This controller augments the quaternion based PD controller by computing the additional torque necessary to exactly follow the desired trajectory.

_{} 
(4.37) 
The computed torque term _{ }
is calculated by using the desired trajectory as input to a system of
equations describing the system dynamics. If the model were exact,
and the thrusters could supply precisely the torque calculated, then
the trajectory would be followed exactly without the need for the PD
controller. Since it is currently impossible to generate a model that
exactly matches the dynamic system, and the thrusters can only
approximate the desired torque, the PD controller is still used to
reduce errors due to these inaccuracies.
For the purpose of analyzing of the boundedness and convergence of
_{ }
and _{ }
it is useful to introduce a single vector _{ }
which is a combination of the two. By noting from (4.35) that _{
},
we can show that

_{} 
(4.38) 
with

_{}. 
(4.39) 
The vector _{ } may be thought of as a type of velocity error since

_{}, 
(4.40) 
where

_{} 
(4.41) 
However it is important to realize that the attitude error is
contained within _{ }.
It is useful to think of _{ }
as a reference angular velocity where the desired angular velocity
(in vehicle frame) is shifted proportionally to the attitude error by
the factor _{ }.
This notational manipulation allows translation of energy related
properties (expressed in terms of estimated angular velocity _{
})
into trajectory control properties (expressed in terms of the virtual
vector _{ }).
Using this notation, (4.37) may be rewritten as

_{} 
(4.42) 
The computed torque _{ } is calculated based on the system dynamics, the chosen physical parameters, and the desired and actual trajectories. The dynamic model chosen takes into account rotational dynamics, drag, and buoyancy moments.

_{} 
(4.43) 
where

_{} 
(4.44) 
and _{ } is evaluated using

_{} 
(4.45) 
as shown in (4.9)(4.11).
In (4.43), _{ }
describes the torques associated with angular acceleration due to the
rigid body inertial parameters, and _{ }
calculates the torques generated due to the current angular velocity
of the vehicle in view of its inertial parameters. The term _{
}
describes the moments applied to the vehicle due to the drag of the
water, and _{ }
describes the moment generated by the buoyancy offset.
When the control law from (4.43) is used to generate the input to the
vehicle system dynamics, the closed loop equations of motion
become

_{}. 
(4.46) 
This is a stable nonlinear differential equation. It can be shown from (4.46), and (4.45) that as _{ }, _{ } and _{ }. [4]