Kinematic Study of the Stewart Platform
Table of Contents
- 1. Jacobian Analysis
- 2. Stiffness Analysis
- 3. Forward and Inverse Kinematics
- 4. Estimation of the range validity of the approximate inverse kinematics
- 5. Estimated required actuator stroke from specified platform mobility
- 6. Estimated platform mobility from specified actuator stroke
- 7. Estimation of the Joint required Stroke
- 8. Functions
The kinematic analysis of a parallel manipulator is well described in (Taghirad 2013):
Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces an torques that cause the motion. In this analysis, the relation between the geometrical parameters of the manipulator with the final motion of the moving platform is derived and analyzed.
The current document is divided in the following sections:
- Section 1: The Jacobian matrix is derived from the geometry of the Stewart platform. Then it is shown that the Jacobian can link velocities and forces present in the system, and thus this matrix can be very useful for both analysis and control of the Stewart platform.
- Section 2: The stiffness and compliance matrices are derived from the Jacobian matrix and the stiffness of each strut.
- Section 3: The Forward and Inverse kinematic problems are presented.
- Section 4: The approximate solution of the Inverse kinematic problem is compared with the exact solution in order to determine the validity of the approximation.
- Section 5: The Inverse kinematic solution is used to estimate required actuator stroke from the wanted mobility of the Stewart platform.
1 Jacobian Analysis
From (Taghirad 2013):
The Jacobian matrix not only reveals the relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities, it also constructs the transformation needed to find the actuator forces from the forces and moments acting on the moving platform.
1.1 Jacobian Computation
If we note:
- \({}^A\hat{\bm{s}}_i\) the unit vector representing the direction of the i’th strut and expressed in frame \(\{A\}\)
- \({}^A\bm{b}_i\) the position vector of the i’th joint fixed to the mobile platform and expressed in frame \(\{A\}\)
Then, we can compute the Jacobian with the following equation (the superscript \(A\) is ignored):
\begin{equation*} \bm{J} = \begin{bmatrix} {\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\ {\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\ {\hat{\bm{s}}_3}^T & (\bm{b}_3 \times \hat{\bm{s}}_3)^T \\ {\hat{\bm{s}}_4}^T & (\bm{b}_4 \times \hat{\bm{s}}_4)^T \\ {\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\ {\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T \end{bmatrix} \end{equation*}
The Jacobian matrix \(\bm{J}\) can be computed using the computeJacobian
function (accessible here).
For instance:
stewart = computeJacobian(stewart);
This will add three new matrix to the stewart
structure:
J
the Jacobian matrixK
the stiffness matrixC
the compliance matrix
1.2 Jacobian - Velocity loop closure
The Jacobian matrix links the input joint rate \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T\) of each strut to the output twist vector of the mobile platform is denoted by \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\):
\begin{equation*} \dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}} \end{equation*}The input joint rate \(\dot{\bm{\mathcal{L}}}\) can be measured by taking the derivative of the relative motion sensor in each strut. The output twist vector can be measured with a “Transform Sensor” block measuring the relative velocity and relative angular velocity of frame \(\{B\}\) with respect to frame \(\{A\}\).
If the Jacobian matrix is inversible, we can also compute \(\dot{\bm{\mathcal{X}}}\) from \(\dot{\bm{\mathcal{L}}}\).
\begin{equation*} \dot{\bm{\mathcal{X}}} = \bm{J}^{-1} \dot{\bm{\mathcal{L}}} \end{equation*}The Jacobian matrix can also be used to approximate forward and inverse kinematics for small displacements. This is explained in section 3.3.
1.3 Jacobian - Static Force Transformation
If we note:
- \(\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T\): vector of actuator forces applied in each strut
- \(\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T\): external force/torque action on the mobile platform at \(\bm{O}_B\)
We find that the transpose of the Jacobian matrix links the two by the following equation:
\begin{equation*} \bm{\mathcal{F}} = \bm{J}^T \bm{\tau} \end{equation*}If the Jacobian matrix is inversible, we also have the following relation:
\begin{equation*} \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}} \end{equation*}2 Stiffness Analysis
Here, we focus on the deflections of the manipulator moving platform that are the result of the external applied wrench to the mobile platform. The amount of these deflections are a function of the applied wrench as well as the manipulator structural stiffness.
2.1 Computation of the Stiffness and Compliance Matrix
As explain in this document, each Actuator is modeled by 3 elements in parallel:
- A spring with a stiffness \(k_{i}\)
- A dashpot with a damping \(c_{i}\)
- A force source \(\tau_i\)
The stiffness of the actuator \(k_i\) links the applied actuator force \(\delta \tau_i\) and the corresponding small deflection \(\delta l_i\):
\begin{equation*} \tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6 \end{equation*}If we combine these 6 relations:
\begin{equation*} \bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}} \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right] \end{equation*}Substituting \(\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}\) and \(\delta \bm{\mathcal{L}} = \bm{J} \cdot \delta \bm{\mathcal{X}}\) gives
\begin{equation*} \bm{\mathcal{F}} = \bm{J}^T \mathcal{K} \bm{J} \cdot \delta \bm{\mathcal{X}} \end{equation*}And then we identify the stiffness matrix \(\bm{K}\):
\begin{equation*} \bm{K} = \bm{J}^T \mathcal{K} \bm{J} \end{equation*}If the stiffness matrix \(\bm{K}\) is inversible, the compliance matrix of the manipulator is defined as
\begin{equation*} \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1} \end{equation*}The compliance matrix of a manipulator shows the mapping of the moving platform wrench applied at \(\bm{O}_B\) to its small deflection by
\begin{equation*} \delta \bm{\mathcal{X}} = \bm{C} \cdot \bm{\mathcal{F}} \end{equation*}
The stiffness and compliance matrices are computed using the computeJacobian
function (accessible here).
3 Forward and Inverse Kinematics
3.1 Inverse Kinematics
For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables \(\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T\).
This problem can be easily solved using the loop closures.
The obtain joint variables are:
\begin{equation*} \begin{aligned} l_i = &\Big[ {}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + \dots\\ &2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i \Big]^{1/2} \end{aligned} \end{equation*}If the position and orientation of the platform lie in the feasible workspace, the solution is unique. Otherwise, the solution gives complex numbers.
This inverse kinematic solution can be obtained using the function inverseKinematics
(described here).
3.2 Forward Kinematics
In forward kinematic analysis, it is assumed that the vector of limb lengths \(\bm{L}\) is given and the problem is to find the position \({}^A\bm{P}\) and the orientation \({}^A\bm{R}_B\).
This is a difficult problem that requires to solve nonlinear equations.
In a next section, an approximate solution of the forward kinematics problem is proposed for small displacements.
3.3 Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix
For small displacements mobile platform displacement \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T\) around \(\bm{\mathcal{X}}_0\), the associated joint displacement can be computed using the Jacobian (approximate solution of the inverse kinematic problem):
\begin{equation*} \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}} \end{equation*}Similarly, for small joint displacements \(\delta\bm{\mathcal{L}} = [ \delta l_1,\ \dots,\ \delta l_6 ]^T\) around \(\bm{\mathcal{L}}_0\), it is possible to find the induced small displacement of the mobile platform (approximate solution of the forward kinematic problem):
\begin{equation*} \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}} \end{equation*}These two relations solve the forward and inverse kinematic problems for small displacement in a approximate way. As the inverse kinematic can be easily solved exactly this is not much useful, however, as the forward kinematic problem is difficult to solve, this approximation can be very useful for small displacements.
The function forwardKinematicsApprox
(described here) can be used to solve the forward kinematic problem using the Jacobian matrix.
4 Estimation of the range validity of the approximate inverse kinematics
The Matlab script corresponding to this section is accessible here.
To run the script, open the Simulink Project, and type run kinematic_study_approximation_validity.m
.
As we know how to exactly solve the Inverse kinematic problem, we can compare the exact solution with the approximate solution using the Jacobian matrix. For small displacements, the approximate solution is expected to work well. We would like here to determine up to what displacement this approximation can be considered as correct.
Then, we can determine the range for which the approximate inverse kinematic is valid. This will also gives us the range for which the approximate forward kinematic is valid.
4.1 Stewart architecture definition
We first define some general Stewart architecture.
stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); stewart = generateGeneralConfiguration(stewart); stewart = computeJointsPose(stewart); stewart = initializeStewartPose(stewart); stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalStruts(stewart); stewart = initializeStrutDynamics(stewart); stewart = initializeJointDynamics(stewart); stewart = computeJacobian(stewart);
4.2 Comparison for “pure” translations
Let’s first compare the perfect and approximate solution of the inverse for pure \(x\) translations.
We compute the approximate and exact required strut stroke to have the wanted mobile platform \(x\) displacement. The estimate required strut stroke for both the approximate and exact solutions are shown in Figure 1. The relative strut length displacement is shown in Figure 2.
Xrs = logspace(-6, -1, 100); % Wanted X translation of the mobile platform [m] Ls_approx = zeros(6, length(Xrs)); Ls_exact = zeros(6, length(Xrs)); for i = 1:length(Xrs) Xr = Xrs(i); L_approx(:, i) = stewart.kinematics.J*[Xr; 0; 0; 0; 0; 0;]; [~, L_exact(:, i)] = inverseKinematics(stewart, 'AP', [Xr; 0; 0]); end
4.3 Conclusion
For small wanted displacements (up to \(\approx 1\%\) of the size of the Hexapod), the approximate inverse kinematic solution using the Jacobian matrix is quite correct.
5 Estimated required actuator stroke from specified platform mobility
The Matlab script corresponding to this section is accessible here.
To run the script, open the Simulink Project, and type run kinematic_study_required_actuator_stroke.m
.
Let’s say one want to design a Stewart platform with some specified mobility (position and orientation). One may want to determine the required actuator stroke required to obtain the specified mobility. This is what is analyzed in this section.
5.1 Stewart architecture definition
Let’s first define the Stewart platform architecture that we want to study.
stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); stewart = generateGeneralConfiguration(stewart); stewart = computeJointsPose(stewart); stewart = initializeStewartPose(stewart); stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalStruts(stewart); stewart = initializeStrutDynamics(stewart); stewart = initializeJointDynamics(stewart); stewart = computeJacobian(stewart);
5.2 Wanted translations and rotations
Let’s now define the wanted extreme translations and rotations.
Tx_max = 50e-6; % Translation [m] Ty_max = 50e-6; % Translation [m] Tz_max = 50e-6; % Translation [m] Rx_max = 30e-6; % Rotation [rad] Ry_max = 30e-6; % Rotation [rad] Rz_max = 0; % Rotation [rad]
5.3 Needed stroke for “pure” rotations or translations
As a first estimation, we estimate the needed actuator stroke for “pure” rotations and translation. We do that using either the Inverse Kinematic solution or the Jacobian matrix as an approximation.
LTx = stewart.kinematics.J*[Tx_max 0 0 0 0 0]'; LTy = stewart.kinematics.J*[0 Ty_max 0 0 0 0]'; LTz = stewart.kinematics.J*[0 0 Tz_max 0 0 0]'; LRx = stewart.kinematics.J*[0 0 0 Rx_max 0 0]'; LRy = stewart.kinematics.J*[0 0 0 0 Ry_max 0]'; LRz = stewart.kinematics.J*[0 0 0 0 0 Rz_max]';
The obtain required stroke is:
From -3.8e-05[m] to 3.8e-05[m]: Total stroke = 76.1[um]
This is surely a low estimation of the required stroke.
5.4 Needed stroke for “combined” rotations or translations
We know would like to have a more precise estimation.
To do so, we may estimate the required actuator stroke for all possible combination of translation and rotation.
Let’s first generate all the possible combination of maximum translation and rotations.
Ps = [2*(dec2bin(0:5^2-1,5)-'0')-1, zeros(5^2, 1)].*[Tx_max Ty_max Tz_max Rx_max Ry_max Rz_max];
| Tx [m] | Ty [m] | Tz [m] | Rx [rad] | Ry [rad] | Rz [rad] |
|---|---|---|---|---|---|
| -5.0e-05 | -5.0e-05 | -5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
| -5.0e-05 | -5.0e-05 | -5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
| -5.0e-05 | -5.0e-05 | -5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
| -5.0e-05 | -5.0e-05 | -5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
| -5.0e-05 | -5.0e-05 | 5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
| -5.0e-05 | -5.0e-05 | 5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
| -5.0e-05 | -5.0e-05 | 5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
