Flexible joint robot tracking control method and system

Abstract

The invention discloses a flexible joint robot tracking control method and system, and relates to the field of robot control, and the method comprises the steps: obtaining a dynamic equation of a flexible joint robot; according to the dynamic equation of the flexible joint robot, a time delay estimation method is adopted to obtain an incremental dynamic equation of a connecting rod end and an incremental dynamic equation of a motor end; determining an increment system according to the increment kinetic equation of the connecting rod end and the increment kinetic equation of the motor end; establishing a stage objective function taking a tracking error and a control variable as objects according to the incremental system and the reference trajectory information; determining an incremental model prediction controller according to the stage objective function and physical constraints for applying a rod end and a motor end; and performing tracking control on the rod end and the motor end of the flexible joint robot according to the incremental model prediction controller. According to the invention, the precision and real-time performance of tracking control of the flexible joint robot can be improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of robot control, in particular to a flexible joint robot tracking control method and system. BACKGROUND

[0002] Due to the passive compliance of the flexible actuator, the flexible joint robot is relatively safe in physics, especially in the process of human-robot interaction. Therefore, in recent years, the flexible joint robot has developed rapidly. The elastic joint torque is transmitted between the motor and the inertia of the connecting rod through the joint stiffness. However, the elasticity between the motor and the connecting rod brings a severe challenge to achieve high-precision control of the robot joint. In addition, in order to further ensure the safe cooperation between human and robot, it is necessary to ensure the high-precision control of the robot and the strict satisfaction of the constraints at the same time. In addition, when the motor constraint cannot be well solved, the oscillation of the system will be excited;

[0003] Among them, the proportional-derivative (PD) feedback controller and the backstepping (BS) are the control methods used in the early design of the flexible joint robot controller. Both methods require full state variables and specific mathematical model information of the system. Although the proportional-derivative feedback controller and the backstepping controller are easy to implement, the modeling uncertainty and external disturbance of the controlled system (flexible joint robot) will inevitably exist, which will cause the tracking control precision of the closed-loop system to be reduced.

[0004] In order to improve the robust performance of the controlled system in response to model uncertainty and external disturbance, scholars have proposed various technical methods such as H ∞ , sliding mode control, neural network, adaptive fuzzy control, etc. for the flexible joint robot. Although the accurate mathematical model of the robot is no longer selected for the controller based on the neural network, the application is still limited due to the need to use a large enough data set and numerous parameters that need to be adjusted for offline training. The time delay estimation method is used for the nonlinear system model of the flexible joint robot, in which the approximate model of the flexible joint robot is obtained mainly through the state value and input value of the flexible joint robot at the latest time. In the existing literature, the time delay estimation method is used to design the controller of the flexible joint robot, and a terminal sliding mode controller is used to enhance the robustness of the controller and realize the fast convergence of the closed-loop system. Generally, the above-mentioned robust control methods can effectively enhance the control precision. However, the high-order derivative of the state variable is generally used in the above-mentioned controller, which is easy to cause noise response in the actual process.

[0005] In order to realize the dynamic order reduction of the flexible joint robot, the singular perturbation method is applied to the controller design of the flexible joint robot. A two-layer time-scale controller is proposed in the prior art, in which the system dynamics is decomposed into fast and slow subsystems and the controllers are designed for the two subsystems independently. In the prior art, the tracking error is effectively reduced by adding a disturbance term to the slow subsystem of the flexible joint robot. However, neither the singular perturbation method nor the above-mentioned robust controller considers the input and state constraints, which are closely related to safety, especially when the robot is close to or interacts with people.

[0006] In contrast, model predictive control is attractive in the design of flexible joint robot control systems, which can achieve optimal control performance while meeting input and state constraints. Influenced by the time-scale separation idea in singular perturbation theory, a flexible joint robot control method for joint angle control is proposed in the prior art. The traditional model predictive control method relies on the accurate mathematical model of the controlled system. As a result, the control accuracy is limited by the modeling error of the controlled system. In order to enhance the robustness of the model predictive control method, high-order extended state observers, learning techniques, and data-driven techniques are used to improve the robustness of the model predictive control method, in which the mathematical model of the controlled system is identified online. Therefore, the accurate mathematical model of the controlled system is no longer required in the controller design process. However, the observer-based model predictive control method still requires the nominal model information of the controlled system. Although the learning and data-driven model predictive control method effectively reduces the dependence on the accurate mathematical model of the controlled system, learning and data-driven techniques further increase the computational complexity of model predictive control.

[0007] Therefore, it is urgent to design a robust and low-complexity model predictive control to meet the requirements of high control accuracy and real-time performance of flexible joint robot control. SUMMARY

[0008] The purpose of the present application is to provide a flexible joint robot tracking control method and system that can improve the accuracy and real-time performance of flexible joint robot tracking control.

[0009] To achieve the above purpose, the present application provides the following solutions:

[0010] In a first aspect, the present application provides a flexible joint robot tracking control method, which comprises:

[0011] obtaining the dynamics equation of the flexible joint robot;

[0012] According to the dynamics equation of the flexible joint robot, the time delay estimation method is used to obtain the incremental dynamics equation of the link end and the incremental dynamics equation of the motor end.

