Fully distributed cooperation for networked uncertain mobile manipulators

09/20/2018
by   Yi Ren, et al.
Technische Universität München
0

This paper investigates the fully distributed cooperation scheme for networked nonholonomic mobile manipulators. To achieve cooperative task allocation in a distributed way, an adaptation-based estimation law is established for each robotic agent to estimate the desired local trajectory. In addition, wrench synthesis is analyzed in detail to lay a solid foundation for tight cooperation task. Together with the estimated task, a set of distributed adaptive control is proposed to achieve motion synchronization of the mobile manipulator ensemble over a directed graph with a spanning tree irrespective of the kinematic and dynamic uncertainties in both the mobile manipulators and the tightly grasped object. The controlled synchronization alleviates the performance degradation caused by the estimation/tracking discrepancy during transient phase. Persistent excitation condition and noisy Cartesian-space velocities are totally avoided. Furthermore, the proposed scheme is independent from the object's center of mass by employing formation-based task allocation and task-oriented strategy. These attractive attributes facilitate its practical application. It is theoretically proved that convergence of the cooperative task tracking error is guaranteed. Simulation results validate the efficacy and demonstrates the expected performance of the proposed scheme.

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I Introduction

Mobile manipulator, which combines the manipulation dexterity in robotic arm and the maneuverability in mobile platform, tends to be far more versatile than the conventional base-fixed counterpart due to its enlarged workspace and the potential for wider application scenarios, e.g, part transfer, rescue and remote maintenance in outdoor environment, etc.  [1]

. Based on that, multiple mobile manipulator ensemble has drawn increasing attention of the research community in recent years owing to its ability to perform more complex tasks such as transporting or assembling large and heavy objects that cannot be achieved by a single mobile robot. While these attractive advantages reveal a major increase of complexity for modelling and controlling such systems, especially for the case considered in this paper that a team of uncertain nonholonomic mobile manipulators cooperate to grasp and manipulate an unknown object. Introduction of the mobile platform, typically a nonholonomic vehicle, not only creates high degree of redundancy but also imposes nonintegrable constraints on the kinematics, which hinders direct control of the whole system and restricts its instantaneous motion capability. In addition, interactions between the mobile platform and the manipulator necessitate the integrated modelling method of both the system dynamics and kinematics. Furthermore, the tightly grasped object and the mobile manipulator ensemble form connected kinematics with star topology, which leads to the imposition of a set of kinematic/dynamic constraints on each mobile manipulator and the degradation of the degree of freedom. This will be accompanied by the generation of internal forces that needs careful regulation. Ignoring the control of these internal forces may result in grasp failure or unrecoverable damage to the end-effector (EE) or the object.

The core problem in multi-robot manipulation besides the modelling complexity mentioned above lies in the establishment of a fully distributed scheme for the inherently centralized cooperation task, especially under certain communication constraint and ubiquitous model uncertainties. Note that the scheme presented in this paper may easily be extended to other cases by releasing some of the considered constraints.

I-a Related Work

Cooperation and coordination of multi-agent system have been well recognized as an important technique to enhance flexibility and improve efficiency [2]. The endeavor to achieve manipulation and transportation of an object by multi-arm system in a cooperative manner generally comprises two control schemes: centralized control and distributed control [3]. Under the centralized architecture, a global coordinator either in leader robot or in another host computer facilitates object-oriented control with the help of available global states of the robotic system. Force/position control [4] and impedance control [5, 6] are both extensively utilized to achieve safe interaction between the dual-arm manipulators and the grasped object. Extension of this cooperative manner to multiple mobile manipulators [7] and its model-based control can benefit from the comprehensive interaction dynamic model from the perspective of kinematic constraint [8]. Although centralized architecture shows sufficient power to control multi-robot system and can easily incorporate single-robot control strategy, assumption of the existence of a central station makes the whole system more vulnerable and its malfunction will lead to the breakdown of the whole system [9]. On the contrary, as a more promising and preferable alternative, distributed approach predominates when robot collectives are subjected to some inevitable physical constraints such as communication limitations. Specifically, it is unreasonable to assume there exists a coordinator for the case studied here since all team members are mobile.

