WHILE the traditional approach to attitude control is based

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JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 4, No., January February Adaptive Control of Nonlinear Attitude Motions Realizing Linear Closed Loop Dynamics Hanspeter Schaub Sandia National Laboratories,
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JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 4, No., January February Adaptive Control of Nonlinear Attitude Motions Realizing Linear Closed Loop Dynamics Hanspeter Schaub Sandia National Laboratories, Albuquerque, New Mexico 8785 Maruthi R. Akella University of Texas at Austin, Austin, Texas 787 and John L. Junkins Texas A&M University, College Station, Texas 7784 An adaptive attitude control law is presented to realize linear closed loop dynamics in the attitude error vector. The Modi ed Rodrigues Parameters (MRPs) are used along with their associated shadow set as the kinematic variables since they form a nonsingular set for all possible rotations. The desired linear closed loop dynamics can be of either PD or PID form. Only a crude estimate of the moment of inertia matrix is assumed to be known. A nonlinear control law is developed which yields linear closed loop dynamics in terms of the MRPs. An adaptive control law is then developed that enforces these desired linear closed loop dynamics in the presence of large inertia and external disturbance model errors. Because the unforced closed loop dynamics are nominally linear, standard linear control methodologiessuch as pole placement can be employed to satisfy design requirements such as control bandwidth. The adaptive control law is shown to track the desired linear performance asymptotically without requiring apriori knowledge of either the inertia matrix or external disturbance. Introduction WHILE the traditional approach to attitude control is based on linear control theory, recent efforts by several authors indicate a shift toward nonlinear control methods. For example, Wie and Barba and Wei et al. develop the rotational equations of motion using the redundant set of Euler parameters. In contrast, Dwyer, 4 outlinesan approachbased on a minimal set of three Euler parameters wherein a nonlinear transformation maps the complete equations of motion into a locally valid linear model that may encounter singular attitudes. It is a well known fact that every threeparameter attitude representation has the problem of singularities. The work of Slotine and Li based on Euler angles also has the same limitation. 5 To address the problem of singular orientations while using a minimal set of three rigid body attitude coordinates, more recently the Modi ed Rodrigues Parameters (MRPs) have been proposed. Any rigid body orientation can be described through two numerically distinct sets of MRPs which abide by the same differential kinematic equation. By switching between the original and alternate sets of MRPs (also referred to as the shadow set), it is possible to achieve a globally nonsingular attitude parameterization for all possible 6 deg rotations. 6 9 Given this advantage, there have been several recent attitude control applications employing MRPs as rotational kinematic variables. A common feature within all these efforts and other developments by Wen and Kruetz-Delgado 4, Wen et al., 5 Meyer, 6,7 Reyhanoglu et al., 8 and Slotine and Li 5 is the control law that is based on a stability analysis driven by an associatedlyapunov analysis. Although such attitude feedback control laws can be found by rst de ning a candidate Lyapunov function and then extracting the corresponding stabilizing nonlinear control, certain Received 9 October 999; revision received 8 February ; accepted for publication 7 April. Copyright c by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Research Engineer, Sandia National Laboratories, Mail Stop-; Member AIAA. Assistant Professor, Department of Aerospace Engineering and Engineering Mechanics; Member AIAA. George J. Eppright Professor, Aerospace Engineering Department; Fellow AIAA. 95 very important concepts from linear control theory, such as closed loop damping and bandwidth, are not very well de ned because the correspondingclosed loop dynamics are generally nonlinear. To achieve a desired closed loop behavior, the closed loop dynamics are linearized about a reference motion in order to use linear control theory techniques to pick the feedback gains. Depending on the nonlinearity of the exact closed loop equations of motion, the desired closed loop performance will be achieved only in a local neighborhood and not globally. Instead of rst nding a feedback control law and then analyzing the closed loop dynamics stability, it is possible to start out instead with a desired (or prescribed) set of stable closed loop dynamics and then extract the corresponding nonlinear control law using a feedback linearization approach 4, 4 common in robotics path planning problems. For example, the closed loop dynamics could be a stable linear differential