2.1. Specifying the navigation objective

When moving along the desired path $\mathscr{P}$ , the robot may arrive at a position where it is entrapped within a narrow “cave” of the exterior $\mathscr{E}$ of the targets: there is no opportunity for further movement, while complying with the requirement not to get too close to the targets by remaining in $\mathscr{E}$ . Such a situation is exemplified in Figure 2(a), where the red dashed circles illustrate attempted escapes from the confined space. The discussed sort of entrapment occurs if the width of a “cave” is less than the diameter $2 \mathcal{R}$ of the sharpest $U$ -turn of the robot or, otherwise stated, if it is impossible to place a $\mathcal{R}$ -disc in the cave. Though the robot might find the possibility to turn back deeper in the cave, not confidently knowing this due to the limited visibility range is a rationale to avoid risks of intruding in areas with even slightest and nebulous signs of being “narrow.”

Figure 2.

(a) Being stuck in a cave; (b) The set Eq. (5) with two connected components.

Figure 2 Long description

The diagram consists of two parts labeled (a) and (b). In part (a), multiple circles are arranged in a pattern with a central circle highlighted by a dashed red line labeled 2R. The circles are interconnected, and an arrow indicates movement. In part (b), two connected components are shown, with circles grouped within each component. The outer perimeter of component 1 is highlighted, and a smaller area within component 2 is marked. The distance between certain circles is labeled as less than 2R, and a triangular area is shaded in red within the circles.

In the dry residue, it is reasonable for the robot to travel so that it is always covered with a continuously moving $\mathcal{R}$ -disc that lies in $\mathscr{E}$ . Such a disc is said to be accompanying, and any robot’s trajectory for which the above condition is satisfied is said to be trustworthy. Among them, the most appealing option is the trajectory nearest to the targets. When looking for it, it is worth taking into account that in general, trustworthy trajectories may cluster into several groups, each with its own group-wide optimal element, whereas competition among various groups does not make sense.

To clarify the last assertion, we firt observe that the center of the accompanying disc moves in the set

(5) \begin{equation} \mathscr{E}^{\,\,\downarrow }_{\mathcal{R}}\,:\!=\,\{{\boldsymbol{c}} \in \mathscr{E}\,:\, \textbf {dist}\left [{\boldsymbol{c}};\mathscr{P}\right ] \geq \mathcal{R}\};\, \end{equation}

see Figure 1a. Since this center moves continuously, it remains in a specific connected component $Q$ of this set. (Figures 2(a) and (b) illustrate scenarios where only one and, respectively, two such components exist.) Consequently, the robot remains in the $\mathcal{R}$ -neighborhood of $Q$ :

(6) \begin{equation} N_{\mathcal{R}}(Q)\,:\!=\, \{{\boldsymbol{r}}\,:\, \textbf {dist}\left [{\boldsymbol{r}};\,Q\right ] \leq \mathcal{R} \} . \end{equation}

As can be shown, the neighborhoods of distinct components $Q$ have no points in common. It follows that the set of trustworthy trajectories is partitioned into non-intersecting classes, where each class consists of all trajectories belonging to the neighborhood Eq. (6) of a specific component $Q$ . The following assumption means that in this paper, we limit ourselves to the case where there is only one component and class.

In turn, this yields the following.

Hence, a trustworthy trajectory is optimal whenever

(7) \begin{equation} \textbf {dist}\left [{\boldsymbol{r}}(t);\,\partial N_{\mathcal{R}}(\mathscr{E}^{\,\,\downarrow }_{\mathcal{R}})\right ] \to 0 \quad \text{as} \, t \to \infty . \end{equation}

The formal control objective is in fact defined in the following.

It is needed to design a controller that ensures this property.

The last remark on the problem statement is bottomed on the note that the above concept of the accompanying disc tacitly accepts feasibility to turn back inside a cave by moving with the minimum turning radius even if the respective circular path multiply touches the boundary of the forbidden region. However, then the robot is unable to counteract accidental disturbances that push it to that region since an even sharper turn might be necessary to avoid this.

To overcome this challenge, the value of $\mathcal{R}$ is in fact taken greater than the robot’s minimum turning radius $\mathcal{R}_{\,\min }\,:\!=\,v/\overline {u}$ in all constructions, considerations, assumptions, and definitions from Section 2.1:

(8) \begin{equation} \mathcal{R} \gt \mathcal{R}_{\,\min }. \end{equation}

When doing so, both Eq. (4) and Asm. 2.2 should be kept valid, which is possible by taking, e.g., $\mathcal{R} \gtrsim \mathcal{R}_{\,\min }$ . As a result, $\mathcal{R}$ gets the status of a parameter of the navigation law.

