Nonlinear Control for Trajectory Tracking of a Nonholonomic RC-Hovercraft with Discrete Inputs

Abstract

This work studies the problem of trajectory tracking for an underactuated RC-hovercraft, the control of which must be done by means of discrete inputs. Thus, the aim is to control a vehicle with very simple propellers that produce only a discrete set of control commands, and with minimal information about the dynamics of the actuators. The control problem is approached as a cascade control problem, where the outer loop stabilizes the position error, and the inner loop stabilizes the orientation of the vehicle. Stability of the controller is theoretically demonstrated and the robustness of the control law against disturbances and noise is established. Simulation examples and experiments on a real setup validate the control law showing the real system to be robust against disturbances, noise, and outdated dynamics.

1. Introduction

Autonomous vehicles (AV) in general, and autonomous surface vehicles (ASV) in particular, are becoming ubiquitous in many fields of engineering research due in part to the flexibility and versatility that a number of them display in the execution of individual and cooperative tasks. These characteristics, coupled with the fact that their use avoids placing human lives at risk, makes them quite attractive in multiple missions. Central to the operation of these autonomous vehicles is the availability of good tracking controllers that allow the AVs to execute demanding tasks in different operational scenarios. Moreover, in the present economical scenario in which many research projects suffer withdrawal of funds, it is of the utmost importance to offer reliable low cost systems.

The three main problems that arise in control theory of autonomous vehicles, according to the classification of [1], are path following [2], point stabilization [3, 4], and trajectory tracking. As part of this trend, the tracking control problem for underactuated vehicles represents a very challenging research topic, being, thus, the problem studied in the present work.

Underactuated vehicles are of special interest due to their reduced cost and the relative easiness of their construction, furthermore, a fully actuated vehicle is only needed for some special operations, making its usage unnecessary for a wide number of actions. In addition, the possibility of using the same vehicle for terrestrial and maritime operations put underactuated hybrid vehicles in a very attractive situation. In this sense, the most common hybrid vehicle is the hovercraft. Hovercraft is usually used as test vehicle for control algorithms of underactuated marine systems due to its dynamic properties, the possibility of being tested on ground, and its relative (and mission dependent) low cost, as in the experimental testbed developed on [5] or [6].

In addition to the low cost and the possible hybrid nature of the vehicle at hand, the nonholonomic condition of the underactuated vehicles is by itself interesting enough and complex for a detailed study. An usual solution for fully actuated vehicles consists in using sliding mode controllers such as the solution described in [7]. The main limitation of this control strategy is the impossibility of acting over all the states of the vehicle at the same time when an underactuated vehicle as an hovercraft is considered. The control problem of underactuated vehicles is much more challenging than the fully actuated one because of the limitation on the control laws that can be implemented. This is a consequence of the famous result stated by Brockett [8] that states that some kind of underactuated vehicles (that includes the hovercraft) cannot be stabilized to a certain pose with a continuous and time invariant control law. The reader is referred to [9] for an interesting survey in control problems for nonholonomic vehicles.

Representative examples of tracking controllers applied to underactuated hovercrafts are stated in [10], where the tracking control problem is solved using backstepping techniques achieving practical stability results, in [11], that solve the same problem exploiting the differential flatness properties of a vessel with respect to its position, and in [12], where the authors extend previous results for the tracking and path following problems to more general marine vehicles, taking into account the parametric uncertainty on the models.

Other remarkable examples are found in [13], where the tracking problem of general underactuated ships is studied, in [14], that explains how to select outputs when generalized forces act on the vehicle, in [15], where Barbalat’s lemma and backstepping techniques are combined to achieve asymptotic tracking, in [16], that solves the tracking problem of a wheeled robot, in [17], where the authors address the tracking problem for a surface craft, and in [18], in which the challenging problem of solving both point stabilization and tracking problem at the same time is studied.

It must be remarked that the control strategies described above achieve a good control performance assuming that force and torque imparted by propellers can be perfectly controlled. Nevertheless, thrust allocation is in general a very complex task in surface crafts (see, e.g., [19]), and moreover, expensive equipments are usually required. Thus, the availability of low cost vehicles, equipped with simple actuators and with a limited knowledge of their dynamics, is of the utmost importance for the problem studied in the present work, and a very interesting alternative from a practical point of view.

In the last few years, the interest on the control of underactuated vehicles where only a discrete set of control inputs is available has increased significantly. See, for example, [20, 21] for a description on the challenges that arise in quantized control of mobile vehicles and remote control of systems over networks with state quantization, respectively. Another remarkable example is found in the works [22, 23], where a nonholonomic underactuated hovercraft is controlled by a discrete set of inputs. In these works, the problem of point stabilization is solved using receding horizon predictive controllers and approaching the continuous output by the closest allowed control action. This strategy works adequately when the optimal control law is bang-bang, but it is not suitable when the control signals take intermediate values between two allowed ones, something impossible to achieve with all-nothing-reverse signals. Unfortunately, this is the actual tracking problem scenario because forces and torques should be close to a continuous reference far away from saturation.

In order to solve the control problem of underactuated vehicles, some authors consider second order sliding-mode control laws. See, for example [24], where this technique is applied to the tracking problem of a surface vessel, and [11], where the trajectory tracking problem for an underactuated hovercraft is solved. These control laws tend to produce control actions that are similar to bang-bang actions. Therefore, they are good candidates to be approximated with all-nothing-reverse thrusters. The later are the kind of thrusters used in the vehicle (hovercraft) dealt in this work due to their simplicity of control and implementation, and also their low cost.

This kind of problem has been partially studied in the ground robotic field. Some interesting results on the field of wheeled robots show that, in the case of Dubin’s vehicles, it is possible to design a tracking controller that makes the vehicle converge to a reference trajectory using only a discrete set of control actions (turn right and left, accelerate, and brake). See, for example [25], that uses a sliding-mode control and [16], where a controller has been developed using optimal synthesis.

An interesting practical problem is the extension of the above results for an underactuated low cost vehicle as a hovercraft. The main difference between a wheeled/ground underactuated vehicle and a marine underactuated vehicle is the nonholonomic restriction on the trajectories that can be followed by each vehicle. The nonholonomic restriction in a vehicle is the relationship between the acceleration and the orientation of the vehicle. The sliding condition for marine vehicles (second-order nonholonomic restriction) is more complex than the nonsliding condition (first-order nonholonomic restriction) for wheeled robots. Thus, the aim of this work is to control a vehicle with very simple propellers that produce only a discrete set of control commands and with a minimal information about the dynamics of the actuators. The availability, simplicity, and low cost of these kind of systems, even the hybrid nature of the hovercraft, make them very attractive to be used in a number of operational scenarios. Furthermore, a group of these vehicles can be used for cooperative and/or coordinated tasks with relatively low price.

The sole conditions that are imposed to the actuators used on this work can be summarized as follows.(1)The available control actions can turn right and left while the vehicle go forward and backwards.(2)The available force and torque are bigger than the force and torque necessary to exactly track the reference (forces and torques can be dominated by the control actions).

Notice that the first condition is important to ensure local controllability (see [26]) while second condition is necessary to ensure feasibility of the trajectory.

The key contribution of the paper is two-fold: (i) we reflect the fact that the inputs are discrete in the controller design process and (ii) we take explicitly into account the existence of unknown bounded input disturbances and measurement noise. Theoretical stability proofs are obtained and, in addition, the control law is successfully tested on a real low cost vehicle. In the literature, to the best knowledge of the authors, there are no attempts to solve the trajectory tracking problem for a second order nonholonomic vehicle as the hovercraft with a discrete set of inputs. Furthermore, in the few cases where a discrete set of inputs is considered in a vehicle of this kind, the effect of this practical limitation and the effect of disturbances and noise have never been studied in detail.

