Rigid-Body Attitude Control

USING ROTATION MATRICES FOR CONTINUOUS, SINGULARITY-FREE CONTROL LAWS

30 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 1066-033X /11/ $ 26.00©2011IEEE igid-body attitude control is motivated by aero- Motivated by the desire to represent attitude both glob- space applications that involve attitude maneu- ally and uniquely in the analysis of rigid-body rotational vers or attitude stabilization. In addition to space- motion, this article uses orthogonal matrices exclusively to craft and atmospheric fl ight vehicles, applications represent attitude and to develop results on rigid-body atti- Rof rigid-body attitude control arise for underwater tude control. An advantage of using orthogonal matrices is vehicles, ground vehicles, and robotic systems. The set of at- that these control results, which include open-loop attitude titudes of a rigid body is the set of 3 3 3 orthogonal matri- control maneuvers and stabilization using continuous ces whose determinant is one. This set is the confi guration feedback control, do not require reinterpretation on the set space of rigid-body attitude motion; however, this confi gu- of attitudes viewed as orthogonal matrices. The main objec- ration space is not Euclidean. Since the set of attitudes is not tive of this article is to demonstrate how to characterize a Euclidean space, attitude control is typically studied using properties of attitude control systems for arbitrary attitude various attitude parameterizations [1]–[8]. These parame- maneuvers without using attitude parameterizations. terizations can be Euclidean, as in the case of Euler angles, which lie in R3, or non-Euclidean, as in the case of quaterni- DESCRIPTIONS OF RIGID-BODY ons, which lie in the non-Euclidean three-sphere S3. ATTITUDE CONFIGURATIONS Regardless of the choice of parameterization, all param- The attitude of a rigid body can be modeled by a linear eterizations [1], [2] fail to represent the set of attitudes both transformation between a reference frame and a body- globally and uniquely. For example, although Euler angles fixed frame that preserves the distance between each pair can represent every attitude, this representationntation of material points in the body and the handedness is not unique at certain attitudes. In fact, the of coocoordinater frames [1], [2], [6], [10]–[13]. time derivatives of the Euler angles at thesee AssAssumingu each frame is defined by three attitudes cannot represent every possible ororthogonalt unit vectors ordered accord- angular velocity. The term gimbal lock is iingn to the right-hand rule, the attitude of used to describe this mathematical sin- ttheh rigid body, as a linear transforma- gularity. Along the same lines, quaterni- ttion,i is represented by a 3 3 3 matrix ons can represent all possible attitudes ththat transforms a vector resolved in the and angular velocities, but this repre- bbody-fixed frame into its representation sentation is not unique; see “Representa- reresolved in the reference frame. The three tions of Attitude” for more details. Since columncolu vectors of the matrix represent the parameterizations such as Euler angles andnd threethree orthogonal unit basis vectors of the quaternions are unable to represent the set of atat-- body-body-fixedfix frame resolved in the reference titudes both globally and uniquely, results obtained frameframe. LikewLikewise, the three rows of the matrix repre- with these parameterizations have to be reinterpreted on sent the three orthogonal unit basis vectors of the reference the set of attitudes described by orthogonal matrices. Ne- frame resolved in the body frame. The resulting matrix, glecting this analysis can result in undesirable behavior referred to as a rotation matrix, represents the physical atti- such as unwinding [9], which refers to the unstable be- tude of the rigid body. The set of all rotation matrices is the havior of a closed-loop attitude control system. In par- special orthogonal group of rigid rotations in R3, which is ticular, for certain initial conditions, the trajectories of denoted by SO132; see “Attitude Set Notation” for a defini- the closed-loop system start close to the desired attitude tion of these and additional related sets. equilibrium in state space and yet travel a large distance In rigid-body pointing applications, the rotation about in the state space before returning to the desired attitude. the pointing direction is irrelevant since these rotations This closed-loop trajectory is analogous to the homo- do not change the direction in which the body points. clinic orbit observed in a simple pendulum that swings Thus, while rotation matrices R [ SO132 can be used to 360° when perturbed from its inverted position at rest. Un- model the attitude of a rigid body, not all the information winding can result in wasted control effort by causing the available in the rotation matrix is required for pointing rigid body to perform a large-angle slew maneuver when applications. In this case, a reduced representation of the a small-angle slew maneuver in the opposite rotational di- rotation matrix can be used for pointing applications. An rection is sufficient to achieve the objective; see “Pitfalls of example of such a reduced representation is the reduced- Using Quaternion Representations for Attitude Control” attitude vector [14]. To develop the reduced-attitude for a description and illustration of unwinding in attitude vector representation of a rigid body, suppose b [ S2, control using quaternions. where S2 is the unit sphere in three-dimensional space, is a unit vector that denotes a fixed pointing direction speci- fied in the reference frame. Then, in the body-fixed frame, ! ^ [ S2 Digital Object Identifier 10.1109/MCS.2011.940459 this direction vector can be expressed as G R b , ^ Date of publication: 16 May 2011 where R denotes the transpose of the rotation matrix R © SHANNON MASH

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 31 Representations of Attitude igid body attitude is often represented using three or four Rparameters. Unit quaternions and the axis-angle represen- Table S1 Properties of attitude representations. tation use four parameters to represent the attitude. Three-pa- rameter representations of attitude include the Euler angles as Attitude Representation Global? Unique? Euler angles No No well as parameters derived from the unit quaternions as in the Rodrigues parameters No No case of the Rodrigues parameters and the modified Rodrigues Modified Rodrigues parameters No No parameters. These three-parameter sets can be viewed as Quaternions Yes No embedded subsets of R3, thus allowing methods of analysis Axis-angle Yes No that are suited to the Euclidean space R3. Euler angles are Rotation matrix Yes Yes kinematically singular since the transformation from their time rates of change to the angular velocity vector is not globally defined. The Rodrigues parameters and modified Rodrigues As shown in Table S1, since the unit quaternions are global parameters are geometrically singular since they are not de- in their representation of attitude, they are often used in practi- fined for 180° of rotation. Therefore, continuous control laws cal applications to represent rigid-body attitude. However, note using these three-parameter representations cannot be glob- that the map from the space S3 of unit quaternions to the space ally defined, and, as such, these representations are limited to SO132 of rotations is not unique. In particular, this map is a two- local attitude maneuvers. fold covering map, where each physical attitude R [ SO132 is Table S1 presents various attitude representations and represented by a pair of antipodal unit quaternions 6q [ S3. If their key properties. While the quaternions, axis-angle vari- not carefully designed, quaternion-based controllers may yield ables, and rotation matrices can represent all attitudes of a undesirable phenomena such as unwinding [9]; see “Pitfalls rigid body, only rotation matrices can represent all attitudes of Using Quaternion Representations for Attitude Control” for uniquely. more details.

representing the attitude of the rigid body. We refer to the lems using continuous feedback control. In particular, we vector G as the reduced-attitude vector of the rigid body. seek continuous attitude control laws that modify the Thus, a reduced-attitude vector describes a direction b dynamics of the spacecraft such that a specified attitude or fixed in the reference frame by a unit vector resolved in reduced attitude is rendered asymptotically stable. For full- the body frame. In pointing applications, a pointing direc- attitude control, this continuous feedback function is tion b [ S2 in the reference frame can be expressed by the defined in terms of the full rigid-body attitude and angular reduced attitude G corresponding to b, and, hence, the velocity, while, for reduced-attitude control, this feedback reduced attitude can be used to study pointing applica- function is defined in terms of the reduced attitude and tions for a rigid body. angular velocity.

FULL- AND REDUCED-ATTITUDE CONTROL LOCAL AND GLOBAL Rigid-body attitude control problems can be formulated ATTITUDE CONTROL ISSUES in terms of attitude configurations defined in SO132 or in Since attitude control problems are nonlinear control prob- terms of reduced-attitude configurations defined in S2. lems, these problems can be categorized into local attitude The former case involves control objectives stated directly control issues and global attitude control issues. In brief, in terms of the full attitude of the rigid body, while the local control issues address changes in the rigid-body atti- latter case involves control objectives stated in terms of tude and angular velocity that lie in an open neighborhood pointing the rigid body. In particular, the latter case is of the desired attitude and angular velocity. Local attitude used when the control objectives can be formulated in control problems typically arise when the rigid body has a terms of pointing a body-fixed imager, antenna, or other limited range of rotational motion. In this case, linear con- instrument in a specified direction in the reference frame, trol methods or local nonlinear control methods based on a where rotations about that specified body-fixed direction convenient attitude representation may suffice, so long as are irrelevant. no singularities occur within the local range of attitude We subsequently study several full-attitude and motion of interest. reduced-attitude control problems. In each case, we begin In contrast to local attitude control, global attitude by treating open-loop attitude control problems defined by control issues arise when arbitrary changes in the specifying initial conditions, terminal conditions, and a rigid-body attitude and angular velocity are allowed. maneuver time. Then we treat attitude-stabilization prob- In this case, no a priori restrictions are placed on the

32 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 possible rotational motion, and nonlinear methods are FRAMEWORK FOR required to address these rotational motions. Global DESCRIBING ATTITUDE MOTION attitude control issues are motivated by applications The space of rigid-body rotations SO132 is a Lie group [34], to highly maneuverable rigid-body systems that may [35]; it has an algebraic group structure based on matrix undergo extreme attitude changes such as rigid-body multiplication as the group operation and a differential tumbling. geometric structure as a compact smooth manifold that enables the use of calculus. Alternatively, the Lie group COMMENTS ON THE SO132 of rigid-body rotations can be viewed as the set of ATTITUDE CONTROL LITERATURE isometries, or length-preserving transformations on R3 The literature on attitude control can be divided into two that leave the origin fixed and preserve the orientation, that categories. In the first category, attitude control is studied is, handedness, of frames. Unlike SO132, the space of rigid- within a geometric framework, as presented in this article, body reduced-attitudes S2 is not a Lie group. However, by viewing rigid-body attitude as an element of SO 132 or S2 each element in SO132, as an orthogonal matrix, is a bijec- [14]–[27]. In the second category [3]–[8], [28], [29], attitude tive mapping from S2 to S2. control is studied using representations of attitude in R3 or S3 ( R4 [1], [2]. Attitude Description Using Rotation Matrices Almost-global attitude stabilization of a rigid body, We now describe the properties of SO132. Every rotation in contrast to global attitude stabilization, is motivated matrix R [ SO132 satisfies

by the fact that there exists no continuous time-invari- ^ ^ R R 5 I 5 RR , det1R2 5 1, (1) ant feedback that globally asymptotically stabilizes a single equilibrium attitude [16]–[18]. This property is where I denotes the 3 3 3 identity matrix, which is also related to the fact that every continuous time-invariant the identity element of the group SO132. This representa- closed-loop vector field on SO132 and S2 has more than tion of the attitude is global and unique in the sense that one closed-loop equilibrium, and hence, the desired each physical rigid-body attitude corresponds to exactly [ 1 2 equilibrium cannot be globally attractive [9]; see “The one rotation matrix. If Rd SO 3 represents the desired Impossibility of Global Attitude Stabilization Using attitude of a rigid body, then attitude control objectives Continuous Time-Invariant Feedback” for more details. can be described as rotating the rigid body so that its atti-

