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Page 58 - Toucpad robotics C11
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    Practical Use: To find the position of the robot’s hand, its controller uses forward kinematics, which involves a chain
                  u
                      of calculations. It starts with the fixed base frame, and then sequentially transforms the coordinates through each
                      moving frame, link by link, until it reaches the end-effector. This chain of transformations allows the robot to precisely
                      know its hand’s position in the fixed frame at all times.
                  The Mathematical Connection: Transformations between Frames

                  The relationship between different frames is defined by a Transformation, which consists of a translation (a shift in
                  position) and a rotation (a change in orientation).

                      Translation: This is simply the vector that points from the origin of one frame to the origin of the other.
                  u
                      Rotation: This is the rotational matrix that describes how the axes of one frame are oriented with respect to the other.
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                  As we discussed in the “Matrix Operations” section, these two concepts are combined into a single  Homogeneous
                  Transformation Matrix. This powerful mathematical tool allows the robot’s controller to perform a series of calculations to:
                  1.  Forward Kinematics: Calculate the position and orientation of the end-effector frame relative to the base frame,
                      given all the joint angles. This involves multiplying all the transformation matrices of the links in a chain.
                  2.  Inverse Kinematics: Calculate the required joint angles to move the end-effector frame to a desired position and
                      orientation in the world frame. This involves working backward from the final desired position to find the necessary
                      joint angles.

                  Some Practical Use from Real World Robotics

                  Understanding frames is not just a theoretical exercise; it’s the foundation of a robot’s ability to perform precise tasks.
                  1.  Industrial Manipulators: A robotic arm in a factory needs to pick up a part from a specific location on a conveyor
                      belt and place it at a specific location on a pallet. Both the conveyor and the pallet’s locations are defined in the
                      world frame. The robot’s controller uses inverse kinematics and the relationships between its base frame and its
                      moving link frames to calculate the required joint angles to accurately place its gripper’s end-effector frame at the
                      destination coordinates.
                  2.  Autonomous Vehicles: A self-driving car needs to know its location on a digital map of Jammu. The map is a large,
                      fixed world frame. The car, with its sensors, constantly tracks its own moving vehicle frame relative to this world
                      frame. When a camera or LIDAR detects a pedestrian, the pedestrian’s position is first measured in the moving sensor
                      frame. This position is then immediately transformed into the vehicle frame and then into the world frame, so the
                      vehicle can share this information with its other control systems for a complete understanding of the environment.
                  3.  Search and Rescue Drones: A drone surveying a disaster zone uses a fixed Global Positioning System (GPS) frame
                      as its primary reference. Its onboard Inertial Measurement Unit continuously tracks its position and orientation in
                      its own moving drone frame, which is then used to update its location in the fixed GPS frame. This allows the drone
                      to provide accurate coordinates of a detected survivor or a collapsed building, even if it loses its GPS signal for a
                      short time.
                  So, we can say that frames are the invisible grid systems that give a robot’s world and its body structure. Fixed frames
                  provide a stable reference point, while moving frames allow the robot to understand its own motion and the dynamic
                  relationship between its parts. By using mathematical transformations to navigate between these frames, a robot can
                  achieve the kind of precise, coordinated, and autonomous movements that we see in industrial automation and advanced
                  mobile robotics. This is the bedrock upon which all sophisticated robotic control is built.
                  Degrees of Freedom: From Robotic Motion to 3D Design

                  The term Degrees of Freedom refers to the number of independent parameters that define the configuration of a system.
                  For a rigid body in three-dimensional space, there are a total of six degrees of freedom: three for translation (movement
                  along an axis) and three for rotation (turning around an axis).
                      Translation: Moving the object left/right on X-axis (Surge), forward/backward on Y-axis (Heave), and up/down on
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                      Z-axis (Sway).
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                  Touchpad Robotics - XI

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