The paper is organized as follows. Section 2 describes the model of the vehicle and the set of trajectories that can be tracked with a discrete set of inputs. The control problem is formulated in Section 3. In Section 4, a nonlinear controller is designed. Practical implementation and the effect of disturbances and noise are discussed in Section 5. The stability and robustness of the controller are analyzed in Section 6. Simulation and experimental tests are shown in Section 7. Finally, conclusions are commented on in Section 8.

2. Vehicle Model

The model of the vehicle operating over a solid surface is defined in two reference frames, one inertial global reference frame, attached to the floor where the RC-hovercraft moves, and a body-fixed reference frame, shown in Figure 1. According to the model (the details of which can be found in [27]), the dynamics of the system can be described in the global reference frame by 𝑥̇=𝑣_𝑥,𝑦̇=𝑣_𝑦,𝑣̇_𝑥=𝐹cos𝜓−𝑑_𝑢𝑣_𝑥+𝑝_𝑣𝑥,𝑣̇_𝑦=𝐹sin𝜓−𝑑_𝑢𝑣_𝑦+𝑝_𝑣𝑦,̇̇𝜓=𝑟,𝑟=𝜏−𝑑_𝑟𝑟+𝑝_𝑟. (1)(2)(3)(4)(5)(6)

Figure 1 

Inertial reference frame and body fixed frame.

The state of the vehicle is given by 𝐱=[𝑥,𝑦,𝑣_𝑥,𝑣_𝑦,𝜓,𝑟]^𝑇, where 𝑥 and 𝑦 are the position coordinates of the centre of mass of the RC-hovercraft, 𝜓 is the orientation, 𝑣_𝑥 and 𝑣_𝑦 are the linear velocities, and 𝑟 is the yaw rate. The parameters 𝑑_𝑢 and 𝑑_𝑟 of the model are the normalized drag coefficients, where 𝑑_𝑢=𝐷_𝑢/𝑚, 𝑑_𝑟=𝐷_𝑟/𝐽, 𝑚 is the mass of the hovercraft, 𝐽 the moment of inertia, and 𝐷_𝑢 and 𝐷_𝑟 are the linear and rotational drag coefficients, respectively. The control actions are the normalized force 𝐹=(𝐹_𝑝+𝐹_𝑠)/𝑚 and torque 𝜏=𝑙(𝐹_𝑝−𝐹_𝑠)/𝐽, where 𝐹_𝑠 and 𝐹_𝑝 are the starboard and port forces produced by the fans, and 𝑙 is the distance between the propellers and the symmetry axis, as Figure 1 shows. The dynamics of the vehicle are also subject to the unknown disturbance vector 𝐩=[𝑝_𝑣𝑥,𝑝_𝑣𝑦,𝑝_𝑟]^𝑇, and the output of the system is corrupted by measurement noise 𝐧=[𝑛_𝑥,𝑛_𝑦,𝑛_𝑣𝑥,𝑛_𝑣𝑦,𝑛_𝜓,𝑛_𝑟]^𝑇, such that the measured state is 𝐲=𝐱+𝐧. Noise and disturbances are unknown but bounded as follows: sup_𝑡≥0‖𝐩(𝑡)‖≤𝑝_𝑚,sup_𝑡≥0𝑑𝐩𝑑𝑡≤𝑝_𝑚,sup_𝑡≥0‖𝐧(𝑡)‖≤𝑛_𝑚. (7)

The value of the parameters for the experimental device are summarized in Table 1.

Table 1 

Model parameters.

2.1. Physical Constraints

The experimental vehicle is under some physical constraints as it is usual in any real device. The control inputs are forces produced by thrusters, the values of which belong to a discrete set, 𝐹_𝑠,𝑝∈{−𝑢_min,0,𝑢_max}, with 𝑢_max>𝑢_min>0. Under these considerations, the control actions, 𝐹 and 𝜏, belong to the discrete set of nine combinations of 𝐹_𝑠 and 𝐹_𝑝 shown in Figure 2. It is important to notice that these control actions can move the hovercraft forward and backwards and right and left. Moreover, it will be shown that it is possible to control the RC-hovercraft with a reduced set of control actions and without knowledge of the exact values of 𝑢_min and 𝑢_max. The key point is that, once the sign of the force and torque is known, then there always exists a proper selection of 𝐹_𝑠 and 𝐹_𝑝 such that the desired values of sign(𝐹) and sign(𝜏) are obtained. As Figure 2 shows, there exist more than one solution. In Table 2, one possible selection scheme is shown.

Table 2 

Force selection table.

Figure 2 

Discrete control inputs on the laboratory setup.

2.2. Feasible Trajectories

Dynamics of the RC-hovercraft are flat with respect to the position coordinates, as demonstrated in [11]. The reader is referred to [28] for further information about flat systems. Thus, when the reference spatial trajectories 𝑥_𝑟(𝑡) and 𝑦_𝑟(𝑡) are set, then the whole state vector and the control actions that must be applied to follow the reference can be computed. For this purpose, considering (3) and (4) in the absence of disturbances and defining 𝐹_𝑟 as the force needed to track the reference, the ideal orientation 𝜓_𝑟 needed to exactly track the reference is given by the following nonholonomic restrictions: 𝐹_𝑟cos𝜓_𝑟=𝑣̇_𝑥𝑟+𝑑_𝑢𝑣_𝑥𝑟,𝐹_𝑟sin𝜓_𝑟=𝑣̇_𝑦𝑟+𝑑_𝑢𝑣_𝑦𝑟. (8) The orientation is well defined if 𝐹_𝑟≠0. It is important to notice that actually there exist two families of solutions, depending on how the trajectory is followed by the vehicle, going forward (𝑛 odd) or backwards (𝑛 even) as follows: 𝜓_𝑟=atan2𝑦̈_𝑟+𝑑_𝑢𝑦̇_𝑟,𝑥̈_𝑟+𝑑_𝑢𝑥̇_𝑟𝐹+𝑛𝜋,_𝑟=(−1)^𝑛𝑥̈_𝑟+𝑑_𝑢𝑥̇_𝑟²+𝑦̈_𝑟+𝑑_𝑢𝑦̇_𝑟². (9) Once the family of solutions of interest is selected, the orientation 𝜓_𝑟 and its derivatives, 𝑟_𝑟 and 𝑟̇_𝑟, can be computed. Straightforward computations yield the requested torque, 𝜏_𝑟=𝑟̇_𝑟+𝑑_𝑟𝑟_𝑟 to track the reference. Now, we consider a spatial trajectory 𝐱_𝑟=[𝑥_𝑟(𝑡),𝑦_𝑟(𝑡)]^𝑇 four times differentiable with 𝐹_𝑟≠0. Two kinds of trajectories of interest can be defined as follows. (1)A feasible trajectory, where 𝐹_𝑟 and 𝜏_𝑟 are such that 𝐹_𝑠,𝑝∈[−𝑢_min,𝑢_max].(2)A D-feasible trajectory, where there always exist four combinations of allowed forces and torques (𝐹₁,𝜏₁), (𝐹₂,𝜏₂), (𝐹₃,𝜏₃), and (𝐹₄,𝜏₄) such that 𝐹_𝑟(𝑡)<min𝐹₁,𝐹₂,−𝐹₃,−𝐹₄,𝜏_𝑟(𝑡)<min−𝜏₁,𝜏₂,−𝜏₃,𝜏₄. (10)On one hand, we consider that a trajectory is feasible when it can be followed by the vehicle; that is, when the necessary forces and torques to track the trajectory can be produced by values of 𝐹_𝑠 and 𝐹_𝑝 smaller than the maximum available. Notice that if a trajectory is not feasible, the only way to track it is to increase the size of the thrusters in order to increment the values of 𝐹_𝑠 and 𝐹_𝑝, allowing the trajectory to be feasible again. The control actions 𝐹_𝑟(𝑡) and 𝜏_𝑟(𝑡) that define a feasible trajectory lie on the dotted region of Figure 3.

Figure 3 

Feasible (dotted) and D-feasible (dashed) regions. A trajectory such as the green one is feasible because (𝐹𝑟(𝑡),𝜏𝑟(𝑡)) lies on the dotted region but is not D-feasible because it lies outside the dashed region.

