Systems and methods for bounded rational game-theoretical modeling of human joint actions with incomplete information

Abstract

The present technology is directed towards methods to model human-human cooperative behaviors in joint actions for applications in eventual human-robot interactions. A computer-implemented bounded rationality game-theoretical model is utilized to model human behavior.

Description

Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b SYSTEMS AND METHODS FOR BOUNDED RATIONAL GAME- THEORETICAL MODELING OF HUMAN JOINT ACTIONS WITH INCOMPLETE INFORMATION GOVERNMENT SUPPORT

[0001] This invention was made with government support under Grant No. 1944833 awarded by the National Science Foundation. The government has certain rights in the invention. CROSS REFERENCE TO RELATED APPLICATIONS

[0002] This is a PCT application that claims benefit to U.S. provisional application serial number 63 / 509,219 filed on June 20, 2023 which is incorporated by reference in its entirety. FIELD

[0003] The present disclosure generally relates to general modeling and predictive computing technologies, and in particular, to a system and associated method to model human-human cooperative behaviors in joint actions for applications in eventual human-robot interactions by leveraging a bounded rationality game theoretical model that is configured to model human behavior. BACKGROUND

[0004] As humans and robots start to collaborate in close proximity, robots are tasked to perceive, comprehend, and anticipate human partners’ actions which demands a predictive model to describe how humans collaborate with each other in joint actions. Previous studies either simplify the collaborative task as an optimal control problem between two agents or do not consider the learning process of humans during repeated interaction. This idyllic representation is thus not able to model human rationality and the learning process.

[0005] It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed. 1 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b BRIEF DESCRIPTION OF THE DRAWINGS

[0006] The present patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

[0007] FIG.1 is an illustration of comparison between rationality models – an irrational agent will choose an action at random, a rational thinker will choose an action with the best outcome, a bounded rational agent will build the theory of mind of the other agent and use bounded rationality to generate action.

[0008] FIG.2 is illustration of an experimental setup during collaborative object pushing: each player observes the current position of the object on the screen and inputs haptic action to move their end of the object using NOVIT falcon input device; in real-time the data is stored in a data log as well as updated through the unity physics engine to create the unity scene.

[0009] FIG.3 is a series of graphs illustrating comparison of the trajectories of the data sets generated with bounded rationality and complete rationality for pair 9 described herein; the bounded rational model is able to generate a trajectory similar to ground truth data in the y direction.

[0010] FIG.4 is a series of graphs illustrating (a) comparison of steady- state errors of the first two trials and the remaining trials for all participants, (b) comparison of the fitting errors of complete rationality and bounded rationality for all pairs of participants, and (c) the goal tracking errors compared with the effort difference between leader and follower.

[0011] FIG.5 is a simplified illustration of an example system / framework for implementing the inventive concepts described herein.

[0012] FIG.6 is a simplified illustration of a computing device that may be implemented with or as part of example systems described herein.

[0013] Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims. 2 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b DETAILED DESCRIPTION

[0014] The present disclosure relates to various embodiments of a system including a processor configured to execute a bounded-rational and game-theoretical human cooperative model; the model developed to describe the cooperative behaviors of the human dyad. An experiment of a joint object pushing collaborative task was conducted with 30 human subjects using haptic interfaces in a virtual environment. The proposed model uses inverse optimal control (IOC) to model the reward parameters in the collaborative task. The collected data verified the accuracy of the predicted human trajectory generated from the bounded rational model excels the one with a fully rational model. Insight is further provided from the conducted experiments about the effects of leadership on the performance of human collaboration. Introduction:

[0015] Physical human-robot interaction (pHRI) has become ubiquitous in robot-assisted rehabilitation, robotic surgery, and collaborative manufacturing. In these applications, an intelligent robot needs to build a human model (often referred to as theory of mind) to anticipate human actions for proactive planning of its own actions. Modeling humans is a challenging task not only because of the inherent uncertainties of human decision-making but also due to the human learning and adaptation of robots mediated by physical interactions. In order to investigate human uncertainty and adaptation mechanisms during the physical collaboration between humans and robots, it is critical to study of how humans collaborate with each other.

[0016] Inverse optimal control (IOC) methods, which assume human motor behavior follows an optimal control law, are promising approaches to describing human physical actions. The oversimplification of human collaboration behavior limited the variety of human motion this model can describe. Inverse reinforcement learning (IRL) emphasizes the need to address the value alignment problem in the human decision- making process, which is trying to identify how humans allocate preferences on the elements of a multi-attribute cost function. Previous work implemented IRL algorithms to model general human-human interaction or collaboration by adding the features that were related to the other human agents in the system to the cost function. Although these studies demonstrated competent results of modeling human collaboration with 3 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b IRL methods, none of these approaches considered the human prediction and adaptation of their partners when executing collaborative tasks.

