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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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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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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