CN113043278A - Mechanical arm track planning method based on improved whale searching method - Google Patents

Mechanical arm track planning method based on improved whale searching method Download PDF

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CN113043278A
CN113043278A CN202110392530.5A CN202110392530A CN113043278A CN 113043278 A CN113043278 A CN 113043278A CN 202110392530 A CN202110392530 A CN 202110392530A CN 113043278 A CN113043278 A CN 113043278A
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mechanical arm
whale
joint
trajectory planning
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CN113043278B (en
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刘杰
卞新宇
吕卓昆
刘洋
李巍
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Dragon Totem Technology Hefei Co ltd
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Nanjing Vocational University of Industry Technology NUIT
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1679—Programme controls characterised by the tasks executed

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Abstract

The invention discloses a mechanical arm track planning method based on an improved whale searching method, which comprises the following steps of 1: importing a three-dimensional model; step 2: establishing a space coordinate system of the mechanical arm six-degree-of-freedom structure body; and step 3: defining motion parameters of the three-dimensional model and acquiring a pose; and 4, step 4: establishing a mechanical arm motion time optimal model; and 5: defining a constraint condition; step 6: the track planning is carried out on the mechanical arm by utilizing the improved whale searching method, the global optimal solution can be obtained more quickly by adopting the method, and the result has higher accuracy.

Description

Mechanical arm track planning method based on improved whale searching method
Technical Field
The invention belongs to the field of industrial robots and computer application, and particularly relates to a mechanical arm track planning method based on an improved whale searching method.
Background
With the rapid development of modern artificial intelligence technology, the manufacturing industry is developing towards integration and automation, and industrial robots serving the mechanical field are rising rapidly. Under the intelligent large environment, the requirements on the automation degree and the precision of the industrial robot are higher and higher. However, the industrial robot body has high investment cost and low efficiency in research and development and actual test, so that a simulation experiment needs to be performed on the designed industrial robot before production to analyze whether the performance of the industrial robot meets the requirements.
Usually, kinematics solution, dynamics solution, trajectory planning and the like are performed on the simulation model in sequence. More and more industrial robot trajectory planning application examples show that trajectory planning only considering continuity and smoothness cannot meet most industrial requirements, and the problems of energy consumption, efficiency, impact and the like need to be considered, so that the performance of the trajectory is improved. Therefore, constraint conditions of trajectory planning need to be provided according to different task requirements so as to achieve the purpose of trajectory optimization. Generally, an intelligent optimization method is adopted to optimize targets, but the intelligent optimization methods commonly used for track planning research have the defects of multiple parameters, complex calculation process and the like. Compared with most intelligent optimization methods, the method for searching the global optimal value by whale provided in recent years has the advantages of few parameters, simple structure and the like, but is easy to fall into a local optimal solution.
In summary, the current mechanical arm trajectory planning method cannot simultaneously meet the requirements of small calculated amount in the analysis process, accurate calculation result, high reliability and the like.
Disclosure of Invention
The purpose of the invention is as follows: in order to solve the problem that the existing mechanical arm track planning method cannot meet the requirements of small calculated amount in the analysis process, accurate calculation result, high reliability and the like at the same time, the invention provides the mechanical arm track planning method based on the improved whale searching method, the method greatly reduces the calculated amount, meets the requirement of high precision at the same time, and can be applied to the track planning of an industrial robot.
The technical scheme is as follows: a mechanical arm track planning method based on an improved whale searching method comprises the following steps:
step 1: acquiring parameters of all connecting rods of the mechanical arm, constructing a three-dimensional model of the mechanical arm and constructing a three-dimensional model of a main body interacting with the motion of the mechanical arm in an actual application scene;
step 2: defining a space coordinate system of each joint of the mechanical arm; wherein the joint i is emptiedEach direction axis of the inter-coordinate system is defined as xi、yi、ziA shaft; thetaiIs xi-1Axial around zi-1The shaft rotates to xiThe angle of the shaft;
and step 3: setting path points between the main body and the manipulator at the tail end of the mechanical arm based on tasks to be completed by the mechanical arm under the space coordinate system in the step 2; the method comprises the steps that tasks are executed through a simulation mechanical arm, and the set pose of a mechanical arm end manipulator corresponding to each path point is obtained;
and 4, step 4: according to task requirements, defining the movement time between two adjacent path points to obtain the total time required for completing the whole task; establishing a mathematical model with optimal mechanical arm movement time based on the total time required for completing the whole task;
and 5: taking the mathematical model established in the step 4 as an objective function of the whale optimization searching method, taking the maximum value of the kinematic parameters of each joint of the mechanical arm as a constraint condition of the whale optimization searching method, and obtaining the optimal movement time by adopting the whale optimization searching method;
step 6: and planning to obtain the motion trail of the mechanical arm based on the optimal motion time.
