LPV Control Simulation of an Underwater Vehicle Using Virtual Reality Technology
Hiroyuki Kajiwara (Kyushu University, Japan)
Rui Gao (Kyushu University, Japan)
Kazuhisa Ohtsubo (Kyushu University, Japan)
Wataru Koterayama (Kyushu University, Japan)
Masahiko Nakamura (Kyushu University, Japan)
Abstract: In this paper, we show a unified framework for control system simulations using virtual reality technology. The controlled object is an underwater vehicle, for which we should solve a collision avoidance problem with large velocity variation. From the viewpoint of advanced control theory, we are interested in applying LPV (Linear Parameter Varying) control methodology. In the evaluation stage of the LPV control systems, it is desirable to simulate the motion behavior avoiding some obstacle in the 3dimensional underwater space with virtual reality. On the other hand, in the design stage of control systems, there are a lot of trials and errors in selecting design parameters. Therefore we need to manage design and evaluation stages effectively, such that an appropriate control system is quickly realized. For this purpose, we demonstrate that it is much effective to integrate both LPV control and virtual reality technologies based on MATLAB^{TM}/Simulink^{TM} and WorldUP^{TM}.
Keywords: underwater vehicle. LPV control, parameter dependent Lyapunov function, virtual reality.
1. INTRODUCTION
Simulation technologies play an important role in control system designs because we can determine design parameters to satisfy the closedloop control specifications without doing real experiments. Furthermore, introducing the virtual reality to simulations would be very useful in the evaluation of the performance, for example, the ability' of collision avoidance.
We are concerned with control system design of an underwater vehicle for collision avoidance, where rapid and large velocity variation is inevitable. From the viewpoint of advanced control theory we are interested in applying LPV (LinearParameter Varying) control methodology to such a problem, which can be solved using LMI (Linear Matrix Inequality) Control Toolbox under MATLAB^{TM} and Simulink^{TM}
In the evaluation stage of the LPV control systems, it is desirable to simulate the motion behavior avoiding some obstacle in the 3dimensional underwater space with virtual reality. On the other hand, in the design stage of control systems, there are a lot of trials and errors. Therefore we need to manage design and evaluation stages effectively, such that an appropriate control system is quickly realized. For the purpose, we choose WorldUP^{TM} as a virtual reality tool, which could be connected with Simulink^{TM} on MATLAB^{TM} in a seamless manner.
The paper is structured as follows. We first give the problem formulation in section 2. Then, we address the LPV control in section 3. We will show the outline of the design methodology, which is appropriate for collision avoidance problem of an underwater vehicle with large velocity variation. Virtual reality techniques are described in section 4, where we will present how to connect WorldUP^{TM} and Simulink^{TM} on MATLAB^{TM}, and demonstrate the usefulness of virtual reality through LPV control simulation. Concluding remarks are given in section 5.
2. PROBLEM FORMULATION
In this section, after a brief introduction of the underwater vehicle, we describe the problem formulation of the paper, i.e. the obstacle avoidance of the underwater vehicle in longitudinal direction. Finally, we give the nonlinear mathematical model of the underwater vehicle.
The considered underwater vehicle is composed of a delta wing, a main body to house the electrical units, a TV camera and a weightshifting mechanism to control the trim, and two propeller thrusters. The mission is to dive to specified depth and measure wide area of the characteristic of water column. The lateral motion is controlled by the differential thruster between the right and left thruster, while the longitude motion is controlled by the right and left thruster together with weight shifter.
Assume the situation that the underwater vehicle avoids obstacle by means of changing the depth rapidly. Because, the size of obstacle is not known in prior, thus it is reasonable to make the vehicle dive (or float upward) as fast as possible and remain stable in wide range of depth. To this end, the advance speed of the vehicle undergoes an abruptly change, so do the hydrodynamics of the vehicle. In this emergent situation. LTI (linear timeinvariant) control of underwater vehicle becomes impossible because the vehicle dynamics deviates far from the equilibrium point.
It has been demonstrated in [9] that for the slow depth control, it is suffice to change the location of the center of gravity weight shifter, but for the rapid depth control the trim moment generated by thrusters is more effective than weight shifter. Therefore, realizing valid obstacle avoidance using two propellers with the weight shifter fixed is our design objective in this paper.
