Development of a Two Link Robotic Manipulator

Development of a two link robotic manipulator - Robots play important roles in day to day 
activities of human endeavour and


INTRODUCTION
The rapid growth in technology prompt for the latest matching devices. However, engineers and scientists are vastly adapting new technologies persistently. The robots perform significant roles in our lives and can carry out complex responsibility faster and accurately when compared to humans. They do not get drained or perform task emotionally. Robots operate in almost all human labours mostly in the fields which are unhealthy or impractical for workers [3]. This fact causes the workers to have more free time to spend on skilled Professions which includes the programming, maintenance, and operation of the robots.
Robotic arms perform simple translational and rotational motion, which has an end effector that performs a specific function [4]. A Robot is a programmable mechanical device that can replace the functions of the human arm. An automated pick and place robot arm, which can reach an object in a given domain or range of space, grip it precisely, and place it to the desired position. This mechanism is used for lifting objects and carrying out tasks that require extreme concentration, expert accuracy and may also be recursive. It can be employed industrially (e.g. cranes). According to [12] Presently industrial arms have risen in its capability and operation through micro-controllers and programming developments, enhanced mechanisms, sensing, and drive systems, which has led to a vast transformation in the robotic industry [4]. Hence, when designing a robot, factors such as artificial intelligence, concept and techniques, and cognitive science are vital to obtaining a viable design.

II.
SYSTEM MODELLING System modelling involves the use of models to conceptualize and construct real-life to analyse, study and simulate the real-life situations or system virtually. This gives us deeper insight into other aspects of the system and this helps in determining the stability and other characteristics of the system. Figure  1 shows that the model has a base upon which other parts of the manipulator is placed upon. Directly on the base is the shoulder link, this link is expected to undergo two different motion patterns first in the 180 0 rotation of the base giving the mechanism the aid to pick and place any desired target within its circumference of operation [5]. In addition to this motion, another actuator will be used to perform a translational motion at an angle which is used to move the next link, the elbow. At the joint between the shoulder and the elbow, another actuator is used making the third that helps to move the elbow link at an angle translational. Thereafter the end-effector is put in place to perform the picking and placing of the target object. The motion of the end-effector is also done with the aid of actuators and gears [6]. This gives a brief overview of the whole mechanism and how the links are connected one to another.   The total kinetic energy of each link in our system is given by the combination of its moving kinetic term Km and its rotating kinetic term Kr as K m = (2) And the potential energy expressed as P=m 3 gl 2 cos θ 2 + m 3 g(L 2 cos θ 2 +i 3 cos θ 3 ) (3) From langrange function If L = K -P (4) Applying eqn (2) and (3) to eqn (4) Finding the derivatives from eqn (5) ⊺ = (j 1 + L 1 2 (m 2 +m 3 )) ϴ 1 +l 1 (m 2 l 2 + m 3 L 2 )cos ϴ 2 ϴ̈2+ l 1 m 3 l 3 cos ϴ 3 ϴ̈3 + b 1 θ̇1L 1 (m 2 l 2 + m 3 L 2 )ϴ 2 1 sin ϴ 2 -L 1 m 3 l 3 ϴ 2 3 sinϴ 3 IV.

THE FORCE AND TORQUE REQUIRED
The mass of the links and the actuators are both considered in computing the torque at each joint. The length and the mass of various components are highlighted below in Figure 4. The torque of each servo motor at no load is obtained to be 10kgcm at 5V Dc input. This is needed to calculate the excess torque the motor has to develop in performing its required task. Moment of inertia is given as At the gripper, the torque developed by the servo motor is 10.5 and the excess torque required is determined to be 9.62987 kg-cm The torque needed at the wrist calculated to be 0.804125 kg-cm and the excess torque is 9.195875 kgcm At the elbow, the torque needed is determined to be 2.76556 kg-cm and the excess torque 7.23444 kg-cm The torque at the shoulder is 3.12366 kg-cm and the excess torque is 6.87634 kg-cm At the bottom which is the base, the torque required is 3.64666 kg-cm and the excess torque is 6.35334 kgcm The maximum torque that the mechanism developed at the gripper should be less than the torque at the base. For the mechanism, the maximum torque at the base is taken to be 5kg.cm giving allowance of 1.3177kg.cm. the radius of operation of the mechanism is the sum of all horizontal link of the mechanism given as 24.3cm.

