Human-Robot Role Arbitration with Cooperative
Game Theory
1st Paolo Franceschi
CNR-STIIMA; UniBs
Milano, Italy; Brescia, Italy
paolo.franceschi@stiima.cnr.it

Abstract—This work studies the role-arbitration between arobot and a human during physical Human-Robot Interaction.
The system is modeled as a Cartesian impedance, with thehuman and the robot interacting by applying two separate
external forces. A reformulation of the problem as a Cooperative
Differential Game allows addressing the bargaining problemby proposing a solution depending on the human interaction
force, interpreted as the will to lead or follow. This defines thearbitration law and assigns the role of leader or follower tothe robot. Experiments show the feasibility and capabilities ofthe proposed control in managing the human-robot arbitration
during a shared-trajectory following task.
Index Terms—physical Human-Robot Interaction, Role Arbitration, Cooperative Differential Game Theory, Adaptive Control,Impedance Control

I. I NTRODUCTION
With the large spread of collaborative robots, collaborative
applications involving a human operator and a robot working
together to achieve a common goal represents the industry’s
most recent trend. In contrast to coexistence (when the humanand the robot are in the same environment but do not interact),
synchronization (when they work in the same workspace, butat different times) and cooperation (when they work in thesame workspace at the same time, though each focuses on
separate tasks), collaboration happens when the human operator and the robot must execute a task together, and the actionof the one has immediate consequences on the other [1]. Collaboration requires communication, which typically happens
through interaction forces, leading to physical Human-Robot
Interaction (pHRI) [2]. In the case the human and the robot
can have complementary roles, Human-Robot Role Arbitration
defines the mechanism that assigns task control to either thehuman or the robot [3]. As discussed in [4], game theory
provides useful tools to analyze complex interactive behaviors
involving multiple agents. It provides mathematical models(cooperative, noncooperative, multi-stages, etc.) of strategicinteraction among rational decision-makers and provides theplayers with ”optimal” policies to minimize their objectives,
taking into account interaction. In [5], [6], and similarly, in [7],for the shared control with an exoskeleton, the continuous role
adaptation is investigated for a Differential Non-Cooperativegame. In these works, the concept of Nash Equilibrium is used
This project has received funding from the European Union’s Horizon
2020 research and innovation program under grant agreement No 101006732(Drapebot).

to update the robot cost function according to the interaction
force. Finally, a general framework for differential gametheoretic modeling of Human-Robot interaction is presented in[8] for the two agents game and extended to multiple agents in[9]. The previous works show that a game-theoretic descriptionof the human-robot interaction can provide optimal behavior
in a non-cooperative framework. Despite this, Nash equilibria(i.e. the solutions of the non-cooperative games) are frequently
not Pareto optimal. Thus, cooperation can often improve
payoffs to all players [10]. Hence, in this work, Cooperative
Game Theory (CGT) is used to describe the interaction model,and Pareto solutions are sought. Cooperation, indeed, canimprove the cost of each agent by agreeing. This work briefly
presents the method introduced in [11]. Please refer to it for
full equations derivation and a deeper work description.
II. M ETHODThis section presents the methodology adopted in this work
to model the system, the interaction and the solution.
A. Cooperative game theoretic system modeling
Working in the Cartesian space is more intuitive and naturalfor the human operator; hence the desired robot motion at theend-effector is implemented as an impedance model in theCartesian space.
Mi ẍ(t) + Di ẋ(t) + Ki (x − x0 )(t) = uh (t) + ur (t)(1)

The system is then reformulated in a classical linear statespace formulation for integration with the Game Theoreticframework, resulting inż = Az + Bh uh + Br ur = Az + BGT uGT(2)with Bgt = [Bh , Br ], and uGT = [uh , ur ]T , and z = [p, v]T
the state vector containing positions and velocities.
For the game-theoretic formulation, the human and the robot
are assumed to minimize a quadratic cost function on the stateand on the control input, defined asZ ∞Ji =(z Qi z + ui Ri ui ) dt(3)0with the subscript i=h,r identifying the human and the robot.
A shared cost function is defined, where the parameter αrepresents the agreement between the two.
Z ∞Jα = α Jh + (1 − α) Jr =(z Qα z + u Rα u) dt (4)
0

