AUGMENTED POLYNOMIAL GUIDANCE FOR TERMINAL VELOCITY CONSTRAINTS Gun-Hee Moon*, Sang-Woo Shim*, and Min-Jea Tah* *Korea Advanced Institute Science and Technology Keywords: Polynomial guidance, terminal velocity, high speed vehicle, Aerodynamic Deceleration Abstract This paper deals with an augmented polynomial guidance law to ulill the terminal velocity requirement o high speed maneuverable reentry vehicle. The guidance command consists o three terms; basis polynomial guidance term, velocity control term, and gravity compensation term. The polynomial guidance can lead the vehicle to the target position in speciic direction, and the augmented guidance command can adjust the terminal velocity without disturbing the basis polynomial guidance. The velocity is predicted by internal simulation under non velocity control command condition. The predicted terminal velocity error is ed bac to velocity control command. Some computer simulations are conducted to show the perormance o the proposed alrithm. 1 Introduction Recently, a guidance law that achieves speciic terminal velocity or an unpowered light vehicle has studied or various purposes, such as saety landing o a space ship or impact eiciency enhancement o a missile. The terminal velocity can be dissipated by maneuver o the vehicle due to aerodynamics, thus it depends on the trajectories o the vehicle. In general, the aerodynamic orce is stronger at the low altitude than the high altitude, as the atmosphere is denser at sea level. Previously, a number o researchers have studied about the impact angle control guidance law. Kim ormulated a terminal guidance system or reentry vehicles with the impact angle constraints as a linear quadratic control problem, and derived a Riccati equation to calculate time-varying eedbac gains [1]. A control energy optimal impact angle guidance law or missiles with a time varying velocity has studied in []. In the paper, the author assumed that velocity o a missile decreases eponentially. A time to weighted control energy optimal guidance law is studied in [3], which results in a generalied vector eplicit guidance law (GENEX). It consists o two parts; one is minimiing ero eort miss and the other minimiing the impact angle error, so, GENEX can control the impact angle o the vehicle, An energy optimal impact angle guidance law or a lag system has been derived rom a linear quadratic optimal control law by Ryoo [4]. He also proposes time-to- weighted optimal impact angle guidance law [5]. A time-to- polynomial guidance law with the impact angle constraint is proposed by Kim, in [6]. The author also proposed a downrange-to- polynomial guidance law that can achieve the impact time and angle together in [7]. The polynomial guidance law is biased with an additional acceleration command which adjusts the trajectory o a vehicle so as to control the impact time. An augmented guidance law using the vector eplicit guidance or the impact angle and velocity constraints is introduced in [8]. It shows a stability analysis o the guidance law on the linearied dynamics. It augments a lateral additional acceleration command to the vector eplicit guidance command to control the terminal velocity. The scales o the augmented lateral guidance command are determined by internal simulations and the secant method. This paper proposes an augmented polynomial guidance law that satisies the speciic terminal velocity. The guidance 1
GUN-HEE MOON, SANG-WOOK SHIM, and MIN-JEA TAHK command consists o an impact angle control polynomial guidance term, a bias guidance term controlling the terminal velocity o the vehicle, and the gravity compensation term. The proposed alrithm predicts the terminal velocity by an internal simulation, and eeds bac the predicted terminal velocity error to the velocity control term. This paper organied as ollows;. The planar dynamic model o maneuverable reentry vehicle is described and problem deinition ollows. 3. The augmented polynomial guidance or the terminal velocity condition is eplained. 4. Simulation results comes. Problem Deinition.1 Planar Guidance Model The planar engagement dynamic model is described as ollows, V cos h Vsin V D / m g sin L / mv g cos / V (1) where is downrange variable, h is altitude, V is the velocity o the vehicle, γ is light path angle, m is the mass o the vehicle, g is gravity acceleration, L is lit orce, and D is drag orce. The aerodynamic orces are oten epressed by non-dimensional coeicient. D C QS L D L C QS 1 Q V re re () where CD is drag coeicient, CL is lit coeicient, Q is dynamic pressure, ρ is ree stream atmosphere density, and Sre is reerence area. Generally CD-CL is related by the polar curve. Simple drag polar model is given as, C C K C C D Dmin ( L L) (3) where CDmin is parasite drag term, it is also minimum drag coeicient, and K is induced drag parameter, and CL is the lit coeicient at minimum drag. In this paper, we employed ollowing atmosphere density model and gravity model. hh / e e, g (4) ( Re h) where ρ is atmosphere density at sea level, hρ is atmosphere density scaling height, μe is earth gravitational parameter, and Re is earth mean radius.. Problem Constraints In this paper, we ocused to ind an acceleration command which let the vehicle have the terminal velocity and light path angle on the inal position. The initial condition o the vehicle is given as, ( t ), ( t ), ( t ), V ( t ) V (5) where t indicates initial time. On the terminal position, the vehicle should have γ. ( t ), ( t ), ( t ), V V ( t ) V min ma (6) Note that the terminal velocity constraint is given by some boundary, not by speciic value. 3 Augmented Polynomial Guidance or Terminal Velocity Constraints 3.1 Polynomial Guidance Kim have introduced a downrange-to- polynomial guidance, which can satisy the impact angle condition and impact time requirement additionally, hitting the target [7].
