Collaborative Target Tracking in Elliptic Coordinates: a Binocular Coordination Approach
Yuan Chang, Zhiyong Sun, Han Zhou, Xiangke Wang, Lincheng Shen, Tianjiang Hu
11 Collaborative Target Tracking in EllipticCoordinates: a Binocular Coordination Approach
Yuan Chang , Zhiyong Sun , Han Zhou , Xiangke Wang , Lincheng Shen , Tianjiang Hu ∗ Abstract —This paper concentrates on the collaborative targettracking control of a pair of tracking vehicles with formationconstraints. The proposed controller requires only distance mea-surements between tracking vehicles and the target. Its noveltylies in two aspects: 1) the elliptic coordinates are used to representan arbitrary tracking formation without singularity, which canbe deduced from inter-agent distances, and 2) the regulationof the tracking vehicle system obeys a binocular coordinationprinciple, which simplifies the design of the control law byleveraging rich physical meanings of elliptic coordinates. Thetracking system with the proposed controller is proven to beexponentially convergent when the target is stationary. When thetarget drifts with a small velocity, the desired tracking formationis achieved within a small margin proportional to the magnitudeof the target’s drift velocity. Simulation examples are provided todemonstrate the tracking performance of the proposed controller.
I. I
NTRODUCTION
This paper investigates the collaborative target trackingproblem and proposes a coordinated control method for apair of tracking vehicles so that the tracking vehicles andthe target eventually form a desired formation. The availablemeasurements are inter-vehicle distances. Both the target rep-resentation and the design of the control law are establishedin the elliptic coordinate system, which encapsulates thecoupled properties between the interacting vehicles, therebysimplifying the controller form.Cooperative target tracking (or so-called circumnaviga-tion/encirclement control) has been studied extensively in theliterature [1]–[4]. The paper [1] proposed a general frameworkthat consists of an outer loop and an inner loop. The outerloop is to design a reference velocity to match the movementof the target, and the inner loop is to control each agent sothat the formation centroid can achieve the desired referencevelocity. Recently, the paper [2] discussed collaborative targettracking using fixed-wing UAV systems with constant-speeds,in tracking a moving target with constant or time-varyingspeeds. Their works are based on the assumption that the targetposition and velocity are known, and the tracking controllerdemands relative position and heading measurements. In com-parison, [4] designed a bearing-based circumnavigation ap-proach, where the relative distance and velocity of the target isobtained by estimation. In this paper, we propose a binocular-based tracking control approach, which does not require theknowledge of estimation of target’ positions, and thus greatly ∗ Corresponding author (e-mail: [email protected]). National University of Defense Technology, Changsha 410073, China. Eindhoven University of Technology, Den Dolech 2, 5612 AZ, Eindhoven,the Netherlands. Sun Yat-sen University, Guangzhou 510725, China. relaxes the assumptions on inter-vehicle measurements fromthe above mentioned works.There are two issues that need to be addressed in cooperativetarget tracking with a binocular coordination approach: thefirst is how to represent formation constraints of the systemconstructed by binocular vehicles and the target, while thesecond is how to design a control law for the vehicles toeliminate the tracking errors. As such, several works on forma-tion control have been addressed based on different availablemeasurements [5]–[7]. One of the most popular methods isdistance-based formation control [5], where the formation isachieved by adjusting inter-agent distances. In contrast, Bishopet al. proposed an angle-based formation control [6], wherethe control law is designed with subtended angles. Later,they extended the angle-based method to mixed-angle-distanceformation [7] that involves scaling control. Our work differsfrom [5]–[7] in that the target does not actively participate inthe formation control. Moreover, the proposed controller obeysa binocular coordination principle [8], which treats binocularnodes as a joint system to design its tracking trajectory.The contributions of this paper are twofold. First, wepropose to describe an arbitrary tracking formation withoutsingularity in elliptic coordinates, which can be viewed as anatural extension of polar coordinates for a binocular system.Second, we present a tracking control law with a succinct formby making full use of physical meanings and geometries ofelliptic coordinates. We have proved that, under the proposedcontrol law, the tracking system is exponentially convergentwhen the target is stationary. Moreover, it is robust to tracka slow-drifting target in the sense that the tracking errorsconverge to a small margin proportional to the drift velocity.II. P
RELIMINARY AND PROBLEM STATEMENT
A. Binocular target tracking
We consider the task of tracking a target with a binocularsystem. The considered binocular system consists of 2 trackingvehicles whose dynamics are described by a single integrator: ˙ p i = u i , i ∈ { , } , (1)where p i ∈ R and u i ∈ R represent the position and controlinput of vehicle i , respectively. First, the following assumptionis naturally established. Assumption 1:
The initial positions of the two trackingvehicles are non-coincident, i.e. p ( t ) (cid:54) = p ( t ) .The measurements from vehicle j available to vehicle i arethe relative distance measurement d ij = (cid:107) p j − p i (cid:107) and relativebearing measurement g ij = ( p j − p i ) /d ij . The target position a r X i v : . [ ee ss . S Y ] S e p Fig. 1. Depiction of an elliptic coordinate system with c = 1 . The red curvesindicate ellipses with a constant ξ , while the blue curves indicate hyperbolaewith a constant η . p t is not directly accessible. Each vehicle i only measuresthe relative distance d it = (cid:107) p t − p i (cid:107) to the target. Besides,the tracking vehicles only know which side of the baselinethe target is located on, but do not have access to accuratebearing information. Remark 1.