[0017] Game-theoretic models have been widely adopted to describe interactive behaviors of humans. Due to the nature of incomplete information of human- involved interactions, modeling how individuals predict their partners is important. In Li et al. (C. Liu, W. Zhang, and M. Tomizuka. “Who to blame? Leaning control strategies with information asymmetry.” in 2016 American Control Conference (ACC). IEEE, 2016. pp.217-228, 2019), the mutual adaption behavior was modeled with a multi-agent recursive least square learning model, and an empathetic intent inference framework was proposed to consider the mutual learning process of intelligent agents. Li et al. (Y. Li, G. Carboni, F. Gonzales, D. Campolo, and E. Burdet, “Differential game theory for versatile physical human-robot interaction,” Nature Machine Intelligence, vol.1, no.1, pp.36-43, 2019) proposed a differential game model for improving the robot controller by understanding the control strategy of human users in pHRI tasks.

[0018] Another key aspect in modeling human behavior is to understand its nature of randomness. Our previous research showed that human participants did not strictly follow the optimal strategies to complete the task. Recent research adopted the idea of suboptimal behavior to describe human motor behavior.

[0019] Prior studies on human-human cooperative motion conducted through haptic interfaces showed the capability of capturing the haptic behavior of human collaboration. The fact that the data sets from these experiments are rarely used for constructing a dynamic model of human collaborative motion generation inspired the present inventive concept and to explore feasible methods to construct dynamic models describing cooperative pHRI scenarios.

[0020] A dynamic model that predicts human motion in a cooperative task would vastly help the robot to achieve safe and efficient collaboration with human partners in pHRI scenarios. The main contribution of the present inventive concept is threefold. First, we propose a novel human collaborative dynamic model based on game theory and bounded rationality, which is the first model to describe human dyadic behavior with the bounded rational theory. Second, we design a haptic interface with a virtual environment and conduct experiments with human subjects. The collected 4 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b human data are used to construct the proposed models and verify their performance of predicting human cooperative behaviors. The third contribution is to observe and report evidence of human cooperative nature, such as the learning process between the human dyads. An interesting relationship between the leadership of the dyads and task performance is also reported in the paper, which asserts that clear role assignment during the collaboration could significantly improve the performance of dyads. Game-Theoretic and Bounded Rational Human Collaborative Model:

[0021] FIG.1 generally demonstrates the difference between bounded rationality, complete rationality, and irrational behavior. In this section, we start by defining the different strategies of a human agent before introducing our game-theoretic and bounded-rational human model. A human agent with complete rationality always chooses the optimal action, which is described as rational thinking in FIG.1.

[0022] In this section, we describe the dynamic human cooperative model as a two-player non-zero-sum game. The optimal solution with complete information is discussed, based on the Nash Equilibrium of the game under the assumption of complete rationality. Then we propose the bounded rational game-theoretic human model, accommodating the rationality assumption of the opponents. The nature of human incomplete information in physical interaction is also discussed in this section. Lastly, we will introduce the IOC method to learn the reward and rationality parameters of our model based on observed data collected through the experiment. Two-player joint object manipulation game

[0023] We set up a game to simulate a human-human cooperative task, where two human players are trying to move a heavy object, such as a couch, to the desired position as shown in FIG.2. Each player is positioned at one end of the large long object which represents the couch in our setup. The grasping points on the object for both the participants are displayed as blue and yellow handles. The direction of the force is limited to be perpendicular to the object. This constraint demands collaboration of two players to complete the game.

[0024] As FIG.2 shows, the position of the object’s center of mass is presented as a vector ^^ = [ ^^^^^^, ^^^^] . Velocity vector is denoted as ^^ = ^^^^, ^^^^^^. The 5 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b orientation of the object is denoted as ^^. Then we can define the state vector ^^ ∈ ℝ6of the system as follows, ^^ = [ ^^, ^^, ^^, ^^̇]^^ (1)

[0025] The system dynamics of the non-cooperative two-player non-zero- sum game in the discrete-time domain is presented as ^^( ^^ + 1) = ^^( ^^( ^^), ^^1( ^^), ^^2( ^^))(2)Where ^^( ^^ + 1) ∈ ℝ6isobservable for both agents, which indicates the game has complete information setup. Rational solution of human collaboration: a Nash equilibrium approach

