c)2001 American Institute of Aeronautics & Astronautics or ...dahleh.lids.mit.edu/wp-content/uploads/2011/09/frazzoli_aiaa.pdf · the execution of rapid attitude maneuvers at a very

c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

AiAA A01-37347

AIAA 2001-4155A RANDOMIZED ATTITUDE SLEWPLANNING ALGORITHM FORAUTONOMOUS SPACECRAFTE. Frazzoli, M. A. Dahleh, E. FeronMassachusetts Institute of Technology, Cambridge, MAR. P. KornfeldJet Propulsion Laboratory, Pasadena, CA

AIAA Guidance, Navigation, and ControlConference and Exhibit

August 6-9, 2001/Montreal, Quebec, CanadaFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344


AIAA 2001-4155

A RANDOMIZED ATTITUDE SLEWPLANNING ALGORITHM FORAUTONOMOUS SPACECRAFT

E. Frazzoli? M. A. Dahlehf E. Feron*Massachusetts Institute of Technology, Cambridge. MA

R. P. Kornfeld8

Jet Propulsion Laboratory, Pasadena, CAThe ability to autonomously generate and execute large angle attitude maneuvers,

while operating under a number of celestial and dynamical constraints, is a key factorin the development of several future space platforms. In this paper we propose a ran-domized attitude slew planning algorithm for autonomous spacecraft, which is able toaddress a variety of pointing constraints, including bright object avoidance and groundlink maintenance, as well as constraints on the control inputs and spacecraft states, andintegral constraints such as those deriving from thermal control requirements. Moreover,through the scheduling of feedback control policies, the algorithm provides a consistentdecoupling between low-level control and attitude motion planning, and is robust withrespect to uncertainties in the spacecraft dynamics and environmental disturbances. Sim-ulation examples are presented and discussed.

Introduction

A key enabling technology that could lead to .greater spacecraft autonomy is the capability to

autonomously and optimally slew the spacecraft to adesired attitude while operating tmder a number of ce-lestial and dynamic constraints. Large angle attitudeslews are typically required for certain space sciencemissions for retargeting of pay load instrumentation,e.g. to point a telescope towards a new celestial ob-ject. In addition, typical mission profiles for futureplatforms , e.g. missile-tracking satellites, will requirethe execution of rapid attitude maneuvers at a veryshort notice.

In most cases the slew maneuver is subject to sev-eral constraints. As an example, consider a space-craft carrying sensitive science or navigation sensors(e.g. cryogenic-ally cooled telescopes, or star track-ers): in this case the attitude reconfiguration mustbe achieved without directing the sensors along theSun vector, or along other "bright" regions of the sky.Other constraints can come from the necessity to main-tain a communication link with the ground station,from pointing requirements imposed by other space-craft subsystem (e.g. thermal control), or even from

'Post-doctoral associate. Laboratory for Information and De-cision Systems, email: frazzoliQalum.mit.edu. AIAA member.

t Professor. Department of Electrical Engineering and Com-puter Science, email: dahlehOmit.edu.

* Associate Professor. Department of Aeronautics and Astro-nautics, email: feronSmit.edu. AIAA senior member.

§ Staff engineer, Avionic Systems Engineering Section, email:richard.p.kornfeldQjpl.nasa.gov. AIAA member.

Copyright © 2001 by the American Institute of Aeronauticsand Astronautics. Inc. All rights reserved.

the necessity to reduce the probability of collision ofsensitive components of the spacecraft with man-madeor natural orbiting objects (e.g. in the case of thecrossing of planetary rings).

The planning of such constrained maneuvers can bevery challenging, and can result in a costly increase inthe workload of the ground segment, and in a reduc-tion of the payload availability and flexibility of thesystem - for example because of a reduced ability totrack targets of opportunity, for which a relatively fastmission replanniiig is required.

