System for guiding aircraft

Abstract

The present invention relates to a system (114) for guiding aircraft (1021), each aircraft being represented by a state (X'i(k)), the guiding system (114) comprising, for each aircraft (102i), with time discretised into successive instants, a reference provision system (118); and - a predictive control follower (120). The reference positions (Pref(l + 1)) are located, when projected onto a plane (PXY), on a reference cycloid curve; the state (X', (k)) of each aircraft (102i) comprises a position of the aircraft (1021) expressed by coordinates referred to as cyclic coordinates ri-(k), (p φi (k)), the cyclic coordinates being related to Cartesian coordinates Xi(k),yi(k) of this position by the equation of the reference cycloid curve; the reference provision system (118) is designed to provide a series of reference phase shifts ( &φrefij(l)) ; a cost function ( / ) is provided to achieve phase-shift consensus by comprising a term minimising phase-shift errors ( Δφij(l)) between predicted phase shifts ( φij(l)), the predicted phase shifts being calculated from the predicted states (X'i(l + 1)), and the reference phase shifts ( φrefij(l)), for all pairs of aircraft.

Description

Description TITLE: Aircraft Guidance System Technical field of the invention

[0001] The present invention relates to an aircraft guidance system, in particular for fixed-wing aircraft, as well as a guidance method and a corresponding computer program. Technological background

[0002] A prior art is known for guiding multiple aircraft, specifically quadcopters, which are highly maneuverable—they can move forward, backward, and hover in a circle around a target. Each quadcopter is represented by a state with state variables including, on the one hand, a radius and an angular velocity relative to the target, and on the other hand, the aircraft's longitudinal velocity. In this guidance system, a reference radius and a reference angular velocity are defined. A predictive control is implemented to minimize a cost function designed so that the radius and angular velocity of each aircraft converge, respectively, to the reference radius and an angular velocity. This latter value is equal to the reference angular velocity adjusted for an angular offset from the next quadcopter and an angular offset from the preceding quadcopter.In this way, the angular offsets between the quadrotors all become equal, so that the quadrotors are evenly distributed around the circle.

[0003] It may be desirable to provide a guidance system that improves aircraft synchronization. Summary of the invention

[0004] Therefore, a guidance system of N is proposed. a aircraft, N a being equal to or greater than two, defining pairs of aircraft, each aircraft being represented by a state, the guidance system comprising for each aircraft, time being discretized into successive instants: a reference supply system designed to provide, for each aircraft, a series of reference states over a horizon depth, each reference state comprising a reference position of the aircraft; and a predictive control solver designed to search for a sequence of predicted controls resulting in a sequence of predicted states over the horizon depth, minimizing a cost function comprising a term minimizing a state error between the predicted state and the reference state, each predicted state being calculated using a discrete model of the aircraft, from a measured state of the aircraft for the first predicted state and from the previous predicted state for the other predicted states; characterized in that: The reference positions are located, in projection onto a plane, on a reference cycloid curve which is one of the four cycloid curves C y , x , C defined by the following equations parameterized by time: x(t) = R x <p(t) + y(t) = + r(t) sin(t) xy x(t) — R x + r(t)cos( <p(t)') ) y(t) = R y(p(t) + r(t) sm(^(t))J y x(t) = R x <p(t) + r(t)cos(<p(t))| y(ü) = R y + r(t)sin((p(ty) ) x(t) - R x + r(t)cos( <p(t))] _ c y(t) ~ R y + r(t)sin( <p(t))J where t is time, R x and R y are constants, r(t) is any function of time or a constant, and <p(t) est une fonction quelconque du temps ou une constante; et The state of each aircraft includes an aircraft position expressed by so-called cyclic coordinates (fc), linked to Cartesian coordinates x i -(k),y i -(k) of this position by the equation of the reference cycloid curve; The reference supply system is designed to provide a series of reference phase shifts between all aircraft over the horizon depth; and the cost function is designed to achieve phase shift consensus by including a term minimizing phase shift errors between predicted phase shifts calculated from predicted states and the reference phase shifts, for all pairs of aircraft.

[0005] Thus, thanks to the invention where the phase-shift consensus is implemented for all pairs of aircraft, and not just for the preceding and following aircraft, aircraft synchronization can be improved.

[0006] The invention may further include one or more of the following optional features, in any technically feasible combination.

[0007] Optionally, the status of each aircraft includes the aircraft's position expressed in Cartesian coordinates.

[0008] Optionally, the status of each aircraft also includes an aerodynamic longitudinal speed of the aircraft and Euler angles of the aircraft.

[0009] Optionally, the cost function / is also as follows: where i, j are identifiers of two aircraft; N a is the number of aircraft; k is a current time; N p is the horizon depth; l ~ 0... N p — 1 is a prediction instant; AX (l + 1) = X' + l + 1) - Xj (l + 1) with X' re a state of aircraft reference i and a predicted state d e aircraft i, where (Z) = U ref - k + l) ■■■ Ui T) with U re f. a reference order and a predicted command; = <prefij(.k + 0 ~ with <p r ef tl unreference phase shift between aircraft i and aircraft j and a predicted phase shift between aircraft i and aircraft j; and Ri(l') eiSij(l) are parameters relating to aircraft 1021 and the interaction between aircraft i and aircraft j.