The method comprises the steps of constructing a three-dimensional model of the mechanical arm in software ROBCAD and constructing a three-dimensional model of a main body interacting with the movement of the mechanical arm in an actual application scene.
Further, step 2 specifically includes:
defining the intersection point of the common perpendicular line of the joints at the two ends of the mechanical arm connecting rod i +1 and the axis of the joint i as an origin, and establishing x along the common perpendicular line and pointing to the direction of the joint i +1iAxis, establishing z along the joint i axisiAxis, determination of y by right techniqueiA shaft.
Further, in step 3, setting a path point between the main body and the end effector of the robot arm specifically includes:
and defining the path point which the manipulator at the tail end of the mechanical arm passes through in the complete task according to the joint motion on the mechanical arm, the speed and the acceleration of the joint motion, the joint axis and the joint motion range.
Further, in step 3, the pose is represented as:
Figure BDA0003017304420000021
wherein n isx、ny、nzNormal unit vectors describing the coordinate system of the end operator in the x, y and z directions respectively; ox、oy、ozDirection unit vectors describing the coordinate system of the end effector in the x, y and z directions respectively; a isx、ay、azRespectively, a near unit vector describing the coordinate system of the end effector in the x, y and z directions; p is a radical ofx、py、pzPosition vectors describing the end effector coordinate system in the x, y, z directions, respectively.
In the invention, the software ROBCAD directly acquires the pose of the manipulator at the tail end of the mechanical arm at the set path point.
Further, in step 4, the mathematical model for optimizing the motion time of the mechanical arm is represented as:
Ttotal＝mint1′+mint′2+mint′3+...... (2)
wherein, t'1、t′2、t′3Respectively representing the running time of the mechanical arm between two adjacent path points.
Further, in step 5, the kinematic parameters of each joint of the mechanical arm include: angular velocity of each joint of mechanical arm
Figure BDA0003017304420000022
Angular acceleration of each joint of mechanical arm
Figure BDA0003017304420000023
And angular acceleration of joints of the arm
Figure BDA0003017304420000024
The constraint conditions of the whale optimization searching method are represented as follows:
Figure BDA0003017304420000031
wherein the content of the first and second substances,
Figure BDA0003017304420000032
is the maximum value of the angular velocity of each joint of the mechanical arm,
Figure BDA0003017304420000033
is the maximum value of the angular acceleration of each joint of the mechanical arm,
Figure BDA0003017304420000034
the maximum value of the angular jerk of each joint of the mechanical arm.
