13 th AAS/AIAA Space Flight Mechanics Meeting

Transcription

1 Paper AAS Multbody Parachute Flght Smulatons for Planetary Entry Trajectores Usng Equlbrum Ponts Ben Raszadeh NASA Langley Research Center Hampton VA th AAS/AIAA Space Flght Mechancs Meetng Ponce, Puerto Rco 9-13 February 2003 AAS Publcatons Offce, P.O. Box 28130, San Dego, CA 92198

2 AAS MULTIBODY PARACHUTE FLIGHT SIMULATIONS FOR PLANETARY ENTRY TRAJECTORIES USING EQUILIBRIUM POINTS INTRODUCTION Ben Raszadeh NASA Langley Research Center Hampton VA ABSTRACT A method has been developed to reduce numercal stffness and computer CPU requrements of hgh fdelty multbody flght smulatons nvolvng parachutes for planetary entry trajectores. Typcal parachute entry confguratons consst of entry bodes suspended from a parachute, connected by flexble lnes. To accurately calculate lne forces and moments, the smulatons need to keep track of the pont where the flexble lnes meet (confluence pont). In prevous multbody parachute flght smulatons, the confluence pont has been modeled as a pont mass. Usng a pont mass for the confluence pont tends to make the smulaton numercally stff, because ts mass s typcally much less that than the man rgd body masses. One soluton for stff dfferental equatons s to use a very small ntegraton tme step. However, ths results n large computer CPU requrements. In the method descrbed n the paper, the need for usng a mass as the confluence pont has been elmnated. Instead, the confluence pont s modeled usng an equlbrum pont. Ths pont s calculated at every ntegraton step as the pont at whch sum of all lne forces s zero (statc equlbrum). The use of ths equlbrum pont has the advantage of both reducng the numercal stffness of the smulatons, and elmnatng the dynamcal equatons assocated wth vbraton of a lumped mass on a hgh-tenson strng. Many planetary entry systems employ a parachute system as a decelerator devce from supersonc to subsonc veloctes. Smulatng the parachute porton of the planetary entry often nvolves modelng the parachute and the suspended bodes as ndvdually at-

3 tached rgd Sx-Degree-Of-Freedom (6 DOF) bodes. The parachute and the suspended bodes are connected by flexble lnes on these planetary entry confguratons. In ths model, the suspenson lnes meet at a confluence pont. The Mars Exploraton Rover (MER) msson uses a parachute system to slow down ts descent and after chute deployment has one such confluence pont (Fg. 1). In current MER smulatons, confluence ponts are modeled usng Three-Degree-Of-Freedom (3 DOF) pont mass bodes. Due to ts small mass, the confluence pont s subject to rapd acceleratons. The rapd acceleratons make the smulatons numercally stff, requrng very small tme ncrements on the order of second. Ths causes the smulaton run tmes to sgnfcantly ncrease. 1 Ths s undesrable n Monte-Carlo smulatons where fast turn-around s requred to support trade studes. In prevous multbody entry models, 1-3 lne masses are lumped nto the confluence ponts. In realty, ths mass s dstrbuted evenly along the suspenson lnes. Therefore, the vbratonal modes wll be those of a dstrbuted mass system, as n a vbratng strng. Thus, the vbratonal modes ntroduced by the confluence pont mass may not be realstc. parachute confluence pont entry capsule Fgure 1 Vehcle entry confguraton. Ths paper outlnes a method to elmnate confluence pont masses, and ntroduces the equlbrum pont. The equlbrum pont s defned as the pont at whch the sum of the lne forces s zero, the pont where the lne forces are n equlbrum. Ths has the same effect as usng a confluence pont wth an nfntely small mass, and allowng all the transents to de out. Instead of ntegratng the confluence pont mass, the equlbrum pont locaton s calculated at every tme step. The soluton s then used to calculate the lne forces. Ths allows the smulatons to run at a larger tme step on the order of 0.01 second (100 tmes larger than wthout), and the hgh frequency vbratons caused by the small confluence mass are elmnated. Ths also reduces the number of dfferental equatons to ntegrate. 2

