# An Adaptive Attitude Control Formulation under Angular Velocity Constraints

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1 An Adaptive Attitude Control Formulation under Angular Velocity Constraints Puneet Singla and Tarunraj Singh An adaptive attitude control law which explicitly takes into account the constraints on individual angular velocity components has been developed for rigid body attitude control problem. Rigorous stability analysis is presented in the paper which guarantee the asymptotic stability of the controller. The performance of the control laws for stable, bounded tracking of attitude trajectories is evaluated. The essential ideas and results from computer simulations are presented to illustrate the performance of the controller developed in this paper. I. Introduction Attitude control is the process of re-orienting a rigid body to a desired attitude or orientation and plays an important role in many applications ranging from various space and air transportation missions (autonomous mid-air re-fueling of an aircraft, International Space Station (ISS) supply and repair, and space systems automated inspection, servicing and assembly), to the control of robotic manipulators. Some of these applications such as mid-air aircraft refueling and space system automated inspection requires very precise rotational maneuvers. These requirements frequently necessitate the use of non-linear rigid body dynamic models for control system design. The attitude motion of a rigid body can be well represented by Euler s equations, 2 for nonlinear relative angular velocity evolution and attitude parameter kinematic equations. The rigid body attitude can be represented by many coordinate choices, 3, 4 but the quaternion representation is an ideal choice for the attitude estimation as it is free of geometrical singularities which is a desirable property when representing large angle amplitude trajectories. Although attitude kinematic and Euler s dynamic equations represent a near-exact dynamical model, for control design purposes, complications may arise from uncertain rigid body inertia which can change due to fuel consumption, solar array deployment, payload variation etc. Furthermore, stability robustness due to model errors and disturbances are primary consideration for design of any autonomous control system. Rigid body attitude control problems have been studied extensively in the literature. 5 Ref. 2 presents a very detailed review of earlier work for rigid-body attitude control problem. In Refs. 6, 8, 9 optimal attitude control laws are presented and in Refs. 5, 6,, 3 Lyapunov analysis based adaptive attitude tracking control Assistant Professor, AIAA, AAS Member, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-426, Professor, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-426, of 2

2 schemes are presented to compensate for the unknown rigid body inertia matrix. Although much progress has been made in rigid body attitude control in the presence of the rigid body inertia matrix uncertainties and external disturbances, there are no means of incorporating constraints on individual angular velocity. Constraints on rigid body angular velocity might be required for many applications such as rendezvous of space shuttle with ISS and mid-air refueling of an aircraft. The main objective of this paper is to develop an adaptive attitude control law to compensate for any errors in inertia matrix which takes into account the constraints on individual components of rigid body angular velocity. The adaptive control formulation in this paper is based upon Lyapunov s direct stability theorem and imposes the exact kinematic equations at the velocity level while taking care of model uncertainties at the acceleration level. An important contribution of the paper is the explicit consideration of constraints on rigid body angular velocity. The structure of this paper is as follows. First, the dynamical models for rigid body rotational motion is set forth followed by the development of adaptive attitude control law. Finally, the controllers designed in this paper are tested using numerical simulations. II. Attitude Dynamics In this section, we set forth the nonlinear rigid body dynamics model that we adopt for attitude kinematics and rotational dynamics. This model is described in order to be specific in the further developments contained in this paper. Rigid body kinematics can be represented by many coordinate choices, 3 but we prefer to use quaternion representation for attitude control since it is free of all geometrical singularities and has linear kinematic differential equations. The attitude motion of a rigid body is represented by nonlinear Euler s equations for angular velocity evolution and quaternion kinematic equations as given below: Quaternion Kinematics: q = 2 Ω(ω)q = 2 B(q)ω Euler s Equations: I ω = u ωiω (a) (b) where, q R 4 is a vector of quaternion that parameterize the rigid body attitude with respect to an inertial frame and ω R 3 represents the rigid body angular velocity expressed in the rigid body frame. u R 3 represents the vector of external torques. Further, I R 3 3 represents the rigid body inertia matrix, and 2 of 2

