—In this paper, we propose sign-elevation angle rigidity and design bearing-only control laws in agents’ local coordinate frames without any orientation synchronization or estimation algorithms. Moreover, we also establish almost global stability results for formation control, with elevation angle measurements that can be obtained using bearing-only sensors in the agents’ local frames. Elevation angle rigidity-based control laws can only guarantee local stability results, and existing results do not address flip, flex, and reflection ambiguities that can arise in elevation angle rigid configurations. Hence, we first establish the theory of sign-elevation angle rigidity to uniquely characterize formation shapes, up to translation and rotation, by employing elevation angle and signed-area constraints for planar frameworks, while for 3-D frameworks, elevation angle constraints along with signed-volume constraints are used. Then, we propose control laws for single integrators and prove almost global stabilization to the desired formation shapes using bearing- only measurements when the desired formation is obtained by triangulated sub-formations for 2-D graphs and tetrahedral sub- formations for 3-D graphs. Control laws for non-holonomic agents are also analyzed. Finally, simulations are provided to validate the obtained results. Index terms: bearing-only control, formation ambiguities, coordinate-free control, formation control I. INTRODUCTION Recent years have witnessed active research on distributed formation control of multi-agent systems, making it an in- triguing area of study [1]–[4]. The primary objective of the formation control problem is to design decentralized control laws for the agents, enabling them to attain a specific geo- metric shape. An associated problem with formation control is identifying geometric shapes using local constraints, which is addressed through rigidity theory and its extensions. When using position-based formation control, every agent is restricted to a specific position, and it must be able to detect its exact position for the strategy to work. On the other hand, displacement-based formation control uses displacement con- straints between agents to determine the shape of the formation [5]–[7]. This method is related to the consensus protocols in multi-agent systems. In a distance-based formation control strategy, the desired formation is specified by inter-agent distance constraints. Distance rigidity theory deals with how to determine unique formation shapes from distance constraints [1], [8]. The formation control laws that were developed based on distance rigidity required both distance and displacement between the agents [1], [3], [8]–[11]. Using distance and angle constraints, studies such as [12]–[14] have defined formation C. Garanayak ( chinmayg22@gmail.com ) and D. Mukherjee (dm@ee.iitb.ac.in ) are with the Department of Electrical Engineer- ing, Indian Institute of Technology Bombay, Mumbai- 400076, India. The work of DM was supported in part by an ISRO funded project bearing code RD/0120-ISROC00-007.shapes, leading to the concept of weak-rigidity. The articles [15], [16] introduced the concept of bearing rigidity theory, which can accurately define unique formation shapes up to a translation and scale through the use of bearing constraints for defining the target formation. Additionally, [17] presented the concept of bearing-ratio-of-distance rigidity theory using bearing and ratio-of-distance constraints. However, the control laws for formation control in [16], [17] necessitated inter-agent displacement measurements. Vision-based applications can benefit from bearing mea- surements, which have the advantage of utilizing low-cost passive optical sensors such as cameras instead of displace- ment measurements. Bearing measurements also tend to be less noisy than displacement and distance measurements [18]. Additionally, they require a less complex sensory setup for the agents, resulting in a lower payload on the agents. Hence, bearing-only formation control has received considerable re- search attention in recent years [4], [18], [19]. The primary objective of bearing-only formation control is to achieve the desired shape of the formation by relying solely on bearing measurements without using inter-agent distance and dis- placement measurements. In [4], the analysis of bearing-only formation control for single integrators was conducted using bearing rigidity theory, and [19] reported tracking control for single and double integrators, as well as unicycles, with constant velocity leaders. Ref. [20], [21] discussed the im- plementation of time-varying bearing-only formation control using the concept of persistence of excitation. It is important to note that bearing measurements rely on a specific coordi- nate system, requiring agents to access a global coordinate system to implement control laws when inter-agent bearing constraints specify the desired formation. However, distance- based formation control laws can be implemented in the agent’s local coordinate system, as distance constraints are not dependent on a specific coordinate system. Therefore, to apply bearing-only control laws in agents’ local coordinate frame, orientation synchronization [22], [23] or orientation estimation algorithms [24], [25] are typically implemented in conjunction with the formation control law. Also, orientation synchronization or estimation algorithms need relative attitude measurements between the agents, which is difficult to obtain in practice or may require extra relative attitude estimation algorithms from local bearing sensing [26]. Refs. [27], [28] studied bearing-only formation control in agents’ local frames based on SE(2)andSE(3)rigidity theories. However, they required extra controllable quantities, which established the relation between agents’ local and global frames. An approach to bearing-only formation control using angle rigidity was suggested for planar agents in [29], [30] to obviate the need for a global coordinate system. This approach incorporated signed angle constraints in the formation speci- fication and established local exponential convergence to the desired formation shape. In [31], unsigned angle constraints were used to describe the desired formation shape. However, the control law in [31] required inter-agent displacement measurements. In refs. [32], [33] a triangular angle rigidity theory was proposed to describe the desired formation shape using triangular angle constraints. However, the control laws in [33] also required relative displacement between the agents. Ref. [34] discussed the specification of planar agents with circular discs and proposes a bearing-only formation control in a two-dimensional setting. Elevation angle rigidity was addressed [35], which involves attaching a rod to each agent in 2D or a ball to agents in 3D to specify elevation angle constraints and the desired formation was described by these constraints. A gradient-based bearing-only formation control law in the agents’ local frames of reference was then proposed, which established local asymptotic stability about the desired formation shape. The constraints on elevation angles were independent of coordinates, so the control laws in [35] did not need information about any global coordinate system. In [36], elevation angle-based control laws were proposed for double integrators and non-holonomic agents. In [37], elevation angle rigidity control laws for single and double integrators were derived in the presence of bounded disturbances. However, it is important to note that setting constraints only on the elevation angles in the formation specification can result in issues such as flip, flex, and reflection ambiguities in the final formation shape, which were not addressed in [35] and only local convergence to the desired formation shape was established in [35]–[37]. In distance-based formation control [38], [39], such ambiguities are also present. In [38], such ambiguities were addressed by using signed area constraints for 2-dimensional scenarios when 3or4agents are employed. Ref. [40] proposed sign-rigidity theory where signed area constraints were used along with distance constraints to address ambiguities in 2- D. Ref. [39] addressed these ambiguities by implementing constraints related to the signed unit area and signed unit volume but only established local convergence results. In view of the above results, the important contributions of this article are