| -5.0e-05 | -5.0e-05 | 5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
| -5.0e-05 | 5.0e-05 | -5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
| -5.0e-05 | 5.0e-05 | -5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
| -5.0e-05 | 5.0e-05 | -5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
| -5.0e-05 | 5.0e-05 | -5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
| -5.0e-05 | 5.0e-05 | 5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
| -5.0e-05 | 5.0e-05 | 5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
| -5.0e-05 | 5.0e-05 | 5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
| -5.0e-05 | 5.0e-05 | 5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
| 5.0e-05 | -5.0e-05 | -5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
| 5.0e-05 | -5.0e-05 | -5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
| 5.0e-05 | -5.0e-05 | -5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
| 5.0e-05 | -5.0e-05 | -5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
| 5.0e-05 | -5.0e-05 | 5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
| 5.0e-05 | -5.0e-05 | 5.0e-05 | -3.0e-05 | 3.0e-05 | 0.0e+00 |
| 5.0e-05 | -5.0e-05 | 5.0e-05 | 3.0e-05 | -3.0e-05 | 0.0e+00 |
| 5.0e-05 | -5.0e-05 | 5.0e-05 | 3.0e-05 | 3.0e-05 | 0.0e+00 |
| 5.0e-05 | 5.0e-05 | -5.0e-05 | -3.0e-05 | -3.0e-05 | 0.0e+00 |
For all possible combination, we compute the required actuator stroke using the inverse kinematic solution.
L_min = 0; L_max = 0; for i = 1:size(Ps,1) Rx = [1 0 0; 0 cos(Ps(i, 4)) -sin(Ps(i, 4)); 0 sin(Ps(i, 4)) cos(Ps(i, 4))]; Ry = [ cos(Ps(i, 5)) 0 sin(Ps(i, 5)); 0 1 0; -sin(Ps(i, 5)) 0 cos(Ps(i, 5))]; Rz = [cos(Ps(i, 6)) -sin(Ps(i, 6)) 0; sin(Ps(i, 6)) cos(Ps(i, 6)) 0; 0 0 1]; ARB = Rz*Ry*Rx; [~, Ls] = inverseKinematics(stewart, 'AP', Ps(i, 1:3)', 'ARB', ARB); if min(Ls) < L_min L_min = min(Ls) end if max(Ls) > L_max L_max = max(Ls) end end
We obtain the required actuator stroke:
From -8.9e-05[m] to 8.9e-05[m]: Total stroke = 177.2[um]
This is probably a much realistic estimation of the required actuator stroke.
6 Estimated platform mobility from specified actuator stroke
The Matlab script corresponding to this section is accessible here.
To run the script, open the Simulink Project, and type run kinematic_study_mobility.m
.
Here, from some value of the actuator stroke, we would like to estimate the mobility of the Stewart platform.
As explained in section 3, the forward kinematic problem of the Stewart platform is quite difficult to solve. However, for small displacements, we can use the Jacobian as an approximate solution.
6.1 Stewart architecture definition
Let’s first define the Stewart platform architecture that we want to study.
stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); stewart = generateGeneralConfiguration(stewart); stewart = computeJointsPose(stewart); stewart = initializeStewartPose(stewart); stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalStruts(stewart); stewart = initializeStrutDynamics(stewart); stewart = initializeJointDynamics(stewart); stewart = computeJacobian(stewart);
Let’s now define the actuator stroke.
L_min = -50e-6; % [m] L_max = 50e-6; % [m]
6.2 Pure translations
Let’s first estimate the mobility in translation when the orientation of the Stewart platform stays the same.
As shown previously, for such small stroke, we can use the approximate Forward Dynamics solution using the Jacobian matrix:
\begin{equation*} \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}} \end{equation*}To obtain the mobility “volume” attainable by the Stewart platform when it’s orientation is set to zero, we use the spherical coordinate \((r, \theta, \phi)\).
For each possible value of \((\theta, \phi)\), we compute the maximum radius \(r\) attainable with the constraint that the stroke of each actuator should be between L_min
and L_max
.
thetas = linspace(0, pi, 50); phis = linspace(0, 2*pi, 50); rs = zeros(length(thetas), length(phis)); for i = 1:length(thetas) for j = 1:length(phis) Tx = sin(thetas(i))*cos(phis(j)); Ty = sin(thetas(i))*sin(phis(j)); Tz = cos(thetas(i)); dL = stewart.kinematics.J*[Tx; Ty; Tz; 0; 0; 0;]; % dL required for 1m displacement in theta/phi direction rs(i, j) = max([dL(dL<0)*L_min; dL(dL>0)*L_max]); end end
Now that we have found the corresponding radius \(r\), we plot the obtained mobility. We can also approximate the mobility by a sphere with a radius equal to the minimum obtained value of \(r\), this is however a pessimistic estimation of the mobility.
L_min [\(\mu m\)] |
L_max [\(\mu m\)] |
R [\(\mu m\)] |
|---|---|---|
| -50.0 | 50.0 | 31.5 |
7 Estimation of the Joint required Stroke
7.1 Estimation with an analytical model
Let’s first define the Stewart Platform Geometry.
stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 150e-3); stewart = generateGeneralConfiguration(stewart); stewart = computeJointsPose(stewart, 'AP', [0; 0; 0]); As_init = stewart.geometry.As;
Tx_max = 60e-6; % Translation [m] Ty_max = 60e-6; % Translation [m] Tz_max = 60e-6; % Translation [m] Rx_max = 30e-6; % Rotation [rad] Ry_max = 30e-6; % Rotation [rad] Rz_max = 0; % Rotation [rad]
Ps = [2*(dec2bin(0:5^2-1,5)-'0')-1, zeros(5^2, 1)].*[Tx_max Ty_max Tz_max Rx_max Ry_max Rz_max];
flex_ang = zeros(size(Ps, 1), 6); Rxs = zeros(size(Ps, 1), 6) Rys = zeros(size(Ps, 1), 6) Rzs = zeros(size(Ps, 1), 6) for Ps_i = 1:size(Ps, 1) Rx = [1 0 0; 0 cos(Ps(Ps_i, 4)) -sin(Ps(Ps_i, 4)); 0 sin(Ps(Ps_i, 4)) cos(Ps(Ps_i, 4))]; Ry = [ cos(Ps(Ps_i, 5)) 0 sin(Ps(Ps_i, 5)); 0 1 0; -sin(Ps(Ps_i, 5)) 0 cos(Ps(Ps_i, 5))]; Rz = [cos(Ps(Ps_i, 6)) -sin(Ps(Ps_i, 6)) 0; sin(Ps(Ps_i, 6)) cos(Ps(Ps_i, 6)) 0; 0 0 1]; ARB = Rz*Ry*Rx; stewart = computeJointsPose(stewart, 'AP', Ps(Ps_i, 1:3)', 'ARB', ARB); flex_ang(Ps_i, :) = acos(sum(As_init.*stewart.geometry.As)); for l_i = 1:6 MRf = stewart.platform_M.MRb(:,:,l_i)*(ARB')*(stewart.platform_F.FRa(:,:,l_i)'); Rys(Ps_i, l_i) = atan2(MRf(1,3), sqrt(MRf(1,1)^2 + MRf(2,2)^2)); Rxs(Ps_i, l_i) = atan2(-MRf(2,3)/cos(Rys(Ps_i, l_i)), MRf(3,3)/cos(Rys(Ps_i, l_i))); Rzs(Ps_i, l_i) = atan2(-MRf(1,2)/cos(Rys(Ps_i, l_i)), MRf(1,1)/cos(Rys(Ps_i, l_i))); end end
And the maximum bending of the flexible joints is: (in [mrad])
1e3*max(max(abs(flex_ang)))
1.1704
1e3*max(max(sqrt(Rxs.^2 + Rys.^2)))
0.075063
1e3*max(max(Rzs))
0.04666
7.2 Estimation using the Simscape Model
7.2.1 Model Init
open('stewart_platform_model.slx')
stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); stewart = generateGeneralConfiguration(stewart); stewart = computeJointsPose(stewart); stewart = initializeStrutDynamics(stewart); stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p'); stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalStruts(stewart); stewart = computeJacobian(stewart); stewart = initializeStewartPose(stewart); stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
ground = initializeGround('type', 'rigid'); payload = initializePayload('type', 'none'); controller = initializeController('type', 'open-loop'); disturbances = initializeDisturbances(); references = initializeReferences(stewart);
7.2.2 Controller design to be close to the wanted displacement
%% Name of the Simulink File mdl = 'stewart_platform_model'; %% Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N] io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m] %% Run the linearization G = linearize(mdl, io); G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};
wc = 2*pi*30; Kl = diag(1./diag(abs(freqresp(G, wc)))) * wc/s * 1/(1 + s/3/wc);
controller = initializeController('type', 'ref-track-L');
7.2.3 Simulations
Tx_max = 60e-6; % Translation [m] Ty_max = 60e-6; % Translation [m] Tz_max = 60e-6; % Translation [m] Rx_max = 30e-6; % Rotation [rad] Ry_max = 30e-6; % Rotation [rad] Rz_max = 0; % Rotation [rad]
Ps = [2*(dec2bin(0:5^2-1,5)-'0')-1, zeros(5^2, 1)].*[Tx_max Ty_max Tz_max Rx_max Ry_max Rz_max];
cl_perf = zeros(size(Ps, 1), 1); % Closed loop performance [percentage] Rs = zeros(size(Ps, 1), 5); % Max Flexor angles for the 6 legs [mrad] for Ps_i = 1:size(Ps, 1) fprintf('Experiment %i over %i', Ps_i, size(Ps, 1)); references = initializeReferences(stewart, 't', 0, 'r', Ps(Ps_i, :)'); set_param('stewart_platform_model','StopTime','0.1'); sim('stewart_platform_model'); cl_perf(Ps_i) = 100*max(abs((simout.y.dLm.Data(end, :) - references.rL.Data(:,1,1)')./simout.y.dLm.Data(end, :))); Rs(Ps_i, :) = [max(abs(1e3*simout.y.Rf.Data(end, 1:2:end))), max(abs(1e3*simout.y.Rf.Data(end, 2:2:end))), max(abs(1e3*simout.y.Rm.Data(end, 1:3:end))), max(abs(1e3*simout.y.Rm.Data(end, 2:3:end))), max(abs(1e3*simout.y.Rm.Data(end, 3:3:end)))]'; end
Verify that the simulations are all correct:
max(cl_perf)
8.1147
7.2.4 Results
| Stroke [mrad] | |
|---|---|
| Rx bot | 1.03 |
| Ry bot | 0.93 |
| Rx top | 1.06 |
| Ry top | 0.95 |
| Rz top | 0.03 |
8 Functions
8.1 computeJacobian
: Compute the Jacobian Matrix
This Matlab function is accessible here.
Function description
function [stewart] = computeJacobian(stewart) % computeJacobian - % % Syntax: [stewart] = computeJacobian(stewart) % % Inputs: % - stewart - With at least the following fields: % - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A} % - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A} % - actuators.K [6x1] - Total stiffness of the actuators % % Outputs: % - stewart - With the 3 added field: % - kinematics.J [6x6] - The Jacobian Matrix % - kinematics.K [6x6] - The Stiffness Matrix % - kinematics.C [6x6] - The Compliance Matrix
Check the stewart
structure elements
assert(isfield(stewart.geometry, 'As'), 'stewart.geometry should have attribute As') As = stewart.geometry.As; assert(isfield(stewart.geometry, 'Ab'), 'stewart.geometry should have attribute Ab') Ab = stewart.geometry.Ab; assert(isfield(stewart.actuators, 'K'), 'stewart.actuators should have attribute K') Ki = stewart.actuators.K;
Compute Jacobian Matrix
J = [As' , cross(Ab, As)'];
Compute Stiffness Matrix
K = J'*diag(Ki)*J;
Compute Compliance Matrix
C = inv(K);
Populate the stewart
structure
stewart.kinematics.J = J; stewart.kinematics.K = K; stewart.kinematics.C = C;
8.2 inverseKinematics
: Compute Inverse Kinematics
This Matlab function is accessible here.
Theory
For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
From the geometry of the manipulator, the loop closure for each limb, \(i = 1, 2, \dots, 6\) can be written as
\begin{align*} l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\ &= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i \end{align*}To obtain the length of each actuator and eliminate \(\hat{\bm{s}}_i\), it is sufficient to dot multiply each side by itself:
\begin{equation} l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right] \end{equation}Hence, for \(i = 1, 2, \dots, 6\), each limb length can be uniquely determined by:
\begin{equation} l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i} \end{equation}If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation. Otherwise, when the limbs’ lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
Function description
function [Li, dLi] = inverseKinematics(stewart, args) % inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A} % % Syntax: [stewart] = inverseKinematics(stewart) % % Inputs: % - stewart - A structure with the following fields % - geometry.Aa [3x6] - The positions ai expressed in {A} % - geometry.Bb [3x6] - The positions bi expressed in {B} % - geometry.l [6x1] - Length of each strut % - args - Can have the following fields: % - AP [3x1] - The wanted position of {B} with respect to {A} % - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A} % % Outputs: % - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A} % - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
Optional Parameters
arguments stewart args.AP (3,1) double {mustBeNumeric} = zeros(3,1) args.ARB (3,3) double {mustBeNumeric} = eye(3) end
Check the stewart
structure elements
assert(isfield(stewart.geometry, 'Aa'), 'stewart.geometry should have attribute Aa') Aa = stewart.geometry.Aa; assert(isfield(stewart.geometry, 'Bb'), 'stewart.geometry should have attribute Bb') Bb = stewart.geometry.Bb; assert(isfield(stewart.geometry, 'l'), 'stewart.geometry should have attribute l') l = stewart.geometry.l;
Compute
Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa));
dLi = Li-l;
8.3 forwardKinematicsApprox
: Compute the Approximate Forward Kinematics
This Matlab function is accessible here.
Function description
function [P, R] = forwardKinematicsApprox(stewart, args) % forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using % the Jacobian Matrix % % Syntax: [P, R] = forwardKinematicsApprox(stewart, args) % % Inputs: % - stewart - A structure with the following fields % - kinematics.J [6x6] - The Jacobian Matrix % - args - Can have the following fields: % - dL [6x1] - Displacement of each strut [m] % % Outputs: % - P [3x1] - The estimated position of {B} with respect to {A} % - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}
Optional Parameters
arguments stewart args.dL (6,1) double {mustBeNumeric} = zeros(6,1) end
Check the stewart
structure elements
assert(isfield(stewart.kinematics, 'J'), 'stewart.kinematics should have attribute J') J = stewart.kinematics.J;
Computation
From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the position and orientation of {B} with respect to {A} using the following formula: \[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
X = J\args.dL;
The position vector corresponds to the first 3 elements.
P = X(1:3);
The next 3 elements are the orientation of {B} with respect to {A} expressed using the screw axis.
theta = norm(X(4:6)); s = X(4:6)/theta;
We then compute the corresponding rotation matrix.
R = [s(1)^2*(1-cos(theta)) + cos(theta) , s(1)*s(2)*(1-cos(theta)) - s(3)*sin(theta), s(1)*s(3)*(1-cos(theta)) + s(2)*sin(theta); s(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta); s(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];


