issue #56:

Enhancement: change equations, change some labels, variables in code, change title picture

doc/chap_examples.rst

@@ -532,11 +532,10 @@ to the green region.
 Two-link planar manipulator
 -------------
 
-.. _hwfig:
+.. _tlpmfig:
 
-.. figure:: /pic/chapter06_section06_task.PNG
-   :alt: highway
-   :width: 30 %
+.. figure:: /pic/chapter06_section06_task.png
+    :width: 30 %
 
 	Two-link planar manipulator.
 
@@ -557,13 +556,13 @@ At first, write the kinetic and potential energy for this system.
 	V_1 =& \ m_1 g s_1 \sin{\phi_1},\\
 	V_2 =& \ m_2 g l_1 \sin{\phi_1} + m_2 g s_2 \sin{\phi_2}
 
-In these equations :math:`m_1, m_2` -- masses of objects, :math:`s_1 = |O_1 C_1|,
-s_2 = |O_2 C_2|`, where :math:`C_1, C_2` -- centers of masses, 
-:math:`\phi_1, \phi_2` -- angles of elements' position,  :math:`l_1, l_2` -- full 
-lengths of elements, :math:`J_{c_1}, J_{c_2}` -- moments of inertia,
+In these equations :math:`m_1, m_2` --- masses of objects, :math:`s_1 = |O_1 C_1|,
+s_2 = |O_2 C_2|`, where :math:`C_1, C_2` --- centers of masses, 
+:math:`\phi_1, \phi_2` --- angles of elements' position,  :math:`l_1, l_2` --- full 
+lengths of elements, :math:`J_{c_1}, J_{c_2}` --- moments of inertia,
 :math:`\left | u_1\right | \leqslant \alpha_1,  \left | u_2\right | \leqslant \alpha_2`
- -- restriction of controls, which are attached to the hinges,
-:math:`M_{tr1}, M_{tr2}` -- torques of friction force and torques are proportional
+ --- restriction of controls, which are attached to the hinges,
+:math:`M_{tr1}, M_{tr2}` --- torques of friction force and torques are proportional
  to the angular speed with coefficients :math:`k_1, k_2`.
 
 .. math::
@@ -572,7 +571,7 @@ lengths of elements, :math:`J_{c_1}, J_{c_2}` -- moments of inertia,
 	x_1 =& \ s_1 \cos{\phi_1},\\
 	y_1 =& \ s_1 \sin{\phi_1},\\
 	x_2 =& \ l_1 \cos{\phi_1} + s_2 \cos{\phi_2},\\
-	y_2 =& \ s_1 \cos{\phi_1}.
+	y_2 =& \ l_1 \sin{\phi_1} + s_2 \sin{\phi_2}.
 
 .. math:: 
    :label: kinetic_def_1
@@ -683,7 +682,7 @@ Make new replacements:
 With new variables we get:
 
 .. math:: 
-   :label: taylor
+   :label: deriv_angles
 
    & \ddot{\phi}_1 = \dot{\phi}_1 y_1 + \dot{\phi}_2 y_2 + u_1 y_3 + u_2 y_4 + y_5,\\
    & \ddot{\phi}_2 = \dot{\phi}_1 x_1 + \dot{\phi}_2 x_2 + u_1 x_3 + u_2 x_4 + x_5.
@@ -731,7 +730,7 @@ Now define new variables:
    
 
 
-where :math:`\phi_1^0, \phi_2^0` -- angles of construction in equilibrium position.
+where :math:`\phi_1^0, \phi_2^0` --- angles of construction in equilibrium position.
 