BackgroundThe problem of planning large angle slew maneuvers

for autonomous spacecraft, avoiding bright objects hasbeen addressed by Mclnnes,1'2 who used an approachbased on the definition of an artificial potential func-tion. The potential function guides the spacecraft tothe target attitude, while avoiding pointing the pay-load boresight along the Sun vector. Such an approachsuffers from the main drawback associated with poten-tial function methods in robot motion planning: theconvergence of the generated trajectory to the desiredattitude is not guaranteed in general, as the spacecraftmight get trapped at a local minimum of the potentialfunction. While possible in principle, the computationof a potential function that is free of local minima (anavigation function3) is a very hard problem from thecomputational point of view, for non-trivial constraintspecifications: as a matter of fact the only constraintthat is taken into account is Sun vector avoidance bya single body-fixed vector (i.e. the boresight of a tele-

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AMERICAN INSTITUTE OF AERONAUTICS AXD ASTRONAUTICS PAPER 2001 4155


scope). Another limitation of this approach (which,as a matter of fact, is common to almost all of theliterature on the subject) is due to the fact that thespacecraft is treated as a pointing device, that is asystem evolving on the sphere 52

? as opposed to arigid body, whose attitude evolves on the group ofrotations hi the three-dimensional space, SO(3), Inother words, pointing constraints are imposed onlyon one axis of the spacecraft; moreover, the controltorques are computed in such a way that only oneaxis of the spacecraft is actually controlled, whereasrotations about this axis can be arbitrary. As a con-sequence, a direct application of such algorithms tocases in which the pointing constraints involve multiplebody-fixed directions is not possible. Dynamical andintegral constraints are also not directly addressed.

A constraint monitoring algorithm was developedfor use on the Cassini spacecraft.4 The purpose of thisalgorithm is to monitor the attitude maneuver requestsfrom the ground for violation of pointing and dynam-ical constraints (such as maximum angular rates oraccelerations), and if necessary provide appropriateconstraint avoidance commands. While the detectionalgorithm is indeed able to check pointing constraintsalong multiple body-fixed directions, as well as dynam-ical and integral (or ;itimed") constraints, the avoid-ance maneuvers are still computed from a pointingdevice perspective. As a matter of fact, the avoid-ance maneuvers are computed in such a way that the"offending axis" Is moved away from the avoidance(-one. In the case in which multiple constraints areviolated at the same time, a linear combination of theavoidance maneuvers is performed. Wlule reportedlysuccessful in many simulations, such a strategy couldfail for apparently simple constraint configurations.Moreover, the avoidance maneuvers are computed toavoid constraint violations over a short period of time,and do not guarantee that progress towards the de-sired attitude is actually made: as in the previous case,the spacecraft might become "trapped" between con-straints.

To avoid such a possibility, a global perspective mustbe retained in the planning process. Hablani5 consid-ers attitude motion planning in the presence of brightobjects, while maintaining a communication link withthe ground station. The proposed approach is basedon a careful analysis of the vectorial kinematics on thesphere, leading to the avoidance of the bright objects.At the same time, the availability of an additionaldegree of freedom is exploited to ensure that the com-munication link with the ground is maintained. Whilevery effective in the case presented by the author, thisapproach is based on a rather complex logic, whichcan be cumbersome in implementation and difficultto extend to more complex scenarios, involving severalbright objects, and integral and dynamical constraints.In particular, the proposed approach is directly appli-

cable only to cases in whidi two pointing constraintsare active at the same time along the desired path.

The above approaches have been specifically devel-oped for spacecraft attitude slew planning. On theother hand, the problem of attitude slew planning isclosely related to the traditional robotics problem ofmotion planning in the plane or in higher dimensionalspace, and as such it should be possible to use tech-niques originally developed in the field of robotics6"9

(notice that this is the same idea behind Mclnnes' ap-proach). The main difference lies in the structure ofthe workspace, which in the case of attitude motion isthe group of rotations SO(3), as opposed to Rn.

The solution of general motion planning problems,is computationally very demanding, even in the mostbasic formulations. The so-called "generalized mover'sproblem", involving motion planning for holonomickinematic robots made of several polyhedral parts,among polyhedral obstacles, has been proven by Reif10

to be PSPACE-hard, and hence at least NP-hard.11

This ••— together with the topological difficulties ofplanning paths on SO(3) — has motivated the de-velopment of heuristic algorithms such as those out-lined earlier in this section which are able to computequickly a solution in many cases of interest. On theother hand, no guarantees are available on the abilityof such algoritlirns to find a solution.