AUGMENTED POLYNOMIAL GUIDANCE FOR TERMINAL VELOCITY CONSTRAINTS In this chapter, we revisit the basis downrangeto-,, polynomial guidance. The polynomial guidance law assumes the guidance command as ollowing polynomial. u c c n m poly n m (7) ( n1)( n) ( m1)( ) ˆ ( m n)ˆ ( )( ˆ ˆ m ) ( m1)( m) ( n1)( ) ˆ cm ( n m)ˆ ( )( ˆ ˆ n ) ˆ (1) where and cm are guidance coeicients, and is downrange-to-. The linearied engagement dynamic model is given as, u poly (8) where upoly=a/v. The apostrophe represents the derivatives by. Thus integrating it gives the linearied trajectory o the vehicle as, n1 cm m1 c n1 m1 c ( n 1)( n ) ( m 1)( m ) n m m c c (9) This trajectory should satisy the boundary conditions. Thus, the terminal constraints determine the integral coeicients as, c c c (1) The initial conditions determine the other unnown coeicients, and cm, n1 cm m1 ˆ ˆ c n1 m1 c ˆ ˆ ( n 1)( n ) ( m 1)( m ) ˆ n m m Solve it or and cm, then, c c (11) Substituting these coeicients to (7) gives u poly ( n)( m) ( n m 3) ( n 1)( m 1) (13) Note that upoly is the normalied command o the linearied geometry, so the dimensional guidance command should be applied with multiplication o V on a guidance rame. In this paper we employed the light path rame o the vehicle as the guidance rame. Thus, ollowing transorm is required. R ( ) ( ) 1 tan Rcos( ) Rsin( ) 3. Augmented Polynomial Guidance. (14) To satisy the terminal velocity constraint and compensate the gravity acceleration, an augmented polynomial guidance is designed. The guidance command has three terms as, a u V a a (15) cmd Poly vel g The irst term is the basis polynomial guidance term. The second term is velocity control acceleration term. The other is gravity compensation term. Gravity compensation term is simply adopted as, 3
GUN-HEE MOON, SANG-WOOK SHIM, and MIN-JEA TAHK a cos g g (16) where g is nominal gravity acceleration. The terminal velocity control term is designed as, v ˆ v avel nma g (17) vˆ where v is nominal terminal velocity and v is predicted terminal velocity, and nma is maimum load actor or avel term. The v is a prediction under the basis polynomial guidance law and gravity compensation. When the terminal velocity prediction is higher than the required value, this terminal velocity control term tends to reshape the trajectory downward so that the vehicle suers rom more aerodynamic orces due to the denser atmosphere. In the opposite case, the avel tends to modiy the trajectory upward so that the vehicle avoids the drag more eectively. Meanwhile, the nominal terminal velocity can be the mean value o the terminal velocity boundary. C L m( u polyv g cos ) () QS Note that this procedure predicts the terminal velocity without the velocity control command. To integrate along the downrange direction, the downrange increments are deined as, re / N (1) where Nsim indicates the number o the loop in the internal simulation. Then the prediction process can be epressed in discrete orm as, 1 1 1 1 sim V V V () Then the inal velocity becomes the predicted terminal velocity. vˆ V (3) N sim v V V min ma (18) 3.3 Terminal Velocity Prediction To predict the terminal velocity o the vehicle, the vehicle ought to simulate its trajectory on its own. However, it is oten cost lots o eort to do so. In order to enhance its prediction eiciency, we change the independent variable o the planar guidance dynamics rom t to as, h tan V D / mv cos g tan / V L / mv cos g / V (19) initial condition terminal condition v h V γ h V γ values = m = 3m = 4 m/s = - deg = 1 m = 1 m = v = deg (15,, 3, 4, 5) m/s As we consider the augment polynomial guidance, the required lit coeicient is calculated as Table 1 numerical simulation scenarios. 4
AUGMENTED POLYNOMIAL GUIDANCE FOR TERMINAL VELOCITY CONSTRAINTS 4 Numerical Simulations 4.1 Simulation Scenarios Numerical simulations are conducted to show the applicability o the proposed alrithm. The vehicle starts rom the given initial condition. It is a maneuverable reentry vehicle, and it needs to modiy its path to the terminal position while satising the terminal velocity constraint. In table 1, the initial and terminal conditions or simulation are depicted. The required inal velocity diers rom 15 m/s to 5 m/s or each case. The other conditions are ied. 4. Simulation Results In Fig. 1, the trajectory o each case is plotted. All vehicles eventually get to the terminal position, but the path is slightly dierent by