CN102514008A

Movatterモバイル変換

Info

Publication number: CN102514008A
Application number: CN2011103716880A
Authority: CN
Inventors: 张雨浓; 郭东生; 李克讷
Original assignee: Sun Yat Sen University
Current assignee: Sun Yat Sen University
Priority date: 2011-11-21
Filing date: 2011-11-21
Publication date: 2012-06-27
Anticipated expiration: 2031-11-21
Also published as: CN102514008B

Abstract

本发明提供了一种冗余度机械臂的不同层性能指标同时优化方法，如下：根据所需要优化的角度层、速度层和加速度层性能指标，通过引入权值调节因子，建立相应的冗余度解析方案，所述的解析方案受约束于速度、加速度的雅可比矩阵等式、机械臂的动力学方程、关节角度极限、关节速度极限、关节加速度极限和关节力矩极限；利用上述三层性能指标的等效性及引入等效性参数，将冗余度解析方案转化为一个统一的二次型规划问题；将二次型规划问题运用二次型规划求解器进行求解；下位机控制器根据求解结果，驱动机械臂使其完成给定的末端任务。本发明通过引入权值调节因子，将不同层的性能指标实现同时优化，同时也使得机械臂完成给定的末端任务。

The present invention provides a method for simultaneously optimizing the performance indexes of different layers of a redundant manipulator, as follows: according to the performance indexes of the angle layer, velocity layer and acceleration layer to be optimized, by introducing a weight adjustment factor, a corresponding redundancy is established Degree analysis scheme, the analysis scheme is constrained by the Jacobian matrix equation of velocity and acceleration, the dynamic equation of the manipulator, joint angle limit, joint velocity limit, joint acceleration limit and joint torque limit; using the above three layers of performance The equivalence of indicators and the introduction of equivalence parameters transform the redundancy analysis scheme into a unified quadratic programming problem; use the quadratic programming solver to solve the quadratic programming problem; the lower computer controller according to Solve the result and drive the robotic arm to complete the given end task. The present invention realizes simultaneous optimization of performance indexes of different layers by introducing weight adjustment factors, and also enables the mechanical arm to complete a given terminal task.

Description

Translated fromChinese

一种冗余度机械臂的不同层性能指标同时优化方法A Simultaneous Optimization Method for Performance Indexes of Different Layers of a Redundant Manipulator

技术领域technical field

本发明涉及冗余度机械臂运动规划及控制领域，具体涉及一种冗余度机械臂的不同层性能指标同时优化方法。The invention relates to the field of motion planning and control of a redundant manipulator, in particular to a method for simultaneously optimizing performance indexes of different layers of a redundant manipulator.

背景技术Background technique

冗余度机械臂是一种自由度大于执行末端任务所需最少自由度的机械装置，其末端运动任务包括焊接、油漆、组装、挖掘和绘图等。冗余度机械臂操作的一个关键问题是冗余度解析问题(包括逆运动学问题和逆动力学问题)，即，通过已知机械臂末端位姿来确定机械臂的关节角度问题。目前有效的冗余度解析方案都是在单一/同层(如速度层、加速度层或力矩层)上进行解析。然而，单一/同层的优化方案存在着不足：单纯的速度层解析方案难以考虑加速度极限和力矩极限；而单纯的加速度层和单纯的力矩层解析方案又容易出现速度发散和末态速度不为零的现象。A redundant manipulator is a mechanical device with more degrees of freedom than the minimum required to perform terminal tasks, such as welding, painting, assembly, digging, and drawing. A key problem in the operation of redundant manipulators is the problem of redundancy analysis (including inverse kinematics and inverse dynamics), that is, the problem of determining the joint angle of the manipulator by knowing the end pose of the manipulator. Currently effective redundancy analysis schemes are all analyzed on a single/same layer (such as velocity layer, acceleration layer or moment layer). However, there are deficiencies in the single/same-level optimization scheme: the pure velocity layer analysis scheme is difficult to consider the acceleration limit and torque limit; and the pure acceleration layer and pure torque layer analysis scheme is prone to velocity divergence and final state velocity. zero phenomenon.