summarized below: We present sign-elevation angle rigidity to characterize unique formation shapes up to a translation and rotation using eleva- tion angle with signed-area or signed-volume constraints for graphs embedded in Euclidian 2-D or 3-D space, respectively. For checking sign-elevation angle rigidity, the sign-elevation angle rigidity matrix is derived based on the sign-elevation angle rigid motions, and a rank condition on the sign-elevation angle rigidity matrix is provided. Then, we establish almost global convergence to the desired formation shape using only local bearing measurements for both planar and three- dimensional formations when the sign-elevation angle rigid framework is generated by a signed-Henneberg construction. Both single integrators and non-holonomic agents are consid- ered for the control design and stability analysis. In contrast with existing results, our current work has the following salient features: (i) Compared with bearing-only control laws proposed in [4],[19]–[21], the agents here do not need to know their orientation information with respect to a global frame. Compared with [22]–[24], no extra orientation synchronization or orientation estimation algorithms are needed for the control implementa- tion. Compared with [16], [17], extra degrees of freedom in terms of orientation control are not required. (ii) Compared with the angle rigidity-based bearing-only con- trol laws in [29], [30], we have used elevation angle constraints in our design. Hence, we do not have any scale ambiguity in the final formation shape. We can obtain a rigid formation in the conventional sense, i.e., the final formation behaves like a rigid body. In contrast, the final formation shape obtained in [29], [30] has translational, rotational, and scale invariance, and due to the scale invariance, the final formation does not behave like a rigid body. (iii) Compared with [35], [37], where only elevation angle constraints were used, we use signed-area or signed-volume constraints to specify the desired formation shape. Hence, flip, flex, and ordering ambiguity can be eliminated. Also, [29], [30], [34], [35], [37] established only local stability results, whereas we established almost global stability. Hence, agents can converge to the desired shape from arbitrary initial positions. Thus, almost global results are proved for bearing- only formation control in agents’ local frame, as a novel contribution of our paper. (iv) Unlike [33], [41] where inter-agent displacement mea- surement is required, here only local bearing information is required. (v) In [42] where a localization approach was adopted to establish global stability, where agents’ desired positions were estimated from angle constraints, and then a position based controller was proposed. Hence, in [42] the controller required agents’ absolute position measurements for its’ implementa- tion, whereas our controller can be implemented directly using vision sensors. (vi) Although the motivation for using a signed-area comes from [40], the control law in [40] is different from the proposed law here. Hence, the structure of the hessian of the potential function is different here than in [40]. Also, [40] has used inter-agent displacements in the control law, whereas our control law uses bearing-only measurements. Also, we have proved stability for the 3-D case as well, whereas [40] was applicable for only planar graphs, i.e., 2-D agents. Some preliminary results on sign-elevation angle rigidity were presented in [43]. However, only local stability was reported therein. In this article, we establish almost global stability results, consider non-holonomic agents in the control design, and present a detailed discussion on sign-elevation angle rigidity along with various proofs that were not given in [43]. Notations: ||.||is used for 2-norm of a vector, |.|represents modulus operation. For x= [x1x2x3]⊤∈R3,[x]×:=  0−x3x2 x3 0−x1 −x2x1 0 ∈R3×3.Symbols ˆi,ˆj,ˆkrepresent the orthogonal unit vectors along the x, y, z -axes of the co- ordinate system, respectively, and ×denotes the cross product. ⟨·,·⟩is the inner product operation. Diag {a1, . . . , a n} ∈Rn×n 1 1′ 2 2′ 3 3′ 4 4′ d12 d14 d13 d23 d34 α12 α13 hc hc hc hc(a) Elevation angles in 2-D 1 1′ 1′′ 2 2′ 2′′ 3 4 4′ 4′′ α32 α34 α31 (b) Elevation angles in 3-D Fig. 1. (a) Elevation angle measurements in 2-D, (b) and in 3-D. 1 1′ 2 2′ 3 3′ 4′ 4 2 2′ 4 4′ 1 1′ 3 3′ Flip ambiguity Flex ambiguity 1 1′ 2 3 3′ 4 4′ 5 5′ 2′ 1 1′ 3 3′ 2 2′ 4 4′ 5 5′ Reflection ambiguity 1 1′ 2 2′ 3 3′ 4 4′ 3 3′ 2 2′ 4 4′ 1 1′ (a) (b) (c) α21 α41 α21 α41 α25 α14 α25 α14 Fig. 2. Formation ambiguities in elevation angle rigid configuration. (a) shows flip ambiguity where elevation angle constraints remain the same when edge (1,2)and(1,4)are flipped over edge (2,4). (b) shows flex ambiguity, and (c) shows reflection ambiguity where there is an ambiguity in the ordering of the agents in configuration. is a diagonal matrix, where a1, . . . , a nare the diagonal ele- ments. J=0 1 −1 0 , and J1,J2,J3∈R3×3correspond to rotations of π/2along the principal axes in R3. II. PRELIMINARIES A. Graph theory This subsection presents some notions from graph theory [44]. We model the interaction among agents by an undirected graph, G(V,E),where V={1, . . . , n }is the node-set, and the set of edges is denoted by E ⊆ V ×V .The neighbor set of the i-th agent is denoted by Ni, with nirepresenting the number of neighbors. The positions of all the agents in Rdare combined to form a realization , denoted by p:= [p⊤ 1, . . . , p⊤ n]⊤∈Rnd. B. Agent dynamics We consider the following systems for control design. 1) Single integrators: Single integrator agents are given by ˙pi=uii∈ V, (1) where ui∈Rd, d∈ {2,3}is the control input. 2) Non-holonomic agents: These are modeled as ˙pi=hi¯vi,˙hi=wi×hii∈ V, (2) where hi∈Rdis the heading direction of agent i,¯vi∈Ris the speed along the heading direction , wi∈Rdis the angular velocity. The control inputs are ¯viandwi. 1 1′ 2 2′ 3 3′ 4 4′ S124 S342 hc hc hc hc(a) signed-area 1 2 3 4 z12 z13 z14 z13×z14 (b) signed-volume Fig. 3. (a) S124 =h−2 c||z12||||z14||sin(θ1 24),and S342 = h−2 c||z34||||z32||sin(θ2 43), (b)V1234 =r−3 cz⊤ 12(z13×z14). C. Elevation angle rigidity These are defined in [35]. The elevation angle is obtained by attaching a rod of height hcwith each agent for the planar case, i.e., for 2-D, and for 3-D, each agent is attached with a rod of height rc. Let zij:=pj−pi, dij:=||zij||and bij:=zij ||zij||=zij dijis the relative displacement, distance, and the bearing vector between agents iandj, respectively. For the 2-D case, the final position of the rod attached with agent iisp′ i:= [p⊤ i0]⊤+ [0 0 hc]⊤.Now, the elevation angle measured by agent itowards agent jis given by ∠j′ij=αij:=arccos (¯b⊤ ijb′ ij) = arccot (dij hc)∈(0,π 2), where ¯bij:= [b⊤ ij0]⊤.For3-D case the elevation angle is defined as ∠j′ij′′=αij:=arccos (b⊤ ij′bij′′) = 2∠j′ij= 2arcsin (rc dij)∈(0,π 3).The points j′andj′′are obtained by drawing tangents from agent ito the ball attached with agent jsuch that i, j, j′, j′′lies on the same plane. Fig. 1 shows elevation α12andα13for2-D case, and α31, α32, α34for3- D case, respectively.The elevation angle function is defined asE(p) :Rdn→Rme, E(p) := [ f1, . . . , f me]⊤, where fk:=cot(αk) =cos(αij) sin(αij)=dij hc, k∈ E,|E|=me,and fk∈(0,∞)for2-D, and fk:=cosec (αk 2) = cosec (αij 2) = dij rc, k:= (i, j)∈ E,|E|=me,andfk∈(2,∞)for3-D. When elevation angle constraints define the desired formation, fk=f∗ k, k:= (i, j)∈ E=⇒E=E∗. (3) Note that a double subscript, such as fij, will be used to de- scribe neighboring nodes/agents involved. A single subscript, likefk, can also be used where kis the edge index. The derivative of the elevation angle constraints yields d fij dt=( h−1 cb⊤ ij(vj−vi) (for2−D) r−1 cb⊤ ij(vj−vi) (for3−D).