To achieve distributed control of a multi-arm system, leader-follower approach is an option generally associated with the schemes that are devoted to reducing the communication burden while trying to achieve as far as possible. Based on the diagram of impedance dynamics and leader-follower scheme, coordinated motion control for multiple mobile manipulators is employed to achieve cooperative object manipulation [10]. Inspired by a team of people moving a table, the followers can implement similar impedance law as the leader’s either by estimating the leader’s desired motion [11] or by taking the contact force as the leader’s motion intention [12]. This innovation enables the whole system to work under implicit communication. Another interesting work [13] that does not require communication achieves force coordination between leader and follower only by measuring the object’s motion as the feedback. However, the assumptions that all followers act passively in the transport task in [10, 11], that desired velocity of the grasped object is constant and available to each agent in [12], that the attachment points of the collectives are centrosymmetric around the center of mass (COM) of the object in [13], act as the primary factors that hind the applications of their works to our case.

The idea that successful object transportation in real application is generally strongly related to the robot formation has also been continuously inspiring related works [14, 15]. Along with the rapid advances in graph theory and control philosophy of the multi-agent systems, distributed scheme under explicit communication plays an important role in the formation control of multiple mobile robots [16, 17]. The challenge existing in formation-based transport task for multiple mobile manipulator ensemble lies in the design of a distributed control law to achieve a global behavior in cooperative manner with limited local information and constrained communication [18]. A typical schema [19] employs a set of distributed controller/observer to achieve relative formation of the multi-arm system. Convergent estimation of the collective states by local observer bridges the gap between the local control and the global cooperative behavior, thus a totally distributed cooperation is achieved [20].

To further maintain high performance when the mobile robot team executes tightly cooperative task whilst suffering from the inevitable uncertainties of the robot dynamics, adaptive mechanism is introduced into the architecture, based on either impedance control [21] or force/position control [19]. More complicate case in which the dynamic uncertainty and communication constraints (e.g. the jointly communication topology [2]) coexist can be easily tackled by embedding the parameter adaptation to the respective control scheme. Recent representative work [22] employs the robust adaptive control to concurrently address dynamic uncertainties and external disturbances. The dependence of the communication network and the costly force/torque sensor is further eliminated by introducing the assumption that all the robot agents know the desired trajectory and exact grasp parameter in advance. In addition, to cope with uncertainties of the grasped object, a distributed approach is presented in [23] to estimate the object’s dynamic/kinematic parameters by properly moving the object or applying specific contact wrenches. Based on this estimation process, the cooperative manipulation of an unknown object can be further expected at the expense of a small bounded tracking error by a two-stage decentralized scheme [24, 25]. While these works either assume that the object’s COM is known to all robotic agents or employ a separate step to estimate the object’s COM, the prerequisite persistent excitation condition may restrict their range of practical application.

In addition, the robotic collectives in the above-mentioned works are free of kinematic uncertainties. However, cooperative transport task is very sensitive to kinematic uncertainties of the interconnected system. Since the manipulators rigidly contact with the object, small kinematic discrepancy may lead to large tracking error and build-up of the internal force. Adaptability to kinematic uncertainties endows the multiple robotic system with improved intelligence and flexibility. All of these demonstrate the necessity of handling the kinematic uncertainties with care.

I-B Contribution

Multi-arm manipulation is associated with tight cooperation in which both the dynamic and kinematic constraints are applied to each of the mobile manipulators. In addition, the cooperative task should be well allocated among the robotic agents also in a distributed way. In light of the above discussions, this study contributes a fully distributed control scheme for a team of networked nonholonomic mobile manipulators (NMMs) to cooperatively transport an unknown object with the following advantages which distinguish our proposed scheme from the existing approaches:

  • Uncertain dynamics/interconnected kinematics and limited communication are addressed comprehensively based on adaptive control.

