Application of Neural Network in Trajectory ... - Semantic Scholar
Application of Neural Network in Trajectory Planning of the Entry Vehicle for Variable Targets Bin Zhang, Shilu Chen, and Min Xu College of Astronautics, Northwestern Polytechnical University, Xi’an, China [email protected]
Abstract. A method for onboard generation of entry trajectory for variable targets is discussed. Conventional trajectory planning algorithms can only be used for the fixed terminal conditions without considering the variable targets. In case the vehicle needs to alert the entry trajectory due to damage or effectors failure, the entry guidance system must real-time design a feasible entry trajectory according to another feasible landing site from current flight conditions. The conventional approaches must be augmented to provide the real-time redesign capability for variable targets, and the redesign trajectory would also satisfy all path constraints and altered terminal conditions. This paper makes use of the neural network as a major controller to overcome this problem. The redesign trajectory problems and control parameter generations online problems can be transformed into the neural network offline training problem, given the initial conditions and the selected terminal conditions. Numerical simulations with a reusable launch vehicle model for various terminal conditions are presented to demonstrate the capability and effectiveness of the approach. Keywords: trajectory planning, neural network, entry, variable targets, onboard, real-time.
1 Introduction An important capability for the next generation entry vehicle is to autonomous return from any initial states, meaning that the guidance system is able to fast design a complete and feasible entry trajectory online from any initial position and velocity without any intervention. This guidance system can provide some benefits unmatched by the current guidance system. It would allow a vehicle can abort any abnormal fly mission during ascent or entry and choice the available landing site from current fly conditions without waiting for the guidance updates from the ground. It would also require significantly less the pre-mission analysis and planning for different missions, reduce dramatically the re-occurring optional costs associated with entry guidance. Conventional approach for entry mission operation consists of offline reference trajectory design and onboard tracking of the reference trajectory [1]. Before every mission, the reference trajectory would be accomplished on the ground and never be altered, which hasn’t apparent satisfactory the requirements of the next generation entry vehicle. Under the supports of the Advanced Guidance and Control (AG&C) Project [2, 3], many advanced guidance technologies have been developed in recent H. Deng et al. (Eds.): AICI 2011, Part III, LNAI 7004, pp. 318–325, 2011. © Springer-Verlag Berlin Heidelberg 2011
Application of Neural Network in Trajectory Planning of the Entry Vehicle
319
years. Gamble at el proposed an analytic predictor-corrector algorithm and a numeric predictor-corrector algorithm for the aero-assist flight experiment [4]. Fury at el presented an atmospheric guidance for steering the Kistler K-1[5], a fully reusable launch vehicle prior to deployment of the stabilization parachute with numeric predictor-corrector technology. Youssef and Chowdhry described a predictor-corrector guidance algorithm for X-33 [6]. Saraf at el designed an acceleration guidance algorithm for entry vehicle [7]. Zimmerman and Dukeman presented an automated method to design entry trajectory with heating constraints [8]. Z Shen and P Lu sought an approach for onboard generation of three dimensional entry trajectories [9]. An ideal algorithm for onboard planning trajectory should satisfy these requirements: guaranteed satisfaction of all inequality constraints, reliability, efficiency, less computational load and precise flight termination. The entry dynamic model is a typical nonlinear multi-input-multi-output (MIMO) system. Numerical predictor-corrector algorithms require repeated integration of the equations of motion, and conventional approaches suffer from the highly constrained nonlinear MIMO system, and don’t consider the trajectory redesign capability for variable targets. Hence, new guidance system that enables fully autonomous and adaptive entry guidance should be searched. In this paper, a real-time redesign trajectory algorithm for variable targets based on artificial neural network (ANN) is described. An onboard three-dimensional entry trajectory generation algorithm is used to obtain a set of training space to train the ANN. Numerical simulations show that trained neural network controller can handle the complicated non-linear MIMO system, design an approximate optimal feasible trajectory, and given a set of landing sites, the neural network controller can generate a series of control commands to maneuver the vehicle to reach the desired terminal conditions while imposing all the trajectory constraints.
