Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006
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Time-Optimal Attitude Reorientation at Constant Angular Velocity Magnitude with Bounded Angular Acceleration Mrityunjay Modgalya Inductis (India) Private Limited Tower-B, Vatika Atrium Sector-53, Main Sector Road Gurgaon 122001, India
Sanjay P. Bhat Department of Aerospace Engineering Indian Institute of Technology Bombay Powai, Mumbai-400076, India 91-22-2576-7142, 91-22-2572-2602 (FAX)
[email protected]
[email protected]
Abstract— This paper considers the problem of steering the orientation of an inertially symmetric rigid body of unit moment of inertia from an initial attitude and nonzero angular velocity to a specified terminal attitude in minimum time under an upper limit on the magnitude of angular acceleration with the magnitude of the angular velocity constrained to remain constant. Optimal control theory is used to show that singular optimal arcs are uniform eigenaxis rotations in which the body rotates at a uniform rate about a body-fixed axis, while nonsingular arcs are coning motions in which the body angular velocity vector rotates at a uniform rate about a body-fixed axis. Symmetries of the problem are exploited to further show that every optimal trajectory consists of at most one coning motion followed either by one uniform eigenaxis rotation or several coning motions of equal duration.
I. I NTRODUCTION Space applications have always been a rich source of optimal control problems. A case in point is the problem of optimal attitude reorientation of a rigid body, which is motivated by the rapid large-angle maneuvering capabilities required by some space missions. Several different kinds of optimal attitude reorientation problems have been considered in the past. References [1], [2] and [3] considered minimization of a quadratic cost functional involving control inputs and/or states subject to initial and terminal constraints on the states. Reference [4] obtained differential equations that determine the optimal angular velocity variation required to achieve a specified attitude change while minimizing a nonquadratic integral cost functional involving even powers of the angular velocity components. The problem of time-optimal reorientation, which is the subject matter of this paper, has also received considerable attention in recent years. References [5] and [6] considered the problem of time-optimal rest-to-rest This work has been supported in part by the ISRO-IITB Space Technology Cell, Indian Institute of Technology Bombay.
1-4244-0171-2/06/$20.00 ©2006 IEEE.
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reorientation of an inertially symmetric rigid body using three independent bounded control torques. Both references showed that, contrary to expectation, an eigenaxis rotation in which the rigid body rotates about a bodyfixed axis, is not time optimal. While [5] used numerical techniques to arrive at this conclusion, [6] used the necessary conditions of optimal control to analyze the optimality of singular arcs of various finite and infinite orders. Other time-optimal reorientation problems considered in the literature include time-optimal maneuvers for spin stabilized symmetric spacecraft using magnetic actuation [7], and minimum-time reorientation of axisymmetric spacecraft using two independent bounded control torques perpendicular to the symmetry axis [8]. An extensive survey of the literature on time-optimal attitude reorientation is presented in [9]. In this paper, we consider the problem of maneuvering an inertially symmetric rigid body from an initial attitude and nonzero angular velocity to a specific terminal attitude in minimum time. Instead of assuming three independent bounded control torques as in [5] and [6], we impose a bound on the Euclidean norm of the total angular acceleration. Thus, instead of requiring the angular acceleration to lie in a body-fixed cube, we constrain it to remain inside a sphere. In addition, we restrict the Euclidean norm of the angular velocity to remain constant throughout the maneuver. Our restriction on the angular velocity magnitude is motivated by the Dubin’s problem [10], a translational analog of our problem involving time-optimal repositioning of a particle translating at a constant speed under an acceleration constraint. Like the constant speed assumption in Dubin’s problem, our restriction on the angular velocity magnitude along with the additional symmetry introduced by our choice of input constraint makes it possible to explicitly characterize time-optimal motions using the maximum principle.