 We have system :math:`\dot{t} = At + Bu + C`, here :math:`t` - vector with 4 coordinates,
 where :math:`t_1 = \phi_1,\ t_2 = \phi_2,\ t_3 = \dot{\phi_3},\ t_4 = \dot{\phi_4}`,
@@ -770,134 +769,107 @@ As a result, we get:
    z_3\\
    z_4\end{array}\right]
    + 
-   \left[\begin{array}{cccc}
-   0 & 0 & 0 & 0 \\
-   0 & 0 & 0 & 0 \\
-   y_{3} & y_{4} & 0 & 0 \\
-   x_{3} & x_{4} & 0 & 0
+   \left[\begin{array}{cc}
+   0 & 0 \\
+   0 & 0 \\
+   y_{3} & y_{4} \\
+   x_{3} & x_{4} 
    \end{array}\right]
    \left[\begin{array}{c}\
    u_1 \\
-   u_2 \\
-   u_3 \\
-   u_4 \end{array}\right],
+   u_2
+   \end{array}\right],
 
-Here :math:`u_3, u_4` -- fictitious variables for right size, 
+Here :math:`u_3, u_4` --- fictitious variables for right size, 
 :math:`|u_1| \leqslant \alpha_1, |u_2| \leqslant \alpha_2`.
 
 
-Now we can compute the reach set of the linear system
+Now we can compute the solvability set of the linear system
 and taking its projection onto :math:`(z_1, z_2)` subspace.
 
 
-.. _hwfig:
-
 .. figure:: /pic/chapter06_section06_pic1.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_3, z_4`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=0, z_2=0,z_3=0,z_4=0`)
 
-
-.. _hwfig:
 
 .. figure:: /pic/chapter06_section06_pic2.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_1, z_2`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=0, z_2=0,z_3=0,z_4=0`)
 
-.. _hwfig:
 
 .. figure:: /pic/chapter06_section06_pic3.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_1, z_3`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=0, z_2=0,z_3=0,z_4=0`)
 
-.. _hwfig:
 
 .. figure:: /pic/chapter06_section06_pic4.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_2, z_4`) of system
    ( evolves in time from t=0 to t=5, end position is
 	:math:`z_1=0, z_2=0,z_3=0,z_4=0`)
 
-.. _hwfig:
 
 .. figure:: /pic/chapter06_section06_pic5.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_3, z_4`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=\frac{\pi}{8}, z_2=\frac{\pi}{8}, z_3=0, z_4=0`)
 
-
-.. _hwfig:
 
 .. figure:: /pic/chapter06_section06_pic6.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_1, z_2`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=\frac{\pi}{8}, z_2=\frac{\pi}{8}, z_3=0, z_4=0`)
-
-.. _hwfig:
+
 
 .. figure:: /pic/chapter06_section06_pic7.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_1, z_3`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=\frac{\pi}{8}, z_2=\frac{\pi}{8}, z_3=0, z_4=0`)
-
-.. _hwfig:
+
 
 .. figure:: /pic/chapter06_section06_pic8.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_2, z_4`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=\frac{\pi}{8}, z_2=\frac{\pi}{8}, z_3=0, z_4=0`)
 
-.. _hwfig:
 
 .. figure:: /pic/chapter06_section06_pic9.png
-   :alt: highway
    :width: 100 %
 
    Now set control restrictions on :math:`\alpha_1 = 10, \alpha_2 = 10`.
    Figure shows solvability set projection onto (:math:`z_1, z_2`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=\frac{\pi}{8}, z_2=\frac{\pi}{8}, z_3=0, z_4=0`)
 
-.. _hwfig:
 
 .. figure:: /pic/chapter06_section06_pic10.png
-   :alt: highway
    :width: 100 %
 
    Figure shows solvability set projection onto (:math:`z_3, z_4`) of system
    ( evolves in time from t=0 to t=5, end position is
    :math:`z_1=\frac{\pi}{8}, z_2=\frac{\pi}{8}, z_3=0, z_4=0`, control limits:
    :math:`\alpha_1 = 10, \alpha_2 = 10`)
 
