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Time-varying Formation Tracking of Multiple Manipulators via Distributed Finite-time Control
^{†}^{†}thanks: Citation Information: Ming-Feng Ge, Zhi-Hong Guan, Chao Yang, Tao Li, & Yan-Wu Wang. Time-varying formation tracking of multiple manipulators via distributed finite-time control. Neurocomputing, 2016, 202: 20-26. http://dx.doi.org/10.1016/j.neucom.2016.03.008

###### Abstract

Comparing with traditional fixed formation for a group of dynamical systems, time-varying formation can produce the following benefits: i) covering the greater part of complex environments; ii) collision avoidance. This paper studies the time-varying formation tracking for multiple manipulator systems (MMSs) under fixed and switching directed graphs with a dynamic leader, whose acceleration cannot change too fast. An explicit mathematical formulation of time-varying formation is developed based on the related practical applications. A class of extended inverse dynamics control algorithms combining with distributed sliding-mode estimators are developed to address the aforementioned problem. By invoking finite-time stability arguments, several novel criteria (including sufficient criteria, necessary and sufficient criteria) for global finite-time stability of MMSs are established. Finally, numerical experiments are presented to verify the effectiveness of the theoretical results.

Keywords time-varying formation tracking, dynamic leader, multiple manipulator systems (MMSs), finite-time stability.

## 1 Introduction

Recently, distributed control problems for a group of dynamical systems have attracted much attentions due to its wide applications, including coordination for multi-agent systems [1]-[5], synchronization in complex networks [6, 7], distributed computing in sensor networks [8]-[10], multi-fingered hand grasping and manipulation [11, 12]. Formation control is a significant issue in the distributed control field. A formation is defined as a special configuration (, desired positions and orientations) formed by a cluster of interconnected autonomous agents, in which a global goal is achieved cooperatively [13]. Many formation control methods have been developed, such as virtual structure methods [14], behavior-based methods [15, 16], leader-follower methods [17], artificial potential field methods [18]. The aforementioned methods can only produce fixed formations for multi-agent systems. However, in a number of real-world applications, the formation of multi-agent systems is always changing to adapt to the dynamical changing environment. It follows that the fixed formations cannot satisfy the practical requirements of many real-world applications. It thus motivates several research on time-varying formations. Time-varying formation control algorithms for a group of unmanned aerial vehicles with its applications to quadrotor swarm systems had been presented based on consensus theory [19]. Coherent formation control of a set of agents, including unmanned aerial vehicles and unmanned ground vehicles, in the presence of time-varying formation had been studied in [20]. Time-varying formation implies that the formation of a multi-agent system can be changing as required without losing system stability, which products the following benefits: i) covering the greater part of complex environments; ii) collision avoidance. However, to the authors’ knowledge, the mathematical formulations of time-varying formation tracking are still not clear, which impedes the development and applications of the relative technologies.

On the other hand, networked robotic systems have been broadly studied due to their various advantages, including flexibility, adaptivity, fault tolerance, redundancy, and the possibility to invoke distributed sensing and actuation [21]. Many control algorithms for global asymptotic tracking of networked robotic systems described by Euler-Lagrange systems can be found in the literature. Adaptive control approaches are proposed to address the leader-follower and leaderless coordination problems for multi-manipulator systems based on graph theory [22, 23]. Distributed containment control had been developed for global asymptotic stability of Lagrangian networks under directed topologies containing a spanning tree [24]. Some distributed average tracking algorithms had been developed invoking extended PI control and applied to networked Euler-Lagrange systems [25]. The task-space tracking control problems of networked robotic systems under strongly connected graphs without task-space velocity measurements had been investigated [26]. In presence of kinematic and dynamic uncertainties, task-space synchronization had been addressed for multiple manipulators under strong connected graphs by invoking passivity control [27] and adaptive control [28]. All of the aforementioned control algorithms produce global asymptotic tracking of robotic manipulators, which implies that the system trajectories converge to the equilibrium as time goes to infinity. Finite-time stabilization of dynamical systems may give rise to fast transient and high-precision performances besides finite-time convergence to the equilibrium, and a lot of work has been done in the last several years [29]-[31].