Further, in step 5, the whale optimization searching method comprises the following steps:
s100: initializing a non-linear convergence factor
Figure BDA0003017304420000035
Coefficient vector
Figure BDA0003017304420000036
Coefficient vector
Figure BDA0003017304420000037
Parameter l, parameter p, nonlinear inertial weight ω and maximum number of iterations tmax(ii) a Setting the population number N and randomly generating an initial population position;
s200: calculating the fitness value of each whale individual in the population, defining the whale individual with the minimum fitness value as the current optimal individual, and using X*Representing its position vector;
s300: updating nonlinear convergence factors of individual whales
Figure BDA0003017304420000038
Coefficient vector
Figure BDA0003017304420000039
Coefficient vector
Figure BDA00030173044200000310
Parameter l, parameter p, nonlinear inertial weight ω;
s400: judging whether the updated parameters meet the following conditions: p < 0.5 and | A | < 1, if satisfied, updating the location of each individual whale according to the following equation:
Figure BDA00030173044200000311
if the updated parameters satisfy that p is less than 0.5 and | A | ≧ 1, updating the position of each whale individual according to the following formula, namely the shortest running time solved currently:
Figure BDA00030173044200000312
if the updated parameters meet that p is more than or equal to 0.5, updating the position of each whale individual according to the following formula, and obtaining the shortest operation time obtained by current solution:
Figure BDA00030173044200000313
wherein the content of the first and second substances,
Figure BDA00030173044200000314
is a nonlinear convergence factor;
Figure BDA00030173044200000315
is a coefficient vector; l is [ -1,1 [ ]]The random number of (2); p is [0,1 ]]The random number of (2);
Figure BDA00030173044200000316
is [0,1 ]]A random vector of (a); omega is a nonlinear inertial weight; t is tmaxIs the maximum iteration number; mu is a constant coefficient; b is a constant defining the shape of the logarithmic spiral.
S500: judging iterationWhether the number of times reaches the maximum iteration number tmaxIf not, returning to S200; if so, outputting the current optimal individual and the position vector X thereof*And obtaining the optimal exercise time. Nonlinear convergence factor in whale searching method
Figure BDA00030173044200000317
The convergence rate is nonlinear, and the convergence rate in the early stage is fast and the convergence rate in the later stage is relatively slow.
Has the advantages that: compared with the prior art, the invention has the following advantages:
(1) the improved whale searching method is applied to track optimization of the mechanical arm, optimal track planning of the movement time of the mechanical arm is achieved, the calculation result has higher precision, and the optimization time is shortened;
(2) according to the invention, the pose of the manipulator tail end corresponding to each path point is obtained through the ROBCAD software, so that the calculation amount of matrix transformation is reduced.
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FIG. 1 is a schematic view of a robot arm configuration;
FIG. 2 is a spatial coordinate system established in the embodiment;
FIG. 3 is a three-dimensional model of an embodiment;
FIG. 4 shows an exemplary embodiment of a standard-based test function F1(x) The improved whale optimization search and other three commonly used intelligent optimization methods are used for comparing the convergence rate;
FIG. 5 shows an exemplary embodiment of a standard-based test function F2(x) The improved whale optimization search and other three commonly used intelligent optimization methods are used for comparing the convergence accuracy;
FIG. 6 shows the result of time optimization in the examples;
FIG. 7 shows the results of the trajectory planning of each joint in the example;
fig. 8 is a general flow chart.
Detailed Description
The technical solution of the present invention will be further explained with reference to the accompanying drawings and examples.
Referring to fig. 8, in the embodiment, a trajectory planning method for a mechanical arm based on an improved whale search method is adopted to plan a trajectory of the mechanical arm for performing a task of clamping raw materials; fig. 3 is a schematic diagram of a robotic arm performing the task of grasping raw meal, as shown in fig. 3, the task being performed by a body comprising: the device comprises a machining center 1, a material taking point 2, a mechanical arm 3, a bin 4 and a raw material point 5; the processing center 1 and the bin 4 are respectively arranged at two sides of the mechanical arm 3, the raw material point 5 is positioned at any position of the bin 4, and the material taking point 2 is positioned at the processing center 1; the end-of-arm manipulator grips the raw meal at the raw meal point 5, moves to the machining center, places the raw meal at the take off point 2, and then returns to the initial position, which is a take off movement.