[0010] A method for guiding N is also proposed. a aircraft, N a being equal to or greater than two, defining pairs of aircraft, each aircraft being represented by a state, the process comprising for each aircraft, time being discretized into successive instants: a provision, for each aircraft, of a series of reference states over a horizon depth, each reference state comprising a reference position of the aircraft; and a search, by a predictive control solver, for a sequence of predicted states over the horizon depth, the predicted states of the sequence minimizing a cost function comprising a term minimizing a state error between the predicted state and the reference state, each predicted state being calculated using a discrete model of the aircraft, from a measured state of the aircraft for the first predicted state and from the previous predicted state for the other predicted states; characterized in that: The reference positions are located, in projection onto a plane, on a reference cycloid curve which is one of the four cycloid curves C xy , Cy, C x , C defined by the following equations parameterized by time: x(t) - +r(r)cos( <p(t)))... y(t) = Ry <p(t) + r(t) sin(y»(£))J xy x(t) = R x + rT)cosT(pT)) ] y(t) = + r(t) sin(ç?(t))J y x(t) = Rx (p(t) + r(t)cos(ç3(t)j) i== c y(t) = R v + r(t)sin( <p(t)) ) x(t) ” R x + r(t)cos( <p(t))^ y(t) = R y + r(t)sm( <p(ü))J where t is time, R x and R y are constants, r(t) is any function of time or a constant, and ^(t) is any function of time or a constant; and the state of each aircraft includes a position of the aircraft expressed by so-called cyclic coordinates r^k), <pi k) reliées à des coordonnées cartésiennes x i (k'),y i(k~) of this position by the equation of the reference cycloid curve; the method further comprises a provision of a series of reference phase shifts between all aircraft over the horizon depth; and the cost function is designed to achieve phase shift consensus by comprising a term minimizing phase shift errors between predicted phase shifts calculated from predicted states and the reference phase shifts, for all pairs of aircraft.

[0011] Also proposed is a computer program downloadable from a communication network and / or stored on a computer-readable medium, characterized in that it includes instructions for executing the steps of a process according to the invention, when said program is run on a computer. Brief description of the figures

[0012] The invention will be better understood with the aid of the following description, given solely by way of example and made with reference to the accompanying drawings in which: Figure 1 is a schematic view of an aeronautical installation comprising several aircraft and an aircraft guidance system; Figure 2 is a graph grouping three examples of cycloid curves. Figure 3 is a graph of an example of a cycloid curve with details of the construction of this curve. Figure 4 is a graph of an example of three aircraft following a cycloidal curve with guidance according to an embodiment of the invention. Figure 5 is a block diagram illustrating the steps of an example of a guiding method according to the invention, and Figure 6 is a schematic view of a computer system for the implementation of part of the guidance system of Figure 1. Detailed description of the invention

[0013] In the following, time is discretized into successive instants.

[0014] With reference to Figure 1, an aeronautical installation 100 in which the invention is implemented will now be described.

[0015] The aeronautical installation 100 initially comprises several aircraft 1021, i ranging from 1 to N a For example, the number N a The number of aircraft (1021) is two or more, preferably at least four. Thus, ■-■■■■■- — pairs of aircraft are defined. Aircraft 102i are, for example, fixed-wing aircraft.

[0016] Each 021 aircraft is located in space by Cartesian coordinates x t! y t! z,- according to a coordinate system R formed by three axes X, Y, Z. The Z axis is for example taken vertical, and this is the situation which is taken for the calculations detailed later.

[0017] It is desired that the aircraft 1021 follow a trajectory T. The movement along the Z-axis will not be detailed further. This movement along the Z-axis is, for example, identical for all aircraft 1021, for example, constant (all aircraft remain at the same altitude) or linear. On the other hand, the trajectory T is on a cycloid curve along the X and Y axes. More precisely, the trajectory T has a projection P onto a plane PXY extending along the two axes X and Y, this projection P having the shape of one of the four cycloid curves C xy , C y , C x> C defined by the following equations parameterized by time: x(£) = R x <p(t + r(t)cos(<jp(£))] >= e (yt) = A y <p(t) + r(ü) sm(^(t))J xy x(t) = R x + r(t)cos'(« / 7(t)) ) y(t) = A y( p(t) + r(ü) sm(^(t))J ' x(t) = R x <p(t) + r t'jcos qj t')')'} y( - R y+ r(t)sin pt)') J C ' x(t) = R x + r(t)cos( >(t))l y(t) = R y + r(t)sm(ry( t)) j where t is time, R x and R y are constants, r(ü) is any function of time (possibly a constant) corresponding to a radius, and (p(t) is any function of time (possibly a constant) corresponding to an angle (also called phase).

[0018] Most of the time, cycioid curves are curves of the type "corkscrew" or "spring" in a plane (the PXY plane in this case).

[0019] The cycloid curve C is a circle of radius r(t) centered at (R x , R y ).

[0020] The cycloid curve C y moves along the Y-axis around an axis collinear with the X-axis and translated by R x It's a trochoid.

[0021] The cycloid curve Cx moves along the X-axis around an axis collinear with the Y-axis and translated by R y : it's a trochoid. For example, with R y = 0: if R x » rt) then the spring-type curve "strets" and there is no longer a loop: we speak of a "shortened" cycloid / trochoid; plus R x The lower the curves are, the more pronounced they are: we then speak of an "elongated" cycloid / trochoid; when R x = 0, the loops are marked until they become circular and centered at (0,0).

[0022] The cycloid curve C x typically has the form of the examples illustrated in Figure 2, obtained for r(t) = 100, R y = 0, <p(t) = 0,05t et pour trois valeurs de R x (R x = 75, R x = 50, and R x = 25).

