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CN107153356A - Decoupling control method between a kind of hydraulic-driven joint type legged type robot joint - Google Patents

Decoupling control method between a kind of hydraulic-driven joint type legged type robot joint Download PDF

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CN107153356A
CN107153356A CN201710407284.XA CN201710407284A CN107153356A CN 107153356 A CN107153356 A CN 107153356A CN 201710407284 A CN201710407284 A CN 201710407284A CN 107153356 A CN107153356 A CN 107153356A
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robot
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高炳微
王思凯
任东
任东一
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Harbin University of Science and Technology
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Harbin University of Science and Technology
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    • G05BCONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

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Abstract

一种液压驱动关节型足式机器人关节间解耦控制方法,将机、液共同耦合作为液压驱动关节型足式机器人耦合行为进行研究,建立机器人腿部关节间的耦合模型;计算腿部关节各支路的耦合度,研究液压驱动关节型足式机器人关节间的耦合特性;根据液压驱动关节型足式机器人腿部关节间的耦合关系,将机器人各运动支链的机构动力学耦合力经过雅可比矩阵变换分解到各关节的液压伺服驱动系统,并将其作为液压系统的干扰力,然后再对电液伺服系统进行解耦控制,以便消除整个机器人系统的交联耦合影响。本发明将机、液共同耦合作为机器人耦合行为进行研究,设计解耦控制方案,提高机器人的控制精度,保持系统具有良好的控制性能,推动液压驱动关节型足式机器人动态性能和自动化水平的提升。

A decoupling control method between the joints of a hydraulically driven articulated footed robot. The joint coupling of machine and fluid is studied as the coupling behavior of a hydraulically driven articulated footed robot, and the coupling model between the leg joints of the robot is established; The coupling degree of the branches is studied to study the coupling characteristics between the joints of the hydraulically driven articulated footed robot; according to the coupling relationship between the leg joints of the hydraulically driven articulated footed robot, the mechanism dynamic coupling force of each branch chain of the robot is passed through the elegant The comparable matrix transformation is decomposed into the hydraulic servo drive system of each joint, and it is used as the disturbance force of the hydraulic system, and then the electro-hydraulic servo system is decoupled to eliminate the cross-link coupling effect of the entire robot system. The invention studies the joint coupling of machine and fluid as the coupling behavior of the robot, designs a decoupling control scheme, improves the control accuracy of the robot, maintains the system with good control performance, and promotes the improvement of the dynamic performance and automation level of the hydraulically driven articulated foot robot .