(4) The elevation angle rigidity matrix is defined by: Re:=∂E ∂p∈ Rme×dn,where d∈ {2,3}is the dimension of ambient space. III. MAIN RESULTS In this section, we first present sign-elevation angle rigidity theory. Then, we design suitable control laws for single integrators and non-holonomic agents and prove stability with the proposed control laws. A. Signed-area and Signed-volume constraints In [35], elevation angle constraints are used to specify the desired formation. Specifically, the vector containing the desired elevation angle constraints was denoted as E∗, and it was set to be equal to the actual elevation angle E. The elevation angle constraints remained constant despite flip, flex, or reflection ambiguities, as described in Fig. 2. Ref. [43, Section III A, B] provides further details on these ambiguities. Motivated by [40], we incorporate signed area constraints for 2-D and signed volume constraints for 3-D to address these ambiguities. The signed area constraint is given by Sijk=h−2 cz⊤ ijJzik=constant =S∗ ijk (5) =⇒Sijk=fijfiksin(θi jk) =S∗ ijk,(i, j),(i, k)∈ E, where sin (θi jk) =−sin(π+θi jk),∀θi jk∈[0, π],where at vertex i, the angle formed by edges (i, j)and(i, k)in the direction from (i, j)to(i, k)is represented byi jk. Fig. 3(a) shows two area constraints S124:=f12f14sin(θ1 24)and S342:=f34f32sin(θ3 42), where θ1 24is the angle from z12to z14in the clockwise sense, and θ3 42is from z34toz32in the anticlockwise sense. Hence, for ambiguities like flip, flex, and reflection (ordering), the sign of the area changes. Similarly, for3-D, the signed-volume constraints are used, which are given by Vijkl=r−3 cz⊤ ij(zik×zil) =V∗ ijkl(i, j),(i, k),(i, l)∈ E, =⇒Vijkl=fijfikfilb⊤ ij(bik×bil) =V∗ ijkl. (6) Here, rcis the radius of the ball attached to each agent. Fig 3(b) illustrates a signed-volume constraint V1234 = r−3 cz⊤ 12(z13×z14) =f12f13f14b⊤ 12(b13×b14),and the direc- tions of different vectors involved in it are shown. We may observe that for ambiguous configurations, the magnitude of the area or volume remains the same, but its sign (positive or negative) will change. Hence, the nomenclature signed- area or signed-volume is used. The multiplication of h−2 cin (3), and r−3 cin (6) will help in avoiding the knowledge of hcandrcby the agents in the control law. To specify the desired formation, a set of elevation angle constraints is used along with an additional set of signed-area or signed-volume constraints that are appropriately selected for 2-D and 3-D, respectively. All the signed-area constraints are stacked to obtain S:= [S1, . . . , S ms]⊤∈Rms, and all the signed-volume constraints are stacked in V= [V1, . . . , V mv]⊤∈Rmv.The subscripts for the area/volume constraints may be expressed as a single variable or as three/four variables (when the nodes involved are specified). Definition 1. (Framework) Define an area index set for 2-D as A={(i, j, k )∈ V3|Sijk∈S}to characterize the subtended angle θi jk∈[0, π]or equivalently the area formed by the edges (i, j)and(i, k). Similarly, in 3-D the volume index set is defined by A={(i, j, k, l )∈ V4|Vijkl∈V}. Now, a framework associated with Gandpis denoted as F(G, p,A), where prepresents a realization. Remark 1. The ordering of the edges matters while defining the angle set, as it is generated by the edges to characterize the signed-area and signed-volume. Hence, (i, j, k )∈ A⇏(j, k, i )∈ A, for 2-D. Similarly, (i, j, k, l )∈ A⇏(j, k, l, i )∈ A, for 3-D. B. Sign-elevation angle rigidity in 2-D Now, we develop the notion of sign-elevation angle rigidity in2-D to characterize geometric shapes of a graph embedded in a plane from elevation angle and signed-area constraints. Definition 2. (Equivalence of formations) Two frameworks, F1(G, p,A)andF2(G, q,A), over the same graph G(V,E) and set A, are equivalent in 2-D if αij(pi, pj) =αij(qi, qj)∀(i, j)∈ E, Sijk(pi, pj, pk) =Sijk(qi, qj, qk)∀(i, j, k )∈ A. Definition 3. (Congruence of formations) Two frameworks F1(G, p,A)andF2(G, q,A), over the same graph G(V,E) and set A, are congruent in 2-D if αij(pi, pj) =αij(qi, qj)i, j∈ V, Sijk(pi, pj, pk) =Sijk(qi, qj, qk)i, j, k∈ V. Definition 4. (Sign-elevation angle rigidity) A framework F(G, p,A)is sign-elevation angle rigid if there exists an ϵ > 0such that, all equivalent frameworks F′(G, q,A)in 2-D satisfying ||p−q||< ϵare congruent to it. Definition 5. (Global sign-elevation angle rigidity) A frame- work,F(G, p,A), is globally sign-elevation angle rigid if any other framework F′(G, q,A)that is equivalent to F(G, p,A), is congruent to it. Next, we introduce the notion of infinitesimal sign-elevation angle rigidity. For this, we first define the following sign- elevation angle rigidity function: Fs e:X →R(me+ms), where X ⊂R2n: Fs e(p) := [ f1. . . , f me, S1, . . . , S ms]⊤, (7) where E= [ f1, . . . , f me]⊤∈Rme, and S= [S1, . . . , S ms]⊤∈Rmsare the vectors containing the elevation angle constraints, and signed-area constraints, respectively. The number of elevation angle constraints and signed area constraints is meandms, respectively. Now, we define the infinitesimal sign-elevation angle rigid motions of a frame- work, which are motions of the framework Fon a differen- tiable path, ϱ(t), such that the sign-elevation angle rigidity function, i.e., Fs e, remains constant, i.e., ˙Fs e≡0.We say that infinitesimal motions of the entire framework ( ˙p) are trivial if they correspond to only translations and rotations of the desired formation shape. Equivalently, the trivial motions of the framework are given by pi(t) =R(t)pi(t0) +T(t), i∈ V, t >0, where R(t)∈R2×2andT(t)∈R2represent rotation and translation motions in R2. The infinitesimal sign-elevation angle rigidity is introduced to characterize all the trivial motions of the framework. Remark 2. From Definition 4, it is clear that a framework, F(G, p,A), is sign-elevation angle rigid if there exists a neighborhood, Up,around psuch that (Fs e)−1(Fs e(p))∩Up= (Fs n(n−1) 2)−1(Fs n(n−1) 2(p))∩ Up, where Fs n(n−1) 2is the sign- elevation angle rigidity function for a complete graph. Definition 6. (Infinitesimal sign-elevation angle rigidity) We say a framework is infinitesimal sign-elevation angle rigid if all of its infinitesimal motions are trivial. Next, we introduce the sign-elevation angle rigidity matrix, Rs e, which is defined as the Jacobian of the elevation angle- sign rigidity function, i.e., Rs e:=∂Fs e ∂p= ∂E ∂p ∂S ∂p = Re Rs ∈R(me+ms)×2n.(8) Taking the derivative of signed area constraints in (5), we get dSijk dt=h−2 cz⊤ ijJ(vk−vi) +h−2 cz⊤ ikJ⊤(vj−vi) = 0 ⇒h−1 c||zij|| hcb⊤ ijJ(vk−vi) +||zik|| hcb⊤ ikJ⊤(vj−vi) = 0 ⇒h−1 cfijb⊤ ijJ(vk−vi) +h−1 cfikb⊤ ikJ⊤(vj−vi) = 0 ,(9) where vi, vj, vkare infinitesimal motions (i.e. velocities) of agents i, j, k , respectively. We combine (4) and (9) to get ˙Fs e=∂Fs e(p) ∂p˙p= ∂E ∂p ∂S ∂p ˙p= 0. (10) The following Lemma and the subsequent theorem provide a way of checking the infinitesimal rigidity of a framework. Lemma 1. For the sign-elevation angle rigidity matrix, we have Lp:= span{1n⊗I2,(In⊗J)p} ⊆ Null(Rs e), and Rank (Rs e)≤2n−3. Proof. From [35, Lemma 1] it is known that span {1n⊗ I2,(In⊗J)p} ⊆Null(Re). AsRs e=Re Rs ,we need to show that span {1n⊗I2,(In⊗J)p} ⊆Null(Rs). Define the new edge set as E′:=E ∪ E s, where Es={(i, j),(i, k)|(i, j, k )∈ A} . Let,mt=|E′|.With the new edge set the induced graph is denoted by G′= (V,E′).We have Rs=∂S ∂p=∂S ∂z′∂z′ ∂p= ∂S ∂z′¯H′,where z′= [z⊤ 1, . . . , z⊤ mt]⊤, zk∈ E′,and ¯H′= (H′⊗I2),withH′being the incidence matrix of graph G′. Now, span {1n⊗I2} ⊆ Null(¯H′⊗I2),implies span {1n⊗ I2} ⊆ Rs e.We consider the l-th element in S, i.e., Sl= h−2 cz⊤ pJzq,where l∈ {1, . . . , m s},andp, q∈ {1, . . . , m t}. Now,∂Sl ∂p(In⊗J)p=∂Sl ∂z′∂z′ ∂p(In⊗J)p=∂Sl ∂z′¯H′(In⊗ J)p=h−2 c∂z⊤ pJzq ∂z′¯H′(In⊗J)p.Also, ¯H′(In⊗J)p= (H′⊗I3)(In⊗J)p= (H′⊗J) = ( Imt⊗J)(H′⊗ I2)p= (Imt⊗J)z′= [(Jz1)⊤, . . . , (Jzmt)⊤]⊤.Hence, ∂z⊤ pJzq ∂z′¯H′(In⊗J)p=∂z⊤ pJzq ∂z′[(Jz1)⊤, . . . , (Jzmt)⊤]⊤= ∂z⊤ pJzq ∂zpJzp+∂z⊤ pJzq ∂zqJzq= (zp)⊤zq−((zq)⊤zp) = 0. Thus, for all l∈ {1, . . . , m s},we have∂Sl ∂p(In⊗J)p= Rs(In⊗J)p= 0,implying span {(In⊗J)p} ⊆Null(Rs(p)). Hence, span {1n⊗I2,(In⊗J)p} ⊆Null(Rs e).Theorem 1. In 2-D, a framework (G, p,A)is infinitesimal sign-elevation angle rigid if and only if rank( Rs e)= 2n−3. Proof. From Lemma 1 it follows that rank of Rs eis less than or equal to 2n−3, and span {1n⊗I2,(In⊗J)p} ⊆Null(Rs e). Observe that 1n⊗I2corresponds to rigid body translation in 2-D (i.e., translation in the direction of ˆiandˆjin a plane), and (In⊗J)pcorresponds to rigid body rotation in 2-D, which are the trivial infinitesimal rigid body motions. Hence, we have rank(Rs e) = 2 n−3if and only if these trivial motions satisfy eqn. (10). Thus, by Definition 6, the framework (G, p,A)is infinitesimal sign-elevation angle rigid. Theorem 2. A framework F(G, p,A)is sign-elevation angle rigid if it is infinitesimal sign-elevation angle rigid. Proof. We know from Definition 6 that if the framework F(G, p,A)is infinitesimal signed-elevation angle rigid, all of the continuous infinitesimal motions of p(t)of the framework are trivial. In other words, these motions only correspond to rigid body translations and rotations. Now, let