Three-Dimensional Trajectory Optimization in Constrained Airspace

Transcription

1 Three-Dimensional Traectory Optimization in Constrained Airspace Ran Dai * and John E. Cochran, Jr. Auburn University, Auburn, Alabama The operational airspace o aerospace vehicles, including airplanes and unmanned aerial vehicles, is oten restricted so that constraints on three-dimensional climbs, descents and other maneuvers are necessary. In this paper, the problem o determining constrained, three-dimensional, minimum-time-to-climb and minimum-uel-to-climb traectories or an aircrat in an airspace deined by a rectangular prism o arbitrary height is considered. The optimal control problem is transormed to a parameter optimization problem. Since a helical geometry appears to be a natural choice or climbing and descending traectories subect to horizontal constraints, helical curves are chosen as starting traectories. A procedure or solving the minimum-time-to-climb and minimum-uel-to-climb problems by using the direct collocation and nonlinear programming methods including Chebyshev Pseudospectral and Gauss Pseudospectral discretization is discussed. Results obtained when dierent constraints are placed on airspace and state variables are presented to show their eect on the perormance index. The question o optimality o the numerical results is also considered. I. Introduction Since the 96 s, the minimum time-to-climb (MTTC) and minimum-uel-to-climb (MFTC) problems have attracted the interest o many researchers. Most early investigations were ocused on two-dimensional (-D) MTTC or MFTC ormulated as an indirect optimal control problem leading to a two-point-boundary-value-problem (TPBVP) model. To obtain numerical solutions to these problems, Bryson and Denham used the steepest-ascent method, while Calise applied singular perturbation techniques, and Ardema 3 used matched asymptotic expansions * Graduate Research Assistant, Department o Aerospace Engineering, Auburn University, Auburn, AL Proessor and Head, Department o Aerospace Engineering, Auburn University, Auburn, AL 36849, Fellow AIAA. American Institute o Aeronautics and Astronautics

2 to get approximate analytical solutions. Although these and other investigators 4, were successul, it is well nown that inding the solutions to indirect TPBVPs is very diicult because the solution is very sensitive to changes in the initial values o the adoint variables. In most cases, when random initial input is used, more iterations are required and/or convergence to a meaningul solution is not achieved. Considering the complexity o the adoint equations or three-dimensional (3-D) MTTC and MFTC problems, it is not surprising that estimating o the initial values o the adoint variables is very diicult. With the advancement o computing power, the use o direct collocation with nonlinear programming (DCLP) to convert TPBVPs into nonlinear programming problems (LPPs) a so-called direct optimal control method is easible and has been applied widely. 6- By discretizing the traectory into mutiple segments, characterized by state and control variables as parameters, a TPBVP is transormed into a problem o determining the parameters that satisy the constraints. In -D MTTC problems, this means inding a parameterized load actor or angle-o-attac as the control, while minimizing the perormance index: the inal time. Hargraves and Paris applied the collocation method to solve a -D MTTC problem. Their results show that in order or an airplane with modest perormance capabilities to climb to a desired altitude, the required horizontal proection o its traectory is similar in magnitude to its inal altitude. In some cases, such long horizontal distances are not available. These restrictions may come rom local terrain constraints, radar coverage constraints, or collision avoidance constraints. Considering these limitations, aerospace vehicles cannot be assumed to be ree to ly anywhere in a given airspace. Instead, a no-ly zone can be deined geometrically. Then, climbs, descents and other maneuvers that satisy the constraints must be determined in three dimensions. When an airplane is constrained to ly in a constrained airspace, it may expend considerably more uel in achieving the desired terminal conditions. Generally, one would expect that a 3-D MTTC or MFTC traectory would have a longer magnitude horizontal proection. That is, considerable turning may be required. Intuitively, a helical traectory is a reasonable initial guess or an optimal path and helical curves have been used in military and transport aircrat landing 3 to eep aircrat within sae areas and prevent collisions with other airplanes. Here, we address the problem o optimal traectories based on helical irst guesses. In what ollows, we present a version o the 3-D MTTC and MFTC problem and our method o solution. Then, we give some results or dierent boundary conditions and constraints. Conclusions based on the results or the MTTC and MFTC problems are then presented. American Institute o Aeronautics and Astronautics