[0023] Within the framework of the cycloid curve C xwith the constant R non-zero and the radius r(t) constant, figure 3 shows the angle <p comme étant l’angle que fait un point M situé sur la courbe cycloïde with a center of rotation denoted Q moving to the right and with coordinates (R x (p(t), R y ). In figure 3, the point M and the center of rotation Q are represented at two different times where iis are designated by respectively, i and M2, Q?.

[0024] The cycloid curve whose projection P has the shape is subsequently called the reference cycloid curve. The reference cycloid curve is thus defined by one of the four equations above and by the values ​​of the constants R x and R y and your functions taken respectively as radius r(t) and phase (p(t)). For example, taking, as the reference cycloid curve, the cycloid curve C with R x - R y - 0, r(t) ~ and ç?(t) - it, the projection P is a circle of radius R o constant distance traveled at constant angular velocity.

[0025] Thus, each point on the reference cycloid curve has coordinates that can be expressed in Cartesian coordinates x, y, or in so-called cycloid coordinates by radius r and phase <p associés aux coordonnées cartésiennes x, y par l’équation de la courbe cycloïde de référence. De manière similaire, des points en dehors de ia courbe cycloïde de référence présentent des coordonnées pouvant être exprimées en coordonnées cartésiennes x, y ou bien en coordonnées cycloïdes r, <p conformément à l’équation de ia courbe cycloïde de référence.

[0026] Each aircraft 102i is represented by a state X'j(k) grouping several state variables, including a position expressed in Cartesian coordinates x é ( ), yi(k), z t (k) in the XYZ frame and also in cycloid coordinates r î:( / c), pi k').

[0027] For example, the state X'iÇk) groups the following state variables: x f (k), V:(k), <l>i(k), &i(k), r^k), fpiÇk'), where ^(k) is an aerodynamic longitudinal velocity of the aircraft 102i (i.e., the velocity along a longitudinal axis of the aircraft 1021), and <p b 9 b Euler angles of aircraft 1021 (roll pitch) and cap

[0028] Returning to Figure 1, each of the aircraft 1021 includes a measurement system 1041 designed to provide, at each instant k, measurements relating to the aircraft 1021. The measurements include, for example: a position measurement p mi ( ) of aircraft 1021 expressed in Cartesian coordinates x m .(k), y mi (k), z m .(k) in the XYZ frame of reference, a measure of longitudinal aerodynamic velocity V m .(k) of aircraft 102i, and an Euler angle measurement of aircraft 1021 (roll m ., plate 0 mi and cap The 1041 measurement system, for example, includes an inertial measurement unit.

[0029] Each of the 1021 aircraft also includes a 1061 conversion system designed to convert the measured Cartesian coordinates x m k), y m ,.(k) in cycloid coordinates measured r mj ( / c), comprising a measured radius r m .(k) and a measured phase (p mi (k). Indeed, as explained previously, the Cartesian coordinates measured x m .(k), y mi (k) are related to the measured cycloid coordinates r m .(k), (p rn (k) by the equation of the reference cycioid curve followed by the projection P of the trajectory T.

[0030] Since there is no analytical solution to this equation to express the measured cycloid coordinates r mi (k), <p mi ( ) as a function of the measured Cartesian coordinates x m .(k), y mi (k), the 1061 conversion system, for example, uses a state estimator to estimate the measured cycloid coordinates

[0031] The 1041 measurement system and the 1061 conversion system thus provide a measured state X' m k) grouping the state variables measured or estimated from the measurements: x m .(k), y mj ( ), z m k), V mj (k), <p mi (k), 6 m . k), ^m^k), r m .(k),Q

[0032] Each aircraft 102I then includes, for example, a wireless transmission system 1081, for example via radio waves, designed to transmit the measured state

[0033] Each of the aircraft 102 also includes a displacement system 110I designed to move the aircraft 102i. To this end, the displacement system 110I is designed to receive, at each instant k, a command Ui k determined, as will be explained later, from the measured state This control Ui(k) includes, for example, an aerodynamic longitudinal speed control l.., a pitch control 0 C and a roll control of 0 C For example, the 1101 displacement system includes, on the one hand, a propulsion system (not shown) to achieve aerodynamic longitudinal speed control V c and, on the other hand, control surfaces (not shown) to achieve the zero pitch control C and the roll control 0 C ..

[0034] For example, aircraft 102i has a wireless receiving system 1121, for example by radio waves, designed to receive the command u; k).

[0035] Aeronautical installation 100 also includes a guidance system 114, for example located on the ground.

[0036] The guidance system 114 includes first of all a wireless receiving system 116, for example by radio waves, designed to receive the measured states X' m . k) aircraft 1021.

[0037] The guidance system 114 further includes a reference supply system 118 designed to supply, for each current instant k a sequence of reference states X' ref .(k. + 1 + 1) over a horizon depth N p (Z = 0... N p — 1), starting at time k + 1 following the current time k (thus, the first reference state ' re f. k + 1) (l ~ 0) is the reference state for the time k + 1); a series of reference phase shifts (p re f tJ ,k + l + 1) between aircraft 1021 on the horizon depth N P , starting at time k + 1 following the current time k (thus, the first reference phase shift tpref^k. + 1) (l = 0) is the reference phase shift for time k + 1); and possibly, a sequence of reference commands U re f.(k + Z) on the horizon depth N p , starting at the current instant k (thus, the reference command U ref k) is the reference order for now k).

[0038] The state variables of the reference state X' ref are noted: x refl , y re .f z ref,^ V refc ^refc ^refc r refi' Vreff

[0039] The reference supply system 118 is further designed to receive a series of positions p b k) placed on the reference cycloid curve. The positions p b (k) of the basic trajectory are expressed in cycloid coordinates r6(fc), (p b (k) according to the equation of the reference cycloid curve.