On the other hand, we consider that a trajectory is D-feasible when the forces and torques selected from Table 2 are greater, in absolute value, than the forces and torques needed to track the trajectory. Notice that all the trajectories that are D-feasible are also feasible, but the opposite is not true. The control actions 𝐹_𝑟(𝑡) and 𝜏_𝑟(𝑡) that define a D-feasible trajectory lie on the dashed region of Figure 3.

3. Problem Formulation

The key objective of the present work is to design and to implement a feedback control law for 𝐹_𝑠 and 𝐹_𝑝 to solve the robust trajectory tracking problem for a vehicle commanded with a discrete set of inputs. In other words, the aim is the development of a controller that is able to track a desired trajectory in the presence of disturbances and noise with only a limited knowledge of the model of the vehicle and the actuators.

The main practical restriction is that the vehicle is underactuated and the control law must only employ a discrete set of inputs produced by the thrusters, 𝐹_𝑠,𝑝∈{−𝑢_min,0,𝑢_max}. Notice that the controller is oriented to be used on small autonomous vehicles equipped with simple thrusters because the addition of more thrusters would add complexity to the system and would increase the power consumption.

Consider the dynamics of the hovercraft described by (1)–(6), a D-feasible spatial reference trajectory 𝐱_𝑒, and the spatial tracking error defined by 𝐞_𝑝=[𝑥−𝑥_𝑟,𝑦−𝑦_𝑟,𝑣_𝑥−𝑣_𝑥𝑟,𝑣_𝑦−𝑣_𝑦𝑟]^𝑇. Then, the requirements for a robust trajectory tracking controller are as follows.(1)The error 𝐞_𝑝 is bounded in the presence of the bounded noise and disturbances described by (7).(2)The final bound of ‖𝐞_𝑝‖ can be made arbitrary small if the noise 𝑛_𝑚 and the disturbances 𝑝_𝑚 are small enough (or alternatively, if force and torque are large enough).

4. Nonlinear Control Design

The control design is based on the cascade structure illustrated in Figure 4. In the outer loop, the force and the orientation are used as virtual control inputs for the position controller to stabilize the position tracking error. The position controller defines an orientation reference 𝜓_𝑐 to be tracked by the inner loop, the orientation controller, that controls the orientation with the torque 𝜏. Therefore, the position controller determines the sign of the force 𝐹 to be applied on the RC-hovercraft, whereas the orientation controller determines the sign of the torque 𝜏. Once sign(𝐹) and sign(𝜏) are computed, the values of 𝐹_𝑠 and 𝐹_𝑝 are selected from Table 2.

Figure 4 

Control architecture.

4.1. Position Control

The objective of this subsection is the design of a controller for the position of the hovercraft. The error variables are defined by: 𝑒_𝑥=𝑥−𝑥_𝑟,𝑒_𝑦=𝑦−𝑦_𝑟,𝑒_𝑣𝑥=𝑣_𝑥−𝑣_𝑥𝑟,𝑒_𝑣𝑦=𝑣_𝑦−𝑣_𝑦𝑟. (11) And, their dynamics can be written as 𝑒_𝑥̇=𝑒_𝑣𝑥,𝑒_𝑦̇=𝑒_𝑣𝑦,𝑒̇_𝑣𝑥=𝐹cos𝜓−𝑑_𝑢𝑒_𝑣𝑥−𝐹_𝑥𝑟+𝑝_𝑣𝑥,𝑒̇_𝑣𝑦=𝐹sin𝜓−𝑑_𝑢𝑒_𝑣𝑦−𝐹_𝑦𝑟+𝑝_𝑣𝑦, (12) where 𝐹_𝑥𝑟=𝐹_𝑟cos(𝜓_𝑟) and 𝐹_𝑦𝑟=𝐹_𝑟sin(𝜓_𝑟) are the nominal forces needed to track exactly the reference in absence of disturbances. At this point, a positive control constant 𝑘₁, and the variables 𝑠_𝑥 and 𝑠_𝑦 are introduced in order to proceed with the error stabilization as follows: 𝑠_𝑥=𝑒_𝑥+𝑘₁𝑒_𝑣𝑥,𝑠_𝑦=𝑒_𝑦+𝑘₁𝑒_𝑣𝑦. (13)

It is easy to check from (13) that the tracking error variables converge exponentially to zero with rate 1/𝑘₁ if 𝑠_𝑥 and 𝑠_𝑦 are both equal to zero. Furthermore, the problem of tracking error stabilization is reduced to the stabilization of 𝑠_𝑥 and 𝑠_𝑦. Computing their dynamics, it is clear that 𝐹 and the orientation 𝜓 are coupled. Consider the following: 𝑠_𝑥̇=𝑘₁𝐹cos𝜓+1−𝑑_𝑢𝑘₁𝑒_𝑣𝑥−𝑘₁𝐹_𝑥𝑟+𝑘₁𝑝_𝑣𝑥,𝑠_𝑦̇=𝑘₁𝐹sin𝜓+1−𝑑_𝑢𝑘₁𝑒_𝑣𝑦−𝑘₁𝐹_𝑦𝑟+𝑘₁𝑝_𝑣𝑦. (14)

Therefore, the stabilization of both variables at the same time is an interesting and difficult problem. To overcome it, both control actions are decoupled by rotating the error vector an angle defined by the orientation angle 𝜓 in the body fixed frame of the vehicle, providing the following new coordinates: 𝑧₁=𝑠_𝑥cos𝜓+𝑠_𝑦𝜓,𝑧sin₂=−𝑠_𝑥sin𝜓+𝑠_𝑦cos𝜓. (15) Then, simple computations yield 𝑧̇₁=𝑘₁𝐹−𝑘₁𝐹_𝑟cos𝜓−𝜓_𝑟+𝑧₂𝑟+1−𝑑_𝑢𝑘₁𝑒_𝑣𝑥cos𝜓+𝑒_𝑣𝑦sin𝜓+𝑘₁𝑝₁,𝑧̇₂=𝑘₁𝐹_𝑟sin𝜓−𝜓_𝑟−𝑧₁𝑟+1−𝑑_𝑢𝑘₁−𝑒_𝑣𝑥sin𝜓+𝑒_𝑣𝑦cos𝜓+𝑘₁𝑝₂, (16)(17) where 𝑝₁ and 𝑝₂ are bounded disturbances given by 𝑝₁=𝑝_𝑣𝑥cos𝜓+𝑝_𝑣𝑦𝜓,𝑝sin₂=−𝑝_𝑣𝑥sin𝜓+𝑝_𝑣𝑦𝜓,‖𝐩‖=𝑝cos₁,𝑝₂^𝑇≤𝑝_𝑚. (18)

Analyzing the dynamics of 𝑧₁ and 𝑧₂, given by (16) and (17), it must be noticed that (17) does not depend on 𝐹, so 𝐹 must be used to control 𝑧₁. In order to solve this problem, the sign of 𝐹 must be opposed to the sign of 𝑧₁; thus, we propose (𝐹)=𝑧sign−sign₁,if𝑧₁(𝐹≥𝜀,sign⁻),if𝑧₁<𝜀, (19) where 𝜀 is a small constant to avoid chattering by holding the sign of the force to its previous value, sign(𝐹⁻), when |𝑧₁| is small. Notice that this condition imposes a minimal time between switches because |𝑧₁| must change from −𝜀 to +𝜀 (or from +𝜀 to −𝜀) before the force 𝐹 changes its sign. In addition, this control law tries to make |𝑧₁|<𝜀.

