CN114800477B

CN114800477B - Minimum flow-based redundant hydraulic mechanical arm motion planning method

Info

Publication number: CN114800477B
Application number: CN202210572618.XA
Authority: CN
Inventors: 丁孺琦; 王振; 李刚; 胡国良
Original assignee: East China Jiaotong University
Current assignee: East China Jiaotong University
Priority date: 2022-05-25
Filing date: 2022-05-25
Publication date: 2023-03-14
Anticipated expiration: 2042-05-25
Also published as: CN114800477A

Abstract

The invention discloses a minimum flow-based redundant hydraulic mechanical arm motion planning method, which comprises the following steps of establishing a redundant hydraulic mechanical arm model; step two, minimum flow optimization: the flow required by the motion of the redundant hydraulic mechanical arm is determined by the motion of each joint, and in order to reduce the flow consumption in the motion process of the mechanical arm, the relation between the cylinder speed of the hydraulic cylinder and the tail end speed of the mechanical arm is deduced from a positive kinematic equation of the hydraulic mechanical arm; and then, a weighting matrix is introduced on the basis of the minimum norm of the cylinder speed, and the instantaneous flow of the mechanical arm is continuously reduced by dynamically optimizing the weight of the weighting matrix, so that the aim of redundant decomposition with the least energy consumption is fulfilled.

Description

Minimum flow-based redundant hydraulic mechanical arm motion planning method

Technical Field

The invention relates to the field of hydraulic control, in particular to a minimum flow-based redundant hydraulic mechanical arm motion planning method.

Background

Compared with an electric driving mechanical arm, the hydraulic mechanical arm adopts a centralized driving mode, the movement throttling loss of the multiple actuators is large, and therefore energy optimization is particularly important. The energy optimization of the hydraulic mechanical arm can be carried out from two aspects of reducing the energy consumption of a hydraulic system and the energy required by movement; much of the existing research has been around the former. For the latter, the redundancy characteristic of multiple joints of the hydraulic mechanical arm can be utilized, and the load energy consumption is reduced through redundancy self-motion planning when the tail end tracks are the same.

In consideration of the hydraulic driving characteristic, the prior art proposes a minimum cylinder speed norm method, namely on the basis of a gradient projection method, the cylinder speed norm minimum is taken as an optimization criterion, and a better energy-saving characteristic can be obtained. However, the minimum cylinder speed norm is the minimum square sum of the solved cylinder speed, and does not directly describe the targets of flow, pressure and the like of the energy consumption of the hydraulic system, so that the minimum hydraulic driving energy consumption has a deviation varying with the position and the time, and the redundant decomposition with optimal energy is difficult to obtain through calculation compensation. The core of the minimum cylinder velocity norm method is Jacobian matrix transformation, and the existing research shows that the weighted matrix weight optimization can further adjust the distribution relation of the hydraulic cylinder velocities on the basis of a primary optimization target (for example, the cylinder velocity norm is minimum), so that the minimum value of the system flow which directly determines the energy consumption of the hydraulic mechanical arm is searched.

Disclosure of Invention

The invention provides a minimum flow-based redundant hydraulic mechanical arm motion planning method aiming at the energy optimization problem under the preset track of a redundant hydraulic mechanical arm; based on the energy suboptimal solution solved by the minimum cylinder velocity norm method, the optimal energy redundancy decomposition is further solved by optimizing a weighted Jacobian matrix with the minimum flow of a hydraulic system as a target; in order to improve the calculation efficiency, a dynamic optimization method of the weighted Jacobian matrix weight is provided, and the online optimal motion planning is realized.

