## 3 Overview of the Controller Architecture  
The fundamental component of the proposed control architecture for a nonholonomic robot is a universal motion controller based on the transverse function (TF) approach [21]. One of the most significant control challenges encountered in nonholonomic kinematics is that a smooth feedback cannot achieve asymptotic point stabilization or asymptotic tracking for any feasible reference trajectory. The TF approach is particularly well-suited for such systems due to its ability to achieve practical stabilization for any reference trajectory--whether feasible or not--persistently excited or reduced to a fixed point. By introducing an additional dynamic system to govern the evolution of the so-called transverse function and relaxing the requirement for asymptotic convergence of posture errors, it becomes possible to overcome the challenges in stabilizing nonholonomic systems, as highlighted by [PERSON] [29].  
Essentially, the controller is designed at the kinematic level, meaning it generates the nominal control input \(u=[u_{1}\ u_{2}]^{T}\) for kinematics (1) based on the current reference posture specified by the trajectory \(x_{r}=[x_{r1}\ x_{r2}\ x_{r3}]^{T}\in\mathcal{X}_{r}\subseteq\mathbb{R}^{3}\), which is assumed to be sufficiently smooth. In this case, since the control strategy is smooth, the nominal signals are also smooth, which facilitates the extension of the control strategy to the dynamic level by incorporating an additional control loop. This can be implemented in a cascade manner such that the nominal kinematic inputs become the desired velocity trajectory, which is assumed to be tracked by the robot's actual velocity. In practice, the velocity of the two-wheeled robot platform considered in this paper is measured and controlled based on the velocities of the left and right wheels. This approach is feasible, however, under operating conditions where wheel slip is negligible, specifically when the assumed velocity constraints are maintained. In such conditions, drive controllers stabilizing velocities of wheels can be employed. Assuming that these controllers ensure velocity tracking with sufficiently small errors, the perturbations affecting the kinematic control law specified for the robot platform become insignificant.  
A motion controller based purely on feedback is usually insufficient to solve more complex tasks in mobile robotics. As a result, it is required to take advantage of planning-based strategies to efficiently shape the reference trajectory \(x_{r}\) in order to ensure the proper operation of the robotic platform. This make it possible to include additional constraints, such as the presence of obstacles and limitations on acceleration and velocity. In the designed architecture, such a trajectory is generated with use of a time elastic band algorithm, which was first presented in [26] and later extended in [30]. The time elastic band algorithm was developed using the solution of a sparse scalar multi-objective optimization problem realized in an open loop. In Figure 1, the architecture responsible for realization of motion tasks is presented. The main blocks can be distinguished, such as the planner generating the reference trajectory based on a preliminary set of waypoints. This trajectory is followed by the feedback-based controller augmented by internal dynamics governing the transverse functions. Next, the drive controller is responsible for the realization of the kinematic control strategy by ensuring that velocity error is reduced significantly.