In order to stabilize 𝑧₂, the sign of 𝐹_𝑟sin(𝜓−𝜓_𝑟) must be the opposite of the sign of 𝑧₂. For this purpose, the following reference for the orientation is defined as follows: 𝜓_𝑐=𝜓_𝑟−𝑘₂tanh𝑧₂sign𝐹_𝑟,𝑟_𝑐=𝜓̇_𝑐=𝜓̇_𝑟−𝑧̇₂𝑘₂1−tanh𝑧₂²sign𝐹_𝑟. (20)(21) The saturation of 𝑧₂ given by the operation tanh in (20) is introduced to guarantee that, when 𝜓=𝜓_𝑐 and 0<𝑘₂<𝜋/2, then 𝐹_𝑟sin(𝜓_𝑐−𝜓_𝑟)=−|𝐹_𝑟|sin(𝑘₂tanh(𝑧₂)), that is, opposed to 𝑧₂.

In general, the orientation reference 𝜓_𝑐 is not perfectly tracked by 𝜓. Thus, the orientation errors are defined by 𝑒_𝜓=𝜓−𝜓_𝑐,𝑒_𝑟=𝑟−𝑟_𝑐, (22)(23) and the dynamics of 𝐳 finally become 𝑧̇₁=𝑘₁𝐹−𝑘₁𝐹_𝑟cos𝑒_𝜓+𝜓_𝑐−𝜓_𝑟+𝑧₂𝑟+𝑘₁𝑝₁+1−𝑑_𝑢𝑘₁𝑒_𝑣𝑥cos𝑒_𝜓+𝜓_𝑐+𝑒_𝑣𝑦sin𝑒_𝜓+𝜓_𝑐𝑧̇₂=−𝑘₁𝐹_𝑟sin𝑘₂tanh𝑧₂−𝑧₁𝑟+𝑘₁𝐹_𝑟𝑤+𝑘₁𝑝₂+1−𝑑_𝑢𝑘₁𝑒_𝑣𝑦cos𝑒_𝜓+𝜓_𝑐−𝑒_𝑣𝑥sin𝑒_𝜓+𝜓_𝑐, (24)(25) where 𝑤 is the disturbance produced by the orientation tracking error, 𝑤=sin(𝜓−𝜓_𝑟)−sin(𝜓_𝑐−𝜓_𝑟), bounded by |𝑤|≤𝜓−𝜓_𝑐=𝑒_𝜓, (26) Theorem 2 will show that for small 𝑒_𝜓, (24)-(25) is a stable system with disturbances 𝑒_𝜓, 𝑝₁, and 𝑝₂.

4.2. Orientation Control

The objective of this subsection is the design of an orientation controller so that 𝜓 can be able to track the control reference 𝜓_𝑐. The error dynamics yields, 𝑒_𝜓̇=𝜓̇−𝜓̇_𝑐=𝑟−𝜓̇_𝑐=𝑒_𝑟,𝑒_𝑟̇=𝜏−𝜏_𝑐−𝑑_𝑟𝑒_𝑟+𝑝_𝑟, (27) where 𝜏_𝑐=𝑑_𝑟𝜓̇_𝑐−𝜓̈_𝑐 is the control torque necessary to track the orientation reference 𝜓_𝑐. At this point, a positive control gain 𝑘₃ is introduced in order to define the variable 𝑠_𝜓 as follows: 𝑠_𝜓=𝑒_𝜓+𝑘₃𝑒_𝑟. (28) It is clear that the orientation error converges exponentially to zero with rate 1/𝑘₃ if 𝑠_𝜓 is equal to zero. The computation of its dynamics yields 𝑠_𝜓̇=𝑘₃𝜏−𝜏_𝑐+𝑝_𝑟+1−𝑘₃𝑑_𝑟𝑒_𝑟. (29) Thus, according to (29), in order to stabilize 𝑠_𝜓, the sign of 𝜏 must be the opposite of the sign of 𝑠_𝜓. This fact motivates the following control law: (𝜏)=𝑠sign−sign_𝜓,if𝑠_𝜓(𝜏≥𝜀,sign⁻),if𝑠_𝜓<𝜀, (30) where chattering is avoided in a similar way as in (19), holding the sign of the torque to its previous value, sign(𝜏⁻), when |𝑠_𝜓| is small. Notice that this control law tries to make |𝑠_𝜓|<𝜀.

5. Practical Implementation and Noise

The positions and velocities of the RC-hovercraft are measured to compute the control law. However, due to the presence of bounded noise 𝐧, only the estimates of the above variables (𝑥, 𝑦, 𝜓, 𝑣_𝑥̂, 𝑣_𝑦̂, and 𝑟̂) are available.

Using the estimates as the actual values of the state, the error variables are computed with (11), then 𝑠_𝑥̂ and 𝑠_𝑦̂ are computed using (13), and rotated with (15) to obtain the estimates 𝑧₁=𝑧₁+𝑛₁ and 𝑧₂=𝑧₂+𝑛₂, where 𝑛₁,𝑛₂≤‖𝐳‖+√21+𝑘₁𝑛_𝑚. (31) The term √2(1+𝑘₁) in (31) comes from the definition of 𝑠_𝑥 and 𝑠_𝑦, (13), while the term ‖𝐳‖ appears from (15) that involves the estimate 𝜓.

The computation of 𝑧₂̇ is carried out with (17) assuming that the unknown disturbance 𝑝₂ is equal to 0. The estimated orientation reference 𝜓_𝑐 is computed using (20), and its derivative 𝜓̇_𝑐 is computed with (21). These values are finally used in (28) to obtain the estimate 𝑠_𝜓̂=𝑠_𝜓+𝑓₃, where 𝑓₃≤𝑘₃𝑘₂𝑘₁𝑝_𝑚+𝑛_𝑚1+𝑘₃+𝑛_𝑚+𝑘₁𝐹_𝑟+|𝑟|+𝑘₂𝑛_𝑚1+𝑘₃𝑧̇₂‖𝐳‖+√21+𝑘₁+2√2𝑛_𝑚1−𝑑_𝑢𝑘₁𝑒²_𝑣𝑥+𝑒²_𝑣𝑦. (32) Thus, the control law (19) and (30) become (𝐹)=𝑧sign−sign₁,if𝑧₁(𝐹≥𝜀,sign⁻),if𝑧₁(𝜏)=𝑠̂<𝜀,sign−sign_𝜓,if𝑠_𝜓̂(𝜏≥𝜀,sign⁻),if𝑠_𝜓̂<𝜀. (33)(34) The control equations (33) and (34) define only the sign of 𝐹 and 𝜏. In order to determine the actual control actions to be applied to the vehicle, 𝐹_𝑒 and 𝐹_𝑝 are selected according to Table 2.

6. Stability Analysis

In this section, theoretical stability results and its practical applications are explained. Detailed proofs of the theorems are given in the Appendix.

Notation. In what follows, we consider that a function 𝛼(𝑥) is a 𝒦 class function if 𝛼(0)=0 and 𝛼 is strictly increasing. A function 𝛽(𝑥,𝑡) is class 𝒦ℒ if it is 𝒦 class with respect to 𝑥 and 𝛽→0 when 𝑡→∞, [29].

Lemma 1. The velocities of the vehicle are globally ultimately bounded. Proof in Section A.1.

The first result states that the velocities are bounded because the control actions and the disturbances that can increase the kinetic energy are bounded, and the terms that tend to reduce the kinetic energy are unbounded.

6.1. Position Stability

Theorem 2. Consider the system (24)-(25) with control action (33) in the presence of bounded disturbances and noise (7). If disturbances and noise are small enough, then, for any D-feasible trajectory 𝐱_𝑟, there exist three positive constants 𝑘₁, 𝑘₂, and 𝑤_𝑚, and two functions, 𝛽 and 𝛾, of class 𝒦ℒ and 𝒦, respectively, such that if |𝑤|≤𝑤_𝑚; then, ‖𝐳(𝑡)‖≤𝛽(‖𝐳(0)‖,𝑡)+𝛾𝑤_𝑚+𝑛_𝑚+𝑝_𝑚+𝜀. (35) Proof on Section A.2.