The redundant hydraulic mechanical arm motion planning method based on the minimum flow comprises the following steps:

the method comprises the following steps: firstly, determining a coordinate transformation relation between a mechanical arm connecting rod i and a connecting rod i-1:

T _i ^i-1 ＝R _X (α _i-1 )D _X (α _i-1 )R _Z (θ _i )D _Z (d _i ) (1)

in the formula: r _X (α _i-1 )、D _X (α _i-1 )、R _Z (θ _i )、D _Z (d _i ) Respectively representing the axial rotation transformation, the displacement transformation along the arm direction, the axial rotation transformation and the axial displacement transformation between the ith joint and the (i-1) th joint;

step two: determining the kinematic relationship between the tail end speed and the joint angle of the mechanical arm:

in the formula (I), the compound is shown in the specification,

is the velocity vector of the tail end of the mechanical arm, J is a Jacobian matrix of i rows and J columns (i < J),

is the joint angular velocity;

step three: converting the Jacobian matrix from the calculated joint angle space to a hydraulic cylinder speed space:

in the formula J _h ^-1 To convert the robot arm tip speed to the pseudo-inverse of the cylinder speed space,

the terminal velocity is obtained according to the set terminal position differential;

step four: obtaining the relation between the energy consumption and the flow of the system:

Q _i ＝v _i A _i (5)

in the formula, Q _i The flow rate of the inlet and outlet of the hydraulic cylinder, A _i The area of the rod cavity, v, of the piston rod _i Is the cylinder speed, P _s The pressure of the constant-pressure system is adopted, and E is the energy consumption of the redundant hydraulic mechanical arm;

step five: determining the relation between the cylinder speed of the hydraulic cylinder and the tail end speed of the mechanical arm:

in the formula (I), the compound is shown in the specification,

the tail end speed of the mechanical arm, v is the hydraulic cylinder speed vector of each joint of the mechanical arm, and R is a transformation matrix for converting the joint speed into the hydraulic cylinder speed;

step six: in order to minimize the redundant hydraulic manipulator hydraulic cylinder velocity norm, the lagrangian multiplier method is used to solve the multivariate function minimum value on L:

min||v|| ² (8)

to velocity norm | v | | non-calculation ² And calculating the partial derivative by the sum coefficient lambda, and making the partial derivative of L to v as 0 to obtain:

the hydraulic cylinder speed v can then be solved:

v＝(JR ^-1 ) ^T λ (11)

substituting equation (11) into equation (7) has:

solving λ from equation (12) and substituting equation (11) yields:

step seven: obtaining a pseudo-inverse matrix from a cylinder speed space to an end effector position space:

J ⁺ ＝(JR ^-1 ) ^T (JR ^-1 (JR ^-1 ) ^T ) ^-1 (14)

wherein J is a Jacobian matrix;

step eight: discretizing the target track into N points, setting the time interval between every two steps to be dt, and jointly determining the joint angle by the solved hydraulic cylinder speed and the angular position of the previous step:

q _i ＝R ^-1 v _i dt+q _i-1 (15)

step nine: introducing a weighting matrix W into the pseudo-inverse matrix solved in the formula (15), wherein W is a diagonal matrix, and giving an exponential form of the weighting matrix, so that the pseudo-inverse matrix for solving the cylinder speed of the hydraulic cylinder is W assigned with an initial value, and can be dynamically adjusted:

J _h ⁺ ＝W ^-2l (J ⁺ ) ^T (J ⁺ W ^-2l (J ⁺ ) ^T ) ^-1 (16)

in the formula, a corner mark l is an index parameter;

step ten: establishing an expression of the angle q of each joint at the k-th moment:

in the formula, q _i，k+1 The angle of the mechanical arm joint at the moment k +1,

the joint velocity at time k, c is the limiting gradient function coefficient,

is a joint angle limiting function, and I is a unit matrix;

step eleven: taking the reduction of the flow consumption of the joint as an optimization target, and solving the reduction direction;

wherein k is ∈ {0,1, …, N-1}, Q _k The flow required by the movement of the mechanical arm at the moment corresponding to k, w _ii Representing the elements of the weighting matrix, wherein the value range of i is 1,2,3; w is a _i Corresponding joint

Weight of (J) _H Corresponding to the pseudo-inverse Jacobian matrix at time k, x _k Corresponding to the tail end position of the mechanical arm at the moment k, gradH is a joint angle limiting function;

step twelve: determining whether the weight of the current solution step can reduce the flow consumption, if so, namely, reducing the flow consumption

