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Page 42 - Toucpad robotics C11
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    Transformations (Rotations and Translations)
                      Robots  constantly  need  to  change  their  position  and orientation  in space.  Transformations are mathematical
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                      operations that describe these changes.
                      Translation: Moving an object from one point to another without changing its orientation (just shifting its position).
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                      This can be represented by adding a vector.
                      Rotation: Changing the orientation of an object around an axis. This is more complex and typically represented by a
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                      rotation matrix.
                      Homogeneous Transformation Matrices: These are special 4x4 matrices that combine both rotation and translation
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                      into a single mathematical operation. They are exceptionally powerful for describing the position and orientation of
                      one coordinate system relative to another.
                      Applications:
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                         Kinematics: Calculating the position and orientation of each link and the end-effector of a robotic arm relative
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                         to its base.
                         Navigation: Transforming sensor readings from the robot’s internal frame to a global map frame.
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                         Object Manipulation: Determining the precise movements needed to grasp an object, which involves transforming
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                         the object’s position into the robot’s gripper frame.
                      Example: Consider a robotic arm with several joints. To determine the exact location of its gripper, we start from
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                      the robot’s base. We apply a matrix operation (a homogeneous transformation matrix) for the first joint’s position
                      and orientation relative to the base. Then, we multiply that by another matrix for the second joint’s position and
                      orientation relative to the first, and so on, until we reach the gripper. This chain of matrix multiplications accurately
                      describes the gripper’s final position and orientation in the robot’s base frame.

                  Vectors and Vector Spaces: Direction, Force, and Position in Space
                  Vectors are fundamental mathematical objects that represent quantities having both magnitude (size) and direction.
                  Vector spaces provide the framework for working with these quantities.
                  Representing Position and Direction

                      Position Vectors: A vector can point from an origin to a specific point in space, representing its position (e.g., the
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                      position of a sensor, an object, or a robot’s joint).
                      Direction Vectors: Vectors are used to represent the direction of movement (e.g., velocity), the orientation of an
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                      object (e.g., which way a camera is pointing), or the normal to a surface.
                      Example: A self-driving car needs to know its current position (using a vector from the map’s origin) and its direction
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                      of travel (using a vector representing its heading).
                  Representing Forces and Velocities
                      Force Vectors: Forces (like gravity, friction, motor thrust) are vector quantities. Their magnitude indicates how strong
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                      the force is, and their direction indicates where they are acting.
                      Velocity  Vectors: A robot’s  speed  and direction  of movement  are represented  by velocity  vectors. Similarly,
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                      acceleration is also a vector.
                      Vector Addition/Subtraction: Used to combine multiple forces acting on a robot or to calculate relative velocities.
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                      Dot Product: Used to calculate work done by a force, or to find the angle between two vectors (useful for checking if
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                      two lines are perpendicular).
                      Cross Product: Used to calculate torque or the normal vector to a surface (useful for determining orientation in 3D).
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                      Example: A drone needs to calculate the resultant force vector from its four propellers to determine its overall
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                      thrust and direction of flight. To ascend, the upward force vector must be greater than the downward force vector
                      due to gravity.

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                  Touchpad Robotics - XI

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