[0013] determine the incremental system according to the incremental dynamics equation of the link end and the incremental dynamics equation of the motor end;

[0014] establish a stage objective function taking the tracking error and the control variable as objects according to the incremental system and the reference trajectory information;

[0015] determine the incremental model predictive controller according to the stage objective function and the physical constraints for applying the link end and the motor end;

[0016] track and control the link end and the motor end of the flexible joint robot according to the incremental model predictive controller.

[0017] Optionally, the flexible joint robot dynamics equation specifically comprises:

[0018]

[0019] wherein, respectively represent the joint angle, the angular velocity and the angular acceleration, represents an n x n inertia matrix of the link, represents a real set, is a Coriolis / centrifugal force vector applied at the link end, represents a gravity term applied at the link end, represents the friction and disturbance force suffered by the link end, represents the torque due to the joint passivity, Γ = K(θ - q), represents a symmetric joint stiffness matrix, represents a diagonal motor inertia matrix with rotor and gear, represents a motor angle vector, is the motor angular acceleration, represents the torque applied at the motor end, represents the friction and disturbance force at the motor end.

[0020] Optionally, the incremental dynamics equation of the link end and the incremental dynamics equation of the motor end are obtained by using a time delay estimation method according to the flexible joint robot dynamics equation, and specifically comprise:

[0021] the incremental dynamics equation of the link end is determined by using the formula

[0022] the incremental dynamics equation of the motor end is determined by using the formula

[0023] wherein, ​​joint angle acceleration at time (t-L), L is time delay, and Δθ is motor angle increment variable Δθ = θ t -θ (t-L) , Δq is joint angle increment variable, θ t is motor angle vector at time t, θ (t-L) is motor angle vector at time (t-L), ε1 is time delay estimation error of function H t and function H (t-L) at link end at time t and (t-L), ε1 = H t -H (t-L) , q (t-L) is joint angle at time (t-L), is joint angle acceleration at time (t-L), is time delay estimation parameter of link end, represents motor angle acceleration at time (t-L), Δτ is increment control variable, Δτ = τ t -τ (t-L) , τ t is torque applied at motor end at time t, τ (t-L) is torque applied at motor end at time (t-L), ε2 is time delay estimation error of motor end, is time delay estimation parameter of motor end.

[0024] Optionally, the determining the increment system according to the increment dynamics equation of the link end and the increment dynamics equation of the motor end further comprises:

[0025] discretizing the increment system by using Euler method;

[0026] transforming the discretized increment system into a standard linear system by using defined stack vectors.

[0027] Optionally, the establishing a stage objective function taking tracking error and control variable as objects according to the increment system and the reference trajectory information comprises:

[0028] determining the stage objective function by using formula

[0029] wherein, X k+i+1|k is state prediction value of the flexible joint robot, Δu k+i|k is control variable, i is intermediate variable, Q is weight matrix, R is weight matrix, E k+i+1|k = SX k+i+1|k -q d (k+i+1), matrix S = [I 0 0 0], I is unit matrix, k is current time value, E k+i+1|kTo track the error, || || Q and || || R Let q be the mean weight norm. d The tracking trajectory signal is used as a reference.

[0030] Optionally, determining the incremental model predictive controller based on the stage objective function and the physical constraints applied to the rod end and motor end specifically includes:

[0031] The constrained optimization control problem is constructed based on the phase objective function and the physical constraints applied to the rod end and the motor end.

[0032] Based on the constrained optimization control problem, an incremental model predictive controller is determined.

[0033] Optionally, the constrained optimization control problem specifically includes the following formula:

[0034]

[0035] stX k+i+1|k =AX k+i|k +BΔu k+i|k ;

[0036]

[0037] in, This is the optimal control sequence. Here, N represents the feasible control sequence, N is the prediction time domain, i and j are intermediate variables, and X is... k+i|k Let A and B be the state variables of a flexible joint robot, where A and B are both weight matrices of a standard linear system, and q k+i+1|k and For the standard linearized joint angles and angular velocities, θ k+i+1|k and Here, τ(k) represents the motor angle and angular velocity after standard linearization, and τ(k) represents the torque applied to the motor terminals after standard linearization. k+j|k X1 is the allowable set of joint angles, X2 is the allowable set of joint angular velocities, and U is a compact set containing the origin.

[0038] Optionally, determining the incremental model predictive controller based on the constrained optimization control problem specifically includes:

[0039] Transform the constrained optimization control problem into a quadratic programming problem;

[0040] The quadratic programming problem is solved using a quadratic programming solver.

[0041] Optionally, the quadratic programming solver includes: the qpOASES solver.

[0042] In a second aspect, the present application provides a flexible joint robot tracking control system, comprising:

[0043] a dynamic equation acquisition module, configured to acquire a flexible joint robot dynamics equation;

[0044] an incremental dynamics equation determination module, configured to acquire, according to the flexible joint robot dynamics equation, an incremental dynamics equation of a link end and an incremental dynamics equation of a motor end by using a time delay estimation method;

[0045] an incremental system determination module, configured to determine an incremental system according to the incremental dynamics equation of the link end and the incremental dynamics equation of the motor end;

[0046] a stage target function establishment module, configured to establish a stage target function taking a tracking error and a control variable as objects according to the incremental system and reference trajectory information;

[0047] an incremental model predictive controller construction module, configured to determine an incremental model predictive controller according to the stage target function and physical constraints for imposing the link end and the motor end;

[0048] a tracking control module, configured to perform tracking control on the link end and the motor end of the flexible joint robot according to the incremental model predictive controller.

