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# Robotics: Cartesian Trajectory Planning

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### Robotics: Cartesian Trajectory Planning

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