3.1. Core idea of the navigation law

This idea is borrowed from refs. [Reference Matveev, Teimoori and Savkin29, Reference Matveev, Wang and Savkin30] and is represented by the following equation:

(9) \begin{align} u = \overline {u} \cdot \textbf {sgn}\, \big \{\,\, \dot {\!\!d}_{\mathcal{R}}(t) + \mu \chi [d_{\mathcal{R}}]\big \} . \end{align}

Here $\mu \gt 0$ is a free parameter and $d_{\mathcal{R}}$ is the distance from the robot to the targeted path $\partial N_{\mathcal{R}}(\mathscr{E}^{\,\,\downarrow }_{\mathcal{R}})$ . To build the controller Eq. (9), a continuous and piece-wise smooth function $\chi (z) \in \mathbb{R}$ of $z \in \mathbb{R}$ should be first chosen so that

(10) \begin{align} z \chi (z) \gt 0 \, \forall z \neq 0, \quad \chi (0) = 0 , \end{align}

(11) \begin{align} \overline {\chi } \,:\!=\, \sup _{z \in \mathbb{R}} |\chi (z)| \lt \infty , \quad \overline {\chi }^\prime \,:\!=\, \sup _{z \in \mathbb{R}} |\chi ^\prime (z\pm )| \lt \infty . \end{align}

Examples include $\chi (z) = a\chi _\ast (z/b)$ , where $a,b\gt 0$ and $\chi _\ast (z) = \arctan (z), \tanh (z), \frac {z}{\sqrt {1+z^2}}$ .

There are two main challenges to implementation of Eq. (9).

1. p.1) In contrast to refs. [Reference Matveev, Teimoori and Savkin29, Reference Matveev, Wang and Savkin30], the targeted path $\partial N_{\mathcal{R}}(\mathscr{E}^{\,\,\downarrow }_{\mathcal{R}})$ does not exists in reality (it is a “mental construct”), making it infeasible to directly measure the distance $d_{\mathcal{R}}$ to it;

2. p.2) The conclusions drawn in refs. [Reference Matveev, Teimoori and Savkin29, Reference Matveev, Wang and Savkin30] strongly rely on the assumption that the robot operates within a region where the distance function remains smooth.

According to ref. [Reference Arnold3], this function is non-differentiable at points where the distance $d_{\mathcal{R}}$ is achieved at multiple locations in $\partial N_{\mathcal{R}}(\mathscr{E}^{\,\,\downarrow }_{\mathcal{R}})$ . Even for a very smooth curve, this occurs due to its concave portions (which are surely present in $\partial N_{\mathcal{R}}(\mathscr{E}^{\,\,\downarrow }_{\mathcal{R}})$ near the concatenation points of $\partial \mathscr{D}_{\cup }^{\,\,\varrho _0}$ ), unless the point of interest is sufficiently close to the curve. It follows that in general, the results of refs. [Reference Matveev, Teimoori and Savkin29, Reference Matveev, Wang and Savkin30] are only valid when the initial state is already in proximity to the desired path.

Before presenting the overall control law, we discuss a way to handle the challenges p.1) and p.2).

3.2. Computation of the distance $d_{\mathcal{R}}$ and its derivative $\dot {d}_{\mathcal{R}}$

The proposed procedure is discussed for locations on a trustworthy trajectory ${\boldsymbol{r}} (t)$ and operates in two modes: $\mathbb{S}$ (single) and $\mathbb{M}$ (multiple). Depending on the mode, the following auxiliary constructs are built.

Mode $\mathbb{S}$ : For any visible target ${\boldsymbol{r}}_i$ that is nearest to the robot $\|{\boldsymbol{r}} - {\boldsymbol{r}}_i\| = \varrho ({\boldsymbol{r}})$ , calculate a point ${\boldsymbol{c}}$ that is located at a distance $\mathcal{R}+\varrho _0$ from the target in the direction to the robot:

(12) \begin{equation} {\boldsymbol{c}} \,:\!=\, {\boldsymbol{r}}_i + (\mathcal{R}+\varrho _0) \frac {{\boldsymbol{r}} - {\boldsymbol{r}}_i}{\|{\boldsymbol{r}} - {\boldsymbol{r}}_i\|}. \end{equation}

Mode $\mathbb{M}$ : An angle $\mathbb{A}$ (a set between two rays) is determined when mode $\mathbb{M}$ is switched on; this angle is stored in memory until the next switch $\mathbb{S} \mapsto \mathbb{M}$ is performed.

The initial mode is $\mathbb{S}$ . The switching logic is as follows:

(13) \begin{align} \mathbb{S} \mapsto \mathbb{M} \quad \text{whenever} \, \left \{\begin{array}{c} \varrho \leq \mathcal{R}+\varrho _0 \, \text{and either} \\[4pt] \text{the visible target ${\boldsymbol{r}}_i$ closest to}\, {\boldsymbol{r}} \, \text{is not unique or} \\[4pt] \text{it is unique but the point (12) has multiple closest targets,} \end{array} \right . \nonumber \\[4pt] \mathbb{M} \mapsto \mathbb{S} \quad \text{whenever the robot exits} \, \mathbb{A}. \end{align}

When $\mathbb{S} \mapsto \mathbb{M}$ , the robot selects a closest target ${\boldsymbol{r}}_i$ , builds the related point $c$ in accordance with Eq. (13), and defines $\mathbb{A}$ as the minimum-angle sector with vertex at $c$ that includes all targets closest to $c$ .

Let $\mathbf{ver}[\mathbb{A}]$ denote the vertex of $\mathbb{A}$ . Finally, $d_{\mathcal{R}}$ is computed as follows

(14) \begin{equation} d_{\mathcal{R}}\,:\!=\, \begin{cases} \varrho - \varrho _0 & \text{in mode} \, \mathbb{S}, \\[5pt] \mathcal{R}-\|\mathbf{ver}[\mathbb{A}] -{\boldsymbol{r}}(t)\| & \text{in mode} \, \mathbb{M}. \end{cases} \end{equation}

In the following Lemmas 3.1 and 3.2, steady targets are assumed. (This will be relaxed in Section 4.)