While [24] treats local attitude stabilization and H` atti- tude R is the same as the desired attitude, that is, R 5 Rd. tude performance issues, [19]–[23] present results for The tangent space at an element in the Lie group various almost-global stabilization problems for the 3D SO132 consists of vectors tangent to all differentiable pendulum; the 3D pendulum has the same rigid-body curves t S R1t2 [ SO132 passing through that element. attitude dynamics as in this article, but the moment due Thus, the tangent space at an element in SO132 is the to uniform gravity is also included [30], [31]. This same vector space of all possible angular velocities that a rigid perspective is used in [25] and [26], where results on body can attain at that attitude. This vector space is iso- almost-global tracking of desired attitude trajectories morphic to the tangent space at the identity element are presented. Rigid-body reduced-attitude control is R 5 I, which is defined as the Lie algebra so132 of the Lie studied in [14], [15], [22], [23], [27], and [32]. group SO132. The algebraic structure of so132 is obtained 1 2 ^ 1 2 1 2 1 2 ^ An alternative approach to attitude control based on from the defining equation R t R t 5 I 5 R t R# t . Let the geometry of SO132 or S2 is to use representations of R1t2 [ SO132 be a curve such that R102 5 I and R102 5 S, ^ attitude in R3 or S3 ( R4 [1], [2], such as Euler angles, Euler where S [ so132. Differentiating R1t2 R1t2 5 I and ^ parameters, or quaternions, Rodrigues parameters, and evaluating at t 5 0, we see that S 1 S 5 0, that is, S is modified Rodrigues parameters [3]–[8], [15], [28], [29], skew symmetric. Thus, so132 is the matrix Lie algebra of [33]. While these representations can be used for local atti- 3 3 3 skew-symmetric matrices. tude control, global results obtained by using this approach Each element of SO132 can be represented by the matrix must be interpreted on the configuration space SO132 of exponential [34] the physical rigid body. Indeed, the global dynamics of the 1 1 c parameterized closed-loop attitude motion on R3 or S3, exp1S2 5 I 1 S 1 S2 1 S3 1 , (2) 2! 3! such as global or almost global asymptotic stability, do not necessarily imply that these closed-loop properties also where S is in the Lie algebra so132 and exp1S2 is in the Lie hold for the dynamics of the physical rigid body evolving group. The Lie algebra so132 is isomorphic to R3, and this on the configuration space SO132. Issues related to contin- vector space isomorphism is given by uous attitude control using these alternative representa- tions of attitude are described in “Representations of 0 2 s3 s2 £ § [ so1 2 Attitude” and “Pitfalls of Using Quaternion Representa- S 5 s3 0 2 s1 3 , (3) tions for Attitude Control.” 2 s2 s1 0

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 33 Pitfalls of Using Quaternion Representations for Attitude Control uaternion representations of attitude are often used in at- In the case of quaternions, continuous, time-invariant state- Qtitude control and stabilization to represent rigid-body at- feedback control using these representations can give rise to titude. Quaternions do not give rise to singularities, but they regions of attraction and repulsion in the neighborhood of double cover the set of attitudes SO132 in the sense that each q [ S3 and 2q [ S3, respectively, where q and 2q represent attitude corresponds to two different quaternion vectors. Spe- the same desired attitude in SO132. Starting from q, we can cifically, a physical attitude R [ SO132 is represented by a pair choose an initial angular velocity such that orbits connecting of antipodal quaternions 6q [ S3. these regions of repulsion and attraction arise. These orbits An implication of this nonunique representation is that run from a neighborhood of q to a neighborhood of 2q in S3, closed-loop properties derived using quaternions might with a terminal velocity that is close to zero. The correspond- not hold for the dynamics of the physical rigid body on ing orbit in SO132 obtained by projection as shown in Figure SO132 3 R3. Thus, almost global asymptotic stability of an S1 starts and ends close to the desired attitude. However, in equilibrium in the quaternion space S3 3 R3 does not guar- the quaternion space, since 2q has a region of repulsion, the antee almost global-asymptotic stability of the correspond- trajectory is repelled from 2q, eventually converging to q. In ing equilibrium in SO132 3 R3. Indeed, a property such as SO132, the corresponding trajectory then follows a homoclin- local asymptotic stability of the closed-loop vector field on ic-like orbit starting close to the desired attitude equilibrium S3 3 R3 holds on SO132 3 R3, only if the property holds for with almost zero angular velocity, gaining velocity as the rigid the projection of the vector field from S3 3 R3 to the space body moves along the orbit, and eventually converging to the SO132 3 R3; see Figure S1. desired equilibrium. Thus, the rigid body unwinds needlessly If the projection of the closed-loop vector field from before converging to the desired attitude. The presence of S3 3 R3 onto SO132 3 R3 is neglected during control design, these homoclinic-like orbits in SO132 indicates that, although then undesirable phenomena such as unwinding can occur the desired closed-loop attitude equilibrium is attractive, it is [9]. In unwinding, a closed-loop trajectory in response to cer- not Lyapunov stable. tain initial conditions can undergo a homoclinic-like orbit that To illustrate unwinding, suppose the quaternion q [ S3 rep- starts close to the desired attitude equilibrium. Unwinding resents the attitude of a rigid body, and let v [ R3 represent its can occur in all continuous closed-loop attitude control sys- angular velocity vector resolved in the body-fixed frame. We ex- 1 2 1 2 [ R R3 tems that are designed using global parameterizations that press q 5 q0, qv , where q0, qv 3 . Then, the attitude are nonunique in their representation of SO132; see “Repre- dynamics of the rigid body in the quaternion space are given by sentations of Attitude” for examples of global parameteriza- # tions that are nonunique in its representation of SO132. Jv 5 Jv3v1u, # 1 q 52 qT v, 2 v # 1 1 3 2 qv 5 qI 1 qv v, Tangent A 2 3 (Quaternion Space) Plane where J is the moment of inertia matrix, I is the 3 3 3 iden- Projection from [ R3 Quaternion to Physical tity matrix, and u is the applied control input. Next, B 1 2 ! Rotation Space consider the PD-type control law u q, v 2kpqv 2 kd v, where k , k . 0. It can be shown that q 5 11, 0, 0, 02 and A′ SO(3) p d e 1 2 B′ (Rotation Space) ve 5 0, 0, 0 is an equilibrium of the resulting closed-loop vector field. In the physical rotation space, this equilibrium cor-

responds to the attitude Re 5 I and ve 5 0. Furthermore, the closed-loop vector field yields almost global asymptotic sta- Tangent Plane 1 2 [ S3 R3 bility for the equilibrium qe, ve 3 . However, almost 1 2 [ S3 R3 global asymptotic stability of qe, ve 3 does not im- FIGURE S1 Projection of the vector field from S3 3 R3 to ply almost global asymptotic stability for the corresponding 1 2 3 SO 3 3 R . Since orthogonal matrices represent the set of 1 2 [ 1 2 R3 equilibrium Re, ve SO 3 3 . attitudes globally and uniquely, results obtained using alterna- To illustrate this failure, we choose J 5 diag11, 1, 12 kg-m2 tive parameterizations of SO132 have to be reinterpreted on the 1 2 1 2 set of attitudes described by orthogonal matrices. Neglecting and kp, kd 5 0.1, 0.237 . Furthermore, choose the initial atti- 1 2 1 2 3 4 this analysis can result in undesirable behavior such as unwind- tude q 0 5 qe and initial angular velocity v 0 5 360/p 0 0 °/s. ing [9]. To interpret the results on SO132, the closed-loop trajec- 1 2 The quaternion element q0 t is shown in Figure S2(a), and tory is projected from the parametric space onto SO132, where the angular velocity is shown in Figure S2(b). Furthermore, the properties of the resulting closed-loop on SO132 are ana- 1 2 lyzed. This figure illustrates the projection of a trajectory from Figure S2(c) shows the error in the attitude in SO 3 using the the space of quaternions S3 onto SO132. eigenangle

34 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 1 U1t2 5 cos21 1trace1R1t222 12 [ 30, 1804, 2 1 which is a scalar measure of the error between R and I. Note

that since Re 5 I, it follows that U 5 0° for R 5 Re. For the cho- 0.5 sen initial conditions, the eigenangle satisfies U102 5 0°. S3 As shown in Figure S2(a) and (b), the trajectory in 0

q 0 approaches 2qe at t 5 17 s with a small angular velocity ≈ of magnitude 0.03 °/s and is then repelled away, eventu- q0 −0.999 −0.5 ally converging to the desired attitude qe. This closed-loop behavior is similar to the response of a simple pendulum with angular velocity damping that is swung up to the un- −1 stable inverted equilibrium starting from the stable hanging 0 20 40 60 80 100 Time (s) equilibrium, where the pendulum eventually converges to (a) 1 2 [ S3 R3 the stable hanging equilibrium. Since qe, 0 3 is 1 2 [ S3 R3 120 a stable equilibrium and 2qe, 0 3 is an unstable equilibrium of the closed-loop dynamics, the closed-loop 100 trajectory in S3 3 R3 shows a closed-loop response similar to the simple pendulum swinging up with angular velocity 80 /s)

damping. ° 3 3 60 However, when the closed-loop trajectory in S 3 R is pro- || ( ||ω|| ≈ 0.03 °/s ω jected onto SO132 3 R3, parts (b) and (c) of Figure S2 show || 40 that the corresponding trajectory in SO132 3 R3 unwinds. Thus, 20 the closed-loop trajectory approaches the desired attitude Re 1U1t2 < 2.75 ) with a small angular velocity 1 7v 1t2 7 < 0.03 /s) ° ° 0 at t 5 17 s, and yet instead of converging to the equilibrium 0 20406080100 1 2 Time (s) attitude Re, the trajectory is repelled away from Re, 0 with an increasing angular velocity. Indeed, U1t2 in Figure S2(c) (b) increases to 180° at t < 50 s, eventually converging to zero. 200 Thus, starting at t 5 17 s, we observe a homoclinic-like orbit 1 2 R3 1 2 in SO 3 3 at Re, 0 , which violates the stability of the 150 equilibrium. The presence of this homoclinic-like trajectory indi-

cates that, while the desired equilibrium is globally attractive in ) °

1 2 R3 ( 100

SO 3 3 , it is not Lyapunov stable and, hence, not asymp- Θ totically stable in SO132 3 R3. 1 2 ! Θ ≈ 2.75° In the above example, the controller u q, v 2kp qv 2 kd v 50 fails to achieve asymptotic stability on the physical space SO132 3 R3 since it does not satisfy u1q, v 2 5 u12q, v 2. Thus, 0 while q and 2q in the quaternion space represent the same 0 20 40 60 80 100 physical attitude in SO(3) the control input u1q, v 2 2 u12q, v 2. Time (s) 1 2 [ 1 2 R3 2 Therefore, for each R, v SO 3 3 , where R Re, the (c) controller yields two distinct control inputs. Hence, the projec- tion of the closed-loop vector field from S3 3 R3 to SO132 3 R3 FIGURE S2 Closed-loop response of the rigid body as observed 3 1 2 yields a multivalued map. Thus, while the quaternion-based in S and SO 3 . As shown in (b) and (c), the rigid body 1 controller results in a closed-loop system that is a continuous approaches the desired attitude U52.75°) with an angular velocity whose magnitude is approximately 0.03 °/s. Thus, vector field on S3 R3, the projected closed-loop system on 3 from asymptotic stability, we expect the trajectory to converge the physical space SO132 3 R3 is not a vector field. Since the 1 2 to the desired equilibrium Re, 0 . On the contrary, the trajec- 1 2 R3 1 2 projected closed loop on SO 3 3 is not a vector field, we tory is repelled from Re, 0 , with an increasing angular veloc- cannot conclude that closed-loop properties on S3 3 R3, such ity. This homoclinic-like orbit occurs since q in (a) approaches as asymptotic stability and almost global-asymptotic stability, the unstable attitude 2qe and is subsequently repelled away, converging eventually to q However, in the configuration 1 2 3 e. imply the same on SO 3 3 R . 1 2 space SO 3 , both qe and 2qe correspond to the same attitude In practical applications, unwinding is often resolved by and hence the homoclinic-like orbit. This phenomenon is changing the sign of the quaternion vector as the closed-loop termed unwinding.

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 35 trajectory passes from one hemisphere to the other. Indeed, the the quaternion-based controller in S3 3 R3. Stability analysis only way to define a closed-loop vector field on SO132 3 R3 for discontinuous closed-loop systems requires set-valued dif- for a continuous time-invariant, state-feedback control- ferential equations [41]. ler using quaternions is to choose a controller that satisfies In contrast to analysis using quaternions, none of the above u1q, v 2 5 u12q, v 2. However, changing the sign of the qua- issues occur for closed-loop vector fields that are defined di- ternion vector as the closed-loop trajectory passes from one rectly on SO132 3 R3 by means of rotation matrices. This prop- hemisphere to the other results in a discontinuous closed-loop erty is due to the fact that, unlike quaternions, rotation matri- vector field on S3 3 R3. Note that almost global asymptotic ces represent rigid-body attitudes both uniquely and globally. stability of the desired attitude for the continuous controller in This uniqueness property is one of the primary motivations for S3 3 R3 does not imply almost global asymptotic stability of studying attitude-control issues on SO132 3 R3 in terms of ro- the desired attitude for the discontinuous implementation of tation matrices.

