A New Path Planner based on Flatness Approach - Application to an atmospheric reentry mission

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In the proceedings of the European Control Conference (ECC'09), august 23-26, 2009, Budapest, Hungary

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A New Path Planner based on Flatness Approach - Application to an atmospheric reentry mission

  1. 1. A New Path Planner based on Flatness Approach Application to an atmospheric reentry mission Vincent MORIO, Franck CAZAURANG, Ali ZOLGHADRI and Philippe VERNIS presented by Vincent MORIO Automatic Control Group IMS lab/Bordeaux University, France http://extranet.ims-bordeaux.fr/aria European Control Conference (ECC'09), 23-26 August 2009, Budapest, Hungary
  2. 2. Outline ² Part I Statement of the TAEM guidance problem ² Part II Flatness-based trajectory planning ² Part III Optimal control problem convexi¯cation ² Part IV Application to US Space Shuttle Orbiter
  3. 3. Part I Statement of the TAEM guidance problem
  4. 4. Part I { Statement of the TAEM guidance problem ² Mission: external tank Insertion in low-Earth orbit of payloads and crews ² First °ight: 04/12/1981, solid rocket boosters ² Total number of °ights: 126 as of 05/11/2009, ² Mean cost per mission: from $300M to $400M (2006), ² 3 operational vehicles until 2010 (°eet retirement). main features symbol value reference area [m2 ] S 249.9 overall mass at injection point [kg] m 89930 wingspan [m] b 23.8 chord length [m] c 12 max. gliding ratio (for M · 3) (L=D)max ¼4 Ixx 1213866 inertial moments [kg=m2 ] Iyy 9378654 Izz 9759518 orbiter Ixz 228209 inertial products [kg=m2 ] Ixy 6136 Iyz 2972 xmrc 17 moments reference center [m] ymrc 0 zmrc -1.2 xcg 27.3 center of gravity [m] ycg 0 zcg 9.5 Space transportation system Orbiter STS-1 main features Slide 4 of 34
  5. 5. Part I { Statement of the TAEM guidance problem vertical rudder/ SRMS stabilizer payload bay speedbrake payload OMS/RCS baydoors OMS thrusters cockpit RCS jets elevons RCS main engines body °ap control surface symbol de°ection limis de°ection min (deg) max (deg) rates (deg/s) elevons pitching ±e -35 20 20 ailerons ±a -35 20 20 rudder ±r -22.8 22.8 10 speedbrake ±sb 0 87.2 5 body °ap ±bf -11.7 22.55 1.3 control surfaces de°ections limits and rates Slide 5 of 34
  6. 6. Part I { Statement of the TAEM guidance problem 3 main phases: ² Hypersonic entry Injection point ² Terminal Area Energy Management (TAEM) ² Autolanding phase (A&L) hypersonic TEP phase TAEM phase Earth horizon Zrwy orbiter A&L phase groundtrack Xrwy ALI Runway Yrwy HAC r adius sketch of an atmospheric reentry mission Slide 6 of 34
  7. 7. Part I { Statement of the TAEM guidance problem TEP dissipation ® ¹ ¯ S-turns lower bound [deg] 0 ¡80 ¡3 upper bound [deg] 25 80 3 HAC max. rate [deg=s] 2 5 2 acquisition HAC guidance inputs bounds and rates HAC3 homing HAC4 HAC1 Zrwy HAC2 HAC2 Objectives: ALI Xrwy ² dissipate the total energy of the wind Yrwy vehicle from entry point (TEP) heading alignment down to nominal exit point (ALI) ² align the vehicle with the extended runway centerline to ensure a safe requirements mechanical constraints autolanding max. load factor ¡max [g] < 2:5 max. dynamic pressure q max [kP a] < 16 2 kinds of constraints: kinematic constraints at ALI Mach number 0:5 altitude [km] 5 ² trajectory constraints: downrange [km] 10 dynamic pressure, load factor crossrange [km] 0 ² mission constraints: ¯nal heading [deg] headwind landing kinematic constraints at ALI °ight path angle [deg] ¡27 TAEM guidance constraints Slide 7 of 34