[0040] For example, the reference cycloid curve can be a circle of radius R o traversed at constant angular velocity. In this case, the positions p b k') of the basic trajectory have the following cycloid coordinates: r b k) = R o , (p b ( ) = ok.

[0041] The reference supply system 118 is further designed to accommodate 5p phase shifts re f.(k desired for each of the aircraft 1021 respectively with respect to this basic trajectory. The reference positions p re r,.(c) successive of each aircraft 102i then present the following cycloid reference coordinates: r rP f i {k') = r b (k~), (p re p.(k) - <p b (k") + ô <p re f.(k').

[0042] The reference supply system 118 is then designed to calculate, at each instant k, for each aircraft 1021, the reference state X' re f. k') and the reference command U re f.(k) from the reference positions p re f .(j<) in cycloid coordinates: r ref .(k), <p re ^.(k).

[0043] For example, the reference positions p re .(k) in Cartesian coordinates x Te ^.(k'), y re f. k) are calculated from the equation of the reference cycloid curve. For example, when the latter is the cycloid curve C xy , the Cartesian coordinates x re f.(k'), y re fk) are calculated as follows: — ^x^Pref^) {jPref (yreffi) = R y (p ref .(k) + r ref .(k)sm(q) refi (kyj

[0044] For the other state variables, the reference supply system 118 first calculates, for example, a reference velocity in Cartesian coordinates x re ^(fc), ÿ re ^.(k), for example, in the following way: xr e f (k} — xr^f (k — 11 >..........:...... -:. 'e ÿ ref i(k) ....... yrefi.t..k....).... "....,^ y,'r..e..ùfA -k ~ 1) ZrefXk) -■ Z ref Xk -■ 1) z ref {k) - - - - -

[0045] Then, the reference supply system 118 calculates, for example, the other state variables from the reference velocity in Cartesian coordinates ÿ re .(A:), z ref (k), for example in the following way: lÿref S^' )\ "1 VW, ( k ) = atan2 Vref 1 / J TT ( \ > ^ref^ 2 Sl9U J SÎ X ref Xk") = 0 We / . C^) = ÿrefi(k " 1) sl *ref XJO = 0 and ÿ ref .(k) = 0 J > V ref i (. k ') = JXef,(k) 2 + ÿ ref XJi) 2 + Z ref .(k) 2 / XJX)\ - V're / .ft " 1)) V„f,(k) < Pref t W ■■■■ T

[0046] For reference order U re f.(kX reference supply system 118 can, for example, take up the reference aerodynamic longitudinal velocity V re f(k), the reference plate 0 ref and the reference roll (p re f, already calculated: ^ reft W = [Vreft W, 9 reft < Pr e f.- (&)] '

[0047] The reference supply system 118 is further designed to calculate, at each instant k, the reference phase shifts (pre^Ça from the reference phases aircraft 102i contained in the reference states. f re / -.( c), for example in the following way: t prefij (^) iprefi (Jt) < Pref j vO

[0048] The guidance system 114 also includes a predictive control solver 120 designed to search, on the horizon depth N p For each of the 102i aircraft, a sequence of predicted commands ^(0 starting at the current time k, these predicted commands U^l') gives rise to a sequence of predicted states XL (7 + 1) starting at time k. + 1 follows the current time k, so that the predicted states X r . l + 1) and possibly the predicted commands 1 / ^(0) minimize a cost function J. The predictive command solver 120 can be based on quadratic sequential programming like the fmincon solver in Matlab (https: / / fr.mathworks.com / help / optim / ug / fmincon.html). As this requires an initial point, it is possible to employ the warm-start procedure (Gros et al., "From linear to non-linear MPC: bridging the gap via the real-time iteration", In: International Journal of Control, 93(1):1-19, Sept 2016).

[0049] Each state predicts X r j(_l + 1) is calculated by the predictive control solver 120 from the previous predicted state X'i l') (or the measured state X' m .(k) of the current instant k for the first predicted state X';(l)) and of the previous predicted order Uj(l"), using each time a state predictor f d . which is a discrete model (linear or non-linear) of aircraft 1021: X'i(l + 1) = ^X'i(l'), with X'i (0) - X' m .(k).

[0050] Thus, the first predicted state X')(l) is obtained in the following way: - f di / AC 0 )), second predicted state X' t (2) is obtained in the following way: X'iÇX) - f dj (xLÇl), Ut (1)), and so on until the last predicted state X ! - N p Who is obtained in the following way: X ! j {N p ) = f d . (x'i(N p - 1), Ui(N p - l)j.

[0051] For example, the state predictor f d . is previously obtained by discretizing a continuous model of the aircraft 1021, for example by discretizing the following continuous non-linear model fi:Vi " s ipi s 8i ^i. Zl = ~~s Gi Vi &i = M 0 q " ®i) 0i = " <&) ■■= x f !: = / [(r t (you £ (t)),j = i,..., Ar a ^ = ^4 i Ry +?jC / p Cg Izj + ('" Rx + ^iSip^Sÿ) SQ V'i C < Pi^y"' s <pi Rr d” ^i ®; ~ -: - -. -: - C < Pi^y ^tpi^-x?i WHERE: - C 0J = cos(0f), c e . = cos(0j), c^. = cos(4'i)> - s^. = sm(0i), s Q . = sin(0 £ ), s^. = sin(ÿ, : ), - 10 = tan( <pi), - g est la gravité, et - k v .,k e ., k. are attitude control loop bandwidths of aircraft 1021 modeled by first-order filters.

