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| [[File:Robot arm model 1.png|thumb|200px|An articulated six [[Degrees of freedom (engineering)|DOF]] [[robotic arm]] uses forward kinematics to position the gripper.]]
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| [[File:Puma Robotic Arm - GPN-2000-001817.jpg|right|thumb|200px|The forward kinematics equations define the trajectory of the end-effector of a PUMA robot reaching for parts.]]
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| '''Forward kinematics''' refers to the use of 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>{{cite book
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| | last = Paul
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| | first = Richard
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| | title = Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators
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| | publisher = MIT Press, Cambridge, MA
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| | date = 1981
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| | url = http://books.google.com/books?id=UzZ3LAYqvRkC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
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| | isbn =978-0-262-16082-7
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| }}
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| </ref> The kinematics equations of the robot are used in [[robotics]], [[computer games]], and [[animation]]. The reverse process that computes the joint parameters that achieve a specified position of the end-effector is known as [[inverse kinematics]].
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| ==Kinematics equations==
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| The kinematics equations for the series chain of a robot are obtained using a [[rigid transformation]] [Z] to characterize the [[relative movement]] allowed at each [[Joint (mechanics)|joint]] and separate rigid transformation [X] to define the dimensions of each link. The result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain to its end link, which is equated to the specified position for the end link,
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| :<math>[T] = [Z_1][X_1][Z_2][X_2]\ldots[X_{n-1}][Z_n],\!</math>
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| where [T] is the transformation locating the end-link. These equations are called the kinematics equations of the serial chain.<ref> J. M. McCarthy, 1990, ''Introduction to Theoretical Kinematics,'' MIT Press, Cambridge, MA.</ref>
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| ==Link transformations==
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| In 1955, Jacques Denavit and Richard Hartenberg introduced a convention for the definition of the joint matrices [Z] and link matrices [X] to standardize the coordinate frame for spatial linkages.<ref>J. Denavit and R.S. Hartenberg, 1955, "A kinematic notation for lower-pair mechanisms based on matrices." ''Trans ASME J. Appl. Mech,'' 23:215–221.</ref><ref>Hartenberg, R. S., and J. Denavit. '''Kinematic Synthesis of Linkages.''' New York: McGraw-Hill, 1964 [http://ebooks.library.cornell.edu/k/kmoddl/toc_hartenberg1.html on-line through KMODDL]</ref> This convention positions the joint frame so that it consists of a screw displacement along the Z-axis
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| :<math> [Z_i] = \operatorname{Trans}_{Z_{i}}(d_i) \operatorname{Rot}_{Z_{i}}(\theta_i),</math>
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| and it positions the link frame so it consists of a screw displacement along the X-axis,
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| :<math> [X_i]=\operatorname{Trans}_{X_i}(a_{i,i+1})\operatorname{Rot}_{X_i}(\alpha_{i,i+1}).</math>
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| Using this notation, each transformation-link goes along a serial chain robot, and can be described by the [[coordinate transformation]],
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| : <math>{}^{i-1}T_{i} = [Z_i][X_i] =
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| \operatorname{Trans}_{Z_{i}}(d_i)
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| \operatorname{Rot}_{Z_{i}}(\theta_i)
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| \operatorname{Trans}_{X_i}(a_{i,i+1})
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| \operatorname{Rot}_{X_i}(\alpha_{i,i+1}),</math>
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| where ''θ<sub>i</sub>'', ''d<sub>i</sub>'', ''α<sub>i,i+1</sub>'' and ''a<sub>i,i+1</sub>'' are known as the [[Denavit-Hartenberg parameters]].
