DP Adaptive Backstepping Controller

This package implements a dynamic positioning (DP) Adaptive backstepping controller for the orca AUV. It estimates the linear and nonlinear damping using adaptive parameters, and compensates for uncertainties and disturbances in real-time in the same manner as a state estimator. The proof for this is done using Lyapunov functions and stability requirements to ensure convergence and stability.

Overview

• Uses the backstepping control method for position and orientation control
• Includes adaptive terms to account for unmodeled dynamics and uncertainties.

Model for AUV

\dot{\eta} = J(\eta)\nu,

M \dot{\nu} + C(\nu)\,\nu - F(\nu, \Theta^\star) = \tau + d^\star

• $\nu$: Body-fixed velocity vector
• $\eta$: Inertial position and orientation vector
• M: Constant mass-inertia matrix
• C($\eta$): Coriolis and centripetal terms
• J($\eta$): Transformation from body to inertial coordinates
• F($\nu$, $\Theta^\star$) = Y($\nu$) $\Theta^\star$: The damping assumed damping function (linear and nonlinear), where Y(*) describes the behaviour
• $d^\star$: Disturbance and uncertainties

File overview

1. dp_adapt_backs_controller.cpp/hpp
   • The controller implementation
   • Implements the main control input, sets gains, references, and mass parameters.
2. dp_adapt_backs_controller_utils.cpp/hpp
   • Provides utility functions: skew-symmetric matrix generation, quaternion-to-Euler, Jacobian and some functions needed for the adaptive functions.
3. dp_adapt_backs_controller_ros.cpp/hpp
   • ROS node wrapper for the controller, subscribing to pose, twist, killswitch, and reference topics.
   • Publishes thrust commands.
4. dp_adapt_backs_controller_node.cpp
   • Entry point for the ROS executable.
5. adapt_params.yaml
   • Tunable controller parameters (K1, K2, adapt_gain, d_gain)
   • Contains the mass matrix (6×6) and other physical parameters:

     M =
     \begin{bmatrix}
     30.0 & 0.0 & 0.0 & \cdots \\
     0.0 & 30.0 & \cdots & \cdots \\
     0.0 & \cdots & 30.0 & \cdots \\
     \cdots & \cdots & \cdots & \cdots
     \end{bmatrix}
     

 (partially shown for brevity)
 This is also added into the orca.yaml file

1. CMakeLists.txt
   • Build configuration, ROS 2 dependencies, executable generation, and installation setup.

Tuning Parameters

• K1 and K2: Gains for the backstepping control laws.
• adapt_gain: Adaptive gain for the linear and nonlinear damping.
• d_gain: Adaptive gain for the disturbances and uncertainties.
• M and I_b: Mass inertia matrix and rotational inertia.
• m: Vehicle mass.

Backstepping controller

Using the Lyapunov proof we get the following functions for the backstepping controller:

Backstepping variables

z_1 = \eta - \eta_d

z_2 = \nu - \alpha

where the $\alpha$ is defined as the function that stabilizes the $z_1$ variable in the Lyapunov function system.

Adaptive parameters

\tilde{\Theta} = \hat{\Theta} - \Theta^\star

\tilde{d} = \hat{d} - d^\star

where:

• $\Theta^\star$ and $d^\star$ are the actual parameters,
• $\hat{\Theta}$ and $\hat{d}$ are the estimated parameters,
• $\tilde{\Theta}$ and $\tilde{d}$ are the estimation errors.

Proof of control law

File truncated at 100 lines see the full file