  8. 8. Part I { Statement of the TAEM guidance problem ² 3 dof model in °at Earth coordinates: position velocity 8 8 > V = ¡ D(®; M) ¡ g sin °; > _ < x = V cos  cos °; _ > > m < : y = V sin  cos °; _ 1 g : _ : °= _ (L(®; M) cos ¹ ¡ Y (¯; M ) sin ¹) ¡ cos °; h = V sin °: > > mV V > > Â= 1 : _ (L(®; M) sin ¹ + Y (¯; M) cos ¹) : mV cos ° where L(®; M) = qSCL0 (®; M); and q = 1 ½V 2 : dynamic pressure, 2 D(®; M) = qSCD0 (®; M); g: constant gravitational acceleration, Y (¯; M) = qSCY0 (¯; M): ½ = ½0 exp (¡h=H0 ): atmospheric density. ² the corresponding optimal control problem is given (in the state space) by: Z tf min C0 (x(t0 ); u(t0 )) + Ct (x(t); u(t)) dt + Cf (x(tf ); u(tf )) x(t);u(t) t0 t.q. x(t) = f (x(t); u(t)) ; _ t 2 [t0 ; tf ]; x(t0 ) = x0 ; u(t0 ) = u0 ; 0 · ¡ (x(t); u(t)) · ¡max ; t 2 [t0 ; tf ]; 0 · q(x(t)) · q max ; t 2 [t0 ; tf ]; umin · u(t) · umax ; t 2 [t0 ; tf ]; x(tf ) = xf ; u(tf ) = uf : Slide 8 of 34
  9. 9. Part II Flatness-based trajectory planning
  10. 10. Part II { Flatness-based trajectory planning ² Consider a nonlinear system de¯ned on a di®erentiable manifold by x(t) = f (x(t); u(t)) ; _ where x : [t0 ; tf ] 7! Rn : state of size n and u : [t0 ; tf ] 7! Rm : control inputs vector of size m. ² We consider that all the the trajectory planning objectives, de¯ned either at the mission" level or at the vehicle" level, may be classically formulated as a constrained optimal control problem (OCP) Z tf min C0 (x(t0 ); u(t0 ; t0 )) + Ct (x(t); u(t); t) dt + Cf (x(tf ); u(tf ); tf ) x(t);u(t) t0 s.t. x(t) = f (x(t); u(t)) ; _ t 2 [t0 ; tf ]; l0 · A0 x(t0 ) + B0 u(t0 ) · u0 ; lt · At x(t) + Bt u(t) · ut ; t 2 [t0 ; tf ]; lf · Af x(tf ) + Bf u(tf ) · uf ; L0 · c0 (x(t0 ); u(t0 )) · U0 ; Lt · ct (x(t); u(t)) · Ut ; t 2 [t0 ; tf ]; Lf · cf (x(tf ); u(tf )) · Uf : Slide 10 of 34
  11. 11. Part II { Flatness-based trajectory planning Advantages of °atness approach for trajectory planning applications ² minimum number of decision variables in the OCP: the optimization variables become the °at output of the system ² integration-free optimization problem: the system dynamics is intrinsically sat- is¯ed ² avoid emergence of unobservable dynamics (which may be potentially unstable) Main drawback: ² often highly nonlinear and nonconvex OCP in the °at output space State space (x(tf ); u(tf )) (x(t0 ); u(t0)) (z(tf ); z(tf ); : : : ; z(¯) (tf )) _ Á Ã (z(t0); z(t0); : : : ; z(¯) (t0)) _ Flat output space equivalence between system trajectories Slide 11 of 34