The half mile is one lap around the course
Pie chart represents the half mile age group makeup. The womans group 30-39 represented the group with the most particpants.
The box plot shows the differences between the finishers of the race. The more compact the box the closer swimmers finished, The bigger the box The more spread out the times were.
The mile is two laps around the course
Pie chart represents the mile age group makeup. The womans group 30-39 represented the group with the most particpants.
The box plot shows the differences between the finishers of the race. The more compact the box the closer swimmers finished, The bigger the box The more spread out the times were.
The Two mile is four laps around the course
Map Placeholder
Pie chart represents the half mile age group makeup. The womans group 50-59 represented the group with the most particpants.
The box plot shows the differences between the finishers of the race. The more compact the box the closer swimmers finished, The bigger the box The more spread out the times were.
Local towns and the number of swimmers from those towns.


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ADP
: adposition
This document is a placeholder for the language-specific documentation
for ADP
.
Treebank Statistics (UD_Gothic)
There are 28 ADP
lemmas (1%), 31 ADP
types (0%) and 4008 ADP
tokens (9%).
Out of 14 observed tags, the rank of ADP
is: 6 in number of lemmas, 10 in number of types and 6 in number of tokens.
The 10 most frequent ADP
lemmas: in, du, bi, us, ana, miþ, fram, at, þairh, af
The 10 most frequent ADP
types: in, du, bi, us, ana, miþ, fram, at, þairh, af
The 10 most frequent ambiguous lemmas: in (ADP 1509, SCONJ 3), du (ADP 717, ADV 2), ana (ADP 194, ADV 1), fram (ADP 151, ADV 2), faura (ADP 39, ADV 4), fairra (ADP 10, ADV 3), neƕa (ADP 9, ADV 5), ufaro (ADP 2, ADV 1), alja (ADV 6, CCONJ 5, ADP 1, SCONJ 1), ūtana (ADV 2, ADP 1)
The 10 most frequent ambiguous types: in (ADP 1505, SCONJ 3), du (ADP 705, ADV 2), ana (ADP 194, ADV 1), faura (ADP 39, ADV 4), fairra (ADP 10, ADV 3), neƕa (ADP 9, ADV 5), ufaro (ADP 2, ADV 1), alja (ADV 6, CCONJ 5, ADJ 1, ADP 1, SCONJ 1), utana (ADV 2, ADP 1)
- in
- du
- ana
- faura
- fairra
- neƕa
- ufaro
- alja
- ADV 6: ni ƕashun þiuþeigs alja ains guþ
- CCONJ 5: alja so managei þaiei ni kunnun witoþ fraqiþanai sind
- ADJ 1: unte ni alja meljam izwis alja þoei anakunnaiþ aiþþau jah ufkunnaiþ
- ADP 1: waila laisari bi sunjai qast þatei ains ist jah nist anþar alja imma
- SCONJ 1: sildaleikja ei swa sprauto afwandjanda af þamma laþondin izwis in anstai Xristaus du anþaramma aiwaggelja þatei nist anþar alja sumai sind þai drobjandans izwis jah wiljandans inwandjan aiwaggeli Xristaus
- utana
Morphology
The form / lemma ratio of ADP
is 1.107143 (the average of all parts of speech is 2.517954).
The 1st highest number of forms (3) was observed with the lemma “us”: ur, us, uz.
The 2nd highest number of forms (2) was observed with the lemma “in”: in, inuh.
The 3rd highest number of forms (2) was observed with the lemma “inuh”: inu, inuh.
ADP
does not occur with any features.
Relations
ADP
nodes are attached to their parents using 8 different relations: got-dep/case (3897; 97% instances), got-dep/mark (78; 2% instances), got-dep/advmod (25; 1% instances), got-dep/conj (3; 0% instances), got-dep/amod (2; 0% instances), got-dep/appos (1; 0% instances), got-dep/nmod (1; 0% instances), got-dep/xcomp (1; 0% instances)
Parents of ADP
nodes belong to 8 different parts of speech: NOUN (2253; 56% instances), PRON (1009; 25% instances), ADJ (287; 7% instances), PROPN (280; 7% instances), VERB (148; 4% instances), ADV (16; 0% instances), NUM (12; 0% instances), ADP (3; 0% instances)
3810 (95%) ADP
nodes are leaves.
183 (5%) ADP
nodes have one child.
14 (0%) ADP
nodes have two children.
1 (0%) ADP
nodes have three or more children.
The highest child degree of a ADP
node is 18.
Children of ADP
nodes are attached using 14 different relations: got-dep/advmod (82; 36% instances), got-dep/det (48; 21% instances), got-dep/appos (44; 19% instances), got-dep/ccomp (19; 8% instances), got-dep/amod (9; 4% instances), got-dep/obl (7; 3% instances), got-dep/cc (6; 3% instances), got-dep/advcl (3; 1% instances), got-dep/case (3; 1% instances), got-dep/obj:dir (3; 1% instances), got-dep/acl (2; 1% instances), got-dep/discourse (1; 0% instances), got-dep/nmod (1; 0% instances), got-dep/nummod (1; 0% instances)
Children of ADP
nodes belong to 10 different parts of speech: ADV (85; 37% instances), DET (48; 21% instances), NOUN (41; 18% instances), VERB (24; 10% instances), ADJ (13; 6% instances), PRON (7; 3% instances), CCONJ (6; 3% instances), ADP (3; 1% instances), NUM (1; 0% instances), PROPN (1; 0% instances)
ADP in other languages: [am] [ar] [bg] [bxr] [ca] [ckb] [cop] [cs] [cu] [da] [de] [el] [en] [es] [et] [eu] [fa] [fi] [fo] [fr] [ga] [gl] [got] [grc] [he] [hi] [hr] [hu] [id] [it] [ja] [kk] [kmr] [ko] [la] [lv] [mr] [nl] [no] [pl] [pt] [ro] [ru] [sa] [sk] [sla] [sl] [so] [sr] [sv] [swl] [ta] [tr] [ug] [uk] [u] [urj] [ur] [vi] [yue] [zh]