发明内容Contents of the invention

本发明的目的在于提供一种操作方便、工作量小的冗余度机械臂的不同层性能指标同时优化方法。The purpose of the present invention is to provide a method for simultaneously optimizing performance indexes of different layers of a redundant mechanical arm with convenient operation and low workload.

为了实现上述发明目的，采用的技术方案如下。In order to realize the object of the above invention, the technical solution adopted is as follows.

一种冗余度机械臂的不同层性能指标同时优化方法，包括以下步骤：A method for simultaneously optimizing performance indexes of different layers of a redundant manipulator, comprising the following steps:

根据角度层、速度层和加速度层性能指标，通过引入权值调节因子，所述权值调节因子用于调节上述三层上需要优化的性能指标在总的优化指标中的权重或比重，建立相应的冗余度解析方案，所述的解析方案受约束于速度、加速度的雅可比矩阵等式、机械臂的动力学方程、关节角度极限、关节速度极限、关节加速度极限和关节力矩极限；According to the angle layer, speed layer and acceleration layer performance index, by introducing the weight adjustment factor, the weight adjustment factor is used to adjust the weight or proportion of the performance index to be optimized on the above three layers in the total optimization index, and establish a corresponding A redundancy analysis scheme, the analysis scheme is constrained by the Jacobian matrix equation of velocity and acceleration, the dynamic equation of the manipulator, the joint angle limit, the joint velocity limit, the joint acceleration limit and the joint torque limit;

利用角度层、速度层和加速度层性能指标的等效性及引入等效性参数，所述等效性参数用于推导不同层性能指标的等效性的过程中，通过设定该参数的值，使到两个在不同层上优化的指标就可以达到性能等效的目的或效果，从而可以将冗余度解析方案转化为一个统一的二次型规划问题；Utilize the equivalence of angle layer, speed layer and acceleration layer performance index and introduce equivalence parameter, described equivalence parameter is used in the process of deriving the equivalence of different layer performance index, by setting the value of this parameter , so that two indicators optimized on different layers can achieve the purpose or effect of equivalent performance, so that the redundancy analysis scheme can be transformed into a unified quadratic programming problem;

通过二次型规划求解器对二次型规划问题进行求解；The quadratic programming problem is solved by the quadratic programming solver;

下位机控制器根据二次型规划问题的求解结果，驱动机械臂使其完成给定的末端任务。The controller of the lower computer drives the manipulator to complete the given terminal task according to the solution result of the quadratic programming problem.

上述技术方案中，所述角度层、速度层和加速度层性能指标同时优化的冗余度解析方案设计为：In the above technical solution, the redundancy analysis scheme for simultaneous optimization of the performance indicators of the angle layer, the velocity layer and the acceleration layer is designed as:

最小化minimize

受约束于 $J \overset{\cdot}{θ} = \overset{\cdot}{r},$ $J \overset{\cdot \cdot}{θ} = \overset{\cdot \cdot}{r} - \overset{\cdot}{J} \overset{\cdot}{θ},$ $τ = H \overset{\cdot \cdot}{θ} + c + g,$ θ^-≤θ≤θ⁺， ${\overset{\cdot}{θ}}^{-} \leq \overset{\cdot}{θ} \leq {\overset{\cdot}{θ}}^{+},$ ${\overset{\cdot \cdot}{θ}}^{-} \leq \overset{\cdot \cdot}{θ} \leq {\overset{\cdot \cdot}{θ}}^{+},$ τ^-≤τ≤τ⁺；subject to $J \overset{&Center Dot;}{θ} = \overset{&Center Dot;}{r},$ $J \overset{&Center Dot; \cdot}{θ} = \overset{\cdot \cdot}{r} - \overset{\cdot}{J} \overset{&Center Dot;}{θ},$ $τ = h \overset{&Center Dot; &Center Dot;}{θ} + c + g,$ θ⁻ ≤ θ ≤ θ⁺ , ${\overset{&Center Dot;}{θ}}^{-} \leq \overset{&Center Dot;}{θ} \leq {\overset{&Center Dot;}{θ}}^{+},$ ${\overset{&Center Dot; &Center Dot;}{θ}}^{-} \leq \overset{&Center Dot; &Center Dot;}{θ} \leq {\overset{&Center Dot; &Center Dot;}{θ}}^{+},$ τ^- ≤ τ ≤ τ⁺ ;

