left-invariant vector field as ξ(g):=TˉeLg(ξ).

    Example 2. Consider the Euclidean Lie group Rn with the sum as group operation. For all gRn, the left translation Lg is the usual translation on Rn, that is, Lg(h)=g+h, hRn. So that, the tangent map to Lg is the identity map on Rn, that is, T0Lg=idT0Rn, where we are using that ThRnRn for all hRn, since Rn is a vector space. Therefore, left-invariant vector fields are constant vector fields, that is, X=v1x1++vnxn for v=(v1,,vn)T0Rn and x=(x1,,xn)Rn.

    Consider a Lie group G, a vector space W and the representation of G on W, given by ρ:G×WW, (g,v)ρg(v), which is a left action, and it is defined by the relation ρg1(ρg2(v))=ρg1g2(v), g1,g2G. Its dual is given by ρ:G×WW, (g,α)ρg1(α), satisfying ρg1(α),v=α,ρg1(v), and it is also a left action of G now on W.

    The infinitesimal generator of the left action of G on W is ρ:g×WW, (ξ,v)ρ(ξ,v)=ddt|t=0ρexp(tξ)(v). For every vW, consider the linear transformation ρv:gW, ξρv(ξ)=ρ(ξ,v) and its dual ρv:Wg, αρv(α). The last transformation defines the momentum map JW:W×Wg, (v,α)JW(v,α):=ρv(α), such that for every ξg, JW(v,α),ξ=ρv(α),ξ=α,ρv(ξ)=α,ρ(ξ,v).

    For ξg, consider the map ρξ:WW, vρξ(v)=ρ(ξ,v) and its dual ρξ:WW, αρξ(α), such that ρξ(α),v=α,ρξ(v). Then, this satisfies JW(v,α),ξ=α,ρ(ξ,v)=α,ρξ(v)=ρξ(α),v. See [33] and [34] for more details on the momentum map.

    Example 3. Let W=g and ρ the adjoint representation of G on W, i.e., ρg=Adg, for any gG. So, for αW=g, ρ is the coadjoint representation of G on W, i.e., ρg=Adg. We also have that the infinitesimal generator for the adjoint representation is ρξ=adξ for any ξW (see [33], Def. 6.4, pp. 225), and it follows that JW(x,α),ξ=α,adξx=α,adxξ=adxα,ξ, for xW=g, which gives JW(x,α)=adxα.

    Similarly, if now W=g and the coadjoint representation of G on W is ρ, i.e., ρg=Adg1, for any gG, then ρξ=adξ for any ξg. So, if x,ξg and αg, it follows that JW(α,x),ξ=adξα,x=α,adxξ=adxα,ξ, which gives JW(α,x)=adxα.

    Denote, by N, a set consisting of s2 free agents, and by EN×N the set describing the interaction between them. The neighboring relationships are illustrated by an undirected graph G=(N,E), where N is the set of vertices and E the set of edges for G. We further assume G is static and connected. For every agent iN the set Ni={jN:(i,j)E} denotes the neighbors of that agent. The agent iN evolves on an n-dimensional Lie group G, and its configuration is denoted by giG. We denote by Gs and by TeGs=:gs the cartesian products of s copies of G and g, respectively, where e=(¯e,¯e,,¯e) is the identity of Gs, and ¯e the identity element of G.

    For each agent iN, there is an associated left-invariant control system described by the kinematic equations

    ˙gi=T¯eLgi(ui),gi(0)=g0i, (3.1)

    where gi(t)C1([0,T],G),TR fixed, uig is the control input and gi0G is considered as the initial state condition. Letting dimg=n, we can have as many basis for the Lie algebra as the number of agents. Thus, for each iN, we have g=span{e1,e2,,en} and the control inputs may be described by coordinates ui=[u1i,u2i,,umi]T, where ui(t)C1([0,T],g), with mn. Hence, the control input for each agent is given by ui(t)=ei0+mk=1uki(t)ek, where ei0g. Thus, the left-invariant control systems (3.1) for each agent iN can be written as

    ˙gi(t)=gi(t)(e0+mk=1uki(t)ek),gi(0)=g0i. (3.2)

    Note that the class of control systems described by (3.2) belongs to the family of underactuated control systems.

    Following with Example 1, consider the agents iN and jNi represented as gk=(Rk,rk), k{i,j}. Note that g1igj=(RTiRj,RTi(rirj)), then, Adg1igj(1,0)=(1,JRTi(rjri)). The inner product on se(2) is given by ξ1,ξ2=tr(ξT1ξ2) for ξ1,ξ2se(2) and, hence, the norm is given by ||ξ||=tr(ξTξ), for any ξse(2). For ξ=(a,b)se(2) we can write the norm of ξ as ||(a,b)||=2a2+bTb. Therefore, ||Adg1igj(1,0)||2=2+|JRTi(rjri)|2=2+|rjri|2, where we have used that R,JSO(2) for the last equality. Hence, it follows that |rirj|=||Adg1igj(1,0)||22.

    The previous computation shows that, if the interaction between agents is determined by a function depending on the distances between them, that is, Vij:G×GR, is such that Vij(gi,gj)=V(|rirj|) for some V:R0R; then, Vij is SE(2)-invariant, that is Vij(hgi,hgj)=Vij(gi,gj). An alternative reasoning of this invariance property has been shown in [36].

