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C01 – 2009.01.20 Advanced Robotics for Autonomous Manipulation

Department of Mechanical Engineering ME 696 – Advanced Topics in Mechanical Engineering. C01 – 2009.01.20 Advanced Robotics for Autonomous Manipulation. Giacomo Marani Autonomous Systems Laboratory, University of Hawaii. http://www2.hawaii.edu/~marani. 1. ME696 - Advanced Robotics.

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C01 – 2009.01.20 Advanced Robotics for Autonomous Manipulation

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  1. Department of Mechanical Engineering ME 696 – Advanced Topics in Mechanical Engineering C01 – 2009.01.20 Advanced Robotics for Autonomous Manipulation Giacomo Marani Autonomous Systems Laboratory, University of Hawaii http://www2.hawaii.edu/~marani 1

  2. ME696 - Advanced Robotics Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Summary • Vectors (definitions and operations) • Coordinate systems • Rotation matrices • Representation of rotation matrices • Transformation matrices • Geometry of robotics structures C01: Geometry of robotics structures 2

  3. k p3 P P` j p1 Q ME696 - Advanced Robotics – C01 Q` i Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Point • Point in a Cartesian system: • Spherical reference system: • Line segment: a part of a line that is bounded by two distinct end points. • Oriented line segment: • Given two oriented line segments [P-Q] and [P`-Q`], they are equipollent if they have the same direction, length and orientation. • Given an oriented segment [P-Q], a correspondent free vector is the whole class of line segments equipollent to [P-Q]. • Note the difference with the bound vector. Vectors: definitions 3

  4. v1 v2 v2 v1  v1  v1 ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Dot (scalar) product • Properties: • Cross (vector) product • Properties: Vectors: operations 4

  5. k k j Oa k < a > < 0 > j i i < b > ME696 - Advanced Robotics – C01 Ob i j Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Coordinate systems • In geometry and kinematics, coordinate systems are used not only to describe the (linear) position of points, but also to describe the angular position of axes, planes, and rigid bodies. In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). Coordinate systems 5

  6. k k j Oa k < a > < 0 > j i i < b > ME696 - Advanced Robotics – C01 Ob i j Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Coordinate systems • < a > { Oa, ia, ja, ka } • < b > { Ob, ib, jb, kb } • Any vector can be uniquely expressed w.r.t either <a> and <b>: • v = c1ia+ c2ja + c3ka • v = h1ib+ h2jb + h3kb Coordinate systems 6

  7. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Orthogonal systems • The reference frame < a >  { Oa, ia, ja, ka } is orthogonal with right-handed orientation if: • orthogonal with left-handed orientation: • If i , j , k are also of unit length the frame are orthonormal. k j Oa < a > i Coordinate systems 7

  8. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Algebraic vector • Given 2 free vectors v and w in an orthonormal-right-handed reference system we have: • Dot product: • Cross product: where: k j Oa k < a > P i < b > Q Ob i j Algebraic vector 8

  9. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Position of reference systems • Given two orthonormal reference systems <a> and <b>, the Rotation Matrix of <b> with respect to <a> is defined by: • Properties: k j Oa k < a > i < b > Ob i j Rotation matrices 9

  10. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Change of reference system • Problem statement: • Given a free vector v with known projection w.r.t. the frame <b>, we want to compute the projection w.r.t. the frame <a>: • Hence: • similarly: k j Oa k < a > i < b > Ob i j Rotation matrices 10

  11. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Change of reference system for the cross-product operator Rotation matrices 11

  12. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Concatenation of rotation matrices • Problem statement: • Given n reference systems, we want to compute the rotation matrix between 2 frames <h> and <k> such as: • Algorithm: • Identify an oriented path from <k> to <h> • Pre-multiply the vector kv with all the rotation matrices encountered (if the arrow is not opposite the rotation matrix is transposed) Rotation matrices 12

  13. ka kb q jb ja ME696 - Advanced Robotics – C01 ia ib Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Exponential representation • Problem statement: • The frame <b>, initially coincident with <a>, is rotated of an angle theta around the axis specified by v. • Expanding the exponential we have: Representation of rotation matrices 13

  14. ka kb q jb ja ME696 - Advanced Robotics – C01 ia ib Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Exponential representation • Special cases: Representation of rotation matrices 14

  15. k2 j2ºj1 k1ºk0 b j1 i2 b (pitch) a i1 g (roll) a (yaw) j3 z0k k3 ME696 - Advanced Robotics – C01 g j0 i3ºi2 i0 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Roll-Pitch –Yaw (Euler) Rotation matrices 15

  16. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Roll-Pitch –Yaw (Euler) Rotation matrices 16

  17. k k j k j i i  P ME696 - Advanced Robotics – C01 i j Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Change of reference system for points in space • Problem statement: • Given a generic point P in the space, compute its coordinates w.r.t. the frames <a> and <b> • We have: Transformation matrices 17

  18. k k j k j i i  P ME696 - Advanced Robotics – C01 i j Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • Change of reference system for points in space • Point P in <b>: Representation of P in homogeneous coordinates: • Transformation matrix: Transformation matrices 18

  19. Giunto Link ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example Multibody systems Geometry of robotics structures 19

  20. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example Denavit-Hartenberg Geometry of robotics structures 20

  21. ME696 - Advanced Robotics – C01 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example Multibody systems Geometry of robotics structures 21

  22. Link 5 Joint 5 Link 3 Joint 3 Link 2 Joint 2 Link 1 Joint 1 Contents 1. Vectors 2. Coordinate sys. 3. Rotation Matrices 4. Representation 5. Transformation M. 6. Geometry 7. Example • RDS: Simple application example • 5 Degrees of freedom linear chain. Simulation Environment Robotics Developer Studio 22

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