其中，

和

分别对应于在角度层、速度层和加速度层上欲优化的性能指标，α₁≥0、α₂≥0和α₃≥0为权值调节因子且保证两个或两个以上权值调节因子同时不为零；θ表示关节角度，表示关节速度，

表示关节加速度，τ表示关节力矩；等式约束对应于机械臂在速度层的末端运动轨迹，J表示机械臂的雅可比矩阵，

表示机械臂末端执行器的速度；等式约束对应于机械臂在加速度层的末端运动轨迹，

表示雅可比矩阵J的时间导数，

表示机械臂末端执行器的加速度；等式约束

为机械臂的动力学方程，H表示机械臂的惯性矩阵，c表示离心力变量，g表示重力变量；θ^±、和τ^±表示关节角度极限、关节速度极限、关节加速度极限和关节力矩极限。in,

and

Corresponding to the performance indicators to be optimized on the angle layer, velocity layer and acceleration layer respectively, α₁ ≥ 0, α₂ ≥ 0 and α₃ ≥ 0 are weight adjustment factors and two or more weight adjustment factors are guaranteed At the same time, it is not zero; θ represents the joint angle, represents the joint velocity,

Represents the joint acceleration, τ represents the joint torque; the equality constraint Corresponding to the end motion trajectory of the manipulator in the velocity layer, J represents the Jacobian matrix of the manipulator,

represents the velocity of the end effector of the manipulator; the equation constraint Corresponding to the end motion trajectory of the manipulator in the acceleration layer,

Denotes the time derivative of the Jacobian matrix J,

represents the acceleration of the end effector of the manipulator; the equation constraints

is 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; θ^± , and τ^± denote joint angle limit, joint velocity limit, joint acceleration limit and joint torque limit.

所述利用角度层、速度层和加速度层性能指标的等效性和引入等效性参数，将不同层性能指标同时优化的冗余度解析方案转化为一个统一的二次型规划问题，其性能指标为x^TOx/2+p^Tx，约束条件为Cx＝d，Ax≤b，x^-≤x≤x⁺，其中，只考虑不同层能量最小化方案时，当α₃＞0时，决策变量x表示关节加速度，Q可由权值调节因子α₁，α₂，α₃对单位矩阵I、惯性矩阵H和H²的加权和获得，p可由权值调节因子α₁，α₂，α₃及等效性参数对

c，g和H的加权和获得，C＝J，

当α₃＝0时，决策变量x表示关节速度，Q可由权值调节因子α₁，α₂对单位矩阵I的加权和获得，p可由权值调节因子α₁，θ和等效性参数的乘积获得，C＝J，

上标^T表示矩阵或向量的转置，Ax≤b用于关节力矩极限和无穷范数约束，x^±表示的x上下限。Using the equivalence of the performance indicators of the angle layer, the velocity layer and the acceleration layer and introducing the equivalence parameters, the redundancy analysis scheme that optimizes the performance indicators of different layers at the same time is transformed into a unified quadratic programming problem, and its performance The index is x^T Ox/2+p^T x, and the constraints are Cx＝d, Ax≤b, x^- ≤x≤x⁺ , where, when only energy minimization schemes of different layers are considered, when α₃ >0, The decision variable x represents the joint acceleration, Q can be obtained by the weighted sum of the weight adjustment factors α₁ , α₂ , α₃ on the identity matrix I, inertia matrix H and H² , p can be obtained by the weight adjustment factors α₁ , α₂ , α₃ and equivalence parameter pairs

c, the weighted sum of g and H is obtained, C=J,

When α₃ =0, the decision variable x represents the joint velocity, Q can be obtained by the weighted sum of the weight adjustment factors α₁ , α₂ on the identity matrix I, p can be obtained by the weight adjustment factors α₁ , θ and the equivalent parameter The product is obtained, C=J,

The superscript^T indicates the transpose of a matrix or vector, Ax ≤ b is used for joint torque limits and infinite norm constraints, and x^± indicates the upper and lower limits of x.