[0026] 1) Cost functions: A cost function for each player should capture these factors: the cumulative sum of the payoff of approaching the goal and the cost of actions. We denote the cost function of agent ^^ ∈ {1,2} in a finite horizon quadratic form as follows, ^^−1^^^^^ ^^ (3)where we define the trajectory of ^^^^^^ ^^ ^^^^( ^^) = { ^^^^( ^^), ^^^^( ^^ + 1), … , ^^^^( ^^ + ^^ − 1)}. ^^ ∈ ^^+denotes the control horizon of the system. ^^^^∈ ℝ^^× ^^is a constant weighting matrix, which represents the preference for each

[0027] 2) Nash equilibrium sets: According to its definition, a Nash equilibrium solution is defined as a pair of action trajectories from both agents〈^^1∗, ^^2∗〉, which satisfies the following conditions 6 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b ^^∗ 1( ^^) =arg ^^ ^^ ^^^^1( ^^1( ^^), ^^2∗( ^^) ) (4)^^1( ^^) ∈ X, ^^( ^^ )(5)

[0028] where X is the permissible action trajectory sets for the two agents. We define a set X∗( ^^) that contains all the Nash equilibrium pairs at time ^^. Then they pick a solution that minimizes its own cost from X∗( ^^). This optimal solution ^^1^^( ^^) is denoted as follows,^^^^1( ^^) =arg min^^ < >∈ X∗ ^^1( ^^1( ^^), ^^2( ^^), ^^( ^^)) (6)

[0029] By solving (6) and (7), we can obtain the optimal control trajectories. Then we choose the first element from each trajectory as their control input for current time step, hence ^^1^^( ^^) = ^^1^^( ^^)[0] and ^^2^^( ^^) = ^^2^^( ^^)[0]. [0] denotes the first element of the set.Imperfect collaboration: rational solution under incomplete information

[0030] In practice, the human players are unaware of the other player’s decision-making and motion generation process, which makes the Nash equilibrium solution provided in the prior subsection inaccessible. Thus, the human subjects need to estimate other players’ decision-making processes. For this reason, from the perspective of agent 1, the optimal solution ^^^^^^( ^^) in Eq. (6) and (7) becomes,^^ ^^ =arg min^^(1)^^ (8)95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b ^^ ^^2 =arg min^^ ( ^^(1), ^^ , ^ ( ) (9)^^ , < ^ (1) (1) ∗(1)2 1 2 ^ ^^ )2^ 1 , ^^ 2 >∈ X

[0031] where ∙(̂1)denotes the estimated values from agent 1’s perspective. The estimatedset is presented as X∗(1)= {< ^^1∗(1), ^^2∗(1)>} where < ^^1∗(1), ^^2∗(1)> satisfies,

[0032] Essentially, agent 1 should estimate the parameter vector, which is denoted as ^^(1)of ag(1)2 ent 2’s cost function ^^2. In the meanwhile, agent 2 would conduct similar process to estimate ^^(12)to generate its estimated optimal control sequence ^^2^^.Bounded Rationality

[0033] The previous sections provided an overview of the game-theoretic model of the human motion generation process with complete rationality under incomplete information scenarios. Previous studies investigated human bounded rational behavior under circumstances where the time and information were limited. Under this assumption, the human decision-making process is no longer deterministic as Eq. (8) shows. For agent 1, the decision is made according to the following probability density function, 8 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b ^^ ^^ ^^ (— ^^ ^^ ( ^ ^^(1) ( )(12) 11 ^ , ^^2, ^^ ^^ ))^^ ^^( ^^1) =1

[0034] ^^1∈ ℝ+is the rational coefficient, which quantifies the level of agent’s rationality. A ^^ value makes the agent better at maximizing his rewardmaking it highly on the other hand when ^^ = 0, the agent is simply choosing actions uniformly at random. The prediction of other agent’s behavior form agent 1’s perspective is denoted as ^^2^^(1).