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| ===Kinematics equations revisited===
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| The kinematics equations of a serial chain of ''n'' links, with joint parameters ''θ<sub>i</sub>'' are given by<ref name='jk'>{{cite web|url=http://elvis.rowan.edu/~kay/papers/kinematics.pdf|title=Introduction to Homogeneous Transformations & Robot Kinematics|author=Jennifer Kay|accessdate=2010-09-11}}</ref>
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| :<math>[T] = {}^{0}T_n = \prod_{i=1}^n {}^{i - 1}T_i(\theta_i),</math>
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| where <math>{}^{i - 1}T_i(\theta_i)</math> is the transformation matrix from the frame of link <math>i</math> to link <math> i-1</math>. In robotics, these are conventionally described by [[Denavit–Hartenberg parameters]].<ref name='fk'>{{cite web|url=http://www.learnaboutrobots.com/forwardKinematics.htm|title=Robot Forward Kinematics|author=Learn About Robots|accessdate=2007-02-01}}</ref>
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| ===Denavit-Hartenberg matrix===
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| The matrices associated with these operations are:
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| : <math>\operatorname{Trans}_{Z_{i}}(d_i)
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| = \begin{bmatrix}
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| 1 & 0 & 0 & 0 \\
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| 0 & 1 & 0 & 0 \\
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| 0 & 0 & 1 & d_i \\
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| 0 & 0 & 0 & 1
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| \end{bmatrix}, \quad
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| \operatorname{Rot}_{Z_{i}}(\theta_i)
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| =
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| \begin{bmatrix}
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| \cos\theta_i & -\sin\theta_i & 0 & 0 \\
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| \sin\theta_i & \cos\theta_i & 0 & 0 \\
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| 0 & 0 & 1 & 0 \\
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| 0 & 0 & 0 & 1
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| \end{bmatrix}.
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| </math>
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| Similarly,
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| : <math>\operatorname{Trans}_{X_i}(a_{i,i+1})
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| =
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| \begin{bmatrix}
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| 1 & 0 & 0 & a_{i,i+1} \\
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| 0 & 1 & 0 & 0 \\
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| 0 & 0 & 1 & 0 \\
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| 0 & 0 & 0 & 1
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| \end{bmatrix},\quad
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| \operatorname{Rot}_{X_i}(\alpha_{i,i+1})
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| =
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| \begin{bmatrix}
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| 1 & 0 & 0 & 0 \\
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| 0 & \cos\alpha_{i,i+1} & -\sin\alpha_{i,i+1} & 0 \\
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| 0 & \sin\alpha_{i,i+1} & \cos\alpha_{i,i+1} & 0 \\
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| 0 & 0 & 0 & 1
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| \end{bmatrix}.
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| </math>
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| The use of the Denavit-Hartenberg convention yields the link transformation matrix, [''<sup>i-1</sup>T<sub>i</sub>''] as
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| : <math>\operatorname{}^{i-1}T_i
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| =
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| \begin{bmatrix}
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| \cos\theta_i & -\sin\theta_i \cos\alpha_{i,i+1} & \sin\theta_i \sin\alpha_{i,i+1} & a_{i,i+1} \cos\theta_i \\
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| \sin\theta_i & \cos\theta_i \cos\alpha_{i,i+1} & -\cos\theta_i \sin\alpha_{i,i+1} & a_{i,i+1} \sin\theta_i \\
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| 0 & \sin\alpha_{i,i+1} & \cos\alpha_{i,i+1} & d_i \\
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| 0 & 0 & 0 & 1
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| \end{bmatrix},
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| </math>
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| <!--=
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| \begin{bmatrix}
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| A & \mathbf{d} \\
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| 0 0 0 & 1
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| \end{bmatrix},
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| where ''A'' is the 3×3 submatrix describing rotation and '''d''' is the 3×1 submatrix describing translation.-->known as the ''Denavit-Hartenberg matrix.''
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| <!--==Simple Introduction==
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| * For a detailed introduction to transformation matrices and forward kinematics, see.<ref name='jk'>{{cite web|url=http://elvis.rowan.edu/~kay/papers/kinematics.pdf|title=Introduction to Homogeneous Transformations & Robot Kinematics|author=Jennifer Kay|accessdate=2010-09-11}}</ref>-->
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| ==See also==
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| * [[Kinematic chain]]
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| * [[Forward kinematic animation]]
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| * [[Robot control]]
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| * [[Mechanical systems]]
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| * [[Robot kinematics]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Forward Kinematics}}
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| [[Category:3D computer graphics]]
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| [[Category:Computational physics]]
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| [[Category:Robot kinematics]]
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Hi there. My name is Sophia Meagher even though it is not the title on my birth certification. Office supervising is my occupation. Some time ago she selected to live in Alaska and her parents reside close by. As a lady what she really likes is fashion and she's been performing it for quite a while.
Here is my web page; real psychic readings - test.jeka-nn.ru -