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In Section III, we use the maximum principle to show that singular optimal arcs are uniform eigenaxis rotations in which the body rotates at a uniform rate about a body-fixed axis. On the other hand, nonsingular arcs are coning motions in which the body angular velocity vector rotates at a uniform rate about a bodyfixed axis. Our result depends very crucially on the existence of two conserved quantities, which makes it possible to eliminate the adjoint variables and obtain the optimal angular acceleration history along singular and nonsingular arcs in feedback form. In Section IV, we reformulate the optimal control problem in terms of a reduced set of variables suggested by the symmetries of the original problem. An application of the maximum principle to the reduced problem allows us to characterize optimal combinations of singular and nonsingular arcs. Specifically, we use the reduced optimal control problem to show that every optimal trajectory of the original problem consists of at most one coning motion followed either by one uniform eigenaxis rotation through an angle not greater than π , or several coning motions of equal duration in which the body angular velocity rotates through an angle that lies in (π , 2π ). We formulate our problem in terms of quaternions, which are widely used to represent orientations. We begin by recalling the necessary mathematical preliminaries in the next section. II. P RELIMINARIES The set Q of quaternions is the four-dimensional real space R4 endowed with the operation of quaternion multiplication. To define quaternion multiplication, it is convenient to write a quaternion q ∈ Q as q = q0 + q1 i + q2 j + q3 k, and define ii = jj = kk = −1, ij = −ji = k, jk = −kj = i, and ki = −ik = j. Since it will be clear from the context whenever quaternion multiplication is intended, we will not introduce a seperate symbol for quaternion multiplication. The real or scalar part of a quaternion q ∈ Q is Re(q) = q0 ∈ R. It is easy to check that if a, b ∈ Q, then Re(ab) = Re(ba). The imaginary or vector part of q ∈ Q is the three-dimensional vector Im(q) = [q1 q2 q3 ]T ∈ R3 . The conjugate of a quaternion q is the quaternion q¯ = q0 − q1 i − q2 j − q3 k. For every quaternion q, qq¯ = qq ¯ = q20 + q21 + q22 + q23 . For every a, b ∈ Q, ¯ ab = ba. ¯ To each vector v ∈ R3 , we associate the quaternion v ∈ Q defined by v = 0 + v1 i + v2 j + v3 k. Thus v is a quaternion with zero real part and imaginary part equal to v, that is, Re( v) = 0, and Im( v) = v. Given v, u ∈ R3 , T it is easy to verify that uv = −v u + (u × v). A unit quaternion is a quaternion q satisfying qq¯ = 1. The set S3 of unit quaternions is a group under quater-
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nion multiplication with the inverse of every quaternion being given by its conjugate, and with the identity element being the unit quaternion having real part 1. As a convenient abuse of notation, we will denote the identity element in S3 by 1. Consider an inertially symmetric body of unit moment of inertia. We represent the instantaneous orientation of the body relative to a reference inertial frame by a unit quaternion q(t) ∈ S3 such that, if xI ∈ R3 and xB ∈ R3 represent the components of a given vector in the inertial frame and body-fixed frame, respectively, then xI = q(t) xB q(t). ¯ If ω (t) ∈ R3 gives the instantaneous body components of the angular velocity of the body frame relative to the reference frame, then the kinematic equations for the attitude motion are given by 1 (t), ω˙ (t) = u(t), q(t) ˙ = q(t)ω 2
(1)
where u(t) ∈ R3 gives the instantaneous body components of the angular acceleration of the body. An eigenaxis rotation is a rotational motion in which the body rotates about a body-fixed axis. A uniform eigenaxis rotation is an eigenaxis rotation at a uniform rotational rate. Every solution of (1) with u ≡ 0 is a uniform eigenaxis rotation. A coning motion is a rotational motion in which the body angular velocity rotates about a body-fixed axis at a uniform rate. A coning motion with its body-fixed coning axis along the unit vector b ∈ S2 and coning rate Λ > 0 is a solution of (1) with u(t) = Λb × ω (t).
(2)
III. A T IME -O PTIMAL ATTITUDE R EORIENTATION P ROBLEM We consider the time-optimal problem of steering the orientation of the body starting from some initial attitude and nonzero body angular velocity to a specified orientation in minimum time under an upper limit on the magnitude of angular acceleration with the magnitude of the angular velocity constrained to remain constant. More precisely, given the desired final quaternion qf = 1, initial unit quaternion qo ∈ S3 , initial angular velocity def ωo ∈ R3 having magnitude Ω = ωo = 0, and an upper limit M > 0 on the magnitude of angular acceleration, we wish to find the angular-acceleration time history ] → R3 that minimizes the cost functional J(u) = u t:f [0,tf 0 1dt, subject to the magnitude constraint u(t) ≤ M, t ∈ [0,tf ], and the constraint that the solution of (1) satisfying q(0) = qo and ω (0) = ωo satisfies the pointwise-in-time state constraint ω (t) = Ω, t ∈ [0,tf ], and the terminal constraint q(tf ) = qf . It should be noted that our constraint on the angular velocity magnitude rules out motions that start and terminate in rest.
