Robotics: Introduction to Kinematics


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Robotics: Introduction to Kinematics

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