Editing Robot kinematics

{{short description|Geometric analysis of multi-DoF kinematic chains that model a robot}}

'''Robot kinematics''' applies [[geometry]] to the study of the movement of [[degree of freedom (mechanics)|multi-degree of freedom]] [[kinematic chain]]s that form the structure of [[robotic]] systems.<ref>{{cite book
  | last = Paul
  | first = Richard
  | title = Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators
  | publisher = MIT Press, Cambridge, Massachusetts
  | date = 1981
  | url = https://books.google.com/books?id=UzZ3LAYqvRkC
  | isbn =978-0-262-16082-7
}}
</ref><ref>J. M. McCarthy, 1990, ''Introduction to Theoretical Kinematics,'' MIT Press, Cambridge, Massachusetts.</ref>  The emphasis on geometry means that the links of the [[robot]] are modeled as [[rigid bodies]] and its [[Kinematic pair|joints]] are assumed to provide pure [[rotation]] or [[Translation (geometry)|translation]].

[[Robot]] [[kinematics]] studies the relationship between the dimensions and connectivity of kinematic chains and the position, [[velocity]] and [[acceleration]] of each of the links in the robotic system, in order to plan and control movement and to compute [[actuator]] forces and [[torque]]s. The relationship between [[mass]] and [[inertia]] properties, motion, and the associated forces and torques is studied as part of [[multibody system|robot dynamics]].  
<!-- this is not exactly true.  One of the most active areas within robot kinematics is the [[screw theory]].-->

==Kinematic equations==
{{See|Kinematic equation}}
A fundamental tool in robot kinematics is the kinematics equations of the kinematic chains that form the robot.  These [[non-linear equation]]s are used to map the joint parameters to the configuration of the robot system. Kinematics equations are also used in [[biomechanics]] of the skeleton and [[computer animation]] of articulated characters.

Forward kinematics uses the kinematic equations of a [[robot]] to compute the position of the [[Robot end effector|end-effector]] from specified values for the joint parameters.<ref>John J. Craig, 2004, Introduction to Robotics: Mechanics and Control (3rd Edition), Prentice-Hall.</ref>  The reverse process that computes the joint parameters that achieve a specified position of the end-effector is known as inverse kinematics.  The dimensions of the robot and its kinematics equations define the volume of space reachable by the robot, known as its workspace.

There are two broad classes of robots and associated kinematics equations: [[serial manipulator]]s and [[parallel manipulator]]s. Other types of systems with specialized kinematics equations are air, land, and submersible mobile robots, hyper-redundant, or snake, robots and [[humanoid robot]]s.

===Forward kinematics===
{{Main|Forward kinematics}}
Forward kinematics specifies the joint parameters and computes the configuration of the chain. For serial manipulators this is achieved by direct substitution of the joint parameters into the forward kinematics equations for the serial chain.  For parallel manipulators substitution of the joint parameters into the kinematics equations requires solution of the a set of [[polynomial]] constraints to determine the set of possible end-effector locations.

===Inverse kinematics===

{{Main|Inverse kinematics}}
Inverse kinematics specifies the end-effector location and computes the associated joint angles. For serial manipulators this requires solution of a set of polynomials obtained from the kinematics equations and yields multiple configurations for the chain.  The case of a general 6R serial manipulator (a serial chain with six [[revolute joint]]s) yields sixteen different inverse kinematics solutions, which are solutions of a sixteenth degree polynomial. For parallel manipulators, the specification of the end-effector location simplifies the kinematics equations, which yields formulas for the joint parameters.

==Robot Jacobian==
The time derivative of the kinematics equations yields the [[Jacobian matrix and determinant|Jacobian]] of the robot, which relates the joint rates to the linear and [[angular velocity]] of the end-effector.  The principle of [[virtual work]] shows that the Jacobian also provides a relationship between joint torques and the resultant force and torque applied by the end-effector.  Singular configurations of the robot are identified by studying its Jacobian.

===Velocity kinematics===

The robot Jacobian results in a set of linear equations that relate the joint rates to the six-vector formed from the angular and linear velocity of the end-effector, known as a [[Twist (screw theory)|twist]].  Specifying the joint rates yields the end-effector twist directly.

The '''inverse velocity''' problem seeks the joint rates that provide a specified end-effector twist.  This is solved by inverting the [[Jacobian matrix]]. It can happen that the robot is in a configuration where the Jacobian does not have an inverse.  These are termed singular configurations of the robot.

===Static force analysis===

The principle of [[virtual work]] yields a set of linear equations that relate the resultant force-torque six vector, called a [[screw theory|wrench]], that acts on the end-effector to the joint torques of the robot.  If the end-effector [[Wrench (screw theory)|wrench]] is known, then a direct calculation yields the joint torques.

The '''inverse statics''' problem seeks the end-effector wrench associated with a given set of joint torques, and requires the inverse of the Jacobian matrix.  As in the case of inverse velocity analysis, at singular configurations this problem cannot be solved. However, near singularities small actuator torques result in a large end-effector wrench. Thus near singularity configurations robots have large [[mechanical advantage]].

==Fields of study==
Robot kinematics also deals with [[motion planning]], '''singularity avoidance''', '''[[Redundancy (engineering)|redundancy]]''', '''collision avoidance''', as well as the kinematic synthesis of robots.<ref>J. M. McCarthy and G. S. Soh, [https://books.google.com/books?id=jv9mQyjRIw4C&q=geometric+design+of+linkages ''Geometric Design of Linkages,''] 2nd Edition, Springer 2010.</ref>
<!--this is described in the article on forward kinematics.  While dealing with the kinematics used in the robots we deal each parts of the robot by assigning a frame of reference to it and hence a robot with many parts may have many individual frames assigned to each movable parts. For simplicity we deal with the single manipulator arm of the robot. Each frames are named systematically with numbers, for example the immovable base part of the manipulator is numbered 0, and the first link joined to the base is numbered 1, and the next link 2 and similarly till n for the last nth link.-->
<!--this is described in the previous section.  ===Forward position kinematics===
The forward position kinematics (FPK) solves the following problem:
"Given the joint positions, what is the corresponding end effector's pose?"
=== Serial chains ===

The solution is always unique: one given joint position vector always corresponds to only one single end effector pose.  The FK problem is not difficult to solve, even for a completely arbitrary kinematic structure.
Methods for a forward kinematic analysis:
* using straightforward geometry
* using transformation matrices
=== Parallel chains (Stewart Gough Manipulators) ===

The solution is not unique: one set of joint coordinates has more different end effector poses. In case of a [[Stewart platform]] there are 40 poses possible which can be real for some design examples. Computation is intensive but solved in closed form with the help of [[algebraic geometry]].{{Citation needed|date=October 2011}} -->
<!--this is described in the previous section.  ===Inverse position kinematics===
The inverse position kinematics (IPK) solves the following problem: "Given the actual end effector pose, what are the corresponding joint positions?"
In contrast to the forward problem, the solution of the inverse problem is not always unique: the same end effector pose can be reached in several configurations, corresponding to distinct joint position vectors.
A 6R manipulator (a serial chain with six [[revolute joint]]s) with a completely general geometric structure has sixteen different inverse kinematics solutions, found as the solutions of a sixteenth order polynomial for best result.
-->

==See also==
{{Portal|Physics}}
*[[Robotics conventions]]
*[[Mobile robot]]
*[[Robot locomotion]]

==References==
{{reflist}}

{{Robotics}}

{{DEFAULTSORT:Robot Kinematics}}
[[Category:Robot kinematics| ]]