[0035] to other works, where the human agents’ cost functionsare merely dependent on their own actions and states, the cost functions in our system rely on both agents’ actions and the shared system states. This tangle of interests of both agents creates challenges for the human agent to evaluate its future actions, which also depend on its partner’s actions. To tackle this difficulty, the human agent needs to develop a mechanism to make a reasonable prediction of its partner. Hence, we propose that the human agent assumes its partner as a complete rational agent, who behaves following the best action in the Nash equilibrium set of the game. Then, we combined the bounded rationality decision-making process that is shown above, with the game-theoretical model that provides a reasonable prediction of the partner’s future actions. One novelty of the model is to reproduce human subjects’ adaptation by considering the equilibrium solution of the game instead of each individual’s optimality, which is akin to the human’s decision-making process. Inverse Optimal Control

[0036] The inverse optimal control (IOC) problem was formed to learn the parameters in the cost functions and probability distributions from the observation data we collected for each pair of participants. In the previous subsections, we have described how human behavior is governed by the bounded rational game-theoretic model. We can generate a series of states that describe the cooperative process of an experiment trial with a bounded rational action human model. We denote the series of states as ^^ =[^^(0), ^^(1), … , ^^(^^)], where ^^ presents the final time step of a trial. 9 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b

[0037] According to the bounded rational game-theoretic model, if a set of parameters Ψ = [ ^^1, ^^1, ^^2, ^^2, ^^1, ^^2] and a proper initial condition ^^(0) are given, acorresponding state trajectory of a trial, which can be denoted as ^^ ^^(Ψ, ^^(0)) =[ ^^ ^^(0), ^^ ^^(1), … , ^^ ^^( ^^)], can be simulated by applying the proposed model.

[0038] Consider the observed system states of a trial in the experiment, to be ^^0=[^^0(0), ^^0(1), ^^0(2), … , ^^0(^^)], where ^^ is the last time step of the collected data. The fitness function (the objective function) is defined as the square of the frobenius norm(‖∙‖2^^)of difference between the observed and simulated trajectory, ^^(Ψ) = ‖ ^^0− ^^^^(Ψ, ^^(0))‖2^^. we constrained all the trials to have the same^^(0), made ^^^^a function of Ψ. The IOC problem is formulated to be an optimization problem, which is to find the Ψ that makes the fitness function minimum, Ψ∗= arg minΨ^^(Ψ).

[0040] By applying the GA algorithm (genetic algorithm) to search for the optimal value of Ψ, eventually, we can learn a set of parameters that contains all the parameters of the bounded rational game-theoretical models for the human dyads. In general, the GA algorithm is generally understood as a “method for solving both constrained and unconstrained optimization problems based on a natural selection process that mimics biological evolution. The algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm randomly selects individuals from the current population and uses them as parents to produce the children for the next generation. Over successive generations, the population "evolves" toward an optimal solution.” (See https: / / www.mathworks.com / discovery / genetic-algorithm.). The GA algorithm generates a population of points at each iteration. The best point in the population approaches an optimal solution. The GA algorithm further selects the next population by computation which uses random number generators. EXPERIMENT SETUP

[0041] In this section, we introduce an experimental study to analyze human behavior in a haptic virtual interface. The hardware setup for the aforementioned 10 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b virtual interface is described. We further explain the participant recruitment detail and the instructions for the experiment. Hardware configuration

[0042] The hardware setup is shown in Fig.2. We used two NOVINT Falcon haptic controllers as the human input devices, which were connected to two PCs via USB connection. A C++ script was running on each PC to collect the raw data from NOVINT Falcon and convert them to the control input of each player in the system. The simulation of the dynamic system was implemented using the UNITY Physics Engine, which receives human inputs from haptic devices. Participant Recruitment and Experiment Process

[0043] The study was approved by the Institutional Review Board (IRB) of Arizona State University (STUDY00011502). A total of 30 participants (randomly assigned to 15 dyads) volunteered to participate in the experiment. None of the participants had experience with haptic devices before. The participants were brought to a quiet room and were asked to sit in front of two isolated PC, each of which connected to a haptic device. The two participants were facing the opposite direction with a makeshift curtain in between. Because they were not able to see or talk to each other during the entire study, they could only communicate with their actions. The participants were briefed about the hardware and joint object manipulation task and a consent form was signed. The participants were asked to move the object with the handle of the haptic device and try to reach the goal position of the object within 10 seconds. After each trial, the participants were given a 30-second break while the system was reset. Each pair were asked to repeat the same task 10 times, which counted as 10 trials of data. Since none of the participants had operated a haptic device before, we set the first two trials as the learning phase, and the data were not used to build human models. RESULTS AND DISCUSSION General Observation: Human Adaption

[0044] We start the discussion of results by highlighting two main observations from human data. The first one is the learning process of the human 11 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b participants, which is indicated by the improvement of goal tracking performance in the first two trials. To check our hypothesis on the learning process, we compared the steady-state goal tracking errors, which contain translation and rotation errors, of the first two trials and the rest of all participants. The mean steady-state error of the first two trials is 0.3834, while the error of the rest trials is 0.1350. The t-test between the two means also suggests a significant difference(^^ = 0.0049). According to Fig.4a, the participants were accustomed to the task after the second trial, which motivated us to mark the first two trials of the experiment as the learning phase as discussed in Sec. III. The human participants could not accomplish the task within the 10-second time limit in the first trial, while they could maintain acceptable performance to accomplish the task in trials 3 to 10. This observation agrees with the collaboration learning phenomenon between the dyads during haptic interactions with incomplete information.

