1. 1 C01 – 2009.01.20 Advanced Robotics for Autonomous Manipulation Department of Mechanical EngineeringME 696 – Advanced Topics in Mechanical Engineering

2. C01: Geometry of robotics structures 2 Summary 1.Vectors (definitions and operations) 2.Coordinate systems 3.Rotation matrices 4.Representation of rotation matrices 5.Transformation matrices 6.Geometry of robotics structures C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example  ME696 - Advanced Robotics

3. Vectors: definitions 3 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. ME696 - Advanced Robotics – C01 i j k p1p1 p3p3 P Q P` Q` C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

4. Vectors: operations 4 Dot (scalar) product Properties: Cross (vector) product Properties: ME696 - Advanced Robotics – C01 v1v1 v2v2   v1v1 v1v1 v 1  v 2 C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

5. Coordinate systems 5 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). ME696 - Advanced Robotics – C01 OaOa k i j i i j j k k ObOb C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

6. Coordinate systems 6  { O a, i a, j a, k a }  { O b, i b, j b, k b } Any vector can be uniquely expressed w.r.t either and : v = c 1 i a + c 2 j a + c 3 k a v = h 1 i b + h 2 j b + h 3 k b ME696 - Advanced Robotics – C01 OaOa k i j i i j j k k ObOb C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

7. Coordinate systems 7 Orthogonal systems The reference frame  { O a, i a, j a, k a } 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. ME696 - Advanced Robotics – C01 OaOa i j k C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

8. Algebraic vector 8 Given 2 free vectors v and w in an orthonormal-right-handed reference system we have: Dot product: Cross product: where: ME696 - Advanced Robotics – C01 OaOa i i j j k k ObOb P Q C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

9. Rotation matrices 9 Position of reference systems Given two orthonormal reference systems and, the Rotation Matrix of with respect to is defined by: Properties: ME696 - Advanced Robotics – C01 OaOa i i j j k k ObOb C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

10. Rotation matrices 10 Change of reference system Problem statement: Given a free vector v with known projection w.r.t. the frame, we want to compute the projection w.r.t. the frame : Hence: similarly: ME696 - Advanced Robotics – C01 OaOa i i j j k k ObOb C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

11. Rotation matrices 11 Change of reference system for the cross-product operator ME696 - Advanced Robotics – C01 C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

12. Rotation matrices 12 Concatenation of rotation matrices Problem statement: Given n reference systems, we want to compute the rotation matrix between 2 frames and such as: Algorithm: 1)Identify an oriented path from to 2)Pre-multiply the vector k v with all the rotation matrices encountered (if the arrow is not opposite the rotation matrix is transposed) ME696 - Advanced Robotics – C01 C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

13. Representation of rotation matrices 13 Exponential representation Problem statement: The frame, initially coincident with, is rotated of an angle theta around the axis specified by v. Expanding the exponential we have: ME696 - Advanced Robotics – C01 kaka iaia jaja ibib jbjb kbkb  v C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

14. Representation of rotation matrices 14 Exponential representation Special cases: ME696 - Advanced Robotics – C01 kaka iaia jaja ibib jbjb kbkb  v C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

15. Rotation matrices 15 Roll-Pitch –Yaw (Euler) ME696 - Advanced Robotics – C01 j0j0 i0i0 z 0k  (yaw)  (pitch)  (roll) k1  k0k1  k0 j1j1 i1i1 i2i2 j2  j1j2  j1 k2k2 j3j3 k3k3 i3  i2i3  i2    C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

16. Rotation matrices 16 Roll-Pitch –Yaw (Euler) ME696 - Advanced Robotics – C01 C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

17. Transformation matrices 17 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 and We have: ME696 - Advanced Robotics – C01 k i j i i j j k k PP C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

18. Transformation matrices 18 Change of reference system for points in space Point P in : Representation of P in homogeneous coordinates: Transformation matrix: ME696 - Advanced Robotics – C01 k i j i i j j k k PP C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

19. Geometry of robotics structures 19 Multibody systems ME696 - Advanced Robotics – C01 Link Giunto C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

20. Geometry of robotics structures 20 Denavit-Hartenberg ME696 - Advanced Robotics – C01 C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

21. Geometry of robotics structures 21 Multibody systems ME696 - Advanced Robotics – C01 C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

22. Simulation Environment Robotics Developer Studio 22 RDS: Simple application example 5 Degrees of freedom linear chain. Link 1 Link 2 Link 3 Joint 1 Joint 2 Joint 3 Link 5 Joint 5 C ontents 1. Vectors 1. Vectors  2. Coordinate sys. 2. Coordinate sys.  3. 3. Rotation Matrices Matrices 4. Representation 4. Representation  5. Transformation5. Transformation M.  6. Geometry 6. Geometry  7. Example 7. Example 

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