11

I have to admit that i haven't seen that specific formula very often, but my guess would be that in case of more than one DOF, you would evaluate it for every joint in every column and then (perhaps?) multiply those results in each column. But let me suggest a simpler apporach to Jacobians in the context of arbitrary many DOFs: Basically, the Jacobian tells ...


11

Back in the day, when I was learning, making this up as I went along, I used simple gradient following to solve the IK problem. In your model, you try rotating each joint each joint a tiny amount, see how much difference that makes to the end point position error. Having done that, you then rotate each joint by an amount proportional to the benefit it gives....


9

From what I understand of your question, you'd like to know if inverse kinematics and reinforcement learning are trying to solve the same problem in the particular case of robotic manipulation. Of course both of these techniques can be applied outside of this particular realm, but let's focus on robot manipulation for now. You're right that inverse ...


9

First, we need to define optimal. Since you do not say what you consider optimal, most people choose a quadratic expression. For example, suppose your current joint angles are given by the vector $\vec{\alpha}$. We can consider minimizing the movement required - with an error $\vec{x} = \vec{\alpha} - \vec{\alpha}_{start}$, you can define a cost function $J=\...


8

Forward kinematics uses joint angles (with known link lengths) to compute the tool position and orientation. Inverse kinematics uses tool position and orientation, to compute joint angles. Note: if your device has prismatic links (the length changes) then those are used just like joint angles in the above. To compute the Jacobian, start with the forward ...


8

Let me give you a mathematician's perspective on the difference between the two kinds problems. Forward kinematics asks the question: given a certain input (i.e. control command), what will be the output (i.e. robot configuration, pose, etc.). Inverse kinematics asks the reverse question: given a certain desired output, what is the necessary input. ...


7

Disclaimer! I will try to solve your problem but it may or may not solve the problem with your code! TL;DR: Two possible mistake in your code. In your pseudo-code, you are using transpose of jacobian instead of pseudo-inverse of jacobian (as suggested in your referenced slide). The other possible mistake is in the calculation of cross product in your ...


7

Yes. As @hauptmech mentioned, you can use your forward kinematics to get the center of mass of each link in the base frame. Then you can simply compute the weighted average of the masses and positions to get the overall center of mass. In other words: $$ M = \sum_{i=0}^n m_i $$ $$ \mathbf{P}_i^0 = pos( \mathbf{T}_i^0(\mathbf{q}) \mathbf{T}_{i_m}^i) $$ $...


7

I think that the main issue is that you're trying to read your rotation matrices from left to right. The sign changes seem random, but actually cycle in an ordinary way. Below follows a more elaborate explanation with some more background information. Let's start with the question: A) Why do minuses appear at all? Followed by: B) Why do minuses appear ...


6

Yes, the Jacobian relates the joint velocities to end-effector velocity through this equation: $$ \mathbf{v}_e = \mathbf{J}(\mathbf{q}) \dot{\mathbf{q}} $$ Where $\mathbf{q}$ is the joint angles, $\dot{\mathbf{q}}$ is the joint velocities, and $\mathbf{v}_e$ is the end-effector velocity. As you can see, the Jacobian, $\mathbf{J}$, is configuration ...


6

I don't know if there is a formal proof to this, but in general, no the set of all possible joint configurations that correspond to a particular end-effector pose is not continuous. I think of the set as islands in joint space. Where each island has some local continuous joint range, but is disconnected from the other islands. I think there are a few ...


6

I would recommend changing the naming convention since it is a bit misleading. In robotics the world Coordinate system (CS) is usually your fixed, absolute coordinate system. Lets call the transformation matrix from your camera to your object $T_{Object,Tool}$ If it cannot include any rotation, then you are right is should have the form as you specified. You ...


6

Writing the equations by hand and deriving them is certainly the best way to understand what is happening "in the background". Generating the equations and deriving them using a syombolics engine, like @SteveO suggested is essentially the same process but someone else, in this case a symbolic engine, is doing the work for you. There are however different ...


6

I agree with SteveO that there is nothing wrong with reinventing the wheel if you want to learn about wheels. And for a single application, 4 DoF arm, the IK is probably not too hard. But I feel like I should mention that most of the kinematics libraries out there are mostly targeted towards Linux. And as such, probably not too hard to compile from ...


6

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 ...


