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Towards Collective Manipulation Without Inter-AgentCommunication
Siamak G. FaalWorcester Polytechnic Institute
100 Institute RdWorcester, MA
Shadi Tasdighi KalatWorcester Polytechnic Institute
100 Institute RdWorcester, MA
Cagdas D. OnalWorcester Polytechnic Institute
100 Institute RdWorcester, MA
ABSTRACTThis article details the formulation of a decentralized con-troller for collective manipulation that does not require anycommunication between agents involved in the task. First,a centralized controller based on the complete system Ja-cobian is discussed as a benchmark. Then, the centralizedcontroller is reformulated to obtain the algorithm for theproposed decentralized control approach. Both, centralizedand decentralized controllers utilize Moore-Penrose pseu-doinverse to distribute a control action through the agents ofthe group. The convergence and stability of both controllersare discussed in detail. Moreover, robustness and effective-ness of the proposed controllers are investigated throughsimulating numerous scenarios, formations and populationsof the agents. We show that, as the population of the groupincreases, the results of the decentralized controller approachto its centralized counterpart with significantly lower com-putational cost.
CCS Concepts•Computer systems organization → Robotic control;Robotic autonomy; •Computing methodologies→Multi-agent systems; Cooperation and coordination; Con-trol methods;
KeywordsCollective manipulation; Decentralized controller; Swarmrobotics
1. INTRODUCTIONCollective manipulation is a solution for manipulating rela-tively large and heavy objects by utilizing agents that are
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DOI: http://dx.doi.org/10.1145/2851613.2851707
Figure 1: Snapshot of three ∆ρ robots carrying afour-leaf clover.
incapable of fulfilling the task individually. This behavioris frequently observed in different insect colonies during theforaging process. The large prey retrieval of Eciton Burchel-lii, also known as army ants, is an impressive example ofcollective manipulation in nature [1]. Collective manipula-tion can be employed in a wide range of applications andhas been a research focus in swarm and multi-robot stud-ies. Applications of collective manipulation include: carry-ing and assembling parts for automated on-site construction;object manipulation and assembly in factories; search andrescue operations in disaster relief actions; and debris collec-tion. Although utilizing a group of simple agents, comparedto a single well-instrumented agent, increases manipulationdexterity, reliability, and robustness, it introduces new chal-lenges on team formation, organization, and control.
Collective manipulation is achievable through two funda-mental control strategies: centralized control [2, 3, 4] anddecentralized control [5, 6, 7, 8, 9, 10, 11]. The central-ized control approaches mostly focus on group formationsand manipulation is performed by keeping form- or force-closure around the manipulated object. While centralizedapproaches can guarantee a form of optimality, they sufferfrom intense internal communication between agents of thegroup. Combined with the large population of the group,the required level of communication make it impractical toutilize centralized control approaches in many real life sce-narios [12].
𝑊𝑊𝑦𝑦𝑑𝑑
𝑂𝑂𝑓𝑓𝑖𝑖
𝑂𝑂𝑟𝑟𝑖𝑖𝑊𝑊𝑦𝑦𝑜𝑜
𝑊𝑊𝑥𝑥𝑑𝑑𝑊𝑊𝑥𝑥𝑜𝑜
𝑊𝑊𝜃𝜃𝑑𝑑
𝑊𝑊𝜃𝜃𝑜𝑜
𝑥𝑥
𝑦𝑦
𝑊𝑊
𝑂𝑂
𝑂𝑂𝑒𝑒𝑑𝑑
Figure 2: Free body diagram of the system and theassociated parameters
Inspired from the rich literature on robotic grasping, graspforce constraints and optimization [13, 14, 15], the collec-tive manipulation strategy proposed in this article takes adifferent path and focuses on manipulation of an object withmultiple agents assuming that the agents are arbitrarily lo-cated around the object. Specifically, this article focuseson the formulation of a Jacobian-based simple decentralizedcontroller that does not require any prior information aboutthe size and formation of the group. Thus, none of theagents is required to communicate any information with thegroup. Stability, convergence, and reliability of the proposedcontroller is discussed in detail and through numerous sim-ulated scenarios. In addition, formulation of a force optimalcentralized controller is also discussed and the responses ofthe two controllers are compared. It is shown that the de-centralized controller response converges to the centralizedcontroller as the population of the team grows larger.
