Obstacle Avoidance via Hybrid Feedback
11 Obstacle Avoidance via Hybrid Feedback
S. Berkane,
Member, IEEE,
A. Bisoffi,
Member, IEEE , D. V. Dimarogonas,
Senior Member, IEEE
Abstract —In this paper we present a hybrid feedback ap-proach to solve the navigation problem of a point mass in the n − dimensional space containing an arbitrary number of ellip-soidal shape obstacles. The proposed hybrid control algorithmguarantees both global asymptotic stabilization to a referenceand avoidance of the obstacles. The intuitive idea of the proposedhybrid feedback is to switch between two modes of control:stabilization and avoidance. The geometric construction of theflow and jump sets of the proposed hybrid controller, exploitinghysteresis regions, guarantees Zeno-free switching between thestabilization and the avoidance modes. Simulation results illus-trate the performance of the proposed hybrid control approachfor 2-dimensional and 3-dimensional scenarios. I. I
NTRODUCTION
For decades, the obstacle avoidance problem has been anactive area of research in the robotics and control communities[1]. In a typical robot navigation scenario, the robot is requiredto reach a given goal (destination) while not colliding with aset of obstacle regions in the workspace. Since the pioneeringwork by Khatib [2], artificial potential fields have been widelyused in the obstacle avoidance problem since they offer thepossibility to combine the solution to the global find-path prob-lem with a feedback controller for the robot, thus, allowing thehigh-level planner to address more abstract tasks. The idea isto generate an artificial potential field that renders the goalattractive and the obstacles repulsive. Then, by consideringtrajectories that navigate along the negative gradient of theartificial potential field, one can ensure that the robot willreach the desired target while avoiding to collide with theobstacles. However, artificial potential field-based algorithmssuffer from 1) the presence of local minima preventing thesuccessful navigation to the target point and 2) arbitrarily largerepulsive potential near the obstacles which are in conflict withthe inevitable actuator saturations.The navigation function-based approach, which was initi-ated by Koditscheck and Rimon [3] for sphere worlds [3,p. 414], solves both problems. It allows to obtain artificialpotential fields with the nice property that all but one ofthe critical points are saddles with the remaining criticalpoint being the desired reference. Since then, the navigationfunction-based approach has been extended in many differentdirections; e.g., for multi-agent systems [4]–[6], for unknownsphere words [7], and for focally admissible obstacles [8].
This research work is partially supported by NSERC-DG RGPIN-2020-04759, the European Research Council (ERC), the EU H2020 Co4Robots, theSSF COIN project, the Swedish Research Council (VR) and the Knut och Al-ice Wallenberg Foundation. S. Berkane ( [email protected] )is with the Department of Computer Science and Engineering, University ofQuebec in Outaouais, Gatineau, Canada. A. Bisoffi ( [email protected] )is with ENTEG and the J.C. Willems Center for Systems and Control,University of Groningen, Groningen, The Netherlands. D. V. Dimarogonas( [email protected] ) is with the Division of Decision and Control Systems,KTH Royal Institute of Technology, Stockholm, Sweden.
The major drawback of navigation functions is that they arenot correct by construction. In fact, navigation functions aretheoretically guaranteed to exist, but their explicit computationis not straightforward since they require an unknown tuningof a given parameter to eliminate local minima.Recently, Loizou [9] introduced the navigation transformthat diffeomorphically maps the workspace to a trivial domaincalled the point world consisting of a closed ball with afinite number of points removed. Once this transformationis found, the navigation problem is solved from almost allinitial conditions without requiring any tuning. In addition,the trajectory duration is explicitly available, which providesa timed-abstraction solution to the motion-planning problem.Similarly, the recent work in [10] uses the so-called prescribedperformance control to design a time-varying control law thatdrives the robot, in finite time, from all initial conditions tosome neighborhood of the target while avoiding the obstacles.Another approach to the navigation problem is through barrierfunctions (see [11] and references therein), which are devel-oped for nonlinear systems with state-space constraints andensure safety. Model predictive control approaches have beenalso used for reactive robot navigation, e.g., [12], [13].However, by using any of the approaches described above,it is not possible to ensure safety from all initial conditionsin the obstacle-free state space. As pointed out in [3], theappearance of additional undesired equilibria is unavoidablewhen considering continuous time-invariant vector fields. Thisis a well known topological obstruction to global asymptoticstabilization by continuous feedback when the free state spaceis not diffeomorphic to an Euclidean space (see, e.g., [14,Thm. 2.2]). Furthermore, this problem is more far-reachingsince, by using a continuous feedback law, it is always possibleto find arbitrarily small adversarial (noise) signals acting on thevector field, such that a set of initial conditions different fromthe target, possibly of measure zero, can be rendered stable[15, Thm. 6.5]. To deal with such limitations, the authorsin [16] proposed a hybrid state feedback controller, usingLyapunov-based hysteresis switching, to achieve robust globalasymptotic regulation in R to a target while avoiding a singleobstacle. This approach has been exploited in [17] to steer aplanar vehicle to the source of an unknown but measurablesignal while avoiding an obstacle. In [18] and [19], a hybridcontrol law was proposed to globally asymptotically stabilizea class of linear systems while avoiding neighbourhoods ofunsafe isolated points in R n . Although such hybrid approachesare promising, they are still challenged by constructing thesuitable hybrid feedback for higher dimensions and with morecomplex obstacles shapes.In this work, we propose a hybrid control algorithm for theglobal asymptotic stabilization of a point mass moving in anarbitrary n − dimensional space while safely avoiding obstacles a r X i v : . [ m a t h . O C ] F e b that have generic ellipsoidal shapes, based on the preliminarytreatment of this problem for a single spherical obstacle in[20]. The ellipsoids provide a tighter bounding volume thanspheres, and in our scheme this volume can be arbitrarily flatand close to the target, which leads to a significant reductionin the level of conservatism compared, for instance, to [21,Thm. 3] (as shown in Section VI).Our proposed hybrid algorithm employs a hysteresis-basedswitching between the avoidance controller and the stabiliz-ing controller in order to guarantee forward invariance ofthe obstacle-free region (corresponding to safety) and globalasymptotic stability of the reference position. We consider tra-jectories in an n − dimensional Euclidean space and we resortto tools from higher-dimensional geometry [22] to provide aconstruction of the flow and jump sets where the differentmodes of operation of the hybrid controller are activated.Furthermore, the hybrid control law guarantees a boundedcontrol input, it matches the stabilizing controller in arbitrarilylarge subsets of the obstacle-free region by a suitable tuning ofits parameters (hence qualifying as minimally invasive), it canbe readily extended to a non-point mass vehicle and enjoyssome level of inherent robustness to perturbations. Structure.
Preliminaries are in Section II. The navigation problem isformulated in Section III. Our proposed hybrid control schemeis discussed in Section IV. Section V presents the mainresult of forward invariance of the obstacle-free space andglobal asymptotic stability of the target, together with otherdesirable complementary properties. Numerical examples arein Section VI. All the proofs are in the Appendix.II. P
RELIMINARIES N , R and R ≥ denote, respectively, the set of nonnegativeintegers, reals and nonnegative reals. R n is the n -dimensionalEuclidean space and S n is the n -dimensional unit sphereembedded in R n +1 . Given the column vectors v ∈ R n and v ∈ R n , ( v , v ) denotes the stack vector (cid:2) v (cid:62) v (cid:62) (cid:3) (cid:62) . TheEuclidean norm of x ∈ R n is defined as (cid:107) x (cid:107) := √ x (cid:62) x andthe geodesic distance between two points x and y on thesphere S n is defined by d S n ( x, y ) := arccos( x (cid:62) y ) for all x, y ∈ S n . For an arbitrary matrix A ∈ R n × n , λ i ( A ) denotesthe i -th eigenvalue of A . If A is a symmetric matrix, then λ min ( A ) and λ max ( A ) denote, respectively, the smallest andlargest eigenvalues of A . Given a closed set A ⊂ R n , wedefine the distance to the set A by | x | A := inf y ∈A (cid:107) x − y (cid:107) .Given two sets A and B , we define the distance from A to B by dist ( A , B ) := inf {| a − b | : a ∈ A , b ∈ B} . Theclosure, interior and boundary of a set A ⊂ R n are denotedas A , A ◦ and ∂ A , respectively. The relative complement of aset B ⊂ R n with respect to a set A is denoted by A\B andcontains the elements of A which are not in B . In particular,we use A c to denote the complement of A in R n , i.e., A c = R n \A . Given the sets A , B and C , the following setidentities [23] will be used A ∪ ( B ∩ C ) = (
A ∪ B ) ∩ ( A ∪ C ) (1a) A ∩ ( B ∪ C ) = (
A ∩ B ) ∪ ( A ∩ C ) (1b) ( A ∩ C ) \ ( B ∩ C ) = (
A ∩ C ) \B = A ∩ ( C\B ) (1c) C\ ( A∪B ) = (
C\A ) ∩ ( C\B ) , C\ ( A∩B ) = (
C\A ) ∪ ( C\B ) (1d) ( A ∪ B ) c = A c ∩B c , ( A ∩ B ) c = A c ∪B c , A\B = A∩B c (1e) A ∪ B = A ∪ B , A ∩ B ⊂ A ∩ B , A\A c = A ◦ (1f) A ⊂ B = ⇒ A ⊂ B , ∂ A c = ∂ A (1g) ∂ ( A ∪ B ) ⊂ ( ∂ A\B ◦ ) ∪ ( ∂ B\A ◦ ) (1h) ∂ ( A ∩ B ) ⊂ ( ∂ A ∩ B ) ∪ ( ∂ B ∩ A ) . (1i)Two sets A and B are said to be disjoint if A∩B = ∅ . Theyare said to be separated if A ∩ B = ∅ = A ∩ B . The notion ofseparated sets is stronger than mere disjointness. If two sets A and B are separated then we have [23, Exercise 1.3.A] ∂ ( A ∪ B ) = ∂ A ∪ ∂ B . (2)The tangent cone to a set K ⊂ R n at a point x ∈ R n , denoted T K ( x ) , is defined as in [24, Def. 5.12 and Fig. 5.4]. A. Projections Maps
For z ∈ R n \{ } , we define the following projection maps: π (cid:107) ( z ) := zz (cid:62) (cid:107) z (cid:107) , π ⊥ ( z ) := I n − zz (cid:62) (cid:107) z (cid:107) , ρ ( z ) := I n − zz (cid:62) (cid:107) z (cid:107) (3)where I n is the n × n identity matrix. The map π (cid:107) ( · ) is theparallel projection map, π ⊥ ( · ) is the orthogonal projectionmap [22], and ρ ( · ) is the reflector map (also called House-holder transformation). Consequently, for any x ∈ R n , thevector π (cid:107) ( z ) x corresponds to the projection of x onto the linegenerated by z , π ⊥ ( z ) x corresponds to the projection of x onto the hyperplane orthogonal to z and ρ ( z ) x corresponds tothe reflection of x about the hyperplane orthogonal to z . For z ∈ R n \{ } , some useful properties of these maps follow: π (cid:107) ( z ) z = z, π ⊥ ( z ) π ⊥ ( z ) = π ⊥ ( z ) , (4a) π ⊥ ( z ) z = 0 , π (cid:107) ( z ) π (cid:107) ( z ) = π (cid:107) ( z ) , (4b) ρ ( z ) z = − z, ρ ( z ) ρ ( z ) = I n . (4c)We also define for z ∈ R n \{ } and θ ∈ R the parametric map π θ ( z ) := cos ( θ ) π ⊥ ( z ) − sin ( θ ) π (cid:107) ( z ) , (5)which can also be written (thanks to π ⊥ ( z ) − ρ ( z ) = 2 π (cid:107) ( z )+ ρ ( z ) = I n ) as π θ ( z ) = ρ ( z ) + cos(2 θ ) I n . (6) B. Geometric Subsets of R n
1) Line:
A line is the one-dimensional subset of R n de-scribed by the set L ( c, v ) := { x ∈ R n : x = c + λv, λ ∈ R } , (7)which corresponds to the line passing by the point c ∈ R n and with direction parallel to v ∈ R n \{ } . If in (7) λ ≥ (respectively λ ≤ ), then we obtain the half-line denoted by L ≥ ( c, v ) (respectively L ≤ ( c, v ) ).
2) Hyperplane:
A hyperplane is the ( n − -dimensionalsubset of R n described by the set P ( c, v ) := { x ∈ R n : v (cid:62) ( x − c ) = 0 } , (8)which corresponds to the hyperplane that passes through apoint c ∈ R n and has normal vector v ∈ R n \{ } . Thehyperplane P ( c, v ) divides the Euclidean space R n into twoclosed subsets P ≥ ( c, v ) and P ≤ ( c, v ) , which are obtained bysubstituting the = in (8) with ≥ and ≤ , respectively.
3) Sphere:
A sphere is the ( n − -dimensional subset of R n described by the set S ( c, r ) := { x ∈ R n : (cid:107) x − c (cid:107) = r } (9)where c is the center of the sphere and r ∈ R ≥ is itsradius. The closed interior (respectively exterior) of the sphere,also called a hyperball and denoted by S ≤ ( c, r ) (respectively S ≥ ( c, r ) ), is obtained from (9) by substituting the = with ≤ (respectively ≥ ).
4) Ellipsoid:
For a positive definite matrix E ∈ R n × n , aellipsoid is the ( n − -dimensional subset of R n described bythe set E ( c, E ) := { x ∈ R n : (cid:107) E ( x − c ) (cid:107) = 1 } (10)where c is the center of the ellipsoid and its i -th principalsemi-axis is the vector λ − i ( E ) v i , with v i the unit eigenvectorcorresponding to the eigenvalue λ i ( E ) . The closed interior(respectively exterior) of the ellipsoid, denoted by E ≤ ( c, E ) (respectively E ≥ ( c, E ) ), is obtained from (10) by substitutingthe = with ≤ (respectively ≥ ). Definition
1: Two ellipsoids E ≤ ( c , E ) and E ≤ ( c , E ) areweakly disjoint if E ≤ ( c , E ) ∩ E ≤ ( c , E ) = ∅ . Explicit algebraic conditions to test weak disjointness of twoellipsoids can be found in [25, Thm. 6] for n = 2 and in [26,Thm. 8] for n = 3 . Definition
2: Two ellipsoids E ≤ ( c , E ) and E ≤ ( c , E ) arestrongly disjoint if ( λ min ( E )) − +( λ min ( E )) − < (cid:107) c − c (cid:107) . Strong disjointness means that the two smallest spherical ballscontaining the ellipsoids are disjoint. Strong disjointness ismore conservative than weak disjointness.