equation. This technique is very general and can be applied to a multitude of systems. However, depending on the nonlinearity of the dynamical system, the nonlinear control laws extracted from such a feedback linearization approach can be potentially very complex. Paielli and Bach 9 present such an attitude control law derived in terms of the Euler parameter components, and that law is remarkably simple. Compared to standard Lyapunov functionderivedattitudecontrollaws, their control law expressionis only slightly more complex. Further, Paielli and Bach illustrate that this type of control law is rather robust for attitude control problems. However, this controllaw feeds back the Gibbs vector 7 as an attitude measure that is singularat 8 deg (error) rotationsabout any axis. As an important contribution of this paper, we develop a feedback linearizing control law based on the MRP vector that achieves the desired set of stable closed loop trajectories without encountering singular orientations. This paper also addresses the issue of uncertainty in the moment of inertia matrix. Even if the attitude control law (based on some nominal value of inertia) is robust with respect to inertia uncertainties, the closed loop dynamics will no longer exhibit the desired performanceif an incorrectinertia matrix is used in the feedback control law. While the inertia matrix is assumed to be essentially unknown in this development, the goal is to ensure that the feedback control law would still produce the desiredclosed loop dynamics. To accomplishthis task, time-varyingupdate laws for the feedback gain matrices are developed that ensure stability for the dynamics of the system adaptively. While classical adaptive control theory due to Narendra and Sastry has also been employed 96 SCHAUB, AKELLA, AND JUNKINS in attitude control problems previously,,, 4 the present MRP vector-basedfeedback linearizationapproach is unique in the sense that it explicitly enables linear closed loop dynamics to be chosen and motivated by useful physical concepts such as damping ratio and loop bandwidth. The paper rst developsall the theorynecessaryto developthe inverse dynamics approach to obtaining stable closed loop rigid body dynamics. In particular, the MRPs are chosen as the attitude parameters. An adaptive control law is presented that includes an integral feedback term in the desired closed loop dynamics and achieves asymptotic stability even in the presence of a class of unmodeled external disturbances. These results are illustrated through various numerical simulations. Linear Closed Loop Dynamics The MRP vector ¾ is adopted as a rigid body attitude measure relative to the target attitude. Note that the vector ¾ contains information about both the principal rotation axis ê and the principal rotation angle U because they are related through ¾ = ê tan(u /4) () Therefore, if ¾!, then the orientation has returned back to the origin. As a complete revolution is performed (i.e. U! 6 deg), this particular MRP set goes singular. As is shown in Refs. 6 and 8, it is possible to map the original MRP vector ¾ to its corresponding shadow counterpart¾ S through ¾ S = (/ r )¾ () where the notation r = ¾ T ¾ is used. By choosing to switch the MRPs whenever r , the MRP vector remains bounded within a unit sphere. Note that there exists no theoretical restriction that MRP vector switching should take place only on the surface of the three-dimensional unit sphere. Switching when the r = surface is penetrated results in the corresponding MRPs always indicating the shortest rotational distance back to the origin. 6,5 Let s assume that we desire the closed loop dynamics to have the following prescribed linear form ¾ + P Ǿ + K ¾ = () where the scalars P and K are the positive velocity and position feedback gains. Observe that both P and K could be chosen to be symmetric, positive de nite matrices. However, doing so greatly complicatesthe resultingalgebra.note that this differentialequation only contains kinematic quantities, and there is no explicit dependence on system properties such as inertia. Linear control theory states that, for any initial ¾ and Ǿ vectors, the resulting motion is asymptotically stable. If desired, one could also easily add an integral feedbackterm with an appropriatelychosengain value K i to the desired closed loop equations and still retain asymptotic stability ¾ + P Ǿ + K ¾ + K i ¾ dt = (4) Note that instead of the MRP vector ¾, any attitude or position vector could have been used. In particular, Paielli and Bach chose to express their linear closed loop equations in terms of the vector components of the Euler parameters. 