CN104483898A - Method for searching Delta robot inscribed cylinder expected work space - Google Patents

Abstract

A method of searching for the expected workspace of an inscribed cylinder of a Delta robot. By determining the height range of the inscribed cylinder, determining the maximum radius of the inscribed circle at a given center position, determining the maximum radius and position of the inscribed circle in the workspace, and determining the workspace The height of the inscribed cylinder with a given radius, the expected workspace of the cylinder with the maximum volume inscribed in the workspace, and other methods search for the expected workspace of the inscribed cylinder in the workspace of the Delta robot. The regular-shaped workspace eliminates the adverse effects of irregular protrusions on the boundary of the actual workspace on the motion and control of the robot. In addition to the search for the expected workspace inscribed in the cylinder, the present invention can also be used for the search for the expected workspace inscribed by other rules.

Description

A Method of Searching the Expected Workspace of the Inscribed Cylinder of Delta Robot

technical field

The invention relates to a method for searching the inscribed rule workspace of a parallel robot, in particular to a method for search for the inscribed cylinder workspace of a Delta robot.

Background technique

A parallel robot is a robot with two or more degrees of freedom that is connected between a moving platform and a fixed platform through at least two independent operating chains. Compared with serial robots, parallel robots have many advantages such as high precision, large bearing capacity, high rigidity, compact structure, and fast response, but at the same time, they have the disadvantage of relatively small working space. The Delta robot with three translational degrees of freedom is a kind of parallel robot that has been successfully used in medical, food, pharmaceutical and other industries. However, the working space of the Delta robot is small, and there are many irregular bumps on the boundary. When the robot runs near the boundary, it is easy to enter a singular state. Therefore, designers and controllers often "expect" to replace the original workspace with a regular-shaped "inscribed desired workspace", which is of great significance to the robot's motion control and path planning. At the same time, determining the size of the expected workspace that can accommodate the rules in the workspace can also be used as an indicator for the design of the Delta robot mechanism, and in order to facilitate automation, the numerical method of application and programming is very necessary.

Contents of the invention

In order to keep Delta away from singular poses so as to ensure its safe work and make it easier to perform motion control and trajectory planning on it, the present invention provides a method for searching the expected workspace of the inscribed cylinder in the Delta workspace. This method is easy to program and implement. In order to realize above function, the present invention adopts following method:

First, establish the geometric model of the Delta robot; then obtain the relational expression between its workspace and the structural parameters of the robot according to the geometric model, and divide the expression into upper and lower bounds for parametric equation expression; Accommodate the maximum height of the expected workspace of the cylinder; then, based on the method of determining the maximum circle that can be accommodated in the workspace by the given circle center coordinates, find the inscribed circle with the maximum radius in the workspace, and obtain the given A method for the maximum height of a cylinder with a fixed radius; finally, a method for determining the expected workspace of a cylinder that can accommodate the maximum volume in the workspace is entered here into the description paragraph of the content of the invention.

Description of drawings

Figure 1 Delta robot structure diagram;

Figure 2 determines the largest circle that can be accommodated in the workspace at (0,0,z);

Figure 3 search strategy;

Figure 4 Determines the tallest cylinder of a given radius that can fit in the workspace.

Detailed ways

This in conjunction with accompanying drawing, the workflow of the present invention is as follows:

(1) Establish the geometric model of the Delta robot and its workspace

Accompanying drawing 1 is Delta robot structure model. The Delta robot is mainly composed of a fixed platform, a moving platform, a motor, a master arm, and a slave arm. The three motors fixed on the fixed platform respectively drive the three active arms, and the three driven arms drive the moving platform, so as to realize the movement of the end. The origin of the coordinate system is located at the center of the regular triangle formed by the connection point F _i ( i = 1, 2, 3, the same below) between the motor and the active arm, the z -axis is perpendicular to the plane where the triangle is located, and the y-axis is perpendicular to F ₂ F ₃ The straight line where it is located deviates from F ₁ , and the three coordinate axes conform to the right-handed coordinate system. The center E ₀ ( x ₀ , y ₀ , z ₀ ) of the triangle formed by the intersection E _i of the movable arm and the moving platform is the reference point of the end effector position, so the z coordinate of E ₀ is always negative. The distance from the coordinate origin O to F _i is f , the distance from the end effector reference point E ₀ to E _i is e , the length of the master arm is F _i J _i = r _f , and the length of the slave arm is E _i J _i = r _e . An OE _i J _i F _i E ₀ constitutes a single branched chain;

According to the established geometric model, the boundary representation of the single branch chain workspace of the Delta robot can be obtained:

Represent it in parametric form:

upper boundary: ,

if , ;otherwise,

Lower boundary: ,

Because the moving platform is jointly driven by three single-branch chains, the intersection of the motion spaces of the three single-branch chains constitutes the working space of the Delta robot. And, it can be known from the parameter form that x ₀ < r _e , so the radius r < r _e of the inscribed cylinder.