We do not impose any restrictions on the targetlocation. This is different from that of [6], which assumes thatthe tracking vehicles and the target cannot be collinear.So far, the triangular formation in the tracking task canbe uniquely represented by d ij and d it . In the followingsubsection, we further electorate formation constraints withtransformations of elliptic coordinates to facilitate the designof tracking control law. B. Formation constraints in elliptic coordinates
The elliptical coordinate system is a two-dimensional or-thogonal coordinate system that has been commonly usedfor modeling and solving problems with binocular nodes,such as [9], [10]. Motivated by their works, we proposeto represent formation constraints using elliptic coordinates.Its coordinate lines are confocal ellipses and hyperbolae, asdepicted in Fig. 1. The two coordinates, ξ and η , indicate thedistance and orientation of a point relative to the binocularsystem, separately. Given the pole locations o l = ( − c, T and o r = ( c, T in the xy -plane, where c ∈ R + , the Cartesiancoordinates are related to the elliptic coordinates by (cid:40) x = c cosh( ξ ) cos( η ) y = c sinh( ξ ) sin( η ) (2)within the domain D := { ( ξ, η ) T | ξ ≥ , ≤ η ≤ π } . Tofacilitate further discussions, we define D + , D − ⊂ D , where D + = { ( ξ, η ) T | ξ ≥ , ≤ η ≤ π } , and D − = D − D + .Without loss of generality, by assigning o l ← p and o r ← p , an elliptical coordinate system can be constructed by thetwo tracking vehicles in a binocular framework, where c = d ij / . Therefore, p t = ( x, y ) T is alternatively represented by e p t = ( ξ, η ) T , where the pre-superscript e refers to ellipticcoordinates. Then, we investigate how to obtain e p t from d it .By eliminating ξ of (2) it derives sin ( η ) x − cos ( η ) y = c sin ( η ) cos ( η ) , (3)which corresponds to a family of hyperbolae with the samefocus distance c . If η = kπ/ for any k ∈ Z , the hyperbola de-generates into a straight line. Then, by properties of hyperbola η is calculated from d it as η = (cid:40) arccos d t − d t c , if e p t ∈ D + ;2 π − arccos d t − d t c , if e p t ∈ D − . (4)Similarly, by eliminating η of (2) it derives sinh ( ξ ) x + cosh ( ξ ) y = c sinh ( ξ ) cosh ( ξ ) , (5)which corresponds to a family of ellipses with the same focusdistance c . If ξ = 0 , the ellipse degenerates into a straight line.Then, by properties of hyperbola ξ is calculated from d it as ξ = arccosh( d t + d t c ) . (6)Together, (4) and (6) constitute the transform equationsfrom d it to e p t . To facilitate the stability analysis in thenext section, we also present the transform equations fromCartesian coordinates p t to elliptic coordinates e p t as follows.By defining p = sin ( η ) and q = − sinh ( ξ ) , we have p ≥ q since ≤ p ≤ , q ≤ . According to [11], we have p = − B + (cid:112) B + 4 c y c , q = − B − (cid:112) B + 4 c y c , (7)where B = x + y − c .As such, η is recovered from the definition of p , dependingon which quadrant the target is located, resulting in η = η , if x ≥ , y ≥ π − η , if x < , y ≥ π + η , if x ≤ , y < π − η , if x > , y < , (8)where η = arcsin √ p .Similarly, ξ is recovered from the definition of q with aunique form as ξ = 12 ln(1 − q + 2 (cid:112) q − q ) . (9) C. Problem statement
So far, we have established an elliptic coordinate systemwith the pole locations designated by the two tracking vehicles.In the following, we simply use p l and p r to represent thepositions of the two vehicles instead of p and p . Note that e p t and c completely characterize the shape and scale of thetriangular formation in the collaborative tracking task. As such,we give a rigorous description of the binocular target trackingproblem as follows. Problem 1 (Binocular target tracking).