[0052] When the reference cycloid curve is a circle, the continuous nonlinear model f simplifies to: %L - c ^j C E> J Z i ÿi ~ • s v , ! s sJ / j ii = -Sf^t V^ k^ -V^ s. = k0(e c - 0,-) s 77 n = Q^.cg.lé + s^.se. Vi ^^9^1 " LC <p, S^S^Vj iï"~~ J

[0053] In the continuous nonlinear model fi, the equations of the derivative of the radius f £ and the derivative of phase 0 £ are, for example, derived from the derivative equations of other state variables and the equation of the reference cycloid. For example, when the reference cycloid curve is the cycloid curve C xy , its equation can be derived to obtain: x = R x <p + rc v —rcps^ = rc* +(R X - rs^tp y = R y <p + r s, p +rc (p q) = rs / p + (R y + rc^)0 which allows us to obtain: (Ry + rcgf)x + (~R X + rs^fy C(pRy $cpRx d* . ~~s <px 4" c (p y ( - <pRy XpRx d" r

[0054] In the case where the reference cycloid curve is a circle, the equations of the derivative of the radius r £ and the derivative of the p- phase are as follows: r = c^x 4- s, p y

[0055] The discretization is, for example, as follows, with T e a sampling period (typically 0.1 s, for example):

[0056] The cost function / first includes a term minimizing an error of 'state at Xj ! (l + 1) = X' ref .(k 4- 14- 1) - Xf(l 4- 1) on the prediction horizon ÄL, and preferably a term minimizing a control error Ay £ ( / .) = U re <,(k + ) ~ Ui l) on the prediction horizon N p .

[0057] Furthermore, in order to achieve a phase shift consensus, the cost function J also includes a minimizing term, over the prediction horizon N p , phase shift errors A® ££ (t) = (p ref k 4- / ) - < ptj(l") for each pair of aircraft 1021, 102j, between the reference phase shift (Pref^ k 4- 1) and a predicted phase shift (pij lf

[0058] To calculate the predicted phase shift <p £ / (0, the predictive control solver 120 is, for example, designed to use a phase shift predictor h d which is a discrete model (linear or non-linear) : 4- 1) ~ h d ..¥ / (£)).

[0059] For example, the phase shift predictor h d is previously obtained by discretization of the continuous nonlinear model H £J£ following, derived from the non-linear models f, f of aircraft 102i, 102j: • + c (pi s ÿt s 8yi __ < PU - < Pi ■■■ < PJ - = h tJ Çx i flfX i fr))'

[0060] The discretization is, for example, as follows, with T e a sampling period (typically 0.1 s, for example): h d .. +

[0061] For example, the cost function / is as follows: AXi'(l + l) 7 Ç;(0âX ; '( i=l N a where Qi(l'), i(l) and Sij(T) are parameters relating to aircraft 102i, for example set by a user, and to the interaction between aircraft 1021 and aircraft 102j.

[0062] The predictive control solver 120 can also take into account internal constraints G int avion (x / Cf), which are for example in the form of a non-linear function that translates a set of internal constraints related to the 102i aircraft: for example: ^(0 ~ ^maxi - 0 1 + Vjnini ° <9 Cf (0 ”■ $maxi 0 ~0 Cj (O + 0 rai£S j < 0 I tpcfo) ~~ 0maxi — 0 ■■■0 C , (D + 0 min i

[0063] Internal constraints may also reflect consideration of a flight zone (safe zone, minimum flight altitude, etc.): X; (7) X max ~ 0 'j -■£; (7) + X mjn < 0 ÿi (J) " y max < 0 I + >'mm 0 | z inax — 0 + Z mjn S oJ

[0064] The predictive control solver 120 can also take into account external constraints G ext obst ...,7^(0,^^ (Z)) which are, for example, in the form of a non-linear function that translates external obstacle avoidance constraints: G ext obst fï, (î), di^G), < 0, for example of the different aircraft 102î between them, for example: ||Pj(Z) ■■■■ ■■■■ A'^ < oj, with R obs a predefined safety distance.

[0065] The synchronism constraint (i.e., maintaining the reference phase shifts (prg^) is carried by the term X' î'i cost function J. This term allows for a consensus operation on the phase shifts between all aircraft 102i. When consensus is reached, then we have assurance in the continuous domain that lim pt) = 0, or in discrete time t->co 1 1 ' ' that limit Y ^Qpitk + 1) ■■■■ (Pi(kJ) ~ 0. The reference phase shift <p r ef i} a 9^ as a phase bias and also allows us to have: Lira [ ^(t) - < / ? / (£)) ~ < Paref- Ce terme est obtenu par une sommation sur i et sur / >i, not on all possible ij because, due to tpi This would amount to considering certain phase shifts twice. For example, for four aircraft: 4 4 VV Sijà(pij = &(pl2S 12 + â <pf3Si3+ à(p^S 14 + ^^23^23 + ^24^24 + &< P 34^34 E=1 j=£+i

[0066] The predictive control solver 120 is thus designed to provide for each aircraft 1021, as a control Ui(k) for the current time k, the first predicted control Â-(0).

[0067] The guidance system 114 further includes a wireless transmission system 122, for example by radio waves, designed to send the commands U^k) to the aircraft 1021, and in particular to their wireless reception system 112i.

[0068] With reference to Figure 4, in the case of three aircraft, an example of synchronized formation flight on a cycloid curve is illustrated.