3 4^ [ R3 [ S2 where s 5 s1 s2 s3 . This isomorphism is expressed For the fixed reference frame direction b , dif- ^ as S 5 s3. The reason for the superscript is related to the ferentiating the reduced attitude G5R b with respect fact for every a, b [ R3, 3a3 4b 5 a 3 b. If we identify to time and substituting the kinematics equation (5) for S [ so132 as s 5aa, where a [ S2 and a [ R, then this iso- the full attitude R [ SO132 yields the kinematics for the morphism gives Euler’s theorem, where a [ S2 is the axis reduced attitude. Since v 3 is skew symmetric, it fol- of rotation and a [ R is the angle, corresponding to the lows that 1 2 [ 1 2 rotation matrix exp S SO 3 . Using this isomorphism, # # ^ ^ ^ ^ we can express the matrix exponential of S [ so132 by G 5 R b 5 1v 3 2 R b 5 1v 3 2 G5 2v3G5G3v. (6) Rodrigues’s formula This kinematics equation is consistent with the fact that the 3 3 2 2 exp1S2 5 I 1 a sin1a2 1 1a 2 11 2 cos1a22 . (4) reduced-attitude vector G evolves on S . Indeed, differenti- ^ ating 7G7 2 ! G G with respect to time and substituting (6) Reduced-Attitude Description yields Using Unit Vectors in R3 ^ # Let G ! R b denote the unit vector b that is fixed in the d ^ ^ ^ ^ 1G G2 5 2G G 5 2G 1G3v2 5 2v 1G3G2 5 0. reference frame, resolved in the body-fixed frame. Then dt G [ S2 describes the reduced attitude of a rigid body. Let ^ [ S2 1 2 [ S2 1 2 [ S2 Gd 5 Rd b represent the desired pointing direction of Therefore, since G 0 , hence G t for every t $ 0. an imager, antenna, or other instrument. Pointing in Note that the component of the angular velocity vector v terms of reduced-attitude control objectives can then be along the vector G does not affect the kinematics (6). described as rotating the rigid body so that its reduced- attitude vector G is aligned with the desired reduced-atti- Attitude Dynamics tude vector Gd, that is G 5 Gd, where both reduced-attitude We now define the attitude dynamics of a rigid body vectors are resolved in the body-fixed frame. Rotation with respect to an inertial reference frame. The rate of about the unit vector b does not change the reduced-atti- change of the rotation matrix R [ SO132 depends on the tude defined by the direction of b. Therefore, all full-body angular velocity through the kinematics given by (5). attitudes that are related by such a rotation result in the The rate of change of the angular velocity expressed in same reduced attitude. the body frame is Euler’s equation for a rigid body, which is given by Attitude Kinematics # Jv 5 Jv3v1u, (7) We denote the attitude of a rigid body by R [ SO132 rela- tive to a reference frame. Let v [ R3 be the angular veloc- where u is the sum of external moments applied to the body ity of the body relative to the reference frame, resolved in resolved in the body frame. We assume that the rigid body the body-fixed frame. Then the full-attitude kinematics is fully actuated, that is, the set of all control inputs that equation that gives the time rate of change of the rigid- can be applied to the rigid body is a three-dimensional body attitude is Euclidean space. # R 5 Rv 3, (5) GLOBAL ATTITUDE CONTROL MANEUVERS We now consider attitude control maneuvers that rotate where v 3 is a skew-symmetric matrix as given in (3). the rigid body from a specified initial attitude and initial

36 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 angular velocity to a specified terminal attitude and termi- nal angular velocity, in a specified time period. The atti- Attitude Set Notation tude control maneuvers are defined by describing the he following set-theoretic notation is used throughout control moment input, the resulting attitude, and the this article. angular velocity as functions of time over the specified T • 7x7 is the Euclidean norm of x. time period. We consider fully actuated open-loop control • S1 5 51x , x 2 [ R2: i 1x , x 2 i 5 16 is the unit circle. problems with a control function u:30, T4 S R3. If the ini- 1 2 1 2 • S2 5 51x , x , x 2 [ R3: i 1x , x , x 2 i 5 16 is the two- tial angular velocity is zero and the terminal angular 1 2 3 1 2 3 sphere. velocity is zero, then the attitude control maneuver is a • S3 5 51x , x , x , x 2 [ R4: i 1x , x , x , x 2 i 5 16 is the rest-to-rest attitude maneuver. 1 2 3 4 1 2 3 4 three-sphere. • SO132 5 5R [ R333: RRT 5 RTR 5 I, det1R2 5 16 is the Full-Attitude Maneuvers special orthogonal group. The attitude maneuver considered here is to transfer the so1 2 5 [ R333 T 6 [ 1 2 • 3 5 U : U 52U is the space of 3 3 3 initial attitude R0 SO 3 and zero initial angular velocity [ 1 2 skew-symmetric matrices, also the Lie algebra of to the terminal attitude Rf SO 3 and zero terminal SO132. angular velocity in the fixed maneuver time T . 0. According to Euler’s theorem [1], there exists an axis a [ S2 and an angle a [ 30, 2p2 that satisfy

3 tude maneuver is required, whereas if they differ by a eaa 5 R R21. (8) f 0 sign, then a is chosen to be an arbitrary direction perpen-

A rest-to-rest attitude maneuver that meets the specified dicular to G0 with a 5 p. A rest-to-rest attitude maneuver boundary conditions is given by that meets the specified boundary conditions is

1 2 u1t2a 3 1 2 a 3 u1t2 R t 5 e R0, G t 5 e G0 3 3 1 2 1 3 2 2 1 1 224 3 3 1 2 1 3 2 2 1 1 224 5 I 1 a sinu t 1 a 1 2 cosu t R , (9) 5 I 1 a sinu t 1 a 1 2 cosu t G0, (15) # 0 # v 1t2 5u1t2a, (10) v 1t2 5u1t2a, (16) $ # $ # u1t2 5u1t2Ja 1u1t2 2 1a 3 Ja2, (11) u1t2 5u1t2Ja 1u1t2 2 1a 3 Ja2, (17)

where the rotation angle u:30, T4 S S1 satisfies where the rotation angle u:30, T4 S S1 satisfies (12) and (13). # To achieve a full-attitude maneuver or a reduced-attitude u 102 5 0, u 102 5 0, (12) maneuver that is not rest to rest, one possible approach is to # a, 0 #a,p, select a sequence of three successive rotations. The first phase u 1T2 5 e u 1T2 5 0. (13) 2 2p1a, p#a,2p, is based on selecting a fixed-axis rotation (with the rotation axis fixed in both the inertial and body frames) that brings the The rotation angle u 1t2 can be chosen as a linear combina- initial angular velocity of the rigid body to rest. The third tion of sinusoids or polynomials with coefficients selected phase is based on selecting a similar fixed-axis rotation that, to satisfy the initial and terminal boundary conditions (12) in reverse time, brings the terminal angular velocity of the and (13). rigid body to rest. The second phase uses the framework pre- sented in (8)–(13) and (14)–(17) to construct the rest-to-rest Reduced-Attitude Maneuvers maneuver that transfers the attitude at the end of phase one to We now consider a reduced-attitude maneuver that trans- the required attitude at the beginning of phase three. [ S2 fers an initial reduced attitude G0 and zero initial The framework presented in (8)–(13) and (14)–(17) can [ S2 angular velocity to a terminal reduced attitude Gf and form the basis for planning attitude maneuvers. For example, zero terminal angular velocity in the fixed maneuver time u 1t2 in (9)–(13) and (15)–(17) can be viewed as depending on T . 0. We follow the strategy of a rotation about an iner- parameter values that can be adjusted to achieve some atti- tially fixed axis. This axis, normalized to lie in S2, is given tude maneuver objective such as minimum time or mini- 7 7 [ S2 by a 5G0 3Gf / G0 3Gf assuming G0 and Gf are not mum integrated norm of the control input using optimization colinear. The angle a [ 30, 2p2 satisfies algorithms. The key is the parameterization of attitude maneuvers in a form that satisfies the boundary conditions. 1 2 ^ cos a 5G0 Gf. (14) Observations on Controllability

If G0 and Gf are colinear, then these vectors are either The constructions given above demonstrate that the fully equal or differ by a sign; if they are equal, then no atti- actuated rigid body is controllable even over arbitrarily

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 37 The Impossibility of Global Attitude Stabilization Using Continuous Time-Invariant Feedback he impossibility of global attitude stabilization using con- Ttinuous time-invariant feedback is illustrated for planar C rotations. The insight obtained for the case of attitude con- figurations on the unit circle S1 is helpful for understanding the analogous results when the attitude configurations are in SO132 or S2. Consider the attitude stabilization of the arm of the clock needle shown in Figure S3. We desire to stabilize A B the needle to the configuration in S1 indicated by point A by applying a force at the tip of the needle, which is tangent to the circle. Typical initial configurations are indicated by points B, C, or D. Each configuration can be moved to configuration D A by either a clockwise force or a counterclockwise force, with the tip of the needle defining trajectories in S1 as shown in FIGURE S3 Illustration of the impossibility of global stabili- Figure S3. This explicit construction of forces indicates con- zation for planar rotations. In this case, the configuration S1 trollability. However, if we attempt to construct a continuous manifold is the circle . To stabilize configuration A, a con- tinuous force vector field that rotates the needle to configu- force vector field that rotates the needle to configuration A from ration A from each configuration or point on the circle is each configuration or point on the circle, then an additional constructed as shown by the vectors tangential to the equilibrium (B) is born. This equilibrium (B) is created because circle. However, since this vector field points in the oppo- on the upper half and lower half of the circle, the force vector site direction in the upper half and lower half of the circle, fields are nonvanishing and point in opposing directions (either the force vector field must vanish at some point B. The van- ishing of the vector field at B creates a second equilibrium clockwise or counterclockwise) toward A. Since the force vec- in addition to A. In a similar fashion, continuous time-invari- tor field is continuous, it must vanish at some point B, thereby ant closed-loop vector fields yield multiple closed-loop creating a second equilibrium in addition to A. equilibria on SO132 and S2.

short maneuver times. This observation is true for both tion. The objective in full-attitude stabilization is to select a full-attitude maneuvers and reduced-attitude maneuvers, feedback control function thereby providing explicit constructions that demonstrate controllability. u : SO132 3 R3 S R3 (18) We now show that the constructed maneuvers (8)– (13) and (14)–(17) are not continuous functions of the that asymptotically stabilizes a desired rigid body attitude boundary conditions. For example, consider a reduced- equilibrium, while the objective in reduced-attitude stabili- attitude maneuver where G0 and Gf lie along the north zation is to select a feedback control function and south pole of S2, respectively. In this case, it follows u : S2 3 R3 S R3 (19) from (12) and (13) that a slight perturbation in the ini- tial condition can result in a control maneuver where that asymptotically stabilizes a desired rigid body reduced- the reduced attitude evolves through either the west or attitude equilibrium. The desired attitude equilibrium or the east hemisphere of S2 depending on the perturba- the reduced-attitude equilibrium is assumed to be speci- tion. An analogous phenomenon occurs for attitude fied, and the desired angular velocity vector is zero. maneuvers on SO132. Thus, minor changes in the values Since we assume full actuation, linear control theory of the initial attitude, initial angular velocity, terminal can be applied to the linearization of the rigid-body equa- attitude, and terminal angular velocity may result in tions of motion around a desired equilibrium to achieve major changes in the attitude maneuver. This disconti- local asymptotic stabilization. However, no continuous nuity is not due to the particular approach used for time-invariant feedback controller can globally asymp- designing open-loop controllers. Rather, this disconti- totically stabilize an equilibrium attitude of a rigid body nuity is a consequence of the non-Euclidean geometry [9], [16]; see “The Impossibility of Global Attitude Stabili- of the special orthogonal group SO132 and the two- zation Using Continuous Time-Invariant Feedback.” In sphere S2. fact, if we construct a continuous time-invariant feedback controller that locally asymptotically stabilizes a desired GLOBAL ATTITUDE STABILIZATION attitude equilibrium on SO132 3 R3 or S2 3 R3, then the We now study fully actuated attitude stabilization and resulting closed-loop vector field must have more than present results for both full- and reduced-attitude stabiliza- one equilibrium.

