Integral sliding mode-based formation control of multiple unertain robots via nonlinear disturbane observer

Abstract

This article proposes a control scheme for formation of maneuvers of a team of mobile robots. The control scheme integrates the integral sliding mode control method with the nonlinear disturbance observer technique. The leader–follower formation dynamics suffer from uncertainties originated from the individual robots. The uncertainties challenge the formation control of such robots. Assuming that the uncertainties are unknown but bounded, an nonlinear disturbance observer-based observer is utilized to approximate them. The observer outputs feed on an integral sliding mode control-based controller. The controller and observer are integrated into the control scheme to realize formation maneuvers despite uncertainties. The formation stability is analyzed by means of the Lyapunov’s theorem. In the sense of Lyapunov, not only the convergence of the approximation errors is guaranteed but also such a control scheme can asymptotically stabilize the formation system. Compared to the results by the sole integral sliding mode control, some simulations are presented to demonstrate the feasibility and performance of the control scheme.

Keywords

Formation control multiple robots uncertainties integral sliding mode control nonlinear disturbance observer

Introduction

Today, robots are changing the way we live and work.¹ They free human being at dangerous places. Manipulation at disaster sites, nuclear plants and warehouses, is often accomplished with robots.² To achieve some complex jobs in rescue, manufacturing, and space exploration, several robots have to cooperate with each other such that a multi-robot system consists of them.³ The multi-robot system can deal with some tasks that are beyond the individual capacities.⁴ Some potential advantages are often referred, such as efficiency, reliability, and flexibility. In the multi-robot system, each robot plays a unique role. To schedule the robots, coordination of such a system rises up.⁵

Formation is one of the typical cooperative tasks. The task can be treated as a special case of consensus problems.^6
–8 Concerning the multi-robot system, the robots have to form up into some predefined shapes in order to achieve a given formation task. The background of formation originates from the real world. For instant, the robots need to maneuver in some formations when they move in very complex environments.

To achieve multi-robot formations, coordinated control mechanisms should be predefined. Up to now, several kinds of coordinated control mechanisms have been reported. These mechanisms can roughly be classified as behavior-based mechanism, virtual structure mechanism, and leader–follower mechanism.³ Among the three categories, the category of leader–follower mechanism has become popular because its dynamics are experimentally modelled, but the internal formation stability can be theoretically guaranteed.⁵ Based on the leader–follower mechanism, various control methods have been developed for multi-robot systems, that is, robust control,⁹ model predictive control,¹⁰ feedback linearization control,¹¹ backstepping control,¹² decentralized control,¹³ to name but a few. A systematic review on this topic is presented by Kala.¹⁴

As a nonlinear feedback design tool, the sliding mode control (SMC) technique¹⁵ is attractive because of its invariance, meaning that matched uncertainties in an SMC system can be suppressed by the invariance. Some SMC-based methods have been addressed to solve the formation control problem of multi-robot systems, that is, integral SMC,^5,16 first-order SMC,¹⁷ terminal SMC,¹⁸ second-order SMC,¹⁹ intelligent SMC,^20,21 and so on. These contributions have verified the feasibility of the SMC methodology for multi-robot formations.

Compared to other SMC methods, the integral SMC design can guarantee an integral SMC system against matched uncertainties from the initial time, indicating that an entire response of the system is of invariance.⁵ Some successful applications of the integral SMC method have been reported in industries, for example, electric vehicle system,²² aerospace engineering,²³ and so on.

Uncertainties exist everywhere. Recall the multi-robot system, where each individual robot may suffer from uncertainties, including but not limited to external disturbances, unmodelled dynamics and parameter perturbations. Affected by the uncertainties, formation dynamics of the multi-robot system become uncertain. Inherently, the formation uncertainties are unmatched such that the invariance of SMC cannot suppress them. In previous works about the SMC-based formations,^5,16

–20 the formation uncertainties are considered because they adversely affect the formation stability. Two solutions can be summarized from the works. One solution is to discuss the formation stability by means of graph theory.^16,21 The other is to analyze the formation stability in light of the Lyapunov’s theorem.^5,17

–20 To guarantee the formation stability, both the solutions need to assume that the formation uncertainties are bounded by a known boundary. Unfortunately, the boundary is rather hard to exactly measure or know in reality. The lack of such a boundary may result in severe problems, that is, decrease of the formation robustness, deterioration of the formation performance as far as deficiency of the formation stability. To obtain the important information, it is desired to adaptively approximate the formation uncertainties.

The technique of nonlinear disturbance observer (NDO) has been proven to be effective in handling uncertainties and improving robustness.²⁴ The applications of NDO have been investigated by some actual cases.²⁵ Such a technique can be considered as an alternative to attack the issue of formation uncertainties. So far, the academic problem of how to eliminate the adverse effects of the formation uncertainties by NDO still remains unsolved and problematic.

