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Modification of Hamiltonian Structure to Stabilize An Underwater Vehicle
Copyright @ IFAC Lagrangian and Hamiltonian Methods for Nonlinear Control, Princeton, New Jersey, USA, 2000
MODIFICATION OF HAMILTONIAN STRUCTURE TO STABILIZE AN UNDERWATER VEHICLE Craig A. Woolsey
.,1,2
Naomi Ehrich Leonard .,1
• Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544
Abstract: This paper presents new results on stabilization of an underwater vehicle using internal rotors. In previous work, a stabilizing control law was derived which preserves the open-loop Hamiltonian structure in the closed-loop system but which modifies the Hamiltonian. The results presented here illustrate the utility of feedback control that not only shapes the energy but also modifies the Hamiltonian structure. Copyright @2000 IFAC Keywords: Lyapunov methods, feedback stabilization, rotors, underwater vehicle
1. INTRODUCTION
2. FEEDBACK STABILIZATION
Internal rotors can provide energy shaping that stabilizes steady forward motion of an underwater vehicle with dynamics described by Kirchhoff's equations (Leonard and Woolsey, 1998; Woolsey and Leonard, 1999b; Bloch et al., 2000). Kirchhoff's equations, which are Hamiltonian (LiePoisson), are stabilized with a feedback law that preserves the Lie-Poisson structure in the closed loop but which modifies the Hamiltonian. Stability requires choosing control gains so that the equilibrium is a maximum of a Lyapunov function constructed from the modified Hamiltonian. Physical dissipation (in this case, viscous drag due to the body's motion through the fluid) decreases the energy and tends to destabilize the equilibrium. The control law proposed here provides stabilization in such a way that physical dissipation enhances stability. The control law provides a Hamiltonian closed-loop system with a modified Hamiltonian structure, as well as a modified Hamiltonian. Modification of Hamiltonian structure is inspired in part by Krishnaprasad (1985).
Consider an ellipsoidal vehicle with three internal rotors and let the ellipsoid principal axes define a body-fixed coordinate frame. Each rotor is axisymmetric and spins about its symmetry axis under the influence of a control torque. The rotors are mounted orthogonally within the vehicle so that each rotor's symmetry axis is aligned with a body coordinate axis. Assume that the vehicle mass is uniformly distributed so that the center of gravity (CG) is also the center of buoyancy (CB). Let the diagonal matrix I = diag(h, h 13 ) represent the inertia of the vehicle without rotors plus the added inertia of the fluid. Similarly, let the diagonal matrix M = diag( m1 , m2, m3) represent the mass of the vehicle multiplied by the identity matrix plus the added mass matrix of the fluid. We assume that the vehicle's I-axis is longest and that its 3-axis is shortest. Then, m1 < m2 < m3. Let diag( J1 ,J4, J~) be the inertia matrix of the rotor which spins about the ith coordinate axis (i = 1,2, or 3). The total inertia, with the rotors locked in place, is A diag(A1' A2' A3) where
=
Aj Research partially supported by NSF grant BES9502477 and ONR grant N00014-98-1-0649 2 Research partially supported by NDSEG Fellowship
= I j + J] + J] + J],
j
= 1,2,3.
1
We also define the matrix J ..
175
= diag(Jl, J?, J:).
Let 0 and v represent the body angular and linear velocity, respectively, in body coordinates. Also, define 0 .. = (nrllnr2,n"s)T, where nr; is the angular rate of the ith rotor relative to the vehicle. The body momenta are given by
The new control law that we propose is
u
= k(P x v).
(2)
This control law is a modification of the original control law (1) and was formulated from physical intuition (see also Leonard (1996)). The closedloop dynamics are
0 P = (AOMO (II) I J .. J..
J .. ) ( 0v) . 0 0 ..
The Hamiltonian is the total kinetic energy, H=
~
(II) (AJ .. 0 J.. )-I (II) P I
.
J .. 0 MOP 0 I
.
This system is almost Lie-Poisson (an implicit generalized Hamiltonian system in the sense of van der Schaft (1998)); that is, the corresponding Poisson bracket does not satisfy the Jacobi identity. Asymptotic stability of steady long-axis translation can be proven by choosing k > 1, using HR to construct a negative semidefinite Lyapunov function, and applying feedback dissipation. In this case, however, fluid drag tends to increase the Lyapunov function, enhancing stability of the desired equilibrium. In addition, when extending these ideas to the case of noncoincident CG and CB, a generalization of the control law (1) requires that the CG be above the CB, whereas a generalization of (2) requires the more practical low CG for stability.
