A time-convergence-based method for preventing over-deformation grip control

Abstract

A time-convergence-based gripping control method to prevent over-deformation addresses the problem of multi-degree-of-freedom manipulators having numerous links and elastic elements, making it difficult for the manipulator to achieve the desired behavior within a specified time. Blindly increasing gain can easily lead to excessive deformation and damage to the object. This invention combines stiffness information of the manipulator when gripping an object with a finite-time convergence control method. First, a simplified dynamic model and Jacobian matrix of the underactuated manipulator are established. Second, convergence functions of the controller on and outside the sliding surface are designed based on Lyapunov functions. Then, the allowable deformation is determined based on the stiffness of the object, and the convergence target is determined based on the overall stiffness of the manipulator and the object. Finally, the overall convergence time of the gripping force is determined based on the stiffness of the grasped object, and convergence parameters are calculated based on the convergence target and convergence time. This method simulates the characteristics of human hand gripping, achieving gripping control of the manipulator within a specified time convergence without over-deformation.

Description

Technical Field

[0001] This invention relates to a gripping control method that prevents over-deformation by converging at a specified time, which can be used by a robotic arm to quickly grip an object within a specified time while preventing over-deformation. Background Technology

[0002] With the development of robotics technology, multi-degree-of-freedom (DOF) robotic hands have been designed to replace single-DOF robotic hands and better meet the needs of dexterity manipulation. However, multi-DOF robotic hands also bring more joints and elastic elements, resulting in a more complex structure and lower overall stiffness. The response of the robotic hand to control commands is also slower. Furthermore, due to the limitations of the robotic hand controller's computing resources, existing robotic hand control methods can only achieve asymptotic stability of the system, making it difficult for the robotic hand to perform the required behavior within a specified time. This leads to significant differences in gripping time when the robotic hand grasps different objects. In this situation, blindly increasing the controller gain to reduce the gripping time can easily result in excessive gripping force or impact that damages fragile objects.

[0003] Finite-time convergence (FTC) algorithms offer a solution to this problem. Originally proposed for first-order systems, this method has been extended to second-order systems. However, the large number of links and closed-chain structure in robotic arms lead to complex dynamic models that require simplification. FTC algorithms, based on these dynamic models, are limited by computational resources, making direct application to robotic arm controllers difficult. Furthermore, current model-based FTC algorithms are primarily used for position control. When the object has high stiffness, directly applying FTC to a robotic arm can result in excessive gripping force to track a target, potentially damaging the object. Even with the addition of force sensors and variable impedance control, the mismatch between gripping force and stiffness—dependent on the operator's intent rather than the object's stiffness—can still damage fragile objects. Therefore, using linearization methods based on the object's true stiffness to achieve time-based convergence control of the robotic arm not only improves its response speed and operational efficiency but also protects fragile objects from damage, further expanding the range of objects and application scenarios that robotic arms can grasp. Summary of the Invention

[0004] The purpose of this invention is to propose a specified-time convergence method for preventing over-deformation gripping. Based on a specified-time convergence algorithm, it combines object stiffness to achieve rapid gripping and anti-deformation control. First, a dynamic model and Jacobian matrix of the manipulator are established, and a simplified kinematic model is obtained by linearizing the contact point. Second, convergence functions of the controller on and outside the sliding surface are designed, and the convergence time is calculated. Then, the deformation of the object is calculated based on its stiffness, and the convergence target is determined. Finally, the overall convergence time of the gripping force is determined based on the stiffness of the gripped object, and the convergence parameters are calculated based on the convergence target and convergence time.

[0005] The technical solution adopted by this invention to solve its technical problem is:

[0006] A method for preventing over-deformation grip control with convergence at a specified time includes the following steps:

[0007] Step 1: The dynamics of the manipulator are established using the Lagrange method, and the Jacobian matrix is ​​established in combination with the contact model. A simplified kinematic and dynamic model is obtained by linearizing the contact points.

[0008] Step 2: On the sliding surface, design the convergence function of the controller based on the Lyapunov function and stability conditions, and calculate the convergence time by solving the differential equation;

[0009] Step 3: Outside the sliding surface, design the control law in conjunction with the dynamic model. By combining the system equations, control law and sliding surface derivatives and substituting them into the Lyapunov equation, the convergence time and convergence function that satisfy Lyapunov stability are obtained.

[0010] Step 4: Calculate the deformation of the object based on its true stiffness, then determine the expected gripping force based on the expected deformation, and determine the convergence target based on the model of the prosthetic hand gripping.