Randomized motion planning algorithmsProbably the most exciting recent advance in motion

planning research is due to the introduction of ran-domized motion planning algorithms.12 Of particularinterest for the purposes of this paper is a new class ofmotion planning algorithms, composed of ProbabilistcRoadMap (PRM) planners.13"18 and their incremen-tal counterparts, pioneered by the Rapidly-exploringRandom Tree algorithm.19"21

This class of algorithms is particularly interestingsince, by relaxing the completeness requirements toprobabilistic completeness (i.e. completeness in expec-tation), it provides computational tractability, whilemaintaining formal guarantees on the behavior of thealgorithm. These algorithms are based on the con-struction of a graph of feasible trajectories (i.e. trajec-tories along which the constraints are satisfied).

Probabilistic RoadMap planners have been provento be complete in a probabilistic sense, that is, theprobability of finding a feasible trajectory to the tar-get attitude (if one exists) approaches unity as thenumber of nodes — and consequently of trajectorysegments in the graph increases. Moreover, per-formance bounds have been derived as a function ofcertain characteristics of the environment (expansive-ness) which prove that the probability of incorrecttermination decreases exponentially fast in the num-ber of nodes.22 Corresponding bounds have also beenfound for the incremental version of such algorithms,

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in the presence of multiple time-varying constraints,dynamical and integral constraints.23'24

In this paper, we will discuss how this new class ofalgorithm can be used to design highly efficient plan-ners for autonomous spacecraft attitude maneuvers;for this purpose we will concentrate on the on-line gen-eration of attitude motion commands, starting fromsome given initial conditions. Since we will build thegraph incrementally, the graph will have the additionalstructure of a tree*, rooted at the initial conditions.Nodes in the tree, also referred to as "milestones" rep-resent states of the system along trajectories, edges canbe thought of as representing "decisions*' (or high-levelcommands) taken in the construction of the motionplan,

The planner presented in this paper is based on analgorithm introduced by the authors in earlier publi-cations, and applied to motion planning of agile au-tonomous vehicles.24'25 Central to this algorithm isthe assumption that an constraint-free guidance loop isavailable, which is able to steer the system to a desiredattitude at rest, assuming that there are no constraintsin the environment. A vast body of literature is avail-able on this subject, as exemplified by the book editedby Wertz.26

Since the ability of the planner-to compute a feasibletrajectory to the target attitude is guaranteed only ina probabilistic sense, it is important to ensure thatconstraint violation is always prevented, even thougha full solution is not found. The algorithm that wewill present does indeed provide safety guarantees inthe sense that all the generated attitude commandswill result, in feasible trajectories, even in the case atrajectory to the target attitude is not found.

Motion planning frameworkThis section introduces the elements used to for-

mulate? the attitude motion planning algorithm, andto generate the simulation examples. Notice that themain assumption is the availability of an obstacle-freeclosed-loop attitude control system that ensures thecapability to steer the spacecraft to the desired atti-tude, if no pointing constraints are present. One ofthe main differences in this paper with respect to ear-lier publications is the fact that we are dealing witha closed-loop, feedback-controlled system, with its re-sulting characteristics of robustness with respect toexternal disturbances and uncertainties.

* A tree is a directed graph, with no cycles, in which all nodeshave several outgoing edges, to each of which corresponds an-other node, called a child. A child node can in turn have severalother children. All nodes (excluding the root) have one andonly one incoming edge, which conies from the parent. The roothas no parent. A common example of a tree is the directorystructure in a computer file system.

System dynamicsFor the sake of simplicity, we will consider a sim-

ple, rigid body model for the attitude dynamics of thespacecraft, in the absence of a gravitational field orother disturbance torques. More complex spacecraftattitude dynamics models can be used, with no signif-icant changes to the planning algorithm (but a stabiliz-ing attitude control law must be available). Under thestated hypothesis, the configuration of the spacecraft isrepresented by an element R of the group of rotationsin the three-dimensional space. SO'(3); in the followingwe will use a matrix representation of R but other rep-resentations are possible — i.e. quaternions or Eulerangles.26 The angular velocity in body axes is givenby the vector u; G R3. The state space of the systemwill consist in the product X = SO(3) x M3, and wecan define the state of the system as x = (/?,u;). Theattitude dynamics of the spacecraft will be describedby the following set of ordinary differential equations(where the explicit dependence on time of R. u; and uis omitted):

R = Ruu = -a? X J(jJ -f M(u). (i)

In the above equations. J is the inertia tensor of thespacecraft, M denotes the moments generated by thecontrols u, and the skew matrix Cj is defined as theunique matrix for which LJV = & x v for all vectorsv e R3.

