ANALYSIS OF LANDING TRAJECTORY USING BACKWARD PROPAGATION

Transcription

1 ANALYSIS OF LANDING TRAJECTORY USING BACKWARD PROPAGATION Melissa M. Onishi Departent of Mechanical Engineering University of Hawai i at Mānoa Honolulu, HI ABSTRACT The past NASA landing issions such as Apollo, Vikings, Mars Pathfinder, the Mars Exploration Rovers and the Mars Science Laboratory projects have led to the necessity of designing the next generation of landers that will achieve safe and precision landing. According to the table Capabilities Roadap: EDL (Entry, Descent and Landing), safe and precision landing is one of any challenges of future issions [1, 2]. In August 6, 2012, NASA Jet Propulsion Laboratory (JPL) led Mars Science Laboratory (MSL) ission successfully landed a robotic rover, known as Curiosity. However pursuing accurate or precision landing was not a priority of this ission. As directed by the NASA EDL Roadap and inspired by this ission, further research has led to continuation of accoplishing this task of precision landing. The ain goal of this research is to study and investigate the possibilities of achieving safe and precision landing with the awareness of hazard avoidance and with efficient usage of fuel [3]. It is expected that the results of this study will lead to iproved siulations of landing trajectories with the creation of anifolds of initial and final points (conditions) for powered descent and landing. These anifolds can be created by a construction of envelopes of the landing trajectories using various ranges for trajectory and lander s paraeters. INTRODUCTION Throughout the seester I studied and worked on creating different envelope constructions using the concept of backward propagation. Soe interesting results regarding the precision landing have been obtained in Ref [4]. Previous studies of NASA related issues in precision landing have slowly decreased the distance between the designated landing site to the actual landing site [3]. It is necessary to consider algoriths which identify and locate landing hazards, react to failed systes and low powered coponents to support terrain tracking [1]. Hopefully, with tie I will be granted the opportunity to accoplish these challenges in achieving these tasks. In early May, I was able to obtain Matlab software and was given a brief tutorial on basic coand functions fro y entor. Within a short period of tie, I was given a crash course on learning the systes of equations of otion and how they pertained to precision landing. I learned how to integrate the differential equations using an ODE call on Matlab which will be further discussed in the ethod section of this report. 66

2 METHOD I created two separate codes, a ain file and a function file using Matlab software. The ain file contains initial conditions and landing selections, while the function file consists of systes of equations that were integrated. The thrust direction is opposite to the velocity direction aking thrust agnitude constant for this approach. The assuption about this direction eans that cos α = v x, cos β = v y, and cos Υ = v z, where cosα, cosβ, and cosγ are V V V the thrust direction cosines, v x, v y, and v z are the velocity coponents, and V is the agnitude of the velocity vector. To ake this application ore realistic, actual Mars Newtonian gravity was used (µ). The landing selection that was used in spherical coordinates contains θ LS = 30, where θ LS = (θ LS ) π 180 andρ LS = 60, where ρ LS = (ρ LS ) π The landing coordinates used are stated as x LS = (r LS )(cos ρ LS )(cos θ LS ), y LS = (r LS )(cos ρ LS )(sin θ LS ), and z LS = (r LS )(sin ρ LS ) where r LS is constant and defined as Mars equatorial radius. Systes of equations of otion used for integration: x = V x v x = µx + r cβ cosα y = V y v y = µy + cβ cosβ r z = V z v z = µz + r cβ cosγ = 0 βt where r = x 2 + y 2 + z 2 Variables defined: x, y, z, are the position vector coponents of the lander s center of ass, µ and β are the constants. µx, µy µz, and are the gravitational acceleration coponents, r r r cβ is the thrust acceleration agnitude. Initial Conditions: Initial tie (t 0 ) = 0 Specific Ipulse (Isp) = 220 s Gravity (g 0 ) = k/s 2 Thrust to Weight Ratio T = 3 W Initial ass ( 0 ) = 2835 kg Gravity on Mars (g ) = 3.7 k/s 2 To obtain assflow rate and final tie I did soe inor algebra to obtain the equations shown above. Below is the calculation I found for assflow rate and final tie. Calculations: c = Isp g 0, T = (c)(β) We know that: W = ( 0 )(g), T = 3 W T = 3W = 3( 0 )(g) = (c)(β) = (Isp)(g 0 )(β) β = 3( T 0)(g) (Isp)(g 0 ) = W 0(g) Isp(g 0 ) Where g is the gravitational acceleration on the surface of Mars. Given in the equations of otion we find that = 0 βt > 0. To obtain final tie we can use 1 = 0 βt 1 > 0 where 0 > 1. Solve for final tie: t 1 = 0 1 β

