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Robotics: Introduction to Kinematics
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Robotics: Introduction to Kinematics

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    is there manual solution of engineering dynamics by:ginsberg?
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  • 1. The Robot System Control System Sensors Kinematics Dynamics Task Planning Software Hardware Mechanical Design Actuators
  • 2. Robot Kinematics.
      • In order to control and programme a robot we must have knowledge of both it’s spatial arrangement and a means of reference to the environment.
      • KINEMATICS - the analytical study of the geometry of motion of a robot arm:
        • with respect to a fixed reference co-ordinate system
        • without regard to the forces or moments that cause the motion.
  • 3. Co-ordinate Frames z y x Right-handed Co-ordinate frame x Base Frame x Tool Frame x Goal Frame x Link Frame x Camera Frame
  • 4. Kinematic Relationship
      • Between two frames we have a kinematic relationship - basically a translation and a rotation.
      • This relationship is mathematically represented by a 4  4 Homogeneous Transformation Matrix.
    z y x z y x
  • 5. Homogeneous Transformations  x  y 3  1 Translation  z 1 Global Scale r1 r2 r3 r4 r5 r6 r7 r8 r9 3  3 Rotational Matrix 0 0 0 1  3 Perspective
  • 6. Kinematic Considerations
    • Using kinematics to describe the spatial configuration of a robot gives us two approaches:
    • Forward Kinematics . (direct)
      • Given the joint angles for the robot, what is the orientation and position of the end effector?
    • Inverse Kinematics .
      • Given a desired end effector position what are the joint angles to achieve this?
  • 7. Inverse Kinematics
    • For a robot system the inverse kinematic problem is one of the most difficult to solve.
    • The robot controller must solve a set of non-linear simultaneous equations.
    • The problems can be summarised as:
      • The existence of multiple solutions.
      • The possible non-existence of a solution.
      • Singularities.
  • 8. Multiple Solutions Goal
    • This two link planar manipulator has two possible solutions.
    • This problem gets worse with more ‘Degrees of Freedom’.
    • Redundancy of movement.
  • 9. Non Existence of Solution
    • A goal outside the workspace of the robot has no solution.
    • An unreachable point can also be within the workspace of the manipulator - physical constraints.
    • A singularity is a place of  acceleration - trajectory tracking.
    Goal
  • 10. Kinematics  Control
    • Kinematics is the first step towards robotic control.
    Cartesian Space Joint Space Actuator Space Kinematics Dynamics Control z y x
  • 11. Joint Space Trajectories
    • For a robot to operate efficiently it must be able to move from point to point in space.
    • A trajectory is a time history of position, velocity and acceleration for each joint.
    • Trajectories are computed at run time and updated at a certain rate - the Path Update Rate. (PUMA robot updates at 36Hz)
  • 12. Joint Space Trajectory Planning Consider a robot with only one link. A B (  0 , t 0 ) (  f , t f )
    • Kinematics gives one configuration for B.
    • Choice of two trajectories to get there.
    • May wish to specify a via point - maybe to avoid an obstacle.
  • 13. Joint Space Schemes.
    • We need to describe path shapes in terms of functions of joint angles.  (t)
    angle  f time 0 t f  0 Lots of choices for continuous functions Cubic Polynomial Splines
  • 14. Cubic Polynomials
      • To move a single revolute joint from A to B in a given time gives four constraints.
    A starts at rest and at angle   B finishes at rest and at angle  f A cubic polynomial has four co-efficients which satisfy the four constraints:
  • 15. An Exercise for you:
    • Place the initial constraints into the formulae for position, velocity and acceleration and prove that the co-effecients are:
  • 16. An exercise for us
      • Given a single link robot arm with a revolute joint. Construct a cubic path function to take it from it’s present rest at 10 degrees to finish at rest at a desired end position of 110 degrees.
  • 17. Making A Spline.
    • A via point gives a constraint with
    angle time A B Via points t via1 t via2
  • 18. More Joint Space Schemes
    • Quintic Polynomials.
      • The cubic polynomial does not specify accelerations at the start and end of the motion. This adds two more constraints which can only be represented by a quintic polynomial. i.e. a 5 t 5
    • Linear Functions with parabolic Blends.
      • Linear function requires an infinite acceleration to get it started so parabolic blends are added at each end of the trajectory.
  • 19. Kinematics  Control
    • Kinematics is the first step towards robotic control.
    Cartesian Space Joint Space Actuator Space Kinematics Dynamics Control z y x