V. MODELLING THE ARMATURE DC SERVO MOTORS FOR THE SYSTEM
The armature controlled dc servo motor is used extensively in the control system for its precision and stability characteristics [9]. The armature controlled dc motor modelling and its mathematical representation is as shown in Figure 5 and equations (9) to (17). It is assumed that the torque generated by the motor is proportional to the air-gap flux and also the armature current. ( ) is armature current (amperes) is armature resistance ( ) is motor torque (Nm) is Motor torque constant is armature inductance (Henries) applied voltage (volts) ( ) is motor angular velocity (rad/sec) is motor voltage constant (v/rad/sec) is total inertia of motor armature plus load α is acceleration (rad/ 2 )

shaft angle in radians
The torque developed by the motor is given as Taking the friction force to be negligible, the back emf is related to the angular velocity by = = Applying Kirchhoff's voltage law around the electrical loop, we have Equation (14) is obtained after taking the Laplace transform of equation (13) ԑ =( s+1) + Also it should be recalled that T= Substitute eqn (15) into eqn (14) ԑ − =( s+1) Hence, becomes   The electronic design for this mechanism at the transmitter involves the use of the two Arduino Atmega 283 microcontroller. It is an 8bit microcontroller and the first step involves configuring the register of the Arduino Atmega 283 to work with the desired pulse width modulation technique used. Fig. 8: Electrical design for the system An external power supply is used to supply the servo motors and also the Arduino Atmega 283 microcontroller. The receiver is however connected to the arduino atmega 283 together with the servo motors. The two DC motors responsible for the movement of the two wheels of the mechanism is connected with the motor controller L293D and both are connected to the Arduino Atmega 283. The system is being powered by a rechargeable 8v battery. A full rank is when the value of the rank is equal to the value of the row or column of a square matrix (i.e a n x n matrix having a rank of n). The controllable matrix for this mechanism is an 8 x 8 matrix and the rank is 4 meaning the rank is not full hence it is not controllable. Fig. 12. The observability matrix and the rank of matrix Fig 12 shows that the rank of the observability matrix is full. A full rank is when the value of the rank is equal to the value of the row or column of the square matrix (i.e a n x n matrix having a rank of n). The observability matrix for this system is an 8 x 8 matrix and the rank is full hence it is controllable. The rise time of the system is 1.01e + 03secs and the settling time is 1.84e + 03secs. The system steady state is at 1e + 08secs. This is considered good for the system. The plot illustrates that the system has a phase margin of 173 0 at a gain cross frequency of 0.75rad/secs. The point on the magnitude plot in which the curve crosses 0dB is traced down in the phase plot and that the corresponding point on the phase plot is the phase margin. Consequently, for the gain margin, the point on the phase plot that the curve crosses -180 0 is traced up to the magnitude plot and corresponding point on the magnitude plot curve is the gain margin. On this plot the phase plot did crosses the -180 0 at the phase crossover frequency of 0.75 rad/secs, the gain margin at this point is -143dB Fig. 15. The polar plot for the system.
The polar plot shows that the phase Margin gotten from the bode plot is 121 0 . The gain Margin of the polar plot is also the same as that of the bode plot and it is -143 0 . This shows that the system is stable. A. Establishing the Robustness with Reference Tracking Fig 17 shows the initial step plot of the system, the response time is 544.5 seconds our system is stable and the robustness is 0.6. The rise time 276 seconds and the settling time is 2210 seconds.

IRE 1701909
ICONIC RESEARCH AND ENGINEERING JOURNALS 20 Fig. 17. Initial System response graph The system is stable but the rise time is too high. PID controller is introduced to tune the system to reduce the rising time and also increase the robustness of the system.
From Fig 18 the response time is increased the 126 seconds and the robustness of the plant is 0.84seconds. The rise time was reduced from 276 seconds to 66.1 seconds and the system is stable. Also, the settling time has reduced. The system is stable and has a robustness of 0.84. this shows that with the aid of the PID controller the system is more stable and its rise time is faster than before and it has good robustness and performance. The assistive device will be able to meet most of the challenges of impaired people in society. The device was designed to fulfil the required tasks without the impaired facing any hindrances due to the lack of muscle activity. The Robotic Arm designed is unique to impaired people specific conditions of Athetoid Quadriplegia. As a result, the controllers of this device are constructed to ease the usage. Assistive robots will help people with impaired arm functions perform ordinary tasks, like eating and drinking, or opening a desk drawer.
The assistive robot construction breakdown involves the use of a microcontroller and the C -Programming language for programming, the radio frequency transmitter, and the receiver together with the servo motor controls. This device will provide the impaired an opportunity to partake in the simple everyday tasks.