Robotics - História úprav

Balogh: /* L2 Position in 2D */

2016-10-03T14:12:26Z

L2 Position in 2D

← Staršia verzia		Verzia z 14:12, 3. október 2016
Riadok 7:		Riadok 7:
	Let's create a 2 dimensional rotation matrix for a certain rotation.		Let's create a 2 dimensional rotation matrix for a certain rotation.

			<div class="toccolours mw-collapsible" style="width:80em">
	<source lang="matlab">		<source lang="matlab">
	rot2(0) % zero rotation		rot2(0) % zero rotation
Riadok 58:		Riadok 59:
	% in the homogeneous form (by appending a 1 to the vector P).		% in the homogeneous form (by appending a 1 to the vector P).
	P1 = inv(T1) * [P ; 1]		P1 = inv(T1) * [P ; 1]
	</source>		</source></div>

	Nakreslime obrazok podobny tomu v prednaske		Nakreslime obrazok podobny tomu v prednaske

Balogh: /* L2 Position in 2D */

2016-10-03T14:11:49Z

L2 Position in 2D

← Staršia verzia		Verzia z 14:11, 3. október 2016
Riadok 95:		Riadok 95:

	Exercise		Exercise
			<div class="toccolours mw-collapsible" style="width:40em">
	<source lang="matlab">		<source lang="matlab">
	% DO NOT CHANGE THE FOLLOWING LINES		% DO NOT CHANGE THE FOLLOWING LINES
Riadok 121:		Riadok 121:
	5. Consider now that T1 represents the pose of the coordinate frame {1}. The coordinate of point B with respect to frame {2} is given by the workspace variable PB_2. 6. Determine the coordinate of point B with respect to the coordinate frame {1}, and place the result in the workspace variable PB_1.		5. Consider now that T1 represents the pose of the coordinate frame {1}. The coordinate of point B with respect to frame {2} is given by the workspace variable PB_2. 6. Determine the coordinate of point B with respect to the coordinate frame {1}, and place the result in the workspace variable PB_1.
	7. Determine the Euclidean distance between point A and point B and place the result in the workspace variable D_AB.		7. Determine the Euclidean distance between point A and point B and place the result in the workspace variable D_AB.
			</div>

	=== L3.10 2-axis representation ===		=== L3.10 2-axis representation ===

Balogh: /* L2 Position in 2D */

2016-10-03T14:06:33Z

L2 Position in 2D

← Staršia verzia		Verzia z 14:06, 3. október 2016
Riadok 93:		Riadok 93:

	[[Súbor:CoordinateFrames2D.png\|200px]]		[[Súbor:CoordinateFrames2D.png\|200px]]

			Exercise

			<source lang="matlab">
			% DO NOT CHANGE THE FOLLOWING LINES
			PA_0 = [6.5; 4.3]; % point A with respect to {0}
			PB_2 = [1.2; -2.7]; % point B with respect to {2}

			% Modify the following lines to return the correct values
			R1 =
			T1 =
			T2 =
			PA_2 =
			PB_1 =
			D_AB =
			</SOURcE>

			1. Compute an SO(2) rotation matrix that corresponds to a rotation of -50 degrees and place the result in the workspace variable R1.
			R1 = rot2(-50, 'deg');
			2. Compute an SE(2) homogenous transformation matrix that corresponds to a translation of 4 in the y-direction and 3 in the x-direction. Place the result in the workspace variable T1.
			T1 = transl2(3, 4); % or e.g.:
			T1 = se2(3,4,0);
			3. Compute an SE(2) homogenous transformation matrix that corresponds to a translation of 5 in the x-direction and 6 in the y-direction and then a rotation of -30 degrees. Place the result in the workspace variable T2.
			T2 = se2(5,6,-30,'deg');
			4. Consider now that T2 represents the pose of the coordinate frame {2}. The coordinate of point A in the world {0} coordinate frame is given by the workspace variable PA_0. Determine the coordinate of the point with respect to frame {2} and place the result in the workspace variable PA_2.

			5. Consider now that T1 represents the pose of the coordinate frame {1}. The coordinate of point B with respect to frame {2} is given by the workspace variable PB_2. 6. Determine the coordinate of point B with respect to the coordinate frame {1}, and place the result in the workspace variable PB_1.
			7. Determine the Euclidean distance between point A and point B and place the result in the workspace variable D_AB.