2 The Entry Dynamics The following normalized three-dimensional point-mass dynamics of the vehicle over a spherical non-rotating Earth are discussed.
r = V sin γ
(1)
V cos γ sinψ r cos φ
(2)
V cos γ cosψ r
(3)
V = − D − sin γ / r 2
(4)
θ = φ =
1 cos γ r Vr
γ = (L / V ) cos σ + V 2 −
(5)
320
B. Zhang, S. Chen, and M. Xu
ψ =
L sin σ V + cos γ sinψ tan φ V cos γ r
(6)
Where r is the radial distance from the center of the earth to the vehicle, normalized by earth radius Re .The longitude and latitude are θ and φ , respectively. The earth-relative velocity V
is
normalized
by
Vc = g 0 Re
.The
time
is
normalized
Re g 0 . σ is the bank angle which is used by the primary control command. L and D are the dimensionless aerodynamic lift and drag accelerations as
by τ
=t
follows:
L = ρ (VcV ) S ref CL / ( 2mg 0 )
(7)
D = ρ (VcV ) Sref CD / ( 2mg 0 )
(8)
2
2
Where
S ref is the reference area of the vehicle, ρ is the atmospheric density, m is the
mass of the vehicle,
CL and CD are the lift and drag coefficients, respectively.
3 Entry Trajectory Constraints A feasible entry trajectory must satisfy path constraints. The typical inequality constraints considered here are: (1) Heating rate constraint:
ρ (VVc ) ≤ Q max 3
(9)
This corresponds to stagnation-point heating rate based on a reference sphere of radius 1m. (2) Normal load factor constraint:
L2 + D 2 ≤ nmax
(10)
It is a constraint on the aerodynamic load in the normal direction. (3) Equilibrium glide constraint:
1 2 1 r − V r − L cos σ ≤ 0
(11)
This constraint serves to reduce the phugoid oscillations in altitudes along the entry trajectory. (4) Dynamic pressure constraint:
q ≤ q max
(12)
Application of Neural Network in Trajectory Planning of the Entry Vehicle
Where
321
qmax is maximum allowable dynamic pressure. Q max , nmax , qmax are the
vehicle-depend parameters which would be enforced strictly.
4 Neural Network The artificial neural network is a mathematical tool inspired in the brain of animals, which is composed of information-processing neuron similar to the neurons of living creatures. The power of ANN lies in their ability to model the nonlinear or constrained systems when conventional analytical methods fail, which determines the ANN is capable of learning more information and is suitable to work as a surrogate model. Giving the learning information, the ANN is trained by iterative process which consists of two steps, the first is forward process which is termed “feed-forward” and the second is error correction process which is called “backpropagation”. In each process, real inputs are processed in a mathematical model to produce model outputs, the same real inputs are processed in a real world simulation to produce real outputs which are the target for the model, the real and model outputs are compared to produce error feedback which is used to correct the model. There are several training algorithms known, among which the Levenberg-Marquardt backpropagation is one of the most efficient for multilayer networks. The most usual model of neuron is the feed-forward ANN. In this model the input signals from other artificial neurons are multiplied by the weights that measure the net connections importance, which results in amplification or attenuation. They are added and the result is compared to the bias, creating the activation input for the activation function. The output of the activation function is the output of the node. The synaptic weights resembling the synapse of biological neurons are adjusted according to selected training algorithm. The process is repeated for every set of input parameters until the network performance reaches an acceptable level as measured by parameters such as error, gradient, or number of epochs. This learning feature of ANN makes the inputs are not included in the training set can generate expected results, the results for these variable inputs is called “generalization” and measures how well the ANN reacts to unforeseen data, and it becomes a criterion for a trained ANN is successful or not. In the paper, the RBF NN is chosen. The RBF neural net is a simply composed of three layers: an input layer, a hidden layer and an output layer. The connections between each layer are associated to synaptic weights. The output of layer y can be described as: No
y ( x ) = wiφ ( x − ci , σ i ) + w0 i =1
(13)
x is input vector to the output nodes, φ (•) represents transfer function of hidden nodes i , ci is the center of radial basis function of hidden nodes i , σ i is the width of the radial basis function, wi is the connection weight between node i and the node in output layer, N o is the number of nodes in hidden layer, w0 is bias.
