Neurodynamics-based formation tracking control of leader-follower nonholonomic multiagent systems

2.1. Algebraic graph theory

Let $$ \mathcal{G}=(\mathcal{V}, \mathcal{E}) $$ denote undirected graph interactions among agents, where $$ \mathcal{V}=\{1, \ldots, N\} $$ is the node set and $$ \mathcal{E} \subseteq \mathcal{V} \times \mathcal{V} $$ is the edge set. The edge between the node $$ i, j \in \mathcal{V} $$ is bidirectional. An edge $$ (j, i) \in \mathcal{E} $$ denotes that the information of node $$ i $$ can be obtained by the node $$ j $$. Let $$ \mathcal{A} = [a_{ij}]_{n \times n} $$ be the adjacency matrix as defined as follows: $$ a_{ij}>0 $$ if $$ (j, i) \in \mathcal{E} $$; otherwise, $$ a_{ij}=0 $$. In this work, we assume that $$ a_{ii}=0 $$ for all agents $$ i $$. The set of neighbors of agents $$ i $$ is denoted by $$ \mathcal{N}_i:=\{j \in \mathcal{V}|(i, j) \in \mathcal{E}\} $$. The Laplacian matrix $$ \mathcal{L} = [l_{ij}] $$ is defined as $$ l_{ij}=-a_{ij} $$ if $$ j \neq i $$, and $$ l_{ij}= \sum_{j=1}^{N}a_{ij} $$ if $$ j=i $$.

In this paper, the followers are represented by nodes $$ \{1, \ldots, N\} $$, and the leader is represented by the node $$ r $$. $$ \bar{\mathcal{G}} $$ is an undirected augmented graph with $$ \bar{\mathcal{V}}=\mathcal{V} \cup \{r\} $$. The interaction between each follower and leader can be expressed as a diagonal matrix $$ \mathcal{B}=diag(b_1, \dots, b_N) $$ with $$ b_i=1 $$ if the leader is a neighbor of follower $$ i $$, and otherwise $$ b_i=0 $$. Finally, define matrix $$ \mathcal{H} $$ as

2.2. Problem Formulation

A multi-robot system with $$ N+1 $$ nonholonomic multiagents, where $$ N $$ followers and one leader are considered. A typical configuration of them is shown in Figure 1. For each follower, the kinematics can be expressed as

Figure 1. Configuration of the nonholonomic mobile robot.

where $$ p_i=[x_i, y_i]^\mathrm{T} \in \mathbb{R}^2 $$ and $$ \theta_i $$ are the position of the center of mass and the orientation of each agent $$ i $$, respectively. $$ v_i\in\mathbb{R} $$ and $$ \omega_i\in\mathbb{R} $$ are the linear velocity and angular velocity of agents $$ i $$, respectively, which also are the control input. The leader's kinematics is given as

where $$ p_r=[x_r, y_r]^\mathrm{T}\in\mathbb{R}^2 $$ is the position coordinates and $$ \theta_r $$ is the orientation, respectively. $$ v_r\in\mathbb{R} $$ is the linear velocity and $$ \omega_r\in\mathbb{R} $$ is the angular velocity.

The objective of the formation control is to develop a formation control law that ensures each follower maintains a specific position and orientation in relation to the leader. In this work, the desired relative position of the agent $$ i $$ with respect to the reference position $$ p_r $$ is defined as $$ \Delta_i = [\Delta_{ix}, \Delta_{iy}]^\mathrm{T}\in\mathbb{R}^2 $$, and we expect all follower agents to have the same orientation as the leader agent.

The formation control problem in this paper is then stated as follows.

Problem 1 Consider $$ N+1 $$ nonholonomic multiagents and the network topology $$ \bar{\mathcal{G}} $$. For each follower $$ i $$ in (2), develop the control law $$ (v_i, \omega_i) $$ to align the orientation with the leader and hold the desired geometrical shape, as follows:

The following lemma and assumptions are used to address the proposed formation control problem.