	=== L3.10 2-axis representation ===		=== L3.10 2-axis representation ===

Balogh: /* L2 Position in 2D */

2016-10-03T14:00:03Z

L2 Position in 2D

← Staršia verzia		Verzia z 14:00, 3. október 2016
Riadok 92:		Riadok 92:
	Vysledok		Vysledok

	[[Súbor:CoordinateFrames2D.png]]		[[Súbor:CoordinateFrames2D.png\|200px]]

	=== L3.10 2-axis representation ===		=== L3.10 2-axis representation ===

Balogh: /* L2 Position in 2D */

2016-10-03T13:59:17Z

L2 Position in 2D

@@ Riadok 59: / Riadok 59: @@
   P1 = inv(T1) * [P ; 1]
 </source>
 === L3.10 2-axis representation ===

Balogh: /* L2 Position in 2D */

2016-10-03T13:39:54Z

L2 Position in 2D

← Staršia verzia		Verzia z 13:39, 3. október 2016
Riadok 7:		Riadok 7:
	Let's create a 2 dimensional rotation matrix for a certain rotation.		Let's create a 2 dimensional rotation matrix for a certain rotation.

			<source lang="matlab">
	rot2(0) % zero rotation		rot2(0) % zero rotation
	R = rot2(pi/2) % pi/2 radian rotation stored in R matrix		R = rot2(pi/2) % pi/2 radian rotation stored in R matrix
Riadok 56:		Riadok 57:
	% use the inverse of the relative pose 1 and multiply it by the points coordinate		% use the inverse of the relative pose 1 and multiply it by the points coordinate
	% in the homogeneous form (by appending a 1 to the vector P).		% in the homogeneous form (by appending a 1 to the vector P).
	P1 = inv(T1) * [P ; 1]		P1 = inv(T1) * [P ; 1]
			</source>

	=== L3.10 2-axis representation ===		=== L3.10 2-axis representation ===

Balogh na 13:38, 3. október 2016

2016-10-03T13:38:59Z

← Staršia verzia		Verzia z 13:38, 3. október 2016
Riadok 1:		Riadok 1:
	Helper page for the Peter Corke: Introduction to robotics MOOC.		Helper page for the Peter Corke: Introduction to robotics MOOC.

			=== L1 Introduction ===

			=== L2 Position in 2D ===

			Let's create a 2 dimensional rotation matrix for a certain rotation.

			rot2(0) % zero rotation
			R = rot2(pi/2) % pi/2 radian rotation stored in R matrix
			rot2(30,'deg') % rotation 30 degrees

			c1 = R(:,1) % first column of the R
			c2 = R(:,2) % second column of R

			dot(c1,c2) % dot product of two vectors
			ans = 0 lebo su pravouhle

			det(R) % determinant
			ans = 1 lebo su ortogonalne

			inv(R) % inverse matrix
			R' % transposition
			% mali by byt rovnake

			trplot2(R) % plot the rotated coordination frame

			transl2(1,2) % creates HOMOGENOUS Transformation matrix for translation
			rot2(30, 'deg') % creates Rotation matrix
			trot2(30, 'deg') % homogeneous transformation matrix equivalent to the two dimensional rotation matrix

			transl2(1,2) * trot2(30, 'deg') % homogenous matrix consisting of both the rotational and translational parts

			se2(1, 2, 30, 'deg') % omogenous matrix with both rotational and translation parts can be created directly

			axis([0 5 0 5])
			axis square
			hold on

			T1 = se2(1, 2, 30, 'deg')
			trplot2(T1, 'frame', '1', 'color', 'b')

			T2 = se2(2,1,0)
			trplot2(T2, 'frame', '2', 'color', 'r')

			T3 = T1 * T2
			trplot2(T3, 'frame', '3', 'color', 'g')

			T4 = T2 * T1
			trplot2(T4, 'frame', '4', 'color', 'c')

			P = [3 2]' % Create a point using the following notation
			plot_point(P, '*')

			% To determine the coordinate of the point with respect to coordinate frame one,
			% use the inverse of the relative pose 1 and multiply it by the points coordinate
			% in the homogeneous form (by appending a 1 to the vector P).
			P1 = inv(T1) * [P ; 1]


