Homogeneous Transformations .

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Homogeneous Transformations. Purpose:
Slide 1

Homogeneous Transformations Purpose: The reason for this part is to acquaint you with the Homogeneous Transformation. This straightforward 4 x 4 change is utilized as a part of the geometry motors of CAD frameworks and in the kinematics model in robot controllers. It is exceptionally valuable for looking at inflexible body position and introduction (stance) of a succession of automated connections and joint casings.

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specifically, you will Examine the structure of the HT (homogeneous change). Perceive how introduction and position are spoken to inside one network. Apply the HT to stance (position and situate) a casing (xyz set of tomahawks) with respect to another reference outline. Look at the HT for basic turns around a pivot. See the impact of duplicating a progression of HT\'s. Translate the request of a result of HT\'s with respect to base and body-altered edges. Perceive how the HT is utilized as a part of apply autonomy.

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B T C B T C Script Notation: Pre super and sub-scripts are regularly used to indicate casings of reference = change of casing C in respect to edge B C p = vector situated in edge C Tsai utilizes a pre and post script documentation = change of edge C with respect to casing B C p = vector situated in edge C Note that we may not utilize the scripting approach, but rather graphically decipher the edge representations.

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a x b x c x p x H = a y b y c y p y a z b z c z p z d 1 d 2 d 3 1 Homogeneous Transformation H can speak to interpretation, pivot, extending or contracting (scaling), and viewpoint changes

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a x b x c x p x H = a y b y c y p y a z b z c z p z 0 1 Interpreting the HT as a casing a b c p R a , b , and c shape an introduction sub-grid indicated by R (3 x 3) , while p (3 x 1) is the casing\'s inception counterbalance.

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a x a = a y a z What do the terms speak to? a will be a vector ( set of bearing cosines a x , a y , and a z ) that situates the edge\'s x pivot in respect to the base X, Y, and Z tomahawks, separately. Comparative translations are made for the edge\'s y and z tomahawks through the heading cosine sets spoke to by vectors b and c . p is a vector of 3 parts speaking to the edge\'s inception with respect to the reference tomahawks. p Frame Base edge

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Interpreting the HT used to find a vector in the base casing Given an altered vector u , its change v is spoken to by v = H u Note that this structure doesn\'t work with the expectation of complimentary vectors ! Outline Interpretation

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u x u y u z u = 1 Transforming vectors The position vector u having segments u x , u y , u z must be extended to a 4 x 1 vector by including a 1. Note: To change an introduction vector, just utilize the introduction sub-framework R, and drop the 1 from the vector with the goal that you are increasing a (3 x 3) lattice times a (3 x 1) vector.

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Interpreting the HT R p u x u y = R u + p v = u z 0 T 1 The 1 includes the edge starting point, while the R determines the vector u into the base casing

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a x b x c x 0 H = a y b y c y 0 a z b z c z 0 1 Special cases: Pure turn

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1 0 p x H = 0 1 0 p y 0 1 p z 0 1 Special cases: Pure interpretation

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1 0 cos q - sin q 0 sin q cos q Rotational structures Pure pivot about x R(x, q ) = q x

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cos q cos q 0 - sin q 0 sin q 0 sin q cos q 1 0 - sin q 0 cos q 1 Rotational structures Pure revolution about y R(y, q ) = Pure pivot about z R(z, q ) =

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Example - Rotate u by 90 o about +Z and 90 o about +Y, where XYZ are the settled base reference tomahawks. What are the last arranges of the vector u after these two turns in the base XYZ tomahawks? On the off chance that the turn request changed, will the last facilitates be the same? Let u T = [0 1 0]. Soln:   v = R (Z,90˚) u "rotate u to v"   w = R (Y,90˚) v "rotate v to w" Thus, w = R (Y,90˚) R (Z,90˚) u

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0 - 1 0 1 0 1 0 - 1 0 1 R(Y, 90 ˚ ) = R(Z, 90 ˚ ) =

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Z, z\', y" y\' w = (0,0,1) ° 90 ° 90 Y, x\', x" (0,1,0) X,z" Graphical elucidation

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0 - 0 1 0 1 0 1 0 1 - 1 0 Change request? w = Not commutative!

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Order of p and R: first R, then p R p

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Order of p and R: first p, then R p Note the distinction in the last network structure. Will you clarify the distinction?

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Understanding HT duplication request If we postmultiply a change (A B) speaking to an edge (in respect to base tomahawks) by a second change (in respect to the casing of the main change), we make the change regarding the edge tomahawks of the principal change. Premultiplying the edge change by the second change (B A) causes the change to be made concerning the base reference outline.

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Example - Given edge and change find outline X = H C and casing Y = C H . Note the distinctions.

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Results : HC H C

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Results : CH H C

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Inverse Transformations Given u and the rotational change R , the directions of u in the wake of being pivoted by R are characterized by v = Ru . The reverse inquiry is given v, what u when pivoted by R will give v ? Answer: u = R - 1 v = R T v

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= H - 1 = Inverse Transformations Similarly for any removal grid H ( R, p ), we can suggest a comparative conversation starter to get u = H - 1 v. What is the backwards of a dislodging change? Without evidence:

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Operational guidelines for square networks of full rank :

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HT synopsis Homogeneous change comprises of three segments: · rotational, orthogonal 3x3 sub-grid which is contained segments of course cosines used to arrange the tomahawks of one edge in respect to another. · segment vector in fourth section speaks to the inception of second edge with respect to first edge, determined in the primary edge. · 0\'s in 4 th column with the exception of 1 in 4,4 position.

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HT synopsis The homogeneous change successfully blends a casing introduction framework and casing interpretation vector into one lattice. The request of the operation ought to be seen as turn to begin with, then interpretation.

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HT outline The homogeneous change can be seen as a position/introduction relationship of one edge in respect to another casing called the reference outline.

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HT outline A B can be translated as casing A depicted with respect to the first or base casing while outline B is portrayed in respect to outline A (typical way). We can likewise decipher B in the base casing changed by An in the base casing. Both understandings give same result.

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