CN110561440A - multi-objective planning method for acceleration layer of redundant manipulator - Google Patents

Abstract

A multi-objective planning method for the acceleration layer of a redundant manipulator, which separately designs the optimization performance index and the inequality criterion of the acceleration layer; combines the planning tasks at the end of the manipulator and the physical limits of the joints of the manipulator to establish a multi-objective planning scheme for the acceleration layer, The scheme is constrained by the Jacobian matrix equation of the acceleration layer, the dynamic equation of the manipulator, the obstacle avoidance inequality, the joint angle limit, the joint speed limit, the joint acceleration limit and the joint torque limit; the above multi-objective programming scheme is transformed into a standard The quadratic optimization problem is solved by numerical algorithm; the lower computer controller drives each joint of the manipulator to complete the given terminal planning task according to the solution result of the algorithm. The acceleration-level multi-objective planning scheme designed by the present invention can enable the mechanical arm to complete a given terminal planning task under the condition of limited joints and obstacles, and achieve the purpose of repeated motion.

Description

A Multi-objective Planning Method for Redundant Manipulator Acceleration Layer

technical field

The invention relates to the field of motion planning and control of redundant manipulators, in particular to a multi-objective planning method for acceleration layers of redundant manipulators.

Background technique

Redundant robotic arms are playing an increasingly important role in industrial automation production activities such as welding, assembly, and painting. Two hotspots in the application research of redundant manipulators are repetitive motion planning and obstacle avoidance: this is related to whether the manipulator can successfully complete repetitive terminal planning tasks in complex industrial environments.

At present, a variety of repetitive motion planning schemes and obstacle avoidance planning schemes have been proposed and effectively applied to manipulators. However, most planning schemes can only achieve a single goal (ie, repetitive motion or obstacle avoidance), and only a few planning schemes can achieve both repetitive motion and obstacle avoidance. These multi-objective planning schemes are designed at the velocity level and studied while ignoring the dynamics of the manipulator. Obviously, they cannot deal with the joint acceleration limit and joint torque limit of the manipulator, nor can they be directly applied to the manipulator related to acceleration or torque control, so the scope of application of the scheme is greatly limited. More importantly, due to the lack of consideration of the dynamics of the manipulator, these schemes are often prone to failure in industrial application practice, resulting in the inability of the manipulator to complete the given end-planning task.

Contents of the invention

The main purpose of the present invention is to overcome the above-mentioned defects in the prior art, and propose a multi-objective planning method for the acceleration layer of redundant manipulators, which can make the manipulators still complete the given tasks when the joints are limited and there are obstacles. end-planning tasks, and achieve the purpose of repetitive motion.

The present invention adopts following technical scheme:

A redundant manipulator acceleration layer multi-objective planning method, is characterized in that, comprises the steps:

1) Design the optimized performance index and obstacle avoidance inequality of the acceleration layer;

2) Combine the planning tasks at the end of the manipulator, the physical limits of the joints of the manipulator and the optimization performance indicators of the acceleration layer to establish a multi-objective planning scheme for the acceleration layer, which is constrained by the Jacobian matrix equation of the acceleration layer, the dynamic equation of the manipulator, and the avoidance Barrier inequality, joint angle limit, joint speed limit, joint acceleration limit and joint torque limit;

3) Transform the multi-objective programming scheme of the acceleration layer into a standard quadratic optimization problem, and use numerical algorithms to solve it;

4) The controller of the lower computer drives each joint action of the manipulator to complete the given terminal planning task according to the solution result.

Preferably, the optimized performance index of the acceleration layer is:

Where ||·|| ₂ represents the two-norm of the vector, Indicates the joint acceleration of the manipulator, Indicates the joint speed of the manipulator, θ indicates the joint angle of the manipulator, and θ(0) indicates the initial state of the joint of the manipulator; the design parameter λ>0 is used to adjust the performance index so that the manipulator can achieve the purpose of repetitive motion planning.

Preferably, the obstacle avoidance inequality is:

Among them, the coefficient matrices M and N are respectively defined as M=-[x _C -x _O , y _C -y _O , z _C -z _O ] ^T J _C and The superscript ^T represents the transpose of the matrix or vector, (x _C , y _C , z _C ) represents the spatial coordinates of the criterion point of the manipulator, (x _O , y _O , z _O ) represents the spatial coordinates of the environmental obstacle point, J _C represents the Jacobian matrix of the criterion point of the manipulator, Denotes the time derivative of _JC .