  • Task allocation and cooperative control are fully distributed. Synchronization idea is fulfilled through the design of the whole control scheme, which can alleviate the performance degradation caused by the estimation and tracking discrepancy during transient phase.

  • Persistent excitation condition and noisy Cartesian-space velocity are totally avoided.

  • Independence from the coordinate attached to the object’s COM by the task-oriented strategy and formation-based idea facilitates the practical implementation.

Fig. 1: Networked mobile manipulators cooperatively transporting and manipulating an unknown tool

Ii Problem Formulation

Ii-a Preliminaries

Consider mobile manipulators tightly grasping a common unknown object, as shown in Fig. 1. Let denote the frame fixed to the end-effector of the ith mobile manipulator. Furthermore, the object frame and the cooperative task frame are two frames both attached to the object and their origin are chosen so as to coincide with the object’s COM and the operational point respectively. All quantities are expressed with respect to the world frame , unless otherwise stated. For each mobile manipulator, the mounted manipulator is considered as a holonomic system while the mobile platform is assumed to be nonholonomic. Throughout this paper, represents null matrix, denotes the identity matrix and () is

column vector with all elements equal to 1 (0). The Cartesian-space variable

can be split into translational part and rotational part in the case of .

Assumption 1: The unknown object is rigid and the networked mobile manipulators grasp the object tightly so that there is no relative motion between the end-effectors and the object. Grasp strategy is not discussed here. For cooperative grasp strategy, the readers are referred to [26, 27].

Assumption 2: Each mobile manipulator has access to its end-effector position. This can be achieved by local on-board camera with localization.

Ii-B Kinematics and dynamics of the interconnected system

Denote by the generalized coordinates of the ith mobile manipulator with representing the position and orientation of the mobile platform and describing the joint angle vector of the mounted manipulator, and .

The nonholonomic constraint acting on the mobile platform can be expressed as [28]:

(1)

where denotes the constraint matrix of the mobile platform. Constraint equation (1) implies that there exists a reduced vector , such that

(2)

where satisfies that . Then a reduced vector can be defined, which will be used in the sequel.

Let denote the EE pose vector of ith mobile manipulator and it is related to the generalized coordinates by

(3)

where is the whole mobile manipulator Jacobian matrix. is the reduced Jacobian matrix that will be defined later. and equality holds for the non-redundant mobile manipulator.

Property 1: The kinematics (3) linearly depends on a kinematic parameter vector , such as joint offsets and link lengths of the manipulator [29]:

(4)

where is the kinematic regressor matrix.

is defined as the coordinate vector of the object’s COM and it is assumed that is related to by

(5)

where denotes the object Jacobian matrix.

is the skew-symmetric operator and

represents the vector from object’s COM to the corresponding contact point. Definitions of for are presented in Fig. 1. Here the object’s COM is introduced to facilitate the force analysis and will be avoided in the controller design.

Assumption 1 imposes the following kinematic constraints on the relative motion of the attached end-effectors.

(6)

where with and respectively representing the relative displacement and orientation, is the rotation matrix of with respect to . denotes the vector pointing from the jth grasp point to the ith grasp point expressed in the frame . The constraint between the ith grasp point and operational point can be expressed in a similar way, i.e. with . is assumed to be known to the robotic agent since it is fixed after the object is grasped.

Dynamics of the ith mobile manipulator in joint space can be expressed as

(7)

where denotes the symmetric positive definite inertial matrix and is the Coriolis and Centrifugal matrix. and represent the interaction inertia torques between manipulator and mobile platform, is the gravitational torque vector; is the input transformation matrix for the whole mobile manipulator; denotes the input torques; in which represents the Lagrange multiplier associated with the nonholonomic constraint and denotes the external wrenches exerted by the holonomic constraint. in which and respectively represent the Jacobian matrices of the mobile base and the mounted manipulator with opportune dimensions.

Considering (2) and its derivative and multiplying both sides of (7) by yields the following reformulation:

(8)

where , , and are given in Appendix A. Then the nonholonomic constraint force in (7) can be eliminated. .