5) Cone:
For a positive definite matrix E ∈ R n × n , a coneis the ( n − -dimensional subset of R n described by the set C ( c,v,θ,E ) := { x ∈ R n : cos( θ ) (cid:107) Ev (cid:107)(cid:107) E ( x − c ) (cid:107) = v (cid:62) E ( x − c ) } (11)where c ∈ R n is its vertex, v ∈ R n \{ } is its axis and θ ∈ [0 , π ] is its aperture. The cone defined here is sometimesreferred to as nappe or half-cone, as opposed to the doublecone. The closed interior (respectively, exterior) of the cone,denoted by C ≤ ( c, v, θ, E ) (respectively C ≥ ( c, v, θ, E ) ), is ob-tained from (11) by substituting the = with ≤ (respectively ≥ ). A normal vector to the cone surface C ( c, v, θ, E ) at x is n ( x ) := Eπ θ ( Ev ) E ( x − c ) , (12)and can be obtained after squaring in (11) and taking thegradient. The next fact will be used. Lemma
1: Let v , v ∈ S n − such that v (cid:62) v = cos θ forsome θ ∈ (0 , π ] . Let ψ , ψ ∈ [0 , π/ with ψ + ψ < θ . Thenfor each c ∈ R n and E ∈ R n × n positive definite, C ≤ ( c, E − v , ψ , E ) ∩ C ≤ ( c, E − v , ψ , E ) = { c } . C. Hybrid Systems Framework
We consider hybrid dynamical systems of the class [24],described through constrained differential and difference in-clusions for state X ∈ R n : (cid:40) ˙ X ∈ F ( X ) , X ∈ F ,X + ∈ J ( X ) , X ∈ J , (13)where the flow map F : R n ⇒ R n governs the continuousevolution, the flow set F ⊆ R n dictates where continuousevolution can occur. The jump map J : R n ⇒ R n governs thediscrete evolution, and the jump set J ⊆ R n defines wherediscrete evolution can occur. The hybrid system (13) is definedby its data and denoted H = ( F , F , J , J ) .A subset T ⊂ R ≥ × N is a hybrid time domain if it is a unionof a finite or infinite sequence of intervals [ t j , t j +1 ] ×{ j } , withthe last interval (if existent) possibly of the form [ t j , T ) with T finite or T = + ∞ . The ordering of points on each hybridtime domain is such that ( t, j ) (cid:22) ( t (cid:48) , j (cid:48) ) if t < t (cid:48) , or t = t (cid:48) and j ≤ j (cid:48) . A hybrid solution is defined in [24, Def. 2.6].A hybrid solution φ is maximal if it cannot be extended andcomplete if its domain dom φ (which is a hybrid time domain)is unbounded. III. P ROBLEM F ORMULATION
We consider a point mass vehicle moving in the n -dimensional Euclidean space containing I ∈ N obstaclesdenoted by O , · · · , O I . For each i ∈ { , · · · , I } =: I , the obstacle O i has an ellipsoidal shape such that O i := E ≤ ( c i , E i ) , for some center c i ∈ R n and some positive definitematrix E i ∈ R n × n defining the orientation and the shape ofthe obstacle. The free workspace (obstacle-free region) is thendefined by the closed set W := (cid:92) i (cid:48) ∈ I E ≥ ( c i (cid:48) , E i (cid:48) ) . (14)The vehicle is moving according to the dynamics ˙ x = u, (15)where x ∈ R n is the state and u ∈ R n is the control input. Thevehicle is required to stabilize its position to a target positionwhile avoiding the obstacles. Without loss of generality weconsider the target position to be x = 0 (the origin). Assumption n ≥ . We consider n ≥ since for n = 1 (i.e., the state space is aline), global asymptotic stabilization with obstacle avoidanceis infeasible. Assumption
2: For all i ∈ I , (cid:107) E i c i (cid:107) > . Assumption 2 requires that the target position x = 0 is notinside any of the obstacle regions O i , otherwise the considerednavigation problem would be infeasible. Assumption {O i } i ∈ I are weakly pairwise disjoint. In Assumption 3 we impose that there is no intersectionregion between the obstacles. Otherwise, the union of the twointersecting obstacles forms another region which might have P ( x, E i ( x − c i )) E i ( x − c i ) κ ( x, i, m ) p i − x O i c i E ( c i , (cid:15)E i ) p i Fig. 1. Illustration of the projection-based avoidance controller. The vehicle isattracted to the auxiliary point p im while sliding on a neighbouring ellipsoid. a different shape than an ellipsoid. Our objectives in designinga control strategy are:i) the obstacle-free region W in (14) is forward invariant,ii) the target x = 0 is globally asymptotically stable.Objective i) guarantees that all solutions of the closed-loopsystem are safely avoiding the obstacles by remaining inthe obstacle-free region W for all times while objective ii)corresponds to global stabilization of the target.IV. H YBRID C ONTROL FOR O BSTACLE A VOIDANCE
In this section, we propose a hybrid controller that switchessuitably between a stabilizing and an avoidance controller. Letus define a discrete variable m ∈ {− , , } =: M . The value m = 0 corresponds to the activation of thestabilizing controller and the values m = − , m = 1 correspond to the activation of one of the two configurationsof the avoidance controller. The avoidance controller dependsalso on the current obstacle O i , as detailed in the next sections. A. Control Input
In this section we propose the feedback law for the controlinput u in (15). u depends on the state x ∈ R n , the obstacle i ∈ I and the control mode m ∈ M as u = κ ( x, i, m ) (16) := (cid:40) − k x, m = 0 , − k m E − i π ⊥ ( E i ( x − c i )) E i ( x − p im ) , m ∈ {− , } , where k − , k , k > are the control gains for each controlmode m ∈ M and the points p im ∈ R n , m ∈ {− , } and i ∈ I ,are design parameters defined below. In the stabilization mode( m = 0 ), the control input in (16) steers x towards the originunder a state feedback. In the avoidance mode depicted inFig. 1, the control input minimizes the distance to the auxiliary attractive point p im while maintaining a constant distance to theobstacle O i . Indeed, the time derivative of (cid:107) E i ( x − c i ) (cid:107) alongsolutions of ˙ x = κ ( x, i, m ) for m ∈ {− , } and i ∈ I , reads ddt (cid:107) E i ( x − c i ) (cid:107) = ( x − c i ) (cid:62) E i κ ( x, i, m )= − k m ( x − c i ) (cid:62) E i π ⊥ ( E i ( x − c i )) E i ( x − p im ) = 0 (17)by (4b). Then, if we activate the avoidance mode sufficientlyaway from the obstacle, the avoidance feedback u = κ ( x, i, m ) guarantees that the vehicle does not hit the obstacle. Whereasthe logic variable i corresponds to obstacle O i , the logic variable m is selected according to a hybrid mechanism thatexploits a suitable construction of the flow and jump sets asdetailed in Section IV-B.In order to clear the obstacle while approaching the desiredtarget position at the origin, we select the points p i and p i − in the region between the obstacle and the origin, see Fig. 1.The motivation is that the avoidance task is equivalent (upto a linear transformation) to a stabilization problem on theunit sphere S n − . Therefore, as pointed out for instance in[27], global asymptotic stabilization cannot be accomplishedby only one continuous time-invariant controller, but it can beby a hybrid feedback with at least two configurations. For thisreason, we consider two avoidance modes with m = − and m = 1 and, hence, the points p i and p i − must be distinct.More precisely, for θ i > (which will be further bounded inLemma 4), the points p i and p i − are selected as p i ∈ C ( c i , − c i , θ i , E i ) \{ c i } , (18a) p i − := − E − i ρ ( E i c i ) E i p i . (18b)By (18), p i − opposes p i diametrically with respect to the coneaxis (for E i = I n , p i − is obtained by an orthogonal reflection)and also belongs to C ( c i , − c i , θ i , E i ) \{ c i } as shown in thenext lemma. Lemma p i − ∈ C ( c i , − c i , θ i , E i ) \{ c i } . Note that the results of the paper hold for any selection of thepoint p i as long as it lies on the surface of the cone as in(18a). An explicit guided choice for those points is given inSection VI for the 2D and 3D cases. Finally, further motivationabout the choice of the avoidance controller mode in (16) isdetailed in Section IV-B and, in particular, in Lemma 3, whichis important for the construction of flow and jump sets. B. Geometric Construction of the Flow and Jump sets
In this section we construct explicitly the flow and jump setswhere the stabilization and avoidance controllers are activated.
1) Safety Helmets:
Our proposed construction of flow andjump sets is based on regions that have the shape of a helmet ,whose construction is now motivated. In the stabilization mode m = 0 , the closed-loop system should not flow when: 1) x isclose enough to any of the obstacle regions E ≤ ( c i , E i ) and 2)the vector field − k x points inside E ≤ ( c i , E i ) . Otherwise, thevehicle ends up hitting the obstacle i . Indeed, by computingthe time derivative of (cid:107) E i ( x − c i ) (cid:107) along solutions of thevector field − k x , we obtain ddt (cid:107) E i ( x − c i ) (cid:107) = − k x (cid:62) E i ( x − c i )= k c (cid:62) i E i c i / − k ( x − c i / (cid:62) E i ( x − c i / k (cid:107) E i ¯ c i (cid:107) (cid:0) − (cid:107) ¯ E i ( x − ¯ c i ) (cid:107) (cid:1) (19)where ¯ c i and ¯ E i are defined as ¯ c i := c i / , ¯ E i := 2 E i / ( (cid:107) E i c i (cid:107) ) . (20)(19) implies that the distance function (cid:107) E i ( x − c i ) (cid:107) decreasesfor all x in the closed set E ≥ (¯ c i , ¯ E i ) . Consider now Fig. 2 fora sketch of the next sets and for obstacle i , define the helmet -shaped set H ∗ i := E ( c i , E i ) ∩ E ≥ (¯ c i , ¯ E i ) . (21) c i E (¯ c i , ν ¯ E i ) E ( c i , (cid:15)E i ) E ( c i , E i ) H i ( (cid:15), ν ) H ∗ i E (¯ c i , ¯ E i ) Fig. 2. The helmet H ∗ i in (21) (red) corresponds to all boundary points wherethe stabilization vector field is pointing inside the obstacle (grey). The safetyhelmet H i ( (cid:15), ν ) in (22) (green) corresponds to an dilated version of H ∗ i . H ∗ i is the set of all points that lie on the boundary of theobstacle O i and generate a vector field pointing towards theobstacle. Then, for obstacle i , we define the safety helmet as: H i ( (cid:15), ν ) := E ≤ ( c i , (cid:15)E i ) ∩ E ≥ ( c i , E i ) ∩ E ≥ (¯ c i , ν ¯ E i ) (22)for some parameters (cid:15), ν > . (cid:15) and ν determine the thicknessof the safety helmet by tuning the dilation/shrinking of theellipsoids E ( c i , E i ) and E (¯ c i , ¯ E i ) , thereby generating a dilatedversion of H ∗ i . The safety helmet H i ( (cid:15), ν ) constitutes the mainingredient of our following constructions.
2) Stabilization Mode m = 0 : Consider from now onFig. 3 for a visualization of the sets we are introducing inour construction. In stabilization mode ( m = 0 ), we createaround each obstacle O i a safety helmet H i ( (cid:15) i , ν i ) whichadds a safety layer to the given obstacle. The controller modemust be switched to the avoidance mode whenever the vehiclereaches this safety helmet. Specifically, we define for each i ∈ I , a jump set J i := H i ( (cid:15) i , ν i ) ∩ W , (23)where (cid:15) i ∈ (0 , (dilating E ≤ ( c i , E i ) to E ≤ ( c i , (cid:15) i E i ) ), and ν i ∈ (1 , ∞ ) (shrinking E ≥ (¯ c i , ¯ E i ) to E ≥ (¯ c i , ν i ¯ E i ) ) and W is the free workspace defined in (14). We emphasize that weconsider the intersection with W in (23) for convenience, butlater we tune the parameters such that H i ( (cid:15) i , ν i ) ⊂ W , whichimplies J i will equal to H i ( (cid:15) i , ν i ) . The selection of J i in (23)leads naturally to the following flow set of the stabilizationmode (corresponding to the closed complement of J i in thefree workspace) F i := (cid:16) E ≥ ( c i , (cid:15) i E i ) ∪ E ≤ (¯ c i , ν i ¯ E i ) (cid:17) ∩ W . (24)Finally, from (23) and (24), we take all the obstacles intoaccount and define the flow and jump sets for the stabilizationmode m = 0 as F := (cid:16) (cid:92) i ∈ I F i (cid:17) × I , J := (cid:16) (cid:91) i ∈ I J i (cid:17) × I . (25)Indeed, the stabilization mode can be selected when the state x belongs to the intersection of the sets F i (and for any obstacleindex i ∈ I ), and a jump to the avoidance mode can occurwhen the state x belongs to the union of the sets J i (andfor any obstacle index i ∈ I ). In other words, if during thestabilization mode the vehicle reaches any one of the safetyhelmets, then the controller jumps to one of the avoidancemodes with m equal to − or . F i J i F i J i − J i ¯ c i p i − p i F i − O j O i Fig. 3. 2D illustration of flow and jump sets considered in Sections IV-V corresponding to obstacle O i (in the presence of a second obstacle O j ).The stabilization-mode jump set J i (hatched red) is constructed by using thehelmet H i ( (cid:15) i , ν i ) , while the corresponding flow set F i is the complement of J i in the free workspace. For the avoidance mode we select p i and p i − tolie on the cone C ( c i , − c i , θ i , E i ) (solid brown line). The avoidance flow set F im , with m ∈ {− , } , corresponds to the helmet H i ( δ i , µ i ) deprived ofthe interior of the the cone region defined by C ( c i , c i − p im , ψ i , E i ) (solidpurple line for m = − and solid orange line for m = 1 ). The correspondingjump set J im is the complement of F im in the free workspace.
