Kinematics study for a spatial manipulator

. For industrial robot design and control problems, it is necessary to determine both the positions of its parts relative to the stationary coordinate system, absolute positions of the parts, and their relative positions, generalised coordinates. The first problem is called the forward problem, and the second is the inverse problem for manipulator positions. The purpose of the paper is to solve the direct problem on manipulator positions, i.e., to determine and study positions of manipulator sections with known generalised coordinates. In the paper a spatial manipulator with four degrees of freedom is considered. Kinematic characteristics of the last link of the manipulator - grabber are defined. The kinematic characteristics are the coordinates of current position, velocity, and acceleration of the grip. Kinematic characteristics are found by applying the vector matrix method, based on the application of transition matrices from one reference system to another and rotation vector matrices. The vector matrix method belongs to the universal methods and is designed for use in mathematical computer simulation systems.


Introduction
A manipulator is an executive mechanism of an industrial robot equipped with actuators and a working organ, by means of which the working functions are performed [1,2].The ability to reproduce motions is achieved by giving the manipulator several degrees of freedom, along which a controlled movement is carried out in order to obtain a given movement of the working body -grip.
The motion of a spatial manipulator with four degrees of freedom (Fig. 1) is given by indicating the time-dependence of the four generalised coordinates (), (),  1 (),  2 ().In a gripper (grabber)  of a solid body is attached to the arm , on which the point is marked.The task is to investigate the kinematics of a spatial manipulator: determine point positions of the body  for any point in time, determine an array of velocity and acceleration values of the point  of the body  for any time moment in global coordinates, and calculate the matrix of angular velocity and angular acceleration of the body  [3,4].
Several coordinate systems are used to set the positions of the manipulator links: global coordinate system and local coordinate systems:

Determining coordinates of a single point 𝑴for the body 𝑻, placed in the arm grip
Let us determine the position of the point  of the body  in the global coordinate system [5,6].The global and local point coordinate matrices are depicted through the radius-vector matrices ̄ with an appropriate ordinal index.
Equation of motion for a point  in global coordinates will be described by a matrix: Here: ), Rotation matrices   (()) and  2 (()) around the axes and  2 respectively: Dependency plots of the global coordinates of a given point  of time in the range from 0 to  1 = 500s, and the trajectory of the point  in global coordinates are shown in Figure 2 and Figure 3:   ) here  0 -an array of point coordinate matrices in a global reference frame ( = 0,1. . ., ).The process of changing the velocity projections of a point  over time is shown in Figure 4. Modulus of velocity of the point  is defined by the expression: The process of changing the velocity modulus of a point over time is illustrated in Figure 5. 0,066 0,132 0,198 0,264 0,33 0,396 0,462 0,528 0,594 0,66 0,726 0,792 0,858 0,924 0,99 Array of point М acceleration matrices is obtained using the numerical rate differentiation procedure: The process of changing the acceleration projections of point M over time is shown in Figure 6.The modulus of acceleration of point M is determined by the expression: The process of changing the acceleration modulus of point M over time is shown in Figure 7. Let us define the angular velocity matrix of the body .The body performs two rotations: around the axis  1 ( 0 ) at an angular velocity  ̇1 () and around the axis  2 at angular velocity  ̇2 ().Absolute angular velocity matrix of the body  can be found as a sum: - here  0 () = ( 0 0  ̇() ) -matrix of transfer angular velocity of a body ,  0 () =   (())  ⋅  2 (())  ⋅  2 () -matrix of the relative angular velocity of the body  on the global coordinate system,  1 () =  2 (())  ⋅  2 () -matrix of the relative angular velocity of the body  in the first local coordinate system; )matrix of the relative angular velocity of the body  in the second local coordinate system [9,10].
Absolute angular velocity matrix of the body  takes the form: The process of changing the projections of the angular velocity of the body over time is shown in Figure 8.The figure shows that the rotation occurs only around the axis  0 ( 1 ).Rotations around the axes  0 ,  0 is not available.The modulus of angular velocity is determined by the formula (9): The process of changing the angular velocity modulus of the body over time is shown in Figure 9.

Determining the angular acceleration of a body 𝑻, placed in the arm grip
Calculate the angular acceleration of the body  by numerically differentiating the angular velocity matrix: (10) The process of changing the projections of the angular acceleration of a body  in time is shown in Figure 10.The figure shows that the angular accelerations around the axes  0 ,  0 are not available.The process of changing the modulus of angular acceleration of a body over time is shown in Figure 11.

Conclusions
A direct problem of manipulator kinematics on the example of a manipulator with four degrees of freedom is solved in the paper by finding position, velocity and acceleration of a grip at any instant of time with known generalised coordinates.The solution of the direct problem is done by vector-matrix method in Mathcad simulation environment.

Fig. 2 .
Fig. 2. Dependency graphs for global point coordinates on time.

2. 2
Determining a single point  velocity of the body , placed in the arm grip To obtain a point velocity matrix  in global coordinates, it is possible to differentiate equation (1) by time, or calculate an array of point velocity values  by numerical differentiation in a sufficiently small step  [7, 8].Array of point velocity matrices in a global reference frame is: 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480

Fig. 4 .
Fig. 4. Diagrams of changes in projection velocity of a point  from time in the global coordinate system.

Fig. 5 .
Fig. 5. Diagram of the change in velocity modulus of a point  for the body  in time.

Fig. 6 .
Fig. 6.Diagrams of changes in projection of the point acceleration from time in the global coordinate system.

Fig. 7 .
Fig. 7. Diagram of the change in velocity modulus of a point  for the body  in time.

Fig.
Fig. The process of changing the angular velocity projections of the bodies  in time.

Fig. .
Fig. .Process of changing projections of angular acceleration of a body  in time.The modulus of angular acceleration is determined by the formula:

Fig. 1 .
Fig. 1 .Diagram of changes in the modulus of angular acceleration of a body in time.
0.5… 0.6… 0.84 1.0… Diagram of changes in the angular velocity modulus of a body in time.