Assumption 1 The undirected augmented graph $$ \bar{\mathcal{G}} $$ has a spanning tree in which the root of the spanning tree is the leader.

Remark 1 Given the aforementioned assumption, matrix $$ H $$ is symmetric positive definite.

Assumption 2 The desired velocity $$ v_r, \omega_r $$ and acceleration $$ \dot v_r, \dot \omega_r $$ are both bounded.

Considering that only a portion of robots can receive the leader's state, an estimator will be developed to estimate the leader's states. To facilitate the estimator design analysis, we provide the technical lemma as follows.

Lemma 1 (Nonsmooth Barbalat's Lemma^[36]) Assume that for all $$ t\geq t_0 $$, there exists a compact set $$ \Omega $$ such that the Filippov solution of $$ \dot{x}=f(x, t) $$ always remains within it. If $$ \dot{\widetilde{U}} $$ is an empty set, there is a regular function of the change over time $$ U:\Omega \rightarrow \mathbb{R} $$ with $$ v \le 0 $$$$ \forall v\in\dot{\widetilde{U}} $$ is ordinary. So in $$ \Omega $$, all trajectories will converge to a fixed point $$ s $$, where $$ s $$ belongs to $$ S=\{x\in\Omega|0\in\dot{\widetilde{U}}\} $$ closure.

3.1. The proposed distributed estimator

The following distributed estimators of estimating the leader's states are defined as

The adaptive weights $$ c_{ij} $$, $$ c_i $$, $$ d_{ij} $$, $$ d_i $$ are updated by

where $$ x_{ir} $$, $$ y_{ir} $$, $$ \theta_{ir} $$, $$ v_{ir} $$, $$ \omega_{ir} $$ are estimates of $$ x_r $$, $$ y_r $$, $$ \theta_r $$, $$ v_r $$, $$ \omega_r $$ obtained by each robot $$ i $$, and $$ \hat{x}_{r} \equiv x_r $$, $$ \hat{y}_{r} \equiv y_r $$, $$ \hat{\theta}_{r} \equiv \theta_r $$, $$ \hat{v}_{r} \equiv v_r $$, $$ \hat{\omega}_{r} \equiv \omega_r $$. Additionally, $$ \sigma_{1} $$, $$ \sigma_{2} $$ are positive control gains.

Theorem 1 Consider nonholonomic multiagent systems (2). Assuming that Assumption 1 holds, by choosing suitable control gains $$ \sigma_{1} $$ and $$ \sigma_{2} $$, the distributed estimator (6) is capable of asymptotically estimating the state of the leader agent, including $$ x_r $$, $$ y_r $$, $$ \theta_r $$, $$ v_r $$, and $$ w_r $$.

Proof. Define the estimate error of each agent as $$ \tilde{x}:=\hat{x}-x_r1_N $$, $$ \tilde{y}:=\hat{y}-y_r1_N $$, $$ \tilde{\theta}:=\hat{\theta}-\theta_r1_N $$, $$ \tilde{v}:=\hat{v}_r1_N $$, $$ \tilde{\omega}:=\hat{\omega}_r1_N $$, where $$ \hat{x}=(x_{1r}, \ldots, x_{Nr})^\mathrm{T} $$, $$ \hat{y}=(y_{1r}, \ldots, y_{Nr})^\mathrm{T} $$, $$ \hat{\theta}=(\theta_{1r}, \ldots, \theta_{Nr})^\mathrm{T} $$, $$ \hat{v}=(v_{1r}, \ldots, v_{Nr})^\mathrm{T} $$, $$ \hat{\omega}=(\omega_{1r}, \ldots, \omega_{Nr})^\mathrm{T} $$.