38 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 Since no continuous time-invariant feedback controller Closed-Loop Equilibria 1 2 can globally asymptotically stabilize a desired attitude Let Re, ve be an equilibrium of the closed-loop system equilibrium, we are faced with the following questions: (24)–(25). Equating the right-hand side of (25) to zero yields 3 1) Since global asymptotic attitude stabilization using a Reve 5 0. Since Re is invertible, we obtain ve 5 0. Substitut- 1 2 continuous time-invariant feedback controller is ing ve 5 0 in (24) and equating it to zero yields Va Re 5 0. impossible, can we obtain closed-loop stabilization It can be shown [23] that there are four closed-loop equilib- 1 2 properties that are nearly global? ria that satisfy Va Re 5 0 and ve 5 0. 2) What are the global properties of the resulting closed- loop system for a continuous time-invariant feedback Proposition 1 1 2 controller? Re, 0 is an equilibrium of the closed-loop attitude equa- 3) Are there closed-loop performance implications for a tions of motion (24)–(25) if and only if continuous time-invariant feedback controller? ^ [ S!53 4 3 4 3 4 Rd Re e1 e2 e3 , e1 2 e2 2 e3 , 2e1 2 e2 e3 , FULL-ATTITUDE STABILIZATION 3 46 2e1 e2 2 e3 . (26) We now present a proportional and derivative feedback 1 2 [ ~ control structure and characterize the resulting closed-loop That is, Re, 0 , where vector field. We use this example to answer the above ques- tions. We consider rigid-body attitude stabilization using ~ ! 51R , 02, 1R , 02, 1R , 02, 1R , 026, (27) the fully actuated rigid-body equations of motion d a b g # Jv 5 Jv3v1u, (20) where # R 5 Rv3, (21) ! 3 4 Ra Rd e1 2e2 2e3 , (28) where R [ SO132 and v [ R3. R ! R 32e 2e e 4, (29) The full-attitude stabilization objective is to select a con- b d 1 2 3 3 3 ! 3 4 tinuous feedback controller u : SO132 3 R S R that Rg Rd 2e1 e2 2e3 . (30) asymptotically stabilizes the desired attitude equilibrium [ 1 2 given by the desired attitude Rd SO 3 and the angular Proof 1 2 velocity v50. The feedback controller is given by Equating Va Re 5 0 yields

1 2 u 52Kvv2KpVa R , (22) ^ ^ ^ a1e1 3 Rd Ree1 1 a2e2 3 Rd Ree2 1 a3e3 3 Rd Ree3 5 0. [ R333 (31) where Kv, Kp are positive definite matrices, 3 4^ a 5 a1 a2 a3 , where a1, a2, and a3 are distinct positive inte- ^ 3 4 [ 1 2 gers, and the map Writing Rd Re 5 ri,j i,j[51,2,36 SO 3 and expanding the

3 cross products, (31) is equivalent to a2r32 5 a3r23, a1r31 5 1 2 ! 1 ^ 2 Va R a ai ei 3 Rd Rei , (23) a3r13, a1r21 5 a2r12. Since a1, a2, and a3 are positive, the rota- i51 ^ tion matrix Rd Re can be written as 3 4 with e1 e2 e3 the identity matrix. The feedback controller (22) requires the full attitude R [ SO132 and angular veloc- r11 r12 r13 ity v [ R3, and it is continuous on SO132 3 R3. The feed- ^ a2 5 E r12 r22 r23U. back controller (22) can be interpreted as introducing a Rd Re a1 (32) potential though the attitude-dependent term and dissipa- a3 a3 r13 r23 r33 tion through the angular-velocity-dependent term. Knowl- a1 a2 edge of the rigid-body inertia is not required by the feedback ^ controller. While the dissipation term and the attitude- Since Rd Re is orthogonal it follows that dependent term in the feedback controller appear linearly,

the feedback controller is nonlinear since the rotation matri- 2 2 a a3 ces are not elements of a vector space. a1 2 2 br2 1 a1 2 br2 5 0, (33) a2 12 a2 13 The closed-loop full-attitude dynamics using the con- 1 1 troller (22) are a2 a2 a 3 b 2 a 3 b 2 1 2 2 r13 1 1 2 2 r23 5 0, (34) a1 a2 # 1 2 2 2 Jv 5 Jv3v2K v2K V R , (24) a a3 # v p a a1 2 2 br2 2 a1 2 br2 5 0. 3 2 12 2 23 (35) R 5 Rv . (25) a1 a2

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 39 Since a1, a2, and a3 are distinct, the solution to (33)–(35) Let the perturbation in the initial attitude be given as 3 ^ 2 1 2 eU0 [ S satisfies either a) r12 5 r13 5 r23 5 0 or b) none of the rij, i j R 0, e 5 Ree , where Rd Re is given in (26) and [ R3 are zero. To see this result, first note that a) is a trivial U0 ; see “Linearization of Nonlinear Models on Euclid- 2 solution of (33)–(35). Suppose r12 0. Then since a1, a2, ean Spaces and Non-Euclidean Spaces” for details. The per- 1 1 2 2 22 2 and a3 are distinct, it follows that 1 2 a2/a1 0, turbation in the initial angular velocity is given as 1 1 2 2 22 2 1 1 2 2 22 2 2 1 2 [ R3 1 2 a3/a1 0, and 1 2 a3/a2 0. Then, since r12 0, v 0, e 5ev0, where v0 . Note that if e50 then 2 1 1 2 1 22 1 2 (33) yields that r13 0, and from (34), it follows that R 0, 0 , v 0, 0 5 Re, 0 , and hence 2 2 r23 0. Similar arguments hold for the case r13 0 and 2 1 1 2 1 22 ; 1 2 r23 0. Thus, every solution to (33)–(35) satisfies either a) R t, 0 , v t, 0 Re, 0 (36) or b). [ 3 2 Next, consider case b). Suppose a1 . a2. Then, for all time t 0, ` , thereby obtaining the equilibrium 1 1 2 2 22 2 2 1 2 a2/a1 r12 . 0 since r12 0, and hence (33) yields solution. 1 1 2 2 22 2 2 1 2 a3/a1 r13 , 0. Since r13 is positive, it follows that Next, consider the solution to the perturbed equations of

a3 . a1. Thus, a3 . a1 . a2. This inequality implies motion for the closed-loop system (24)–(25). These equa- tions are given by 2 2 a3 a3 a1 2 br2 , 0, a1 2 br2 , 0. # a2 13 a2 23 Jv 1t, e2 5 Jv 1t, e2 3v1t, e2 2 K v 1t, e2 2 K V 1R1t, e22, (37) 1 2 # v p a R1t, e2 5 R1t, e2v 1t, e23. (38) Therefore, the left-hand side of (34) is negative, yielding a contradiction. Similar arguments show that for a1 , a2, To linearize (24)–(25), we differentiate both sides of

(33)–(35) yields a contradiction for case b). Hence since a1, (37) and (38) with respect to e and substitute e50, a2, and a3 are distinct, case b) yields a contradiction, and yielding the only solution to (33)–(35) is case a), that is, # r 5 r 5 r 5 0. Jv 1t, 02 52K v 1t, 02 2 K V 1R 1t, 022, (39) 12 13 23 # e v e p a e Then, substituting r12 5 r13 5 r23 5 0 into (32), it follows R 1t, 02 5 R v 1t, 023, (40) ^ [ S e e e that r11, r22, r33 each have value 1 1 or 2 1. Thus, Rd Re 1 2 [ ~ u 1 2 ! 1' 1 2 1' 22 1 2 ! 1' 1 2 as given in (26), and hence Re, 0 in (27). where ve t, 0 v t, e / e e50 and Re t, 0 R t, e / 1' 22 [ R3 Proposition 1 shows that the continuous feedback con- e e50. Now we define the linearized states Dv, DU 1 2 ! 1 2 1 2 3 ! ^ 1 2 troller results in a closed-loop system with four distinct as Dv t ve t, 0 and DU t Re Re t, 0 . Then from equilibria. One of these equilibria is the desired equilib- (40) we obtain rium specified by the attitude Rd, while the remaining # # 1 2 3 ^ 1 2 1 23 1 23 equilibria are designated by Ra, Rb, and Rg. These attitudes DU t 5 Re Re t, 0 5ve t, 0 5Dv t .

differ from the desired attitude Rd by 180° of rotation about each of the three body-fixed axes. Suppressing the time dependence, we obtain # Local Structure of the Closed-Loop System DU 5Dv. (41) We next analyze the stability of the desired equilibrium and each of the three additional equilibria, and we deduce From (39), we obtain the local structure of the closed-loop vector field near # 1 3 2 each of these equilibria using the Lie group properties of JDv 52KvDv 2 KpVa ReDU . (42) SO132. Since dim3SO132 3 R3 4 5 6, each linearization evolves on R6. To obtain the local structure of the closed- Now, loop attitude dynamics, we linearize the closed-loop 1 3 2 1 ^ 3 2 1 ^ 3 2 equations about each equilibrium. While it is possible to Va ReDU 5 a1e1 3 Rd ReDU e1 1 a2e2 3 Rd ReDU e2 study the linearized dynamics in local coordinates, the 1 ^ 3 2 1 a3e3 3 Rd ReDU e3 , closed-loop vector field (24)–(25) can also be linearized ^ ^ 521a e3R R e3 1 a e3R R e3 using the Lie-group properties of the non-Euclidean 1 1 d e 1 2 2 d e 2 3 ^ 3 2 1 2 1 2 [ ~ 1 a3e3 Rd Ree3 DU, (43) manifold SO 3 . Let Re, 0 be an equilibrium solu- tion of the closed-loop system (24)–(25). Consider a per- 1 2 [ ~ ^ S turbation of the equilibrium Re, 0 in terms of a where Rd Re belongs to as given in (26) in Proposition 1. perturbation parameter e [ R. We can express the per- Combining (41)–(43), we obtain the linearization of the turbation as a rotation matrix in SO132 using exponential closed-loop system (24)–(25) about each equilibrium in (27) coordinates [34]–[37]. This representation guarantees as that the perturbation is a rotation matrix for all values of $ # the perturbation parameter. JDU 1 KvDU 1 KDU 5 0, (44)

40 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 Linearization of Nonlinear Models on Euclidean Spaces and Non-Euclidean Spaces inearization of a nonlinear model is defined from the re- similar analysis can be used to linearize the reduced rotational L sponses to an infinitesimal perturbation of initial conditions dynamics of a rigid body. from a given equilibrium. For rigid-body rotational dynamics, In contrast to the nonlinear perturbation function (S1) for we require that the perturbations in initial conditions lie on the rigid-body rotational dynamics, we can choose F as a linear manifold SO132 3 R3 so that the solution evolves on the mani- additive vector perturbation for models whose states evolve fold SO132 3 R3. on Euclidean spaces. For attitude dynamics, while a linear Let F: SO132 3 R3 3 R S SO132 3 R3 be the mapping additive perturbation is used for the angular velocity, the ex- F1 2 such that Re, ve, e represents the perturbation from the ponential function is used for the attitude perturbation in (S1). 1 2 equilibrium Re, ve , where e denotes the perturbation param- The reason behind this choice is that, while linear additive eter. Thus e 5 0 implies no perturbation from the equilibrium perturbations guarantee that, for all values of the perturba- solution. If 1R1t, e2, v 1t, e22 represents the unique solution tion parameter e, the resulting angular velocity perturbations corresponding to the perturbation e, then the linearization is lie in the Euclidean space of angular velocities, adding rota- obtained by differentiating this perturbed solution with respect tion matrices does not guarantee that the resulting matrix is to e, where the perturbation function is defined as also a rotation matrix. Since the exponential map (2) of SO132 guarantees that the resulting matrix is a rotation matrix [34],

3 F1 2 ! 1 eU0 2 Re, ve, e Ree , ve 1ev0 (S1) [14], [36], [37], an exponential function is used to construct the attitude perturbation as shown in (S1). Thus, the attitude 1 2 [ R3 R3 1 2 and U0, v0 3 . This computation yields the lineariza- perturbation in (S1) lies in SO 3 for all values of the perturba- tion of the rigid-body rotational dynamics at the equilibrium. A tion parameter e.