us con- sider another framework F′(G, p′,A)which is equivalent to F(G, p,A), and||p−p′||< ϵ. In this case, the continuous motions that take ptop′while maintaining Fs e= (Fs e)∗are also trivial motions, meaning they only involve translation and rotation of the framework. Therefore, the relation in Definition 3 is satisfied, and we can conclude that F′(G, p′,A) is congruent to F(G, p,A). Definition 7. [40, Definition 2.2](Strong Distance rigidity) A distance rigid framework F(G, p)in2-D is called strongly distance rigid if we have (pi−pj)and(pi−pk)are independent for all (i, j),(i, k)∈ E. Proposition 1. If a framework F(G, p)is strongly distance rigid in R2, then the signed framework F1(G1, p,A), where G1=G(i.e., sensing topology for distances in distance rigid graph Gis the same as the elevation angle constraints in G1), is infinitesimally sign-elevation angle rigid in R2. Proof. Consider the signed framework F1(G1, p,A)with its elevation angle constraints replaced by distance constraints. As we have F(G, p)strongly distance rigid in 2-D, using [10, Proposition 1] F(G, p)is infinitesimal distance rigid. Now, a framework is elevation angle rigid if and only if it is distance rigid [35, Proposition 2]. Hence, the framework with elevation angle constraints F(G1, p)is elevation angle rigid. So, there exists a non zero (2n−3)×(2n−3)minor of Re[35]. Now, from the definition of sign-elevation angle rigidity matrix Rs e=Re Rs there exists a 2n−3×2n−3non zero minor for Rs e. Then, the proof follows from Theorem 3. 1) sign-elevation-Henneberg construction in 2-D: (Con- structing globally sign-elevation angle rigid frameworks:) To generate rigid frameworks with global sign elevation angle, we adopt a method named sign-elevation-Henneberg con- struction. The steps of this method are as follows. First, we assume the initial framework has |V|= 3,|E|= 3 , andA= 1,i.e., we have 3elevation angle and 1signed- area constraints. For a given globally sign elevation angle 1 2 3 S123 1 2 3 4 S123 S413 1 2 3 4 5 S123 S413 S543 1 2 3 4 5 6 S123 S413 S543 S652Fig. 4. Example of sign-elevation-Henneberg construction method for gener- ating globally sign-elevation angle rigid frameworks in 2-D. rigid framework F(G, p,A), an agent vis added to it such that it has two more elevation angle constraints and one signed-area constraint. Now the new framework ¯F(¯G,¯p,¯A) consists of triangular frameworks and ¯G= (¯V,¯E),¯V= V ∪ { v},¯E=E ∪ { (i, v),(j, v)},¯A=A ∪ { (¯i,¯j,¯k)},and p= [p⊤p⊤ v]⊤∈R2(n+1),where i, j∈ V,¯i,¯j,¯k∈ {i, j, v} and¯i̸=¯j̸=¯k,alsoi, j, v should not be co-linear. Fig. 4 demonstrates the steps used in this method. Proposition 2. An infinitesimally sign-elevation angle rigid framework F(G, p,A)is globally sign elevation angle rigid if it results from a sign-elevation-Henneberg construction. Proof. As the framework, F(G, p,A), is infinitesimally sign- elevation angle rigid, from Theorem 1 we have rank (Rs e) = 2n−3. Hence, from Lemma 1, Rs eis of maximal row rank, and using [8, Proposition 2] there exists a neighbourhood Upsuch that(Fs e)−1(Fs e(p))∩Upis a3-dimensional smooth manifold which corresponds to 2-D rigid transformations. Further, from Definition 4, it follows that F(G, p,A)is sign-elevation angle rigid. In addition, using signed Henneberg construction, we can uniquely determine the position of each agent, as the position of each agent in a sequence is uniquely determined by2elevation angle and 1area constraints. Hence, from Definition 5, we conclude that F(G, p,A)is globally sign- elevation angle rigid. C. Sign-elevation angle rigidity in 3-D In this subsection, sign-elevation angle rigidity is developed in3-D. We have for x, y∈R3, x×y= [x]×y. Definition 8. (Equivalence of formations) Two frameworks in 3-D, F1(G, p,A)andF2(G, q,A), over the same graph G(V,E), and angle set A, are equivalent if αij(pi, pj) =αij(qi, qj)∀(i, j)∈ E, Vijkl(pi, pj, pk, pl) =Vijkl(qi, qj, qk, ql)∀(i, j, k, l )∈ A. Definition 9. (Congruence of formations) Two frameworks in 3-D, F1(G, p,A)andF2(G, q,A), over the same graph G(V,E), and angle set A, are congruent if αij(pi, pj) =αij(qi, qj)i, j∈ V Vijkl(pi, pj, pk, pl) =Vijkl(qi, qj, qk, ql)i, j, k, l ∈ V. Definition 10. (Sign-elevation angle rigidity) A framework F(G, p,A)is sign-elevation angle rigid if there exists an ϵ >0such that, all frameworks F′(G, q,A)in3-D satisfying ||p−q||< ϵare congruent to it. Letmerepresent the number of elevation angle constraints andmvrepresent the number of signed volume constraints. Weintroduce the following elevation angle-sign rigidity function Fv e:X →R(me+mv), where X ⊂R3n. Fv e(p) := [ f1. . . , f me, V1, . . . , V mv]⊤. (11) Combining (3) and (6), we get ˙Fv e=∂Fv e(p) ∂p˙p= ∂E ∂p ∂V ∂p ˙p= 0, where E= [f1, . . . , f me]⊤∈RmeandV= [V1, . . . , V mv]⊤∈ Rmvare the vectors that comprise the elevation angle con- straints and signed-volume constraints, respectively. The sign- elevation angle rigidity matrix, denoted as Rv e, is defined as the Jacobian of the elevation angle-sign rigidity function, i.e., Rv e:=∂Fv e ∂p= ∂E ∂p ∂V ∂p = Re Rv ∈R(me+mv)×3n. Taking the derivative of the volume constraint (6) gives, dVijkl dt=r−3 cz⊤ ij(zik×(vl−vi))−r−3 cz⊤ ij(zil×(vk−vi)) + (zik×zil)⊤(vj−vi) =r−1 c(fijfikb⊤ ij([bik]×(vl−vi)−fijfilb⊤ ij([(bil]×(vk−vi)) −fikfilb⊤ il([bik]×(vj−vi)), (12) where vi, vj, vk, vlare velocities of agents i, j, k, l . Lemma 2. For the sign-elevation angle rigidity matrix ( Rv e), we have Lp:=span{1n⊗I3,(In⊗J1)p,(In⊗J2)p,(In⊗ J3)p} ⊆Null(Rv e), and Rank (Rv e)≤3n−6. Proof. The proof is similar to that of Lemma 1. Theorem 3. In 3-D, a framework (G, p,A)is infinitesimal sign-elevation angle rigid if and only if rank( Rs e)= 3n−6. Proof. From Lemma 2 it is clear that rank of Rv eis less than or equal to 3n−6, and span {1n⊗I3,(In⊗J1)p,(In⊗J2)p,(In⊗ J3)p} ⊆Null(Re v).Observe that 1n⊗I3corresponds to rigid body translations in 3-D (i.e. translations along ˆi,ˆj, and ˆk), and(In⊗Jβ)p, β= 1,2,3correspond to rigid body rotations, which are the trivial infinitesimal rigid body motions in 3-D. Hence, clearly rank (Rv e) = 3 n−6if and only if these trivial motions satisfy eqn. (10). Thus, according to Definition 6, framework (G, p,A)is infinitesimal sign-elevation angle rigid. Theorem 4. A framework F(G, p,A)is sign-elevation angle rigid if it is infinitesimal sign-elevation angle rigid. Proof. The proof is similar to that of Theorem 2. 1) Sign-elevation-Henneberg construction in 3-D: (Con- structing globally sign-elevation angle rigid frameworks:) The following steps are followed to generate global sign elevation angle rigid frameworks in 3-D. For convenience, we adopt a similar nomenclature for it as for the 2-D case and denote this sign-elevation-Henneberg construction in 3-D. Here, new frameworks are generated by adding tetrahedral frameworks with one new signed-volume constraint. First, we assume the initial framework has |V|= 4,|E|= 6, andA= 1.For a given globally sign elevation angle rigid framework F(G, p,A), an 1 2 3 4 V1234 1 2 4 V1234 V5234 5 3 1 2 3 4 5 6 V1234 V5234 V6345Fig. 5. Example of sign-elevation-Henneberg construction method for gener- ating globally sign-elevation angle rigid frameworks in 3-D. agent vis added to it such that it has three more elevation angle constraints and one signed-volume constraint. Now, the new framework ¯F(¯G,¯p,¯A)consists of tetrahedral frameworks and ¯G= (¯V,¯E),¯V=V ∪ { v},¯E=E ∪ { (i, v),(j, v),(k, v)},¯A= A ∪ { (¯i,¯j,¯k,¯l)},andp= [p⊤p⊤ v]⊤∈R3(n+1),where i, j, k∈ V,with (i, j),(j, k),(k, i)∈ E,¯i,¯j,¯k,¯l∈ {i, j, k, v } and¯i̸=¯j̸=¯k̸=¯l.Fig. 5 demonstrates these steps. It is important to note that here we cannot add the new edges to any arbitrary nodes i, j, k∈ V,but need to add the edges such that(i, j),(j, k),(k, i)∈ E is satisfied, e.g., in Fig. 5 node 6 is added with edges to 3,4,5; however, the addition of agent 6to nodes 1,3,5is not allowed as (1,5)̸∈ E.The reason behind such a construction will be clarified later. Proposition 3. An infinitesimally sign-elevation angle rigid framework F(G, p,A)in3-D is globally signed elevation angle rigid if it has sign-elevation-Henneberg construction. Proof. Proof is similar to that of Proposition 2. Definition 11. (Strong Distance rigidity in 3-D) A distance rigid framework F(G, p)in3-D is called strongly distance rigid if we have (pi−pj),(pi−pk), and (pi−pl)independent for all (i, j),(i, k),(i, l)∈ E. Proposition 4. If a framework, F(G, p), is strongly distance rigid in R3, then the signed framework F1(G1, p,A)where G1=G,(i.e., sensing