3) Avoidance Mode m ∈ {− , } : We consider now theconstruction of flow and jump sets for the avoidance modes m ∈ {− , } and the specific obstacle i ∈ I with the aid ofFig. 3. To highlight their motivation, we first define such flowsets and state later the corresponding jump sets (see (28)). Foreach i ∈ I and m ∈ {− , } , the avoidance flow set is F im := H i ( δ i , µ i ) ∩ C ≥ ( c i , c i − p im , ψ i , E i ) ∩ W , (26)where δ i ∈ (0 , (cid:15) i ) (dilating E ≤ ( c i , (cid:15) i E i ) to E ≤ ( c i , δ i E i ) ), µ i ∈ ( ν i , ∞ ) (shrinking E ≥ (¯ c i , ν i ¯ E i ) to E ≥ (¯ c i , µ i ¯ E i ) ), and ψ i ∈ (0 , π/ . In the two configurations m ∈ {− , } of theavoidance of obstacle i ∈ I , we want the vehicle to slide on thesafety helmet H i ( δ i , µ i ) while maintaining a constant distanceto the obstacle. By selecting δ i ∈ (0 , (cid:15) i ) and µ i ∈ ( ν i , ∞ ) , oneobtains a dilated version of H i ( (cid:15) i , ν i ) used in J i and, thus,creates a hysteresis region useful to prevent infinitely manyconsecutive jumps (Zeno behavior). However, the avoidancevector field κ ( x, i, m ) in (16) has some undesirable equilibria, TABLE IS
ELECTION OF THE DESIGN PARAMETERS OF (30),
WITH i ∈ I .Parameter Selection Parameter Selection δ i ( δ i , (cid:15) i ( δ i , µ i (1 , ¯ µ i ( δ i )) ν i (1 , µ i ) θ i (0 , ¯ θ i ( δ i , µ i )) ¯ ψ i (0 , θ i ) k , k , k − (0 , + ∞ ) ψ i (0 , ¯ ψ i ) p i , p i − as in (18) which we need to rule out from the flow sets F i and F i − .These are characterized in the next lemma. Lemma
3: Let c ∈ R n , p ∈ R n \{ c } and E ∈ R n × n positivedefinite. For each x ∈ R n \{ c } , π ⊥ ( E ( x − c )) E ( x − p ) = 0 ifand only if x ∈ L ( c, p − c ) . For each m ∈ {− , } , i ∈ I , we want solutions to eventuallyleave the set F im of the avoidance mode, so it is necessary toselect point p im and flow set F im such that L ( c i , p im − c i ) ∩F im = ∅ based on Lemma 3, otherwise solutions could stayin avoidance mode indefinitely. This motivates the intersectionwith the cone in (26), and the next lemma. Lemma
4: For each i ∈ I , define the quantities δ i := (cid:107) E i c i (cid:107) − (27a) ¯ µ i ( δ i ) := (cid:0) − δ i (1 − δ i /δ i ) (cid:1) − (27b) ¯ θ i ( δ i , µ i ) := arccos (cid:18) δ i δ i + 14 δ i (cid:18) − µ i (cid:19)(cid:19) (27c) and select the parameters δ i , µ i , θ i , ψ i as in Table I so that ¯ µ i ( δ i ) and ¯ θ i ( δ i , µ i ) are well-defined. Then, for each m ∈{− , } , L ( c i , p im − c i ) ∩ F im = ∅ . From the flow set in (26), we suitably define the jump setfor the avoidance mode, of an obstacle i ∈ I with configuration m ∈ {− , } , to be the closed complement of F im in the freeworkspace. For i ∈ I and m ∈ {− , } , J im := (cid:16) E ≥ ( c i , δ i E i ) ∪ E ≤ (¯ c i , µ i ¯ E i ) (28) ∪ C ≤ ( c i , c i − p im , ψ i , E i ) (cid:17) ∩ W . Finally, the avoidance mode has overall flow and jump sets F := (cid:91) i ∈ I (cid:0) F i ×{ i } (cid:1) , J := (cid:91) i ∈ I (cid:0) J i ×{ i } (cid:1) , (29a) F − := (cid:91) i ∈ I (cid:0) F i − ×{ i } (cid:1) , J − := (cid:91) i ∈ I (cid:0) J i − ×{ i } (cid:1) , (29b)where F im and J im ( m ∈ {− , } ) are defined in (26) and (28).Indeed, each obstacle i gives rise, for the avoidance mode, toa specific flow (jump) set with two configurations F i and F i − ( J i and J i − ), as we motivated in this section. C. Hybrid Mode Selection
In this section we define the hybrid switching strategy thatpermits a Zeno-free transition between the different controlmodes. The hybrid selection of the logical variables i ∈ I and m ∈ M is implemented in the hybrid system ˙ x = κ ( x, i, m )˙ (cid:122)(cid:123) i = 0˙ m = 0 ( x, i, m ) ∈ F (30a) (cid:40) x + = x (cid:104) i + m + (cid:105) ∈ L ( x, i, m ) ( x, i, m ) ∈ J (30b)where κ ( x, i, m ) is the control input as defined in (16) andthe flow and jump sets are given by F := (cid:91) m ∈ M ( F m ×{ m } ) , J := (cid:91) m ∈ M ( J m ×{ m } ) . (30c)with F m and J m being defined in (25) for m = 0 and in(29a)-(29b) for m ∈ {− , } . We define now the (set-valued)jump map L in (30b). To this end, for i ∈ I and m ∈ {− , } ,define the sets C im as C im := C ≥ ( c i , c i − p im , ¯ ψ i , E i ) (30d)which corresponds to the region outside the cone with vertexat c i , axis c i − p im and aperture ψ i , where ¯ ψ i is a designparameter selected below. The jump map L for m ∈ {− , } is then defined as L ( x, i, −
1) := L ( x, i,
1) := { [ i ] } , (30e)i.e., when jumping to stabilization mode, the obstacle index i is not used in the control law κ in (16) and consequently isnot updated. The jump map L for m = 0 is L ( x, i,
0) := (cid:110)(cid:2) i (cid:48) m (cid:48) (cid:3) : x ∈ J i (cid:48) , m (cid:48) ∈ M ( x, i (cid:48) ) (cid:111) (30f)where M is defined, based on (30d), as M ( x, i ) := {− } x ∈ C i − \C i { } x ∈ C i \C i − {− , } x ∈ C i − ∩ C i . (30g) L ( · , · , captures that when jumping from the stabilizationmode m = 0 , the suitable avoidance mode of obstacle i (cid:48) ∈ I with configuration m (cid:48) ∈ {− , } is selected based on theposition x of the vehicle ( m (cid:48) , in particular, is selected based onwhether x is within the cone region C i (cid:48) − or C i (cid:48) ). A necessarycondition to implement our hybrid controller is that the jumpmap is nonempty, for which we have the next lemma. Lemma
5: Select the parameters ¯ ψ i and ψ i as in Table I.Then, the set L ( x, i, m ) is nonempty for all ( x, i, m ) ∈ J . For compact notation, we write flow and jump maps as ( x, i, m ) (cid:55)→ F ( x, i, m ) := ( κ ( x, i, m ) , , (30h) ( x, i, m ) (cid:55)→ J ( x, i, m ) := ( x, L ( x, i, m )) , (30i)and the overall state of the hybrid system as ξ := ( x, i, m ) ∈ R n × I × M . (30j)This completes the description of the hybrid controller in (30).The selections we made in this section for the parametersof (30) are summarized in Table I.V. M AIN R ESULT
In this section, we show that the hybrid controller achievesforward invariance (Section V-A) and global asymptotic sta-bility (Section V-B) (related to the objectives in Section III),as well as some complementary properties (Section V-C).
The mild regularity conditions satisfied by the hybrid sys-tem (30), as in the next lemma, allows us to invoke usefulresults on hybrid systems in the proof of our results.
Lemma
6: The hybrid system with data ( F , F , J , J ) satisfies the hybrid basic conditions in [24, Assumption 6.5].A. Forward Invariance In this section, we show that all generated solutions arecomplete and safe. Since the state x must evolve always withinthe free workspace W in (14) regardless of the logic variables i and m , we seek forward invariance of the set K defined as: K := (cid:92) i (cid:48) ∈ I E ≥ ( c i (cid:48) , E i (cid:48) ) × I × M = W × I × M . (31)The next lemma shows that the union of flow and jumpsets covers exactly the obstacle-free state space K and thatsolutions cannot leave K through jumps. Lemma F ∪ J = K and J ( J ) ⊂ K . Forward invariance of K is proven in the next theorem. Theorem
1: Under Assumptions 1-3, consider the hybridsystem (30) with parameters selected as in Table I. Assumealso that the controller parameters δ i are tuned so thatthe ellipsoids {E ≤ ( c i , δ i E i ) } i ∈ I are weakly pairwise disjoint.Then, the obstacle-free set K in (31) is forward invariant. The existence of tuning parameters δ , . . . , δ I satisfying theweak pairwise disjointness of the sets {E ≤ ( c i , δ i E i ) } i ∈ I isguaranteed by Assumption 3, which implies that weak pairwisedisjointness holds when δ i = 1 for all i ∈ I . Hence, by acontinuity argument, we can always tune each δ i sufficientlyclose to in order to guarantee the weak pairwise disjointnessof the dilated obstacles {E ≤ ( c i , δ i E i ) } i ∈ I . Note that algebraictests of weak pairwise disjointness (provided in [25, Thm. 6]for n = 2 and in [26, Thm. 8] for n = 3 ) can be used for thistuning purpose. B. Global Asymptotic Stability
In this section we show that from all initial conditions in thefree workspace, all solutions converge asymptotically to theorigin. To this end, we define the notion of sufficient disjoint-ness of a set of ellipsoids, which is slightly stronger than weakdisjointness but less conservative than strong disjointness, andguarantees that each obstacle is avoided at most one time. Themotivation behind the assumption of sufficient disjointness isthat the ellipsoids considered here can be arbitrarily large andflat, which might lead to long detours during the avoidancemode that take the vehicle far away from the origin. In thiscase, specific configurations of the obstacles exist such thatfrom a set of initial conditions, the vehicle does not convergeto the origin although it remains safe. Similarly, in the Bug planning algorithm [28], termination (i.e., convergence to thetarget) is not always guaranteed since the algorithm is designedto “walk toward the target whenever you can” [28]. Ourhybrid feedback shares a similar philosophy since the vehiclejumps from avoidance to stabilization mode whenever thestabilization controller generates a vector field not pointingtowards the obstacle (see (19)). To proceed, the next lemmacharacterizes the intersection of two ellipsoids of interest. O i (cid:48) O i R ∗ i (cid:48) R ∗ i weakly disjoint sufficiently disjoint strongly disjoint Fig. 4. Different types of disjointness introduced in the paper with set R ∗ i (orange, see (36)). For global attractivity, sufficient disjointness is asked. Lemma
8: Consider an arbitrary i ∈ I . For δ i , δ (cid:55)→ ¯ µ i ( δ ) and ( δ, µ ) (cid:55)→ ¯ θ i ( δ, µ ) defined in (27) , let δ ∈ [ δ i , , µ ∈ [1 , ¯ µ i ( δ )] and ϑ i ( δ, µ ) be such that cos( ϑ i ( δ, µ )) := 1 − cos(¯ θ i ( δ, µ )) δ i (cid:113) (1 + µ − ) / − δ − δ i . (32) The expression in (32) is well-defined and positive, and E ( c i , δE i ) ∩ E (¯ c i , µ ¯ E i ) ⊂ C (0 , c i , ϑ i ( δ, µ ) , E i ) . (33)Let us consider for each obstacle i ∈ I the sphere S (0 , ¯ r i ) withcenter at the origin and radius ¯ r i defined by the next quadraticoptimization problem ¯ r i := min (cid:107) x (cid:107) subject to x ∈ H ∗ i (34)where H ∗ i is the helmet defined in (21). The radius ¯ r i definesthe minimum distance from the helmet H ∗ i to the origin. Let x be a point belonging to the intersection of the two ellipsoids E ( c i , E i ) and E (¯ c i , ¯ E i ) . Taking δ and µ equal to in Lemma 8,one obtains x ∈ C (0 , c i , ¯ ϑ i , E i ) with cos( ¯ ϑ i ) := cos( ϑ i (1 , (cid:112) − (cid:107) E i c i (cid:107) − , (35)from (32), (27c) and (27a). Now, let us define the set R ∗ i := C (0 , c i , ¯ ϑ i , E i ) ∩ S ≥ (0 , ¯ r i ) ∩ E ≥ ( c i , E i ) ∩ E ≤ (¯ c i , ¯ E i ) , (36)whose geometry is sketched in Fig. 4. Intuitively speaking, itis a subset of all points on the cone C (0 , c i , ¯ ϑ i , E i ) that have adistance to the origin greater than the distance ¯ r i of the helmet H ∗ i to the origin. The idea is that the vehicle should not tostart avoiding another obstacle while it is still in R ∗ i , otherwisethere is no guarantee that the number of times the vehicleavoids the obstacles is bounded and that global attractivityholds. This motivates the next definition. Definition
3: The ellipsoids {E ( c i , E i ) } i ∈ I are sufficientlypairwise disjoint if they are weakly pairwise disjoint and ∀ i, i (cid:48) ∈ I with i (cid:54) = i (cid:48) , R ∗ i ∩ E ≤ ( c i (cid:48) , E i (cid:48) ) = ∅ . (37)Now, let us introduce the ingredients for a dilated versionof R ∗ i as in (39) below and refer to Fig. 5. First, considerthe escape annulus cone where solutions escape from theavoidance mode by applying the stabilization vector field.This region lies between the two cones C (0 , c i , ϑ i (1 , µ i ) , E i ) and C (0 , c i , ϑ i ( δ i , µ i ) , E i ) which are related, according toLemma 8, to the intersections E ( c i , E i ) ∩ E (¯ c i , µ i ¯ E i ) and E ( c i , δ i E i ) ∩ E (¯ c i , µ i ¯ E i ) , respectively. Second, consider foreach obstacle i ∈ I the ball S ≥ (0 , r i ) where the radius r i isdefined by the quadratic optimization problem r i := min (cid:107) x (cid:107) subject to x ∈ H i ( δ i , µ i ) . (38) R i ( δ i , µ i ) S (0 , r i ) E (¯ c i , ¯ E i ) H i ( δ i , µ i ) C (0 , c i , ϑ i (1 , µ i ) , E i ) C (0 , c i , ϑ i ( δ i , µ i ) , E i ) Fig. 5. Safety helmet H i ( δ i , µ i ) (green) and the corresponding escape region R i ( δ i , µ i ) (orange). The region R i ( δ i , µ i ) must not intersect with any otherjump set J i (cid:48) , i (cid:48) (cid:54) = i , to avoid starting another avoidance while the distanceto target has not yet decreased. Note the following on (38). 1) The safety helmet H i ( δ i , µ i ) iscompact and, hence, the solution to (38) exists. 2) For each i ∈ I , r i > . Indeed, for each i ∈ I , (cid:107) δ i E i c i (cid:107) = δ i δ − i > δ i δ − i > by Assumption 2 and the selection of δ i in Table I, so that / ∈ E ≤ ( c i , δ i E i ) and in turn / ∈ H i ( δ i , µ i ) ( H i ( δ i , µ i ) ⊂E ≤ ( c i , δ i E i ) ). Hence, since H i ( δ i , µ i ) is compact there exists r i > such that (cid:107) x (cid:107) ≥ r i for all x ∈ H i ( δ i , µ i ) . Finally, wecan define the considered dilated version of R ∗ i as R i ( δ i , µ i ) := S ≥ (0 , r i ) ∩ E ≥ ( c i , δ i E i ) ∩ E ≤ (¯ c i , ¯ E i ) ∩ C ≥ (0 , c i , ϑ i (1 , µ i ) , E i ) ∩ C ≤ (0 , c i , ϑ i ( δ i , µ i ) , E i ) . (39) Lemma
9: Assume that the obstacles {O i } i ∈ I are suffi-ciently pairwise disjoint. Then, for each i ∈ I , there exist δ ∗ i , µ ∗ i such that for all δ i ∈ ( δ ∗ i , and µ i ∈ (1 , µ ∗ i ) , we have ∀ i (cid:48) , i (cid:48)(cid:48) ∈ I , i (cid:48) (cid:54) = i (cid:48)(cid:48) , R i (cid:48) ( δ i (cid:48) , µ i (cid:48) ) ∩E ≤ ( c i (cid:48)(cid:48) , δ i (cid:48)(cid:48) E i (cid:48)(cid:48) ) = ∅ . (40)Property (40) of Lemma 9 will be used to show global attrac-tivity. Intuitively, we require that after avoiding an obstacle, thedistance (cid:107) x (cid:107) to the target decreases before the vehicle reachesthe proximity of another obstacle. Although the bounds δ ∗ i and µ ∗ i are not defined explicitly for generic ellipsoids, theparameters δ i and µ i can be tuned offline. Now, we are readyto state our main result for this section. Theorem
2: Consider the hybrid system (30) under the sameassumptions as Theorem 1. Assume also that the obstacles {O i } i ∈ I are sufficiently pairwise disjoint, and δ i and µ i aretuned such that (40) holds. Then, the set A := { } × I × M is globally asymptotically stable for (30) and the number ofjumps is bounded. For spherical obstacles, we show next that the extra tuningof the parameters to satisfy (40) is not needed.