4 SYMBOLS AND ABBREVIATIONS MER Mars Exploraton Rover POST Program to Optmze Smulated trajectores DOF Degree of Freedom p Poston vector of the attach pont per lne p ep Poston vector of the equlbrum pont e Lne stran per lne e Lne stran rate per lne L 0 Free length per lne K Stffness coeffcent per lne C Dampng coeffcent per lne û Unt vector n drecton of the lne f Lne force vector per lne f k Lne force vector due to stffness per lne f c Lne force vector due to dampng per lne L t Lne length n current tme step L t 1 Lne length n prevous tme step [ x y z ] Attach pont coordnates [ x ep yep zep ] Equlbrum pont coordnates Ixx Moment of nerta about the roll axs Iyy Moment of nerta about the ptch axs Izz Moment of nerta about the yaw axs C D Drag coeffcent Cp Center of pressure S ref Reference surface area Ω Mars gravtatonal constant R e Mars equatoral radus R p Mars polar radus ω Mars rotaton rate J Mars gravty zonal harmoncs BACKGROUND The underlyng smulaton used to test the equlbrum ponts concept s the Program to Optmze Smulated Trajectores II (POST II). POST II s the latest major upgrade to POST. 4 POST was orgnally developed for the Space Shuttle program to optmze ascent and entry trajectores. Over the years t has been upgraded and mproved to 3

5 nclude many new capabltes. POST II reles on most of the techncal elements establshed by POST, but the executve structure has been reworked to take advantage of today s faster computers. The new executve routnes allow POST II to smulate multple bodes smultaneously, and to mx Three-Degree-Of-Freedom (3DOF) bodes wth Sx- Degree-Of-Freedom (6DOF) bodes n a sngle smulaton. The already establshed and verfed multple body capablty usng confluence pont masses allows POST II to smulate parachutes. Ths s done by connectng the spacecraft and the parachute by massless sprng-dampers. The sprngs can be attached at any pont on the body. No moments are appled except those due to force applcaton away from the center of mass. Each lne connects an attach pont on one body to an attach pont on another body and provdes a tenson-only force. When the lnes are stretched, tenson n the lnes s determned as functon of stran and stran rate. In the current multple body POST II, the dfferental equatons of moton for all bodes are explctly ntegrated numercally, and the lne forces are calculated based on relatve poston and velocty of the bodes attached. APPROACH Frst, the algorthm to calculate the equlbrum pont s descrbed. Ths algorthm needs to be robust snce t has to solve the equlbrum pont at every tme step, and also be able to account for specal cases, such as when the lnes are slack. Although ths capablty has been added to POST II, t could be used wth any multple body smulaton programs such as ADAMS. 5 To verfy the current multple body capablty of POST II (wthout equlbrum ponts ), a seres of more than 35 tests of ncreasng complexty were performed. These tests were ntended to prove that the POST II model was mplemented correctly by evaluatng ts performance on problems that could be verfed by other means. The test cases started wth a smple vertcal drop from rest of the fully deployed parachute and entry capsule and gradually ncreased n complexty to nclude parachute openng, non-zero ntal condtons, lne deployments, wnd gusts, and other effects. The POST II model was compared to both MATLAB-based and ADAMS-based mult-degree-of-freedom smulatons. In each case the agreement between the smulatons was excellent. These tests have valdated the general parachute model wthn POST II. Some of the valdaton s reported more fully n Reference 1. To verfy the method usng equlbrum pont some of the same test cases are used, except the confluence pont mass s replaced by the equlbrum pont, and the results are compared. ALGORITHM The equlbrum pont s defned as the pont at whch the sum of all lne forces s zero. Note that n the general case an arbtrary number of lnes can be connected at the equlbrum pont. In ths algorthm the equlbrum pont poston s the unknown. [ ] p = x y zep ep ep ep The poston of the attach pont s fxed n each tme step. p x y z ] = [ 4

