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Abstract

This paper presents the robustness in formation control of multiple mobile robots using leader-follower method. The uncertainty considered is the measured error which is included in the relative state. The robust stability conditions against the relative state error are derived by Lyapunov’s stability theory (direct method). We also obtain the formation steady-state deviation when formation stability is ensured. The formation control environment is constructed on Simulink. The validity of the stability condition and the steady-state deviation is demonstrated by numerical simulation. It is seen that the L-F method provides a robust control law against the relative state error although large formation steady-state deviation is occurred in some cases.

Keywords

Formation control multiple mobile robots leader-follower method robustness

Introduction

Formation control of multi-mobile robots has attracted much attention in recent years. In this background, the formation is one of efficient transportation forms such as automobiles and airplanes, and it can become a system that brings about more efficient work achievement than a single robot.^1–4 Reactive behavior-based method,^5–7 multi-agent system method,^8–11 virtual structural method,^12,13 and leader-follower (L-F) method^14–18 are well known as control approaches to realize formation control. This paper is a study on the L-F method for mobile robots. In this method, multiple mobile robots are classified into a leader and followers. The objective of the leader is to lead the fleet of the followers, while that of the followers is to keep the specified relative positional relation with the leader. Therefore, the issue to be examined in the L-F method is basically to design control systems for the followers.

The authors have been studying several issues in formation control based on the L-F method by using real mobile robots; autonomous decentralized control law, obstacle avoidance, and shape transition.^19–23 The control system designs were based on feedback linearizaton.^24,25 The parameters of the control law were given while checking real operation status of the mobile robots. Since the decentralized control law^19,21 was constructed by the relative states; that is, the relative distance and the relative angle, which were measured by the followers, the relative states included the measured errors. Fortunately, the designed control law was acceptable without troubles in carrying out the experiments mentioned above. However, the control system of the L-F method is a nonlinear system with respect to the relative states, and the relative state errors may have influences on the stability and the control performance of the closed-loop system. The L-F method is one of the well-known control methods not only for mobile robots but also for UMV/UAV formation control such as multi-rotor vehicles.^26–28 However, there are few researches for investigating the robustness against the relative state errors. It is therefore worth verifying the robustness of the L-F method.

This paper discusses the robustness in the formation control of mobile robots using the L-F method. First, the problem setting of formation control considered in this paper is clearly defined. Since the L-F method treats the kinematics of mobile robots, the uncertainty in the controlled system is the measured error which is included in the relative state. This paper derives the stability conditions based on Lyapunov’s stability theory^29,30 for the closed-loop system with the relative state errors containing control law. The steady-state deviation in formation control is presented when the stability is ensured. Its approximate representations are also derived. The formation control environment is constructed on Simulink. The validity of the stability condition and the steady-state deviation is demonstrated by numerical simulations. Through these analysis and simulation, this paper provides the effectiveness and the robustness of the L-F method from the theoretical point of view.

Leader-follower method of mobile robots

Statements of problem

Let us consider a mobile robot with nonholonomic constraints. The position of a robot moving in a two-dimensional plane is represented by a stationary Cartesian coordinate system $(x (t), y (t))$ , where t is continuous-time. The azimuth of the robot is $θ (t)$ ( $- π \leq θ (t) < π$ [rad]), which is measured from the positive x-axis. The translational and angular velocities are denoted as v(t) and ω(t), respectively, which are the control inputs of the mobile robot. Figure 1 shows a situation where two mobile robots are controlled by the L-F method. The subscripts “l” and “f” to variables mean the leader and the follower, respectively. The relative states between them are given by

d_{f} ≜ \sqrt{{(x_{l} (t) - x_{f} (t))}^{2} + {(y_{l} (t) - y_{f} (t))}^{2}} (= d_{l})

(1)

γ_{f} (t) ≜ \tan^{- 1} (y_{l} (t) - y_{f} (t), x_{l} (t) - x_{f} (t)) - θ_{f} (t)

(2)

γ_{l} (t) ≜ \tan^{- 1} (y_{f} (t) - y_{l} (t), x_{f} (t) - x_{l} (t)) - θ_{l} (t)

(3)

χ_{f} (t) ≜ θ_{l} (t) - θ_{f} (t)

(4)

Where $\tan^{- 1} (\cdot, \cdot)$ is defined as follows:

\tan^{- 1} (y, x) ≜ {\begin{matrix} \tan^{- 1} (y / x) & (x \geq 0) \\ \tan^{- 1} (y / x) + π & (x < 0, y > 0) \\ \tan^{- 1} (y / x) - π & (x < 0, y \leq 0) \end{matrix}

(5)

Figure 1.

Relative position and relative orientation between leader and follower robot.

$d_{f}, γ_{f}$ , and $χ_{f}$ are the relative distance, angle, and azimuth of the follower to the leader, respectively. When the number of mobile robots is greater than two, a formation is constructed by sequentially continuing the relation between the leader and follower shown in Figure 1.

To make the formation control issue of mobile robots clear, this paper defines application policies of the L-F method as follows.

(M-1) Multiple mobile robots with nonholonomic constraints form a formation in the forward direction.

(M-2) The leader is used to lead the fleet of the followers to the target, while the followers follow the leader with keeping the relative positional relation specified in advance.

(M-3) The formation trajectory, also called the leader trajectory, is given by a trajectory with a smaller limit on the relative azimuth $χ_{f}$ .

(M-4) The transitional velocity of the leader v_l is given within the specified range.

Based on these application policies, the objective of the formation control in this paper is that the followers are controlled by their translational and the angular velocities v_f and ω_f so as to keep the relative states $d_{f}$ and $γ_{f}$ at their constant references $d_{f}^{ref}$ and $γ_{f}^{ref}$ .

Relative state-space equation

The kinematics of a mobile robot is given by the following equation.