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def
Let q0 = Re(q) and qi = Im(q). Putting q = q0 + qi in = −ω T qi + q (1), using qi ω i × ω , and equating real and imaginary parts yields 1 (3) q˙0 (t) = − ω T (t)qi (t), 2 1 1 q0 (t)ω (t) + (qi (t) × ω (t)). (4) q˙i (t) = 2 2 The pointwise-in-time state constraint is equivalently given by ω T (t)ω (t) − Ω2 = 0, t ∈ [0,tf ], while the terminal constraint is given by q0 (tf ) − 1 = 0 and qi (tf ) = 0. In order to write down the necessary conditions for time optimality, we introduce the Hamiltonian H : R7 × R7 × R3 → R and the Lagrangian function L : R7 × R7 × R1 × R3 → R given by H(q0 , qi , ω , λ0 , λi , λω , u) = −1 − 12 λ0 (ω T qi ) + λω T u + 12 λi T (q0 ω + qi × ω ) and L(q0 , qi , ω , λ0 , λi , λω , µ , u) = H(q0 , qi , ω , λ0 , λi , λω , u) + µ (ω T ω − Ω2 ), where λ0 ∈ R, λi ∈ R3 and λω ∈ R3 are the adjoint vectors corresponding to q0 , qi and ω , respectively, while µ ∈ R is a multiplier. In the rest of the paper we denote the optimum time by tf∗ , the time-optimal angular acceleration history by u∗ : [0,tf∗ ] → R3 , and the resulting time-optimal solution of (3)-(4) by q0 : [0,tf∗ ] → R and qi : [0,tf∗ ] → R3 , and the corresponding angular velocity variation by ω : [0,tf∗ ] → R. According to Pontryagin’s maximum principle [11, Thm. 4.1], [12, Thm. 6-3], there exist piecewise absolutely continuous adjoint time histories λ0 : [0,tf∗ ] → R, λi : [0,tf∗ ] → R3 and λω : [0,tf∗ ] → R3 satisfying the differ∗ ential equations λ˙ 0 (t) = − ∂∂ Lq = − 12 λi T (t)ω (t), λ˙ i (t) = 0 ∗ ∗ − ∂∂Lq = 12 [λ0 (t)ω (t) + λi (t) × ω (t)], λ˙ ω (t) = − ∂∂Lω = i 1 2 [λ0 (t)qi (t) − q0 (t)λi (t) + qi (t) × λi (t)] − 2 µ (t)ω (t), for almost every t ∈ [0,tf∗ ], and the transversality condition λω (tf∗ ) = 2β ω (tf∗ ), for some scalar β ∈ R, such that the optimal angular acceleration satisfies 0 = H(q0 (t), qi (t), ω (t), λ0 (t), λi (t), λω (t), u∗ (t)) ≥ H(q0 (t), qi (t), ω (t), λ0 (t), λi (t), λω (t), a), (5) for almost every t ∈ [0,tf∗ ] and every a ∈ R3 satisfying a ≤ M. While the transversality condition requires the terminal adjoint vector λω (tf ) to be parallel to the terminal angular velocity ω (tf ), the maximum principle yields no information on the terminal values of λ0 and λi since the terminal quaternion is specified. Define λ : [0,tf∗ ] → Q by λ (t) = λ0 (t) + λi (t). The expressions for λ˙ 0 and λ˙ i can be combined to yield 1 (t), λ˙ (t) = λ (t)ω (6) 2 for almost every t ∈ [0,tf∗ ]. Next, define v : [0,tf∗ ] → R3 and c : [0,tf∗ ] → R3 by v(t)q(t)]. ¯ The exv(t) = Im[q(t) ¯ λ (t)] and c(t) = Im[q(t) pression for λ˙ ω can now be rewritten as 1 λ˙ ω (t) = − v(t) − 2µ (t)ω (t). (7) 2
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The following lemma gives key properties of optimal trajectories that will enable us to characterize the optimal acceleration time history in the sequel. Lemma 3.1: The function c is constant on [0,tf∗ ]. Moreover, for every t ∈ [0,tf∗ ], λω (t) × ω (t) + 12 (v(t) − c) = 0. Proof: For notational convenience, we suppress the dependence on t at various places in this proof. We have v = Im( q¯λ ) = 12 [q¯λ − λ¯ q], so that c = q vq¯ = 12 [λ q¯ − qλ¯ ]. ˙=0 Direct differentiation along with (1) and (6) yields c ∗ for almost every t ∈ [0,tf ]. Thus c is constant. Next, we ˙ − vω ˙ = d (q ]q, v+2v ¯ = 12 q[ω ¯ use (1) to compute 0 = c dt vq) T T ˙ + ω v− v which yields −v ω + ω × v+2v × ω = 0. Thus ˙ = v v × ω , which yields v(t) ˙ = v(t) × ω (t),