[0049] According to the specific embodiments provided by the present application, the present application has the following technical effects:

[0050] The present application provides a flexible joint robot tracking control method and system, which approximates the flexible joint robot dynamics equation by using a time delay estimation method without using model parameters according to the flexible joint robot dynamics equation, acquires an incremental dynamics equation of a link end and an incremental dynamics equation of a motor end, establishes a stage target function taking a tracking error and a control variable as objects according to an incremental system and reference trajectory information, and considers physical constraints for imposing the link end and the motor end to construct a constraint optimization control problem, and then constructs an incremental model predictive controller, and then realizes real-time control of the flexible joint robot under a high control frequency condition through the incremental model predictive controller. BRIEF DESCRIPTION OF DRAWINGS

[0051] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed in the embodiments will be briefly introduced as follows. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor on the basis of these drawings.

[0052] Figure 1A flowchart of a flexible joint robot tracking control method according to an embodiment of the present application;

[0053] Figure 2 A control structure and signal flow diagram of a flexible joint robot incremental model predictive controller according to an embodiment of the present application;

[0054] Figure 3 An experimental setup diagram of a DLRSofty flexible joint robot according to an embodiment of the present application;

[0055] Figure 4 Tracking error diagrams of TDSMC, PD and IMPC according to an embodiment of the present application;

[0056] Figure 5 Control signal diagrams of TDSMC, PD and IMPC according to an embodiment of the present application;

[0057] Figure 6 A calculation time diagram of IMPC in Scheme 1 according to an embodiment of the present application;

[0058] Figure 7 A tracking control effect diagram of IMPC in Scheme 2 Case 1 with strict torque constraint according to an embodiment of the present application (wherein "b1" and "b2" represent the minimum and maximum torque values, and "S2-C1" represents Scheme 2 Case 1);

[0059] Figure 8 A tracking control effect diagram of IMPC in Scheme 2 Case 1 with strict joint angular velocity constraint according to an embodiment of the present application (wherein "b1" and "b2" represent the minimum and maximum joint angular velocity values, "S1" represents Scheme 1, and "S2-C2" represents Scheme 2 Case 2, and the gray background describes when the robot arm velocity constraint is limited). DETAILED DESCRIPTION

[0060] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the present application.

[0061] The above purposes, features and advantages of the present application will be more apparent and easy to understand. The present application will be further described in detail below with reference to the drawings and specific embodiments.

[0062] In one exemplary embodiment, as shown in Figure 1 a flexible joint robot tracking control method is provided, which comprises the following S101 to S106. Wherein:

[0063] S101, Obtain the dynamic equations of the flexible joint robot;

[0064] The dynamic equations of the flexible joint robot specifically include:

[0065]

[0066] in, These represent joint angle, angular velocity, and angular acceleration, respectively. This represents the n×n inertia matrix of the link. Represents the set of real numbers. The Coriolis / centrifugal force vector applied to the end of the connecting rod. This represents the gravity term applied to the end of the connecting rod. This represents the frictional and interference forces experienced at the end of the connecting rod. The torque generated due to the passive nature of the joint is represented by Γ = K(θ - q). Represents the symmetric joint stiffness matrix. This represents the inertia matrix of a diagonal motor with a rotor and gears. Represents the motor angle vector. For the angular acceleration of the motor, This indicates the torque applied at the motor end. This represents the friction and interference forces at the motor end.

[0067] Due to modeling errors, the M(q), D, and... parameters in the dynamic equations of flexible joint robots are affected. w1 and w m It is assumed to be an unknown / uncertain function. Furthermore, for flexible joint robots, the following properties are satisfied:

[0068] Positive definite inertia matrices: The unknown link and motor inertia matrices M(q) and D satisfy the consistent positive definiteness, that is, there exist positive numbers m1, m2, n1, and n2 such that for every eigenvalue (λ) of the inertia matrices M(q) and D... i (M(q)) and λ i (D) represents this, where i∈I [1,n] I [1，n] (where the integers are positive integers) satisfying the inequality m1≤λ i (M(q))≤m2 and n1≤λ i (D)≤n2.

[0069] The parameters m1, m2, n1, and n2 indicate that the inertia matrices M(q) and D are uniformly positive definite, and their precise values ​​are not required when designing the controller. The existence of constants in the positive definite inertia matrices is the theoretical basis for determining the parameters of the incremental model predictive controller, i.e., the delay estimation parameters. and delay estimation parameters

[0070] S102, according to the flexible joint robot dynamics equation, using time delay estimation method, get the incremental dynamics equation of the link end and the incremental dynamics equation of the motor end;

[0071] In order to deal with modeling error and eliminate the dependence on accurate mathematical model of dynamics equation, time delay estimation method is used to approximate the uncertain dynamics equation of the system. S102 mainly contains two steps:

[0072] Step 1: separation. Introducing a time delay estimation parameter The uncertain dynamics equation of the link end (1) can be decomposed into known and unknown two parts.

[0073]

[0074] In the formula, the function H is the term containing all uncertain and unmodeled dynamics equation.