The time derivative $\boldsymbol{\dot {d}_{\mathcal{R}}}$ can be obtained through numerical differentiation of $d_{\mathcal{R}}$ with respect to time. The following lemma presents an alternative to this method.

Lemma 3.2. Let the assumptions of Lem. 3.1 be true. For almost all $t$ , the following relations hold:

(15) \begin{align} \dot {d}_{\mathcal{R}} = \begin{cases} \underset {i:\, \|{\boldsymbol{r}}-{\boldsymbol{r}}_i\| = d_{\mathcal{R}}+ \varrho _0 }{\min } \left \langle \dfrac {{\boldsymbol{r}}-{\boldsymbol{r}}_i}{\|{\boldsymbol{r}}-{\boldsymbol{r}}_i\|};\, v\vec {{e}}(\theta ) \right \rangle & \text{in mode} \, \mathbb{S}, \\[12pt] \left \langle \dfrac {\mathbf{ver}[\mathbb{A}]-{\boldsymbol{r}}}{\|\mathbf{ver}[\mathbb{A}]-{\boldsymbol{r}}\|};\, v\vec {{e}}(\theta ) \right \rangle & \text{in mode} \, \mathbb{M}. \end{cases} \end{align}

3.3. A scheme to avoid singularities of the distance function $d_{\mathcal{R}}$

The term “singularity” denotes a point ${\boldsymbol{r}}$ where the function is non-differentiable. So long as we deal with the distance to the smooth curve $\partial N_{\mathcal{R}}(\mathscr{E}^{\,\,\downarrow }_{\mathcal{R}})$ , singularities do not occur in close proximity to the curve, though they may exist far away from it [Reference Arnold3].

Passing through singularities is inadvisable. The point is that the control law Eq. (9) relies on the expectation that sliding motion over its discontinuity surface commences sooner or later and is maintained subsequently. From then on, $S\,:\!=\,\dot {d}_{\mathcal{R}}(t) + \mu \chi [d_{\mathcal{R}}] =0$ , which ODE implies that $d_{\mathcal{R}} \to 0$ as $t \to \infty$ thanks to Eq. (10), i.e., the control objective Eq. (7) is attained. However, the gradient of $d_{\mathcal{R}}$ abruptly changes when passing through a singularity [Reference Arnold3]. This typically causes an abrupt jump of the first addend $\dot {d}_{\mathcal{R}}=v \left \langle \nabla d_{\mathcal{R}} ; \vec {{e}}(\theta ) \right \rangle$ in the definition of $S$ , whereas the second addend $\mu \chi [d_{\mathcal{R}}]$ varies continuously. This ruins the sliding mode regime $S=0$ , contrary to what is desired.

To erase the harm from singularities, the following is done:

1. d.1) The control law Eq. (9) is fed by the primary distance $d_{\mathcal{R}}$ only in proximity to the targeted path;

2. d.2) Otherwise, it is fed by the distance $d_R$ to a similar but yet different path $\partial N_{R}(\mathscr{E}^{\,\,\downarrow }_{R})$ , where

   (16) \begin{equation} R\gt \mathcal{R} . \end{equation}

The idea behind d.1) and d.2) is based on the fact that as can be shown, the latter path $\partial N_{R}(\mathscr{E}^{\,\,\downarrow }_{R})$ converges to the boundary of the convex hull $H$ of the outer perimeter $\mathscr{P}$ as $R \to \infty$ . So $d_R$ goes to the distance to this hull, which distance function has no singularities thanks to the convexity of $H$ [Reference Arnold3]. In turn, this is a reason for the singularities of $d_R$ to move to infinity as $R \to \infty$ . Hence, they can be “shipped out” of any bounded workspace of the robot by taking $R$ large enough.

Combinability of d.1) and d.2) is principally founded on two facts. First, the properly tuned control law Eq. (9) does succeed in the absence of en-route singularities and so $d_R \to 0$ at phase d.2). Second, the primary $\partial N_{\mathcal{R}}(\mathscr{E}^{\,\,\downarrow }_{\mathcal{R}})$ and auxiliary $\partial N_{R}(\mathscr{E}^{\,\,\downarrow }_{R})$ paths have common pieces so that convergence to the latter entails any degree of closeness to the former at some moments of circumnavigation. At the moment when a pre-specified (and high enough) degree is detected, d.2) is switched to d.1).

3.4. Final formulation of the navigation law

In summary, a hybrid control law with two discrete modes $\mathfrak{A}$ (approaching) and $\mathfrak{M}$ (main) is advocated. They correspond to d.2) and d.1), respectively. Mode $\mathfrak{A}$ uses adjustable parameters $R \gt \mathcal{R}$ and $\beta \in (0,1)$ , where $\beta$ sets the above “degree of closeness” and is used to trigger the switch $\mathfrak{A} \mapsto \mathfrak{M}$ .

In both modes, the control rule is defined by formula Eq. (9), which is interpreted literally in mode $\mathfrak{M}$ and is modified via substituting $R$ in place of $\mathcal{R}$ in mode $\mathfrak{A}$ . The results from Section 3.2 remain valid for calculation of $d_R$ and $\dot {d}_R$ , provided that $\mathcal{R}$ is changed to $R$ everywhere in Section 3.2.