topology for distances in distance rigid graph Gis same as elevation angle constraints in G1), is infinitesimally sign-elevation angle rigid in R3. Proof. Consider the signed framework F1(G1, p,A)and re- place its elevation angle constraints with distance constraints to obtain G. As we have F(G, p)strongly distance rigid in 3- D, using similar induction-based arguments in [10, Proposition 1], it can be proved that F(G, p)is infinitesimal distance rigid. Now, a framework is elevation angle rigid if and only if it is distance rigid [35, Proposition 2]. Hence, the framework with elevation angle constraints F(G1, p)is elevation angle rigid. So, there exists a non zero (3n−6)×(3n−6)minor of Re [35]. Now, from the definition of sign-elevation angle rigidity matrix Rv e=Re Rv there exists a 3n−6×3n−6non zero minor for Rv e. Then, the proof follows from Theorem 3. D. Bearing-only formation control law using sign-elevation angle rigidity for nagents: This section presents bearing-only control laws to achieve the desired formation specified by elevation angle and signed- area constraints considering nsingle integrators.1) Control law for 2-D: We have the following elevation- angle constraints and the desired elevation-angle constraints specifying the formation in 2-D: E(p) = [. . . , f ij, . . .]⊤∈Rme E∗= [. . . , f∗ ij, . . .]⊤∈Rme, where fij=cot(αij)andf∗ ij=cot∗(αij)for(i, j)∈ E.Also, we have the following signed-area and desired area constraints: ¯S=κ[. . . , S ijk, . . .]⊤∈Rms ¯S∗=κ[. . . , S∗ ijk, . . .]⊤∈Rms, where (i, j, k )∈ A , and κ > 0is a positive constant. The formation error is defined by e(p) = [E⊤(p)¯S⊤(p)]⊤−[E∗⊤(¯S∗)⊤]⊤. (13) Letu= [u⊤ 1, . . . , u⊤ n]⊤∈R2ncontain the control inputs of all agents. The following gradient-based control law is proposed for single integrators: ˙p=u=−hc∇Φ =−hc¯Rs e(p)⊤e(p), (14) where Φ =1 2e⊤e, and ¯Rs e= [R⊤ eκR⊤ s]⊤,which we call the modified sign-elevation angle rigidity matrix. Although κis multiplied with the part corresponding to the signed area of the sign-elevation angle matrix, observe that Null (¯Rs e) =Null(Rs e) and Null (¯Rv e) =Null(Rv e), hence the rank of Rs eis same as that of rank of ¯Rs e. From (14), the control law for i-th agent is: ˙pi=ui=−hc(¯Rs e)⊤(i,:)e(p), (15) where (¯Rs e)⊤(i,:)is the i-th row of (¯Rs e)⊤. Using (4) and (9), thei-th column of ¯Rs eis given by: ¯Rs e(:, i) = [0 , . . . ,−h−1 cb⊤ ij, . . . ,−h−1 cb⊤ ik,0, . . . , −κ(h−1 cfijb⊤ ijJ+h−1 cfikb⊤ ikJ⊤), . . . , 0]⊤,(16) where (i, j),(i, k)∈ E , and (i, j, k )∈ A.Here, fij:= cot(αij) =cos(αij) sin(αij)=¯b⊤ ijbij′ q 1−(b⊤ ijbij′)2.Hence, each term in(¯Rs e(:, i))⊤only requires bearing measurements, i.e., bij andbij′. Also, the signed area constraints are Sijk= h−2 cz⊤ ijJzik=fijfikb⊤ ijJbik,and hence each term in Fs ecan also be calculated using bearing-only measurements. There- fore, the control law uirequires bearing-only information. Also, from (16) and (14), it is clear that the control law (14) is free from hc. 2) Control law for 3-D case: Similarly, we have the fol- lowing signed volume and desired volume constraints for 3-D: ¯V=κ[. . . , V ijkl, . . .]⊤∈Rmv ¯V∗=κ[. . . , V∗ ijkl, . . .]⊤∈Rmv, where (i, j, k, l )∈ A, and κ >0is a positive constant. The formation error is defined by e(p) = [E(p)⊤¯V⊤]⊤−[E∗⊤(¯V∗)⊤]⊤(17) Letu= [u⊤ 1, . . . , u⊤ n]⊤contain the control inputs of all agents. The following gradient-based control law is proposed for the single integrators: ˙p=u=−rc∇Φ =−rc(¯Rv e(p))⊤e(p), (18) where Φ =1 2e⊤e, and ¯Rs e= [R⊤ eκR⊤ v]⊤,which we call the modified sign-elevation angle rigidity matrices. Al- though κis multiplied with the part corresponding to signed volume of the sign-elevation angle matrix, observe that Null(¯Rv e) =Null(Rv e), hence the rank of Rv eis same as that of the of ¯Rv e. From (18), the control law for i-th agent is: ˙pi=ui=−rc(¯Rv e)⊤(i,:)(Fv e−(Fv e)∗)for3D,(19) where (¯Rv e(:, i))⊤are the i-th row (¯Rv e)⊤. Using (4) and (12), thei-th column of ¯Rv eis given by: ¯Rv e(:, i) = [0 , . . . ,−r−1 cb⊤ ij,0, . . . ,−r−1 cb⊤ ik,0. . . ,−r−1 cb⊤ il, 0, . . . , 0,−κr−1 cfijfikb⊤ ij[bik]×+κr−1 cfijfilb⊤ ij[bil]×+ κr−1 cfikfilb⊤ il[bik]×, . . . , 0]⊤, (20) where (i, j),(i, k),(i, l)∈ E,(i, j, k, l )∈ A.Here, fij:= cosec (αij 2) =1 sin(αij 2)=1 √ (1−cos(αij))/2=1q (1−b⊤ ij′bij′′)/2. Also, signed-volume constraints can be represented as Vijkl:= r−3 cz⊤ ij(zik×zil) =fijfikfilb⊤ ij(bik×bil).Hence, the control lawuiin3-D also requires bearing-only information. Also, additional information on rcis not required, which is clear from (18) and (20). Remark 3. Observe that for the control design, we have modified the signed-area and signed-volume constraints using a positive gain κ. The error function and the potential function are defined with these constraints. As discussed above, as the rank of the modified sign-elevation angle matrix ( ¯Rs eor¯Rv e) is the same as Rs eorRv e, introducing κdoes not affect the rigidity properties of the frameworks. However, κwill play a crucial role in proving almost global stability with the gradient control laws (14) and (18). Theorem 5. Suppose Qdenotes a set of pcorresponding to formations of single integrators (described by (1)) that are congruent to a desired infinitesimal sign-elevation angle rigid formation. If the initial position p(0)∈ Bp∗,where Bp∗is a neighbourhood of p∗, and p∗∈ Q,using the control law (14) for2-D or (14) for3-D,p(0)locally exponentially converges to a desired formation pdes∈ Q. Proof. This can be proved using the arguments of center manifold theorem [11], [39], [45], and hence is omitted. From Theorem 5, it is clear that under the control law (14) and (18), the desired configuration is a stable equilibrium point. However, it only establishes local convergence results. In the subsequent subsections, we establish almost global convergence for the frameworks generated by a sign-elevation- Henneberg construction with large κ, implying that for suffi- ciently large κ, we can not have other stable equilibriums. Remark 4. Observe that we have adopted a sign-elevation- Henneberg construction to construct a global sign-elevation angle rigid framework. However, we cannot conclude directly i j k Sijk(a)F1(G1, β1,A1) i j k Sijk(b)Fl(Gl, βl,Al), l∈ {2,3, . . . , ψ } i j k l Vijkl(c)Fw(Gw, βw,Aw), w∈ {1,2,3, . . . , ψ } Fig. 6. Examples of sub-frameworks Fl(Gl, βl,Al)for2-D case for (a) l= 1, (b)l∈ {1,2, . . . , ψ }, (c) Examples of sub-frameworks Fw(Gw, βw,Aw) for3-D case for w∈ {1,2, . . . , ψ } from this fact that under the control law (14) and (18), we can achieve the desired configuration almost globally. Now, we state the following Lemma, which will be helpful in our subsequent analysis. Lemma 3. Letβj(∈R6or∈R12depending on 2-D or 3- D case, i.e., d= 2 or3) be a regular point1forFs ej(or Fv ej)j∈ {1,2, . . . , ψ }, then there exists a derivative map Dζi of a map ζi(βi)|Gjforβiis in the neighborhood of βj, i.e., βi∈ Bβj, close to Fj(Gj, βj,Aj)such that Dζi(vi)|Gj=( vi∈Rd|Vi|ifi=j vi∈Tζi(βi)Fs ej−1Fs ej(βj)otherwise , where TxMis the tangent space to a manifold Mat point x. Proof. The proof follows from [40, Lemma A1]. Remark 5. From Lemma 1 (or 2 ) and Theorem 1 (or 3), we have βj, j∈ {1,2, . . . , ψ }is a regular point of Fs ej(orFv ej)implies the sub-frameworks are sign-elevation angle rigid. Hence, from Lemma 3 and [8, Proposition 2] Fs ej−1Fs ej(βj) (orFv ej−1Fv ej(βj)), j∈ {1,2, . . . , ψ }are the motions of Fj(Gj, βj,Aj)inSE(2)(orSE(3)). E. Almost global stabilization for 2-D case In this subsection, we establish almost global convergence result using the gradient control law (14) for 2-D case for suf- ficiently large κwith a sign-elevation-Henneberg construction. As we have a sign-elevation-Henneberg construction, we split the framework F(G, p,A)intoψsub frameworks. Here, F(G, p,A)is divided into Fl(Gl, βl,Al), l∈ {1,2, . . . , ψ }. Each partition, Fl(Gl, βl,Al), consists of three agents, and hence for the sub-graph Gl= (Vl,El)we have Vl={i, j, k} andβl= [p⊤ ip⊤ jp⊤ k]⊤∈R6, i, j, k ∈ Vlis a realization asso- ciated with the sub-framework. Hence, p= [β⊤ 1. . . β⊤ ψ]⊤∈ R2n. Further, we have E=Sψ l=1ElandEl∩ El′=∅,when l, l′∈ {1,2, . . . , ψ }, l̸=l′,and|El|= 3,forl= 1, and |El|= 2 otherwise. Also, A=Sψ l=1AlwithAl∩ Al′=∅ when l, l′∈ {1,2, . . . , ψ }andl̸=l′with|Al|= 1.It follows thatV=Sψ l=1Vland|Vl|= 3,and if βlandβl′are adjacent forl, l′∈ {1,2, . . . , ψ }, l̸=l′, then Vl∩ Vl′̸=∅,else Vl∩ Vl′=∅.This is illustrated in Fig. 6. 