# 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
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
• 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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