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'''Kinematics equations''' refers to the constraint equations of a mechanical system such as a robot manipulator that define how input movement at one or more joints specifies the configuration of the device, in order to achieve a task position or end-effector location.<ref>{{cite book
The London schedule has long had cheerleaders for its talent on home territory, but in the past it has been squeezed out by more commercially robust cities.<br><br>
  | last = Paul
  | first = Richard
  | title = Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators
  | publisher = MIT Press, Cambridge, MA
  | year = 1981
  | url = http://books.google.com/books?id=UzZ3LAYqvRkC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
  | isbn =978-0-262-16082-7
}}
</ref><ref>J. M. McCarthy, 1990, ''Introduction to Theoretical Kinematics,'' MIT Press, Cambridge, MA.</ref> Kinematics equations are used to analyze and design articulated systems ranging from four-bar linkages to serial and parallel robots.


Kinematics equations are constraint equations that characterize the geometric configuration of an articulated mechanical system.   Therefore, these equations are assume the links are rigid and the joints provide pure rotation or translation.  Constraint equations of this type are known as [[holonomic constraints]] in the study of the [[dynamics (mechanics)|dynamics]] of multi-body systems.
Now it seems the capital is holding its ground against the bigger boys, as more and more international press, and buyers, make London an important stop in their schedule - blizzards in New York notwithstanding.<br>This in part has been due to brands like Burberry Prorsum, Mulberry and Tom Ford, which, with the weight of their advertising budgets behind them, command an international audience. Topshop, led by creative director Kate Phelan, also of British Vogue, has also helped provide financial support and a professional framework in which the creativity of London�s emerging designers can be showcased.<br><br>This situation has not gone unnoticed in Milan, where Giorgio Armani has taken Stella Jean under his wing, hosting her spring/summer 2014 show last September. �The new generation of Italian designers needs our support,� said Armani, echoing the sentiment that has long made London great<br>
LVMH (the parent company of brands such as Louis Vuitton, Christian Dior and Celine) is also taking lessons from London: on Thursday it announced a shortlist of 30 designers for its inaugural LVMH Young Fashion Designer Prize. With �300,000 (�246,000) at stake, plus a year of mentoring, the number of London designers shortlisted, including Simone Rocha, Thomas Tait, Meadham Kirchhoff and J JS Lee, is a clear indication of the capital�s fertile groun<br><br>
Fittingly, the nominees - many of which have come through support programmes Newgen and Fashion East - are some of the most exciting shows on the schedul<br>
Korean-born J JS Lee, who trained at Central Saint Martins, took the prized opening slot yesterday in rain-swept central London. For autumn/winter 2014 she showed an [http://wordpress.org/search/elegant+collection elegant collection] of brushed mohair, plaid overcoats and formal dresses - experimenting with cocoon shapes, biker jackets and rounded shoulders for the former, while playing with different lengths for the latte<br><br>
Cobalt blue houndstooth as a laser-cut detail and d�grad� printed silk provided a jolt of colour in a predominantly cream, navy and forest-green palette. Slim, long-line trousers were an elegant proposition for evenings, while chunky, cable knits were covetably co<br>.
The London College of Fashion MA Show at the Waldorf Astoria Hotel on day one of London Fashion Week yesterday (PA) [http://www.pcs-systems.co.uk/Images/celinebag.aspx http://www.pcs-systems.co.uk/Images/celinebag.aspx] With a fit-to-burst schedule of shows, presentations and parties over the next four days, there�s plenty to inspire and excite the fashion world - both on and off the catwa<br><br>


==Loop equations==
JW Anderson�s first womenswear collection since LVMH bought a minority stake in his business last September is sure to be as controversial as ever: the designer�s gender-blurring take on dressing has its detractors, but just as many fans. The designer was also appointed creative director of Loewe, a Spanish leather brand within the stable, so an excellent accessories offering is to be expect<br><br>
The kinematics equations for a mechanical system are formed as a sequence of rigid transformations along links and around joints in an mechanical system. The principle that the sequence of transformations around a loop must return to the identity provides what are known as the ''loop equations.''  An independent set of kinematics equations is assembled from the various sets of loop equations that are available in a mechanical system.