3 II. Aircrat Mathematical Model The state equations or a three-dimensional point-mass aircrat model 4 that is commonly used or ormulating MTTC and MFTC problems in a lat Earth-ixed reerence rame are listed in Eq. () and illustrated in Fig. : V& = & γ = [( T D) / W sin γ ] ( g / V )[ nv cosγ ] ( g / V )[ n / cosγ ] & χ = h& = V sin γ x& = V cosγ cos χ y& = V cosγ sin χ h g () Here, V is the light speed; γ is the light path angle; χ is the heading angle, h is the altitude, x is the down range o the airplane, and y is the cross range o the aircrat. µ L r T r Also, T is the magnitude o the thrust, which is assumed to be aligned with the velocity and is determined by Mach D r γ number and altitude; D is the drag; W is the weight o the W r aircrat; and M is the light Mach number. The two control variables are n and n, the vertical and horizontal load V h actors, respectively. The resultant load actor is = nv n + n h and the ban angle o the aircrat is χ Fig. Three-Dimensional Aircrat Model. ν = arctan( n h / n V ). The lit and drag are assumed to be given by: L = ρv SC α L α () and D = ρv S + [ η α ] C D C L α (3) 3 American Institute o Aeronautics and Astronautics

4 L = nw respectively. By setting the zero-lit axis in Eq. (3) by:, we may replace α, the angle o attac which is measured rom the velocity to D = ρv n W + η C V SCD L ρ α S (4) The lit and drag coeicients and C, drag due to lit actor CL α D η, together with the airplane s weight and wing area S that were used to obtain the numerical results given in this paper. The atmospheric density, ρ, is derived rom the 976 U.S. Standard Atmosphere. The thrust magnitude and aerodynamic data is given by tables in Re. [4] and reproduced in Table and o this paper. Table Thrust as a unction o altitude and Mach number rom Re. 4 or Aircrat Thrust T (thousands o lb) Mach Altitude h (thousands o t) o. M Table Lit and drag coeicients as a unction o angle o attac and Mach number or aircrat M C L α C D S = t, W = 34lb, η = III. Direct Collocation and onlinear Programming A. Chebyshev Pseudospectral Discretization Method The basic idea o Direct Collocation (DC) is to discretize a continuous solution to a problem represented by state and control variables by using linear interpolation to satisy the dierential equations. In this way an optimal control 4 American Institute o Aeronautics and Astronautics

5 problem (OCP) is transormed into a nonlinear programming problem (LPP). Since the exact solution to the OCP is in terms o an ininitely many values o state and control variables, DC is an approximation. The well-nown discretization methods are trapezoidal, Hermite-Simpson and Runge-Kutta methods, but these methods and some higher order-discretization techniques 9 put the constraints on the deect phase between two adacent nodes and the distribution o the nodes is arbitrary. That is, it can be dense in one area and sparse in another. On the other hand, we expect that the ideal distribution o the nodes should be airly uniorm within the time interval considered. Thus, the Chebyshev Pseudospectral method (CPM) 6-8, which uses Chebyshev-Gauss- Lobatto (CGL) collocation to locate these points, may perorm this tas better. The standard interval considered here is denoted as τ [,] with collocation points τ set as τ = cos( π / ), =, K, () By using a linear transormation, the actual time t can be expressed as a unction o τ via t = [( t t ) τ + ( t + t )] / (6) where t is the initial time and t is the inal time. The state variables x (τ) and control variables u(τ) can then be approximated by th-order Lagrange interpolating polynomials x u ( τ) x φ ( τ) = ( τ) u φ ( τ) = (7) (8) where φ ( τ) = τ τ l= τ τ l l (9) By using Eq. (7), we can express the time derivative o x τ ) as the ollowing product o a ( + ) ( + ) ( matrix D and x (τ ) American Institute o Aeronautics and Astronautics