5

Not by merely looking at Jacobian but by looking at the Singular Value Decomposition of the Jacobian, one can see the degrees of freedom that are lost, if lost. Of course it technically somehow turns up to finding the null space but yet I guess it is somewhat familiar and easier. For example let the Jacobian be: $$J = \begin{bmatrix} -50 &...


5

Kawato M. Developped the technique called the Minimum Commanded Torque and Minimum Commanded Torque Change to optimize joint torques (see article from Kaneko Y., Kawato M. and al., 2005). A similar approach in the kinematics domain is the Minimum Jerk (MJ) model. It have been proven to be the domain in which humans plan movements, see article from Flash T. ...


5

It is rather straightforward to implement inverse kinematics for a particular manipulator in C++. Of course, you need to begin with the inverse kinematic equations themselves. Putting those into code will only involve a few trigonometric functions such as acos, asin, and atan2 (use atan2 instead of atan), and probably a couple of square and square root ...


5

It depends on the method that you use for computing an IK solution. If you have an analytic formula for IK solutions then you do not need the current joint values of the robot. You just plug in the translation and rotation of the end-effector to the formula and you get a solution (or a set of solutions). (Sometimes knowing the current joint angles is better, ...


5

The formulation is typical for redundant robots, in which there are an infinite number of joint velocity vectors that could satisfy the $\dot{r}_{t}$ goal. In the version you cite, the $Q$ matrix would allow you to weight the different joint velocities in order to create an optimal solution that matches that $Q$. Other formulations of this approach ...


5

You will find it helpful to keep the physical robot and the math separate. The kinematics equations map joint parameters (which are often grouped as a vector $q$) to Cartesian coordinates ($x$,$y$,$z$). If you take the first-order partial derivative of the kinematics equations, you get a set of equations that map a change in the joint parameters to a ...


4

For these sorts of problems, I always like to attempt to solve them myself. And they're surprisingly simple when you think about them geometrically. Parallel mechanisms are especially easy because each part of the mechanism can be considered separately, and each part is very simple. The base part (at the top of the image) contains three servo motors. Think ...


4

You are tackling two non trivial problem at the same time 1. Inverse kinematics of an overactuated manipulator 2. Obstacle avoidance using the null space By definition of the null-space projection the solution you want will only be able to avoid obstacles which are not on the desired Cartesian trajectory to be followed during the task. Think about sliding ...


4

Nowadays we no longer employ exact solutions for the IK problem, simply because the number of degrees of freedom so as the number of constraints the final configuration needs to comply with make the so called geometric approach intractable. By contrast, iterative methods are used to converge to the most plausible solution. Therefore, the reachability is ...


4

The analytic inverse kinematics solutions you found do depend on those $0$ terms in your transformation matrices. Those values are, as you've implied, based on the $0$ and $90$ degree values for the axes relationships. The kinematic mapping process works for other $a_i$ and $d_i$ values, too, but it becomes difficult to find closed-form inverse solutions ...


4

To compare two different arm configurations, Euclidean distance in joint space is usually sufficient. $$ d = \sqrt{\sum_{i=0}^n(q_i - q_{i_{ref}})^2} $$ Where $q_i$ is joint $i$ of the test configuration, and $q_{i_{ref}}$ is the same joint in the reference configuration. This will work even if the end-effector has a different pose. But its usefulness ...


4

There are more approaches to solve the inverse kinematics equations. If you want to continue to use the pseudo-inverse based approach and still obtain more then 1 solution you can flip the sign of joint angle 1 for example (if it is a puma type robot arm) in the initial guess and run the iterative solver again. Same for joint angle 3. You can have ...


4

I suggest you look at Craig's book Introduction to Robotics Mechanics and Control. In the inverse kinematics chapter he addresses the multiple closed-form solutions obtained analytically. Many other texts address this issue also. In general, if the wrist is spherical (i.e., all three axes intersect), you can enumerate all of the various closed-form ...


4

You will need to provide target coordinates (x,y,z, etc) to your inverse kinematics equations. As long as they have the same scale as your DH parameters, the joint angles calculated will be the same. But since you don't know the scale factor, you don't know how to scale your inputs.


4

The best way to understand forward and inverse kinematics is to write a library for its own. A good starting point is to implement Cyclic Coordinate Descent. So called "ready-to-run" librarys like OpenRave or OMPL have an extensive C++ tutorial section too but are notorius difficult to install. Most of them only run under linux, and are not even part of ...


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