In most cases of interest, the dynamics of the space-craft are limited by constraints due to actuator sat-uration. Among these we can mention limits on themaximum impulse or maximum thrust available fromthrusters, and on the maximum torque and maximumstored momentum for reaction wheel or control mo-ment gyros. Other constraints of the same form mightbe driven by sensor requirements. For example, lim-its on the angular rates can derive from star-trackerrequirements. These constraints do not depend onthe attitude of the spacecraft, and we can call them"flight envelope" constraints. Such constraints can beencoded by inequalities of the form:

;, u) (2)

or, as in the case of maximum stored momentum, byintegral inequalities of the form:

(3)

Constraint-free guidance systemThe main assumption on which our algorithm is

based is the availability of a guidance algorithm whichprovides feasible trajectories in environments with no"obstacles". or pointing constraints. More specifically,we assume knowledge of a guidance law that can steer

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the system, in the absence of pointing constraints,to a rest condition at a desired target attitude Rd.This guidance law will be a feedback control policyparametrized by the desired attitude, wliich will bedenoted by 7r(-,/?d) (i.e. the control input will be

Based on the definition of the control policy TT, wealso require that a Lyapimov function be available,which provides a measure of the "distance'' of the ini-tial condition x = (R,u) from the target (/J^.O); sucha Lyapimov function can be given for example by the"cost", in terms of time or of control effort, whichis necessary to reach the target (or a neighborhoodthereof), using the policy TT. We will denote such acost function as J* — J*(x, Rd)* Last, we assume thata simulator of the closed-loop behavior of the space-craft is available. Such a simulator can be obtainedby integrating the differential equations (1) under theaction of the policy TT.

As an additional remark, notice that in general, itwill not be possible to ensure global stability of theclosed-loop system using a smooth, static, feedbackcontrol law, because of the topological structure ofthe configuration space. 50(3); however, most of theavailable continuous control laws have a large region ofattraction, and the knowledge of this region, denotedby V(Rd) C SO(3) xR3, will be enough for the purposeof this paper.

In this paper, we will only consider a simpleconstraint- free attitude control law, stated in by Bulloand Murray, 2i which has the benefits of a simplemathematical formulation, a relatively large region ofattraction, and a readily available Lyapunov function.This control law is appropriate for the case in whichthe spacecraft is equipped with reaction wheels, pro-viding independent torques on three axes. In practicalapplications, other attitude control laws — e.g. theones proposed by Wie et aL2S for control of agilespacecraft — can, and should, be implemented, withno changes to the planning algorithm.

Assume that the objective of the (constraint-free)guidance law is to slew the spacecraft to an attitudeRft, with zero angular rate. The guidance law is basedoil the definition of a Lyapunov function composedof a term penalizing the attitude error, and a termpenalizing angular rates:

4- (4)

where kp is a positive constant (a "proportional7*gain), and Tr(jR) denotes the trace of the matrix R,that is the sum of its diagonal elements (or. equiv-alent Iy5 of its eigenvalues). It is easy to check thatthe derivative of the above function can be madenegative (semi)-definite in a set V(Rd) = {(R,&) £SO(S) x R(3) | V(R,u,Rd) < 2kp}. by setting the

control moments such that:

M(u) = w x Jo/ - (5)

where kd is a positive constant (a "derivative" gain),Skew(J?) := l/2(R - RT), and the notation (:)v in-dicates the inverse of the "hat" operator introducedbefore.

Environment characterizationWe consider an environment in which both fixed and

time varying pointing constraints are present, and weassume that the future evolution of the constraints isknown in advance. This is not a restrictive hypothe-sis in the spacecraft case, since the ephemerides of allthe celestial bodies of interest are known with greataccuracy.