The robot starts in mode $\mathfrak{A}$ ; the modes are switched only once and according to the following rule:

(17) \begin{equation} \mathfrak{A} \xrightarrow {\quad d_R({\boldsymbol{r}})\leq \varrho ({\boldsymbol{r}}) -\varrho _0 \leq (1-\beta ) \mathcal{R}, t \geq 3 \pi /\overline {u} \quad} \mathfrak{M} . \end{equation}

We recall that the distance to the targets $\varrho ({\boldsymbol{r}})$ is defined by Eq. (2).

For the proposed control law to be implementable, the robot should have access to the distances $\varrho ({\boldsymbol{r}}), d_{\mathcal{R}}$ , and $d_R$ . By taking into account the procedure from Section 3.2, which includes observation of all targets nearest to the point Eq. (12), we see that the law is well defined whenever the visibility range is large enough and the robot operates not excessively far from the targets:

\begin{equation*} 2(R+\varrho _0) \lt \mathcal{R}_{\,\text{vis}}, \qquad \varrho ({\boldsymbol{r}}) \lt \begin{cases} R+\varrho _0 & \text{in mode} \, \mathfrak{A}, \\ \mathcal{R}+\varrho _0 & \text{in mode} \, \mathfrak{M}. \end{cases} \end{equation*}

The proposed control law is discontinuous. So the closed-loop solutions are understood in the Filippov’s convexification sense [[Reference Filippov14], a), Section 4]. Since the control $u$ is bounded, existence of solutions for all $t \geq 0$ is guaranteed by Lem. 1 and 3 [[Reference Filippov14], Section 6] and Thm. 1 [[Reference Filippov14], Section 7]. Uniqueness of the solutions is a more intricate matter. We omit its discussion and consider all solutions except non-viable ones. They are those going (even temporarily) over a repulsive discontinuity surface. They are practically non-existant due to extreme finite-horizon instability: virtually any infinitesimal perturbation causes a significant deviation from these trajectories.

4.1. Computation of $d_{\mathcal{R}}$ and $\dot {d}_{\mathcal{R}}$ in the case of moving targets

As for the procedure from Section 3.2, the robot should memorize the angle $\mathbb{A}$ in the frame $\mathscr{F}$ . A way to accomplish this is as follows. Whenever $\mathbb{S} \mapsto \mathbb{M}$ and so an angle $\mathbb{A}$ should be computed and memorized, the robot remembers two targets $i$ and $j$ that define the bounding rays of $\mathbb{A}$ , as well as the location of the vertex $\mathbf{ver}[\mathbb{A}]$ relative to ${\boldsymbol{r}}_i$ and ${\boldsymbol{r}}_j$ . The latter may be done via calculating the height or the length of the legs of the isosceles triangle with the vertices ${\boldsymbol{r}}_i, {\boldsymbol{r}}_j$ , and $\mathbf{ver}[\mathbb{A}]$ . At any time $t$ during the subsequent mode $\mathbb{M}$ the robot restores the isosceles triangle based on the current locations ${\boldsymbol{r}}_i(t), {\boldsymbol{r}}_j(t)$ and defines $\mathbb{A}$ as the angle opposite to the base.

This saves the validity of Lem. 3.1. Lem. 3.2 remains valid only if the second multiplier $v\vec {{e}}(\theta )$ is replaced by $v\vec {{e}}(\theta ) - \dot {{\boldsymbol{r}}}_i$ and $v\vec {{e}}(\theta ) - \frac {d}{dt} \mathbf{ver}[\mathbb{A}]$ in the first and second line from Eq. (15), respectively. So practical use of Lem. 3.2 assumes that the robot is able to assess the relative velocities of the visible targets. Since this is a technically demanding, we rely on numerical differentiation of $d_{\mathcal{R}}$ when computing $\dot {d}_{\mathcal{R}}$ .

6.1. Simulation environment

It was developed based on the physics library MuJoCo. The employed robot’s model is the TurtleBot, a classic robot with a differential drive. To meet the Dubins car (1) model employed for the theoretical design of the control law, a desired constant longitudinal speed is chosen by the experimenter, whereas the angular velocity of the robot’s body is generated by the proposed controller, and finally, these speeds and velocity are recalculated into the rotational velocities of the left and right wheels. When starting the simulator, the robot’s model is loaded in the Unified Robot Description Format. The scene example is illustrated in Figure 3. Despite the algorithm is based on a planar both scene and robot models, the simulation handles 3D models, partly in order to obtain feasible results for real-world experiments. The moving targets are created so that they maintain a pre-specified formation. While the simulator is running, the minimum distance to the targets within the visibility range is constantly calculated. These computations are based on modeling a LIDAR operation, including measurement noises. Noise characteristics are aligned with the parameters of some mainstream robotic devices, which are presented in Table I. The range measurement accuracy is commonly higher for industrial units than for entry-level ones. In the simulation tests, noise parameters are based on the performance of entry-level LIDARs for medium measurement distances: an additive zero-mean Gaussian noise with the standard deviation $ \sigma = 0.015$ m is used. This noise is injected into the sensor measurements immediately in the simulator engine.

Table I.

Parameters of some mainstream LIDAR sensors.

