Optimal Variable-Structure Control racking of Spacecraft Maneuvers John L. Crassiis 1 Srinivas R. Vaali F. Lanis Markley 3 Introuction In recent years, much effort has been evote to the close-loop esign of spacecraft with large angle slews. Vaali an Junkins 1 an Wie an Barba erive a number of simple control schemes using quaternion an angular velocity (rate) feeback. Other full state feeback techniques have been evelope that are base on variable-structure (sliing-moe) control, which uses a feeback linearizing technique an an aitional term aime at ealing with moel uncertainty. A variable-structure controller has been evelope for the regulation of spacecraft maneuvers using a Gibbs vector parameterization, 3 a moifie-rorigues parameterization, 4 an a quaternion parameterization. 5 In both [] an [5], a term was ae so that the spacecraft maneuver follows the shortest path an requires the least amount of control torque. he variable-structure control approach using a quaternion parameterization has been recently expane to the attitue tracking case. 6,7 However, these controllers o not take into account the shortest possible path as shown in Refs. [] an [5]. his note expans upon the results in Ref. [5] to provie an optimal control law for asymptotic tracking of spacecraft maneuvers using variable-structure control. It also provies new insight using a simple term in the control law to prouce a maneuver to the reference attitue trajectory in the shortest istance. Controllers are erive for either external torque inputs or reaction wheels. 1 Assistant Professor, exas A&M University, Department of Aerospace Engineering, College Station, X 77843-3141. Senior Member AIAA. Professor, exas A&M University, Department of Aerospace Engineering, College Station, X 77843-3141. Associate Fellow AIAA. 3 Engineer, NASA-Goar Space Flight Center, Guiance, Navigation, an Control Center, Coe 571, Greenbelt, MD 0771. Fellow AIAA. 1
Backgroun In this section, a brief review of the kinematic an ynamic equations of motion for a threeaxis stabilize spacecraft is shown. he attitue is assume to be represente by the quaternion, q q, with q [ ] n ( ) an ( ) efine as º é 13 q ù ë 4 û 13 º q1 q q3 = ˆ sin F/ q 4 = cos F /, where ˆn is a unit vector corresponing to the axis of rotation an F is the angle of rotation. he quaternion kinematic equations of motion are erive by using the spacecraft s angular velocity (w ), given by where W( w ) an X() q are efine as [ w ] 1 1 q& = W ( w) q = X() q w (1) [ q ] é- M w ù éqi 4 3 3+ 13 ù ê ú ê ú W ( w) º ê LL M LLú, X () q º ê LLLLLL ú ê ú êë -w M 0 úû êë -q13 úû an I n n represents an n n ientity matrix. he 3 3 () imensional matrices [ w ] an [ q 13 ] are referre to as cross prouct matrices since a b= [ a ] b (see Ref. [8]). Since a three egree-of-freeom attitue system is represente by a four-imensional vector, the quaternion components cannot be inepenent. his conition leas to the following normalization constraint quaternions, q an q, is efine by 13 13 q4 1 q q = q q + =. Also, the error quaternion between two é q13ù -1 q º ê ú= qäq (3) êë q4 úû where the operator Ä enotes quaternion multiplication (see Ref. [8] for etails), an the inverse quaternion is efine by by =X ( ) q13 q q an q4 q 1 - = é-q -q -q q ù ë 1 3 4 q 4» 1, an q13 correspons to the half-angles of rotation. û. Other useful ientities are given = q q. Also, if Equation (3) represents a small rotation then he ynamic equations of motion, also known as Euler s equations, for a rotating spacecraft are given by