A first class of pointing constraints formulates thenecessity to avoid pointing the boresight of sensitiveinstruments towards the Sun or other bright objects.In this case the constraint is usually formulated as aminimum offset angle amzn between the instrument'sline of sight and a celestial object. If the boresightof the instrument in body axes is represented by theunit vector v&, and the direction of the celestial ob-ject in inertia! coordinates is given by the unit vectorWi, such a constraint can be encoded by the followinginequality:

Rvb ' Wi < COSOroin. (6)

A second kind of pointing constraints addresses themaintenance of communication links. In this case, thecommunication antenna must be kept within an an-gle amax from the ground station direction. Such aconstraint can be encoded by the following inequality:

It is easy to see that Eq. (7) can be reduced to theform (6) by replacing the body-fixed vector vt> by — •*?&,and the avoidance angle by TT — CKmax -

The pointing constraints described above must bechecked pointwise in time. In some applications, itmight be required to impose integral constraints: thisis the case, for example, when the heat input to radi-ators, or other spacecraft surfaces, must be limited tomaintain thermal control. Such constraints can be for-mulated as integral inequalities, which must hold forall times t (these have also been called "timed" con-straints.4 A simple formulation can be the following:

max {0, cos (/&;?, • w^} dr < H. (8)

If the environment is time-varying, constraint viola-tions must be checked on (state x time) pairs (or, t). Aconstraint violation checking algorithm is assumed tobe available, via trajectory sampling or other appro-priate method.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2001-4155


Problem formulationThe motion planning problem can now be stated as

follows: Given an initial state x& — (,Ro, o;0) € SO(3) xR3. at time tD? and a target attitude Rd € 5O(3)? finda control input u(t) (within the saturation limits of theactuators), that can steer the system the system fromXQ to (#d,Q), while ensuring that constraints (6),(7),and (8) are satisfied. While the usual formulation ofthe motion planning problem is concerned only withfinding a feasible trajectory, in most applications weare also interested in finding a trajectory minimizingsome cost. In this paper, we will assume a cost func-tional of the following form:

J = (9)

Such cost functional express the cost associated withthe time required to complete the maneuver, the to-tal length of the generated attitude path, the requiredcontrol effort, or combinations thereof.

Constrained attitude maneuverplanning

The motion planning algorithm in the presence ofpointing constraints is based on the determination of atime-parameterized sequence of 'Virtual targets'- Rv(t}that effectively steers the system to the desired con-figuration while avoiding constraint violations. Suchan approach casts the location of the virtual targetas a function of time as a "pseudo-control" input forthe system. Since the actual control inputs can becomputed from the knowledge of the control policy^(-,7^;), this means that the low-level control layer(the layer actually interacting with the vehicle) andthe high-level, maneuver planning layer are effectivelydecoupled, while at the same time ensuring full consis-tency between the two levels. As a consequence, unlikeother approached in constrained motion planning, orconstraint monitoring, the planning algorithm can berun at a rate that is much slower than the rate requiredfor the low-level control laws.

Note also that the ideas outlined above can be seenas a motion planning technique through schedulingof Lyapunov functions, where the Lyapunov functionscharacterize the closed-loop stability of the system un-der the attitude control laws introduced earlier. Whilethe concept is not entirely new in control theory.29""*1

to the authors' knowledge, this is the first applicationattitude slew planning with pointing constraints . Afundamental difference can also be seen in the fact thatin our algorithm the ordering of Lyapunov functionsis performed on-line, whereas in the references the or-dering was determined a priori.

The basic idea of the algorithm is the following:starting from the initial conditions (#0,0/0). we incre-mentally build a tree of feasible trajectories, trying

to explore efficiently the reachable set (i.e. the set of(x, t) couples which can be reached with the availablecontrols, and maintaining fesibility with respect to theconstraints).

At each step, we will add a new branch (edge) anda new milestone (node) to the tree. For this purpose,at each tree expansion step we have to address twopoints: (i) Which node do we want to expand? (ii)In which "direction" shall we explore?. The first in-cremental randomized motion planning algorithm wasrecently introduced by LaValle and Kufrher,20 basedon previous work by Kavraki, Svestka et a/.,13 withthe name of Rapidly Exploring Random Tree (RRT).Another algorithm was introduced by Hsu, KindeL eta/.23 In this paper we will use an algorithm devel-oped by the authors, and detailed in a forthcomingpaper.24 which recovers the most desirable character-istics of both the above mentioned algorithms. It is outof the scope of this paper to give a detailed descriptionof the algorithm, and as a consequence we will give aquick overview of the main features of the algorithm,especially for what pertains to the spacecraft attitudemaneuver planning problem.

