Precise positioning method for the end of a hazardous chemical transportation robotic arm based on a dual neural network

Abstract

A precise positioning method for the end of a hazardous chemical transportation robotic arm based on a dual neural network. Considering the self-joint angle constraints during the movement of the hazardous chemical transportation robotic arm and the uncertainty of the initial position of the end effector, the dual neural network model has finite-time convergence, which can not only improve the convergence speed but also achieve a high convergence accuracy. Eventually, all joints return to the initial desired position within a finite time to achieve repetitive tasks. The entire motion system of the hazardous chemical transportation robotic arm of the present invention has better position error convergence and steady-state characteristics.

Description

Technical Field

[0001] The present invention relates to a method for positioning the end of a hazardous chemical transportation robotic arm based on a dual neural network, particularly a method for controlling the end of a hazardous chemical transportation robotic arm with joint angle constraints and initial position offsets. Background Art

[0002] As a robotic arm with a special function, the hazardous material transportation robotic arm adjusts the movement trajectory of the robotic arm through a controller to complete complex and dangerous repetitive tasks. The hazardous material transportation robotic arm is a mechatronic device with the functions of a human arm, wrist, and hand. Its end tasks include handling, welding, assembly, etc. A robotic arm generally has 3 or more rotational degrees of freedom. When using the robotic arm to complete a specific task, due to its redundant rotational degrees of freedom, this special-purpose robotic arm has a larger operating space, such as physical limit avoidance and environmental obstacle avoidance.

[0003] Recurrent neural networks are widely used in the solution of time-varying problems. Compared with traditional solution methods, neural networks have higher convergence and finite time during the solution process. When traditional neural networks are applied to the repetitive motion trajectory planning of robotic arms, they cannot overcome the physical limits of the joints of the robotic arm itself, even if they can achieve convergence within a limited time. To solve this problem, a finite-time dual neural network is proposed. This neural network can not only achieve finite-time convergence but also overcome the joint angle constraint problem of the robotic arm when solving time-varying calculation problems with equality constraints.

[0004] During the movement of the hazardous chemical transportation robotic arm, there are self-joint angle constraints and uncertainties in the initial position. When the movement trajectory of the end effector is closed, after the robotic arm completes the end task, the trajectories of each joint angle variable in the motion space are not necessarily closed. This non-repetitive problem may generate unexpected joint configurations, causing unexpected situations in the repetitive operation of the end closed trajectory of this type of robotic arm, and even leading to accidents and dangerous situations. Therefore, based on the traditional neural network, a dual neural network is designed to effectively plan the movement trajectory of the hazardous chemical transportation robotic arm. Summary of the Invention

[0005] In order to overcome the deficiencies of the prior art and considering the self-joint angle constraints and uncertainties in the initial position of the end effector during the movement of the hazardous chemical transportation robotic arm, the present invention provides a method for motion planning of a hazardous chemical transportation robotic arm based on a dual neural network, enabling all joints to return to the initial desired position within a limited time and realizing repetitive tasks. This dual neural network model has finite-time convergence, which can not only improve the convergence speed but also achieve a high convergence accuracy.

[0006] To solve the above technical problems, the present invention provides the following technical solutions:

[0007] A precise positioning method for the end of a hazardous chemical transportation robotic arm based on a dual neural network, comprising the following steps:

[0008] Step 1, establish the desired target trajectory r d and the desired joint angles θ * (0) of the end effector of the hazardous chemical transportation robotic arm, establish the kinematic equation of the hazardous chemical transportation robotic arm, and describe the coordinate directions and parameters between adjacent linkages by the D-H (Denavit-Hartenberg) parameter method. Each joint and rod of the hazardous chemical transportation robotic arm is a rigid object. Analyze each joint of the hazardous chemical transportation robotic arm, establish a global coordinate system and a local coordinate system. The global coordinate system is a coordinate system based on the ground, and the local coordinate system is a reference coordinate system established according to the D-H parameters. Through the homogeneous transformation matrix T i-1i of each joint angle in the local coordinate system, obtain the position of the end effector of the hazardous chemical transportation robotic arm in the global coordinate system;

[0009] Step 2, establish the motion relationship between the direction vector r(t) ∈ R n of the end effector of the hazardous chemical transportation robotic arm and the joint angle vector θ(t) ∈ R m ;

[0010] Step 3, construct a repeated motion quadratic programming scheme with constraint conditions;

[0011] Step 4, construct a dual neural network solution model.