Concerning the formation problem of an uncertain multi-robot system, this article proposes a control scheme that integrates one controller with one observer, where the controller is designed by the integral SMC method and the observer is structured by the NDO technique. One of crucial issues induced by the integration is how to ensure the integral SMC-based formation maneuvers with guaranteed stability in spite of the formation uncertainties. To settle the issue, an integral SMC law of the controller is developed by the Lyapunov’s theorem as well as the estimation errors of the observer. Some results are presented to demonstrate that the formation maneuvers are of asymptotic stability.

Modelling

One single robot

Consider a unicycle-like robot in Figure 1. The round robot is with a radius of r. It is equipped with two driven wheels that are symmetrically mounted on both sides of an axis crossing the center. Each wheel of the robot is independently controlled by one DC motor. Define a vector $τ = {[x y θ]}^{T}$ to describe the robot’s posture, where (x y) are the center in the inertial frame and θ denotes the orientation angle of a mobile frame.

Figure 1.

One single robot.

In a multi-robot system, there are N identical robots displayed in Figure 1. Considering the pure rolling and nonslip conditions, the kinematic model describing the n th robot has the form of

{\dot{τ}}_{n} = [\begin{matrix} {\dot{x}}_{n} \\ {\dot{y}}_{n} \\ {\dot{θ}}_{n} \end{matrix}] = [\begin{matrix} \cos θ_{n} & 0 \\ sin θ_{n} & 0 \\ 0 & 1 \end{matrix}] \cdot [\begin{matrix} v_{n} \\ ω_{n} \end{matrix}]

The conditions indicate (1) subjects to $[\begin{matrix} - sin θ_{n} & \cos θ_{n} & 0 \end{matrix}] \cdot {\dot{τ}}_{n} = 0$ . v_n in equation (1) is the linear velocity of this robot. Accordingly, ω_n means the angular velocity. Taking uncertainties into considerations, the dynamic model of the robot can be depicted by

[\begin{matrix} {\ddot{x}}_{n} \\ {\ddot{y}}_{n} \\ {\ddot{θ}}_{n} \end{matrix}] = [\begin{matrix} - {\dot{y}}_{n} {\dot{θ}}_{n} \\ {\dot{x}}_{n} {\dot{θ}}_{n} \\ 0 \end{matrix}] + [\begin{matrix} \cos θ_{n} & 0 \\ sin θ_{n} & 0 \\ 0 & 1 \end{matrix}] u_{n} + [\begin{matrix} \cos θ_{n} & 0 \\ sin θ_{n} & 0 \\ 0 & 1 \end{matrix}] Δ_{n} u_{n} + π_{n} (τ_{n}, {\dot{τ}}_{n})

In (2), $u_{n} = {[α_{n} β_{n}]}^{T}$ is the control inputs vector. Here, $α_{n} = F_{n} / m_{n}$ and $β_{n} = σ_{n} / J_{n}$ are the acceleration and the angular acceleration, respectively. m_n and J_n indicate the mass and moment of inertia of this robot and their values are nominal. F_n and σ_n mean the force and torque. Δ_n indicates the internal uncertainties from the robot, that is, parameter perturbations, described by $diag {ε_{n}, {ε^{'}}_{n}}$ , here ${ε^{'}}_{n}$ and ε_n represent the variations in the moment of inertia and the mass, respectively. $π_{n} (τ_{n}, {\dot{τ}}_{n})$ includes all the external disturbances, that is, slipping and skidding, defined by ${[π_{n x} π_{n y} π_{n θ}]}^{T}$ .

Leader–follower mechanism

Having modelled the n th robot, a formation mechanism of the group should be defined to coordinate the robots and achieve their formation maneuvers. According to the leader–follower mechanism, appoint arbitrary one and name it leader. The remaining N − 1 robots are titled followers. Such a hierarchical mechanism creates N − 1 pairs, where the sole leader traces a predefined trajectory and all the followers takes charge of tracking the leader. To form up into a given formation, each follower has to keep track of its desired distance and relative angle from the leader. Figure 2 illustrates the mechanism by a pair, where indexes i and k represent the leader and the follower, respectively.

Figure 2.

A leader–follower pair.

Given a receiver located at the leader’s center and a sensor equipped at the follower’s front castor, the distance l_ik is defined by $l_{i k} = \sqrt{{(x_{i} - x_{k} - r \cos θ_{k})}^{2} + {(y_{i} - y_{k} - r sin θ_{k})}^{2}}$ and the relative angle ψ_ik is determined by $ψ_{i k} = π - θ_{i} + ζ_{i k}$ , where ζ_ik has a form of $ζ_{i k} = \arctan \frac{y_{i} - y_{k} - r sin θ_{k}}{x_{i} - x_{k} - r \cos θ_{k}}$ .

