KINEMATIC CHAINS

Kinematic Chains This lecture continues the discussion on a body that cannot be treated as a single particle but as a combination of a large number of particles. Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the discussion on the analysis of kinematic chains with focus on robots. After this lecture, the student should be able to: Appreciate the concept of kinematic pairs (joints) between rigid bodies Define common (lower) kinematic pairs Distinguish between open and closed kinematic chains Appreciate the concept of forward and inverse kinematics and dynamics analysis Express a finite motion in terms of the transformation matrix

General Rigid-Body Motion General Motion = Translation + Rotation The general motion is equivalent to that of the action of a screw, which can be described using 4 “screw” parameters: Axis of rotation The angle of rotation  Axis of rotation passes through the point The displacement component parallel to the direction of rotation

Kinematic Pair A kinematic pair is the coupling or joint between 2 rigid bodies that constraints their relative motion. The kinematic pair can be classified according to the contact between the jointed bodies: Lower kinematic pairs: there is surface contact between the jointed bodies Higher kinematic pairs: the contact is localized to lines, curves, or points

Lower Kinematic Pairs  Revolute/pin joint s  Cylindrical joint  x y Planar joint  Spherical joint   Prismatic joint s

Kinematic Chain A kinematic chain is a system of rigid bodies which are joined together by kinematic joints to permit the bodies to move relative to one another. Kinematic chains can be classified as: Open kinematic chain: There are bodies in the chain with only one associated kinematic joint Closed kinematic chain: Each body in the chain has at least two associated kinematic joints A mechanism is a closed kinematic chain with one of the bodies fixed (designated as the base) In a structure, there can be no motion of the bodies relative to one another

Open and Closed Kinematic Chains Base (fixed) Open kinematic chain Kinematic joint Rigid bodies Structure

Robots Robots for manipulation of objects are generally designed as open kinematic chains These robots typically contain either revolute or prismatic joints

X Notation Z-axis C B “O” Y-axis A X-axis The position of point “C” expressed in frame {a} is denoted by For example, is the position of point “C” relative to frame {b} expressed in terms of frame {a}

X Transformation matrix Consider the 2 frames {a} and {b}. Notice that for a vector Example: X A B C X-axis Y-axis Z-axis “O”

X Example: Transformation matrix What is the transformation matrix for the case below? X A B C X-axis Y-axis Z-axis “O”

Example: Transformation matrix B C X-axis Y-axis Z-axis “O”

Transformation matrix and Position If R is the identity matrix, then there is no rotation and the transformation matrix will represent the pure displacement of the origins of the frames of reference:

Summary Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the the discussion on the analysis of kinematic chains with focus on robots. The following were covered: The concept of kinematic pairs (joints) between rigid bodies Definition of common (lower) kinematic pairs Open and closed kinematic chains The concept of forward and inverse kinematics and dynamics analysis Finite motion in terms of the transformation matrix