[0045] Summary: From Fig.4a, we can observe the task completion performance of the first two trials is longer than the rest of the trial and is termed as the learning phase. The effect of bounded rationality

[0046] To justify the advantage of the bounded rational model, we applied the IOC method described herein on the observed data to generate the bounded rational game-theoretical model for each of the participants. For each experiment, we combined their trial 3 to 10 data into a data set. Then we learned two bounded rational models, each of which is associated with one of the participants. We simulate the game with the two bounded rational human cooperative models to generate another data set, which contains the states’ trajectories of 100 trials of the game. The difference between each generated trajectory to the corresponding participant’s mean trajectory is defined as the fitting error.

[0047] In the meanwhile, we adopted a similar IOC process to learn two complete rational game-theoretic models, which we discussed herein. An example of the comparison of the model fitting results is presented in Fig.3. In this figure, we choose pair 9 to demonstrate the advantage of the bounded rational model. The sub- figures of Fig.3 represent the displacement of x and y directions and orientation of the object during an experimental trial, where the colored regions represent the mean and 12 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b variance of the trajectories for all the trials. As Fig.3 shows, the trajectory band for the bounded rational models is closer to the original data than the one of complete rational models, especially in the y-direction where the majority of the motion happened. We ran a t-test for the comparison of the mean steady-state errors between the complete rational model and the bounded rational model. As a result of the one-tailed student t-test, the fitting error of the bounded rational model was significantly lower than the complete rational model with ^^ < 0.003(^^ = 4.6636 ^^ − 6).

[0048] The main reason why complete rational models had larger steady- state errors than the bounded rational ones is the assumption of human optimal behavior. The complete rationality assumption of human, which expected that human always chooses the optimal actions, would attribute the suboptimal actions of human participants to their lack of motivation, which diminishes the weighting factor on goal tracking in the learned human models. In the last subsection, we discussed the human suboptimal behavior, which is ubiquitous despite the human participants’ strong desires. From the comparison of the simulated game playing under different assumptions of human mode, we could draw a conclusion that the bounded rational model can better describe human suboptimal behaviors when they interact with someone they are not familiar with.

[0049] Summary: Figure 4b suggests that the bounded rationality models are closer to the ground truth cooperative behavior of the human dyads. Leadership and performance of collaboration

[0050] In this subsection, we will report an important observation between the difference in the effort each individual invested in playing the game to the goal tracking errors, which evaluate the performance of the game. Before demonstrating the results, it is critical to define the roles in the game. In this experiment, we observed that the participants never started the game simultaneously; one participant, which we define as the leader, initiated the game by pushing the object first. Then the other participant, who was defined as the follower, would start his or her motion to collaborate. For a specific pair of participants, the roles assigned to each individual during every trial could be different. Due to the limited time, the roles were fixed within each trial, which indicated that the leader’s actions would dominate the collaboration 13 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b between the participants. To evaluate the effort each participant invested in each trial, we introduce the effort metric ^^^^for agent ^^ ∈ {“Leader”, “Follower”}. We define ^^^^as follows, ^^^^ = ∫^^^^(13) ^^‖^^(^^)‖^^ ^^ ^^^^

[0051] where ^^0and ^^^^are the starting time and final time of the trial. ^^^^( ^^) is the control input of agent ^^ at time ^^. The goal tracking error of each trial was measured by calculating the distance between the actual final states of the object and the goal, which evaluated the quality of the collaboration. A lower goal tracking error indicated better performance during the specific trial. The difference in the effort metric between the leader and follower, hence ^^^^ ^^ ^^ ^^= ^^^^ ^^ ^^ ^^ ^^ ^^− ^^^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^, could reveal the style of leadership.

