Sliding Mode Control - Nonlinear Control Mechanical System | AOE 5344, Study notes of Aerospace Engineering

Download Sliding Mode Control - Nonlinear Control Mechanical System | AOE 5344 and more Study notes Aerospace Engineering in PDF only on Docsity! Sliding Mode Control Example Remotely operated vehicles (ROVs) are commonly used to perform scientific or other tasks at great depths. Typically, power and an operator’s control commands are transmitted to the ROV through a tether while sensor data are transmitted from the ROV back to the mother ship. Figure 1 shows Jason II, an ROV operated by the Woods Hole Oceanographic Institute. Jason II is representative of many ROVs. Its blockish shape facilitates storage and deployment but makes hydrodynamic modeling difficult or impossible. The presence of a tether further complicates any attempt at dynamic modeling. Any model-based control algorithm must therefore allow for a great deal of model uncertainty. Figure 1: The ROV Jason II. Jason II has six powerful thrusters which provide actuation in surge, sway, yaw, and heave. (Vehicles may also be actuated in roll and pitch, although Jason II is not.) These thrusters can be used to dominate any unmodeled dynamics and drive the system to a desired state. Sliding mode control is well-suited to the problem of ROV control. Indeed, this is the application which inspired much of the original work on sliding mode control. R x 1 b 2 b 3 b i i i 1 2 3 Figure 2: Coordinate frames. We will model the ROV as depicted in Figure 2. The vehicle’s attitude, with respect to an inertially fixed coordinate frame, is given by the rotation matrix R. Its position is given by the vector x. Our goal is to specify three components of force and three components of torque to drive the vehicle to a desired attitude Rd and position xd. Without loss of generality, we will assume that Rd = I and position xd = 0. That is, we assume that the inertial coordinate frame is fixed at the desired location in the desired orientation. Thus, the error rotation matrix is Re = R T d R = R. We parameterize R using the unit quaternion q = ( q0 qv ) . 1 Thus, we have R(q) = I − 2(q0I − q̂v)q̂v, where â =   0 −a3 a2 a3 0 −a1 −a3 a1 0   for any vector a = [a1, a2, a3] T . Recall that q = [1, 0, 0, 0]T when the attitude error vanishes, i.e., when Re = R = I. The vehicle kinematic equations are ẋ = Rv Ṙ = Rω̂ where v is the body translational velocity (surge, sway, and heave rate) and ω is the the body angular velocity (roll, pitch, and yaw rate). In terms of the error quaternion, we have ( ẋ q̇ ) = ( R(q)v 1 2 Q(q)ω ) (1) where Q(q) = [ −qv T q0I + q̂v ] . At low speeds, the vehicle dynamics can be crudely modeled by v̇ = M−1 (Mv × ω − Dvv + F ) ω̇ = I−1 (Iω × ω + Mv × v − Dωω + T ) (2) where T is a vector of control torques and F is a vector of control forces. The matrix I represents the total (true plus added) inertia and M represents the total mass. Because of added mass and inertia, the matrices I and M are practically impossible to compute or measure. It is common to estimate these matrices as diagonal. We define the nominal inertia and mass matrices Ī and M̄ . The linear damping matrices Dω and Dv are poorly known, as well; we define nominal values D̄v and D̄ω. Also, define M = ( M 0 0 I ) and M̄ = ( M̄ 0 0 Ī ) . The dynamics equations may be rewritten as ( v̇ ω̇ ) = M̄−1 ( ( M̄v × ω − D̄vv ) ( Īω × ω + M̄v × v − D̄ωω ) ) + M̄−1 (u + δ) where u = ( F T ) and δ = − ( ( M̄v × ω − D̄vv ) ( Īω × ω + M̄v × v − D̄ωω ) ) + M̄M−1 ( (Mv × ω − Dvv) (Iω × ω + Mv × v − Dωω) ) + ( M̄M −1 − I ) u. 2