A non-direct physical contact human-computer interaction method based on a differential game framework

Abstract

The application relates to the field of non-direct physical contact human-computer interaction, and discloses a non-direct physical contact human-computer interaction method based on a differential game framework. The method constructs a human input model to predict human control input; defines solving robot and human control input satisfying differential game equilibrium to achieve optimal efficiency of a human-computer interaction system as an original problem; converts a robot dynamics model and a cost function representing control efficiency of the robot and the human into a differential flat space to obtain an equivalent problem; solves the equivalent problem to obtain the robot control input; and controls the human-computer interaction system under non-direct physical contact through the human control input and the robot control input. The application avoids solving complex differential equations; can reduce the influence caused by human input inaccuracy and noise, so as to improve the interaction effect.

Description

Technical Field

[0001] This invention relates to the field of human-computer interaction control, and more specifically to a human-computer interaction method based on a differential game framework under non-direct physical contact. Background Technology

[0002] Human-computer interaction (HCI) technology aims to fully leverage the high precision of robots and the adaptability of humans to complex environments, achieving superior results compared to simple manual operation or automated robot execution of specific tasks. In recent years, the application of differential game theory in the field of Physical Contact Human-Computer Interaction (PHRI) has developed rapidly, and the differential flatness method has matured in analyzing and processing complex nonlinear dynamic systems. Methods for solving differential game problems are divided into those based on differential equations and those based on algebraic equations. The former obtains the optimal solution by solving the Hamilton-Jacobi-Bellman equations (HJB equations), while the latter transforms the original problem into an algebraic Riccati equation form under certain assumptions.

[0003] Existing human-computer interaction scenarios within the framework of differential game theory are typically limited to problems involving direct physical contact. Humans and robots generate force feedback and transmission through direct physical contact, and the robot adjusts in real time based on the human's torque input. Within this framework, by solving the algebraic Riccati equation, the optimal control input for the robot to satisfy Nash equilibrium is obtained.

[0004] Existing human-computer interaction scenarios use raw human input as the system's input. Human input is often of low precision, and it follows the rule that the further the control input deviates from human expectations, the lower the probability, resulting in low control precision and poor control performance.

[0005] Traditional human-computer interaction technologies require relatively strict requirements on the dynamic system, such as whether it is a linear system or an affine nonlinear system. These assumptions greatly limit the widespread application of this technology.

[0006] Existing methods require solving a differential equation, the Hamilton-Jacobi-Bellman equation (HJB equation), in the original space, which demands high computational power. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention proposes a human-computer interaction method based on a differential game framework under indirect physical contact. This method enables human-computer interaction to handle dynamic systems that satisfy differential flatness while avoiding the inability to guarantee real-time performance due to the complexity of solving the original problem. This invention also proposes a problem transformation scheme based on differential flatness and a technique for transforming HJB equations into Hamilton-Jacobi-Bellman equations (SDRE equations). These two aspects significantly improve the applicability and practicality of indirect physical contact human-computer interaction based on differential game theory.

[0008] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0009] Step A: Construct a human input model based on a Gaussian model to predict human control input u. h The model is trained using people's historical input data;

[0010] Step B: Define and solve for robot control that satisfies differential game equilibrium. and the control input of the person who satisfies differential game equilibrium The original problem is defined as achieving the optimal efficiency of the human-computer interaction system.

[0011] Step C: Model the robot dynamics in non-direct physical contact human-computer interaction problems. The cost function c represents the robot control efficiency. r (t) and the cost function c representing human control efficiency h (t) Transformed into a differentially flat space, we obtain the equivalent problem of the original problem in the differentially flat space, where x is the state of the robot, u r is the control input of the robot, and f is a dynamic system that satisfies differential flatness;

[0012] Step D: Solve the equivalent problem to obtain the robot's control input u. r ;

[0013] Step E: Input u via human control h Control input u of the robot r Controlling human-computer interaction systems that do not involve direct physical contact.

[0014] Furthermore, step A specifically includes:

[0015] Using Gaussian model As the backbone of the human input model, it describes the human control input u. h t represents a continuous sampling time, u h σ(t) represents the mean of the human control input within the time period corresponding to t, and σ(t) represents the standard deviation of the human control input within the time period corresponding to t.