In our algorithm we roughly proceed as follows:

• pick a target attitude Rv at random, and try toexpand all the nodes in the tree in sequence, inorder of increasing cost J^(Rj,,Rv) (i.e. startingfrom the closest node, using the measure of dis-tance provided by J*(*. RV))>

• Apply the optimal control policy ^(-,RV). untilthe system gets to R» (or to a neighborhood of

If a feasible trajectory can not be found for all thenodes in the current tree, the candidate milestone isdiscarded.

The milestones generated according to the aboveprocedure can be regarded as primary milestones. Weassume that Rrand is generated from a uniform distri-bution on the workspace SO (3). Notice that the factthat the algorithm involves building a tree rooted atsome initial condition, allows easy implementation ofan integral constraint checker, by simple bookkeepingat the tree nodes and edges.

The above procedure is repeated, and new mile-stones are added to the tree, until the original targetattitude can be reached using the available constraint-free control law (e.g. the control law (5)), at whichpoint the problem is solved. If needed, it is possibleto continue the computations, with the objective ofimproving the total cost of the resulting motion plan.This can be done in a particularly efficient manner ifthe constraint-free control law is actually optimal (inthe absence of constraints) with respect to the costfunctional (9).

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Initial stateRn

Fig. 1 Example of tree expansion procedure

Improving performanceAs it can be easily recognized, the algorithm out-

lined above consists of jumps from rest conditions toother rest conditions and as such is unlikely to providesatisfactory performance.

Performance may be restored by realizing that theavailable guidance policy will stabilize the spacecraftabout the (virtual) target attitude for all initial condi-tions in the region V(RV). This suggests introducingthe following step: consider the tree at some pointin time and a newly added milestone to the tree. Asecondary milestone is defined to be any state of thespacecraft along the path leading from the parent nodein the tree to the newly added milestone.

These secondary milestones are added to the tree,and can be selected as the tree node to be expandedin later iterations. Note that all secondary milestones,by construction, have a primary milestone in a childsubtree (see Fig. 1), which ensures the fact that thespacecraft can come to rest at an attitude which satis-fies all the given constraint. In other words, safety ofthe generated path is always guaranteed, even in thecase a full solution to the target attitude is not found.(More technical details on the safety characterizationof generated motion plans in the presence of movingobstacles are available in other papers.24)

This procedure is perhaps better understood

through an example, given in Figure 1. The currenttree is depicted as a set of thick lines, the squares repre-sent the primary milestones, and the circles representsecondary milestones. The target attitude Rj is notreachable from any of the milestones currently hi thetree by using the policy TT(-, #<*). Thus, a new can-didate milestone R» is randomly generated, and thenattempts are made to join the nodes of the tree toRv. The closest milestone, in the sense induced by thecost-to-go function J*(-,J?V) is indicated by the num-ber one. Application of the policy TT(-, RV) with initialcondition corresponding to this milestone results in aviolation of one of the constraints. The same can besaid from the second-closest milestone, indicated bythe number two. With the third-closest milestone wehave better luck, and a new collision-free trajectoryis generated. A new primary milestone is created atAy, a new secondary milestone is created at a ran-dom point on the newly generated trajectory, and thetwo new milestones are connected to the tree throughedges encoding the new trajectory. Finally, we checkwhether or not the target point is reachable from thenew milestones. In this case the answer is indeed pos-itive, and the feasibility problem is solved. If this isnot the case (or a icbetter" solution is sought), the it-eration would is started again with a new randomlychosen attitude J^.