-
-.. raw:: latex
-
-    \newpage	
+
 
 .. literalinclude:: ../products/+elltool/+doc/+snip/s_chapter06_section06_snippet01.m
    :language: matlab

products/+elltool/+doc/+snip/s_chapter06_section06_snippet01.m

@@ -1,108 +1,75 @@
-
-% in this block we initialize all variables for this model
-m_1 = 1;
-m_2 = 2;
-
-J_1 = 0;
-J_2 = 0;
-
-phi_10 = 0;
-phi_20 = 0;
-
-L_1 = 5;
-L_2 = 5;
-
-s_1 = 3;
-s_2 = 3;
-
-alpha_1 = 3;
-alpha_2 = 10;
-
-k_1 = 0;
-k_2 = 0;
-
+% initialize all variables for this model
+m = [1; 2];
+J = [0; 0];
+phi_0 = [0; 0];
+L = [5; 5];
+s = [3; 3];
+alpha = [3; 10];
+k = [0; 0];
 g = 9.8;
 
-a_1 = m_1 * s_1^2 + J_1 + m_2 * L_1^2;
-a_2 = L_1 * s_2 * cos(phi_10 - phi_20);
-b_1 = L_1 * s_2 * cos(phi_10 - phi_20);
-b_2 = m_2 * s_2^2 + J_2;
+a_1 = m(1) * s(1)^2 + J(1) + m(2) * L(1)^2;
+a_2 = L(1) * s(2) * cos(phi_0(1) - phi_0(2));
+b_1 = L(1) * s(2) * cos(phi_0(1) - phi_0(2));
+b_2 = m(2) * s(2)^2 + J(2);
 
-d_1 = (m_1 * a_1 + m_2 * L_2) * g;
-d_2 = m_2 * a_2 * g;
+d_1 = (m(1) * a_1 + m(2) * L(2)) * g;
+d_2 = m(2) * a_2 * g;
 
-X_1 = -k_1 * a_2 / (a_2 * b_1 - a_1 * b_2);
-X_2 = k_2 * a_1 / (a_2 * b_1 - a_1 * b_2);
+X_1 = -k(1) * a_2 / (a_2 * b_1 - a_1 * b_2);
+X_2 = k(2) * a_1 / (a_2 * b_1 - a_1 * b_2);
 X_3 = a_2 / (a_2 * b_1 - a_1 * b_2);
 X_4 = -a_1 / (a_2 * b_1 - a_1 * b_2);
 X_5 = -(d_1 * a_2 + d_2 * a_1) / (a_2 * b_1 - a_1 * b_2);
 
-Y_1 = -k_1 * b_2 / (a_2 * b_1 - a_1 * b_2);
-Y_2 = k_2 * b_1 / (a_2 * b_1 - a_1 * b_2);
+Y_1 = -k(1) * b_2 / (a_2 * b_1 - a_1 * b_2);
+Y_2 = k(2) * b_1 / (a_2 * b_1 - a_1 * b_2);
 Y_3 = b_2 / (a_2 * b_1 - a_1 * b_2);
 Y_4 = -b_1 / (a_2 * b_1 - a_1 * b_2);
 Y_5 = (d_2 * b_1 - d_1 * b_2) / (a_2 * b_1 - a_1 * b_2);
 
-X_11 = d_1 * a_2 * sin(phi_10) / (a_2 * b_1 - a_1 * b_2);
-X_12 = d_2 * a_1 * sin(phi_20) / (a_2 * b_1 - a_1 * b_2);
-X_13 = (-d_1 * a_2 * cos(phi_10) - d_1 * a_2 * sin(phi_10) * phi_10) / ...
-    (a_2 * b_1 - a_1 * b_2) + (-d_2 * a_1 * cos(phi_20) - ...
-    d_2 * a_1 * sin(phi_20) * phi_20) / (a_2 * b_1 - a_1 * b_2);
+X_11 = d_1 * a_2 * sin(phi_0(1)) / (a_2 * b_1 - a_1 * b_2);
+X_12 = d_2 * a_1 * sin(phi_0(2)) / (a_2 * b_1 - a_1 * b_2);
+X_13 = (-d_1 * a_2 * cos(phi_0(1)) - d_1 * a_2 * sin(phi_0(1)) * phi_0(1)) / ...
+    (a_2 * b_1 - a_1 * b_2) + (-d_2 * a_1 * cos(phi_0(2)) - ...
+    d_2 * a_1 * sin(phi_0(2)) * phi_0(2)) / (a_2 * b_1 - a_1 * b_2);
 