[0075] Step 2: approximation. The value of function H at t time can be approximated by its value at (t-L) time:

[0076]

[0077] Then, the passive torque Γ of the joint is regarded as the intermediate control variable, and the incremental dynamics equation of the link end can be obtained by combining equation (2) and equation (3):

[0078]

[0079] Similarly, introducing a time delay estimation parameter The dynamics equation of the motor end Can be transformed into the following form:

[0080]

[0081] Where, Indicates the joint angular acceleration at (t-L) time, L is the time delay time, Δθ is the motor angle incremental variable Δθ = θ t -θ (t-L) , Δq is the joint angle incremental variable, θ t Is the motor angle vector at t time, θ (t-L) Is the motor angle vector at (t-L) time, ε1 is the time delay estimation error of function H t At t time and function H (t-L) At (t-L) time of the link end, ε1 = H t -H (t-L) , q (t-L)joint angle at time (t-L), joint angle acceleration at time (t-L), time delay estimation parameter at the link end, represents the motor angle acceleration at time (t-L), Δτ is the incremental control variable, Δτ=τ t -τ (t-L) , τ t is the torque applied at the motor end at time t, τ (t-L) is the torque applied at the motor end at time (t-L), ε2 is the time delay estimation error at the motor end, time delay estimation parameter at the motor end.

[0082] The dynamics equations of the link end and the motor end are approximated using the time delay estimation method, without the need of specific mathematical models, only using the latest state measurement values of the system.

[0083] S103, according to the incremental dynamics equation of the link end and the incremental dynamics equation of the motor end, determine the incremental system;

[0084] According to the design of the previous time delay estimation method, the sampling period T S is usually selected as the delay time L, to ensure that the delay time is small enough. In practice, when the sampling rate is faster than 30 times the system bandwidth, the digital system is regarded as a continuous system. Therefore, according to the function continuity property, when a small enough delay time L is selected, the time delay estimation errors ε1 and ε2 can be ignored. When ε1=0 and ε2=0, the nominal incremental system of the dynamics equation (1) can be obtained:

[0085]

[0086] In order to determine the incremental model predictive controller which is linear and can robustly approximate the nonlinear function of the system containing uncertainties and modeling errors, the incremental system (6) is discretized using the Euler method, and then a discrete system suitable for executing the model predictive controller is generated. Since the sampling time of the system is small enough, the discretization error can be ignored. Therefore, the following linear system can be obtained:

[0087] x(k+1)=A1x(k)+A2x(k-1)+B1Δu(k)(7)

[0088] wherein, col is an operator for splicing multiple vectors into a vector.

[0089] matrix

[0090] matrix matrix

[0091] Define a stacked vector X(k) = col(x(k), x(k-1)), equation (7) can be transformed into the standard linear system as follows:

[0092] X(k+1) = Ax(k) + BΔu(k) (8)

[0093] where, In the following model predictive controller design process, the incremental system (8) will be used to generate the state predictions in the prediction horizon.

[0094] To verify the accuracy of the time delay estimation, the following comparison with the nominal system is made; only the latest system state measurement value is used to derive the incremental system for the system dynamics equation approximation. However, due to the inevitable time delay estimation error, there is a certain difference between the actual nonlinear dynamics equation and its incremental approximation. In this section, the analysis will prove that the derived incremental system has higher approximation accuracy than the nominal system.

[0095] Take the dynamics equation of the link end as an example, by comparing with the nominal system, the approximation accuracy of the incremental system will be analyzed. If the nominal inertia matrix, Coriolis / centrifugal vector and friction vector are represented by M n ,C n ,G n and w 1,n , respectively, the system nominal dynamics equation can be obtained according to equation (1) as follows:

[0096]

[0097] where q n is the approximation of the joint angle under the nominal system condition. In order to obtain the system dynamics equation approximation error, equation (1) can be transformed into the following form:

[0098]

[0099] where, is the modeling error of the inertia matrix M. Combining equations (9) and (10), the approximation error of the system nominal dynamics equation can be obtained as follows:

[0100]

[0101] where, and are the modeling errors of C, G, w1, respectively.

[0102] The approximation error of the incremental system will be analyzed. Combining equation (4) and equation (5), the approximation error δ of the incremental system is directly related to the time delay estimation error ε1, i.e.:

[0103]

[0104] where ε d = ε c + ε g + ε l . ε c , ε g and ε l denote the time delay estimation errors of C, G and w1, respectively.

[0105] Comparing equation (11) and equation (12), the main difference between the system dynamics approximation error of the delta system and the corresponding modeling error of the nominal system is the inertia modeling error. For the delta system, if the sampling period is small enough, the inertia modeling error can be effectively compensated by its time delay value. However, for the nominal system, the inertia modeling error cannot be compensated. In addition, when a sufficiently high sampling frequency is used, ε d can be small enough. Therefore, when the sampling period is small enough, the delta system can exhibit higher approximation accuracy.

[0106] To ensure the higher approximation accuracy of the delta system, a sufficiently small sampling period is usually required. However, a too high sampling frequency will increase the computational burden, making it difficult to ensure the real-time performance of the control. However, there is no systematic method to determine the allowable sampling period for the time delay estimation method. The existing technology has verified that when the sampling period of the robot system is selected as 1ms and 2ms, the time delay estimation method can exhibit very high approximation accuracy. Therefore, the sampling period of the flexible joint robot considered in this chapter is selected as 1ms.