Table I Long description

The table compares parameters of mainstream LIDAR sensors, including Model, Field of View, Range, Range Accuracy, and Horizontal Resolution. The models listed are SLAMTECH RPLIDAR A1, Waveshare STL-27L, Velodyne VLP-16, and Livox Mid-360. The Field of View varies, with some models offering 360 degrees horizontally. The Range spans from 0.15 to 12 meters for SLAMTECH RPLIDAR A1 and up to 100 meters for Velodyne VLP-16. Range accuracy is highest for Velodyne VLP-16 and Livox Mid-360 at 30 millimeters. Horizontal resolution ranges from 0.09 degrees to 0.2 degrees.

$^1$ Field of view (horizontal $\times$ vertical). $^2$ Under a scanning frequency of 10 Hz.

Figure 3.

Scene with the blue robot in MuJoCo simulator. The targets are the small red cylinders, and the yellow segments show the LIDAR rays that detect the targets.

Figure 3 Long description

A blue robot in a MuJoCo simulator detects small red cylinders using yellow LIDAR rays. The scene shows the robot emitting multiple yellow rays that intersect with the red cylinders, indicating detection. The background consists of a grid pattern in varying shades of blue. The robot is positioned on the left side of the image, while the red cylinders are scattered to the right, with some rays extending beyond the cylinders. The setup illustrates the robot's ability to localize and track multiple targets simultaneously.

To compute the forward dynamics in MuJoCo, implicitfast integrator was used with the integration step $0.005$ s. The following values of the robot and scenario parameters are used in all simulation tests:

\begin{gather*} v = 0.45 \ \text{m/s}, \quad \bar u = 0.325 \ \text{rad/s}, \quad \mathcal{R}_{\,\text{vis}} = 10.0 \ \text{m}, \quad \mathcal{R}_{\,\min } = 1.38 \ \text{m}, \quad \mu = 0.5 \ \text{m/s}, \quad \beta = 0.615 . \end{gather*}

The parameter $R$ from Eq. (16) (used in the approaching mode) is chosen individually for every experiment. In Eq. (9), the following linear function with saturation is employed:

(37) \begin{equation} \xi (z) = \left \{\begin{matrix} \gamma z, & \ \text{if} \ |z| \leq \Delta , \\ \gamma \Delta \textbf {sgn}\, z, & \ \text{if} \ |z| \gt \Delta , \end{matrix}\right . \qquad \text{where} \quad \gamma = 0.25 \ \text{1/m}, \, \Delta = 0.75 \ \text{m}. \end{equation}

Figure 4.

Architecture of the controlled system.

Figure 4 Long description

The flowchart illustrates the architecture of a controlled system. It begins with a time source labeled 't' feeding into the platform, which includes the chassis, sensors, and compute unit. The process starts with scanning once, producing outputs labeled as rho_n and theta_n. These outputs feed into the calculation of ri, which then leads to the calculation of phi(r). A check labeled (17) follows, leading to the setting of the target distance. Parallelly, local odom provides input 'r' to a check labeled (13), which sets the operational mode. The operational mode influences the calculation of d_R and d_dot_R, which further feed into the calculation labeled (9). The result 'u' from this calculation is used to apply control. The flowchart includes decision points and conditional paths, illustrating the flow of data and control within the system.

6.2. Overall architecture of the controlled system and some implementation issues

This architecture is shown in Figure 4 as a computation graph and is applied equally in our simulation tests and real-world experiments. In Figure 4, Platform block denotes either the physical robotic platform or its digital twin in the simulation environment and provides the external inputs fed to the algorithm. In particular, Platform provides (i) a time signal $t$ , (ii) LIDAR measurements acquired in a single scan cycle (represented by the set $\{\rho _n,\theta _n\}$ ), and (iii) local odometry, which yields the current estimate of the robot’s state and, in particular, the current position vector $r$ .

The left branch of the graph summarizes the “perception” pipeline: Scan once block provides a set of detected targets $\{r_i\}$ that are processed to compute the distance Eq. (2). This value is then evaluated against the condition from (17), which determines whether the transition between the approaching $\mathfrak{A}$ and main $\mathfrak{M}$ modes must be triggered.

The middle branch implements the logics Eq. (13) of switching between the modes $\mathbb{S}$ and $\mathbb{M}$ . Odometry-derived quantities are checked against the conditions from Eq. (13) to set the current operating mode, which is subsequently used to carry out the distance-related computations in the upper part of the graph.

The right branch describes control synthesis and actuation. Based on the selected mode, the value of the distance $d_R$ and its rate of change $\dot d_R$ (evaluated via either numerical differentiation or applying Lemma3.2) are computed and forwarded to the block (9), which calculates the control signal $u$ in accordance with Eq. (9). Apply Control block represents the low-level chassis controller that converts $u$ into actuator commands. Its output affects the platform motion and therefore closes the feedback loop through subsequent odometry and LIDAR updates.

6.3. Description of the simulation scenarios

Simulation tests for the following scenarios were performed:

1. 1. A rigid formation of seven point-wise objects is dynamically controlled to move with a small linear speed and a constant angular velocity, see Figure 5(a);

2. 2. Five point-wise objects move with common and constant linear speed and angular velocity of rotation around individual centers so that they maintain a rigid $\Pi$ -shaped formation moving without any self-rotation, see Figure 6(a);

3. 3. Eleven point-wise objects are arranged into a three-pointed star-like shape and move as a whole with constant speeds and angular velocities, see Figure 7(a);

4. 4. Twelve point-wise objects are arranged into a three-pointed star-like shape with a “tail” moving with quasi-constant speeds and angular velocities, see Figure 8(a).