[0011] Step 5: Determine the overall convergence time of the gripping force based on the stiffness of the object being gripped, and calculate the convergence parameters based on the convergence target and the convergence time. Attached Figure Description

[0012] Figure 1 This is a schematic diagram of the grip control algorithm of the present invention.

[0013] Figure 2 This is a simplified structural diagram of the object to be implemented in this invention.

[0014] Figure 3 This is a comparison chart of the control effects of the present invention under different convergence times.

[0015] Figure 4 This is a comparison chart of the convergence performance of the present invention and other methods.

[0016] Figure 5 This invention demonstrates the control effect when gripping an object with a stiffness of 0.379 N / mm.

[0017] Figure 6 This invention demonstrates the control effect when gripping an object with a stiffness of 0.832 N / mm.

[0018] Figure 7 This invention demonstrates the control effect when gripping an object with a stiffness of 1.246 N / mm.

[0019] Figure 8 This invention demonstrates the control effect when gripping an object with a stiffness of 3.589 N / mm. Detailed Implementation

[0020] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0021] like Figure 1 The specific steps of the over-deformation grip control method with specified time convergence are as follows:

[0022] Step 1, with Figure 2 Taking a multi-link robot as an example, this paper explains how to obtain the kinematic and dynamic models of the robot in this method. Based on the uniquely determined triangle ABE:

[0023]

[0024] The triangle OBC uniquely determines the following:

[0025]

[0026] Among them l n Let θ be the length of link n. n Let n be the rotation angle of link n;

[0027] Furthermore, based on geometric relationships, we have:

[0028]

[0029] Where x n ,y n Let i be the geometric center coordinates of link n, where i is the imaginary unit;

[0030] The relationship between the angles of each link and the angle of the drive link can be obtained:

[0031] θ n =f n (θ1)(4) A simplified kinematic model can be obtained by performing a first-order Taylor expansion of the kinematic relationship at the contact point:

[0032] Establish the dynamic equations of the thumb hand:

[0033]

[0034] Where L is the Lagrange function. Let I be the centroid coordinates. i Let m be a Lagrangian function. i Let g be the mass of link i, g be the acceleration due to gravity, τ1 be the driving torque of link i, J be the Jacobian matrix, F be the fingertip grip force, and f be the friction force.

[0035] When the prosthetic hand grasps, it contacts the object at the fingertips. Let the point of contact be p. The equation for the point of contact is: Combined with the contact point equation, the Jacobian matrix can be obtained from the velocity relationship:

[0036]

[0037] Among them l i Let i be the length of link i;

[0038] Finally, the simplified model of the fingertip contact dynamics of the robotic arm body can be obtained as follows:

[0039]

[0040] Where M is the mass matrix of the robot, C is the velocity matrix of the robot, and G is the gravity matrix of the robot;

[0041] This modeling method is not limited to underactuated manipulators. In addition to the manipulator used in this example, it can be applied to any manipulator with an equivalent rigid body structure. Finally, the dynamic formula shown in (11) can be obtained.

[0042] Step 2: Design the convergence function on the sliding surface.

[0043] On the sliding surface, by adjusting the control parameters, the non-singular terminal sliding surface can converge in a finite time. First, using...

[0044] The following is the terminal sliding surface:

[0045]

[0046] Where s is the sliding surface function, e is the control error, α and k are arbitrary positive real numbers, β is the time convergence parameter variable, and p and q are arbitrary positive integers with p > q;

[0047] In this example, we take α = 2, p = 9, q = 5, and k = 100.

[0048] When the system moves on the sliding surface, s = 0, at which point:

[0049]

[0050] Choose the Lyapunov function V1 as:

[0051]

[0052] Differentiating the Lyapunov function, we have:

[0053]

[0054] Since q and p are odd numbers

[0055]

[0056] when When s≡0, the Lyapunov stability condition is satisfied;

[0057] when And when the error e ≥ ε:

[0058]

[0059] The convergence time can be calculated by solving the differential equation (17):

[0060]

[0061] That is when After t e After that, the error e can converge stably to ε << 1.

[0062] Step 3: Design the convergence function outside the sliding surface based on the dynamic model.