(8)
for almost every t ∈ [0,tf∗ ]. First, assume that the interval (t1 ,t2 ) ⊆ [0,tf∗ ] corresponds to a singular segment of the optimal trajectory. Then, λω ≡ 0 on (t1 ,t2 ), so that λ˙ ω ≡ 0 on (t1 ,t2 ). Equations (7) and (8) imply that v(t) = −4µ (t)ω (t) and v(t) ˙ = 0 for every t ∈ (t1 ,t2 ). Hence, dtd [λω × ω + 12 (v − ˙ = 0 on (t1 ,t2 ). c)] = dtd [0 + 12 (v − c)] = 12 (v) Next, assume that (t1 ,t2 ) ⊆ [0,tf∗ ] corresponds to a nonsingular segment of the optimal trajectory, so that λω (t) = 0 for almost every t ∈ (t1 ,t2 ). Equation (5) implies that u∗ (t) = Mλω (t)−1 λω (t),
(9)
for almost every t ∈ (t1 ,t2 ). Direct differentiation along with (7), (8) and (9) yields dtd [λω × ω + 12 (v − c)] = 0, for almost every t ∈ (t1 ,t2 ). Since dtd [λω × ω + 12 (v − c)] = 0 almost everywhere on singular as well as nonsingular segments of the optimal trajectories, it follows that dtd [λω × ω + 12 (v − c)] = 0 for almost every t ∈ [0,tf∗ ]. Since q(tf∗ ) = 1, it follows that c(tf∗ ) = v(tf∗ ). Also, the transversality condition implies that λω (tf∗ ) is parallel to ω (tf∗ ). Hence λω (tf∗ ) × ω (tf∗ ) + 1 ∗ ∗ 2 (v(tf ) − c (tf )) = 0. Thus, 1 λω (t) × ω (t) + (v(t) − c) = 0, 2
(10)
for every t ∈ [0,tf∗ ]. Our first main result characterizes singular and nonsingular segments of the optimal trajectory. Theorem 3.1: The optimal acceleration time-history along a singular segment of the optimal trajectory is identically zero. Along a nonsingular segment of the optimal trajectory defined on the interval (t1 ,t2 ) ⊆ [0,tf∗ ], the optimal acceleration is given by u∗ (t) = −
Mr(t) (ω (t) × (qi (t) × ω (t))), 2Ωqi (t) × ω (t) (11)
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where r(t) : [t1 ,t2 ] → {−1, +1}. Furthermore, singular optimal arcs are uniform eigenaxis rotations, while nonsingular optimal arcs are coning motions. Proof: First, assume that the interval (t1 ,t2 ) ⊆ [0,tf∗ ] corresponds to a singular segment of the optimal trajectory. Then, λω ≡ 0 on (t1 ,t2 ), so that λ˙ ω ≡ 0 on (t1 ,t2 ). Equations (7) and (8) imply that v(t) = −4µ (t)ω (t) and v(t) ˙ = 0 for every t ∈ (t1 ,t2 ), so that v is constant on (t1 ,t2 ). Since ω is constant in magnitude, it follows from v(t) = −4µ (t)ω (t) that ω is constant on (t1 ,t2 ). Consequently, u∗ ≡ 0 on (t1 ,t2 ). Next, assume that (t1 ,t2 ) ⊆ [0,tf∗ ] corresponds to a nonsingular segment of the optimal trajectory, so that λω (t) = 0 and (9) holds for almost every t ∈ (t1 ,t2 ). Now, differentiating ω (t)2 = Ω2 and using (9), we get λωT (t)ω (t) = 0 for every t ∈ (t1 ,t2 ). By Lemma 3.1, (10) holds on [0,tf∗ ], so that v(t) − c2 = 4λω (t) × ω (t)2 = λω (t)2 ω (t)2 = 0 for almost every t ∈ (t1 ,t2 ). Hence v(t) − c = 0 for almost every t ∈ (t1 ,t2 ). Now, (10) implies that the vectors ω (t) and λω (t) are orthogonal to v(t) − c for almost every t ∈ (t1 ,t2 ). Also, for every c − v)) = Re((q vq)q) ¯ − t ∈ (t1 ,t2 ), qi T (v − c) = Re(q( Re(q v) = 0. Thus qi (t) is perpendicular to (v(t) − c) for every t ∈ (t1 ,t2 ). Hence, it follows that ω (t), λω (t) and qi (t) lie in the two-dimensional plane orthogonal to (v(t) − c) for almost every t ∈ (t1 ,t2 ). It can also be shown that, under the angular velocity magnitude constraint, qi × ω is identically zero on an interval only if v − c is identically zero on the same interval. Since v(t) − c = 0 for almost every t ∈ (t1 ,t2 ), it follows that qi (t) and ω (t) = 0 are linearly independent for almost every t ∈ (t1 ,t2 ). Since λωT (t)ω (t) = 0 for every t ∈ (t1 ,t2 ), it follows that, for every t ∈ (t1 ,t2 ), λω = Ω−2 (ω T ω )λω = Ω−2 (ω × (λω × ω )). Equation (10) now implies that λω (t) = − 2Ω1 2 ω (t) × (v(t) − c) for almost every t ∈ (t1 ,t2 ). Next, on using equation (9) we get u∗ (t) = −