where the constant matrix K, depending on the equilib- exponential convergence. Since the feedback controller is rium attitude, is defined as continuous, the closed-loop vector field has the additional 1 2 1 2 1 2 equilibria Ra, 0 , Rb, 0 , and Rg, 0 . 1 2 Kp diag a2 1 a3, a1 1 a3, a1 1 a2 ,if Re 5 Rd, We next want to understand the global structure of the 1 2 µ 2 Kpdiag a2 1 a3, a1 2 a3, a1 2 a2 ,if Re 5 Ra, solutions of the closed-loop system. Consider again the K 5 1 2 2 Kpdiag a3 2 a2, a3 2 a1, a1 1 a2 ,if Re 5 Rb, closed-loop linearizations (44). For each linearization about 1 2 1 2 1 2 1 2 2 Kpdiag a2 2 a3, a1 1 a3, a2 2 a1 ,if Re 5 Rg. the unstable equilibria Ra, 0 , Rb, 0 , and Rg, 0 , the linear- (45) ization has at least one eigenvalue with a negative real part Thus, the local attitude dynamics near each of the four equi- and at least one eigenvalue with a positive real part. Further- librium solutions are given by a linear second-order more, the four equilibria are hyperbolic, that is, the lineariza- mechanical system with positive damping. The stiffness tion has no imaginary eigenvalues. Thus, local properties of depends on the equilibrium; near the desired equilibrium, the closed-loop vector field near each of these equilibria are the stiffness is positive definite, and hence, the local attitude as follows. There exist stable and unstable manifolds for each 1 2 1 2 1 2 dynamics near the desired equilibrium are locally exponen- of the equilibria Ra, 0 , Rb, 0 , and Rg, 0 , denoted by 1W s W u 2 1W s W u 2 1W s W u 2 tially stable. Around the additional three equilibria corre- a, a , b, b , and g, g such that all solutions sponding to the attitudes Ra, Rb, and Rg, the stiffness matrix that start on the stable manifold converge to the correspond- has at least one negative eigenvalue resulting in a saddle ing unstable equilibrium, and all solutions that start on the point. Therefore, the closed-loop dynamics near each equi- unstable manifold locally diverge from the corresponding librium solution are unstable. Furthermore, these equilibria equilibrium [38]. Furthermore, the dimensions of the stable have no zero eigenvalues, yielding the following result. and unstable manifolds of each unstable equilibrium are the same as the number of eigenvalues with negative and posi- Proposition 2 tive real parts, respectively [38]. 1 2 The equilibrium Rd, 0 of the closed-loop equations (24)– Since the union of the stable manifolds has dimension (25) is asymptotically stable with locally exponential conver- less than six, it can be shown [23] that the set has measure 1 2 1 2 gence. The additional closed-loop equilibria Ra, 0 , Rb, 0 , zero and is nowhere dense. Furthermore, using Lyapunov 1 2 and Rg, 0 are hyperbolic and unstable. analysis [23], it can be shown that all solutions converge to one of the four equilibria. Since all solutions that converge Global Analysis of the to the three unstable equilibria lie in a nowhere dense set, Full-Attitude Closed-Loop System almost all closed-loop solutions converge to the desired 1 2 The feedback controller (22) renders the desired equi- equilibrium Rd, 0 . This result is precisely stated in the fol- 1 2 librium Rd, 0 locally asymptotically stable with local lowing theorem.

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 41 Theorem 1 dent term and dissipation through the angular Consider the closed-loop rigid-body attitude equations velocity-dependent term. Knowledge of the rigid-body of motion given by (24) and (25) for the controller (22), inertia is not required by the feedback controller. Note where Kp and Kv are positive definite, and a1, a2, and a3 that while the dissipation term and the reduced-attitude- are distinct positive integers. Then the closed-loop equi- dependent term in the feedback controller appear lin- 1 2 librium Rd, 0 is asymptotically stable with locally early, the feedback controller is nonlinear since the exponential convergence. Furthermore, there exists a reduced-attitude vectors lie in S2 and, hence, are not ele- nowhere dense set M such that all solutions starting in ments of a vector space. M converge to one of the unstable equilibria The closed-loop reduced-attitude equations of motion 1 2 1 2 1 2 Ra, 0 , Rb, 0 , and Rg, 0 , and all remaining solutions are 1 2 # converge to the desired equilibrium Rd, 0 . Thus, the 1 2 1 2 Jv 5 Jv3v1kp Gd 3G 2 Kvv, (49) domain of attraction of the desired equilibrium Rd, 0 is # almost global. G 5G3v. (50) 1 2 Since the domain of attraction of Rd, 0 is the comple- ment of the union of the stable manifolds for each of the Closed-Loop Equilibria three unstable equilibria, the eigenspace corresponding Consider the equilibrium structure of the closed-loop to the stable eigenvalues of an unstable equilibrium pro- equations (49) and (50). Equating the right-hand side of 1 2 vides explicit local information on the complement of the (49) and (50) to zero yields that an equilibrium at Ge, ve 1 2 domain of attraction of Rd, 0 near that unstable equilib- satisfies ve 5mGe, where m is a constant, and rium. Note that the structure and characteristic of the 2 1 2 unstable equilibria depends on the particular choice of m JGe 3Ge 1 kp Gd 3Ge 2mKvGe 5 0. feedback control. However, as shown in [9], the comple- 1 2 ^ ^ ment of the domain of attraction of Rd, 0 is necessarily Premultiplying both sides by Ge yields mGe KvGe 5 0. Since 7 7 nonempty for every continuous time-invariant feedback Kv is positive definite and Ge 5 1, it follows that m 5 0. 1 2 R3 controller on SO 3 3 . Thus, the existence of a nowhere Thus, ve 5 0. Then, we see that Ge 3Gd 5 0, and hence

dense set, which is the complement of the domain of Ge 56Gd. We thus have the following proposition. 1 2 attraction of Rd, 0 , is not a consequence of the specific controller (22). Since global stabilization using continu- Proposition 3 ous time-invariant feedback is impossible, almost global The reduced closed-loop attitude equations of motion (49) 1 2 stabilization is the best possible result for this stabiliza- and (50) have two equilibrium solutions given by Gd, 0 1 2 tion problem. and 2Gd, 0 . Proposition 3 shows that the continuous time-invari- REDUCED-ATTITUDE STABILIZATION ant feedback control results in a closed-loop system with Rigid-body reduced-attitude stabilization is analyzed two distinct equilibria given by the desired equilibrium 1 2 using the fully actuated rigid-body equations of motion Gd, 0 and an additional closed-loop equilibrium 1 2 # 2Gd, 0 . The existence of the two reduced-attitude equi- Jv 5 Jv3v1u, (46) libria implies that there is a manifold of full-attitude # ^ G 5G3v, (47) equilibria, all of which satisfy R b 5Gd, and another dis- joint manifold of full-attitude equilibria, all of which sat- [ S2 [ R3 ^ where G and v . isfy R b 52Gd; see “Relation Between Full Attitude and The objective of reduced-attitude stabilization is to select a Reduced Attitude.” continuous feedback controller u : S2 3 R3 S R3 that asymptotically stabilizes the desired reduced-attitude Local Structure of the Closed-Loop System [ S2 equilibrium defined by the reduced-attitude vector Gd We study linearizations of the closed-loop equations (49) 1 2 and angular velocity vector v50. The feedback controller and (50) about the equilibrium Gd, 0 and about the equi- 1 2 3S2 R3 4 is given by librium 2Gd, 0 . Since dim 3 5 5, each lineariza- tion evolves on R5. Similar to the analysis presented in [23], 1 2 u 5 kp Gd 3G 2 Kvv, (48) it can be shown that each equilibrium solution has the fol- lowing local properties. where kp is a positive real number and Kv is a positive definite matrix. This feedback controller is based on Proposition 4 ^ [ S2 1 2 feedback of the reduced attitude G5R b and the The equilibrium Gd, 0 of the closed-loop reduced-attitude angular velocity v [ R3, and it is continuous on S2 3 R3. equations (49) and (50) is asymptotically stable and the con- The feedback controller (48) can be interpreted as intro- vergence is locally exponential. Furthermore, the equilib- 1 2 ducing a potential through the reduced-attitude-depen- rium 2Gd, 0 is hyperbolic and unstable.

42 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 Global Analysis of the Reduced-Attitude Closed-Loop System Relation Between Full Attitude In this section, we study the global behavior of the closed- and Reduced Attitude loop system defined by (49) and (50). The results are paral- or a rigid body, one reduced attitude corresponds to a lel to those stated above for the full-attitude closed-loop Fmanifold of full attitudes. To see this connection, note that system. In particular, we characterize the domain of attrac- the definition of the reduced attitude G [ S2 implies that there 1 2 tion of the desired equilibrium Gd, 0 and describe the [ 1 2 T exists a full attitude R0 SO 3 such that R0b 5G. Then global structure of the solutions that converge to the unsta- Rodrigues’s formula shows that every full attitude given by 1 2 ble equilibrium 2Gd, 0 . 1 3 2 [ R T R 5 R0 exp aG with a also satisfies R b 5G. In physi- It can be shown using Lyapunov analysis that all cal terms, all full attitudes that differ by a rotation about G are closed-loop solutions converge to one of the two closed- equivalent in the sense that they have the same reduced at- 1 2 R3 loop equilibria (see “Lyapunov Analysis on SO 3 3 titude, namely, G. S2 R3 1 2 and 3 ”). Since 2Gd, 0 is unstable and the linear- 1 2 ization about 2Gd, 0 has no imaginary or zero eigenval- ues, it follows that there exists a stable invariant manifold M 1 2 r [38] of the closed-loop equations (49) and (50) such of Gd, 0 , is not a consequence of the specific controller M 1 2 that all solutions that start in r converge to 2Gd, 0 . that is studied in this article. Rather, the complement of 1 2 1 2 The tangent space to this manifold at 2Gd, 0 is the the domain of attraction of Gd, 0 is necessarily non- stable eigenspace corresponding to the negative eigen- empty for every continuous time-invariant feedback con- values. Since the linearization of the closed-loop dynam- troller on S2 3 R3 [9]. Thus, since global stabilization 1 2 ics at 2Gd, 0 has no eigenvalues on the imaginary axis, using continuous time-invariant feedback is impossible, the closed-loop dynamics has no center manifold, and almost global stabilization is the best possible result for 1 2 every closed-loop solution that converges to 2Gd, 0 lies this stabilization problem. M in the stable manifold r. M It can be shown that r is a lower dimensional subset in CLOSED-LOOP PERFORMANCE LIMITATIONS the five-dimensional manifold S2 3 R3 and is nowhere The difference between global asymptotic stability and dense [23]. Hence, the closed-loop dynamics partition almost global asymptotic stability of the desired equilib- S2 R3 M 3 into two sets. The first set r is defined by solu- rium of a closed-loop attitude control system leads to cer- 1 2 tions that converge to the unstable equilibrium 2Gd, 0 , tain closed-loop attitude performance limitations. Thus, we and its complement is defined by the dense set of all solu- now address the third question posed in the section “Global 1 2 tions that converge to the desired equilibrium Gd, 0 . Thus, Attitude Stabilization” related to performance implications 1 2 Gd, 0 is an asymptotically stable equilibrium of the closed- of almost global stabilization. loop system (49) and (50) with an almost global domain of While all solutions that lie in the domain of attraction attraction. asymptotically approach the desired equilibrium, solutions that encounter a sufficiently small neighborhood of the Theorem 2 stable manifold of one of the unstable hyperbolic equilibria Consider the reduced rigid-body attitude dynamics can remain near that stable manifold for an arbitrarily long given by the closed-loop equations (49) and (50). Assume that k is positive, and K is symmetric and positive defi- p v 1 2 3 R3 nite. Then the closed-loop equilibrium 1G , 02 is asymp- Lyapunov Analysis on SO 3 and d S2 3 R3 totically stable with local exponential convergence. Furthermore, there exists a lower dimensional manifold losed-loop full-attitude dynamics evolve on the non- M CEuclidean manifold SO132 3 R3, while closed-loop re- r that is nowhere dense such that closed-loop solu- M duced-attitude dynamics evolve on the non-Euclidean mani- tions starting in r converge to the unstable equilib- 1 2 fold S2 3 R3. Since these manifolds are locally Euclidean, rium 2Gd, 0 and all remaining closed-loop solutions 1 2 local stability properties of a closed-loop equilibrium solution converge to the desired equilibrium Gd, 0 . Thus, the domain of attraction of the desired equilibrium is can be assessed using standard Lyapunov methods. In ad- almost global. dition, the LaSalle invariance result and related Lyapunov 1 2 results apply to closed-loop vector fields defined on these Since the domain of attraction of Gd, 0 is the comple- 1 2 2 ment of the stable manifold of the unstable equilibrium manifolds. However, since the manifolds SO 3 and S are 1 2 compact, the radial unboundedness assumption cannot be 2Gd, 0 , the eigenspace corresponding to the stable 1 2 satisfied; consequently, global asymptotic stability cannot fol- eigenvalues of 2Gd, 0 provides explicit local informa- tion on the complement of the domain of attraction of low from a Lyapunov analysis on Euclidean spaces [40], and 1 2 1 2 therefore must be analyzed in alternative ways [19]–[23]. Gd, 0 near 2Gd, 0 . The presence of the nowhere dense set, which is the complement of the domain of attraction