3 ω R 3 3 is a skew-symmetric matrix given as: ω = ω 3 ω 2 ω 3 ω ω 2 ω (2) Finally, Ω(ω) = ω 3 ω 2 ω ω 3 ω ω 2 ω 2 ω ω 3 ω ω 2 ω 3, B(q) = q 4 q 3 q 2 q 3 q 4 q q 2 q q 4 q q 2 q 3 (3a) III. Controller Formulation In this section, the velocity bounded adaptive control law will be derived for attitude control, using Lyapunov s direct stability theorem. The novel feature of the control law developed in this paper is that it explicitly accounts for bounds on uncertain rigid body inertia matrix and rigid body angular velocity. First, we will develop nominal attitude control for bounded rigid body angular velocity and later we will generalize the controller for bounds on uncertain rigid body inertia matrix. A. Feedback Control Formulation For Attitude Motion In this section, we seek to design a feedback control law for the system described by Eqs. (a) and (b) to regulate the rigid body attitude parameterized by reference quaternion q f such that ω (t) k, ω 2 (t) k 2, ω 3 (t) k 3, t (4) For this purpose, we define the error quaternion δq which represents the departure from the reference attitude trajectory q f. δq(t) = q(t) q f = q f 4 I q T f 3 q f3 q f3 q f4 q(t) (5) Making use of attitude kinematic Eq. (a), the attitude error kinematics can be written as: δ q = 2 Ωδq = B(δq)ω (6) 2 3 of 2

4 Now, to find an expression for a stabilizing controller, let us consider a candidate Lyapunov function: V = ( δq 4 ) 2 + δq T 3δq k 4 log 3 ki 2 i= (7) 3 (ki 2 ω2 i ) where, k, k 2 and k 3 are positive constant. We mention that the log-term in our Lyapunov function is motivated by the Lyapunov function introduced in Ref. [4]. Now, differentiating V with respect to time and making use of the fact that Ω(ω) is a skew-symmetric matrix leads to following expression for i= V : V = δq T 3ω + k 4 ω T k 2 ω2 k 2 2 ω2 2 k 2 3 ω2 3 ω (8) Now, substituting for ω from Eq. (b) in the above expression leads to V = δq T 3ω + k 4 ω T k 2 ω2 k 2 2 ω2 2 k 2 3 ω2 3 I ( ωiω + u) (9) Now, making use of the fact that ω is a skew-symmetric matrix, we get the following expression for V Now, if we choose control law to be V = δq T 3ω + k 4 ω T k 2 ω2 k 2 2 ω2 2 k 2 3 ω2 3 I u () u = IK ω (δq 3 + k 8 ω), K ω = k 4 k 2 ω2 k 4 k 2 2 ω2 2 k 4 k 2 3 ω2 3 () then V reduces to the following negative semi-definite function V = k 8 ω T ω (2) 4 of 2

5 It is clear that V is positive definite for the domain where ω has to lie within the hyper-rectangle with sides 2k i, i =, 2, 3. Thus, we can state that: ω i k i, i =, 2, 3 (3) Since V and V >, V is only negative semi-definite. However, we can easily show that δq, ω L. 5, 6 Further from the integral of Eq. (2), it follows that ω L L 2 and therefore from Barbalat s Lemma ω as t. Finally, using LaSalle s invariance principle 6 3, 5 we can show that δq as t. IV. Adaptation Law for Uncertain Inertia Matrix In this section, we seek to develop adaptation laws for inertia matrix I along with control law developed in the previous section to take care of any uncertainties in the inertia matrix. Let Î be the estimated value of the inertia matrix and I the inertia error matrix defined as follows: I = Î I (4) Further, we define a new variable J = I and analogous to I, we define: J = Ĵ J (5) Furthermore to enforce the symmetry constraint of J, we define a 6 vector: Θ = { J, J 2, J 3, J 22, J 23, J 33, } (6) Further, we make use of principle of equivalence 6 and assume that the expression for control vector of Eq. () is still valid. As a consequence of this, the applied control can be written as: u = ÎK ω (δq 3 + k 8 ω) (7) where, K ω = k 4 k 2 ω2 k 4 k 2 2 ω2 2 k 4 k 2 3 ω2 3 (8) 5 of 2

