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Department of Mechanical Engineering ME 696 – Advanced Topics in Mechanical Engineering. C02,C03 – 2009.01.22,27,29 Advanced Robotics for Autonomous Manipulation. Giacomo Marani Autonomous Systems Laboratory, University of Hawaii. http://www2.hawaii.edu/~marani. 1.
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Department of Mechanical Engineering ME 696 – Advanced Topics in Mechanical Engineering C02,C03 – 2009.01.22,27,29 Advanced Robotics for Autonomous Manipulation Giacomo Marani Autonomous Systems Laboratory, University of Hawaii http://www2.hawaii.edu/~marani 1
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 Kinematics – Part A 2
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
ME696 - Advanced Robotics – C02 Example: spherical joint Example: Kinematics of the joints 22
ka kb jb ja ME696 - Advanced Robotics – C02 ia ib Example: spherical joint Finally: Kinematics of the joints 23