M −1 Ω (v(t) − c)−1 (ω (t) × (v(t) − c)), (12) 2
for almost every t ∈ (t1 ,t2 ). Since the vectors ω (t) and qi (t) lie in the plane orthogonal to (v(t) − c) for almost every t ∈ (t1 ,t2 ), it follows that (v(t) − c)−1 (v(t)−c) ∈ {±qi (t) × ω (t)−1 (qi (t) × ω (t))}, which yields (11). For every t ∈ (t1 ,t2 ), such that v(t) − c = 0, define def a(t) = − 12 ω (t) + M2 Ω−1 (v(t) − c)−1 (v(t) − c). Equations (8) and (12) as well as the fact that c is constant can be used to verify that a˙ = 0 on (t1 ,t2 ), that is, a is a constant. Also, a × ω = − M2 Ω−1 (v − c)−1 (ω × (v − c)) = u∗ . Thus, on any interval on which v(·) − c = 0, u∗ is of the form (2) with Λ = a and b = a−1 a. The resulting motion is a coning motion. On the other hand, on a singular segment of the optimal trajectory, u∗ ≡ 0. It follows that a singular segment of the optimal trajectory
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is a uniform eigenaxis rotation. Theorem 3.1 implies that the time-optimal angular acceleration history is given by (11) with r(t) ∈ {1, −1, 0}. The time-optimal problem thus reduces to finding a switching strategy for r that causes the solution of (1) to meet the terminal condition in the least possible time. The results that we present in the next section essentially identify switchings which are non-optimal. In order to motivate the analysis presented in the next section, we take a small digression to present a geometric perspective on the development thus far. def Let Q = {ω ∈ R3 : ω = Ω}. We seek time-optimal def trajectories on the five-dimensional manifold M = S3 × 3 Q. The group S acts on M through two different actions given by G1 : (s, (q, ω )) → (sq, ω ) and G2 : s)). ¯ Im(sω ¯ The optimal trajectories for (s, (q, ω )) → (sqs, q, ω , λ , λω form trajectories of a Hamiltonian system on the cotangent bundle T ∗ M of M , with the Hamiltonian function being the maximized Hamiltonian function [13, Prop. 12.3]. The maximized Hamiltonian function is 2 of G1 1 and G invariant under the symplectic lifts G and G2 to T ∗ M . The functions J1 : (q, ω , λ , λω ) → c and J2 : (q, ω , λ , λω ) → λω × ω + 12 (v − c) are the momentum 2 , respectively. Lemma 3.1, 1 and G maps of the actions G which states that J1 and J2 are constant along the optimal trajectories, is simply a consequence of Noether’s theorem [14]. Equations (1) and (11) describe the evolution of the optimal trajectories on the intersection of the level sets of the momentum maps J1 and J2 . Closer observation reveals that the group action G2 is a symmetry of the dynamics represented by (1) and (11). Indeed G2 is a symmetry of our optimal control problem, since G2 leaves the input and state constraints as well as the Hamiltonian function invariant. It is well known, especially in classical mechanics, that symmetries make reduction possible. In the next section, we consider