Introduction

This section provides an introduction to the fundamental concepts of robotics, specifically focusing on the kinematics and control of robotic manipulators. Understanding these concepts is essential for designing and operating robotic systems efficiently.

General Terminology in Robotics (KUKA IIWA 14):

Workspace:

The reachable workspace of the KUKA IIWA 14 robot's end-effector is the manifold of reachable frames, which determines the range of motion and tasks it can perform.

Accuracy:

Accuracy on the KUKA IIWA 14 refers to its ability to position its end-effector precisely at a desired target point within the work volume, influenced by factors such as sensor technology and control algorithms.

Repeatability:

Repeatability on the KUKA IIWA 14 is a measure of its ability to consistently return to a specific position when performing repetitive tasks under similar conditions. It is crucial for maintaining precision and reliability in industrial applications.

Safety:

Ensuring safety is paramount in the design and operation of the KUKA IIWA 14. Its collaborative features and advanced control systems aim to minimize the risk of human-robot collisions and ensure a safe working environment for human-robot interaction.

Forward Kinematics

Forward kinematics involves calculating the position and orientation of the end-effector of a robotic manipulator based on the given joint parameters. This is achieved by using the known link lengths and joint angles to determine the end-effector's pose in the workspace.

DH - Table :

θ_i: Joint angle for the i-th joint, which is variable.
d_i: Offset along the previous z-axis to the common normal.
a_i: Length of the common normal (distance between z_i-1 and z_i along x_i).
α_i: Angle between the previous z-axis and the current z-axis around the common normal (along x_i).

Schematic Diagram :

D-bs = 340mm

D-se = 400mm

D-ew = 400mm

D-wf = 126mm

Transformation Matrix:

Formula:

Calculations :

The Final Transformation Matrix :

Inverse Kinematics

Inverse kinematics is the process of determining the joint parameters that provide a desired position and orientation of the robotic manipulator's end-effector. This involves solving complex mathematical equations and is crucial for tasks that require precise positioning.

Advanced Algorithm Description :

Advanced algorithms for robotic manipulators include various techniques for optimizing motion planning, trajectory planning, and adaptive control. These algorithms enhance the performance and efficiency of robotic systems, enabling them to handle more complex tasks and operate in dynamic environments.

Particle Swarm Algorithm :

PSO is a population-based stochastic optimization technique inspired by the social behavior of birds flocking or fish schooling. In PSO, a population of potential solutions (particles) is initialized randomly, and these particles move through the search space, adjusting their positions based on their own best-known position and the global best-known position of the entire swarm. This movement is guided by mathematical formulas derived from the behavior of social groups.

PSO can be applied to solve various optimization problems, such as inverse kinematics (IK). Inverse kinematics is the process of determining the joint configurations of a robot to achieve a desired end-effector position. PSO can be used to search for optimal joint configurations that minimize the difference between the desired end-effector position (target position) and the actual end-effector position. By using PSO, the robot can efficiently explore the joint space and find solutions that satisfy the given constraintsand objectives. PSO is used to solve the inverse kinematics problem for a robotic manipulator (represented by the imported robot model). PSO is compared with another method, Jacobian-based IK, to evaluate its performance in terms of accuracy and computation time.

Analysis :

When to Prefer PSO for Inverse Kinematics:

1. Highly Nonlinear Systems:
PSO: Effective for highly nonlinear systems where the relationship between joint angles and end-effector position is complex.
Jacobian: May struggle with convergence if the system is highly nonlinear and if the Jacobian matrix is difficult to compute or becomes singular.

2. Discontinuous or Non-Differentiable Objective Functions:
PSO: Does not require the objective function to be continuous or differentiable, making it suitable for problems where the objective function has discontinuities or is not differentiable.
Jacobian: Requires the objective function to be smooth and differentiable to compute gradients.

3. Global Optimization:
PSO: Capable of global search and can avoid getting trapped in local minima, making it suitable for finding global solutions in complex search spaces.
Jacobian: Primarily a local search method, which can get stuck in local minima and may not find the global optimum.

4. Complex Constraints:
PSO: Handles complex constraints naturally, as it evaluates the fitness of each particle considering all constraints.
Jacobian: Handling complex constraints may require additional modifications or constraints handling techniques.

5. Initialization Sensitivity:
PSO: Less sensitive to initial conditions due to its population-based nature, which explores multiple regions of the search space simultaneously.
Jacobian: Highly sensitive to initial conditions, and poor initialization can lead to slow convergence or failure to converge.