1[8] A realization βjis a regular point of a function Fs ejif rank∂Fs ej ∂βj =max rank∂Fs ej ∂βj βj∈Rd|Vj| The following functions correspond to different sub- frameworks. The vectors contain the actual and the desired elevation angle constraints El(βl) = [. . . f ij. . .]⊤∈R|El|(21) E∗ l(β∗ l) = [. . . f∗ ij. . .]⊤∈R|El|, (22) where l∈ {1,2, . . . , ψ }and(i, j)∈ El. The actual and desired signed-area constraints for a sub-framework are Sl(βl) =κSijk∈R, S∗ l=κS∗ ijk∈R, (23) where l∈ {1,2, . . . , ψ }and(i, j, k )∈ A l.Hence, the formation error associated with each framework is el(βl) = [E⊤ l(βl)Sl(βl)]⊤−[E∗ l⊤S∗ l]⊤:= [e⊤ ele⊤ sl]⊤, (24) where eel=E⊤ l−E∗ l⊤, esl=Sl−S∗ l.The potential function for the sub-framework is given by Φl=1 2||el(βl)||2, l∈ {1,2, . . . , ψ }and the corresponding Hessian matrix is Hβl=∂2Φl ∂β2 l∈R6×6, (25) while the sign-elevation angle rigidity matrix is ¯Rs el=∂Fl ∂βl∈R(|El|+1)×6, l∈ {1,2, . . . , ψ }, (26) with Fl:= [ E⊤ l(βl)Sl(βl)]⊤∈R|El|+1being the sign-elevation angle rigidity function associated with sub- framework Fl(Gl, βl,Al). Lemma 4. Consider a system of nsingle integrators with a framework F(G, p,A),generated by a sign-elevation- Henneberg construction. If p=p∗is a configuration cor- responding to a stable equilibrium under the control law (14), then for a sufficiently large κ >0, the following statements hold (i) At equilibrium, the signed-area has the same sign as that of the desired area, i.e., sign of the equilibrium area must match the sign of the desired area. (ii)At equilibrium, the framework F(G, p∗,A),must be in- finitesimally sign-elevation angle rigid. Proof. The Hessian of Φis also the negative Jacobian of the matrix in (14): Hp=∂2Φ ∂p2∈R2n×2n. (27) The Hessian associated with each sub-framework in given by: Hβl= (Rs el)⊤Rs el+M2⊗J+¯M1. (28) Letp†be an equilibrium realization with an incorrect sign as that of the desired signed- area. This means that there exists a signed-area Sijksuch that at equilibrium, Sijk≤0, while S∗ ijk>0,(i, j, k )∈ A.So we may observe that for a sub- framework Fl(Gl, βl,Al), l∈ {1,2, . . . , ψ }we have Sijk− S∗ ijk̸= 0,(i, j, k )∈ A latp†. Now, we will show that when this happens, a negative eigenvalue for Hpatp†results for sufficiently large κ, implying p=p†is unstable for sufficiently large κ.We will use a contradiction-based argument to prove this. First, assume that for sufficiently large κ,p=p†is a stable equilibrium. As we have ¯Rs el= [R⊤ elκR⊤ sl]⊤, where Rel= ∂Fs el ∂βlandRsl=∂Sijk ∂βlfor(i, j, k )∈ A l, l∈ {1,2, . . . , ψ }, using (44) we may write Hβl=R⊤ elRel+κ2R⊤ slRsl+M2⊗J+¯M1. Next, we define a vector as y:= [y⊤ 1ϵy⊤ 20]⊤∈R6,withy1∈ R2being unit vector perpendicular to J⊤zjkwhere (i, j, k )∈ Alandl∈ {1,2, . . . , ψ },y2∈R2is a non-zero vector with ϵ >0being a small positive number. Hence, using the fact that ∂Sijk ∂βl= [z⊤ jkJ z⊤ ikJ z⊤ ijJ],(i, j, k )∈ Al, l∈ {1,2, . . . , ψ }, we have y⊤(κ2R⊤ slRsl+M2⊗J)y (29) =κ2ϵ2y⊤ 2J⊤zikz⊤ ikJy2| {z } =y⊤R⊤slRsly+κϵy⊤ 1Jy2esijk|{z} =y⊤(M2⊗J)y, (30) where esijk=κ(Sijk−S∗ ijk). Observe that (from Appendix (A)) the two terms R⊤ elReland¯M1are independent of κ. Thus for sufficiently large κ y⊤Hβly=κ2(ϵy⊤ 1Jy2(Sijk−S∗ ijk) +O(ϵ2) +O(¯ϵ2)), where ¯ϵ=1 κ. Hence, we may choose yso that y⊤ 1Jy2(Sijk− S∗ ijk)is negative and y⊤Hβlybecomes negative definite at p†.Thus, for sufficiently large κ, when an incorrect sign for (i, j, k )∈ A loccurs, a negative eigenvalue exists for Hβl. Also, we have Φ =Pψ l=1Φl, which implies Hp=Pψ l=1ˆHβl, where ˆH=∂2Φl ∂p2∈R2n×2n,and hence ˆHβlis constructed by adding zero rows and columns to Hβl. This implies that there exists ˆy∈R2n, which can be constructed by adding zeros to ysuch that ˆy⊤ˆHβlˆy <0, and ˆHβlhas at least one negative eigenvalue at p†. Also, note that the sub-frameworks Fl(Gl, βl,Al), l∈ {1,2, . . . , ψ }has at least one agent common with the adja- cent sub-framework F′ l(G′ l, β′ l,A′ l), l′∈ {1,2, . . . , ψ }\{l}. Hence, Hpis not a block diagonal matrix consisting of Hβl, l∈ {1,2, . . . , ψ }as the diagonal blocks. Hence, we cannot directly conclude that Hphas a negative eigenvalue atp†,from the fact that ˆHβlhas a negative eigenvalue atp†.However, now we invoke Lemma 3 to reach this conclusion. From the above analysis, it is clear that with sufficiently large κat equilibrium each of the sub-frameworks Fl(Gl, βl,Al), l∈ {1,2, . . . , ψ }, which consists of three agents, are infinitesimally sign-elevation angle rigid config- uration. Hence, from Lemma 1 and Theorem 1, ¯Rs elis of maximum row rank, which implies βl, l∈ {1,2, . . . , ψ }is a regular point for Fs el. Also, we have span( Lp)⊂Null(ˆHβl) [10, Lemma 2], [46] . Hence, it follows from Lemma 3 that there exists a vector Dζl(ˆy)|Glsuch that (Dζl(ˆy)|Gl′)⊤ˆHβl(p†)Dζl(ˆy)|Gl′=( θ <0ifl=l′ 0otherwise .(31) Now, using the fact that H=Pψ l=1ˆHβl, we have (Dζl(ˆy)|Gl′)⊤Hp(p†)Dζl(ˆy)|Gl′=θ <0. Hence, we conclude for sufficiently large κ > 0, we can not have an incorrect signed area at the equilibrium, i.e., we will have Sijk>0 ifS∗ ijk>0,and hence part (i)is proved. Further, note that this means we cannot have Sijk= 0at the equilibrium, which implies at equilibrium, the framework F(G, p†,A)is strongly distance rigid. Using Proposition 1, we have the equilibrium framework as sign-elevation angle rigid, proving part (ii). We next present the following Theorem pertaining to the almost global convergence to a desired formation shape. Theorem 6. Consider a sign-elevation angle rigid framework constructed by sign-elevation-Henneberg construction in 2-D. Using control law (14) for single integrators as in (1), there exists sufficiently large κ, such that the formation tracking error ealmost globally converges to zero. Proof. We rewrite the control law in (14) as ˙p=−(¯Rs e)⊤(p)e(p) =−R⊤ e(p)ee(p)−κR⊤ s(p)es(p)(32) =−R⊤ e(p)ee(p)−κ2R⊤ s(p)es(p), (33) where ee(p) := E(p)−E∗, es(p) := ¯S(p)−¯S∗,¯es(p) := S(p)−S∗.Now consider the Lyapunov candidate L:=1 2e⊤e, which is positive definite and radially unbounded. Taking its derivative along (14) yields ˙L=e⊤˙e=−e⊤¯Rs e¯Rs e⊤e=−||¯Rs e⊤e||2≤0. (34) From, (34), we have ˙L ≤0. Hence, all the signals are bounded. Also, ˙L ≡0, implies, ¯Rs e⊤e≡0.Hence, the largest invariant set is given by the solution to ¯Rs e⊤e≡0.Let, there exists non- zeroR⊤ e(p)ee(p),andR⊤ s(p)es(p)at equilibrium p=p∗and ¯Rs e⊤e= 0,which implies from (32) that −R⊤ e(p∗)ee(p∗)− κ2R⊤ s(p∗)¯es(p∗) = 0 .However, as eis bounded and hence ReandRsare also bounded, we may choose sufficiently large κ > 0such that −R⊤ e(p∗)ee(p∗)−κ2R⊤ s(p∗)¯es(p∗)̸= 0, which is a contradiction. Hence, at equilibrium, we have R⊤ e(p∗)ee(p∗) =R⊤ s(p∗)¯es(p∗) = 0 . (35) From Lemma 4, we already know that for sufficiently large κ we have at equilibrium, the framework (G, p)being strongly distance rigid, implying rank (Re(p∗)) = 2 n−3. Hence, using the fact that we have a sign-elevation-Henneberg construction and|E|= 2n−3,Re(p∗)is of full row rank. Therefore, we have R⊤ e(p∗)ee(p∗) = 0 , implying ee(p∗) = 0 .Also, from Lemma 4 for sufficiently large κ, note that the signed area at equilibrium has the same sign as that of the desired area. Now, asee= 0,implies the desired elevation angles, and hence, the desired distances are achieved, and the desired areas are also achieved, i.e., es(p∗) = 0 .Hence, the largest invariant set is given by e= 0,hence using Lassal’s invariance principle [47] we have egoes to zero asymptotically. Remark 6. From Lemma 4 and Theorem 6, it is clear that there exists sufficiently large κ > 0such that for a stable equilibrium p∗of the system (14) we will obtain the desired formation shape. However, system (14) may also have unstable equilibriums which is a set of measure zero and nowhere dense. Hence, the result of stability is almost global. Due to the nonlinear nature of gradient system (14), finding these unstable equilibrium points is intractable.Remark 7. Note that although the motivation for using a signed-area comes from [40], the control law in [40] is different than that of the proposed law here, and hence the structure of hessian in [40] is different here than in [40]. Also, [40] has used inter-agent displacements in the control law, whereas our control law uses bearing-only measurements. Also, in the next section, we will be proving stability for the 3- D case, whereas [40] was applicable for only planar graphs, i.e.,2-D agents. F . Almost global stabilization for 3-D In this subsection, we establish almost global convergence in3-D, using the gradient control law in (18) for sufficiently large values of κwith a sign-elevation-Henneberg construction for3-D case. As we have a sign-elevation-Henneberg construction, we may split the signed framework, F(G, p,A), into ψsub frame- works. Here, F(G, p,A)is divided into Fw(Gw, βw,Aw), w∈ {1,2, . . . , ψ }.Each partition, Fw(Gw, βw,Aw), consists of four agents, and hence for the sub-graph Gw= (Vw,Ew) we have Vw={i, j, k, l }andβw= [p⊤ ip⊤ jp⊤ kp⊤ l]⊤∈ R12, i, j, k, l ∈ Vwis a realization associated with the sub- framework. Hence, p= [β⊤ 1. . . β⊤ ψ]⊤∈R3n. Also, we haveE=Sψ w=1EwandEw∩ Ew′={(i, j),(j, k),(k, i)}, when w, w′∈ {1,2, . . . , ψ }, w̸=w′,and|Ew|= 6. Also, A=Sψ w=1AwwithAw∩ Aw′=∅when w, w′∈ {1,2, . . . , ψ }andw̸=w′with|Aw|= 1.It follows that V=Sψ w=1Vwand|Vw|= 4,and if βwandβw′are adjacent forw, w′∈ {1,2, . . . , ψ }, w̸=w′, then Vw∩ Vw′̸=∅, otherwise Vw∩ Vw′=∅.Examples of sub-frameworks are given in Fig. 6 (c). Observe that, unlike in 2-D, where there is no edges common for the sub-frameworks, for the 3-D case, there are always 3edges that are common for any two adjacent sub-frameworks. The following functions correspond to different sub- frameworks. The vectors contain the actual and the desired elevation angle constraints Ew(βw) = [. . . f ij. . .]