[0107] S104, according to the delta system and the reference trajectory information, a stage objective function taking the tracking error and the control variable as objects is established;

[0108] The control objective in this application is to enable the flexible joint robot to accurately track the given reference trajectory in the joint space. In order to ensure the safety of the flexible joint robot, it is required not to violate the physical constraints of the flexible joint robot, such as joint position, velocity and torque, etc., that is, the following constraints are imposed on the state and input of the flexible joint robot:

[0109]

[0110] where X i (i∈I [1,4]) and U are both compact sets containing the origin, X1 is the admissible set of joint angles, X2 is the admissible set of joint angular velocities, X3 is the admissible set of motor rotation angles, X4 is the admissible set of motor rotation angular velocities, and X5 is the admissible set of motor torques. For a selected flexible joint robot, the following constraints are imposed on the link / motor positions, velocities, and torques:

[0111] q min ≤q≤q max ,

[0112] θ min ≤θ≤θ max ,

[0113] τ min ≤τ≤τ max ;

[0114] where ·min and ·max are the lower and upper bounds.

[0115] The reference trajectory information q d satisfies the following conditions:

[0116] The reference trajectory q is bounded and smooth: there exist bounded constants such that the reference trajectory q d satisfies the inequality

[0117] Since a non-smooth reference trajectory will cause sharp changes in the actuators, which will cause great damage to the mechanical structure. Therefore, in actual processes, a bounded and smooth reference trajectory is usually given.

[0118] The stage target function is determined by the formula

[0119] where X k+i+1|k k+i is the state prediction value of the flexible joint robot, Δu k+i|k is the control variable, i∈I is the intermediate variable, Q is the weight matrix, R is the weight matrix, E k+i+1| k=SX k+i+1| k-q d (k+i+1), the matrix S=[I 0 0 0], I is the unit matrix, k is the current time value, E k+i+1|k is the tracking error, || || Q is the weight norm, || || R is the weight norm, q d is the reference tracking trajectory signal.

[0120] S105, as Figure 2 ​As shown, the incremental model predictive controller is determined based on the stage objective function and the physical constraints applied to the rod end and motor end.

[0121] S105 specifically includes:

[0122] S51, construct a constrained optimization control problem based on the stage objective function and the physical constraints used to apply to the rod end and motor end;

[0123] The constrained optimization control problem specifically includes the following formulas:

[0124]

[0125] stX k+i+1|k =AX k+i|k +BΔu k+i|k ;

[0126] q k+i+1|k ∈X1,

[0127] θ k+i+1|k ∈X1,

[0128]

[0129] in, This is the optimal control sequence. This is a feasible control sequence. N represents the prediction time domain, i and j are intermediate variables, and X... k+i|k Let A and B be the state variables of a flexible joint robot, where A and B are both weight matrices of a standard linear system, and q k+i+1|k and For the standard linearized joint angles and angular velocities, θ k+i+1|k and Here, τ(k) represents the motor angle and angular velocity after standard linearization, and τ(k) represents the torque applied to the motor terminals after standard linearization. k+j|k X1 is the allowable set of joint angles, X2 is the allowable set of joint angular velocities, and U is a compact set containing the origin.

[0130] This is the optimal control sequence. The first column A new control law is generated based on the current control variable u(k) and applied to the controlled system, that is,

[0131] S52, Based on the constrained optimization control problem, determine the incremental model predictive controller.

[0132] The specific components of S52 include:

[0133] transforming the constrained optimization control problem into a quadratic programming problem;

[0134] solving the quadratic programming problem by using a quadratic programming solver to solve the optimization control problem, so as to achieve the purpose of real-time control of the controlled system under the condition of high control frequency.

[0135] The quadratic programming solver includes but is not limited to a qpOASES solver.

[0136] S106, tracking control is performed on the rod end and motor end of the flexible joint robot according to the incremental model predictive controller.

[0137] In order to verify the effectiveness of the proposed method, a series of real-time hardware experiments are carried out on the flexible joint robot. The experimental results verify that the proposed incremental model predictive controller has high control accuracy, can achieve optimal performance, and can meet the physical constraints of the robot input and state.

[0138] As shown in Figure 3 , the Softy flexible joint robot will be used for experimental verification in this experiment. Softy uses the lightweight robot III drive unit of the German Aerospace Center (DLR). In Softy, the joint angle position q and the motor angle position θ can be directly measured, and the corresponding angular velocity and angular acceleration are calculated by using a first-order digital differentiator according to the angle position measurement value.

[0139] For the considered flexible joint robot, the system state sampling period is T S = 1 ms, and the fixed joint stiffness matrix K = 362 Nm / rad. In addition, the parameters of the proposed incremental model predictive control method are selected as follows: the control parameters and the weighting matrix Q = diag{3000, 400, 300, 40}, R = 1, and the prediction horizon N = 10. The constrained optimization control problem is solved by the active set quadratic programming solver qpOASES. In the experiment, the reference signal for angle tracking is a sine curve with an amplitude of 0.15 rad and a frequency of 0.2π Hz.