The individual uncertainties in equation (2) have unfavorable effects on the leader–follower mechanism, leading to the formation uncertainties. Such derivative uncertainties adversely affect the formation stability. Owning to their randomness and complexity, the formation uncertainties are rather hard to be modelled so that they challenge the formation control design very much. Motivated by the challenge, the article develops a control scheme under a mild assumption that the formation uncertainties are bounded by an unknown boundary. In order to concentrate upon the development, some ideal conditions in the pair are taken into considerations: (a) no collision, (b) no communication delay, and (c) the follower knows the leader’s posture as well as its own.

The leader–follower mechanism in Figure 2 can be modelled by differentiating l_ik and ψ_ik twice with respect to time. Finally, the formation dynamics can have the form of

{\dot{x}}_{i k} = A_{i k} x_{i k} + B_{i k,1} u_{k} + B_{i k,2} d_{i k} (x_{i k}, τ_{i}, τ_{k}, {\dot{τ}}_{i}, {\dot{τ}}_{k})

In equation (3), $x_{i k} = {[x_{1} x_{2} x_{3} x_{4}]}^{T}$ where $x_{1} = l_{i k}$ , $x_{2} = {\dot{l}}_{i k}$ , $x_{3} = ψ_{i k}$ , and $x_{4} = {\dot{ψ}}_{i k}$ . Moreover, $A_{i k} = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}]$ , $B_{i k,1} = [\begin{matrix} 0 & 0 \\ \cos ϕ_{i k} & r sin ϕ_{i k} \\ 0 & 0 \\ - \frac{sin ϕ_{i k}}{l_{i k}} & \frac{r \cos ϕ_{i k}}{l_{i k}} \end{matrix}]$ , $B_{i k,2} = {[\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]}^{T}$ , and $d_{i k} (x_{i k}, τ_{i}, τ_{k}, {\dot{τ}}_{i}, {\dot{τ}}_{k})$ abbreviated by d_ik is given by $G_{i k} Δ_{k} u_{k} + L_{i k} (I_{2} + Δ_{i}) u_{i} + F_{i k} + P_{i k}$ , where I₂ denotes a 2 × 2 identity matrix, $G_{i k} = [\begin{matrix} \cos ϕ_{i k} & r sin ϕ_{i k} \\ \frac{- sin ϕ_{i k}}{l_{i k}} & \frac{r \cos ϕ_{i k}}{l_{i k}} \end{matrix}]$ , $L_{i k} = [\begin{matrix} - \cos ψ_{i k} & 0 \\ \frac{sin ψ_{i k}}{l_{i k}} & - 1 \end{matrix}]$ , $ϕ_{i k} = ψ_{i k} + θ_{i k}$ , $θ_{i k} = θ_{i} - θ_{k}$ , $F_{i k} = {[F_{1} F_{2}]}^{T}$ , $P_{i k} = {[P_{1} P_{2}]}^{T}$ , F₁, F₂, P₁, and P₂ are formulated by

F_{1} = {({\dot{ψ}}_{i k})}^{2} l_{i k} + 2 {\dot{ψ}}_{i k} {\dot{θ}}_{i} l_{i k} + {({\dot{θ}}_{i})}^{2} l_{i k} - r \cos ϕ_{i k} {({\dot{θ}}_{k})}^{2} - ({\dot{y}}_{k} {\dot{θ}}_{k} - {\dot{y}}_{i} {\dot{θ}}_{i}) \cos (ψ_{i k} + θ_{i}) - ({\dot{x}}_{i} {\dot{θ}}_{i} - {\dot{x}}_{k} {\dot{θ}}_{k}) sin (ψ_{i k} + θ_{i})

\begin{array}{l} F_{2} = & \frac{- ({\dot{y}}_{k} {\dot{ϕ}}_{i k} - {\dot{ψ}}_{i k} {\dot{y}}_{i}) sin (ψ_{i k} + θ_{i}) - ({\dot{x}}_{k} {\dot{ϕ}}_{i k} - {\dot{ψ}}_{i k} {\dot{x}}_{i}) \cos (ψ_{i k} + θ_{i}) - r {\dot{θ}}_{k} {\dot{ϕ}}_{i k} sin ϕ_{i k}}{l_{i k}} \\ + \frac{{\dot{l}}_{i k} [({\dot{y}}_{i} - {\dot{y}}_{k}) \cos (ψ_{i k} + θ_{i}) - ({\dot{x}}_{i} - {\dot{x}}_{k}) sin (ψ_{i k} + θ_{i}) - r {\dot{θ}}_{k} \cos ϕ_{i k}]}{l_{i k}^{2}} \end{array}

P_{1} = - (π_{i x} - π_{k x}) \cos (ψ_{i k} + θ_{i}) - (π_{i y} - π_{k y}) sin (ψ_{i k} + θ_{i}) + r π_{k θ} sin ϕ_{i k}

P_{2} = \frac{(π_{i x} - π_{k x}) sin (ψ_{i k} + θ_{i}) - (π_{i y} - π_{k y}) \cos (ψ_{i k} + θ_{i}) + r π_{k θ} \cos ϕ_{i k} - l_{i k} π_{i θ}}{l_{i k}}