Theorem
3: (Spherical obstacles) Let E i = λ i I n for all i ∈ I . Under the same assumptions as Theorem 1, the set A := { } × I × M is globally asymptotically stable for (30) and the number of jumps is bounded.C. Complementary Properties In this section we present four relevant complementaryproperties of the proposed hybrid law for obstacle avoidance.
1) Bounded Control:
First, we can show that x remains al-ways in a given ball. Indeed, let S ≤ (0 , r b ) , with r b > , be thesmallest ball containing all the dilated ellipsoids E ( c i , δ i E i ) (which must exist since these ellipsoids are compact). Dur-ing stabilization mode the distance (cid:107) x (cid:107) is decreasing andduring avoidance mode the vehicle stays within the dilatedellipsoids E ( c i , δ i E i ) . Then, it is guaranteed that from all x (0 , ∈ S ≤ (0 , r b ) , x ( t, j ) ∈ S ≤ (0 , r b ) for all ( t, j ) ∈ dom x .Moreover, since the projection matrix π ⊥ ( E i ( x − c i )) haseigenvalues in and , it follows that we can upper boundthe control input in (16) by (cid:107) u (cid:107) ≤ kα ( r b + p ) where k = max { k , k , k − } , α = max i ∈ I ( λ max ( E i ) /λ min ( E i )) and p = max i ∈ I (cid:107) p i (cid:107) . The control gains can then be tunedto satisfy the inherent practical saturation of the actuators.
2) Semiglobal Preservation:
The second property is theso-called semiglobal preservation property [18, §II]. Thisproperty is desirable when the original controller parametersare optimally tuned and the controller modifications imposedby the presence of the obstacles should be as minimal aspossible. Such a property is also accounted for in the quadraticprogramming formulation of [29, III.A.]. We summarize thisproperty for our case in the next proposition.
Proposition
1: Be (cid:15) ∈ (0 , and W (cid:15) := (cid:84) i (cid:48) ∈ I E ≥ ( c i (cid:48) , (cid:15)E i (cid:48) ) .There exist controller parameters such that the control lawmatches, in W (cid:15) , the stabilization feedback u = − k x ( k > )used in the absence of obstacles.3) Non-point Mass Vehicles: There is no loss of generalityin considering a point-mass vehicle in this work. Let us ratherconsider that the vehicle has some volume, e.g., bounded by S ≤ ( x, r v ) . Then, for the navigation scenario to be feasible, theradius r v of the vehicle needs to be smaller than the smallestdistance between the obstacles, i.e., for all i, i (cid:48) ∈ I with i (cid:54) = i (cid:48) , r v < dist ( E ≤ ( c i , E i ) , E ≤ ( c i (cid:48) , E i (cid:48) )) . For the safety of thevehicle during the stabilization mode, selecting the parameter (cid:15) i as (cid:15) i < (1 + λ max ( E i ) r v ) − is sufficient (in addition toTable I) to guarantee that the vehicle starts the avoidance modeaway from the obstacle. Indeed, under this condition, it iseasy to show that for all x ∈ E ≥ ( c i , (cid:15) i E i ) ( i.e., the vehiclecenter is outside the dilated ellipsoid E ( c i , (cid:15) i E i ) ) and for all x (cid:48) ∈ S ≤ ( x, r v ) , one has x (cid:48) ∈ E ≥ ( c i , E i ) , which guaranteessafety of the whole volume of the vehicle.
4) Robustness:
The constructed hybrid controller guaran-tees some level of robustness to perturbations (e.g., in theform of measurement noise). Hysteresis switching is one ofthe typical ways to ensure robustness to measurement noise,and hysteresis switching is indeed behind the designed hybridfeedback, in particular the hysteresis regions of flow and jumpsets in Section IV-B and the logical selections of the jump setsin Section IV-C. More generally, fundamental results in [24,Chap. 7] guarantee structurally that global asymptotic stabilityof A in Theorem 2 is also uniform (by [24, Thm. 7.12]) androbust (by [24, Thm. 7.21]) with respect to perturbations since A is a compact set and the hybrid basic conditions are satisfiedas in Lemma 6. VI. S IMULATIONS
We illustrate the effectiveness of the proposed hybrid controlstrategy through two simulation scenarios. The first scenario
Fig. 6. Plot (at time t = 0 . and t = 30 seconds) of the -dimensionaltrajectory of the vehicle starting at different initial conditions. considers obstacles in D (see Fig. 6) while the second oneconsiders obstacles in D (see Fig. 7). For both cases, Table Iprovides a suitable order to choose the parameters for each i ∈ I , as follows.1) For δ i in (27a), select δ i and (cid:15) i so that δ i < δ i < (cid:15) i < ;2) For δ i and ¯ µ i ( δ i ) in (27b), select ν i and µ i so that <ν i < µ i < ¯ µ i ( δ i ) (possibly iterating steps 1) and 2) so that δ i and µ i satisfy (40));3) For δ i , µ i and ¯ θ i ( δ i , µ i ) in (27c), select ψ i , ¯ ψ i and θ i sothat < ψ i < ¯ ψ i < θ i < ¯ θ i ( δ i , µ i ) .Any parameter selection according to this guideline guaranteesour results, and can be carried out keeping in mind the physicalinterpretation illustrated in Section IV-B for these parameters.The gains are k = k = k − = 1 / and determine the speedof convergence of the scheme. By (18a), the point p i can beselected arbitrarily as long as it is on C ( c i , − c i , θ i , E i ) \{ c i } .A suitable choice is given by p i = π ⊥ ( E − i R ( θ i ) E i c i ) c i (41)where R ( θ i ) is the standard × rotation matrix with angle θ i or the standard × axis-angle rotation matrix with angle θ i and an arbitrary vector of S as axis. The idea behind (41)is to project c i on the plane orthogonal to a rotated version of c i , in order to obtain the point lying on the cone and closest tothe origin. Having all points p im close enough to the origin isan effective way so that k , k , k − can take the same valuesand yield comparable speeds for avoidance and stabilization,independently of the obstacles.Fig. 6 (Fig. 7, respectively) shows that the solution gen-erated by the closed loop hybrid system avoids the Dobstacles ( D obstacles, respectively) and Fig. 8 shows theconvergence of solutions to the origin. Complete simulationvideos for the D and D cases can be found at https://youtu.be/CnXJlhzlzd8, https://youtu.be/4mzTXPR6D9Y.Finally, we note that for the very obstacle configuration ofthe 2D scenario, the state-of-the-art approach of navigationfunctions [3], [21] cannot be applied since the condition [21,Thm. 3, Eq. (23)] is violated for all obstacles except obstacle O , where [21, Eq. (23)] intuitively corresponds to the factthat obstacles are not too flat and not too close to the targetposition. ([21, Eq. (23)] is violated for all obstacles of the 3Dscenario.) Moreover, navigation function approaches requiretuning a parameter sufficiently large ( k in [21, Eq. (17) andRemark 5]), which may conflict with actuator limitations.Instead, our approach provides a clear tuning guideline for Fig. 7. Plot (at time t = 30 seconds) of the -dimensional trajectory of thevehicle starting at different initial conditions.Fig. 8. Plot of the position norm (cid:107) x (cid:107) versus time showing the convergenceof the solutions to the origin from the considered initial conditions. all parameters (given in this section) and actuator limitationscan be taken into account (see Section V-C1).VII. C ONCLUSIONS
We proposed a novel hybrid feedback on R n to solvethe obstacle avoidance problem for arbitrarily flat ellipsoidalobstacles. Our control strategy ensures global asymptotic sta-bilization to the target and safety (thus, successful navigationfrom all initial conditions) while guaranteeing a Zeno-freeswitching between the avoidance and stabilization modes.Moreover, the control input remains bounded (in particular,arbitrarily close to any obstacle) and matches semi-globally inthe free-state space the nominal feedback used in the absenceof obstacles.Future work will be devoted to considering more com-plex vehicle dynamics ( e.g. , under-actuated and second-orderdynamics) and more generic obstacle shapes ( e.g. , convexobstacles). Furthermore, although our scheme considers staticobstacles to obtain formal guarantees for global asymptoticstability and safety, extending this approach to deal withunknown environments is an interesting research direction thatwe aim at pursuing in the future.A PPENDIX
In the appendix, an equation number over an (in)equality in-dicates which equation has been used to obtain the (in)equality.
1) Proof of Theorem 1:
Define S H ( K ) as the set ofall maximal solutions φ to H with φ (0 , ∈ K . Each φ ∈ S H ( K ) has range rge φ ⊂ K = F ∪ J by Lemma 7and the definition of hybrid solution [24, p. 124], so K isforward pre-invariant [30, Def. 3.3]. The set K is in factforward invariant [30, Def. 3.3] if for each ξ ∈ K there exists one solution and each φ ∈ S H ( K ) is complete, which weshow in the rest of the proof through [24, Prop. 6.10]. In therest of the proof, let F ∗ := (cid:92) i ∈ I F i , J ∗ := (cid:91) i ∈ I J i . (42) Lemma
10: Under the assumptions of Theorem 1, we havefor each i ∈ I and m ∈ {− , }J i = H i ( (cid:15) i , ν i ) , (43a) F im = H i ( δ i , µ i ) ∩ C ≥ ( c i , c i − p im , ψ i , E i ) , (43b) ∂ F ∗ \J ∗ ⊂ (cid:91) i ∈ I (cid:0) E ( c i , E i ) \E ≥ (¯ c i , ¯ E i ) (cid:1) , (43c) ∂ F im \J im ⊂E ( c i , E i ) \ ( E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p im , ψ i , E i )) . (43d)First, let us show that the viability condition F ( x, i, m ) ∩ T F ( x, i, m ) (cid:54) = ∅ (44)holds for all ( x, i, m ) ∈ F\J . Let ( x, i, m ) ∈ F\J , whichimplies by (30c) that ( x, i ) ∈ F m \J m for some m ∈ M ,and divide into the cases m = 0 and m ∈ {− , } . When m = 0 , from (25) there exists i ∈ I such that x ∈ F ∗ \J ∗ .If x ∈ ( F ∗ ) ◦ \J ∗ (hence, x is in the interior of F ∗ ), then T F ∗ ( x ) = R n , so that T F ( ξ ) = R n × { } × { } and (44)holds. If x ∈ ∂ F ∗ \J ∗ , which satisfies the set inclusion (43c),the weak pairwise disjointness of {E ( c i , E i ) } i ∈ I yields: x ∈ E ( c i , E i ) , i ∈ I T F ( x, i,
0) = P ≥ (0 , E i ( x − c i )) × { } × { } . (45)By (19) and x / ∈ E ≥ (¯ c i , ¯ E i ) by (43c), we obtain − k x (cid:62) E i ( x − c i ) = k (cid:107) E i ¯ c i (cid:107) (cid:0) −(cid:107) ¯ E i ( x − ¯ c i ) (cid:107) (cid:1) > , (46)hence, κ ( x, i, ∈ P ≥ (0 , E i ( x − c i )) in (45), and (44) holdsfor m = 0 . When m ∈ {− , } , we have i ∈ I and x ∈ ∂ F im \J im , which satisfies the set inclusion (43d), and so T F ( x, i, m ) = P ≥ (0 , E i ( x − c i )) × { } × { } . (47) κ ( x, i, m ) ∈ P ≥ (0 , E i ( x − c i )) in (47) because − k m ( x − p im ) (cid:62) E i π ⊥ ( E i ( x − c i )) E i ( x − c i ) = 0 , (48)so the viability condition (44) holds for m ∈ {− , } as well.Second, we apply [24, Prop. 6.10]. By it and (44), thereexists a nontrivial solution to H from each initial conditionin K . Finite escape times can only occur through flow. Theycan neither occur for x in the set F i − ∪ F i ( F i − and F i are bounded by their definitions in (26)) nor for x in theset F ∗ because they would make x (cid:62) x grow unbounded, andthis would contradict that ddt ( x (cid:62) x ) ≤ by the definition of κ ( x, i, and by (30a). So, all maximal solutions do not havefinite escape times. By Lemma 7, J ( J ) ⊂ K = F ∪J . Hence,by [24, Prop. 6.10], all maximal solutions are complete.