如果方案含有最小力性能指标时，根据α₃取值的不同，决策变量x为上述对应变量加上辅助变量s的向量增广，辅助变量s是二次型规划问题中用于辅助对应最小力性能指标变量求解的变量，同时，它是一个非负数，并且取值为最小力性能指标中的最大分量的绝对值。系数矩阵Q和C以及系数向量p也对应加0进行增广。If the scheme contains the minimum force performance index, according to the value of_α3 , the decision variable x is the vector augmentation of the above corresponding variable plus the auxiliary variable s, and the auxiliary variable s is used to assist the corresponding minimum force in the quadratic programming problem The performance index variable is the variable to be solved. At the same time, it is a non-negative number, and its value is the absolute value of the largest component in the minimum force performance index. The coefficient matrices Q and C and the coefficient vector p are also correspondingly augmented by adding 0.

通过二次型规划求解器对二次型规划问题进行求解，具体为：将所述二次型规划问题进一步变换为分段线性投影方程，从而构造相应的二次型规划求解器(如，二次型规划数值算法)进行求解；The quadratic programming problem is solved by a quadratic programming solver, specifically: the quadratic programming problem is further transformed into a piecewise linear projection equation, thereby constructing a corresponding quadratic programming solver (such as a quadratic programming solver) Subtype programming numerical algorithm) to solve;

本发明与先有技术相比，具有以下优点：Compared with the prior art, the present invention has the following advantages:

本发明能有效克服单一/同层优化方案存在的不足，提供了一种操作方便、工作量小的冗余度机械臂的不同层性能指标同时优化方法。The invention can effectively overcome the shortcomings of the single/same-layer optimization scheme, and provides a method for simultaneously optimizing performance indexes of different layers of a redundant mechanical arm with convenient operation and small workload.

附图说明Description of drawings

图1为本发明的流程图。Fig. 1 is a flowchart of the present invention.

具体实施方式Detailed ways

下面结合附图对本发明做进一步的说明。The present invention will be further described below in conjunction with the accompanying drawings.

图1所示的一种冗余度机械臂的不同层性能指标同时优化方法主要由建立不同层性能指标同时优化的冗余度解析方案1、转为二次型规划问题2、二次型规划求解器3、下位机控制器4、冗余度机械臂5组成。A simultaneous optimization method for different layers of performance indicators of a redundant manipulator shown in Figure 1 is mainly composed of establishing a redundancy analysis scheme for simultaneous optimization of different layers of performance indicators 1, converting to a quadratic programming problem 2, quadratic programming Thesolver 3, the lower computer controller 4, and the redundant mechanical arm 5 are composed.

首先根据所需要优化的不同层性能指标，通过引入权值调节因子，建立相应的冗余度解析方案；然后利用不同层性能指标的等效性及引入等效性参数，将该方案转化为一个统一的二次型规划问题；从而构造相应的二次型规划求解器(如，二次型规划数值算法)来求解该问题；最后将求解结果用于驱动机械臂的各个关节电机使机械臂完成给定的末端任务。First, according to the performance indicators of different layers to be optimized, a corresponding redundancy analysis scheme is established by introducing weight adjustment factors; then, the scheme is transformed into a A unified quadratic programming problem; thereby constructing a corresponding quadratic programming solver (such as a quadratic programming numerical algorithm) to solve the problem; finally, the solution result is used to drive each joint motor of the manipulator to make the manipulator complete given terminal task.