    Next, suppose that we wish to write the distance from an arbitrary point rR2 to a fixed point x0R2 in terms of the adjoint action. Consider ξ0=(1,Jx0)se(2). Then, for any (R,r)SE(2), we have Ad(R,r)1(1,Jx0)=(1,RTJ(x0r)), and, therefore, ||Ad(R,r)1(1,Jx0)||2=2+|x0r|2. Next, assume we have an obstacle avoidance function V0i:SE(2)R for each agent iN which can be written as V0i(Ri,ri)=V(|ri|) with V:R0R. Note that under this assumption, V may be chosen arbitrarily. Then, V0i is not SE(2)-invariant, but it is SO(2) invariant, i.e., V0i(R0Ri,ri)=V0i(Ri,ri) for any R0SO(2). Note also that SO(2) is the isotropy group for the coadjoint action, that is, SO(2){gSE(2)Adg(ξ0)=ξ0}, therefore, the obstacle avoidance potential functions V0i are invariant under the left action of the isotropy group.

    In this situation, one can redefine the potential function V0i to make it SE(2)-invariant as follows. Consider x0=0, so, ξ0=(R0,x0)=(J,0)se(2), and Ad(Ri,ri)1ξ0=2+|ri|2. Hence, V0i(Ri,ri)=V(|ri|)=V(||Ad(Ri,ri)1ξ0||22). This gives a motivation to define an extended obstacle avoidance function Vexti,0:GWR with GW=SE(2)se(2) as Vexti,0(gi,ξ):=V(||Adg1iξ||22).

    Note that the extended obstacle avoidance function possesses an SE(2)-symmetry (i.e., Vexti,0 is invariant under a left action of SE(2)) since Vexti,0˜Φ(h,(g,ξ))=Vexti,0(Lhg,Adhξ)=Vexti,0(g,ξ), for any hSE(2), with left action ˜Φ given by

    ˜Φ:SE(2)×(SE(2)×se(2))SE(2)×se(2),(h,(g,ξ))(Lh(g),Adh(ξ)). (3.3)

    The problem under study consists on finding reduced necessary conditions for optimality in an optimal control problem for (3.1) (or equivalently (3.2)). These solution curves should minimize a cost function and prevent collisions among agents, while they should also avoid static obstacles in the workspace.

    Problem (collision and obstacle avoidance): Find reduced optimality conditions on g(t)=(g1(t),,gs(t))Gs and the controls u(t)=(u1(t),,us(t))gs avoiding collision among the agents and obstacles (which will be defined shortly) in the workspace, and such that (g(t),u(t))Gs×gs minimize the following cost function,

    min(g,u)si=1T0(Ci(gi(t),ui(t))+V0i(gi)+12jNiVij(gi(t),gj(t)))dt, (3.4)

    subject to the kinematics ˙gi(t)=T¯eiLgi(t)(ui(t)), boundary conditions g(0)=(g1(0),,gn(0))=(g01,,g0s) and g(T)=(g1(T),,gs(T))=(gT1,,gTs) with TR+ the final time, and under the following assumptions:

    (i) There is a left representation ρ of G on a vector space V.

    (ii) Ci:G×gR are G-invariant functions for each iN (under a left action of G on G×g, which is given below) and are also differentiable almost everywhere.

    (iii) Vij:G×GR (collision avoidance potential functions) satisfying Vij=Vji are G-invariant functions under Φ, defined by

    Φ:G×(G×G)G×G,(g,(g1,g2))(Lg(g1),Lg(g2)), (3.5)

    i.e., VijΦg=Vij, for any gG, that is, Vij(Lg(gi),Lg(gj))=Vij(gi,gj), for any (gi,gj)E, jNi and they are also differentiable almost everywhere.

    (iv) V0i:GR (obstacle avoidance potential functions) are not G-invariant functions and they are also differentiables almost everywhere, for iN.

    (v) The obstacle avoidance functions V0i depend on a parameter α0W. Hence, we can define the extended potential function as Vexti,0:G×WR, with Vexti,0(,α0)=V0i, by making the parameter evolve - due to the group action - with initial value α0.

    (vi) The extended obstacle avoidance functions are G-invariant under ˜Φ, defined by

    ˜Φ:G×(G×W)G×W,(g,(h,α))(Lg(h),ρg1(α)), (3.6)

    where ρg1W is the adjoint of ρg1W, i.e., Vexti,0˜Φg=Vexti,0, for any gG, or Vexti,0(Lg(h),ρg1(α))=Vexti,0(g,α) where αW.

    (vii) The obstacle avoidance potential functions are invariant under the left action of the isotropy group

    Gα0={gGρg(α0)=α0}. (3.7)

    Note that G×g is a trivial vector bundle over G and, in order to respect assumption (ⅱ), we define the left action of G on G×g as follows,

    Ψ:G×(G×g)(G×g),(g,(h,u))(Lg(h),u). (3.8)

    We further assume that each Ci:G×gR is G-invariant under (3.8), i.e., CiΨg=Ci, for any gG. In addition, each agent iN occupies a disk of radius ¯r on G. This radius is consider to be small enough in order that all agents can be packed on G and, hence the potential functions are well defined and feasible for d(gi(t),gj(t))>2¯r for all t, where d(,):G×GR denotes a distance function on G.

    We next study reduced optimality conditions for extrema for the OCP. We address the problem as a constrained variational problem and obtain the Euler-Poincaré equations that extremal must satisfy in Theorem 4.1 and Proposition 4.2.