From equation (3), $rank [B_{i k,1} A_{i k} B_{i k,1}] = 4$ is of full rank, indicating that equation (3) is controllable so that the poles of the closed-loop formation system can be arbitrarily assigned. d_ik aggregates all the formation uncertainties and is inherently unmatched. It is mild that d_ik is hypothetically bounded by a unknown boundary. Unfortunately, the assumption contradicts the formation control design because the formation stability depends on the value of boundary. Usually, a conservative boundary can be empirically predefined to guarantee the formation stability. However, such a conservative boundary may adversely affect the formation performance. Especially, a drawback is that the chattering phenomenon may be induced. To fill the gap between the unknown boundary and the formation performance, the NDO technique is employed so that such an observer serves as a compensator to approximate d_ik.

Formation design

Nonlinear disturbance observer

Assumption 1

${‖ d_{i k} ‖}_{\infty} ⩽ d_{i k}^{*}$ , where $d_{i k}^{*} > 0$ is constant but unknown and ${‖ \cdot ‖}_{\infty}$ denotes ∞-norm.

Assumption 2

d_ik is slowly time-varying, indicating ${\dot{d}}_{i k} ≃ 0_{2} {= [00]}^{T}$ .

Consider the formation dynamics (3) and design the following NDO-based observer (4).²⁴

{\begin{array}{l} {\dot{p}}_{i k} = - L_{i k} B_{i k,2} p_{i k} - L_{i k} (B_{i k,2} L_{i k} x_{i k} + A_{i k} x_{i k} + B_{i k,1} u_{k}) \\ {\hat{d}}_{i k} = p_{i k} + L_{i k} x_{i k} \end{array}

In (4), $p_{i k} \in ℜ^{2 \times 1}$ is the internal states of the observer, the gain matrix $L_{i k} \in ℜ^{2 \times 4}$ is designed such that all the eigenvalues of H _ik are positive (Here, $H_{i k} = L_{i k} B_{i k,2}$ ) and ${\hat{d}}_{i k}$ is the approximation of d_ik.

Define an error vector $e_{d i k} = d_{i k} - {\hat{d}}_{i k}$ , differentiate the error vector with respect to time, substitute equation (4) into the derivative of e _dik , and take assumptions 1 and 2 into account. We can obtain equation (5).

\begin{matrix} {\dot{e}}_{d i k} = {\dot{d}}_{i k} - {\dot{\hat{d}}}_{i k} ≃ - {\dot{p}}_{i k} - L_{i k} {\dot{x}}_{i k} \\ = H_{i k} p_{i k} + L_{i k} (B_{i k,2} L_{i k} x_{i k} + A_{i k} x_{i k} + B_{i k,1} u_{k}) - L_{i k} (A_{i k} x_{i k} + B_{i k,1} u_{k} + B_{i k,2} d_{i k}) \\ = H_{i k} ({\hat{d}}_{i k} - L_{i k} x_{i k}) + L_{i k} (B_{i k,2} L_{i k} x_{i k} + A_{i k} x_{i k} + B_{i k,1} u_{k}) - L_{i k} (A_{i k} x_{i k} + B_{i k,1} u_{k} + B_{i k,2} d_{i k}) \\ = H_{i k} ({\hat{d}}_{i k} - d_{i k}) = - H_{i k} e_{d i k} \end{matrix}

The solution of equation (5) is $e_{d i k} = \exp (- H_{i k} t) e_{d i k} (0)$ , where $e_{d i k} (0)$ is the initial conditions at t = 0. Owing to $λ_{min} (H_{i k}) > 0$ (here $λ_{min} (H_{i k})$ is the minimum eigenvalue of $(H_{i k})$ , this fact indicates that the error vector e _dik is exponentially convergent to 0₂ as t → ∞ (better performance than asymptotic stability).

Integral sliding mode controller

To coordinate the pair in Figure 2 by the integral SMC method, a nominal integral sliding surface vector (6) has to be developed at first.

δ_{i k} = C_{i k} [(x_{i k} - x_{i k}^{0}) - \int_{0}^{t} (A_{i k} - B_{i k,1} K_{i k}) \cdot x_{i k} (ς) d ς]

where $δ_{i k} \in ℜ^{2 \times 1}$ defined by ${[δ_{i k,1 \times 1} δ_{i k,2 \times 1}]}^{T}$ denotes the nominal integral sliding surfaces. $x_{i k}^{0}$ means the initial states at t = 0; $C_{i k} \in ℜ^{2 \times 4}$ is a positive constant matrix and K _ik is calculated by pole assignment to make all the eigenvalues of matrix $(A_{i k} - B_{i k,1} K_{i k})$ rigidly located at the left-half complex plane.

