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AlAA 90-2829
Optimal Multiple Vehicle Trajectories
C. R. Hargraves and S. W. Paris
Three approaches for solving multiple vehicle trajectory optimization problems are
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described. All three approaches are options that are available in a trajectory optimization computer
code called OTIS (Optimal Trajectories by Implicit Simulation). Three examples of multiple
vehicle problems solved by OTIS are described.
Many contemporary trajectory optimization problems involve more than one vehicle.
Examples are:
Determine the minimum time intercept of a target flying a specified path.
Study orbit injections where one or more stages are to return to the launch site.
Optimal control of an aircraft to evade a missile with proportional navigation g u i d a n.c. ~
The "standard" approach for solving these problems is to treat each one separately and to
develop a specialized procedure for solving it. In this paper we are interested in demonstrating
techniques that allow solution of a wide class of multiple vehicle problems as well as the more
usual problems involving a single vehicle. The methods described below have been incorporated
into a general purpose trajectory optimization code called OTIS. The OTIS code is based on a
direct optimization approach using cubic splines, collocation and nonlinear programming.
A direct optimization method using collocation and nonlinear programming was described
by the authors in Reference 1. These procedures were developed into a computer code called
OTIS which is documented in Reference 2.
Traiectory Modeling
The concept of decomposing a trajectory into a linked sequence of arcs, called phases, is
fundamental to providing a flexible simulation. The initial time and the final time for each phase
are called events. An event may be thought of as an interruption of the current phase. Any time
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the data describing a phase is to be changed, an event must occur. The desired simulation
characteristics for the next phase must then be specified.
The information that must be specified for each phase is:
starting conditions;
aerodynamic data;
propulsion data;
number of vehicles;
jettison weight;
environmental model;
stopping condition;
value of stopping condition;
control angle functional form;
numerical integration type;
guidance type;
integration coordinates (Cartesian, flight path or body);
physical constraints;
launch options;
integration step size or accuracy;
print step size;
output options.
The required number of phases is a function of the trajectory. In most cases phases are
sequential. This is not required, e.g., two phases may start at the same time and end at different
times. This provides the mechanism to model multiple vehicle problems A trajectory description
by phase is illustrated in Figure 1.
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1 I
I Phase I Phase
Figure 1 Trajectory Description by Phase
Mathematical Problem Definition
We want to determine the trajectories for one or more aerospace vehicles. The start and
end times of each phase are defied as Ej and Ej+l, where j is equal to the phase number and NP
is equal to the total number of phases. The parameters Ej, j = 1, NP + 1 are called events.
Solutions are sought to the set of differential state constraints (the equations of motion,
mass-flow equation, heat-flux etc.),
where xl = (xi: i = 1, NS(j)) and NS(j) equals the number of states for phase j. In addition to the
state variables, x, there are auxiliary variables, y, (dynamic pressure, heat flux etc.) defined as
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where yj = (xk: k = 1, NAG)) and NAG) is equal to the number of auxiliary parameters for phase j.
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This allows path constraints of the form
where NPCG) equals the number of path constraints for phase j. The control vector, u, (angle of
attack, throttle factor, etc.) is defined as,
where NCCj) equals the number of control variables for phase j. The discrete design parameters
D = ( q j : p = l,NDP(j), j = 1,NP)
where NDPG) equals the number discrete or design variables in phase j.
The intervals (phases) are established as,
[Ej, Q+l], j=l,NP
At each event, boundary conditions of the form
tj E
can be imposed. In addition, discontinuities in the states can be imposed at the events
Denote the collection of events by E = (El, E2,...,E ~ p + l )the
, collection of states by
x = (xl, x2,..., xNP+l), and the collection of controls by u = (ul, u2, ..., uNP+l). The optimal
solution to Equations (1) to (5) is defined to be the set of vector time histories x(t), u(t) and vector
parameters D and E which satisfy (1) to (5) and minimize the payoff function, J,
which without loss of generality is assumed to depend only on the boundary values (i.e.,
conditions at events). Integral payoff functions are easily handled by introducing new states. We
choose to always minimize the payoff. To maximize $, minimize J equal - $.
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Outimal Control Solution Method
The method we use to solve an optimal control problem is outlined below.