The weight is adjusted according to the following equation (19), and the calculation is repeated after the adjustment in the substitution step eleven if the flow consumption cannot be reduced, that is, the weight is adjusted

Keeping the weight of the previous solving step;

in the formula, w _ii Represents the elements of the weighting matrix, Δ d represents the search step size; w is a _ini Representing the initial value of the current step weight value;

step thirteen: and C, calculating the system track according to the weight value kept in the step twelve, finishing the current step if the system track can be finished, substituting the value of W into the step nine if the expected system motion track cannot be finished, covering the initial assignment of W, and performing the operation from the step nine to the step eleven again.

Based on the above steps, further, in the second step:

in the formula, a _i-1 The movement distance between the ith joint and the (i-1) th joint of the mechanical arm along the direction of the (i-1) th section arm rod is obtained; alpha is alpha _i-1 The angle of the ith joint of the mechanical arm around the rotating shaft of the ith-1 joint is determined; d _i The moving distance of the rotation shaft direction of the ith joint and the i-1 joint of the mechanical arm is calculated; theta _i The angle of the ith joint of the mechanical arm rotated by taking the ith-1 joint as an axis.

Based on the above steps, further, in the third step:

J _h ^-1 ＝J×R ^-1 (24)

wherein i represents a joint number, r _i The force arm of the hydraulic cylinder is calculated by the formula (25).

Based on the above steps, further, in the ninth step, the weighting matrix W is assigned with an initial value, the weighting matrix W is represented by a diagonal matrix, and the weighting matrix is assigned with an initial value setting W ₁₁ ＝1，w ₂₂ ＝1，w ₃₃ =1, representing the weight of each joint motion speed in the motion of the tail end of the mechanical arm;

based on the above steps, further, the step size of Δ d search in step twelve is a closed interval with a range of 0.001 to 0.05.

The beneficial effect of above-mentioned scheme is:

the energy consumption of each step of the preset track is reduced by dynamically adjusting the weight of the weighting matrix, and the flow demand of joint motion is reduced under the condition of not influencing the motion track of the tail end of the mechanical arm, so that the energy consumption of the system is reduced; compared with the existing optimization methods such as a gradient projection method, a minimum cylinder velocity norm method and the like, the method has better energy-saving effect; the test result of the hydraulic mechanical arm shows that the energy consumption of the elliptical track is reduced by 5.4% on average, and the energy consumption of the triangular track is reduced by 5% on average.

Drawings

FIG. 1 is a plot of joint limit function;

FIG. 2 is a table of structural parameters of the robot arm and the hydraulic cylinder;

FIG. 3 is a schematic diagram of an energy optimization algorithm;

FIG. 4 is a schematic view showing the geometrical relationship between the joints of the robot arm;

FIG. 5 is an end triangular trajectory (time in the pose picture is start time);

FIG. 6 is a triangular trajectory plan angle;

FIG. 7 is an end elliptical trajectory (time in pose picture is start time);

FIG. 8 is an elliptical trajectory plan view;

FIG. 9 is a redundant hydraulic robotic arm control system;

FIG. 10 comparison of triangular trace tests;

FIG. 11 comparison of elliptical trace experiments;

FIG. 12 triangular trace flow curve;

FIG. 13 is an elliptical trajectory flow curve;

FIG. 14 comparison of energy consumption for elliptical trajectories;

FIG. 15 triangular trace energy consumption comparison;

FIG. 16 is a flow chart of the present invention.

Detailed Description

The invention will be further clarified and fully described in the following with reference to the accompanying drawings, without limiting the scope of the invention thereto.

