C03 – 2009.02.05 Advanced Robotics for Autonomous Manipulation

1. Department of Mechanical Engineering ME 696 – Advanced Topics in Mechanical Engineering C03– 2009.02.05 Advanced Robotics for Autonomous Manipulation Giacomo Marani Autonomous Systems Laboratory, University of Hawaii http://www2.hawaii.edu/~marani 1

2. ME696 - Advanced Robotics – C02 Contents 1. Vectors deriv. 2. Angular velocity 3. Derivative for P. 4. Generalized Vel. 5. Derivative for R 6. Joint Kinematics 7. Simple kin. Joint • Summary • Vectors derivatives • Angular velocity • Derivative for points • Generalized velocity • Derivative of orientation matrix • Joint kinematics • Simple kinematic joint • Parameterization of simple kinematical joint • Kinematic equation of simple joints • Kinematics of robotics structures Kinematics – Part A 2

3. ME696 - Advanced Robotics – C02 • Vector Derivatives • Time derivative of geometrical vector , computed w.r.t. frame <a>: • (2.1) • Same time derivative but in the different reference frame <b>: • In general: k j Oa k < a >  i < b > Ob i j Vector derivative 3

4. ME696 - Advanced Robotics – C02 • Vector Derivatives • Proof: • (2.2) • Hence the result (very important): k j Oa k < a >  i < b > Ob i j Vector derivative 4

5. ME696 - Advanced Robotics – C02 • Vector Derivatives • If we project the (2.1) over the frame <b> we have: • FIRST derive THEN project (not allowed the reverse) • Meaning: An observer integral with <a>sees the change of the components over <b> of . These components change independently from the place of the observer. • This the definition of derivative of algebraic vector. k j Oa k < a >  i < b > Ob i j Vector derivative 5

6. ka kb q jb ja ME696 - Advanced Robotics – C02 ia ib • Angular Velocity • Since the rotation matrix between <a> and <b> is time dependent, we can define Angular Velocityof the frame <a>w.r.t. the frame <b> the vector b/a which, at any instant, gives the following information: • Its versor indicates the axis around which, in the considered time instant, an observer integral with <a> may suppose that <b> is rotating; • The component (magnitude) along its versor indicates the effective instantaneous angular velocity (rad/sec.) • To the vector Angular Velocity we can associate the following differential form: • The above relationship does not coincide with any exact differential. w(t) Angular Velocity 6

7. ka kb q jb ja ME696 - Advanced Robotics – C02 ia ib • Angular Velocity • We want not to write in a different form the (2.2): • We need Poisson formulae: • Thus we have: • (2.3) • If  is constant: (rigid body) w(t) Angular Velocity 7

8. ME696 - Advanced Robotics – C02 • Angular Velocity • Properties: • b/a = - a/b • Given n frames, the angular velocity of <k> w.r.t. <h> if given by adding the successive ang. Velocities encounteredalong any path. • In this example: Angular Velocity 8

9. ME696 - Advanced Robotics – C02 • Time derivative for points in space • We define: • “velocity of P computed w.r.t. the frame <a>”: • “velocity of P computed w.r.t. the frame <b>”: • It is possible to proof that: • where vp/b is the velocity of the origin of the frame <b> w.r.t <a> Angular Velocity 9

10. ME696 - Advanced Robotics – C02 • Time derivative for points in space • Proof: • We define vb/a the velocity of the origin on the frame <b> w.r.t. <a>: • Using the (2.3) with the opportune indexes we have: Angular Velocity 10

11. ME696 - Advanced Robotics – C02 Generalized velocity In order to completely describe the relative motion between 2 frames we organize the angular velocity and the velocity of the origin within a vector called Generalized Velocity : We can project the G.V. in any frame: This definition is valid forany point integral with the frame <b>: where Angular Velocity 11

12. ka kb q jb ja ME696 - Advanced Robotics – C02 ia ib Derivative of the orientation matrix Problem: we want to compute the relationship between the derivative of the orientation matrix and the angular velocity: Remember that: Deriving w.r.t. time: w(t) Derivative of the Orientation matrix 12

13. ka kb q jb ja ME696 - Advanced Robotics – C02 ia ib Derivative of the orientation matrix Finally: (2.4) Remembering the transformation of the cross-prod operator: the previous equation becomes: (2.5) The (2.4) and (2.5) are very useful in computing the time evolution of the orientation matrix: w(t) Derivative of the Orientation matrix 13