[0063] The control rate can be designed as follows:

[0064]

[0065] Γ(e)=(q / p)tanh(ke 1-q / p ) / e 1-q / p +k(1-q / p)[1-tanh(ke 1-q / p )](20)

[0066] Where φ and γ are any positive real numbers, p0 and q0 are any positive integers, and p0 > q0;

[0067] In this example, we take φ = 100, γ = 100, p0 = 3, q0 = 1;

[0068] Choose the Lyapunov function V2 as:

[0069]

[0070] To prove stability, we first differentiate the sliding surface:

[0071]

[0072] Combining the system equations, control law, and sliding surface derivative, we have:

[0073]

[0074] Differentiating the Lyapunov function, we have:

[0075]

[0076] Since q0 and p0 are odd numbers, when When s≡0, the Lyapunov stability condition is satisfied;

[0077] The convergence time outside the sliding surface of differential equation (23) can be calculated as follows:

[0078]

[0079] Therefore, this system can be used at t s The time interval converges from s(t0)≠0 to s(t) s ) = 0.

[0080] Step 4: When the robotic arm grasps an object, the gripping force needs to be adjusted according to the object's rigidity.

[0081] Based on the formula shown in the robotic arm model, it can be summarized as follows:

[0082]

[0083] The error is defined as:

[0084] e = θ1 - θ 1d (27)

[0085] Where θ 1d For the desired position;

[0086] When the stiffness of the object is known, first determine the desired gripping force F. d Divide by the object's stiffness K1 to calculate the expected deformation of the object, from

[0087] This determines the corrected grip force F. 1d :

[0088]

[0089] Where Δδ max To determine the maximum allowable deformation, Δδ is taken in the example.max =2.5mm, F d =4N, K1 = 1N / mm;

[0090] Then the expected position θ 1d Determined by the ratio of desired force to overall stiffness K1:

[0091] Step 5: Calculate the convergence parameters based on the convergence time and convergence target.

[0092] The total convergence time is the time t outside the sliding surface. s and time t on the sliding surface e Sum::

[0093]

[0094] Considering that it takes longer for the human hand to grasp softer objects or apply greater force, the following gripping time rule is adopted:

[0095]

[0096] Where T max F represents the longest gripping time, ζ is the gripping time coefficient, and F d For decoding power, F max For maximum grip strength;

[0097] Take T in the instance max =2s, ζ=0.5, F max =4N;

[0098] Once the convergence time T is calculated, the parameters α, γ, and φ are known, and the parameter β is a variable. The following formula can then be used for calculation:

[0099]

[0100] By setting different gripping convergence times, such as 0.5s, 1.0s, 1.5s, and 2.0s, different parameters can be calculated. These parameters, when substituted into the control law, achieve gripping control with convergence within a specified time. The results are as follows: Figure 3 As shown.

[0101] Figure 4 The figure shows a comparison of the convergence performance of this invention and three mainstream methods. Method 1 is a linear sliding mode control method, Method 2 is a terminal sliding mode control method, and Method 3 is a non-singular terminal sliding mode control method. The experimental results show that this algorithm has a faster convergence speed compared with the current mainstream control methods.

[0102] Figure 5 This is the control result of the present invention when gripping an object with a stiffness of 0.379 N / mm. Figure 6This is the control result of the present invention when gripping an object with a stiffness of 0.832 N / mm. Figure 7 This is the control result of the present invention when gripping an object with a stiffness of 1.246 N / mm. Figure 8 The results of this invention demonstrate the control achieved when gripping an object with a stiffness of 3.589 N / mm. The experimental results show that this method can adjust the gripping force to control the convergence time to 1.886 s, 1.75 s, 1.626 s, and 0.923 s, and the gripping force to 0.973 N, 2.080 N, 3.115 N, and 4.000 N, respectively, based on the stiffness of the four objects. This simulates the human hand and achieves over-deformation gripping control with convergence at a specified time.