${paymentRequest.listing.title} ${paymentRequest.listing.description} ${paymentRequest.user.name} USD ${paymentRequest.amount | usdFormat} Approve Reject Details Submitted: ${paymentRequest.created_at} Requested amount: ${paymentRequest.amount | usdFormat} Status: ${paymentRequest.completed ? 'Completed' : 'Not yet completed'}


Robert Peston's girlfriend drops sex bombshell about naughty way he passed driving test
Columnist Charlotte Edwardes claimed boyfriend Robert Peston slept with his driving instructor at the age of 18
by Emmeline SaundersRobert Peston's journalist girlfriend has revealed the X-rated way he managed to get his driving licence: by bedding his 30-something instructor when he was just 18.
Charlotte Edwardes, who writes a column for the Sunday Times, claimed her ITV political editor boyfriend slept with the much older woman as a teenager while talking about the question Boris Johnson refused to answer .
"My boyfriend's naughtiest thing is that he slept with his driving instructor — a woman in her late thirties — when he was 18," she wrote in her column.
"This has always made complete sense to me, because if you've ever had the misfortune to get into a car with my boyfriend, you'll know there is literally no other explanation for him passing his test."
While Robert is yet to confirm or deny his girlfriend's bombshell, he did retweet Charlotte's column to his one million followers.
It comes two years after the couple first got together after meeting in 2017.
Charlotte hit the headlines in September this year when she went public with her allegation that the prime minister had once groped her thigh during a lunch party in 1999.
Johnson, who was then editor of the Spectator, had allegedly placed his hand on Charlotte's thigh under the table - and when she complained to the female guest sitting on his other side, the woman replied: "Oh god, he did exactly the same to me."
Number 10 denied Charlotte's claims, but Robert is believed to have pulled out of a scheduled interview with the prime minister in the wake of Charlotte's explosive column.


PHP staking interface for coinshielding nodes.
Read it completely before using.
Important General Information
'VRSC Wallet and Data location` on Linux: `~/.Komodo/VRSC`
Necessary files:
Link 1: Download latest Wallet
Link 2: Download Verus-Staking-CLI
Link 3: Go to Verus-Staking-CLI
Procedure:
These tools will help you stake on your permanently running machine using Verus CLI, instead of running the Agama GUI wallet. Features automatic shielding and unshielding of coinbase (staking, solo mining) automatic tracking of total coins generated (also present in e-mail notifications) optional e-mail notifications on newly generated coins optional periodical wallet.dat backups, also as an encrypted ZIP archive optional periodical wallet balance e-mail notifications Initial Verus daemon configuration Follow the guide in link 3 above to set-up your daemon for the first time. You will need a transparent address, a zs address and the daemon running with correct parameters (-mint -cheatcatcher=zs...). Do not stake with the same wallet.dat on multiple nodes.
Submitted by @kbs1


This is mainly a page for testing stuff.
| Word | Definition |
|---|---|
| HTML | HTML (HyperText Markup Language) is used to give content to a web page and instructs web browsers on how to structure that content. |
| HTML Element | An HTML element is a piece of content in an HTML document. |
| HTML Tag | The syntax for a single HTML tag is an opening angle bracket < followed by the element name and a closing angle bracket > . |
| Element Content | The content of an HTML element is the information between the opening and closing tags of an element. |
| Body Element | The <body> element represents the content of an HTML document. Content inside body tags are rendered on the web browsers. |
| Heading Elements | HTML can use six different levels of heading elements. The heading elements are ordered from the highest level to the lowest level |
| Padding | The padding property in CSS defines the innermost portion of the box model, creating space around an element's content, inside of any defined margins and/or borders. |
| Div Element | The <div> element is used as a container that divides an HTML document into sections and is short for “division”. elements can contain ﬂ o w c o n t e n t such as headings, paragraphs, links, images, etc.
|
| Paragraph Element | The <p> paragraph element contains and displays a block of text |
| Span Element | The <span> element is an inline container for text and can be used to group text for styling purposes. However, as span is a generic container to separate pieces of text from a larger body of text, its use should be avoided if a more semantic element is available. |
| Emphasis Element | The <em> emphasis element emphasizes text and browsers will usually it emphasize the emphasized text by default. |
| Strong Element | The strong element highlights important, serious, or urgent text and browsers will normally render this highlighted text in bold by default. |
| Line Break Element | The <br> line break element will create a line break in text and is especially useful where a division of text is required, like in a postal address. The line break element requires only an opening tag and must not have a closing tag. |
| Unordered List Element | The <ul> unordered list element is used to create a list of items in no particular order. Each individual list item will have a bullet point by default. |
| List Item Element | The <li> list item element create list items inside <ol> and <ul> |
| Ordered List Element | The <ol> ordered list element creates a list of items in sequential order. Each list item appears numbered by default. |
| Image Element | HTML image <img> elements embed images in documents. The src attribute contains the image URL and is mandatory. 'img' is an empty element meaning it should not have a closing tag. . |


Before Change
Returns a list of paths.
return self.lazy_tool_classpath_from_products(products, key, scope)()
def lazy_tool_classpath_from_products(self, products, key, scope):
Get a lazy classpath for the tool previously registered under the key in the given scope.
After Change
Returns a list of paths.
callback_product_map = products.get_data("jvm_build_tools_classpath_callbacks") or {}
callback = callback_product_map.get(scope, {}).get(key)
if not callback:
raise TaskError("No bootstrap callback registered for {key} in {scope}"
.format(key=key, scope=scope))
return callback()


Faculty Coordinator
9418340317
Hill'ffair Secretary
9805121081
Hill'ffair Manager
9882218585
Club Management Secretary
9882202177
Club Secretary For Performing Arts
9817824154
Finance Secretary
8894603751
Finance & Treasury
9882936887
Quality Control Secretary
7018622434
Logistics Secretary
9816454577
Organization Secretary
Music Club
9805076240
Pixonoids
88949 59393
Dramatics
9460334508
Discipline
9816677089
Organisation
9882554180
INS and Control
9805620685
Technical
94592 01957
Design and Deco
8350837457
Web Team
98050 75250
83509 29741
9736345533
Fine Arts
8167 31826
9882998832
Music
7895863354
PR
8219868534
94598 41148
7832855237
English Club
9882550776
Informals
9882550826
9816049958
9779993747
Hindi Samiti
98825 49612


Jun 20, 2012. now it's official: Johnny Depp and Vanessa Paradis have broken up. and actress have been together for 14 years and have two children, Jul 16, 2013 Yep, they're definitely a couple. now: Johnny Depp and Amber Heard. lunch on October 29, amid reports the Jonas Brothers have broken up
Jul 9, 2013. Johnny Depp is considered one of the most attractive entertainers on Earth Johnny said: 'When you have children, there is no room for lies, Being with Vanessa and having children has made things very easy and clear for me There's nothing dark
A documentary about wounded veterans that Roger championed gets a screening at the Rehabilitation Institute of Chicago and screenings in New York and Los Angeles…
Jul 18, 2013 Amber Heard is getting to know the kids. The 27-year-old actress, who has been dating her "Rum Diary" co-star Johnny Depp on and off since
Films featuring Depp have grossed over $3 1 billion at the United. States box office. Depp was born in Owensboro, Kentucky, as. the youngest of four children of
May 11, 2013 Johnny Depp and Amber Heard are apparently so "serious" they have "They've
talked about children - that's how serious it is," a source told
Jul 4, 2013 Johnny Depp believes there is simply no room for lies when you have children. The actor recently opened up about. his breakup with long-time
Jul 25, 2013. Lainey Gossip|No Johnny Depp in Paris while his kids are back with. if he was
there that he would have gone unphotographed on the carpet Jul 20, 2013 Johnny Depp chats to the paparazzis protecting his kids and Amber Heard. Alert icon You need Adobe Flash Player to watch this video.