As shown in fig. 3, 4 and 16, the specific steps are as follows:

T _i ^i-1 ＝R _X (α _i-1 )D _X (α _i-1 )R _Z (θ _i )D _Z (d _i ) (1)

in the formula: r _X (α _i-1 )、D _X （α _i-1 )、R _Z (θ _i )、D _Z (d _i ) Respectively representing the axial rotation transformation, the displacement transformation along the arm direction, the axial rotation transformation and the axial displacement transformation between the ith joint and the (i-1) th joint;

in the formula, a _i-1 The movement distance between the ith joint and the (i-1) th joint of the mechanical arm along the direction of the (i-1) th section arm rod is obtained; alpha is alpha _i-1 The angle of the ith joint of the mechanical arm around the rotating shaft of the (i-1) th joint is defined; d _i The moving distance of the ith joint of the mechanical arm and the direction of the rotating shaft of the i-1 joint is obtained; theta.theta. _i The angle of the ith joint of the mechanical arm which rotates by taking the ith-1 joint as an axis;

step two: determining the kinematic relationship between the end velocity and the joint angle of the mechanical arm as follows:

in the formula (I), the compound is shown in the specification,

is the velocity vector of the tail end of the mechanical arm, J is a Jacobian matrix of i rows and J columns (i)<j)，

For angular velocity of jointDegree;

in the formula J _h ^-1 To convert the end-of-arm velocity to the pseudo-inverse of the cylinder velocity space,

J _h ^-1 ＝J×R ^-1 (24)

Step five: and (3) obtaining that the energy consumption of the system is determined by the flow:

Q _i ＝v _i A _i (5)

in the formula, Q _i The flow rate of the inlet and outlet of the hydraulic cylinder, A _i The area of the rod cavity, v, with or without the rod _i For cylinder speed, P _s The pressure of the constant-pressure system is shown, and E is the energy consumption of the redundant hydraulic mechanical arm.

Step six: determining the relation between the cylinder speed of the hydraulic cylinder and the speed of the tail end of the mechanical arm from the cylinder speed space of the hydraulic cylinder;

in the formula (I), the compound is shown in the specification,

the tail end speed of the mechanical arm, v is the hydraulic cylinder speed vector of each joint of the mechanical arm, and R is a transformation matrix for converting the joint speed into the hydraulic cylinder speed; in order to minimize the speed norm of the redundant hydraulic mechanical arm hydraulic cylinder, a Lagrange multiplier method is used for solving the minimum value of a multivariate function related to L;

min||v|| ² (8)

to velocity norm | | v | | non-conducting phosphor ² And calculating the partial derivative by the sum coefficient lambda, and making the partial derivative of L to v as 0 to obtain:

the hydraulic cylinder speed v can then be solved:

v＝(JR ^-1 ) ^T λ (11)

substituting formula (11) into formula (7) has:

solving λ from equation (12) and substituting equation (11) yields:

step seven: based on the sixth step, a pseudo-inverse matrix from the cylinder velocity space to the end effector position space can be obtained as follows:

J ⁺ ＝(JR ^-1 ) ^T (JR ^-1 (JR ^-1 ) ^T ) ^-1 (14)

where J is a Jacobian matrix.

Step eight: the target trajectory is discretized into N points, and the time interval between every two steps is set to dt. The joint angle can then be determined jointly from the solved cylinder speed of the hydraulic cylinder and the angular position of the previous step:

q _i ＝R ^-1 v _i dt+q _i-1 (15)

step nine: introducing a weighting matrix W, which is generally a diagonal matrix, into the pseudo-inverse matrix solved in the formula (15), and giving an exponential form of the weighting matrix, so that the pseudo-inverse matrix for solving the cylinder speed of the hydraulic cylinder is as follows:

J _h ⁺ ＝W ^-2l (J ⁺ ) ^T (J ⁺ W ^-2l (J ⁺ ) ^T ) ^-1 (16)

in the formula, a corner mark l is an index parameter;

in the existing research, the weight of a weighting matrix W does not change along with the position and the posture of a joint generally; however, as the pose of the mechanical arm changes during movement, different weighting matrixes W can change the flow required by the movement; therefore, the weight of the weighting matrix W is dynamically adjusted, so that the flow required by the system can be reduced under the condition that the tail end track of the mechanical arm is not changed; the weighting matrix W may be represented by a diagonal matrix, with the weighting matrix being assigned an initial value that generally sets W ₁₁ ＝1，w ₂₂ ＝1，w ₃₃ =1, representing the weight of each joint movement speed in the movement of the end of the mechanical arm;

step ten: the angle q for each joint at time k may be expressed as:

the joint velocity at time k, c is the limiting gradient function coefficient,

is a joint angle limiting function, and I is an identity matrix. When the joint angle does not exceed the joint limit, the limit function value is close to 0; after the joint angle exceeds the limit term, the limit function value tends to be infinite; an example of a joint limit function curve is shown in figure 1 below.