    The optimal control problem (3.4) can be resolved as a constrained variational problem by utilizing the Lagrangian multipliers λgi=TgiLg1i(λi)TgiG, where λiC1([0,T],g) into the cost functional.

    Let {e1,,em,em+1,,en} be a basis of g dual of the basis {ei} of g. Then λi=nk=m+1λikek, where λik are the components of the vector λi in the given basis of the Lie algebra g. Thus, we define the Lagrangian L:Gs×gs×(TgiG)sR by

    L(g,u,λ)=si=1[Ci(gi(t),ui(t))+λgi,T¯eiLgiui+V0i(gi(t))+Vi(g)], (4.1)

    where Vi:GsR,

    Vi(g)=12jNiVij(gi(t),gj(t))

    By assumption (v), the obstacle avoidance potential functions V0i:GR depend on a parameter α0W, so we can extend {them} to Vexti,0:G×WR by making the parameter evolve under the Lie group action with α(0)=α0, and, therefore, we can consider the extended Lagrangian function on Gs×gs×(TgiG)s×W,

    Lext(g,u,λ,α)=si=1[Ci(gi(t),ui(t))+λgi,T¯eiLgiui+Vexti,0(gi,αi)+Vi(g)]

    where Lext(g,u,λ,α0)=L(g,u,λ).

    By assumptions (ⅰ) to (ⅳ), and by taking advantage of the G-invariance of Ci, Vexti,0 and Vij (and so Vi), we can define the reduced extended Lagrangian :Gs1×gs×(g)s×WR by

    (g,u,λ,α)=si=1[Ci(ui)+λi,ui+Vexti,0(¯e,αi)+Vi(g)],

    defining g1 to be the identity ˉe1 on G. This is equivalent to consider the Lagrangian ˜Lext(h,u,λ,α)=Lext(ˉe1,h,u,λ,α) with hGs1. Then (g,u,λ,α) is obtained by considering a left action in each term of the sum. In fact, if Liext is each term in the sum from 1 to s in the definition of Lext, then

    (g,u,λ,α)=ni=1Liext(Lg1ig,u,TgiLg1iλgi,αi)

    under the assumption that g1=ˉe1 and αi=ρgi(α). Note here the slight abuse of notation regarding the positions g. In the definition of the reduced Lagrangian gGs1, while in that of the Lagrangian Lext, gGs.

    Theorem 4.1. For s2, an extremal for the OCP (3.4) satisfies the following Euler-Poincaré equations

    ddt(Ciui+λi)=adui(Ciui+λi)+JW(Vexti,0αi,αi)+Θi1sk=1T¯eLgi(Vkgi), (4.2)
    ˙αi=ρui(αi),αi(0)=ρg0i(α0), (4.3)

    where JW:TWW×Wg is the momentum map that corresponds to the left action of G on W, and it is defined through the left representation ρ of G on W, and where Θi1=0 if i=1, otherwise Θi1=1.

    Proof. Consider the variational equation

    δT0Lext(g(t),u(t),λ(t),α)dt=0,

    which holds for variations of g, that vanish at the endpoints, and u. Also, consider the constrained variational equation

    δT0(g(t),u(t),λ(t),α(t))dt=0, (4.4)

    that holds for variations of ui and αi with δui=˙ηi+aduiηi and δαi=ρηi(αi), where ηi is a path of gs that vanishes at the endpoints, i.e. ηi(0)=ηi(T)=0.

    The two variational equations are equivalent since the cost functions Ci and the extended potential functions Vexti,0 are G-invariant, i.e. CiΨg,Vexti,0˜Φ=Vexti,0 and λgi,TˉeiLgiui=TgiLg1iλgi,ui=λi,ui. The variations δgi of gi induce and are induced by variations δui=˙ηi+aduiηi with ηi(0)=ηi(T)=0, where ηi=TgiL1gi(δgi) and δui=TgiLg1i(˙gi). Variations of αi are given by δαi=ρηi(αi). Thus, we have

    δT0(g,u,λ,α)dt=si=1T0Ciui,δui+λi,δui+Vexti,0αi,δαi+sk=2Vigk,δgkdt. (4.5)

    Using the variations of ui, which are given by δui=˙ηi+aduiηi, applying integration by parts and by the definition of the co-adjoint action the first two terms, yield:

    T0ddt(Ciui+λi)+adui(Ciui+λi),ηidt.

    From the variations δαi=ρηi(αi), the third term gives

    Vexti,0αi,δαi=Vexti,0αi,ρηi(αi)=αi,ρηi(Vexti,0αi)=JW(Vexti,0αi,αi),ηi.

    Taking into account that T(LgiLg1i)=TLgiTLg1i is equivalent to the identity map on TGi and ηi=TgiLg1i(δgi), the fourth term can be written as

    sk=2Vigk,δgk=sk=2Vigk,(T¯eLgkTgkLg1k)(δgk)=sk=2Vigk,T¯eLgk(ηk)=sk=2T¯eLgk(Vigk),ηk.

    Therefore, after performing a change of variables between indexes i and k in the fourth term, the above variational equation (4.4) yields

    ddt(Ciui+λi)=adui(Ciui+λi)+JW(Vexti,0αi,αi).

    for i=1. Otherwise,

    ddt(Ciui+λi)=adui(Ciui+λi)+JW(Vexti,0αi,αi)+sk=1T¯eLgi(Vkgi).