  12. 12. Part II { Flatness-based trajectory planning ² Di®erential °atness concept introduced in 1991 by Fliess, L¶vine, Martin and e Rouchon: deals with pseudo" nonlinear systems Nonlinear systems True" nonlinear pseudo" systems nonlinear systems ² speci¯c tools, ² equivalent to linear trivial systems, ² predictive control, ² feedback linearization techniques, ² nonlinear H1 , ... ² di®erential °atness. De¯nition (Di®erential °atness (Fliess et al., 1995))). The nonlinear system x = f (x; u) is di®erentially °at (or, shortly °at) if and only if there exists _ a collection z of m variables, whose elements are di®erentially independant, de¯ned by: ³ ´ (®) z = Á x; u; u; : : : ; u _ ; such that ½ ¡ ¢ x = Ãx ¡z; z; : : : ; z (¯¡1) _ ¢ u = Ãu z; z; : : : ; z (¯) _ where Ãx and Ãu are smooth applications over the manifold X, and ® = (®1 ; : : : ; ®m ), ¯ = (¯1 ; : : : ; ¯m ) are ¯nite m-tuples of integers. The collection z 2 Rm is called a °at output (or linearizing output). Slide 12 of 34
  13. 13. Part II { Flatness-based trajectory planning ² the equivalent optimal control problem in the °at output space is given by Z tf min C0 (Ãx (z(t0 )); Ãu (z(t0 )); t0 ) + Ct (Ãx (z(t)); Ãu (z(t)); t) dt z(t) t0 +Cf (Ãx (z(tf )); Ãu (z(tf )); tf ) s.t. l0 · A0 z(t0 ) · u0 ; lt · At z(t) · ut ; t 2 [t0 ; tf ]; lf · Af z(tf ) · uf ; L0 · c0 (Ãx (z(t0 )); Ãu (z(t0 ))) · U0 ; Lt · ct (Ãx (z(t)); Ãu (z(t))) · Ut ; t 2 [t0 ; tf ]; Lf · cf (Ãx (z(tf )); Ãu (z(tf ))) · Uf : where the °at output ³ ´ (®) z = Á x; u; u; : : : ; u _ satis¯es 8 ³ ´ < x = Ãx z; z; : : : ; z _ (¯¡1) ; ³ ´ : u = Ãu z; z; : : : ; z (¯) : _ (2) (2) ² OCP decision variables: z = (z1 ; : : : ; zm ; z1 ; : : : ; zm ; : : : ; z1 ; : : : ; zm ; : : :) _ _ Slide 13 of 34
  14. 14. Part II { Flatness-based trajectory planning 1) parametrization of the OCP decision variables by means of B-spline curves q1 X z1 (t; p1 ) = c1 Bi;k1 (t) i for the knot breakpoint sequence ´1 ; i=0 q2 X z2 (t; p2 ) = c2 Bi;k2 (t) i for the knot breakpoint sequence ´2 ; i=0 . . . qm X zm (t; pm ) = cm Bi;km (t) i for the knot breakpoint sequence ´m ; i=0 where Bi;kj (t) is the zero order derivative of the i-th function associated to the B-spline basis of order kj , built on the knot breakpoint sequence ´j , and cj is i the corresponding vector of control points. 2) discretization of the optimal control problem over the time partition t0 = ¿1 < ¿2 < ¿N = tf ; where N is a prede¯ned number of collocation points. The cost functional is approximated by means of a quadrature rule. Slide 14 of 34
  15. 15. Part II { Flatness-based trajectory planning ¡ ¢ ² by setting ui , ci ; ci ; : : : ; cii (ki ¡si )+si 2 Rli (ki ¡si )+si , the set of all control points 1 2 l of the B-splines can be de¯ned by u , (u1 ; : : : ; um ) : ² the OCP constraints, evaluated at every collocation points are given by ³ ´ 1 N 1 N ¤(u) = ¤li (u); ¤nli (u); ¤lt (u); : : : ; ¤lt (u); ¤nlt (u); : : : ; ¤nlt (u); ¤lf (u); ¤nlf (u) ; 8 j > ¤lt (u) > j = At z(tj ); j = 1; : : : ; N; > > ¤ (u) > nlt > = ct (Ãx (z(tj )); Ãu (z(tj ))) ; j = 1; : : : ; N; < ¤li (u) = A0 z(t0 ); > ¤lf (u) > = Af z(tf ); > > ¤nli (u) > > = c0 (Ãx (z(t0 )); Ãu (z(t0 ))) ; : ¤nlf (u) = cf (Ãx (z(tf )); Ãu (z(tf ))) : ² the B-splines control points become the new decision variables of the nonlinear programming (NLP) problem min J (u) u2RM s.t. Lb · ¤(u) · Ub ; Xm where M = li (ki ¡ si ) + si : i=1 ² the NLP problem can be solved onboard by using NPSOL, SNOPT, KNITRO, ... Slide 15 of 34