根据所需要优化的不同层性能指标，通过引入权值调节因子，不同层性能指标同时优化的冗余度解析方案可设计为：According to the performance indicators of different layers that need to be optimized, by introducing weight adjustment factors, the redundancy analysis scheme for simultaneous optimization of performance indicators of different layers can be designed as:

最小化：

minimize:

约束条件： $J \overset{\cdot}{θ} = \overset{\cdot}{r}, - - - (2)$ Restrictions: $J \overset{&Center Dot;}{θ} = \overset{&Center Dot;}{r}, - - - (2)$

$J J \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} = = \overset{\cdot &Center Dot; \cdot &Center Dot;}{r r} - - \overset{\cdot &Center Dot;}{J J} \overset{\cdot &Center Dot;}{θ θ},, - - - - - - ((33))$

$τ τ = = H h \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} + + c c + + g g,, - - - - - - ((44))$

θ^-≤θ≤θ⁺，(5)θ⁻ ≤ θ ≤ θ⁺ , (5)

${\overset{\cdot &Center Dot;}{θ θ}}^{- -} \leq \leq \overset{\cdot &Center Dot;}{θ θ} \leq \leq {\overset{\cdot &Center Dot;}{θ θ}}^{+ +},, - - - - - - ((66))$

${\overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ}}^{- -} \leq \leq \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} \leq \leq {\overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ}}^{+ +},, - - - - - - ((77))$

τ^-≤τ≤τ⁺，(8)τ^- ≤ τ ≤ τ⁺ , (8)

其中，

和分别对应于在角度层、速度层和加速度层上欲优化的性能指标，α₁≥0、α₂≥0和α₃≥0为权值调节因子且保证两个或两个以上权值调节因子同时不为零；θ表示关节角度，

表示关节速度，

表示关节加速度，τ表示关节力矩；等式约束

对应于机械臂在速度层的末端运动轨迹，J表示雅可比矩阵，表示机械臂末端执行器的速度；等式约束

对应于机械臂在加速度层的末端运动轨迹，

表示雅可比矩阵J的时间导数，

表示机械臂末端执行器的加速度；等式约束

为机械臂的动力学方程，H表示机械臂的惯性矩阵，c表示离心力变量，g表示重力变量；θ^±、

和τ^±分别表示关节角度极限、关节速度极限、关节加速度极限和关节力矩极限。in,

and Corresponding to the performance indicators to be optimized on the angle layer, velocity layer and acceleration layer respectively, α₁ ≥ 0, α₂ ≥ 0 and α₃ ≥ 0 are weight adjustment factors and two or more weight adjustment factors are guaranteed At the same time, it is not zero; θ represents the joint angle,

represents the joint velocity,

Represents the joint acceleration, τ represents the joint torque; the equality constraint

Corresponding to the end motion trajectory of the manipulator in the velocity layer, J represents the Jacobian matrix, represents the velocity of the end effector of the manipulator; the equation constraint

Corresponding to the end motion trajectory of the manipulator in the acceleration layer,

Denotes the time derivative of the Jacobian matrix J,

is 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; θ^± ,

and τ^± represent joint angle limit, joint velocity limit, joint acceleration limit and joint torque limit, respectively.

利用不同层性能指标等效性原理以及引入等效性参数，上述带物理约束的冗余度机械臂的不同层性能指标同时优化方法(1)-(8)便可描述为如下的二次型规划问题：Using the principle of equivalence of performance indicators of different layers and introducing equivalence parameters, the simultaneous optimization method (1)-(8) of the performance indicators of different layers of the above-mentioned redundant manipulator with physical constraints can be described as the following quadratic form Planning questions:

最小化：x^TQx/2+p^Tx，(9)Minimize: x^T Qx/2+p^T x, (9)

约束条件：Cx＝d，(10)Constraints: Cx=d, (10)

Ax≤b，(11)Ax≤b, (11)

x^-≤x≤x⁺，(12)x⁻ ≤ x ≤ x⁺ , (12)