The equations of motion are
where u = (UI,U2,U3)T is the control and Ui is the torque applied to the ith internal rotor about its spin axis. With u = 0, it can be verified that steady translation of the vehicle along its long axis is an unstable relative equilibrium (Lamb, 1932). The control law developed in our earlier work is u =
kTI = k(II
x 0
+P
x v),
(1)
where k is a control gain. Let 3. REFERENCES
1
<: = -l-k (I-kIl)
(II,
Bloch, A. M., N. E. Leonard and J. E. Marsden (2000). Controlled Lagrangians and the stabilization of Euler-Poincare mechanical systems. Preprint. Krishnaprasad, P. S. (1985). Lie-Poisson structures, dual-spin spacecraft and asymptotic stability. Nonlinear Anal., Theory, Meth. f3 App. 9(10), 1011-1035. Lamb, H. (1932). Hydrodynamics. 6th ed.. Dover. Leonard, N. E. (1996). Stability and stabilization of underwater vehicle dynamics. In: Proc. GISS. Princeton, NJ. pp. 771-775. Leonard, N. E. and C. Woolsey (1998). Internal actuation for intelligent underwater vehicle control. In: Proc. 10th Yale Workshop on Adaptive and Learning Sys.. pp. 295-300. van der Schaft, A.J. (1998). Implicit Hamiltonian systems with symmetry. Reports on Mathematical Physics 41(2), 203-221. Woolsey, C. and N. E. Leonard (1999a). Global asymptotic stabilization of an underwater vehicle using internal rotors. In: Proc. IEEE CDC. Vol. 38. pp. 2527-2532. Woolsey, C. and N. E. Leonard (1999b). Underwater vehicle stabilization by internal rotors. In: Proc. ACe. pp. 3417-3421.
(II,
and change coordinates from P, I) to P, <:). Note that <: is conserved, by construction. The closed-loop equations of motion are Lie-Poisson with respect to the energy HR =
(II - <:) . le - III(-)<: + 21 P.
M-Ip
21
where le
,
= I~kj. The equations are
Conditions for stability may be determined by applying the energy-Casimir method. The method proves stability of steady long-axis translation if we choose k > 1 (le < 0). The method also provides a Lyapunov function, Hcf>, constructed from the energy HR and other conserved quantities. The desired equilibrium is a maximum of Hcf>. Fluid drag, which is not included in this model, tends to destabilize the stabilized equilibrium by decreasing Hcf>. Asymptotic stability thus requires additional feedback dissipation to dominate drag (Woolsey and Leonard, 1999a).
176
MODIFICATION OF HAMILTONIAN STRUCTURE TO STABILIZE AN UNDERWATER VEHICLE Craig A. Woolsey
.,1,2
Naomi Ehrich Leonard .,1
• Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544
Abstract: This paper presents new results on stabilization of an underwater vehicle using internal rotors. In previous work, a stabilizing control law was derived which preserves the open-loop Hamiltonian structure in the closed-loop system but which modifies the Hamiltonian. The results presented here illustrate the utility of feedback control that not only shapes the energy but also modifies the Hamiltonian structure. Copyright @2000 IFAC Keywords: Lyapunov methods, feedback stabilization, rotors, underwater vehicle
1. INTRODUCTION
2. FEEDBACK STABILIZATION
Internal rotors can provide energy shaping that stabilizes steady forward motion of an underwater vehicle with dynamics described by Kirchhoff's equations (Leonard and Woolsey, 1998; Woolsey and Leonard, 1999b; Bloch et al., 2000). Kirchhoff's equations, which are Hamiltonian (LiePoisson), are stabilized with a feedback law that preserves the Lie-Poisson structure in the closed loop but which modifies the Hamiltonian. Stability requires choosing control gains so that the equilibrium is a maximum of a Lyapunov function constructed from the modified Hamiltonian. Physical dissipation (in this case, viscous drag due to the body's motion through the fluid) decreases the energy and tends to destabilize the equilibrium. The control law proposed here provides stabilization in such a way that physical dissipation enhances stability. The control law provides a Hamiltonian closed-loop system with a modified Hamiltonian structure, as well as a modified Hamiltonian. Modification of Hamiltonian structure is inspired in part by Krishnaprasad (1985).