    Finally, by taking the time derivative of αi=ρgi(α0), we have ˙αi=ρui(αi), together with α(0)=ρgi0(αi0).

    Note that the above Euler-Poincaré equations (4.2) cannot, in fact, describe the motion properly because there are more unknowns than equations. In particular, observe that equations (4.2) together with (3.1) (or equivalently (3.2)), give rise to only two equations for the three unknown variables ui, λi and gi. However, we provide an additional structure to the Lie algebra, g, that allows one to decouple equations (4.2) into two equations. The next Proposition describes this process.

    Proposition 4.2. If the structure of the Lie algebra permits a decomposition g=rs where r=span{e1,,em} and s=span{em+1,,en} such that

    [s,s]s,[s,r]r,[r,r]s, (4.6)

    then the Euler-Poincaré equations of motion (4.2) are given by the following equations:

    ddtCiui=aduiλi+JW(Vexti,0αi,αi)|r+Θi1sk=1T¯eLgi(Vkgi)|r,˙λi=aduiCiui+JW(Vexti,0αi,αi)|s+Θi1sk=1T¯eLgi(Vkgi)|s, (4.7)

    where uir and the restrictions |r and |s give the projection onto r and s, respectively, with respect to the associated splitting of the dual space g=rs.

    Remark 4. Note that this is the case for semisimple Lie algebras, they admit a Cartan decomposition, i.e., if g is semisimple, then g=rs such that [r,r]s,[s,r]r,[s,s]s, where the set r={xgθ(x)=x} is the 1 eigenspace of the Cartan involution θ, whereas the set s={xgθ(x)=x} is the +1 eigenspace of the Cartan involution θ. Additionally, the Killing form is proven to be positive definite on r and negative definite on s (see, e.g., [15]). Thus, suitable candidates that satisfy the assumption of Proposition 4.2 are connected semisimple Lie groups. On the other hand, a Cartan decomposition determines a Cartan involution θ (see, e.g., [37]). In particular, the proposed decomposition for the Lie algebra is not restrictive in the sense that the usual manifolds/work-spaces used in applications as SO(n) and SE(n) allow such a decomposition.

    Proof. Given g=rs we get g=rs, where r=span{e1,,em} and s=span{em+1,,es}. Thus, from (4.6) we have that adsss,adsrr,adrsr,adrrs and given that uir, Ciuir and λis by definition we conclude that aduiCiuis and aduiλir. Also, JW(Vexti,0αi,αi)g and sk=1T¯eiLgi(Vkgi)g hence, they have a decomposition into r and s. Thus, the equations (4.2) split into the following equations

    ddtCiui=aduiλi+JW(Vexti,0αi,αi)|r+Θi1sk=1T¯eLgi(Vkgi)|r,˙λi=aduiCiui+JW(Vexti,0αi,αi)|s+Θi1sk=1T¯eLgi(Vkgi)|s, (4.8)

    Remark 5. For the initial value problem guaranteeing a solution for the previous system of equations, we look for a solution to the equations with the initial condition ui(0)=Tg(0)Lg1(0)(˙gi(0)) and the kinematic equation ˙gi(t)=T¯eLgi(t)(ui(t)) with g(0)=(g1(0),,gs(0)).

    Remark 6. Assuming the reduced Lagrangian is hyperregular, we use the Legendre transformation and we can define the reduced Hamiltonian h:Gs1×(g)s×(g)s×WR given by

    h(g,μ,λ,α)=μ,u(g,u,λ,α),

    where μ=u=(Cu+λ)(g)s. Under this assumption, the Euler-Poincare equations (4.2), (4.3) can be written as the Lie-Poisson equations (see, e.g., [22])

    ˙μi=aduiμi+JW(Vexti,0αi,αi)+Θi1sk=1T¯eLgi(Ukgi), (4.9)
    ˙αi=ρui(αi),αi(0)=ρg0i(α0). (4.10)

    In this section, we study the discrete-time reduction by symmetries for necessary conditions in the collision and obstacle avoidance optimal control problem. The goal is to construct a variational integrator based on the discretization of the augmented cost functional. Such integrator inherits discrete-time symmetries from its continuous counterpart and generates a well-defined (local) flow for reduced necessary conditions characterizing (local) extremal in the optimal control problem.

    Given the set T={tkR+,tk=khk=0,,N}, Nh=T, with T fixed (recall that TR+ is the end point of the cost functional - see for instance equation (3.4)), a discrete trajectory for the agent i is determined by a set of N+1 points equally spaced in time, g0:Ni={g0i,,gNi}, where gkigi(kh)G, and h=T/N is the time step. The path between two adjacent points gki and gk+1i must be given by a curve lying on the Lie group G. To construct such a curve we make use of a retraction map R:gG.

    Definition 5.1. A retraction map R:gG is an analytic local diffeomorphism assigning a neighborhood Og of 0g to a neighborhood of the identity ¯eG.

