Recently I'm reading 《Morion Control of Rendundant Robots under Joint Constraints:Saturation in the Null Space》 and have a few questions.
1.In the end of Section 2, the author mentioned that they can obtain the largest possible reduncing factor s=0.9091 with their algorithm. I don't understand why they can get this result by using a null-space velocity vector Qndot=[-0.4913 0.8537 -0.6093 -0.9007].
2.The pseudocode of SNS algorithm at the velocity level, how to choose or calculate an appropriate null-space velocity vector? What's the difference between this vector and several objective functions?
I'd like to appreciate so much if someone can help me to get through this part. It confused me a lot.
Actually, I think that one of the situation we have to use SNS algorithm is that when the robotic arm is close to the singular position, because singularity of Jacobian matrix may cause large velocity and over constraints. Is this thoughts correct?
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I have a time-dependent complex matrix A(t), and I want to follow its eigenvalues over time. In other words, in the time-dependent list of eigenvalues a[1](t), ..., a[n](t), I want each entry to change continuously over time.
One approach is to find the eigendecomposition of A(t+ε) iteratively, using the eigendecomposition of A(t) as an initial guess. Since the guess is almost correct, the iteration should only change it slightly, giving the desired continuity.
I think the LOBPCG and SVD solvers in IterativeSolvers.jl can do this, because they let you store the iterator state. Unfortunately, they only work for matrices with real eigenvalues. (The SVG solver also requires real entries.) The solvers in ArnoldiMethod.jl can handle complex eigenvalues, but doesn't seem to allow an initial guess. Is there any available eigensolver that has both the features I need?
I'm watching a lecture about estimating the fundamental matrix for use in stereo vision using the 8 point algorithm. I understand that once we recover the fundamental matrix between two cameras we can compute the epipolar line on one camera given a point on the other. To my understanding this epipolar line (after it's been rectified) makes it easy to find feature correspondences, because we are simply matching features along a 1D line.
The confusion comes from the fact that 8-point algorithm itself requires at least 8 feature correspondences to estimate the Fundamental Matrix.
So, we are finding point correspondences to recover a matrix that is used to find point correspondences?
This seems like a chicken-egg paradox so I guess I'm misunderstanding something.
The fundamental matrix can be precomputed. This leads to two advantages:
You can use a nice environment in which features can be matched easily (like using a chessboard) to compute the fundamental matrix.
You can use more computationally expensive operations like a sequence of SIFT, FLANN and RANSAC across the entire image since you only need to do that once.
After getting the fundamental matrix, you can find correspondences in a noisy environment more efficiently than using the same method when you compute the fundamental matrix.
I have a problem that I have expressed as the minimization of a convex quadratic program with linear constraints. The problem is that I want to disallow any point that is strictly interior (i.e. I only find the answer useful if it is on a vertex of the feasible region.
I'd like to do this without modifying the objective function. I have already considered several modifications that would make this a non-issue, but they all have the unfortunate result of making the program non-convex.
By my estimation my only option for an efficient solution would be a solver that uses a penalty method to approach a solution from the outside of the feasible region. Does anyone know a decent solver for this?
My current objective function is a sum of parabolic cylinders.
Can you just find the vertices of the feasible region and then take the one which minimizes the objective function? This should just involve a bit of linear algebra and then a limited number of evaluations of the objective function.
I'm trying to solve a 6th-order nonlinear PDE (1D) with fixed boundary values (extended Fisher-Kolmogorov - EFK). After failing with FTCS, next attempt is MoL (either central in space or FEM) using e.g. LSODES.
How can this be implemented? Using Python/C + OpenMP so far, but need some pointers
to do this efficiently.
EFK with additional 6th order term:
u_t = d u_6x - g u_4x + u_xx + u-u^3
where d, g are real coefficients.
u(x,0) = exp(-x^2/16),
ux = 0 on boundary
domain is [0,300] and dx << 1 since i'm looking for pattern formations (subject to the values
of d, g)
I hope this is sufficient information.
All PDE solutions like this will ultimately end up being expressed using linear algebra in your program, so the trick is to figure out how to get the PDE into that form before you start coding.
Finite element methods usually begin with a weighted residual method. Non-linear equations will require a linear approximation and iterative methods like Newton-Raphson. I would recommend that you start there.
Yours is a transient solution, so you'll have to do time stepping. You can either use an explicit method and live with the small time steps that stability limits will demand or an implicit method, which will force you to do a matrix inversion at each step.
