Manipulator KinematicsRead Chapter 3Kinematics – Science of Motion• Determine position & orientation of linkage in a manipulator arm (or a leg in the legged robot)• Key problem: direct kinematics (forward kinematics) determining end-effector (foot) position/orientation as “f” of joint angles• Assign a frame to each link- Use homogeneous transformation to relate them- Use Denavit-Hartenberg convention for attaching the frame• Use the OSU Hexapod leg as an example- Describe the motion of leg- 3 revolute joins (3 DOF)I. Assign joint axes and number links and joints• Joints – numbered from 1 to n• Links – numbered from 0 to n; #0 = baseII. Define link parameters• Link – a rigid body which defines the relationship between two neighboring joint axes of a manipulator• Ignore external shape, mass, etc., for now•= link length: common normal distance•α= link twist: angle between the axis in a plane perpendicular to .II. Define link parameters - continued• Note values for link #1 of the OSU leg–= 2.5–α= - 90• Joint – that connects 2 links• Two joint parameters (for interconnections)– joint offset (the book calls link offset)– θ join angle• In summary: there are four parameters when two links are connected by simple joints– 3 fixed link parameters (, α󰇜 plus one of the following two– 1 joint variable (θor 󰇜RevolutePrismaticOSU HexapodOSU Hexapod LegZ1Z0Z2Z3ZFXFYFX2X3X1X0#1#2#312 + 9/16 2 + 1/2 2 + 7/1617Z3Z2X3X21 + 7/16 X0X1Z1Z0l4l1l2l3l5#0.III. Assign frame to links•Frame  - rigidly attached to link • Original at the point where the common normal across link  intersects with joint axis • Zialong axis  (joint )• Xialong common normal of link # • Yi= Zi×Xi• Take OSU leg as an example– Frame {1} see origin– Frame {2} see origin• Parallel axes: the origin is chosen to make the joint offset (d2) in this case = 0– Frame {0} coincident with adjacent frame {1}– Frame {3} parallel with adjacent frame {2}^^^^^• Use the following table• Think how the above parameters are obtained? For example: for link 21. Rotate about X1by α2. Translate along X1by 3. Translate Z2by  4. Rotate about Z2by θIV. Specify the link parametersLink αθ1000θ2-902.5 0θ3 0 12.8 0θ^^^^V. Determine link transformation• Task: determine the general form of the transformation of i related to i-1 for the Denavit –Hartenberg convention• It is equivalent to the product of 4 transformsα󰇛) θ󰇛) Screwx(,α) Screwz(,θ) 1. Involves 2 translations and 2 rotations2. Pair of translation+rotation = screw3. Note ordering and rotation: α,from link 1 to define  across link ; θ, at joint Result θθ0  θαθαα  αθα0θα0α α0 1 Equation 3.6 in the