    The retraction map (see Figure 1) is used to express small discrete changes in the group configuration through unique Lie algebra elements given by uk=R1((gk)1gk+1)/h, where ukg (see [38,39] for further details). That is, if uk were regarded as an average velocity between gk and gk+1, then R is an approximation to the integral flow of the dynamics. The difference ξk,k+1:=(gk)1gk+1G, which is an element of a nonlinear space, can now be represented by a vector space element uk. For the derivation of the discrete equations of motion, the right trivialized tangent retraction map will be used. It is the function dR:g×gg given by

    TR(ξ)η=TRR(ξ)dRξ(η), (5.1)
    Figure 1.  Retraction map.

    where ηg and R:GG the right translation on G and TR(ξ)η is the directional derivative of R at ξ in the direction of η (see [38,39] for the derivation of such a map). Here we use the following notation, dRξ:=dR(ξ):gg. The function dR is linear, but only on one argument.

    Remark 7. The natural choice of a retraction map is the exponential map at the identity ¯e of the group G, exp¯e:gG. Bear in mind that, for a finite-dimensional Lie group, exp¯e is locally a diffeomorphism and gives rise to a natural chart [40]. Then, there exists a neighborhood U of ¯eG such that exp1¯e:Uexp1¯e(U) is a local Cdiffeomorphism. For an element gG a chart is given by Ψg=exp1¯eLg1.

    Generally, it is not an easy task to work with the exponential map since the differential of the exponential map involves power series expansions with iterated Lie-brackets. Consequently, it will be more convenient to use a different retraction map. More concretely, the Cayley map, which is usually used in numerical integration with matrix Lie-groups configurations (see [38,39] for further details), will provide to us a proper framework in the application shown in the next Section.

    Next, we consider a discrete cost function to construct variational integrators in the same way as in discrete mechanics [41]. In other words, consider the continuous-time Lagrangian L:Gs×gsR defined by the cost functional (3.4), that is,

    L(g,u)=si=1(Ci(gi(t),ui(t))+V0i(gi)+12jNiVij(gi(t),gj(t))),

    and for a given h>0, we define the discrete Lagrangian Ld:Gs×gsR as an approximation of the cost functional (3.4) along each discrete segment between gk and gk+1=gkR(huk), that is,

    Ld(gk,uk)=hL(κ(gk,uk),ζ(gk,uk))(k+1)hkhL(g,u)dt,

    where κ and ζ are functions of (gk,uk)Gs×gs which approximate the configuration g(t) and the control input u(t), respectively, in that interval. In the following, for simplicity, κ will be the projection onto Gs and ζ will be the projection onto gs. Thus, we consider

    Ld(gk,uk)=hsi=1(Ci(gki,uki)+V0i(gki)+Vi(gk)). (5.2)

    We remark that different choices of κ and ζ could result in higher-order numerical methods, such as, using the middle point rule with κ(gk,uk)=gkR(h2uk).

    Next, we are going to define the optimal control problem for discrete-time systems and derive a variational integrator for Ld:Gs×gsR, in a similar fashion as the variational equation presented in Theorem 4.1.

    Problem: Consider the discrete-time optimal control problem for collision and obstacle avoidance of left-invariant multi agent control systems, which is given by finding the discrete configurations {gk}Nk=0={(gk1,,gks)}Nk=0 and discrete control inputs {uk}Nk=0={(uk1,,uks)}Nk=0 minimizing the discrete cost functional

    min(gk,uk)si=1N1k=0h(Ci(gki,uki)+V0i(gki)+Vi(gk)) (5.3)

    subject to gk+1i=gkiR(huki) (i.e., a first order approximation of equation (3.1)) with given boundary conditions g0 and gN, where h>0 denotes the time step, R:gG is a retraction map, ui(0) and ui(T) are given, and each cost function Ci:G×gR, potential functions V0i and Vi satisfy properties (ⅰ) - (ⅶ).

    The discrete-time optimal control problem (5.3) can be considered as a discrete constrained variational problem by introducing the Lagrange multipliers μkig into the cost functional. Consider the augmented discrete Lagrangian Ld:Gs×Gs×gs×(g)sR given by

    Ld(gk,gk+1,uk,μk)=hsi=1(Ci(gki,uki)+V0i(gki)+Vi(gk)+μki,1hR1(ξk,k+1i)uki) (5.4)

    where ξk,k+1i=(gki)1gk+1iG, for each iN. Note that the last term in the augmented Lagrangian represents a first-order discretization of the kinematic constraint paired with a Lagrange multiplier in analogy with the variational equation presented in Section 4.

    Now, extending the potential V0i, we obtain an extended Lagrangian Lext,d:Gs×Gs×gs×(g)s×(W)sR given by

    Lext,d(gk,gk+1,uk,μk,α)=hsi=1(Ci(gki,uki)+V0,exti(gki,αi)+Vi(gk)+μki,1hR1(ξk,k+1i)uki), (5.5)

    which is invariant under the left action of G on Gs×Gs×gs×(g)s×(W)s given by ˜Φg(h1,h2,u,μ,α)=(gh1,gh2,u,μ,ρg1(α)) by assumption (ⅵ). In particular, under assumptions (ⅳ)-(ⅵ), the extended discrete Lagrangian Lext,d(,,,,α0)=:Lext,d,α0 is Gα0-invariant under ˜Φg.

    The following result (Theorem 5.2) derives a variational integration for reduced optimality conditions for the discrete-time optimal control (5.3) in analogy with the results presented in Section 4. To derive the numerical algorithm, first we need the following result describing variations for elements on the Lie algebra and its relation with variations on the Lie group by using the retraction map, in addition to a property used in the proof of Theorem 5.2. Though it is a well-known result in the literature, we include it here to make the paper self-contained.

