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1/31Kinematics
Kinematics analyzes the geometry of a manipulator, robot ormachine motion. The essential concept is a position.
Statics deals with the forces and moments which are aplied on themechanism at rest. The essential concept is a stiffness.
Dynamics analyzes the forces and moments which result frommotion and acceleration of a mechanism and a load.
The terms and laws studied can be applied to robot-industrialmanipulator as well as to any other machine with movingcomponents. We will refer here to robot and will use some terms usedin robotics (like end effector) but any machine could and shall bestudied in similar way when position, stiffness or dynamics of thesystem is important.
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Kinematics – Terminology
J5 axisJ4 axis
Fore armElbow block
J3 axis
Upper arm
J2 axis
Base
J1 axis
Shoulder
J6 axis
Link is the rigid part of the robot body (e.g.forearm).
Joint is a part of the robot body whichallows controlled or free relative motionof two links (connection element).
End effector is the link of the manipulatorwhich is used to hold the tools (gripper,spray gun, welding gun...).
Base is the link of the manipulator which isusually connected to the ground and isdirectly connected to the world coordi-nate system.
Kinematic pair is a pair of links which rela-tive motion is bounded by the joint con-necting them (e.g. base and shoulderconnected by J1 axis).
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Kinematics – Terminology II
Kinematic chain is a set of links connected by joints. Kinematic chain can berepresented by a graph. The vertices represent links and edges represent joints.
Mechanism is a kinematic chain when one of its links is fixed to the ground.
Open kinematic chain Mixed kinematic chain Parallel manipulatorsis the chain which can be de-scribed by acyclic graph. contains in its graph a loop. consist of equivalent loops.
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Kinematics – Degrees of Freedom
Degrees of freedom (less formal definition) is a number ofindependent parameters needed to specify the position of themechanism completely. Examples:
Rigid body in a 2D space e.g. plane has 3 DOF.
Rigid body in a 3D space has 6 DOF.
A point in a plane has 2 DOF.
A point in a 3D space has 3 DOF.
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Kinematics – Degrees of Freedom
The DOF is important notion not only in robotics. Few more definitions are related to it:
Ambient space - the space robot/mechanism lives in, usually E2 (the plane - planarmanipulator) or E3 (space). It is Euclidean space.
Operational space.
is the subspace of the ambient space occupied by any of therobot part during any of possible robot motions. The operatio-nal (3D, physical) space shall be guarded by fence, doors orinvisible bariers to prevent injuries of both robot and humans.
Work envelope (working space).
is the subspace of the ambient space wherethe robot can reach by the end effector. Thecuts through this space are usually drawn inthe technical specifications by manufactu-rers. The work envelope has actually littleuse in the practice, it can just provide basicnotion where the robot can work.
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Kinematics – Degrees of Freedom
We usually need to study the position of the end effector or the tool fixed to it. Letus assume the end effector or tool is a rigid body.
The rigid body position in the 3D ambient space can be described by sixparameters. The semantics and values depend on the chosen parametrization,e.g. position of the reference point on it (3 parameters) and 3 angles.
The “position” space is the 6D (3D for planar case) space representing allpossible positions of rigid body in 3D (2D) ambient space.
The end effector position can be studied in this 6D space position space.
The working space is a subspace of the position space containing positionswhich can be reached by end effector (tool). All required end effector positionsshall of course lie in the working space, so the feasibility of the particular robotuse (its reach) shall be studied in this space.
The joint space a space of all possible??????????????????
joints space trajectories
The DOF is a dimension of the subspace which contains ??????????????
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Types of kinematic pairs
Symbol Name has/removes DOF
Spherical 3 / 3
Revolute 1 / 5
Prismatic 1 / 5
Cylindrical 2 / 4
Flat 3 / 3
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Typical structure of manipulators II
Angular SCARA
Animations taken from Masud Salimian’s web page
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Body in the coordinate system
Point in 3D - described by three coordinates.
Rigid body in 3D - described by 6 coordinates:
� 3 coordinates of reference point x0 =æçççççççè
x0y0z0
ö÷÷÷÷÷÷÷ø
� orientation could be described e.g. by:
ê coordinates of vectors attached to the body (n, t,b),
ê Euler angles (Φ, Θ,Ψ),
ê rotational matrix R.
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Rotation Matrix Resulting from Euler Anglesæçççççççç
è
Cos(Φ) Cos(Ψ) - Cos(Θ) Sin(Φ) Sin(Ψ) -Cos(Θ) Cos(Ψ) Sin(Φ) - Cos(Φ) Sin(Ψ) Sin(Φ) Sin(Θ)Cos(Ψ) Sin(Φ) + Cos(Φ) Cos(Θ) Sin(Ψ) Cos(Φ) Cos(Θ) Cos(Ψ) - Sin(Φ) Sin(Ψ) -Cos(Φ) Sin(Θ)
Sin(Θ) Sin(Ψ) Cos(Ψ) Sin(Θ) Cos(Θ)
ö÷÷÷÷÷÷÷÷
ø
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Working space
Working spaceis a space of all points, where robot can place a held object or a tool.
J5 axisJ4 axis
Fore armElbow block
J3 axis
Upper arm
J2 axis
Base
J1 axis
Shoulder
J6 axis
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