A Nonlinear Contact Force Control Method and System for Robots Based on Maxwell-Fractional Impedance Model

Abstract

This invention belongs to the technical field of robot force control and discloses a nonlinear contact force control method and system for robots based on the Maxwell-fractional-order impedance model. The method includes the following steps: using a force-position controller based on the Maxwell-fractional-order impedance model to control a robot end effector to scan the surface of a soft material to be scanned in a horizontal plane according to a preset scanning trajectory, maintaining constant force contact between the robot end effector and the soft material to be scanned in the height direction. This constant force contact is achieved as follows: a preset initial contact position between the robot end effector and the soft material to be scanned is established; the optimal position of the robot end effector in the height direction is calculated in real time when the steady-state error of the contact force tracking during the scanning process is zero; and the position of the robot end effector in the height direction is adjusted in real time according to this optimal position. This invention solves the problem of safe force control during the scanning of soft material surfaces.

Description

Technical Field

[0001] This invention belongs to the field of robot force control technology, and more specifically, relates to a nonlinear contact force control method and system for robots based on the Maxwell-fractional-order impedance model. Background Technology

[0002] To achieve stability in scanning tasks using robots in soft material environments (such as silicone, human tissue, and foam), the robot's end effector requires a more stable and precise force control method. Considering robot control safety, most robots do not expose their underlying force / torque control interfaces; therefore, position-based control methods must be chosen to design the force controller. Position-based impedance control is widely used in compliant control tasks for collaborative robots, but traditional impedance control methods suffer from drawbacks such as large position fluctuations, poor disturbance rejection, low force tracking accuracy, and susceptibility to force impacts. Furthermore, when the robot's end effector establishes contact with the environment, it is difficult to simultaneously ensure both rapid contact establishment and safety.

[0003] For interactive tasks in soft material environments, a certain understanding of the environment's characteristics is required, typically described using model representation. Therefore, the parameters of the environmental model also need to be identified; inaccurately estimated environmental parameters cannot accurately describe the environment's characteristics, thus affecting the effectiveness of contact force control in robot interactive tasks. The force control effect of the impedance controller is highly correlated with the input reference position. Therefore, using an online real-time reference position update method can significantly improve force tracking accuracy. Moreover, real-time reference position updates ensure the safety of robot-environment interaction under changing environmental conditions.

[0004] Therefore, in order to improve the force control accuracy and safety of robots in scanning tasks in soft material environments, it is necessary to improve the traditional impedance controller model and propose a force control method suitable for soft silicone material environments. Summary of the Invention

[0005] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a nonlinear contact force control method and system for robots based on the Maxwell-fractional-order impedance model, which solves the problem of force safety control when scanning soft material shells.

[0006] To achieve the above objectives, according to one aspect of the present invention, a nonlinear contact force control method for a robot based on a Maxwell-fractional-order impedance model is provided, the method comprising the following steps:

[0007] A force-position controller based on the Maxwell-fractional-order impedance model is used to control the robot's end-effector to scan the surface of the soft material to be scanned in the horizontal plane according to a preset scanning trajectory. A constant force contact is maintained between the robot's end-effector and the soft material to be scanned in the height direction. This constant force contact is achieved in the following manner:

[0008] The initial contact position between the robot end-effector and the soft material to be scanned is preset. The optimal position of the robot end-effector in the height direction is calculated in real time when the steady-state error of the contact force tracking of the robot end-effector is zero during the scanning process. The position of the robot end-effector in the height direction is adjusted in real time according to the optimal position.

[0009] More preferably, the steady-state error of the contact force tracking is calculated in the following manner:

[0010] The force-position controller is used to calculate the actual position of the robot's end effector in the height direction;

[0011] The nonlinear contact force model of the scanned soft material is transformed into a linear contact force model, and the contact force deviation between the robot end tool and the soft material to be scanned is calculated using the linear contact force model.

[0012] A steady-state error relationship for the contact force is established by combining the actual position of the robot end effector in the height direction and the contact force deviation.

[0013] More preferably, the formula for the actual position of the robot end effector in the height direction is as follows:

[0014]

[0015] Where, x a x is the calculated actual height position. r This is the reference position in the height direction. Δf is the position deviation, Δf is the difference between the actual measured force and the preset expected contact force, and Φ(s) is the transfer function of the force-position controller.

[0016] More preferably, the linear contact force model of the soft material to be scanned is as follows:

[0017]

[0018] Among them, F z It is the contact force in the height direction. These represent the penetration depth and penetration velocity of the end-effector into soft materials, respectively. d K is the preset desired contact force. e λ and β are the stiffness coefficient, damping coefficient, and exponential coefficient identified by the nonlinear contact force model of soft materials, respectively.