Now, with reference to this task, the steps of the planning method used in this embodiment are described, including the following steps:
step 1: importing a three-dimensional model: acquiring main parameters of each connecting rod of the mechanical arm, wherein the main parameters comprise the size, the material density, the centroid position, the connection mode and the like of each connecting rod including the base, and importing three-dimensional models of the mechanical arm and other main bodies related to the mechanical arm in an actual application scene into software ROBCAD; see fig. 1;
step 2: referring to fig. 2, a space coordinate system of the six-degree-of-freedom structural body of the robot arm is established, wherein each directional axis of the space coordinate system of the joint i is respectively defined as xi、yi、ziA shaft; thetaiIs xi-1Axial around zi-1The shaft rotates to xiAngle of the shaft: defining a base coordinate system {0} and a space coordinate system of each joint of the mechanical arm by a standard D-H parameter method; the method specifically comprises the following steps: defining the intersection point of the common perpendicular line of the joints at the two ends of the connecting rod i +1 and the axis of the joint i as an origin, and establishing x along the common perpendicular line and pointing to the direction of the joint i +1iShaft, along which joint axis z is establishediAxis, determining y by right-hand ruleiA shaft;
and step 3: defining model motion parameters, and acquiring a pose: based on step 2, defining the path points of the moving mechanism (including joint motion, joint motion speed and acceleration, joint axis, motion range and the like) and the mechanical arm end manipulator to complete specific tasks in the ROBCAD software, which will be based on the following stepsDefining the secondary passing path point as P0、P1、P2、P1、P3、P4、P3、P0Wherein P is0At an initial position of the end-of-arm manipulator, P1Is at any position on the storage bin close to the raw material point, P2At the location of the charging point, P3Is at any position on the machining center close to the material taking point, P4Is the position of a material taking point. Simulating the process that the mechanical arm clamps raw materials in an actual application scene, moves in a machining center, and then returns to an initial position, wherein the software ROBCAD can directly acquire the pose of the mechanical arm end operator corresponding to each set path point:
Figure BDA0003017304420000051
Figure BDA0003017304420000052
Figure BDA0003017304420000053
Figure BDA0003017304420000054
Figure BDA0003017304420000055
and 4, step 4: time-optimal modeling: defining the motion time between two adjacent path points according to the practical application requirement, namely t'1、t′2、t′3、t′4、t′5、t′6、t′7Then the total time required to complete the entire specific task is:
Ttotal＝mint′1+mint′2+mint′3+mint′4+mint′5+mint′6+mint′7
and 5: defining a constraint condition: adding and researching angular velocity of each joint of mechanical arm
Figure BDA0003017304420000056
Angular acceleration
Figure BDA0003017304420000057
Sum angular jerk
Figure BDA0003017304420000058
And (3) taking the maximum value of the isokinetic parameters as a constraint condition of the optimized search of the improved whale:
Figure BDA0003017304420000059
in the present embodiment, the constraint conditions of each joint are set as follows:
Figure BDA0003017304420000061
step 6: planning a track: taking the mathematical model in the step 4 as an objective function of the method, and carrying out trajectory planning on the mechanical arm based on the constraint conditions defined in the step 5 and an improved whale searching method, wherein the improved whale searching method comprises the following calculation steps:
(1) setting the number of the populations to be N-30, and randomly generating the positions of the initial populations; initialization parameters
Figure BDA0003017304420000062
l, p, ω and tmaxWherein:
Figure BDA0003017304420000063
Figure BDA0003017304420000064
Figure BDA0003017304420000065
Figure BDA0003017304420000066
wherein the content of the first and second substances,
Figure BDA0003017304420000067
is a nonlinear convergence factor;
Figure BDA0003017304420000068
is a coefficient vector; l is [ -1,1 [ ]]The random number of (2); p is [0,1 ]]The random number of (2);
Figure BDA0003017304420000069
is [0,1 ]]A random vector of (a); omega is a nonlinear inertial weight; t is tmaxThe maximum number of iterations is 500 in this example; mu is a constant coefficient, and the value in the embodiment is 0.01 after multiple simulation experiments.
(2) Calculating the fitness value of each whale individual (namely a search agent) in the population, selecting the whale individual with the smallest fitness value, defining the whale individual as the current optimal individual, and using X*Represents its position vector:
(3) once per iteration, the relevant parameters for each search agent are updated:
Figure BDA00030173044200000610
l, p, ω. If p < 0.5 and | A | < 1, the location update for each search agent is calculated as in equation (8):
Figure BDA00030173044200000611
otherwise, if p is less than 0.5 and | A | ≧ 1, the calculation is performed according to equation (9):
Figure BDA00030173044200000612
if p is greater than or equal to 0.5, the position is updated according to equation (10):
Figure BDA00030173044200000613
(4) and comparing the individuals in the population after the position is updated, and determining the globally optimal individual and the current position.
