Optimal solutions to the path planning problem can be reached by using multiple solutions of the Inverse Kinematic (IK) problem. Furthermore, in addition to path planning, control of robot manipulators requires the mapping from the end-effector Cartesian space coordinates into corresponding joint positions. This mapping is referred to as the Inverse Kinematics of the robot. Finding the position and orientation of the end-effector from the joint angles is called the Forward Kinematics (FK) problem. FK of a robot manipulator can easily be formulated if the link parameters and joint variables of a robot are known, while IK is a nonlinear configuration-dependent problem that may have multiple solutions. Although extensive literature for single-solutions IK solvers exist, an unaddressed niche for IK solvers specifically suitable for Modular and Reconfigurable robots and the problem of Task-Based Configuration Optimization (TBCO) problem exists. Such a solver should have the following features:

• It should be general. Such an algorithm is used for all the possible kinematic configurations which can be assembled from the inventory of modules.

• The method should be complete. In order to incorporate all the IK solutions in TBCO and for optimum path planning of Modular and Reconfigurable Robots, the desired solver should be able to find all the solutions of the IK problem.

• The algorithm should obtain the IK solutions without any priori knowledge of the number of solutions. Since the number of possible kinematic configurations is very large, no information about the number or approximate locations of the IK solutions in the joints space exist.

• The solver should be applied to distinct kinematic configurations with minimum modifications. Since such an IK solver is used for solving IK problem of a wide range of distinct kinematic configurations, modifying the algorithm symbolically or analytically each time it is being applied to a new kinematic configuration can prove to be an obstacle.

• The method should be fast. Since in the TBCO problem the IK problem should be solved thousands of times, a higher speed of the IK solver is directly translated into a faster TBCO. Hence, the speed of such an algorithm is of prominent importance.

In this research, two distinct methods with these characteristics are proposed:

1. Niching Genetic Algorithm based IK solver:

A modified Genetic Algorithm for solving the inverse kinematics of a serial robotic manipulator is presented. The algorithm is capable of finding multiple solutions of the inverse kinematics through niching methods. Despite the fact that the number and position of solutions in the search space depends on the position and orientation of the end-effector as well as the Kinematic Configuration (KC) of the robot, the number of GA parameters that must be set by a user are limited to a minimum through the use of an adaptive niching method. The only requirement of the algorithm are the Forward Kinematics equations which can be easily obtained from the Denavit-Hartenberg (DH) link parameters and joint variables of the robot. For identifying and processing the outputs of the proposed GA, a modified filtering and clustering phase is also added to the algorithm. For the post-processing stage, a numerical inverse kinematics solver is used to achieve convergence to the desired accuracy.

2. Joint Reflection Operator based IK solver:

This approach efficiently obtains the multiple solutions of the IK problem. The method is developed for non-redundant Modular and Reconfigurable Robots. The algorithm requires no symbolic or analytical modifications, when the robot configuration changes. Also, the novel algorithm is proven to be computationally efficient and fast.

The approach is a generalization of numerical IK solvers, designed to incorporate the results of analytical methods for 2 degree-of-freedom manipulators to obtain multiple solutions of the IK problem. Such an operation is performed using the so called Joint Reflection Operator (JRO). The solver operates by finding all the positioning solutions and utilizing these solutions to determine the desired positioning/orienting IK solutions. The performance of the algorithm is verified by solving for multiple solutions of the IK problem for three distinct 6 degree-of-freedom kinematic configurations. For one of the tested kinematic configurations, no closed-form IK solutions exists.