Differentiate the vector δ_ik in equation (6) with respect to time, substitute equation (3) into the derivative of δ_ik and ${\dot{δ}}_{i k}$ becomes ${\dot{δ}}_{i k} = C_{i k} B_{i k,1} K_{i k} x_{i k} + C_{i k} B_{i k,1} u_{k} + C_{i k} B_{i k,2} d_{i k}$ . Suggested by Utkin,¹⁵ the equivalent control law u_keq independently governs the integral SMC system on the sliding mode stage. The expression of u_keq can be derived from ${\dot{δ}}_{i k} = 0_{2}$ , formulated by $- {(C_{i k} B_{i k,1})}^{- 1} [C_{i k} B_{i k,1} K_{i k} x_{i k} + C_{i k} B_{i k,2} d_{i k}]$ . Since d_ik is completely unknown except its boundedness, the unknown term d_ik has to be replaced by its approximation ${\hat{d}}_{i k}$ . Finally, u _keq is rearranged by

u_{k eq} = - {(C_{i k} B_{i k,1})}^{- 1} [C_{i k} B_{i k,1} K_{i k} x_{i k} + C_{i k} B_{i k,2} {\hat{d}}_{i k}]

Concerning the uncertain term, a real sliding mode surfaces vector is defined as

s_{i k} = δ_{i k} + {\hat{d}}_{i k} = C_{i k} [(x_{i k} - x_{i k}^{0}) - \int_{0}^{t} (A_{i k} - B_{i k,1} K_{i k}) \cdot x_{i k} (ς) d ς] + {\hat{d}}_{i k}

where $s_{i k} \in ℜ^{2 \times 1}$ defined by ${[s_{i k,1 \times 1} s_{i k,2 \times 1}]}^{T}$ denotes the real integral sliding surfaces.

According to the designed control scheme, the control law is defined by $u_{k} = u_{k eq} + u_{k sw}$ , where $u_{k sw}$ is the switching control law. Design $u_{k sw} = - {(C_{i k} B_{i k,1})}^{- 1} [{\bar{κ}}_{i k} sign (s_{i k}) + {\bar{η}}_{i k} s_{i k}]$ , where $\bar{κ}$ and $\bar{η}$ are positive constants and $sign (s_{i k}) \in ℜ^{2 \times 1}$ defined by ${[sign (s_{i k,1 \times 1}) sign (s_{i k,2 \times 1})]}^{T}$ is the signum functions vector. Finally, the control law u _k can be formulated by

u_{k} = - {(C_{i k} B_{i k,1})}^{- 1} [C_{i k} B_{i k,1} K_{i k} x_{i k} + J_{i k} {\hat{d}}_{i k} + {\bar{κ}}_{i k} sign (s_{i k}) + {\bar{η}}_{i k} s_{i k}]

where $J_{i k} = C_{i k} B_{i k,2}$ .

System structure

The system structure is presented in Figure 3. From Figure 3, the state variables, composed of l_ik, ψ_ik, and their first derivatives, are located at the feedback channel. They feed on the NDO-based observer to estimate the approximation ${\hat{d}}_{i k}$ . Furthermore, the state variables construct the vector s _ik . The designed control scheme employs s _ik , x _ik , and ${\hat{d}}_{i k}$ to generate the final control inputs vector u _k . Finally, u _k is applied to the follower k in order to maintain the desired distance and relative angle with respect to the leader i during formation maneuvers.

Figure 3.

Structure of the control scheme.

System stability

Assumption 3

${‖ e_{d i k} ‖}_{\infty} ⩽ e_{d i k}^{*}$ , where $e_{d i k}^{*} > 0$ is constant but unknown.

Theorem 1

Consider the formation dynamics (3), take assumptions 2 and 3 into account, design the NDO-based observer (4), define the real sliding surfaces (8), and utilize the integral SMC-based control law (9). The closed-loop formation control system structured by Figure 3 is asymptotically stable if ${\bar{κ}}_{i k} > | | H_{i k} + J_{i k} | |_{\infty} e_{d i k}^{*}$ .

Proof

Differentiate s _ik in equation (8) with respect to time. Substituting equation (3) into the derivative of s _ik yields

{\dot{s}}_{i k} = C_{i k} B_{i k,1} K_{i k} x_{i k} + C_{i k} B_{i k,1} u_{k} + J_{i k} d_{i k} + {\dot{\hat{d}}}_{i k}

Considering the control law (9), equation (10) becomes

{\dot{s}}_{i k} = - {\bar{κ}}_{i k} sign (s_{i k}) - {\bar{η}}_{i k} s_{i k} + J_{i k} (d_{i k} - {\hat{d}}_{i k}) + {\dot{\hat{d}}}_{i k}

From equation (5), ${\dot{e}}_{d i k} (t) + H_{i k} e_{d i k} (t) = 0$ holds true which means ${\dot{\hat{d}}}_{i k} = - H_{i k} ({\hat{d}}_{i k} - d_{i k})$ . Finally, ${\dot{s}}_{i k}$ has a form of