6 s = d dt x d ( τ ) = x φ ( τ ) = D x dt = =, =, K, () Appropriately, the matrix D is called the dierentiation matrix and has the ollowing explicit orm or the Chebyshev spectral dierentiation matrix: D ( c = / c )[( ) ( ( + /( τ τ )], + ) / 6, τ / ( τ ), + ) / 6, = = = = otherwise () c = c = c = = c = where and K. Unlie the trapezoidal and Hermite-Simpson collocation methods which are based on orcing the deect vector o mid points to be zero to satisy the system dierential equations, the CPM enorces constraints directly at the CGL points selected by d t t = s ( x, u, t ) = =, K, () Then, the derivative o the state variables can be calculated using these nodes themselves with the dierentiation matrix. In this way, the CPM generally achieves a higher degree o accuracy using orthogonal polynomials instead o the numerical integration polynomials. B. Gauss Pseudospectral Discretization Method The Gauss Pseudospectral Method 9, (GPM) also uses Lagrange interpolating polynomials to approximate the state and control variables. The GPM diers rom the CPM in that its discretization nodes are not exactly the collocation points. In GPM, collocation points are evaluated at the Legendre-Gauss points that lie on the universal time interval τ [,]. The discretization points include the collocation points, initial point at t starting time and inal point at ending time t. Then, the state and control variables are approximated by x( τ) = x φ ( τ) (3) u( τ) = u φ ( τ) (4) 6 American Institute o Aeronautics and Astronautics

7 and the system equality constraints at these collocation points are expressed as = t t d Di xi ( x, u, t ) = =, K, = i () There are additional constraints that must be enorced at the inal time state variable x by the Gauss Quadrature integration o the dynamics over the entire time interval d t t = x x w ( x, u, t ) = = (6) Here, w is the Gauss weights and is deined as w = [ P& ( )] τ τ (7) where P & is the derivative o the Legendre polynomial o degree. As illustrated in reerences [] and [], the costate mapping when GPM is used has higher accuracy than when other pseudospectral methods are applied. Hence, it was introduced above to evaluate the costates and show, latter, that certain optimality conditions are satisied. C. LP Solver: SOPT 6. The nonlinear programming (LP) solver used to solve the LPP considered in this wor is based on a Sequential Quadrature Programming (SQP) algorithm and is called SOPT, 3. SOPT can be used to solve problems lie the ollowing: Minimize a perormance index J (x), subect to constraints on individual state and/or control variables x x L x U (8) constraints deined by linear combinations o state and/or control variables: b Ax L b U (9) and/or constraints deined by nonlinear unctions o state and/or control variables: c c( x) L c U () 7 American Institute o Aeronautics and Astronautics

8 The 3-D MTTC problem may be transormed into this standard orm. The problem is to minimize J = t t with all LP variables x = [ x, x, x, Lx, x+, Lx, x, u, u, u, Lu, u+, L, u, u, t, t ] [ x, x, x, Lx, x+, L, x 3, x, x, u, u, Lu, u+, L, u 3, u, t, t ] where: lower bound x upper bound, subect to the nonlinear constraints: incpm ingpm () d = cu = c L = () where =, K, in CPM and =, K, in GPM. In this way, the dynamic system equations will be deined as nonlinear constraints together with state variable constraints. IV. Two-Dimensional Minimum-Time-To-Climb Problem To illustrate the dierences in -D and 3-D traectories, it is best to start with the -D MTTC problem. Here, the state variables are reduced to our, V, γ, h and x. The one control variable is n, the load actor, the reormulated equations are [( T ( M, h) D( M, h, n) )/ W sin γ ] ( g / V )[ n cosγ ] V& = & γ = h& = V sin γ x& = V cosγ g (3) The other properties are the same as or the 3-D problem. An example optimal traectory was calculated using the DCLP method with initial and inal conditions deined as V () = 8. t / sec h() = γ () = ht = 3, t (4) 8 American Institute o Aeronautics and Astronautics