    Lemma 5.1 (adapted from [38,39]). The following properties hold

    (i)

    1hδ(R1(ξk,k+1))=1hdR1(huk)(ηk+AdR(huk)ηk+1),

    where ηk=TgkL(gk)1(δgk)g and ξk,k+1=(gk)1gk+1.

    (ii)

    (dR1(huk))μk=AdR(huk)(dR1(huk))μk,

    where μk(g) and dR1 is the inverse right trivialized tangent of the retraction map R defined in (5.1).

    Theorem 5.2. Under assumptions (i)-(vii), an extremal for the discrete-time optimal control problem (5.3) satisfies the following equations

    gk+1i=gkiR(huki), (5.6)
    (dR1(huki))μki=(dR1(huk1i))μk1i+JW(hV0,extiˉαki,ˉαki)+hΘi1sl=1T¯eLgki(Vlgki), (5.7)
    μki=(Ciuki), (5.8)
    ˉαk+1i=ρR(huki)(ˉαki),ˉα0i=ρg0i(α0i), (5.9)

    for k=1,,N1; where Θi1=0 if i=1, otherwise Θi1=1.

    Proof. Since the cost functions and the potential functions satisfy assumptions (ⅰ) - (ⅶ), as in the continuous-time case, it is possible to induce the reduced augmented discrete Lagrangian ext,d:Gs1×Gs×gs×(g)s×(W)sR as

    ext,d(gk,ξk,k+1,uk,μk,ˉαk)=hsi=1(Ci(uki)+V0,exti(ˉαki)+μki,1hR1(ξk,k+1i)uki+Vi(gk),

    where ˉαki=ρgki(α0i) for a fixed α0iW satisfying α0i=ρg0i(α0i) and, with a slight abuse of notation, Ci(uki)=Ci(¯e,uki) and V0,exti(αki)=V0,exti(¯e,αki). Notice that, also here, gk1 is set to be the identity element, so that we have gkGs1.

    As in the proof for Theorem 4.1, the technical part is to show that an extremal of the reduced variational equation

    δN1k=0ext,d(gk,ξk,k+1,uk,μk,ˉαk)=0 (5.10)

    satisfies equations (5.6)-(5.8) for all variations of R1(ξk,k+1) (induced by variations of gk vanishing at the endpoints), uk and ˉαk of the form ρηk(ˉαk), where ηkgs vanishes at the endpoints. Then, similarly as in the proof for Theorem 4.1, it follows that an extremal for the optimal control problem (5.3) satisfies the variational equation

    δN1k=0Ld(gk,gk+1,uk,μk)=0,

    for all variations of gk (vanishing at the endpoints), R1(ξk,k+1) (induced by variations of gk) and uk.

    Note that

    0=δN1k=0ext,d(gk,ξk,k+1,uk,μk,ˉαk)=N1k=0si=1h[Ciukiμki,δuki+V0,extiˉαk,δˉαk+sl=2Vigkl,δgkl+μki,1hdR1huki(ηki+AdR(huki)ηk+1i)]

    where we used Lemma 5.1 to obtain the last term. Since variations δuki are arbitrary, we obtain μki=Ciuki. As for the second term, we have that

    V0,extiˉαk,δˉαk=V0,extiˉαk,ρηki(ˉαk)=JW(V0,extiˉαk,ˉαk),ηki.

    As we have show along the proof for Theorem 4.1, we have that

    sl=2Vigkl,δgkl=sl=2T¯eLgkl(Vigkl),ηkl.

    and the indexes might be interchanged. The last term to obtain equation (5.7) may be dealt with, using integration by parts in discrete-time, which is just rearranging the indexes, together with the second statement in Lemma 5.1 and the fact that η0i=ηNi=0. Therefore,

    N1k=0si=1μki,dR1huki(ηki+AdR(huki)ηk+1i)=N1k=1si=1(dR1(huk1i))μk1i(dR1(huki))μki,ηki

    and the equations (5.6)-(5.8) follow.

    Remark 8. Equations (5.6)-(5.8) are as a discrete approximation of the Lie-Poisson equations for the Hamiltonian version of the optimal control problem considered in [22]. The equation μki=(Ciuki) represents the discrete time version of the reduced Legendre transformation and the equation gk+1i=gkiR(huki) is the analogous of the reconstruction equation in the discrete time counterpart. These three equations are used to compute uki,μki and gk+1i given uk1i, μk1i, gk1i and gki from k=1 to k=N1.

    To compute the discrete-time reduced necessary condition for the optimal control problem (5.3) we must enforce boundary conditions given by the continuous-time quantities. More precisely, we must set

    (dR1(hu0i))μ0i=Ciui(ui(0))+hΘi1sl=2T¯eLg0i(Vlg0i)+hJW(V0,extiˉα0,ˉα0),Ciui(ui(T))=(dR1(huN1i))μN1i, (5.11)

    relating the momenta at the initial and final times, and use to transform boundary values between the continuous and discrete representation. They follow from the principle that any variation with free boundary points of the action (5.10) along a solution of equations (5.6)-(5.9) equals the change in momentum Ciui(ui(T)),gNiδgNiCiui(ui(0)),g0iδg0i (see [39] for a discussion in the single agent case). Therefore, we must enforce

    δN1k=0ext,d(gk,ξk,k+1,uk,μk,ˉαk)=Ciui(ui(T)),gNiδgNiCiui(ui(0)),g0iδg0i

    where the variation is taken along a family of solutions of the discrete equations (5.6)-(5.9) whose boundary values are not fixed. By taking the variations of the discrete action, we eventually get a vanishing term corresponding to the fact that this family of sequences satisfies the discrete equations together with boundary terms multiplying η0i and ηNi, and simplifying, to the above equations.

