Comments on \Hamiltonian Adaptive Control of Spacecraft" Thor I. Fossen
Division of Engineering Cybernetics The Norwegian Institute of Technology N-7034 Trondheim, NORWAY Telephone: +47 7 594361 Telefax: +47 7 594399 E-mail:
[email protected]
IEEE Transactions on Automatic Control Vol. 38, TAC-5, pp. 671{672, April 1993.
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Comments on \Hamiltonian Adaptive Control of Spacecraft"
where ?1 H (x) = J ?T HJ h i C (x; x_ ) = J ?T C ? HJ ?1 J_ J ?1 g (x) = J ?T g
Thor I. Fossen
Abstract ? In the adaptive scheme presented by Slotine and Benedetto (1990) for attitude tracking control of rigid spacecraft, the spacecraft is parameterized in terms of the inertial frame. This paper shows how a parameterization in body coordinates considerably simpli es the representation of the adaption scheme. The new symbolic expression for the regressor matrix is easy to nd even for 6 degrees of freedom (DOF) Hamiltonian systems with a large number of unknown parameters. If the symbolic expression for the regressor matrix is known in advance, the computational complexity is approximately equal for both representations. In the scheme presented by Slotine and Benedetto (1990) this is not trivial because the transformation matrix between the inertial frame and the body coordinates is included in the expression for the regressor matrix. Hence implementation for higher DOF systems is strongly complicated. An example illustrates the advantage of the new representation when modelling a simple 3 DOF model of the lateral motion of a space shuttle.
Assume the desired trajectory: x d , x_ d and xd to be bounded. The tracking error vector x~ is de ned as x~ = x ? xd while: ~_ + x~ s=x
(2) is used as a measure of tracking. is a strictly positive constant which may be interpreted as the control bandwidth. It is convenient to rewrite Eq. 2 as: s = x_ ? x_ r
It is important to notice that the terms H and C are linear in their parameters. These unknown parameters may be lumped together into a parameter vector . Let ^ be the time-varying parameter vector estimate and let ~ = ^ ? be the parameter error vector. To prove global stability Slotine and Benedetto (1990) suggest to use a Lyapunov-like function candidate: V (s; ~ ; t) = 21 sT H s + 21 ~ T ?~ where ? is a positive de nite weighting matrix of appropriate dimension. Di erentiating V with respect to time and using the skew-symmetry property: x_ T (H_ ? 2C )x_ = 0 yields the following expression:
I. Introduction Consider a Hamiltonian systems written as:
H q + C (q_ ; x)q_ + g(x) = x_ = J (x)q_
where x_ r = x_ d ? x~
V_ = sT (J ?T ? H x r ? C x_ r ? g ) + ~_ ?~ (3) T
(1)
This suggests that the control law can be selected as:
where q 2