Motivated by our preliminary work on distributed control [32, 33], the time-varying formation tracking of multiple manipulator systems (MMSs) is taken into account. Distributed finite control is developed to drive the centroid of the MMS to follow the leader at a distance and to achieve the desired time-varying formation of the MMS meanwhile. The main contributions are summarized as following: i) Comparing with the existing work based on multi-agent systems with single-integrator and double-integrator dynamics [21], we consider MMSs described by Euler-Lagrange systems. ii) Comparing with the existing fixed formation tracking algorithms for multi-agent system [34], we consider the time-varying formation tracking problems with a dynamic leader and present an explicit mathematical formulation of time-varying formation based on its practical characteristics. iii) Some novel estimator-based finite-time control algorithms are developed for the above time-varying formation tracking problems. For the presented control algorithms, some conditions (including sufficient conditions, necessary and sufficient conditions) are derived to guarantee the achievement of time-varying formation tracking.

The rest of this paper is organized as follows: system formulation and some preliminaries are presented in Section 2. The control algorithms and conditions of time-varying formation tracking are given in Section 3. In Section 4, the simulation results are presented. The conclusions are provided in Section 5.

## 2 Preliminaries

### 2.1 System formulation

The dynamics of the ith manipulator in the MMS is given as following [35]:

(1) |

where , , is the initial time, and are the position, velocity and acceleration vectors of generalized coordinates, and are the inertia and the Coriolis/centrifugal force matrices, and denote the gravitational torque and the input torque respectively.

The leader for the MMS is given as following:

where are the position, velocity and acceleration vectors of generalized coordinates respectively.

We invoke a directed graph to describe the interaction of the MMS,
where denotes the node set given right after (1), is the edge set,
represents the adjacency matrix.
The ith node denotes the ith manipulator in the MMS.
An edge denotes that the ith node can access information from the jth node.
The adjacency weight is defined as if , and otherwise.
Besides, self-edges are not allowed in this paper, , .
A directed path from the ith node to the jth node is an ordered sequence of edges
in the directed graph.
The neighbor set of the jth manipulator is denoted by .
is said to be undirected if and only if ,
, , .
Throughout this paper, is supposed to be undirected.
Let be the nonnegative weight vector between the n nodes and the leader,
where if the information of the leader is available to the ith node,
namely, the ith node is pinned; otherwise.
The Laplacian matrix for is defined as
and .
Two assumptions throughout this paper are presented as following:

A1) The leader has a directed path to the nodes in the set under and ;

A2) ,
where represents the Euclidean norm and is a positive constant.

By Assumption A2, the derivative of the acceleration of the leader is bounded,
which happens to be the actual characteristics of the trajectories that can be reachable by the manipulators
described by Euler-Lagrange system [36].

###### Lemma 1.

[37] Suppose that Assumption A1 holds. is symmetric positive definite, where denotes the Kronecker product and represents the identity matrix.

### 2.2 Problem Statement

In a number of real-world applications, the desired formation for the MMS is required to be time-varying and switching according to task demands. In this section, the explicit mathematical definition of time-varying formation tracking is presented.

Let be a finite set of desired formations, where denotes the sth desired formation, denotes the local coordinate of the ith manipulator in the m-dimensional Euclidean space with respect to , . Note that becomes a desired geometric pattern in 2D plane if . Let denote the index set of . A switching signal is introduced with a sequence of time points , satisfying , at which the desired formation changes. Let be the desired formation at time . Then for any , the desired formation . Besides, we assume that the desired formation is closed at each time instant, , , .

The control objective is to design distributed control for the ith manipulator by invoking its information (, , and ) and its neighbour node’s states (, , and for ) such that for any , the time-varying formation tracking is said to be achieved for the MMS, ,

(2) |

where denotes the settle time. In this paper, we assume that the minimum switching interval is large enough such that can be included in the half-open interval , .

###### Remark 1.

Note that (2) means that for any , the MMS converge to the desired formation and the centroid of the MMS follows the leader before time . By designing time-varying formations, the obstacle and collision avoidance can be achieved while the centroid follows the leader. It is worthy to point out that the control problem addressed in [34] is a special case of (2).

### 2.3 Finite-time stability

Some concepts for finite-time stability and homogeneous systems are introduced in this section [40]. Consider a k-dimensional system

(3) |

where is an arbitrary positive integer. The continuous vector field is homogeneous of degree with dilation , if for any ,

where . System (3) is said to be homogeneous if its vector field is homogeneous. Additionally, the following k-dimensional system

(4) |

is called being locally homogeneous of degree with dilation , if system (3) is homogeneous and the continuous vector field satisfies

Based on the above presentations, some results and lemmas in [40]-[42] which will be used in this paper are proposed here.

###### Lemma 2.

(LaSalle’s Invariance Principle) Let be a solution of , where is the initial time, is continuous with an open subset of , and be a locally Lipschitz function such that , where denotes the upper Dini derivative. Then is contained in the union of all solutions that remain in , where denotes the positive limit set.