-Y_11 = d_1 * b_2 * sin(phi_10) / (a_2 * b_1 - a_1 * b_2);
-Y_12 = -d_2 * b_1 * sin(phi_20) / (a_2 * b_1 - a_1 * b_2);
-Y_13 = (d_2 * b_1 * cos(phi_20) + d_2 * b_1 * sin(phi_20) * phi_20) / ...
-    (a_2 * b_1 - a_1 * b_2) + (-d_1 * b_2 * cos(phi_10) - ...
-    d_1 * b_2 * sin(phi_10) * phi_10) / (a_2 * b_1 - a_1 * b_2);
+Y_11 = d_1 * b_2 * sin(phi_0(1)) / (a_2 * b_1 - a_1 * b_2);
+Y_12 = -d_2 * b_1 * sin(phi_0(2)) / (a_2 * b_1 - a_1 * b_2);
+Y_13 = (d_2 * b_1 * cos(phi_0(2)) + d_2 * b_1 * sin(phi_0(2)) * phi_0(2)) / ...
+    (a_2 * b_1 - a_1 * b_2) + (-d_1 * b_2 * cos(phi_0(1)) - ...
+    d_1 * b_2 * sin(phi_0(1)) * phi_0(1)) / (a_2 * b_1 - a_1 * b_2);
 
 %initialize A and B relying on theoretical materials
 aMat = [0 0 1 0; 0 0 0 1; Y_11 Y_12 Y_1 Y_2; X_11 X_12 X_1 X_2];
-bMat = [0 0 0 0; 0 0 0 0; Y_3 Y_4 0 0 ; X_3 X_4 0 0];
-
-x_01=0;
-x_02=0;
-x_03=0;
-x_04=0;
-myTime=5;
+bMat = [0 0; 0 0; Y_3 Y_4; X_3 X_4];
 
+x_0 = [0; 0; 0; 0];
+endTime=5;
 % after initializing we create ellipsoid for control function
-diagonal = diag([alpha_1, alpha_2, 1, 1]);
-uEllObj = ellipsoid(diagonal);
-
+diagMat = diag([alpha(1), alpha(2)]);
+uEllObj = ellipsoid(diagMat);
 % define linear systems
 linSys=elltool.linsys.LinSysContinuous(aMat, bMat, uEllObj);
-
 % initialize direction matrix
 nShape = 4;
 mSize = 10;
 dirMat = 2*(rand(nShape, mSize) - 0.5);
 for iElem = 1 : mSize
     dirMat(:, iElem) = dirMat(:, iElem) ./ norm(dirMat(:, iElem));
 end;
-
 % end position
-x_0v = [x_01; x_02; x_03; x_04];
 
 % set of start coordinates
-x0EllObj = 1E-2 * ell_unitball(4) + x_0v;
-
+x0EllObj = 1E-2 * ell_unitball(4) + x_0;
 % calculate solvability set
-timeVec = [myTime, 0];
+timeVec = [endTime, 0];
 rsTube = elltool.reach.ReachContinuous(linSys, x0EllObj, dirMat, ...
     timeVec, 'isRegEnabled', true, 'isJustCheck', false, 'regTol', 1e-3);
-
 % create projection matrix and draw projection on current axis (z_3, z_4)
 v1v2Mat = [0 0 1 0 ; 0 0 0 1]';
 prTube = rsTube.projection(v1v2Mat);
 prTube.plotByIa();
 hold on;
 ylabel('z_3'); zlabel('z_4');
-rotate3d on;
-
-% create projection matrix and draw projection on current axis (z_1, z_2)
-v1v2Mat = [1 0 0 0 ; 0 1 0 0]';
-prTube = rsTube.projection(v1v2Mat);
-prTube.plotByIa();
-hold on;
-ylabel('z_1'); zlabel('z_2');
 rotate3d on;