Corollary 3. Under the conditions of Theorem 2, there exist two functions 𝛽₂ and 𝛾₂, of class 𝒦ℒ and 𝒦, respectively, such that for |𝑤|≤𝑤_𝑚𝐞_𝑝(𝑡)≤𝛽₂𝐞_𝑝(0),𝑡+𝛾₂𝑤_𝑚+𝑛_𝑚+𝑝_𝑚+𝜀. (36) Proof in Section A.3.

Equation (35) means that for a small orientation error (and thus a small 𝑤), if noise and disturbances are not too large compared with the control action 𝐹, then, 𝐳 is bounded. Furthermore, as (36) states, the steady state tracking error (when 𝛽₂ vanishes) is a growing function depending on the orientation error, disturbances and noise. Thus, while the noise, disturbances, and orientation error increase, the tracking performance decreases.

6.2. Orientation Stability

Theorem 4. Consider the dynamics described by (29) with control action (34) in the presence of bounded disturbances and noise (7). If disturbances and noise are small enough, then there exist five positive constants 𝑧_𝑚, 𝑘₃, 𝑎, 𝑏, and 𝑐, and a 𝒦 class function 𝛾₃ such that for any D-feasible trajectory 𝐱_𝑟 and any positive gain 𝑘₂, if ‖𝐳‖≤𝑧_𝑚; then, 𝑠_𝜓(𝑡)≤max𝑠_𝜓(0)−𝑎𝑡,𝛾₃𝑏+𝑧_𝑚+𝑝_𝑚𝑛_𝑚+𝑐𝑝_𝑚𝜀. (37) Proof in Section A.4.

Corollary 5. Under conditions of Theorem 4, there exists another constant 𝑑 such that for ‖𝐳(𝑡)‖≤𝑧_𝑚, 𝑒_𝜓(𝑡)≤𝑑⋅𝑒^{−𝑡/𝑘3}+𝛾₃𝑏+𝑧_𝑚+𝑝_𝑚𝑛_𝑚+𝑐𝑝_𝑚+𝜀. (38) Proof in Section A.5.

Equation (37) means that if the tracking error is bounded by 𝑧_𝑚, and disturbances and noise are not too large compared with the control input 𝜏, then the variable 𝑠_𝜓 is stabilized in finite time. Thus, according to (38), the orientation error converges exponentially. Notice that the performance decreases while noise and disturbance levels increase, as it occurs with the position tracking.

6.3. Overall Convergence

Previous results have demonstrated that the orientation error, 𝑒_𝜓, converges exponentially to a neighborhood of the origin while ‖𝐳‖ is bounded by 𝑧_𝑚. Thus, (26) implies that 𝑤 converges to a neighborhood of the origin also. Furthermore, when 𝑤 is smaller than an arbitrary maximum value 𝑤_𝑚, then 𝐳 converges to a neighborhood of the origin too.

The following result states that control laws (33) and (34) can stabilize the tracking error when both are used at the same time.

Theorem 6. Consider the dynamics of the hovercraft, (1)–(6), in the presence of bounded disturbances and noise, (7), and subject to control actions (33)-(34). For any positive constant 𝑀, if the initial condition of the hovercraft is such that ‖𝐳(0)‖<𝑀, and the reference trajectory 𝐱_𝑟 is D-feasible, then it is possible to choose the control constants 𝑘₁, 𝑘₂, 𝑘₃, and 𝜀 such that 𝐞_𝑝(𝑡)≤𝛽₃𝐞_𝑝(0),𝑡+𝛾₄𝑛_𝑚+𝑝_𝑚+𝜀, (39) where 𝛾₄ is a class 𝒦 function and 𝛽₃ is a class 𝒦ℒ function. Proof in Section A.6.

This result states that given any bound for the initial condition 𝑀 and a trajectory that is D-Feasible; then the constants of the tracking controller can be set in order to stabilize the tracking error.

There exists a compromise between the tracking performance and the control effort imposed by 𝜀. If 𝜀 is too large, then the tracking performance is poor, as shown in (39). If 𝜀 is too small, then (33) and (34) will produce fast switches in the control law.

Remark 7. It must be noticed that the control gains, and thus, the tracking performance, depend on the bounds of the initial conditions (see (A.24) of the Appendix for more details). This result is conservative in the sense that if we want to increase the region of attraction (increasing 𝑀), then the gain 𝑘₂ must be small, and thus the effect of noise and disturbances is larger. The result is also conservative in the sense that it restricts its applicability to D-feasible trajectories, but in practice the system is stable from any initial condition with a good selection of the control gains.

7. Results

In this section, simulations and experiments on a real setup have been developed in order to test and illustrate the potential of the control design of Section 4.

7.1. Simulation Results

The forthcoming simulation examples consist in tracking a circular reference with and without noise and disturbances. Then, a more general trajectory tracking problem is studied to show that the control design can overcome some of the limitations that are assumed on the stability demonstrations. The control parameters used in the simulations are 𝑘₁=6 s, 𝑘₂=1 s, 𝑘₃=1 s, and 𝜀=0.01.

7.1.1. Circular Trajectory without Noise and Disturbances

The reference trajectory is a circumference with its center at the origin of the inertial coordinate frame and with a radius, 𝑅, of 2 meters. The trajectory is tracked with a speed of 𝑉=0.5 m/s. The force and torque needed to track perfectly this trajectory (computed as described in Section 2.2) are constants (𝐹_𝑟=0.2179 m/s² and 𝜏_𝑟=0.4339 rad/s²), so the trajectory is D-feasible because they do not saturate actuators. The trajectory of the hovercraft is depicted in Figure 5. The vehicle starts from the origin, without disturbances and noise. This figure shows how the trajectory followed by the hovercraft converges and tracks almost perfectly the reference trajectory.

Figure 5 

Simulated trajectory (solid) and circular reference (dashed).

The evolution of the position is shown in Figure 6 and velocities are shown in Figure 7. It must be noticed that the actual trajectory converges to the reference trajectory in a time close to 20 s.

(a)

(b)

Figure 6 

Simulated position (solid) versus references (dashed).

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(b)

Figure 7 

Simulated velocities (solid) versus references (dashed).

The orientation 𝜓 and its control reference 𝜓_𝑐 are shown in Figure 8. The orientation converges faster than the position (less than 5 seconds). The reason is intuitively clear because the position only converges when the orientation error, 𝑒_𝜓, is small enough. Once 𝑒_𝜓 is smaller than 𝑤_𝑚, 𝐳 starts to converge to zero and the control references 𝜓_𝑐 and 𝑟_𝑐 converge to the references 𝜓_𝑟 and 𝑟_𝑟, too.

(a)

(b)

Figure 8 

Simulated orientation and yaw rate (solid) vs references (dashed).

7.1.2. Effect of Noise and Disturbances

Noise and disturbances are introduced in the simulations to study their effects on the control performance. In order to make the simulation more realistic, the controller is simulated with a sampling period of 1/17 s, that corresponds to the frame rate of the camera in the real setup. In addition, a disturbance 𝐩=[0m/s²,0.02m/s²,0.02m/s²] and random noise with an amplitude of 0.05 (meters for 𝑥 and 𝑦, m/s for 𝑣_𝑥 and 𝑣_𝑦, rad for 𝜓, and rad/s for 𝑟) are introduced in the simulation.

The effect of noise and disturbances on the performance depends strongly on the reference trajectory that must be tracked. To illustrate this effect, Figure 9 shows the trajectories followed by the system when the reference speed is fixed and the radius varies from 1 to 4 meters. This figure shows how the performance decreases for small radii for two reasons. The first one is that the reduction of the radius reduces the signal/noise ratio. The second one deals with centripetal forces. While the radius decreases, centripetal forces increase with a factor 𝑉²/𝑅. This implies that forces increase and the trajectories are pushed to the unfeasible region. In this case, the impact of noise and disturbance becomes more evident because the control authority is degraded.

Figure 9 

Simulated trajectories with noise and disturbances for different circle radius 𝑅=1,2,3,4 meters in red, violet, green and blue, respectively.

