# Path Optimization Using Metaheuristic Techniques for a Surveillance Robot
## 3 Description of the Experimental Setup and Simulation Scenarios
### Differential Robot Model  
Furthermore, to obtain useful results close to reality, a simulation environment was created to replicate the real movement of the Hussar robot in the defined spaces. The construction of this environment required, on the one hand, the definition of the type of robot and its movement, as well as its modeling and parameters to control. On the other hand, the surveillance sensor and its range were modeled.  
The kinematic of this robot is shown in Figure 3 (adapted from [31]).  
In this figure the position of the robot is represented by its Cartesian coordinates \(p_{I}=\{x,y,\theta\}\) in the inertial frame \(\{X_{I},Y_{I}\}\). The transverse velocity \(Y_{R}\) is assumed to be 0 because there is no lateral slip.  
Figure 3: Kinematic model of the Hussar robot.  
Figure 2: Low-complexity room (**a**), medium-complexity room (**b**), and high-complexity room (**c**).  
In this model, the longitudinal velocity \(V\) and angular velocity W are the result of the combination of the linear velocities of each wheel, \(V_{R}\) and \(V_{L}\) (the subindexes correspond to right and left, respectively).  
\[V=\frac{V_{R}+V_{L}}{2},W=\frac{V_{R}-V_{L}}{L} \tag{1}\]  
In addition, it must be considered that the left and right wheel speed are adjusted by a controller, to follow a longitudinal and angular speed profile. This can be easily modelled by Equations (2) and (3).  
\[\left\{\begin{array}{ll}V(t)=V_{0}+\alpha_{V}(t-t_{0})&V<V_{ref}\\ V(t)=V_{ref}&V=V_{ref}\end{array}\right. \tag{2}\]  
\[\left\{\begin{array}{ll}W(t)=W_{0}+\alpha_{W}(t-t_{0})&W<W_{ref}\\ W(t)=W_{ref}&W=W_{ref}\end{array}\right. \tag{3}\]  
It is assumed that the speed (longitudinal and angular) follows a trapezoidal profile. The longitudinal acceleration is denoted by \(\alpha_{V}\) and the angular acceleration is \(\alpha_{W}\). These values can be different for acceleration and deceleration. This way the speed increases or decreases linearly until it reaches the reference speed, \(V_{ref}\) for longitudinal and \(W_{ref}\) for angular speed.  
The speed profiles defined by Equations (2) and (3) are shown in Figure 4 (red line, longitudinal speed, and blue line, angular speed). The black dashed line indicates the acceleration starting time.  
It should be noted that this way of inserting and modeling speed profiles is common among mobile robots, but any other profile could be used. To do this, it would be enough to modify the acceleration and maximum speed parameters (longitudinal and angular).  
If \(p_{I}\) is considered an arbitrary position in the global inertia frame, the kinematic model can be given by Equation (4).  
\[\dot{p}_{I}=\begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{bmatrix}=\begin{pmatrix}\cos(\theta)&-\sin(\theta)&0\\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{pmatrix}\times\begin{bmatrix}V\\ 0\\ W\end{bmatrix} \tag{4}\]  
The state vector is determined by \([x,\ y,\ \theta]\). All Cartesian coordinates are in meters and angles are in radians. Likewise, linear velocities are expressed in meters per second and angular velocities in radians per second.  
Figure 4: Speed profiles of the robot.