    Remark 9. If we choose the midpoint rule to discretize the potential Vi, then we would obtain the following boundary conditions

    (dR1(hu0i))μ0i=Ciui(ui(0))+h2Θi1sl=1T¯eLg0i(Vlg0i)+hJW(V0,extiˉα0,ˉα0),Ciui(ui(T))=(dR1(huN1i))μN1i+h2Θi1sl=1T¯eLgNi(VlgNi).

    The boundary condition gs(T) for agent s is enforced by the relation

    R1((gNs)1gs(T))=0. (5.12)

    Recalling that R(0)=¯e, this last expression just means that gNs=gs(T). Moreover, by computing recursively the equation gk+1s=gksR(huks) for k=1,,N1, using that g0s=gs(0) and R(0)=¯e, it is possible to translate the final configuration gNs in terms of uks, such that there is no need to optimize over any of the configurations gks. In that sense, (5.7) for i=s, l=0 together with

    R1[(R(huN1s))1(R(hu0s))1(gs(0))1gs(T)]=0, (5.13)

    form a set of (nN)-equations (since dimg=n) where nN unknowns are for u0:N1s, denoting the entire sequence of controls {u0s,,uN1s} for each agent s.

    The numerical algorithm to compute the reduced optimality conditions is summarized in Algorithm 1.

    Algorithm 1 Reduced conditions for optimality
    1: Data: Lie group G, its Lie algebra g, cost functions Ci, artificial potential functions Vij, V0i,ext, final time T, # of steps N.
    2: inputs: gi(0), gi(T), ui(0), ui(T), α0i, u0i for all i=1,,s and h=T/N.
    3: for i=1s do
    4:  Fix g0i=gi(0) and α0i
    5:  solve (5.6) and (5.9) for k=0.
    6: outputs: g1s, α1s
    7: for k=1N1 do
    8:  for i=1s do
    9:    solve (5.6)-(5.9) subject to (5.11).
    10: outputs: g0:N1s, u0:N1s, μ0:N1s,α0:N1s.
    11: Compute g0:N1s, u0:N1s, μ0:N1s,α0:N1s subjected to (5.13).

    Note also that the exact form of equations (5.6)-(5.8) depends on the choice of R. This choice will also effect the computational efficiency of the optimization framework in the case, the above equalities are imposed as constraints. For instance, in Section 6, we will employ the Cayley transform on the Lie group SE(2) as a choice of R to write in a compact form the numerical integrator [38], [17], but another natural choice would be to employ the exponential map, as we explained in Section 5.1.

    In this case study, we apply the proposed reduction by symmetry strategy to an optimal control for autonomous surface vehicles (ASVs). The configuration space whose elements determine the motion of each ASV is SE(2)SO(2)×R2. An element giSE(2) is given by gi=(cosθisinθixisinθicosθiyi001), where (xi,yi)R2 represents the center of mass of a planar rigid body describing the ASV and θi represents the angular orientation of the ASV. The control inputs, for each ASV, are given by ui=(u1i,u2i), where u1i denotes the speed of the center of mass for the ASV and u2i denotes the angular velocity of the ASV.

    The kinematic equations for the multi-agent system are:

    ˙xi=u2icosθi,˙yi=u2isinθi,˙θi=u1i,i=1,,s. (6.1)

    Using the notation of Example 1, the Lie algebra se(2) is identified with R2 through the isomorphism (aJb00)(a,b). The elements of the basis of the Lie algebra se(2) are e1=(010100000),e2=(001000000),e3=(000001000),

    which satisfy [e1,e2]=e3,[e2,e3]=0,[e3,e1]=e2. Thus, the kinematic equations (6.1) take the form ˙gi=giui=gi(u1ie1+u2ie2) and give rise to a left-invariant control system on SE(2)s×se(2)s. The inner product on se(2) is given by ξ1,ξ2:=tr(ξT1ξ2) for ξ1,ξ2se(2) and hence, the norm is given by ||ξ||=tr(ξTξ), for any ξse(2). The dual Lie algebra se(2) of SE(2) is defined through the dual pairing, α,ξ=tr(αξ), where αse(2) and ξse(2), hence, the elements of the basis of se(2) are e1=(01201200000),e2=(000000100),e3=(000000010).

    Consider the cost function Ci(gi,ui)=12ui,ui and the artificial potential function Vij:SE(2)×SE(2)R given by Vij(gi,gj)=σij2((xixj)2+(yiyj)24¯r2), where σijR0 and ¯r is the radius of the disk each agent occupies as defined at the end of Section Ⅲ. Consider a spherical obstacle with unit radius and without loss of generality let it be centered at the origin. Hence, consider the obstacle avoidance potential function V0i:SE(2)R, V0i(gi)=σi02(x2+y2(¯r+1)2), where σi0R>0.

    Note that the obstacle avoidance potential functions are not SE(2)-invariant but SO(2)-invariant, so they break the symmetry. Using the norm of se(2) and for ¯r=1, Vij and V0i are equivalently given by Vij(gi,gj)=σij2(||Adg1igje1||26) and V0i(gi)=σi02(||Adg1ie1||26).