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 43 loop properties of a continuous time-invariant vector field 1 2 R3 S2 R3 3 on SO 3 3 or 3 . In contrast to continuous feedback controllers, 2.5 switched or discontinuous feedback controllers can also be considered for attitude control. Controllers based on 2 nonsmooth feedback may not suffer from the perfor- 1.5 mance limitations observed in smooth controllers [27]. || (rad) θ

|| However, discontinuous controllers suffer from chatter- 1 ing at the surface of the discontinuity of the closed-loop 0.5 vector field. Chattering in discontinuous controllers can be resolved by using hysteresis [39]. However, this dis- 0 continuity can render standard Lyapunov theory inappli- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t /T cable due to the non-Lipschitz nature of the closed-loop (a) vector field [40], [41]. 0.9 0.8 EXAMPLES OF RIGID-BODY ATTITUDE 0.7 MANEUVERS AND ATTITUDE STABILIZATION 0.6 We now illustrate the attitude control results by construct- 0.5 ing a rest-to-rest full-attitude maneuver, a rest-to-rest reduced-attitude maneuver, a full-attitude stabilizing feed-

|| (rad/s) 0.4

ω back controller, and a reduced-attitude stabilizing feedback || 0.3 controller. The rigid body is assumed to be an asymmetric 0.2 body. The body-fixed frame is defined by the principal axes 0.1 of the body, and the body inertia matrix is assumed to be 0 1 2 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 J 5 diag 3, 4, 5 kg-m . t /T (b) A Rest-to-Rest Full-Attitude Maneuver 2.4 We construct a rest-to-rest full-attitude maneuver that 2.2 aligns the three principal axes of the rigid body with the 2 three inertial axes. Specifically, the full-attitude maneuver 1.8 transfers the initial attitude R to a terminal attitude R , 1.6 0 f where 1.4 || (N-m)

τ 1.2 || 2 0.3995 0.8201 0.4097 100 1 £ § £ § 0.8 R0 5 0.1130 2 0.3995 0.9097 , Rf 5 010, 0.6 0.9097 0.4097 0.0670 001 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 in a maneuver time of T 5 5 s. Equation (8) can be solved to t /T 5 1 2 a5 p/ (c) obtain a 0.5, 0.5, 0.707 , and the angle 5 6 rad. Expressions for the attitude, angular velocity, and control input that satisfy the specified boundary conditions are FIGURE 1 (a) Principal angle, (b) norm of angular velocity, and given by (9)–(11). In this instance, the rotation angle is (c) norm of the control torque plotted versus normalized time u 1t2 5 1a/2211 2 cos11p/T2t22 rad. for the rest-to-rest, full-attitude, open-loop maneuver for a For the rest-to-rest full-attitude maneuver, Figure 1 shows rigid body with principal moment of inertia matrix J 5 diag(3, 4, 5) kg-m2 within the maneuver time T 5 5 s. The principal the time responses for the principal angle u, the norm of the angle u(t) is obtained as a sinusoidal interpolation between angular velocity 7v 7, and the norm of the control torque 7u7 the given initial attitude R (0) and the desired final attitude versus normalized time t/T. Numerical solutions for these R (T ) for this rest-to-rest maneuver. The angular velocity and open-loop maneuvers are obtained using Lie group varia- control torque are obtained from the principal angle and resulting attitude profile. tional integrators, as described in “Numerical Integration of Attitude Control Systems.” With this control scheme and fixed initial and final attitudes, the normalized time plots for period of time [38]. Such solutions in the domain of attrac- the principal angle, the angular velocity norm, and control tion approach the desired equilibrium arbitrarily slowly. norm for every final time T qualitatively look similar to This property of the closed-loop system is an attitude per- those shown in Figure 1, except that larger T requires less formance limitation that arises due to the inherent closed- control effort.

44 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 Numerical Integration of Attitude Control Systems quations (24) and (25) describe the full-attitude control sys- called variational integrators. In [S2]–[S4], Lie group variation- Etem, whereas (49) and (50) describe the reduced-attitude al integrators are developed and used to numerically simulate control system, which evolve on SO132 3 R3 and on S2 3 R3, attitude dynamics and control systems. Lie group variational respectively. Traditional numerical integration schemes, de- integrators incorporate the desirable features of both geometric veloped for differential equations that evolve on Rn, do not integration schemes and variational integration schemes. guarantee that the computed solution lies in the appropriate manifold [S1]. In contrast to traditional numerical integration REFERENCES [S1] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integra- schemes, geometrical numerical integration schemes guaran- tion. New York: Springer-Verlag, 2000. tee that the computed solution lies in the appropriate manifold [S2] T. Lee, M. Leok, and N. H. McClamroch, “Computational geometric [S1]. Geometric integration schemes can be applied to mani- optimal control of rigid bodies,” Commun. Inform. Syst., vol. 8, no. 4. pp. 435–462, 2008. S3 folds such as without requiring renormalization or projec- [S3] T. Lee, M. Leok, and N. H. McClamroch, “Lagrangian mechanics tion. If the manifold under consideration is a Lie group such and variational integrators on two-spheres,” Int. J. Numerical Methods as SO132, then such integration schemes are called Lie group Eng., vol. 79, pp. 1147–1174, 2009. [S4] A. K. Sanyal and N. Nordkvist, “A robust estimator for almost glo- methods. Integration methods that are obtained by using the bal attitude feedback tracking,” in Proc. AIAA Guidance, Navigation and discrete version of the variational principle of mechanics are Control Conf., Toronto, Ontario, Aug. 2010, AIAA-2010-8344.

A Rest-to-Rest Reduced-Attitude Maneuver tity matrix. The feedback controller is given by (22), where We construct a rest-to-rest reduced-attitude maneuver that we select the vector a 5 11, 2, 32, and the nominal gains 1 2 1 2 aligns the third principal axis of the body frame with the Kp 5 diag 1, 2, 3 and Kv 5 diag 5, 10, 15 . third axis of the inertial frame. This reduced-attitude The closed-loop eigenvalues of the linearized closed- 1 2 maneuver transfers the initial reduced attitude G0 5 loop system at the equilibrium Rd, 0 are given 1 2 ^ 1 2 5 6 R 0 e3 5 0.9097, 0.4097, 0.0670 to the terminal reduced by 214.3739,29.1231, 23.618, 21.382, 20.8769, 20.6261 , 1 2 attitude Gf 5 0, 0, 1 in a maneuver time of T 5 5 s. The thereby de scribing the local convergence rate near the axis and angle variables are given as a 5 10.9118, desired equilibrium. The system has three additional 2 0.4107, 02 and a51.5038 rad, respectively. Expressions equilibria corresponding to the attitudes 5diag11, 21, 212, for the reduced attitude, the angular velocity, and the con- diag121,21, 12, diag121, 1, 2126 as described by (28)–(30) trol input that satisfy the specified boundary conditions in Proposition 1. are given by (15)–(17). In this instance, the rotation angle is We refer to the gains given above as defining a nominal u 1t2 5 1a/2211 2 cos11p/T2t22 rad. controller. Two alternative sets of gains define a stiff con- The full-attitude maneuver studied above also satis- troller and a damped controller. The stiff gains are obtained

fies the boundary conditions imposed for the reduced- by increasing Kp and decreasing Kv by 25% each. In a simi- attitude maneuver for the specified initial full rigid-body lar manner, the damped gains are obtained by decreasing

attitude. Since the reduced-attitude maneuver is less con- Kp and increasing Kv by 25% each. In each case, the desired 1 2 strained than the full-attitude maneuver, it typically full-attitude equilibrium Rd, 0 is almost globally asymp- requires less control effort. For this rest-to-rest reduced- totically stable on SO132 3 R3. attitude maneuver, Figure 2 gives the (a) time responses The three sets of gains yield nominal, stiff, and damped of the principal angle, (b) norm of the angular velocity, closed-loop responses locally near the desired equilibrium. and (c) norm of the control torque, all plotted versus nor- These closed-loop responses are shown in Figure 4 for the malized time. Figure 2(d) shows the reduced-attitude initial conditions vector components versus normalized time for this maneuver. Figure 3 gives three snapshots of the reduced- 0 0.2887 0.4082 2 0.8660 1 2 £ § 1 2 £ § attitude configuration, and the path traced out on the v 0 5 0 , R 0 5 2 0.8165 0.5774 0 . (51) two-sphere at different instants in this maneuver, includ- 0 0.5000 0.7071 0.5000 ing the initial and final time instants. In Figure 3, the axes shown in the snapshots of the reduced-attitude maneu- The error in the attitude is represented by the eigenangle vers are the body axes. 1 U1t2 5 cos21 1trace1R1t22 2 12, (52) Full-Attitude Stabilization 2 We construct a feedback controller that asymptotically which is a scalar measure of the error between R and I. aligns the three principal axes of the rigid body with the Thus, U102 5 80°. From Figure 4(a), we see that, com- three inertial axes. Thus the desired attitude Rd is the iden- pared to the nominal controller, the stiff controller acts

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 45 1.6 0.5 1.4 0.45 1.2 0.4 0.35 1 0.3 0.8 0.25 || (rad) || (rad/s) θ 0.2

0.6 ω || || 0.15 0.4 0.1 0.2 0.05 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t /Tt/T (a) (b) 1.4 1.2 1.2 1 1 Γ 0.8 1 Γ 0.8 0.6 2 Γ 3 || (N-m) 0.6

τ 0.4 ||

0.4 Vector) (Unit Γ 0.2 0.2 0 0 −0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t /Tt/T (c) (d)

FIGURE 2 (a) Principal angle, (b) norm of angular velocity, (c) norm of the control torque, and (d) components of the reduced-attitude vector versus normalized time for the rest-to-rest reduced-attitude open-loop maneuver for a rigid body with inertia matrix J 5 diag(3, 4, 5) kg-m2, within the maneuver time T 5 5 s. The reduced-attitude vector is the direction of the third principal axis of the rigid body, and the maneuver aligns the reduced-attitude vector with the third coordinate axis of the inertial coordinate frame. The principal axis of rotation is the cross product of the given initial reduced attitude G(0) and the desired final reduced attitude G(T), while the principal angle u(t) is obtained by a sinusoidal interpolation between the boundary conditions. The angular-velocity and control-torque profiles are obtained from the principal angle and the resulting reduced-attitude profile. Although the norms of the angular-velocity and control- torque profiles look similar in shape to the corresponding profiles for the full-attitude maneuver, their values are considerably lower. faster in reducing the error to zero, while the damped to the stable and unstable manifolds at the equilib- 1 2 controller is slower in its response. However, as shown rium Rg, 0 . in Figure 4(b) and (c), the stiff controller requires a To demonstrate performance limitations, we choose a higher control torque and also leads to larger transient family of initial conditions that lie close to the stable mani- 1 2 angular velocities compared to the nominal controller, fold of the unstable equilibrium Rg, 0 . We choose 3 4 3 4 while the damped controller requires less control torque va 5 DUa Dva and vb 5 DUb Dvb as a linear combination and has lower transient angular velocities. The param- of eigenvectors from the stable and unstable eigenspaces, 1 2 eters a1, a2, and a3 distribute the control effort among the respectively, of the equilibrium Rg, 0 , where three body axes, and these gains can be modified accordingly based on the moment of inertia and the 0.4021 0.5939 £ § £ § desired response. DUa 5 0.3037 , Dva 5 2 0.9528 , To illustrate the closed-loop performance limitations, 0.2993 2 0.9542 we consider the linearization of the closed-loop system, as 0 0 given in (44) for the nominal gains, at the unstable equilib- DU 5 £ 0.8432 § , Dv 5 £ 0.5375 § . rium corresponding to the attitude R 5 diag121, 1, 212. b b g 0.9827 0.1849 The local closed-loop dynamics near this unstable equ i librium are described by the positive eigen- Consider the initial conditions values 10.1882, 0.63752, and negative eigenvalues 1 2 1 2 1 2 1 3 3 2 20.2324, 21.4343,23.1882, 23.1375 . These eigenvalues R 0 5 diag 21, 1, 21 exp dDUa 1eDUb , (53) 1 2 confirm that the equilibrium Rg, 0 is unstable, and the 1 2 v 0 5dDva 1eDvb, (54) stable manifold has dimension four, while the unstable manifold is two dimensional. The eigenspaces correspond- which depends on the parameter e. If we choose 0 , d V 1, ing to the negative and positive eigenvalues are tangent then, for e50, Figure 5 shows that the initial conditions

46 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 t = 0.0 80 Nominal Damped 60 Stiff 1

