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First of all, singularities are not configurations that have the same end-effector position and orientation. Those configurations are inverse kinematic (IK) solutions to that end-effector pose (position and orientation). The formal definition of singularities is the configurations that the Jacobian loses its rank. At such configurations, the manipulator may ...

Singular configurations are configurations at which the Jacobian is rank-deficient. In this case $J$ is a square matrix, you can find conditions for singularity by solving $\det(J) = 0$. The last row of $J$ being all ones means that no matter the configuration, you can always generate some angular velocity. This actually implies that the conditions you get ...

Due to singularity in matrix (i.e. the condition of matrix during which determinant of matrix becomes zero), matrix cannot be inverted. So you can possibly try pseudo-inverse. It gives most approximated form of inverse matrix. For more information visit this link: https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse

Think of those "terrible to look at" terms as diamonds in the rough. Yes, they look complicated at first. But you will find an amazing number of them will combine using basic trigonometric identities. For example, look for repeating patterns of squares of the sines and cosines of the various angles, and use those to factor out terms and combine them to ...

I think you need to step back a bit and think beyond the math. An (E)KF is used to estimate the true value of a signal in the presence of noise; it's only because of this noise that we even need the algorithm. When you set R to zero you are saying "I have a perfect measurement". In this case there is no need for an estimate. In practice, I think you have ...

Quite easily, by applying the definition of the Null Space straight away, you have to solve for: $$ \mathbf{J} \left( q_1,q_2=0\right) \cdot \left[\dot{q_1}, \dot{q_2} \right]^T = 0. $$ You'll come up with the following relation: $$ \frac{\dot{q_1}}{\dot{q_2}} = - \frac{l_2}{l_1+l_2}, $$ which in turn can be summarized by $\mathcal{N} \left( \mathbf{J} \...

As others have already pointed out, there must be an issue with your implementation of the IK algorithm since there shall not be any singular behavior in the descriptions you gave. Now you have two alternatives: either you start off debugging the code or you might want to exploit the fact that the problem can be easily broken down in two subproblems for ...

One possible way to use the inverse kinematics equations. For obtaining the inverse kinematics equations you can either use the Jacobian matrix or the geometric approach. For a robotic arm that has 3 DOF using the geometric approach is easier. But since your robot has 7 DOF, you should use the Jacobian Matrix. Please refer to the link below. Computing the ...

If $\Omega$ is singular, you cannot avoid a matrix singularity. This is like saying you want to solve the system of equations $$x = 1$$ $$x = 7$$ There are many techniques to avoid robotic singularities, but if the math represents a pose whereby a degree of freedom is lost in the mechanism, there is no way to get around a singular (and therefore non-...

If $J$ is not square, solve $$|J^TJ| = 0$$ for $\theta_i$.

According to Pai and Leu, 1989, for a generic manipulator (which includes all 6R robots with a spherical wrist), the set of singular points of rank r are smooth manifolds in joint space of codimension (j - r)(k - r), where j is the dimension of joint space and k is the dimension of task space.