{\begin{matrix} \overset{\cdot}{x} (t) = v (t) \cos θ (t) \\ \overset{\cdot}{y} (t) = v (t) \sin θ (t) \\ \overset{\cdot}{θ} (t) = ω (t) \end{matrix}

(6)

When differentiating both sides of equations (1) and (2) by t and using equation (6) corresponding to the leader and the follower, the derivatives of the relative distance and angle of the follower can be expressed collectively as.^19,20

{\overset{\cdot}{ξ}}_{f} = A_{f} η_{f} + d_{u}

(7)

\begin{array}{l} A_{f} ≜ [\begin{matrix} - \cos γ_{f} & 0 \\ \frac{\sin γ_{f}}{d_{f}} & - 1 \end{matrix}], d_{u} ≜ [\begin{matrix} - \cos γ_{l} \\ \frac{\sin γ_{l}}{d_{l}} \end{matrix}] v_{l}, \\ ξ_{f} ≜ [\begin{matrix} d_{f} \\ γ_{f} \end{matrix}], η_{f} ≜ [\begin{matrix} v_{f} \\ ω_{f} \end{matrix}] \end{array}

(8)

Equation (7) is called the relative state-space equation, which means the controlled object in this paper. The control objective described in the previous section is to design the control input of the follower η_f which achieves ξ_f→ξ_f^ref for t→∞ in equation (7). The control input by feedback linearization is given as follows.

η_{f} = A_{f}^{- 1} (μ_{f} - d_{u})

(9a)

μ_{f} = K (ξ_{f}^{ref} - ξ_{f})

(9b)

µ_f is a linear input whose gain matrix K is given by

K ≜ diag {k_{d}, k_{γ}} (k_{d}, k_{γ} > 0)

(10)

Equation (9a) is used to cancel out the nonlinear terms in equation (7). However, when $γ_{f} = \pm π / 2$ , $| A_{f} | = 0$ . Then η_f cannot be constructed; that is, the control input becomes singular. This situation must be avoided in formation control. d_u is a term with respect to the leader. In the experiments using real mobile robot Pioneer,³¹ the control system was constructed using an estimated control law, called Self Made Input,^19,21 in which d_u was estimated by the follower. Since this paper however focuses on the robustness of the L-F method, d_u is assumed to be known. When $| A_{f} | \neq 0$ and substituting equation (9a) into equation (7), we have

{\overset{\cdot}{ξ}}_{f} = K (ξ_{f}^{ref} - ξ_{f})

(11)

This achieves the control objective ξ_f→ξ_f^ref when t→∞.

Relative state-space equation with relative state error

To use the control input (9a), the follower needs to measure the relative distance and angle by its own sensor. Denoting these measurements as d_s and $γ_{s}$ , the differences between the measurements and their true states are defined as

ε_{d} (t) ≜ d_{s} (t) - d_{f} (t)

(12a)

ε_{γ} (t) ≜ γ_{s} (t) - γ_{f} (t)

(12b)

ε_d and ε_γ are called the relative distance and the relative angle errors, respectively. For convenience in the following descriptions, the following vector notations are used collectively as the relative state error and the measured relative state.

e_{f} (t) ≜ [\begin{matrix} ε_{d} (t) \\ ε_{γ} (t) \end{matrix}], ξ_{s} (t) ≜ [\begin{matrix} d_{s} (t) \\ γ_{s} (t) \end{matrix}],

(13)

When the relative state error e_f is contained in ξ_f , the control inputs by feedback linearization are given by

η_{f} = A_{s}^{- 1} (μ_{f} - d_{u}), A_{s} ≜ [\begin{matrix} - \cos γ_{s} & 0 \\ \frac{\sin γ_{s}}{d_{s}} & - 1 \end{matrix}]

(14a)

μ_{f} = K (ξ_{f}^{ref} - ξ_{s}) = K (ξ_{f}^{ref} - ξ_{f} - e_{f})

(14b)

Equation (14a) is not constructed when $γ_{s} = \pm π / 2$ because of $| A_{s} | = 0$ . When $| A_{s} | \neq 0$ substituting equation (14a) into equation (7), we have

{\overset{\cdot}{ξ}}_{f} = A_{f} A_{s}^{- 1} K (ξ_{f}^{ref} - ξ_{f} - e_{f}) + (I - A_{f} A_{s}^{- 1}) d_{u}

(15)

Equation (15) shows the closed-loop system with the relative state error. In the following sections, we will discuss the stability of the closed-loop system and the steady-state deviation when it is stable. Equations (12a) and (12b) are generally time-varying, but in this paper, we consider a case that they are constant. Note that the equilibrium point of equation (15) is the relative state whose derivative is equal to zero; that is, $\overset{\cdot}{ξ}$ f = 0. In general, there may be more than one. The equilibrium point of interest is asymptotically stable in the closed-loop system. That is, it is an equilibrium point that the relative state converges from a given initial state. It is denoted as $ξ_{f}^{e q} ≜ {[d_{f}^{e q} γ_{f}^{e q}]}^{T}$ hereafter.

Stability condition with relative state error

This section presents a stability condition for the closed-loop system with the relative state error by using Lyapunov’s stability theory (direct method).^29,30 Since the range of the relative state should be taken into consideration (see Section “Range of relative state”), the stability condition to be derived in this section is the local asymptotic stability. At the beginning, the following variable transformation

z (t) ≜ ξ_{f} (t) - ξ_{f}^{eq} ≜ {[\begin{matrix} z_{1} (t) & z_{2} \end{matrix} (t)]}^{T}

(16)

is applied to equation (15). The closed-loop system (15) is then rewritten as follows.