0.6 ) °

( 40

0.2 Θ z –0.2 –0.6 20 –1 0 0 2 4 6 8 ––1 –11 –0.66 Time (s) –0.66 –0.22 –0.2 00.2.2 (a) 020.2 0.606 x y 060.6 1 1 (a) 0 /s) ° (

t = 2.5 1 −20 ω −40 0 2 4 6 8 1 40

/s) Nominal 0.6 ° 20

( Damped 2 Stiff

0.2 ω 0 z –0.2 02468 –0.6 30

/s) 20

–1 °

( 10 3

ω 0 –1– 02468 –11 –0.–0.66 –0.66 –0.22 Time (s) –0.20.2 00.2.2 y 0.6 1 1 0.6 x (b) (b) 10 Nominal t = 5.0 8 Damped Stiff 6 1 0.6 4 0.2 z –0.2 2 Two−Norm of u (N-m) Two−Norm –0.6 0 –1 02468 Time (s) ––1 (c) –11 –0.66 –0.66 –0.22 –0.20.2 0.2 060.6 060.66 x y 1 1 FIGURE 4 (a) Eigenangle, (b) angular velocity components along (c) each body-fixed axis, and (c) the two-norm of the control torque are plotted for three controllers, namely, a nominal controller, a stiff FIGURE 3 Snapshots of the rest-to-rest, open-loop, reduced-atti- controller, and a damped controller. As shown in (a), the error con- tude maneuver for a rigid body with inertia matrix J 5 diag(3, 4, 5) verges to zero faster in the case of the stiff controller since the gain 2 kg-m . The reduced-attitude vector is denoted by a bold green line Kp that acts on the error is 25% larger than the nominal controller. from the origin to a point on the surface of the two-sphere. The On the other hand, the error converges more slowly for the damped path traced by the tip of the reduced-attitude vector on the two- controller where the gain Kp is 25% lower than the nominal case. sphere at several instants during the maneuver time T 5 5 s is While the stiff controller yields faster convergence, the angular given by a bold red line. The principal axis of rotation, which is velocity as shown in (b) has larger transients in the case of the stiff inertially fixed, is given by the blue line. The indicated axes are the controller compared to the nominal or damped controller. Further- body axes and not the inertial axes. (a) shows the initial reduced more, as shown in (c), the magnitude of the torque for the stiff attitude, (b) is the reduced attitude at time t 5 2.5 s, and (c) is the controller is higher than the nominal controller, while the magni- final reduced attitude at time t 5 5 s. tude of the control torque is lower for the damped controller.

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 47 if va is not zero, then, whether or not e50, the initial con- 2D-Unstable ditions (53), (54) lie close to the stable manifold but never Manifold 1 2 exactly at the equilibrium Rg, 0 . In the simulations, we choose d51024. vb vb In Figure 6, we plot the scalar attitude error given by the eigenangle (52) for four initial conditions corresponding to ε 4D-Stable four different values of e in (53) and (54) given by v Manifold δ a e [ 51025, 1024, 1023, 10226. As shown in Figure 6, the time of convergence increases with decreasing values of e. Fur- (Rγ , 0) thermore, the increase in the transient period occurs when the attitude error is near 180°, indicating that the solutions 1 2 remain close to the unstable equilibrium Rg, 0 for an extended period. This behavior demonstrates that the FIGURE 5 Initial conditions to illustrate closed-loop performance closer the solution starts to the complement of the domain limitations. The unstable equilibrium (Rg, 0) has a four-dimen- 1 2 sional stable manifold and a two-dimensional unstable manifold. of attraction of the desired equilibrium Rd, 0 , the longer it 1 2 The vectors va and vb are tangent to the stable and the unstable takes for the solution to converge to Rd, 0 . manifolds, respectively, at (Rg, 0). The initial conditions are pertur- bations from (Rg, 0) such that, for e 5 0, the initial condition lies Reduced-Attitude Stabilization close to the stable manifold, and, as increases, the initial condi- e We construct a feedback controller that asymptotically tion lies farther away from the stable manifold. A fixed nonzero d guarantees that the initial condition is near but not exactly at the aligns the third principal axis of the body frame with the 1 2 equilibrium (Rg, 0) for every sufficiently small e. third axis of the inertial frame. Thus, b 5 0, 0, 1 , and the 1 2 goal is to asymptotically stabilize Gd 5 0, 0, 1 , that is, to ^ asymptotically align G5R b with Gd. The feedback con- troller, given by (48), is chosen to have the gains 180 K 5 diag15, 10, 152 and k 5 4. 160 ε = 1e−5 v p ε 140 = 1e−4 The closed-loop eigenvalues of the linearized closed- ε = 1e−3 1 2 5 120 ε = 1e−2 loop system at Gd, 0 are given by 20.8333 6 0.7993i, 6 ) 20.5, 22, 23 , thereby describing the local conver- ° 100 (

Θ 80 gence rate near the desired equilibrium. There also 1 2 60 exists one additional equilibrium given by 2Gd, 0 40 that is unstable. 20 We refer to the gains given above as defining a nominal 0 0 5 10 15 20 25 30 35 40 controller. Two alternative sets of gains define a stiff con- Time (s) troller and a damped controller. The stiff gains are obtained by increasing kp and decreasing Kv by 25% each. In a similar FIGURE 6 The error U(t) for the nominal controller for four initial manner, the damped gains are obtained by decreasing kp conditions corresponding to four different values of e in (53) and and increasing Kv by 25% each. In each case, the desired 1 2 (54). The time for the error to converge to zero increases as e is reduced-attitude equilibrium Gd, 0 is almost globally decreased. Most of the increase in the transient time occurs in the asymptotically stable on S2 3 R3. early part of the trajectory, where the scalar error is close to 180°. The three sets of gains yield nominal, stiff, and damped This behavior indicates that, as e decreases, the trajectory dwells closed-loop responses locally near the desired equilib- close to the unstable equilibrium (Rg, 0) and hence close to the error of 180°, since the solution starts closer to the four-dimen- rium. Closed-loop responses are plotted in Figure 7 for the sional stable manifold as illustrated in Figure 5. However, since same initial conditions as above for the full-attitude case. the initial condition does not lie on the stable manifold, it soon Thus the initial condition for the reduced attitude is diverges along the two-dimensional unstable manifold, finally con- 1 2 1 2 ^ 1 2 G 0 5 R 0 e3 5 0.5, 0.7071, 0.5 . The error in the reduced verging to the desired equilibrium (Rd, 0). The divergence along the unstable manifold results in a sharp drop in the attitude error attitude is to zero as shown above. The smaller the parameter e, the longer 1 2 21 1 ^ 1 22 the trajectories dwell close to the unstable equilibrium (Rg, 0). U t 5 cos Gd G t , (55)

which is a scalar measure of the error between G and Gd rep- (53), (54) lie close to the stable manifold of the unstable resenting the angle between the two unit vectors. Thus, equilibrium. As e increases, the distance to the stable U102 5 60°. From Figure 7(a), we see that, compared to the 1 2 manifold of the unstable equilibrium Rg, 0 increases. nominal controller, the stiff controller is faster in reducing Therefore, as e decreases, we expect the time of conver- the error to zero, while the damped controller is slower in its gence to the desired attitude Rd 5 I to increase. Note that response. However, as shown in Figure 7(b) and (c), the stiff

48 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 controller requires a higher control torque and also leads to larger transient angular velocities compared to the nominal 60 controller, while the damped controller requires less control Nominal Damped torque and has lower transient angular velocities. Stiff To illustrate performance limitations in reduced-atti- 40 )

tude feedback, we consider the linearization of the closed- ° (

loop system for the nominal controller about the unstable Θ 1 2 equilibrium 2Gd, 0 . It can be shown that the linearized 20 closed-loop dynamics are described by the positive eigen- values 10.3508, 0.59072 and the negative eigenvalues 1 2 22.2573, 22.8508, 23 . These eigenvalues indicate that 0 the equilibrium at 12G , 02 is unstable, and the stable man- 0510 d Time (s) ifold has dimension three, while the unstable manifold is (a) two dimensional. The eigenspaces corresponding to the negative and positive eigenvalues are tangent to the stable 0 /s)

1 2 °

and unstable manifolds at the equilibrium 2Gd, 0 . (

1 −20

To demonstrate performance limitations, we choose a ω −40 family of initial conditions that lie close to the stable mani- 02468 10 1 2 20 fold of the unstable equilibrium 2Gd, 0 . We choose /s) 3 4 3 4 ° 10 va 5 DUa Dva and vb 5 DUb Dvb as a linear combination ( 2 0 of eigenvectors from the stable and unstable eigenspaces, ω −10 1 2 02468 10 respectively, of the equilibrium 2Gd, 0 , where 1

/s) Nominal °

0 0 ( 0 Damped 3 Stiff £ § £ § ω DUa 5 0 , Dva 5 0 , −1 2 0.3162 0.9487 02468 10 Time (s) 0.8610 0.5086 (b) DU 5 £ 0 § , Dv 5 £ 0 § . b b 10 0 0 Nominal 8 Damped Stiff Consider the family of initial conditions 6 1 2 1 3 3 21 2 G 0 5 exp 2dDUa 2eDUb 2Gd , (56) 4 1 2 v 0 5dDva 1eDvb, (57) 2 which depend on the parameter e. of u (N-m) Two−Norm V If we choose 0 , d 1, then, for e50, the initial condi- 0 tions (56), (57) lie close to the stable manifold of the unsta- 0510 1 2 Time (s) ble equilibrium 2Gd, 0 . As e increases, the distance to the 1 2 (c) stable manifold of the unstable equilibrium 2Gd, 0 e increases. Therefore, as decreases, we expect the time of FIGURE 7 (a) The angle between the desired and actual reduced- convergence to the desired attitude Gd to increase. Note attitude vectors, (b) angular velocity components along each that, if va is not zero, then, whether or not e50, the initial body-fixed axis, and (c) the two-norm of the control torque plotted conditions (56), (57) lie close to the stable manifold but for three controllers, namely, a nominal controller, a stiff controller, never exactly at the equilibrium 12G , 02. In the simula- and a damped controller. As shown in (a), the error converges to d zero faster in the case of a stiff controller since the gain k that acts d5 24 p tions, we choose 10 . on the error is 25% larger than the nominal controller. On the other In Figure 8, we plot the scalar reduced-attitude error hand, the error converges more slowly for the damped controller,

defined in (55) for four initial conditions corresponding to where the gain kp is 25% lower than the nominal case. While the values of e in (56) and (57) given by e [ 51025, 1024, stiff controller yields faster convergence, the angular velocity as 1023, 10226. As shown in Figure 8, the time of convergence shown in (b) has larger transients in the case of the stiff controller compared to the nominal and damped controllers and hence can e increases as decreases. Furthermore, the increase in the excite unmodeled dynamics. Furthermore, as shown in (c), the transient period occurs when the reduced-attitude error is magnitude of the torque for the stiff controller is higher than the near 180°, indicating that the solutions remain close to the nominal controller, while the magnitude of the control torque is 1 2 lower for the damped controller. unstable equilibrium 2Gd, 0 for an extended period. This

JUNE 2011 « IEEE CONTROL SYSTEMS MAGAZINE 49 the domain of attraction lies in a lower dimensional set of 180 the state space that is nowhere dense. Some features of the ε = 1e−5 160 ε = 1e−4 geometry of the complement of the domain of attraction of 140 ε = 1e−3 ε = 1e−2 the desired equilibrium are described, identifying it as the 120 union of the undesired closed-loop equilibria and their )