Complete homogeneity of the team members and not requir-ing any information on the team formation are the main dif-ferences between the proposed decentralized controller andsimilar approaches discussed in [4, 5, 6]. The proposed con-troller assumes that agents can apply forces to the objectin any direction. Three holonomic robots that are carryinga four-leaf clover shaped object are shown in Fig.1. Due totheir holonomic structure, the robots are capable of applyingforces in any arbitrary planar direction parallel to the sub-strate surface without utilizing any active arm mechanisms.This characteristic makes them a suitable and inexpensiveplatform for experimental validation of the proposed algo-rithm.
The rest of the article is organized as follows. Section 2discusses mathematical modeling of the system and intro-duces the parameters used to define the model. Section 3covers the centralized and decentralized controller designsfollowed by a discussion on stability and convergence of theproposed controllers. The results obtained by applying theproposed controller to different simulated scenarios and com-parison between the centralized and decentralized controllersare presented in Section 4. The paper is concluded with dis-cussions and future work in Section 5.
2. MODELINGThis section covers the derivation of the differential equa-tions of motion (DEM) for the system under consideration.Although a reformulation of the proposed controller maymake it applicable to non-rigid and/or 3-D systems, for thesake of brevity, the manipulated object is considered to be arigid body that is constrained to planar motion. Throughoutthis manuscript, the vector and coordinate frame notationsare adopted form [16]. Based on these notations, a trans-formation A from coordinate frame {i} to coordinate frame{j} is designated by j
iA. Similarly, iv represents the vectorv that defined in coordinate frame {i}. A free body dia-gram of the overall system is illustrated in Fig. 2. Note thatthe coordinate frame {O} is attached to the Center of Mass(CoM) of the object. The vector Ori defines the position ofthe applied force by the ith agent with respect to the CoMof the object. The position of the ith agent with respect tothe CoM of the object is defined in coordinate frame {O}.Similar to Ori,
Ofi defines the force applied by the ith agentto the object in the same coordinate frame. The vector Oedis the linear error between the desired position and the ob-ject position. The total error vector, that includes linearand angular error values, is described by (1).
Oe = OWR× W e, (1)
where OWR is the rotation matrix that maps the world co-
ordinate frame {W} to the object frame {O}. Since theobject is constrained to planar motions, O
WR is equal to theelemental rotation about z axis for θo radians, Rz(θo). Thetotal error vector in world coordinate frame, W e , is definedas:
W e =W [(xd − xo).i+ (yd − yo).j + (θd − θo).k]. (2)
Equations (3) and (4) describe the DEM of an object withmass of mo and mass moment of inertia of Io written in thebody fixed (non-inertial) reference frame {O}. The vectorsOvo and Ofi represent the velocity of the object and theforces applied by each agent, respectively. The position ofthe applied force by the ith agent with respect to the CoMof the object is defined by Ori. This position is also referredas the agent position throughout this article.
mo(O vo + θo · Ok ×O vo) =
N∑i=1
Ofi (3)
Ioθo · Ok =
N∑i=1
Ori ×O fi, (4)
where N represents the total number of agents that are in-volved in the task. Unless noted otherwise, throughout therest of this article, all the vectors are defined in coordinateframe {O}. Thus, for the sake of brevity, the superscript Ois dropped from vector names in the following sections.
3. CONTROLLER DESIGNThis section starts with formulation of a centralized con-troller that uses the full Jacobian of the system and Moore-Penrose pseudoinverse [17] method to distribute a controlfunction output between the agents of the group. The dis-cussion is followed by reformulating the proposed central-
Figure 3: Snapshots of a simulation with the decen-tralized controller. Total of 4 agents are located at0◦, 90◦, 100◦, 220◦ with respect to Ox axis around theobject (Illustrated with red lines). The desired x,y and θ values and are illustrated by black dashedlines. The color-bar indicates time in seconds.
ized controller and converting it into a decentralized con-troller. Since the decentralized controller does not requireany knowledge on the group population and formation, itdoes not need any information exchange between the mem-bers of the group. Note that the terms “pseudoinverse”and “Moore-Penrose pseudoinverse” are used interchange-ably throughout the text.