6 Now, let us consider a candidate Lyapunov function: V = ( δq 4 ) 2 + δq T 3δq k 4 log 3 ki 2 i= + 3 (ki 2 ω2 i ) i= 2 ΘT Γ Θ (9) Differentiating the above expression w.r.t. t and using the fact that ω is a skew-symmetric matrix leads to the following expression for V V = δq T 3ω + ω T K ω I u + Θ T Γ Θ (2) Now, substituting for u from Eq. (7) leads to [ ] V = δq T 3ω + ω T K ω I ÎK ω (δq 3 + k 8 ω) + Θ T Γ Θ (2a) Now, making use of Eq. (5), we have: ) I Î = JĴ = (Ĵ J Ĵ = JÎ, = (22a) Now, substitution of Eq. (22a) in Eq. (2a) leads to ) V = δq T 3ω ω T K ω ( JÎ K ω (δq 3 + k 8 ω) + Θ T Γ Θ = k 8 ω T ω + ω T K ω JÎK ω (δq 3 + k 8 ω) + Θ T Γ Θ (23a) Further, we can write: y y 2 y 3 JK ω ω = MΘ, M = y y 2 y 3, y = K ω ω (24) y y 2 y 3 Now, substitution of Eq. 24 in Eq. 23a leads to V = k 8 ω T ω + Θ T M T ÎK ω (δq 3 + k 8 ω) + Θ T Γ Θ (25a) 6 of 2

7 So, if we choose adaptation law to be Θ = ΓM T ÎK ω (δq 3 + k 8 ω) (26) Then, V reduces to the following negative semi-definite function: V = k 8 ω T ω (27) Since V and V >, V is only negative semi-definite. Once again, we can easily show that δq, ω, Θ L. 5, 6 Further from the integral of Eq. (27), it follows that ω L L 2 and therefore from Barbalat s Lemma ω as t. Finally, using LaSalle s invariance principle 6 3, 5 we can show that δq as t. Further, notice that it is easy to construct Ĵ = J from the expression for Θ and further, an expression for Î can be obtained by making use of the fact that ÎĴ = : ÎĴ + Î Ĵ = Î = Î ĴÎ (28a) (28b) Finally, notice that we can only guarantee that Θ L which means that estimated inertia matrix is bounded and there is no guarantee that estimated inertia matrix will converge to the true inertia matrix. Furthermore, one can invoke parameter projection 6 to guarantee that Ĵ is non-singular and positive-definite. V. Numerical Results The control laws presented in this paper are illustrated in this section for a particular attitude regulating maneuver. We consider following two simulation test cases with different initial conditions to demonstrate the effectiveness of the adaptive control laws. Test Case : q(t ) = {,,, } T, q(t f ) = {,.5, } T.75, ω(t ) = {,, } T, ω(t f ) = {,, } T Test Case 2: q(t ) = {,,, } T, q(t f ) = {,.5, } T.75, ω(t ) = {3, 3, 3} T, ω(t f ) = {,, } T 7 of 2

9 q.5 Adaptation off Adaptation on q 2 q 3 q 4 (a) q vs. ω 3 ω ω Adaptation off Adaptation on (b) ω vs. 2 u u u (c) u vs Δ I ij (d) I ij vs. Figure. Simulation Results for Test Case. 9 of 2