  16. 16. Part III Optimal control problem convexi¯cation
  17. 17. Part III { Optimal control problem convexi¯cation Main objective: Convexi¯cation of the optimal control problem by deformable shapes. Motivations: ² the OCP described in the °at output space is often highly nonlinear and nonconvex (Ross, 2006) ² to guarantee global convergence of NLP solvers How? ² the convexi¯cation problem is solved by a genetic algorithm in order to get a global solution ² development of a Matlab software library (by the author): OCEANS (Op- timal Convexi¯cation by Evolutionary Algorithm aNd Superquadrics) initial feasible convex superquadric domain shape Convexi¯cation Slide 17 of 34
  18. 18. Part III { Optimal control problem convexi¯cation Superquadrics: ² generalization in 3 dimensions of the superellipses (Barr, 1981) ² used to perform a trade-o® between the complexity of the shapes and the numerical tractability in high order °at output spaces Convex "1 = 0:1 "1 = 1:0 "1 = 2:0 "1 = 2:5 "2 = 0:1 "2 = 1:0 "2 = 2:0 "2 = 2:5 examples of 3D superquadrics morphing of a 3D superquadric Drawbacks: Advantages: ² limited number of shapes ² compactness of the representation ² symetric shapes only ² an explicit parametrization exists Slide 18 of 34
  19. 19. Part III { Optimal control problem convexi¯cation Introduction of n-D transformations: rotation, translation and linear pinching Rotation initial 3D superquadric e®ect of a 3D rotation Pinching initial 3D superquadric e®ect of a linear pinching along z axis The set ª contains the sizing parameters needed to obtain a positioned, oriented and bended superquadric shape ª = f a1 ; : : : ; an ; "1 ; : : : ; "n¡1 ; ©1 ; : : : ; ©n(n+1)=2 ; d1 ; : : : ; dn ; v1 ; : : : ; vn¡1 g | {z } | {z } | {z } | {z } | {z } semi-major axes roundness par. rotation par. translation par. pinching par. Slide 19 of 34
  20. 20. Part III { Optimal control problem convexi¯cation Proposition (trigonometric parametrization of a bended n-D superellipsoid (Morio,2008)). Let S a superquadric ellipsoid of size n, described by the vector ª. Then, the corresponding trigonometric parametrization, with cartesian coordinates xi , i = 1; : : : ; n, is de¯ned by 8 n¡1 n¡1 > > Y Y > a1 (v1 sin > "p ¡1 µ "j cos µj + 1) cos"k µk ; i = 1; > > p¡1 > > j=p k=1 > > > > n¡1 Y n¡1 Y > > ai (vi sin"p ¡1 µp¡1 "j "i¡1 > > cos µj + 1) sin µi¡1 cos"k µk ; i = 2; : : : ; n ¡ 1; i 6= p; < j=p k=i xi = n¡1 > > Y > a sin"p¡1 µ > p cos"j µj ; i = p; > > p¡1 > > j=p > > > > n¡1 Y > > a (v sin"p ¡1 µ > n n > : p¡1 cos"j µj + 1) sin"n¡1 µn¡1 ; i = n; j=p where p is the pinching direction (vp = 0). In addition, the vector of anomalies µ satis¯es µi 2 [¡¼; ¼[ if i = 1 and µi 2 [¡ ¼ ; ¼ ] if i = 2; : : : ; n ¡ 1. 2 2 No. of anomalies 3D trigonometric parametrization variation of the number of anomalies Slide 20 of 34
