Prof. Fanny Ficuciello Robotics for Bioengineering Trajectory planning

Transcription

1 Trajectory planning to generate the reference inputs to the motion control system which ensures that the manipulator executes the planned trajectories path and trajectory joint space trajectories operational space trajectories

2 Path and trajectory path: the locus of points in the joint space, or in the operational space, which the manipulator has to follow in the execution of the assigned motion; a path is then a pure geometric description of motion trajectory: is a path on which a timing law is specified, for instance in terms of velocities and/or accelerations at each point trajectory planning algorithm inputs: path description path constraints constraints imposed by manipulator dynamics outputs: end-effector trajectories in terms of a time sequence of the values attained by position, velocity and acceleration

3 Path and trajectory a reduced number of parameters is specified extremal points possible intermediate points geometric primitives the motion timing law total trajectory time constraints on the maximum velocities and accelerations Eventually the assignment of velocity and acceleration at points of particular interest trajectory planning in the operational space naturally description of the task path constraints to be accounted (obstacles) neighbourhood of singular configurations presence of redundant DOFs trajectory in joint space inverse kinematics control action

4 Trajectory in the joint space the planning algorithm generates a function q(t) interpolating the given vectors of joint variables at each point, in respect of the imposed constraints generated trajectories should be not very demanding from a computational viewpoint joint positions and velocities should be continuous functions of time continuity of accelerations may be imposed, too undesirable effects should be minimized, e.g., nonsmooth trajectories point-to-point motion final points on the path and the traveling time are specified motion through a sequence of points final points, intermediate points on the path and the traveling time are specified

5 Point-to-point motion generation of from an initial value to a final value in cubic polynomial computation of coefficients initial and final values of acceleration, polynomial of at least fifth order

6 Example

7 Trapezoidal velocity profile

8 Trapezoidal velocity profile assigned trajectory

9 Trapezoidal velocity profile assigned trajectory

10 Motion through a sequence of points for each joint variable there are N constraints, and then one might want to use an (N 1)-order polynomial it is not possible to assign the initial and final velocities oscillatory behaviour increases and this may lead to trajectories which are not natural numerical accuracy for computation of polynomial coefficients decreases the resulting system of constraint equations is heavy to solve polynomial coefficients depend on all the assigned points, if it is desired to change a point, all of them have to be recomputed

11 Solution low-order interpolating polynomials, continuous at the path points, are considered in place of a single high-order polynomial interpolating polynomials with imposed velocities at path points interpolating polynomials with computed velocities at path points interpolating polynomials with continuous accelerations at path points (splines) Interpolating linear polynomials with parabolic profile (via points)

12 Operational space trajectories the time sequence of the values attained by the operational space variables is utilized in real time to obtain the corresponding sequence of values of the joint space variables inverse kinematics algorithm trajectories in the joint space linear microinterpolation for given path points the corresponding components can be interpolated with a sequence of cubic polynomials, polynomials with parabolic blends, and so on on the other hand, if the end-effector motion has to follow a prescribed trajectory of motion (primitives of motions), this must be expressed analytically

13 Path primitives parametric representation of the path Γ

14 Rectilinear path

15 Circular path

16 Position polinomial interpolation Segment

17 Position circle

18 Sequence of points

19 Orientation interpolation of? interpolation of

20 equivalent axes of rotation Orientation

21 Orientation