Where
322
B. Zhang, S. Chen, and M. Xu
In the hidden layer, the Gaussian kernel function is described as:
φ ( x − ci , σ i ) = exp −
x − ci σ i2
It is the most commonly used radial basis function. The closer the input
(14)
x p is to be
center of the Gaussian function ci , the larger is the output of the basis function.
5 Training The center purpose of trajectory onboard planning algorithm is to develop a high-performance controller to generate approximate control commands with respect to the nonlinear system imposed by nonlinear constraints. In order to purpose this goal, a set of training samples would be created to train the weight matrix W. The next generation reusable launch vehicles (RLV) would be high lift-to-drag ratio entry vehicles. This feature makes the entry trajectory has some obvious characteristics, it consists of three different phases: initial descent phase, quasi-equilibrium glide (QEG) phase, and pre-TAEM (terminal area energy management) phase [9]. The characteristic of initial descent phase is low atmospheric density and the vehicles have not enough control authority to steer the RLV, which is called the “controlled fall” that takes the RLV to the QEG interface where the dynamic pressure can shape the trajectory. The QEG phase is most critical flight phase, the trajectory must observe all of the inequality path constraints, the achieved range would reach the landing site, and the control commands suffer from the highly constrained nonlinear MIMO dynamics system. In the pre-TAEM phase, the trajectory must satisfy all of the equality constrains, meaning that the trajectory would meet the TAEM conditions at the end of this phase. Determining the real-time trajectory for high nonlinear entry vehicles has been of considerate research interest in past years. Conventional approaches can not alter the entry trajectory, and can not also deal the abort missions which need redesign the new trajectory for feasible new landing sites. Continued efforts have been made to search for adaptive entry guidance methods that enable fully autonomous capability of guidance system to provide precise flight termination. Ref [9] has developed a completely new and efficient onboard algorithm for RLV. It makes novel use of quasi-equilibrium glide phenomenon in lifting entry as a center piece for efficient enforcement of the inequality constraints, and the longitudinal and lateral trajectory profiles planning problem is reduced to one parameter search problem. This approach can generate a complete and feasible 3DOF entry trajectory in about 2 to 3 seconds on a desktop computer. In this text, this approach is used to create the training samples. A three layer RBF neural network is used to approximate the dynamics model of the entry trajectory discussed above. The network uses the structure of 6 neurons in the input layer, 6 neurons in the hidden layer and one neuron in the output layer. The input to the network is the state variables
[ r , φ ,θ , v, γ ,ψ ]
T
, and its output is bank angle σ ,
the angle of attack profile does not change significantly from mission to mission, including the abort missions. This paper uses the nominal angle of attack profile. The
Application of Neural Network in Trajectory Planning of the Entry Vehicle
323
ANN training process is repeated for each selected initial condition and accomplished via Levenberg-Marquardt backpropagation algorithm.