Figure 10 confirms the behavior predicted by Theorem 6. First, the trajectories experiment a transient that depends on the initial conditions. This transient is of approximately 25 s during which the tracking error decreases to the steady state value. After this time, only a residual tracking error remains. It shows how the tracking error increases for small radii because the trajectory is more difficult to track (close to be unfeasible).

Figure 10 

Tracking error for different circle radius 𝑅=1,2,3,4 meters in red, violet, green, and blue, respectively.

7.1.3. General Trajectory

For this third scenario, a new reference trajectory is set, 𝑥_𝑟(𝑡)=2sin(3/8𝑡), 𝑦_𝑟(𝑡)=3cos(2/8𝑡), in order to test the control design. The necessary forces and torques to track the trajectory are depicted in Figure 11. It can be noticed that the forces lie outside the D-feasible region; however, the trajectory is feasible. This example is used to illustrate that the D-feasible condition (10) is conservative. Furthermore, Figure 12 shows that the same control design used for the circular trajectory converges to this non-D-feasible reference too.

Figure 11 

Force map for a general trajectory that is feasible but not D-feasible.

Figure 12 

General trajectory followed by the vehicle (blue) and reference (red).

This behavior can be explained from an intuitive point of view. D-feasible conditions (10) indicate that the reference force and torque must be smaller, in absolute value, that any force and torque than can be selected by the control law from Table 2. However, this is a sufficient condition but not always a necessary condition. For example, suppose that the trajectory is such that 𝐹_𝑟 is positive for all the time (the current scenario, as Figure 11 shows), then it is possible to select a positive force that is larger than 𝐹_𝑟, with a larger torque, in absolute value, than 𝜏_𝑟. This is enough to stabilize the trajectory because during the trajectory tracking a negative force is never needed.

7.2. Experimental Results

7.2.1. Experimental Setup

The experimental setup consists in an underactuated radio controlled RC-hovercraft, a CCD camera to capture images from a zenithal point of view, and a computer to send the control commands via a radio control interface. The whole system is shown in Figure 13. The laboratory has similar capacities to the experimental testbed developed on [5] or [6].

Figure 13 

Experimental setup.

The CCD camera is equipped with wide angle optics that cover a region of approximately 3m×4m when the camera is placed at 4 meters over the floor. The frame rate of the computer-camera system is 17 fps (frames per second).

The position of the vehicle is tracked by the camera by detecting three LEDs that are attached to the RC-hovercraft. These 3 LEDs compose an isosceles triangle to determine the vehicle orientation. The triangle size is 25cm of length in its major edge and 16 cm in its minor edge.

The control signals are sent to the vehicle through a communication interface that connects the parallel port of the computer with the radio controller. The control law is implemented in LabVIEW programming language. The control algorithm works as follows.(1)Once a frame is captured, the image is binarized and the centroid of each individual LED is computed.(2)The barrel distortion introduced by the lenses is corrected and the undistorted positions of the LEDs are obtained by solving the perspective transformation.(3)The centre of the mass of the vehicle and its orientation are computed from the positions of the triangle vertex. The velocities are estimated by a finite difference.(4)Finally, the control law computes the forces 𝐹_𝑠 and 𝐹_𝑝, and the corresponding commands are sent to the radio controller.

For a more detailed description of the experimental setup, see [30].

7.2.2. Real Tracking Test

The control design is now tested on the experimental setup described above. To check the robustness of the controller, a worst case analysis has been realized. A trajectory with a very small radius, 𝑅=0.6 m, is tracked with a constant speed of 𝑉=0.5 m/s. Notice that this trajectory is very hard to track because of its small radius as above mentioned in Section 7.1.2. Figure 14 shows that the vehicle is able to track this very challenging trajectory with an acceptable tracking error. As far as the authors know, the performance achieved is comparable to the best results obtained with a laboratory hovercraft (see [10]). Notice the added hard limitation imposed by the all-nothing actuators whose dynamics are unknown. Figure 16 shows the evolution of the position of the system and demonstrates that the control law is able to track the trajectory. Figure 17 shows the velocities. Notice that the measured velocity is noisy due to the all-nothing-reverse action of the propellers. This problem can be reduced by filtering the position measurements, but the aim of this example is to show that this controller can deal with these problems and the vehicle is able to track the velocity reference with great accuracy. Figure 15 shows that the final bound of the tracking error is about 0.4 m and thus compatible with simulation results shown on Figure 10. Finally, Figure 18 shows the input thrust applied to the vehicle.

Figure 14 

Actual tracking. Trajectory (solid) and references (dashed).

Figure 15 

Tracking error.

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(b)

Figure 16 

Actual positions (solid) versus references (dashed) in a real setup.

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(b)

Figure 17 

Actual velocities (solid) versus references (dashed) in a real setup.

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(b)

Figure 18 

Control actions 𝐹𝑠 and 𝐹𝑝 applied in a real setup.

8. Conclusions

In this paper, a tracking control law for an underactuated nonholonomic vehicle with a discrete set of inputs has been designed and tested. The control law has been designed taking into account the fact that the actuators of the vehicle can only produce thrust in an all-nothing reverse fashion.

Despite this hard limitation on the real setup, this paper shows how this control design can overcome these difficulties. Furthermore, this work demonstrates that a low cost underactuated vehicle with very simple actuators (capable of moving the vehicle forward and backwards, and simultaneously turn right and left) can be controlled in order to track any D-feasible trajectory under the sole assumption that control actions are large enough compared with noise and disturbances.

Theoretical proofs of stability and robustness against noise and disturbances have been obtained showing that the control law makes the position tracking error to be semiglobal finitely bounded if the vehicle starts with a bounded tracking error and the trajectory is D-feasible.

Simulations and experimental results state that the control strategy is robust enough to be applied to a real environment with disturbances, noise, and unmodeled dynamics. In addition, this control law is very suitable to be applied to low cost and simple vehicles where precise models of the actuators and dynamics are not available.

Appendix

A. Proofs

This section contains the proofs of the theoretical results.

A.1. Proof of Lemma 1

Consider the functions 𝐿₁=𝑣²_𝑥/2, 𝐿₂=𝑣²_𝑦/2, and 𝐿₃=𝑟²/2 and compute the derivative along the trajectories of the vehicle 𝐿̇₁=−𝑣_𝑥(𝑑_𝑢𝑣_𝑥−𝐹cos(𝜓)+𝑝_𝑣𝑥). Defining 𝐹_max and 𝜏_max as the maximum values of |𝐹| and |𝜏| that can be selected from Table 2, then 𝐿̇₁ can be bounded by 𝐿̇₁<−𝑑_𝑢𝑣²_𝑥𝐹1−^max+𝑝_𝑚𝑑_𝑢𝑣_𝑥. (A.1) Equation (A.1) shows that 𝐿̇₁ is strictly negative when 𝑑_𝑢|𝑣_𝑥|>(𝐹_max+𝑝_𝑚). This implies that 𝑣_𝑥 is ultimately bounded by (𝐹_max+𝑝_𝑚)/𝑑_𝑢. The same analysis can be done with 𝐿₂ and 𝐿₃ showing that the set Ω=𝐱∈ℜ⁶∶𝑣_𝑥,𝑣_𝑦≤𝐹_max+𝑝_𝑚𝑑_𝑢,|𝑟|≤𝜏_max+𝑝_𝑚𝑑_𝑟 (A.2) is a positively invariant set of the system (1)–(6).