Then, rewrite (6) in the matrix-vector form, as given below

Since the sign functions in (6d) and (6e) are discontinuous, a nonsmooth analysis will be performed to investigate their stability. For this purpose, the Filippov solution of (6d) and (6e) is defined as to be an absolutely continuous differential inclusion solution, as (9d) and (9e), respectively. $$ C_1 \equiv diag(c_{ij}) $$, $$ C_2\equiv diag(c_i) $$, $$ D_1\equiv diag(d_{ij}) $$, $$ D_2\equiv diag(d_i) $$.

Choose the Lyapunov function candidate for (9d)

where the positive constants $$ c_e^* $$ and $$ c_p^* $$ will be determined later. The set-valued Lie derivative of $$ V_1 $$ along the solution of (9d) is

Notice that by using $$ \delta sign(\delta ) = {\left\| \delta \right\|_1} $$^[32], we can get

In addition

Then, combining (11)-(14) yields, we can get

Choosing $$ c_e^* > \frac{\rho }{{{\lambda _{\min }}(H)}} $$, $$ c_p^* > \frac{\rho }{{{\lambda _{\min }}(H)}} $$, such that $$ {\dot V_1}\; \le - {\tilde v^T}H\tilde v $$, $$ H $$ is positive definite matrix, so $$ \tilde v $$ asymptotically converges to $$ {0_N} $$.

Likewise, consider Lyapunov function $$ {V_2} = \frac{1}{2}{\tilde \omega ^T}\tilde \omega + \frac{1}{2}\sum\limits_{i = 1}^N {\sum\limits_{j \in {N_i}}^{} {\frac{{{{({d_{ij}} - d_e^*)}^2}}}{\sigma_2 }} } + \frac{1}{2}\sum\limits_{i = 1}^l {\frac{{{{({d_i} - d_p^*)}^2}}}{\sigma_2 }} $$. The set-valued Lie derivative of $$ V_2 $$ along the solutions of (9e) also can be derived, yielding $$ {\dot V_2} \le - {\tilde \omega ^T}H\tilde \omega \le 0 $$. So, $$ \tilde \omega $$ converges to $$ {0_N} $$ asymptotically.

Next, we consider the estimation dynamics (9c) for $$ \theta_{ir} $$. The estimation error's time derivative is provided by

We can get the solutions as $$ \tilde \theta = {e^{ - Ht}}\tilde \theta (0) + \int_0^t {{e^{ - H(t - s )}}\tilde \omega (s )ds } $$. Because the matrix H is Hurwitz and $$ \tilde \omega $$ asymptotically converges to $$ {0_N} $$, we have

Therefore, it can be said that $$ \tilde \theta $$ converges to $$ {0_N} $$ asymptotically.

In addition, we can obtain from (9a),

Inspired by the above, let $$ \bar \omega : = diag\left( {\cos \hat \theta } \right)\left( {\hat v - {v_r}{1_N}} \right) + {v_r}diag\left( {\cos \hat \theta - \cos {\theta _r}{1_N}} \right) $$. Then, $$ \tilde x = {e^{ - Ht}}\tilde v(0) + \int_0^t {{e^{ - H(t - s )}}\bar \omega (s )ds} $$. Since $$ \tilde{\theta} $$ and $$ \tilde{v} $$ asymptotically converge to $$ {0_N} $$, we have

Consequently, it can be verified that $$ \tilde{x} $$ converges to $$ {0_N} $$ asymptotically. In the same way, it can be demonstrated that $$ \tilde{y} $$ converges to $$ {0_N} $$ asymptotically. This completes the proof.

