1. 1 C03 – 2009.02.05 Advanced Robotics for Autonomous Manipulation Department of Mechanical EngineeringME 696 – Advanced Topics in Mechanical Engineering

2. Kinematics – Part A 2 Summary 1.Vectors derivatives 2.Angular velocity 3.Derivative for points 4.Generalized velocity 5.Derivative of orientation matrix 6.Joint kinematics 7.Simple kinematic joint 8.Parameterization of simple kinematical joint 9.Kinematic equation of simple joints 10.Kinematics of robotics structures C ontents 1. Vectors1. Vectors deriv.  2. Angular velocity 2. Angular velocity  3. Derivative for P. 3. Derivative for P.  4. Generalized Vel4. Generalized Vel.  5. Derivative for R 5. Derivative for R  6. Joint Kinematics 6. Joint Kinematics  7. Simple kin. Joint 7. Simple kin. Joint  ME696 - Advanced Robotics – C02

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

4. Vector derivative 4 Vector Derivatives Proof: (2.2) Hence the result (very important): ME696 - Advanced Robotics – C02 OaOa i i j j k k ObOb 

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

6. Angular Velocity 6 Since the rotation matrix between and is time dependent, we can define Angular Velocity of the frame w.r.t. the frame the vector  b/a which, at any instant, gives the following information: 1)Its versor indicates the axis around which, in the considered time instant, an observer integral with may suppose that is rotating; 2)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. ME696 - Advanced Robotics – C02 kaka iaia jaja ibib jbjb kbkb   t 

7. Angular Velocity 7 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) ME696 - Advanced Robotics – C02 kaka iaia jaja ibib jbjb kbkb   t 

8. Angular Velocity 8 Properties: 1)  b/a = -  a/b 2)Given n frames, the angular velocity of w.r.t. if given by adding the successive ang. Velocities encounteredalong any path. In this example: ME696 - Advanced Robotics – C02

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

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

11. Angular Velocity 11 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 : where ME696 - Advanced Robotics – C02

12. Derivative of the Orientation matrix 12 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: ME696 - Advanced Robotics – C02 kaka iaia jaja ibib jbjb kbkb   t 

13. Derivative of the Orientation matrix 13 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: ME696 - Advanced Robotics – C02 kaka iaia jaja ibib jbjb kbkb   t 

14. Kinematics of the joints 14 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: 1.Closure. For all a, b in G, the result of the operation a b is also in G. 2.Associativity. For all a, b and c in G, the equation (a b) c = a (b c) holds. 3.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. 4.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. ME696 - Advanced Robotics – C02

15. Kinematics of the joints 15 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. ME696 - Advanced Robotics – C02

16. Kinematics of the joints 16 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: ME696 - Advanced Robotics – C02

17. Kinematics of the joints 17 Joint Kinematics The joint can be characterized by a relationship that involves the generalized velocity of the frame w.r.t. : (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. ME696 - Advanced Robotics – C02

18. Kinematics of the joints 18 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: ME696 - Advanced Robotics – C02

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

20. Kinematics of the joints 20 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. ME696 - Advanced Robotics – C02

21. Kinematics of the joints 21 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: ME696 - Advanced Robotics – C02

22. Kinematics of the joints 22 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: ME696 - Advanced Robotics – C02

23. Kinematics of the joints 23 Example: spherical joint Example: ME696 - Advanced Robotics – C02 kaka iaia jaja ibib jbjb kbkb

24. Kinematics of the joints 24 Example: spherical joint Finally: ME696 - Advanced Robotics – C02 kaka iaia jaja ibib jbjb kbkb

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

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

27. Kinematics of the joints 27 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 Lj asfollows: ME696 - Advanced Robotics – C02

28. Kinematics of the joints 28 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) ME696 - Advanced Robotics – C02

29. Kinematics of the joints 29 Example: spherical joint Example: [ sin(q1) sin(q2) cos(q1) sin(q2)] [1 --------------- ---------------] [ cos(q2) cos(q2) ] [ ] [0 cos(q1) -sin(q1) ] [ ] [ sin(q1) cos(q1) ] [0 ------- ------- ] [ cos(q2) cos(q2) ] ME696 - Advanced Robotics – C02 kaka iaia jaja ibib jbjb kbkb

30. Kinematics of the joints 30 ME696 - Advanced Robotics – C02 Kinematics of robotics structures

31. Kinematics of the joints 31 ME696 - Advanced Robotics – C02 Kinematics of robotics structures

32. Kinematics of the joints 32 ME696 - Advanced Robotics – C02

33. End of presentation