[0016] Define the kernel function of the Gaussian model as the radial basis function k:

[0017]

[0018] Among them, u h Represents the current user's control input, u h′ represents the human control input predicted by the human input model, φ1 and φ2 are the parameters of the radial basis function k, exp represents the exponential function, and |||| represents the Euclidean distance;

[0019] The human input model is trained by using historical input data, fitting the historical input data over time, and updating φ1 and φ2 using the maximum likelihood method to complete the training of the human input model.

[0020] Furthermore, step C, which describes obtaining the equivalent problem of the original problem in a differentially flat space, specifically includes:

[0021] Step C1: Based on the robot dynamics model Establish a differential flat output z to describe the current robot state x and the robot's control input u. r Human control input u h And the relationship between the first derivative of the current robot state x;

[0022] Step C2: Based on the robot's state x and the human control input u predicted by the human input model... h Constructing the trajectory of human expectations x d ;

[0023] Step C3: Based on the robot's state x and the human's desired trajectory x in the original problem... d With robot control input u r Human control input u h The robot's trajectory corresponds to the trajectory z in a differentially flat space. r The trajectory of a person corresponds to the trajectory z in a differential flat space. h The relationship between the original problem is transformed into a differential flat space, and represented by the differential flat output.

[0024] Furthermore, a differential game theory framework is used to describe the original problem defined in step B, specifically including:

[0025] The dynamic model of the robot in the original problem is as follows: Let x be the derivative of the robot's state.

[0026] The dynamic model of the human in the original problem is: u h =k h x e k h x is a constant e =x d -x, x e x represents the error state after human and robot operations in a human-computer interaction system. d The representative's expected trajectory;

[0027] The cost function c representing the robot control efficiency in the original problem r (t) and the cost function c representing human control efficiency h (t) is:

[0028]

[0029]

[0030] Where T represents the matrix transpose operation, u d Q represents the desired robot input. r Q h R r1 R h1 R r2 and R h2 It is a positive definite matrix;

[0031] In the original problem, J represents the cumulative cost of robot control efficiency. r (u r u h The cumulative cost of representative control efficiency J h (u r u h )for:

[0032]

[0033]

[0034] The optimization objective of the original problem is to find the optimal robot control that satisfies differential game equilibrium. and the control input of the optimal person satisfying differential game equilibrium The cumulative cost of the robot J r The cumulative cost of people J h satisfy:

[0035]

[0036]

[0037] Furthermore, step C3, which involves transforming the original problem into a differentially flat space, specifically includes:

[0038] Constructing a robot dynamics model in a differentially flat space:

[0039]

[0040] z1=Φ(x d u d u d(1) , ..., u d (p) );

[0041] Where p and q are non-negative integers, z represents the differentially flat output of the robot's dynamics model, z1 represents the desired state of the human in the differentially flat space, and z d z represents the state vector that humans expect to achieve in a human-computer interaction scenario. (q) Let z1 be the q-th derivative of z. (q) U represents the q-th derivative of z1. d (p) The robot input u represents the desired input. d The p-th derivative, z r The state vector of a human-computer interactive linear control system, v r The control input vector of a human-computer interactive linear control system, I m Let A be an m-order identity matrix. z B is the state transition matrix of a human-computer interactive linear control system. z Here is the control matrix of a human-computer interactive linear control system, where Φ is a smooth function;

[0042] The dynamic model of the equivalence problem is: in For the augmented state vector, To enhance control input, For the augmented state transition matrix, To augment the control matrix;

[0043] The cost function for robot control efficiency in a differentially flat space:

[0044]

[0045] The cost function of human control efficiency in a differentially flat space:

[0046]

[0047] in, and It is a positive definite matrix that depends on z. r and z d The values ​​of are as follows:

[0048]

[0049]

[0050]

[0051]

[0052]

[0053] in, and For a smooth matrix function, Φ x and Both are smooth functions, k h x is a constant reflecting the sensitivity of human control. d The trajectory of humanity representing expectations;

[0054] In the original problem, J represents the cumulative cost of robot control efficiency. zr The cumulative cost of representative control efficiency J zh for:

[0055]

[0056]

[0057] Solving for Nash equilibrium and The cumulative cost J to robot control efficiency zr The cumulative cost of human control efficiency J zh satisfy:

[0058]

[0059]

[0060] in, This represents the optimal control output of the robot in a differentially flat space. This represents the optimal control output desired by humans in a differentially flat space.