[0069] We thus see three planes A1, A2 and A3. The phases of the three planes evolve between times t1 and t2, but the phase differences between the planes are constant: HAS t1, we have: ^(A.) ~ -170°, = -20 °, (psÇt^) = 160° À 12, o na: ~ 0°, ” 150°, <p3(t2) ~ 230° (=-130°)

[0070] At t1 and t2, the phase shift values ​​are conserved: q) 12 (-1) = < P12 (^2)' ^13 ( f l) ^13 ^2) » ^23(^1) ”■■ ^23(^2)

[0071] We will now describe an example of centralized guidance of two aircraft on a circular formation of radius R0 with phase opposition constraint and constant angular velocity coo, at constant altitude ho, altitude which must remain above h min .

[0072] Model definition:

[0073] In this case, we have two planes and the collaborative model is written: ^•1 1 ÿi = Vi = kv^ Cï ■■ 14)

[0074] = / i(.¥[, ui) - k 01 (0 C1 - 0i) Tl J — C / CTpj-L C-. ^it1 See t?j! 17 -11 1" S (( ~, ô->n tfi Pi J - _ + S ip] KL *2 ~ C lp2 C 02^2 y 2. = S.p2s e y2z2= se2V2

[0075] ^ = ^2(0 C2 "02) / 2 (^2' ^2) 02- k ti2 (e.,2-■ e2) - ^4

[0076] S <p 12 = = toi2(XLXD) '1 r 2

[0077] Reference calculation:

[0078] The circular formation is a particular cycloid characterized by Rx = Ry =

[0080] Furthermore, we have the following references:

[0081] Plane 1:

[0082] , n fn ....4, ^Pref^Q) ^0^ S're / \(O = ^ocos(a) r j.),x rei r (l') -R o <y o sin("y o Z)

[0083] yre / ., (0 7? QSin.(6> QZ), ÿref^CO ” J? Q67QCOS(<1! QZ) ^re / ^CO "”^0- z ref^ (l) ~ 0 Vref^J-) " ^0 ÙJ 0 1st fl (J-) = 0

[0084] ÿ, re ^(l) — —atan2' = ~-stcyne(tan(a)0Z))" + 600Z < Pref, (]■") = ~~ (V'reA (0 ~ V'reA "!)) ==: (-sj s gne(tczn(to0Z)) | + sign [tan(a)0(l - 1)))

[0085] X r ' efL (l) = [^ / / 0-WjO^reA®] 7 with* re / - - [■Ae / v yi-efj i z refj, ^refa > ®rej\> (ferefj. ^Pre / J

[0086] HER reh (I) = [iZ re / 1 (I), 8 refi (Lead) reh (0] '

[0087] Plane 2: ref-, " ^0

[0088] z n ,, (pref ^ref 2^-^ — / ?()Cos(^CL)gZ "b 7ï) f ([11 Children)ÿSinÿg)Ql "F YOU)

[0089] yref2^ ” j Ro s Ol(ûJ0( + 7ï), ÿ re f r ( < i') " / ? Q <t>oCOs(û)0Z + ÎT) ^ref 2^-^ ^-0' ^ref 2^-^ ® ^ e / 2(0 = «0^0

[0090] 0 king / 2(O 0 (prefSO ~ ~atan2 ~ ~-signe(tan((i) o l + rr))- + in o l + rr

[0091] 0 re / 2 ( / .) = "(V're / JO ” ^ref2( l ~~ 1)) = ”” (~slgneÇtan(a) o l + TT)) ~ + sicyne(tan(à)0(Z — 1) + TT)) ~ + a>0^

[0092] X,'. e / . (Z) = [X^(Z),r re / z (Z),i?? reÆ (Z)]' avecX re / ? = [■*7-0 / 2 ' yref2> z ref2> ^ref2> @ref2> (Pref2> ^Pref2]

[0093] U ref? (Z) - [ v ref? (I), and ref? (l), $ ref? ( / )] 1

[0094] Déphasage avion 1 / avion 2:

[0095] Yes? re / 12 (I) = ~7F

[0096] Definition of the centralized optimization problem:

[0097] Collaborative guidance thus consists of solving the following optimization problem:Pi(o), 01(1;, -,y1(Np-i)) p ' ■'■ ' [U2(0), O2(l),-, U2( / V p -l)J subject to AXP(l + 1) T Q ] (î)AA , 1 , ( I f + 1) + &x2''(l + 1) T Q2(Z)AX2'(Z l yy?- l^ a (X , 1(_k), X'2(k). ^(k)y. 1 2 Years / 1(Z) T R1(0At / 1(Z) + AJ72(0 T / ?2(0A[ / 2(Z) + '"p Acpy (_l) T If j(l)has(pjj (Z) AX| f (Z) — X ref x (- k ~i~ 0 — -^1 / (7) AX / G) = X re / 2 '(k + 1) − X2(l) Acf 32 ( Z ) = 5pts 12f , gf (k + Z) — 5 <p 12 (I) TO THE CO - U refl (k + Z) − U r (l) Ay2a) − L^(fc + / ) − L / 2a) + (X2M ^2(D) x2(t + 1) − f d? (x2′(Z), i / 2(0) S <p 12 (Z + 1) − / i dl2 (X / (Z), X / ( / ))