9 The minimum time obtained is 7.8 sec with boundary constraints and system equality constraints well satisied. The MTTC traectory o Altitude-Down Range plot is shown in. 3 x 4 Fig.. It can be seen rom the ootprint that in order or an aircrat with modest perormance most o the time to climb rom sea level to the desired altitude o 3, t, its horizontal displacement will be almost hal o the altitude gained even Altitude (t) Down Range (t) with a ast initial speed. That is because most o the time the aircrat s light path angle is less than π / 3, maing the Fig. -D MTTC Altitude and Down Range Traectory Using DCLP Method proection o the traectory as long as almost hal o the magnitude o the altitude change. I the initial speed is lower, the down range proection will be even longer. In some cases, such long horizontal distance is not available. In those cases it is necessary or the aircrat to mae some turns to avoid the violating the airspace restrictions. Hence, the 3-D aircrat model above is used to determine the maneuvers and types o traectories needed or climbs within constrained airspace. V. Initial LP Variable Inputs Although in most cases the choice o initial input o the LP h variables is arbitrary, a good guess will improve the rate o convergence and probability o getting a good result. Assuming that the aircrat is climbing inside a cylinder with square proection in x-y plane, to allow or a large enough horizontal light distance, the inal traectory will mae a good use o the constrained airspace as much as possible. From the D optimal traectory, the horizontal proection is almost hal o its inal χ γ X altitude. Due to the space constraints, the 3D traectory horizontal proection is expected to be greater than that o a D Y Fig. 3 Circular helix curve wrapped on a cylinder. traectory that had the same inal altitude. That is because part o the lit orce in D traectory is spent on generating the turning orce to avoid collision with the bounds, which 9 American Institute o Aeronautics and Astronautics

10 maes the aircrat travels longer horizontal distance to achieve the same altitude. The 3D horizontal proection on the x-y plane is a circular curve with the same radius as one-hal a side o the base i the aircrat is expected to travel as ar as possible in designed turning angle. Expanding the horizontal proection in vertical direction turns to be a helical curve wrapped on a right-circular cylinder with radius R enclosed within the square cylinder, as shown in Fig. 3. As a starting point, we assume that the helix curve is transversed with constant velocity and inclination angle. Then, the system equations o motion can be simpliied to get V& = V = VI & γ = γ = γ I & χ = VI cosγ / R h& = V sin γ h = V t sin γ x = R sin γ y = R cosγ I () where VI and γ I are the pre-assumed constant velocity and constant light path angle o the aircrat. ormally, they are assumed to be the tae-o velocity and light path angle, separately. I the time intervals are chosen as Chebyshev or Gauss points, then, all the initial LP variables can be ixed according to Eq. (). In all o the ollowing cases, the irst initial guess uses discretization nodes selected according to this simpliied model. Ater that the number o nodes is doubled and these nodes are interpolated using the results obtained in the irst calculation. Finally, the traectory is reined with a high number o nodes and relatively ast convergence. VI. Results or 3-D MTTC and MFTC problems A. 3-D MTTC problem In this section, we present some results or 3-D MTTC problems in the orm o a collection o traectories in dierent types o constrained airspaces. The boundary constraints and perormance limitations are those listed in Table 3. Case : Set the airspace constraints on x and y as ±, t. The aircrat is required to ly rom sea level to an altitude o h = 3, t with an initial light speed o Mach =., an initial light path angle o γ =.6 o, and an initial heading angle o χ =. The minimum time obtained or this case is sec using CPM and sec using GPM. The three-dimensional traectory is shown in Fig. 4 and the time histories o vertical and horizontal American Institute o Aeronautics and Astronautics