[0019] More preferably, the formula for the contact force deviation is as follows:

[0020] Among them, F z The contact force F is in the height direction. d The preset desired contact force, These are the environmental stiffness and damping parameters after linearization of the nonlinear contact force model for soft materials, x e and These are the position and rate of change of the soft material surface, x a and These are the actual spatial position and velocity of the end-effector's center point, respectively.

[0021] More preferably, the contact force tracking steady-state error is calculated in the following manner:

[0022]

[0023] Where B is the control damping gain of the force-position controller. and , x represents the environmental stiffness and damping parameters after linearization of the nonlinear contact force model of soft materials. r and These are the desired position and desired velocity of the end-effector center.

[0024] More preferably, the formula for calculating the optimal position is as follows:

[0025]

[0026] in, and These are the environmental stiffness and damping parameters after linearization of the nonlinear contact force model for soft materials, x e For the location of the soft material surface, F is the velocity at the center point of the end-effector. d This is the preset desired contact force.

[0027] More preferably, the The formulas for β are as follows:

[0028]

[0029]

[0030] Among them, K e λ and β are the stiffness coefficient, damping coefficient, and exponential coefficient identified by the nonlinear contact force model of soft materials, respectively. d The preset desired contact force, These are the iterative parameters during the identification process.

[0031] More preferably, the formula for the force position controller is as follows:

[0032]

[0033] Where M, B, and K are the desired mass, damping, and stiffness coefficients of the controller, respectively, and β... D For fractional differential operators, λ I Let E be a fractional integral operator, s be a Laplace operator that transforms the control law from the time domain to the complex frequency domain, and E be a fractional integral operator. x (s) and E f (s) represent the Laplace transforms of position error and force error, respectively.

[0034] According to another aspect of the present invention, a system for nonlinear contact force control of a robot based on a Maxwell-fractional-order impedance model is provided. The system includes an actuator for performing the aforementioned nonlinear contact force control method for a robot based on a Maxwell-fractional-order impedance model.

[0035] In summary, the technical solutions conceived by this invention have the following beneficial effects compared with the prior art:

[0036] 1. This invention employs a force-position controller based on the Maxwell-fractional-order impedance model to ensure that the steady-state error of the contact force tracking at the robot end effector during the scanning process is zero, thereby achieving constant force contact between the robot end effector and the soft material to be scanned during scanning and solving the problem of force safety control during the scanning process of soft material surfaces.

[0037] 2. This invention uses a force-position controller to control the robot to move along a preset trajectory in the horizontal plane on the one hand, and to maintain constant force contact with the soft material to be scanned in the vertical direction on the other hand, thereby achieving two-dimensional control of the robot's movement with high control precision.

[0038] 3. This invention utilizes a nonlinear contact force model of the soft material to be scanned to calculate the contact force deviation between the robot end tool and the soft material to be scanned. It fully considers the physical properties of the material itself. The deviation calculated by the force model includes the contact characteristics of the soft material and describes the relationship between the scanning position of the end tool and the contact force change caused by the deformation of the soft material. Based on this, the control position of the force position control output can be adjusted in a timely manner.

[0039] 4. The contact force tracking steady-state error established in this invention is analyzed by combining the end-effector control position output by the force position controller and the contact force model of the soft material. In order to ensure that the force tracking steady-state error is as close to zero as possible, the optimal position of the robot end in the height direction is obtained in real time as a reference trajectory and adjusted online in real time. The robot can improve the accuracy of force tracking by tracking this trajectory. Attached Figure Description

[0040] Figure 1 This is a flowchart of a nonlinear contact force control method for a robot based on a Maxwell-fractional-order impedance model, constructed according to a preferred embodiment of the present invention.

[0041] Figure 2 This is a schematic diagram of the calibrated end TCP coordinate system according to a preferred embodiment of the present invention.

[0042] Figure 3 This is a schematic diagram of the initial reference trajectory generated by teaching points according to a preferred embodiment of the present invention, wherein (a) is a schematic diagram of the initial reference trajectory being a straight line, and (b) is a schematic diagram of the initial reference trajectory being a curve.

[0043] Figure 4 This is a diagram of a robot control framework based on the Maxwell-fractional impedance model according to a preferred embodiment of the present invention.

[0044] Figure 5 This is a robot excitation trajectory diagram estimated by the parameters of the nonlinear environmental force HC model according to a preferred embodiment of the present invention.

