Forward Kinematics: It involves determining the position and orientation of the end effector of a robotic manipulator based on known joint parameters (displacements for prismatic joints and angles for revolute joints).
It solves for both:
Position: Where the end effector is located.
Orientation: How the end effector is oriented with respect to a reference frame (e.g., the ground).
Example: To pick up a glass from a table, the robot must not only reach the position of the glass but also orient its gripper correctly.
Inputs for Forward Kinematics
Kinematic Structure of the Robot: Includes link lengths, joint parameters, and angles.
Denavit-Hartenberg (DH) Representation: A method to systematically describe the geometry of a robot.
Other methods exist, particularly in computer graphics for transformations.
Frames and Transformations
Ground Frame: The fixed reference frame, typically defined at the origin (0, 0, 0).
Link Transformations: Each link has a transformation characterized by:
Translation: Movement from one frame to another.
Rotation: Orientation of the link in three-dimensional space.
Homogeneous Transformation Matrix: 4x4 matrix that represents the transformation from one frame to another, combining rotation and translation.
Denavit-Hartenberg Parameters
Link Length (a_i): Distance between the Z-axes of consecutive links.
Joint Offset (d_i): Distance measured along the Z-axis between the previous link and the current link.
Joint Angle (θ_i): The angle between the X-axis of the previous link and the X-axis of the current link.
Twist Angle (α_i): The angle between the Z-axes of consecutive links, measured about the X-axis.
Important Concepts
Frame Assignment: Each link is assigned a frame based on its position and orientation relative to the previous link.
Fixed Parameters: Link length (a_i), joint offset (d_i), and twist angle (α_i) do not change, while joint angle (θ_i) varies depending on the configuration (revolute or prismatic joint).
Transformation Steps
Rotation about Z-axis by angle θ_i.
Translation along Z-axis by distance d_i.
Translation along X-axis by distance a_i.
Rotation about X-axis by angle α_i.
Applying the Transformations
Each transformation can be performed in sequence to determine the end effector's position and orientation:
Start with the ground frame and apply transformations for each link.
The final transformation matrix represents the end effector's position and orientation relative to the ground frame.
Summary of DH Parameters
Overall, the four DH parameters (a_i, d_i, θ_i, α_i) provide a compact representation of the robot's kinematic structure and are crucial for calculating forward kinematics.
In practice, these transformations enable the calculation of the position and orientation of the robot's end effector efficiently.