On-Line Trajectory Optimization Including Moving Threats and Targets

Transcription

1 AIAA Guidance, Naviation, and Control Conerence and Ehibit 6-9 Auust 00, Providence, Rhode Island AIAA On-Line rajector Optimization Includin Movin hreats and arets hannon wi *, Anthon Calise and Eric Johnson chool o Aerospace Enineerin, Georia Institute o echnolo, Atlanta, GA Abstract Dierent methods o optimizin terrain ollowin trajectories have been investiated as an etension o earlier reduced order ormulations. his paper incorporates wind, movin threats -- such as other aircrat to be avoided -- and a movin taret -- such as a rendezvous situation -- into the basic constant velocit ormulation. Consequentl, this will allow a more realistic model to be simulated. In addition, chanin inormation, such as a threat appearin durin liht, are considered. he reduced order ormulation results in a sstem o our dierential equations with two unknown initial conditions. he solution is ound with the use o a enetic alorithm in conjunction with a conjuate-radient numerical search method. I. Introduction Hih-lin unmanned reconnaissance and surveillance sstems are now bein used etensivel in the United tates militar. Current development prorams are producin demonstrations o net -eneration unmanned liht sstems that are desined to perorm combat missions. heir use in irst-strike combat operations will dictate operations in densel cluttered environments that include unknown obstacles and threats, and will require the use o terrain or maskin. he demand or autonom o operations in such environments dictates the need or rapid onboard trajector optimization capabilit. In the earl 990s, P. K. Menon and Eulon Kim researched methods o optimal trajector path plannin or terrain ollowin and terrain maskin liht. his research produced a reduced order ormulation based on a constant velocit approach., his paper epands on the work done b Menon and Kim. wo ormulations are presented: one usin local tanent plane equations o motion and one usin simpliied equations o motion. In addition, the eects o wind, movin threats and a movin taret are added. he movin threat could consist o an object, such as an aircrat, that should be avoided durin liht. he movin taret would be a movin destination, or a rendezvous problem. Also, the addition o a pop-up threat is eamined. II. Optimal rajector Formulations he eects o wind as well as havin a movin taret and movin threat are added to the basic problem ormulated in Reerences and. his will be demonstrated in two dierent ormulations one usin the local tanent plane equations o motion and one utilizin simpliied equations o motion. Fiure depicts a sample terrain proile with the X-Y-H coordinate sstem and a local - -z coordinate sstem. he movin local coordinate sstem has its oriin on the terrain surace at a current, position with the - plane bein the tanent plane. he local tanent plane ormulation incorporates the constraint that the vehicle lies tanentiall to the local terrain directl into the equations o motion and can be written as V V = + + u(, ) A AA VA = + v(, ) A () () *AIAA member, Graduate Research Assistant, Georia Institute o echnolo, Atlanta, Georia AIAA Fellow, Proessor, Georia Institute o echnolo, Atlanta, Georia AIAA member, Assistant Proessor, Georia Institute o echnolo, Atlanta, Georia American Institute o Aeronautics and Astronautics Copriht 00 b authors. Published b the American Institute o Aeronautics and Astronautics, Inc., with permission.

2 he simpliied equations o motion are an approimation written in the local level rame and nelect the eects o the terrain slope in the position kinematics. = V + u(, ) = V sin + v(, ) () () Fiure : Relationship Between Inertial Frame and Local anent Plane A. Local anent Plane Equations o Motion In this ormulation, the equations o motion can be seen above in equations () and (). hese equations embod the constraint that at all times the vehicle lies tanentiall to the local terrain. Here, and are the north and east components, respectivel. V is the total aircrat velocit while u and v are the wind velocities in the and -directions, respectivel. he headin o the vehicle is represented b -- the headin anle measured with respect to the local tanent plane. Also, and are the partial derivatives o the terrain proile. A and A are iven b A = + (5) A + + = (6) he cost unction or this problem can be seen in the ollowin equation. J t = [( K ) K(,, t) ] + dt (7) 0 In this equation, the combined threat and terrain unction, (,,t), is iven as a unction o time as well as the position and can be deined as ollows. (,, t) (, ) + (,, t) = (8) Here, (,) is the unction or the terrain proile and (,,t) is the unction denotin the movin threat. he weihtin parameter, K, can var between 0 and and determines the relative importance o time and terrain maskin/threat avoidance used in the optimization. When K = 0, the equations are optimized with respect to time. American Institute o Aeronautics and Astronautics