11 Table 3 Boundary Conditions and Perormance Limitations or 3-D MTTC Problem Constraints Values Initial Coordinate Constraints x = x, y = y, h = Initial V,γ AD χ Constraints V = V, γ = γ, χ = χ Final Altitude h = h x x x y y y Airspace Constraints Maximum and Minimum Vertical Load Factor = v max Maximum and Minimum Horizontal Load Factor = h max n load actors and, velocity V, light path angle V n h min max, min max n, n v = min n, n h = min γ and heading angle χ are shown in Fig. or CPM. The corresponding results or GPM are shown in Fig. 6 and Fig. 7. From the plots, it is obvious that the two discretization methods produce very similar results. For conciseness, in the additional cases, only the CPM plots are presented and both the CPM and GPM results are summarized and compared in Table 4. The perormance index, minimum time, increased 6.9 sec that is 3.8% o the time required in D MTTC results. So the constraints on the airspace volume caused obvious dierence on the perormance index. 3 n v n h. - - Altitude(t)... Y(t) X(t). v (t/sec) heading angle (rad) light path angle (rad) Thrust (lb) - 4 x 4 Fig. 4 3-D MTTC traectory or case using CPM Fig. Control and State Variables History or case using CPM Case : When the inal 3, t altitude was reached under the same initial condition o Case and with the constraints on both x and y o ±7, t, the minimum time calculated here is.4 sec. The traectory and corresponding state and control variable time histories are shown in Fig. 8 and Fig. 9, respectively. It can be seen American Institute o Aeronautics and Astronautics

12 when the wide square base is not available, instead the width is constrained to 7% o the original one, the minimum time increases by 39.3%. 3 n v n h Altitude(t) Y(t) X(t). v (t/sec) heading angle (rad) - light path angle (rad) Thrust (lb) x 4 Fig. 6 3-D MTTC traectory or case using GPM Fig. 7 Control and State Variables History or case using GPM n v n h Altitude(t).... Y(t) X(t). v (t/sec) heading angle (rad) light path angle (rad) Thrust (lb) - 4 x 4 Fig. 8 3-D MTTC traectory or case using CPM Fig. 9 Control and State Variables History or case using CPM Case 3: When the inal light speed constraint o Mach =.8, is added to Case, the optimal climb time changes to 7.8 sec. The traectory and corresponding state and control variable time histories are shown in Fig. and Fig. respectively. The speed proile in Fig. 9 o Case approaches to zero at the ending point, the constraint on the inal speed avoids this stall point but results in the sacriice o spending 39.96% more time in the climb than in Case. American Institute o Aeronautics and Astronautics

13 Altitude(t) 3... n v v (t/sec) - n h light path angle (rad) - -. Y(t) X(t). heading angle (rad) Thrust (lb) 4 x 4 Fig. 3-D MTTC traectory or case 3 using CPM Fig. Control and State Variables History or case 3 using CPM B. 3-D MFTC Problem In the 3-D MTTC problem, the aircrat s weight is treated as a constant. Actually, the consumption o the uel during this time interval maes the weight o the aircrat a unction o time. The change o the weight is omitted in above discussion because the climb tas is completed in a couple o minutes which maes this change o weight small. However, multiple climbs may be necessary in a given mission and uel consumption may be the principal concern. Here, the variation o weight is included in the model by assuming that m & = = T / cg (6) c where =,8 sec and is the uel consumption rate and the mass is added as additional state variable to the pervious 6 state variable vector. The other system equations are symbolically the same with the weight considered W as a unction o time (t) with the initial value, =34, lb. The obective unction is the inal weight W J = W (7) Results or some cases are tested with the same boundary constraints and perormance limitations used in previous MTTC problems were obtained or comparison. Case 4: The same initial and inal conditions and airspace constraints as in Case with unspeciied climb time were used. Transversal o the MFTC traectory required similar time, 93.3 sec, when compared to Case with inal weight o 33,46. lb and uel consumption o 63. lb. I other aircrats are involved in this constrained space at the same time, the volume speciied here is not available all the time due to other traic occupying part o this 3 American Institute o Aeronautics and Astronautics

