1. Trajectory Generation How do I get there? This way!

2. Chapter Objectives By the end of the Chapter, you should be able to: Specify simple trajectories Design path in joint space using polynomials Design path in joint space using linear functions with parabolic blends Understand Cartesian motion Bibliography: Craig’s book

3. Trajectories Problem: compute trajectory in n-dimensional space Trajectory = Time history of position, velocity and acceleration for each DOF Problem includes: –Trajectory specification –Trajectory representation –Trajectory generation

4. Path Description & Generation Path specification: motion of tool frame w.r.t. basis frame Basic problem: move manipulator from {T initial } to {T final } Usually initial and final frames are not enough. We add: – Via Points: Intermediate frames constraining position and orientation during motion –Temporal attributes: Bounds in velocity –Smoothness: motion should be smooth to be “feasible”

5. Trajectories in Joint Space Take {T initial } and {T final } and compute  (0),  (t f ) Compute intermediate points  (t i ) Find a smooth function  (t) verifying: –initial, final and intermediate constraints –velocity constraints Resulting function is a candidate trajectory “Natural” choice for  (t) : low order polynomials.

6. 4 Constraints:  (0)=  0  (t f )=  f  ’(0)=0  ’(t f )=0. Use 3rd order polynomial:  (t)=a 0 +a 1 t+a 2 t 2 +a 3 t 3 T-in-JS: Cubic Polynomials tftf t0t0 00 ff.

7. Cubic Polynomials with Via Points One way to deal with VP: stop at each point and use result above Better one: constraint velocity at VP but not necessarily to 0:  (t f )=  ’(t f ). Generalize previous result Options: –Specify velocity in terms of Cartesian velocity –Algorithm chooses velocity using heuristics –Algorithm chooses velocity via constrains on acceleration

8. Linear Functions with Blends Idea: interpolate between points using lines Difficulty: velocity is discontinuous when beginning /ending the motion Solution: smooth the path blending regions Smooth blending: use constant acceleration and same duration blending segments

9. Blend Computation I Putting the equations together: t f -t b tbtb thth hh