# robotics: cartesian trajectory planning

Post on 22-Nov-2014

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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

- 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

- 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

- 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

- 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

- 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

- 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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