6 Testing The nominal vehicle model, X-33, is used in this paper. The corresponding aerodynamic parameters can be cited from [6]. The nominal initial entry states are:
[ r0 ,θ0 ,φ0 ,V0 , γ 0 ]
120km, 208D , 76D , 7650m / s, −0.5D . The desired D longitudinal terminal states are: [ r0 , V0 , γ 0 ] = 25km,1760m / s, −4 . Random =
landing sites are chosen to generate a set of training data. There are five cases to demonstrate the generalization capability of the well trained neural network. The corresponding longitudes and latitudes for all of the cases are random generated based on the acceptable limit of the vehicle maneuverability. Table 1 shows the TAEM comparative errors for all the cases. The results illuminate that these errors are small and the terminal landing sites generated by ANN can be accepted. Fig.1 shows the altitude history, Fig.2 shows the velocity history. These two figures describe that the trajectories generated by ANN are very close. The three-dimensional trajectories can be shown in Fig.3. All these figures and table reveal that the RBF ANN can successfully design the entry trajectory and the entry vehicle can reach the terminal states with an acceptable error range. Table 1. TAEM Comparative errors
Case1 Case2 Case3 Case4 Case5
Δrf (km)
ΔV f (m/s)
Δθ f (deg)
Δφ f (deg)
0.276% 0.171% 0.114% 0.152% 0.147%
0.285% 0.313% 0.152% 0.396% 0.354%
0.225% 0.364% 0.146% 0.254% 0.312%
0.387% 0.413% 0.345% 0.263% 0.154%
140 120 100
km
） 80
r
（ 60 40 20 0
200
400
600
800
1000
1200
1400
Time（ s）
Fig. 1. Trajectories altitudes for different terminal conditions
324
B. Zhang, S. Chen, and M. Xu
8000 7000 6000
m/s
）5000
v
（4000 3000 2000 1000 0
200
400
600
800
1000
1200
1400
Time（ s）
Fig. 2. Trajectories velocities for different terminal conditions
r炷km炸
150
100
50
0 80 60
Φ炷°炸
40 20 150
160
170
180
190
200
210
θ炷°炸
Fig. 3. Three-dimensional trajectories for different terminal conditions
7 Conclusion In this paper, an artificial neural network trained by the three-dimensional onboard trajectory generation algorithm is applied to real-time design different entry trajectory according to different landing sites. Theoretical analysis and simulation results show that the well trained neural network has good generalization capability, and the each trajectory generated by neural network is feasible solution with acceptable terminal states, meaning that a well trained artificial neural network augments the entry vehicles autonomous redesign trajectory capability and is more suitable for complex entry dynamics system.
Application of Neural Network in Trajectory Planning of the Entry Vehicle
325
References 1. Harpold, J.C., Graves, C.A.: Shuttle entry guidance. Journal of the Astronautical Sciences 27, 26–29 (1979) 2. Hanson, J.: A plan for advanced guidance and control technology for 2nd generation reusable launch vehicles. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, pp. 1–9. AIAA Press, New York (2002) 3. Hanson, J.: Advanced guidance and control project for resuable launch vehicles. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, pp. 1–10. AIAA Press, New York (2000) 4. Gamble, J.D., Cerimele, C.J., Moore, T.E., Higgins, J.: Atmospheric guidance concepts for an aeroassist flight experiment. Journal of the Astronautical Sciences 36, 45–71 (1988) 5. Fury, D.P.: Adaptive atmospheric entry guidance for the kistler K-1 orbital vehicle. In: AIAA Guidance, Navigation, and Control Conference, pp. 1275–1288. AIAA Press, New York (1999) 6. Youssef, H., Chowdhry, R.S., Lee, H., Rodi, P., Zimmerman, C.: Predictor-corrector entry guidance for reusable launch vehicles. In: AIAA Guidance, Navigation, and Control Conference, pp. 1–8. AIAA Press, New York (2001) 7. Saraf, A., Leavitt, J.A., Chen, D.T., Mease, K.D.: Design and evaluation of an acceleration guidance algorithm for entry. Journal of Spacecraft and Rockets 41, 986–996 (2004) 8. Zimmerman, C., Dukeman, G., Hanson, J.: Automated method to compute orbital reentry trajectories with heating constraints. Journal of Guidance, Control, and Dynamics 26, 523–529 (2003) 9. Shen, Z., Lu, P.: On-board generation of three-Dimensional constrained entry trajectories. Journal of Guidance, Control, and Dynamics 26, 111–121 (2003)
Abstract. A method for onboard generation of entry trajectory for variable targets is discussed. Conventional trajectory planning algorithms can only be used for the fixed terminal conditions without considering the variable targets. In case the vehicle needs to alert the entry trajectory due to damage or effectors failure, the entry guidance system must real-time design a feasible entry trajectory according to another feasible landing site from current flight conditions. The conventional approaches must be augmented to provide the real-time redesign capability for variable targets, and the redesign trajectory would also satisfy all path constraints and altered terminal conditions. This paper makes use of the neural network as a major controller to overcome this problem. The redesign trajectory problems and control parameter generations online problems can be transformed into the neural network offline training problem, given the initial conditions and the selected terminal conditions. Numerical simulations with a reusable launch vehicle model for various terminal conditions are presented to demonstrate the capability and effectiveness of the approach. Keywords: trajectory planning, neural network, entry, variable targets, onboard, real-time.