Properties that are not initialized in the constructor should be annotated as nullable.
class User { /** @var string */ // <-- Property is not annotated correctly. Add null type private $name; public function getName() { } public function setName(string $name) { } }


The following four characteristics are common for successful online students:
Use this self-assessment to gauge your readiness for online learning. To get a good idea of your chances for success as an online student, rate how well each item below describes yourself.'


You can vote for the Google Play Users’ Choice awards right now
by Jared PetersIt’s the end of the year, and that means it’s time to vote for your favorite apps and games on the Play Store from 2019. Google has opened up voting for the User Choice awards where everyone will be casting their vote for several different categories of apps and contents.
There are four categories, including the typical best app, game, movie, and book. You’ll have to select from preset nominees, so you can’t just write anything in, but now’s a great chance to check out some new high-quality apps if there’s anything on the list that you haven’t used before.
The voting is open until November 25th, and you can change your vote as many times as you want up until that point. Winners will be announced on December 3rd, not long after voting closes.


Notice: The new YouTube breaks SubTube; as a workaround, set your User-Agent to an older browser like Internet Explorer 7.
Mozilla/4.0 (compatible; MSIE 7.0; Windows NT 5.1)
About SubTube
SubTube is a self-contained webpage that gets the latest videos from YouTube channels you like. Unlike YouTube's subscription feature, SubTube doesn't require you to be logged in. You can save this webpage as a plain HTML file and use it locally.
Get Started
For each user or channel on YouTube that you want to subscribe to, add the channel name to the text area above. To figure out the channel name, go to the channel's YouTube video page and look at the URL:
If the URL contains "user", enter the channel name that follows it without any changes. For "youtube.com/user/apple/videos", the channel name is "apple".
If the URL contains "channel", enter the channel name prepended by a "#". For "youtube.com/channel/ABCDEFGHIJKLM/videos", the channel name is "#ABCDEFGHIJKLM"
Seperate each channel name with a comma and no space.


Please note that Win 98 CSS is still in development and only a few features are available. If you're still ready to use it,this guide explains how do you use it. It must be noted that the developer is not responsible for any unexpected behaviours due to other CSS frameworks. It is best advised to use Win 98 CSS Standalone. If you want to file a bug or add a feature open an issue.
To get started download the Win 98 CSS. You can download Win 98 CSS uncompressed or compressed. Include that in your HTML file. Next, Download the Win 98 JS, uncompressed or compressed and also include it in you HTML file and you're ready to go.
<!doctype html>

<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">

<link rel="stylesheet" href="win98.min.css">
<title>Hello, world!</title>

Hello, world!


--- layout: page title: about ---
The Access and Benefit-sharing Clearing-House (ABS Clearing-House, ABSCH) is a platform for exchanging information on access and benefit-sharing established by Article 14 of the Nagoya Protocol, as part of clearing-house mechanism under Article 18, paragraph 3 of the Convention. The ABS Clearing-House is a key tool for facilitating the implementation of the Nagoya Protocol by enhancing legal certainty, clarity and transparency on procedures for access and for monitoring the utilization of genetic resources along the value chain. By making relevant information regarding ABS available, the ABS Clearing-House helps users access genetic resources and associated traditional knowledge, and providers fairly and equitably share in the benefits arising from their utilization.
The primary goal of the ABS Clearing-House is to share information in order to:
- Assist users in finding information on how to access genetic resources and associated traditional knowledge; and
- Assist providers in receiving information related to the utilization of their genetic resources once they leave the provider country’s jurisdiction.
The Nagoya Protocol on Access and Benefit-sharing (ABS) was negotiated in order to provide greater legal certainty and transparency for both providers and users of genetic resources and associated traditional knowledge by:
Researchers and businesses need legal clarity, certainty and transparency when accessing genetic resources and/or associated traditional knowledge. Without such legal certainty, potential users of genetic resources and/or associate traditional knowledge could be less eager to invest in bioprospecting and researching activities, as their activities may lead to controversy and allegations of misappropriation or misuse of these resources or knowledge.
Providers of genetic resources and associated traditional knowledge, on the other hand, want to ensure that benefits arising from the utilization of their resources are shared in a fair and equitable manner once the resource or knowledge leaves the country.
The ABS Clearing-House can help everyone get what they want
What Providers want:
- Control over access to their own genetic resources (GR) and traditional knowledge (TK)
- Ensure that users comply with all conditions set in mutual agreed terms (MAT)
- Assurance that the entitled benefits arising from utilization are received in accordance with MAT
What Users want:
- Access to GR and associated TK
- A clear understanding about how to access GR and TK
- Legal certainty for utilizing GR and TK when due process has been followed
What everybody wants:
- Fair and equitable sharing of benefits
- Sustainable use of biodiversity
- Conservation of biodiversity
The Nagoya Protocol's provisions on access, benefit-sharing and compliance provide a framework that aims to address the concerns of both users and providers. However, in order to translate the Nagoya Protocol into practice, Parties need to create the necessary conditions and take the measures required by the Protocol to ensure that a national framework is in place to implement ABS at the national level and enable the development of ABS agreements for the benefit of all involved in the process. It is in this context and with this aim that Parties to the CBD included Article 14 in the Nagoya Protocol, which establishes the ABS Clearing-House.
The Nagoya Protocol establishes the ABS Clearing-House, as part of the clearing-house mechanism of the Convention, as a means for sharing information related to access and benefit-sharing, and in particular its goal is to provide access to information made available by each Party relevant to the implementation of the Protocol.
The ABS Clearing-House allows countries to share information on procedures for accessing genetic resources and monitor the utilization of the resources along the value chain. The ABS Clearing-House plays a key role in enhancing the legal certainty and transparency that both providers and users of genetic resources, as well as associated traditional knowledge, are looking for.
The ABS Clearing-House is helping to make the "ABC's" of ABS a reality
The "ABC's" of ABS:
- Access – Enhancing legal certainty, transparency and clarity on how to access genetic resources and associated traditional knowledge.
- Benefit-sharing – Contributing to increase opportunities for sharing benefits from the use of genetic resources and associated traditional knowledge.
- Compliance – Assisting to ensure compliance with ABS measures and transparency in monitoring the utilization of genetic resources through the value chain.
A fully functional ABS Clearing-House also represents a major step in achieving Aichi Biodiversity Target 16, which provides that by 2015, the Nagoya Protocol is in force and operational, consistent with national legislation.
More information:
- The ABS portal on the CBD website
- Key developments related to the ABS Clearing-House
- Relevant decisions and documents on the ABS Clearing-House
- Text of the Nagoya Protocol and courtesy translations
- How to become a Party: States and regional economic integration organizations that are Parties to the Convention on Biological Diversity are eligible to become a Party to the Nagoya Protocol. The United Nations Secretary-General acts as depository for the Nagoya Protocol through the United Nations Treaty Section in New York.
- Information on the current membership of the COP-MOP Bureau
- Nagoya Protocol COP-MOP Decisions


Click the man with the mustache to make him crawl in and out!
Click the basketball to make it go in the hoop