其中，只考虑不同层能量最小化方案时，当α₃＞0时，决策变量x表示关节加速度，Q可由权值调节因子α₁，α₂，α₃对单位矩阵I、惯性矩阵H和H²的加权和获得，p可由权值调节因子α₁，α₂，α₃及等效性参数对θ，c，g和H的加权和获得，C＝J，

如果方案含有最小力性能指标时，根据α₃取值的不同，决策变量x为上述对应变量加上辅助变量s的向量增广，系数矩阵Q和C以及系数向量p也对应加0进行增广。上标^T表示矩阵或向量的转置，Ax≤b用于关节力矩极限和无穷范数约束，x^±表示x的上下限。为了容易理解，考虑一个性能指标为“最小化

”且受约束于(2)-(8)的冗余度解析方案，其中α₂＞0和α₃＞0，||·||₂表示向量的二范数，利用不同层性能指标等效性及引入等效性参数，可将该方案转化为如(9)-(12)所描述的二次型规划问题，且相应的参数定义如下：Among them, when only the energy minimization schemes of different layers are considered, when α₃ >0, the decision variable x represents the joint acceleration, and Q can be adjusted by the weight adjustment factors α₁ , α₂ , α₃ on the identity matrix I, inertia matrices H and H² , p can be obtained by the weighted sum of weight adjustment factors α₁ , α₂ , α₃ and equivalent parameters to θ, c, g and H, C=J,

If the scheme contains the minimum force performance index, according to the different values of α₃ , the decision variable x is the vector augmentation of the above corresponding variable plus the auxiliary variable s, and the coefficient matrices Q and C and the coefficient vector p are correspondingly augmented with 0 . The superscript^T indicates the transpose of a matrix or vector, Ax ≤ b is used for joint torque limits and infinite norm constraints, and x^± indicates the upper and lower limits of x. For ease of understanding, consider a performance metric as "minimizing

” and is constrained by the redundancy analysis scheme of (2)-(8), where α₂ >0 and α₃ >0, ||·_|| and introducing equivalence parameters, the scheme can be transformed into a quadratic programming problem as described in (9)-(12), and the corresponding parameters are defined as follows:

$x = \overset{\cdot \cdot}{θ},$ Q＝(α₂+α₃)I， $p = α_{2} λ \overset{\cdot}{θ},$ $x = \overset{&Center Dot; &Center Dot;}{θ},$ Q=(α₂ +α₃ )I, $p = α_{2} λ \overset{\cdot}{θ},$

C＝J， $d = \overset{\cdot \cdot}{r} - \overset{\cdot}{J} \overset{\cdot}{θ},$ $A = [\begin{matrix} H \\ - H \end{matrix}],$ $b = [\begin{matrix} τ^{+} - c - g \\ c + g - τ^{-} \end{matrix}],$ C=J, $d = \overset{\cdot \cdot}{r} - \overset{\cdot}{J} \overset{\cdot}{θ},$ $A = [\begin{matrix} h \\ - h \end{matrix}],$ $b = [\begin{matrix} τ^{+} - c - g \\ c + g - τ^{-} \end{matrix}],$

${x x}^{- -} = = max max {{{κ κ}_{p p} (({θ θ}^{- -} + + &upsi; &upsi; - - θ θ)),, {κ κ}_{V V} (({\overset{\cdot \cdot}{θ θ}}^{- -} - - \overset{\cdot \cdot}{θ θ})),, {\overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ}}^{- -}}},,$ ${x x}^{+ +} = = min min {{{κ κ}_{p p} (({θ θ}^{+ +} - - &upsi; &upsi; - - θ θ)),, {κ κ}_{V V} (({\overset{\cdot \cdot}{θ θ}}^{+ +} - - \overset{\cdot &Center Dot;}{θ θ})),, {\overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ}}^{+ +}}},,$

其中，I表示单位矩阵，等效性参数λ为正数且远大于0，关节极限转换参数κ_p＞0和κ_v＞0，关节极限转换裕量υ＞0。Among them, I represents the identity matrix, the equivalence parameter λ is a positive number and much greater than 0, the joint limit conversion parameters κ_p >0 and κ_v >0, and the joint limit conversion margin υ>0.