Remark 1 In this paper, we assume that only part of the followers can obtain the state information $$ x_r $$, $$ y_r $$, $$ \theta_r $$, $$ v_r $$, $$ \omega_r $$ of the leader. This part is reflected as the neighbors of the leader in the paper, and $$ b_i $$ is used to represent the interaction relationship between the follower and the leader. If follower $$ i $$ is the neighbor of the leader, $$ b_i=1 $$, otherwise $$ b_i=0 $$. The variables $$ x_r $$, $$ y_r $$, $$ \theta_r $$, $$ v_r $$, $$ \omega_r $$ in estimator (6) are used on the premise that $$ b_i=1 $$, meaning that the agent is a neighbor of the leader and can obtain the state $$ x_r $$, $$ y_r $$, $$ \theta_r $$, $$ v_r $$, $$ \omega_r $$ of the leader, satisfying our assumption. For clarity of presentation, we use $$ \hat{x}_r $$, $$ \hat{y}_r $$, $$ \hat{\theta}_r $$, $$ \hat{v}_r $$, $$ \hat{\omega}_r $$ instead of $$ x_r $$, $$ y_r $$, $$ \theta_r $$, $$ v_r $$, $$ \omega_r $$, where $$ \hat{x}_{r} \equiv x_r $$, $$ \hat{y}_{r} \equiv y_r $$, $$ \hat{\theta}_{r} \equiv \theta_r $$, $$ \hat{v}_{r} \equiv v_r $$, $$ \hat{\omega}_{r} \equiv \omega_r $$. This notation makes the expression of the estimator more rigorous.

Remark 2 Noteworthy is the fact that equations (7) and (8) present an adaptive dynamic control law that depends entirely on the estimation errors. Therefore, it is not necessary to determine the desired velocity's bound. In equation (6), the sign function enables the asymptotically estimation of the leader's information. However, if you substitute a sigmoid or saturation function for the sign function, you can avoid chattering. The downside is that you will lose the ability to reconstruct the leader's state asymptotically.

3.2. Tracking error and error dynamics

This section discusses the scenario where certain follower agents are only aware of their intended relative position in relation to other follower agents. Using the leader's estimated states, the following defines the formation tracking errors

Then, the tracking error system is given as

where $$ R({\theta _i}) $$ and $$ \varrho_i $$ are given by

The neighborhood formation tracking error is designed in the control law to prevent relying solely on the tracking error $$ \tilde p_i $$. Define

Below is the distributed formation controller based on a typical backstepping technique.

where $$ k_1 $$, $$ k_2 $$, and $$ k_3 $$ are positive constants. However, the current formation controller, which is based on the backstepping technique, has an issue with velocity jumps. After analyzing it, we found that the velocity jumps occur due to abrupt changes in the tracking error, specifically $$ \tilde{\theta}_i $$. To enhance the controller's efficiency and performance, we have introduced a BIN model into the backstepping control.

4.1. The bioinspried neurodynamics model

Using circuit elements, Hodgkin and Huxley developed a membrane block model of a biological neural system. In their membrane block model, the state equation of voltage across the membrane $$ V_m $$ is given as follows.

where $$ C_m $$ is the membrane capacitance, and $$ E_k $$, $$ E_{Na} $$, and $$ E_p $$ are the Nernst potentials for the potassium ions, the sodium ions, and the passive leak current, respectively. In addition, the parameters $$ g_k $$, $$ g_{Na} $$ and $$ g_p $$ are the Conductance of potassium, Conductance of sodium and Conductance of passive channels, respectively. By setting $$ C_m=1 $$ and substituting $$ x_i = E_p+V_m $$, $$ A_i = g_p $$, $$ B_i = E_{Na}+E_p $$, $$ D_i = E_k-E_p $$, $$ S_i^ + = g_{Na} $$, and $$ S_i^ - = g_k $$. Later, a typical shunting neural dynamic model was derived as follows

The i-th neuron's neuronal activity is represented by the equation, involving the membrane potential $$ x_i $$, nonnegative constants $$ A_i $$, $$ B_i $$, and $$ D_i $$ representing passive decay rate, upper bounds and lower bounds. Variables $$ S_i^+ $$ and $$ S_i^- $$ express excitatory and inhibitory inputs. The BIN model has the following properties: (1) For arbitrary excitatory and inhibitory inputs, the neural activity will eventually stay in this region $$ [–D, B] $$; (2) The neural dynamic system is stable in the Lyapunov sense. We use the BIN technique to address the velocity jump problem in backstepping control. This method helps to eliminate the challenges and provides smooth velocity commands.

