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Direct Kinematics. Link Description. The concept of Direct Kinematics. Choosing wisely the coordinate systems on the links If the wise choice was made, each link can be represented with 4 parameters
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The concept of Direct Kinematics • Choosing wisely the coordinate systems on the links • If the wise choice was made, each link can be represented with 4 parameters • When the parameters are found, the transformation matrices between the links can be found from a closed formula
DK Algorithm • 1) Draw sketch • 2) Identify and number robot links. Base = 0, Last = n • 3) Draw axis Zi for joint i. For rotating joint, Zi is the rotation axis. For prismatic (translating) joint, Zi can merge with the DOF axis or be perpendicular to it. • 5) Determine joint length ai-1between Zi-1 and Zi • 6) Draw axis Xi-1 along the shortest distance between Zi-1 and Zi. If the distance is 0, choose the direction of Xi-1 to be a normal to the plane that they create.
DK Algorithm (2) • 7) Determine joint twist i-1 measured around Xi-1 (between Zi-1 and Zi) • 8) Determine the joint offset di • 9) Determine joint angle i around Zi • 10) Write DH table • 11+12) Write link transformations and calculate the common transformation
Kinematics Parameters of a link Link length Link twist
What are the kinematics parameters of this link? • a = 7 • = 450
Link offset d Joint angle Kinematics Parameters of a link
Summary of the link parameters in terms of link frames • ai = the distance from Zi to Zi+1 measured along Xi • i = the angle between Zi and Zi+1 measured about Xi • di = the distance from Xi-1 to Xi measured along Zi • i = the angle between Xi-1 and Xi measured about Zi • We usually choose ai > 0 since it corresponds to a distance; • However,i , di , i are signed quantities.
There is no unique attachment of frames to links: • 1. When we align Zi axis with joint axis i, two choices of the Zi direction. • 2. When we have intersecting joint axes (ai=0), two choices of the Xi direction, corresponding to choice of signs for the normal to the plane containing Zi and Zi+1. • 3. When axes i and i+1 are parallel, the choice of origin location for {i} is arbitrary (generally chosen in order to cause di to be zero).
Three link Arm : RPR mechanism • “Cylindrical” robot – 2 joints analogous to polar coordinates when viewed from above. • Schematic: point – axes intersection; prismatic joint at minimal extension • Find coordinate systems and a, , d, (i=3)