Considering a pairof vehicles with agent model described by (1), for any given e p ∗ t ∈ D and c ∗ ∈ R + , design a control law such that e p t ( t ) → e p ∗ t and c ( t ) → c ∗ as t → ∞ . Fig. 2. Combine collective velocities v ξ , v η and v c to realize target trackingwith formation constraints. III. B
INOCULAR T ARGET T RACKING
A. Tracking control law
The design of the tracking law obeys the binocular co-ordination principle , which treats two vehicles as a jointsystem to control their collective behaviors. The control inputsare delivered to both tracking vehicles simultaneously. Fora binocular system, it can be arbitrarily configured throughthree affine behaviors: translation, rotation, and scaling, asdepicted in Fig. 2. To achieve target tracking with formationconstraints, the translation speed v ξ , rotation speed v η , andscaling speed v c are designed to eliminate the tracking andformation errors in the directions/dimensions of ξ , η and c ,respectively. Therefore, the control component is given by v c = κ c ( c ∗ − c ) v η = κ η ( η ∗ − η ) v ξ = κ ξ ( ξ ∗ − ξ ) , (10)where κ c , κ ξ and κ η are positive constants.Denote R ( · ) ∈ R × as the rotation matrix of angle ( · ) . Theoverall control inputs for the two vehicles are given by (cid:40) u l = A l ( p r − p l ) u r = A r ( p r − p l ) , (11)where (cid:40) A l = c ( v ξ R ( ϕ ) + v η R ( π/
2) + v c R ( π )) A r = c ( v ξ R ( ϕ ) + v η R ( − π/
2) + v c R (0)) . (12) Remark 2.
Note that (10) is given in the form of a proportionalcontroller. Such simplicity is due to the delicate definition of ξ and η , where some new terms that encapsulate the couplingproperties between the interacting vehicles in the Euclideanspace, B = x + y − c and (cid:112) B + 4 c y , have emerged.As a result, the tracking errors are decoupled in the new spacewith clear physical meanings. B. Stability analysis for tracking a stationary target
Define the tracking error as e = c − ( c ∗ ) ) η − η ∗ ξ − ξ ∗ = e e e , (13) where c = ( p l − p r ) T ( p l − p r ) / . The error dynamics is derivedas ˙ e = ( p l − p r ) T ( ˙ p l − ˙ p r )˙ η ˙ ξ = ˙ e ˙ e ˙ e . (14)Now we are ready to present the first main result. Theorem 1.
Consider a pair of tracking vehicles described by(1). Then, for any given e p ∗ t ∈ D and c ∗ ∈ R + , by applyingthe proposed controller (10) to (12), e = 0 is an exponentiallystable equilibrium point of the tracking error dynamics (14). Proof.