###### Lemma 3.

###### Lemma 4.

If the equilibrium of a closed-loop system is global asymptotic stable and local finite-time stable, then it is also global finite-time stable.

## 3 Time-varying formation tracking of multiple manipulators

In this section, we are concerned with the time-varying formation tracking problems where the formations of the MMS is time-varying and the leader has varying vectors of generalized coordinate derivatives.

Before moving on, some auxiliary variables are given. Let the th manipulator’s estimated value of be , . For any and , some auxiliary variables are defined as follows:

(5) |

###### Remark 2.

The variable presented in (5) contains the information of the time-varying formations and switches at the time sequence . Besides, means that the formation described by is obtained for the MMS.

Let , , and

(6) |

where are positive constants, and are continuous odd vector fields satisfying , , and around for some positive constants and , is the th entry of the adjacency matrix , is the weight between the leader and the th manipulator, , is the signum function, . We then propose the following distributed estimator-based control

(7a) | |||

(7b) |

where is presented in Assumption A2, , .

###### Remark 3.

As shown in (6), the sliding-mode estimator (7b) provides a distributed estimated value to construct the auxiliary variable . Moreover, inspired by the inverse dynamics control technology proposed in [43]-[45], the input torque presented in (7a) is developed by using . Thus, the control law (7) is called distributed estimator-based control.

###### Theorem 1.

###### Proof.

The proof proceeds in the following three steps. First, the simplification of the close-loop system is derived from the finite-time stability of sliding-mode estimators. Second, the global asymptotic stability is proved based on the LaSalle’s Invariance Principle. Thirdly, the global finite-time stability is demonstrated using finite-time stability arguments for homogeneous systems.

For the first presentation, the simplification of the close-loop system is carried out. Substituting (6) and (7a) into (1) gives

(8) |

The positive definiteness of implies that the eigenvalues of is greater than . Then the combination of (7b) and (8) yields the following cascade system:

(9) |

Let be the column stack vector of , . The sliding-mode estimator (7b) can be rewritten as

(10) |

where denotes the n-dimensional column vector whose elements are all one. By Lemma 1, is symmetric positive definite. Take the Lyapunov function ca. . ndidate for system (10). By the similar analysis in Theorem 3.1 of [37], we get that

Therefore, for the sliding-mode estimator (7b), there exists a bounded settle time given by

such that when , . We then show that for bounded initial values and , invoking (7) for (1), the states and remain bounded when , . The distributed sliding-mode estimator (7b) implies that remain bounded for any initial value when . For bounded states and , , equation (5) implies that , , and remain bounded when , . It thus follows from (9) that is bounded with respect to bounded states , , , and . Thus, we can obtain that and remain bounded for bounded initial values and when , . Thus, using (6) and (7) for (1), when , the closed-loop dynamics of system (1) can be rewritten as

(11) |

where . It thus follows from (5) that and are the first-order and second-order derivatives of , . Let , and be the column stack vectors of , and respectively, . System (11) can be rewritten as

(12) |

The first presentation shows that for bounded initial values , and , the states , and remain bounded when , and the close-loop dynamics of (1) under the control algorithms (6) and (7) is equivalent to equation (12) when .

For the second presentation, the global asymptotic stability of system (12) is analyzed. Let an auxiliary variable . Then and . When , for (12), consider the Lyapunov function candidate with

where denotes the th element of the vector . By Lemma 1, is symmetric positive definite. It thus follows from the definition of that the Lyapunov function candidate is positive definite. Taking the derivatives of and along (12) renders that

It thus follows that

Considering that is continuous odd function, we can conclude that . Besides, gives that . It thus follows from the positive definiteness of that if and only if , which implies that . It thus follows from (12) that , which means that . By LaSalle’s Invariance Principle in Lemma 2, for any bounded and , the states and as . Hence, the second presentation shows that the equilibrium of system (12) is global asymptotic stable.

For the third presentation, the global finite-time stability of system (12) is analyzed. First, the local finite-time stability is proven by invoking Lemma 3 and 4. To this end, let , and . By the definition of and right after (6), we can get that system (12) can be written as

(13) |

where

It visibly follows that () is the equilibrium of system (13). Considering that , we can conclude that system (13) is locally homogeneous of degree with respect to dilation , where and are -dimensional column vectors whose elements are and respectively. Hence, the third presentation shows that the equilibrium of system (12) is finite-time asymptotic stable.

By Lemma 4, the second and third presentations show that for bounded and , there exists a time point that the states and as . By the first presentation, and remain bounded for bounded initial value , and . Hence, for bounded initial value , and , the states and as . This completes the proof. ∎

Note that the following necessary and sufficient condition can be easily obtained by some simple transformation for Theorem 1.