6 Lne stran s found by subtractng the poston vectors, calculatng the magntude, and dvdng by free length. The force s n the drecton of the lne connectng the end ponts. e = p pep L 0 2 L 0 The unt vector n the drecton of the lne force s found by subtractng the poston vectors and dvdng by the magntude. uˆ = ( p p ep ) / p p ep 2 Force vector due to stffness s found by the followng equaton: f k = L K e uˆ 0 In calculatng the dampng force we do not have drect access to velocty of the equlbrum pont n nertal space. Thus stran rate of the lne s not readly avalable. Instead, the stran rate s approxmated numercally by subtractng the lne length from the prevous tme step and dvdng by tme ncrement. The drecton of dampng force s along the lne, and opposte to the stran rate of the lne. e L Lt = dt L t 1 0 Force due to dampng s then found usng the followng relatonshp: f c = L C e uˆ 0 The net lne force s the sum of force due to stffness and dampng force f f = L = L 0 0 K e uˆ + L C e uˆ 0 ( K e + C e ) uˆ At the equlbrum pont sum of all the lne forces s equal to zero. n = 1 f = 0 At the ntal guess for the equlbrum pont there wll most lkely be a non zero resultant force vector. The objectve s to fnd a unque locaton n space where the summaton all lne forces s zero. The only varable here s the poston of the equlbrum pont, and the output s the resultant force vector. Ths problem can be solved usng one of many avalable methods for solvng system of equatons. Here, the Newton method s chosen to solve ths system. In summary, the Newton method uses the slope of the output varable (n ths case force vector), by varyng the nput varable (n ths case the equlbrum pont poston vector) to converge to a pont where the output s zero (statc equlbrum). 5

7 RESULTS As mentoned earler, some of the same test cases used prevously to verfy multple body capablty of POST are used agan to evaluate the equlbrum pont soluton. For a detaled descrpton of the test cases see Reference 1. Test cases 2 and 3 are chosen for ths study. Comparson s made between the soluton usng the equlbrum poston, and the soluton utlzng the confluence pont mass. All test cases are performed n a Mars envronment. But ths method can be appled to parachute entry smulatons on any planet. A constant atmospherc densty of kg/m 3 s assumed for all runs. Aerodynamc drag acts on the parachute only. Mars gravty and an oblate planet model have been used. The planet s assumed to be non-rotatng. All smulatons start at zero lattude and zero longtude at a heght of approxmated 8.4 klometers. Tables 1, 2 and 3 summarze the nputs used. Table 1 LINE PROPERTIES Lne Parameter Value Sngle rser L m K 60,000 N/m C 600 N/(m/s) Trple rsers L m K 47,000 N/m C 470 N/(m/s) Table 2 PLANET MODEL Parameter Value Ω e 13 m 3 /s 2 R e e 6 m R p e 6 m ω 0.0 rad/s J terms 0.0 Table 3 PARACHUTE AND ENTRY CAPSULE INPUTS Body Parameter Value Parachute DOF 6 Mass 16.0 kg Ixx kg.m 2 Iyy kg.m 2 Izz kg.m 2 C D 0.46 Cp 1.57 m S ref m 2 Backshell/lander DOF 6 Mass 761 kg Ixx kg.m 2 Iyy kg.m 2 Izz kg.m 2 6

8 chute cg sngle rser trple rsers 1.23 chute attach pont swvel/confluence pont entry capsule cg Fgure 2 Test cases confguraton. Note: Dmensons are typcal (not based on any specfc Mars msson). Test case 2 s repeated here wth the equlbrum pont and results are compared. In ths test case a parachute system s dropped from rest. To ntroduce dynamcs nto the system we ncluded a slack of one centmeter n the Sngle Rser (see Fgure 2). Note that the aerodynamc drag acts on the parachute only. The entry capsule ntally drops faster than the parachute. Eventually, the sngle rser runs out of slack, thus exctng the system. In ths smulaton t takes the system about sxty seconds to reach termnal velocty. It takes approxmately two seconds for vbratonal dynamcs to damp out. Fgure 3a s the plot of force n the Sngle Rser. Note that the smulaton started wth a one-centmeter slack n the Sngle Rser. It takes about 0.7 second for the slack to run out. The entry bodes then undergo an oscllatory moton. The oscllatons damp out approxmately 0.5 second after they start (Fgure 3b). After ths pont, the lne force gradually bulds up untl t reaches a steady state value. Note that n Fgure 3a the curves are ndstngushable. In Fgure 3b the dfferences become more vsble. In the soluton wth the equlbrum pont, an ntegraton tme step of 0.01 second was used as opposed to second for the smulaton usng confluence pont mass. 7