[0140] Scheme 1

[0141] In order to verify the high accuracy and optimal performance of the incremental model predictive control, a comparative experiment will be carried out with a proportional-derivative controller and a time-delay sliding mode controller. For convenience, they are referred to as PD, TDSMC and IMPC respectively. For the PD algorithm, the outer loop control frequency w n = 14 rad / s, and the damping rate ζ = 0.7. For TDSMC, the design parameters K P = 100, KD = 20, the time delay estimation parameter is the same as IMPC. For IMPC, the angle, angular velocity and torque value constraints of the bar / motor are set as 20 deg, 20 deg / s and 20 N m, respectively. The high control accuracy and optimality of the controller are verified by checking the tracking error and control signal. In addition, the computational efficiency of IMPC is verified by measuring the computation time of each iteration period.

[0142] The experimental results of Scheme 1 are shown in Figures 4-6 The root mean square error of the angle position tracking of the PD controller designed using the dynamic model of the robot arm is 78.91 x 10 -2 deg, which is the highest error among the three controllers. For TDSMC and IMPC, the tracking errors are 3.09 x 10 -2 deg and 5.95 x 10 -2 deg, respectively.

[0143] Although the control accuracy of TDSMC is the highest among the three controllers, the incremental model predictive control method proposed in the present application still exhibits excellent control performance in terms of optimality, mainly because the torque curve generated by IMPC is smoother than that of TDSMC, as shown in Figure 5 This is mainly because in TDSMC, the time delay estimation method combined with terminal sliding mode control uses high-order derivatives of the joint angle position to calculate the control torque for the sole purpose of reducing error. In contrast, the proposed IMPC not only considers tracking control accuracy but also considers control efficiency as shown in the constrained optimization control problem. In addition, the proposed IMPC no longer requires high-order derivatives of the joint position state, making the noise response generated by the control smaller.

[0144] In Figure 6 , it can be observed that the computation time of IMPC is significantly less than the sampling time (1 ms), thereby verifying the real-time performance of IMPC control in actual work.

[0145] Scheme 2

[0146] More stringent input and state constraints are applied to verify that the proposed IMPC can guarantee strict satisfaction of the constraints. Two cases will be considered, in which the limit values of the input and output state constraints are different. As shown below: the position, velocity and input torque constraint values are 20 deg, 20 deg / s and 12 Nm, respectively.

[0147] Case 1: 20 deg, 20 deg / s, and 12 Nm;

[0148] Case 2: 20 deg, 4 deg / s, and 20 Nm;

[0149] The experimental results of Case 1 are shown in Figure 7 FromFigure 7 It can be observed that although the tracking error is slightly larger than that in Scheme 1, the input constraints can be satisfied and the system can still remain stable.

[0150] The experimental results for scenario 2 are as follows Figure 8 As shown. From Figure 8 As can be seen from the results, the proposed incremental model predictive controller can satisfy the joint angular velocity constraints. However, due to the neglect of time delay estimation errors and measurement resolution biases during the IMPC design process, a certain state prediction error exists. As a result, the state constraints cannot be strictly satisfied. During real-time hardware experiments, it can be observed that the joint angular velocity constraints are difficult to satisfy, mainly because the joint angular velocities are obtained through numerical calculations, which inevitably introduce some bias. Although there are constraint violations near the constraint boundaries, the proposed IMPC still possesses a significant ability to regulate the state, and the closed-loop system remains stable.

[0151] This application has the following effects:

[0152] 1) Improving Controller Robustness and Computational Efficiency: The dynamic equations of the flexible joint robot are approximated using a time delay estimation method without employing model parameters, and the linear incremental equations of the system are obtained through discretization. Therefore, by combining a quadratic cost function to achieve reference trajectory tracking, a linear incremental model predictive controller capable of robustly approximating the nonlinear functions of a system containing uncertainties and modeling errors is designed. Traditional nonlinear model predictive control schemes typically require model parameter identification for the complex dynamic equations of the flexible joint robot system, and the algorithm computational complexity is high. In contrast, the incremental model predictive control proposed in this application significantly improves computational efficiency. This application employs a computationally efficient quadratic programming solver and implements real-time hardware experiments at a control frequency of 1kHz.

[0153] 2) Considering physical constraints: Due to the underactuated nature of the linkage and motor dynamics, it is difficult to achieve satisfactory control performance while ensuring that physical constraints related to operational safety, such as torque and state constraints, are met. Treating the passive torque of the joint as an intermediate control variable, an incremental model predictive controller was designed for the flexible joint robot system, where all physical constraints are treated as inequality constraints.

[0154] 3) Avoiding high-order derivatives of system states: Existing control algorithms for flexible-joint robots usually require high-order derivatives of states, as can be seen in traditional feedback linearization-based methods and more modern methods (such as elastic holding structure control and TDE-based methods, etc.). In order to achieve an optimal tracking control effect, the joint reference trajectory should be sufficiently smooth. However, these high-order derivative signals are extremely susceptible to significant influences of measurement noise, and high-resolution low-noise sensors need to be used for state measurement. Nevertheless, oscillation responses at high frequencies are still unavoidable, which will cause mechanical noise and also shorten the service life of the machine. Therefore, the proposed method aims to calculate the control signals by solving a constrained constrained optimization control problem, thereby avoiding the necessity of high-order derivatives, and effectively reducing the negative effects of noise responses.