2) Proof of Theorem 2:
We prove global asymptotic sta-bility of A by [24, Def. 7.1]. For each i ∈ I , (cid:107) δ i E i c i (cid:107) = δ i δ − i > δ i δ − i > by Assumption 2 and the selection of δ i in Table I, so / ∈ E ≤ ( c i , δ i E i ) . As a consequence, there exists ε ∗ > such that the ball S ≤ (0 , ε ∗ ) does not intersect with anyof the dilated obstacles E ≤ ( c i , δ i E i ) . It can be shown easilythat for each ε ∈ [0 , ε ∗ ] , the set S := S ≤ (0 , ε ) × I × M isforward invariant because S ≤ (0 , ε ) is disjoint from J ∗ andthe component x of solutions evolves, after at most one jump,with the stabilization mode ˙ x = − k x . Thanks to forwardinvariance of S , stability of A for (30) is immediate from [24,Def. 7.1]. Let us prove global attractivity of A .Before that, weneed the next intermediate result. Lemma
11: There exists σ > such that for all solutions ξ = ( x, i, m ) with ξ ( t, j ) ∈ F l ×{ l } for some l ∈ {− , } and ( t, j ) ∈ dom ξ , there exists ( s, (cid:96) ) ∈ dom ξ such that ( s, (cid:96) ) (cid:23) ( t, j ) and (cid:107) x ( s, (cid:96) ) (cid:107) ≤ (cid:107) x ( t, j ) (cid:107) − σ. (49)Now, for each solution ξ to (30), there exists a finite time ( T, J ) (cid:23) (0 , after which the solution does not evolve withthe avoidance controller any longer, i.e. , m ( t, j ) = 0 for all ( t, j ) (cid:23) ( T, J ) . Otherwise, there would exist a sequence ofhybrid times { ( t k , j k ) } ∞ k =0 such that ξ ( t k , j k ) ∈ F l k × { l k } with l k ∈ {− , } and this would imply by Lemma 11 that (cid:107) x ( t k +1 , j k +1 ) (cid:107) ≤ (cid:107) x ( t k , j k ) (cid:107) − σ for all k ∈ N . This isindeed a contradiction as it would lead to (cid:107) x ( · , · ) (cid:107) becomingnegative. Then, the solution ξ enters the stabilizing mode m =0 after ( T, J ) and its flow map ˙ x = − k x guarantees in turnglobal attractivity. Moreover, J is the maximum number ofjumps of the hybrid system since any extra jump will cause m to take values in {− , } , which is not possible after ( T, J ) .
3) Proof of Theorem 3:
To prove the theorem, it is sufficientto show that for spherical obstacles the result of Lemma11 holds. The proof of Lemma 11 under the assumptionsof Theorem 3 is the same up to (85). From (85) we have x ( t (cid:48) , j + 1) = x ( t (cid:48) , j ) ∈ ˜ P i ( t (cid:48) ,j ) m ( t (cid:48) ,j ) , ( δ, ψ ) , i ( t (cid:48) , j + 1) = i ( t (cid:48) , j ) =: ι and m ( t (cid:48) , j ) = 0 . However, since x ( t, j ) and x ( t (cid:48) , j + 1) both belong to E ( c ι , δE ι ) , we can write (cid:107) x ( t, j ) − c ι (cid:107) = (cid:107) x ( t (cid:48) , j + 1) − c ι (cid:107) (since E ι = λ ι I n )and hence (cid:107) x ( t, j ) (cid:107) − c (cid:62) ι x ( t, j ) = (cid:107) x ( t (cid:48) , j + 1) (cid:107) − c (cid:62) ι x ( t (cid:48) , j + 1) . (50) x ( t (cid:48) , j + 1) ∈ ˜ P ι , ( δ, ψ ) ⊂ E (¯ c ι , µ ι ¯ E ι ) implies also that c (cid:62) ι x ( t (cid:48) , j + 1) = (cid:107) x ( t (cid:48) , j + 1) (cid:107) + (1 − µ − ι ) (cid:107) ¯ c ι (cid:107) , thus, with (50), we have c (cid:62) ι x ( t, j ) = (cid:107) x ( t, j ) (cid:107) + (cid:107) x ( t (cid:48) , j + 1) (cid:107) − µ − ι ) (cid:107) ¯ c ι (cid:107) . However, since x ( t, j ) ∈ E ≥ (¯ c ι , µ ι ¯ E ι ) , we have c (cid:62) ι x ( t, j ) ≤ (cid:107) x ( t, j ) (cid:107) + (1 − µ − ι ) (cid:107) ¯ c ι (cid:107) , and, hence, (cid:107) x ( t (cid:48) , j + 1) (cid:107) ≤ (cid:107) x ( t, j ) (cid:107) must hold. Also,by (85), x ( t (cid:48) , j + 1) ∈ E ( c ι , δE ι ) ∩ E (¯ c ι , µ ι ¯ E ι ) , and, byLemma 8, x ( t (cid:48) , j + 1) ∈ C (0 , c ι , ϑ ι ( δ, µ ι ) , E ι ) . In view ofStep 2 of the proof of Lemma 11, both the sets E ≤ (¯ c ι , ¯ E ι ) and C (0 , c ι , ϑ ι ( δ, µ ι ) , E ι ) are forward invariant under thestabilization flow map for x , i.e., − k x . Since the obstaclesare weakly disjoint, the solution then flows in E ≤ (¯ c ι , ¯ E ι ) ∩C (0 , c ι , ϑ ι ( δ, µ ι ) , E ι ) until it reaches the set E ( c ι , δ ι E ι ) at ( t (cid:48)(cid:48) , j + 1) . Since flow with stabilization mode decreases thedistance to the origin we have (cid:107) x ( t (cid:48)(cid:48) , j + 1) (cid:107) ≤ (cid:107) x ( t (cid:48) , j + 1) (cid:107) ≤ (cid:107) x ( t, j ) (cid:107) Also, the solution must flow after ( t (cid:48)(cid:48) , j + 1) up to some ( s, j + 1) with the stabilization mode (since obstacles areweakly disjoint) such that at least σ in (86) is traversed, i.e., (cid:107) x ( s, j + 1) (cid:107) + σ ≤ (cid:107) x ( t (cid:48)(cid:48) , j + 1) (cid:107) ≤ (cid:107) x ( t, j ) (cid:107) . This proves Lemma 11 and in turn Theorem 2.
4) Proof of Proposition 1:
Note preliminarily that thanks to (cid:15) < , W (cid:15) ⊂ W in (14). It is sufficient to show that the closedloop system under the proposed hybrid feedback cannot flowexcept with stabilization mode m = 0 when x ∈ W (cid:15) . Indeed, ifin Table I we further constrain δ i as δ i ∈ (max( δ i , (cid:15) ) , for all i ∈ I , then F im ⊂ H ( δ i , µ i ) ⊂ E ≤ ( c i , δ i E i ) ⊂ E ≤ ( c i , (cid:15)E i ) and E ≤ ( c i , δ i E i ) (cid:54) = E ≤ ( c i , (cid:15)E i ) . Therefore, we have F im ∩ W (cid:15) = ∅ for all i ∈ I and m ∈ {− , } . This implies that solutionscannot flow with the avoidance mode when x belongs to W (cid:15) and must then flow with the stabilization mode.
5) Proof of Lemma 1:
Let x l ∈ C ≤ ( c, E − v l , ψ l , E ) \{ c } , l = 1 , , and be otherwise arbitrary. Define then z l := E ( x l − c ) / (cid:107) E ( x l − c ) (cid:107) ∈ S n − for l = 1 , . Hence, z l ∈ S l := C ≤ (0 , v l , ψ l , I n ) ∩ S n − , l = 1 , . For l = 1 , , z l satisfies, by (11), cos( ψ l ) ≤ z (cid:62) l v l , and consequently d S n − ( v l , z l ) ≤ ψ l . It follows from the triangle inequalitythat θ = d S n − ( v , v ) ≤ d S n − ( v , z ) + d S n − ( z , z ) + d S n − ( v , z ) ≤ d S n − ( z , z ) + ψ + ψ . Hence, in viewof the condition ψ + ψ < θ , d S n − ( z , z ) > . Thisfact implies that the compact sets S and S (and in turn C ≤ ( c, E − v l , ψ l , E ) \{ c } , l = 1 , ) are disjoint.
6) Proof of Lemma 2:
First, by (18b) and (3) we have p i − − c i = − E − i ρ ( E i c i ) E i ( p i − c i ) , hence (51) (cid:107) E i ( p i − − c i ) (cid:107) (51) = ( p i − c i ) (cid:62) E i ρ ( E i c i ) ρ ( E i c i ) E i ( p i − c i ) (4c) = ( p i − c i ) (cid:62) E i ( p i − c i ) = (cid:107) E i ( p i − c i ) (cid:107) (52)(so that p i (cid:54) = c i implies p i − (cid:54) = c i ). Based on (11) for C ( c i , − c i , θ i , E i ) , one has − c (cid:62) i E i ( p i − − c i ) (51) = c (cid:62) i E i ρ ( E i c i ) E i ( p i − c i ) (4c) = − c (cid:62) i E i ( p i − c i ) (18a) = cos( θ i ) (cid:107) E i ( − c i ) (cid:107)(cid:107) E i ( p i − c i ) (cid:107) (52) = cos( θ i ) (cid:107) E i ( − c i ) (cid:107)(cid:107) E i ( p i − − c i ) (cid:107) . This concludes by (11) that p i − ∈ C ( c i , − c i , θ i , E i ) \{ c i } .
7) Proof of Lemma 3:
As for the = ⇒ implication, let x ∈ R n \{ c } be such that π ⊥ ( E ( x − c )) E ( x − p ) = 0 , which isequivalent to π ⊥ ( E ( x − c )) E ( p − c ) = 0 . By substituting thedefinition of π ⊥ ( · ) in (3), one obtains (cid:107) E ( x − c ) (cid:107) ( p − c ) = (cid:0) ( p − c ) (cid:62) E ( x − c ) (cid:1) ( x − c ) . This very equation excludes that ( p − c ) (cid:62) E ( x − c ) = 0 since E is positive definite, x (cid:54) = c ,and p (cid:54) = c by assumption. So, letting λ = (cid:107) E ( x − c ) (cid:107) / (cid:0) ( p − c ) (cid:62) E ( x − c ) (cid:1) in (7), one deduces that x ∈ L ( c, p − c ) . The ⇐ = implication is straightforward.
8) Proof of Lemma 4:
The quantities in (27b)-(27c) arewell-defined. Indeed, we have for (27b) that − δ i (1 − δ i /δ i ) = (2 δ i − +4 δ i ( δ − i − > thanks to δ i ∈ ( δ i , .Moreover, by µ i ∈ (1 , ¯ µ i ( δ i )) , the argument of the arccos in (27c) belongs to (0 , , so ¯ θ i ( δ i , µ i ) is also well-defined.Now, define ˆ F im := E ≤ ( c i , δ i E i ) ∩ E ≥ (¯ c i , µ i ¯ E i ) ∩C ≥ ( c i , c i − p im , ψ i , E i ) ∩ E ≥ ( c i , E i ) ⊃ F im . (53)By proving that for each i ∈ I and m ∈ {− , }L ( c i , p im − c i ) ∩ ˆ F im = ∅ , (54)the claim of the lemma is also proven. We prove then (54)for an arbitrary i ∈ I and an arbitrary m ∈ {− , } . For thisproof, select the following angle ψ (cid:48) i as any angle ψ (cid:48) i ∈ (0 , ψ i ) .First, let us show that the following set inclusions hold L ≤ ( c i , p im − c i ) ⊂ C ≤ ( c i , c i − p im , ψ (cid:48) i , E i ) , (55a) L ≥ ( c i , p im − c i ) ⊂ C ( c i , − c i , θ i , E i ) . (55b)Let x ∈ L ≤ ( c i , p im − c i ) . Then there exists λ ≤ such that x − c i = λ ( p im − c i ) . Such x − c i verifies the condition cos( ψ (cid:48) i ) (cid:107) E i ( c i − p m ) (cid:107)(cid:107) E i ( x − c i ) (cid:107) ≤ ( c i − p im ) (cid:62) E ( x − c i ) corresponding to C ≤ ( c i , c i − p m , ψ (cid:48) i , E i ) by simple computa-tions for any < ψ (cid:48) i < ψ i (since cos( ψ (cid:48) i ) ≤ ). This proves(55a). Now, let x ∈ L ≥ ( c i , p im − c i ) . Then there exists λ ≥ such that x − c i = λ ( p im − c i ) . Such x − c i verifies thecondition cos( θ i ) (cid:107) E i ( − c i ) (cid:107)(cid:107) E i ( x − c i ) (cid:107) = − c (cid:62) i E i ( x − c i ) corresponding to C ( c i , − c i , θ i , E i ) by simple computationsusing that cos( θ i ) (cid:107) E i ( − c i ) (cid:107)(cid:107) E i ( p im − c i ) (cid:107) = − c (cid:62) i E i ( p im − c i ) (corresponding to p im ∈ C ( c i , − c i , θ i , E i ) from (18a) andLemma 2). This proves (55b). Second, from (55) one has ˆ F im ∩ L ( c i , p im − c i )= ( ˆ F im ∩ L ≤ ( c i , p im − c i )) ∪ ( ˆ F im ∩ L ≥ ( c i , p im − c i )) ⊂ ( ˆ F im ∩C ≤ ( c i , c i − p im , ψ (cid:48) i , E i )) ∪ ( ˆ F im ∩C ( c i , − c i , θ i , E i )) (56)and we prove that the two intersections in (56) are empty.Since < ψ (cid:48) i < ψ i , one obtains readily from the definition ofthe cone in (11) that C ≥ ( c i , c i − p im , ψ i , E i ) ∩ C ≤ ( c i , c i − p im , ψ (cid:48) i , E i ) = { c i } . This relationship and the definition of ˆ F im in (53) implythat the first intersection in (56) is empty. We show nowthat the second intersection in (56) is also empty. Let x ∈ ˆ F im ∩ C ( c i , − c i , θ i , E i ) . So, c (cid:62) i E i ( x − c i ) (20) = − ( x − c i ) (cid:62) E i ( x − c i )+ c (cid:62) i E i c i ( x − ¯ c i ) (cid:62) ¯ E i ( x − ¯ c i ) − c (cid:62) i E i c i = −(cid:107) E i ( x − c i ) (cid:107) + (cid:107) E i c i (cid:107) (cid:107) ¯ E i ( x − ¯ c i ) (cid:107) − (cid:107) E i c i (cid:107) ≥ − δ i − (cid:107) E i c i (cid:107) (cid:16) − µ i (cid:17) (57)where the bound holds since x ∈ ˆ F im implies x ∈ E ≤ ( c i , δ i E i ) and x ∈ E ≥ (¯ c i , µ i ¯ E i ) . We continue (57) as c (cid:62) i E i ( x − c i ) ≥ − δ i − (cid:107) E i c i (cid:107) (cid:16) − µ i (cid:17) , (27a) = − cos(¯ θ i ( δ i , µ i )) /δ i (27a) = − cos(¯ θ i ( δ i , µ i )) (cid:107) E i c i (cid:107)≥ − cos(¯ θ i ( δ i , µ i )) (cid:107) E i c i (cid:107)(cid:107) E i ( x − c i ) (cid:107) (58)since cos(¯ θ i ( δ i , µ i )) ≥ and (cid:107) E i ( x − c i ) (cid:107) ≥ ( x ∈ ˆ F im implies x ∈ E ≥ ( c i , E i ) ). It is also x ∈ C ( c i , − c i , θ i , E i ) . So, c (cid:62) i E i ( x − c i ) = − cos( θ i ) (cid:107) E i c i (cid:107)(cid:107) E i ( x − c i ) (cid:107) < − cos(¯ θ i ( δ i , µ i )) (cid:107) E i c i (cid:107)(cid:107) E i ( x − c i ) (cid:107) (59)from the bound on θ i in Table I. (58) and (59) contradict eachother, so the second intersection in (56) is also empty. Then,(54) is proven.
