ne formulation mkm
DESCRIPTION
NE Formulation MKMTRANSCRIPT
DEPARTMENT OF INSTRUMENTATION AND CONTROL ENGINEERING,
MANIPAL INSTITUTE OF TECHNOLOGY, MANIPAL.
Saturday, October 3, 2015 1
ICE 425
Presentation Created By:MUKUND KUMAR MENON,ASSISTANT PROFESSOR,DEPT. OF INSTRUMENTATION & CONTROL ENGG.,MIT, MANIPAL.
Robotic Systems & Control
NEWTON EULER {NE} FORMULATION
NEWTON EULER {NE} FORMULATION
DEPARTMENT OF INSTRUMENTATION AND CONTROL ENGINEERING,
MANIPAL INSTITUTE OF TECHNOLOGY, MANIPAL.
αc,i = acceleration of the centre of mass of link i
αe,i = acceleration of the end of link i (which is the origin of frame i + 1)
ωi = angular velocity of frame i with respect to frame 0
αi = angular acceleration of frame i with respect to frame 0
zi = axis of actuation of frame i with respect to frame 0
gi = acceleration due to gravity
fi = force exerted by link (i − 1) on link i
τi = torque exerted by link (i − 1) on link i
LET
Saturday, October 3, 2015 2NEWTON EULER {NE} FORMULATION
DEPARTMENT OF INSTRUMENTATION AND CONTROL ENGINEERING,
MANIPAL INSTITUTE OF TECHNOLOGY, MANIPAL.
Rii+1 = rotation matrix from frame ‘i’ to frame ‘i + 1’
mi = the mass of link i
Ii = inertia-tensor of link i about a frame parallel to frame ‘i’, whose origin is
at the centre of mass of link ‘i'
r i−1,ci = vector from the origin of frame (i − 1) to the centre of mass of link i
r i−1,i = vector from the origin of frame (i − 1) to the origin of frame i
r i,ci = vector from the origin of frame i to the centre of mass of link i
LET
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DEPARTMENT OF INSTRUMENTATION AND CONTROL ENGINEERING,
MANIPAL INSTITUTE OF TECHNOLOGY, MANIPAL.
Fig: Forces and torques acting on a random link ‘i’
Saturday, October 3, 2015 4NEWTON EULER {NE} FORMULATION
By the law of action and reaction, f i is the force exerted by link i − 1 on link i, and −f i+1 is the forceexerted by link i+1 on link i. According to the definitions above, fi is expressed in frame i while −f i+1 isexpressed in frame i + 1. Thus, to express both forces in frame i, it is required to post-multiply the latterwith Ri
i+1 . The same apply to the torque, again by the law of action and reaction.
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MANIPAL INSTITUTE OF TECHNOLOGY, MANIPAL.
Saturday, October 3, 2015 5
1. FORCE BALANCE EQUATION:
1 1 ,
i
i i i i i i c if m g R f m a
, 1 1
i
i i c i i i i if m a R f m g
At link ‘i’:
NEWTON EULER {NE} FORMULATION
c
link
f ma
DEPARTMENT OF INSTRUMENTATION AND CONTROL ENGINEERING,
MANIPAL INSTITUTE OF TECHNOLOGY, MANIPAL.
Saturday, October 3, 2015 6
First, the moment exerted by a force ‘f’ about a point is given by (f × r), where
‘r’ is the radial vector from the point where the force is applied to the point
where the moment is computed.
Second, the vector migi does not appear in the moment balance since it is
applied directly at the centre of mass.
2. MOMENT BALANCE EQUATION:
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link
I I
1, 1 1 1 1 ,
i i
i i i ci i i i i i ci i i i i if r R R f r I I
1, 1 1 1 1 ,
i i
i i i i i i i i ci i i i i i ciI I f r R R f r
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Because of the fact that the angular velocity of frame i equals the angular
velocity of frame ‘i−1’ PLUS the added rotation from joint i, using rotation
matrices this leads to:
the rotation of joint i expressedin frame i
NOW,
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Saturday, October 3, 2015 NEWTON EULER {NE} FORMULATION 9
It is interesting & also vitally important to note that:
which means
ii
αi is the derivative of
the angular velocity of
frame i, but expressed
in frame i.
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MANIPAL INSTITUTE OF TECHNOLOGY, MANIPAL.
Saturday, October 3, 2015 NEWTON EULER {NE} FORMULATION 10
The time derivative of:
becomes:
and expressed in frame ‘i', it directly becomes:
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Now it only remains to find an expression for ac,i .
The linear velocity of the centre of mass of link i is expressed as:
constant in frame i
Differentiating both sides, we get:
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Multiplying with rotation matrices and using the fact that:
Then, the final expression for the acceleration of the centre of mass of link i,
expressed in frame i, becomes:
and,
THE ACCELERATION OF THE END OF THE LINK,
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Saturday, October 3, 2015 NEWTON EULER {NE} FORMULATION 13
This completes the recursive formulation, and The Newton-Euler
formulation of an n-link manipulator can be stated as follows :
2. Forward recursion:
1. Start with the initial conditions: ω0 = α0 = ac,0 = ae,0 = 0, fn+1 = τn+1 = 0
For increasing i, from 1 to n, solve in the following order:
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3. Backward recursion:
For decreasing i, from n to 1, solve in the following order:
1,
1 1 1 1 ,
i i i i i i i i ci
i i
i i i i i ci
I I f r
R R f r
, 1 1
i
i i c i i i i if m a R f m g
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MANIPAL INSTITUTE OF TECHNOLOGY, MANIPAL.
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Qn:Analyze the dynamics of the planar elbow manipulator using NE RecursiveMethod.
Hint:Pg 222-224, Sect.6.7, Mark W. Spong, Seth Hutchinson,and M. Vidyasagar , “Robot Modeling and Control”, FirstEdition, Wiley.
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REFERENCE:
1. Herman Høifødt, “Dynamic Modeling and Simulation of Robot
Manipulators: The Newton-Euler Formulation”, Thesis, Master of
Science in Engineering Cybernetics, Norwegian University of Science
and Technology, Department of Engineering Cybernetics, June 2011.
2. Mark W. Spong, Seth Hutchinson, and M. Vidyasagar , “Robot
Modeling and Control”, First Edition, Wiley.
3. John r. Taylor, “Classical mechanics”, University Science Books, 2005.
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