States and controls are represented by piecewise polynomials.
Differential equations (1) are always satisfied at nodes (Hermite Interpolation).
Implicit integration is used to enforce the differential equations at the segment centers.
Nonlinear programming is used to:
minimize the objective;
integrate the differential equations;
satisfy the boundary conditions;
and impose physical constraints.
The free parameters are:
the states and controls at nodes (X,U);
design variables (D);
events (or phase times) (E).
When using implicit integration the phases are subdivided into a sequence of segments. The end
points of segments are called nodes. All events are nodes but only a few (in general) nodes are
events. This situation is illustrated in Figure 2. Within each segment, all states and controls are
represented by polynomials in time. The time length of each segment is a user specified fraction
of its corresponding phase time length. Segment lengths should be selected so that shorter
segments occur where accelerations are high and longer segments are placed where they are low.
A consequence of implicit integration is that an accurate integrated trajectory is not obtained until
the optimization code has converged.
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Events E
Time --b
Figure 2 Piecewise Polynomial Representation
We may integrate the equations of motion by quadratures of the form,
f(x, t) dt
We impose the following conditions:
x is a cubic polynomial in time on each segment,
f is to be evaluated at nodes and segment centers only,
any conditions to be satisfied must be local, i.e. involve only adjacent nodes.
If x is known at node 1, then at node 2 we must have,
x2 = xl
f(x, t)dt
where T (=Tpt ) is the time length of the segment.
If x is given at the first node, the above equation applied to each segment defines a set of
conditions which when satisfied, define x as the integral o f f over one phase. Linear conditions are
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used to hook the phases together.
Since x is to be cubic in time, it is sufficient that p be quadratic in time. Function values a
two nodes and the segment center are sufficient to define a quadratic in normalized time E =t /T on
the segment,
This integrates to,
A little algebra shows that we must have,
where F1, F2 and FC are the values o f f at the left and right nodes and the segment center. Thus
we have,
Moving x2 to the right in the above gives a nonlinear equation which is to be driven to zero. We
can multiply this by a constant 312T. Doing this, rearranging the F terms and calling the result A
We call A the "defect" at the segment center. If the cubic polynomial is capable of representing the
solution on the given segment, then selecting x l , x2 and Tsi to drive A to zero will produce an
accurate solution to equation (I). Note that A can also be defined as the difference between ~ ( X C )
and XC'.
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The above procedure is illustrated in Figure 3. The defects for each state are evaluated at
the center of each segment and equated to zero. This constitutes a set of nonlinear algebraic
equations which are functions of the states and controls at each node, the events and the design
parameters . The values and slopes of the controls at each node are free. Constraints are imposed
to make the second derivative of the controls continuous at interior nodes and zero at the boundary
nodes of each phase. Cubic interpolation is used to obtain the controls at the segment centers. The
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boundary conditions (4) and the constraints (3) evaluated at both the nodes and the centers of the
segments provide additional equations (constraints) to be satisfied. Equation ( 5 ) constitutes a set
of linear
Select XI, X 2 and Ts so that A=O
Segment Center
Figure 3 Implicit Integration
equations which connects all the phases. All of the independent variables are collected into a single
vector P defined by
Collecting all of the nonlinear equations into a single vector equation yields
Dij is the defect for ith statement at jth segment center.
CN is the collection of all nonlinear constraints from (3).
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is the collection of all nonlinear boundary conditions from (5).
Note that the payoff function J is a function of only P. The trajectory optimization problem
stated above can be expressed as
minimize J = c#P)
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subject to
Np is the dimension of P. AP is comprised of all linear relationships from equations (3) and (5).
1 and u are lower and upper bounds. To handle equality constraints, li is set to ui. Single sided
inequality constraints are handled by setting li= -
or ui= +-.
The partial derivatives of the constraints (defects, physical constraints and boundary
conditions) with respect to the independent variables (states, controls, phase times and design
parameters) are required. This is accomplished by using finite differences. We note that the task
of computing the derivatives is greatly simplified by the sparsity of the Jacobian which results
from the fact that the defects depend only on the states at adjacent nodes.
We consider the following three methods for describing multiple vehicle problems:
(1) dumb targets,
(2) linked phases,
(3) extended state vector.
