I have a boat with two trusters and a trajectory planner which is based on the bicycle model. In order to control the boat, the linear velocity of each truster has to be provided.
The following information is known for trajectory tracker: position $P_d$, linear velocity $v_d$,and desired yaw angle $\theta_d$ and desired yaw rate $\omega_d$
The following information is known for the boat: current position $P_a$ and current yaw angle $\theta_a$ and yaw rate $\omega_a$
The objective is to navigate the boat according to the trajectory planner. I was trying to develop a PID controller. But still no luck. The following steps have been carried out, not sure exactly what is the correct way to achieve this?
$v = k_p(P_d - P_a) - k_d v_a, \quad \omega = k_a(\theta_d-\theta_a) - k_v \omega_a$
Control signals (left and right velocities):
$v_l = v - k_1\omega, \quad v_r = v + k_1 \omega$, where $k_1$ is a constant.
If a PD controller can not be used for the boat, could someone suggest to me what kind of way to control the boat?