3 When K is set to, the path is optimized with respect to the threats and the terrain. he Hamiltonian equation can then be iven as H V V VA sin = A u + + v A AA A (9) In this epression, and are the costate equations and the coeicient A can be seen in the ollowin equation. (, t) A K K, = (0) + he movin threat and taret equations o motion are: = V = V = V = V () () In each epression, it is assumed that the respective velocit and headin anle are known at all times. he movin taret then results in a new boundar condition. ( t) ( t) Ψ( t ) = t t () ( ) ( ) t= t In this epression, it can be seen that Ψ(t ) has an eplicit dependence on the inal time as a consequence o the act that the taret coordinates are assumed to satis equation (). hereore, or a ree inal time, the Hamiltonian equation satisies [ + sin ] t= t Ψ H( t ) () = = V t t= t Due to the movin threat, the Hamiltonian equation, (9), is eplicitl dependent on time. Given this, the optimalit condition or a solution alon an etremal arc shows that H H = = K t t (5) where t denotes the partial derivative o the penalt unction with respect to time. Assumin that the threat is constant when epressed in a coordinate sstem that is attached to the movin threat, then with the threat coordinates satisin (). hus (,, t) [ ( t), ( t) ] = (6) H = KV ( cos + ) (7) American Institute o Aeronautics and Astronautics

4 Because the inal time is ree, the boundar condition or this epression is deined in (). he optimalit condition or this problem is deined as Evaluatin this epression results in the ollowin relationship H = 0 (8) V V A = A A A (9) VA Equation (9) can then be substituted into the Hamiltonian equation, (7), to determine equations deinin the two costates, and as ollows. ( A H ) A Den A A H = (0) ( A H ) ( ) Den = () where Den = VA + A u + v Avsin () hese new epressions or the costates can then be inserted into () to result in a new boundar condition or the Hamiltonian at the inal time. H ( t ) V = V A ( A + A ) ( A + A ) Den t= t Dierential equations or the costates can be ound usin () = H = H () his ields where D + + D + Du D Dv = K D D (5) D5 + D6 + Du D7 + Dv = K D D (6) D A A V A D = (7) = (8) D VA A B VA VA B = (9) American Institute o Aeronautics and Astronautics

5 D VA B VA A = (0) D = V () 5 A D6 VA A B VA VA B = () D = VA B VA A () B = + 7 () B = + (5) B = + (6) B = + (7) Net, the time derivative o either equation (0) or () is taken and set equal to its counterpart in equation (5) or (6). he resultin epression is + u + v + u + 5u + 6v + 7v = (8) 8 where + V ( A H ) + ( A H ) 5 + ( A H ) 6 ( A H ) 7 ( A H ) 8 ( A H ) 9 ( A H ) 0 ( A H ) A KV K K = (9) = (0) = () = () = () 5 = () 6 = (5) 7 = (6) [ ( ) ] A A + + ( A ) A [ ( ) ] A sin + + cos A A ( A + ) cos + ( A A + ) ( A ) + + A A ( A ) cos A A ( + A ) cos + ( + A A ) A A ( A ) 8 A = A (7) + = (8) = A A (9) 5 = + A = + A A 6 A A = + A 7 (50) (5) (5) = (5) 5 8 = A A cos (5) 5 American Institute o Aeronautics and Astronautics

6 ( ) sin cos A + A A ( A ) A A ( ) 9 A A cos = (55) = (56) cos 0 A his solution consists o our dierential equations,,, H and, and requires two initial conditions to be ound or H and. he inal value o the Hamiltonian is known, via equation (). he solution is reached when the inal values o the Hamiltonian and position are met and the cost is minimized. When there are no movin threats, the Hamiltonian is constant in value so there are onl three dierential equations and the inal value is still known. When there is no movin taret, the inal value o the Hamiltonian is zero. B. impliied Equations o Motion he equations o motion used in the simpliied ormulation are stated above in equations () and () and are restated here = V + u(, ) = V + v(, ) hese equations are written in the local level plane and nelect the eects o the terrain slope. he cost equation or this case is the same as earlier and can be ound in equation (7). he correspondin Hamiltonian equation is thereore [ V + u] + [ V sin v] H = A + (59) + he equations overnin the movin taret and movin threat can be seen above in equations () throuh (). Evaluatin the optimalit condition stated in equation (8) or this ormulation results in the epression = (60) ubstitutin this into the Hamiltonian equation results in the ollowin costate equations ( A H ) = (6) V + u + vsin A H sin ( ) = (6) V + u + vsin hereore, the Hamiltonian evaluated at the inal time will be H ( t ) = V cos V A cos( ) ( ) ( V + u + vsin ) t= t (6) he costate dierential equations can then be ound to be = H = K u v (6) = H = K u v (65) 6 American Institute o Aeronautics and Astronautics

