robotics_cource_kinematics

Robotics Kinematics: Understanding Robot Motion

Robotics kinematics is the study of robot motion without considering the forces and torques that cause it. It focuses on the relationships between the robot's joint positions and its end-effector's position and orientation. This chapter explores the fundamental concepts of robotics kinematics, including forward and inverse kinematics, and their applications in robot control.

1. Fundamental Concepts:

Degrees of Freedom (DOF): The number of independent parameters that define a robot's configuration.
Joints: Connections between robot links that allow for relative motion. Common joint types include revolute (rotational) and prismatic (linear).
Links: Rigid bodies connected by joints.
End-Effector: The last link of a robot arm, typically used for grasping or manipulating objects.
Configuration Space: The space of all possible joint positions of a robot.
Workspace: The space of all positions and orientations that the end-effector can reach.

2. Forward Kinematics:

Forward kinematics determines the end-effector's position and orientation given the robot's joint positions.
It involves applying a series of transformations to relate the joint coordinates to the end-effector coordinates.
The Denavit-Hartenberg (DH) convention is a widely used method for representing these transformations.
- The DH convention uses four parameters to describe the relative position and orientation between consecutive links:
  - $d$ : Link offset
  - $θ$ : Joint angle
  - $a$ : Link length
  - $α$ : Link twist
- These parameters are used to construct homogeneous transformation matrices, which are then multiplied to obtain the overall transformation from the base frame to the end-effector frame.
Forward kinematics is essential for simulating robot motion and visualizing the robot's workspace.

3. Inverse Kinematics:

Inverse kinematics determines the robot's joint positions required to achieve a desired end-effector position and orientation.
It is a more challenging problem than forward kinematics, as there may be multiple solutions or no solution at all.
Methods for solving inverse kinematics include:
- Analytical Solutions: Deriving closed-form equations that relate the end-effector coordinates to the joint coordinates. These solutions are often specific to certain robot geometries.
- Numerical Solutions: Using iterative algorithms, such as Jacobian-based methods, to find approximate solutions. These methods are more general but may be computationally intensive.
- Geometric Solutions: Using geometric insights to derive solutions for specific robot configurations.
Inverse kinematics is crucial for robot control, enabling robots to perform tasks that require precise end-effector positioning.

4. Jacobian Matrix:

The Jacobian matrix relates the joint velocities to the end-effector velocities.
It is used in Jacobian-based methods for solving inverse kinematics and for controlling robot motion.
$< 0 > J = \frac{\partial x}{\partial θ _{1}} \frac{\partial y}{\partial θ _{1}} \frac{\partial z}{\partial θ _{1}} \frac{\partial ϕ}{\partial θ _{1}} \frac{\partial ψ}{\partial θ _{1}} \frac{\partial ω}{\partial θ _{1}} \frac{\partial x}{\partial θ _{2}} \frac{\partial y}{\partial θ _{2}} \frac{\partial z}{\partial θ _{2}} \frac{\partial ϕ}{\partial θ _{2}} \frac{\partial ψ}{\partial θ _{2}} \frac{\partial ω}{\partial θ _{2}} \dots \dots \dots \dots \dots \dots \frac{\partial x}{\partial θ _{n}} \frac{\partial y}{\partial θ _{n}} \frac{\partial z}{\partial θ _{n}} < /0 > \frac{\partial ϕ}{\partial θ _{n}} \frac{\partial ψ}{\partial θ _{n}} \frac{\partial ω}{\partial θ _{n}}$
- Where x,y,z,phi,psi,omega are the end effectors cartesian and rotational velocities, and theta are the joint angles.
Singularities occur when the Jacobian matrix loses rank, resulting in a loss of controllability.

5. Applications:

Robot Control: Kinematics is essential for controlling robot motion, enabling robots to perform tasks such as pick-and-place operations and welding.
Path Planning: Kinematics is used to plan robot trajectories that avoid obstacles and achieve desired end-effector positions.
Robot Simulation: Kinematics is used to simulate robot motion and visualize the robot's workspace.
Robot Calibration: Kinematics is used to calibrate robot parameters and improve the accuracy of robot motion.

Robotics kinematics is a fundamental aspect of robot control and motion planning.¹ By understanding the relationships between joint positions and end-effector positions, we can design and control robots that perform complex tasks with precision and accuracy.