In Figure 5–8, all units of measurement are given in the SI system. In every of these figures, the subfigure (a) describes the initial deployment of the objects. The subfigure (b) presents the scene’s snapshots taken at various time instants during the experiment, and the paths of the robot and targets are shown in blue and gray, respectively. The subfigure (c) illustrates the performance of the navigation law by providing the graph of the distance to the group target as a function of time. More precisely, this is the distance computed by the robot based on the noisy LIDAR measurements. In this subfigure, the vertical dashed line indicates the moment when the controller switches from the approaching mode $\mathfrak{A}$ to the main one $\mathfrak{M}$ . The subfigure (d) exhibits the output of the digital processing unit generated in accordance with the discontinuous control law Eq. (9).

6.4. First simulation scenario

The robot has to circumnavigate a formation of seven moving targets at a distance of $\varrho _0 = 2.5$ m. In the approaching mode $\mathfrak{A}$ , the targeted distance $R$ from Eq. (16) is set to $4.0$ m. The targets maintain a rigid formation illustrated in Figure 5(a); as a whole, this formation slowly drifts to the right with a speed of $0.01$ m/s and rotates with an angular velocity of $0.05$ rad/s.

Since the employed model assumes that LIDAR is mounted rather low on the robot, additional noise due to shaking occurs (especially at long measurement distances) whenever the chassis turns quickly. This effect is clearly noticeable at the beginning of the simulation, when the mobile robot is at a considerable distance from the targeted group of objects ( $ t \lt 45$ s). This shaking occurs since the chassis may slightly stagger around the wheels axis, which in turn affects the LIDAR position and orientation and entails an increase of inaccuracy in distance measurements. Despite this, the robot reaches the targeted equidistant curve with confidence, switches to the mode $\mathfrak{M}$ , and confidently accompanies the objects at the desired distance.

Figure 5.

First simulation scenario: (a) Layout of the formation; (b) Snapshots of the scene top view; (c) Minimum distance to the targets; (d) Control signal.

Figure 5 Long description

The image contains four graphs depicting a simulation scenario for autonomous robotic platforms. The first graph, labeled (a), shows the layout of the formation with five red stars arranged in a vertical line, each separated by 1.4 meters horizontally and 0.7 meters vertically, spanning a total height of 5.6 meters. The second graph, labeled (b), displays eight snapshots of the scene top view at different time intervals, showing the movement and positioning of a blue dot within a grid. The third graph, labeled (c), illustrates the minimum distance to the targets over time, with a blue line representing the actual distance and an orange line representing a constant reference distance. The fourth graph, labeled (d), presents the control signal over time, with a blue bar graph indicating the desired angular velocity and red dashed lines marking the upper and lower bounds of the control signal. The graphs collectively demonstrate the effectiveness of autonomous robotic platforms in localization, following, and circumnavigation of group objects.

6.5. Second simulation scenario

In this scenario, a scene with five point-wise objects forming the letter $\Pi$ is considered. The relative positions of objects in the formation are illustrated in Figure 6(a). The group moves along an arc with constant linear $\bar v = 0.006$ m/s and angular $\bar \omega = 0.005$ rad/s speeds without changing its orientation relative to the global coordinate frame. The robot has to accompany the group at a distance of $\rho _0 = 1.75$ m. In the approaching mode, the preliminary distance $R$ is set to $3$ m. As can be seen in Figure 6(c), the robot quickly caught up with the targets and then escorts them with a safe margin.

Figure 6.

Second simulation scenario: (a) Deployment of the objects in the formation; (b) Snapshots of the scene top view; (c) Distance to the group target; (d) Control signal.

Figure 6 Long description

The image contains four graphs illustrating a simulation scenario. The first graph shows the deployment of objects in a formation, with objects positioned in a rectangular grid. The second set of graphs provides snapshots of the scene from a top view at different time intervals, showing the movement and interaction of the objects. The third graph displays the minimum distance to the group target over time, with a blue line representing the distance and an orange line indicating a threshold. The fourth graph illustrates the desired control signal in radians per second over time, with a blue line showing the signal and red dashed lines indicating upper and lower bounds. The graphs collectively demonstrate the dynamics of the simulation scenario.

6.6. Third simulation scenario

Eleven point-wise objects are arranged in the shape of a triangular star; their relative positions are illustrated in Figure 7(a). This formation moves with a high linear velocity $\bar v = 0.085$ m/s, while quickly rotating about the star’s center $\bar \omega = 0.0075$ rad/s. The robot has to accompany the objects at a distance of $\rho _0 = 2$ m; in the approaching mode, $R = 3.5$ m. The level of the noise in the sensory data is fairly large at the beginning of the experiment, partly due to the reasons discussed in Section 6.4. Nevertheless, the robot copes with the mission. Moreover, the static tracking error appears to be smaller than in the previous scenarios, in spite of the more rapid dynamics of the objects.

Figure 7.

Third simulation scenario: (a) Deployment of the objects in the formation; (b) Snapshots of the scene top view; (c) Distance to the group target; (d) Control signal.