{\dot{s}}_{i k} = - {\bar{κ}}_{i k} sign (s_{i k}) - {\bar{η}}_{i k} s_{i k} + (J_{i k} + H_{i k}) (d_{i k} - {\hat{d}}_{i k})

Select a Lyapunov candidate function $V (t) = {‖ s_{i k} ‖}_{2}$ where ${‖ \cdot ‖}_{2}$ means 2-norm. Differentiate $V (t)$ with respect to time. The derivative of V can be written by $\dot{V} (t) = \frac{s_{i k}^{T} {\dot{s}}_{i k}}{{‖ s_{i k} ‖}_{2}}$ . Replace $\dot{s}$ in the derivative of V by equation (12). Equation (13) can be obtained

\begin{matrix} \dot{V} (t) = \frac{s_{i k}^{T}}{{‖ s_{i k} ‖}_{2}} [- {\bar{κ}}_{i k} sign (s_{i k}) - {\bar{η}}_{i k} s_{i k} + (J_{i k} + H_{i k}) (d_{i k} - {\hat{d}}_{i k})] = - {\bar{κ}}_{i k} \frac{s_{i k}^{T} sign (s_{i k})}{{‖ s_{i k} ‖}_{2}} - {\bar{η}}_{i k} \frac{s_{i k}^{T} s_{i k}}{{‖ s_{i k} ‖}_{2}} + \frac{s_{i k}^{T}}{{‖ s_{i k} ‖}_{2}} (J_{i k} + H_{i k}) (d_{i k} - {\hat{d}}_{i k}) \\ = - {\bar{κ}}_{i k} \frac{{‖ s_{i k} ‖}_{1}}{{‖ s_{i k} ‖}_{2}} - {\bar{η}}_{i k} \frac{{‖ s_{i k} ‖}_{2}^{2}}{{‖ s_{i k} ‖}_{2}} + \frac{s_{i k}^{T}}{{‖ s_{i k} ‖}_{2}} (J_{i k} + H_{i k}) e_{d i k} ⩽ - {\bar{κ}}_{i k} \frac{{‖ s_{i k} ‖}_{1}}{{‖ s_{i k} ‖}_{2}} - {\bar{η}}_{i k} \frac{{‖ s_{i k} ‖}_{2}^{2}}{{‖ s_{i k} ‖}_{2}} + \frac{s_{i k}^{T}}{{‖ s_{i k} ‖}_{2}} [\begin{matrix} {‖ (J_{i k} + H_{i k}) e_{d i k} ‖}_{\infty} \\ {‖ (J_{i k} + H_{i k}) e_{d i k} ‖}_{\infty} \end{matrix}] \\ ⩽ - {\bar{κ}}_{i k} \frac{{‖ s_{i k} ‖}_{1}}{{‖ s_{i k} ‖}_{2}} - {\bar{η}}_{i k} \frac{{‖ s_{i k} ‖}_{2}^{2}}{{‖ s_{i k} ‖}_{2}} + \frac{{‖ s_{i k} ‖}_{1}}{{‖ s_{i k} ‖}_{2}} {‖ (J_{i k} + H_{i k}) e_{d i k} ‖}_{\infty} ⩽ - {\bar{κ}}_{i k} \frac{{‖ s_{i k} ‖}_{1}}{{‖ s_{i k} ‖}_{2}} - {\bar{η}}_{i k} \frac{{‖ s_{i k} ‖}_{2}^{2}}{{‖ s_{i k} ‖}_{2}} + \frac{{‖ s_{i k} ‖}_{1}}{{‖ s_{i k} ‖}_{2}} {‖ (J_{i k} + H_{i k}) ‖}_{\infty} e_{d i k}^{*} \\ = - [{\bar{κ}}_{i k} - {‖ (J_{i k} + H_{i k}) ‖}_{\infty} e_{d i k}^{*}] \frac{{‖ s_{i k} ‖}_{1}}{{‖ s_{i k} ‖}_{2}} - {\bar{η}}_{i k} \frac{{‖ s_{i k} ‖}_{2}^{2}}{{‖ s_{i k} ‖}_{2}} \end{matrix}

where ${‖ \cdot ‖}_{1}$ means 1-norm.

Pick up ${\bar{κ}}_{i k} > | | H_{i k} + J_{i k} | |_{\infty} e_{d i k}^{*}$ such that $\dot{V} (t) < 0$ exists. Concerning $V (t) \geq 0$ , the closed-loop formation system is asymptotically stable in the sense of Lyapunov.