10 that the only difference between the two test cases is the initial conditions for the angular velocity vector ω. These results completely support the theoretical result that performance of the controller can be improved with the adaptation of unknown inertia matrix. VI. Concluding Remarks An asymptotically stable adaptive controller has been designed for rigid body attitude control which explicitly takes into consideration the bounds on angular velocity. The adaptive control formulations in this paper is based upon Lyapunov s direct stability theorem and imposes the exact kinematic equations at the velocity level while taking care of model uncertainties at the acceleration level. The proposed control law is shown to work well in the presence of bounded angular velocity constraints fully consistent with the asymptotic stability analysis presented. While, the simulation results presented in this paper merely illustrate formulations for a particular attitude maneuver, further testing would be required to reach any conclusions about the efficacy of the control and adaptation laws for tracking arbitrary maneuvers. References Schaub, H. and Junkins, J. L., Analytical Mechanics of Space Systems, AIAA Education Series, AIAA, Junkins, J. L. and Turner, J. D., Optimal Spacecraft Rotational Maneuvers, Elsevier Science Publishers, Shuster, M. D., A Survey of Attitude Representations, Journal of the Astronautical Sciences, Vol. 4, No. 4, October December 993, pp Junkins, J. L. and Singla, P., How Nonlinear Is It? A Tutorial on Nonlinearity of Orbit and Attitude Dynamics, Journal of Astronautical Sciences, Vol. 52, No. -2, 24, pp. 7 6, keynote paper. 5 Schaub, H., Akella, M., and Junkins, J. L., Adaptive Control of Nonlinear Attitude Motions Realizing Linear Closed Loop Dynamics, AIAA Journal of Guidance, Control, and Dynamics, Vol. 24, No., 2, pp Carrington, C. K. and Junkins, J. L., Optimal nonlinear feedback control for spacecraft attitude maneuvers, AIAA Journal of Guidance, Control, and Dynamics, Vol. 9, No., 986, pp Tsiotras, P., Further Passivity Results for the Attitude Control Problem, IEEE Transactions on Automatic Contorl, Vol. 43, No., Nov 998, pp Krstic, M. and Tsiotras, P., Inverse optimality results for the attitude motion of a rigid spacecraft, American Control Conference, 997. Proceedings of the 997, Vol. 3, 4-6 Jun 997, pp vol.3. 9 Tsiotras, P., Stabilization and optimality results for the attitude control problem, AIAA Journal of Guidance, Control, and Dynamics, Vol. 9, No. 4, 996, pp Tsiotras, P., A passivity approach to attitude stabilization using nonredundant kinematic parameterizations, Decision and Control, 995., Proceedings of the 34th IEEE Conference on, Vol., 3-5 Dec 995, pp vol.. Junkins, J. L., Akella, M. R., and Robinett, R. D., Nonlinear Adaptive Control of Spacecraft Maneuvers, AIAA Journal of Guidance, Control, and Dynamics, Vol. 2, No. 6, 997, pp Wen, J. T. Y. and Kreutz-Delgado, K., The attitude control problem, Automatic Control, IEEE Transactions on, Vol. 36, No., Oct 99, pp of 2

11 q q 2 q 3 q 4 Adaptation off Adaptation on (a) q vs. ω ω 2 ω Adaptation off Adaptation on (b) ω vs. u 5 Adaptation off Adaptation on u u (c) u vs Δ I ij (d) I ij vs. Figure 2. Simulation Results for Test Case 2. of 2

12 3 Tanygin, S., Generalization of Adaptive Attitude Tracking, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Monterey, CA, USA, August Ngo, K., Mahony, R., and Jiang, Z.-P., Integrator backstepping design for motion systems with velocity constraint, Control Conference, 24. 5th Asian, Vol., July 24, pp Vol.. 5 Sastry, S., Nonlinear Systems: Analysis, Stability and Control, Springer-Verlag, NY, USA, Ionnaou, P. A. and Sun, J., Robust Adaptive Control, Prentice Hall Inc., NJ, of 2

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