Optimization of Inverse Kinematic Solution of a T4R Robotic Manipulator

Corresponding Author: Shravan Anand Komakula Kakatiya Institute of Technology and Science, Warangal, India Email: shravan.komakula@gmail.com Abstract: Inverse kinematics has always been a major subject of extensive research filled with redundancy. A need for meticulous functioning and fast computation is always high in any mechatronic system and is growing with new advances and human requirements. In this paper, a T4R (T-Twisting, R-Rotary) robot with constrained motion and joint precision of 1 degree is analyzed for its workspace and kinematics. While the redundant inverse kinematic solution is being optimized for precision and limiting errors, further can be used for programming and motion planning a Robotic Manipulator or Arm. The strategies developed in this paper could also be useful for solving the inverse kinematics problems of other similar types of robotic arms.


Introduction
Robotic arm actions are executed in the joint angles while robot motions are specified in the Cartesian coordinates. Conversion of the position and orientation of a robot manipulator end-effector from Cartesian space to joint angles called as inverse kinematics problem, which is of fundamental importance in calculating desired joint configurations for robot manipulator design and control. For robotic manipulators that are redundant or with high degrees of freedom (dof), an analytical solution to the inverse kinematics is very difficult or impossible as there may be more than one solutions being possible. There is no effective solution to its inverse kinematics to date (Gan et al., 2005). For a manipulator with n degree of freedom, at any instant of time joint variables is denoted by θ i = θ(t), i = 1, 2, 3,..., n and position variables x j = x(t), j = 1, 2, 3,...,m. The relations between the end-effector position x(t) and joint angle θ(t) can be represented by a forward kinematic equation: where, f is a nonlinear, continuous and differentiable function.
On the other hand, with the given desired end-effector position, the problem of finding the values of the joint variables is inverse kinematics. This can be solved by: The different techniques used for solving inverse kinematics can be classified as algebraic (Craig, 1989), geometric (Lee, 1982) and iterative (Korein et al., 1982). The algebraic methods do not guarantee closed form solutions. In the case of geometric methods, closed form solutions for the first three joints of the manipulator must exist geometrically. The iterative methods converge to only a single solution or multiple solutions depending on the end-effector criterions (Alavandar and Nigam, 2008). In this paper, an iterative method is used in the computation of inverse kinematics and the solution is optimized for errors in result, this type of strategies can be used for motion planning and programming robots efficiently. This method is also developed for faster computation and further optimized for compatibility.

Kinematic Description of the Arm
Model Consider the robot configuration as shown in Figs. 1, 2.
Where: θ 1 ,θ 2 ,θ 3 ,θ 4 ,θ 5 Are the Angles of links relative to horizontal (Degree) L 1 ,L 2 ,L 3 ,L 4 ,L 5 Are Link lengths (cm) (x 1 ,y 1 ,z 1 ) is the point of origin of the chain (x 5 ,y 5 ,z 5 ) is the end-effector point The side view of the planar kinematic chain is shown above in Fig. 1 with vertical Z-axis and horizontal Xaxis where we can observe angles θ 2 ,θ 3 ,θ 4 ,θ 5 which are taken with respect to the X-Y plane.
The top view of the planar kinematic chain is shown above in Fig. 2 with vertical Y-axis and horizontal Xaxis where we can observe angle θ 1 which are taken with respect to X-Z plane. Angles taken in an anti-clockwise direction are taken as positive and clockwise directions are taken as negative.

Degrees of Freedom
AT4R robot has 5 degrees of freedom i.e., it requires five parameters to specify a particular position of the chain. To control a robot the input parameters required are in the form of angles to drive servo motors and the output being the positioning of end-effector in 3D space.

Constraints and Limits
Considering that joints actions are performed using servo motors which have a precision of 1 degree, 5 such kinds of joints are present in T4R robot and each servo motor has 180 degrees of constrained motion only, this limits can be mathematically represented in the range of angles as:

Methodology
The methodology process is as follows: a. Defining a virtual model of the robot. b. Computing inverse kinematics using iterations in Matlab for different end-effector positions. c. Plotting the results and observing the errors for optimizing.

Computation Process
The inputs for the program are three coordinates of end-effector along with the angle made by fourth link θ 5 o with horizontal and the limit of error in cm. The process begins with calculating the coordinates of joint 4 in the three-dimensional system with θ 1 o using X5, Y5 coordinates and tangent rule. Then the joint 4 is rotated through the Z-axis through an angle θ 1 o to place the kinematic chain in the plane of X-Z axis. In the next step an array locus of points tracing the path of the circle with L 1 as radius and X 1 ,Z 1 as the center, then another array locus of points tracing the path of joint 3 i.e., the circle with L 3 as radius and rotated point of joint 4 as the center are computed. Then the distance between points on the circles with each other is iterated and when the distance is equal to L 2 with a precision of 0.1 cm there forms a feasible position of the chain. This precision is required as the mathematical results are not always accurate to higher decimal points and rounded off to 6 decimal places in Matlab. Thus joint angles made by links in this position are calculated using the points on circles and the values are floored to the nearest integer as our consideration constraint, the servos accept only integer angles. The floored angles undergo forward kinematics using trigonometric equations on sine and cosine rules and the error in 3D point obtained to the required end-effector point is calculated if the distance is less than or equal to the error limit defined at the starting of the program then it is plotted and the link angles are printed in the command window. This data can be further used for motion planning.
This type of process is adapted to decrease the time for computation so that the algorithm can be used for real-time programming of a robot control unit.

Z=z; End
The equations given below convert joint angles which are with reference to the horizontal axis to with reference to links, these angles can be used as angle inputs to servo motors directly for a specific function: where, ⌀ 1 , ⌀ 2 , ⌀ 3 , ⌀ 4 , ⌀ 5 are angles relative to Links(Degree).

Results
The circles are plotted in red color with radii of 0.01 cm, 0.05 cm, 0.1 cm, 0.15 cm, 0.2 cm and center as the required end-effector point for reference. The study has been divided into 3 cases with different endeffector positions and computation is done with an error limit of 0.2 cm.

Case Study -I
Input: X5=15; Y5=15; Z5=20; θ 5 o =45; The Figs. 3, 4 and 5 shows the projections of obtained end-effector points in three planes of axes obtained for this case, which can be used as a reference for observing the outcome of the robot being programmed.
From the figures, we can observe that there are six feasible configurations for the robot possible within the error limits of 0 to 0.08 cm in this case, with θ 1 obtained as 45 o and θ 5 also being 45 o . The feasible link configuration with least error is highlighted red color in Table 1.

Case Study -II
Input: X 5 =15; Y 5 =20; Z 5 =25; θ 5 o =60; The Figs. 6, 7, 8 shows the projections of obtained end-effector points in three planes of axes obtained for this case.   From the above figures, we can observe that there are five feasible configurations for the robot possible within the error limits of 0 to 0.06 cm in this case, with θ 1 obtained as 53 o and θ 5 being 60 o . The feasible link configurations with least error are highlighted red color in Table 2.
From the figures, we can observe that there are ten feasible configurations for the robot possible within the error limits of 0 to 0.06 cm in this case, with θ 1 obtained as 45 o and θ 5 being -45 o . The feasible link configuration with least error is highlighted red color in Table 3.

Conclusion
Even if there is no perfect solution to an inverse kinematic problem with redundancy, there are methods which can be used to compute an effective solution and can be optimized for the requirement. These type of developments in computation processes contribute to a reduction in computation time and also can be adopted in high precise robots and program with efficiency.