14. ME696 - Advanced Robotics – C02 Group definition A group is a set, G, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. The symbol "•" is a general placeholder for a concretely given operation, such as the addition. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms: Closure. For all a, b in G, the result of the operation a • b is also in G. Associativity. For all a, b and c in G, the equation (a • b) • c = a • (b • c) holds. Identity element. There exists an element e in G, such that for all elements a in G, the equation e • a = a • e = a holds. Inverse element. For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element. The order in which the group operation is carried out can be significant. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation a • b = b • a may not always be true. Kinematics of the joints 14

15. ME696 - Advanced Robotics – C02 Rotation Group In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e. handedness) of space. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it is in fact a Lie group. Kinematics of the joints 15

16. ME696 - Advanced Robotics – C02 Joint Kinematics In general, the set of all the relative positions between two free bodies constitutes a group that may be represented by the matrix: SO(3) is the Special Euclidian group. Kinematics in G can be represented as an object belonging to its Lie algebra: Kinematics of the joints 16

17. ME696 - Advanced Robotics – C02 Joint Kinematics The joint can be characterized by a relationship that involves the generalized velocity of the frame <b> w.r.t. <a>: (2.6) where q is the “configuration”. This means: If the distribution q è integrable, the constraint is Holonomic. In case that the axis are integral with at least one body, the matrix A is constant. Will name this kind of joints as Simple Kinematic Joints. Kinematics of the joints 17

18. ME696 - Advanced Robotics – C02 Simple Kinematic Joint s In this case, the solution of the (2.6) is given by: where the column of H creates a base for the kernel of A and r is the number of degreesof freedom of the joint: Kinematics of the joints 18

19. ME696 - Advanced Robotics – C02 Simple Kinematic Joint s H is the Joint Matrix. Often p is known as quasivelocity. Examples of joint matrices: Kinematics of the joints 19

20. ME696 - Advanced Robotics – C02 Parameterization of Simple Kinematic Joint s In general, the joint configuration is defined by the previous differential equation: which can be re-written as: We can now integrate the above equation, obtaining the evolution of the transformation matrix T. Kinematics of the joints 20

21. ME696 - Advanced Robotics – C02 Parameterization of Simple Kinematic Joint s Example: r=1 H1 is the direction of the rotation axis, hence: H2 is the direction of the translation, so we have: which, integrated, gives: If H has more columns: Kinematics of the joints 21

22. ME696 - Advanced Robotics – C02 Parameterization of Simple Kinematic Joint s Summary r=1 h1 is the direction of the rotation axis h2 is the direction of the translation, so we have: If h has more columns: Kinematics of the joints 22

23. ka kb jb ja ME696 - Advanced Robotics – C02 ia ib Example: spherical joint Example: Kinematics of the joints 23

24. ka kb jb ja ME696 - Advanced Robotics – C02 ia ib Example: spherical joint Finally: Kinematics of the joints 24

25. ME696 - Advanced Robotics – C02 Example: translational joint Example: Find the transformation matrix parameterized by q1: Solution: Kinematics of the joints 25

26. ME696 - Advanced Robotics – C02 Example: Screw Example: Find the transformation matrix parameterized by q1: Solution: Kinematics of the joints 26

27. ME696 - Advanced Robotics – C02 Kinematic equation of simple joints Problem statement: Find a relationship between the quesivelocities (p) and the derivative of the joint parameters (q). Consider a simple joint described by the matrix: We can define Kinematic Equation the following relationship: where the matrix Gamma is defined by the following recursive algorithm: 1) For j=1..r define the matrices Rj and Ljasfollows: Kinematics of the joints 27

28. ME696 - Advanced Robotics – C02 Kinematic equation of simple joints 1) For j=1..r define the matrices Rj and Lj as follows: 2) Build a matrix B as follows: 3) Finally compute Gamma as follows: where B* is a right-inverse of B(q) Kinematics of the joints 28

29. ME696 - Advanced Robotics – C02 Homework E01 Describe the joint matrix for a free body and find the associate parameterization and transformation matrix. Kinematics of the joints ka kb jb ja ia ib 29

30. ka kb jb ja ME696 - Advanced Robotics – C02 ia ib Example: spherical joint Example: Kinematics of the joints 30

31. ME696 - Advanced Robotics – C02 Homework E02 Find the transformation matrix of the following PUMA structure: Kinematics of the joints 31

32. ME696 - Advanced Robotics – C02 Homework E01 Use 2 different methods for positioning the frames: D-H convention All frames parallel to the base frame and compare the results. Kinematics of the joints 32

33. ME696 - Advanced Robotics – C02 Kinematics of robotics structures Kinematics of robotic structures 33

34. ME696 - Advanced Robotics – C02 Direct kinematic problem (Forward kinematics) Direct kinematic problem 34

35. ME696 - Advanced Robotics – C02 Direct kinematic problem 35

36. ME696 - Advanced Robotics – C02 Kinematics of the joints 36

37. End of presentation