===Transformations===
It was all change at Mulberry last season too, as Emma Hill departed after five years as creative director of the British brand. Hill seemed to have hit on a winning formula: balancing design and the right kind of celebrity endorsements fortified the brand�s finances, but recent profit warnings are cited as the reason for her departu<br><br>
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 frames 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
:<math> [Z_i] =  \begin{bmatrix}
    \cos\theta_i & -\sin\theta_i & 0 & 0 \\
    \sin\theta_i &  \cos\theta_i & 0  & 0 \\
    0 & 0 & 1 & d_i \\
    0 & 0 & 0 & 1
  \end{bmatrix},
</math>


and it positions the link frame so it consists of a screw displacement along the X-axis,
This season, the brand will not be hosting its usual catwalk extravaganza in Claridge�s ballroom. Instead it has announced a photocall at which Cara Delevingne, its campaign face, will reveal the brand�s next big move. Speculation is rife that she will be announced as creative director, but the brand�s success with bags named after It girls makes the debut of the �Cara� a far more likely prosp<br><br>
:<math> [X_i]=\begin{bmatrix}
    1 & 0 & 0 & a_{i,i+1} \\
    0 & \cos\alpha_{i,i+1} & -\sin\alpha_{i,i+1}& 0 \\
    0 & \sin\alpha_{i,i+1} & \cos\alpha_{i,i+1} & 0 \\
    0 & 0 & 0 & 1
  \end{bmatrix}.
</math>


The kinematics equations 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. 
Joseph, celebrating 25 years since its first west London boutique opened, is presenting on schedule for the first time too. It may not have the budgets of its blockbuster digital rivals, but it is held in esteem by serious shoppers and the fashion cr<br><br><br>


The result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain around a loop back to the base to obtain the loop equation,
While there is a respectable coterie of British luxury brands, it�s the commercial pull of the high street that is unrivalled, something the British Fashion Council - responsible for co-ordinating the schedule - has cottoned on to. As well as Topshop�s premium offering Unique, now a regular on the schedule, the BFC has courted brands such as River Island, which debuted its collaboration with Rihanna in London this time last y<br>r.
:<math>[Z_1][X_1][Z_2][X_2]\ldots[X_{n-1}][Z_n]=[I].\!</math>
Now Whistles is presenting on the schedule for autumn/winter 2014; with a mens collection rumoured to be on the horizon and a flagship store on Mayfair�s Dover Street, the brand is certainly looking to exp<br><br>
The series of transformations equate to the identify matrix because they return to the beginning of the loop.


==Serial chains==
So too are Christopher Kane and Roksanda Ilincic, who plan to open boutiques on Mount Street this year.
The kinematics equations for a serial chain robot are obtained by formulating the loop equations in terms of a transformation [T] from the base to the end-effector, which is equated to the series of transformations along the robot. The result is,
:<math>[T] = [Z_1][X_1][Z_2][X_2]\ldots[X_{n-1}][Z_n],\!</math>
These equations are called the kinematics equations of the serial chain.
 
==Parallel chains==
The kinematics equations for a parallel chain, or parallel robot, formed by an end-effector supported by multiple serial chains are obtained from the kinematics equations of each of the supporting serial chains.  Suppose that ''m'' serial chains support the end-effector, then the transformation from the base to the end-effector is defined by ''m'' equations,
:<math>[T] = [Z_{1,j}][X_{1,j}][Z_{2,j}][X_{2,j}]\ldots[X_{n-1,j}][Z_{n,j}],\quad j=1,\ldots, m.\!</math>
These equations are the kinematics equations of the parallel chain.
 
==Forward kinematics==
The kinematics equations of serial and parallel robots can be viewed as relating parameters, such as joint angles, that are under the control of actuators to the position and orientation [T] of the end-effector. 
 
From this point of view the kinematics equations can be used in two different ways.  The first called ''forward kinematics'' uses specified values for the joint parameters to compute the end-effector position and orientation.  The second called ''inverse kinematics'' uses the position and orientation of the end-effector to compute the joint parameters values.
 
Remarkably, while the forward kinematics of a serial chain is a direct calculation of a single matrix equation, the forward kinematics of a parallel chain requires the simultaneous solution of multiple matrix equations which presents a significant challenge.
 