[0098] v in VJ’ maxi — u -Key(D + v min l < or N n - 1 6cl (0 -■ Smax i < 0 -0 C1 (Z) + 0 min 1 < 0 ^cl C' ) 0max l — 0 "0cl (0 + 0min 1 0 K:2 (0 ~ Vmax2 0 -■V C 2 (l) + V mi n 2 < 0 &c2(l) ~~ $max 2 5= 0 ”^c2(0 + ^min 2 ~ ® 0c2(O '”■ 0max 2 — 0 ~~0c2(O + 0min2~ 0 2 l(0 ~ k max ^2(0 — ""h m ax (x1( / ) - z2(n) 2 + (y i ( / ) - y2(n) 2 obs X / (0)::: X / (k) 17(0) = fc )i S C1 (k) xwi pi (0)1 0 C1 (fc) «2(«) [U2(0)] v c2(k) -0c2 i J <y

[0099] We can then choose the following typical sizing:

[0100] T e = 0.1 s

[0101] N P = 50

[0102] V max 1 = V max 2 = 40 m / s

[0103] V min 1 = V min 2 = 10 m / s

[0104] θ max 1 = θ max 2 = 30°

[0105] i min 1 = θ min 2 = −10°

[0106] 0 maxi "0max2" 30

[0107] = 0 m in 2 ~ ""30"

[0108] k V = k V = 1 / (2π0,1)

[0109] k θ = k θ = 1 / (2π0.3)

[0110] k φ = k φ = 1 / (2π0.4)

[0111] z ref = k ref = −25 m

[0112] ω0= 2π / 6 rad / s

[0113] R o = 25 m

[0114] (Pref^ = 0, (p re / ? "Tt, ( Pref r2 ~ ~~ 7T

[0115] Çi = Q2= dîa#(4.10~ 5 , 4.10” 5 , 0.05, 0.001, 0.002, 1.10” 4 , 0, 4.10~ 5 , 0.002)

[0116] R1= R2= diag(0.01, 5.10 −4 , 0.01 )

[0117] S 12 - 0.002

[0118] With reference to Figure 5, an example of method 500 according to the invention will now be described.

[0119] During a step 502, the reference supply system 118 provides, for each aircraft 1021, the sequence of reference states X' re f.(l + 1) on the horizon depth N p .

[0120] During step 504, the reference supply system 118 provides the series of reference phase shifts ô'ç3 ref ..(0 between all aircraft 1021 on the horizon depth N P .

[0121] During step 506, the predictive control solver 120 searches, for each aircraft 1021, for the sequence of predicted commands giving rise to the sequence of predicted states + 1) on the horizon depth N P , then provides the first predicted command U - T) as the next to be applied by aircraft 1021.

[0122] With reference to Figure 6, an example of the implementation of system 118 and system 120 will now be described.

[0123] In this example, these systems are implemented by a computer system comprising a data processing unit 602 (such as a microprocessor) and a main memory 604 (such as RAM, from the English "Random Access Memory") accessible by the processing unit 602. The computer system further comprises, for example, a network interface and / or a computer-readable medium, such as a local medium (such as a local hard disk 606) or a remote medium (such as a remote hard disk accessible via the network interface through a communication network) or a removable medium (such as a USB flash drive, from the English "Universal Serial Bus", or a CD, from the English "Compact Disc" or a DVD, from the English "Digital Versatile Disc") readable by means of an appropriate reader of the computer system (such as a USB port or a CD and / or DVD disc drive).A computer program 608 containing instructions for the processing unit 602 is stored on the storage medium 606 and / or downloadable via the network interface. This computer program 608 is, for example, intended to be loaded into the main memory 604, so that the processing unit 602 can execute its instructions.

[0124] The instructions of computer program 608 are designed, when executed by processing unit 602, for the computer system to carry out the steps of process 500.

[0125] In conclusion, it should be noted that the invention is not limited to the embodiments described above; indeed, it will appear to a person skilled in the art that various modifications can be made to the embodiments described above, in light of the teaching which has just been disclosed to him.

[0126] For example, it would be possible to add other couplings between aircraft besides phase shift. In this case, the functions f(X : '(i) u,(t)) would be replaced by functions fn(X, '(t), u>(t), X(t), Uj(t)), which would not change the principle of the invention in any way.

[0127] In addition, the aircraft model could be different to accommodate other types of aircraft, such as VTOLs.

[0128] Furthermore, it is possible to work with controlled thrust instead of controlled speed.

[0129] In the detailed presentation of the invention given above, the terms used shall not be interpreted as limiting the invention to the embodiments set forth in this description, but shall be interpreted as including all equivalents which can be foreseen by a person skilled in the art by applying their general knowledge to the implementation of the teaching which has just been disclosed to them.< / t> < / l>