Weight of (J) _H Corresponding to the pseudo-inverse Jacobian matrix at time k, x _k And corresponding to the tail end position of the mechanical arm at the moment k, gradH is a joint angle limiting function.

Keeping the weight of the previous solving step;

in the formula, w _ii Represents the elements of the weighting matrix, Δ d represents the search step size; w is a _ini Indicating the initial value of the current step weight.

Experimental example 1: because the three-joint hydraulic mechanical arm has redundancy property during planar motion, and the energy optimization method of more joint mechanical arms only increases matrix dimension compared with the three-joint redundant mechanical arm, the energy optimization conclusion of the planar three-degree-of-freedom redundant mechanical arm can be extended to the redundant hydraulic mechanical arms with more joints, the energy optimization algorithm is researched by taking the planar three-joint hydraulic mechanical arm shown in fig. 3 as an example, as shown in fig. 4, the

joints

1,2 and 3 of the mechanical arm are hydraulic cylinders to drive the joints to rotate, the length and the joint angle of the hydraulic cylinder can be represented by a triangular geometric relationship, and the relationship between the joint angle and the length of the hydraulic cylinder is as follows:

wherein i is a joint number, L _i1 And L _i2 Is the distance between the rotation center of the mechanical arm joint 1 and the hinge points at the two ends of the hydraulic cylinder.

β ₁₁ Indicating the angle beta between the connecting line of two hinge points of the base and the vertical direction ₁₂ Represents the included angle beta between the connecting line of the piston hinge point of the joint 1 and the rotating center and the connecting line of the rotating centers of the joint 1 and the joint 2 ₁₃ The included angle between the connecting line of the two hinge points of the base and the connecting line of the rotating centers of the joint 1 and the joint 2 is shown. The structural parameters of the joint 2 and the joint 3 can be obtained in the same way.

In the formula v _i And a _i Respectively the velocity and acceleration of the joint i.

β ₂₃ ＝π+β ₂₁ -β ₂₂ +q ₂ (32)

β ₃₃ ＝π-β ₃₁ -β ₃₂ +q ₃ (33)

In the formula r _i Is the arm of force of a hydraulic cylinder, beta _ij The geometrical significance of (1) is shown in fig. 4.

And (3) taking the initial angle of the joint 3 as a variable, obtaining an inverse kinematics analysis solution of the initial position of the tail end of the mechanical arm and the initial angle of the joint 3, and then respectively optimizing the triangular track and the elliptical track by using a minimum flow optimization (min _ Q), a minimum cylinder velocity norm method (min _ v) and a gradient projection method (min _ a) on the basis of the obtained analysis solution.

Wherein the intermediate variable n ₁ 、α、n ₂ 、n ₃ Can be expressed as follows:

wherein L is ₁ 、L ₂ 、L ₃ The geometrical dimensions q of the arm lever 1, the arm lever 2 and the arm lever 3 respectively ₁ 、q ₂ 、q ₃ The angle of the joint 1 rotating relative to the horizontal base, the angle of the joint 2 rotating relative to the joint 1, and the angle of the joint 3 rotating relative to the joint 2 are respectively. d _x 、d _y Respectively the displacement of the mechanical arm tail end in the x direction and the y direction.

And finally, setting an optimization effect of a typical track verification minimum flow optimization method in an accessible space of the mechanical arm, and comparing the minimum cylinder velocity norm method and the gradient projection method in the conventional research, wherein joint angle limit is added in all three methods. The structural dimensions of the mechanical arm and the structural parameters of the hydraulic cylinder are shown in figure 2.