[0012] Furthermore, in the above Step 1, the homogeneous transformation matrix T i-1i is as follows:

[0013]

[0014] where θ i , α i , a i and d i respectively represent the moving joint angle, link rotation angle, link length and link offset of the i-th joint of the robotic arm; calculate the transformation matrices T 01 ~T 67 between the joints of the PA10 robotic arm, and multiply these transformation matrices to obtain the homogeneous transformation matrix of the end effector relative to the base coordinate system, which is specifically expressed as follows:

[0015]

[0016] where n ∈ R 3 is the normal vector of the end effector in the base coordinate system, s ∈ R3 , a ∈ R 3 and p = [p x , p y , p z T are the sliding vector, approaching vector, and position vector of the end effector respectively. Then, the position vector r(t) of the end effector of the PA10 robotic arm belongs to R 3 and the joint angle vector θ(t) ∈ R 7 are related as follows:

[0017] r(t) = f(θ(t)) = [p x p y p z T (3)

[0018] Differentiate the above equation to calculate the Jacobian matrix of the hazardous chemical transportation robotic arm as follows:

[0019]

[0020] Furthermore, in step 2, the motion relationship expression is as follows:

[0021] g(θ(t)) = r(t) (5)

[0022] where: g(·): R m → R n is a non - linear continuous function mapping;

[0023] The process of step 2 is as follows:

[0024] 2.1 Define the velocity - layer optimization performance index

[0025] To achieve the motion task objective, eliminate the joint angle deviation by minimizing the displacement between the current joint position and the initial position. The obtained velocity - layer optimization performance index is described as follows:

[0026]

[0027] where: θ(0) ∈ R m is the initial value of the joint angle variable; β > 0 is a design parameter used to adjust the joint displacement amplitude. Since a is a decision variable in equation (6), a is a constant relative to Therefore, the performance index to be optimized is further transformed into

[0028]

[0029] ​​2.2 Establish the joint physical constraints of the hazardous chemical transportation manipulator itself to form a minimum optimization scheme with constraints. The physical limits of the joints of this type of manipulator itself are, that is

[0030]

[0031] Among them: θ ± and are the upper and lower bounds of the joint angle vector θ(t) and the joint velocity vector respectively. Transforming equation (8) into the velocity layer gives:

[0032]

[0033] Among them: ∈>0 is used to adjust the feasible region of the joint velocity. Combining equation (8) and equation (9), the inequality constraints are further transformed into

[0034]

[0035] Among them, the i-th elements of η - and η + are respectively expressed as

[0036]

[0037] Based on the above analysis, the motion planning problem of the hazardous chemical transportation manipulator with joint physical constraints is described as the following time-varying quadratic programming problem with constraints:

[0038]

[0039] Furthermore, in step 3, based on the constrained problem (12) in step 2, let in equation (12) to obtain the following repeated motion quadratic programming scheme with constraints:

[0040]

[0041] Among them: M = I, I is the m-dimensional identity matrix, and J is a row full-rank matrix, that is, rank(J) = n.

[0042] In step 4, according to the Karush-Kuhn-Tucker (KKT) conditions, the optimal solution of the repeated motion planning scheme (13) with constraints should also satisfy the following conditions:

[0043]

[0044] Among them: γ ∈ R n and μ ∈ R m are the dual variables of the equality constraint (13) respectively;

[0045] By defining a projection function

[0046] wherein:

[0047]

[0048] f i (y i ) represents the processing function for each element. We can obtain: x = f(x + μ). Combining Equation (14), the constrained repeated motion planning scheme (13) is transformed into the following dual problem for solution:

[0049]

[0050] wherein: K = M -1 -M -1 J T (JM -1 J T ) -1 JM -1 , d = M -1 J T (JM -1 J T ) -1 (c + JM -1 a) - M -1 a;

[0051] The following dual neural network model (18) is obtained: that is

[0052]

[0053] wherein: α ∈ R and α > 0; Φ: R m → R m is the activation function, specifically defined as

[0054] Φ(z) = [φ1(z1), φ1(z2), …, φ i (z i )] T , z ∈ R m , φ i (z i ) = |z i | p sgn(z i ),

[0055] wherein: p ∈ R and 0 < p < 1, sign(·) is the sign function.