Consider the Lyapunov function candidate V = 12 e T e = 12 ( e + e + e ) . (15)The time-derivative of V is ˙ V = ˙ e T e = e ˙ e + e ˙ e + e ˙ e . (16)Next, we will discuss the dynamics of e , e and e ,separately, to show the convergence of the tracking errors. • Step 1: analysis on ˙ e Substituting the obtained expressions in (14) with the pro-posed controller (10) to (12) yields a quadratic form ˙ e = ( p l − p r ) T ( A r − A l )( p l − p r ) , (17)where A r − A l = 1 c ( v c I + (cid:20) −
11 0 (cid:21) ) . (18)Note that the second item in (18) is a skew symmetricmatrix, whose quadratic form is 0. Utilizing c = (cid:107) p l − p r (cid:107) / and v c = κ c ( c ∗ − c ) , (17) is further simplified to ˙ e = 4 cκ c ( c ∗ − c ) = − cκ c c + c ∗ e := − F e , (19)where F is positive definite for c > , so that e ( t ) → as t → ∞ , which infers c ( t ) → c ∗ monotonously. Therefore, c ( t ) ≥ min { c ( t ) , c ∗ } > for all t ≥ t . Moreover, F is afunction of e according to (14), and the error dynamics givenby (19) constitute an autonomous system. Note that F is aclass K function with repect to e . Thus, we have F > k ,where k = 2 min κ c (cid:113) e ( t )2 + ( c ∗ ) (cid:113) e ( t )2 + ( c ∗ ) + c ∗ , κ c (20)is a positive constant. • Step 2: analysis on ˙ e By partial differentiation of η and combining with (14), thefollowing expression is obtained: ˙ e = ∂η∂x ˙ x + ∂η∂y ˙ y = sgn( xy ) G (cid:18) ( BG − x ˙ x + ( B + 2 c G − y ˙ y (cid:19) , (21)where G = (cid:112) B + 4 c y , G = c √ p − p . The signfunction sgn( xy ) is used because η has different definitionsin different quadrants. The relative target motion dynamics by the proposed con-troller is given by (cid:20) ˙ x ˙ y (cid:21) = sgn( xy ) v ξ G ϕ (cid:20) y cos ηx sin η (cid:21) + v η c (cid:20) − yx (cid:21) , (22)where G ϕ = (cid:112) x sin η + y cos η > . The sign function sgn(xy) is used to indicate the direction of vehicle movementswhen the target is in different quadrants.Substituting (22) in (21), combining with sin η = ( G − B ) / (2 c ) and v η = − κ η e yields ˙ e = − cκ η | xy | G G e := − F e , (23)We prove that there exists a positive constant < µ < c such that F ≥ k , where k = κ η c (cid:113) µ µ + c is a positiveconstant (see proof A in Appendix). • Step 3: analysis on ˙ e Similarly, by partial differentiation of ξ and combining with(14), we have ˙ e = ∂ξ∂x ˙ x + ∂ξ∂y ˙ y = G (cid:18) (1 + BG ) x ˙ x + (1 + B + 2 c G ) y ˙ y (cid:19) , (24)where G = c √ q − q .Substituting (22) again and combining with v η = − κ η e , v ξ = − κ ξ e , we have ˙ e = − | xy | κ ξ G G ϕ e − cxyκ η G G e . (25)As discussed above, the second term is exponentially con-vergent. Therefore, we only investigate the convergence of thefirst term. Define F = 2 | xy | κ ξ G G ϕ . We prove that there existsa positive ν > c such that F ≥ k , where k = κ ξ c √ ν + c isa positive constant (see proof B in Appendix). • Step 4:
Combining Steps 1-3, we have ˙ V < − k V , where k := 2 min { k , k , k } is a positive constant. Then, by ([12], Theorem 4.10), one concludes that the origin e = 0 isexponentially stable, and the proof is complete. Remark 3.
In the above discussions, we exclude the casewhere − c ≤ x ≤ c, y = 0 . However, it is easy to check thatsuch a case would not result in singularity. For example, if y = 0 , y ∗ > , then it is equivalent to ξ = 0 , ξ ∗ > , leadingto a positive v ξ , which will drive the vehicles away from x -axisso that y (cid:54) = 0 immediately. C. Tracking a moving target under slow drift
So far we have established the exponential stability for thetracking system for a stationary target, where ˙ p ∗ t = 0 . Now weconsider the tracking control with a moving target p ∗ t ( t ) , byassuming that there exists ε ∈ [0 , ∞ ) such that for all t ∈ R , (cid:107) ˙ p ∗ t ( t ) (cid:107) ≤ ε. (26)Herein, we regard the drift-free error system (14) as thenominal system and rewrite it as ˙ e = f ( t, e ) . The Lyapunovfunction of the norminal system V ( t, e ) : [0 , ∞ ) × D → R + is defined by (15), where D = { e ∈ R | (cid:107) e (cid:107) < r } . Then, the moving target’s slow drift is equivalently regarded as an addi-tional translational movement of the binocular system due tothe motion relativity, which acts as a drift term g ( t, e ) additiveto the nominal system, resulting in a perturbed tracking errorsystem ˙ e = f ( t, e ) + g ( t, e ) . (27)The following result is herein presented. Theorem 2.