Description

Decoupling control method between a kind of hydraulic-driven joint type legged type robot joint
Technical field
The present invention relates to hydraulic-driven prosthetic robot control technology field, more particularly to a kind of hydraulic-driven articulated type Decoupling control method between joint of robot.
Background technology
Hydraulic-driven joint type every leg of legged type robot is a multivariant serial mechanism of multi-joint, leg system With non-linear and strong coupling, the multifreedom motion coupling being not only present between each movement branched chain of leg mechanism is coupled, also It is present in the hydraulic drive mechanism in each joint, therefore, the coupling between each joint of hydraulic pressure quadruped robot is machine, liquid collective effect Result.Coupling between this joint produced jointly by mechanical structure and the multiple variables of hydraulic-driven so that system Control becomes sufficiently complex.During robot actual motion, want to be controlled one of joint, it is necessary to by leg Other all joints in portion are all locked, once these joints are unlocked, mutual infection is there is between each joint, because This, in robot normal motion, each joint motion can by other joints coupling influence.This gives hydraulic pressure Coordinated movement of various economic factors control brings very big difficulty between quadruped robot multi-joint, thus, between reduction hydraulic pressure quadruped robot joint Mutual infection, carrying out uneoupled control to robot each joint becomes very necessary.
The content of the invention
Decoupling control method between a kind of hydraulic-driven joint type legged type robot joint
Decoupling control method is realized according to the following steps between hydraulic-driven joint type legged type robot joint:
Step A, set up between joint and be crosslinked coupling model:
The application is studied the coupling behavior of machine, liquid coupled in common as hydraulic-driven joint type legged type robot, Based on the leg mechanical structure to hydraulic-driven joint type legged type robot and the analysis of hydraulic servo driving system, according to mechanism Dynamics and hydraulic principle, set up the kinetic model and leg hydraulic servo driving system of leg mechanism of robot respectively Model, finally set up the interarticular coupling model of hydraulic-driven joint type legged type robot, be that the uneoupled control of robot is carried For foundation.
Because in motion process, the motion in each joint of hydraulic-driven joint type legged type robot belongs to low-speed motion, Therefore, interarticular motion is influenceed smaller by coriolis force and centripetal force, can be neglected, but influenceed by inertia force and gravity It is larger, for the ease of analyzing and solving problem, it is necessary to catch the principal contradiction of problem, therefore only consider the shadow of inertia force and gravity Ring, ignore the influence of coriolis force and centripetal force, so, hydraulic-driven joint type legged type robot leg power model be changed into as Lower form:
Then, the kinetics relation exerted oneself between hydraulic cylinder displacement in any two joint is as follows:
In formula,For inertial matrix,For gravity.
Laplace transformation is carried out to above formula, coupled relation formula can be obtained as follows:
All there is obvious coupling between hydraulic-driven joint type legged type robot any two joint it can be seen from formula Close.
Then, kinetic model is brought into hydraulic system model, the model that just can obtain whole control system is as follows:
In formula, the elements in a main diagonal product represents the transmission function of single joint respectively, rather than diagonal term is joint Between coupling terms.
Step B, the degree of coupling for calculating each branch road in leg joint:
It is provided with n input, the n coupling object exported:
Y (s)=C (s) R (s)
In formula:
Y (s)=[y1(s),y2(s)…yn(s)]T
R (s)=[r1(s),r2(s)…rn(s)]T
Wherein, R (s) is the input of object, and Y (s) is the output of object, and C (s) is coupling object matrix.
Object C (s) is divided into four parts, obtained:
In formula:
Wherein, C11(s) it is [y1(s) r1(s) controlling brancher].
If K is C (s) static gain matrix, i.e.,:
Then [y1(s) r1(s)] degree of coupling of branch road is defined as:
Similarly, [yi(s) rj(s)] degree of coupling of branch road is defined as:
In formula,K is scratched for the i-th row vector of matrix KijFormer tactic row vector is pressed afterwards;For matrix K Jth column vector scratches KijFormer tactic column vector is pressed afterwards;KPFor matrix K scratch after the i-th row and jth row remaining element by Former tactic matrix.
The summation of all pairing coupling factor of branch is the degree of coupling of system, then the degree of coupling of system is:
η=Σ Pij
Asked for the degree of coupling of each branch road, it is possible to understand the coupled characteristic of system, judged according to its coupled characteristic be Whether system needs to be decoupled.The coupled characteristic that the degree of coupling is reflected can be summarized as follows:
Work as PijWhen=0, no coupling, it is not necessary to use decoupling measure;
Work as PijDuring > 0, there is coupling, coupling reduces the control action of the passage, it is necessary at this moment to take decoupling measure 's;
Work as PijDuring=∞, superpower coupling, system is not normally functioning.
Decoupling control method between step C, design hydraulic-driven joint type legged type robot joint:
Under motion state, the coupling between each joint of hydraulic-driven joint type legged type robot is actually that machine, liquid are common The result of same-action, according to the coupling model set up, the application is converted by Jacobian matrix, by the mechanism of robot system Dynamics Coupling power decomposes each joint fluid cylinder pressure, by the Dynamics Coupling masterpiece of mechanism be hydraulic system outer perturbed force, so Uneoupled control is carried out to electrohydraulic servo system again afterwards, to reduce the crosslinking coupling influence of whole robot system.
