Lecture15-hierarchical-animation

4 1 1 4 2 1 θ ϖ θ ϖ θ θ θ θ z z y x x x v v v

This preview shows page 19 - 29 out of 40 pages.

= 4 1 1 4 2 1 θ ϖ θ ϖ θ θ θ θ z z y x x x v v v v J Change in position Change in orientation Joint position velocities Relates a joint position change to end effector change Check dimensions: V = 6x1 J(θ) = 6x4 θ = 4x1
Image of page 19

Subscribe to view the full document.

Where we stand ( 29 θ θ J V = Can figure this out. Can figure this out. Want this. Instantaneous positional change vectors Desired change vector
Image of page 20
Solve for θ ( 29 θ θ = - V J 1 Instantaneous positional change vectors Desired change vector Small problem: J(θ) may not be easy to invert.
Image of page 21

Subscribe to view the full document.

IK – Transpose of the Jacobian Compute how much the change vector contributes to the desired change vector: Project joint change vector onto desired change vector Dot product of joint change vector and desired change vector is what we want and it can be computed using the transpose of the Jacobian
Image of page 22
IK – Transpose of the Jacobian θ = V J T = z y x z y x v v v V ϖ ϖ ϖ = 4 4 2 1 1 1 θ ϖ θ θ θ ϖ θ θ z x x z y x v v v v J
Image of page 23

Subscribe to view the full document.

Transpose of the Jacobian. Project change vector for joint i onto desired change. Those relative lengths determine degree of rotation applied to each joint. Joints who’s change vector are closer to parallel w/ desired change vector get rotated more. Then scale that by a constant It is just an approximation after all.
Image of page 24
The big picture
Image of page 25

Subscribe to view the full document.

Computing Joint Positions Have: joint angles and segment lengths Have: location of base in world coordinates Need: joint position in world coordinates Traverse the arm computing positions in world coordinates as you go.
Image of page 26
Computing Joint Positions (Example) Angle 1 = 92.95 degrees Seg. 1 is at (-0.05,0.998) Angle 2 = 68.5 deg. Seg 2 is at (-0.53, 1.16) Angle 3 = 37.0 Seg 3 is at (-1.00, 1.00) angle 1 angle 2 angle 3 Seg. 1 posn. x axis for joint 1 x a x i s f o r j o i n t 2 x axis for joint 3
Image of page 27

Subscribe to view the full document.

Computing Joint Positions Angle 1 = 92.95 degrees Seg. 1 is at (-0.05,0.998) Angle 2 = 68.5 deg.
Image of page 28
Image of page 29
You've reached the end of this preview.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern