robotics: cartesian trajectory planning

Download Robotics: Cartesian Trajectory Planning

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  • 1. Cartesian Trajectory Planning
    • Trajectories can include via points
      • Pass close to but not necessarily pass through
      • (knot points in b-splines: graphics)
    • E.g. straight line paths connected by via point
      • Both position and orientation have to be interpolated
    p 0 p(t 1 p 2 p 1 p(t 1
  • 2. Positional Translation
    • Shape of the interpolated region ?
      • RH Taylor 1979
      • Start at p 0 , arrive at p 1 in time t 1 under constant velocity
      • End at p 2 , from at p 1 in time t 2 under constant velocity
      • At time before arrival at p 1 begin the curved transition thus p(t 1 - ) and p(t 1 + ) are the two transition points
      • The curved segment is a parabola with constant acceleration
    p 0 p(t 1 p 2 p 1 p(t 1
  • 3.
    • The initial conditions p 0 , p 1 , p 2 , t 1 and t 2 are all specified.
    • We now consider the position and velocity constraints of the transition points
  • 4.
    • Integrating the acceleration equation we get
    • Rearrange and express in terms of the position function p(t)
    • Solve for a p at the second transition point t=t 1 +
    • Substitute into the equation for p(t)
  • 5.
    • The solution to the trajectory reduces to:
    • Note the solution does not pass through point p 1
  • 6. Rotational Transition
    • The rotational transition is found by finding the equivalent axis of rotation k
    • R 0 is the start orientation, R 1 the orientation at the via point and R 2 the orientation at the goal.
  • 7.
    • The rotation about k 1 and k 2 can be made a linear function of time
    • Rotation along the straight lines
    • The rotations can be derived in a similar way to the positions
  • 8. Velocity and Acceleration
    • We need to describe the velocities and accelerations of tools or of grasped objects
    • Position of a link rotating about and origin
    • Joint angle velocity
    • Same as swinging ball velocity perpendicular to position vector
    O o x 1 y 1 a 1 x 0 1 r v
  • 9.
    • More general form based on rotation matrices
    • Y is found by rotating x by /2
  • 10.
    • Consider the more general case where the link length is not fixed
    R( + /2)p 1 O 1 O 0 p 0 p 1 1 d 01 y 0 x 0 x 1 y 1 P
  • 11. 3D-Motion
    • Derivatives of Rotational Matrices
    • Euler angle rates
      • Representational singularities: Some valid velocities cannot be represented by Euler angles
    • Quaternion Rates
      • Convert from to q and integrate to get q
  • 12. Manipulator Jacobian
    • Matrix of differentials
    • Describe the motion of the tool in terms of changes in the joints
    • Jacobian calculated by differentiating the Forward Kinematic transform
    Cartesian Velocities Joint Velocities
  • 13. Inverse Kinematic Velocities and Accelerations
    • Given a tool speed. Find angle rates
    • Inverse Jacobian Method
      • Assumes that the Jacobian is non-singular(has an inverse at all points)
      • Not true at Singularities
      • Very Computationally expensive
    • Block Matrix Method
      • Split the Jacobian up into components exploiting the geometries of the robot arm
  • 14. Joint Force and Torque
    • Gravity acts at the centre of mass
    • Force/torque equations for link 2
    • Force /torque equations for link 1
    2 1 a 1 a 2 O 2 O 1 O 0 x 1 x 0 x 2 y 0 m 1 g m 2 g z 0 z 1
  • 15.
    • Rearranging the equations
    • Joint 1 gravity compensation torque compensates for its own weight plus the torque due to link 2
    • Forces and torques are generated at the end effector
      • Again the Jacobian is used to compute the transmitted forces and torques
  • 16. Dynamics
    • Newton-Euler equations
  • 17.
    • Calculating the dynamic Joint torques
      • For the planar manipulator the Newton Euler equations can be derived
      • Equations used to determine the driving forces and torques
    • Forward Dynamics
      • Joint Force and Torque Joint Motion
    • Inverse Dynamics
      • Joint Motion Joint Forces and Torques
    • General Manipulator dynamics
      • Recursive application of Newton-Euler dynamics
  • 18. Position Control
    • Proportional Derivative(PD) control
      • Damp the energy out of the motion to stop at end point of path
      • K is the proportional part, B is the damping part
      • Motion can be
        • under damped: oscillations
        • over-damped: sluggish response
        • or critically damped: best response without ossilations
  • 19. Trajectory following
    • Proportional Velocity(PV) control
      • Remove damping except where there are deviations from the path
    • For manipulator each joint can have an independent PV controller
  • 20. Other Schemes
    • Computed Torque Control
      • Feedback at one joint affects the others
      • Take these effects into accout
    • Resolved Acceleration Control
      • Use quaternions to allow the use of cartesian controller
    • Resolved Acceleration Force Control
      • Combine the above with the transmitted forces as detailed by the J matrix
  • 21. Manipulator Robotics
    • Haptics and Interaction
      • Force Control and Compliance
    • Non Rigid manipulators
      • Space and surgery
    • Task Based and Shared Control
      • Tele-manipulation

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