[0045] Figure 6 The environmental parameters are obtained according to the preferred embodiment of the present invention based on the exponentially weighted least squares method, wherein (a) is the estimated environmental stiffness, (b) is the estimated environmental damping, and (c) is the coefficient of the exponential term.

[0046] Figure 7 This is a diagram showing the actual trajectory position of the end of a robot soft silicone material scanning experiment according to a preferred embodiment of the present invention, wherein (a) is the actual trajectory under a straight reference trajectory, and (b) is the actual trajectory under a circular reference trajectory.

[0047] Figure 8 This is a schematic diagram of an experimental setup for a robotic soft silicone material scanning experiment according to a preferred embodiment of the present invention.

[0048] Figure 9 The diagram shows the actual force tracking effect of the robot according to a preferred embodiment of the present invention, where (a) is the contact force under a straight reference trajectory and (b) is the contact force under a circular reference trajectory. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0050] like Figure 1 As shown, a nonlinear contact force control method for robots based on the Maxwell-fractional-order impedance model specifically includes the following steps:

[0051] (a) Perform end-effector calibration and real-time gravity compensation for force sensors, and generate the robot's initial motion trajectory based on the teaching points.

[0052] Connect the robot and the computer controller, and use the UR robot teach pendant to perform TCP calibration on the end effector to determine the coordinates of the end effector's center point. In an embodiment of the invention, the transformation matrix of the end effector's center point relative to the robot flange is:

[0053]

[0054] like Figure 2 As shown, this figure is a schematic diagram of the coordinate system of the calibrated TCP. Gravity compensation is applied to the end effector, and an initial robot scanning trajectory is generated based on the position of the soft silicone surface. Figure 3 The initial reference trajectory generated by the trajectory generator.

[0055] (b) A robot force-position controller is designed based on the Maxwell-fractional-order impedance model to simultaneously control the scanning trajectory and contact force, ensuring accurate and stable scanning on soft silicone material.

[0056] Based on the Maxwell model, a modified classical impedance controller was designed to simultaneously achieve robot contact force and position tracking. The control equations are as follows:

[0057]

[0058] Among them, X a X r Let the robot's actual position and the input reference position be the two values, and the difference between them be defined as the position disturbance E = X. a -X r ; ΔF=F d -F e is the difference between the expected contact force and the actual contact force; M, B, and K are the inertia coefficient, damping coefficient, and stiffness coefficient, respectively.

[0059] Based on the modified Maxwell's impedance control equations, and because fractional calculus possesses memory and inheritance properties in describing nonlinear systems, it can better describe the dynamic characteristics of soft material interactions. Therefore, the force error integrator is transformed into a fractional integrator. The controller equations are as follows:

[0060]

[0061] In the above formula, The output of the fractional integrator for the force error can be obtained according to the fractional Grünwald-Letnikov definition:

[0062]

[0063] If the step size h is chosen to be sufficiently small during control, then...

[0064] Where t0 is the initial memory time of the fractional integrator, t is the current time, and λ I Here, h is the integral coefficient, and h is the step size. To take the closest integer, φ j The coefficients are binomial coefficients. Since Γ(x+1)=xΓ(x), where Γ(·) is the Gamma function, and the binomial coefficients φ... j The recurrence relation is as follows:

[0065]

[0066] The quadratic coefficients calculated using recursive relationships have higher computational accuracy, avoiding the large errors caused by direct calculation using the Gamma function. Therefore, the binomial coefficients φ j Calculated according to the following rules:

[0067]

[0068] For the impedance control equations of position quantity analysis, the position error differential is transformed into a fractional-order differentiator. Since real-time performance and complexity affect the effectiveness of the fractional-order differentiator in practical applications, a superposition of spring-damped and mass-spring-damped models is designed for approximation. Using the superposition method for equivalent processing achieves the same effect with better real-time performance. The Maxwell-fractional-order control approximation model is as follows:

[0069]

[0070] Where, β D E is the differential operator for a fractional-order differentiator. x (s), E f(s) represent the Laplace transforms of the position error and force error, respectively. The schematic diagram of the control framework in step (b) is shown below. Figure 4 As shown.

[0071] (c) Construct a nonlinear environmental force model and use the exponential weighted least squares method to identify the model parameters online.

[0072] The environmental modeling parameters are identified using the weighted least squares method, with a time-varying forgetting factor θ used in the identification process. k =1-γ k ,in u is the rate of change factor of the forgetting factor. During the identification process, u = 2, and the sensor data output rate is 500 Hz. The time-varying forgetting factor can improve the speed and stability of the identification process.

