differential kinematics and statics ref: ...
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Differential Kinematics and Statics Ref: 理论力学,洪嘉振,杨长俊,高等教育出版社, 2001. Incremental Motion. What small (incremental) motions at the end-effector ( D x, D y, D z) result from small motions of the joints ( Dq 1 , Dq 2 , …, Dq n )? - PowerPoint PPT PresentationTRANSCRIPT
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Differential Kinematics and Statics
Ref: 2001
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Incremental MotionWhat small (incremental) motions at the end-effector (Dx, Dy, Dz) result from small motions of the joints (Dq1, Dq2, , Dqn )? Alternatively, what velocities at the end-effector (vx, vy, vz) result from velocities at the joints (w1, w2, wn)?
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Some DefinitionsLinear Velocity: The instantaneous rate-of-change in linear position of a point relative to some frame.v=(vx, vy, vz)TAngular Velocity: The instantaneous rate-of-change in the orientation of one frame relative to another.Angular Velocity depends on the way to represent orientation (Euler Angles, Rotation Matrix, etc.)Angular Velocity Vector and the Angular Velocity Matrix.
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Some DefinitionsAngular Velocity Vector: A vector whose direction is the instantaneous axis of rotation of one frame relative to another and whose magnitude is the rate of rotation about that axis.
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Free VectorLinear velocity are insensitive to shifts in origin but are sensitive to orientation.{D}xx
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Free VectorAngular velocity are insensitive to shifts in origin but are sensitive to orientation.{D}xxxx
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Velocity Framesframe of reference: this is the frame used to measure the objects velocityframe of representation.: this is the frame in which the velocity is expressed.
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X0Y0x0y00q1Y1X10x2a1vvvvq2Ra2y2Figure 2.13: Two-Link Planar Robot
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X0Y0x0y00w10vvvvEnd-effector velocity for w1r0n
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X0Y0x0y00w20vvvvEnd-effector velocity for w2r1n
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Two-Link Planar RobotDirect kinematics equation
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Incremental Motiontaking derivatives of the position equation w.r.t. time we have
note that
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Incremental Motionwritten in the more common matrix form,
or in terms of incremental motion,
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Differential KinematicsFind the relationship between the joint velocities and the end-effector linear and angular velocities.Linear velocityAngular velocityfor a revolute jointfor a prismatic joint
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Differential KinematicsDifferential kinematics equation
Geometric Jacobian
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Relationship with T(q)Direct kinematics equation
Linear velocity
Angular velocity?
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Vector (Cross) ProductVector product of x and y
Skew-symmetric matrix
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Vector (Cross) ProductSkew-symmetric matrix
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Derivative of a Rotation MatrixdefineS(t) is skew-symmetric
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Interpretation of S(t)
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Interpretation of S(t)Given R(t)
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Example 3.1: Rotation about Z
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