7 As beore, the time derivative o (6) or (6) is ound and equated to either (6) or (65). his epression can then be rearraned to result in the ollowin headin dierential equation. with ( u v ) R + Ru + Rv + R + R5u + R6v = (66) R R R R 7 KV ( ) ( ) ( ) = (67) = K (68) = K (69) R A H ( ) ( A ) H ( A ) H R ( A H ) = (70) R5 cos R6 sin = (7) = (7) = 7 (7) Aain, the inclusion o a movin taret and movin threat results in a sstem o our dierential equations with two initial parameters to be ound. C. olvin the stem o Equations As stated earlier, the problem, in its most comple orm havin both a movin taret and a movin threat, consists o our dierential equations with two unknown initial conditions. he dierential equations include the two describin the position as well as the Hamiltonian equation and the headin anle. he initial conditions or both the Hamiltonian and the headin anle are unknown. he problem is solved when the inal positions equal the movin taret inal positions, the actual inal Hamiltonian value matches its inal condition -- rom equations () and (6) -- and the cost is minimized. o solve this problem, two numerical solvin techniques were emploed. First a enetic alorithm was used to ind initial conditions that are close to the actual initial conditions needed. Net, these values are emploed in a conjuate-radient search method to ind the actual desired initial conditions. he enetic alorithm was accomplished b irst creatin a random population o ten chromosomes each consistin o nine diits -- the irst ive representin the initial Hamiltonian value and the last our determinin the initial headin anle. hen, twent more chromosomes were created b randoml mutatin the oriinal ten chromo somes. Ater the ull population o thirt was created, each was tested in the routines to determine a cost. he cost, C, was valued throuh the equation C = c dist + c J + c err (7) Here, c, c and c are weihtin values, dist is the distance between the inal position and the inal taret position, J is the cost o the run ound rom equation (7) and err is the dierence between the actual and desired inal Hamiltonian. he ten chromosomes with the lowest costs were then kept to bein the net eneration. his process was repeated or thirt enerations and the chromosome at the end with the lowest cost was used as the startin position o the conjuate search. he conjuate search method is a orm o a steepest descent search. First the radient o the surace deined b the cost -- as seen in equation (7) -- at a test point is determined. he search direction is then deined as the direction havin the steepest radient. Various points are checked alon this direction until a minimum is ound. Ater the irst step, this basic process is repeated; however, to determine the new search directions the radient inormation rom the previous run is included such that the new search directions are conjuate. 6 7 American Institute o Aeronautics and Astronautics

8 III. Numerical Results A. Real errain Data Real terrain data was acquired rom the United tates Geoloical urve to incorporate into this model. he data was ound in tabular ormat relatin the altitude to the locations lonitude and latitude, with data points spaced approimatel ever 8 eet. his data was then converted to matri orm, rom which it could then be used as. Because o the distance between the sampled altitude points in the matri, the data was then smoothed to (, ) appear more continuous and to remove discontinuities in altitude. he radients o this matri, alon both the and directions, were calculated numericall to orm matrices representin (, ) and (, ). he radients o these two matrices ielded matrices or (, ), (, ) and (, ). For this portion o the testin, it was decided to use a section o terrain near Columbus, Ohio. A proile o this terrain can be seen in Fiure. In this raph, the and -aes depict the position coordinates, measured in eet, such that the ais point north and the ais points east. he altitude o the terrain is measured alon the z-ais and is also iven in eet. his plot depicts a square plot o land, with 0,000 eet to a side. he measurements alon the and -aes are relative to a set oriin or the terrain data collected; this plot is just one small portion o the database. Fiure : errain Plot o an Area Near Columbus Ohio B. Wind Eects Net, the eects o a wind blowin were investiated. For this, a mostl lat plane with a sinle hill was used, as shown in Fiure. his terrain was ormulated usin the eponential unction r b = Ae (75) where A is the amplitude, b is a scalin actor to adjust the width and r is the distance rom an position to the center o the threat. 8 American Institute o Aeronautics and Astronautics

9 Fiure : errain with threats ormulated as an eponential unction. On the riht is an overhead view. For these lihts, the initial and inal points are (500,800) and (500,00). hereore the hill is directl between the two endpoints. With K set to, the optimal path ound will curve around the hill. ince this is a smmetric ield, there are two possible optimal paths when there is no wind. Fiure : Wind manitudes o a circulatin pattern. Here, a circulatin wind pattern was introduced to the problem. In this case, the wind moved in a circular pattern centered at the top o the hill with a decreasin speed movin awa rom the hill. he manitudes o the wind can be seen in Fiure. his plot was enerated usin equation (75); however, in this case, A and b were set to 0 and , respectivel. With the winds added, the optimal path is the option where the aircrat moves in the same direction as the circulatin wind low. Fiure 5 shows the solutions ound with the winds movin in a clockwise direction on the let and in a counterclockwise direction on the riht. In both cases, the local tanent plane equations o motion were utilized. 9 American Institute o Aeronautics and Astronautics