Dynamics of the grasped object can be described by:

(9)

where denotes the inertial matrix and is assumed to be bounded and symmetric positive definite, , where and

represent the minimum and maximum eigenvalues of

; is the Coriolis and Centrifugal matrix and represents the gravitational vector. is the resultant wrench acting on the object’s COM by the multiple mobile manipulator ensemble.

By virtual of kineto-statics duality, the resultant wrench acting on the object’s COM satisfies the following relation:

(10)

where is the well-known grasp matrix, is the collective vector consisting of all generalized wrenches exerted by the mobile manipulators on the object, which can be measured by the force/torque sensor mounted at the wrist of each manipulator. can also be expressed as where is the wrench indirectly applied on the object’s COM by ith end-effector and it can be decomposed into two orthogonal components, one is motion-induced wrench (MW) which contributes to the motion of the grasped object, the other one is the internal wrench (IW) which contributes to the build-up of the stress in the object. This relation can be expressed as: .

Since the internal wrenches cancel each other, i.e. , the object dynamics (9) can be rewritten as

(11)

Suppose that a desired load distribution among the robot team is described by a set of positive definite diagonal matrices for with the assumption that ( is a positive constant) [30]:

(12)

An important physical property regarding these load distribution matrices is that . Then incorporating (12) and above-mentioned properties into (8) gives the complete dynamical equations of the interconnected system in a distributed fashion:

(13)

where

Property 2: is a skew symmetric matrix so that for all . Please see Appendix A for the proof.

Then the following inequality holds since the robot work in finite joint space

(14)

where is a positive constant that denotes the upper bounds of .

Property 3: The nonlinear dynamics linearly depends on a dynamic parameter vector which is composed of physical parameters of the object and mobile manipulator, which gives rise to

(15)

where is the dynamic regressor matrix.

Remark 1.

To avoid input transformation uncertainty which is associated with kinematic uncertainty of the mobile base, can be defined to coincide with the actuation space of the mobile manipulator, i.e. for a two-wheel mobile platform where and are the two independent driving wheels. Thus, and the uncertainties in the input transformation matrix are eliminated.

Ii-C Communication topology

As is commonly done in the distributed control, a graph is employed here to describe the communication topology among the mobile manipulator systems where the vertex set represents all the robots in the network and the edge set denotes the communication interaction between the robots. The edge in directed graph indicates that robot can receive information from robot but not vice versa. Neighbors of robot form a set . The adjacency matrix associated with this graph is defined as if and otherwise. Additionally, self-loops are not contained, i.e. . Then the Laplacian matrix can be defined as

(16)

Several properties associated with the Laplacian matrix are listed in the following lemma.

Lemma 1: The matrix is nonsingular and is positive definite with following definitions: and where and is employed to indicate that the is accessible to the ith () robotic agent, otherwise  [31].

Assumption 3: In this paper, the graph that the mobile manipulators interact on is directed with a spanning tree whose root node has direct access to the desired trajectory of the operational point, i.e. .

This assumption implies that only a small subset of the robotic agents has direct access to the desired cooperative trajectory . It is a more general case with less restriction in real application scenarios, yet, may complicate the controller design since the agents that cannot obtain should estimate it to achieve cooperative task by utilizing only locally available signals.

Based on the models, assumptions and interaction topology discussed above, the control objective is to design a set of distributed controllers for the multiple mobile manipulator system to cooperatively transport an object irrespective of uncertain dynamics and closed-chain kinematics such that:

  • Operational point on the manipulated object could track a desired designated trajectory.

  • Estimation and motion tracking of the networked mobile manipulators are synchronized.

  • All the signals in the closed-loop interconnected system remain bounded.

Fig. 2: Fully distributed cooperation scheme for the networked mobile manipulator

Iii Distributed Adaptive Cooperation

This section is devoted to the formulation of the fully distributed cooperation scheme, whose main structure is shown in Fig. 2. To enable the networked robotic system to perform a tight cooperation task, i.e. object transportation discussed in this paper, the cooperative task should be well allocated to every agent in a distributed way. In addition, internal wrench emerging from constraint dynamics and the motion tracking control of each agent in the presence of kinematic/dynamic uncertainties both need careful treatment. Discussions of these three topics constitute the proposed scheme and are respectively presented in the following subsections.