Triangle trajectory planning

The joint angle of the triangular trace in fig. 5 optimized by the minimum flow method is shown in fig. 6. The joint motion angles planned by the three methods are different remarkably, but the tail end tracks of the three methods are consistent. In addition, the calculation solving time of the three methods is shorter and is far lower than that of a dynamic planning algorithm in the existing research.

Elliptical trajectory planning

The robot arm in fig. 7 is set to complete an elliptical trajectory at a speed in the form of a fifth-order polynomial, and the comparison result of the minimum flow optimization angle is shown in fig. 8. There are also significant differences in the articulation angles for all three methods.

Test verification

A hydraulic robotic arm test platform for validating the algorithms herein is shown in fig. 9. The hydraulic system consists of a hydraulic power source, a hydraulic cylinder, a hydraulic slip ring and a servo valve. The maximum flow of the hydraulic power source is determined by the variable displacement pump, and the system pressure is regulated by the proportional overflow valve. The flowmeter is arranged at a high-pressure oil inlet of the mechanical arm, and can detect the system flow of the hydraulic mechanical arm in real time. The computer control system adopts a Matlab/Simulink XPC-Target real-time control system and works in a master-slave dual-machine mode of a host machine and a Target machine. The control system obtains feedback information such as pressure, flow, angle and the like of the pressure sensor, the flow sensor and the angle encoder through the bus system, the acquisition card, the target machine and the host machine and outputs a calculation control signal to the servo valve so as to realize high-precision track tracking control of the hydraulic mechanical arm.

As shown in fig. 5 and 7, the designed triangular track and the elliptical track take 30s to run; fig. 10 and 11 are diagrams illustrating the trajectory of the end of the actuator in the cartesian coordinate system obtained by substituting the actual angular displacement into the positive kinematic solution of the redundant hydraulic manipulator, and drawing the actual trajectory in the XOY plane. Therefore, in the test of the three methods, the hydraulic mechanical arm accurately tracks the planned joint angle, and the tail end of the hydraulic mechanical arm completely moves along the preset track.

The system flow data of the three methods are collected as shown in fig. 12 and fig. 13. Due to different poses, the extension/retraction displacement of the hydraulic cylinder, the position change of the tail end of the arm rod and the like are different. The flow optimization method provided by the invention aims at the time-varying pose of the mechanical arm in motion to find the solution with least flow consumption. The piston area of the hydraulic cylinder of the joint 3 in the test model is less than 2. After the minimum flow method is optimized, the motion amplitude of the joint 2 is smaller than the solving results of the minimum cylinder velocity norm method and the gradient projection method, and the motion amplitude of the joint 3 is larger than the solving results of the minimum cylinder velocity norm method and the gradient projection method. This shows that the minimum flow optimization method has the capability of reducing the flow required by the movement of the hydraulic mechanical arm under any track. For a triangular track, when the joint 3 moves to the limit position (16 s), since the joint 3 cannot cross the limit position, the joint 2 starts to move to complete the set track, and the movement of the joint 1 needs more flow, so that the flow of the minimum flow optimization algorithm is temporarily higher than that of the minimum cylinder velocity norm method and the gradient projection method.

In a constant pressure system that does not take into account pressure loss and flow leakage, the system energy consumption is calculated by equation (38).

In the formula t ₁ 、t ₂ Start and end times, respectively, P representing system pressure, Q _p (t) is a function of flow rate over time.

The experimentally verified energy consumption of the different algorithms under the two trajectories is found according to equation (40), as shown in fig. 14 and 15. Compared with a gradient projection method and a minimum cylinder velocity norm method, the minimum flow optimization method is proved to save 5.4% and 5% of energy on average through tests. The energy-saving effect of the minimum flow method is verified through experiments.

The embodiments of the present invention are disclosed as the preferred embodiments, but not limited thereto, and those skilled in the art can easily understand the spirit of the present invention and make various extensions and changes without departing from the spirit of the present invention.