[0073] The dynamics of the robot's interaction with the environment are modeled using the Hunt–Crossley (HC) model, and the contact forces are represented as follows: Where δx(t) = x(t) - x e (t). The robot excitation trajectory input during the experiment is as follows: Figure 5 As shown. After logarithmic linearization and discretization, we can obtain... The meanings of each parameter are as follows:

[0074]

[0075] The intermediate quantities for identification using the EWRLS method are calculated as follows: L k+1 P k+1 These are the adaptive gain vector and the covariance matrix, respectively.

[0076]

[0077] The parameters estimated by the EWRLS method are iteratively updated, and the update equation is as follows.

[0078]

[0079] The final parameters of the HC model are as follows, and the parameter estimation results are as follows. Figure 6 As shown.

[0080]

[0081] (d) Analyze the steady-state error of the robot force position controller and design an online trajectory generator based on a nonlinear environmental force model.

[0082] Analyzing the interaction between the robot's end effector and the environment, we assume that the interaction between the robot's end effector and the environment tends to be in equilibrium, satisfying δx = δx sThe HC model is approximated by linearization using Taylor expansion.

[0083]

[0084] At the equilibrium position, it satisfies By calculating F e The partial differential terms show that the contact force between the robot end effector and the environment approximately satisfies a linear relationship, as shown in the following equation.

[0085]

[0086] The steady-state error of contact force tracking is obtained as follows:

[0087] The force position controller is used to calculate the actual position of the robot's end effector in the height direction;

[0088]

[0089] Where x a x is the calculated actual height position. r This is the reference position in the height direction. Let Δf be the position deviation, Δf be the difference between the actual measured force and the preset desired contact force, and Φ(s) be the transfer function of the designed force-position controller control law. A nonlinear contact force model of the soft material to be scanned is established, and the contact force deviation between the robot end-effector and the soft material to be scanned is calculated using the force model.

[0090]

[0091] Among them, F z F represents the contact force in the height direction. d The preset desired contact force, The environmental stiffness and damping parameters are obtained by linearizing the above nonlinear contact force model for soft materials. x represents the position and rate of change of the soft material surface, respectively. a and These represent the actual spatial position and velocity of the end-effector's center point, respectively.

[0092] A steady-state error relationship for the contact force is established by combining the actual position of the robot end effector in the height direction with the contact force deviation.

[0093] By considering the impedance controller as adjusting the position deviation on the reference trajectory and combining the approximately linear relationship of the environmental force, the force tracking steady-state error of the Maxwell-fractional-order impedance controller can be derived as follows.

[0094]

[0095] To ensure zero force tracking error during the robot's end effector scanning of the soft silicone material environment, the following requirement must be met: t→∞ If Δf(t) = 0, then the calculation of the reference trajectory satisfies The reference trajectory velocity is approximated to the actual end velocity at the previous moment to avoid abrupt changes in contact force caused by drastic velocity variations. An online robot reference trajectory generator is constructed. The reference position is input in real time to the robot's Maxwell-fractional impedance controller. The robot's end-effector trajectory is as follows: Figure 7 As shown.

[0096] In one specific embodiment of the present invention, the soft material used in the experiment is silicone, specifically Blue Butterfly LDX / V6. A coupling agent is evenly applied to the surface of the area to be scanned for lubrication and tight adhesion. The end is manually dragged above the silicone plate, and the initial environmental position x is recorded. e Parameter identification of the nonlinear HC model is performed, and the online reference trajectory generator outputs the reference trajectory in real time. The updated reference trajectory is input into the Maxwell-fractional impedance controller, which outputs the pose information of the robot's intended trajectory points. The pose information is then used to calculate the robot's joint angles q using inverse kinematics, and these target joint angles are sent to the robot motion controller to control the robot's motion. For example... Figure 8 The diagram shown is a schematic of the experimental setup.

[0097] The parameters used in the actual experiment are as follows, and the impedance control parameters are:

[0098] M = 20, B = 1000, K = 4000

[0099] The parameters of the fractional-order controller are:

[0100] λ I =β D =0.7, h=0.02, L=2000

[0101] The desired force is set to 10N, and the maximum safe force threshold is 30N. For example... Figure 9 As shown in the figure, this figure is a diagram of the contact force variation of a robot in a nonlinear environment based on the Maxwell-fractional order impedance model. MFO-IC (Maxwell-Fractional Order Impedance Control) represents the method proposed in this invention.