[0061] Furthermore, step D specifically includes:

[0062] The optimal control output of the robot in the differential flat space Substitute into the dynamic equation v in r Solving for the given information yields the following results. This represents the robot's optimal state vector in a differentially flat space.

[0063] The optimal robot control input that satisfies differential game equilibrium. for:

[0064]

[0065] Compared with the prior art, the beneficial technical effects of the present invention are:

[0066] 1. This invention proposes a method for predicting human behavior based on Gaussian process regression. This technique reduces the impact of insufficient precision in human operation, while providing future predictions and uncertainty estimates, thereby improving the accuracy and effectiveness of the interaction.

[0067] 2. This invention proposes a method for analyzing and solving dynamic systems based on differential flatness, which enhances the scalability of the proposed method. This method expands its applicability while reducing computational complexity and time, thereby promoting the implementation of human-computer interaction algorithms.

[0068] 3. This invention proposes a method to map the cost function of the original problem to a differential flat space. The new cost function is equivalent to the cost function of the original problem, thus ensuring that the solution to the new problem is the solution to the original problem. Simultaneously, it guarantees that the new cost function is quadratic, facilitating the solution by existing solvers. Solving for Nash equilibrium in a differential flat space can greatly simplify the analysis and solution costs and time. Attached Figure Description

[0069] Figure 1 This is a flowchart illustrating the indirect physical contact human-computer interaction method based on the differential game framework of the present invention.

[0070] Figure 2 This is an algorithm structure diagram of the indirect physical contact human-computer interaction method based on the differential game framework of the present invention.

[0071] Figure 3 This is a schematic diagram illustrating the transformation of the original problem proposed in this invention into an equivalent problem;

[0072] Figure 4 This is a schematic diagram illustrating the solution of the equivalent problem proposed in this invention. Detailed Implementation

[0073] Figure 1 This is a flowchart illustrating the indirect physical contact human-computer interaction method based on a differential game framework provided by the present invention. Figure 2 This is a diagram illustrating the algorithm structure of the indirect physical contact human-computer interaction method based on a differential game framework, as described in this invention. The following section combines... Figure 1 and Figure 2 A preferred embodiment of the present invention will be described in detail below.

[0074] This invention utilizes the mathematical tool of differential flatness to enable the differential game framework to effectively handle non-physical contact human-computer interaction problems. The human interaction description in this invention is based on the Gaussian process assumption, according to the human's historical input sequence {(t... y u h (t y))} describes human interactive input and obtains real-time estimates of the human input model. Where t y U represents the y-th sampling time. h The control input representing the person, u h (t y ) represents in t y The human-computer interaction system obtains control input from the user at all times. For a Gaussian process, u h (t) represents the control input u for all people. h (t y The mean of ) and σ(t) is the control input u of all people. h (t y The standard deviation of the standard deviation; the selected kernel function is the radial basis function k:

[0075]

[0076] Among them, u h Represents the current user's control input, u h ' represents the human input model's prediction of human control input. φ1 and φ2 are the parameters of the radial basis function kernel. The larger φ1 is, the greater the influence of past human input on the current input in this human-computer interaction scenario. The larger φ2 is, the greater the variance of human input in this human-computer interaction scenario. Variance can reflect the accuracy of the original human control. The larger the variance, the worse the perceived control. exp represents the exponential function, and |||| represents the Euclidean distance.

[0077] By fitting the historical input sequence of humans in the time dimension, the parameters φ1 and φ2 of the radial basis kernel function k are updated using the maximum likelihood method, and finally the training of the human input model is completed, reducing the impact of the inaccuracy of human input on subsequent control.

[0078] This invention will solve for optimal robot control that satisfies differential game equilibrium. and the control input of the optimal person satisfying differential game equilibrium The original problem is defined as achieving the optimal efficiency of the human-computer interaction system. Optimal efficiency refers to the optimal performance of the entire system during the process of humans and robots collaborating to complete a task, including indicators such as task completion time, accuracy, and stability.

