c03 – 2009.02.05 advanced robotics for autonomous manipulation

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Department of Mechanical EngineeringME 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. ME696 - Advanced Robotics – C02. - PowerPoint PPT Presentation

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A real-time approach for singularity avoidance in RMRC of Robotic Manipulators

Vector derivative5Vector 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 C02Oa< a >iijjkk< b >ObKinematics Part A2SummaryVectors derivativesAngular velocityDerivative for pointsGeneralized velocityDerivative of orientation matrixJoint kinematicsSimple kinematic jointParameterization of simple kinematical jointKinematic equation of simple jointsKinematics of robotics structures Contents1. Vectors deriv.2. Angular velocity3. Derivative for P.4. Generalized Vel.5. Derivative for R6. Joint Kinematics7. Simple kin. JointME696 - Advanced Robotics C02Vector derivative4Vector DerivativesProof: (2.2)Hence the result (very important):ME696 - Advanced Robotics C02Oa< a >iijjkk< b >ObAngular Velocity6Angular VelocitySince 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:Its versor indicates the axis around which, in the considered time instant, an observer integral with may suppose that 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.ME696 - Advanced Robotics C02kaiajaibjbkbqw(t)Angular Velocity7Angular VelocityWe 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 C02kaiajaibjbkbqw(t)Angular Velocity8Angular VelocityProperties: b/a = - a/b 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 C02Angular Velocity9Time derivative for points in spaceWe define:velocity of P computed w.r.t. the frame :velocity of P computed w.r.t. the frame :It is possible to proof that:where vp/b is the velocity of the origin of the frame w.r.t ME696 - Advanced Robotics C02Angular Velocity10Time derivative for points in spaceProof:We define vb/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 C02Angular Velocity11Generalized velocityIn 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 : whereME696 - Advanced Robotics C02Derivative of the Orientation matrix12Derivative of the orientation matrixProblem: 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 C02kaiajaibjbkbqw(t)Derivative of the Orientation matrix13Derivative of the orientation matrixFinally: (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 C02kaiajaibjbkbqw(t)Kinematics of the joints14Group definitionA 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.ME696 - Advanced Robotics C02Kinematics of the joints15Rotation GroupIn 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 C02Kinematics of the joints16Joint KinematicsIn 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 C02Kinematics of the joints17Joint KinematicsThe 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 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 C02Kinematics of the joints18Simple Kinematic Joint sIn 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 C02Kinematics of the joints19Simple Kinematic Joint sH is the Joint Matrix. Often p is known as quasivelocity.Examples of joint matrices:ME696 - Advanced Robotics C02Kinematics of the joints20Parameterization of Simple Kinematic Joint sIn 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 C02Kinematics of the joints21Parameterization of Simple Kinematic Joint sExample:r=1H1 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 C02Kinematics of the joints22Parameterization of Simple Kinematic Joint sSummaryr=1h1 is the direction of the rotation axish2 is the direction of the translation, so we have:If h has more columns:ME696 - Advanced Robotics C02Kinematics of the joints23Example: spherical jointExample:ME696 - Advanced Robotics C02kaiajaibjbkbKinematics of the joints24Example: spherical jointFinally:ME696 - Advanced Robotics C02kaiajaibjbkbKinematics of the joints25Example: translational jointExample:Find the transformation matrix parameterized by q1:Solution:ME696 - Advanced Robotics C02Kinematics of the joints26Example: ScrewExample:Find the transformation matrix parameterized by q1:Solution:ME696 - Advanced Robotics C02Kinematics of the joints27Kinematic equation of simple jointsProblem 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 C02Kinematics of the joints28Kinematic equation of simple joints1) 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 C02Kinematics of the joints29Example: spherical jointExample: [ 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 C02kaiajaibjbkbKinematics of the joints30ME696 - Advanced Robotics C02Kinematics of robotics structuresKinematics of the joints31ME696 - Advanced Robotics C02Kinematics of robotics structuresKinematics of the joints32ME696 - Advanced Robotics C02End of presentationsfericoGiunto)z(asse a viteGiunto(asse x)traslazionaleGiunto)z(asse rotazionaleGiunto10001000100000000000100k000100001000Consider first a linear chain of k links e k joints, each one characterized by: DOF; Joint matrix ; configuration vector quasivelocity vector Lets consider a point of a gene

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