Preferably, the acceleration layer multi-objective planning scheme is designed as:

minimize

subject to

where the equality constraints Corresponding to the motion planning task at the end of the manipulator at the acceleration layer, α and β represent the design parameters and their values satisfy α>0 and β>0, J represents the Jacobian matrix of the manipulator, Represents the time derivative of J, e represents the position error of the end of the manipulator and is defined as e=f(θ)-r _d , f( ) represents a nonlinear mapping function, r _d represents the expected trajectory of the end of the manipulator, ε represents the velocity error at the end of the manipulator and is defined as Indicates the time derivative of r _d , express The time derivative of ; the equality constraint Represents the dynamic equation of the manipulator, H represents the inertia matrix of the manipulator, c represents the centrifugal force variable, g represents the gravity variable, τ represents the joint torque of the manipulator arm; inequality constraints It is used to realize the avoidance of environmental obstacles by the robotic arm; and τ ^± represent joint angle limit, joint velocity limit, joint acceleration limit and joint torque limit, respectively.

Preferably, the acceleration layer multi-objective programming scheme is converted into a standard quadratic optimization problem, its performance index is x ^T x/2+p ^T x, and the constraint conditions are Cx=d, Ax≤b, η ^- ≤x ≤η ⁺ , where, η ^± means simultaneous joint angle θ, joint speed and joint acceleration After the upper and lower limits, and, k ₁ >0 and k ₂ >0 represent the joint limit conversion parameters, and δ>0 represents the joint limit conversion margin.

Preferably, a numerical algorithm is used to solve the quadratic optimization problem, specifically: the solution of the quadratic optimization problem is converted into a piecewise linear projection equation, and the corresponding numerical algorithm is used to calculate the fractional Solutions of segmental linear projection equations and optimal solutions of quadratic optimization problems.

As can be seen from the above description of the present invention, compared with the prior art, the present invention has the following beneficial effects:

The present invention can effectively overcome the shortcomings of the existing methods, and provides a redundant mechanical arm that can complete a given terminal planning task and achieve the purpose of repeated motion even when the joints are limited and there are obstacles. The acceleration layer multi-objective planning method is of great significance and value to the application practice of manipulators in complex industrial environments.

Description of drawings

Fig. 1 is a flowchart of the present invention.

The present invention will be described in further detail below in conjunction with the accompanying drawings and specific embodiments.

Detailed ways

The present invention will be further described below through specific embodiments.

A multi-objective programming method for the acceleration layer of redundant manipulators shown in Figure 1 is mainly composed of designing performance indicators and inequality criteria that can realize repetitive motion and obstacle avoidance, establishing a multi-objective planning scheme for the acceleration layer, and turning it into a standard It consists of six parts: quadratic optimization problem, numerical algorithm solver, lower computer controller, and redundant manipulator. Specifically, the following steps are included:

1) First, in order to achieve the purpose of repetitive movement of the manipulator and obstacle avoidance, the optimization performance index of the acceleration layer and the obstacle avoidance inequality are respectively designed.

The performance index of the acceleration layer that can realize repeated motion is designed as:

Where ||·|| ₂ represents the two-norm of the vector, Indicates the joint acceleration of the manipulator, Indicates the joint speed of the manipulator, θ indicates the angle of the joint of the manipulator, and θ(0) indicates the initial state of the joint of the manipulator; the design parameter λ>0 is used to adjust the performance index so that the manipulator can achieve the purpose of repetitive motion.

The inequality criterion that can realize obstacle avoidance is designed as

Among them, the coefficient matrices M and N are respectively defined as M=-[x _C -x _O , y _C -y _O , z _C -z _O ] ^T J _C and The superscript ^T represents the transpose of the matrix or vector, (x _C , y _C , z _C ) represents the spatial coordinates of the criterion point of the manipulator, (x _O , y _O , z _O ) represents the spatial coordinates of the environmental obstacle point, J _C represents the Jacobian matrix of the criterion point of the manipulator, Denotes the time derivative of _JC .