Simulation example

In the following example, the case of a (purely imag-inary) deep-space spacecraft (see Fig. 2). For thepurpose of this example, we assume that the space-craft carries a telescope with a fixed boresight in bodyaxes, directed along the Z axis. A star tracker ismounted so that its boresight is directed along the Yaxis. In order to protect the optics, the telescope axiscannot be less than 30 degrees off any bright object(i.e. the Sun, the Earth, and other bodies hi the So-lar System), whereas the star tracker axis must stayat least 20 degrees away from the Sun. Moreover, atelemetry antenna is mounted hi the X direction: inorder to maintain the communication link, the the an-tenna must be within 60 degrees from the directionof the Earth. Facing in the — Z direction is a radia-tor, for which the maximum exposure to the Sun isconstrained by H = 120s in Eq. (8).

We consider a case in which the spacecraft is com-manded to point the telescope to a new target, froman initial condition which would result in a constraintviolation if the constraint-free control law were used.In the example, the "bright celestial objects" will becalled Sun, Earth and Moon, even though their po-sitions are fictitious. Here we present the results ofa challenging scenario, in which the cones of Sun andMoon avoidance (for the telescope) overlap. Moreover,the Earth and the Sun are separated in such a way thatthe Sun avoidance maneuver would result in the acti-

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again, all the trajectories in the tree satisfy the point-r ing constraints. This time, the axis of the antenna

must be within 60 degrees from the Earth. Notice hownon-trivial maneuvering is required to satisfy both thetelescope and antenna constraints. In this case we aremaking full use of the capability to plan motions onthe rotation group.

60N

• Antenna

Star tracker

180W 90W 0 90ELongitude (degrees)

180E

Fig. 2 Spacecraft configuration

vation of the communication link constraint. In otherwords, an arbitrary maneuver to avoid pointing thetelescope to the Sun would result in an interruption ofthe communications.

In Fig. 3 the trace of pointing direction of the tele-scope on the tree of attitude plans is depicted. Thesolution (i.e. the trajectory that is eventually com-manded to the spacecraft) is in a darker color, andjoins the initial condition with the target point, as ex-pected. It can be noticed that all the trajectories inthe tree satisfy the pointing constraints (as well as thedynamical and integral constraints), since the axis ofthe telescope is always at least 30 degrees away fromeach of the celestial bodies under consideration.

Fig. 4 Trace of the antenna pointing direction.The pointing constraints on the antenna requirethat it never exceeds a 60 degree separation fromthe Earth.

For completeness, we also report the trace of thestar tracker pointing direction in Fig. 5, even thoughit is not influential in the determination of the requiredmaneuver (as is the integral constraint in this case).

90W 0 90ELongitude (degrees)

180E

180W 90 W 90E 180ELongitude (degrees)

Fig. 3 Trace of the telescope pointing direction.The pointing constraints on the telescope requirethat it maintains a 30 degrees separation with eachone of the bright celestial bodies.

In Fig. 4 the trace of the antenna direction is shown:

Fig. 5 Trace of the star tracker pointing direction.The pointing constraints on the star tracker requirethat it maintains a 20 degree separation from theSun.

A few remarks about performance. The algorithmwas implemented in C+-f, using the LEDA library,32

on a 300 MHz Pentium II running Linux. Several runswith different constraint and initial/final condition ge-ometries were performed, and a feasible solution wasfound by the proposed algorithm in all cases, in about1.1 seconds on average (standard deviation 0.6 sec-onds). This is believed to be due to the fact that inmost realistic attitude motion planning problems, theenvironment has a large expansiveness, and as a con-sequence the environment can be explored rapidly bythe randomized algorithm.

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ConclusionsIn this paper a randomized attitude slew planning

algorithm was presented, based on using a constraint-free guidance systems as local planners in a probabilis-tic roadinap framework. The main advantage of thealgorithm is the capability to address in an efficientand natural fashion the dynamics of the system, mul-tiple pointing constraints (including constraints on thepointing of several different body-fixed vectors), andintegral constraints wliile at the same time providinga consistent decoupling between attitude slew plan-ning and the low-level controls. Formal guarantees onthe probabilistic completeness of the algorithm, andon the convergence rate to unity of the probabilityof correct termination, have already been established,and application to spacecraft attitude motion planningproblems yielded a very efficient and reliable planner,in all the cases which have been examined.

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c)2001 American Institute of Aeronautics & Astronautics or ...dahleh.lids.mit.edu/wp-content/uploads/2011/09/frazzoli_aiaa.pdf · the execution of rapid attitude maneuvers at a very