There is the kinetic model of coupling according to hydraulic-driven joint type legged type robot leg, will be main in M (y), G (y) Cornerwise non-coupled item removes, then can obtain corresponding bonding force and be:
In formula,For inertial matrix,For gravity.
FdAs bonding force decomposes the perturbed force in each fluid power system.
So converted by Jacobian matrix, regard the mechanism dynamic bonding force of each movement branched chain of robot as outer interference Power is transformed into each joint fluid cylinder pressure, and control variable is transformed into joint space from working space, then hydraulic system carried out again Uneoupled control, to realize the decoupling of whole robot system.
Invention effect:
The present invention is directed to the crosslinking coupled problem existed between each joint in hydraulic-driven joint type legged type robot leg, will Machine, liquid coupled in common are studied as the coupling behavior of hydraulic-driven joint type legged type robot, are set up between joint of robot Coupling model;Calculate each branch road in leg joint the degree of coupling, illustrate hydraulic-driven joint type legged type robot uneoupled control must The property wanted;The interarticular coupled characteristic of hydraulic-driven joint type legged type robot is studied, the robot that changes commanders is become by Jacobian matrix The mechanism dynamic bonding force of system is controlled as the outer perturbed force of hydraulic system, realizes the sufficient formula of hydraulic-driven joint type Each interarticular uneoupled control of robot leg, improves the control accuracy of robot, and holding system has good control performance, Promote the lifting of hydraulic pressure quadruped robot dynamic property and automatization level.
Brief description of the drawings
Fig. 1 is thigh and calf articular couple relation principle block diagram;
Fig. 2 is Decoupling Control figure between each joint in hydraulic-driven joint type legged type robot leg;
Embodiment
With reference to embodiments and accompanying drawing is described in further detail to the present invention, but the present invention embodiment Not limited to this.
Embodiment:Uneoupled control problem between hydraulic-driven joint type legged type robot size leg joint
Step A, set up coupling model is crosslinked between thigh and calf joint:
The application is studied the coupling behavior of machine, liquid coupled in common as hydraulic-driven joint type legged type robot, Based on the leg mechanical structure to hydraulic-driven joint type legged type robot and the analysis of hydraulic servo driving system, according to mechanism Dynamics and hydraulic principle, set up the kinetic model and leg hydraulic servo driving system of leg mechanism of robot respectively Model, finally set up the coupling model in two joints of robot thigh and shank, foundation provided for the uneoupled control of robot, The theory diagram of its coupled relation is as shown in Figure 1.
In accompanying drawing 1, u2For thigh joint servo valve input voltage;u3For calf joint servo valve input voltage;y2For thigh The output displacement in joint;y3For the output displacement of calf joint.
From accompanying drawing 1 as can be seen that the output y of big leg joint2By the output y of calf joint3Influence, and calf joint Output y3It is similarly subjected to the output y of big leg joint2Influence, therefore, the big leg joint of hydraulic pressure quadruped robot and calf joint it Between there is obvious coupling, moreover, from coupled relation, coupling influence between two joint not only with mechanism dynamic parameter It is relevant, it is also relevant with hydraulic system parameters.
Step B, the degree of coupling for calculating each branch road in leg joint:
In order to understand the coupled characteristic of system, it is necessary to calculate the degree of coupling of each branch road, judged according to the size of the degree of coupling Whether system, which needs, is decoupled.
The coupling object that can draw hydraulic-driven joint type legged type robot leg thigh and calf according to step A is:
Then static gain matrix K is:
So, obtaining degree of coupling matrix P by calculating is:
From above-mentioned degree of coupling matrix, the robot leg system uses [y2(s) u2(s)]、[y3(s) u3 (s) matching method], the motion with actual robot thigh and calf is consistent, and the degree of coupling of system is η=P11+P22=1.28, coupling It is right larger, it is necessary to carry out uneoupled control, to improve the Control platform of system.
Decoupling control method between step C, design hydraulic-driven joint type legged type robot thigh and calf joint:
Under motion state, the coupling between hydraulic-driven joint type legged type robot thigh and shank be actually machine, The coefficient result of liquid, according to the coupling model set up, the mechanism for becoming robot system of changing commanders using Jacobian matrix is moved Coupling with Mechanics power decomposes each joint fluid cylinder pressure, and the Dynamics Coupling masterpiece of mechanism is come for the outer perturbed force of electrohydraulic servo system It is controlled, uneoupled control then is being carried out to electrohydraulic servo system, to realize the uneoupled control of whole robot system.It is controlled The principle of system is as shown in Figure 2.
According to the coupling model set up, the non-coupled item of leading diagonal in M (y), G (y) is removed, be can obtain corresponding Bonding force is:
In formula,For inertial matrix,For gravity.
FdAs bonding force decomposes the perturbed force in each fluid power system.
Therefore, the bonding force that can further obtain between robot thigh, shank is:
So converted by Jacobian matrix, regard the mechanism dynamic bonding force of each movement branched chain of robot as outer interference Power is transformed into each joint fluid cylinder pressure, and control variable is transformed into joint space from working space, then hydraulic system carried out again Uneoupled control, to realize the decoupling of whole robot system.