The control scheme of the present study is modelbased, thus the nonlinear mathematical model is necessary for both controller design and controlled system evaluation. The following model in six degreeoffreedom for underwater vehicle dynamics is considered.
MξB + C (ξB) ξB + D (ξB) ξB + G (ξE) = τ (ξE・ξ) (1)
ξE = J (ξE) ξB (2)
Eq.(1) is the dynamics equation of motion, and (2) is the kinematics differential equation. The vector ξB is a composite vector of the velocity [ξu. ξv. ξw]^{T} and the angular velocity [ξp. ξq. ξr]^{T} both in the bodyfixed reference frame. The vector ξE is a vector of vehicle position [ξX. ξY. ξZ]^{T} and Euler angles [ξφ. ξθ. ξψ]^{T}. The components of ξB and ξE correspond to the motion variables in surge, sway, heave, roll, pitch, and yaw, respectively. The vector τ∈R^{2} is the control vector. The block diagonal transformation matrix J(ξE) relates the bodyfixed frame to the inertial reference frame. The inertia matrix M includes hydrodynamic added mass. The matrix C(ξB) consists of Coriolis and centrifugal terms, and D(ξB) is a matrix of hydrodynamic damping terms. The matrices M. C. and D have the following properties [5]:
1. M = M^{T} < 0, M = 0
2. C(ξB) = C^{T} (ξB)・∀ξB∈>R^{6}
3. D(ξB) < 0, ∀ξB∈>R^{6}・ξB≠0
Control variables are voltage to two thrusters: ξ = [I'I,I'F]. And measured output is [ξz, ξφ, ξθ, ξψ]. In the sequel, we restrict us to the longitudinal dynamics.
3. LPV CONTROL
In this section, we first derive the LPV model of the underwater vehicle. Then we present the outline of the methodology we used for controller design where parameter dependent Lyapunov function (PDLF) based approach is focused upon. The design procedure is described in the end.
3.1 LPV Modeling
After removing the position variables ξX, ξY from the state vector, we derive the explicit LPV model of 10order based on the advance speed. We define the advance speed as
Because ξv = 0 for longitudinal motion, and following relationships hold in the motion:
ξu = U cosξθ, ξw = U sinξθ
It is likely to choose the advance speed U the scheduling variable [7]. The affined type LPV model is then given as
x = (A0 + U × A1)x + (B0 + U × B1)u (3)
where
Details on the value of coefficient matrices see [8]. When the advance speed of the underwater vehicle ranges in [Umin, Umax] and the acceleration in [Umin, Umax], the affined type LPV model can be translated into a polytopic LPV model as
x = (λmin (U) Amin + λmax (U) Amax) x + (λmin (U) Bmin + λmax (U) Bmax)u (4)
where
Amin = A0 + UminA1, Amax = A0 + UmaxA1,
Bmin = B0 + UminB1, Bmax = B0 + UmaxB1,
3.2 Solution Conditions for LPV control
In order to realize effective motion control for emergent obstacle avoidance, the controller is designed to satisfy the following objectives for all admissible velocity variation:
S1) the closedloop system is internally stable.
S2) the controlled system tracks the command signal robustly.
S3) the controller must be implementable with a minimum sampling interval.
S4) the above S1)S3) are expected to hold in the largest range of speed variation.
For the LPV system in (3) or (4), the above design objectives can be interpreted into designing an output feedback LPV controller,
xK(t) = AK(ρ)xK(t) + BK(ρ)y(t)
u = CK(ρ)xK(t) + DK(ρ)y(t) (5)
such that the induced L2norm of the controlled system achieves some performance of γ, and at the same time the poles of the frozen LPV closed loop system locate in some prescribed region. The controlled system admits the form
ζ(t) = Acl(ρ)ζ(t) + Bcl(ρ)w(t) (6)
z(t) = Ccl(ρ)ζ(t) + Dcl(ρ)w(t)
The closedloop input/output relationship defined by (6) with zero initial conditions is denoted as
The induced norm is defined as
A controller is said to achieve a performance of γ if the unforced closedloop system is exponentially stable for all admissible parameter trajectories and if the induced norm of the closedloop system (6) satisfies
‖Twd‖i, 2 < γ