(2) Determine the maximum height of the expected workspace of the cylinder that can be accommodated in the workspace of the Delta robot

When the radius of the cylinder is 0, the height is the largest, that is, the intersection of the workspace and the z- axis is the upper and lower bottom surfaces of the cylinder. therefore,

z _min =-| r _e + r _f |cos( θ ), θ =-arcsin( L /| r _e + r _f |);

If | r _e - r _f |<| L |, Z _max =0; otherwise, Z _max =-| r _e - r _f |cos( θ ), θ =-arcsin( L /| r _e - r _f | );

The maximum height is: H = Z _max - Z _min .

(3) Determine the radius of the inscribed circle whose center is at (0,0, z ) in the working space of the Delta robot

Accompanying drawing 2 is the flow chart of the method for determining the radius of the inscribed circle whose center is located at (0,0, z ) in the working space of the Delta robot. First, according to the input structural parameters of the Delta robot, determine the maximum height of the cylinder that can be accommodated in its workspace, and judge whether the coordinates of the given center of the circle are within this range. If not, then R = 0, and end the program; otherwise, proceed to the next step, with 0 as the lower bound of the radius and r _e as the upper bound for binary search. When a radius is obtained by the dichotomy method, in order to determine that the circle of the radius is completely within the workspace, a point within the circle is randomly generated and compared with the upper and lower bounds of the workspace to determine whether it is within the workspace. If the point is not within the workspace, a binary search is performed using the radius as a new upper bound. Until N points are generated and all points are located in the work space, it is determined that the circle is completely located in the work space, and the binary search is performed with the radius as the new lower bound. When the difference between the upper bound and the next is less than a given threshold, the mean value of the lower bound on the radius at this time is taken as the final result;

 In order to reduce the amount of calculation, when it is determined that the circle in this cycle is completely within the working space, the next cycle will not check the points inside the circle, that is, to determine whether the circle with a radius range of [ r _min , r _max ] is completely within the workspace.

(4) Determine the inscribed circle with the largest radius in the working space of the Delta robot

Accompanying drawing 3 is the search strategy for determining the inscribed circle with the largest radius in the working space of the Delta robot. When the radius of the bottom surface of the cylinder is the largest, the height is 0, and the upper and lower bottom surfaces coincide. This radius provides an upper bound on the range for the search of cylinder radii. Because within the above height range, the radius of the bottom surface of the cylinder has a maximum value and is unique, and is monotonous on both sides of the maximum value, the following search strategy shown in Figure 3 is adopted:

It is known that R(z) has a maximum value among [ a , b ] and is unique. Take two different points c and d in [ a , b ], and assume c < d , then compare their function values:

If R ( c )> R ( d ), the maximum value is located in [ a , c ];

If R ( c )< R ( d ), the maximum value is located in [ d , b ];

 Then, perform the above operations in the new interval until the length of the interval is less than a given threshold, then the midpoint of the interval is considered to be the maximum value of the function.

(5) Determine the maximum height of an inscribed cylinder of a given radius in the workspace of the Delta robot

Accompanying drawing 4 is the flow chart of the method for determining the maximum height of an inscribed cylinder with a given radius in the working space of a Delta robot. The input information is the structural parameter parameter of the Delta robot, given the radius Radius of the bottom surface of the desired cylindrical workspace, and the maximum inscribed circle radius r ( r < Radius ) and center coordinates ( 0,0, z ). Determine the height range [ Z _min , Z _max ] that can accommodate the cylinder according to the input, and then use the dichotomy method to search for the coordinate position Z of the center of the inscribed circle with a radius of Radius in [ z , Z _max ] and [ Z _min , z ] respectively _up and Z _down . If Z _up is not found, it is judged whether Z _max is 0. If yes, Z _up =0; otherwise Z _up = r .

(6) Determine the largest inscribed cylinder in the working space of the Delta robot

Because the function of the volume of the cylinder that can accommodate the largest volume in the working space with respect to the radius of the base has only one maximum value; and on both sides of the maximum value, the function is monotonous. This case is the same as the exact method for the maximum base radius of a cylinder, so the search method in (4) is used.

Claims (4)

1. A search method for the expected workspace of the maximum inscribed cylinder with a given radius that can be accommodated in the workspace of a Delta robot, the method is by determining the maximum height and the center of the circle that can accommodate the expected workspace of a cylinder in the workspace of a Delta robot The radius of the inscribed circle at (0,0, z ), the radius of the largest inscribed circle and its position can be obtained. 2. A search method for the expected workspace of the inscribed cylinder that can accommodate the largest volume in the workspace of a Delta robot, the method is located at ( 0,0, z ), the radius of the largest inscribed circle and its position, the largest inscribed cylinder with a given radius, etc. 3. according to the described method of claim 1 or 2, wherein determine the determination of the radius of the inscribed circle at (0,0, z ) in the workspace of the Delta robot, mainly by randomly generating points in the garden and by comparing with the upper and lower boundaries Determine whether it is located in the workspace. 4. A method of searching for the maximum value of a function. The function has a unique maximum (small) value within a given interval and is monotonous on both sides of the maximum value. By determining two points in the interval and comparing the function values of the two points , thereby narrowing the range of intervals.