Underwater Robots - Gianluca Antonelli.pdf

7.6 Sliding Mode Control
7.6 Sliding Mode Control 151
Robust techniques such asSliding Mode Control have been successfully applied
in control of awide class of mechanical systems. In this Section the
application of aSliding Mode based approach tomotion control of UVMSs
is discussed.
The basic idea of this approach isthe definition of asliding surface
s ( x ,t)= ˙˜x + Λ ˜x = 0 (7.9)
where asecond order mechanical system has been assumed, x is the state
vector, t is the time, ˜x = x d − x and Λ is apositive definite matrix. When
the sliding condition issatisfied, the system is forced to slide toward the
value ˜x = 0 with an exponential dynamic (for the scalar case, itis ˙˜x = − λ ˜x ).
The control input, thus, has the objective toforce the state laying in the
sliding surface. With aproper choice ofthe sliding surface, n -order systems
can be controlled considering a1 st -order problem in s .
The only information required to design astable sliding mode controller
is abound on the dynamic parameters. While this is an interesting property
of the controller, one must pay the price ofanhigh control activity. Typically,
sliding mode controllers are based onaswitching term that causes chattering
in the control inputs.
While the first concepts on the sliding surface appeared in the Soviet
literature inthe end of the fifties, the first robotic applications of sliding
mode control are given in [293, 313]. An introduction on Sliding Mode Control
theory con be found in [268].
Control law. The vehicle attitude control problem has been addressed
among the others in the paper [121] which extends the work in [108] and [265]
to obtain asingularity-free tracking control of an underwater vehicle based
on the use of the unit quaternion. Inspired by the work in [121], acontrol law
is presented for the regulation problem of an UVMS. Toovercome the occurrence
of kinematic singularities, the control law isexpressed in body-fixed
and joint-space coordinates soastoavoid inversion of the system Jacobian.
Further, to avoid representation singularities of the orientation, attitude control
of the vehicle is achieved through aquaternion based error. The resulting
control law isvery simple and requires limited computational effort.
Let us recall the dynamic equations in matrix form (2.71):
M ( q ) ˙ ζ + C ( q , ζ ) ζ + D ( q , ζ ) ζ + g ( q , R I B )=Bu, (2. 71)
the control law is
u = B † [ K D s + ˆg ( q , R I B )+K S sign(s )] , (7.10)
where B † is the pseudoinverse of matrix B , K D is apositive definite matrix
of gains, ˆg ( q , R I B )isthe estimate of gravitational and buoyant forces, K S