  21. 21. Part III { Optimal control problem convexi¯cation Proposition (angle-center parametrization of a bended n-D superellipsoid (Morio,2008)). Let S be a superquadric ellipsoid of size n, described by the vector ª. Then, the corresponding angle-center parametrization, with cartesian coordinates xi , i = 1; : : : ; n, is de¯ned by 8 0 1 > n¡1 Y n¡1 Y > r(µ) @ v1 r(µ) sin µ > > cos µj + 1A cos µk ; i = 1; > > p¡1 > > ap j=p > > 0 1 k=1 > > n¡1 n¡1 > Y Y > r(µ) @ vi r(µ) sin µ > > cos µj + 1 A sin µi¡1 cos µk ; i = 2; : : : ; n ¡ 1; i 6= p; > < p¡1 ap j=p k=i xi = > > n¡1 Y > r(µ) sin µ > cos µj ; i = p; > > p¡1 > > > > 0 j=p 1 > > n¡1 > > Y > r(µ) @ vn r(µ) sin µ > cos µj + 1A sin µn¡1 ; i = n; > : p¡1 ap j=p 1 where p is the pinching direction (vp = 0). The radius r(µ) = Ân;n is given by 8 2Ã Q 3 "2 1 > ! 2 Ã Q n¡1 ! 2 > > n¡1 cos µk "1 sin µ1 cos µk "1 > Â > n;2 = 4 k=1 + k=2 5 ; j = 2; > > < a1 a2 > 2 Ã Q n¡1 !" 2 3 "j¡1 > > 2 > > 2 sin µj¡1 k=j cos µk j¡1 > Ân;j > = 4 (Ân;j¡1 ) "j¡1 + 5 ; j = 3; : : : ; n; : aj with µi 2 [¡¼; ¼[ if i = 1 and µi 2 [¡ ¼ ; 2 ¼ 2 ] if i = 2; : : : ; n ¡ 1. Slide 21 of 34
  22. 22. Part III { Optimal control problem convexi¯cation No. of anomalies 3D angle-center parametrization variation of the number of anomalies The angle-center parametrization results in a better sampling of the superquadric surface for smooth convex shapes Proposition (inside-outside function of a bended n-D superellipsoid (Morio,2008)). Let S be a superellipsoid of size n, described by the vector ª. Then, the corresponding (implicit) inside-outside function Fn (ª; x) = ¤n;n (ª; x), is de¯ned by the recursive expression Fn(ª; x) > 1 8 0 1 2 0 1 2 > > "1 "1 > > ¤ x1 x2 Fn(ª; x) = 1 > n;2 (ª; x) = @ ³ > ´A +@ ³ ´A ; > < v1 v2 a1 a xp + 1 a2 a xp + 1 p p > 0 1 2 > > ¡ "k¡2 ¢" "k¡1 Fn(ª; x) < 1 > > ¤ xk > n;k (ª; x) = ¤n;k¡1 (ª; x) k¡1 + @ > ³ ´A ; : a vk x +1 k ap p where vp = 0 in the pinching direction p. Slide 22 of 34
  23. 23. Part III { Optimal control problem convexi¯cation d(ª; x0 ) Proposition (n-D euclidean radial distance (Morio,2008)). The euclidean radial distance Q d (ª; x0 ) is de¯ned as being the distance between a point Q with coordinates x0 , and a point P P with coordinates xs , corresponding to the projection of Q onto the superellipsoid, along the direction de¯ned by the point Q and the center of the geometric shape. For an arbitrary n-D superellipsoid, described by the vector ª, the expression of the radial euclidean distance O d (ª; x0 ) = jx0 ¡ xs j is given by ¯ ¯ ¯ "n¡1 ¯ ¯1 ¡ (Fn (ª; x0 ))¡ 2 ¯ ; d (ª; x0 ) = jx0 j ¢ ¯ ¯ Proposition (volume of a bended n-D superellipsoid (Morio,2008)). Let S be a bended superellipsoid of size n, described by the vector ª. The volume Vn (ª) of S is de¯ned by 2 3 2 3 n¡1 ³" ´7 n¡1 µ ¶ 6 Y i "i X j®j + 1 p¡1 Vn (ª) = 2an 64 ai " i B ; i + 1 7 ¢4 ap¡1 "p¡1 5 v® B "p¡1 ; "p¡1 + 1 5 ; i=1 2 2 2 2 j®j=0 i6=p¡1 where the multi-index ® = (®1 ; : : : ; ®p¡1 ; 0; ®p+1 ; : : : ; ®n ) satis¯es n Y n X ® ® v = vk k ; j®j = ®j ; ®i 2 f0; 1g; i = 1; : : : ; n; k=1 j=1 In addition, the Beta function B(x; y) is linked to the Gamma function by Z ¼=2 ¡(x)¡(y) B(x; y) = 2 sin2x¡1 Á cos2y¡1 ÁdÁ = ; 0 ¡(x + y) the Gamma being typically de¯ned by Z 1 ¡(x) = exp¡t tx¡1 dt; 0 Slide 23 of 34