\overset{\cdot}{z} = A_{cl} (z) z + g (z; d_{u})

(17)

A_{c l} (z) ≜ - A_{f} A_{s}^{- 1} K = [\begin{matrix} - \frac{k_{d} \cos γ_{f}}{\cos γ_{s}} & 0 \\ \frac{- k_{d}}{\cos γ_{s}} (\frac{\sin γ_{s}}{d_{s}} - \frac{\sin γ_{f}}{d_{f}}) & - k_{γ} \end{matrix}] ≜ [\begin{matrix} a (z) & 0 \\ c (z) & b (z) \end{matrix}]

(18)

g (z; d_{u}) ≜ A_{f} A_{s}^{- 1} K (ξ_{f}^{ref} - e_{f} - ξ_{f}^{eq}) + (I - A_{f} A_{s}^{- 1}) d_{u}

(19)

By equation (16), the equilibrium point is moved to the origin (z^eq = 0). The last notation of equation (18) is used for Lyapunov equation which will be presented in Section “Lyapunov equation with respect to A_cl(z).”g(z;d_u) in equation (19) satisfies the following constraints:

g (0; d_{u}) = 0

(20a)

| | g (z; d_{u}) | | \leq c_{g} ∥ Z ∥ (c_{g} > 0)

(20b)

Equation (20a) is obvious from the definition of the equilibrium point ξ_f^eq. Equation (20b) is a constraint on the magnitude of g(z;d_u), where c_g may be constant or varying (see Section “Verification by simulation”).

Range of relative state

The range of the relative state should be taken into consideration in the formation control from the viewpoints of the definition of the variables, the constraints on the control law and the practical application. As for the relative distance d_f , it is a positive number from the definition. There exists an upper limit due to the distance sensor. There also exits a lower limit for avoiding collisions between robots. As for the relative angle $γ_{f}$ , on the other hand, the relative angle is limited within $| γ_{f} | \leq π$ from the definition. It is furthermore limited in $| γ_{f} | \leq π$ /2 from (M-1) and (M-2) and singularity avoidance in the control law (14a). Therefore, in this paper, we define the ranges with respect to the relative states as follows.

R_{d} ≜ {d_{f} | 0 < \underline{d} < d_{f} < \bar{d}}

(21a)

R_{γ} ≜ {γ_{f} | | γ_{f} | < \frac{π}{2} - δ_{r}, 0 < δ_{r} << \frac{π}{2}}

(21b)

Where $\bar{d}$ and $\underline{d}$ are the upper and lower limits of the relative distance, respectively. δ_r is a margin of angle. In addition,

D_{FC} ≜ R_{d} \times R_{γ} \subset R^{2}

(22)

is defined as the formation region. Thus, $ξ_{f} \in D_{FC}$ means

d_{f} \in R_{d}, γ_{f} \in R_{γ} \Leftrightarrow \underline{d} < d_{f} < \bar{d}, | γ_{f} | < \frac{π}{2} - δ_{r} .

(23)

This is also used for the transformed variable z. That is, $z \in D_{FC}$ means

\begin{matrix} z_{1} \in R_{d}, z_{2} \in R_{γ} \Leftrightarrow \underline{d} - d_{f}^{eq} < z_{1} < \bar{d} - d_{f}^{eq}, \\ | z_{2} | < \frac{π}{2} - δ_{r} - γ_{f}^{eq} . \end{matrix}

(24)

Lyapunov equation with respect to A_cl(z)

As a preparation for applying Lyapunov’s stability theory to equation (17), the following Lyapunov equation

P A_{cl} (z) + A_{cl}^{T} (z) P + Q (z) = 0

(25)

is introduced in this section, where P is a constant matrix and Q(z) is a z-dependent matrix. They are positive definite. For existence of a unique solution in equation (25), A_cl(z) must be a stable matrix. Eigenvalues of A_cl(z) are $a (z) = - \frac{k_{d} \cos γ_{f}}{\cos γ_{s}}$ and $b (z) = - k_{γ}$ . Therefore, if either of

| γ_{f} | < \frac{π}{2}, | γ_{s} | < \frac{π}{2}

(26a)

| γ_{f} | > \frac{π}{2}, | γ_{s} | > \frac{π}{2}

(26b)

is satisfied, A_cl(z) is a stability matrix. Then, the existence of Lyapunov equation (25) is expressed in the following theorem.

Theorem 1. If A_cl(z) is a stable matrix, a pair of P > 0 and Q(z) > 0 that satisfies equation (25) is given by

P = [\begin{matrix} α & 0 \\ 0 & β \end{matrix}], Q (z) = [\begin{matrix} - 2 α a (z) & - β c (z) \\ - β c (z) & - 2 β b (z) \end{matrix}]

(27)

where α and $β$ are positive constants.

Proof. When P and Q(z) defined as

P = [\begin{matrix} p_{1} & p_{3} \\ p_{3} & p_{2} \end{matrix}], Q (z) = [\begin{matrix} q_{1} (z) & q_{3} (z) \\ q_{3} (z) & q_{2} (z) \end{matrix}]

(28)

are used in equation (25), the following relationship is obtained.

p_{2} = - \frac{q_{2}}{2 b}, p_{3} = - \frac{q_{3} + p_{2} c}{a + b}, p_{1} = - \frac{q_{1}}{2 a} - \frac{p_{3} c}{a}

(29)

When some elements of P and Q(z) are given as

p_{3} = 0, q_{1} = - 2 α a, q_{2} = - 2 β b

(30)

the rest of the elements are obtained as

p_{2} = β, p_{3} = - β c, p_{1} = α

(31)

Thus, equation (27) is obtained. Furthermore, P > 0 because of α, $β$ > 0. As for Q(z), we have

| Q (z) | = (4 α a (z) b (z) - β c^{2} (z)) β > 0

(32)

for α, $β$ > 0. Therefore, Q(z) > 0 from the Silvester’s criterion.³²

Equation (26a) is suitable for (M-1) and (M-2), whereas equation (26b) is not. Equation (26a) also corresponds to $R_{γ}$ that makes up the formation region $D_{FC}$ . For these reasons, equation (26a) is only considered for the stability condition of A_cl(z).

Stability condition

The following is a stability condition which is obtained by applying Lyapunov’s stability theory to the closed-loop system (17).