[0155] Based on the same inventive concept, the embodiments of the present application also provide a flexible-joint robot tracking control system for implementing the above-mentioned flexible-joint robot tracking control method. The implementation scheme for solving the problem provided by the system is similar to the implementation scheme described in the above method, and therefore the specific limitations in one or more flexible-joint robot tracking control system embodiments provided below can refer to the limitations of the flexible-joint robot tracking control method described above, which will not be described here again.

[0156] In an exemplary embodiment, a flexible-joint robot tracking control system is provided, comprising:

[0157] a dynamic equation acquisition module configured to acquire a flexible-joint robot dynamics equation;

[0158] an incremental dynamics equation determination module configured to obtain an incremental dynamics equation of a link end and an incremental dynamics equation of a motor end by using a time delay estimation method according to the flexible-joint robot dynamics equation;

[0159] an incremental system determination module configured to determine an incremental system according to the incremental dynamics equation of the link end and the incremental dynamics equation of the motor end;

[0160] a stage objective function establishment module configured to establish a stage objective function taking a tracking error and a control variable as objects according to the incremental system and reference trajectory information;

[0161] an incremental model predictive controller construction module configured to determine an incremental model predictive controller according to the stage objective function and physical constraints for applying to the link end and the motor end;

[0162] a tracking control module configured to perform tracking control on the link end and the motor end of the flexible-joint robot according to the incremental model predictive controller.

[0163] In an example embodiment, a computer device is provided, which can be a server or a terminal. The computer device comprises a processor, a memory, an input / output interface (I / O) and a communication interface. The processor, the memory and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. The processor of the computer device is configured to provide computing and control capabilities. The memory of the computer device comprises a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for running the operating system and the computer program in the non-volatile storage medium. The input / output interface of the computer device is configured to exchange information between the processor and external devices. The communication interface of the computer device is configured to communicate with external terminals through a network connection. The computer program is executed by the processor to implement a flexible joint robot tracking control method.

[0164] In an example embodiment, a computer readable storage medium is provided, which stores a computer program. The computer program is executed by a processor to implement the steps in the above method embodiments.

[0165] In an example embodiment, a computer program product is provided, which comprises a computer program. The computer program is executed by a processor to implement the steps in the above method embodiments.

[0166] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data for analysis, stored data, displayed data, etc.) involved in the present application are all information and data authorized by the user or authorized by all parties, and the collection, use and processing of the relevant data need to comply with relevant regulations.

[0167] Those skilled in the art can understand that all or part of the processes in the above-mentioned embodiment methods can be completed by instructing the relevant hardware through a computer program. The computer program can be stored in a non-volatile computer readable storage medium, and when the computer program is executed, the processes of the above-mentioned embodiments of the methods can be included. Any reference to memory, database or other medium used in the embodiments provided in the present application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (Read-Only Memory, ROM), magnetic tape, floppy disk, flash memory, optical storage, high-density embedded non-volatile memory, resistive memory (ReRAM), magnetoresistive random access memory (Magnetoresistive Random Access Memory, MRAM), ferroelectric memory (Ferroelectric Random Access Memory, FRAM), phase change memory (Phase Change Memory, PCM), graphene memory, etc. Volatile memory can include random access memory (Random Access Memory, RAM) or external cache memory, etc. As an illustration but not limitation, RAM can be in various forms, such as static random access memory (Static Random Access Memory, SRAM) or dynamic random access memory (Dynamic Random Access Memory, DRAM), etc.

[0168] The database involved in the embodiments provided in the present application can include at least one of a relational database and a non-relational database. The non-relational database can include a distributed database based on a blockchain, etc., without being limited thereto. The processor involved in the embodiments provided in the present application can be a general-purpose processor, a central processing unit, a graphics processing unit, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., without being limited thereto.

[0169] In the present application, all actions of obtaining signals, information or data are performed in accordance with the corresponding data protection regulations and policies of the country where the device is located, and with the authorization of the owner of the corresponding device.

[0170] The technical features of the above embodiments can be combined arbitrarily. In order to make the description simple, all possible combinations of the technical features in the above embodiments are not described, but as long as the combinations of the technical features do not exist, they should be considered as the scope of the present application.

[0171] The principles and implementation manners of the present application are described herein by using specific examples, and the above examples are only used to help understand the method of the present application and its core idea; meanwhile, for those skilled in the art, according to the idea of the present application, the specific implementation manners and application ranges will have changes. In conclusion, the content of the specification should not be understood as a limitation of the present application.

Claims

1. A tracking control method for a flexible joint robot, characterized in that, The flexible joint robot tracking and control method includes: Obtain the dynamic equations of the flexible joint robot; Based on the dynamic equations of the flexible joint robot, the incremental dynamic equations at the link end and the motor end are obtained by using the time delay estimation method. The incremental system is determined based on the incremental dynamic equations at the connecting rod end and the motor end. Establish a stage objective function based on the incremental system and reference trajectory information, taking tracking error and control variables as the objects; The incremental model predictive controller is determined based on the stage objective function and the physical constraints applied to the rod end and motor end. The incremental model predictive controller performs tracking control on the lever end and motor end of the flexible joint robot.

