Trajectory Generation • Goal: – Turn a specified Cartesian-space trajectory of P e into appropriate joint position reference values • Steps: – Use inverse kinematics of a robot manipulator arm to find joint values for any particular location of P e – Use sampling and curve fitting to reduce computation • Output: – a series of joint position/velocity reference values to send to the controller
Trajectory Generation. Goal: Turn a specified Cartesian-space trajectory of P e into appropriate joint position reference values Steps: Use inverse kinematics of a robot manipulator arm to find joint values for any particular location of P e - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Trajectory Generation• Goal:
– Turn a specified Cartesian-space trajectory of Pe into appropriate joint position reference values
• Steps:– Use inverse kinematics of a robot
manipulator arm to find joint values for any particular location of Pe
– Use sampling and curve fitting to reduce computation
• Output:– a series of joint position/velocity reference
values to send to the controller
5 Step Process for Trajectory Generation
1. Obtain function for workspace path
2. Sample function to get discrete joint pts
3. Apply IK & Jacobian calculations
4. Fit function to joint points5. Sample to get discrete
reference points
1. C
2. D
3. D
4. C
5. D
C=continuous
D=discrete
Step One: Continuous Fcn
• Obtain an analytic function to describe motion with respect to the base frame
• Obtain rate of change of location
Step 2: Sample
• Sample the trajectory to obtain a finite number, m, of sample points on the continuous trajectory:
• Sample rate of change
Step 3: IK & J
• (a) Use inverse kinematics to convert each Cartesian trajectory sample point vector, into a corresponding joint space vector,– Handle multiple solutions,
admissibility, etc.
Step 3: IK & J
• (b) Use the inverse Jacobian relation to convert each velocity vector, into a corresponding joint speed vector, – Handle singular configurations
Step 4: Fit Continuous Curve to Joint Points
• Use the sequence of vectors and i=1,…,m to generate continuous expressions for each joint and j=1,…,dof which pass through or sufficiently near to each of joint space sample points, and rate of change sample points, to produce continuous joint space trajectories for each joint.
Step 4: Fit Continuous Curve to Joint Points
Spline or Polynomial Fit
& derivatives:
Step 4: Fit Continuous Curve to Joint Points
Let’s look at fitting a curve to one interval
Step 4. Fit Continuous Curve to Joint Points
• Fit a continuous function, q(t) to the points:
• Time info – from original sampling• For now use notation (get rid of
subscripts i and i+1):
t=t0
t=tf
Step 4. Fit Continuous Curve to Joint Points
• Splines, polynomials,…• To match position, velocity and
acceleration at end points use a quintic polynomial (6 parameters to match the 6 unknowns):
Step 4. Fit Continuous Curve to Joint Points
• Note: To match only position and velocity at end points use a cubic polynomial (4 parameters to match the 4 unknowns):
Step 4. Fit Continuous Curve to Joint Points
• Use endpoints and time values in quintic polynomial (6 linear equations, 6 unknowns)
Step 4. Fit Continuous Curve to Joint Points
• In matrix form:
Step 4. Fit Continuous Curve to Joint Points
• In matrix form:
Step 4: Computational Thoughts
• Need to perform fit for each joint but…
on each TIME interval, this matrix is the same for each joint – compute inverse only once
Step 4: Fit Continuous Curve to Joint Points
Piecewise polynomials: one polynomial for each joint for each timte interval (and we can easily take derivatives)
…..
one for each time interval(i, i+1)
Step 5: Sample Joint Curve
• Sample each continuous joint trajectory to generate a sequence of discrete reference values for each joint, where ttotal/N is the sampling period used.