( J ) Jw& =- w w +u (4) where J is the inertia matrix of the spacecraft, an u is the total external torque input. If the spacecraft is equippe with 3 orthogonal reaction or momentum wheels, then Euler s equations become: ( J J) w& w ( Jw Jw) - = - + -u ( w w) (5a) J & + & =u (5b) where J is the iagonal inertia matrix of the wheels, J now inclues the mass of the wheels, w is the wheel angular velocity vector relative to the spacecraft, an u is the wheel torque vector. Selection of Switching Surfaces Optimal Control Analysis We first consier a sliing manifol for a purely kinematic relationship, with w as the control input. he variable-structure control esign is use to track a esire quaternion q an corresponing angular velocity w. As shown previously for regulation, 5 uner ieal sliing conitions, the trajectory in the state-space moves on the sliing manifol. For tracking, the following loss function is minimize to etermine the optimal switching surfaces: 1 P ( w ) = é r 13 13 + ( - ) ( - ) ù t ò êë q q w w w w úû (6) t s subject to the bilinear system constraint given in Equation (1). Note that r is a scalar gain an t s is the time of arrival at the sliing manifol. Minimization of Equation (6) subject to Equation (1) leas to the following two-point-bounary-value-problem: q& 1 = X () q w 1 l& =-rx( q) X ( q) q+ X( l) w (7a) (7b) where l is the co-state vector. he optimal w is given by 1 w =- X () q l + w (8) Next, the following sliing vector is chosen: 3
( ) k ( q ) w - w + X q= 0 (9) where k is a scalar gain. he sliing vector is optimal if the solution of Equation (9) minimizes Equation (6). his can be proven by first substituting Equation (9) into Equation (8), yieling k l =- q (10) Next, using the fact that the esire quaternion can be obtaine from the following leas irectly to 1 q& = X( q) w (11) ( ) l& =-kx q w (1) Comparing Equation (1) to Equation (7b), an using Equation (9) now leas to the following relationship: ( ) r ( ) ( ) ( ) ( ) ( ) -kx q w = - X q X q q-kx q w + k X q X q q (13) Equation (13) is satisfie for k an optimal solution (i.e., minimum P in Equation (6)). =± r. herefore, the sliing conition in Equation (9) leas to For this special case, it can be shown that the value of the loss function in Equation (6) is given by * P = ké ë1- q ts ù û (14) 4 ( ) where k must now be strictly positive. Since q4 correspons irectly to the cosine of half the angle error of rotation, both q an - q represent the same rotation; however, the value of the loss function in Equation (14) is significantly ifferent for each rotation. One rotation ( q) gives the shortest istance to the sliing manifol, while the other (- q) gives the longest istance. In orer to give the shortest possible istance the following sliing vector is chosen: For a iscussion on using é q ( t ) ( ) ksgn é q ( t ) ù ( ) w - w + ë 4 s ûx q q= 0 (15) ë 4 s ùû in a control law see Refs. [] an [5]. Using this sliing conition leas to the following value for the loss function: 4
P * = k é ë 1- q ts ù û 4 ( ) (16) which yiels a minimal value for any rotation. Lyapunov Analysis he sliing vector shown in Equation (15) can also be shown to be stable using a Lyapunov analysis. he time erivative of q13 can be shown to be given by 1 1 q& 13 = q4 w - w + 13 w + w ( ) [ q ]( ) (17) Next, the following caniate Lyapunov function is chosen: 1 V s = q13 q13 (18) Using the sliing vector in Equation (15), the time erivative of Equation (18) is given by 1 V & s =- k q4 q13 q13 0 (19) Hence, V s is inee a Lyapunov function for k > 0. his analysis generalizes the results shown in Ref. [7], where the spacecraft s attitue is restricte in the workspace efine with q 4 > 0. Variable-Structure racking ë 4 s ù û in the sliing manifol. In this section, we consier a variable-structure controller for the complete system he previous section showe the effectiveness of using sgn é q ( t ) (i.e., incluing the ynamics). he goal of the variable-structure controller is to track a esire quaternion q an corresponing angular velocity with external torques only is given by (note that sgn éë q4 () t ùû sgn é ë q t ù û ) 4 ( s) w. he variable-structure control esign is now being use instea of ì1 [ ] sgn ü u= w Jw + Jí k ( q4) éx () q X( q) w -X ( q) X () q w ù+ w& -GJ ý (0) î ë û þ where G is a 3 3 positive efinite, iagonal matrix, an the i th component of J is given by ( s e ) Ji = sat i, i, i = 1,,3 (1) 5