Claims (3)

1.一种液压驱动关节型足式机器人关节间解耦控制方法,其特征在于将机、液共同耦合作为液压驱动关节型足式机器人耦合行为进行研究,建立机器人关节间的耦合模型;计算腿部关节各支路的耦合度,研究液压驱动关节型足式机器人关节间的耦合特性;通过雅可比矩阵变换将机器人系统的机构动力学耦合力作为液压系统的外干扰力来进行控制,实现液压驱动关节型足式机器人腿部各关节间的解耦控制。1. A decoupling control method between joints of a hydraulically driven articulated footed robot, characterized in that the joint coupling of machine and fluid is studied as the coupling behavior of a hydraulically driven articulated footed robot, and the coupling model between robot joints is established; The coupling degree of each branch of the internal joints is studied, and the coupling characteristics between the joints of the hydraulically driven articulated legged robot are studied; the mechanism dynamic coupling force of the robot system is controlled as the external disturbance force of the hydraulic system through the Jacobian matrix transformation, and the hydraulic pressure is realized. Decoupling control between the joints of the driven leg of an articulated legged robot. 2.根据权利要求1所述的一种液压驱动关节型足式机器人关节间解耦控制方法,其特征在于液压驱动关节型足式机器人在运动状态下各关节之间的耦合实际上是机、液共同作用的结果;根据液压伺服控制原理先建立伺服阀阀芯位移和液压缸出力之间的关系,然后采用拉格朗日方法建立机器人腿部机构驱动力矩和关节转角之间的动力学关系,最终利用空间几何关系理论建立机器人关节间的耦合模型;2. The decoupling control method between joints of a hydraulically driven articulated footed robot according to claim 1, wherein the coupling between each joint of the hydraulically driven articulated footed robot is actually a machine, According to the principle of hydraulic servo control, the relationship between the displacement of the servo valve spool and the output force of the hydraulic cylinder is first established, and then the dynamic relationship between the driving torque of the robot leg mechanism and the joint rotation angle is established by using the Lagrangian method , and finally use the theory of spatial geometric relationship to establish the coupling model between the robot joints; 液压驱动关节型足式机器人任意两个关节之间的耦合模型如下:The coupling model between any two joints of the hydraulically driven articulated leg robot is as follows: <mrow> <msub> <mi>k</mi> <mn>0</mn> </msub> <mi>J</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mn>21</mn> </msub> <msup> <mi>s</mi> <mn>3</mn> </msup> <mo>+</mo> <msub> <mi>BM</mi> <mn>21</mn> </msub> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mn>21</mn> </msub> <mo>+</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> <mi>s</mi> <mo>+</mo> <msub> <mi>BG</mi> <mn>21</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>M</mi> <mn>22</mn> </msub> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mn>22</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>M</mi> <mn>31</mn> </msub> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mn>31</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mn>32</mn> </msub> <msup> <mi>s</mi> <mn>3</mn> </msup> <mo>+</mo> <msub> <mi>BM</mi> <mn>32</mn> </msub> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mn>32</mn> </msub> <mo>+</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> <mi>s</mi> <mo>+</mo> <msub> <mi>BG</mi> <mn>32</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> <mrow> <msub> <mi>k</mi> <mn>0</mn> </msub> <mi>J</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mn>21</mn> </msub> <msup> <mi>s</mi> <mn>3</mn> </msup> <mo>+</mo> <msub> <mi>BM</mi> <mn>21</mn> </msub> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mn>21</mn> </msub> <mo>+</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> <mi>s</mi> <mo>+</mo> <msub> <mi>BG</mi> <mn>21</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>M</mi> <mn>22</mn> </msub> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mn>22</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>M</mi> <mn>31</mn> </msub> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mn>31</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mn>32</mn> </msub> <msup> <mi>s</mi> <mn>3</mn> </msup> <mo>+</mo> <msub> <mi>BM</mi> <mn>32</mn> </msub> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mn>32</mn> </msub> <mo>+</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> <mi>s</mi> <mo>+</mo> <msub> <mi>BG</mi> <mn>32</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> 式中,主对角线元素乘积分别表示单个关节的传递函数,而非对角线项即为关节间的耦合项。In the formula, the product of the main diagonal elements represents the transfer function of a single joint, and the non-diagonal items are the coupling items between joints. 3.根据权利要求1所述的一种液压驱动关节型足式机器人关节间解耦控制方法,其特征在于通过雅可比矩阵变换,将机器人各运动支链的机构动力学耦合力作为外干扰力转化到各关节液压缸,将控制变量从工作空间转化到关节空间,然后再对液压系统进行解耦控制,以实现整个机器人系统的解耦;3. The decoupling control method between joints of a hydraulically driven articulated footed robot according to claim 1, wherein the mechanism dynamic coupling force of each branch chain of the robot is used as an external disturbance force through Jacobian matrix transformation Convert to each joint hydraulic cylinder, convert the control variable from the work space to the joint space, and then decouple the hydraulic system to realize the decoupling of the entire robot system; 根据所建立的耦合模型,将M(y)、G(y)中主对角线的非耦合项去掉,可得到相应的耦合力如下:According to the established coupling model, the uncoupling items of the main diagonal in M(y) and G(y) are removed, and the corresponding coupling force can be obtained as follows: <mrow> <msub> <mi>F</mi> <mi>d</mi> </msub> <mo>=</mo> <msub> <mi>M</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>G</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>F</mi> <mi>d</mi> </msub> <mo>=</mo> <msub> <mi>M</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>G</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> 式中,为惯性矩阵,为重力项;In the formula, is the inertia matrix, is the gravity item; Fd即为耦合力分解到各液压驱动系统上的干扰力。F d is the interference force decomposed from the coupling force to each hydraulic drive system.
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