Considering assumption 3, $e_{d i k}^{*}$ is unknown, meaning that it is hard to determine $\bar{κ}$ . To guarantee the system stability, a conservative value of $e_{d i k}^{*}$ should be assigned. On this aspect, there seem no benefits earned from such an NDO-based integral SMC scheme. However, $e_{i k}$ originated from the scheme is exponentially convergent, indicating that a small value of $e_{d i k}^{*}$ can be chosen. According to theorem 2, ${\bar{κ}}_{i k}$ can be decreased to improve the chattering phenomenon. Thus, the presented control scheme can improve the formation performance.

Simulation results

To demonstrate the performance of the integral SMC-based control scheme, this section implements some numerical simulations on a multi-robot platform and discusses the results. The platform is composed of three mobile robots. The robots are identical (see Figure 1) with radius 0.05 m, where one leader is numbered by 1 and two followers are numbered by 2 and 3, respectively.

The leader in the formation mechanism is to take charge of tracking its predefined trajectories; its control can be treated as tracking control of a single robot and it is equipped with a kinematic-based sliding mode controller.²⁶ The designed control scheme is applied to the other two followers in order that the two robots can maintain the desired distances and the desired relative angles.

Δ_i and Δ_k in equation (3) are considered as $diag {0.3 rand () - 0.15 0.3 rand () - 0.15}$ , where i = 1 delegates the leader, k = 2, 3 means the followers, and rand() is a MATLAB R2010a function to generate a random number. Such consideration indicates that the largest parameter variations are up to 15% of the nominal ones. P _ik contains $π_{i x} - π_{k x}$ and $π_{i y} - π_{k y}$ in equation (3) which shows that it is not representative to define $π_{i} = π_{k}$ . Here, we consider $π_{i x} = π_{i y} = π_{i θ} = 0.5 sin (2 π t)$ and $π_{k x} = π_{k y} = π_{k θ} = 0.8 sin (π t)$ .

The parameter matrices K _ik and C _ik of the real sliding surface vector s _ik in equation (8) have to be predefined. K _ik needs to be designed such that the eigenvalues of matrix $A_{i k} - B_{i k} K_{i k}$ in equation (8) are those in the specified vector ${[- 1.7500 \pm 0.6614 i - 1.4500 \pm 0.8930 i]}^{T}$ . Finally, K _ik is determined by $K_{i k} = [\begin{matrix} 3.5cos (ϕ_{i k}) & 3.5cos (ϕ_{i k}) & - 2.9 l_{i k} sin (ϕ_{i k}) & - 2.9 l_{i k} sin (ϕ_{i k}) \\ \frac{3.5sin (ϕ_{i k})}{d} & \frac{3.5sin (ϕ_{i k})}{d} & \frac{2.9 l_{i k} cos (ϕ_{i k})}{d} & \frac{2.9 l_{i k} cos (ϕ_{i k})}{d} \end{matrix}]$ . The other matrix C _ik is determined by $[\begin{matrix} 120 & 88 & 0 & 0 \\ 0 & 0 & 90 & 30 \end{matrix}]$ . The two parameters ${\bar{κ}}_{i k}$ and ${\bar{η}}_{i k}$ of the switching control law are determined by 1 and 15, respectively. Concerning the observer, its gain matrix is chosen as $[\begin{matrix} 0 & 15 & 0 & 18 \\ 0 & 15 & 0 & 20 \end{matrix}]$ by trial and error.

Consider the following formation task. The robots move along straight lines and form up into a string formation. Their formation maneuvers take place at t = 8 s. Thereafter, the alternative formation becomes triangular. The initial postures of the three mobile robots are located at $τ_{1} = {[0 m 0 m 0 rad]}^{T}$ , $τ_{2} = {[0 m 0.4 m 0 rad]}^{T}$ , and $τ_{3} = {[0 m - 0.3 m 0 rad]}^{T}$ , respectively. Accordingly, the initial states of the followers can be calculated by $x_{12}^{0} = {[0.4 m 0 m / s \frac{π}{2} rad 0 rad / s]}^{T}$ and $x_{13}^{0} = {[0.3 m 0 m / s - \frac{π}{2} rad 0 rad / s]}^{T}$ . To achieve the given formation maneuvers, the desired states during the first 8 s are $x_{12}^{d} = {[0.2 m 0 m / s \frac{π}{2} rad 0 rad / s]}^{T}$ and $x_{13}^{d} = {[0.15 m 0 m / s - \frac{π}{2} rad 0 rad / s]}^{T}$ . The desired states during the next 8 s are $x_{12}^{d} = {[0.5 m 0 m / s \frac{2 π}{3} rad 0 rad / s]}^{T}$ and $x_{13}^{d} = {[0.5 m 0 m / s - \frac{2 π}{3} rad 0 rad / s]}^{T}$ .

