4.4.4 Quaternion based PD Plus Adaptive Nonlinear Compensation

The quaternion based PD plus adaptive nonlinear dynamics controller uses the same basic control law as in the previous section. However, this controller is augmented by an equation which refines the estimates of the physical system parameters based on its response to control inputs. Although simulation and other methods help to produce estimates of the values of the physical parameters, it is clearly unrealistic to assume that all of the parameters of such a complex system could be known precisely. Additionally, since ranger is reconfigurable, both in manipulator configuration and payloads, it is impossible to know all of the possible parameter sets for all configurations. These facts make an adaptive method desirable, where the controller refines its parameter estimates based on the response of the vehicle to its commands.
The parameters to be estimated are the 3 diagonal elements of the inertia matrix, 3 off diagonal elements of the inertia matrix (the inertia matrix is symmetric), the three diagonal elements of the drag matrix (off diagonal drag elements are currently not included in the controller), and three buoyancy offset parameters. Given this parameterization of the system physical properties, the system dynamics from (4.43) become linear with respect to these parameters, and may be written as


(4.47)

where is the estimate of the 12 system parameters.


(4.48)

Nominally, the parameter values are initialized to the values used in the dynamic flight simulator. These values model the vehicle in the stowed configuration and were determined by observing the motion of the vehicle.


(4.49)

The system dynamics may now be expressed as elements of a 12x3 matrix .


,

(4.50)

where is the gravity vector expressed in the vehicle reference frame.
Using this new expression for the system dynamics allows the quaternion based PD plus nonlinear controller to be simply expressed as


,

(4.51)

where is the quaternion based PD portion of the controller, and represents the nonlinear computed torque using estimates , of the true vehicle parameters .
It is unrealistic to assume that the parameters of the system are precisely known. The controller may improve its performance by observing the response of the vehicle to its commands, and refining the parameter estimates in response. The equation describing the rule for refining the parameter estimates based on the error metric is


.

(4.52)

is a constant specifying the rate of adaptation.
When the control law from (4.51) is used to generate the input to the vehicle system dynamics and the adaptation law from (4.52) is used to refine the estimates of the system parameters, the closed loop equations of motion can be written as


,

(4.53)

where


(4.54)

represents the difference between the actual and estimated system parameters. This is a stable nonlinear differential equation. It can be shown from (4.53) and (4.45) that the closed loop dynamics produced by the coupled control and adaptation laws are stable, and guarantee that as , and . [4]