并且，上述的二次型规划问题(9)-(12)等价于如下的分段线性投影方程：Moreover, the above quadratic programming problems (9)-(12) are equivalent to the following piecewise linear projection equations:

P_Ω(y-(My+q))-y＝0，(13)P_Ω (y-(My+q))-y=0, (13)

其中P_Ω(·)表示分段线性投影算子。分段线性投影方程(13)中的原对偶决策变向量y，增广系数矩阵M和向量q分别定义如下：where P_Ω (·) represents the piecewise linear projection operator. The original dual decision variable vector y, the augmented coefficient matrix M and the vector q in the piecewise linear projection equation (13) are respectively defined as follows:

$y the y = = [\begin{matrix} x x \\ u u \\ v v \end{matrix}],,$ $M m = = [\begin{matrix} Q Q & - - {C C}^{T T} & {A A}^{T T} \\ C C & 00 & 00 \\ - - A A & 00 & 00 \end{matrix}],,$ $q q = = [\begin{matrix} p p \\ - - d d \\ b b \end{matrix}],,$

其中，对偶决策变量u和v分别对应于等式约束(10)和不等式约束(11)。对于上述的分段线性投影方程(13)和二次型规划问题(9)-(12)，可采用如下的二次型规划数值算法(即，二次型规划求解器)来求解：Among them, the dual decision variables u and v correspond to equality constraints (10) and inequality constraints (11), respectively. For the above-mentioned piecewise linear projection equation (13) and quadratic programming problems (9)-(12), the following quadratic programming numerical algorithm (ie, quadratic programming solver) can be used to solve:

e(y^k)＝y^k-P_Ω(y^k-(My^k+q))，e(y^k )=y^k -P_Ω (y^k -(My^k +q)),

y^k+1＝y^k-ρ(y^k)φ(y^k)，y^k+1 = y^k -ρ(y^k )φ(y^k ),

φ(y^k)＝(M^T+I)e(y^k)，φ(y^k )=(M^T +I)e(y^k ),

$ρ ρ (({y the y}^{k k})) = = {| | | | e e (({y the y}^{k k})) | | | |}_{22}^{22} / / {| | | | φ φ (({y the y}^{k k})) | | | |}_{22}^{22},,$

其中，迭代次数k＝0，1，2，...。给定初始值y⁰，通过该算法的不断迭代，便可得到分段线性投影方程(13)的解，从而得到二次型规划问题(9)-(12)的最优解，也即不同层性能指标同时优化的冗余度解析方案(1)-(8)的最优解。Wherein, the number of iterations k=0, 1, 2, . . . Given the initial value y⁰ , through continuous iteration of the algorithm, the solution of piecewise linear projection equation (13) can be obtained, thereby obtaining the optimal solution of quadratic programming problems (9)-(12), that is, different The optimal solution of redundancy resolution schemes (1)-(8) with simultaneous optimization of layer performance indicators.

通过二次型规划求解器得到该二次型规划问题的解后，再将求解结果传递给下位机控制器驱动机械臂的运动，从而使得机械臂完成给定的末端任务。After obtaining the solution of the quadratic programming problem through the quadratic programming solver, the solution result is passed to the lower computer controller to drive the movement of the manipulator, so that the manipulator can complete the given end task.

Claims

1. A method for simultaneously optimizing performance indexes of different layers of a redundant manipulator is characterized by comprising the following steps:

according to performance indexes of an angle layer, a speed layer and an acceleration layer, establishing a corresponding redundancy analysis scheme by introducing weight adjusting factors, wherein the redundancy analysis scheme is restricted by a Jacobian matrix equation of speed and acceleration, a kinetic equation of a mechanical arm, a joint angle limit, a joint speed limit, a joint acceleration limit and a joint moment limit;

converting the redundancy resolution scheme into a uniform quadratic programming problem by utilizing the equivalence of performance indexes of an angle layer, a speed layer and an acceleration layer and introducing equivalence parameters;

solving the quadratic programming problem through a quadratic programming solver;

and the lower computer controller drives the mechanical arm to complete a given end task according to the solution result of the quadratic programming problem.