1 Introduction An important capability for the next generation entry vehicle is to autonomous return from any initial states, meaning that the guidance system is able to fast design a complete and feasible entry trajectory online from any initial position and velocity without any intervention. This guidance system can provide some benefits unmatched by the current guidance system. It would allow a vehicle can abort any abnormal fly mission during ascent or entry and choice the available landing site from current fly conditions without waiting for the guidance updates from the ground. It would also require significantly less the pre-mission analysis and planning for different missions, reduce dramatically the re-occurring optional costs associated with entry guidance. Conventional approach for entry mission operation consists of offline reference trajectory design and onboard tracking of the reference trajectory [1]. Before every mission, the reference trajectory would be accomplished on the ground and never be altered, which hasn’t apparent satisfactory the requirements of the next generation entry vehicle. Under the supports of the Advanced Guidance and Control (AG&C) Project [2, 3], many advanced guidance technologies have been developed in recent H. Deng et al. (Eds.): AICI 2011, Part III, LNAI 7004, pp. 318–325, 2011. © Springer-Verlag Berlin Heidelberg 2011
Application of Neural Network in Trajectory Planning of the Entry Vehicle
319
years. Gamble at el proposed an analytic predictor-corrector algorithm and a numeric predictor-corrector algorithm for the aero-assist flight experiment [4]. Fury at el presented an atmospheric guidance for steering the Kistler K-1[5], a fully reusable launch vehicle prior to deployment of the stabilization parachute with numeric predictor-corrector technology. Youssef and Chowdhry described a predictor-corrector guidance algorithm for X-33 [6]. Saraf at el designed an acceleration guidance algorithm for entry vehicle [7]. Zimmerman and Dukeman presented an automated method to design entry trajectory with heating constraints [8]. Z Shen and P Lu sought an approach for onboard generation of three dimensional entry trajectories [9]. An ideal algorithm for onboard planning trajectory should satisfy these requirements: guaranteed satisfaction of all inequality constraints, reliability, efficiency, less computational load and precise flight termination. The entry dynamic model is a typical nonlinear multi-input-multi-output (MIMO) system. Numerical predictor-corrector algorithms require repeated integration of the equations of motion, and conventional approaches suffer from the highly constrained nonlinear MIMO system, and don’t consider the trajectory redesign capability for variable targets. Hence, new guidance system that enables fully autonomous and adaptive entry guidance should be searched. In this paper, a real-time redesign trajectory algorithm for variable targets based on artificial neural network (ANN) is described. An onboard three-dimensional entry trajectory generation algorithm is used to obtain a set of training space to train the ANN. Numerical simulations show that trained neural network controller can handle the complicated non-linear MIMO system, design an approximate optimal feasible trajectory, and given a set of landing sites, the neural network controller can generate a series of control commands to maneuver the vehicle to reach the desired terminal conditions while imposing all the trajectory constraints.