[0079] In this invention, the dynamic characteristics of the interactive objects only need to satisfy differential flatness. The process of transforming the original problem into an equivalent problem in a differentially flat space is as follows: Figure 3 As shown, a robot dynamics model is needed in the original indirect physical contact human-computer interaction problem. The cost function c, representing the robot control efficiency, is the objective function of the original problem. r(t) and the cost function c representing human control efficiency h (t) represents the cumulative cost J of robot control efficiency. r The cumulative cost of representative control efficiency J h The current state of the robot, x, and the human's control input, u. h Where f is a dynamic system that satisfies differential flatness, such as common physical systems like drones and unmanned vehicles; the current state x of the robot is a state vector describing the robot's dynamic behavior, which is a vector composed of all factors related to the robot's motion. Let u be the derivative of x. r Represents the control input for the robot, and the control input for the human, u. h The dynamic model is calculated based on human input. It is based on known models summarized from human-computer interaction systems, representing the current robot state x and the robot's control input u. r Human control input u h And the relationship between the first derivative of the current robot state x. Robot dynamics model It only needs to satisfy differential flatness, and is not limited to linear or affine nonlinear systems. Cumulative cost J r and J h Define the optimal efficiency (Nash equilibrium) in the original problem. The mathematical form of the equivalent problem is consistent with the original problem, but it is easier to solve.

[0080] The specific process of problem transformation is as follows:

[0081] Based on the robot's dynamic model Choose the differential flat output z to establish the robot's state x and the robot's control input u. r and differential flat output and u r The functional relationship between finite-order derivatives is z = Φ(x, u). r u r (1) u r (2) , ..., u r (p) ), where p is a non-negative integer, u r (p) The control input u of the robot r The p-th derivative of , where Φ is a smooth function. This can be determined from x and u. r Obtain the state vector z of the human-computer interactive linear control system r , where u r The derivatives of each order are obtained through numerical methods. A dynamic model of the robot in a differentially flat space is constructed:

[0082]

[0083]

[0084] Where q is a non-negative integer, z (q) The q-th derivative of the differentially flattened output z of the robot's dynamic model, z r The state vector of a human-computer interactive linear control system, v r The control input vector of a human-computer interactive linear control system, I m Let A be an m-order identity matrix. z B is the state transition matrix of a human-computer interactive linear control system. z It is the control matrix of a human-computer interactive linear control system.

[0085] z represents the differentially flat output of the robot's dynamics model, and z1 corresponds to z, where z1 represents the desired state of the human in the differentially flat space. d With z r Corresponding to z d z represents the state vector that humans expect to achieve in a human-computer interaction scenario. (q) Let z1 be the q-th derivative of z. (q) U represents the q-th derivative of z1. d (p) The robot input u represents the desired input. d The p-th derivative, z r The state vector of a human-computer interactive linear control system, v r The control input vector of a human-computer interactive linear control system, I m Let A be an m-order identity matrix. z B is the state transition matrix of a human-computer interactive linear control system. z Let Φ be the control matrix of a human-computer interactive linear control system, where Φ is a smooth function.

[0086] Generate human control input u using a human input model. h According to human control input u h With the current robot state x and the human's desired trajectory x d The relationship between the two is used to obtain the expected trajectory x of the person. d Under the guarantee of certain theorems, it is possible to establish x. d The corresponding trajectory z in the same differential flat space d The relationship between them, that is in For a smooth function; further establish the future u h With the future z r and z d The relationship between them, that is in Let be a smooth function. The function consisting of the differentially flat output z and its finite-order derivatives can represent the state variables and control inputs in the original space, and its dimension is equal to u. r The dimension of z is independent of z and its derivatives:

[0087] z = Φ(x, u) r u r (1) , ..., u r (p) );

[0088] x = Φ x (z,z (1) , ..., z (q) )=Φ x (z r );

[0089]

[0090] Where, Φ x and Both are smooth functions; in particular, if Φ x (0)≠0, so the origin of the original coordinate system needs to be translated so that Φ x (0) = 0, Similarly.