a reduced optimal control problem related to the equations (1) and (11), and use it to characterize optimal combinations of singular and nonsingular arcs. For a rigorous treatment of symmetry, conservation, and reduction in optimal control problems, see [15]-[17]. IV. A R EDUCED T IME -O PTIMAL P ROBLEM Define g : S3 × R3 → R3 by g1 (q, ω ) = Re(q), ) = −Ω−1 ω T qi , g3 (q, ω ) = g2 (q, ω ) = Ω−1 Re(qω −1 −Ω qi × ω , where qi = Im(q) and Ω = ω . It is easy to check that g(q, ω ) ∈ S2 for every (q, ω ) ∈ S3 × R3 . The function g maps the trajectories of the control system on S3 × R3 obtained by applying (11) to the system (1) to the trajectories of a control system on S2 . To see this, we let ξ = [ ξ1 ξ2 ξ3 ]T = g(q, ω ), and use (3), (4) and (11) to compute Ω r(t)M ˙ e1 × ξ (t), ξ (t) = − e3 − (13) 2 Ω
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where e1 = [ 1 0 0 ]T and e3 = [ 0 0 1 ]T . Since g(1, ω ) = e1 , for every ω ∈ R3 , the terminal constraint q(tf ) = 1 is equivalent to ξ (tf ) = e1 . Therefore, the problem of steering the solutions of the closed-loop system comprising (1) and (11) to the terminal constraint q(tf ) = 1 is equivalent to the problem of steering the solutions of (13) to the terminal condition ξ (tf ) = e1 . Hence we seek to find an input time history r : [0,tf ] → [−1, +1] that minimizes the cost functional Jr (r) = tf 1dt, subject to the constraint that the solution of 0 (13) satisfying ξ (0) = ξo satisfies the terminal constraint ξ (tf ) − e1 = 0. In order to write down the necessary conditions for time optimality, we introduce the Hamiltonian Hr : R3 × R3 × [−1, +1] → R for the the system (13) given × , where e ξ by Hr (ξ , λξ , r) = −1 + λξT − Ω2 e3 − rM Ω 1 3 λξ ∈ R is the adjoint vector corresponding to ξ . In the rest of the paper we denote the optimum time by tf∗ , the time-optimal input history by r ∗ : [0,tf∗ ] → [−1, +1], and the resulting time-optimal solution of (13) by ξ : [0,tf∗ ] → R3 . According to Pontryagin’s maximum principle [11, Thm. 4.1], [12, Thm. 6-3], there exists a piecewise absolutely continuous adjoint time history λξ : [0,tf∗ ] → R3 satisfying ∗ ˙λ (t) = − ∂ Hr = − Ω e3 − r (t)M e1 × λ (t), (14) ξ ξ ∂ξ 2 Ω for almost every t ∈ [0,tf∗ ], such that the optimal input satisfies 0 = Hr (ξ (t), λξ (t), r∗ (t)) ≥ Hr (ξ (t), λξ (t), a) for almost every t ∈ [0,tf∗ ] and every a ∈ [−1, +1]. Since ξ (tf ) is specified, the maximum principle yields no information on the terminal value of λξ . Before stating our next result, we denote p : [0, def ∗ 3 t f ] → R by p(t) = λξ (t) × ξ (t) and introduce Λ = Ω2 4
2
+M > 0, Θ ∈ [0, π2 ) defined by tan Θ = Ω2
2M , Ω2 def a+ =
and unit vectors a+ ∈ S2 , a− ∈ S2 defined by def − cos Θe3 −sin Θe1 and a− = − cos Θe3 +sin Θe1 . Using (13) and (14), we compute p(t) ˙ = Λ(− cos Θe3 − r∗ (t) sin Θe1 ) × p(t), (15) for almost every t ∈ [0,tf∗ ]. Our first result of this section characterizes singular and nonsingular optimal arcs of the reduced optimal control problem. Proposition 4.1: Along every nonsingular subarc of the optimal trajectory, r ∗ takes the value ±1 almost everywhere. Along every nonsingular subarc for which r∗ = 1 (−1), ξ rotates about a+ (a− ) at a constant rate Λ. Along every singular subarc of the optimal trajectory, r∗ ≡ 0. Moreover, every singular subarc lies along the great circle {η ∈ S2 : eT3 η = 0}.