  24. 24. Part III { Optimal control problem convexi¯cation We assume that the nonconvex domain may be described by means of one or more analytical expressions de¯ned by fmin · fnc (x) · fmax ; where x is a set of variables of size n. Problem (superellipsoidal annexion problem (Morio,2008)). Let S be a superellipsoid of size n, described by the vector ª. The superellipsoidal annexion problem (or convexi¯cation problem) consists then in ¯nding the optimal parameters ª¤ associated to the biggest superel- lipsoid Sopt contained inside the feasible domain (supposed to be nonconvex) de¯ned by the analytical expression fnc , such that max e Vn (ª) ª 8 < Fn (ª; x) · 1; s.t. fmin · fnc (x) · fmax ; : l xi · xi · xu ; i = 1; : : : ; n: i 1 e e where the normalized superquadric volume Vn (ª) is de¯ned by Vn (ª) = Vn (ª) n , and Fn (ª; x) is the inside-outside function.The variables x are the cartesian coordinates as- sociated to a prede¯ned number of sampling points at the supersuadric surface. initial feasible convex superquadric domain shape Convexi¯cation Slide 24 of 34
  25. 25. Part III { Optimal control problem convexi¯cation start stop Initialization Best individual yes Criteria OK? no Migration Selection Reinsertion Crossover Generation of new population Fitness evalutation Mutation Multi-population extended genetic algorithm adapted to the problem at hand Slide 25 of 34
  26. 26. Part III { Optimal control problem convexi¯cation ² the convex optimal control problem in the °at output space is given by Z tf min C0 (Ãx (z(t0 )); Ãu (z(t0 )); t0 ) + Ct (Ãx (z(t)); Ãu (z(t)); t) dt convex superquadric z(t) t0 shape + Cf (Ãx (z(tf )); Ãu (z(tf )); tf ) s.t. l0 · A0 z(t0 ) · u0 ; lt · At z(t) · ut ; t 2 [t0 ; tf ]; lf · Af z(tf ) · uf ; trajectory L0 · c0 (Ãx (z(t0 )); Ãu (z(t0 ))) · U0 ; 0 · Fn (ª¤ ; z(t)) i · 1; t 2 [t0 ; tf ]; Lf · cf (Ãx (z(tf )); Ãu (z(tf ))) · Uf : where Fn (ª¤ ; z(t)), i = 1; : : : ; ns , are the inside-outside functions associated to i the optimized convex shapes. ² boundary constraints must be met: Fn (ª¤ ; z(t0 )) · 1 and Fn (ª¤ ; z(tf )) · 1. It is possible to check if the extremal points of the trajectory are lying inside the convex envelopes by computing the associated n-D radial euclidean distances ² a convex cost functional may be obtained by using the same process. Slide 26 of 34
  27. 27. Part IV Application to US Space Shuttle Orbiter
  28. 28. Part IV { Application to US Space Shuttle Orbiter Assumptions: ² °at Earth: coriolis and centrifugal forces neglected, ² symetric °ight: ¯ = 0 (typical guidance assumption), ² no cost functional considered: feasibility problem only Tabulated aerodynamic force coe±cients in clean con¯guration are approxi- mated by means of: ² principal component analysis (PCA): results in a decoupling of angle-of- attack and Mach number variables, ² analytical neural networks (ANN): parcimonious approximators of smooth multivariate functions lift coe±cient CL0 gliding ratio CL0 =CD0 drag coe±cient CD0 Slide 28 of 34