[0091] Combining differential flattening process and x=Φ x (zr) represents the cost function in the objective function of the original problem. and Transform into a new cost function and Where, x e =x d -x represents the error state after human-robot interaction in the human-computer interaction system, T represents matrix transpose, and u d Q represents the desired robot input. r Q h R r1 R h1 R r2 and R h2As a positive definite matrix, the cost function often reflects certain indicators of interest during the control process, such as minimizing energy consumption and minimizing path length. The larger the weight of the matrix, the more important the indicator. In human-computer interaction, both humans and robots tend to minimize their cumulative costs. Therefore, game theory is needed to achieve an equilibrium state and maximize the effectiveness of the human-computer interaction system. and It is a positive definite matrix that depends on z. r and z d The values ​​of are as follows:

[0092]

[0093]

[0094]

[0095]

[0096]

[0097] in, and Let k be a smooth matrix function. h Let x be a constant. d Represents the expected human trajectory.

[0098] The augmented dynamic model for this problem is: in For the augmented state vector, To enhance control input, For the augmented state transition matrix, To augment the control matrix.

[0099] Therefore, the original problem involves finding the optimal robot control that satisfies differential game equilibrium. and the control input of the optimal person satisfying differential game equilibrium To achieve the cumulative cost J of the robot r The cumulative cost to people J h Satisfying Nash Equilibrium and Transform into: Solving and The cumulative cost J to achieve robot control efficiency zr The cumulative cost of human control efficiency J zh Satisfying the new Nash equilibrium and in This represents the optimal control output of the robot in a differentially flat space. This represents the optimal control output desired by humans in a differentially flat space.

[0100] The Nash equilibrium in the original problem indicates that when both the human and the robot are in an equilibrium (optimal) state, neither party can obtain a higher benefit by unilaterally changing its control strategy. Therefore, based on the assumption that humans are rational, we believe that the human will adopt the optimal strategy. In this way, the robot only needs to retrieve This means that the human-computer interaction system can reach this equilibrium state. The new equilibrium shows that when both the human and the robot are in an equilibrium (optimal) state, changing the control strategy of either party alone will not yield higher benefits.

[0101] If the equivalence problem is solved, then it is only necessary to... Substitute into the dynamic equation Solving for the results Substitute again To obtain the optimal robot control input that satisfies differential game equilibrium

[0102]

[0103] The method for solving equivalent problems in this invention transforms the solution of differential equations into the solution of algebraic equations, which can reduce the amount of computation and improve the real-time performance of the algorithm.

[0104] Specifically, the original problem is described by constructing a Hamiltonian-Jacobi-Bellman equation in a differential flat space based on the differential game framework. This equation is then transformed into a state-dependent Riccati equation. Finally, a solver for the state-dependent Riccati equation is used to obtain the differential flat output, which is then used to obtain the robot's control input u. r .

[0105] The specific solution process is as follows: Figure 4 As shown:

[0106] Write the SDRE equations for humans according to their optimization objective function; write the SDRE equations for robots according to their optimization objective function.

[0107] Call the SDRE solver to transform the augmented state transition matrix. Augmented control matrix and and Input the solutions into the solver and solve them.

[0108] The SDRE solver can solve for... and Will Substitute into the dynamic equation Solving for the results Substitute again have to:

[0109]

[0110] in To obtain the optimal robot control input that satisfies the equilibrium of the differential game, the robot's control input can be obtained. Ultimately, this allows the operator to control the robot in the original space without needing to focus on its specific motion characteristics (dynamic characteristics). The operator can concentrate on their own goal and control the robot according to their desired trajectory, ensuring the robot moves along the trajectory expected by the human.

[0111] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention, and no reference numerals in the claims should be construed as limiting the scope of the claims.

[0112] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A non-direct physical contact human-computer interaction method based on a differential game framework, characterized in that, Includes the following steps: Step A: Construct a human input model based on the Gaussian model to predict human control input. The model is trained using people's historical input data; Step B: Define and solve for robot control that satisfies differential game equilibrium. and the control input of the person who satisfies differential game equilibrium The original problem is defined as achieving the optimal efficiency of the human-computer interaction system. Step C: Model the robot dynamics in non-direct physical contact human-computer interaction problems. The cost function representing the robot control efficiency and the cost function of representative control efficiency Transforming the problem into a differentially flat space yields the equivalent problem of the original problem in a differentially flat space, where... It is the robot's state. It is the control input of the robot. For a dynamical system that satisfies differential flatness; step C describes obtaining the equivalent problem of the original problem in the differentially flat space, specifically including: Step C1: Based on the robot dynamics model Establish differential flat output Describe the current state of the robot. Robot control input Human control input and the current status of the robot. The relationship between the first derivatives; Step C2: Based on the robot's state Human input model predicts human control input Constructing the trajectory of human expectations ; Step C3: Based on the robot's state in the original problem Human Expectations With robot control input Human-controlled input The robot's trajectory corresponds to the trajectory in a differential flat space. The trajectory of a person corresponds to the trajectory in a differential flat space. The relationship between the two problems transforms the original problem into a differentially flat space, and the output is represented by the differentially flatness. Step D: Solve the equivalent problem to obtain the robot's control input. ; Step E: Through human control input Control input to the robot Controlling human-computer interaction systems that do not involve direct physical contact.