Dumb Targets
Dumb targets are vehicles that move as a specified function of time or some primary state
variables. The OTIS code provides three alternative ways of propagating the position vectors for
dumb targets:
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(1) motion along great circle paths,
(2) tabular functions of position and velocity.
(3) satellite orbits (two body propagation),
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Linked Phases
The idea of linked phases is that all trajectory problems be described by one or more
phases that can be totally independent of each other. Each has its own set of gravitational,
aerodynamic and propulsive forces and each may have independent boundary conditions.
Information is then provided as to how these phases should be linked together in both time and
state. In the simplest situation, the phases may be different stages of a single vehicle (e.g., a 3
stage booster for putting a satellite in orbit). However, much more complex situations are
possible. Each phase can be totally different vehicle with position and velocity evolving at the
same time or at different times. They can also represent a single vehicle which splits into two or
more vehicles (e.g., a flyback booster). The linking conditions specify that the initial time and state
for each phase may be:
(1) free,
(2) fixed,
(3) equal to final time or state of another phase,
(4) be a function of the final time and state of another phase. (Reference 3)
Option (3) can specify, for example, complete continuity in time and state as would occur
if all phases represent the same vehicle. Discontinuities in mass alone can represent spent stage
jettisons. Option (4) can allow, for example, analytic propagation between phases. This can make
certain orbit propagation problems much more efficient since only the thrusting phases need be
Extended State Vector
Dumb targets and linked phases have the disadvantage of limited influence of one vehicle
on another. For example, neither of these approaches allow simulation of an airplane optimally
evading a missile with proportional guidance. For these and other more general multi-vehicle
problems, we extend the state vector to represent the position, velocity and mass for two or more
vehicles. OTIS allows simulation of two vehicles using this approach. Controls for one vehicle
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may be used for optimization while the other must follow some fixed guidance law (e.g.,
proportional navigation, tabular etc.).
(1) Glider Minimum Time to Intercept
This problem is involved with generating the minimum time intercept of a target flying at
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constant speed, altitude, and heading. The interceptor was an unpowered aircraft with the
following initial conditions:
V= 20,000 ft/sec
h = 90 degrees (due east)
y = 0 degrees
h= 200,000 ft
longitude = 0 degrees
latitude = 0 degrees
The target state vector at time 0 was
V= 1,000 fdsec
h = 180 degrees (due south)
y = 0 degrees
h= 50,000 ft
longitude = 30 degrees
latitude = 0 degrees
The final interceptor position was constrained to be within 1000 feet of the target. This
boundary condition was imposed using one of two methods. The first method constrained the
slant range between the interceptor and the target to be 1000 feet. This method did not converge
well. The second method involves placing constraints on the differences between the interceptor
and target's altitude, longitude, and latitude. This approach was superior in terms of convergence.
The solution for this problem is shown in figures 4 and 5. The trajectory for this type of
problem is rather simple. The interceptor turns and flies to a point in space.
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Time, t, sec
Time, t, sec
- 10- 15203
Time, t, sec
2 a
Angle of Attack.
Time, t, sec
Figure 4 Glider Minimum Time Intercept Trajectory
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Longitude, 8, degrees
Longitude, 9, degrees
Figure 5 Dumb Target Intercept
(2) Flyback Booster
This is a booster problem where the final weight injected into a specified orbit is
maximized. The vehicle is composed of two stages (booster and orbiter) which are similar in
exterior shape and size. The rocket engines are initially crossfed from the booster stage until its
fuel and oxidizer are consumed. At that point, the booster is separated and the orbiter (second
stage) continues to orbit injection.
Two optimal trajectory problems were solved. The first trajectory was involved in
maximizing the weight injected into a polar circular orbit whose altitude is 100. n.mi. No
considerations were made for the recovery of the booster. The second trajectory was also
concerned with the maximization of the final weight into a 100 n.mi. circular orbit, with the
constraint that the booster glide back to the launch point.
