# A Heterogeneous Robotic Swarm for Long-Term Monitoring of Marine Environments
## 4 Agent Capabilities
### Guidance and Control  
A detailed examination and explanation of the dynamic and kinematic models of the platform, as well as developed low-level control structures, are given in ([14]). A brief overview is given here for completeness. The dynamic model of the platform in the horizontal plane can be described using a velocity vector and a vector of actuating forces and moments acting on the platform. The velocity vector is given by \(\mathbf{v}=\left[ \begin{array}{ccc}u&v&r\end{array} \right]^{\mathrm{T}}\) where \(u\), \(v\) and \(r\) are surge, sway and yaw speed, respectively. The vector of actuating forces and moments acting on the platform is given by \(\boldsymbol{\tau}=\left[ \begin{array}{ccc}X&Y&N\end{array} \right]^{\mathrm{T}}\) where \(X\), \(Y\) are surge and sway forces and \(N\) is yaw moment. Both of these vectors are defined in the body-fixed (mobile) coordinate frame. The platform is designed to be symmetrical with respect to the \(x\) and \(y\) axes in the body-fixed frame. Thus, the uncoupled dynamic model in the horizontal plane is given with (2) where \(\mathbf{M}\) is a diagonal matrix with mass and added mass terms for each component expressed with \(\mathbf{M}=diag\left( \begin{array}{ccc}\alpha_{u},&\alpha_{u},&\alpha_{r} \end{array} \right)\), and \(\mathbf{D}\left(\mathbf{v}\right)\) is a diagonal matrix consisting of nonlinear hydrodynamic damping terms, component-wise \(\mathbf{D}\left(\mathbf{v}\right)=diag\left( \begin{array}{ccc}\beta_{u} \left(u\right),&\beta_{u}\left(v\right),&\beta_{r}\left(r\right)\end{array} \right)\). \[\mathbf{M}\dot{\mathbf{v}}=-\mathbf{D}\left(\mathbf{v}\right)+\boldsymbol{\tau} \tag{2}\]  
The kinematic translatory equations for the platform motion in the horizontal plane on the sea surface are given with (3), where \(x\) and \(y\) are the position and \(\psi\) is the orientation of the platform in the Earth-fixed coordinate frame and \(\mathbf{R}\left(\psi\right)\) is the rotation matrix. \[\left[ \begin{array}{c}\dot{x}\\ \dot{y}\end{array} \right]=\underbrace{\left[ \begin{array}{cc}\cos\psi&- \sin\psi\\ \sin\psi&\cos\psi\end{array} \right]}_{\mathbf{R}\left(\psi\right)}\left[ \begin{array}{c}u\\ v\end{array} \right] \tag{3}\]  
An additional equation present in the kinematic model is \(\dot{\psi}=r\). The platform is overactuated, i.e., it can move in any direction in the horizontal plane by modifying its surge and sway speed, while attaining arbitrary orientation. For the low-level speed controller on the aPad, a PI controller was chosen, given with (4), where \(\mathbf{v}^{*}=\left[ \begin{array}{ccc}u^{*}&v^{*}&r^{*}\end{array} \right]^{ \mathrm{T}}\) are the desired linear and angular speeds of the platform, \(\mathbf{K_{Pv}}=diag\left( \begin{array}{ccc}K_{Pu},&K_{Pv},&K_{Pr}\end{array} \right)\) and \(\mathbf{K_{Iv}}=diag\left( \begin{array}{ccc}K_{Iu},&K_{Iv},&K_{Ir}\end{array} \right)\) are diagonal matrices with proportional and integral gains for individual degrees of freedom, respectively. \[\boldsymbol{\tau}=\mathbf{K_{Pv}}\left(\mathbf{v}^{*}-\mathbf{\hat{v}}\right) +\mathbf{K_{Iv}}\int\left(\mathbf{v}^{*}-\mathbf{\hat{v}}\right)\mathrm{d}t+ \boldsymbol{\tau_{F}} \tag{4}\]  
The tilde sign marks estimated values--the vehicle's speeds are often estimated since they are either difficult to measure or any available measurements are fairly unreliable. The \(\boldsymbol{\tau_{F}}\) term represents additional action introduced in the controller to improve the closed loop behaviour [19]. This action can be in the form \(\boldsymbol{\tau_{F}}=\mathbf{D}\left(\mathbf{v}\right)\mathbf{v}\) which results in the feedback linearisation procedure, in which measured or estimated speeds are used to compensate for the nonlinearity in the process. It is more usual and convenient to use the feedforward term \(\boldsymbol{\tau_{F}}=\mathbf{D}\left(\mathbf{v}^{*}\right)\mathbf{v}^{*}\). Controller parameters \(\mathbf{K_{Pv}}\) and \(\mathbf{K_{Iv}}\) can be calculated based on the desired closed loop characteristic equation as shown in [19].