1.5

0.5

150

6100

40.50

CGT HIC

LIC LQR(a) Tracking error

W [J]

0.3

F [N]

||e|| [10 -2 m]||e|| [10 -2 m]

0.4

10.2

50

20.10CGT HIC0LIC LQR(b) X-correlated error

CGT HIC

LIC LQR0(c) ForceCGT HICLICLQR(d) Mechanical workFig. 1: Indexes comparison

with Qα and Rα representing weighted cost matrices. Theminimization of Jα is the same as an LQR minimization,
which provides the system with the optimal feedback gain
−1 Tmatrix K f b = Rα
BGT P and the feedforward gain matrix
−1  A
BGT
0defined as K f f = [K f b I].
CDI
The control input is computed asuGT = uf b + uf f = −K f b z(t) + K f f zref (t)TABLE I: The mass, spring and damping parameters used forthe experiments.
KDD/DcrM

CGT55 ÷ 550
53 ÷ 135
1.13 ÷ 0.92
10HIC
100
570.9
10LIC
2025.5
0.9
10LQR
22284
0.89
10(5)

and finally, to obtain the robot control input take the bottom
rows of (5). Note that the gain matrices multiply positionsand velocities, the robot contribution can be seen as additional
stiffness and damping to the system in (1), leading the system
to a variable impedance control.
B. Bargaining solution
In a cooperative environment, there are infinite Pareto
optimal solutions and the bargaining theory addresses whichis the most effective one. The bargaining problem is how todefine the appropriate α. In this work, the two players bargain
on who leads and follows the task; in this sense, one canaccept a higher cost if he is the follower. Hence, α is used
as a weighting factor to move the control authority from therobot to the human and vice-versa. Each situation in between
represents a cooperative solution where the control authority
is shared appropriately. The selection of the weight parameter
α depends on the force applied by the human and is processedby the sigmoid function:a(6)α=d−−b(||uh ||−c)1+ewhere the constant parameters a, b, c, d are used to shape thefunction properly.
III. E XPERIMENTS
An experiment is designed to test the proposed control
method for sharing control authority in human-robot collaboration. The robot has to follow a planar circular trajectory,
while the human has a different path to follow, which partiallyoverlaps with the robot one. The presented control is comparedwith other three similar controllers, High stiffness ImpedanceController (HIC), Low stiffness Impedance Controller (LIC),and LQR impedance controller. Table I shows the mass, spring,and damping values used. Note that spring and damping vary
according to interaction, so a range is shown. It is interesting
to observe that the damping ratio varies from values lower togreater than 1, allowing fast tracking with robot leading, andhigh damping with human leading. Four indices are evaluatedfor comparison: (i) trajectory following error: which measures
the quality of tracking the human reference trajectory, (ii)path tracking following error: which measures the capacity tofollow the geometrical path, (iii) measured interaction force:measures the force required by the human to follow its target
trajectory, and (iv) mechanical work: measures the energy
required by the human to follow its target trajectory. Figure 1
shows the results as the mean and standard deviation of themeasured indexes for the four controllers. Despite the CGTcontroller is not the best one in all the cases, it can be seenthat it is the one that overall minimizes all indexes together.
Particularly interesting is also the variable damping behavior(see table I), which switches between values above and below
the critical one, allowing fast motion when the robot is leading,and damped motion when the human is.
IV. C ONCLUSIONSIn conclusion, the proposed framework shows good capabilities in describing a human-robot cooperative task. It
allows easy integration of different modules (human cost function identification, human intention identification) and easy
implementation of different robot behaviors (leader-follower
adaptation, co-manipulation, co-transportation), which will beinvestigated in future works. Finally, such an approach will
be extended to the cooperative human-robot transportation ofcarbon fiber plies and implemented in the context of the EU
project DrapeBot.

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