2. The indirect physical contact human-computer interaction method based on a differential game framework according to claim 1, characterized in that, Step A specifically includes: Using Gaussian model As the backbone of the human input model, it describes human control input. , Represents a continuous sampling time period. for The average value of human control input over the corresponding time period. for The standard deviation of human control input over the corresponding time period; Define the kernel function of the Gaussian model as the radial basis function. : ； in, Represents the current person's control input. The representative input model predicts the control input of the person. and Radial basis kernel function The parameters, Represents an exponential function. Represents Euclidean distance; The human input model is trained using historical input data, fitting the model to historical input over time, and then updated using the maximum likelihood method. and The training of the human input model is completed.

3. The indirect physical contact human-computer interaction method based on a differential game framework according to claim 1, characterized in that, The original problem defined in step B is described using a differential game framework, specifically including: The dynamic model of the robot in the original problem is as follows: , For the robot's state The derivative; The dynamic model of the human in the original problem is as follows: , It is a constant. , This represents the error state after human and robot operations in a human-computer interaction system. The representative's expected trajectory; The cost function representing the robot control efficiency in the original problem and the cost function of representative control efficiency for: ； ； in, Represents the matrix transpose operation. Represents the desired robot input, , , , , and It is a positive definite matrix; The cumulative cost representing robot control efficiency in the original problem The cumulative cost of representative control efficiency for: ; ; The optimization objective of the original problem is to find the optimal robot control that satisfies differential game equilibrium. and the control input of the optimal person satisfying differential game equilibrium The cumulative cost of robots The cumulative cost of people satisfy: ； 。 4. The indirect physical contact human-computer interaction method based on the differential game framework according to claim 1, characterized in that, Step C3, which involves transforming the original problem into a differentially flat space, specifically includes: Constructing a robot dynamics model in a differentially flat space: ； ， ； in, and It is a non-negative integer. The differential flattened output represents the dynamic model of the robot. This represents the desired state in a differentially flat space. This represents the state vector that humans expect to achieve in human-computer interaction scenarios. represent of First derivative, represent of First derivative, Robot input representing expectations of First derivative, The state vector of a human-computer interactive linear control system The control input vector of a human-computer interactive linear control system. for An identity matrix of order 1. The state transition matrix of a human-computer interactive linear control system The control matrix of a human-computer interactive linear control system It is a smooth function; The dynamic model of the equivalence problem is: ;in For the augmented state vector, To enhance control input, For the augmented state transition matrix, To augment the control matrix; The cost function for robot control efficiency in a differentially flat space: ； The cost function of human control efficiency in a differentially flat space: ； in, and It is a positive definite matrix, depending on and The values ​​of are as follows: ； ； ； ； ； ； ； in, , , , and For smooth matrix functions, and Both are smooth functions. It is a constant that reflects the sensitivity of human control. The trajectory of humanity representing expectations; The cumulative cost representing robot control efficiency in the original problem The cumulative cost of representative control efficiency for: ; ; Solving for Nash equilibrium and The cumulative cost of robot control efficiency The cumulative cost of human control efficiency satisfy: ， ； ， ； in, This represents the optimal control output of the robot in a differentially flat space. This represents the optimal control output desired by humans in a differentially flat space.

5. The indirect physical contact human-computer interaction method based on the differential game framework according to claim 4, characterized in that, Step D specifically includes: The optimal control output of the robot in the differential flat space Substitute into the dynamic equation In Solving for the given information yields the following results. , This represents the robot's optimal state vector in a differentially flat space. The optimal robot control input that satisfies differential game equilibrium. for: .

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