==References==
{{reflist}}
 
 
 
[[Category:Robot kinematics]]

Latest revision as of 13:04, 12 March 2014

The London schedule has long had cheerleaders for its talent on home territory, but in the past it has been squeezed out by more commercially robust cities.

Now it seems the capital is holding its ground against the bigger boys, as more and more international press, and buyers, make London an important stop in their schedule - blizzards in New York notwithstanding.
This in part has been due to brands like Burberry Prorsum, Mulberry and Tom Ford, which, with the weight of their advertising budgets behind them, command an international audience. Topshop, led by creative director Kate Phelan, also of British Vogue, has also helped provide financial support and a professional framework in which the creativity of London�s emerging designers can be showcased.

This situation has not gone unnoticed in Milan, where Giorgio Armani has taken Stella Jean under his wing, hosting her spring/summer 2014 show last September. �The new generation of Italian designers needs our support,� said Armani, echoing the sentiment that has long made London great
LVMH (the parent company of brands such as Louis Vuitton, Christian Dior and Celine) is also taking lessons from London: on Thursday it announced a shortlist of 30 designers for its inaugural LVMH Young Fashion Designer Prize. With �300,000 (�246,000) at stake, plus a year of mentoring, the number of London designers shortlisted, including Simone Rocha, Thomas Tait, Meadham Kirchhoff and J JS Lee, is a clear indication of the capital�s fertile groun

Fittingly, the nominees - many of which have come through support programmes Newgen and Fashion East - are some of the most exciting shows on the schedul
Korean-born J JS Lee, who trained at Central Saint Martins, took the prized opening slot yesterday in rain-swept central London. For autumn/winter 2014 she showed an elegant collection of brushed mohair, plaid overcoats and formal dresses - experimenting with cocoon shapes, biker jackets and rounded shoulders for the former, while playing with different lengths for the latte

Cobalt blue houndstooth as a laser-cut detail and d�grad� printed silk provided a jolt of colour in a predominantly cream, navy and forest-green palette. Slim, long-line trousers were an elegant proposition for evenings, while chunky, cable knits were covetably co
. The London College of Fashion MA Show at the Waldorf Astoria Hotel on day one of London Fashion Week yesterday (PA) http://www.pcs-systems.co.uk/Images/celinebag.aspx With a fit-to-burst schedule of shows, presentations and parties over the next four days, there�s plenty to inspire and excite the fashion world - both on and off the catwa

JW Anderson�s first womenswear collection since LVMH bought a minority stake in his business last September is sure to be as controversial as ever: the designer�s gender-blurring take on dressing has its detractors, but just as many fans. The designer was also appointed creative director of Loewe, a Spanish leather brand within the stable, so an excellent accessories offering is to be expect

It was all change at Mulberry last season too, as Emma Hill departed after five years as creative director of the British brand. Hill seemed to have hit on a winning formula: balancing design and the right kind of celebrity endorsements fortified the brand�s finances, but recent profit warnings are cited as the reason for her departu

This season, the brand will not be hosting its usual catwalk extravaganza in Claridge�s ballroom. Instead it has announced a photocall at which Cara Delevingne, its campaign face, will reveal the brand�s next big move. Speculation is rife that she will be announced as creative director, but the brand�s success with bags named after It girls makes the debut of the �Cara� a far more likely prosp

Joseph, celebrating 25 years since its first west London boutique opened, is presenting on schedule for the first time too. It may not have the budgets of its blockbuster digital rivals, but it is held in esteem by serious shoppers and the fashion cr


While there is a respectable coterie of British luxury brands, it�s the commercial pull of the high street that is unrivalled, something the British Fashion Council - responsible for co-ordinating the schedule - has cottoned on to. As well as Topshop�s premium offering Unique, now a regular on the schedule, the BFC has courted brands such as River Island, which debuted its collaboration with Rihanna in London this time last y
r. Now Whistles is presenting on the schedule for autumn/winter 2014; with a mens collection rumoured to be on the horizon and a flagship store on Mayfair�s Dover Street, the brand is certainly looking to exp

So too are Christopher Kane and Roksanda Ilincic, who plan to open boutiques on Mount Street this year.

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