2. The flexible joint robot tracking control method according to claim 1, characterized in that, The dynamic equations of the flexible joint robot specifically include: in, These represent joint angle, angular velocity, and angular acceleration, respectively. This represents the n×n inertia matrix of the link. Represents the set of real numbers. The Coriolis / centrifugal force vector applied to the end of the connecting rod. This represents the gravity term applied to the end of the connecting rod. This represents the frictional and interference forces experienced at the end of the connecting rod. The torque generated due to the passive nature of the joint is represented by Γ = K(θ - q). Represents the symmetric joint stiffness matrix. This represents the inertia matrix of a diagonal motor with a rotor and gears. Represents the motor angle vector. For the angular acceleration of the motor, This indicates the torque applied at the motor end. This represents the friction and interference forces at the motor end.

3. The flexible joint robot tracking control method according to claim 2, characterized in that, The incremental dynamic equations for the link end and the motor end are obtained by using a time delay estimation method based on the dynamic equations of the flexible joint robot, specifically including: Using formula Determine the incremental dynamic equations at the connecting rod end; Using formula Determine the incremental dynamic equations at the motor end; in, This represents the joint angular acceleration at time (tL), where L is the time delay, and Δθ is the motor angle increment variable Δθ = θ. t -θ (t-L) Δq is the joint angle increment variable, θ t Let θ be the motor angle vector at time t. (t-L) Let ε1 be the motor angle vector at time (tL), and H be the function of the connecting rod end at time t. t The function H at time (tL) (t-L) The time delay estimation error, ε1=H t -H (t-L) , q (t-L) Let (tL) be the joint angle at time (tL). Let (tL) be the joint angular acceleration. The time delay estimation parameters are for the link end. Let Δτ represent the motor angular acceleration at time (tL), where Δτ is the incremental control variable, and Δτ = τ. t -τ (t-L) , τ t Let τ be the torque applied to the motor terminals at time t. (t-L) Let εL be the torque applied to the motor terminals at time (tL), and ε2 be the time delay estimation error at the motor terminals. These are the time delay estimation parameters for the motor.

4. The flexible joint robot tracking control method according to claim 1, characterized in that, The incremental system is determined based on the incremental dynamic equations at the connecting rod end and the motor end, and then the process further includes: The incremental system is discretized using the Euler method; Using the defined stack vector, the discretized incremental system is transformed into a standard linear system.

5. The flexible joint robot tracking control method according to claim 4, characterized in that, The establishment of a stage objective function based on the incremental system and reference trajectory information, taking tracking error and control variables as objects, specifically includes: Using formula Determine the stage objective function Among them, X k+i+1|k Δu is the predicted state value of the flexible joint robot. k+i|k Let i be the control variable, i be the intermediate variable, Q be the weight matrix, R be the weight matrix, and E be the weight matrix. k+i+1| k = SX k+i+1| kq d (k+i+1), matrix S = [I 0 0 0], where I is the identity matrix, k is the current value, and E k+i+1|k For tracking error, || || Q and || || R Let q be the mean weight norm. d The tracking trajectory signal is used as a reference.

6. The flexible joint robot tracking control method according to claim 5, characterized in that, The determination of the incremental model predictive controller based on the stage objective function and the physical constraints applied to the rod end and motor end specifically includes: The constrained optimization control problem is constructed based on the phase objective function and the physical constraints applied to the rod end and motor end. Based on the constrained optimization control problem, an incremental model predictive controller is determined.

7. The flexible joint robot tracking control method according to claim 6, characterized in that, The constrained optimization control problem specifically includes the following formulas: s.t.X k+i+1|k =AX k+i|k +BΔu k+i|k ; q k+i+1|k ∈X1, i k+i+1|k ∈X1, in, This is the optimal control sequence. Here, N represents the feasible control sequence, N is the prediction time domain, i and j are intermediate variables, and X is... k+i|k Let A and B be the state variables of a flexible articulated robot, where A and B are both weight matrices of a standard linear system, and q k+i+1|k and For the standard linearized joint angles and angular velocities, θ k+i+1|k and Here, τ(k) represents the motor angle and angular velocity after standard linearization, and τ(k) represents the torque applied to the motor terminals after standard linearization. k+j|k X1 is the allowable set of joint angles, X2 is the allowable set of joint angular velocities, and U is a compact set containing the origin.

8. The flexible joint robot tracking control method according to claim 6, characterized in that, The step of determining the incremental model predictive controller based on the constrained optimization control problem specifically includes: Transform the constrained optimization control problem into a quadratic programming problem; The quadratic programming problem is solved using a quadratic programming solver.

9. The flexible joint robot tracking control method according to claim 8, characterized in that, The quadratic programming solver includes the qpOASES solver.

10. A tracking and control system for a flexible joint robot, characterized in that, The flexible joint robot tracking and control system includes: The dynamic equation acquisition module is used to acquire the dynamic equations of the flexible joint robot; The incremental dynamics equation determination module is used to obtain the incremental dynamics equations at the link end and the motor end based on the dynamics equations of the flexible joint robot and using the time delay estimation method. The incremental system determination module is used to determine the incremental system based on the incremental dynamic equations at the link end and the motor end. The phase objective function establishment module is used to establish a phase objective function based on the incremental system and reference trajectory information, taking the tracking error and control variables as objects. The incremental model predictive controller building module is used to determine the incremental model predictive controller based on the stage objective function and the physical constraints applied to the rod end and motor end; The tracking control module is used to perform tracking control on the lever end and motor end of the flexible joint robot based on the incremental model predictive controller.

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