⊤∈R|Ew|(36) E∗ w(β∗ w) = [. . . f∗ ij. . .]⊤∈R|Ew|, (37) where w∈ {1,2, . . . , ψ }and(i, j)∈ Ew. The actual and desired signed-volume constraints for a sub-framework are Vw(βw) =κVijkl∈R, V∗ w=κV∗ ijkl∈R, (38) where w∈ {1,2, . . . , ψ }and(i, j, k, l )∈ A w.The sign- elevation angle rigidity matrix is ¯Rv ew=∂Fw ∂βw∈R(|Ew|+1)×12, w∈ {1,2, . . . , ψ }, (39) where Fw:= [E⊤ w(βw)Vl(βw)]⊤∈R|Ew|+1is the sign- elevation angle rigidity function associated with a sub- framework, Fw(Gw, βw,Aw).Hence, the formation error as- sociated with each framework is ew(βw) = [E⊤ w(βw)Vw(βw)]⊤−[E∗ w⊤V∗ w]⊤. (40) Now, we modify the edge errors associated with the sub- frameworks such that the new error is divided by the number of sub-frameworks connected with that edge, i.e., ˆeeij:= 1√ϱijeeij, where eeij:=fij−f∗ ij, and ϱijis the number of sub-frameworks associated with edge (i, j).According to this definition, from Fig. 5, we have ˆee34:=1√ 3ee34,i.e., ϱ34= 3, and ˆee12:=ee12,i.e.,ϱ12= 1.Thus, we have the modified formation error associated with each sub-framework w∈ {1,2, . . . , ψ }given by ˆew(βw) =Kew(βw)∈R|Ew|+1, (41) where K=Diag{. . . ,1√ϱij. . . ,1} ∈R(|Ew|+1)×(|Ew|+1), (i, j)∈ Ew.The potential function for the sub-framework is Φw=1 2||ˆew(βw)||2, w∈ {1,2, . . . , ψ }and its Hessian is Hβw=∂2Φw ∂β2w∈R12×12. (42) Using this modified definition of edge errors, we have Φ =Pψ w=1Φw. Lemma 5. Consider a system of nsingle integrators and a framework F(G, p,A),generated by a sign-elevation- Henneberg construction in 3-D. If p=p∗is a configuration corresponding to a stable equilibrium under the control law (18), then with sufficiently large κ >0, the following hold: (i) At equilibrium, the signed volume has the same sign as the desired volume, i.e., the sign of the equilibrium volume must match with the sign of the desired volume. (ii)At equilibrium, the frame work F(G, p∗,A),must be in- finitesimally sign-elevation angle rigid. Proof. Let us calculate the Hessian of Φ, which is also the negative Jacobian matrix of (18): Hp=∂2Φ ∂p2∈R3n×3n. (43) The hessian associated with each sub-framework is given by: Hβl= (Rv ew)⊤K4Rv ew+¯N2+¯N1. (44) Letp†be a realization with an incorrect sign than that of the desired signed volume. This means there exists a signed-volume Vijklsuch that at equilibrium, Vijkl≤, while V∗ ijkl>0,(i, j, k, l )∈ A.Observe that for a sub-framework Fw(Gw, βw,Aw), l∈ {1,2, . . . , ψ }we will have Vijkl− V∗ ijkl̸= 0,(i, j, k, l )∈ A watp†. Now, we prove that when this happens, we have a negative eigenvalue for Hpatp†for sufficiently large κ, making p=p†an unstable equilibrium. We use a contradiction-based argument. Assume that for sufficiently large κ,p=p†is a stable equilibrium. As ¯Rv ew= [R⊤ ewκR⊤ vw]⊤, where Rew=∂Few ∂βwandRvw=∂Vijkl ∂βwfor (i, j, k, l )∈ Aw, w∈ {1,2, . . . , ψ },using (44) we may write Hβw=R⊤ ew¯KRew+κ2R⊤ vwRvw+¯N1+¯N2, where ¯K:=Diag{. . . ,1 ϱij2, . . .} ∈R|Ew|×|Ew|,(i, j)∈ Ew. Define y:= [y⊤ 1ϵy⊤ 20 0]⊤∈R12,with y1∈R3being a unit vector perpendicular to (zkl×zkj)where (i, j, k, l )∈ Awandw∈ {1,2, . . . , ψ },y2∈R3is a non-zero vector withϵ >0being a small positive number. Using the fact that∂Vijk ∂βl=r−3 c[(zkl×zkj)⊤(zik×zil)⊤(zil×zij)⊤(zij× zik)⊤],(i, j, k, l )∈ Aw, w∈ {1,2, . . . , ψ }, we have y⊤(κ2R⊤ vwRvw+¯N2)y (45) =κ2ϵ2y⊤ 2(zik×zil)(zik×zil)⊤y2| {z } =y⊤R⊤vwRvwy+κϵy⊤ 1∆y2evijkl|{z } =y⊤(¯N2)y,(46) where evijkl=κ(Vijkl−V∗ ijkl), and ∆ :=−[zkl]×+ [zik]×− [zil]×=−2[zkl]×∈R3×3(because [zik]×−[zil]×= −[zkl]×). Observe (from Appendix (B)) that these two terms R⊤ ew¯KRewand¯N1are independent of κ, hence y⊤Hβwy=κ2(ϵy⊤ 1∆y2(Vijkl−V∗ ijkl) +O(ϵ2) +O(¯ϵ2)), where ¯ϵ=1 κ. We may thus can always choose yso that y⊤ 1∆y2(Vijkl−V∗ ijkl)is negative for sufficiently large κ. This is because due to our assumption (Vijkl−V∗ ijkl)>0and y⊤ 1∆y2=⟨y2,∆⊤y1⟩, we have to choose y2such that it makes an obtuse angle with the vector ∆⊤y1. Also, note that ∆⊤y1= 2[zkl]×y1= 2(zkl×y1), and hence, ∆⊤y1is always a non- zero vector. Hence, y⊤Hβlyis negative definite at p†.So, for sufficiently large κ, when an incorrect sign for (i, j, k, l )∈ Awoccurs, Hβwhas a negative eigenvalue. We have Φ =Pψ w=1Φw, which implies Hp=Pψ w=1ˆHβw,where ˆH= ∂2Φw ∂p2∈R3n×3n.Hence ˆHβwis constructed by padding zero rows and columns to Hβw. This implies that there exists ˆy∈ R3n, which can be constructed by adding zeros to ysuch that ˆy⊤ˆHβwˆy <0, and ˆHβwhas at least one negative eigenvalue at p†. Now, using Proposition 4 and Lemma 3 and using similar arguments as in the proof of Lemma 4, the proof follows. Theorem 7. Consider a sign-elevation angle rigid framework constructed by a sign-elevation-Henneberg construction in 3- D. Using control law (18) for single integrator agents (1), for sufficiently large κ, the formation tracking error ealmost globally converges to zero. Proof. Proof follows similar lines as that of Theorem 6. G. Control design for non-holonomic agents In this subsection, we propose control laws for non- holonomic agents, as described in equation (2), and demon- strate their stability using our proposed laws. The following control laws are proposed ¯vi=h⊤ iui wi=hi×uii∈ Vi, (47) where ui∈Rdis the control input proposed for single integrators, i.e., the expression for uican be found in (14) for planar graphs ( 2-D) and (18) in 3-D. Theorem 8. Consider a sign-elevation angle rigid framework constructed by a sign-elevation-Henneberg construction in 2- D or 3-D. Using control law (47) for non-holonomic agents in(2), with sufficiently large κ, the formation tracking error, e, almost globally converges to zero. Proof. Using the proposed control law in (47), the overall dynamics of nagents becomes ˙p=HH⊤u,˙h=H⊥(H⊥)⊤u, (48) where H:= diag(h1, . . . , h n)∈Rnd×nd, H⊥:= diag(h⊥ 1, . . . , h⊥ n)∈Rnd×nd.Now, let the Lyapunov candi- date be L=1 2e⊤e,which is positive definite and radially unbounded. The time derivative of L,along the system trajec- tory (48) is given by ˙L=e⊤˙e=( e⊤¯Rs e˙p=−e⊤¯Rs eHH⊤(¯Rs e)⊤efor2−D e⊤¯Rv e˙p=−e⊤¯Rv eHH⊤(¯Rv e)⊤efor3−D =−||H⊤u||2≤0. (49) Form (49), it is clear that ˙Lis negative semi-definite, and hence, all the signals are bounded. Now, when ˙L ≡0,either (i)ui≡0,∀i∈ V,or(ii)we have hi⊥ui, ui̸≡0∀i∈ V. Clearly, case (ii)cannot occur because if (ii)occurs, wiwould become non-zero and hence hiwill change and the condition in(ii)will not be satisfied. Hence, we have case (i)as the only possibility, and from Theorem 6 (and Theorem 7), it implies eapproaches zero asymptotically. IV. ILLUSTRATIVE EXAMPLES A. Single integrators in 2-D Here, we first consider 6single integrator agents (1) in 2-D. The desired configuration is generated by a sign-elevation- Henneberg construction in 2-D, with the edge set is E= {(1,2),(2,3),(3,4),(4,1),(1,3),(5,3),(5,4),(6,1),(6,2)} and the area index set is A = {(4,3,1),(2,1,3),(5,3,4),(6,1,2)}. The desired formation shape is a hexagonal shape, with the desired elevation angle constraints f∗ 12=f∗ 34=f∗ 53=f∗ 54=f∗ 61=f∗ 62= 10, f∗ 23= f∗ 41= 20, f∗ 13= 10√ 5,and the desired signed-area constraints S∗ 431=S∗ 213= 200 , S∗ 534=S∗ 612= 86.6.The height of the rod is set as hc= 2.The initial position of the agents are set to be p1(0) = [1 .5−9.1]⊤, p2(0) = [5 .4−10.7]⊤, p3(0) = [1.1,−14.7]⊤, p4(0) = [ −7.6 15.5]⊤, p5(0) = [8 .9 1.8]⊤and p6(0) = [15 .7 10.6]⊤. First, we set κ= 0, i.e., no