A.2. Proof of Theorem 2

Without loose of generality, the analysis is restricted to the region defined by Ω. If noise and disturbances are small enough, then there exists a positive constant 𝛿 such that |𝐹|−|𝐹_𝑟|−𝑝_𝑚≥𝛿, by definition of a D-feasible trajectory. Then, for any positive 𝜀₁, it is always possible to find a constant 𝑘₁ such that √21−𝑑_𝑢𝑘₁𝑑_𝑢𝑘₁2𝐹_max+𝑝_𝑚≤𝛿𝜀₁. (A.3) Moreover, if disturbances are small enough, then it is also possible to select 𝑤_𝑚 so that √2𝑝_𝑚+𝛿𝜀₁+𝐹_𝑟𝑤_𝑚<min𝛿,𝐹_𝑟sin𝑘₂. (A.4) Consider the following Lyapunov candidate function 𝑉=(𝑧²₁+𝑧²₂)/2. The computation of its derivative yields 𝑉̇=𝑘₁𝑧₁𝐹−𝐹_𝑟cos𝜓−𝜓_𝑟+𝑝₁−𝑘₁𝑧₂𝐹_𝑟sin𝑘₂tanh𝑧₂−𝐹_𝑟𝑤−𝑝₂+1−𝑑_𝑢𝑘₁𝑠₁𝑒_𝑣𝑥+𝑠₂𝑒_𝑣𝑦. (A.5) For an appropriate stability analysis, two cases must be taken into account.

Case A ( |𝑧₁|>𝑛₁+𝜀). In this analysis, our aim is to bound all the terms in (A.5). The term |𝐹_𝑟cos(𝜓−𝜓_𝑟)| can be bounded by |𝐹_𝑟|, and |𝑧₁| can be bounded by ‖𝐳‖. Then, we define the following function: 𝑔(𝑥)𝛿=min√2,𝐹_𝑟√2sin𝑘₂𝑥tanh√2. (A.6) Condition (A.3), in combination with (A.6), implies that −𝛿|𝑧₁|−|𝐹_𝑟𝑧₂|sin(𝑘₂tanh(|𝑧₂|))≤−‖𝐳‖𝑔(‖𝐳‖), so 𝑉̇≤−𝑘₁‖𝐳‖𝑔(‖𝐳‖)−𝛿𝜀₁+𝐹_𝑟|𝑤|+𝑝_𝑚. (A.7) Equation (A.7) is negative when 𝑔(‖𝐳‖)>𝜀₁+|𝐹_𝑟|𝑤_𝑚+𝑝_𝑚. This condition holds when ‖𝐳‖>𝑧_max1, where 𝑧_max1 is defined as 𝑧_max1=√2tanh⁻¹1𝑘₂sin⁻¹√2𝑝_max+𝛿𝜀𝐹_𝑟+𝑤_𝑚. (A.8) Note that condition (A.4) implies that 𝑧_max1 is finite and 𝑉 decreases when ‖𝐳‖>𝑧_max1.

Case B ( |𝑧₁|≤𝑛₁+𝜀). In this case, we have to establish the bound of ‖𝐳‖. On one hand, we suppose that |𝑧₁|>|𝑧₂|, and with (31), we find ‖𝐳‖≤√2𝑧₁≤√2𝑛₁+𝜀≤√2‖𝐳‖+√21+𝑘₁𝑛_𝑚+𝜀. (A.9) So, for any positive 𝜀₂, if 𝑛_𝑚 is small enough, it yields ‖𝐳‖≤√2√21+𝑘₁𝑛_𝑚+𝜀1−√2𝑛_𝑚=𝜀₂. (A.10)

On the other hand, if |𝑧₁|≤|𝑧₂| then ‖𝐳‖≤√2|𝑧₂|, so we find in both cases that ‖𝐳‖≤√2|𝑧₂|+𝜀₂ and |𝑧₁|≤2(|𝑧₂|+1−𝑘₁)𝑛_𝑚+𝜀. Thus, the derivative of 𝑉 can be bounded as 𝑉̇≤𝑘₁2𝑧₂+1−𝑘₁𝑛_𝑚+𝜀𝐹_max+𝐹_𝑟+𝑝_𝑚−𝑘₁𝑧₂𝐹_𝑟sin𝑘₂tanh𝑧₂−𝐹_𝑟𝑤−𝑝_𝑚+2√2𝑧₂+𝜀₂𝑑_𝑢𝑘₁𝛿𝜀₁. (A.11)

Grouping all terms that can be made arbitrary small by reducing noise and disturbances in 𝜀₃ and 𝜀₄, (A.11) can be rewritten as 𝑉̇≤−𝑘₁𝑧₂𝐹_𝑟sin𝑘₂tanh𝑧₂−𝐹_𝑟𝑤−𝑝_𝑚−𝜀₃+𝜀₄. (A.12) Now, we define 𝑧_max2=tanh⁻¹1𝑘₂sin⁻¹𝑝_𝑚𝐹_𝑟+𝜀₃+𝑤_𝑚+𝛿. (A.13) Combining (A.13) with 𝑉̇, and grouping all the small terms in 𝜀₅; then 𝑉̇≤𝑘₁(−|𝑧_max2|𝛿+𝜀₅) that is negative if 𝜀₅ is small enough compared with 𝛿.

Conclusion. Defining 𝑧_max=max(𝑧_max1,𝑧_max2), 𝑉 decreases outside the bounded region ‖𝐳‖≤𝑧_max for small enough 𝑛_𝑚. Since 𝑉 is definite positive and radially unbounded, then 𝐳 is globally finitely bounded by 𝑧_max1. Furthermore, as 𝑧_max1 and 𝑧_max2 are class 𝒦 functions of 𝑤_𝑚, 𝑛_𝑚, 𝑝_𝑚, and 𝜀, then (35) holds.

A.3. Proof of Corollary 3

The bound of ‖𝐳‖ implies that ‖𝐬‖ is also bounded because (15) shows that ‖𝐬‖=‖𝐳‖. Then, by definition of 𝑠_𝑥 and 𝑠_𝑦, 𝑘₁𝑒_𝑥̇=𝑠_𝑥−𝑒_𝑥,𝑘₁𝑒_𝑦̇=𝑠_𝑦−𝑒_𝑦. (A.14) This system is linear and stable; thus, it is ISS with respect to 𝐬, that is globally ultimately bounded since ‖𝐬‖=‖𝐳‖. Therefore, 𝐞_𝑝 is ISS with respect to 𝑛_𝑚, 𝑝_𝑚, 𝑤_𝑚, and 𝜀; then (36) holds.