(5) If the iteration times reach the maximum value, namely the termination condition of the loop part in the WOA is reached, outputting a result, namely the optimal movement time; otherwise, returning to the step (2) and continuing to calculate until the termination condition is met.
In order to verify the convergence performance of the improved whale optimization algorithm, the improved whale optimization algorithm is calculated based on a standard test function and compared with the convergence performance of other three commonly used intelligent optimization algorithms, and fig. 4 is a diagram of an embodiment based on a standard test function F1(x) The improved whale optimization search and other three commonly used intelligent optimization methods are used for comparing the convergence rate; standard test function F1(x) Expressed as:
Figure BDA0003017304420000071
FIG. 5 shows an exemplary embodiment of a standard-based test function F2(x) The improved whale optimization search and other three commonly used intelligent optimization methods are used for comparing the convergence accuracy; standard test function F2(x) Expressed as:
Figure BDA0003017304420000072
as shown in fig. 4 and 5,. The improved whale optimization algorithm is better than a genetic algorithm GA, a particle swarm algorithm PSO and a basic whale optimization algorithm WOA in convergence speed and convergence precision through simulation experiment result analysis.
The optimal movement time of the mechanical arm obtained by the searching method in the embodiment is shown in the following table and fig. 6, and the final result of the trajectory planning is shown in fig. 7.
Figure BDA0003017304420000073
The mechanical arm trajectory planning result which is obtained by the method of the embodiment and aims at time optimization meets the requirements of small calculated amount, reliable calculated result and high precision.

Claims (7)

1. A mechanical arm track planning method based on an improved whale searching method is characterized by comprising the following steps: the method comprises the following steps:
step 1: acquiring parameters of all connecting rods of the mechanical arm, constructing a three-dimensional model of the mechanical arm and constructing a three-dimensional model of a main body interacting with the motion of the mechanical arm in an actual application scene;
step 2: defining a space coordinate system of each joint of the mechanical arm;
and step 3: setting path points between the main body and the manipulator at the tail end of the mechanical arm based on tasks to be completed by the mechanical arm under the space coordinate system in the step 2; the method comprises the steps that tasks are executed through a simulation mechanical arm, and the set pose of a mechanical arm end manipulator corresponding to each path point is obtained;
and 4, step 4: according to task requirements, defining the movement time between two adjacent path points to obtain the total time required for completing the whole task; establishing a mathematical model with optimal mechanical arm movement time based on the total time required for completing the whole task;
and 5: taking the mathematical model established in the step 4 as an objective function of the whale optimization searching method, taking the maximum value of the kinematic parameters of each joint of the mechanical arm as a constraint condition of the whale optimization searching method, and obtaining the optimal movement time by adopting the whale optimization searching method;
step 6: and planning to obtain the motion trail of the mechanical arm based on the optimal motion time.
2. The mechanical arm trajectory planning method based on the improved whale search method as claimed in claim 1, wherein the mechanical arm trajectory planning method comprises the following steps: the step 2 specifically comprises:
defining the intersection point of the common perpendicular line of the joints at the two ends of the mechanical arm connecting rod i +1 and the axis of the joint i as an origin, and establishing x along the common perpendicular line and pointing to the direction of the joint i +1iAxis, establishing z along the joint i axisiAxis, determination of y by right techniqueiA shaft.
3. The mechanical arm trajectory planning method based on the improved whale search method as claimed in claim 1, wherein the mechanical arm trajectory planning method comprises the following steps: in step 3, the setting of the path point between the main body and the manipulator at the end of the mechanical arm specifically includes:
and defining the path point which the manipulator at the tail end of the mechanical arm passes through in the complete task according to the joint motion on the mechanical arm, the speed and the acceleration of the joint motion, the joint axis and the joint motion range.