[0056] In the present invention, during the solution process of the dual neural network model, the joint angle limitations of the hazardous chemical transportation robotic arm can be considered, enabling all joints to return to the initial expected positions within a limited time and realizing repetitive tasks. This dual neural network model has finite-time convergence, which can not only improve the convergence speed but also achieve a high convergence accuracy, making the entire motion system of the hazardous chemical transportation robotic arm have better position error convergence and steady-state characteristics.

[0057] The beneficial effects of the present invention are as follows: It has better position error convergence and steady-state characteristics. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 It is a flowchart of the solution of the dual neural network provided by the present invention.

[0059] Figure 2 It is a mechanical configuration diagram of the hazardous chemical transportation robotic arm PA10 adopting the present invention.

[0060] Figure 3 It is a motion trajectory diagram of the PA10 robotic arm.

[0061] Figure 4 It is an end effector tracking error diagram of the PA10 robotic arm.

[0062] Figure 5 It is a joint angle diagram of the end effector of the PA10 robotic arm tracking a circular trajectory.

[0063] Figure 6 It is a joint velocity diagram of the end effector of the PA10 robotic arm tracking a circular trajectory.

[0064] Figure 7 It is the convergence error J of the solution of the dual neural network model for the constrained motion scheme E diagram. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0065] The present invention will be further described below with reference to the accompanying drawings.

[0066] Referring to Figures 1 to 7 , a method for precise positioning of the end of a hazardous chemical transportation robotic arm based on a dual neural network includes the following steps:

[0067] Step 1, establish the expected target trajectory r d of the end effector of the hazardous chemical transportation robotic arm and the expected joint angles θ *(0), establish the kinematic equation of the hazardous chemical transportation manipulator. A common method is to use the D-H (Denavit-Hartenberg) parameter method to describe the coordinate directions and parameters between adjacent links. Each joint and rod of the hazardous chemical transportation manipulator is a rigid object. Analyze each joint of the hazardous chemical transportation manipulator, establish a global coordinate system and a local coordinate system. The global coordinate system is a coordinate system based on the ground, and the local coordinate system is a reference coordinate system established according to the D-H parameters. Through the homogeneous transformation matrix T of each joint angle in the local coordinate system i-1i , obtain the position of the end effector of the hazardous chemical transportation manipulator in the global coordinate system. The homogeneous transformation matrix T i-1i is as follows:

[0068]

[0069] where θ i , α i , a i and d i respectively represent the moving joint angle, link rotation angle, link length, and link offset of the i-th joint of the manipulator. Calculate the transformation matrices T 01 ~T 67 between each joint of the PA10 manipulator. Multiply these transformation matrices to obtain the homogeneous transformation matrix of the end effector relative to the base coordinate system, which is specifically expressed as follows:

[0070]

[0071] where, n ∈ R 3 is the normal vector of the end effector in the base coordinate system, s ∈ R 3 , a ∈ R 3 and p = [p x , p y , p z T are the sliding vector, approaching vector, and position vector of the end effector respectively. Then the position vector r(t) ∈ R 3 of the end effector of the PA10 manipulator and the joint angle vector θ(t) ∈ R 7 are related as follows:

[0072] r(t) = f(θ(t)) = [p x p y p z T (3)

[0073] Differentiate the above equation to calculate the Jacobian matrix of the hazardous chemical transportation manipulator, as follows:

[0074] ​​

[0075] Step 2, establish the motion relationship between the direction vector r(t) ∈ R of the end effector of the hazardous chemical transportation robotic arm n and the joint angle vector θ(t) ∈ R m Therein.

[0076] g(θ(t)) = r(t) (5)

[0077] where: g(·): R m → R n is a non - linear continuous function mapping;

[0078] 2.1 Define the optimization performance index of the velocity layer

[0079] To achieve the motion task goal, by minimizing the displacement between the current joint position and the initial position to eliminate the joint angle deviation, the obtained optimization performance index of the velocity layer is described as follows:

[0080]

[0081] where: θ(0) ∈ R m is the initial value of the joint angle variable; β > 0 is a design parameter used to adjust the joint displacement amplitude. Since a is a decision variable in equation (6), then a is a constant relative to Therefore, the performance index to be optimized can be further transformed into

[0082]

[0083] 2.2 Establish the joint physical constraints of the hazardous chemical transportation robotic arm itself to form a constrained minimum optimization scheme. The joint physical limits of this type of robotic arm itself, that is

[0084]

[0085] where: θ ± and are the upper and lower bounds of the joint angle vector θ(t) and the joint velocity vector respectively. Transforming equation (8) into the velocity layer gives:

[0086]

[0087] where: ∈ > 0 is used to adjust the feasible region of the joint velocity. Combining equation (8) and equation (9), the corresponding inequality constraint is further transformed into

[0088]

[0089] where the i - th elements of η - and η + are respectively expressed as

[0090]

[0091] Based on the above analysis, the motion planning problem of the hazardous chemical transportation manipulator with joint physical constraints is described as the following time-varying quadratic programming problem with constraints:

[0092]

[0093] Step 3: Construct a repeated motion quadratic programming scheme with constraint conditions;

[0094] Based on the constrained problem (12) in Step 2, let in Equation (12) The following repeated motion quadratic programming scheme with constraint conditions is obtained:

[0095]

[0096] where: M = I, and I is the m-dimensional identity matrix. It should be noted that in this paper, we assume that J is a row full-rank matrix, i.e., rank(J) = n;

[0097] Step 4: Construct a dual neural network solution model;

[0098] According to the Karush-Kuhn-Tucker (KKT) conditions, the optimal solution of the constrained repeated motion planning scheme (13) should also satisfy the following conditions:

[0099]

[0100] where: γ ∈ R n and μ ∈ R m are the dual variables of the equality constraint (13) respectively.

[0101] By defining a projection function

[0102] where:

[0103]

[0104] f i (y i ) represents the processing function of each element, and we get: x = f(x + μ). Combining Equation (14), the constrained repeated motion planning scheme (13) is transformed into the following dual problem for solution:

[0105]

[0106] where: K = M -1 -M -1 J T (JM-1 J T ) -1 JM -1 ,d = M -1 J T (JM -1 J T ) -1 (c + JM -1 a) - M -1 a。

[0107] Therefore, the following dual neural network model (18) is obtained: That is

[0108]

[0109] where: α ∈ R and α > 0; Φ: R m → R m is the activation function, specifically defined as

[0110] Φ(z) = [φ1(z1), φ1(z2), …, φ i (z i )] T , z ∈ R m , φ i (z i ) = |z i | p sgn(z i ),

[0111] where: p ∈ R and 0 < p < 1, sgn(·) is the sign function.

[0112] The hazardous chemical transportation robotic arm PA10 used to implement the precise positioning solution of the present invention is as Figure 2 shown, and the D - H parameters of the robotic arm PA10 and the physical constraints of each joint are as follows in Table 1:

[0113]

[0114] Table 1

[0115] Set the desired joint angles for the robotic arm PA10 to retract as θ * (0) = [0; π / 4; π / 4; 2π / 3; 0; -π / 4; 0]. Considering that there will be a certain deviation in the initial position of the PA10 robotic arm, the initial values of the seven joint angles of the robotic arm PA10 are set as:

[0116] θ(0) = [0; π / 4; π / 4; 2π / 3; 0; -π / 4; 0].

[0117] To verify the feasibility of the proposed method, the present invention presents the simulation results of the robotic arm with constrained repetitive motion planning scheme based on the finite-time dual neural network on the MATLAB platform:

[0118] Set the target task path as a circular trajectory task, and the desired path of the end effector of the robotic arm is

[0119]

[0120] where L = 0.1 m, e = (sin(πt / 2 / T)) 2 , θ = π / 6, t = [0, T], T = 10 s is the motion period of the mobile robotic arm. The remaining parameters are given as follows: α = 10, β = 5, p = 0.7, and ∈ = 20.

[0121] The content described in the embodiments of this specification is only a list of the implementation forms of the inventive concept and is only for illustrative purposes. The protection scope of the present invention should not be regarded as limited to the specific forms stated in this embodiment. The protection scope of the present invention also extends to equivalent technical means that can be conceived by those of ordinary skill in the art based on the inventive concept of the present invention.