3.1 Centralized ControllerTo derive the centralized controller, the first step is to definethe full Jacobian of the system. As explained in (5), the fullJacobian is formulated by defining the transformation thatmaps all the forces applied by all the agents into a resultantforce and moment that is applied to the CoM of the object.
F = J[fT1 fT
2 · · · fTN
]T. (5)
In the above equation, F =[Fx Fy Mz
]Tis the vector of
the resultant forces along the x and y axes and the momentabout the z axis of the object coordinate frame, respectively.The matrix J is the full Jacobian of the system:
J =
1 0 1 0 · · · 1 00 1 0 1 · · · 0 1
−ry1 rx1 −ry2 rx2 · · · −ryN rxN
, (6)
where rxi and ryi are the x and y components of the vectorri, respectively. Since the size of J is 3×2N , the overall sys-tem can either be under-actuated (N = 1) or over-actuated({N ∈ Z|N ≥ 2}). To hold the focus of the article on multi-robot and swarm scenarios, it is assumed that N ≥ 2. Notethat one can also derive the Jacobian by writing the differ-entials of ri vectors with respect to time as a vector-valuedfunction and taking partial derivatives of this function withrespect to xo, yo and θo.
The control objective is to define all the fi values such thatthe norm of the error approaches to zero in finite time. Thiscan be achieved by equating F to output of a control func-tion φ(e). Considering the dynamic behavior of the sys-tem and to simplify the overall formulation, a Proportional-Derivative (PD) controller is considered to serve as the con-trol function φ(e). Note that any linear or nonlinear controlfunction can be utilized as φ(e) based on the complexity ofthe object’s dynamics. Substituting a PD controller for φ(e)yields:
F = φ(e) = Kpe+Kde. (7)
The next step is to convert the values obtained for F to theforces applied by each agent. Since J is not a square ma-trix, there is no unique solution for the system of equation.Thus, Moore-Penrose pseudoinverse is utilized to computethe minimum Euclidean norm solution for fi. Doing so, thefinal centralized controller is formulated as:
f = J+φ(e) = J+(Kpe+Kde), (8)
where J+ is the Moore-Penrose pseudoinverse of the Jaco-bian matrix J .
3.2 Decentralized ControllerThe fundamental idea for the formulation of the decentral-ized controller is that each agent only relies on a virtualagent that is located at an arbitrary location around the ob-ject. It is assumed that each agent only knows a minimalset of information: its point of attachment to the object,ri, and the error vector e [5]. Although the virtual agenti, associated with ith agent, does not necessarily representsthe effect of the rest of the group, it allows the members ofthe group to define the necessary control forces to fulfill themanipulation task. As a result, each agent will have a lo-cally defined Jacobian Ji. The locations of the virtual agentsneed to be defined such that the pseudoinverse of the localJacobian matrices yield to a valid solution, as discussed indetail in the following section. A possible location for thevirtual agent i is on the mirror position of the location ofthe ith agent with respect to the CoM of the object. Thelocal Jacobian, Ji, for each agent is defined as:
Ji =
1 0 1 00 1 0 1
−ryi rxi −ayi axi
, (9)
where Ji is the Jacobian of the system assumed by agenti. Parameters axi and ayi are the x and y components ofthe position of the ith virtual agent. After defining all thelocal Jacobian matrices, the control formulation follows thediscussion in Section 3.1. By applying a PD control functionas φ(e), the final control law for each agent is defined as:
fi = Ji+φ(e) = Ji
+(Kpe+Kde), (10)
where Ji+ is the Moore-Penrose pseudoinverse of the lo-
cal Jacobian matrix Ji. A simulation demonstrating theproposed decentralized collective manipulation algorithm isshown in Fig. 3.
