IK EQUATIONS & MODELING FOR SCARA ROBOTS

The paper presents the backward kinematics analysis, modeling equations for a 4 axis robot. Index Terms — SCARA Robot, Computer Control


I. INTRODUCTION
Robotics is an interdisciplinary field that mixes various engineering disciplines such as electrical, electronics, instrumentation, mechanical, computer science, control engineering, mathematics, communications and many other fields into one.In this, work a unique 4 axes system is designed and fabricated with indigenous components with a brief kinematic analysis of the designed robot.
The kinematically modeled designed robot is used for performing some PNP operations and was named as a Selective Compliance Assembly Robot Arm (SCARA).The primary motive behind the work was to develop a modular educational robotic system, the CRUST 2002 (Computerized Robotic Unit with Selective Tractability system) with the help of locally available components and sub-systems as shown in Fig. 1 and also to develop a user friendly GUI to control it [1].

II. PHYS ICAL STRUCTURE DES IGN
A four axis / four DOF designed SCARA robot arm as shown in Fig. 1.A SCARA robot is a 4 DOF stationary robot arm having base, elbow, vertical extension and tool roll and consisting of both rotary and prismatic joints [2].There is no tool yaw and tool pitch (only tool roll) [1].There are 4 joints, 4 axis (three major axes -base, elbow, vertical extension and one minor axis -tool roll).The 4 DOF's are given by Base, Elbow, Vertical Extension and Tool Roll, i.e., there are three rotary joints and one prismatic joint.Since n = 4; 16 KP's are to be obtained and 5 RHOCF's are to be attached to the various joints [3].
The vector of joint variables is a combination of  and d, i.e., q = {  , d } T .

Vector of link twist angles are
All the 4 joint axes are vertical in nature (all the z -axes can be pointing down or up) as shown in Fig. 3.The first three (B, E, VE) axes are called as the major axes and are used for positioning the wrist, while the last one, the minor axes ( TR ) is used to orient the tool in the direction of the object [4].The 1 st 3 major axes determines the shape and size of work envelope.It consists of a L shaped structure, to the end of which the second link is attached.There are two links a 1 and a 2 which move parallel to the work surface ; The vertical extension d 3 is variable and moves in a direction  r to the work surface ; the length of the gripper / tool / EE is d 4 [2].

IV. BLOCK DIAGRAM OF THE INVERSE KINEMATIC MODELING
{ , } = ?{ q , q , q , q } = ,d , The analysis of the robot is developed in 2 passes, viz., development of the link coordinate diagram as shown in Fig. 4 and development of the kinematic parameter table as shown in the Table 1 [10].The link coordinate diagram is developed as shown in the Fig. 6 with the kinematic parameter table as shown in Table 1 [2].

V. COMPUTATION OF THE JOINT VARIABLES
The output of the direct kinematics of the SCARA is [ The Tool Configuration Vector (TCV) is given by [2] [9] Note : To find out q 1 , q 2 , q 3 , q 4 ; apply row operations used in mathematics to the components of w using various trigonometric identities [2], [11].

To Extract Elbow Joint Variable
Eq n 3.12:4 q 2 =  arc cos (w 1 2 + w 2 2a 1 2a 2 2 , 2a 1 a 2 ) Eq n 3.12:5 From the above equations , we see that the IK solution is not unique and hence we get two solutions.Hence , solution to IKP to the shoulder is not unique [10].

To Extract Base Joint Variable
Expand C 12 and S 12 using sum of sines and cosines ; isolate C 1 , S 1 write in matrix form , collect all C 1 terms and S 1 terms , find A -1 and | A | , solve for q 1 [2], [15] .
This solution given by this gives the values of the base angle q 1 over the complete range (- , + ) ; since , we had used the arc tan 2 function .Hence, if we use the arc tan 2 function , we can recover the base angles over the complete range , i.e., 360° [17], [18] To Extract Vertical Extension Joint Parameter , q 3 = d 3 From the 3 rd component of TCV ; w , we have w 3 = d 1q 3d 4 The 3 rd variable q 3 is extracted as [13] q 3 = d 1d 4w 3 To Extract Tool Roll Angle , q 4 =  4 [2] The tool roll angle is computed from the last component of the TCV , i.e., w 6 [14].From TCV, we have ; w 4 = w 5 = 0 , i.e., there is no yaw and pitch .Using the equation [15] q n =  ln w 4 2 + w 5 2 + w 6 2 =  ln 0 2 + 0 2 + w 6 2 q 4 =  ln  w 6 

VI. CONCLUS ION
A four axes inverse kinematic analysis was performed for the designed robot and was successfully implemented in the laboratory.The robot was controlled using a GUI developed in visual basic language in various modes.A number of pick and place operations were successfully performed by the developed robot by using teaching mode, manual mode and programming modes and the inverse kinematic model.