[0052] By examining the correlation between the effort differences and the goal tracking errors for all the participants, we found a significant negative correlation between them with ^^ ≤ 0.003( ^^ = 0.0029), which is shown in Fig.4c. This observation strongly indicates that the performance of the dyad highly depends on the effort of the leader. If the leader took more responsibility for completing the game, which was reflected by a larger effort difference between the leader and the follower, a better result could be achieved by the dyad. This observation also agrees with the results reported in Messeri et al.

[0053] Summary: Figure 4c shows an inverse relationship between the tracking error and the difference in effort between the agents. Discussion: from bounded rationality to pHRI

[0054] The results of this study reveal the stochastic nature of human cooperative behavior. As the ultimate objective is to help the robot comprehend the human partner’s intent better, we believe the proposed model has the following contributions for the robot to befittingly collaborate with human partners. 14 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b

[0055] From the observation and verification, we have shown that the human participants’ behavior follows bounded rationality under cooperative tasks. Unlike other research where the human manipulation was modeled as an optimization process, the human partners show suboptimal behavior when they acknowledged that their partners were also human participants. When we design the cooperative algorithm for the robot in pHRI scenarios, this study suggests that it is critical for the robot to determine its role during the collaboration. As a leader, the robot should be more active during the interaction, because stronger leadership could result in better performance. CONCLUSIONS AND RELATED WORK

[0056] In this inventive disclosure, we explored novelty methods to model human-human cooperative behaviors in joint actions. We designed a virtual joint object translation game, which required collaboration between human participants. A bounded rationality game-theoretical model was introduced to model human behavior during the game. By comparing results with bounded rational and complete rational assumptions of human actions, we found that the bounded rationality model fits the experimental data better, which suggested that the robots should be aware of suboptimal human behaviors while working with human partners. Another test demonstrated that if the leader paid less effort during the game, the performance of the pair could be worse. This observation highlighted the importance of leadership during collaboration.

[0057] The concepts described herein can be expanded to investigate how bounded rationality affects the pHRI scenarios with different goal positions. We plan to test how the bounded rationality game-theoretical human model could help the robot improve its performance while working with human partners. Exemplary Examples:

[0058] Referring to FIG.5, embodiments of a system described herein may take the form of a computer-implemented system, designated system 100, configured for anticipating human actions for use in physical human-robot interaction as described herein. In general, as indicated, the system 100 includes at least one processor 102 or processing element that is configured for executing functions / operations described herein; e.g., the processor 102 can execute instructions 15 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b 104 stored in a memory 103 including any form of machine-readable medium. In general, the processor 102, via instructions, accesses input data and is configured to execute a bounded rational and game-theoretical model using the plurality of parameters corresponding to the cost function and the probability function to predict a system state trajectory that occurs at a position along a discrete time domain, among other functions described herein.

[0059] The instructions 104 may be implemented as code and / or machine- executable instructions executable by the processor 102 that may represent one or more of a procedure, a function, a subprogram, a program, a routine, a subroutine, a module, an object, a software package, a class, or any combination of instructions, data structures, or program statements, and the like. In other words, one or more of the features for processing described herein may be implemented by hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware or microcode, the program code or code segments to perform the necessary tasks (e.g., a computer- program product) may be stored in a computer-readable or machine-readable medium (e.g., the memory 103 and / or the memory of computing device 1200), and the processor 102 performs the tasks defined by the code. In some embodiments, the processor 102 is a processing element of a cloud such that the instructions 104 may be implemented via a cloud-based web application.

[0060] In some examples, the processor access input data from an end user device 108 in operable communication with a display 110. An end-user, via a user interface 112 rendered along the display 110, can provide input elements 114 to the processor 102 for executing functionality herein. In addition, examples of the system 100 include one or more data source devices 120 for accessing by the processor 102 datasets, images, and other input data as described herein.

[0061] Referring to FIG.6, a computing device 1200 is illustrated which may be configured, via the instructions 104 and / or other computer-executable instructions, to execute functionality described herein. More particularly, in some embodiments, aspects of the system and / or methods described herein may be translated to software or machine-level code, which may be installed to and / or executed 16 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b by the computing device 1200 such that the computing device 1200 is configured to functionality described herein. It is contemplated that the computing device 1200 may include any number of devices, such as personal computers, server computers, hand- held or laptop devices, tablet devices, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronic devices, network PCs, minicomputers, mainframe computers, digital signal processors, state machines, logic circuitries, distributed computing environments, and the like.

[0062] The computing device 1200 may include various hardware components, such as a processor 1202, a main memory 1204 (e.g., a system memory), and a system bus 1201 that couples various components of the computing device 1200 to the processor 1202. The system bus 1201 may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. For example, such architectures may include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, and Peripheral Component Interconnect (PCI) bus also known as Mezzanine bus.

