Analysis of the base motion model of industrial robots moving

3. Dynamic equation of an industrial robot with a fixed base.
It is known that the motion of an industrial robot can be represented by three different Lagrange-Euler, Newton-Euler, and D’Alamber equations. These equations have their own conveniences. For example, one is convenient for analytical representation of the motion of an industrial robot, the other is convenient for calculating the coefficients involved in the equation, and the third is convenient for obtaining a numerical solution. [9-12].
Suppose that the motion of an industrial robot is expressed using the Lagrange-Euler equation.
(1)
here
- (nxn) is a dimensional symmetric matrix and is related to the configuration of the industrial robot;
, - (nx1) as a dimensional matrix, coriolits, centrifugal and gravitational force vectors;
- (nx1) dimensional generalized force vector.
(4) given in the equation coefficients will vary depending on the kinematic pair and the following views of the links [13-14]:
I-form is a thin, thin weightless sternum;
II- form - mass omogeneous thin, slender sternum;
III- form - mass homogeneous thin, slender tube.

1. 
   I- for links in form
   

II- for links in form

2. 
   III- for links in form
   

In some cases, these three equations are generalized to a single equation.

4. The motion model of basic moving industrial robots. How the links are connected to each other, i.e. the kinematic pair, plays a key role in performing the complex manipulation operation of the industrial robot’s working arm. There are four different views of the kinematic pair [1-5,14]..
1st view of the kinematic pair. A place where one side is fastened with a link.
2nd view of the kinematic pair. One-sided fastened link rotation.
View 3 of the kinematic pair. Linear migration of the link.
Figure 4 of the kinematic pair. Ball hinge.
Let us give the kinematic and dynamic equations of complex spatial motion on these pairs. The appearance of the kinematic pair is less important when constructing a kinematic equation. There can be two cases related to the basis in the kinematics of an industrial robot. The basis of an industrial robot is movable and immovable. The immobility of the robot base is widely covered in [1-5,14,16]. According to them, the motion matrix of each link.
(1)
Radius vector representing the position of an industrial robot relative to the base of the grip device and using the motion matrices of the links, it is possible to give the equation of spatial motion [1-5]:
, (2)
here,
.
For the moving state of the robot base, the matrix representing the motion of the base is first introduced as follows [6-9,14-16]:

(2) is the appearance of the formula
(3)
Using formula (2) it is possible to put the problem of speed on a stationary industrial robot [12-14].
In kinematic Euler angles play a key role. Euler angles determine the target when aiming an industrial robotic grip device at an object. These three angles characterize the operation of the three cranes, coinage and ryskanie (search, search) on the grip device. The angles are in Euclidean space in a rotating system (sometimes called a counting system) [16 - 19]. There are three systems of Euler angles.

1. 
   For the first system.
   

around the axis turn the corner.

1. 
   around the axis turn the corner.
   
2. 
   around the axis turn the corner.
   

II. For the second system.

1. 
   around the axis turn the corner.
   

2. around the axis turn the corner.
3. turn the corner turn the corner.
III. For the third system.
1. around the axis turn the corner .
2. around the axis turn the corner .
3. around the axis turn the corner .
Deflection matrix for the first system is as follows:


Deflection matrix for the second system is as follows:


Deflection matrices for the third system are as follows:

M maneuverability of an industrial robot is as follows when the catching device is fixed:

Concept of service angle is introduced in increasing the positional accuracy of the movement of an industrial robot. The service is viewed in three-dimensional space, and the number of free levels can be 6, 7, or more. This is because it is necessary not only to lower the grip device to a given point, but also to direct it to the desired point. The service angle is determined as follows [15-17]:
,
here spherical surface of the last link holding device.
length of the holding device.
Relative size called the service coefficient.
Number of mobility of a three-link industrial robot

Maneuverability
Suppose, lengths of the links, to be comfortable . Let the holding device stand at a point in space.
The maximum distance of the base relative to the base coordinate system
Minimal distance

It is known that the service coefficient will be . To determine the full service area, proceed as follows:
when zone:
First zone

Second zone

Comparison for it

Last zone

A comparison for this zone


Service ratio for the second zone

Service ratio for the last zone

Although these equations are simple to write, they take some time to calculate. (Table 1) [15-17].

Table 1
Comparison of dynamic equations of various forms of industrial robot motion by calculation

	Lagrange-Euler equation	Newton-Euler equation '	D'Alamber equation
Number of multiplication operations
Number of add operations
Kinematic writing	4x4 size homogeneous matrix	Status vectors and deflection matrix	Status vectors and deflection matrix
Form of the equation of motion	Closed system of differential equations	Рекуррент тенглама	Closed system of differential equations

Of the Lagrange, Newton-Euler, and D'Alamber equations, the Lagrange equation is more inconvenient for calculating dynamic coefficients.