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Proof: For every t ∈ [0,tf∗ ] such that, eT1 (λξ (t) × ξ (t)) = 0, the maximum principle yields r∗ (t) = sign(eT1 p(t)) ∈ {+1, −1},
(16)
that is, r∗ is +1 or −1 for every t ∈ [0,tf∗ ] satisfying eT1 p(t) = 0. The first statement thus follows. The second statement follows by using (16) in (13). The last two statements can be proved easily by differentiating the switching function t → eT1 p(t). e1 p(t) − a+ γ
B e2
A
e3
− a− Fig. 1.
Motion of p along nonsingular arcs.
Equations (15) and (16) imply that, along a nonsingular trajectory, p rotates about a+ whenever eT1 p > 0, and about a− whenever eT1 p < 0. Figure 1 shows a typical curve traced out by p along a nonsingular trajectory. Points A and B in Figure 1 represent switching instants. Because of symmetry, the curve is closed and symmetric about the e2 -e3 plane. Consequently, the time duration between successive switching instants is the same. Figure 1 also shows that the angle γ of rotation of p (and hence of ξ ) between successive switching instants lies in (π , 2π ). Proposition 4.1 asserts that r ∗ ≡ 0 along an optimal singular segment. Since eT1 p, and hence, eT1 p˙ ≡ 0 along a singular segment, (15) implies that e3 × p ≡ 0 along an optimal singular segment. Hence, along any nonsingular optimal arc that is followed or preceded by a singular optimal arc, p must trace a circular arc that intersects the e3 - axis and whose center lies along a+ or a− . Symmetry further implies that the circular arc must be tangential to the e3 -axis. Figure 2 shows such a circular arc traced by p for r∗ ≡ 1. In the case depicted in Figure 2, if C represents an instant where r∗ either switches from +1 to −1, or continues to remain +1, then the resulting motion will involve a complete rotation of ξ through an angle 2π before the next switching opportunity. Since this is
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e1 p(t) − a+
C
e2
e3
Fig. 2. Motion of p along a nonsingular arc that terminates in a singular arc.
clearly nonoptimal, it follows that C represents an instant after which r∗ remains 0, thus implying that a singular arc is part of a terminal arc. The last argument also suggests that r∗ can switch between +1 and −1 only in the case depicted in Figure 1, thus indicating that a trajectory that involves switching between +1 and −1 is completely nonsingular. The preceding observations lead to our next two results, which we state without proof. Proposition 4.2: If there exists a nonsingular subarc along which r∗ switches between +1 and −1, then the whole trajectory is nonsingular. In this case, the time duration between any two successive switching instants is the same and is contained in Λπ , 2Λπ . Our last result states that a singular arc can only occur as part of a terminal singular arc. Taken together with Proposition 4.2, the next result implies that an optimal trajectory having a singular arc has at most two subarcs, of which the terminal one is singular and involves rotation through an angle not greater than π . Proposition 4.3: Every singular subarc is contained in a terminal singular subarc of the optimal trajectory. Moreover, on an optimal singular subarc, the angle of rotation of ξ cannot be greater than π . The main result of this section stated below characterizes those combinations of eigenaxis rotations and coning motions that make up optimal trajectories. The result follows easily from Theorem 3.1 and propositions 4.1, 4.2 and 4.3. Theorem 4.1: Every optimal trajectory consists of at most one coning motion followed either by one uniform eigenaxis rotation through an angle that lies in (0, π ], or n coning motions, all involving equal angular rotations of the body angular velocity through an angle that lies in (π , 2π ), where n ≥ 0. V. C ONCLUSION We have considered the time-optimal problem of steering the orientation of an inertially symmetric rigid body
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