weights are given to the area constraints, and therefore, the control law becomes identical to the only elevation angle-based control laws proposed in [35]. Fig. 7 (a) shows the trajectories of the agents with κ= 0.The initial position of the agents are marked with circles and the final positions are marked with squares. The final formation shape is not the desired shape, since using only elevation angle constraints, the agents 5 and6have flipped from their desired positions. Hence, the final shape becomes ambiguous using the control law in [35]. Now, we set κ= 0.02and Fig. 7 (b) shows the trajectories of the agents in this setup. Although agent 6has gone to its desired position, still agent 5has gone to the flipped position giving rise to an ambiguous shape. This is due to very low value of κ.Next, we increase κto0.1, and Fig. 7 (c) shows the trajectories of the agents. We may observe that now the desired formation shape is achieved. B. Single integrators in 3-D Next, we consider four agents in 3-D, with the edge set E= {(1,2),(2,3),(3,1),(4,1),(4,2),(4,3),(5,1),(5,2),(5,3)} and the volume index set A={(4,1,2,3),(5,1,2,3)}.The desired formation shape is a hexahedron with triangularfaces, while the desired elevation angle constraints are given byf∗ ij= 10 ,∀(i, j)∈ E,and the desired signed-volume constraints are V∗ 4123=−707.1068, V∗ 5123=−707.1068.The radius of the ball is rc= 0.1.The initial positions of the agents arep1(0) = [1 −0.1 0.1,]⊤, p2(0) = [ −1.2 0.7 0.9]⊤, p3(0) = [1.1 1.5 1]⊤, p4(0) = [0 .8 0.2 2]⊤, p5(0) = [0 .6 0.5 1]⊤.First we set κ= 0and then the control law becomes identical with [35], and Fig. 8 (a) plots the trajectories of the agents in this case. We may observe that agents 4and5are coinciding to one position giving rise to a final shape that is not congruent to the desired formation shape. Now, we set κ= 0.02and the plot for this case is given in Fig. 8 (b), which indicates that the desired shape is achieved with our proposed law. C. Non-holonomic agents For non-holonomic agents, we consider the same de- sired formation shape as for the case of single integra- tors in 2-D, with the same desired elevation angle and signed-area constraints. The initial position of the agents arep1(0) = [ −1.7−4.1]⊤, p2(0) = [1 .2 3.3]⊤, p3(0) = [2.1 5.1]⊤, p4(0) = [ −6.4−7.4]⊤, p5(0) = [11 .6 9.7,]⊤ andp6(0) = [ −3.1 7.1]⊤, and the initial headings of the agents are h1(0) = [1 0]⊤, h2(0) = [ −1 0]⊤, h3(0) = [1√ 21√ 2]⊤, h4(0) = [0 1]⊤, h5(0) = [0 −1]⊤andh6(0) = [1√ 3√ 2√ 3]⊤. The gain κ= 0.1, and Fig. 7 (d) shows the trajectories for the non-holonomic agents with our proposed control law (47), and the desired formation shape is achieved. V. CONCLUSIONS In this work, we have proposed sign-elevation angle rigidity for uniquely characterizing formation shapes using elevation angle and signed-area constraints or signed-volume constraints for2-D and 3-D graphs, respectively. Then, bearing-only control laws are derived using only agents’ local bearing information. Compared with the elevation angle rigidity-based control laws in [35], [37], which can only ensure local stability results, we have proved almost global results for the frame- works generated by sign-elevation-Henneberg construction, in which the overall framework consists of triangulated sub- frameworks for 2-D and tetrahedral sub-frameworks for 3-D. A gain parameter κwas introduced for signed-area and signed- volume constraints, and it was proved that there always exists sufficiently large κsuch that agents can achieve the desired formation shape almost globally. Determining the exact lower bound for κis a future topic for research. The results presented here can be extended for obtaining bearing-only formation maneuvers using similar maneuvering methods as in [35]. APPENDIX (A)Expression of Hessian for 2-D sub-frameworks: The Hessian associated with the sub-framework for 2-D is Hβl=∂2Φl ∂β2 l=∂ ∂βl(¯Rs el⊤el)∈R6×6, (50) where el(βl) = [ E⊤ l(βl)Sl(β1)]⊤−[E∗ l⊤S∗ l]⊤,Φl= 1 2||el(βl)||2,andβl= [p⊤ ip⊤ jp⊤ k]⊤.Consider l= 1,hence -20 -10 0 10 20 30 X-axis-25-20-15-10-505101520 Y-axis Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6(a) Single integrators with κ= 0 (using [35]) -30 -20 -10 0 10 20 30 40X-axis-25-20-15-10-505101520 Y-axis Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6(b) Single integrators with κ= 0.02 -40 -30 -20 -10 0 10 20 30 40 50 X-axis-25-20-15-10-505101520 Y-axis Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 (c) Single integrators with κ= 0.1 -40 -30 -20 -10 0 10 20 30 40 X-axis-20-15-10-505101520 Y-axisAgent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 (d) Non-holonomic agents with κ= 0.1 Fig. 7. (a) Trajectories of single integrator agents with κ= 0, i.e., control law used is same as in [35], (b) Trajectories of single integrator agents with κ= 0.02, (c) Trajectories of single integrator agents with κ= 0.1, (d) Trajectories of non-holonomic agents with κ= 0.1. (a) Trajectories of agents for κ= 0 (using [35]) (b) Trajectories of agents for κ= 0.02 Fig. 8. Trajectories of single integrators in 3-D for (a) κ= 0, (b)κ= 0.02. Fs e= [fijfikfjkκSijk]⊤(Fig. 6(a)). The corresponding sign-elevation angle rigidity matrix is calculated as: ¯Rs el=∂Fs el ∂βl=h−1 c −b⊤ ij b⊤ ij 0 −b⊤ ik 0 b⊤ ik 0 −b⊤ jk b⊤ jk κh−1 cX⊤−κh−1 cz⊤ ikJ⊤−κh−1 cz⊤ ijJ , where X:= (J⊤zij+Jzik), Fs el= [E⊤ l(βl)Sl(βl)]⊤∈ R|El|+1.Hence, (¯Rs el)⊤elis calculated as (¯Rs el)⊤el=h−1 c −eeijbij−eeikbik−h−1 cκXe sijk eeijbij−eejkbjk+h−1 cκ(Jzik)esijk eeikbik−eejkbjk−h−1 cκ(J⊤zij)esijk , where eeij:=fij−f∗ ijfor(i, j)∈ Elandesijk:=κ(Sijk− S∗ ijk),for(i, j, k )∈ Al.Now, the Hessian calculated as Hβl=∂ ∂βl(¯Rs el⊤el)∈R6×6=¯M1+¯M2+ [Cp1Cp2Cp3], where Cp1= h−2 cbijb⊤ ij+h−2 cbikb⊤ ik+h−4 cκ2XX⊤ −h−2 cbijb⊤ ij−h−4 cκ2JzikX⊤ −h−2 cbikb⊤ ik−h−4 cκ2J⊤zijX⊤  Cp2= −h−2 cbijb⊤ ij−h−4 cκ2(J⊤zij+Jzik)(Jzik)⊤ h−2 cbijb⊤ ij+h−2 cbjkb⊤ jk+h−4 cκ2(Jzik)(Jzik)⊤ −h−2 cbjkb⊤ jk+h−4 cκ2(J⊤zij)(Jzik)⊤  Cp3= −h−2 cbikb⊤ ik−h−4 cκ2(J⊤zij+Jzik)(J⊤zij)⊤ −h−2 cbjkb⊤ jk+h−4 cκ2(Jzik)(J⊤zij)⊤ h−2 cbikb⊤ ik+h−2 cbjkb⊤ jk+h−4 cκ2(J⊤zij)(J⊤zij)⊤ ,and[Cp1Cp2Cp3] = ( ¯Rs el)⊤¯Rs elholds. Here, ¯M2=M2⊗I2, M2= 0 −h−2 cκesijkh−2 cκesijk h−2 cκesijk 0 h−2 cκesijk −h−2 cκesijk h−2 cκesijk 0 and ¯M1=h−1 c Y1 −eeijPbij ||zij||−eeikPbik ||zik|| −eeijPbij ||zij||Y2 −eejkPbjk ||zjk|| −eeikPbik ||zik||−eejkPbjk ||zjk||Y3  where Y1=eeijPbij ||zij||+eeikPbik ||zik||, Y2=eeijPbij ||zij||+eejkPbjk ||zjk||, andY3=eeikPbik ||zik||+eejkPbjk ||zjk||,withPbijis the projection matrix defined by Pbij:=I2−bijb⊤ ij. Forl∈ {2, . . . , ψ },we have Fs e= [fijfikSijk]⊤(Fig. 6(b)). Hence the sign elevation angle rigidity matrix ( Rs e) for these sub-frameworks can be obtained by deleting row 4from the expression of Rs ecalculated for l= 1.For calculating the Hessian for this case, expression for ¯M2remains the same as that of calculated for w= 1, and ¯M1becomes ¯M1=h−1 c Y1 −eeijPbij ||zij||−eeikPbik ||zik|| −eeijPbij ||zij||eeijPbij ||zij||0 −eeikPbik ||zik||0 eeikPbik ||zik|| (51) where Y1=eeijPbij ||zij||+eeikPbik ||zik||. (B)Expression of Hessian for 3-D sub-frameworks : Now, we calculate the Hessian associated with the sub- framework in the case of 3-D, which is given by Hβw=∂2Φw ∂β2w=∂ ∂βw(¯Rv ew⊤K2ew)∈R12×12, (52) where ew(βw) = [ E⊤ w(βw)Sw(βw)]⊤−[E∗ w⊤S∗ w]⊤,Φw= 1 2||ˆew(βw)||2,andβw= [p⊤ ip⊤ jp⊤ kp⊤ l]⊤.Consider w∈ {1, . . . , ψ },hence Fv e= [fijfikfilfkjfklfjlκVijkl]⊤(Fig. 6(c)). The sign-elevation angle rigidity matrix is calculated as: ¯Rv ew=∂Fv ew ∂βw=1 rc −b⊤ ij b⊤ ij 0 0 −b⊤ ik 0 b⊤ ik 0 −b⊤ il 0 0 b⊤ il 0 b⊤ kj −bkj 0 0 0 −b⊤ kl b⊤ kl 0 −b⊤ jl 0 b⊤ jl r−2 cκZ⊤ 1r−2 cκZ⊤ 2r−2 cκZ⊤ 3r−2 cκZ⊤ 4 , where Z1=zkl×zkj, Z2=zik×zil, Z3=zil×zij, Z4= zij×zik.Note that Z1=−Z2−Z3−Z4.Now, the Hessian is calculated as Hβw=∂ ∂βw(¯Rv ew⊤K2ew) =¯N1+¯N2+ (¯Rv ew)⊤K4¯Rv ew, where ¯N2=κevijkl r3cN2∈R12×12,andN2=  0 −[zkl]×[zkl]×−[zkj]×[zkj]× [zik]×−[zil]×0 [ zil]×−[zik]× [zil]×−[zij]×−[zkl]×0 [ zij]× [zij]×−[zik]×[zik]×−[zij]×0  and¯N1= r−1 c Y1 −Pbij ||zij||˜eeij−Pbik ||zik||˜eeik−Pbil ||zil||˜eeil −Pbij ||zij||˜eeij Y2 −Pbkj ||zkj||˜eekj−Pbjl ||zjl||˜eejl −Pbik ||zik||˜eeik−Pbkj ||zkj||˜eekj Y3 −Pbkl ||zkl||˜eekl −Pbil ||zil||˜eeil−Pbjl ||zjl||˜eejl−Pbkl ||zkl||˜eekl Y4 , where Y1=Pbij ||zij||˜eeij+Pbik ||zik||˜eeik+Pbil ||zil||˜eeil, Y2=Pbij ||zij||˜eeij+ Pbkj ||zkj||˜eekj+Pbjl ||zjl||˜eejl, Y3=Pbik ||zik||˜eeik+Pbkj ||zkj||˜eekj+ Pbkl ||zkl||˜eekl, Y4=Pbil ||zil||˜eeil+Pbjl ||zjl||˜eejl+Pbkl ||zkl||˜eekl,withPbij:= I3−bijb⊤ ij∈R3×3is the projection matrix, and ˜eeij:= 1 ϱijeeij,(i, j)∈ Ew.