[0063] The computing device 1200 may further include a variety of memory devices and computer-readable media 1207 that includes removable / non- removable media and volatile / nonvolatile media and / or tangible media, but excludes transitory propagated signals. Computer-readable media 1207 may also include computer storage media and communication media. Computer storage media includes removable / non-removable media and volatile / nonvolatile media implemented in any method or technology for storage of information, such as computer-readable instructions, data structures, program modules or other data, such as RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium that may be used to store the desired information / data and which may be accessed by the computing device 1200. Communication media includes computer-readable instructions, data structures, program modules, or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term 17 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b "modulated data signal" means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. For example, communication media may include wired media such as a wired network or direct-wired connection and wireless media such as acoustic, RF, infrared, and / or other wireless media, or some combination thereof. Computer-readable media may be embodied as a computer program product, such as software stored on computer storage media.

[0064] The main memory 1204 includes computer storage media in the form of volatile / nonvolatile memory such as read only memory (ROM) and random access memory (RAM). A basic input / output system (BIOS), containing the basic routines that help to transfer information between elements within the computing device 1200 (e.g., during start-up) is typically stored in ROM. RAM typically contains data and / or program modules that are immediately accessible to and / or presently being operated on by processor 1202. Further, data storage 1206 in the form of Read-Only Memory (ROM) or otherwise may store an operating system, application programs, and other program modules and program data.

[0065] The data storage 1206 may also include other removable / non- removable, volatile / nonvolatile computer storage media. For example, the data storage 1206 may be: a hard disk drive that reads from or writes to non-removable, nonvolatile magnetic media; a magnetic disk drive that reads from or writes to a removable, nonvolatile magnetic disk; a solid state drive; and / or an optical disk drive that reads from or writes to a removable, nonvolatile optical disk such as a CD-ROM or other optical media. Other removable / non-removable, volatile / nonvolatile computer storage media may include magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The drives and their associated computer storage media provide storage of computer-readable instructions, data structures, program modules, and other data for the computing device 1200.

[0066] A user may enter commands and information through a user interface 1240 (displayed via a monitor 1260) by engaging input devices 1245 such as a tablet, electronic digitizer, a microphone, keyboard, and / or pointing device, commonly referred to as mouse, trackball or touch pad. Other input devices 1245 may include a joystick, game pad, satellite dish, scanner, or the like. Additionally, voice inputs, gesture 18 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b inputs (e.g., via hands or fingers), or other natural user input methods may also be used with the appropriate input devices, such as a microphone, camera, tablet, touch pad, glove, or other sensor. These and other input devices 1245 are in operative connection to the processor 1202 and may be coupled to the system bus 1201 but may be connected by other interface and bus structures, such as a parallel port, game port or a universal serial bus (USB). The monitor 1260 or other type of display device may also be connected to the system bus 1201. The monitor 1260 may also be integrated with a touch-screen panel or the like.

[0067] The computing device 1200 may be implemented in a networked or cloud-computing environment using logical connections of a network interface 1203 to one or more remote devices, such as a remote computer. The remote computer may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computing device 1200. The logical connection may include one or more local area networks (LAN) and one or more wide area networks (WAN), but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet.

[0068] When used in a networked or cloud-computing environment, the computing device 1200 may be connected to a public and / or private network through the network interface 1203. In such embodiments, a modem or other means for establishing communications over the network is connected to the system bus 1201 via the network interface 1203 or other appropriate mechanism. A wireless networking component including an interface and antenna may be coupled through a suitable device such as an access point or peer computer to a network. In a networked environment, program modules depicted relative to the computing device 1200, or portions thereof, may be stored in the remote memory storage device.

[0069] Certain embodiments are described herein as including one or more modules. Such modules are hardware-implemented, and thus include at least one tangible unit capable of performing certain operations and may be configured or arranged in a certain manner. For example, a hardware-implemented module may comprise dedicated circuitry that is permanently configured (e.g., as a special-purpose 19 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b processor, such as a field-programmable gate array (FPGA) or an application-specific integrated circuit (ASIC)) to perform certain operations. A hardware-implemented module may also comprise programmable circuitry (e.g., as encompassed within a general-purpose processor or other programmable processor) that is temporarily configured by software or firmware to perform certain operations. In some example embodiments, one or more computer systems (e.g., a standalone system, a client and / or server computer system, or a peer-to-peer computer system) or one or more processors may be configured by software (e.g., an application or application portion) as a hardware-implemented module that operates to perform certain operations as described herein.