Consider an ellipsoidal vehicle with three internal rotors and let the ellipsoid principal axes define a body-fixed coordinate frame. Each rotor is axisymmetric and spins about its symmetry axis under the influence of a control torque. The rotors are mounted orthogonally within the vehicle so that each rotor's symmetry axis is aligned with a body coordinate axis. Assume that the vehicle mass is uniformly distributed so that the center of gravity (CG) is also the center of buoyancy (CB). Let the diagonal matrix I = diag(h, h 13 ) represent the inertia of the vehicle without rotors plus the added inertia of the fluid. Similarly, let the diagonal matrix M = diag( m1 , m2, m3) represent the mass of the vehicle multiplied by the identity matrix plus the added mass matrix of the fluid. We assume that the vehicle's I-axis is longest and that its 3-axis is shortest. Then, m1 < m2 < m3. Let diag( J1 ,J4, J~) be the inertia matrix of the rotor which spins about the ith coordinate axis (i = 1,2, or 3). The total inertia, with the rotors locked in place, is A diag(A1' A2' A3) where
=
Aj Research partially supported by NSF grant BES9502477 and ONR grant N00014-98-1-0649 2 Research partially supported by NDSEG Fellowship
= I j + J] + J] + J],
j
= 1,2,3.
1
We also define the matrix J ..
175
= diag(Jl, J?, J:).
Let 0 and v represent the body angular and linear velocity, respectively, in body coordinates. Also, define 0 .. = (nrllnr2,n"s)T, where nr; is the angular rate of the ith rotor relative to the vehicle. The body momenta are given by
The new control law that we propose is
u
= k(P x v).
(2)
This control law is a modification of the original control law (1) and was formulated from physical intuition (see also Leonard (1996)). The closedloop dynamics are
0 P = (AOMO (II) I J .. J..
J .. ) ( 0v) . 0 0 ..
The Hamiltonian is the total kinetic energy, H=
~
(II) (AJ .. 0 J.. )-I (II) P I
.
J .. 0 MOP 0 I
.
This system is almost Lie-Poisson (an implicit generalized Hamiltonian system in the sense of van der Schaft (1998)); that is, the corresponding Poisson bracket does not satisfy the Jacobi identity. Asymptotic stability of steady long-axis translation can be proven by choosing k > 1, using HR to construct a negative semidefinite Lyapunov function, and applying feedback dissipation. In this case, however, fluid drag tends to increase the Lyapunov function, enhancing stability of the desired equilibrium. In addition, when extending these ideas to the case of noncoincident CG and CB, a generalization of the control law (1) requires that the CG be above the CB, whereas a generalization of (2) requires the more practical low CG for stability.
The equations of motion are
where u = (UI,U2,U3)T is the control and Ui is the torque applied to the ith internal rotor about its spin axis. With u = 0, it can be verified that steady translation of the vehicle along its long axis is an unstable relative equilibrium (Lamb, 1932). The control law developed in our earlier work is u =
kTI = k(II
x 0
+P
x v),
(1)
where k is a control gain. Let 3. REFERENCES
1
<: = -l-k (I-kIl)
(II,
Bloch, A. M., N. E. Leonard and J. E. Marsden (2000). Controlled Lagrangians and the stabilization of Euler-Poincare mechanical systems. Preprint. Krishnaprasad, P. S. (1985). Lie-Poisson structures, dual-spin spacecraft and asymptotic stability. Nonlinear Anal., Theory, Meth. f3 App. 9(10), 1011-1035. Lamb, H. (1932). Hydrodynamics. 6th ed.. Dover. Leonard, N. E. (1996). Stability and stabilization of underwater vehicle dynamics. In: Proc. GISS. Princeton, NJ. pp. 771-775. Leonard, N. E. and C. Woolsey (1998). Internal actuation for intelligent underwater vehicle control. In: Proc. 10th Yale Workshop on Adaptive and Learning Sys.. pp. 295-300. van der Schaft, A.J. (1998). Implicit Hamiltonian systems with symmetry. Reports on Mathematical Physics 41(2), 203-221. Woolsey, C. and N. E. Leonard (1999a). Global asymptotic stabilization of an underwater vehicle using internal rotors. In: Proc. IEEE CDC. Vol. 38. pp. 2527-2532. Woolsey, C. and N. E. Leonard (1999b). Underwater vehicle stabilization by internal rotors. In: Proc. ACe. pp. 3417-3421.
(II,
and change coordinates from P, I) to P, <:). Note that <: is conserved, by construction. The closed-loop equations of motion are Lie-Poisson with respect to the energy HR =
(II - <:) . le - III(-)<: + 21 P.
M-Ip
21
where le
,
= I~kj. The equations are
Conditions for stability may be determined by applying the energy-Casimir method. The method proves stability of steady long-axis translation if we choose k > 1 (le < 0). The method also provides a Lyapunov function, Hcf>, constructed from the energy HR and other conserved quantities. The desired equilibrium is a maximum of Hcf>. Fluid drag, which is not included in this model, tends to destabilize the stabilized equilibrium by decreasing Hcf>. Asymptotic stability thus requires additional feedback dissipation to dominate drag (Woolsey and Leonard, 1999a).
176