Iii-a Distributed cooperative task allocation

To achieve motion tracking control of the operational point, the task allocation for the end-effector of each mobile manipulator under limited communication can be interpreted as a rigid formation control problem, i.e. the operational point can be treated as a virtual leader that generates a desired trajectory and virtual followers are associated with the mobile manipulators that should be controlled to maintain the rigid formation in consideration of the rigid grasp.

A continuous function usually can be approximated by a linear combination of a set of prescribed basis functions [32], i.e. . Then the desired trajectory for the operational point of the grasped object can be represented as

(17)

where is the collective basis function that is assumed to be known to all robotic agents. is the constant parameter vector for the ith component of . denotes the constant parameters which is only available to the robotic agents marked by . This statement coincides with Assumption 3 and the definition of the matrix .

Remark 2.

It is worth mentioning here that employing (17

) is reasonable to approximate the desired trajectories, especially in the context of the robot trajectory planning since they are typically planned as an interpolating polynomial function to guarantee the continuity of the velocity and acceleration.

To facilitate the distributed estimation of the desired trajectory, the following two error variables are defined:

(18)
(19)

where is intermediate error variable. denotes the actual estimation error between each mobile manipulator and the desired trajectory. represents the distributed estimation of the desired trajectory allocated to the end-effector of ith mobile manipulator, which together achieve ideal cooperative trajectory under rigid interconnection condition; and are the estimates of the trajectory parameter vector and by ith mobile manipulator. is the estimate of .

To achieve our control objective, the definition of standard local neighborhood consensus error [33] is redesigned as:

(20)

where is the estimate of . can be split into and .

Together with (19),  (20) can be reformulated as:

(21)

where is the estimation error of .

Then, the distributed estimation law of the local desired trajectory and its parameter update law are proposed as:

(22)
(23)

where and the positive constant. is the derivative of and it equals to . denotes the first derivative of and this nomenclature applies to the definition of and .

Remark 3.

Since is associated with , thus is also not applicable to the robotic agents with . It should be noted that the pose (including the position and orientation) formation tracking control is achieved here, which distinguishes the above estimation law from the standard consensus error defined in [33].

Remark 4.

Unlike other decentralized or distributed schemes [21, 30] in which the desired cooperative trajectory for each robotic agent is assumed to be known to all, only locally available signals are utilized in the above estimation/update law (21)- (23), thus endowing the whole mobile manipulator ensemble with the ability to work in a fully distributed way. Furthermore, the proposed estimation law does not require any persistent excitation condition of the desired trajectory and it is independent from the frame of the object’s COM. These two wonderful properties that are also the objectives pursued through the following adaptive control design facilitate the whole scheme’s practical implementation.

Iii-B Internal wrench regulation

Before formulating the distributed internal wrench regulation, three constraints that a physically plausible internal wrench must obey are presented first [34]:

(24)

where is the internal wrench mapped to the contact point. denotes the wrench sensed by the force/torque sensor mounted on the ith manipulator and it is equivalent to the interaction wrench . is the torque induced by and denotes component internal torque induced by .

Extracting internal wrench component from the interaction wrench (Wrench Decomposition) needs full contact wrench information and thus is inherently centralized. However, definition of the internal wrench is still a controversial and active research topic in multi-arm cooperation. Three typical wrench decomposition approaches, i.e. nonsqueezing specific pseudoinverse [35], generalized Moore-Penrose inverse [7] and parametrized pseudoinverse [36, 37], have been widely used in the internal wrench control. The above physical constraints are further taken into consideration to limit the range of the internal wrench [34].