9) Proof of Lemma 5:
Given (30e)-(30f), we just need toshow that L ( x, i, (cid:54) = ∅ for all ( x, i ) ∈ J . This holds ifwe show, as we do in the rest of the proof, that for each x ∈ R n and i ∈ I , M ( x, i ) (cid:54) = ∅ . First, we show that ∩ m = − , C ≤ ( c i , c i − p im , ¯ ψ i , E i ) = { c i } for each i ∈ I . Tothis end, note that p i ∈ C ( c i , − c i , θ i , E i ) by (18a), and thisimplies cos ( θ i ) (cid:107) E i c i (cid:107) (cid:107) E i ( p i − c i ) (cid:107) = ( c (cid:62) i E i ( p i − c i )) or, equivalently, ( p i − c i ) (cid:62) E i π θ i ( E i c i ) E i ( p i − c i ) = 0 . (60)Introduce then v im := E i ( c i − p im ) / (cid:107) E i ( c i − p im ) (cid:107) for m ∈{− , } , and compute ( v i ) (cid:62) v i − = ( p i − c i ) (cid:62) E i ( p i − − c i ) (cid:107) E i ( p i − c i ) (cid:107)(cid:107) E i ( p i − − c i ) (cid:107) (52) = − ( p i − c i ) (cid:62) E i ρ ( E i c i ) E i ( p i − c i ) (cid:107) E i ( p i − c i ) (cid:107) (6) = − ( p i − c i ) (cid:62) E i (2 π θ i ( E i c i ) − cos(2 θ i ) I n ) E i ( p i − c i ) (cid:107) E i ( p i − c i ) (cid:107) (60) = cos(2 θ i )( p i − c i ) (cid:62) E i ( p i − c i ) (cid:107) E i ( p i − c i ) (cid:107) = cos(2 θ i ) Then, by Lemma 1 and ψ i < θ i , ∩ m = − , C ≤ ( c i , c i − p im , ¯ ψ i , E i ) = { c i } . Second, note that ∪ m = − , C > ( c i , c i − p im , ¯ ψ i , E i ) = (cid:0) ∩ m = − , C ≤ ( c i , c i − p im , ¯ ψ i , E i ) (cid:1) c = R n \{ c i } . (61)Therefore, we have ∪ m = − , C im (30d) = ∪ m = − , C ≥ ( c i , c i − p im , ¯ ψ i , E i ) = R n (62)since ∪ m = − , C im is a superset of the set in (61) and contains c i . So, for each x ∈ R n and i ∈ I , M ( x, i ) (cid:54) = ∅ in (30g).
10) Proof of Lemma 6: F and J are closed subsets of R n ×{− , , } . F is continuous on F . J ( x, i, m ) (cid:54) = ∅ for each ( x, i, m ) ∈ J thanks to Lemma 5 and J has a closed graphrelative to J because, in particular, the construction in (30g)allows M to be set-valued whenever x ∈ ∩ m = − , C im . Then, J is outer semicontinuous and locally bounded relative to J .
11) Proof of Lemma 7:
If we prove that ∀ i ∈ I , m ∈{− , } (cid:16) (cid:92) i (cid:48) ∈ I F i (cid:48) (cid:17) ∪ (cid:16) (cid:91) i (cid:48) ∈ I J i (cid:48) (cid:17) = W = F im ∪ J im (63)then (25), (29a), (29b) imply straightforwardly F ∪ J = W × I = F ∪ J = F − ∪ J − and, in turn, (30c) implies F ∪ J = W × I × M =: K . Therefore, we just need to prove(63) in the remainder. For each i ∈ I and m ∈ {− , } , F im ∪ J im (26) , (28) = (cid:16) E ≤ ( c i , δ i E i ) ∩ E ≥ (¯ c i , µ i ¯ E i ) ∩C ≥ ( c i , c i − p im , ψ i , E i ) ∩W (cid:17) ∪ (cid:16)(cid:0) E ≥ ( c i , δ i E i ) ∪E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p im , ψ i , E i ) (cid:1) ∩ W (cid:17) = W . (64)We are left with proving (cid:0) (cid:84) i (cid:48) ∈ I F i (cid:48) (cid:1) ∪ (cid:0) (cid:83) i (cid:48) ∈ I J i (cid:48) (cid:1) = W in (63).First, note that for each i (cid:48) , H i (cid:48) ( (cid:15) i (cid:48) , ν i (cid:48) ) ∩W = E ≤ ( c i (cid:48) , (cid:15) i (cid:48) E i (cid:48) ) ∩E ≥ (¯ c i (cid:48) , ν i (cid:48) ¯ E i (cid:48) ) ∩ W and, hence, (cid:0) (cid:92) i (cid:48) ∈ I F i (cid:48) (cid:1) ∪ (cid:0) (cid:91) i (cid:48) ∈ I J i (cid:48) (cid:1) = (cid:92) i (cid:48) ∈ I (cid:16)(cid:0) E ≥ ( c i (cid:48) , (cid:15) i (cid:48) E i (cid:48) ) ∪ E ≤ (¯ c i (cid:48) , ν i (cid:48) ¯ E i (cid:48) ) (cid:1) ∩ W (cid:17) ∪ (cid:91) i (cid:48) ∈ I (cid:16) H i (cid:48) ( (cid:15) i (cid:48) , ν i (cid:48) ) ∩ W (cid:17) = (cid:16) (cid:92) i (cid:48) ∈ I (cid:0) E ≥ ( c i (cid:48) , (cid:15) i (cid:48) E i (cid:48) ) ∪ E ≤ (¯ c i (cid:48) , ν i (cid:48) ¯ E i (cid:48) ) (cid:1) ∪ (cid:91) i (cid:48) ∈ I (cid:0) E ≤ ( c i (cid:48) , (cid:15) i (cid:48) E i (cid:48) ) ∩E ≥ (¯ c i (cid:48) , ν i (cid:48) ¯ E i (cid:48) ) (cid:1)(cid:17) ∩W = R n ∩W = W . From
F ∪ J = K , the definition of K in (31) and x + = x inthe jump map J , it follows immediately that J ( J ) ⊂ K .
12) Proof of Lemma 8:
The intersection of E ( c i , δE i ) and E (¯ c i , µ ¯ E i ) corresponds to the two quadratic equations (cid:40) δ (cid:107) E i ( x − c i ) (cid:107) = 1 µ (cid:107) ¯ E i ( x − ¯ c i ) (cid:107) = 1 . (65)By expanding squares and using (20), (65) is equivalent to (cid:40) (cid:107) E i x (cid:107) − c (cid:62) i E i x + (cid:107) E i c i (cid:107) − δ − = 0 (cid:107) E i x (cid:107) − c (cid:62) i E i x + (cid:107) E i c i (cid:107) (1 − µ − ) / . Solving for (cid:107) E i x (cid:107) and c (cid:62) i E i x , we obtain using (27a) (cid:40) (cid:107) E i x (cid:107) = (cid:107) E i c i (cid:107) (cid:0) (1 + µ − ) / − δ − δ i (cid:1) c (cid:62) i E i x = (cid:107) E i c i (cid:107) (cid:0) (3 + µ − ) / − δ − δ i (cid:1) (66)and both right-hand sides of (66) are positive because (1 + µ − ) / − δ − δ i ≥ (1 + ¯ µ i ( δ ) − ) / − δ − δ i (27b) = 1 − δ i + δ − δ i = ( δ i − + δ i ( δ − − ≥ ( δ i − > (67a) (3 + µ − ) / − δ − δ i ≥ (3 + ¯ µ i ( δ ) − ) / − δ − δ i = 1 − δ i > , by Assumption 2 . (67b)From (66), one obtains with some computations c (cid:62) i E i x (cid:107) E i c i (cid:107)(cid:107) E i x (cid:107) (66) = (cid:107) E i c i (cid:107) (cid:0) (3+ µ − ) / − δ − δ i (cid:1) (cid:107) E i c i (cid:107)(cid:107) E i c i (cid:107) (cid:113) (1+ µ − ) / − δ − δ i (27c) = 1 − cos(¯ θ i ( δ, µ )) δ i (cid:113) (1 + µ − ) / − δ − δ i . (68)For µ ∈ [1 , (cid:0) − δ i (1 − δ i /δ ) (cid:1) − ] and δ < , we can prove − cos(¯ θ i ( δ, µ )) δ i (cid:113) (1 + µ − ) / − δ − δ i = (3 + µ − ) / − δ − δ i (cid:113) (1 + µ − ) / − δ − δ i < (69) ( e.g. , set χ := (1 + µ − ) / , obtain the bounds of χ from thebounds of µ , substitute χ in (69), and note that the obtainedquadratic inequality holds true for such bounds of χ due to δ < ). Because of (67a), (67b) and (69), the expression in (32)is well-defined and positive. Since (68) yields c (cid:62) i E i x (cid:107) E i c i (cid:107)(cid:107) E i x (cid:107) = 1 − cos(¯ θ i ( δ, µ )) δ i (cid:113) (1 + µ − ) / − δ − δ i (32) = cos( ϑ i ( δ, µ )) , (33) holds as well by the cone definition in (11).
13) Proof of Lemma 9:
The obstacles {O i } i ∈ I are suffi-ciently pairwise disjoint, so (37) holds. Moreover, the sets R i (cid:48) ( δ i (cid:48) , µ i (cid:48) ) in (39) and E ≤ ( c i (cid:48)(cid:48) , δ i (cid:48)(cid:48) E i (cid:48)(cid:48) ) are bounded, and for δ i (cid:48) , δ i (cid:48)(cid:48) , µ i (cid:48) → , R i (cid:48) ( δ i (cid:48) , µ i (cid:48) ) and E ≤ ( c i (cid:48)(cid:48) , δ i (cid:48)(cid:48) E i (cid:48)(cid:48) ) reduce re-spectively to R ∗ i (cid:48) and E ≤ ( c i (cid:48)(cid:48) , E i (cid:48)(cid:48) ) . By a continuity argument,there exist parameters δ ∗ i and µ ∗ i such that the lemma holds.