Theorem 2. Let $z (0), z^{eq} \in D_{FC} .$ The equilibrium point z = z^eq = 0 (ξ_f = ξ_f^eq) of the closed-loop system (17) is locally asymptotically stable if $\tilde{Q}$ (z) defined by

\tilde{Q} (z) = [\begin{matrix} - 2 (α a (z) + \bar{λ} c_{g}) & - β c (z) \\ - β c (z) & - 2 (β b (z) + \bar{λ} c_{g}) \end{matrix}]

(33)

is positive definite, where $\bar{λ}$ is the largest eigenvalue of P given by equation (27). Lyapunov function in this case is V (z) = z^TPz.

Proof. Using equations (17), (20b), and (25), the derivative of V (z) is

\begin{matrix} \overset{\cdot}{V} (z) = 2 z^{T} P \overset{\cdot}{z} = z^{T} {P A_{cl} (z) + A_{cl}^{T} (z) P} z + 2 z^{T} Pg (z; d_{u}) \\ \leq z^{T} Q (z) z + 2 \bar{λ} c_{g} {| | z | |}^{2} = - z^{T} \tilde{Q} (z) z . \end{matrix}

(34)

Thus, if $\tilde{Q}$ (z) > 0, the equilibrium point z = z^eq = 0 (ξ_f = ξ_f^eq) is locally asymptotically stable.

Theorem 2 expresses the stability condition of the closed-loop system (17) as $\tilde{Q}$ (z) > 0. Conditions for concretely judging $\tilde{Q}$ (z) > 0 are summarized as the following theorem.

Theorem 3. The necessary and sufficient condition for $\tilde{Q}$ (z) > 0 is to satisfy either of the following conditions

(PD - 1) : ϕ (z) > 0, D (z) > 0, (c_{g} - ψ (z)) ϕ (z) - \sqrt{D (z)} < 0

(35)

(PD - 2) : ψ (z) > 0, c^{2} (z) - 4 ϕ (z) ψ (z) < 0

(36)

where

\begin{matrix} ϕ (z) ≜ - a (z) - c_{g}, ψ (z) ≜ - b (z) - c_{g}, \\ D (z) ≜ b^{2} (z) ϕ^{2} (z) - c_{g} c^{2} (z) ϕ (z) \end{matrix}

(37)

Proof. The necessary and sufficient condition for $\tilde{Q}$ (z) > 0 from Sylvester’s criterion is that the following three inequalities hold, where the following notation $ζ = α / β$ is introduced.

ζ a + \frac{\bar{λ}}{β} c_{g} < 0

(38a)

b + \frac{\bar{λ}}{β} c_{g} < 0

(38b)

| \tilde{Q} / β | = 4 (ζ a + \frac{\bar{λ}}{β} c_{g}) (b + \frac{\bar{λ}}{β} c_{g}) - c^{2} > 0

(38c)

(i) $α \geq β$ (ζ ≥ 1): We have λ¯ = α. Equations (38a), (38b), and (38c) become

a + c_{g} < 0

(39a)

b + ζ c_{g} < 0

(39b)

4 c_{g} ϕ ζ^{2} + 4 b ϕ ζ + c^{2} < 0

(39c)

ζ which the left-hand side of equation (39c) becomes zero is given by

ζ_{1, 2} = \frac{- ϕ b \pm \sqrt{D}}{4 ϕ c_{g}} (ζ_{1} : +, ζ_{2} : -)

(40)

When D ≤ 0, there is no ζ satisfying equation (39c). When D > 0, it exists in the range of

0 < ζ_{2} < ζ < ζ_{1}

(41)

For existence of ζ ≥ 1, ζ₁ has to be greater than one. To summarize the above, the conditions for $\tilde{Q}$ (z) > 0 are to satisfy equation (39a), D > 0 and ζ₁> 1; that is, (PD-1).

(ii) $α < β$ (ζ < 1): We have $\bar{λ} = β$ . Equations (38a), (38b), and (38c) are

ζ a + cg < 0

(42a)

b + cg < 0

(42b)

4 c_{g} ψ ζ + {4 ψ c_{g} + c^{2}} < 0

(42c)

ζ satisfying equation (42c) is given by

ζ > ζ_{3} = \frac{4 ψ c_{g} + c^{2}}{- 4 c_{g} ψ} < 0

(43)

For existence of ζ < 1, ζ₃ has to be smaller than one. To summarize the above, the conditions for $\tilde{Q}$ (z) > 0 are to satisfy equation (42b) and ζ₃ < 1; that is, (PD-2).

Equations (38a) and (38b) are redundant in the condition of $\tilde{Q}$ (z) > 0 according to the Silvester’s criterion. In the case of (i), equation (38b) is omitted from (PD-1). In the case of (ii), equation (38a) is omitted from (PD-2). If the former is satisfied, then ζ ≥ 1 ( $α \geq β$ ). If the latter is satisfied, then 0 < ζ < 1 ( $α < β$ ). If both are satisfied at the same time, then ζ > 0 ( $α - β$ ) exists. Theorem 2 gives one of conditions for the locally asymptotically stable closed-loop system (17), and Theorem 3 expresses the conditions in the form of inequalities. Generally speaking, Lyapunov’s stability theory represents a sufficient condition, so it may be stable even if these conditions are not satisfied. Taking this into consideration, we also refer to the closed-loop system (17) as formation stable if the relative state converges to the equilibrium point from a given initial position. Thus, the conditions presented in Theorem 3 are sufficient to be formation stable.

Steady-state deviation in formation control

This section presents evaluation of the steady-state deviation when the relative state error e_f is included. As the leader trajectory is given according to (M-3) and (M-4), this paper deals with a case where e_f and d_u are constant.

Formation steady-state deviation (FSSD)

In the formation control, the following deviations are defined.