14 space. To prevent collision with other traic, the aircrat taing-o at latter time has to mae horizontal circles or climb with constrained light path angle to allow or the earlier taing-o aircrats climbing irst. In order to estimate the eect o this constraint on the uel consumption, a maximum light path angle o 36 o is added to Case 4 and the traectory, corresponding state and control variable time histories are shown in Fig. and Fig. 3 together with the results o ree light path angle, respectively. The inal weight o the aircrat with light path angle constraints is 33,439.6 lb and the uel consumption is 76.4 lb which is 6.36% more than the single aircrat consumption. 3 n v n h. - - Altitude(t).. v (t/sec) light path angle (rad) -. Y(t) X(t). heading angle (rad) Thrust (lb) 4 x 4 Fig. 3-D MTTC traectory or case 4 with ( L ) and without (o ) light path angle constraint using CPM Fig. 3 Control and State Variables History or case 4 with (L ) and without ( ) light path angle constraint using CPM Case : The above MFTC case considers only unspeciied inal time, the LP solver will choose the optimized time to get the minimum uel results. When the inal time is speciied as sec in Case 4, the consumption o uel is more than previous results and the inal weight is reduced to 33,7. lb and the uel consumption increased to 7. lb. The traectory with light path angle constraints and the designated light time o sec will end with 33,48. lb and a uel consumption o lb which is 4.9% more than the consumption without light path angle constraints. The MFTC traectories, corresponding state and control variable time histories with and without light path angle constraints are shown in Fig. 4 and Fig., respectively. 4 American Institute o Aeronautics and Astronautics

15 3 n v n h Altitude(t).... Y(t) X(t). v (t/sec) heading angle (rad) - light path angle (rad) Thrust (lb) x 4 Fig. 4 3-D MTTC traectory or case with ( L ) and without (o ) light path angle constraint using CPM Fig. Control and State Variables History or case with (L ) and without ( ) light path angle constraint using CPM Table 4 CPM and GPM optimal results comparison Perormance Case Case Case 3 Case 4 Case CPM t min = sec t.4 min = sec t min = 7. 8 sec Wmax = 33,46. lb W max = 33,7. lb GPM t min = sec t min =. 78 sec t min = sec W max = 33,46. lb W max = 33, lb C. Optimality In order to show that the results obtained rom the DCLP method are optimal within the precision o the numerical calculations, we consider the Hamiltonian ormulation o the optimal control problem. I the solutions are optimal, then the DCLP discrete state and control variables should be good approximations to the solutions to the indirect, 4 optimal control problem at the collocation points. There are ways to estimate the costates and Lagrange multipliers related to the path constraints at the collocation points according to dierent discretization methods. In the two methods used above, only the GPM has been used to address the problem o estimating the costate. However, since the results obtained or the states and controls using the CPM and the GPM methods are very close, proving those rom one are optimal should be suicient. In the ollowing, the GPM traectory optimality proo will be provided. The Hamiltonian or the 3-D MTTC problem may be written as American Institute o Aeronautics and Astronautics

16 H T D = + λv ( W + λ χ h ( g / V )[ n /cosγ ] h sinγ ) g + λ y x γ + λ V + λ V sinγ λ V cosγ sinχ +υ C ( g / V )[ n cosγ ] V cosγ cosχ, =, K, (8) = [ λv, λ γ, λ χ, λ h, λ x, λ y ] T The costate vector is λ is the costate vector and υ is the Lagrange multiplier vector associated with path constraints vector C at collocation point. When an optimal solution is obtained by the LP solver, it will provide costates ~ λ together with Lagrange multipliers υ ~ at each collocation point which are called Karush-Kuhn-Tucer (KKT) multipliers. These KKT multipliers satisy the necessary optimal conditions o the LP problem ormulated above. Then, the costates and the Lagrange multipliers at collocation points can be estimated by using the KKT multipliers in the ollowing simple transormation ~ λ ~ λ = + λ w ~ λ( t ) = λ, υ ~ υ = t t w ~ λ( t ) = λ (9) Pontryagin s Minimum Principle requires the control variables to minimize the Hamiltonian along the optimal traectory at every point in time. Also, the Hamiltonian inal value conditions indicate that or a minimum time problem H ( t ) =. From Eq. (9), it can be seen that the Hamiltonian is not explicitly a unction o time, so it is constant. Hence, the Hamiltonian should be identically zero. For the Case 3-D MTTC problem, the costates and the Hamiltonian were calculated and are shown in Fig. 6. lambda v lambda ai lambda g ama lambda h x Fig. 6 Costates and Hamiltonian or case using CPM H. In this case, the Hamiltonian is very close to zero at all discretization points. This result along with the approximate equality o the states is suicient evidence o the optimality o the DCLP solution. The other cases can be veriied in the same way. D. Discussion o Results 6 American Institute o Aeronautics and Astronautics