As state previously, the term sgn( q 4 ) is use to rive the system to the esire trajectory in the shortest istance. he saturation function is use to minimize chattering in the control torques. his function is efine by ì1 for si > ei ïs sat ( s, i i ei) º í for si ei i = 1,,3 ï e ï î - 1 for si <- ei () where e is a small positive quantity. he sliing manifol is given by ( ) ksgn( q ) ( ) s = w - w + X q q (3) he stability of the close-loop system using this controller can be evaluate using the following caniate Lyapunov function 1 V = s s (4) Using Equations (4), (0) an (3) the time-erivative of Equation (4) can be shown to be given 4 by V& =-s GJ, which is always less than or equal to zero as long as G is positive efinite. Hence, stability is proven. If wheels are use to control the spacecraft, then the sliing moe controller is given by the following: ì1 ( ) sgn ü u = J -J í k ( q4) éx ( q) X() q w -X () q X( q) w ù - w& + GJ ý î ë û þ [ w ]( Jw Jw) - + (5) he stability of the control system with wheels in Equations (5) an (5) can also be easily proven using the Lyapunov function in Equation (4). Analysis In this section an analysis of using sgn( q 4 ) for all times in the control law is shown. We first assume that the esire angular velocity is zero ( w = 0 ) an that the matrix G is given by a scalar times the ientity matrix ( gi 3 3). We further assume that the thickness of the bounary layer e an the gain g are sufficiently large so that 6
where b ( ) GJ = bw + b ksgn q4 q13 (6) = g e. Using Equations (4), (17) an (0), the close-loop ynamics for w now become w& =- 1 sgn ( ){ [ ]} sgn ( ) k q q I + q w - b w - b k q q 4 4 3 3 13 4 13 (7) aking two time erivatives of q4 = q q, an using both Equation (7) an the quaternion constraint equation q q = 1 yiels æ1 ö æ1 1 ö 1 q&& 4 + ç k q4 + b q& 4 + ç b k q4 + w w q4 = b ksgn( q4) (8) è ø è 4 ø Equation (8) represents a secon-orer nonlinear spring-mass-amper type system with an exogenous step input. he stability of Equation (8) can be evaluate by consiering the following caniate Lyapunov function 1 1æ1 ö 1 V q = q& ( ) 4 4 + ç w w q4 + b ké1-q4sgn q4 ù è4 ø ë û (9) he last term in Equation (9) is always greater or equal to zero since b > 0, k > 0, an ( q ) 0 q4sgn 4 1. aking the time-erivative of Equation (9), an using Equations (7) an (8) gives ( ) 1 1 1 V & æ ö æ ö q =- k q 4 ç 4 + b q4 - b + k q4 q è ç 4 ø & 4è ø w w (30) Hence, since b > 0 an k > 0, Equation (9) is inee a Lyapunov function. he avantage of using sgn( q 4 ) in the control law at all times (even before the sliing manifol is reache) now becomes clear. he step input in Equation (8) is a function of sgn( q 4 ). herefore, the response for q4 will approach ( ) sgn q 4 for any initial conition. his tens to rive the system to the esire location in the shortest istance. Furthermore, this inherently takes into account the rate errors as well. For example say that q ( t ) 4 0 is positive, an that a high initial rate is given which tens to rive the system away from q4 = 1. he control law will automatically begin to null the rate. But, if the initial rate is large enough an the control ynamics are relatively slow, then q4 may become negative. Since ( ) sgn q 4 is use in the 7