Δ ξ_{f} (t) ≜ ξ_{f}^{ref} - ξ_{f} (t)

(44)

Δ ξ_{s} (t) ≜ ξ_{f}^{ref} - ξ_{s} (t)

(45)

Δξ_f (t) and Δξ_s(t) are called true and virtual formation deviations, respectively. The latter is known and is used in the linear input (14b) µ_f = KΔξ_s. Both deviations are related as

Δ ξ_{f} (t) ≜ e_{f} (t) + Δ ξ_{s} (t)

(46)

These steady-state deviations, taken as t→∞, are written as follows.

Δ ξ_{f}^{s s} ≜ ξ_{f}^{r e f} - ξ_{f} (\infty) ≜ [\begin{matrix} Δ d_{f}^{ss} \\ Δ γ_{f}^{ss} \end{matrix}] ≜ [\begin{matrix} d_{f}^{ref} - d_{f} (\infty) \\ γ_{f}^{ref} - γ_{f} (\infty) \end{matrix}]

(47)

Δ ξ_{s}^{s s} ≜ ξ_{s}^{r e f} - ξ_{s} (\infty) ≜ [\begin{matrix} Δ d_{s}^{ss} \\ Δ γ_{s}^{ss} \end{matrix}] ≜ [\begin{matrix} d_{f}^{ref} - d_{s} (\infty) \\ γ_{f}^{ref} - γ_{s} (\infty) \end{matrix}]

(48)

Δξ_f^ss and Δξ_s^ss are the true and the virtual formation steady-state deviations (FSSDs), respectively. In equation (15) for the case of the leader trajectory with constant d_u, they are expressed by the following equations by $\overset{\cdot}{ξ}$ f = 0.

Δ ξ_{f}^{ss} = e_{f} (\infty) + Δ ξ_{s}^{ss}

(49)

Δ ξ_{s}^{ss} = - K^{- 1} (A_{s} A_{f}^{- 1} - I) d_{u}

(50)

where the relative states included in the elements of A_s and A_f in equation (50) are the steady-state when t→∞. Equation (49) means that the true FSSD is the sum of the relative state error and the virtual FSSD. Using equations (4), (8), and (18), the elements of Δξ_s^ss are derived as follows:

Δ d_{s}^{ss} = \frac{v_{l}}{k_{d}} (\cos γ_{f} - \cos γ_{s}) \frac{\cos (γ_{f} - χ_{f})}{\cos γ_{f}}

(51a)

Δ γ_{s}^{ss} = \frac{v_{l}}{k_{γ}} (\frac{\sin γ_{s}}{d_{s}} - \frac{\sin γ_{f}}{d_{f}}) \frac{\cos (γ_{f} - χ_{f})}{\cos γ_{f}}

(51b)

where all of the relative states in the above equations are the steady-state when t→∞.

Approximation of virtual FSSD

This section presents two approximations to simplify virtual FSSD.

Approximation 1. For the leader trajectories that satisfy (M-3), the following approximation holds.

\frac{\cos (γ_{f} - χ_{f})}{\cos γ_{f}} ≃ \frac{\cos γ_{f} + χ_{f} \sin γ_{f}}{\cos γ_{f}} = 1 + χ_{f} \tan γ_{f}

(52)

Using this into equations (51a) and (51b), the virtual FSSD is approximated as

Δ d_{s}^{ss} ≃ \frac{v_{l}}{k_{d}} (\cos γ_{f} - \cos γ_{s}) (1 + χ_{f} \tan γ_{f})

(53a)

Δ γ_{s}^{ss} ≃ \frac{v_{l}}{k_{γ}} (\frac{\sin γ_{s}}{d_{s}} - \frac{\sin γ_{f}}{d_{f}}) (1 + χ_{f} \tan γ_{f})

(53b)

Approximation 2. For the leader trajectories that satisfy (M-3) and smaller relative state error,

| ε_{d} | << d_{f}

(54a)

\sin ε_{γ} ≃ ε_{γ}, \cos ε_{γ} ≃ 1

(54b)

the following approximation also holds.

\frac{\sin γ_{s}}{d_{s}} - \frac{\sin γ_{f}}{d_{f}} ≃ ε_{γ} \frac{\cos γ_{f}}{d_{f}} - ε_{d} \frac{\sin γ_{f}}{d_{f}^{2}}

(55)

Using this into equations (53a) and (53b), the virtual FSSD is approximated as

Δ d_{s}^{ss} ≃ \frac{v_{l}}{k_{d}} ε_{γ} \sin γ_{f} (1 + χ_{f} \tan γ_{f})

(56a)

Δ γ_{s}^{ss} ≃ \frac{v_{l}}{k_{γ}} (ε_{γ} \frac{\cos γ_{f}}{d_{f}} - ε_{d} \frac{\sin γ_{f}}{d_{f}^{2}}) (1 + χ_{f} \tan γ_{f})

(56b)

These two approximations provide the following considerations for the virtual FSSD. Δd_s^ss and Δγ_s^ss are proportional to v_l and inversely proportional to k_d and k_γ. The relative angle error ε_γ affects both Δd_s^ss and Δγ_s^ss proportionally. On the other hand, the relative distance error ε_d affects Δγ_s^ss but not Δd_s^ss because it is not explicitly included.

Verification by simulation

This section presents simulation results in which formation control by a leader and two followers, referred to as L, F₁, and F₂ hereafter, was performed on Simulink. Since the purpose of this simulation was to confirm the stability conditions and the FSSDs derived in Sections “Stability condition with relative state error” and “Steady-state deviation in formation control,” the results for typical cases are presented, rather than simulations for all cases. Therefore, more detailed studies, such as the boundary of stability range and techniques to improve the FSSD, are left to the other reports in future. The simulation conditions are shown below.