Although it is in general not possible to achieve full decomposition of interactions into IWs leading to wrenches compensating each other and MWs contributing to the resulting wrench only without compensating parts [38], the idea of a non-squeezing pseudoinverse [35] can be employed in wrench synthesis problem. Different from the statement in [35] that the non-squeezing load distribution is unique, we will show a more general formulation with non-squeezing effect in desired MWs. From the view of physical plausibility, the desired internal wrench cannot be arbitrary assigned once the desired load distribution is given. To avoid virtual wrenches that may appear in most of the internal wrench regulation schemes, the wrench decomposition constraints in wrench analysis [34, 38] are further incorporated in our proposed wrench synthesis.

To achieve desired grasp, wrench synthesis is discussed here and every solution mentioned above for the wrench decomposition can be straightforwardly applied. For optimized load distribution, readers are referred to [39]. Here we adopt the nonsqueezing pseudoinverse [35] to formulate the wrench synthesis scheme. The desired MW can be given as

(25)

where is the desired net wrench that should be applied to the object’s COM. denotes the desired MW mapped to the contact point. As to wrench synthesis, there is no need to keep consistency of the pseudoinverse matrix utilized in MW and IW, which leads to a more general synthesis formulation.

In this way, the designation of the desired MW and IW for each mobile manipulator is totally decoupled, which contributes to the reduction in the number of constraint equations and the ease of implementation. The constraints imposed on desired load distribution and desired internal wrench are reformulated as the following inequalities in terms of the wrench synthesis.

(26)

where represents the desired internal wrench.

The following linearization is introduced to facilitate the internal wrench regulation:

(27)

where is the regressor matrix associated with the desired internal wrench and is the linearized parameters.

The internal wrench regulation law , that will be used in the sequel, can be given as:

(28)

where is the estimate of the Jacobian matrix transpose , is the robust term and is a dynamically regulated positive gain whose range will be given in the sequel.

Remark 5.

With above regulation law, the internal wrench tracking error only approaches to a neighborhood of the zero point if the P.E. condition is not satisfied. This is acceptable in most applications, especially for the multi-arm grasp, since internal wrench is only required to be regulated around a constant value to either avoid excessive stress or to provide sufficient grasp force [40].

Remark 6.

The decentralized schemes in the related literature that cope with the internal wrench regulation can be classified into two categories. In the first one 

[30, 41], the robotic agents cannot communicate with each other but are assumed to have access to the global force information, so these schemes can only be termed as semi-decentralized scheme. The other approach [19]

uses the moment-based observer to estimate the net wrench acting on the grasped object exerted by the mobile manipulator ensemble. However, this is not applicable to our case because it requires the accurate dynamics and kinematics of the grasped object. Different from the above-mentioned works in which internal force is calculated or estimated in a centralized way and an extra integral feedback of the internal force is applied to reduce the overshoot and limit the upper bound of the interaction force, we adopt the synchronization mechanism and small control gains instead. To maintain high tracking performance under multiple uncertainties even with small gains, adaptability studied in the next subsection is effective. This compromise endows the scheme with an attractive attribute that calculation of the internal force is avoided, thus maintaining the distributed manner of the whole scheme. For tight connection case considered in this paper, the internal wrench should be bounded as small as possible to avoid potential damage to the whole system, i.e.

.

Iii-C Distributed adaptive control

In this section, a distributed adaptive control is presented to achieve allocated task tracking and task motion synchronization of the interconnected multiple mobile manipulator under uncertain close-chain kinematics and dynamics. This can be interpreted as and after , where denotes the Cartesian-space tracking error of the ith mobile manipulator. Different from state-of-art works [19, 25], motion synchronization should be achieved here to alleviate the transient performance degradation since multiple uncertainties exist in our case and the mobile manipulator ensemble are tightly interconnected through the object.

To avoid the noisy Cartesian-space velocities, a distributed observer is presented as follows [42]:

(29)

where denotes the observed Cartesian -space velocity, is a positive design constant. is the estimated Cartesian-space velocity in the presence of kinematic uncertainties and it can be expressed by

(30)

where is the estimated kinematic parameter vector and is the estimated Jacobian matrix.

Substituting (30) into (29) and using (3) yields the following closed-loop dynamics of the observer:

(31)

where denotes the observer error.