3.3 Stability and convergenceFor clarity, the proof of the convergence and stability of thecentralized controller is discussed first. Based on (5), the
𝑥𝑥
𝑦𝑦
18°
36°
𝑁𝑁 × 18°
Figure 4: Formation of the agents around the object.The first agent is located on the intersection of Oxaxis with the circumference of the object. The restof the agents are located at 18◦ increments.
total force applied to the CoM of the object is equal to themultiplication of the full Jacobian of the system to the forcesapplied by each agent. As described in Section 3.1, the pseu-doinverse of the full Jacobian matrix is used to distributethe output of the control function φ(e) between the agents.Thus, substituting (5) and (8) into (3) and (4) yields:[
mo(vo + θo · k × vo)
Ioθo · k
]= J [J+φ(e)]. (11)
Since the rows of the matrix J are linearly independent,the multiplication of J by its pseudo-inverse results in anidentity matrix (JJ+ = I). Thus, equation (11) reduces to:[
mo(vo + θo · k × vo)
Ioθo · k
]= φ(e). (12)
Thus, the convergence and stability of the system directlyrelated to the definition of the control function φ(e). If φ(e)guarantees the stability of the object dynamics (as definedin (3) and (4)), the centralized controller will also be stableand its response will be the same as the response of thesystem that is directly controlled by φ(e).
A similar approach can be followed to analyze the stabilityand convergence of the decentralized controller. Rewriting(12) for the decentralized controller yields:
[mo(vo + θo · k × vo)
Ioθo · k
]= J
K1
K2
...KN
φ(e), (13)
where Ki is equal to the rows 1 to 2 and columns 1 to 3 ofJi
+ and it is equal to:
Ki =1
2<i
[ki11 ki12 ki13ki21 ki22 ki23
], (14)
where: ki11 = <i+ayi2−ryi
2, ki12 = −(axi+rxi)(ayi−ryi),ki13 = 2(ayi − ryi), ki21 = −(axi − rxi)(ayi + ryi),ki22 = <i + axi
2 − rxi2, ki23 = −2(axi − rxi), and
<i = (axi − rxi)2 + (ayi − ryi)
2.
Since[K1 K2 · · · KN
]Tis not the pseudoinverse of J ,
the right hand side of (13) will not reduce to φ(e). Thus,the outputs of the control function φ(e) are mapped to theforces applied to the CoM of the object by a resultant trans-
Table 1: System parameters
Parameter Value Dimension Descriptionmo 1 [Kg] MassIo 1 [Kg ·m2] Mass moment of inertiaR 0.2 [m] Radius of the objectKp 10 [N/m] Proportional gainKd 5 [Ns/m] Derivative gain
formation T . In other words:
F = J
K1
K2
...KN
φ(e) = Tφ(e). (15)
As long as the transformation matrix T preserves the conver-gence and stability of φ(e), the decentralized controller willbe stable and convergent. A necessary, but not sufficient,condition on matrix T to preserve the behavior of φ(e) isthat all the eigenvalues of T must be positive. Based onthe formulation presented in Section 3.2, the ai vectors canhave any arbitrary value, as long as they generate a valid Tmatrix. Here, to reduce the complexity of the system, it isassumed that a virtual agent for each agent i is located atits mirror position with respect to the CoM of the object.Thus: −→ai = −−→ri , yielding:
Ki =1
2
[1 0 −ryi/(rxi
2 + ryi2)
0 1 rxi/(rxi2 + ryi
2)
]. (16)
Note that the K matrix for a system composed of only twoagents which are located on the mirror position of each otherwill be equal to the pseudo-inverse of the complete Jacobianof that system. Thus, in this specific case, the decentralizedcontroller will behave similarly to the centralized controller.The final step is to substitute the results obtained in (16)into (15) and perform the multiplication to calculate T . Thetransformation matrix T for a group of N agents is equal to:
T =1
2
N 0 −
N∑i=1
ryi/(rxi2 + ryi
2)
0 NN∑i=1
rxi/(rxi2 + ryi
2)
−N∑i=1
ryiN∑i=1
rxi N
, (17)
and the corresponding eigenvalues of the matrix T are:
λ =[N N + δ N − δ
], (18)
where δ is defined as:
δ =
√√√√ N∑i=1
ryirxi2 + ryi2
N∑i=1
ryi +
N∑i=1
rxirxi2 + ryi2
N∑i=1
rxi (19)
All the eigenvalues of T are positive if and only if −N < δ <N . Since the perimeter of the object is finite and bounded,and the physical agents have a finite and nonzero perimeter,the growth in population of the group causes the matrix Tto approach to a scaled identity matrix. As a result, all
Figure 5: The response of the system to the decen-tralized controller based on formations depicted inFig.4. The response of the system to the centralizedcontroller is illustrated with dashed black line. Thecolor map indicates the number of agents.
the eigenvalues of T remain greater than zero and a stablemapping is obtained between φ(e) and and F .