2 The Entry Dynamics The following normalized three-dimensional point-mass dynamics of the vehicle over a spherical non-rotating Earth are discussed.
r = V sin γ
(1)
V cos γ sinψ r cos φ
(2)
V cos γ cosψ r
(3)
V = − D − sin γ / r 2
(4)
θ = φ =
1 cos γ r Vr
γ = (L / V ) cos σ + V 2 −
(5)
320
B. Zhang, S. Chen, and M. Xu
ψ =
L sin σ V + cos γ sinψ tan φ V cos γ r
(6)
Where r is the radial distance from the center of the earth to the vehicle, normalized by earth radius Re .The longitude and latitude are θ and φ , respectively. The earth-relative velocity V
is
normalized
by
Vc = g 0 Re
.The
time
is
normalized
Re g 0 . σ is the bank angle which is used by the primary control command. L and D are the dimensionless aerodynamic lift and drag accelerations as
by τ
=t
follows:
L = ρ (VcV ) S ref CL / ( 2mg 0 )
(7)
D = ρ (VcV ) Sref CD / ( 2mg 0 )
(8)
2
2
Where
S ref is the reference area of the vehicle, ρ is the atmospheric density, m is the
mass of the vehicle,
CL and CD are the lift and drag coefficients, respectively.
3 Entry Trajectory Constraints A feasible entry trajectory must satisfy path constraints. The typical inequality constraints considered here are: (1) Heating rate constraint:
ρ (VVc ) ≤ Q max 3
(9)
This corresponds to stagnation-point heating rate based on a reference sphere of radius 1m. (2) Normal load factor constraint:
L2 + D 2 ≤ nmax
(10)
It is a constraint on the aerodynamic load in the normal direction. (3) Equilibrium glide constraint:
1 2 1 r − V r − L cos σ ≤ 0
(11)
This constraint serves to reduce the phugoid oscillations in altitudes along the entry trajectory. (4) Dynamic pressure constraint:
q ≤ q max
(12)
Application of Neural Network in Trajectory Planning of the Entry Vehicle
Where
321
qmax is maximum allowable dynamic pressure. Q max , nmax , qmax are the
vehicle-depend parameters which would be enforced strictly.
4 Neural Network The artificial neural network is a mathematical tool inspired in the brain of animals, which is composed of information-processing neuron similar to the neurons of living creatures. The power of ANN lies in their ability to model the nonlinear or constrained systems when conventional analytical methods fail, which determines the ANN is capable of learning more information and is suitable to work as a surrogate model. Giving the learning information, the ANN is trained by iterative process which consists of two steps, the first is forward process which is termed “feed-forward” and the second is error correction process which is called “backpropagation”. In each process, real inputs are processed in a mathematical model to produce model outputs, the same real inputs are processed in a real world simulation to produce real outputs which are the target for the model, the real and model outputs are compared to produce error feedback which is used to correct the model. There are several training algorithms known, among which the Levenberg-Marquardt backpropagation is one of the most efficient for multilayer networks. The most usual model of neuron is the feed-forward ANN. In this model the input signals from other artificial neurons are multiplied by the weights that measure the net connections importance, which results in amplification or attenuation. They are added and the result is compared to the bias, creating the activation input for the activation function. The output of the activation function is the output of the node. The synaptic weights resembling the synapse of biological neurons are adjusted according to selected training algorithm. The process is repeated for every set of input parameters until the network performance reaches an acceptable level as measured by parameters such as error, gradient, or number of epochs. This learning feature of ANN makes the inputs are not included in the training set can generate expected results, the results for these variable inputs is called “generalization” and measures how well the ANN reacts to unforeseen data, and it becomes a criterion for a trained ANN is successful or not. In the paper, the RBF NN is chosen. The RBF neural net is a simply composed of three layers: an input layer, a hidden layer and an output layer. The connections between each layer are associated to synaptic weights. The output of layer y can be described as: No
y ( x ) = wiφ ( x − ci , σ i ) + w0 i =1
(13)
x is input vector to the output nodes, φ (•) represents transfer function of hidden nodes i , ci is the center of radial basis function of hidden nodes i , σ i is the width of the radial basis function, wi is the connection weight between node i and the node in output layer, N o is the number of nodes in hidden layer, w0 is bias.