2. The method of claim 1, wherein the redundancy resolution scheme for simultaneously optimizing the performance indexes of different layers of the manipulator is designed as follows:

minimization，

Wherein

Figure 2011103716880100001DEST_PATH_IMAGE004

、

Figure 2011103716880100001DEST_PATH_IMAGE006

And

Figure 2011103716880100001DEST_PATH_IMAGE008

respectively corresponding to performance indexes to be optimized on an angle layer, a speed layer and an acceleration layer,

Figure 2011103716880100001DEST_PATH_IMAGE010

、

Figure 2011103716880100001DEST_PATH_IMAGE012

and

Figure 2011103716880100001DEST_PATH_IMAGE014

is a weight value adjustment factorTwo or more weight value adjusting factors are not zero at the same time;

Figure 2011103716880100001DEST_PATH_IMAGE016

the angle of the joint is represented by,

the velocity of the joint is represented by,the acceleration of the joint is represented by,representing the joint moment.

3. The method of simultaneously optimizing performance indexes of different layers of a redundant manipulator according to claim 2, wherein the redundancy resolution scheme for simultaneously optimizing performance indexes of an angle layer, a velocity layer and an acceleration layer is constrained by:

，

，

，

，

，

，

;

wherein the equality constrains

Corresponding to the motion track of the mechanical arm at the tail end of the speed layer,

a jacobian matrix representing the mechanical arm,

representing a velocity of the end effector of the robotic arm; constraint of equality

Corresponding to the motion track of the mechanical arm at the tail end of the acceleration layer,

representing a Jacobian matrix

The time derivative of (a) of (b),

representing an acceleration of the end effector of the robotic arm; constraint of equality

Is a kinetic equation of the mechanical arm,

a matrix of inertia representing the arm of the robot,which represents a variable of the centrifugal force,

to indicate gravityA variable;

、

、

and

respectively representing joint angle limit, joint velocity limit, joint acceleration limit and joint moment limit.

4. The method of claim 3, wherein the quadratic programming problem has a performance index of

With the constraint condition of

，，

Wherein only different layer energy minimization schemes are considered when

Time, decision variables

The acceleration of the joint is represented by,

adjusted by weightFactor(s)

，

，

For unit matrixInertia matrix

And

the weighted sum of (a) and (b) is obtained,

adjusting the factor by the weight

，，

And equivalence parameter pairs

，

，

And

the weighted sum of (a) and (b) is obtained,

，

(ii) a When in useTime, decision variablesThe velocity of the joint is represented by,adjusting the factor by the weight

,

For unit matrix

The weighted sum of (a) and (b) is obtained,adjusting the factor by the weight

，

And an equivalence parameter, and obtaining the product of the equivalence parameter,

，

(ii) a Upper label

Representing a transpose of a matrix or a vector,

for joint moment limit and infinite norm constraints,

to represent

The upper and lower limits of (2).

5. The method of claim 4, wherein the simultaneous optimization of performance indexes of different layers of the redundant manipulator is performed according to the minimum force performance index if the simultaneous optimization of performance indexes of different layers includes the minimum force performance index

Difference in value, decision variable

Adding auxiliary variables to the above-mentioned correspondent variables

Vector augmentation of, the auxiliary variable

The value is the absolute value of the maximum component in the minimum force performance index, and the coefficient matrix

And

and coefficient vector

Also correspondingly addAnd (4) carrying out augmentation.

6. The method for simultaneously optimizing performance indexes of different layers of a redundant manipulator according to claims 1 to 5, wherein a quadratic programming problem is solved by a quadratic programming solver, specifically: and further transforming the quadratic programming problem into a piecewise linear projection equation, thereby constructing a corresponding quadratic programming solver for solving.