17 The inal traectories are similar to a helical curve wrapped on a cylinder, which maes the guess o the helical traectory as an initial input reasonable. The CPM and GPM methods showed very close results. Case and reached the same inal altitude but had dierent constrained airspace, it was shown that smaller constraints will cost more time or an aircrat to climb to a desired altitude. The optimized traectory can be treated as a process o climb-dive-climb. The aircrat stays at sea level initially to gain speed and then maes a ast climb. In some cases the aircrat s speed will reduce to close to zero beore it reaches the inal altitude. To complete the climb tas, it is necessary to mae a dive to gain speed and then climb again. Cases and 3 illustrate this relation. When higher altitudes are required, more dives and climbs may be perormed. That s why the traectory is a repeat process o climb-dive-climb. In MFTC problem, the traectory o Case 4 is similar to the traectory o Case and the mass o the aircrat changes in a small scale which can be negligible. In order to obtain the obective o minimum uel consumption in the climb procedure, it is expected that the aircrat can reach the inal altitude in the least time to consume as little as possible uel. From this point o view, it is reasonable that the MTTC traectory and MFTC traectory are close to each other i the tas i completed in a short period o time. The results or Case show that any speciied time larger than that or the unspeciied inal time result will cost more uel consumption. From all o the presented traectories and corresponding perormance index data, it can be seen that a smaller volume o the airspace constraints or additional bounds o the state variables will mae the previous traectory deviate rom the original optimized solution and cost more time or uel to achieve the same altitude. The time increase percentage between Case and, Case and 3 shows the eect o airspace constraints and inal speed constraints on the climbing time. The uel consumption increase percentage o Case 4 and Case with and without light path angle constraints shows the eect o the availability o the speciied airspace on the uel consumption. All o these data illustrates that a small dierence o the speciied constraints will cost relatively high percentage o perormance index to achieve the same obective while satisying the new constraints. VII. Conclusions The contribution o this paper includes two sides. Theoretically, it expands rom two dimensional aircrat model to three dimension and starts rom helical curve as initial guess, then uses the Chebyshev Pseudospectral method, Gauss Pseudospectral method and nonlinear programming solver to solve the three-dimensional minimum-time-to- 7 American Institute o Aeronautics and Astronautics

18 climb and minimum-uel-to-climb problems under dierent assumption conditions. The optimality o the traectories was considered and numerical evidence o the optimality was obtained by estimating the costate variables and the Hamiltonian. The results show that the perormance index, the climb time, may be ound while system equality constraints, boundary constraints and control constraints are satisied. Practically, this paper considers dierent constraints eect on the perormance index and illustrates their importance on maintaining a high perormance maneuver which will improve the eiciency in aircrat tas perorming. Future research will ocus on a moving target in a small area under same constraints, which means a eedbac o the sensed target state is required to orm a closed loop system to catch this moving target. Reerences [] Bryson, A. E. and Denham, W. F., A Steepest-Ascent Method or Solving Optimum Programming Programs, Journal o Applied Mechanics, Vol. 9, June 96, pp [] Calise, A. J., Extended Energy Management or Flight Perormance Optimization, AIAA Journal, Vol., o.3, March 977, pp34-3. [3] Ardema, M. D., Solution o the Minimum-Time-to-Climb Problem by Matched Asymptotic Expansions, AIAA Journal, Vol. 4, o. 7, 976, pp [4] Bryson, A. E. and Desai, M.. and Homan, W. C., Energy-State Approximation in Perormance Optimization o Supersonic Aircrat, Journal o Aircrat, Vol. 6, o. 6, 969, pp [] Rao, A. V. and Mease, K. D., Minimum Time-to-Climb Traectories Using a Modiied Sweep Method, AIAA Paper CP, 99. [6] Ringertz, U., Optimal Traectory or a Minimum Fuel Turn, Journal o Aircrat, Vol. 37, o.,, pp [7] orsell M., Multistage Traectory Optimization with Radar-Range Constraints, Journal o Aircrat, Vol. 4, o. 4,, pp [8] Williams, P., Sgarioto, D. and Trivailo, P. M., Constrained Path-Planning or an Aerial-Towed Cable System, Aerospace Science and Technology, Vol., o., 8, pp [9] Herman, A. L. and Spencer, D. B., Optimal, Low-Thrust Earth-Orbit Transers Using Higher-Order Collocation Methods, Journal o Guidance, Control, and Dynamics, Vol., o.,, pp [] Horie, K. and Conway, B. A., Optimal, Aeroassisted Orbital Interception, Journal o Guidance, Control, and Dynamics, Vol., o., 999, pp American Institute o Aeronautics and Astronautics