Where
322
B. Zhang, S. Chen, and M. Xu
In the hidden layer, the Gaussian kernel function is described as:
φ ( x − ci , σ i ) = exp −
x − ci σ i2
It is the most commonly used radial basis function. The closer the input
(14)
x p is to be
center of the Gaussian function ci , the larger is the output of the basis function.
5 Training The center purpose of trajectory onboard planning algorithm is to develop a high-performance controller to generate approximate control commands with respect to the nonlinear system imposed by nonlinear constraints. In order to purpose this goal, a set of training samples would be created to train the weight matrix W. The next generation reusable launch vehicles (RLV) would be high lift-to-drag ratio entry vehicles. This feature makes the entry trajectory has some obvious characteristics, it consists of three different phases: initial descent phase, quasi-equilibrium glide (QEG) phase, and pre-TAEM (terminal area energy management) phase [9]. The characteristic of initial descent phase is low atmospheric density and the vehicles have not enough control authority to steer the RLV, which is called the “controlled fall” that takes the RLV to the QEG interface where the dynamic pressure can shape the trajectory. The QEG phase is most critical flight phase, the trajectory must observe all of the inequality path constraints, the achieved range would reach the landing site, and the control commands suffer from the highly constrained nonlinear MIMO dynamics system. In the pre-TAEM phase, the trajectory must satisfy all of the equality constrains, meaning that the trajectory would meet the TAEM conditions at the end of this phase. Determining the real-time trajectory for high nonlinear entry vehicles has been of considerate research interest in past years. Conventional approaches can not alter the entry trajectory, and can not also deal the abort missions which need redesign the new trajectory for feasible new landing sites. Continued efforts have been made to search for adaptive entry guidance methods that enable fully autonomous capability of guidance system to provide precise flight termination. Ref [9] has developed a completely new and efficient onboard algorithm for RLV. It makes novel use of quasi-equilibrium glide phenomenon in lifting entry as a center piece for efficient enforcement of the inequality constraints, and the longitudinal and lateral trajectory profiles planning problem is reduced to one parameter search problem. This approach can generate a complete and feasible 3DOF entry trajectory in about 2 to 3 seconds on a desktop computer. In this text, this approach is used to create the training samples. A three layer RBF neural network is used to approximate the dynamics model of the entry trajectory discussed above. The network uses the structure of 6 neurons in the input layer, 6 neurons in the hidden layer and one neuron in the output layer. The input to the network is the state variables
[ r , φ ,θ , v, γ ,ψ ]
T
, and its output is bank angle σ ,
the angle of attack profile does not change significantly from mission to mission, including the abort missions. This paper uses the nominal angle of attack profile. The
Application of Neural Network in Trajectory Planning of the Entry Vehicle
323
ANN training process is repeated for each selected initial condition and accomplished via Levenberg-Marquardt backpropagation algorithm.