4. RESULTSThe system responses to the proposed controllers are simu-lated by numerical integration of the nonlinear DEM usingthe Dormand-Prince method available in MATLAB. The nu-merical values for system parameters that are used for sim-ulations are depicted in Table 1. Figure 3 illustrates snap-shots of a simulation with the decentralized controller and 4agents that are located at 0◦, 90◦, 100◦, 220◦ with respectto Ox axis around the object. The desired position is de-fined to be 1 for all x, y and θ values that are illustrated byblack dashed lines in the figure. The color mapping is usedto depict the passage of time in seconds. As illustrated inthis figure, the decentralized controller can successfully ma-nipulate the object to the desired position and orientation.
Several formations of the agents around the object are simu-lated to show the reliability and robustness of the proposeddecentralized controller. The results are also compared withthe centralized controller. As discussed in Section 3.3, aslong as there is no limitation on the forces produced byeach agent, the centralized controller will behave indepen-dent of the formation and population of the group. Fig.5shows the simulation results for different formations of theagents around the object that are controlled by the decen-tralized controller. As depicted in Fig.4, the agents are lo-cated around the object with 18◦ incremental angles betweenthem. The trajectories of the object and its orientations forthe same formations are illustrated in Fig.6. In both Fig.5and Fig.6, the response and the trajectory of the central-ized controller are illustrated with black dashed lines. Thecolor map in these two figures defines the population of theagents, N . As observed in the figures, As the populationof the agents grows, the response of the decentralized con-
Figure 6: The trajectories of the object for as it ismanipulated by the decentralized controller. Theformation of the agents around the object is de-picted in Fig.4. The trajectory of the object that iscontrolled by the centralized controller is illustratedwith dashed black lines. The color map indicatesthe number of agents surrounding the object. Thearrows indicate the orientations of the Ox axis.
troller approaches to the centralized one. This effect is alsoobservable in (17).
To further analyze the behavior of the proposed controller,a series of simulations conducted with completely randomformation of the agents around the object. The simulationsstarted with a group of 4 agents and continued to a group of20 agents. For each group, total of 20 uniformly distributedrandom formations are considered. For each simulation, thesettling time of the response is calculated. Fig.7 illustratesthe results of these simulations. The central red mark ofeach box represents the the median of each settling time foreach group population, the edges of the box are the 25th
and 75th percentiles and the whiskers are extended to themost extreme data points without including outliers. Theoutliers are illustrated by red plus markers. As shown inthe figure, as the population of the group gets larger, thenumber of outliers reduces. Also the extreme data pointsapproach to the 25th and 75th percentiles. This indicatesthat the reachability and robustness of the system increasesby the growth of the population of the group. The settlingtime is defined to be the time where the norm of error vectorremains less than or equal to 0.01 (which is equivalent to0.57% of the initial norm of the error vector).
5. CONCLUSIONSThis paper described formulation of a centralized and a de-centralized controllers that are utilizing the jacobian of thesystem to distribute the output of a control function be-tween the agents that are performing a collective manipula-tion. The proposed decentralized controller does not requireany information about the formation and the population ofthe group. As a result, agents do not need to exchange any
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Population of the group
τ s [s
ec]
Figure 7: The median, 25th and 75th percentiles, ex-treme data points and outliers for the settling timesof 20 random formation for each group population.
information and there is no need to have a communicationchannel between them. This can highly improve the appli-cability of the proposed controller for real case scenarios.The stability and convergence for both controllers is dis-cussed in detail; and it is shown that as the population of thegroup increases, the response of the decentralized controllerapproaches to the centralized counterpart. The robustnessand effectiveness of the proposed controller are also studiedthrough numerous simulations with different group popula-tions and formations. Although the results are satisfactory,it is still possible to further increase the efficiency of con-troller by updating the local jacobians based on feedbackobtained from the dynamics of the object. A simple alter-ation of the algorithm will make it applicable to 3-D systemsas well. Utilizing real robots to experimentally validate theperformance and stability of the proposed algorithms andfinding a suitable update rule for the local jacobinas aresome of the future works of this research.
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