14) Proof of Lemma 10:
We prove the claim for arbitrary i ∈ I and m ∈ {− , } . Let us prove (43a)-(43b). Thanks tothe weak pairwise disjointness of {E ≤ ( c i , δ i E i ) } i ∈ I we have E ≤ ( c i , δ i E i ) ∩ E ≥ ( c i (cid:48) , δ i (cid:48) E i (cid:48) ) = E ≤ ( c i , δ i E i ) for all i (cid:48) (cid:54) = i .Then, E ≤ ( c i , δ i E i ) ∩ W = E ≤ ( c i , δ i E i ) ∩ E ≥ ( c i , E i ) , whichimplies by (26) that F im satisfies (43b). By a similar argument, J i satisfies (43a) as well.Let us prove (43c). Write the complement of F ∗ as ( F ∗ ) c (42) = (cid:0) (cid:92) i ∈ I F i (cid:1) c (1) = (cid:91) i ∈ I (cid:0) F i (cid:1) c (24) = (cid:91) i ∈ I (cid:16)(cid:0) E ≥ ( c i , (cid:15) i E i ) ∪ E ≤ (¯ c i , ν i ¯ E i ) (cid:1) ∩ W (cid:17) c (1) = (cid:91) i ∈ I (cid:0) E ≥ ( c i , (cid:15) i E i ) ∪ E ≤ (¯ c i , ν i ¯ E i ) (cid:1) c ∪ W c (14) = (cid:91) i ∈ I E < ( c i , (cid:15) i E i ) ∩ E > (¯ c i , ν i ¯ E i ) ∪ (cid:91) i ∈ I E < ( c i , E i )= (cid:91) i ∈ I (cid:0) E < ( c i , (cid:15) i E i ) ∩ E > (¯ c i , ν i ¯ E i ) (cid:1) ∪ E < ( c i , E i )= (cid:91) i ∈ I ( E < ( c i ,(cid:15) i E i ) ∪E < ( c i , E i )) ∩ ( E > (¯ c i , ν i ¯ E i ) ∪E < ( c i ,E i ))= (cid:91) i ∈ I E < ( c i , (cid:15) i E i ) ∩ ( E > (¯ c i , ν i ¯ E i ) ∪ E < ( c i , E i )) (70)because (cid:15) i < . Thanks to the weak pairwise disjointnessof {E ( c i , δ i E i ) } i ∈ I and δ i < (cid:15) i , the sets {E < ( c i , (cid:15) i E i ) ∩ ( E > (¯ c i , ν i ¯ E i ) ∪ E < ( c i , E i )) } i ∈ I can actually be proven tobe pairwise separated. Then, we can use (2) to obtain theboundary of the set F ∗ as ∂ F ∗ = ∂ (cid:0) ( F ∗ ) c (cid:1) (70) = ∂ (cid:16) (cid:91) i ∈ I E < ( c i , (cid:15) i E i ) ∩ ( E > (¯ c i , ν i ¯ E i ) ∪ E < ( c i , E i )) (cid:17) (2) = (cid:91) i ∈ I ∂ (cid:0) E < ( c i , (cid:15) i E i ) ∩ ( E > (¯ c i , ν i ¯ E i ) ∪ E < ( c i , E i )) (cid:1) (1) ⊂ (cid:91) i ∈ I (cid:16) ∂ E < ( c i , (cid:15) i E i ) ∩ E > (¯ c i , ν i ¯ E i ) ∪ E < ( c i , E i ) ∪ ∂ (cid:0) E > (¯ c i , ν i ¯ E i ) ∪ E < ( c i , E i ) (cid:1) ∩ E < ( c i , (cid:15) i E i ) (cid:17) (1) ⊂ (cid:91) i ∈ I (cid:18) E ( c i , (cid:15) i E i ) ∩ (cid:0) E ≥ (¯ c i , ν i ¯ E i ) ∪ E ≤ ( c i , E i ) (cid:1) ∪ (cid:16)(cid:0) E (¯ c i , ν i ¯ E i ) \E < ( c i ,E i ) (cid:1) ∪ (cid:0) E ( c i ,E i ) \E > (¯ c i , ν i ¯ E i ) (cid:1)(cid:17) ∩E ≤ ( c i ,(cid:15) i E i ) (cid:19) = (cid:91) i ∈ I (cid:16)(cid:0) E ( c i , (cid:15) i E i ) ∩E ≥ (¯ c i , ν i ¯ E i ) (cid:1) ∪ (cid:0) ( E (¯ c i , ν i ¯ E i ) \E < ( c i , E i )) ∩ E ≤ ( c i , (cid:15) i E i ) (cid:1) ∪ (cid:0) E ( c i , E i ) \E > (¯ c i , ν i ¯ E i ) (cid:1)(cid:17) =: (cid:91) i ∈ I P i (43c) is finally proven in (74), for which we note that: (a) J ∗ is simplified into ∪ i ∈ I H i ( (cid:15) i , ν i ) thanks to (43a); (b) P i ∩H i (cid:48) ( (cid:15) i (cid:48) ,
1) = ∅ for all i (cid:54) = i (cid:48) ; (c) most of the sets in the next-to-last expression are empty, so the last expression follows.Let us prove (43d). First, note that by (43b) and (22) F im = E ≤ ( c i , δ i E i ) ∩ E ≥ ( c i , E i ) ∩ E ≥ (¯ c i , µ i ¯ E i ) ∩ C ≥ ( c i , c i − p im , ψ i , E i ) , which is an intersection of four closed sets. By successiveapplications of (1i) and (1f), the boundary of F im satisfies ∂ F im ⊂ (cid:91) k ∈{ , , , } P im,k (71)with the following definitions P im, := E ( c i , δ i E i ) ∩ E ≥ ( c i , E i ) ∩ E ≥ (¯ c i , µ i ¯ E i ) ∩ C ≥ ( c i , c i − p im , ψ i , E i ) (72a) P im, := E ≤ ( c i , δ i E i ) ∩ E ( c i , E i ) ∩ E ≥ (¯ c i , µ i ¯ E i ) ∩ C ≥ ( c i , c i − p im , ψ i , E i ) (72b) P im, := E ≤ ( c i , δ i E i ) ∩ E ≥ ( c i , E i ) ∩ E (¯ c i , µ i ¯ E i ) ∩ C ≥ ( c i , c i − p im , ψ i , E i ) (72c) P im, := E ≤ ( c i , δ i E i ) ∩ E ≥ ( c i , E i ) ∩ E ≥ (¯ c i , µ i ¯ E i ) ∩ C ( c i , c i − p im , ψ i , E i ) . (72d)Second, note that since F im is closed, ∂ F im ⊂ F im ⊂ W , andhence ∂ F im \W ⊂ W\W = ∅ . By this fact, we can write that ∂ F im \J im (28) = ∂ F im \ (cid:18)(cid:16) E ≥ ( c i , δ i E i ) ∪ E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p im , ψ i , E i ) (cid:17) ∩ W (cid:19) (1) = ∂ F im \W ∪ ∂ F im \ (cid:16) E ≥ ( c i , δ i E i ) ∪ E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p im , ψ i , E i ) (cid:17) = ∂ F im \ (cid:16) E ≥ ( c i , δ i E i ) ∪E ≤ (¯ c i , µ i ¯ E i ) ∪C ≤ ( c i , c i − p im , ψ i , E i ) (cid:17) (71) ⊂ (cid:16) ∪ k ∈{ , , , } P im,k (cid:17) \ (cid:16) E ≥ ( c i , δ i E i ) ∪ E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p im , ψ i , E i ) (cid:17) (1) = ∪ k ∈{ , , , } (cid:16) P im,k \ (cid:16) E ≥ ( c i , δ i E i ) ∪ E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p im , ψ i , E i ) (cid:17)(cid:17) (1) ⊂ ∪ k ∈{ , , , } (cid:16)(cid:0) P im,k \E ≥ ( c i , δ i E i ) (cid:1) ∩ (cid:0) P im,k \E ≤ (¯ c i , µ i ¯ E i ) (cid:1) ∩ (cid:0) P im,k \C ≤ ( c i , c i − p im , ψ i , E i ) (cid:1)(cid:17) . Finally, we simplify this expression through the following facts P im, \E ≥ ( c i , δ i E i ) = ∅ , P im, \E ≤ (¯ c i , µ i ¯ E i ) = ∅P im, \C ≤ ( c i , c i − p im , ψ i , E i ) = ∅ , (73)which are an immediate consequence of (72) and yield ∂ F im \J im (73) ⊂ P im, \ (cid:16) E ≥ ( c i , δ i E i ) ∪ E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p im , ψ i , E i ) (cid:17) (1) = P im, ∩ (cid:16) E ≥ ( c i ,δ i E i ) ∪E ≤ (¯ c i ,µ i ¯ E i ) ∪C ≤ ( c i ,c i − p im ,ψ i ,E i ) (cid:17) c (1) = P im, ∩ E < ( c i ,δ i E i ) ∩ E > (¯ c i ,µ i ¯ E i ) ∩ C > ( c i ,c i − p im ,ψ i ,E i )= E < ( c i ,δ i E i ) ∩E ( c i ,E i ) ∩E > (¯ c i ,µ i ¯ E i ) ∩C > ( c i , c i − p im ,ψ i ,E i )= E ( c i , E i ) ∩ E > (¯ c i , µ i ¯ E i ) ∩ C > ( c i , c i − p im , ψ i , E i ) (1) = E ( c i , E i ) \ ( E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p im , ψ i , E i )) .
15) Proof of Lemma 11:
We divide the proof into steps.
Step 1: For each i ∈ I and m ∈ {− , } , δ ∈ [ δ i , and ψ ∈ [ ψ i , ¯ ψ i ) , consider the sets ˜ F im ( δ, ψ ) := E ≤ ( c i , δE i ) ∩ E ≥ ( c i , E i ) ∩ E ≥ (¯ c i , µ i ¯ E i ) ∩ C ≥ ( c i , c i − p im , ψ, E i ) . (75a) ˜ P im, ( δ, ψ ) := E ≤ ( c i , δE i ) ∩ E ≥ ( c i , E i ) ∩ E (¯ c i , µ i ¯ E i ) ∩ C ≥ ( c i , c i − p im , ψ, E i ) (75b) Each maximal solution x to the flow-only hybrid system ˙ x = κ ( x, i, m ) =: u ( x ) , x ∈ ˜ F im ( δ, ψ ) (76) has T = sup t dom x < + ∞ and x ( T ) ∈ ˜ P im, ( δ, ψ ) . Wefirst prove that T is finite. Consider the following nonnegativefunction V ( x ) := (cid:107) E i ( x − p im ) (cid:107) . Simple computations,(16), and (4a) yield that for all x ∈ ˜ F im ( δ, ψ ) (cid:104)∇ V ( x ) , κ ( x, i, m ) (cid:105) = − k m ( x − p im ) (cid:62) E i π ⊥ ( E i ( x − c i )) E i ( x − p im )= − k m (cid:107) π ⊥ ( E i ( x − c i )) E i ( x − p im ) (cid:107) < − e < , where e > follows from π ⊥ ( E i ( x − c i )) E i ( x − p im ) vanishingonly for x ∈ L ( c i , p im − c i ) (Lemma 3), and L ( c i , p im − c i ) isseparated by a positive distance from ˜ F im ( δ, ψ ) (Lemma 4).Then, T is finite, otherwise V evaluated along solutions wouldbecome negative. In order to show x ( T ) ∈ ˜ P im, ( δ, ψ ) , weresort to a viability argument based on tangent cones. To thisend, we need the next lemma. Lemma
12: For all x ∈ ˜ F im ( δ, ψ ) \ ˜ P im, ( δ, ψ ) , u ( x ) ∈ T E ≤ ( c i ,δE i ) ( x ) (77a) u ( x ) ∈ T E ≥ ( c i ,E i ) ( x ) (77b) u ( x ) ∈ T E ≥ (¯ c i ,µ i ¯ E i ) ( x ) (77c) u ( x ) ∈ T C ≥ ( c i ,c i − p im ,ψ,E i ) ( x ) . (77d) ˜ F im ( δ, ψ ) in (75a) is the intersection of four closed sets: if u ( x ) ∈ T E ≤ ( c i ,δE i ) ( x ) ∩ T E ≥ ( c i ,E i ) ( x ) ∩ T E ≥ (¯ c i ,µ i ¯ E i ) ( x ) ∩ T C ≥ ( c i ,c i − p im ,ψ,E i ) ( x ) , (78)then u ( x ) ∈ T ˜ F im ( δ,ψ ) ( x ) by the next fact, which is animmediate corollary of [31, Thm. 5]. Fact v ∈ A ∩ B with A , B closedsubsets of R n . Suppose T A ( v ) ∩ ( T B ( v ) ◦ ) (cid:54) = ∅ . Then, T A∩B ( v ) ⊃ T A ( v ) ∩ T B ( v ) . The condition (78) has been checked in Lemma 12 for each x ∈ ˜ F im ( δ, ψ ) \ ˜ P im, ( δ, ψ ) , hence u ( x ) ∈ T ˜ F im ( δ,ψ ) ( x ) ∀ x ∈ ˜ F im ( δ, ψ ) \ ˜ P im, ( δ, ψ ) . (79)Then, it can only be x ( T ) ∈ ˜ P im, ( δ, ψ ) , otherwise the solutioncould be further extended by viability results such as [24,Lemma 5.26(b)]. Step 2: For each i ∈ I , both the sets E ≤ (¯ c i , ¯ E i ) and C (0 , c i , ϑ i ( δ, µ i ) , E i ) ( δ ∈ [ δ i , ) are forward invariant underthe vector field − k x . For x ∈ E (¯ c i , ¯ E i ) , we have − k x (cid:62) ¯ E i ( x − ¯ c i ) (20) = − k (cid:107) ¯ E i x (cid:107) − k x (cid:62) ¯ E i ( x − c i ) (19) = − k (cid:107) ¯ E i x (cid:107) + k (cid:0) − (cid:107) ¯ E i ( x − ¯ c i ) (cid:107) (cid:1) = − k (cid:107) ¯ E i x (cid:107) ≤ , (80)where the last equality follows from x ∈ E (¯ c i , ¯ E i ) . There-fore, {− k x } ∈ P ≤ (0 , ¯ E i ( x − ¯ c i )) = T E (¯ c i , ¯ E i ) ( x ) . For x ∈ C (0 , c i , ϑ i ( δ, µ i ) , E i ) , we have − k x (cid:62) E i π ϑ i ( δ,µ i ) ( E i c i ) E i x = 0 . (81)Therefore, {− k x } ∈ P (0 , E i π ϑ i ( δ,µ i ) ( E i c i ) E i x ) = T C (0 ,c i ,ϑ i ( δ,µ i ) ,E i ) ( x ) (cf. (12)). Forward invariance followsthen from the classical Nagumo’s theorem. Step 3: Proof of (49) . Let ξ ( t, j ) =( x ( t, j ) , i ( t, j ) , m ( t, j )) ∈ F l × { l } with l ∈ {− , } and ( t, j ) ∈ dom ξ . Hence, by (38), (cid:107) x ( t, j ) (cid:107) ≥ r i ( t,j ) > (82)thanks to the discussion below (38). We further divide intomutually exclusive subcases. Step 3a: ξ ( t, j ) ∈ ( F l ∩ J l ) × { l } and the solutionjumps. By (30b) and (30e), ξ ( t, j + 1) = ( x ( t, j ) , i ( t, j ) , .Depending on x ( t, j ) , either the solution never jumps againor reaches the set J ∗ in (42) at some ( t (cid:48) , j + 1) . Consider theformer case. The flow map for x in (30a) ensures x ( τ, j +1) =exp( − k ( τ − t )) x ( t, j ) for all τ ≥ t . By (82), s ≥ t existssuch that (cid:107) x ( s, j + 1) (cid:107) = min i (cid:48) ∈ I r i (cid:48) / σ > . So, (cid:107) x ( s, j + 1) (cid:107) + σ = min i (cid:48) ∈ I r i (cid:48) ≤ r i ( t,j ) ≤ (cid:107) x ( t, j ) (cid:107) (83)and the claim of the proposition is proven. Consider the lattercase, i.e., there exist