6 Testing The nominal vehicle model, X-33, is used in this paper. The corresponding aerodynamic parameters can be cited from [6]. The nominal initial entry states are:
[ r0 ,θ0 ,φ0 ,V0 , γ 0 ]
120km, 208D , 76D , 7650m / s, −0.5D . The desired D longitudinal terminal states are: [ r0 , V0 , γ 0 ] = 25km,1760m / s, −4 . Random =
landing sites are chosen to generate a set of training data. There are five cases to demonstrate the generalization capability of the well trained neural network. The corresponding longitudes and latitudes for all of the cases are random generated based on the acceptable limit of the vehicle maneuverability. Table 1 shows the TAEM comparative errors for all the cases. The results illuminate that these errors are small and the terminal landing sites generated by ANN can be accepted. Fig.1 shows the altitude history, Fig.2 shows the velocity history. These two figures describe that the trajectories generated by ANN are very close. The three-dimensional trajectories can be shown in Fig.3. All these figures and table reveal that the RBF ANN can successfully design the entry trajectory and the entry vehicle can reach the terminal states with an acceptable error range. Table 1. TAEM Comparative errors
Case1 Case2 Case3 Case4 Case5
Δrf (km)
ΔV f (m/s)
Δθ f (deg)
Δφ f (deg)
0.276% 0.171% 0.114% 0.152% 0.147%
0.285% 0.313% 0.152% 0.396% 0.354%
0.225% 0.364% 0.146% 0.254% 0.312%
0.387% 0.413% 0.345% 0.263% 0.154%
140 120 100
km
） 80
r
（ 60 40 20 0
200
400
600
800
1000
1200
1400
Time（ s）
Fig. 1. Trajectories altitudes for different terminal conditions
324
B. Zhang, S. Chen, and M. Xu
8000 7000 6000
m/s
）5000
v
（4000 3000 2000 1000 0
200
400
600
800
1000
1200
1400
Time（ s）
Fig. 2. Trajectories velocities for different terminal conditions
r炷km炸
150
100
50
0 80 60
Φ炷°炸
40 20 150
160
170
180
190
200
210
θ炷°炸
Fig. 3. Three-dimensional trajectories for different terminal conditions
7 Conclusion In this paper, an artificial neural network trained by the three-dimensional onboard trajectory generation algorithm is applied to real-time design different entry trajectory according to different landing sites. Theoretical analysis and simulation results show that the well trained neural network has good generalization capability, and the each trajectory generated by neural network is feasible solution with acceptable terminal states, meaning that a well trained artificial neural network augments the entry vehicles autonomous redesign trajectory capability and is more suitable for complex entry dynamics system.
Application of Neural Network in Trajectory Planning of the Entry Vehicle
325
References 1. Harpold, J.C., Graves, C.A.: Shuttle entry guidance. Journal of the Astronautical Sciences 27, 26–29 (1979) 2. Hanson, J.: A plan for advanced guidance and control technology for 2nd generation reusable launch vehicles. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, pp. 1–9. AIAA Press, New York (2002) 3. Hanson, J.: Advanced guidance and control project for resuable launch vehicles. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, pp. 1–10. AIAA Press, New York (2000) 4. Gamble, J.D., Cerimele, C.J., Moore, T.E., Higgins, J.: Atmospheric guidance concepts for an aeroassist flight experiment. Journal of the Astronautical Sciences 36, 45–71 (1988) 5. Fury, D.P.: Adaptive atmospheric entry guidance for the kistler K-1 orbital vehicle. In: AIAA Guidance, Navigation, and Control Conference, pp. 1275–1288. AIAA Press, New York (1999) 6. Youssef, H., Chowdhry, R.S., Lee, H., Rodi, P., Zimmerman, C.: Predictor-corrector entry guidance for reusable launch vehicles. In: AIAA Guidance, Navigation, and Control Conference, pp. 1–8. AIAA Press, New York (2001) 7. Saraf, A., Leavitt, J.A., Chen, D.T., Mease, K.D.: Design and evaluation of an acceleration guidance algorithm for entry. Journal of Spacecraft and Rockets 41, 986–996 (2004) 8. Zimmerman, C., Dukeman, G., Hanson, J.: Automated method to compute orbital reentry trajectories with heating constraints. Journal of Guidance, Control, and Dynamics 26, 523–529 (2003) 9. Shen, Z., Lu, P.: On-board generation of three-Dimensional constrained entry trajectories. Journal of Guidance, Control, and Dynamics 26, 111–121 (2003)