[0070] Accordingly, the term “hardware-implemented module” encompasses a tangible entity, be that an entity that is physically constructed, permanently configured (e.g., hardwired), or temporarily configured (e.g., programmed) to operate in a certain manner and / or to perform certain operations described herein. Considering embodiments in which hardware-implemented modules are temporarily configured (e.g., programmed), each of the hardware-implemented modules need not be configured or instantiated at any one instance in time. For example, where the hardware-implemented modules comprise a general-purpose processor configured using software, the general-purpose processor may be configured as respective different hardware-implemented modules at different times. Software may accordingly configure the processor 1202, for example, to constitute a particular hardware- implemented module at one instance of time and to constitute a different hardware- implemented module at a different instance of time.

[0071] Hardware-implemented modules may provide information to, and / or receive information from, other hardware-implemented modules. Accordingly, the described hardware-implemented modules may be regarded as being communicatively coupled. Where multiple of such hardware-implemented modules exist contemporaneously, communications may be achieved through signal transmission (e.g., over appropriate circuits and buses) that connect the hardware-implemented modules. In embodiments in which multiple hardware-implemented modules are configured or instantiated at different times, communications between such hardware- 20 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b implemented modules may be achieved, for example, through the storage and retrieval of information in memory structures to which the multiple hardware-implemented modules have access. For example, one hardware-implemented module may perform an operation, and may store the output of that operation in a memory device to which it is communicatively coupled. A further hardware-implemented module may then, at a later time, access the memory device to retrieve and process the stored output. Hardware-implemented modules may also initiate communications with input or output devices.

[0072] Computing systems or devices referenced herein may include desktop computers, laptops, tablets e-readers, personal digital assistants, smartphones, gaming devices, servers, and the like. The computing devices may access computer- readable media that include computer-readable storage media and data transmission media. In some embodiments, the computer-readable storage media are tangible storage devices that do not include a transitory propagating signal. Examples include memory such as primary memory, cache memory, and secondary memory (e.g., DVD) and other storage devices. The computer-readable storage media may have instructions recorded on them or may be encoded with computer-executable instructions or logic that implements aspects of the functionality described herein. The data transmission media may be used for transmitting data via transitory, propagating signals or carrier waves (e.g., electromagnetism) via a wired or wireless connection.

[0073] It is believed that the present disclosure and many of its attendant advantages should be understood by the foregoing description, and it should be apparent that various changes may be made in the form, construction, and arrangement of the components without departing from the disclosed subject matter or without sacrificing all of its material advantages. The form described is merely explanatory, and it is the intention of the following claims to encompass and include such changes.

[0074] It should be understood from the foregoing that, while particular embodiments have been illustrated and described, various modifications can be made thereto without departing from the spirit and scope of the invention as will be apparent to those skilled in the art. Such changes and modifications are within the scope and teachings of this invention as defined in the claims appended hereto. 21 95684789.1

Claims

Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b CLAIMS What is claimed is:

1. A system for anticipating human actions for use in physical human-robot interaction, comprising: a processor configured to execute one or more processes; and a machine-readable storage medium storing instructions executable by the processor such that the processor is operable to: access a data set including a series of system states; and determine a plurality of parameters corresponding to a cost function and a probability function of a bounded rational and game- theoretical model by applying a genetic algorithm to solve an inverse optimal control (IOC) problem of the bounded rational and game-theoretical model; and run the bounded rational and game-theoretical model using the plurality of parameters corresponding to the cost function and the probability function to predict a system state trajectory that occurs at a position along a discrete time domain.

2. The system of claim 1, wherein the bounded rational and game-theoretical model is configured to be trained by: collecting an actual series of system states from a collaborative experiment; determining the plurality of parameters corresponding to the cost function and the probability function of the bounded rational and game theoretical model based on the actual series of system states collected; running the bounded rational and game theoretical model using the plurality of parameters corresponding to the cost function and the 22 95684789.1Attorney Docket No.: 055743-794112 Client’s Ref.: M23-174P^-WO1-b probability function to predict a series of system state trajectories that occur along the discrete time domain; comparing the predicted series of system state trajectories to the actual series of system states; and adjusting the plurality of parameters according to the difference between the predicted series of system state trajectories and the actual series of system states.

3. The system of claim 1, wherein the bounded rational and game-theoretical model is configured to simulate an imperfect collaboration of humans through a modified Nash equilibrium set. 23 95684789.1

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