[0102] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A nonlinear contact force control method for robots based on the Maxwell-fractional-order impedance model, characterized in that, The method includes the following steps: A force-position controller based on the Maxwell-fractional-order impedance model is used to control the robot's end-effector to scan the surface of the soft material to be scanned in the horizontal plane according to a preset scanning trajectory. A constant force contact is maintained between the robot's end-effector and the soft material to be scanned in the height direction. This constant force contact is achieved in the following manner: The initial contact position between the robot end-effector and the soft material to be scanned is preset. The optimal position of the robot end-effector in the height direction is calculated in real time when the steady-state error of the contact force tracking of the robot end-effector is zero during the scanning process. The position of the robot end-effector in the height direction is adjusted in real time according to the optimal position.

2. The nonlinear contact force control method for a robot based on the Maxwell-fractional-order impedance model as described in claim 1, characterized in that, The steady-state error of the contact force tracking is calculated in the following manner: The force-position controller is used to calculate the actual position of the robot's end effector in the height direction; The nonlinear contact force model of the scanned soft material is transformed into a linear contact force model, and the contact force deviation between the robot end tool and the soft material to be scanned is calculated using the linear contact force model. A steady-state error relationship for the contact force is established by combining the actual position of the robot end effector in the height direction and the contact force deviation.

3. The nonlinear contact force control method for a robot based on the Maxwell-fractional-order impedance model as described in claim 2, characterized in that, The formula for the actual position of the robot's end effector in the height direction is as follows: Where, x a x is the calculated actual height position. r This is the reference position in the height direction. Δf is the position deviation, Δf is the difference between the actual measured force and the preset expected contact force, and φ(s) is the transfer function of the force-position controller.

4. A nonlinear contact force control method for a robot based on a Maxwell-fractional-order impedance model as described in claim 2 or 3, characterized in that, The linear contact force model of the soft material to be scanned is as follows: Among them, F z It is the contact force in the height direction. These represent the penetration depth and penetration velocity of the end-effector into soft materials, respectively. d K is the preset desired contact force. e λ and β are the stiffness coefficient, damping coefficient, and exponential coefficient identified by the nonlinear contact force model of soft materials, respectively.

5. The nonlinear contact force control method for a robot based on the Maxwell-fractional-order impedance model as described in claim 4, characterized in that, The formula for the contact force deviation is as follows: Among them, F z The contact force F is in the height direction. d The preset desired contact force, These are the environmental stiffness and damping parameters after linearization of the nonlinear contact force model for soft materials, x e and These are the position and rate of change of the soft material surface, x a and These are the actual spatial position and velocity of the end-effector's center point, respectively.

6. A nonlinear contact force control method for a robot based on a Maxwell-fractional-order impedance model as described in claim 1 or 5, characterized in that, The contact force tracking steady-state error is calculated in the following manner: Where B is the control damping gain of the force-position controller. and , x represents the environmental stiffness and damping parameters after linearization of the nonlinear contact force model of soft materials. r and These are the desired position and desired velocity of the end-effector center.

7. A nonlinear contact force control method for a robot based on a Maxwell-fractional-order impedance model as described in claim 1 or 5, characterized in that, The formula for calculating the optimal position is as follows: in, and These are the environmental stiffness and damping parameters after linearization of the nonlinear contact force model for soft materials, x e For the location of the soft material surface, F is the velocity at the center point of the end-effector. d This is the preset desired contact force.

8. The nonlinear contact force control method for a robot based on the Maxwell-fractional impedance model as described in claim 7, characterized in that, The The formulas for β are as follows: Among them, K e λ,β are the stiffness coefficients, damping coefficients, and exponential coefficients identified by the nonlinear contact force model of soft materials, F d The preset desired contact force, These are the iterative parameters during the identification process.

9. A nonlinear contact force control method for a robot based on a Maxwell-fractional-order impedance model as described in claim 1 or 5, characterized in that, The formula for the force position controller is as follows: Where M, B, and K are the desired mass, damping, and stiffness coefficients of the controller, respectively, and β... D For fractional differential operators, λ I Let E be a fractional integral operator, s be a Laplace operator that transforms the control law from the time domain to the complex frequency domain, and E be a fractional integral operator. x (s) and E f (s) represent the Laplace transforms of position error and force error, respectively.

10. A system for nonlinear contact force control of a robot based on the Maxwell-fractional-order impedance model, characterized in that, The system includes an actuator for performing a nonlinear contact force control method for a robot based on the Maxwell-fractional-order impedance model as described in claims 1-9.

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