My research interests lie in the fields of Information Theory, Information Processing, Machine Learning, and their applications
Miguel Rodrigues is a Professor of Information Theory and Processing at University College London; he leads the Information, Inference and Machine Learning (IIML) Lab at University College London; and he also leads the MSc in Integrated Machine Learning Systems at University College London. He is also a Turing Fellow with the Alan Turing Institute — the UK National Institute of Data Science and Artificial Intelligence.
He has also previously held various appointments with various institutions worldwide including Cambridge University, Princeton University, Duke University, and the University of Porto, Portugal. He obtained the undergraduate degree in Electrical and Computer Engineering from the Faculty of Engineering of the University of Porto, Portugal and the Ph.D. degree in Electronic and Electrical Engineering from University College London.
Dr. Rodrigues’s research lies in the general areas of information theory, information processing, and machine learning. His most relevant contributions have ranged from the information-theoretic analysis and design of communications systems, information-theoretic security, information-theoretic analysis and design of sensing systems, and the information-theoretic foundations of machine learning.
Dr. Rodrigues’s work has led to (a) 200+ publications with 6000+ citations in leading journals and conferences in the field; (b) a book on “Information-Theoretic Methods in Data Science” published by Cambridge University Press; (c) 5+ keynotes in leading conferences and venues in the field; (d) various prizes and awards; (e) the supervision of 20+ postdoctoral and doctoral researchers; and (e) media coverage. His work — attracting £4.5M+ in research funding and fellowships his team alone — is supported by UKRI, Innovate UK, Royal Society, EU H2020, industry, among others.
Dr. Rodrigues serves/served as Editor for IEEE BITS – The Information Theory Magazine, IEEE Transactions on Information Theory, the IEEE Open Journal of the Communications Society, the IEEE Communications Letters, and Lead Guest Editor / Guest Editor of the Special Issues on “Information-Theoretic Methods in Data Acquisition, Analysis, and Processing” and “Deep Learning for High Dimensional Sensing” of the IEEE Journal on Selected Topics in Signal Processing.
He also serves/served as Co-Chair of the Technical Programme Committee of the IEEE Information Theory Workshop 2016, Cambridge, UK (with H. Bolcskei and AR Calderbank), and as member of the organizing / technical programme committee of various IEEE conferences in the area of information theory and processing. He is a member of the IEEE Signal Processing Society Technical Committee on “Signal Processing Theory and Methods”, the EURASIP SAT on Signal and Data Analytics for Machine Learning (SiG-DML), and the EPSRC Full Peer Review College.
Dr. Rodrigues also consults widely in the area of machine learning and artificial intelligence technologies with government institutions, funding agencies, and industry. He is also a member of the FCT-PARSUK “Scientific Council” advising Portuguese Government Institutions on research strategy and opportunities.
Dr. Rodrigues is the recipient of the Prize Eng. Antonio de Almeida, Prize Eng. Cristiano Spratley, the Merit Prize from the University of Porto, Portugal, fellowships from the Portuguese Foundation for Science and Technology as well as the Foundation Calouste Gulbenkian, and the IEEE Communications and Information Theory Societies Joint Paper Award 2011.
Abstract: Real-world data processing problems often involve various image modalities associated with a certain scene, including RGB images, infrared images or multi-spectral images. The fact that different image modalities often share certain attributes, such as edges, textures and other structure primitives, represents an opportunity to enhance various image processing tasks. This paper proposes a new approach to construct a high-resolution (HR) version of a low-resolution (LR) image given another HR image modality as guidance, based on joint sparse representations induced by coupled dictionaries. The proposed approach captures complex dependency correlations, including similarities and disparities, between different image modalities in a learned sparse feature domain in lieu of the original image domain. It consists of two phases: coupled dictionary learning phase and coupled super-resolution phase. The learning phase learns a set of dictionaries from the training dataset to couple different image modalities together in the sparse feature domain. In turn, the super-resolution phase leverages such dictionaries to construct a HR version of the LR target image with another related image modality for guidance. In the advanced version of our approach, multi-stage strategy and neighbourhood regression concept are introduced to further improve the model capacity and performance. Extensive guided image super-resolution experiments on real multimodal images demonstrate that the proposed approach admits distinctive advantages with respect to the state-of-the-art approaches, for example, overcoming the texture copying artifacts commonly resulting from inconsistency between the guidance and target images. Of particular relevance, the proposed model demonstrates much better robustness than competing deep models in a range of noisy scenarios.
LinkAbstract: Magnetic resonance (MR) imaging tasks often involve multiple contrasts, such as T1-weighted, T2-weighted and Fluid-attenuated inversion recovery (FLAIR) data. These contrasts capture information associated with the same underlying anatomy and thus exhibit similarities in either structure level or gray level. In this paper, we propose a Coupled Dictionary Learning based multi-contrast MRI reconstruction (CDLMRI) approach to leverage the dependency correlation between different contrasts for guided or joint reconstruction from their under-sampled k-space data. Our approach iterates between three stages: coupled dictionary learning, coupled sparse denoising, and enforcing k-space consistency. The first stage learns a set of dictionaries that not only are adaptive to the contrasts, but also capture correlations among multiple contrasts in a sparse transform domain. By capitalizing on the learned dictionaries, the second stage performs coupled sparse coding to remove the aliasing and noise in the corrupted contrasts. The third stage enforces consistency between the denoised contrasts and the measurements in the k-space domain. Numerical experiments, consisting of retrospective under-sampling of various MRI contrasts with a variety of sampling schemes, demonstrate that CDLMRI is capable of capturing structural dependencies between different contrasts. The learned priors indicate notable advantages in multi-contrast MR imaging and promising applications in quantitative MR imaging such as MR fingerprinting.
LinkAbstract: Quantization plays a critical role in digital signal processing systems. Quantizers are typically designed to obtain an accurate digital representation of the input signal, operating independently of the system task, and are commonly implemented using serial scalar analog-to-digital converters (ADCs). In this work, we study hardware-limited task-based quantization, where a system utilizing a serial scalar ADC is designed to provide a suitable representation in order to allow the recovery of a parameter vector underlying the input signal. We propose hardware-limited task-based quantization systems for a fixed and finite quantization resolution, and characterize their achievable distortion. We then apply the analysis to the practical setups of channel estimation and eigen-spectrum recovery from quantized measurements. Our results illustrate that properly designed hardware-limited systems can approach the optimal performance achievable with vector quantizers, and that by taking the underlying task into account, the quantization error can be made negligible with a relatively small number of bits.
LinkAbstract: Quantizers take part in nearly every digital signal processing system which operates on physical signals. They are commonly designed to accurately represent the underlying signal, regardless of the specific task to be performed on the quantized data. In systems working with high-dimensional signals, such as massive multiple-input multiple-output (MIMO) systems, it is beneficial to utilize low-resolution quantizers, due to cost, power, and memory constraints. In this work we study quantization of high-dimensional inputs, aiming at improving performance under resolution constraints by accounting for the system task in the quantizers design. We focus on the task of recovering a desired signal statistically related to the high-dimensional input, and analyze two quantization approaches: We first consider vector quantization, which is typically computationally infeasible, and characterize the optimal performance achievable with this approach. Next, we focus on practical systems which utilize hardware-limited scalar uniform analog-to-digital converters (ADCs), and design a taskbased quantizer under this model. The resulting system accounts for the task by linearly combining the observed signal into a lower dimension prior to quantization. We then apply our proposed technique to channel estimation in massive MIMO networks. Our results demonstrate that a system utilizing low-resolution scalar ADCs can approach the optimal channel estimation performance by properly accounting for the task in the system design.
LinkAbstract: This paper studies how to optimally capture side information to aid in the reconstruction of high-dimensional signals from low-dimensional random linear and noisy measurements, by assuming that both the signal of interest and the side information signal are drawn from a joint Gaussian mixture model. In particular, we derive sufficient and (occasionally) necessary conditions on the number of linear measurements for the signal reconstruction minimum mean squared error (MMSE) to approach zero in the low-noise regime; moreover, we also derive closed-form linear side information measurement designs for the reconstruction MMSE to approach zero in the low-noise regime. Our designs suggest that a linear projection kernel that optimally captures side information is such that it measures the attributes of side information that are maximally correlated with the signal of interest. A number of experiments both with synthetic and real data confirm that our theoretical results are well aligned with numerical ones. Finally, we offer a case study associated with a panchromatic sharpening (pan sharpening) application in the presence of compressive hyperspectral data that demonstrates that our proposed linear side information measurement designs can lead to reconstruction peak signal-to-noise ratio (PSNR) gains in excess of 2 dB over other approaches in this practical application.
LinkAbstract: The generalization error of deep neural networks via their classification margin is studied in this work. Our approach is based on the Jacobian matrix of a deep neural network and can be applied to networks with arbitrary non-linearities and pooling layers, and to networks with different architectures such as feed forward networks and residual networks. Our analysis leads to the conclusion that a bounded spectral norm of the network's Jacobian matrix in the neighbourhood of the training samples is crucial for a deep neural network of arbitrary depth and width to generalize well. This is a significant improvement over the current bounds in the literature, which imply that the generalization error grows with either the width or the depth of the network. Moreover, it shows that the recently proposed batch normalization and weight normalization re-parametrizations enjoy good generalization properties, and leads to a novel network regularizer based on the network's Jacobian matrix. The analysis is supported with experimental results on the MNIST, CIFAR-10, LaRED and ImageNet datasets.
LinkAbstract: We address the problem of compressed sensing (CS) with prior information: reconstruct a target CS signal with the aid of a similar signal that is known beforehand, our prior information. We integrate the additional knowledge of the similar signal into CS via l 1 -l 1 and l 1 -l 2 minimization. We then establish bounds on the number of measurements required by these problems to successfully reconstruct the original signal. Our bounds and geometrical interpretations reveal that if the prior information has good enough quality, l 1 -l 1 minimization improves the performance of CS dramatically. In contrast, l 1 -l 2 minimization has a performance very similar to classical CS, and brings no significant benefits. In addition, we use the insight provided by our bounds to design practical schemes to improve prior information. All our findings are illustrated with experimental results.
LinkAbstract: An information-theoretic projection design framework is proposed, of interest for feature design and compressive measurements. Both Gaussian and Poisson measurement models are considered. The gradient of a proposed information-theoretic metric (ITM) is derived, and a gradient-descent algorithm is applied in design; connections are made to the information bottleneck. The fundamental solution structure of such design is revealed in the case of a Gaussian measurement model and arbitrary input statistics. This new theoretical result reveals how ITM parameter settings impact the number of needed projection measurements, with this verified experimentally. The ITM achieves promising results on real data, for both signal recovery and classification.
LinkAbstract: In support of art investigation, we propose a new source separation method that unmixes a single X-ray scan acquired from double-sided paintings. In this problem, the X-ray signals to be separated have similar morphological characteristics, which brings previous source separation methods to their limits. Our solution is to use photographs taken from the front-and back-side of the panel to drive the separation process. The crux of our approach relies on the coupling of the two imaging modalities (photographs and X-rays) using a novel coupled dictionary learning framework able to capture both common and disparate features across the modalities using parsimonious representations; the common component captures features shared by the multi-modal images, whereas the innovation component captures modality-specific information. As such, our model enables the formulation of appropriately regularized convex optimization procedures that lead to the accurate separation of the X-rays. Our dictionary learning framework can be tailored both to a single- and a multi-scale framework, with the latter leading to a significant performance improvement. Moreover, to improve further on the visual quality of the separated images, we propose to train coupled dictionaries that ignore certain parts of the painting corresponding to craquelure. Experimentation on synthetic and real data - taken from digital acquisition of the Ghent Altarpiece (1432) - confirms the superiority of our method against the state-of-the-art morphological component analysis technique that uses either fixed or trained dictionaries to perform image separation.
LinkAbstract: This paper offers a characterization of fundamental limits on the classification and reconstruction of high-dimensional signals from low-dimensional features, in the presence of side information. We consider a scenario where a decoder has access both to linear features of the signal of interest and to linear features of the side information signal; while the side information may be in a compressed form, the objective is recovery or classification of the primary signal, not the side information. The signal of interest and the side information are each assumed to have (distinct) latent discrete labels; conditioned on these two labels, the signal of interest and side information are drawn from a multivariate Gaussian distribution that correlates the two. With joint probabilities on the latent labels, the overall signal-(side information) representation is defined by a Gaussian mixture model. By considering bounds to the misclassification probability associated with the recovery of the underlying signal label, and bounds to the reconstruction error associated with the recovery of the signal of interest itself, we then provide sharp sufficient and/or necessary conditions for these quantities to approach zero when the covariance matrices of the Gaussians are nearly low rank. These conditions, which are reminiscent of the well-known Slepian-Wolf and Wyner-Ziv conditions, are the function of the number of linear features extracted from signal of interest, the number of linear features extracted from the side information signal, and the geometry of these signals and their interplay. Moreover, on assuming that the signal of interest and the side information obey such an approximately low-rank model, we derive the expansions of the reconstruction error as a function of the deviation from an exactly low-rank model; such expansions also allow the identification of operational regimes, where the impact of side information on signal reconstruction is most relevant. Our framework, which offers a principled mechanism to integrate side information in high-dimensional data problems, is also tested in the context of imaging applications. In particular, we report state-of-theart results in compressive hyperspectral imaging applications, where the accompanying side information is a conventional digital photograph..
LinkAbstract: This paper studies the classification of high-dimensional Gaussian signals from low-dimensional noisy, linear measurements. In particular, it provides upper bounds (sufficient conditions) on the number of measurements required to drive the probability of misclassification to zero in the low-noise regime, both for random measurements and designed ones. Such bounds reveal two important operational regimes that are a function of the characteristics of the source: 1) when the number of classes is less than or equal to the dimension of the space spanned by signals in each class, reliable classification is possible in the low-noise regime by using a one-vs-all measurement design; 2) when the dimension of the spaces spanned by signals in each class is lower than the number of classes, reliable classification is guaranteed in the low-noise regime by using a simple random measurement design. Simulation results both with synthetic and real data show that our analysis is sharp, in the sense that it is able to gauge the number of measurements required to drive the misclassification probability to zero in the low-noise regime.
LinkAbstract: This paper considers the classification of linear subspaces with mismatched classifiers. In particular, we assume a model where one observes signals in the presence of isotropic Gaussian noise and the distribution of the signals conditioned on a given class is Gaussian with a zero mean and a low-rank covariance matrix. We also assume that the classifier knows only a mismatched version of the parameters of input distribution in lieu of the true parameters. By constructing an asymptotic low-noise expansion of an upper bound to the error probability of such a mismatched classifier, we provide sufficient conditions for reliable classification in the low-noise regime that are able to sharply predict the absence of a classification error floor. Such conditions are a function of the geometry of the true signal distribution, the geometry of the mismatched signal distributions as well as the interplay between such geometries, namely, the principal angles and the overlap between the true and the mismatched signal subspaces. Numerical results demonstrate that our conditions for reliable classification can sharply predict the behavior of a mismatched classifier both with synthetic data and in a motion segmentation and a hand-written digit classification applications.
LinkAbstract: Repurposing tools and intuitions from Shannon theory, we derive fundamental limits on the reliable classification of high-dimensional signals from low-dimensional features. We focus on the classification of linear and affine subspaces and suppose the features to be noisy linear projections. Leveraging a syntactic equivalence of discrimination between subspaces and communications over vector wireless channels, we derive asymptotic bounds on classifier performance. First, we define the classification capacity, which characterizes necessary and sufficient relationships between the signal dimension, the number of features, and the number of classes to be discriminated, as all three quantities approach infinity. Second, we define the diversitydiscrimination tradeoff, which characterizes relationships between the number of classes and the misclassification probability as the signal-to-noise ratio approaches infinity. We derive inner and outer bounds on these measures, revealing precise relationships between signal dimension and classifier performance.
LinkAbstract: We consider the characterization of the asymptotic behavior of the average minimum mean-squared error (MMSE) in scalar and vector fading coherent channels, where the receiver knows the exact fading channel state but the transmitter knows only the fading channel distribution, driven by a range of inputs. In particular, we construct expansions of the quantities that are asymptotic in the signal-to-noise ratio (snr) for coherent channels subject to Rayleigh, Ricean or Nakagami fading and driven by discrete inputs and continuous inputs. The construction of these expansions leverages the fact that the average MMSE can be seen as an ψ-transform with a kernel of monotonic argument: this offers the means to use a powerful asymptotic expansion of integrals technique-the Mellin transform method-that leads immediately to the expansions of the average MMSE and- via the I-MMSE relationship-to expansions of the average mutual information, in terms of the so called canonical MMSE of a standard additive white Gaussian noise (AWGN) channel. We conclude with applications of the results to the optimization of the constrained capacity of a bank of parallel independent coherent fading channels driven by arbitrary discrete inputs.
LinkAbstract: A generalization of Bregman divergence is developed and utilized to unify vector Poisson and Gaussian channel models, from the perspective of the gradient of mutual information. The gradient is with respect to the measurement matrix in a compressive-sensing setting, and mutual information is considered for signal recovery and classification. Existing gradient-of-mutual-information results for scalar Poisson models are recovered as special cases, as are known results for the vector Gaussian model. The Bregman-divergence generalization yields a Bregman matrix, and this matrix induces numerous matrix-valued metrics. The metrics associated with the Bregman matrix are detailed, as are its other properties. The Bregman matrix is also utilized to connect the relative entropy and mismatched minimum mean squared error. Two applications are considered: 1) compressive sensing with a Poisson measurement model and 2) compressive topic modeling for analysis of a document corpora (word-count data). In both of these settings, we use the developed theory to optimize the compressive measurement matrix, for signal recovery and classification.
LinkAbstract: This paper determines to within a single measurement the minimum number of measurements required to successfully reconstruct a signal drawn from a Gaussian mixture model in the low-noise regime. The method is to develop upper and lower bounds that are a function of the maximum dimension of the linear subspaces spanned by the Gaussian mixture components. The method not only reveals the existence or absence of a minimum mean-squared error (MMSE) error floor (phase transition) but also provides insight into the MMSE decay via multivariate generalizations of the MMSE dimension and the MMSE power offset, which are a function of the interaction between the geometrical properties of the kernel and the Gaussian mixture. These results apply not only to standard linear random Gaussian measurements but also to linear kernels that minimize the MMSE. It is shown that optimal kernels do not change the number of measurements associated with the MMSE phase transition, rather they affect the sensed power required to achieve a target MMSE in the low-noise regime. Overall, our bounds are tighter and sharper than standard bounds on the minimum number of measurements needed to recover sparse signals associated with a union of subspaces model, as they are not asymptotic in the signal dimension or signal sparsity.
LinkAbstract: This paper determines to within a single measurement the minimum number of measurements required to successfully reconstruct a signal drawn from a Gaussian mixture model in the low-noise regime. The method is to develop upper and lower bounds that are a function of the maximum dimension of the linear subspaces spanned by the Gaussian mixture components. The method not only reveals the existence or absence of a minimum mean-squared error (MMSE) error floor (phase transition) but also provides insight into the MMSE decay via multivariate generalizations of the MMSE dimension and the MMSE power offset, which are a function of the interaction between the geometrical properties of the kernel and the Gaussian mixture. These results apply not only to standard linear random Gaussian measurements but also to linear kernels that minimize the MMSE. It is shown that optimal kernels do not change the number of measurements associated with the MMSE phase transition, rather they affect the sensed power required to achieve a target MMSE in the low-noise regime. Overall, our bounds are tighter and sharper than standard bounds on the minimum number of measurements needed to recover sparse signals associated with a union of subspaces model, as they are not asymptotic in the signal dimension or signal sparsity.
LinkAbstract: This paper considers the problem of filter design with secrecy constraints, where two legitimate parties (Alice and Bob) communicate in the presence of an eavesdropper (Eve), over a Gaussian multiple-input-multiple-output (MIMO) wiretap channel. This problem involves designing, subject to a power constraint, the transmit and the receive filters which minimize the mean-squared error (MSE) between the legitimate parties whilst assuring that the eavesdropper MSE remains above a certain threshold. We consider a general MIMO Gaussian wiretap scenario, where the legitimate receiver uses a linear Zero-Forcing (ZF) filter and the eavesdropper receiver uses either a ZF or an optimal linear Wiener filter. We provide a characterization of the optimal filter designs by demonstrating the convexity of the optimization problems. We also provide generalizations of the filter designs from the scenario where the channel state is known to all the parties to the scenario where there is uncertainty in the channel state. A set of numerical results illustrates the performance of the novel filter designs, including the robustness to channel modeling errors. In particular, we assess the efficacy of the designs in guaranteeing not only a certain MSE level at the eavesdropper, but also in limiting the error probability at the eavesdropper. We also assess the impact of the filter designs on the achievable secrecy rates. The penalty induced by the fact that the eavesdropper may use the optimal non-linear receive filter rather than the optimal linear one is also explored in the paper.
LinkAbstract: In this paper, we investigate the linear precoding and power allocation policies that maximize the mutual information for general multiple-input-multiple-output (MIMO) Gaussian channels with arbitrary input distributions, by capitalizing on the relationship between mutual information and minimum mean-square error (MMSE). The optimal linear precoder satisfies a fixed-point equation as a function of the channel and the input constellation. For non-Gaussian inputs, a nondiagonal precoding matrix in general increases the information transmission rate, even for parallel noninteracting channels. Whenever precoding is precluded, the optimal power allocation policy also satisfies a fixed-point equation; we put forth a generalization of the mercury/waterfilling algorithm, previously proposed for parallel noninterfering channels, in which the mercury level accounts not only for the non-Gaussian input distributions, but also for the interference among inputs.
LinkAbstract: This paper considers the transmission of confidential data over wireless channels. Based on an information-theoretic formulation of the problem, in which two legitimates partners communicate over a quasi-static fading channel and an eavesdropper observes their transmissions through a second independent quasi-static fading channel, the important role of fading is characterized in terms of average secure communication rates and outage probability. Based on the insights from this analysis, a practical secure communication protocol is developed, which uses a four-step procedure to ensure wireless information-theoretic security: (i) common randomness via opportunistic transmission, (ii) message reconciliation, (iii) common key generation via privacy amplification, and (iv) message protection with a secret key. A reconciliation procedure based on multilevel coding and optimized low-density parity-check (LDPC) codes is introduced, which allows to achieve communication rates close to the fundamental security limits in several relevant instances. Finally, a set of metrics for assessing average secure key generation rates is established, and it is shown that the protocol is effective in secure key renewal-even in the presence of imperfect channel state information.
Link