I'd do a Fourier analysis first of the linear piece to get an idea of the stability requirements.
The only term in that equation that makes it non-linear is the last one: -u^3. Have you tried starting by leaving that term off and solving the linear equation that remains?
UPDATE: Some additional thoughts prompted by comments:
I understand how important the u^3 term is. Diffusion is a 2nd order derivative w.r.t. space, so I wouldn't be so certain that a 6th order equation will follow suit. My experience with PDEs comes from branches of physics that don't have 6th order equations, so I honestly don't know what the solution might look like. I'd solve the linear problem first to get a feel for it.
As for stability and explicit methods, it's dogma that the stability limits placed on time step size makes them likely to fail, but the probability isn't 1.0. I think map reduce and cloud computing might make an explicit solution more viable than it was even 10-20 years ago. Explicit dynamics have become a mainstream way to solve difficult statics problems, because they don't require a matrix inversion.
If I have a system of a springs, not one, but for example 3 degree of freedom system of the springs connected in some with each other. I can make a system of differential equations for but it is impossible to solve it in a general way. The question is, are there any papers or methods for filtering such a complex oscilliations, in order to get rid of the oscilliations and get a real signal as much as possible? For example if I connect 3 springs in some way, and push them to start the vibrations, or put some weight on them, and then take the vibrations from each spring, are there any filtering methods to make it easy to determine the weight (in case if some mass is put above) of each mass? I am interested in filtering complex spring like systems.
Three springs, six degrees of freedom? This is a trivial solution using finite element methods and numerical integration. It's a system of six coupled ODEs. You can apply any form of numerical integration, such as 5th order Runge-Kutta.
I'd recommend doing an eigenvalue analysis of the system first to find out something about its frequency characteristics and normal modes. I'd also do an FFT of the dynamic forces you apply to the system. You don't mention any damping, so if you happen to excite your system at a natural frequency that's close to a resonance you might have some interesting behavior.
If the dynamic equation has this general form (sorry, I don't have LaTeX here to make it look nice):
Ma + Kx = F
where M is the mass matrix (diagonal), a is the acceleration (2nd derivative of displacements w.r.t. time), K is the stiffness matrix, and F is the forcing function.
If you're saying you know the response, you'll have to pre-multiply by the transpose of the response function and try to solve for M. It's diagonal, so you have a shot at it.
Are you connecting the springs in such a way that the behavior of the system is approximately linear? (e.g. at least as close to linear as are musical instrument springs/strings?) Is this behavior consistant over time? (e.g. the springs don't melt or break.) If so, LTI (linear time invariant) systems theory might be applicable. Given enough measurements versus the numbers of degrees of freedom in the LTI system, one might be able to estimate a pole-zero plot of the system response, and go from there. Or something like a linear predictor might be useful.
Actually it is possible to solve the resulting system of differential equations as long as you know the masses, etc.
The standard approach is to use a Laplace Transform. In particular you start with a set of linear differential equations. Add variables until you have a set of first order linear differential equations. (So if you have y'' in your equation, you'd add the equation z = y' and replace y'' with z'.) Rewrite this in the form:
v' = Av + w
where v is a vector of variable, A is a matrix, and w is a scalar vector. (An example of something that winds up in w is gravity.)
Now apply a Laplace transform to get
s L(v) - v(0) = AL(v) + s w
Solve it to get
L(v) = inv(A - I s)(s w + v(0))
where inv inverts a matrix and I is the identity matrix. Apply the inverse Laplace transform (if you read up on Laplace transforms you can find tables of inverse of common types of functions - getting a complete list of the functions you actually encounter shouldn't be that hard), and you have your solution. (Be warned, these computations quickly get very complex.)
Now you have the ability to take a particular setup and solve for the future behavior. You also have the ability to (if you do things really carefully) figure out how the model responds to a small perturbation in parameters. But your problem is that you don't know the parameters to use. However you do have the ability to measure the positions in the system at repeated times.
If you put this together, what you can do is this. Measure your position at a number of points. First estimate all of the initial values of the parameters, and then all of the values a second later. You can adjust your parameters (using Newton's method) to come close enough to the values a second later. Take the measurements from 5 seconds later and use that initial estimate as your starting point to refine your calculations for what is happening 5 seconds later. Repeat with longer intervals to get all of your answers.
Writing and debugging this should take you some time. :-) I would strongly recommend investigating how much of this Mathematica knows how to do for you already...