THE USE OF THE THEORY OF NONHOLONOMIC CONSTRAINTS IN THE PROCESS OF AUTOMATIC CONTROL OF A MANIPULATING MACHINE

The presented study contains a sample of utilization of the control laws treated as kinematic relations of parameter deviations and realized in the process of ordered automatic control of a manipulating machine. Movement of the grasping end is considered in an inertial reference standard rigidly joined with an immobile working environment of the manipulator. The specificity of the control's choice required creating program relations constituting the ordered parameters describing the movement of the manipulator's elements. During work, the ordered parameters are compared to the parameters realized in the process of the grasping end's work. This was deviations are determined, which thanks to properly prepared control laws are leveled by the manipulator's control executive system.


Introduction
Manipulating machines due to their accuracy and repetitive positioning properties are nowadays an essential part of different industries such as industrial processes, medical fields, and automotive industries. They are used to help in dangerous, monotonous, and tedious jobs. Typical Thus, the purpose of this topic is to compile control rights treated as non-holonomic relations imposed on the manipulator's arms' movement. Mobile objects, such as manipulators and mobile robots, have a limitation of degrees of freedom through control. The imposed relations limiting free movement, which are non-holonomic relations, are considered control rights. It was assumed that a manipulator moves with program movement, thus the control rights were determined as geometric and kinematic relations of deviations between the ordered and actual trajectory of movement of the grasping end. This distinguish the presented algorithm from among other control algorithms, which can be found in polish and foreign literature.

Kinematic model -realized parameters of the manipulator's movement
When modeling of the movement of the manipulator frames of reference locating the object in space were assumed (Fig.1). The primary frame of reference, in relation to which the manipulator's movements were considered, is the rigid inertial system related with its immobile foundation O1x1y1z1. The grasping end is related with the gravitational system OCxgygzg with axes parallel to appropriate axes of the immobile system O1x1y1z1 as well as the grasping end's own system OCxyz. It is a dextrorotary system, with start in the C joint and axis OCx directed along the line connecting point C with the center of mass K of the grasping end along with the weight carried by it.
The study takes into account the example of a fictional manipulator, the model of which was selected in such a way, that it is possible to analyze a great variety of its kinematic pairs. And so (Fig.1) two rotational parts were discriminated O1A and AB ( , ), rotations of which are described appropriately by angles 1 and 2 as well as angular velocities 1 and 2. The third arm -BC -is the moved arm ), which can extend itself with angular velocity of 3 l   . However, the manipulator's grasping end, the length of which rC is treated as the distance from point C to the center of mass K of the grasping end along with the weight carried by it, may perform any spherical movements in relation to joint C with angular velocity 3 (Fig.1). Spherical movements of the grasping end were described with the use of quasi-Euler movements , ,  (NIZIOŁ 2005), combining the gravitational system OCxgygzg with the grasping end's own system OCxyz. The vector of temporary location of the working end in the system O1x1y1z1  The vector of temporary linear velocity of the end 1 1 1 3 This way the kinematic relations, which provide information on the linear and angular location of the working end, realized during its work, were compiled. They were formulated in an inertial frame of reference O1x1y1z1 rigidly related with an immobile foundation of the manipulator and they constitute parameters realized by the automatically controlled manipulator.

Program relations -ordered parameters of the manipulator's movement
A necessary element of each control system is the targeting algorithm realized by it. Such an algorithm imposes boundaries on the object's movement. Thus, selection of the control method is very important.
In the case of a manipulator, in the situation, in which the grasping end is to perform a task specified in advance we assume, that the trajectory it should move along should be pre-ordered by the operator (Fig.2). Thus it is a program movement, which is carried out along a spatial trajectory ordered in advance.

Ordered program relations
where: xz,yz,zz -ordered parameters of the grasping end's movement In this case we are dealing, on one hand, with program geometrical relations (Eq.7), which impose limitations on the spatial location of the manipulator's grasping end, and on the other hand with kinematic ties (Eq.8), which impose limitations on its velocity vector -tangent to the ordered trajectory of movement. Thus the program ties constitute the ordered parameters of the manipulator's movement, which are compared to the parameters realized during targeting.

Control rights of the working end
Automatic control systems perform numerous types of tasks, which include: improvement of the dynamic properties of the controlled object, stabilization of the selected state parameters or directive changes of the ordered movement, performing the selected maneuvers, automatic realization of the object' complete movement along with all of its phases.
The proposed automatic control of the manipulating machine, according to the general concept show in the flowchart (Fig.3), functions based on the previously calculated movement program and automatic stabilization utilizing the compiled control rights. Fig. 3. Flowchart of the manipulator's control executive system The control law were established in such a way that during movement of the manipulator deviations, which are the differences between the current state parameters (x, y, z, Vx, Vy, Vz, Ωx, Ωy, Ωz) and the ordered parameters (xz, yz, zz, Vx, Vy, Vz, Ωx, Ωy, Ωz) are read. These deviations after proper amplification (with the use of Kj i amplification coefficient) are transferred to the executive system of the control system, causing the ordered movements of the manipulator's arms. 10 where: T1 itime constants, Kj i -control signals' amplification coefficients.
The set control laws are strongly non-linear functions of time. These are kinematic ties, nonintegrable, not directly brought down to geometric ties, this is why they constitute non-holonomic ties imposed on the movement of the manipulator's arms (NEJMARK and FUFAJEW 1971). In order to verify such control laws the non-linear model of the manipulator's dynamics shall be adopted

Conclusions
The study presents the kinematics model and control laws describing non-holonomic constraints imposed on the movement of manipulator's joints. The resulting equations are kinematic ties of deviations between the realized and ordered parameters of movement of the grasping end of the manipulator. They determine the relations between the kinematic parameters of the grasping end, and the possible deviations of the individual manipulator's joints, which are time-determined thanks to T i time constants. This way the control laws, along with movement equations, determine the behavior of the manipulator on the route during tracking, in order to enable its optimum control. They allow to introduce automatic control of the grasping end moving along a programmed route.