10 Fiure 5: Optimal path rom (500,800) to (500,00). he iure on the let shows the optimal path with a clockwise wind while the iure on the riht shows the optimal path with a counterclockwise path. C. Movin aret and hreats he simple terrain depicted above in Fiure consistin o a lat plain with a sinle hill was used to test the movin threat and taret. he initial position is aain located at (500, 800). he movin taret beins at the point (900, 00) and travels south while the two threats bein at (600, 00) and (600, 00), respectivel, and travel in a south-easterl direction. Fiure 6:Paths Generated with Consideration o a Movin aret and wo Movin hreats Usin Local anent Plane Equations o Motion Fiure 6 shows the three results usin the local tanent plane equations o motion. he beinnin position or each trajector is marked with a red circle. he movin taret is indicated with a purple line while the movin 0 American Institute o Aeronautics and Astronautics

11 threats are represented with blue lines. he irst case -- Path -- is usin just a movin taret, but no movin threats. his trajector is depicted b the red dashed line. he ellow marks the spot where Path intersects hreat at the same instant. Path represents the trajector when the movin taret and the irst movin threat are considered and is portraed b a red dot-dash line. he reen star shows the moment when the Path and hreat collide. he third path shows the trajector when both the threats are considered and is illustrated b a solid red line. Net is the case with one movin threat that intersects the oriinal trajector determined. he inal path results rom the addition o a second threat. Fiure 7:Paths Generated with Consideration o a Movin aret and wo Movin hreats Usin impliied Equations o Motion In a similar manner, Fiure 7 shows the results rom the same scenario when the simpliied equations o motion are implemented. It can be seen that the results enerated usin these two dierent sets o equations o motion are quite similar. D. Pop-up hreats he case o pop-up threats durin liht was also investiated. In this case, the optimal path is in mid-liht when a stationar threat appears. A new trajector must then be calculated. o test this, a liht throuh the Columbus terrain shown in Fiure was used, utilizin the constant velocit, local tanent plane equations o motion. In this case, a threat was added to the terrain as a sinle hill, as shown above in Fiure, with a heiht o 00 eet above the level o the terrain at that point. he results or this section can be seen in Fiure 8. In this plot, the black line depicts the oriinal trajector ound; here it oes directl throuh the new threat. hree dierent new trajectories are then shown as maenta lines. hese depict the results or three dierent times at which the threat is ound; these times are at 7. seconds, 9.6 seconds and at 9.79 seconds into the approimatel 90 second liht. In each case, this point is marked on the plot with a red star. American Institute o Aeronautics and Astronautics

12 Fiure 8: rajectories Found with a Pop-up hreat Usin Local anent Plane Equations o Motion. IV. Future Research For the time bein it has been decided to continue usin both the local tanent plane and simpliied equations o motion. Once the trajectories can be compared usin the liht simulator, the dierences between the two ormulations will be evaluated and a decision will be made about which ormulation to use in the remainder o the project. he lon-rane oal is to imbed as much o the ull vehicle dnamics as possible into the ormulation, while maintainin a tractable solution process. hose dnamics that cannot be directl accounted or will be treated usin sinular perturbation methods o analsis. V. Reerences () Kim, Eulon. Optimal Helicopter rajector Plannin or errain Followin Fliht. hesis. Georia Institute o echnolo () Menon, P.K., E. Kim, V.H.L. Chen. "Optimal rajector Plannin or errain Followin Fliht" Journal o Guidance, Control and Dnamics, Vol., No., Jul Auust 99, pp () United tates Geoloical urve. errain Data. tp://edctp.cr.uss.ov/pub/data/dem/50/ () Calise, A.J., inular Perturbation echniques or On-Line Optimal Fliht-Path Control, AIAA Journal o Guidance and Control, Vol., No., 98. (5) Brson, A.E., Jr. and Ho, Y-C. Applied Optimal Control. Hemisphere Publishin Corporation, 975. (6) Vanderplaats, Garret N. Numerical Optimization echniques or Enineerin Desin: With Application. New York: McGraw-Hill Publishin Compan, 98. American Institute o Aeronautics and Astronautics