Replacing $$ {A_i} = A $$, $$ {B_i} = B $$, $$ {D_i} = D $$, $$ {x_i} = {\beta _i} $$, $$ S_i^ + (t) = {f_{1i}}({\tilde \theta _i}) $$, $$ S_i^ - (t) = {g_{1i}}({\tilde \theta _i}) $$ in (21), we can get

Similarly, replacing $$ {A_i} = A $$, $$ {B_i} = B $$, $$ {D_i} = D $$, $$ {x_i} = {\gamma _i} $$, $$ S_i^ + (t) = {f_{2i}}({\tilde \theta _i}) $$, $$ S_i^ - (t) = {g_{2i}}({\tilde \theta _i}) $$ in (21), we can obtain

where $$ {f_{1i}}({\tilde \theta _i}) $$, $$ {g_{1i}}({\tilde \theta _i}) $$, $$ {f_{2i}}({\tilde \theta _i}) $$ and $$ {g_{2i}}({\tilde \theta _i}) $$ are defined as

where $$ k_2 $$, $$ k_3 $$, $$ k_5 $$ and $$ k_6 $$ are positive constants.

4.2. Tracking control algorithm

We design a distributed formation control law to accomplish formation objectives (4) and (5) by utilizing the neighborhood formation tracking error and the leader's estimated state. The control law is designed using a combination of a BIN approach and a backstepping model. By replacing the variable $$ \tilde{\theta}_i $$ in equation (22) with $$ \beta_i $$, we obtain the proposed formation control law for each follower agent.

where $$ k_1 $$, $$ k_2 $$, and $$ k_3 $$ are positive control gains, and $$ \alpha_1 $$, $$ \alpha_2 $$ are constant vectors given by $$ {\alpha _1} = {\left[ {1, 0} \right]^T} $$ and $$ {\alpha _2} = {\left[ {0, 1} \right]^T} $$.

It is evident from equation (22) that the value of $$ \beta_i $$ relies on the tracking error $$ \tilde{\theta}_i $$, and as the input state changes, so does the shunting neural dynamic model's output state. The shunting model is dynamic in nature, and thus, the proposed control law (26) is a smooth function. This ensures that the output state of the shunting neural dynamic model changes gradually without any abrupt jumps, even in cases of abrupt input changes.

Remark 3 The control laws (26) have a similar structure to the tracking controllers in^[17]. However, there are two main differences: (1) instead of the actual leader states $$ v_r $$, $$ \dot{\theta}_r $$, the estimated leader states $$ v_{ir} $$, $$ \dot{\theta}_{ir} $$ are used; and (2) the control law is developed using the followers' coordination error $$ {p_{j}} - {p_i} + {\Delta _{ij}} $$ and the tracking error $$ \tilde{p_i} $$. In this way, each follower robot does not need to know the leader's full state. The reliance on the state estimation of the leader is reduced, and the decentralization of the control is enhanced.

In the following, in order to facilitate the stability analysis, we define

Thus, $$ e_i $$ given in (18) can be expressed as

Substituting (26) into (17), the closed-loop system becomes

where $$ {m_{1i}} \in {R^{2 \times 2}} $$ and $$ {n_{1i}} \in {R^{2 \times 2}} $$ are calculated using

4.3. Stability analysis

This part will discuss stability conditions using the Lyapunov functional technique. A technical lemma for the analysis of closed-loop system stability is proposed before the main conclusions.

Lemma 1^[37] For a continuously differentiable function $$ V:\mathbb{R}^+ \rightarrow \mathbb{R}^+ $$ and a uniformly continuous function $$ W:\mathbb{R}^+ \rightarrow \mathbb{R}^+ $$, if they satisfy the following condition for any $$ t\geq 0 $$

where $$ p_1(t) $$ and $$ p_2(t) $$ are nonnegative functions in $$ L_1 $$ space, and V (t) is bounded, then there is a constant $$ c $$, so that as $$ t \rightarrow \infty $$, $$ W(t) \rightarrow 0 $$ and $$ V(t) \rightarrow c $$.