  29. 29. Part IV { Application to US Space Shuttle Orbiter ² time being not a relevant parameter during atmospheric reentry, the 3 dof model is reparameterized wrt. free trajectory duration parameter ¸ t d(:) d(:) ¿= , with 0 · ¿ · 1: normalized time (:)0 = =¸ ; ¸ d¿ dt ² the new point-mass model is given by position velocity 8 µ ¶ D > V 0 = ¸ ¡ ¡ g sin ° ; 8 0 > > < x = ¸V cos  cos °; > > µ m ¶ < : y 0 = ¸V sin  cos °; L cos ¹ g : 0 : °0 = ¸ ¡ cos ° ; h = ¸V sin °: > > mV V > > 0 :  = ¸ L sin ¹ : > mV cos ° ² this model is not °at since ¯ = 0, but the autonomous observable may be parameterized wrt. z1 = x, z2 = y and z3 = h and the parameter ¸ p 0 00 0 00 0 00 02 02 02 z1 + z2 + z3 0 z1 z1 + z2 z2 + z3 z3 states: V = ; V = p ; 02 02 02 ¸ ¸ z1 + z2 + z3 à ! 0 00 02 02 0 0 00 0 00 z3 0 z3 (z1 + z2 ) ¡ z3 (z1 z1 + z2 z2 ) ° = arctan p ; ° = p ; 02 02 02 02 02 02 z1 + 02 z2 (z1 + z2 + z3 ) z1 + z2 µ 0 ¶ 00 0 0 00 z2 z1 ¡ z2 z1  = arctan z2 ; Â0 = 02 02 : 0 z1 z1 + z2 Slide 29 of 34
  30. 30. Part IV { Application to US Space Shuttle Orbiter 0 1 µ 0 ¶ Â0 cos ° A 2m ° g cos ° a0 inputs: ¹ = arctan @ g cos ° ; ®= + ¡ ; °0 + ¸ a1 fCL0 (M)½SV cos ¹ ¸ V a1 V where CL0 (®; M) = (a0 + a1®)fCL0 (M); V0 1 ½SV 2 CD0 (®; M) equality constraint: ¤¿ (x; u) = + g sin ° + = 0; ¸ 2 m The corresponding optimal control problem in the °at output space is given by ¯nd (z(t); ¸) s.t. Ãx (z(¿0 ); ¸) = x0 ; Ãu (z(¿0 ); ¸) = u0 ; ¤¿ (Ãx (z(¿ ); Ãu (z(¿ ); ¸) = 0; ¿ 2 [¿0 ; ¿f ]; 0 · ¡ (Ãx (z(¿ ); ¸); Ãu (z(¿ ); ¸)) · ¡max ; ¿ 2 [¿0 ; ¿f ]; 0 · q(Ãx (z(¿ ); ¸) · q max ; ¿ 2 [¿0 ; ¿f ]; umin · Ãu (z(¿ ); ¸) · umax ; ¿ 2 [¿0 ; ¿f ]; Ãx (z(¿f ); ¸) = xf ; Ãu (z(¿f ); ¸) = uf ; where z = (z1 ; z2 ; z3 ; z_1 ; z_2 ; z_3 ; z1 ; z2 ; z3 ), ¿0 = 0 and ¿f = 1. Ä Ä Ä Slide 30 of 34
  31. 31. Part IV { Application to US Space Shuttle Orbiter ² example: dynamic pressure constraint along the TAEM trajectory, expressed wrt. °at outputs µ ¶ p 02 1 z3 z1 + z202 + z302 0 · ½0 exp ¡ S · q max : 2 H0 ¸ ² nonconvex constraint: exponentially decreasing spherical shape ² Inner approximation by a 5-D superellipsoid described by ª = f a1 ; : : : ; a5 ; "1 ; : : : ; "4 ; ©1 ; : : : ; ©15 ; d 1 ; : : : ; d5 ; v1 ; : : : ; v4 g: | {z } | {z } | {z } | {z } | {z } semi-major axes roundness par. rotation par. translation par. pinching par. 0 z3 qmax Vmin °>0 z3 0 0 (z1 ; z2 ) geometric interpretation Slide 31 of 34
  32. 32. Part IV { Application to US Space Shuttle Orbiter ² simple genetic algorithm tuning parameters provide good results ² the inside-outside function Fq (ª¤ ; z) is given by " µ ¶20 #0:1 µ ¶2 ¡ ¢20 0 0 z1 z2 Fq (ª¤ ; z) = 0:8:10¡4 z3 ¡ 1:2 + + 3:2:104 + 5:3z3 3:5:104 + 5:9z3 µ 0 ¶2 µ ¶2 z3 ¸ + +: ; 3:1:104 + 5:3z3 45:7 + 0:76:10¡2 z3 where ª¤ are optimal de¯ning parameters and z = (z3 ; z1 ; z2 ; z3 ; ¸). 0 0 0 individuals ¯tnesses wrt. generations approximating convex shape ² other nonconvex trajectory constraints convexi¯ed by using the same process Slide 32 of 34
  33. 33. Part IV { Application to US Space Shuttle Orbiter 3D reference trajectory projection in the horizontal plane optimized superellipsoid superellipsoid inside-outside function optimized superellipsoid Slide 33 of 34
  34. 34. Part IV { Application to US Space Shuttle Orbiter Atmospheric reentry guidance: TAEM and Autolanding phases Slide 34 of 34
  35. 35. THANK YOU FOR YOUR ATTENTION

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