4. The mechanical arm trajectory planning method based on the improved whale search method as claimed in claim 1, wherein the mechanical arm trajectory planning method comprises the following steps: in step 3, the pose is expressed as:
Figure FDA0003017304410000011
wherein n isx、ny、nzNormal unit vectors describing the coordinate system of the end operator in the x, y and z directions respectively; ox、oy、ozDirection unit vectors describing the coordinate system of the end effector in the x, y and z directions respectively; a isx、ay、azRespectively, a near unit vector describing the coordinate system of the end effector in the x, y and z directions; p is a radical ofx、py、pzPosition vectors describing the end effector coordinate system in the x, y, z directions, respectively.
5. The mechanical arm trajectory planning method based on the improved whale search method as claimed in claim 1, wherein the mechanical arm trajectory planning method comprises the following steps: in step 4, the mathematical model for optimizing the motion time of the mechanical arm is represented as:
Ttotal＝min t′1+min t′2+min t′3+......(2)
wherein, t'1、t′2、t′3Respectively representing the running time of the mechanical arm between two adjacent path points.
6. The mechanical arm trajectory planning method based on the improved whale search method as claimed in claim 1, wherein the mechanical arm trajectory planning method comprises the following steps: in step 5, the kinematic parameters of each joint of the mechanical arm include: angular velocity of each joint of mechanical arm
Figure FDA0003017304410000021
Angular acceleration of each joint of mechanical arm
Figure FDA0003017304410000022
And angular acceleration of joints of the arm
Figure FDA0003017304410000023
The constraint conditions of the whale optimization searching method are represented as follows:
Figure FDA0003017304410000024
wherein the content of the first and second substances,
Figure FDA0003017304410000025
is the maximum value of the angular velocity of each joint of the mechanical arm,
Figure FDA0003017304410000026
maximum angular acceleration of each joint of the mechanical armThe value of the one or more of,
Figure FDA0003017304410000027
the maximum value of the angular jerk of each joint of the mechanical arm.
7. The mechanical arm trajectory planning method based on the improved whale search method as claimed in claim 6, wherein the mechanical arm trajectory planning method comprises the following steps: in step 5, the whale optimization searching method comprises the following steps:
s100: initializing a non-linear convergence factor
Figure FDA0003017304410000028
Coefficient vector
Figure FDA0003017304410000029
Coefficient vector
Figure FDA00030173044100000210
Parameter l, parameter p, nonlinear inertial weight ω and maximum number of iterations tmax(ii) a Setting the population number N and randomly generating an initial population position;
s200: calculating the fitness value of each whale individual in the population, defining the whale individual with the minimum fitness value as the current optimal individual, and using X*Representing its position vector;
s300: updating nonlinear convergence factors of individual whales
Figure FDA00030173044100000211
Coefficient vector
Figure FDA00030173044100000212
Coefficient vector
Figure FDA00030173044100000213
Parameter l, parameter p, nonlinear inertial weight ω;
s400: judging whether the updated parameters meet the following conditions: p is less than 0.5 and | A | is less than 1; if the calculated running time is less than the preset running time, updating the position of each whale individual according to the following formula:
Figure FDA00030173044100000214
if the updated parameters satisfy p < 0.5 and | A | ≧ 1, the location of each individual whale is updated according to the following equation:
Figure FDA00030173044100000215
if the updated parameters satisfy that p is more than or equal to 0.5, updating the position of each whale individual according to the following formula:
Figure FDA00030173044100000216
wherein the content of the first and second substances,
Figure FDA00030173044100000217
is a nonlinear convergence factor;
Figure FDA00030173044100000218
is a coefficient vector; l is [ -1,1 [ ]]The random number of (2); p is [0,1 ]]The random number of (2);
Figure FDA00030173044100000219
is [0,1 ]]A random vector of (a); omega is a nonlinear inertial weight; t is tmaxIs the maximum iteration number; μ is a constant coefficient, b is a constant defining the shape of a logarithmic spiral;
s500: judging whether the iteration number reaches the maximum iteration number tmaxIf not, returning to S200; if so, outputting the current optimal individual and the position vector X thereof*And obtaining the optimal exercise time.
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CN114654464B (en) * 2022-03-22 2024-07-02 杭州景吾智能科技有限公司 Cleaning robot positioning position selection method and system based on time optimization

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