Claims

Demands [1] N guidance system (114) a aircraft (102i), N a being equal to or greater than two, defining — pairs of aircraft (102i), each aircraft being represented by a state (X'i(k)), the guidance system (114) comprising for each aircraft (102i), time being discretized into successive instants: a reference supply system (118) designed to provide, for each aircraft (102i), a series of reference states ( X' re f (l + 1)) over a horizon depth N P ), each reference state X' re f.(J + 1)) comprising a reference position (Pref l + 1)) of the aircraft (1021); and a predictive control solver (120) designed to search for a sequence of predicted commands (t / j(O) giving rise to a sequence of predicted states (X'itl + 1)) on the horizon depth ( / V p), minimizing a cost function (j) including a term minimizing a state error (A'iQ + 1)) between the predicted state (X''j(t + 1)) and the reference state X' re f (l + 1)), each predicted state (X r . (I + 1)) being calculated using a discrete model (M £ ) of the aircraft (1021), from a measured state of the aircraft (1021) for the first predicted state (X'j(l)) and from the previous predicted state (X ( / )) for the other predicted states (X'j(l + 1)); characterized in that: the reference positions + 1)) are located, in projection onto a plane (PXY), on a reference cycloid curve which is one of the four cycloid curves C xy , C y , C x , C defined by the following equations parameterized by time: x(t) = R x φ(t) + r(t)cos(φ(t))] y(t) = R y φ(t) + r(t) sin(φ(t))] := Cxy x(t) = R x + r(t)cos(φ(t))] y(t) = R y φ(t) + r(t) sin(φ(t))] := C y x(t) = R x φ(t) + r(t)cos(φ(t))] y(t) = R y + r(t)sin(φ(t))] := C x x(t) = R x + r(t)cos(φ(t))] y(t) = R y + r(t)sin(φ(t))] := C where t is time, R x and R y are constants, r(t) is any function of time or a constant, and <p(t) est une fonction quelconque du temps ou une constante; et the state (À"j( c)) of each aircraft (102i) includes a position of the aircraft (1021) expressed by so-called cyclic coordinates ri(k),(pi(k) related to Cartesian coordinates Xi(k),yi(k) of this position by the equation of the reference cycloid curve; the reference supply system (118) is designed to provide a series of reference phase shifts (pre ^ ( / )) between all aircraft (1021) on the horizon depth (A' p ); And the cost function ( / ) is designed to achieve a phase shift consensus by including a term minimizing phase shift errors (Ay? ; / (0) between predicted phase shifts (< Pi7(0) calculated from the predicted states X'iÇl + 1)) and the reference phase shifts for all pairs of aircraft. [2] Guidance system according to claim 1, wherein the state (X') of each aircraft (1021) comprises the position of the aircraft (1021) expressed in Cartesian coordinates x b y t ). [3] Guidance system according to claim 1 or 2, wherein the state (A")) of each aircraft (1021) comprises an aerodynamic longitudinal velocity (l) of the aircraft (1021) and Euler angles ( 9 ib,) of the aircraft (1021). [4] A guidance system according to any one of claims 1 to 3, wherein the cost function J is as follows: ∑ ΔX i '(l + 1) T Q i (l)ΔX i '(l + 1) + i=l ∑ ΔU i (L) T R i (l)ΔU i (l) + l = 0 i = 1 N a N a where i, j are identifiers of two aircraft; N a is the number of aircraft; k is a current time; N p is the horizon depth; l = 0... N p - 1 is a moment of prediction; at Xj\l + 1) - X' re f.(_k + 1 + 1) - X^l + 1) with X' refl a reference state of aircraft i and X a predicted state d e aircraft i, where ~ U r efS^ + "U t (l) with U ref a reference command and [ / f (0 a predicted order; A(Pij(l) - <p ref (k + 0 - with (pref^ a reference phase shift between aircraft i and aircraft j and <p t j is a predicted phase shift between aircraft i and aircraft j; and ^i(D and S^-ft) are parameters relating to aircraft 1021 and the interaction between aircraft i and aircraft j. [5] N guiding method (500) a aircraft (1021), N a being equal to or greater than two, defining pairs of aircraft (102i), each aircraft being represented by a state (X' ; ( / <)), the process comprising for each aircraft (1021), time being discretized into successive instants: a supply (502), for each aircraft (1021), of a series of reference states + 1)) over a horizon depth (N P ), each reference state ( X' re / 1(^ + 1)) having a reference position p re f,(- + 1)) of the aircraft (1021); and a search (506), by a predictive control solver (120), for a sequence of predicted states + 1)) on the horizon depth N p ), the predicted states + 1)) of the sequence minimizing a cost function ( / ) including a term minimizing a state error (A. X' £ (+1)) between the predicted state (' £ (Z + 1)) and the reference state (X' re ■_.(<' + 1)), each predicted state ( f j(i + 1)) being calculated using a discrete model (M £ ) of the aircraft (1021), from a measured state (X' m .(fc)) of the aircraft (1021) for the first predicted state (X' £ (l)) and from the previous predicted state (X' £ (i)) for the other predicted states ( ''j(l + 1)); characterized in that: The reference positions Preffi + 1)) are located, in projection onto a plane (PXY), on a reference cycloid curve which is one of the four cycloid curves C xy , C ys C x , C defined by the following equations parameterized by time: x(t) = R x φ(t) + r(t)cos(φ(t))] y(t) = R y φ(t) + r(t) sin(φ(t))] := C xy x(t) = R x + r(t)cos(φ(t))] y(t) = R y φ(t) + r(t) sin(φ(t))] := C y 2 / x(t) = R x φ(t) + r(t)cos(φ(t))] y(t) = R y + r(t)sin((p(ty) ) ' " x x(t) = R x + r(t)cos(φ(t))] y(t) = R y + r(t)sin(φ(t))] := C where t is time, R x and R yare constants, r(t) is any function of time or a constant, and pt) is any function of time or a constant; and the state ( 'iQ<)) of each aircraft (1021) includes a position of the aircraft (1021) expressed by so-called cyclic coordinates (A:), (fc) related to Cartesian coordinates Xj(fc),yj(fc) of this position by i equation of the reference cycloid curve; the process further comprises supplying (504) a series of reference phase shifts (Sp re f..(l ) between all aircraft (1021) on the horizon depth (N p ); And The cost function ( / ) is designed to achieve phase shift consensus by including a term minimizing phase shift errors (&(Pij(T)) between predicted phase shifts ( x ; (T)) calculated from the states predicted + 1)) and the reference phase shifts (p rof ( / )), for all pairs of aircraft. [6] Computer program (608) downloadable from a communication network and / or stored on a computer-readable medium, characterized in that it includes instructions for carrying out the steps of a process (500) according to claim 5, when said program is executed on a computer.

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