Claims

1. A time-convergence-based anti-over-deformation gripping control method, applicable to robotic arms during object gripping, enabling rapid gripping while preventing excessive deformation and damage to the object, characterized in that: First, a dynamic model and Jacobian matrix of the manipulator are established, and a simplified kinematic model is obtained by linearizing the contact point. Second, the convergence function of the controller on and off the sliding surface is designed, and the convergence time is calculated. Then, the allowable deformation is determined based on the object's stiffness, and the convergence target is defined. Finally, the overall convergence time of the gripping force is determined based on the stiffness of the grasped object, and the convergence parameters are calculated based on the convergence target and convergence time to ensure that the controller completes the gripping action within the set time. The specific steps of the over-deformation gripping control method with specified time convergence are as follows: The first step is to establish the dynamic equations of the robotic arm: Where L is the Lagrange function, I i Let θ be the moment of inertia of link i. i Let m be the rotation angle of link i. i Let x be the mass of link i. i y i Let be the coordinates of the center of mass of link i, g be the acceleration due to gravity, τ1 be the driving force of link i, J be the Jacobian matrix, F be the fingertip contact force, and f be the friction force. When the robotic arm grasps an object, it makes contact with the object at its fingertips. Let the point of contact be p. The Jacobian matrix can be obtained from the velocity relationship: A simplified kinematic model can be obtained by performing a first-order Taylor expansion on the kinematic relationships: Where f n-1 (θ1) represents the functional relationship between link n and link one; Finally, the simplified model of the fingertip contact dynamics of the robotic arm body can be obtained as follows: Where M is the mass matrix of the robot, C is the velocity matrix of the robot, and G is the gravity matrix of the robot; Step 2: On the sliding surface, by adjusting the control parameters, the non-singular terminal sliding surface can converge in a finite time. The following terminal sliding surface is used first: Where s is the sliding surface function, e is the control error, α, β, k are arbitrary positive real numbers, and p, q are arbitrary positive integers where p > q; and when the system moves on the sliding surface, s = 0, then: The Lyapunov function is chosen as follows: Differentiating the Lyapunov function, we have: Since q0 and p0 are odd numbers when When s≡0, the Lyapunov stability condition is satisfied; when And when the error e ≥ ε: Convergence time t on the sliding surface e It can be calculated by solving differential equation (10): That is when After t e After that, the error e can converge stably to ε<<1; The third step is to design a convergence function outside the sliding surface, combining the dynamic model, and design the control law as follows: Γ(e)=(q / p)tanh(ke 1-q / p ) / e 1εq / p +k(1-q / p)[1-tanh(ke 1-q / p )] (13) Where φ and γ are any positive real numbers, p0 and q0 are any positive integers, and p0 > q0; The Lyapunov function is chosen as follows: To prove stability, we first differentiate the sliding surface: Combining the system equations, control law, and sliding surface derivative, we have: Differentiating the Lyapunov function, we have: Since q0 and p0 are odd numbers, when When s≡0, the Lyapunov stability condition is satisfied; By solving differential equation (16), the convergence time t outside the sliding surface is obtained. s It can be calculated as follows: Therefore, this system can be used at t s The time interval converges from s(t0)≠0 to s(t) s ) = 0; Fourth, when the robotic arm grasps an object, the gripping force needs to be adjusted according to the object's stiffness. According to formula (5), it can be summarized as follows: The error is defined as: e=θ1-θ 1d (20) Where θ 1d For the desired position; When the stiffness of the object is known, first determine the desired gripping force F. d Divide the object's stiffness K1 to calculate the expected deformation, and then determine the corrected gripping force F. 1d : Where δ max The maximum allowable deformation; Then the desired position θ1 1d Determined by the ratio of the desired force to the object's stiffness K1: in The angle of link one when the robotic arm comes into contact with the object; Fifth, calculate the convergence parameters based on the convergence time and convergence objective. The total convergence time is the time t outside the sliding surface. s and time t on the sliding surface e sum: Considering that it takes longer for a human hand to grasp softer objects or apply greater force, the following grasping time rules are adopted to more realistically simulate the human hand: Where T max F represents the longest gripping time, ζ is the gripping time coefficient, and F d For decoding power, F max For maximum grip strength; Once the convergence time T is calculated, with α, γ, and φ parameters chosen as fixed values ​​and β parameter as a variable, the convergence parameter β can be obtained by calculating using the following formula: The convergence parameter is based on t s The feedback error at each moment is calculated, and by incorporating the calculated convergence parameters into the control law, grasp control that converges at a specified time is achieved.

2. The method for preventing over-deformation grip control with convergence at a specified time according to claim 1, characterized in that... The Lyapunov functions and their derivatives at different convergence stages use the same convergence structure, which can solve for the overall convergence time and the exact numerical solution of the convergence parameters.

3. The method for preventing over-deformation grip control with convergence at a specified time according to claim 1, characterized in that... The maximum deformation of an object is estimated based on its stiffness, and then the gripping force is adjusted based on the maximum deformation. The gripping force adjustment relationship based on the object's deformation includes linear functions, exponential functions, polynomial functions, trigonometric functions, and piecewise functions.

4. The method for preventing over-deformation grip control with convergence at a specified time according to claim 1, characterized in that... The convergence parameters are solved based on the calculated convergence time, including those based solely on t. e Time, based solely on t s Time, and calculation of convergence parameters based on the overall time T.

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