"Simply Irresistible" begins with Sarah Michelle Gellar picking up a case of crabs.
Things go downhill from there.
Actually, that's not entirely true. Despite being almost two hours long, "Irresistible" never really goes anywhere. It just dawdles along, never becoming interesting enough to pay attention to or bad enough to walk out on.
It's a perfect example of a "wallpaper movie," the kind of film that serves as pleasant background noise while you're doing something more productive, such as making out or learning all the latest gossip. After all, how dazzling can a romantic comedy be when Cupid is a crustacean?
"Irresistible" sets up Gellar as Amanda Shelton, a restaurant owner who somehow finds the dough to dress in Todd Oldham creations, even though her business is going down the tubes. Salvation arrives in the form of Tom Bartlett (Sean Patrick Flanery), a go-getter who's just spent $4 million to open a swank new eatery on the top floor of the Henri Bendel department store in New York. Tom and Amanda are brought together when one of the aforementioned crabs crawls out of its case - with the help of some laughably obvious wires - and attaches itself to a leg of Tom's expensive pants.
A conversation with the rich and hunky Tom puts Amanda's hormones into overdrive, and when she returns to the kitchen she quickly shatters her reputation as a third-rate chef. In no time at all, she's gone from making soups that taste "like bleu cheese and dirt" to whipping up steamed cockles with grapefruit sauce.
The elegant entrees are the real stars of "Irresistible" and receive much more flattering photography than either Gellar or Flanery, both of whom often look like they've just returned from an all-night frat party. "Irresistible" strives for the kind of effortless sparkle of a "Breakfast at Tiffany's," but its weirdly harsh lighting and dull camera work would be better-suited to a documentary on heroin addicts.
That the movie has any allure whatsoever is due entirely to the comfortable chemistry Gellar and Flanery generate, particularly inthe scenes that have nothing to do with the silly story. Every so often they have a moment together that's so charming you wish the rest of the film could magically rise to the same level, much in the same way Amanda and Tom float onto the ceiling of her kitchen at one point.
Judith Roberts' screenplay is an uncertain brew of equal parts "Witches of Eastwick," "Pretty Woman" and "Big Night," although its most obvious inspiration is director Alfonso Arau's 1992 classic "Like Water For Chocolate." At least three scenes from that film are reprised almost verbatim - including the one in which the heroine's tears fall into a sauce and provoke crying from everyone who tastes the dish - with no credit given whatsoever to Laura Esquivel, who wrote both the novel "Chocolate" was based on and the screenplay. That's simply reprehensible.
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#include <AggOOSpline.h>
Public Member Functions | |
| Segment (const Segment &seg1, const Segment &seg2) | |
| Segment & | Merge (const Segment &seg2) |
| Segment | CreateMerger (const Segment &seg2) const |
| Segment & | operator= (const Segment &src) |
| REAL | Length () const |
| void | SetSegmentType (SegmentType type) |
| PointF | Position (REAL ratio) const |
| PointF | Tangent (REAL ratio) const |
Static Protected Member Functions | |
| static REAL | lengthLinear (const Segment *segment, REAL delta=0.001) |
| static REAL | lengthQuadratic (const Segment *segment, REAL delta=0.001) |
| static REAL | lengthCubic (const Segment *segment, REAL delta=0.001) |
| static PointF | positionLinear (const Segment *segment, REAL ratio) |
| static PointF | positionQuadratic (const Segment *segment, REAL ratio) |
| static PointF | positionCubic (const Segment *segment, REAL ratio) |
| static PointF | tangentLinear (const Segment *segment, REAL ratio) |
| static PointF | tangentQuadratic (const Segment *segment, REAL ratio) |
| static PointF | tangentCubic (const Segment *segment, REAL ratio) |
Constructor which creates a segment by merging two existing segments
| REAL AggOO::Spline::Segment::lengthLinear | ( | const Segment * | segment, | |
| REAL | delta = 0.001 | |||
| ) | [static, protected] |
Calculate the length of a linear segment
| REAL AggOO::Spline::Segment::lengthQuadratic | ( | const Segment * | segment, | |
| REAL | delta = 0.001 | |||
| ) | [static, protected] |
Calculate the length of a quadratic segment
| REAL AggOO::Spline::Segment::lengthCubic | ( | const Segment * | segment, | |
| REAL | delta = 0.001 | |||
| ) | [static, protected] |
Calculate the length of a cubic segment
| PointF AggOO::Spline::Segment::positionLinear | ( | const Segment * | segment, | |
| REAL | ratio | |||
| ) | [static, protected] |
Calculate the position along a linear segment
| PointF AggOO::Spline::Segment::positionQuadratic | ( | const Segment * | segment, | |
| REAL | ratio | |||
| ) | [static, protected] |
Calculate the position along a quadratic segment
| PointF AggOO::Spline::Segment::positionCubic | ( | const Segment * | segment, | |
| REAL | ratio | |||
| ) | [static, protected] |
Calculate the position along a cubic segment
| PointF AggOO::Spline::Segment::tangentLinear | ( | const Segment * | segment, | |
| REAL | ratio | |||
| ) | [static, protected] |
Calculate the tangent along a linear segment
| PointF AggOO::Spline::Segment::tangentQuadratic | ( | const Segment * | segment, | |
| REAL | ratio | |||
| ) | [static, protected] |
Calculate the tangent along a quadratic segment
| PointF AggOO::Spline::Segment::tangentCubic | ( | const Segment * | segment, | |
| REAL | ratio | |||
| ) | [static, protected] |
Calculate the tangent along a cubic segment
| Spline::Segment & AggOO::Spline::Segment::Merge | ( | const Segment & | seg2 | ) |
Merge this segment with another segment, storing the merger in this segment.
| Spline::Segment AggOO::Spline::Segment::CreateMerger | ( | const Segment & | seg2 | ) | const |
Merge this segment with another segment. The returned segment is a new segment and the originals are unaltered.
| Spline::Segment & AggOO::Spline::Segment::operator= | ( | const Segment & | src | ) |
Assignment operator
| REAL AggOO::Spline::Segment::Length | ( | ) | const [inline] |
Get the length of the segment
| void AggOO::Spline::Segment::SetSegmentType | ( | SegmentType | type | ) |
Set the segment type
| PointF AggOO::Spline::Segment::Position | ( | REAL | ratio | ) | const |
Get the position along the segment at a given ratio
| PointF AggOO::Spline::Segment::Tangent | ( | REAL | ratio | ) | const |
Get the tangent along the segment at a given ratio