3 DATA AND RESULTS In these studies, the realistic results that could be potentially used for future issions have been used. Below are the figures with different independent and dependent variables. One out of the five figures contains six subplots with six different states. Each plot in figures 2-6 is a representation of a trajectory envelope that covers acceptable ranges of landing trajectory and its paraeters. Notice that the ass vs. tie plot in Fig. 5 looks different fro the rest of the other figures. This particular plot is linear due to the liner dependence of ass on tie ( t = 0 βt). Fig. 2 displays envelope constructions of six subplots of the first six states stated in the function file. There is a seven kiloeters difference found for these plots which is in a suitable range for achieving altitude between 5 k h 10 k where h is altitude. Fig. 3 displays the envelope construction of position vs. tie. I found that there was a seventeen kiloeters distance between the highest and lowest trajectory for position. Fig. 4 displays the envelope construction of velocity vs. tie. For velocity vs. tie, there was about a 1 k/s difference between the highest and lowest trajectory. Fig. 6 displays the envelope construction of altitude vs. position. Interestingly, this particular plot stops at a certain position once it achieves a certain altitude. I found that the difference between the highest and lowest trajectory was approxiately 1 k. Figure 2 Envelope construction containing six subplots tie vs. state 68

4 Figure 3 Envelope construction Position vs. Tie Figure 4 Envelope construction Velocity vs. Tie Figure 5 Envelope construction Mass vs. Tie Figure 6 Envelope construction Altitude vs. Position DISCUSSION AND CONCLUSIONS The siulations that were found fro MATLAB software justified the purpose of using backward propagation for this application. By starting fro the designated landing site and creating ultiple trajectories fro that point, we were able to create a variety of different envelopes. With the help of a working code, we saw that each envelope construction had different distances between the highest to lowest point of trajectory. Initially, the proposed research started off as creating trajectory siulations and creating an envelope construction. There was no specific way of how I was going to create trajectory siulations. During id-seester, I decided to use backward propagation in order to achieve precision landing. There are three tasks entioned in the proposal for future studies. The next step is to achieve forward propagation using the results that were found in backward propagation using the envelopes of the trajectories and anifolds of the initial and final points of powered descent and landing trajectories. By conducting studies for forward propagation, we can increase the accuracy of envelopes that could be potentially used for future issions. In the future, I hope that y results will be useful for the next ission stated in the table year 2020 Capabilities Roadap: EDL [1,3]. 69

5 Throughout the seester, there were any challenges and strides ade to accoplish the task proposed. The ain coponent of this research was to understand the physics behind landing trajectories. It was not siple considering I had no prior knowledge or experience in understanding EDL trajectories. Secondly, getting accustoed to MATLAB was another challenge in itself such as learning new coand functions. Overall we found that backward propagation allow us to review the behavior state of vector coponents, and to create landing trajectory envelopes. By creating different variations of landing site conditions we are able to create anifolds of landing points and initial points of powered descent trajectories. The anifolds consist of initial and final points that will be used for the future studies to increase landing accuracy [5]. It is expected that optiization of landing trajectory allows us to iniize the fuel consuption while providing precision landing. REFERENCES Aziov, D.M. Extreal Analytical and Guidance Solutions for Powered Descent and Precision Landing. 23rd International Syposiu on Space Flight Dynaics. Pasadena, California, October 29- Noveber 2, Johnson, A., Klupp, A., Collier, J., Wolf, A., Lidar-based Hazard Avoidance for Safe Landing on Mars, California Institute of Technology, Jet Propulsion Laboratory. NASA Draft Entry, Descent and Landing Roadap. Technology Area 09. Noveber, Rivellini, T. The Challenges of Landing on Mars. Wolf, A. A. and etc. Perforance Trades for Mars Pinpoint Landing. IEEEAC paper #1661, Version 2. Updated January 9,