Nonholonomic System Modelling and Control with Single Degree of Freedom

This paper considers nonholonomic system modelling and control of a single degree of freedom. The model is based on a linear ordinary differential equation using the principles of vibrations in the area of feedback control system which is applied in many industrial applications. In this field, actual motion deviate significantly from the desired motion, and as a result of this deviation, performance, precision and accuracy of the system may not be acceptable. The problem is solved using the principles of PID and Routh-Hurwitz criterion of stability. At end the system was stable and the actual motion is the same as desired motion. The system was controllable and observable.


Introduction
In applications of theories in solving of problems of motion and equilibrium of mechanical systems, one may make use of the constraints to unearth a hidden fact about the systems under the discussion. One in which the mechanical state of a given system is defined by a finite number of parameters that can completely describe the position of the system at any given time. In this description certain conditions arises which are handicap of the system. These handicaps conditions are termed constrains. In terms of this, systems are classified as holonomic and nonholonomic. Mathematically, Holonomic system are systems in which all constraints are integrable into positional constraints of the form 1 2 ( , ,..., , ) 0 n n i f q q q t q     and t is time. In such a system, it can be used to reduce the number of degrees of freedom in the system. In case of the nonholonomic systems, it cannot be used to reduce the number of degrees of freedom in the system and can be defined as systems which have constraints that are nonintegrable into positional constraints. Intuitively, Holonomic system where a robot can move in any direction in the configuration space whereas Nonholonomic systems are systems where the velocities (magnitude and or direction) and other derivatives of the position are constraint and can not move in any direction in the configuration space. Moreover, the controllable degree of freedom is less than the total degrees of freedom, then it is known as non-Holonomic drive. A car has three degrees of freedom; i.e. its position in two axes and its orientation. However, there are only two controllable degrees of freedom which are acceleration (or braking) and turning angle of steering wheel. This makes it difficult for the driver to turn the car in any direction (unless the car skids or slides). Hence a car is nonholonomic system.
The origin of nonholonomic system can be divided into two classes namely: • Bodies of motion in contact with each other as they roll/move without slippage. • Conservation of moments in a multi-body system associated with under-actuated control (Neimark & Fufaev, 1972) .
For the Unicycle example nonholonomy arises because at the touching point between disk and surface, the velocity are confined to be aligned with the heading angle, no slippage is allowed. For UAV model, since the engine thrust is always aligned with body's longitudinal direction, it can be considered approximately that there is no side slippage. For the car-like model the two nonholonomic constraints arise because there are no side slippage at both front and rear wheel. For the hopping robot model nonholonomy arises because when it flies in the air, the angular moment is conserved since there is no external force applied to the system . (Neimark & Fufaev, 1972). This paper we shall deal with nonholonomic systems with a single degree of freedom and the focus was robot. An important application of the theory of vibration is the area of feedback control system. In many industrial applications, a robot system is designed to perform, with high precision of a specified task or follow a desired motion. However, due to disturbances or the effect of unknown parameters such as friction, wear, clearances in the joint etc., the desired motion of the system cannot be achieved. The actual motion deviates significantly from the desired motion, and as a result of this deviation, performance, precision and accuracy of the system is not be acceptable. It is therefore important to be able to deal with this problem by proper design of a control system that automatically reduces this deviation and if possible, eliminates it.

2.Concept of Ordinary Differential Equations in Relation to Vibration Mechanics
In this section mathematical concept underpinning vibration mechanics will be considered. Below are the strands showing the flow of discussion:  Second-order ordinary linear differential equation with constant coefficient  Vibration Mechanics

Second-Order Ordinary Linear Differential Equation with Constant Coefficient
Application of Newton's second law to the study of motion of physical systems leads to second-order ordinary differential equations. Generally, in order to examine, understand, and analyse the behaviour of physical systems, we must first solve differential equations that governs vibrations of systems.
Considering nth  order ordinary linear differential equation with constant coefficient of the form where A B C and D are respectively n n n n n matrices The sufficiently condition for complete state controllability is that the n n  controllability matrix, is non-singular having a non-zero determinant.

Vibration Mechanics
The coefficient of second order ordinary differential equation in application to mechanics are acceleration, velocity and displacement. As already mentioned in the previous section, the coefficients also represent physical parameters such as inertia, damping, and restoring elastic forces. These coefficients not only have a significant effect on the response of the mechanical and structural system, but they also affect the stability as well as the speed response of the system to a given excitation. Changes in these coefficients may result in a stable and unstable system, and /or oscillatory or non-oscillatory system. In all this situations, control of vibration is critical and important (Inman, 2017)

Mathematical model of Robotic Arm
In order to demonstrate the use of control system theory based on mathematical principles in solving the problem stated, we consider industrial single robotic arm as illustrated figure 1 below.
where v k and p k represent velocity and position gain respectively which is selected in such a manner that the deviation from the desired displacement is eliminated.
, d d (2) d  are assumed to be known because of the desired motion is assumed to be specific.

  and
ii. The inertia properties and dimension of the robotic arm are assumed to be known. Using proper sensors to measure the actual displacement a  and its derivatives due to the fact that the sensor measure the angular orientation, velocity and acceleration of the robotic arm during the actual motion. At this point, equation 3 can be calculated and proper signal is given to motion in order to produce this torque. Equating equations 2 and 3, we have,   Figure 2 below illustrate the block diagram of the equation 3, Figure 2. Block diagram In this section, control action concept which is based on control law was used to further analyse the system. Considering equation 18, it is clear the control action is proportional -derivative (PD). Despite PD control action improves the transient response of the system, very effective when all the system parameter are known. Moreover, when there are no disturbances. Since some of the system parameters are not known in addition to existence of disturbances, the PD control will result in non-zero state error. Hence not appropriate. For Proportional -integral Control Theory and Informatics www.iiste.org ISSN 2224-5774 (Paper) ISSN 2225-0492 (Online) Vol.10, 2020 28 (PI) control action improves the steady state of the system. Due to this reason the combination of the two i.e. proportional -integral -derivative (PID) control action improves overall time response of the robotic arm system and also, solves all the problems the other two control actions could not perform. Hence PID was used in this situation. To be able to used it, an additional term i k  was added to the torque T in equation 18 given as (2) (1

Data Analysis
The researcher used a of mass 7.2kg and length of the rod ( ) l is 10m . Hence mass of inertia is given as    figure 4, it is clear the control action is proportional -derivative (PD). Despite PD control action improves the transient response of the system very effective when all the system parameter are known and when there are no disturbances. Since some of the system parameters are not known in addition to existence of disturbances, the PD control will result in non-zero state error. Hence not appropriate. For Proportional -integral (PI) control action improves the steady state of the system. Due to this reason the combination of the two i.e. proportional -integral -derivative (PID) control action improves overall time response of the robotic arm system. Hence PID was used in this situation. To be able to used it, an additional term i k  was added to the torque T in equation 6. Therefor using equation 26 figure 6, it is clear that the errors in the system has been eliminated as time elapses which means that actual motion is the same as desired motion. The graph also indicates the system is stable. Moreover, reliability of a system is a function of errors in the system. Since the error function moving to zero as time elapses implies that the system is reliable. Furthermore, it also implies the system is controllable and observable. The researcher moves further to verify whether the system is totally controllable and observable or not.