###### Corollary 1.

###### Proof.

The sufficiency of Corollary 1 is proved as the same as in Theorem 1. Next we show the necessity part by contradiction. If Assumption A1 dose not hold, there exists an isolated subset of manipulators, which cannot obtain any information of the leader directly or mediately. It follows that the evolution of the close-loop dynamics of these manipulators is carried out without any information of the leader. Thus, these manipulators cannot necessarily follow the trajectory of the leader. This ends the proof. ∎

Let a switching graph describe the interaction of the MMS), where represents the weight adjacency matrix. Let be the switching nonnegative weight vector between the n nodes and the leader. Then the following corollary can be obtained for the case, in which the communication topology is switching.

###### Corollary 2.

Suppose that A2 holds and the leader are reachable to the MMS under and . Let the control algorithms be replaced by

then (2) holds (, the time-varying formation tracking is achieved for the MMS) if and only if Assumption A1 holds.

###### Proof.

###### Remark 4.

Note that the functions and can be easily selected, such as , , and , where and denote the saturation function and the hyperbolic tangent function respectively. Besides, by the boundedness of and , we can conclude that the control law in this paper is bounded by the boundedness of the dynamic terms in system (1).

###### Remark 5.

The dynamics of the leader can also be described by the Euler-Lagrange equation , which gives a additional task for designing . In this case, the MMS has a master-slave structure, in which the master manipulator acts as the leader while the slave manipulators act as followers [38, 39]. By designing suitable such that Assumption A2 holds following [24], the main results presented in this paper can still be effective.

###### Remark 6.

Comparing with [24, 25], in which global asymptotic stability is achieved, we study the global finite-time stability for time-varying formation tracking which is more practical and challenging than traditional global asymptotic stability, especially for robotic systems. Different from [27, 28], in which the constant agreement value is taken into account, we consider the time-varying formation tracking problem of multi-robot systems with a dynamic leader.

## 4 Simulations

In this section, simulations are presented to illustrate the effectiveness of the proposed algorithms. We consider the time-varying formation tracking problem for a MMS containing six manipulators (, agents) with three desired formations. Each agent is assumed to be a planer robotic manipulator with two revolute joints, , . The dynamic model and the physical parameters presented in [45] are invoked. For simplify, in our simulation, we choose if agent can access the information of agent , otherwise; if agent can obtain the information of the leader directly, otherwise. The interaction topology is shown in Fig.1. The Laplacian matrix is

and the nonnegative weight vector is given by . The elements of the initial values , and are randomly selected from .

The finite set of desired formations is shown in Fig.2, the local coordinates in the 2D plane is given as , . The details of , , are presented in Table 1. The sampling period is adopted to be . The simulation time span is selected as . The time-varying formation are given as following

The trajectory of the leader is given by , and , can be calculated easily, where . Without loss of generality, let and . The other control parameters are selected as follows: , , and can be easily computed.

The simulation results are presented in Fig.3 and Fig.4. Fig.3 shows that the tracking errors and defined in (5) converge to zero in finite time at each dwell time interval, which means the time-varying formations of the MMS in the 2D plane and the tracking of the leader can be achieved simultaneously, , the time-varying formation tracking is accomplished. Additionally, the trajectory of the manipulators in 2D space is illustrated in Fig.4. It follows that the robots can reach the desired time-varying formation and the geometric center of the MMS follows the leader as required. It is clear in Fig.3 and Fig.4 that using the control algorithm (7) under the aforementioned configurations, the time-varying formation tracking can be achieved for the MMS.

###### Remark 7.

It is shown from picture and in Fig.3 that the second elements of and do not change at the switching time instant . Note that the time-varying formation changes from to . By the set of and in Table 1, stays the same at the switching time instant , which thus gives that and do not change at the switching time instant.

## 5 Conclusion

For multiple manipulator systems (MMSs) under fixed and switching graphs, the time-varying formation tracking problem is addressed using inverse dynamics control technologies. Based on the functional characteristics of MMSs, an explicit formulation of time-varying formation is presented. The conditions (including sufficient conditions, necessary and sufficient conditions) on the interaction topology and control parameters are derived. Simulation results are presented to verify the effectiveness of the proposed algorithms. A few interesting issues, which are not addressed in this paper, concern the time-varying formation tracking problems of uncertain Euler-Lagrange systems and the extension of the presented approaches to the case of the polynomial trajectories. These issues will be considered in our future work.

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