A.4. Proof of Theorem 4

Without loose of generality, consider the initial state of the system (1)–(6) lying inside the set Ω. From (23), |𝑒_𝑟|≤|𝑟|+|𝑟_𝑟|≤|𝜏_𝑚|/𝑑_𝑟+|𝑟_𝑟|. The velocities are bounded in Ω, so (24)-(25), imply that the first derivatives of 𝐳 can also be bounded, and differentiating (25) it is straightforward to check that there exist positive constants 𝑐₁--𝑐₆ such that ‖𝐳̇‖≤𝑧̇₁+𝑧̇₂≤𝑐₁+𝑐₂𝑧_𝑚+𝑐₃𝑝_𝑚,𝑧̈₂≤𝑐₄+𝑐₅𝑧_𝑚+𝑐₆𝑝_𝑚. (A.15) By the definition of 𝜏_𝑐, it yields 𝜏_𝑐=𝜏_𝑟−𝑘₂sign𝐹_𝑟1−tanh²𝑧₂𝑑_𝑟𝑧̇₂+𝑧̈₂+2𝑘₂sign𝐹_𝑟1−tanh²𝑧₂tanh𝑧₂𝑧̇²₂. (A.16) Substituting (A.15) into (A.16) and (32), the effect of disturbances and noise on the computation of 𝑠_𝜓 and 𝜏_𝑐 can be bounded with a positive constant, 𝑐₇, and two polynomial bounded functions, 𝛾₄ and 𝛾₅, of class 𝒦 as follows: 𝜏_𝑐≤𝜏_𝑟+𝑘₂𝛾₄𝑧_𝑚+𝑝_𝑚,𝑓₃≤𝛾₅𝑐₇+𝑧_𝑚+𝑝_𝑚𝑛_𝑚+𝑘₃𝑘₂𝑘₁𝑝_𝑚. (A.17)(A.18) Then, |𝜏_𝑟|<|𝜏| because the reference trajectory is D-feasible and thus there exists a positive constant 𝛿 such that |𝜏|−|𝜏_𝑟|≤𝛿. Now, we define 𝑧_𝑚 as 𝑧_𝑚1=−2𝛾₄⁻¹𝛿4𝑘₂. (A.19) When ‖𝐳‖<𝑧_𝑚, the disturbances are small enough such that 𝑘₂𝛾₄(2𝑝_𝑚)+𝑝_𝑚<𝛿/4, and with (A.17), then |𝜏_𝑖|−|𝜏_𝑐|−|𝑝_𝑚|≥𝛿−𝛿/4−𝛿/4=𝛿/2. Now, we select 𝑘₃ such that 1−𝑘₃𝑑_𝑟𝑘₃𝑑_𝑟𝜏_𝑚+𝑑_𝑟𝑟_𝑟≤𝛿4. (A.20) And we define the Lyapunov function 𝑉₂=𝑠²_𝜓/2, whose time derivative is 𝑉̇₂=𝑠_𝜓(𝑘₃(𝜏−𝜏_𝑐+𝑝_𝑟)+(1−𝑘₃𝑑_𝑟)𝑒_𝑟). Consider first that |𝑠_𝜓|≥𝑓₃+𝜀; then the control law implies that the sign of 𝜏 is the opposite of the sign of 𝑠_𝜓. Condider the following: 𝑉̇₂=𝑠_𝜓𝑠_𝜓̇≤−𝛿𝑘₃12−14𝑠_𝜓⟶𝑠_𝜓̇≤−𝛿𝑘₃4sign𝑠_𝜓. (A.21) Solving (A.21) for the equality and applying the comparison lemma (see chapter 3 of [29]), we find that |𝑠_𝜓(𝑡)|≤|𝑠_𝜓(0)|−𝛿𝑘₃/4𝑡. If 𝑠_𝜓 lies inside the region |𝑠_𝜓|<𝑓₃+𝜀, (A.21) does not hold. Nevertheless, 𝑉₂ decreases on the frontier of this region, preventing 𝑠_𝜓 from becoming larger than 𝑓₃+𝜀. Then, in any case |𝑠_𝜓(𝑡)|≤max(|𝑠_𝜓(0)|−𝛿𝑘₃/4𝑡,𝑓₃+𝜀), and substituting the bound (A.18); then (37) is obtained.

A.5. Proof of Corollary 5

By the definition of 𝑠_𝜓 in (28), the orientation error dynamics become 𝑘₃𝑒_𝜓̇=𝑠_𝜓−𝑒_𝜓, whose solution is 𝑒_𝜓(𝑡)=𝑒_𝜓(0)𝑒^{−𝑡/𝑘3}+1𝑘₃^𝑡₀𝑒^{−(𝑡−𝜏)/𝑘3}𝑠_𝜓(𝜏)𝑑𝜏. (A.22) If the terms inside the integral are bounded by (37), then 𝑒_𝜓(𝑡)≤𝑒_𝜓(0)𝑒^{−𝑡/𝑘3}+𝑠_𝜓(0)𝑒^{|𝑠𝜓(0)|/𝑎𝑘3}−1𝑒^{−𝑡/𝑘3}+𝛾₃𝑏+𝑧_𝑚+𝑝_𝑚𝑛_𝑚+𝑐𝑝_𝑚+𝜀. (A.23) That takes the form of (38) with 𝑑=sup_Ω{|𝑒_𝜓|+|𝑠_𝜓|𝑒^{|𝑠𝜓|/𝑎𝑘3}}.

A.6. Proof of Theorem 6

The key point of the proof is that if the vehicle starts from a bounded initial condition ‖𝐳(0)‖<𝑀, where 𝑀 is an arbitrary constant, it is always possible to choose the control gain 𝑘₂ such that during the time that 𝑤 needs to converge to 𝑤<𝑤_𝑚, ‖𝐳‖ never reaches 𝑧_𝑚. Therefore the conditions of Theorem 4 always hold and the tracking error is finitely bounded.

Select 𝑘₂ so that (A.19) of Theorem 4 is satisfied for 𝑧_𝑚=2𝑀 as follows: 𝑘₂=𝛿4𝛾₄().4𝑀 (A.24) As ‖𝐳(0)‖<𝑀, it will take sometime for 𝑇 to go from ‖𝐳‖=𝑀 to ‖𝐳‖=2𝑀. This time 𝑇 can be lower bounded by considering the maximum derivative of 𝑉. 𝑉̇≤𝑘₁𝑧₂𝐹_𝑟𝑤+𝑝₂+1−𝑑_𝑢𝑘₁𝑠₁𝑒_𝑣𝑥+𝑠₂𝑒_𝑣𝑦≤𝑏₁|𝐳|+𝑏₂, (A.25) for some constants 𝑏₁ and 𝑏₂. Select 𝑀>𝑏₂/𝑏₁ then ‖𝐳‖>𝑏₂/𝑏₁ and thus 𝑉̇=𝐳𝐳̇≤2𝑏₁𝐳→‖𝐳̇‖≤2𝑏₁→‖𝐳‖≤‖𝐳(0)‖+2𝑏₁𝑡. Hence, ‖𝐳‖ can only grow with a finite rate of 2𝑏₁ and therefore 𝑇>𝑀/2𝑏₂.

Consider noise and disturbances small enough such that 𝛾₃(𝑏+𝑀+𝑝_𝑚)𝑛_𝑚+𝑐𝑝_𝑚+𝜀≤𝑑𝑒^{−𝑇/𝑘3}. From (38), it yields that |𝑒_𝜓(𝑇)|≤2𝑑⋅𝑒^{−𝑇/𝑘3}. In addition, for small enough disturbances compared with the control action, consider the following: 𝑝max^𝑚+𝜀₅𝐹_𝑟,𝑝_𝑚+𝛿𝜀𝐹_𝑟<𝑑⋅𝑒^{−𝑇/𝑘3}. (A.26) Substituting (A.26) into (A.8) and (A.13), it yields 𝑧_max(𝑇)=√2tanh⁻¹4𝛾₄(4𝑀)𝛿sin⁻¹3√2𝑑⋅𝑒^{−𝑀/2𝑐2𝑘3}. (A.27) Function 𝛾₄ is a polynomial order function in ‖𝐳‖, bounded by 𝑀. Therefore, when 𝑀→∞, then 𝑧_max→0, and for a large enough 𝑀, then 𝑧_max(𝑇)≤2𝑀=𝑧_𝑚.

The result is now proved by contradiction. Suppose that there exists some time 𝑡₁ for which ‖𝐳(𝑡₁)‖>2𝑀, then, there must exist another time 𝑡₂ for which ‖𝐳(𝑡₂)‖=2𝑀 and 𝑉 grows in 𝑡₂. But 𝑉 only grows if ‖𝐳‖<𝑧_max, so, according to (26), 𝑧_max(|𝑤(𝑡₂)|)≤𝑧_max(|𝑒_𝜓(𝑡₂)|)=𝑀. However, |𝑒_𝜓(𝑡₂)|≤2𝑑𝑒^𝑇/𝑘3 and therefore ‖𝐳(𝑡₂)‖<𝑧_max(|𝑤(𝑡₂)|)≤𝑧_max(|𝑒_𝜓(𝑡₂)|)=𝑀, because ‖𝐳‖<𝑧_𝑚 for all 𝑇≤𝑡₁≤𝑡₂. This drives to a contradiction because ‖𝐳(𝑡₂)‖=2𝑀.

Therefore, ‖𝐳‖ never reaches 𝑧_𝑚 and conditions of Theorem 4 always hold. This means that before some time 𝑡₂ conditions of Theorem 2 holds, thus we can conclude by (39) that ‖𝐞_𝑝‖ is finitely bounded.

Acknowledgments

The authors would like to acknowledge the Spanish Ministry of Economy and Competitiveness for its financial support under Grant DPI 2009-14552-C02-02 and the UNED University for the support under Research Project 2012V/PUNED/0003.