19 [] Geiger, B. R., Horn, J. F. DeLullo, A. M. AD Long, L.., Optimal Path Planning o UAVs Using Direct Collocation with onlinear Programming, AIAA Guidance, aviagtion, and Control Conerence and Exhibit, Keystone, Colarado, AIAA Paper 6-699, 6. [] Hargraves, C. R. and Paris, S. W., Direct Traectory Optimization Using onlinear Programming and Collocation, Journal o Guidance, Control and Dynamics, Vol., o. 4, 987, pp [3] Craword, D. J. and Bwoles, R. L., Automatic Guidance and Control o A Transport Aircrat During A Helical Landing Approach, ASA Technical Report, ASA T D-798, 97. [4] Kumar, R. K., Seywald, H., and Cli, E. M., ear-optimal Three-Dimensional Air-to-Air Missile Guidance Against Maneuvering Target, Journal o Guidance, Control, and Dynamics, Vol. 8, o. 3, 99, pp [] Betts, J. T. and Human, W. P., Path-Constrained Traectory Optimization Using Sparse Sequential Quadratic Programming, Journal o Guidance, Control, and Dynamics, Vol. 6, o., 993, pp [6] Fahroo, M. and Ross, I. M., Direct Traectory Optimization by a Chebychev Pseudospectral Method, Journal o Guidance, Control, and Dynamics, Vol., o.,, pp [7] Treethen, L.., Spectral Methods in MATLAB, Society or Industrial and Applied Mathematics, Philadelphia,. [8] Pietz, J. A., Pseudospectral Collocation Methods or the Direct Transcription o Optimal Control Problems, M.A. Thesis, Dept. o Computational and Applied Mathematics, Rice University, Houston, TX, 3. [9] Ben, D., A Gauss Pseudospectral Transcription or Optimal Control, Ph.D. Thesis, Department o Aeronautics and Astronautics, MIT, ov. 4. [] Benson, D. A., Huntington, G. T., Thorvaldsen, T.P., and Rao, A. V., Direct Traectory Optimization and Costate Estimation via an Orthogonal Collocation Method, Journal o Guidance, Control and Dynamics, Vol. 7, o. 3, 4, pp [] Huntington, G. T., Benson, D. A. and Rao, A. V., A comparison o Accuracy and Computational Eiciency o Three Pseudospectral Methods, AIAA Guidance, avigation, and Control Conerence and Exhibit, Hilton Head, SC, AIAA Paper 7-64, August, 7. [] Gano, S. E., Perez, V. M. and Renaud, J. E., Development and Veriication o A MATLAB Driver For The SOPT Optimization Sotware, AIAA Paper -6,. [3] Holmstrom, K., Goran, A.O. and Edvall, M. M, User s Guide For TOMLAB /SOPT, Tomlab Optimization Inc.,, [4] Fahroo, F. and Ross, I.M., Costate Estimation By a Legendre Pseudospectral Method, Journal o Guidance, Control and Dynamics, Vol. 4, o.,, pp American Institute o Aeronautics and Astronautics