The primary result is now expressed as follows.

Theorem 1 Under Assumptions 1 and 2, the closed-loop systems (29) and (30) are asymptotically stable with the distributed estimator (6) and the formation controllers (26) by choosing the appropriate control gains $$ k_1 $$, $$ k_2 $$, and $$ k_3 $$; therefore, Problem 1 can be solved.

Proof. Consider the Lyapunov function candidate

The $$ \dot{V}_3 $$ along (29) and (30) is

Then, we have by set $$ B = D $$

According to $$ {f_{1i}}\left( {{{\tilde \theta }_i}} \right) $$ and $$ {g_{1i}}\left( {{{\tilde \theta }_i}} \right) $$ in (24), whenever $$ {\tilde \theta _i} \ge 0 $$ or $$ {\tilde \theta _i} < 0 $$, we can get

Hence, equation (36) becomes

where $$ W_1 $$, $$ W_2 $$ are denoted as

We can get $$ {f_{1i}}\left( {{{\tilde \theta }_i}} \right) \ge 0 $$ and $$ {g_{1i}}\left( {{{\tilde \theta }_i}} \right) \ge 0 $$ from (24). Furthermore, $$ A $$ and $$ B $$ are nonnegative constants. Thus, we have

We obtain that $$ s_i $$ denoted by (27) satisfies the following via the Cauchy-Schwarz inequality

and the Lyapunov function $$ V_3 $$ can be demonstrated to satisfy

One can derive the inequality as follows

where $$ h_1 = 4\left( {n + 1} \right)\mathop {\max }\limits_{1 \le i \le n, 1 \le j \le n} \left\{ {{a_{ij}}} \right\} $$. Similarly, applying the Cauchy-Schwarz inequality with respect to $$ \tilde{\theta}_i $$ can obtain $$ {\left( {\sum\limits_{i = 1}^n {\left| {{{\tilde \theta }_i}} \right|} } \right)^2} \le n\sum\limits_{i = 1}^n {{{\tilde \theta }^2}_i} $$, and along with the fact $$ {V_3} \ge \frac{1}{{2{k_3}}}\sum\limits_{i = 1}^n {{{\tilde \theta }^2}_i} $$, we can get

Using inequalities (44) and (45), we have

Since $$ [{x_{ir}}, {y_{ir}}, {\theta _{ir}}, {v_{ir}}, {\omega _{ir}}] $$ asymptotically converges to the leader's state $$ [{x_r}, {y_r}, {\theta _r}, {v_r}, {\omega _r}] $$, respectively, it can be verified that both $$ {\left\| {{m_{1i}}} \right\|_2} $$ and $$ \left| {{n_{1i}}} \right| $$ asymptotically converge to zero. Therefore, the Lyapunov function candidate $$ V_3 $$ satisfies Lemma 2's requirements, and we can obtain that $$ W_1 $$ tends to zero, or

By applying Barbalat's lemma to (30), and considering that $$ \tilde{\theta}_i $$ tends to zero, along with the facts that $$ \mathop {\lim }\limits_{t \to 0} \frac{{\sin {{\tilde \theta }_i}}}{{{{\tilde \theta }_i}}} = 1 $$ and $$ \mathop {\lim }\limits_{t \to 0} \left| {{v_r}\left( t \right)} \right| \ge 0 $$ by Assumption1, we have

Combining (47) and (49), we can get

Denote $$ s = \left[ {{s_1}^T, {s_2}^T, \cdots , {s_n}^T} \right] $$ and $$ \tilde p = \left[ {{{\tilde p}_1}^T, {{\tilde p}_2}^T, \cdots , {{\tilde p}_n}^T} \right] $$, and given $$ s = H \tilde p $$ as denoted by (27). Thus, following (50), we can get

Since the formation tracking error $$ \tilde{p}_i $$ and $$ \tilde{\theta}_i $$ asymptotically converge to zero, we can achieve the control goals (4) and (5). This completes the proof.