Leader trajectory

Straight line on x-axis with constant velocity v_l = 2 [m/s]

Formation shape and referenced relative states

\begin{matrix} F_{1} : {d_{f}}_{1}^{ref} = 2 [m], {γ_{f}}_{1}^{ref} = 30 [\deg] \\ F_{2} : {d_{f}}_{2}^{ref} = 2 [m], {γ_{f}}_{2}^{ref} = - 30 [\deg] \end{matrix}

Initial position and azimuth

\begin{matrix} L : (x_{l} (0), y_{l} (0)) = (0, 0) [m], θ_{l} (0) = 0 [\deg] \\ F_{1} : ({x_{f}}_{1} (0), {y_{f}}_{1} (0)) = (- 4, 4), \Rightarrow {d_{f}}_{1} (0) = 5.7 [m], \\ {γ_{f}}_{1} (0) = 53 [\deg] \\ F_{2} : ({x_{f}}_{2} (0), {y_{f}}_{2} (0)) = (- 3, 5), \Rightarrow {d_{f}}_{2} (0) = 5.8 [m], \\ {γ_{f}}_{2} (0) = - 69 [\deg] \end{matrix}

Constraints on control input

{\bar{v}}_{f} = \infty, {\bar{ω}}_{f} = \infty

Gain of linear input

K = diag {k_{d}, k_{γ}} = diag {1, 1}

Relative state errors

\begin{matrix} ε_{d} = {- 1.4 : 0.2 : 1.4} [m] \\ ε_{γ} = {- 40 : 10 : 40} [\deg] \end{matrix}

Parameter values in the formation region

\underline{d} = 0, \bar{d} = \infty, δ_{r} = 0

Some explanations are given on the above condition setting. The leader trajectory was given by a straight line (referred as Line hereafter) on x-axis with a constant velocity as a simple setting. Formation shape was given by a triangle. As for the initial position of the mobile robots, L was placed at the origin and faced to the direction of x > 0. F₁ and F₂ were placed at the appropriate coordinates and directions. The linear input gain was the same for F₁ and F₂. It was assumed that the relative state errors ε_d and ε_γ were only added to F₁ within the specified ranges shown above. Thus, the control law for F₁ was equation (14), whereas that for F₂ was equation (9). Since F₂ was always formation stable and the FSSD was consistently zero, the simulation results shown below will be the stability and the FSSD of F₁. Furthermore, since the simulation here was to examine the robustness, the constraints of the control input were not imposed. The formation region $D_{FC}$ was set in the range that the mobile robot regarded as a point. The upper and the lower limits for the relative distance were not imposed. The singular points of the relative angle were not included in $R_{γ}$ .

Formation stability

Table 1 summarizes the formation stability when constant relative state errors ε_d and ε_γ were included in the control law of F₁. In these tables, “S” means formation stable and “Un” means unstable. The relative state errors examined were equivalent to the referenced relative states from 10% to 70%. However, the closed-loop system was formation stable for wide ranges of the relative state errors.

Table 1.

Formation stability region of F₁ with respect to relative state errors.

ε_d [m]	ε_γ [deg]
ε_d [m]	−40	−30	−20	−10	0	10	20	30	40
−1.4	S	S	S	S	S	S	S	S	Un
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
0.0	S	S	S	S	S	S	S	S	Un
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
0.6	S	S	S	S	S	S	S	S	Un
0.8	S	S	S	S	S	Un	S	S	Un
1.0	S	S	S	S	S	Un	S	S	Un
1.2	S	S	S	S	Un	Un	S	S	Un
1.4	S	S	S	Un	Un	Un	Un	S	Un

Table 2 shows the details of six typical cases with respect to the formation stability. “Cond. of Theorem 3” shows the conditions of Theorem 3, where c_g in equation (20b) was examined by the following two cases; 1: c_g = max(||g(z; d_u)||/||z||) (constant) and 2: c_g = ||g(z; d_u)||/||z|| (varying). “⊙” means that (PD-1) or (PD-2) was satisfied over the entire simulated time-range, “ $Δ$ ” means that it was satisfied in a part, and “×” means that it was not satisfied over the entire simulated time-range. Moreover, the sign of a(z) which is one of eigenvalues of A_cl indicates as follows; “>0”/“<0”: positive/negative over the entire simulated time-range and “>*0”: positive in a part. Figures 2 to 4 show the time histories of Cases A, C and E. “Trajectory” shows the trajectories of L, F₁, and F₂. In Figures 2 and 3, “Condition of Theorem 3” is the time histories of (PD-1) and (PD-2) of Theorem 3, where “Qt1” is the third equation in (PD-1) and “Qt2” is the second equation in (PD-2), in which the inequalities held when they were negative. In addition, “sat-flag” was introduced to judge whether (PD-1) or (PD-2) was satisfied or not; that is, sat-flag = 1: satisfied and 0: not satisfied.

Table 2.

Summary of typical cases with respect to formation stability of F₁.

Case (ε_d, ε_γ)	Cond. of Th 3	Relative angle [deg]			a(z)	Formation stability
Case (ε_d, ε_γ)	c_g: {1, 2}	$γ_{f 1}^{eq}$	$γ_{f 1}^{eq} + ε_{γ}$	$γ_{f 1} (0) + ε_{γ}$	a(z)	Formation stability
A (0.4, −20)	$1, 2 : ⊙$	$66.1 \in R_{γ}$	$44.1 \in R_{γ}$	$33.0 \in R_{γ}$	<0	S
B (1.0, −30)	$1 : Δ, 2 : ⊙$	$85.9 \in R_{γ}$	$55.9 \in R_{γ}$	$33.0 \in R_{γ}$	<0	S
C (1.4, −20)	$1, 2 : Δ$	$96.6 \notin R_{γ}$	$76.6 \in R_{γ}$	$33.0 \in R_{γ}$	>*0	S
D (1.4, −10)	$1, 2 : Δ$	$107.6 \notin R_{γ}$	$97.6 \notin R_{γ}$	$43.0 \in R_{γ}$	>*0	Un
E (0.8, 10)	$1, 2 : Δ$	$84.9 \in R_{γ}$	$94.9 \notin R_{γ}$	$63.0 \in R_{γ}$	>*0	Un
F (0.0, 40)	$1, 2 : \times$	$- 39.9 \in R_{γ}$	$0.1 \in R_{γ}$	$93.0 \notin R_{γ}$	>0	Un

Figure 2.