The specified (fixed) boundary conditions for these problems are as follows:
Initial condition2
Velocity = 0. feedsecond
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Flight Path Angle = 90 degrees
Altitude = 0. feet
Longitude = - 120. degrees
Latitude = 34.5 degrees
Weight = 1,249,318 pounds
Stage One Separation
Weight of Stage One and Stage Two = 67 1,590 pounds
Staoe Two Ignition
Weight = 532,931 pounds
Stage One Start of Glide
Weight = 138,659 pounds
b g . e Two Bum Out
Altitude > 300,OO feet
Apogee Altitude > 100. n.mi.
Perigee Altitude > 50. n.rni.
Orbital Inclination = 90. degrees
End of Stage One Glide
200 feetkecond < Velocity < 700. feedsecond
Altitude > 0 feet
Longitude = - 120. degrees
Latitude = 34.5 degrees
The trajectory was modeled using four phases. The first phase covers a short vertical rise
where the vehicle is pitched over at a rate of -.001 radians per second. The second phase models
the the two mated stages. The third phase models the flight of the orbiter. The fourth and final
phase models the booster's glide from the staging point to the launch site. The payoff was the
weight injected into a circular orbit whose altitude is equal to the the apogee at the end of third
phase. The controls for this problem were chosen to be angle of attack and bank angle. The bank
angle was constrained to be zero for phases 1,2 And 3. The initial conditions for phase 4 are
specified to be the same as the final conditions for phase 2. The final conditions for the glide were
constrained using nonlinear constraints on velocity, altitude and flight path angle. Equality
constraints were use to specify the values for longitude and latitude.
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The final optimal trajectory and several plots displaying the flyback and non-flyback cases
are given as Figures 6, 7 and 8. The maximum injected weight for the flyback case was 117,445
lb, while the value for the non-flyback case was 122,356 lb.
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Time, t, sec
Time, t, sec
Figure 6 Booster Trajectories
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Figure 7 Booster Controls
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Downrange, d, n.mi
Downrange, d, n.mi.
Figure 8 Booster Ground Track
(3) Optimal Evasion of Missile with Proportional Guidance
This problem involves an aircraft evading an incoming missile. The initial conditions for
the aircraft and the missile are given below.
Altitude = 5,000 feet
Altitude = 2,000 feet
Velocity = 500 feet/sec
Velocity = 2,000 feet/sec
Flight Path Angle = 0.0 degrees
Weight = 22,000 pounds
Heading = -90.0 degrees (W)
Latitude 30.0 degrees (N)
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Longitude = 50.0 degrees (E)
The following optimization problems was solved: Maximize the miss distance by flying
an optimal evasive maneuver. The missile chases the aircraft.
The aircraft is modeled as the first vehicle and OTIS will determine optimal angle of attack
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and bank angle to maximize the miss distance at the point of closest approach, which is forced to
be the final time as described below.
The second vehicle is modeled as the missile and its angle of attack and bank angle are determined
by a proportional navigation algorithm.
The state vector for this problem contains the states for first vehicle followed by the states for
second vehicle followed by first vehicle controls and control rates (18 total - no second vehicle
control information).
A final constraint on the relative position and velocity ensures that the final point is the
point of closest approach.
Plots for this problem are shown as Figures 9, and 10. For the initial conditions chosen
the missile flies through the Earth. From a practical point of view, the simulation ended when the
missile hit the ground.
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9 49'. 91
49'. 95
Longitude, 8,degrees
Longitude, 8, degrees
Figure 9 Optimal Missile Evasion
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P ; I W m
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Time, t, sec
Time, t, sec
Figure 10 Missile Evasion Controls
Efficient simulation of multi-vehicle trajectory optimization problems is greatly facilitated
by use of a variety of simulation methods. A computer code (OTIS) which uses collocation and
nonlinear programming has incorporated many of the procedures described. Three different
examples of OTIS solutions are described to demonstrate the different approaches.
(1) Hargraves, C.R. and Paris, S.W., "Direct Trajectory Optimization using Nonlinear
Programming and Collocation", Journal of Guidance, Control and Dynamics, July-August 1987.
(2) Paris, S.W., Ilgenfritz, D. H., and Hargraves, C.R., "Trajectory Design for Hypemelocity
Research Vehicles", AIAA paper 88-4344.
(3) Conway, B.A., Presentation on Trajectory Optimization Procedures, Kent, Wa. January 1990.
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