6.1. Unconstrained controller

Consider the communication link for a leader agent and three follower agents, as shown in Figure 2. The leader's linear velocities and the angular velocities are set as $$ v_0 = 2-0.5cos(t) $$ and $$ \omega_0 = 0.5cos(t) $$. The initial state values of the three followers and one leader are $$ (x_1, y_1, \theta_1) = (1.5, 2, 0, ) $$, $$ (x_2, y_2, \theta_2) = (0, 3, -0.5) $$, $$ (x_3, y_3, \theta_3) = (0, 0, -1) $$ and $$ (x_0, y_0, \theta_0) = (2, 2.5, 0.6) $$. Moreover, desired relative positions are $$ \Delta_{1x} = -2, \Delta_{2x} = -4, \Delta_{3x} = 4, \Delta_{1y} = 4, \Delta_{2y} = 2, \Delta_{3y} = -2 $$ for each robot $$ i $$. The parameters are chosen in estimation laws $$ \sigma_1 = 5, \sigma_2 = 2 $$. Similarly, the parameters chosen for control law are chosen to be $$ k_1 = 1 $$, $$ k_2 = 2.5 $$, $$ k_3 = 0.5 $$, $$ A = 5 $$, $$ B = 1 $$, and $$ D = 1 $$. Then, by exploiting the estimator (6), Figures 3 and 4 demonstrate that the estimation errors $$ (x_{ir}-x_r), (y_{ir}-y_r), (\theta_{ir}-\theta_r) $$ and $$ (v_{ir}-v_r), (\omega_{ir}-\omega_r) $$ converge to zero. Furthermore, Figures 5-8 display the simulation of the UFC (26) using this estimation term. It is evident from Figure 5 that the three follower robots guided by the leader robot form the desired triangular formation in a certain time, and in Figure 6, the position and orientation tracking errors converge to zero; i.e., the control objectives (4) and (5) are achieved. Figures 7 and 8 depict the evolution of linear velocities and angular velocities of one leader and three follower agents. It is evident that all three of the following agents will ultimately be able to match the leader agent's angular and linear velocities.

Figure 2. Communication topology graph between one leader and three followers.

Figure 3. Estimation error $$ (x_{ir}-x_r) $$, $$ (y_{ir}-y_r) $$ and $$ (\theta_{ir}-\theta_r) $$.

Figure 4. Estimation error $$ (v_{ir}-v_r) $$ and $$ (\omega_{ir}-\omega_r) $$.

Figure 5. The evolution trajectory of one leader and three follower robots' position in the $$ (x, y, t) $$ plane.

Figure 6. Evolution of position and orientation tracking error.

Figure 7. Evolution of position and orientation tracking error.

Figure 8. Evolution of angular velocity of one leader and three follower agents.

6.2. Constrained controller

Define the following formation tracking errors as

Based on the above formation tracking errors, a constrained formation tracking controller with saturated velocities for the follower agents is given as

where $$ k_7 $$, $$ k_8 $$, and $$ k_9 $$ are positive gains and satisfy the following

The velocity constraints are defined as $$ v_i \in [-4, 4] $$ and $$ \omega_i \in [-3.5, 3.5] $$. The group reference velocities have bounds of $$ v_r^+ = 2.5 $$, $$ v_r^- = 1.5 $$, and $$ \omega_r^+ = 0.5 $$, $$ \omega_r^- = -0.5 $$. The leader's linear velocities and the angular velocities are set as $$ v_0 = 2-0.5cos(t) $$ and $$ \omega_0 = 0.5cos(t) $$, which satisfy Assumptions 2. Based on (76) and (77), adjust the control law parameters $$ k_4 = 1.5 $$, $$ k_5 = 6.8 $$, $$ k_6 = 0.6 $$, $$ A = 8 $$, $$ B = 1 $$, $$ D = 0.2 $$. The leaders' initial state value is considered as $$ (x_0, y_0, \theta_0) = (2, 2.5, 0.6) $$. Moreover, desired relative positions are $$ \Delta_{1x} = -2, \Delta_{2x} = -4, \Delta_{3x} = 4, \Delta_{1y} = 4, \Delta_{2y} = 2, \Delta_{3y} = -2 $$ for each robot $$ i $$. For CFC (53), Figure 9 displays the path of all mobile robots during the 0-20 s, indicating that they converge to the desired formation. Figure 10 demonstrates that the formation tracking errors converge to zero, thereby achieving objectives (4) and (5).