Time histories in straight line formation, Case A (ε_d = 0.4 [m], ε_γ = −20 [deg]): (a) trajectory of L, F₁, and F₂ and (b) condition of Theorem 3, c_g: 1.

Figure 3.

Time histories in straight line formation, Case C (ε_d = 1.4 [m], ε_γ = −20 [deg]): (a) trajectory of L, F₁, and F₂ and (b) condition of Theorem 3, c_g: 2.

Figure 4.

Time histories in straight line formation, Case E (ε_d = 0.8 [m], ε_γ = 10 [deg]): (a) trajectory of L, F₁, and F₂, (b) relative state, and (c) control input.

In Cases A and B, a(z) which is one of the eigenvalues of A_cl was negative over the entire simulated time-range, and there existed a Lyapunov solution P. In Case A, both (PD-1) and (PD-2) of Theorem 3 were satisfied. The equilibrium point was locally asymptotically stable (Figure 2(b)). By “c_g: 1” in Case B, it was not satisfied in the latter half of the simulation. In “c_g: 2,” on the other hand, (PD-1) was not temporarily satisfied, but was almost satisfied. The constraint of “c_g: 1” was more conservative than that of “c_g: 2.”

In Cases C, D, and E, a(z) was partly positive and neither (PD-1) nor (PD-2) was satisfied in the latter half of the simulation. The closed-loop system was formation stable in Case C but unstable in Cases D and E. This depended on whether $γ_{f 1}^{eq} + ε_{γ}$ belonged to $R_{γ}$ or not. Since $γ_{f 1} (t) + ε_{γ} = γ_{s 1} (t)$ in F₁, $γ_{f 1} (t) \to γ_{f 1}^{eq}$ and $γ_{s 1} (t) \to γ_{f 1}^{eq} + ε_{γ}$ when t→∞. Even in Cases D and E, F₁ was controlled so as to be formation stable until the middle of the simulated time-range. However, in the process to converge to the equilibrium point, As used in the control law (14a) fell into the singular point, which caused the divergence of the control input and caused the destabilization (Figure 4(b) and (c)). Since $γ_{f 1}^{eq} \notin R_{γ}$ but $γ_{f 1}^{eq} + ε_{γ} \in R_{γ}$ in Case C as shown in Table 2, it was stayed in formation stable without becoming singular. However, since the converged equilibrium point was violated from the application policy (M-2), it was not acceptable formation control. In Case F, since $γ_{f 1} (0) + ε_{γ} = γ_{s 1} (0) \notin R_{γ}$ at the initial time, it became unstable from the initial regardless of the equilibrium point. This was the reason that the cases with ε_γ = 40 [deg] were unstable in Table 1.

Summarizing the above discussion of the formation stability, the closed-loop system of the formation control by the L-F method was able to ensure the formation stability even for relatively large relative state errors. In particular, when (PD-1) or (PD-2) in Theorem 3 was satisfied, the L-F method guarantees the formation stability and achieves allowable formation performance along with the application policies. Moreover, when the relative angle was near the singular point of the control law and the measured initial relative angle was outside of $R_{γ}$ , destabilization due to the input divergence was caused.

FSSD

Figure 5 shows the FSSD with respect to the relative state error ε_d and ε_γ. The dotted-squares in the figures are $| ε_{d} | \leq 0.4$ [m] and $| ε_{γ} | \leq 20$ [deg]; that is, it represents a region of “small relative state error.” In Figure 5(a) and (c) which show the virtual FSSD $Δ ξ_{s}^{ss} = {[Δ d_{s}^{ss} Δ γ_{s}^{ss}]}^{T}$ , $Δ d_{s}^{ss}$ was almost negative, while $Δ γ_{s}^{ss}$ was proportional to ε_γ. ε_d did not affect both $Δ d_{s}^{ss}$ and $Δ γ_{s}^{ss}$ because the slope of these graphs was small. $Δ d_{s}^{ss}$ reduced $Δ d_{f}^{ss}$ , but $Δ γ_{s}^{ss}$ increased $Δ γ_{f}^{ss}$ because of equation (49).

Figure 5.

FSSD of F₁: (a) virtual FSSD of relative distance, (b) true FSSD of relative distance, (c) virtual FSSD of relative angle, and (d) true FSSD of relative angle.

Since the leader trajectory in this simulation was a straight line, Approximation 1 given by equation (53) was consistent with equation (51). Table 3 shows the range of the virtual FSSD given by equations (51) and (56) (Approximation 2). It is seen that Approximation 2 was acceptable in the region of “small relative state error.”

Table 3.

Range of virtual FSSD given by equations (51) and (56) (Approximation 2) within “small relative state error”: |ε_d| ≤ 0.4 [m] and |ε_γ| ≤ 20 [deg].

	Equation (51)	Equation (56)
$Δ d_{s}^{ss}$ [m]	[−0.58, 0.11]	[−0.64, 0.08]
$Δ γ_{s}^{ss}$ [deg]	[−16.1, 22.6]	[−16.3, 27.7]

Summarizing the above discussion, the true FSSD was caused by the relative state error and the virtual FSSD derived from the leader’s state. ε_d did not affect both $Δ d_{s}^{ss}$ and $Δ γ_{s}^{ss}$ very much. The formation shape affected $Δ d_{s}^{ss}$ but not $Δ γ_{s}^{ss}$ . Approximation 2 was also an acceptable approximate expression within the region of “small relative state error.”

Further simulations

To enhance the validity of the stability condition and the formation steady-state deviation, which were described in Sections “Stability condition with relative state error” and “Steady-state deviation in formation control,” this section presents the following additional simulations: sinusoidal leader trajectory (referred as Sinusoid hereafter) and five follower formation.