Claims

1. The minimum flow-based redundant hydraulic mechanical arm motion planning method is characterized by comprising the following steps of:

in the formula: r _X (α _i-1 )、D _X (α _i-1 )、R _Z (θ _i )、D _Z (d _i ) Respectively shows the axial rotation transformation, the displacement transformation along the arm direction, the axial rotation transformation and the axial displacement transformation between the ith joint and the (i-1) th jointChanging;

step two: determining the kinematic relationship of the tail end speed and the joint angle of the mechanical arm:

in the formula (I), the compound is shown in the specification,

Joint angular velocity;

step three: joint angle Q calculated from jacobian matrix _i Space conversion to hydraulic cylinder speed space:

a terminal velocity obtained from a set terminal position differential;

Q _i ＝v _i A _i (5)

in the formula, Q _i The flow rate of the inlet and outlet of the hydraulic cylinder, A _i The area of the rod cavity, v, with or without the rod _i For cylinder speed, P _s The pressure of the constant-pressure system is adopted, and E is the energy consumption of the redundant hydraulic mechanical arm;

in the formula (I), the compound is shown in the specification,

step six: in order to minimize the speed norm of the hydraulic cylinder of the redundant hydraulic mechanical arm, the minimum value of a multivariate function related to L is solved by using a Lagrange multiplier method:

min||v|| ² (8)

to velocity norm | | v | | non-conducting phosphor ² And calculating a partial derivative by using the sum coefficient lambda, and enabling the partial derivative of L to v to be 0 to obtain:

the hydraulic cylinder speed v can then be solved:

v＝(JR ^-1 ) ^T λ (11)

substituting formula (11) into formula (7) has:

solving λ from equation (12) and substituting equation (11) yields:

J ⁺ ＝(JR ^-1 ) ^T (JR ^-1 (JR ^-1 ) ^T ) ^-1 (14)

wherein J is a Jacobian matrix;

step eight: discretizing the target track into N points, setting a time interval dt between every two steps, and jointly determining the joint angle by the solved cylinder speed of the hydraulic cylinder and the angular position of the previous step:

qi＝R ^-1 v _i dt+q _i-1 (15)

J _h ⁺ ＝W ^-2l (J ⁺ ) ^T (J+W ^-2l (J ⁺ ) ^T ) ^-1 (16)

in the formula, a corner mark l is an index parameter;

step ten: establishing an expression of the angle q of each joint at the kth time:

the joint velocity at time k, c is the limiting gradient function coefficient,

is a joint angle limiting function, and I is a unit matrix;

Weight of (C), J _H Corresponding to the pseudo-inverse Jacobian matrix at time k, x _k Corresponding to the tail end position of the mechanical arm at the moment k, gradH is a joint angle limiting function;

Keeping the weight of the previous solving step;

2. The minimum flow based redundant hydraulic manipulator motion planning method according to claim 1, wherein: in the second step:

in the formula, a _i-1 The movement distance between the ith joint and the (i-1) th joint of the mechanical arm along the direction of the (i-1) th section arm rod is obtained; alpha is alpha _i-1 The angle of the ith joint of the mechanical arm around the rotating shaft of the (i-1) th joint is defined; d is a radical of _i The moving distance of the ith joint of the mechanical arm and the direction of the rotating shaft of the i-1 joint is obtained; theta.theta. _i The angle of the ith joint of the mechanical arm rotated by taking the ith-1 joint as an axis.

3. The minimum flow based redundant hydraulic manipulator motion planning method according to claim 1, wherein: in the third step:

J _h ^-1 ＝J×R ^-1 (24)

4. The minimum flow based redundant hydraulic manipulator motion planning method according to claim 1, wherein: in the ninth step, a weighting matrix W is assigned with an initial value, the weighting matrix W is represented by the following diagonal matrix, and the weighting matrix is assigned with an initial value setting W ₁₁ ＝1，w ₂₂ ＝1，w ₃₃ =1, representing the weight of each joint motion speed in the motion of the tail end of the mechanical arm;

5. the minimum flow based redundant hydraulic manipulator motion planning method according to claim 1, wherein: step twelve, searching a closed interval with the step length range of 0.001 to 0.05.