9 Let the vector u be an external control torque vector which is applied to a rigid body with the inertia matrix [I]. The vector F e is the unmodeled torque vector due to in uences such as atmospheric or solar drag or bearing friction. The vector! is the body angular velocity vector. Euler s rotational equations of motion state that [I] Ç! + [!][I ]! = u + F e (5) where the tilde matrix [!] is the vector cross product operator de- ned as x x! = 4 x x 5 (6) x x It is desired to nd a nonlinear control law u that will render the closed loop dynamics to be of the stable form in Eq. () or (4), assuming the system inertia matrix is perfectly known. To achieve this, we treat the body angular acceleration vector Ç! as the control variable in the following development.once the necessary vector Ç! is found,then the physicalcontroltorqueis found througheq. (5). To extract Ç! from eithereq. () or (4), all velocitiesand accelerationsin these closed loop equations must be expressed in terms of the body angular velocity vector. Assume the target attitude is stationary. Then, the MRP kinematic differential equations can be written as are 6 9 Ǿ = [B(¾)]! (7) 4 where the matrix [B] =[B(¾)] is conveniently expressed as 6,7 [B] = ( ¾ T ¾) I + [ ¾] + ¾¾ T (8) with the skew-symmetric matrix operator being de ned in Eq. (6). Differentiating the MRP kinematic differential equation in Eq. (7) we nd ¾ = 4 [B] Ç! + 4 [ ÇB]! (9) Substituting Eqs. (7) and (9) into the desired linear closed loop dynamics in Eq. (), the following constraint condition is found. ¾ + P Ǿ + K ¾ = = 4 [B][ Ç! + P! + [B] ([ ÇB]! + 4K ¾)] () The following algebra is greatly simpli ed by making use of the 5 7 explicit expression of the matrix inverse of [B] given by [B] = [/( + r ) ][B] T () This expression is valid for all nonsingular values of ¾ and can readily veri ed by using it to con rm that [B] [B] = I. More importantly,the matrix [B] is simply a scalar factor multiplied by [B] T. As suggestedby Bach in a privatecommunication,yet another interesting property of the [B] matrix is that the MRP Vector ¾ is an eigenvector and + r an eigenvalue of both [B] and [B] T. By virtue of switching between MRP and the shadow MRP sets, we have j ¾j, and the matrix [B] is always invertible. As a result, using Eq. (), the following expression must be true Ç! + P! + [B] ([ ÇB]! + 4K ¾) = () Eq. () yields the necessary Ç! term to calculate the actual torque vector u in Eq. (5). The vector Ç! is written as Ç! = P! [B] ([ ÇB]! + 4K ¾) = Á () where the expression of the right hand side of Eq. () is set equal to the new state vector Á. Using the vector product de nition of the [B] matrix in Eq. (8), the product [ ÇB]! is expressed as [ ÇB]x = ¾ T!( r )! ( + r )(x /)¾ ¾ T![!]¾ + (¾ T!) ¾ (4) where the shorthand notation x =! T! is used. The expression in Eq. (4) is obtained after considerable algebraic manipulations using the identities [ã]a = and [ã][ã] = aa T a T ai, any a R (5) Using this [ ÇB]! expression and Eq. () together, we obtain the following r x [B] ([ ÇB]! + 4K ¾) =!! T + [4K / ( + ) /]I ª ¾ (6) SCHAUB, AKELLA, AND JUNKINS 97 Making use of this result in Eq. (), the vector Á is nally given by the elegantly simple expression ª Á = P!!!T + [4K / ( + r ) x / ]I ¾ (7) Therefore the desired Linear Closed Loop Dynamics (LCLD) in Eq. () can be rewritten as ¾ + P Ǿ + K ¾ = [B]( Ç! Á) = (8) 4 Substituting Ç! = Á into Euler s rotational equations of motion in Eq. (5) yields the required nonlinear feedback control law vector u. u = [!][I]! + [I]Á F e (9) We remark that these developments are parallel to those in Ref. 9 where the three vector components of the Euler parameters are used instead of the MRP vector used in this paper. However, it can easily be seen that the singularityat 8 degrees is removed by using the MRP vector. It will also be recognized that this control law contains the inertia matrix [I ] linearly. When the inertia matrix is unknown, we cannot directly implement Eq. (9). In the following section, we develop an adaptive controller for such situations. An attractive component of this methodology when dealing with known system parameters is that the structure of the closed loop equationscan easily be modi ed usingstandardlinearcontroltheory techniques by appropriate choice of the constants P and K. If it is necessary that the feedback control reject external disturbances, an integral measure of the attitude error is added to the closed loop equations as shown in Eq. (4). Following similar steps as were done previously in this section, the linearizing body angular acceleration vector Ç! = Á for closed loop dynamics with an attitude integral measure are written as Á = P!!! T + 4K + r x I ¾ 4K i [B] ¾ dt () For this choice of Á, the correspondingphysical control vector u is of the same form as shown in Eq. (9). Adaptive Control Formulation While the vector Á is a kinematic quantity depending only on the state vectors ¾ and!, to compute the proper linearizing control vector u, the system inertia matrix[i ] and the externaltorque vector F e must be known precisely. In the following development it is assumed that only very crude estimates of the inertia matrix and external torque vector are known. In this case, the vector Á is no longer equal to Ç!, and the actual closed loop dynamics will not be linear. The following adaptive control law requires that the unknown states appear linearly in the control formulation. Therefore, we rewrite Eq. (9) as u = [L ] g + [M ]Á F