2) Then, combined with the planning tasks at the end of the manipulator and the physical limits of the joints of the manipulator, a scheme that can achieve multiple planning objectives at the same time is established, that is, the acceleration layer multi-objective planning scheme.

Based on the above performance index (1) and inequality criterion (2), combined with the planning tasks at the end of the manipulator and the physical limits of the manipulator joints, the following acceleration layer multi-objective planning scheme can be established:

minimize

Restrictions:

θ ^- ≤ θ ≤ θ ⁺ (7)

τ ⁻ ≤τ≤τ ⁺ (10)

where the equality constraints Corresponding to the motion planning task at the acceleration layer at the end of the manipulator, J represents the Jacobian matrix of the manipulator, Represents the time derivative of J, e represents the position error of the end of the manipulator and is defined as e=f(θ)-r _d , f( ) represents a nonlinear mapping function, r _d represents the expected trajectory of the end of the manipulator, ε represents the velocity error at the end of the manipulator and is defined as Indicates the time derivative of r _d , express The time derivative of ; the equality constraint Represents the dynamic equation of the manipulator, H represents the inertia matrix of the manipulator, c represents the centrifugal force variable, g represents the gravity variable, τ represents the joint moment of the manipulator arm; inequality constraints It is used to realize the avoidance of the manipulator to the environmental obstacles; θ ^± , and τ ^± represent joint angle limit, joint velocity limit, joint acceleration limit and joint torque limit, respectively.

3) Furthermore, transform the planning scheme into a standard quadratic optimization problem, and use numerical algorithms to solve it, as follows:

For the above acceleration layer multi-objective planning schemes (3)-(10), it can be transformed into the following standard quadratic optimization problem:

Minimize ^xTx /2+ ^pTx (11)

Constraints: Cx=d (12)

Ax≤b (13)

η ^- ≤ x ≤ η ⁺ (14)

in, A=[M ^T -H ^T H ^T ] ^T , η ^± means simultaneous joint angle θ, joint speed and joint acceleration After the upper and lower limits, and, k ₁ >0 and k ₂ >0 represent the joint limit conversion parameters, and δ>0 represents the joint limit conversion margin.

The solutions to the above quadratic optimization problems (11)-(14) can be transformed into equivalent solutions to the following piecewise linear projection equations:

uP _Ω (u-(Mu+q))＝0 (15)

where P _Ω (·) represents the piecewise linear projection operator. The original dual decision vector u, the coefficient matrix M and the vector q in the piecewise linear projection equation (15) are respectively defined as follows:

where I represents the identity matrix, and the dual variables w and v correspond to the equality constraints (12) and inequality constraints (13), respectively. In order to solve the above piecewise linear projection equation (15) and the corresponding quadratic optimization problems (11)-(14), the following numerical algorithm can be used:

σ(u ^k )＝u ^k -P _Ω (u ^k -(Mu ^k +q))

where the superscript k represents the number of iterations and k=0, 1, 2, . . . Given an initial value u ⁰ , the theoretical solution of the piecewise linear projection equation (15) can be obtained through continuous iterative calculation of the numerical algorithm, and thus the optimal solution of the quadratic optimization problem (11)-(14) can be obtained , that is, the optimal solution of the acceleration layer multi-objective planning scheme (3)-(10) mentioned above.

4) Finally, the solution results are used to drive each joint of the manipulator so that the manipulator can still complete the given terminal planning task when the joints are limited and have obstacles, and achieve the purpose of repeated motion, as follows:

After calculating the optimal solutions of acceleration layer multi-objective planning schemes (3)-(10) by using the above numerical algorithm solver, the corresponding solution results are passed to the lower computer controller to drive the movement of each joint of the manipulator, so that The robotic arm can still complete a given terminal planning task with limited joints and obstacles, and achieve the purpose of repetitive motion.

The above is only a specific embodiment of the present invention, but the design concept of the present invention is not limited thereto, and any non-substantial changes made to the present invention by using this concept should be an act of violating the protection scope of the present invention.