Figure 4 displays the group of mobile robots forms up into a triangular formation from a string one while moving in straight lines, where those filled circles denote the initial positions of the robots and those unfilled circles indicate the robots’ positions in the dynamic process. In each circle, the arrow means the orientation angle of that robot. To demonstrate their postures and formation, the solid lines in Figure 4 bond the robots together at the same moment. In the simulation, the desired linear velocity $v_{1}^{d}$ and the desired angular velocity $ω_{1}^{d}$ of the leader are set by $0.5 m \cdot s^{- 1}$ and $0 rad \cdot s^{- 1}$ , respectively. From Figure 4, the two followers can follow the leader and the given formation task has been achieved. Concerning the motor load, the acceleration and the angular acceleration of the two followers are confined to $∥ α_{k} ∥ \leq \frac{π}{2} m \cdot s^{- 2}$ and $∥ β_{k} ∥ \leq 30 rad \cdot s^{- 2}$ , respectively.

Figure 4.

Maneuvers of the multi-robot system from a string formation to a triangular one while moving along straight lines

In the process of achieving the formation task, some comparisons between the adaptive integral SMC and the sole integral SMC are shown in Figures 5, 6, and 7, where the uncertainties in the sole integral SMC-based formation control system are given a conservative boundary.

Figure 5.

Performance comparisons between the integral SMC-based formation systems. (a) Distance l₁₂, (b) relative angle ψ₁₂, (c) distance l₁₃, and (d) relative angle ψ₁₃. SMC: sliding mode control.

Figure 6.

Performance comparisons of u₂. (a) Acceleration of follower 2 α₂ by NDO-based integral SMC, (b) angular acceleration of follower 2 β₂ by NDO-based integral SMC, (c) acceleration of follower 2 α₂ by integral SMC, and (d) angular acceleration of follower 2 β₂ by integral SMC. SMC: sliding mode control; NDO: nonlinear disturbance observer.

Figure 7.

Performance comparisons of u₃. (a) Acceleration of follower 3 α₃ by NDO-based integral SMC, (b) angular acceleration of follower 3 β₃ by NDO-based integral SMC, (c) acceleration of follower 3 α₃ by integral SMC, and (d) angular acceleration of follower 3 β₃ by integral SMC. SMC: sliding mode control; NDO: nonlinear disturbance observer.

Figure 5 depicts the system states of the two followers, that is, the distance l_ik and the relative angle ψ_ik, where the dot curves are desired, the solid curves denote the performances by the presented control scheme (short for NDO-based integral SMC), and the dash curves delegate the performances by the sole integral SMC methd (short for integral SMC). Both the SMC systems share the same integral sliding surfaces. Although the control system can achieve the formation maneuvers by the sole integral SMC, the control system is more sensitive to uncertainties because it has no ability to adaptively resist unknown uncertainties. In contrast, the NDO-based integral SMC-based formation system is more robust and its curves are smoother in the dynamic process.

Figure 6 shows the control inputs of the follower 2 by the two integral SMC-based methods. In Figure 6, the curves by the sole integral SMC dramatically jump back and forth. The phenomenon originates from chattering, a drawback of SMC. The jumping points in Figure 6 are different in each of simulations because Δ_i and Δ_k are random. Compared to the results of the two methods, the curves by the NDO-based integral SMC are much smoother because the NDO-based observer can suppress uncertainties so that the control performances can be improved. Some similar results can be draw from Figure 7.

Figure 8 displays each elements of the real sliding surface vector s _ik . As proven, the real sliding surface vectors are of asymptotic stability. The designed control scheme combines the integral SMC method and the NDO technique. The integral SMC-based controller and the NDO-based observer work together to realize the robust formation control for the group of uncertain multiple robots. Figure 9 illustrates the approximation errors ${\hat{e}}_{i k}$ . From Figure 9, the NDO-based observer has noticeable errors at the outset. However, the errors are exponentially convergent to guarantee the formation stability and to improve the formation performance in the presence of uncertainties.

Figure 8.

Sliding-surface vectors and their elements. (a) s_12,1, (b) s_{12, 2}, (c) s_{13, 1}, and (d) s_{13, 2}.

Figure 9.

Approximation errors e _ik . (a) e_12,1, (b) e_{12, 2}, (c) e_{13, 1}, and (d) e_{13, 2}.

Conclusions

This article has investigated the formation control problem for swarms of uncertain robots. By the leader–follower mechanism, individual uncertainties from the robots chain to the formation dynamics. The formation uncertainties bounded by an unknown boundary trouble the formation control problem. To resist the formation uncertainties when forming up the robots, the integral SMC-based control scheme is addressed. The control scheme consists of an integral sliding mode controller and a NDO. The theoretical analysis proves that the formation control system in the presence of uncertainties is of asymptotic stability. The control scheme has achieved the formation maneuvers by a multi-robot platform. The simulation results have demonstrated the effectiveness of the control scheme.

The control scheme benefits us a lot because it can approximate the boundary of uncertainties so that the formation control system can have the guaranteed stability. The effort to consider the communication topology of the multi-robot systems via the control scheme is our continuous interest and this field is still in progress.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the National Natural Science Foundation of China Project under the grants 60904008 and 61473176, and the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities (ZR2015JL021).

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