Figure 9. The evolution trajectory of one leader and three follower agents' position in the $$ (x, y) $$ plane.

Figure 10. Evolution of position and orientation tracking error.

Then, we choose Follower 2 as the comparison object. The higher gains of the UFC (26) are taken as $$ k_1 = 2 $$, $$ k_2 = 4 $$, and $$ k_3 = 1 $$, and the lower gains of the UFC (26) are taken as $$ k_1 = 0.7 $$, $$ k_2 = 1 $$, and $$ k_3 = 0.2 $$. The gains of the CFC (53) are taken as $$ k_4 = 1.5 $$, $$ k_5 = 6.8 $$, and $$ k_6 = 0.6 $$. The gains of the SFC (78) are taken as $$ k_7 = 0.7 $$, $$ k_8 = 2.5 $$, and $$ k_9 = 1.5 $$. The rule of parameter selection for the SFC (78) is to make the response curves of formation errors obtain better dynamic and steady-state performance as far as possible under the premise that the gains $$ k_7 $$, $$ k_8 $$, and $$ k_9 $$ meet the constraints in (79) and (80) while comparing. Figures 11-13 show the status errors of follower 2 respectively. The curve of the control input is depicted in Figures 14 and 15. For a clearer comparison of the performance of the various controllers, tracking error adjustment times are listed in Table 1.

Figure 11. Time evolution histories of $$ x_r-x_2+\Delta_{2x} $$ of follower2.

Figure 12. Time evolution histories of $$ y_r-y_2+\Delta_{2y} $$ of follower2.

Figure 13. Time evolution histories of $$ \theta_r-\theta_2 $$ of follower2.

Figure 14. Time evolution histories of linear velocity of follower2.

Figure 15. Time evolution histories of angular velocity of follower2.

According to Table 1,Figures 14 and 15, the tracking error convergence time of UFC (26) with a higher gain is almost the same as that of CFC (53), but CFC (53) is effective against input saturation and can keep the maximum linear velocity and angular velocity not exceeding the threshold. The tracking error convergence time of the CFC (53) is shorter than that of the UFC (26) with lower gain while ensuring the same control input amplitude. In addition, through simulation comparison, both CFC (53) and SFC (78) satisfy the velocity constraint in equation (52), and CFC (53) guarantees better dynamic performance than controller (78).

According to the above simulation results, compared with UFC (26) and SFC (78), CFC (53) ensures that the control input meets the saturation constraint and does not lead to a significant increase in adjustment time. At the same time, it is obvious that the convergence time of the UFC (26) is much smaller than that of the SFC (78) regardless of whether the parameters of the UFC (26) are chosen as high or low gain. The superiority of the proposed controller is demonstrated.

Authors' contributions

Made substantial contributions to conception and design of the study and performed data analysis and interpretation: Zhao XW, Li MK

Performed data acquisition and provided administrative, technical, and material support: Lai Q, Liu ZW

Availability of data and materials

Not applicable.

Financial support and sponsorship

The authors are grateful for the support of the National Natural Science Foundation of China under grant 62173121.

Conflicts of interest

Lai Q is a Junior Editorial Board Member of the journal Intelligence & Robotics, while the other authors have declared that they have no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

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Copyright

© The Author(s) 2024.