Claims (6)

1. a kind of redundancy mechanical arm acceleration layer multi-objective planning method, is characterized in that, comprises the steps: 1) Design the optimized performance index and obstacle avoidance inequality of the acceleration layer; 2) Combine the planning tasks at the end of the manipulator, the physical limits of the joints of the manipulator and the optimization performance indicators of the acceleration layer to establish a multi-objective planning scheme for the acceleration layer, which is constrained by the Jacobian matrix equation of the acceleration layer, the dynamic equation of the manipulator, and the avoidance Barrier inequality, joint angle limit, joint speed limit, joint acceleration limit and joint torque limit; 3) Transform the multi-objective programming scheme of the acceleration layer into a standard quadratic optimization problem, and use numerical algorithms to solve it; 4) The controller of the lower computer drives each joint action of the manipulator to complete the given terminal planning task according to the solution result. 2. a kind of redundancy mechanical arm acceleration layer multi-objective planning method as claimed in claim 1, is characterized in that, the optimization performance index of described acceleration layer is: Where ||·|| ₂ represents the two-norm of the vector, Indicates the joint acceleration of the manipulator, Indicates the joint speed of the manipulator, θ indicates the joint angle of the manipulator, and θ(0) indicates the initial state of the joint of the manipulator; the design parameter λ>0 is used to adjust the performance index so that the manipulator can achieve the purpose of repetitive motion planning. 3. a kind of redundant manipulator acceleration layer multi-objective planning method as claimed in claim 2, is characterized in that, described obstacle avoidance inequality is: Among them, the coefficient matrices M and N are respectively defined as M=-[x _C -x _O , y _C -y _O , z _C -z _O ] ^T J _C and The superscript ^T represents the transpose of the matrix or vector, (x _C , y _C , z _C ) represents the spatial coordinates of the criterion point of the manipulator, (x _O , y _O , z _O ) represents the spatial coordinates of the environmental obstacle point, J _C represents the Jacobian matrix of the criterion point of the manipulator, Denotes the time derivative of _JC . 4. a kind of redundancy mechanical arm acceleration layer multi-objective planning method as claimed in claim 3, is characterized in that, described acceleration layer multi-objective planning scheme is designed as: minimize subject to θ ⁻ ≤ θ ≤ θ ⁺ , τ ^- ≤ τ ≤ τ ⁺ ; where the equality constraints Corresponding to the motion planning task of the end of the manipulator at the acceleration layer, α and β represent the design parameters and their values satisfy α>0 and β>0, J represents the Jacobian matrix of the manipulator, Represents the time derivative of J, e represents the position error of the end of the manipulator and is defined as e=f(θ)-r _d , f( ) represents a nonlinear mapping function, r _d represents the expected trajectory of the end of the manipulator, ε represents the velocity error at the end of the manipulator and is defined as Indicates the time derivative of r _d , express The time derivative of ; the equality constraint Represents the dynamic equation of the manipulator, H represents the inertia matrix of the manipulator, c represents the centrifugal force variable, g represents the gravity variable, τ represents the joint torque of the manipulator arm; inequality constraints It is used to realize the avoidance of the manipulator to the environmental obstacles; θ ^± , and τ ^± represent joint angle limit, joint velocity limit, joint acceleration limit and joint torque limit, respectively. 5. a kind of redundancy mechanical arm acceleration layer multi-objective programming method as claimed in claim 4, is characterized in that, the acceleration layer multi-objective programming scheme is transformed into a standard quadratic optimization problem, and its performance index is x ^T x/2+p ^T x, the constraints are Cx=d, Ax≤b, η ^- ≤x≤η ⁺ , where, C=J, A=[M ^T -H ^T H ^T ] ^T , η ^± means simultaneous joint angle θ, joint speed and joint acceleration After the upper and lower limits, and, k ₁ >0 and k ₂ >0 represent the joint limit conversion parameters, and δ>0 represents the joint limit conversion margin. 6. A kind of redundant manipulator acceleration layer multi-objective programming method as claimed in claim 1 or 2 or 3 or 4 or 5, is characterized in that, adopts numerical algorithm to solve quadratic optimization problem, specifically : The solution of the quadratic optimization problem is equivalent to the solution of a piecewise linear projection equation, and the corresponding numerical algorithm is used to calculate the solution of the piecewise linear projection equation and the optimal solution of the quadratic optimization problem.