cases. Fig. shows the light path history o each case. All cases satisied the terminal light path angle. The light time is dierent case by case, the case with low terminal velocity showed longer light time. In cases o v = 15 and m/s, the light path angle smoothly attached to deg. In the other cases, it sharply converged to deg. Fig. 3 shows the velocity history o each simulation. In cases o v = 15,, and 3 m/s, the terminal velocity error is less than 1 m/s. On the other hand, the other cases showed s about 1 % terminal velocity error. Fig. 4 depicts the prediction history o the terminal velocity. The ive cases show similar prediction at the beginning as their paths are similar. However, as the velocity control accelerations aect their paths, the predicted terminal velocity diers case by case. The predicted values at the inal moment are identical to the inal velocities. In Fig. 5, guidance command history is shown in the load actor unit. The acceleration command is as much as -1 g at the beginning, and converges to 1 g nearby the inal moment due to the gravity compensation. Fig. light path angle history o each case. 5 15 V [m/s] 1 Fig. 1 trajectory o each case. X: 88.9 Y: 553.8 X: 16. 5 Y: 39. X: 173.6 X: 11.7 Y: 151.1 Y: 441.6 X: 16 Y:.6 4 6 8 1 1 14 16 18 t [sec] Fig. 3 velocity history o each case. 5
GUN-HEE MOON, SANG-WOOK SHIM, and MIN-JEA TAHK 5 Conclusion The augmented polynomial guidance is studied to control the terminal velocity o the high speed maneuverable reentry vehicle. The guidance command is composed o three parts, basis polynomial guidance, velocity control term and gravity compensation term. The terminal velocity without the velocity control command is predicted by the internal simulation. The internal simulation integrates along variable so that it can estimate the terminal velocity quicly. The predicted terminal velocity error is ed bac to the velocity control term. Numerical simulation shows that the proposed alrithm can satisy the terminal velocity within 1% error at worst. Angle/Acceleration Constraints, The 18th IFAC World Congress, Milano, Italy, August 8 September, 11. [7] Kim, T.-H., and Tah, M.-J., Augmented Polynomial Guidance with Impact Time and Angle Constraints, IEEE Transactions on Aerospace and Electronic Systems, vol. 49, no. 4, October 13. [8] Yu, W., and Chen, W., "Trajectory-Shaping Guidance with inal speed and load actor constraints." ISA transactions, 14. Predicted V [m/s] 8 7 6 5 4 3 6 Acnowledgments This wor was conducted at High-Speed Vehicle Research Center o KAIST with the support o Deense Acquisition Program Administration and Agency or Deense Development. (Contract number: PD1413 CD) 1 4 6 8 1 1 14 16 18 Fig. 4 terminal velocity prediction history. t [s] 7 Contact Author Email Address mailto : mjtah@dcl.aist.ac.r Reerences [1] Kim, M., and Grider, K. V., Terminal Guidance or Impact Attitude Angle Constrained Flight Trajectories, IEEE Transactions on Aerospace and Electronic Systems, vol. AES-9, no. 6, November 1973. pp. 85-859. [] Song, T.-L., Shin, S. J., and Cho, H.-J., Impact Angle Control or Planar Engagements, IEEE Transactions on Aerospace and Electronic Systems, vol. 35, no. 4, October 1999. pp. 1439-1444 [3] Ohlmeyer, E. J., and Phillips, C. A., Generalied Vector Eplicit Guidance, Journal o Guidance, Control, and Dynamics, vol. 9, no., March- April 6. [4] Ryoo, C.-K., Cho, H.-H., and Tah, M.-J., Optimal Guidance Laws with Terminal Impact Angle Constraints, Journal o Guidance, Control, and Dynamics, vol. 8, no. 4, July-August 5. [5] Ryoo, C.-K., Cho, H.-J., and Tah, M.-J., Time to Go Weighted Optimal Guidance with Impact Angle Constraints, IEEE Transactions on Control Systems Technology, vol. 14, no. 3, May 6. [6] Kim, T.-H., Lee, C.-H., and Tah, M.-J., Time to Polynomial Guidance Laws with Terminal Impact Fig. 5 non-dimensional guidance command. Copyright Statement The authors conirm that they, and/or their company or organiation, hold copyright on all o the original material included in this paper. The authors also conirm that they have obtained permission, rom the copyright holder o any third party material included in this paper, to publish it as part o their paper. The authors conirm that they give permission, or have obtained permission rom the copyright holder o this paper, or the publication and distribution o this paper as part o the ICAS proceedings or as individual o-prints rom the proceedings. 6