Figure 7 Long description

The image contains four graphs depicting a simulation scenario involving autonomous sensor-based circumnavigation of group targets. The first graph (a) shows the deployment of objects in the formation with concentric circles labeled with different radii. The second graph (b) presents snapshots of the scene top view at different time intervals, illustrating the movement of objects. The third graph (c) displays the distance to the group target over time, with a blue line representing the minimum distance and an orange line representing a constant value. The fourth graph (d) shows the control signal over time, with a blue area representing the desired angular velocity and red lines indicating upper and lower bounds. The graphs collectively illustrate the dynamic interaction and control of the robot in navigating around group targets.

6.7. Fourth simulation scenario

Twelve point-wise objects are arranged in the shape shown in Figure 8a. The formation is not rigid: it moves as if the objects are connected to one another by spring-and-damper links as is illustrated in Figure 8a (where every link is represented by only its spring part). The central object plays the role of the pacemaker, and its motion is set exogenously, whereas the other objects are driven by the link forces. The force from the $i$ th link is given by

\begin{equation*} F_i(t) = k_p \cdot \big [ || r_j(t) - r_k(t) || - \rho _{i}^0 \big ] + k_d \cdot \frac {d}{dt} || r_j(t) - r_j(t) ||. \end{equation*}

Here $j$ and $k$ are the serial numbers of the objects at the ends of this link, $r_j$ and $r_k$ are their respective locations, $\rho _i^0$ is the natural length of the spring in the $i$ th link, and $k_p\gt 0,k_d\gt 0$ are given coefficients.

The central pacemaker moves rectilinearly from right to left with a speed of $0.085$ m/s; the initial velocities of the other objects correspond to virtual rotation about the initial location of the pacemaker with an angular velocity of $0.0075$ rad/s. The robot has to accompany the targets at a distance of $\rho _0 = 2$ m.; in the approaching mode, $R = 3.5$ m. By comparing Figures 7 and 8, we see that the overall performance of the navigation law is now similar to that in the previous experiment, whereas the control signal $u$ switches more frequently. This is presumably due to the more sophisticated motion generally inherent to non-rigid formations as compared with rigid ones. This in turn is expected to increase the diversity of events and frequency of their changes and so calls for more frequent alteration of the control decisions. Nevertheless, the robot copes with the mission. Moreover, the averaged steady state error surprisingly looks as the best among all four simulation tests.

Figure 8.

Fourth simulation scenario results. (a) Distribution of objects in target formation; (b) Snapshots of the scene top view; (c) Distance to the target group as a function of time; (d) Control signal generated by navigation algorithm.

Figure 8 Long description

The image contains four subfigures illustrating the results of a simulation scenario. Subfigure (a) shows the distribution of objects in a target formation, with objects represented as red stars and distances labeled. Subfigure (b) presents snapshots of the scene's top view at different time intervals, showing the movement and positioning of objects. Subfigure (c) is a line graph depicting the distance to the target group as a function of time, with the blue line representing the distance and the orange line indicating a reference distance. Subfigure (d) displays a bar graph of the control signal generated by the navigation algorithm over time, with different shades of blue representing various signal components. All values are approximated.

7.1. Hardware experimental setup

To test the robustness of the navigation law against various uncertainties, noises, disturbances, and other real-world imperfections, experiments were performed with a physical robot. Specifically, two mobile robots were used, as is shown in Figure 9. The first of them has to perform the circumnavigation mission, is driven by the proposed control law, and uses the LIDAR RPLIDAR A1 as a sensor device. In this case, Dubins car (1) was emulated by using the classic Diffbot platform: the experimenter sets a desired constant longitudinal speed, the angular velocity of the robot’s body is generated by the proposed controller, and finally, these speeds and velocity are recalculated into the rotational velocities of the left and right wheels. The second robot was mounted on the Pololu Zumo chassis and served as a carrier of rigidly fixed thin vertical plastic tubes, which played the role of a rigid group of point-wise objects. The distance gap between them is $10.0$ cm. The carrier robot moves over a given trajectory with a given speed; practical implementation of this motion was achieved by using lower-level PD feedback controllers fed by data from built-in encoders.

Figure 9.

Mobile robots employed for real-world experiments.

Figure 9 Long description

The image shows two mobile robots positioned on a wooden floor. The robot on the left has a white structure with three vertical antennas and two horizontal bars. The robot on the right has a circular top with a black component and a red button, connected by a blue cable. It also has yellow wheels with star-shaped hubs. The robots are placed on a wooden floor with visible grain patterns.