Consider the perturbed system described by (27)with a slow-moving target constainted by (26). Then, for all (cid:107) e (0) (cid:107) < r , there exists a positive constant K such that lim t →∞ (cid:107) e ( t ) (cid:107) ≤ Kε.
Proof.
The target dynamics in the local xy -frame with respectto a stationary binoculr system is described by (cid:40) ˙ x t = v cos θ t ˙ y t = v sin θ t , (28)where v = (cid:107) ˙ p ∗ t ( t ) (cid:107) , θ t ∈ [0 , π ] denotes the direction of ˙ p ∗ t ( t ) .Specifically, the drift term can be equivalently describedby a vector given by g ( t, e ) = [0 , g η ( t, e ) , g ξ ( t, e )] T . Then,by following a procedure similar to Subsection III-B, wewill derive the expressions and upper bounds of g η ( t, e ) and g ξ ( t, e ) separately to investigate the upper bound of | g ( t, e ) | .Substituting (28) in (21) yields g η ( t, e ) = G (cid:18) ( BG − x ˙ x t + ( B + 2 c G − y ˙ y t (cid:19) . (29)Applying Cauchy’s inequality and combining with (26), wehave | g η ( t, e ) | ≤ vG (cid:115) ( BG − x + ( B + 2 c G − y ≤ εc √ h , (30)where h = G /c . Similarly, substituting (28) in (24) yields g ξ ( t, e ) = G (cid:18) (1 + BG ) x ˙ x t + (1 + B + 2 c G ) y ˙ y t (cid:19) . (31)By applying Cauchy’s inequality again and combining with(26), we have | g ξ ( t, e ) | ≤ vG (cid:115) (1 + BG ) x + (1 + B + 2 c G ) y ≤ εc √ h . (32)Note that (30) and (32) share the same expression, which hasthe same monotonicity property with F . Then, suppose that ( x − c ) + y ≤ δ , where < δ < c , we have | g η ( t, e ) | ≤ ¯ Kε and | g ξ ( t, e ) | ≤ ¯ Kε , where ¯ K = 1 / √ δ − cδ .Now we have proven that there exists < θ < and ¯ K ∈ R + such that (cid:107) g ( t, e ) (cid:107) ≤ ¯ Kε < k θr (33)Then, the result immediately follows by applying ( [12],Lemma 9.2), which describes the bounded stability of per-turbed systems, and we have K = ¯ K/ ( k θ ) . Fig. 3. Coordinated tracking of a stationary target under different initial positions: (a) p t ∈ D + ; (b) p t ∈ D − ; (c) p t is located on the midpoint of twovehicles; (d) p t is on the x-axis and x > c . The upper figures show the tracking vehicles’ trajectories and the final formations in Euclidean space. Figures inthe below show the tracking errors of individual channels, which all converge to 0 exponentially fast.Fig. 4. Coordinated tracking of a circular moving target with e p ∗ t =(1 . , π/ T and c ∗ = 20 . After t , the tracking error converges to a smallinterval proportional to the target speed. Remark 4.
Theorem 2 shows that the proposed controller isable to track a moving target and the tracking errors convergeto a small region proportional to the target speed, whichindicates that small perturbation will not result in large steady-state deviations from the origin. In fact, we only require thetarget speed to not exceed a sphere without explicit restrictionson the moving trajectory (26), and a specific moving target canbe regarded as a special case of (26). IV. R
ESULTS AND D ISCUSSIONS
In this section, we show several examples to illustrate thetracking performance of the proposed controller.
