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1 C02,C03 – 2009.01.22,27,29 Advanced Robotics for Autonomous Manipulation Department of Mechanical EngineeringME 696 – Advanced Topics in Mechanical Engineering

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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 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Kinematics of the joints 22 Example: spherical joint Example: ME696 - Advanced Robotics – C02

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

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