	=== L3.10 2-axis representation ===		=== L3.10 2-axis representation ===

Balogh: /* L3.10 2-axis representation */

2016-10-03T09:37:34Z

L3.10 2-axis representation

← Staršia verzia		Verzia z 09:37, 3. október 2016
Riadok 56:		Riadok 56:
	>> oa2r( [-1.8833, 6.1427, 1.0000], [1.0000, -0.4925, 4.9085])		>> oa2r( [-1.8833, 6.1427, 1.0000], [1.0000, -0.4925, 4.9085])
	>> oa2r( [2.9914, 1.0000, -0.5091], [-1.8833, 6.1427, 1.0000])		>> oa2r( [2.9914, 1.0000, -0.5091], [-1.8833, 6.1427, 1.0000])



			=== L3.12 Quaternions ===


			Exercise

			* In the MATLAB workspace create a rotation matrix equivalent to a rotation of 30° about the x-axis, and store it in the workspace as the variable R1.
			* Create another which is equivalent to a rotation of 60° about the y-axis and store that in the variable R2.
			* Create a unit-quaternion equivalent to the rotation R1 and store it in the workspace as the variable q1.
			* Use the notation q1.R to convert the quaternion to a rotation matrix. Compare this to the original rotation matrix R1.
			* Create another quaternion, call this one q2, equivalent to the rotation matrix R2.
			* Inspect the vector part of the quaternions, what axes are these parallel to?
			* Multiply quaternion q1 by quaternion q2, using the standard MATLAB multiplication operation.
			* Compute the inverse of quaternion q1, using the MATLAB function inv.
			* Use the .R notation from above to convert the quaternion to a rotation matrix. Compare this result to directly computing the inverse of R1.
			* Multiply quaternion q1 by its inverse and inspect the result. Convert the result back to a rotation matrix.

			Hints/Solutions to Exercise

			R1 = rotx(30, 'deg')
			R2 = roty(60, 'deg')

			q1 = Quaternion(R1)

			q1.R

			q2 = Quaternion(R2)

			q1*q2

			ans.R
			R1*R2

			inv(q1)
			ans.R
			inv(R1)

			q1*inv(q1)
			ans.R

Balogh: /* L3.10 2-axis representation */

2016-10-03T09:11:32Z

L3.10 2-axis representation

← Staršia verzia		Verzia z 09:11, 3. október 2016
Riadok 45:		Riadok 45:
	1. How can matrices R=oa2r(o,a) and r=[n' o' a'] proved to be equivalent?		1. How can matrices R=oa2r(o,a) and r=[n' o' a'] proved to be equivalent?
	2. Why is a difference appearing in the yaw angle calculated from matrix R and from matrix r?		2. Why is a difference appearing in the yaw angle calculated from matrix R and from matrix r?



			Explanation

			We can really only do this by trial error, try each of the potential answers and see if it gives the desired rotation matrix. Note that the arguments to the oa2r() function are given in the order O then A, the reverse of what's above.

			>> oa2r( [1.0000, -0.4925, 4.9085], [-1.8833, 6.1427, 1.0000])
			>> oa2r( [2.9914, 1.0000, -0.5091], [1.0000, -0.4925, 4.9085])
			>> oa2r( [-1.8833, 6.1427, 1.0000], [1.0000, -0.4925, 4.9085])
			>> oa2r( [2.9914, 1.0000, -0.5091], [-1.8833, 6.1427, 1.0000])

Balogh: /* L3.10 2-axis representation */

2016-10-03T09:10:54Z

L3.10 2-axis representation

← Staršia verzia		Verzia z 09:10, 3. október 2016
Riadok 13:		Riadok 13:
	* A=(−1.8833,6.1427,1.0000),O=(1.0000,−0.4925,4.9085)		* A=(−1.8833,6.1427,1.0000),O=(1.0000,−0.4925,4.9085)
	* A=(1.0000,−0.4925,4.9085),O=(2.9914,1.0000,−0.5091)		* A=(1.0000,−0.4925,4.9085),O=(2.9914,1.0000,−0.5091)
	* A=(1.0000,−0.4925,4.9085),O=(−1.8833,6.1427,1.0000)		* A=(1.0000,−0.4925,4.9085),O=(−1.8833,6.1427,1.0000) (CORRECT)
	* A=(−1.8833,6.1427,1.0000),O=(2.9914,1.0000,−0.5091)		* A=(−1.8833,6.1427,1.0000),O=(2.9914,1.0000,−0.5091)