The choice of the controller parameters was mainly based on the theoretical recommendations refined and specified using the above computer simulation tests, although minor modifications were also made particularly for real-world experiments. In Eq. (9), the linear function with saturation Eq. (37) (where $\gamma = 0.275$ and $\Delta = 0.625$ ) was employed. Unlike the experiments in simulation environments, the frequencies of the control and LIDAR data updates were equalized and were equal to $10$ Hz. Other key parameters are as follows:

\begin{gather*} v = 0.2 \ \text{m/s}, \ \bar u = 0.7 \ \text{rad/s}, \ \rho _0 = 0.2 \ \text{m}, \mathcal{R}_{\,\text{vis}} = 3.0 \ \text{m} \\ \mathcal{R}_{\,\min } = 0.286 \ \text{m}, \ \mu = 0.52, \ \beta = 0.725, R = 0.5 \ \text{m} . \end{gather*}

7.2. Hardware test scenario and results

In the experiment, the carrier robot moves rectilinearly with a constant speed of $0.05$ m/s without any rotation; hence, so does the group of three thin targets mounted atop. Figure 10(a)–(f) offer consecutive snapshots of the scene during the experiment, where the carrier travels from right to left. More detailed and clearer representation of the spatial information is given in Figure 11, which is built upon the telemetry data. In this figure, the targets are depicted as three colored small stars moving rectilinearly from bottom to top; the gray segments behind them represent the trails they traveled until the current time instant. The robot performing the basic mission of circumnavigation is shown as a larger blue dot; the track it has covered is curved and shown in blue. Overall, Figure 11 provides a general evidence that the robot does repeatedly encircle the group target, while moving close to it.

Figure 10.

Successive snapshots of the top view of the scene.

Figure 10 Long description

The image consists of six photos arranged in a grid, each capturing a different moment in time as a robot moves around a stationary object. The photos are labeled with time stamps indicating intervals of 10 seconds. In the first photo, the robot is positioned near the object with its arm extended. In the second photo, the robot has moved slightly to the right. In the third photo, the robot continues to move around the object, with its arm now pointing downward. In the fourth photo, the robot is positioned to the left of the object. In the fifth photo, the robot moves further to the left, and in the sixth photo, it completes the circumnavigation, returning to a position near its starting point. The background is a wooden floor, providing a consistent setting for the sequence.

Figure 11.

Top views of the scene built based on the telemetry data.

Figure 11 Long description

The image contains eight separate graphs arranged in a two-by-four grid, each representing a different time snapshot of a robot's movement around a group of targets. The graphs show the robot's position and orientation at various time intervals, with the x-axis representing the horizontal position and the y-axis representing the vertical position. Each graph includes a blue contour representing the targets, a blue dot indicating the robot's position, and green and orange stars marking specific points related to the robot's orientation. The time intervals range from 7.15 seconds to 56.65 seconds, showing the progression of the robot's movement around the targets. The graphs illustrate the robot's ability to navigate and circumnavigate the group of targets efficiently. All values are approximated.

Figure 12.

The distance $d$ to the nearest target as a function of time.

Figure 12 Long description

The line graph displays the minimum distance to the nearest target as a function of time. The x-axis represents time in seconds, ranging from 0 to 60 seconds. The y-axis represents the minimum distance in meters, ranging from 0 to 1.5 meters. The blue line represents the distance to the nearest target, which starts at approximately 1.5 meters and gradually decreases to around 0.5 meters over time. The orange line represents a constant value, likely a reference distance, which remains steady throughout the time period. A vertical dashed line at around 5 seconds marks a significant point on the x-axis. All values are approximated.

In more detail, the algorithm’s performance is illustrated in Figure 12. It shows that the pursuer robot promptly finds and approaches the targets: starting at a distance of $\approx 160.0$ cm from them, it reduces this distance to $\approx 60.0$ cm as quickly as in $\approx 4.0$ s and then moves away no further than this distance. As for the requested distance $\rho _0 = 20.0$ cm to the currently closest and (nearly) point-wise target, it should be remarked that perfectly maintaining such a distanceFootnote ² means moving over a circle with a radius of $\rho _0 =20.0$ cm. This is, however, infeasible for the robot with the minimal turning radius $\mathcal{R}_{\,\min } = 28.6$ cm. This gives an idea that the posed control objective with $\rho _0 =20.0$ cm is in fact unattainable. Meanwhile, it can be seen in Figure 12 that despite this challenge, after a transient the average distance to the nearest target equals a quite decent value $\approx 30.0$ cm (which is close to $\mathcal{R}_{\,\min }$ ), never exceeds $50.0$ cm, and only rarely and slightly reduces below the safety margin $20.0$ cm. Thus, we see that the algorithm demonstrates a fairly good performance in a somewhat difficult situation in spite of the low frequency of the controller and sensor and associated delays in updating the control signals and sensory data. We recall that this holds with the controller parameters elaborated by using theoretical and computer simulation models. This provides an extra evidence in favor of the robustness of the proposed control law against model’s uncertainties.

Figure 13.

Control signal generated by navigation algorithm.

Figure 13 Long description

A line graph displays the control signal generated by a navigation algorithm over a period of 60 seconds. The y-axis represents the desired angular velocity in radians per second, ranging from negative to positive values, while the x-axis represents time in seconds. The graph features three distinct lines: a solid blue line representing the control signal, a dashed red line indicating the upper bound, and a dashed black line indicating the lower bound. The control signal oscillates rapidly between the upper and lower bounds throughout the time period, maintaining a consistent amplitude. All values are approximated.

During the experiment, the robot once switches from the mode $\mathfrak{A}$ to $\mathfrak{M}$ . At the same time, it consistently alternates the modes $\mathbb{S}$ and $\mathbb{M}$ since while moving around the targets, the robot recurrently passes by singular points of the targeted equidistant curve. Figure 13 illustrates a typical behavioral pattern of the control signal generated at the output of the digital processing unit in accordance with the discontinuous control law (9).

Multimedia of real-life experiments are available at “ https://youtu.be/PhyzMy9-TSA ”