Sinusoid formation

Figures 6 to 8 show the time histories in sinusoid formation where the relative errors were given by Cases A, C, and E in Table 2. As shown in the figures, Cases A and C were formation stable but Case E was unstable. The results shown in Figures 6 to 8 were approximately similar to those in Figures 2 to 4 in which the leader trajectory was given by a straight line. Thus, as mentioned in Section “Formation stability,” the conditions of Theorem 3, the equilibrium and the initial of the relative angle were closely related to the formation stability.

Figure 6.

Time histories in sinusoid formation, Case A (ε_d = 0.4 [m], ε_γ = −20 [deg]): (a) trajectory of L, F₁, and F₂ and (b) condition of Theorem 3, c_g: 1.

Figure 7.

Time histories in sinusoid formation, Case C (ε_d = 1.4 [m], ε_γ = −20 [deg]): (a) trajectory of L, F₁, and F₂ and (b) condition of Theorem 3, c_g: 2.

Figure 8.

Time histories in sinusoid formation, Case E (ε_d = 0.8 [m], ε_γ = 10 [deg]): (a) trajectory of L, F₁, and F₂, (b) relative state, and (c) control input.

Table 4 shows the virtual FSSD given by equations (51) and (56) (Approximation 2), where the relative state error was Case A which was within the region of “small relative state error.” The leader trajectory was given by Line and Sinusoid. It is seen that Approximation 2 was acceptable in both leader trajectories.

Table 4.

Virtual FSSD given by equations (51) and (56) (Approximation 2), Case A (ε_d = 0.4 [m], ε_γ = −20 [deg]).

Leader trajectory	Equation (51)		Equation (56)
Leader trajectory	$Δ d_{s}^{ss}$ [m]	$Δ γ_{s}^{ss}$ [deg]	$Δ d_{s}^{ss}$ [m]	$Δ γ_{s}^{ss}$ [deg]
Line	−0.58	−16.1	−0.64	−16.3
Sinusoid	−0.59	−15.5	−0.65	−15.5

Five follower formation

This section presents a formation control where the number of the followers was five. The formation shape was given by a triangle structure which is shown in Figure 9. The references of the followers were given as

\begin{matrix} {d_{f}}_{1}^{ref} = {d_{f}}_{2}^{ref} = {d_{f}}_{3}^{ref} = {d_{f}}_{4}^{ref} = {d_{f}}_{5}^{ref} = 2 [m] \\ {γ_{f}}_{1}^{ref} = {γ_{f}}_{3}^{ref} = 30 [\deg], {γ_{f}}_{2}^{ref} = {γ_{f}}_{4}^{ref} = {γ_{f}}_{5}^{ref} = - 30 [\deg] \end{matrix}

Figure 9.

Structure of five follower formation.

The leader for F₁ and F₂ was L, the one for F₃ and F₄ was F₁ and the one for F₅ was F₂. The orange dashed-line in Figure 9 indicates that the measured relative states of F₁, F₃, and F₅ included the following relative state errors:

\begin{matrix} F_{1} : ε_{d} = 0.4 [m], ε_{γ} = - 20 [\deg] \\ F_{3} : ε_{d} = - 0.8 [m], ε_{γ} = - 10 [\deg] \\ F_{5} : ε_{d} = 0.6 [m], ε_{γ} = 10 [\deg] \end{matrix}

Figure 10 shows the trajectory of five follower formation where the leader trajectory was given by Line. Although some FSSDs occurred, the formation stability was established by satisfying the stable conditions between each leader and follower.

Figure 10.

Trajectory of five follower formation, Line.

As mentioned above, it is concluded from further simulations that the stability conditions and the FSSD described in Sections “Stability condition with relative state error” and “Steady-state deviation in formation control” were also effective in curved leader trajectory and increase of the followers.

Concluding remarks

This paper has presented the robustness of the formation control of mobile robots using the L-F method. The uncertainty considered was the relative state error which was included in the relative state. The robust stability conditions against the relative state error were derived by Lyapunov’s stability theory (direct method). We also obtained the formation steady-state deviation (FSSD) when the formation stability was ensured. The formation control environment was constructed on Simulink. The validity of the stability condition and the FSSD was demonstrated by numerical simulation. Consequently, it was seen that the closed-loop system was formation stable even for large relative state error; that is, the L-F method provided a robust control law against the relative state error. On the other hand, for the FSSD, the validity of the derived approximation was shown in the region in which the relative state errors were small. These results were similar to those obtained in which the leader trajectory is curved and the number of followers was increased. However, large FSSD was occurred in some cases although it was formation stable. These cases should be avoided in practical application. Some techniques are required to reduce the large FSSD.

The knowledge obtained in this paper clarified that the L-F method has robust characteristics from a theoretical point of view, and it may be said that we showed the superiority of using the L-F method for formation control. This paper considered the relation of one leader and one follower. When the number of followers is increased in the L-F method, the formation is developed by constructing the relation of leader and follower continually, and the control law is installed on each follower in the decentralized control approach. Therefore, the robust stability and the FSSD presented in this paper are effective in each relation of leader and follower. This was demonstrated in further simulations.

Since the simulation presented in this paper was the examination in the case of constant relative state error, it is required to obtain more enough knowledge by varying relative state error. Furthermore, it seems to be necessary for the practical application of the L-F method to examine a reduction technique on the FSSD. These are future research subjects.

Footnotes

Handling Editor: Chenhui Liang

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Atsushi Fujimori

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Robustness in formation control of mobile robots using leader-follower method

Abstract

Keywords

Introduction

Leader-follower method of mobile robots

Statements of problem

Relative state-space equation

Relative state-space equation with relative state error

Stability condition with relative state error

Range of relative state

Lyapunov equation with respect to Acl(z)

Stability condition

Steady-state deviation in formation control

Formation steady-state deviation (FSSD)

Approximation of virtual FSSD

Verification by simulation

Formation stability

FSSD

Further simulations

Sinusoid formation

Five follower formation

Concluding remarks

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Lyapunov equation with respect to A_cl(z)