9 confluence pont mass equlbrum pont sngle rser force (N) tme (sec) 3a Entre smulaton confluence pont mass equlbrum pont sngle rser force (N) tme (sec) 3b Intal two seconds. Fgure 3 Sngle Rser force. An nterestng set of results s obtaned by zerong out lne dampng, and leavng all other nputs unchanged. The comparson of the Sngle Rser force s presented n Fgure 4. In Fgure 4a, the overall force-tme hstory of the lne appears to produce a good comparson. But a closer nspecton shows that the model wth confluence pont mass has an addtonal hgh frequency mode (Fgures 4b and 4c). Ths s caused by the confluence pont mass vbratng back-and-forth n between the lnes n tenson. Ths example llustrate how the confluence pont mass tends to ntroduce unwanted frequency contents nto the smulaton, and how t can be elmnated usng the equlbrum pont. 8

10 confluence pont mass equlbrum pont sngle rser force (N) tme (sec) 4a Entre smulaton confluence pont mass equlbrum pont sngle rser force (N) tme (sec) 4b Intal two seconds confluence pont mass equlbrum pont sngle rser force (N) tme (sec) 4c Mdway range. Fgure 4 Sngle Rser force, no lne dampng. 9

11 The next test case s dentcal to the prevous wth the excepton that the entry capsule s gven an ntal horzontal velocty of 1 m/s. In ths test case, more degrees of freedom are excted, as opposed to the prevous test case where the bodes were excted n the vertcal drecton only. Fgure 5 shows the plot of the Sngle Rser lne force as a functon of tme. The overall force curve shows good agreement (Fgure 5a), but when observed n detal the model usng confluence pont mass has addtonal frequency contents caused by the pont mass (Fgure 5b). The smulaton wth confluence pont mass ran usng ntegraton tme step of second, and the smulaton usng equlbrum pont used an ntegraton tme step of 0.01 second, resultng n substantal savngs n run tme (at least by a factor of ten) confluence pont mass equlbrum pont sngle rser force (N) tme (sec) 5a Entre smulaton sngle rser force (N) 240 confluence pont mass equlbrum pont tme (sec) 5b Mdway range Fgure 5 Sngle Rser, non-zero ntal condtons 10

12 CONCLUSION Modelng the confluence ponts usng a pont mass s a numercal challenge for all the multbody parachute entry smulatons (past and present). The method descrbed n ths paper reduces numercal stffness assocated wth modelng the confluence pont as a pont mass. Ths method also elmnates unwanted hgh frequency oscllatons caused by the confluence pont mass. Reduced stffness s desrable because t allows larger ntegraton tme steps whch should lead to shorter run tmes. Also, by elmnatng the confluence pont mass, the smulaton has fewer equatons of moton to ntegrate. These speed enhancements are somewhat offset by the need to numercally solve for the equlbrum pont poston. Effectveness of ths scheme can be mproved by refnng numercal technques used to solve for the equlbrum pont. REFERENCES [1] Ben Raszadeh, Erc M. Queen, Partal Valdaton of Multbody Program to Optmze Smulated Trajectores II (POST II) Parachute Smulaton Wth Interactng Forces, NASA/TM , Aprl 2002 [2] Erc M. Queen, Ben Raszadeh, Mars Smart Lander Parachute Smulaton Model, AIAA Paper , August 2002 [3] Kenneth S. Smth, Cha-Yen Peng, Al Behboud, Multbody Dynamc Smulaton of Mars Pathfnder Entry, Descent and Landng, JPL D-13298, Aprl 1995 [4] Program to Optmze Smulated Trajectores: Volume II, Utlzaton Manual, prepared by: R.W. Powell, S.A. Strepe, P.N. Desa, P.V. Tartabn, E.M. Queen; NASA Langley Research Center, and by: G.L. Brauer, D.E. Cornck, D.W. Olson, F.M. Petersen, R. Stevenson, M.C. Engel, S.M. Marsh; Lockheed Martn Corporaton, Verson G, May 2000 [5] ADAMS, Software Package for Smulatng Force and Moton Behavor of Mechancal System, Property of Mechancal Dynamcs Inc. 11