--- title: Common Lisp Quick Reference layout: post tags: [lisp, link, reddit] uuid: d74ba1a7-25bb-37f4-7b2e-46d2d56bfe36 ---
I found this Common Lisp Quick Reference the other day from r/lisp, and I think it's fantastic. It's a comprehensive, libre booklet of the symbols defined by the Common Lisp ANSI standard. Very slick!
The main version is meant to be printed out and nested with a vertical fold, and it works quite well. If I ever get a chance to use Common Lisp at work (a man can dream), probably at a location without Internet access, this could come in handy. So I printed out one for myself,


Black Lives Matter is a movement started in response to the acquittal of George Zimmerman in the shooting of African-American teen Trayvon Martin. All throughout history, Black people have been supressed and treated as less than equal. THAT IS NOT OK. Black lives are absolutely lives and they deserve to be treated as such. It is up to every one of us to be the force of change and to say that enough is enough! Black Americans are more than twice as likely than white Americans to be killed by excessive use of police force (31 fatal shootings per million for black Americans and 13 fatal shootings per million for white Americans). We urge you to be an agent of change and always remember to be empathetic towards others.
Black Lives Matter Die-In protest in St. Paul, Minnesota. Photo courtesy of Fibonacci Blue. This is licensed under a Creative Commons Attribution License.
ACLU - The American Civil Liberties Union's website that highlights what's at stake, the current issues, the latest information, and how you can act.


A simple calendar app for scheduling your work day.
This app is designed to provide a centralized resource hub for bootcamps students while better facilitating more direct communication and scheduling between Bootcamp tutors and students.
This is an online community for video game enthusiasts.
This is a Star Wars series fanpage and information database with a unique user experience.


Scenes¶
Intro¶
There is nothing much. There are two sprites, cursor and background. When window size changes, background streches, cursors’ position is relative to window’s size so it does not change.
The game has 2 themes, so there is a change theme option.
Pre-Stage¶
This is shown to user everytime before the stage, it shows current level, score and health. All components are relative to window size.
Stage¶
This is the only complex scene in game. It is the game session. It contains two main component sets, map and HUD. HUD is simple, on left there is a bar says how much diamond you need to collect to open the door, on right there are score bar and time bar shows you how much time left, on middle of the top there is a pause button and thats it. The map part shows a part of the all map near to player, every block takes 30px30p space, this is not changing, block sizes are fixed, when you change window size, it only effects how much block is shown on screen. Everytime a stage starts, it initializates audio device and sound and closing them when stage ends.
End¶
It says gameover and shows your final score. If you passed all levels it’s green, if you failed it’s red and thats it.


Taeyeon singing 'Speechless' at the fan meeting has fans anticipate her official cover
by olmalGirls' Generation's Taeyeon singing Naomi Scott's 'Speechless' at the fan meeting has fans excited about the official cover of the song.
At the event, Taeyeon said she thought fans would love to listen to her version of 'Speechless' so she practiced it for a couple of days. She has shared she would have liked more time practicing the song but she is singing it now for the fans attending the event. Now, this short bit has fans anticipate her beautiful rendition of this famous song.
Check out the clip below and share your thoughts in the comments! Would you also like to see her official cover?


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