A. Tracking control with a stationary target
The first example illustrates how the tracking vehicles mov-ing from an initial position to collaboratively track an arbitraryplaced stationary target with formation constraints. In allsimulations, the initial positions are set as p ( t ) = ( − , T and p ( t ) = (10 , T . The desired formation configuration isgiven by e p ∗ t = (1 . , π/ T and c ∗ = 40 . The parameters areset as κ ξ = 1 . , κ η = 1 . and k c = 0 . .Fig. 3 shows four rounds of simulations with differenttarget positions p t . As expected, in all the simulations thesystem converges to the desired formation w. r. t. the target,and the tracking errors converge to 0 exponentially fast. Thisdemonstrates the global convergence of the tracking systemwith the proposed controller. The tracking errors provide anintuitive description of the underlying mechanism. B. Tracking control with a moving target
Now we consider tracking a moving target with an initialposition p t ( t ) = (100 , T . In the first example, the targetcircles around the point (100 , T with a radius of 50 anda speed of 5. The values of the setup p ( t ) , p ( t ) , e p ∗ t and c ∗ are the same as those in the above subsection. The resultsare illustrated in Fig. 4. Note that the tracking vehicles andthe target eventually approach the desired formation after t .As the time approaches infinity, the tracking errors convergeto a small interval, as stated in Theorem 2.In the next example, we consider a target trajectory withsharp turns, which is common in adversarial tracking tasks.The results are illustrated in Fig. 5. Note that the trackingvehicles actively adjust their movements to track the target.The tracking errors converge to a small value, which can befurther reduced by adjusting control gains. Fig. 5. Coordinated tracking of a moving target trajectory with sharp turnsunder e p ∗ t = (1 . , π/ T and c ∗ = 20 . The target trajectory is continuousin stages, resulting in some oscillations in the tracking errors but eventuallyconverge to a limited interval. These two examples demonstrate that the proposed con-troller enables successful tracking of a moving target withformation constraints.V. C
ONCLUDING R EMARKS
In this paper, we have proposed a novel control method forcollaborative target tracking using a pair of tracking vehicles.The formation constraints in the tracking task are representedin elliptic coordinates and the control objective is achieved byregulating the rotation, translation, and scaling of the targettracking system. A detailed stability analysis, as well as arich set of simulations, have been provided to demonstrate thetracking performance with a stationary or slow-drifting target.The proposed collaborative target tracking control approachcan also be extended to a multi-robot system, which will beaddressed in the future.VI. A
PPENDIX
Proof A (lower bound of F ). Recall that F = 2 cκ η | xy | G G .Let x + y = gc , G = hc , it is easy to check that g −
Proc. of the American Control Conference (ACC) ,IEEE, 2006, pp. 5269-5275.[2] Z. Sun, H. Garcia de Marina, B. D. O. Anderson, and C. Yu., “Collabora-tive target-tracking control using multiple autonomous fixed-wing UAVswith constant speeds,”
Journal of Guidance, Control and Dynamics ,accepted and in press. 2020. Also available at arXiv: 1810.00182.[3] I. Shames, S. Dasgupta, B. Fidan, and B. D. O. Anderson, “Cir-cumnavigation using distance measurements under slow drift,”
IEEETransactions on Automatic Control , vol. 57, no. 4, pp. 889-903, 2011.[4] Y. Yu, Z. Li, X. Wang, and L. Shen, “Bearing-only circumnavigationcontrol of the multi-agent system around a moving target,”
IET ControlTheory & Applications , vol. 13, no. 17, pp. 2747-2757, 2019.[5] U. Helmke, S. Mou, Z. Sun, and B. D. O. Anderson, “Geometricalmethods for mismatched formation control,” in
Proc. of the IEEE 53rdAnnual Conference on Decision and Control (CDC) , 2014, pp. 1341-1346.[6] M. Basiri, A. N. Bishop, and P. Jensfelt, “Distributed control oftriangular formations with angle-only constraints,”
Systems & ControlLetters , vol. 59, no. 2, pp. 147-154, 2010.[7] A. N. Bishop, T. H. Summers, and B. D. O. Anderson, “Control oftriangle formations with a mix of angle and distance constraints,”
IEEEInternational Conference on Control Applications , IEEE, 2012.[8] Y. Chang, H. Zhou, X. Wang, L. Shen and T. Hu, “Cross-Drone Binocu-lar Coordination for Ground Moving Target Tracking in Occlusion-RichScenarios,”
IEEE Robotics and Automation Letters , vol. 5, no. 2, pp.3161-3168, 2020.[9] J. C. Guti´errez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,”
Journal of the Optical Society of America A Optics Image Science &Vision , vol. 22, no. 2, pp. 289-98, 2005.[10] R. A. Blaya, R. A. Avila, J. B. Reyes, and Ramn M. RodrguezDagnino.“2D quaternionic time-harmonic maxwell system in elliptic coordinates,”
Advances in Applied Clifford Algebras , vol. 25, no. 2, pp. 250-270, 2014.[11] C. Sun, “Explicit Equations to Transform from Cartesian to EllipticCoordinates,”
Mathematical Modelling and Applications , Vol. 2, No. 4,2017, pp. 43-46, 2017.[12] H. K. Khalil,