    Let W=se(2), so we define the extended potential functions Vexti,0:SE(2)×se(2)R by Vexti,0(gi,α)=σi02(||Adg1iα||26), which are SE(2)-invariant under the action of ˜Φ given by (3.6), i.e. Vexti,0˜Φ=Vexti,0, for any gSE(2). Since, W=se(2) we have JW(Vexti,0αi,αi)=adαi(Vexti,0αi), and equations (4.2) and (4.3) yield

    ddt(ui+λi)=adui(ui+λi)+adαi(Vexti,0αi)+Θi1jNiT¯eLgi(Vijθie1+Vijxie2+Vijyie3),

    together with ˙αi=aduiαi and αi=Adg1iα0.

    Note also that ui=u1ie1+u2ie2, αi=α1ie1+α2ie2+α3ie3 and λi=λi3e3 thus,

    adui(ui+λi)=(0u2iλi320u2iλi3200u1iλi3u1iu2i0),aduiαi=(00u1iα3i00u1iα2iu2iα1i000),adαi(Vexti,0αi)=(000000Γ31i,0Γ32i,00)=σi0α1i(||αi||26)2(000000α3iα2i0),T¯eLgi(Vijgi)=(000000Γ31ijΓ32ij0)=σij((xij)2+(yij)24)2(000000xijyij0),

    where xij=xixj,yij=yiyj and Adg1iα0=(01xisinθiyicosθi10xicosθi+yisinθi000).

    Therefore, by applying Proposition 4.2 for r=span{e1,e2} and s=span{e3}, Euler-Lagrange equations for the OCP (3.4) are

    ˙u1i=u2iλi32,˙u2i=u1iλi3+Γ31i,0+Θi1jNiΓ31ij,˙λi3=u1iu2i+Γ32i,0+Θi1jNiΓ32ij,

    with

    ˙α1i=0,α1i(0)=1,˙α2i=u1iα3i,α2i(0)=x0isinθ0iy0icosθ0i,˙α3i=u1iα2i+u2iα1i,α3i(0)=x0icosθ0i+y0isinθ0i.

    For the discrete-time setting, one would choose

    Ci(gki,uki)=h2ukij,ukij, Vi(gki)=hσi2(Ad(gki)1e126),

    where gki=[cosθkisinθkixkisinθkicosθkiyki001]SE(2) and uki=3j=1ukijejse(2). Also, in the discrete-time setting, the extended potential function Vd,ext:SE(2)×se(2)R can be constructed in exactly the same way as in the above example, and is given by

    V0,exti(gki,αi)=hσi2(Ad(gki)1αi26),

    where αki=3j=1αkijej. We do not give all the details again and leave it up to reader to verify that the assumptions (ⅰ) - (ⅶ) from (3.3) are satisfied. The discrete-time equations are

    (dR1huki)μki=(dR1huk1i)μk1i+adˉαkiV0,extiˉαki=Θi1jNiT¯eLgi(Vijθie1+Vijxie2+Vijyie3),ˉαk+1i=AdR(huki)1ˉαki,ˉα0i=Ad(g0i)1α0i,

    where

    adˉαkiV0,extiˉαki=hσiˉαki1(ˉαki26)2[000000ˉαki3ˉαki20].

    For numerical purposes, we first choose a suitable retraction map, like the Cayley map or the exponential map, and then compute the quantities (dR1huki)μki and (dR1huk1i)μk1i. As an example, if we choose the Cayley map cay:se(2)SE(2) as the retraction map, (see [38] and [17] for instance) then we have

    [dcay1huki]μki=[012γi012γi00μki2huki1μki32huki1μki22+μki30]

    where

    γi=(h2(uki1)24+1)μki1+(h2uki1uki24huki32)μki2+(h2uki1uki34+huki22)μki3

    and μki=3j=1μkijej. Note that for v=3i=1vieig, the matrix representation for dcay1v is given by

    [dcay1v]=[1+(v1)2400v1v24v321v12v1v34+v22v121].

    We studied the reduction by symmetry for optimality conditions of extremal in an OCP for collision and obstacle avoidance of left-invariant multi-agent control system on Lie groups, by exploiting the physical symmetries of the agents and obstacles. Reduced optimality conditions are obtained using techniques from variational calculus and Lagrangian mechanics on Lie groups, in the continuous-time and discrete-time settings. We applied the results to an OCP for multiple unmanned surface vehicles. The method proposed in this work allows the construction of position and velocity estimators, by discretizing the variational equation given in Theorem 4.1 - instead of discretizing the equations of motion - and by deriving variational integrators - see Theorem 5.2. The reduction of sufficient conditions for optimality will be also studied by using the notion of conjugate points, as in [42], in future work, as well as the reduction by symmetry of the variational obstacle avoidance problems [43] on semidirect products of Lie groups endowed with a bi-invariant metric on a Riemannian manifold.

    L. Colombo is very grateful to A. Bloch, R. Gupta and T. Ohsawa for many useful comments and stimulating discussions during the last years on the topic of this paper, which is inspired by our common previous work.

    The authors acknowledge financial support from the Spanish Ministry of Science and Innovation, under grants PID2019-106715GB-C21, MTM2016-76702-P.

    We the authors declare that there are no conflicts of interest.



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