e () where the matrices [L ] and [M ] are de ned as I I [L ] = 4 I I 5 () I I I I I I [L ] = 4 I I I I 5 () I I I I [L ] [L... L ], [M ] [I] (4) the vector F e is the true external torque vector, and the 6 vector g is de ned as g x T x x x x x x x x (5) The control vector expressionin Eq. () is rewritten by introducing the matrix [Q ] and the into the compact form [Q ] = L... M. F e (6) state vector x g x = 4 Á 5 (7) u = [Q ] x (8) Note that Eq. (8) still assumesthat all plant parametersare perfectly known. From here on, we assume that the inertia matrix and the external torque vector are not known precisely. The actual control vector u that is implemented is then given by u = [Q(t)] x (9) where [Q(t)] =[L(t)... M(t)... Fe (t)] contains the time-varying adaptive estimates of the unknown system parameters. The difference between the adaptive estimates and true system parameters is expressed through the matrix [ Q] as [ Q] [Q(t)] [Q ] () Assume that the desired LCLD are to be of the linear PID form given in Eq. (4), then the actual closed loop dynamics, due to the imperfect control vector u in Eq. (9), are found to be ¾ + P Ǿ + K ¾ + K i ¾ dt = [B]( Ç! Á) 4 = 4 [B][I ] [L ] g + u + F e [M ]Á = 4 [B][I ] ([Q(t)] x [Q ] x) = 4 [B][I] [ Q] x () A key feature of this method is that the desired LCLD do not depend on the unknown inertia matrix. This makes it possible to design a desired performance without any knowledge of the actual system parameters. The goal of the following adaptive control law is to nd learning laws for the inertia matrix quantities [L] and [M], and if necessary for the external torque vector F e, such that the actual closed loop dynamics asymptotically approaches the desired linear form. The main advantage of this control law is that standard linear feedback gain techniquescan be employed to nd appropriatefeedbackgains P, K, and K i that meet system requirements such as control bandwidth and performance. These quantities are typically dif cult to enforce with general nonlinear control laws. With the adaptation superimposed on the linearizing control law, we will be guaranteed that the desired closed loop performanceis achieved asymptotically, even in the presence of large parameteric uncertainty. Let the vector ¾ r be the solution of the differential equation ¾ r + P Ǿ r + K ¾ r + K i ¾ r dt = () where ¾ r (t ) = ¾(t ), and Ǿ r (t ) = Ǿ(t ). Thus the trajectory ¾ r (t) representsthe desiredclosedloop performance.any deviationsfrom this performance are assumed to be due to system model errors [ Q]. Let the augmented 9 state vector ² express the difference between the actual states and the reference states. ² R t (¾ ¾ r ) dt ¾ ¾ r A () Ǿ Ǿ r 98 SCHAUB, AKELLA, AND JUNKINS Using Eq. () and (), note that Dz is given by I Dz = 4 I 5² A K i I K I {z P I } {z} [ A] b with the vector being de ned as (4) Studying Eq. (44), it is evident that if we set the system parameter learning rate [ ÇQ] to be [ ÇQ] = [C 4 ] [I ] [B] T [S ]² x T [c ] (46) the Lyapunov rate function is guaranteed to be of the negative de - nite form ÇV = ² T [R]² (47) = 4 [B][I] [ Q] x (5) We then de ne the following positive de nite Lyapunov function V around the desired reference performance. V = ² T [S]² + tr([ Q] T [C ][ Q][c ] ) (6) where [S] and [C ] are yet to be determined positive de nite gain matrices,and[c ] is a diagonalmatrix containingthe variouslearning rates c i. Note that the trace operator in Eq. (6) can be written as tr([ Q] T [C ][ Q][c ] ) = X c i = i Q i Q i Q i T C B A [C Q i Q i Q i C A (7) whichis clearlya positivede nitefunctionin [ Q]. Takingthe derivative of Eq. (6) and using Eq. (4), we nd ÇV = ² T ([S][A] + [A] T [S])² + ² T [S] b + tr([ Q] T [C ][ Ç Q][c ] ) (8) By partitioning the 9 9 matrix [S] into three 9 sub-matrices [S i ], the Lyapunov rate ÇV is rewritten as [S] = [S... S... S ] (9) ÇV = ² T ([S][A] + [A] T [S])² + ² T [S ] + tr([ Q] T [C ][ Ç Q][c ] ) (4) Since [ A] is a stable matrix, Lyapunov s stability theorem for linear systems states that for any symmetric, positive de nite matrix [ R], we are guaranteed that there exists a correspondingsymmetric, positive de nite matrix [S] such that 8 [S][A] + [A] T [S] = [R] (4) Therefore, we can pick [R] and numerically solve for a corresponding positive de nite matrix [S] for a given stable matrix [A]. Using Eqs. (5) and (4), the Lyapunov rate ÇV is reduced to ÇV = ² T [R]² + ² T [S ] 4 [B][I ] [ Q]x + tr([ Q] T [C ][ Ç Q][c ] ) (4) Using severalmatrix identitieslisted in Ref. 9, it can be shown that 4 ²T [S ][B][I ] [ Q] x = tr x T [ Q] T [I ] [B] T [S 4 ] T ² = 4 tr [ Q] T [I] [B] T [S ] T ² x T (4) Using Eq. (4), the Lyapunov rate is expressed as ÇV = ² T [R]² + tr [ Q] T 4 [I] [B] T [S ]² x + [C ][ Ç Q][c ] (44) Assuming that the true external torque vector F e is constant, then [ Ç Q] = [ ÇQ] [ ÇQ ] = [ ÇQ] (45) Since ÇV in Eq. (47) is negative semide nite in the state vector, ² L. The adaptive system parameter estimate errors [ L], [ M] and F e are stable. Further, it may be shown
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