° 100

( associated stable manifolds. A key aspect of the analysis

Θ 80 given in this article is that we perform both local and global 60 40 analysis of the feedback controllers without resorting to 1 2 S2 20 coordinates for SO 3 or . 0 0 5 10 15 20 25 30 35 40 ACKNOWLEDGMENTS Time (s) We appreciate the many conversations and suggestions made by Taeyoung Lee and Melvin Leok, who helped to FIGURE 8 The error U(t) for the nominal controller for four initial clarify some of the attitude control issues addressed in this conditions corresponding to four different values of e in (56) and (57). The time for the error to converge to zero increases as e is article. We also thank the reviewers for suggestions that decreased. Most of this increase in the transient time occurs in the helped to improve the presentation of this article. early part of the trajectory, where the scalar error is close to 180°. This behavior indicates that, as e decreases, the trajectory dwells AUTHOR INFORMATION close to the unstable equilibrium (2Gd, 0), and hence close to the Nalin A. Chaturvedi ([email protected]) is error of 180°, since the solution starts closer to the three-dimen- sional stable manifold. However, since the initial condition does a senior research engineer at the Research and Technol- not lie on the stable manifold, it soon diverges along the two- ogy Center of the Robert Bosch LLC, Palo Alto, California. dimensional unstable manifold, finally converging to the desired He received the B.Tech. and the M.Tech. in aerospace engi- reduced-attitude equilibrium (Gd, 0). The divergence along the neering from the Indian Institute of Technology, Bombay, unstable manifold results in a sharp drop in the attitude error to in 2003, receiving the Institute Silver Medal. He received zero as shown above. The smaller the parameter e, the longer the trajectories dwell close to the unstable equilibrium manifold. the M.S. in mathematics and the Ph.D. in aerospace engi- neering from the University of Michigan, Ann Arbor, in example illustrates that the closer the solution starts to the 2007. He was awarded the Ivor K. McIvor Award in recog- complement of the domain of attraction, the longer it takes nition of his research in applied mechanics. His current 1 2 to converge to Gd, 0 . interests include the development of model-based control for complex physical systems involving thermal-chemical- CONCLUSIONS fluid interactions, nonlinear dynamical systems, state and This article summarizes global results on attitude control parameter estimation, and adaptive control with applica- and stabilization for a rigid body using continuous time- tions to systems governed by PDE, nonlinear stability the- invariant feedback. The analysis uses methods of geomet- ory, geometric mechanics, and nonlinear control. He is a ric mechanics based on the geometry of the special member of the Energy Systems subcommittee within the orthogonal group SO132 and the two-sphere S2. Mechatronics technical committee of the ASME Dynamic We construct control laws for full- and reduced-atti- Systems and Controls Division and an associate editor on tude maneuvers for arbitrary given boundary condi- the IEEE Control Systems Society Editorial Board. He is a tions and an arbitrary maneuver time. These results visiting scientist at the University of California, San Diego. demonstrate controllability properties for both the full- He can be contacted at Robert Bosch LLC, Research and and for reduced-attitude models, assuming full actua- Technology Center, 4005 Miranda Avenue, Suite 200, Palo tion. The open-loop controllers have a discontinuous Alto, CA 94304 USA. dependence on the boundary conditions, thus suggest- Amit K. Sanyal received the B.Tech. in aerospace engi- ing the impossibility of global stabilization using con- neering from the Indian Institute of Technology, Kanpur, tinuous feedback. in 1999. He received the M.S. in aerospace engineering Both full- and reduced-attitude feedback stabilization from Texas A&M University in 2001, where he obtained results are provided, assuming full actuation. Since no con- the Distinguished Student Masters Research Award. He tinuous time-invariant feedback controller can achieve received the M.S. in mathematics and Ph.D. in aerospace global asymptotic stabilization of an equilibrium, global engineering from the University of Michigan in 2004. properties of the closed-loop attitude dynamics using the From 2004 to 2006, he was a postdoctoral research asso- continuous feedback controllers (22) and (48) are character- ciate in the Mechanical and Aerospace Engineering De- ized. The particular feedback controllers given here are partment at Arizona State University. From 2007 to 2010, globally defined and continuous, and they achieve almost- he was an assistant professor in mechanical engineering global full-attitude stabilization or almost-global reduced- at the University of Hawaii and is currently an assistant attitude stabilization in the sense that the complement of professor in mechanical and aerospace engineering at

50 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2011 New Mexico State University. His research interests are ics and Control of Multibody Systems, J. E. Marsden, P. S. Krishnaprasad, and J. in geometric mechanics, geometric control, discrete vari- C. Simo, Eds. Providence, RI: American Mathematical Society. [17] S. Bharadwaj, M. Osipchuk, K. D. Mease, and F. Park, “A geometric ap- ational mechanics for numerical integration of mechani- proach to global attitude stabilization,” in Proc. AIAA/AAS Astrodynamics cal systems, optimal control and estimation, geometric/ Specialist Conf., San Diego, 1996, pp. 806–816. algebraic methods applied to nonlinear systems, space- [18] S. Bharadwaj, M. Osipchuk, K. D. Mease, and F. C. Park, “Geometry and inverse optimality of global attitude stabilization,” J. Guid. Control Dyn., craft guidance and control, and control of unmanned vol. 21, no. 6, pp. 930–939, 1998. vehicles. He is a member of the IEEE Control Systems [19] N. A. Chaturvedi, F. Bacconi, A. K. Sanyal, D. S. Bernstein, and N. H. Society Technical Committee on Aerospace Control and McClamroch, “Stabilization of a 3D rigid pendulum,” in Proc. American Control Conf., Portland, OR, 2005, pp. 3030–3035. the AIAA Guidance, Navigation, and Control Technical [20] N. A. Chaturvedi and N. H. McClamroch, “Global stabilization Committee. of an inverted 3D pendulum including control saturation effects,” in N. Harris McClamroch received the Ph.D. in engineer- Proc. 45th IEEE Conf. Decision and Control, San Diego, CA, Dec. 2006, pp. 6488–6493. ing mechanics from the University of Texas at Austin. [21] N. A. Chaturvedi, N. H. McClamroch, and D. S. Bernstein, “Stabiliza- Since 1967 he has been with the University of Michigan, tion of a specified equilibrium in the inverted equilibrium manifold of the Ann Arbor, Michigan, where he is a professor emeritus 3D pendulum,” in Proc. American Control Conf., 2007, pp. 2485–2490. [22] N. A. Chaturvedi and N. H. McClamroch, “Asymptotic stabilization in the Department of Aerospace Engineering. During the of the hanging equilibrium manifold of the 3D pendulum,” Int. J. Robust past 25 years, his primary research interest has been in Nonlinear Control, vol. 17, no. 16, pp. 1435–1454, 2007. nonlinear control. He has worked on control engineering [23] N. A. Chaturvedi, N. H. McClamroch, and D. S. Bernstein, “Asymptotic smooth stabilization of the inverted 3D pendulum,” IEEE Trans. Automat. problems arising in robotics, automated systems, and aero- Contr., vol. 54, no. 6, pp. 1204–1215, 2009. space flight systems. He is a Fellow of the IEEE, received [24] W. Kang, “Nonlinear H-infinity control and its application to rigid the IEEE Control Systems Society Distinguished Member spacecraft,” IEEE Trans. Automat. Contr., vol. 40, no. 7, pp. 1281–1285, 1995. [25] A. K. Sanyal and N. A. Chaturvedi, “Almost global robust attitude Award, and is a recipient of the IEEE Third Millennium tracking control of spacecraft in gravity,” in Proc. AIAA Conf. Guidance, Medal. He has served as an associate editor and editor of Navigation, and Control, Honolulu, HI, Aug. 2008, AIAA-2008-6979. IEEE Transactions on Automatic Control, and he has held [26] A. K. Sanyal, A. Fosbury, N. A. Chaturvedi, and D. S. Bernstein, “Iner- tia-free spacecraft attitude trajectory tracking with disturbance rejection numerous positions in the IEEE Control Systems Society, and almost global stabilization,” AIAA J. Guid. Control Dyn., vol. 32, pp. including president. 1167–1178, 2009. [27] N. A. Chaturvedi and N. H. McClamroch, “Asymptotic stabilization of the inverted equilibrium manifold of the 3D pendulum using non-smooth feedback,” IEEE Trans. Automat. Contr., vol. 54, no. 11, pp. 2658–2662, 2009. REFERENCES [28] J. L. Junkins and J. D. Turner, Optimal Spacecraft Rotational Maneuvers. [1] M. D. Shuster, “A survey of attitude representations,” J. Astronaut. Sci., New York: Elsevier, 1986. vol. 41, no. 4, pp. 439–517, 1993. [29] M. J. Sidi, Spacecraft Dynamics and Control—A Practical Engineering Ap- [2] J. Stuelpnagel, “On the parameterization of the three-dimensional rota- proach. Cambridge, U.K.: Cambridge Univ. Press, 2000. tion group,” SIAM Rev., vol. 6, no. 4, pp. 422–430, 1964. [30] J. Shen, A. K. Sanyal, N. A. Chaturvedi, D. S. Bernstein, and N. H. Mc- [3] R. E. Mortensen, “A globally stable linear attitude regulator,” Int. J. Con- Clamroch, “Dynamics and control of a 3D pendulum,” in Proc. IEEE Conf. trol, vol. 8, no. 3, pp. 297–302, 1968. Decision and Control, Bahamas, 2004, pp. 323–328. [4] B. Wie and P. M. Barba, “Quaternion feedback for spacecraft large angle [31] N. A. Chaturvedi, T. Y. Lee, M. Leok, and N. H. McClamroch, “Non- maneuvers,” AIAA J. Guid. Control Dyn., vol. 8, no. 3, pp. 360–365, 1985. linear dynamics of the 3D pendulum,” J. Nonlinear Sci., vol. 21, no. 1, pp. [5] B. Wie, H. Weiss, and A. Arapostathis, “Quaternion feedback regulator for 3–32, 2011. spacecraft eigenaxis rotation,” J. Guid. Control Dyn., vol. 12, pp. 375–380, 1989. [32] N. A. Chaturvedi, N. H. McClamroch, and D. S. Bernstein, “Stabiliza- [6] J. T. Wen and K. K. Delgado, “The attitude control problem,” IEEE Trans. tion of a 3D axially symmetric pendulum,” Automatica, vol. 44, no. 9, pp. Automat. Cont., vol. 36, no. 10, pp. 1148–1162, 1991. 2258–2265, 2008. [7] R. Bach and R. Paielli, “Linearization of attitude-control error dynam- [33] P. Tsiotras, M. Corless, and J. Longuski, “A novel approach for the atti- ics,” IEEE Trans. Automat. Cont., vol. 38, no. 10, pp. 1521–1525, 1993. tude control of an axisymmetric spacecraft subject to two control torques,” [8] S. M. Joshi, A. G. Kelkar, and J. T.-Y. Wen, “Robust attitude stabilization Automatica, vol. 31, no. 8, pp. 1099–1112, 1995. of spacecraft using nonlinear quaternion feedback,” IEEE Trans. Automat. [34] A. M. Bloch, Nonholonomic Mechanics and Control. New York: Springer- Cont., vol. 40, no. 10, pp. 1800–1803, 1995. Verlag, 2003. [9] S. P. Bhat and D. S. Bernstein, “A topological obstruction to continuous [35] V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations. global stabilization of rotational motion and the unwinding phenomenon,” New York: Springer-Verlag, 1984. Syst. Control Lett., vol. 39, no. 1, pp. 63–70, 2000. [36] R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Ro- [10] P. C. Hughes, Spacecraft Attitude Dynamics. New York: Wiley, 1986. botic Manipulation. Boca Raton, FL: CRC Press, 1994. [11] B. Wie, Spacecraft Vehicle Dynamics and Control. Reston, VA: AIAA, 1998. [37] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. New [12] G. Meyer, “Design and global analysis of spacecraft attitude control York: Springer-Verlag, 2005. systems,” NASA Tech. Rep. R-361, Mar. 1971. [38] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Sys- [13] P. E. Crouch, “Spacecraft attitude control and stabilization: Applica- tems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983. tions of geometric control theory to rigid body models,” IEEE Trans. Au- [39] C. G. Mayhew, R. G. Sanfelice, and A. R. Teel, “Robust global asymp- tomat. Contr., vol. 29, no. 4, pp. 321–331, 1984. totic attitude stabilization of a rigid body by quaternion-based hybrid feed- [14] F. Bullo, R. M. Murray, and A. Sarti, “Control on the sphere and reduced back,” in Proc. 48th IEEE Conf. Decision and Control, Shanghai, Dec. 2009, pp. attitude stabilization,” in Proc. IFAC Symp. Nonlinear Control Systems, Tahoe 2522–2527. City, CA, 1995, vol. 2, pp. 495–501. [40] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice-Hall, [15] P. Tsiotras and J. M. Longuski, “Spin-axis stabilization of symmetrical 2002. spacecraft with two control torques,” Syst. Control Lett., vol. 23, no. 6, pp. [41] J. Cortés, “Discontinuous dynamical systems: A tutorial on solutions, 395–402, 1994. nonsmooth analysis, and stability,” IEEE Control Syst. Mag., vol. 28, no. 3, [16] D. E. Koditschek, “The application of total energy as a Lyapunov func- pp. 36–73, 2008. tion for mechanical control systems,” in Proc. AMS-IMS-SIAM Joint Sum- mer Research Conf., Bowdin College, Maine, 1988, vol. 97, pp. 131–157. Dynam-

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