Claims

1. A precise positioning method for the end of a hazardous chemical transportation robotic arm based on a dual neural network, characterized in that, The method includes the following steps: Step 1, establish the desired target trajectory r of the end effector of the hazardous chemical transportation robotic arm d and the desired folded joint angles θ * (0), establish the kinematic equation of the hazardous chemical transportation robotic arm, and describe the coordinate directions and parameters between adjacent connecting rods through the D-H parameter method. Each joint and rod of the hazardous chemical transportation robotic arm are rigid bodies. Analyze each joint of the hazardous chemical transportation robotic arm, establish a global coordinate system and a local coordinate system. The global coordinate system is a coordinate system based on the ground, and the local coordinate system is a reference coordinate system established according to the D-H parameters. Through the homogeneous transformation matrix T of each joint angle in the local coordinate system i-1i , obtain the position of the end effector of the hazardous chemical transportation robotic arm in the global coordinate system; Step 2, establish the motion relationship between the end effector direction vector r(t) ∈ R of the hazardous chemical transportation robotic arm n and the joint angle vector θ(t) ∈ R m ; Step 3: Construct a repeated motion quadratic programming scheme with constraint conditions; Step 4: Construct a dual neural network solution model; In step 2, the motion relationship expression is as follows: g(θ(t)) = r(t) (5) where: g(·):R m →R n is a non - linear continuous function mapping; The process of step 2 is as follows: 2.1 Define the optimization performance index of the velocity layer To achieve the motion task objective, the displacement between the current joint position and the initial position is minimized to eliminate the joint angle deviation. The obtained optimization performance index of the velocity layer is described as follows: where: θ(0) ∈ R m is the initial value of the joint angle variable; β > 0 is a design parameter used to adjust the joint displacement amplitude. Since a is a decision variable in equation (6), a is a constant relative to is a constant. Therefore, the performance index to be optimized is further transformed into 2.2 Establish the joint physical constraints of the hazardous chemical transportation manipulator itself to form a constrained minimum optimization scheme, which is the joint physical limit of the hazardous chemical transportation manipulator itself, that is Wherein: θ ± and are the upper and lower bounds of the joint angle vector θ(t) and the joint velocity vector respectively. Transforming equation (8) into the velocity layer gives: where: ∈>0 is used to adjust the feasible region of the joint velocity. Combining equation (8) and equation (9), the inequality constraint for is further transformed into where η - and η + the i-th elements of are respectively denoted as Based on the above analysis, the motion planning problem of the hazardous chemical transportation manipulator with joint physical constraints is described as the following constrained time-varying quadratic programming problem: In step 3, based on the constrained problem (12) in step 2, let in equation (12) The following repeated motion quadratic programming scheme with constraint conditions is obtained: where: M = I, I is the m-dimensional identity matrix, and J is a row full-rank matrix, that is, rank(J) = n; In step 4, according to the Karush-Kuhn-Tucker (KKT) conditions, the optimal solution of the constrained repeated motion planning scheme (13) should also satisfy the following conditions: where: γ ∈ R n and μ ∈ R m are the dual variables of the constrained repetitive motion planning scheme (13), respectively; By defining a projection function where: f i (y i ) represents the processing function for each element, obtaining: x = f(x + μ). Combining equation (14), the constrained repeated motion planning scheme (13) is transformed into the following dual problem for solution: where: K = M -1 -M -1 J T (JM -1 J T ) -1 JM -1 , d = M -1 J T (JM -1 J T ) -1 (c + JM -1 a) - M -1 a; the following dual neural network model (18) is obtained: namely Where: α ∈ R and α > 0; Φ: R m → R m is an activation function, specifically defined as Φ(z) = [φ1(z1), φ1(z2), …, φ i (z i )] T , z ∈ R m , φ i (z i ) = |z i | p sgn(z i ), where: p ∈ R and 0 < p < 1, sgn(·) is the sign function.

2. The method for precise positioning of the end of a hazardous chemical transportation robotic arm based on a dual neural network according to claim 1, wherein In the said step 1, the homogeneous transformation matrix T i-1i is as follows: where θ i , α i , a i and d i represent the moving joint angle, link rotation angle, link length, and link offset of the i-th joint of the robotic arm respectively; calculate the transformation matrices T 01 ~T 67 between the joints of the PA10 robotic arm, and multiply these transformation matrices to obtain the homogeneous transformation matrix of the end effector relative to the base coordinate system, which is specifically expressed as follows: where \(n\in R\) 3 is the normal vector of the end effector in the base coordinate system, \(s\in R\) 3 , \(a\in R\) 3 and \(p = [p\) x , \(p\) y , \(p\) z T are respectively the sliding vector, approaching vector and position vector of the end effector. Then, the position vector \(r(t)\in R\) 3 of the end effector of the PA10 robotic arm and the joint angle vector \(\theta(t)\in R\) 7 are related as follows:​ r(t) = f(θ(t)) = [p x p y p z T (3)​ Differentiate equation (3) to calculate the Jacobian matrix of the hazardous chemical transportation manipulator as follows:

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