control system, then from Equation (8) the control law will subsequently rive the system towars q4 1 =-. herefore, using ( ) type of initial conition error. Example sgn q 4 at all times prouces an optimal response for any A variable-structure controller has been use to control the attitue of the Microwave Anisotropy Probe (MAP) spacecraft from quaternion observations an gyro measurements. Details of the spacecraft parameters an esire attitue an rates can be foun in Ref. [9]. he control system has been esigne to bring the actual attitue to the esire attitue in less than 0 minutes. For the simulations the spacecraft has been controlle using reaction wheels. he gains use in the control law given by Equation (5) are: k = 0.015, G = 0.0015 I 3 3, an e = 0.01. o illustrate the importance of using ksgn éë q4 () t ùû, a number of simulations have been run. he initial quaternion is given by q( t ) = é ( F ) ( F ) ù Äq ( t ) ë0 0 sin cos û 0 0 an the actual velocity has been set to zero. est cases have been execute using F= 10 o, 40 o, 70 o, 300 o, an 330 o. able 1 summarizes the results of using ksgn éë q4 () t ùû an k only in Equation (5). he final time for the simulation runs is given by t f = 0 minutes. in the control law, better performance is achieve in the close- Clearly, by using ksgn éë q4 () t ùû loop system than using just k. Conclusions A new variable-structure controller for optimal spacecraft tracking maneuvers has been shown. he new controller was formulate for both external torque inputs an reaction wheel inputs. Global asymptotic stability was shown using a Lyapunov analysis. New insight was shown for the avantage of using a simple term in the control law (i.e., proucing a maneuver to the reference attitue trajectory in the shortest istance). he sliing motion was also shown to be optimal in the sense of a quaratic loss function in the multiplicative error quaternions an angular velocities. Simulation results inicate that the aition of the simple term in the control law always provies an optimal response, so that the reference attitue motion is achieve in the shortest possible istance. 8
References [1] Vaali, S.R., an Junkins, J.L., Optimal Open-Loop an Stable Feeback Control of Rigi Spacecraft Attitue Maneuvers, he Journal of the Astronautical Sciences, Vol. 3, No., April-June 1984, pp. 105-1. [] Wie, B., an Barba, P.M., Quaternion Feeback for Spacecraft Large Angle Maneuvers, Journal of Guiance, Control an Dynamics, Vol. 8, No. 3, May-June 1985, pp. 360-365. [3] Dwyer,.A.W., an Sira-Ramirez, H., Variable Structure Control of Spacecraft Reorientation Maneuvers, Journal of Guiance, Control an Dynamics, Vol. 11, No. 3, May-June 1988, pp. 6-70. [4] Crassiis, J.L., an Markley, F.L., Sliing Moe Control Using Moifie Rorigues Parameters, Journal of Guiance, Control an Dynamics, Vol. 19, No. 6, Nov.-Dec. 1996, pp. 1381-1383. [5] Vaali, S.R., Variable-Structure Control of Spacecraft Large-Angle Maneuvers, Journal of Guiance, Control an Dynamics, Vol. 9, No., March-April 1986, pp. 35-39. [6] Robinett, R.D., an Parker, G.G., Least Squares Sliing Moe Control racking of Spacecraft Large Angle Maneuvers, Journal of the Astronautical Sciences, Vol. 45, No. 4, Oct.-Dec. 1997, pp. 433-450. [7] Lo, S.-C., an Chen, Y.-P., Smooth Sliing-Moe Control for Spacecraft Attitue racking Maneuvers, Journal of Guiance, Control an Dynamics, Vol. 18, No. 6, Nov.- Dec. 1995, pp. 1345-1349. [8] Shuster, M.D., A Survey of Attitue Representations, he Journal of the Astronautical Sciences, Vol. 41, No. 4, Oct.-Dec. 1993, pp. 439-517. [9] Crassiis, J.L., Vaali, S.R., an Markley, F.L., Optimal racking of Spacecraft Using Variable-Structure Control, Proceeings of the Flight Mechanics/Estimation heory Symposium, (Greenbelt, MD), NASA-Goar Space Flight Center, May 1999, NASA/CP- 1999-0935, pp. 01-14. 9
able 1 Cost Function Values for Various F Value of 1 t f ò s s 0 F (eg) Gain = k Gain = ksgn é q () t 10 1.0787 0.9313 40 0.8988 0.6543 70 0.6136 0.3566 300 0.3185 0.1300 330 0.1095 0.0194 t ë 4 ùû