t (cid:48) ≥ t and i (cid:48) ∈ I such that x ( t (cid:48) , j +1) ∈ J i (cid:48) ⊂ J ∗ . J i (cid:48) is a shrinking of the set F i (cid:48) ∪ F i (cid:48) − by construction of the flow and jump sets (cf. (23) and (26),where the union of the cones from (26) gives R n from thesame arguments yielding (62)). Then, by Lemma 5, there exist l (cid:48) such that ξ ( t (cid:48) , j + 2) ∈ ( F l (cid:48) \J l (cid:48) ) × { l (cid:48) } and the solution isforced to flow as in Step 3b below. Step 3b: ξ ( t, j ) ∈ ( F l \J l ) × { l } and the solution flows. If x ( t, j ) ∈ F il \J il , there exists δ ∈ ( δ i , and ψ ∈ ( ψ i , ¯ ψ i ) suchthat x ( t, j ) ∈ ˜ F il ( δ, ψ ) as defined in (75a). From the facts H i ( δ, µ i ) ∩ E ≥ ( c i , δ i E i ) = ∅ (84a) C ≥ ( c i , c i − p il , ψ, E i ) ∩ C ≤ ( c i ,c i − p il , ψ i , E i )= { c i } / ∈ E ≥ ( c i , E i ) , (84b) ∂ F ∗ \J ∗ ⊂ (cid:16) (cid:91) i ∈ I P i (cid:17) \ (cid:16) (cid:91) i ∈ I H i ( (cid:15) i , ν i ) (cid:17) = (cid:91) i ∈ I (cid:16) P i \ (cid:91) i (cid:48) ∈ I (cid:0) H i (cid:48) ( (cid:15) i (cid:48) , ν i (cid:48) ) (cid:1)(cid:17) = (cid:91) i ∈ I (cid:16) (cid:92) i (cid:48) ∈ I (cid:0) P i \H i (cid:48) ( (cid:15) i (cid:48) , ν i (cid:48) ) (cid:1)(cid:17) = (cid:91) i ∈ I (cid:16) P i \H i ( (cid:15) i , ν i ) ∩ (cid:92) i (cid:48) ∈ I ,i (cid:48) (cid:54) = i P i (cid:17) = (cid:91) i ∈ I (cid:0) P i \H i ( (cid:15) i , ν i ) (cid:1) = (cid:91) i ∈ I (cid:16)(cid:0) E ( c i , (cid:15) i E i ) ∩ E ≥ (¯ c i , ν i ¯ E i ) (cid:1) ∪ (cid:0) ( E (¯ c i , ν i ¯ E i ) \E < ( c i , E i )) ∩ E ≤ ( c i , (cid:15) i E i ) (cid:1) ∪ ( E ( c i , E i ) \E > (¯ c i , ν i ¯ E i )) (cid:17) \H i ( (cid:15) i , ν i )= (cid:91) i ∈ I (cid:16)(cid:0) E ( c i , (cid:15) i E i ) ∩ E ≥ (¯ c i , ν i ¯ E i ) (cid:1) ∪ (cid:0) ( E (¯ c i , ν i ¯ E i ) \E < ( c i , E i )) ∩ E ≤ ( c i , (cid:15) i E i ) (cid:1) ∪ ( E ( c i , E i ) \E > (¯ c i , ν i ¯ E i )) (cid:17) ∩ H i ( (cid:15) i , ν i ) c = (cid:91) i ∈ I (cid:16)(cid:0) E ( c i ,(cid:15) i E i ) ∩E ≥ (¯ c i ,ν i ¯ E i ) (cid:1) ∪ (cid:0) ( E (¯ c i ,ν i ¯ E i ) \E < ( c i ,E i )) ∩E ≤ ( c i ,(cid:15) i E i ) (cid:1) ∪ ( E ( c i ,E i ) \E > (¯ c i ,ν i ¯ E i )) (cid:17) ∩ (cid:0) E > ( c i ,(cid:15) i E i ) ∪E < (¯ c i ,ν i ¯ E i ) ∪E < ( c i ,E i ) (cid:1) = (cid:91) i ∈ I (cid:16) E ( c i , (cid:15) i E i ) ∩ E ≥ (¯ c i , ν i ¯ E i ) ∩ E > ( c i , (cid:15) i E i ) (cid:17) ∪ (cid:16) E ( c i , (cid:15) i E i ) ∩ E ≥ (¯ c i , ν i ¯ E i ) ∩ E < (¯ c i , ν i ¯ E i ) (cid:17) ∪ (cid:16) E ( c i , (cid:15) i E i ) ∩ E ≥ (¯ c i , ν i ¯ E i ) ∩ E < ( c i , E i ) (cid:17) ∪ (cid:16)(cid:0) E (¯ c i , ν i ¯ E i ) \E < ( c i , E i ) (cid:1) ∩ E ≤ ( c i , (cid:15) i E i ) ∩ E > ( c i , (cid:15) i E i ) (cid:17) ∪ (cid:16)(cid:0) E (¯ c i , ν i ¯ E i ) \E < ( c i , E i ) (cid:1) ∩ E ≤ ( c i , (cid:15) i E i ) ∩ E < (¯ c i , ν i ¯ E i ) (cid:17) ∪ (cid:16)(cid:0) E (¯ c i , ν i ¯ E i ) \E < ( c i , E i ) (cid:1) ∩ E ≤ ( c i , (cid:15) i E i ) ∩ E < ( c i , E i ) (cid:17) ∪ (cid:16)(cid:0) E ( c i , E i ) \E > (¯ c i , ν i ¯ E i ) (cid:1) ∩ E > ( c i , (cid:15) i E i ) (cid:17) ∪ (cid:16)(cid:0) E ( c i , E i ) \E > (¯ c i , ν i ¯ E i ) (cid:1) ∩ E < (¯ c i , ν i ¯ E i ) (cid:17) ∪ (cid:16)(cid:0) E ( c i , E i ) \E > (¯ c i , ν i ¯ E i ) (cid:1) ∩ E < ( c i , E i ) (cid:17) = (cid:91) i ∈ I E ( c i , E i ) \E ≥ (¯ c i , ν i ¯ E i ) . (74) ˜ F il ( δ, ψ ) ∩ J il is a subset of ˜ P il, ( δ, ψ ) because ˜ F il ( δ, ψ ) ∩ J il = E ≤ ( c i , δE i ) ∩ E ≥ ( c i , E i ) ∩ E ≥ (¯ c i , µ i ¯ E i ) ∩ C ≥ ( c i , c i − p il , ψ, E i ) ∩ J il ⊂ H i ( δ, µ i ) ∩ C ≥ ( c i , c i − p il , ψ, E i ) ∩ (cid:0) E ≥ ( c i , δ i E i ) ∪ E ≤ (¯ c i , µ i ¯ E i ) ∪ C ≤ ( c i , c i − p il , ψ i , E i ) (cid:1) = (cid:0) H i ( δ, µ i ) ∩ C ≥ ( c i , c i − p il , ψ, E i ) ∩ E ≥ ( c i , δ i E i ) (cid:1) ∪ (cid:0) H i ( δ, µ i ) ∩ C ≥ ( c i , c i − p il , ψ, E i ) ∩ E ≤ (¯ c i , µ i ¯ E i ) (cid:1) ∪ (cid:0) H i ( δ, µ i ) ∩C ≥ ( c i , c i − p il , ψ, E i ) ∩C ≤ ( c i , c i − p il , ψ i , E i )) (cid:1) (84) = H i ( δ, µ i ) ∩ C ≥ ( c i , c i − p il , ψ, E i ) ∩ E ≤ (¯ c i , µ i ¯ E i )= E ≤ ( c i , δE i ) ∩E ≥ ( c i , E i ) ∩E (¯ c i , µ i ¯ E i ) ∩C ≥ ( c i , c i − p il , ψ, E i ) (75b) = ˜ P il, ( δ, ψ ) . (85)This fact combined with Step 1, shows that the solution leavesthe set F il in finite time through the set ˜ P il, ( δ, ψ ) , where itjumps. Then, we have x ( t (cid:48) , j +1) = x ( t (cid:48) , j ) ∈ ˜ P i ( t (cid:48) ,j ) m ( t (cid:48) ,j ) , ( δ, ψ ) , i ( t (cid:48) , j + 1) = i ( t (cid:48) , j ) =: ι and m ( t (cid:48) , j ) = 0 . Then, by (85), x ( t (cid:48) , j + 1) ∈ E ( c ι , δE ι ) ∩ E (¯ c ι , µ ι ¯ E ι ) , and, by Lemma 8, x ( t (cid:48) , j + 1) ∈ C (0 , c ι , ϑ ι ( δ, µ ι ) , E ι ) . We have shown inStep 2 that both the sets E ≤ (¯ c ι , ¯ E ι ) and C (0 , c ι , ϑ ι ( δ, µ ι ) , E ι ) are forward invariant under the stabilization flow map for x , i.e., − k x . Since the obstacles are weakly disjoint, thesolution then flows in E ≤ (¯ c ι , ¯ E ι ) ∩ C (0 , c ι , ϑ ι ( δ, µ ι ) , E ι ) untilit reaches the set E ( c ι , δ ι E ι ) at ( t (cid:48)(cid:48) , j + 1) . We either have (cid:107) x ( t (cid:48)(cid:48) , j + 1) (cid:107) < r ι or (cid:107) x ( t (cid:48)(cid:48) , j + 1) (cid:107) ≥ r ι . Consider theformer case. Define σ := min i,i (cid:48) ∈ I , i (cid:54) = i (cid:48) dist ( E ≤ ( c i , δ i E i ) , E ≤ ( c i (cid:48) , δ i (cid:48) E i (cid:48) )) > , (86)which is positive because obstacle are compact, pairwisedisjoint sets. Since x ( t (cid:48)(cid:48) , j +1) ∈ E ( c ι , δ ι E ι ) and the obstaclesare weakly pairwise disjoint, the solution can only flow up tothe time ( s, j + 1) such that σ is traversed, i.e., (cid:107) x ( s, j + 1) (cid:107) + σ ≤ (cid:107) x ( t (cid:48)(cid:48) , j + 1) (cid:107) < r ι ≤ (cid:107) x ( t, j ) (cid:107) , (87)and the claim of the proposition is proven. Consider the lattercase, i.e., (cid:107) x ( t (cid:48)(cid:48) , j + 1) (cid:107) ≥ r ι . Then, the definition of theset R ι ( δ ι , µ ι ) in (39) implies that x ( t (cid:48)(cid:48) , j + 1) ∈ R ι ( δ ι , µ ι ) . However, thanks to (40), the solution can only flow withstabilization mode while in R ι ( δ ι , µ ι ) , so that ( t (cid:48)(cid:48)(cid:48) , j +1) existssuch that (cid:107) x ( t (cid:48)(cid:48)(cid:48) , j + 1) (cid:107) = r ι . Define σ := min i,i (cid:48) ∈ I ,i (cid:54) = i (cid:48) dist ( R i ( δ i , µ i ) , E ≤ ( c i (cid:48) , δ i (cid:48) E i (cid:48) )) > , (88)which is positive because the considered sets are compact andpairwise sufficiently disjoint. Before a jump to avoidance modeat ( s, j + 1) can occur, we have (cid:107) x ( s, j + 1) (cid:107) + σ ≤ (cid:107) x ( t (cid:48)(cid:48)(cid:48) , j + 1) (cid:107) = r ι ≤ (cid:107) x ( t, j ) (cid:107) . (89) Step 3c: ξ ( t, j ) ∈ ( F l ∩J l ) ×{ l } and the solution flows. Thesolution cannot flow forever, as established in Step 1. If it flowsuntil its component x reaches the set P i ( t,j ) l, ( δ i ( t,j ) , ψ i ( t,j ) ) ,the second part of the argument of Step 3b still applies, inparticular (87) or (89). If it jumps beforehand, Step 3a applies.Then we do not have circularity. By combining (83), (87) and(89), (49) is proven with σ := min { σ , σ , σ } > .
16) Proof of Lemma 12:
As for (77a), we have that T E ≤ ( c i ,δE i ) ( x ) is either R n for x ∈ E < ( c i , δE i ) or P ≤ (0 , E i ( x − c i )) for x ∈ E ( c i , δE i ) . For all x ∈E ≤ ( c i , δE i ) ⊃ ˜ F im ( δ, ψ ) , u ( x ) ∈ P (0 , E i ( x − c i )) (see(48)), so (77a) is proven. A similar argument yields (77b).As for (77c), we note that ˜ F im ( δ, ψ ) \ ˜ P im, ( δ, ψ )= E ≤ ( c i ,δE i ) ∩E ≥ ( c i ,E i ) ∩E ≥ (¯ c i ,µ i ¯ E i ) ∩C ≥ ( c i , c i − p im , ψ, E i ) \ (cid:16) E ≤ ( c i , δE i ) ∩E ≥ ( c i , E i ) ∩E (¯ c i , µ i ¯ E i ) ∩C ≥ ( c i , c i − p im , ψ, E i ) (cid:17) (1c) = E ≤ ( c i , δE i ) ∩ E ≥ ( c i , E i ) ∩ E > (¯ c i , µ i ¯ E i ) ∩ C ≥ ( c i , c i − p im , ψ, E i ) . (90)Then, for all x ∈ ˜ F im ( δ, ψ ) \ ˜ P im, ( δ, ψ ) , T E ≥ (¯ c i ,µ i ¯ E i ) ( x ) = R n thanks to (90), and this proves (77c). As for (77d), we havethat T C ≥ ( c i ,c i − p im ,ψ,E i ) ( x ) is either R n for x ∈ C > ( c i , c i − p im , ψ, E i ) or P ≥ (0 , n im ( x )) for x ∈ C ( c i , c i − p im , ψ, E i ) with n im ( x ) := E i π ψ ( E i ( c i − p im )) E i ( x − c i ) by (12). If we prove that u ( x ) (cid:62) n im ( x ) ≥ for all x ∈ C ( c i , c i − p im , ψ, E i ) ⊃ ˜ F im ( δ, ψ ) , then (77d) holds. Indeed, this last step follows from u ( x ) (cid:62) · n im ( x ) = − k m ( x − p im ) (cid:62) E i π ⊥ ( E i ( x − c i )) E − i · E i π ψ ( E i ( c i − p im )) E i ( x − c i )= k m ( p im − c i ) (cid:62) E i π ⊥ ( E i ( x − c i )) · π ψ ( E i ( c i − p im )) E i ( x − c i ) (5) , (3) = k m ( p im − c i ) (cid:62) E i π ⊥ ( E i ( x − c i )) · (cid:0) cos ( ψ ) I n − π (cid:107) ( E i ( c i − p im )) (cid:1) E i ( x − c i )= − k m ( p im − c i ) (cid:62) E i π ⊥ ( E i ( x − c i )) · π (cid:107) ( E i ( c i − p im )) E i ( x − c i ) (3) = k m ( c i − p im ) (cid:62) E i π ⊥ ( E i ( x − c i )) E i ( c i − p im ) · ( c i − p im ) (cid:62) E i ( x − c i ) (cid:107) E i ( c i − p im ) (cid:107) − ≥ (91)because for all x ∈ C ( c i , c i − p im , ψ, E i ) , one has x ∈P ≥ ( c i , E i ( c i − p im )) . R EFERENCES[1] M. Hoy, A. S. Matveev, and A. V. Savkin, “Algorithms for collision-free navigation of mobile robots in complex cluttered environments: asurvey,”
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Soulaimane Berkane received his Engineering and M.Sc. degrees in Au-tomatic Control from Ecole Nationale Polytechnique, Algeria, in 2013, andhis PhD in Electrical Engineering from the University of Western Ontario,Canada, in 2017. He held postdoctoral positions at the University of WesternOntario, Canada, and at KTH Royal Institute of Technology, Sweden, between2018 and 2019. He is currently an assistant professor at the Departmentof Computer Science and Engineering, University of Quebec in Outaouais,Canada. His research interests are in the area of nonlinear control theory withapplications to robotic and autonomous systems.