9 votes
Accepted

Understanding the Robot Jacobian

Let's start from the forward kinematics equation $$x = f(q),$$ where $x \in \mathbf{R}^6$ is the end-effector position, $q$ is the joint angles, and $f$ is a (usually highly nonlinear) forward ...
user avatar
8 votes
Accepted

Forward kinematic and inverse kinematic... When to use what?

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 ...
user avatar
  • 1,238
8 votes

Forward kinematic and inverse kinematic... When to use what?

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 ...
user avatar
  • 4,326
7 votes
Accepted

Can a Jacobian be used to determine required joint angles for end effector velocity/position?

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, $\...
user avatar
  • 5,215
7 votes
Accepted

Cartesian Velocity Control between Two 3D Poses

You essentially want to find the time derivative of a linear interpolation between two rotations. The easiest way to obtain this would probably to convert the rotation matrix between the two ...
user avatar
  • 921
6 votes

Computing the Jacobian Matrix -- chain rule?

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, ...
user avatar
  • 6,322
6 votes
Accepted

D(q) Inertia Matrix and the Jacobian Matrix

I think this is a matter of notations. In the given formula for $D(q)$, the matrices $J_{vi}$ and $J_{\omega i}$ are not simply the direct extraction of columns of the Jacobian of the system. $J_i$ ...
user avatar
6 votes

Relationship between the velocity twist Jacobian and the spatial velocity Jacobian

There are a lot of definitional problems and inconsistencies in this area. Geometric Jacobian. I'm not sure this has a precise and agreed upon meaning. But across the more classical robotics books (...
user avatar
  • 1,522
6 votes
Accepted

Why with the pseudo-inverse it is possible to invert the Jacobian matrix even in a singular configuration?

The pseudoinverse gives a “least squared error, minimum-norm” solution: Out of all $\dot{q}$ vectors at your current $q$, the vector $$\dot{q}_{s} = J^{+}(q)\dot{p}_{\text{in}}$$ satisfies two ...
user avatar
  • 609
5 votes
Accepted

Jacobian-based trajectory following

The math required to deal with the orientation error (in terms of error with respect to both desired position and desired velocity) is well described in the book of Sciavicco-Siciliano. See section 3....
user avatar
5 votes
Accepted

How can serial manipulator have unique condition number for given end effector position?

Condition number and manipulability are measured at a specific joint configuration, not end-effector location. You already understand it correctly that the values change according to the robot ...
user avatar
5 votes
Accepted

Confusion about Jacobians stemming from class notes

Short answer Robot Dynamics and Control by Spong et al. (especially Chapter 5) can definitely help you on this matter. Long answer First of all, you are partially correct about a Jacobian. It is ...
user avatar
5 votes

Screw based Jacobian

The geometric Jacobian provides all the information you need for singularity or manipulability analysis. Linearly dependent columns correspond to joints with parallel axes. More information about ...
user avatar
  • 1,522
5 votes

Jacobian of a 6DOF arm

Write the forward kinematic equations $$\vec(x) = F\vec(\theta)$$ Taking the partial derivatives of each $\vec (x)$ term with respect to each joint variable $\vec(\theta)$ will give you $J$.
user avatar
  • 4,326
5 votes
Accepted

How do I compute the derivative of the Jacobian with Matlab?

To answer your solution, specifically, all you need to do is: ...
user avatar
  • 324
4 votes
Accepted

Robot arm reachability of a pose in Cartesian space

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 ...
user avatar
4 votes
Accepted

Robotic manipulator Jacobian by product of exponentials

The Jacobian in that equation is from the joint velocity to the "spatial velocity" of the end effector. The spatial velocity of an object is a somewhat unintuitive concept: it is the velocity of a ...
user avatar
  • 609
4 votes

Solving Inverse Kinematics with Gradient Descent

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 ...
user avatar
  • 6,322
4 votes
Accepted

Velocity Relation for Parallel Robots

If you can write the forward kinematics equations of a parallel robot in an explicit form, you can derivate those equations and you get the formula for the velocities. This is generally valid ...
user avatar
  • 6,322
4 votes
Accepted

Integrating Forward Kinematics Map

(EDITED TO CLARIFY PARENTHETICAL ABOUT CARTESIAN MANIPULATORS) Your equation is true in general only for those manipulators in which $J_a$ is independent of $\theta$ (such as with Cartesian ...
user avatar
  • 4,326
3 votes
Accepted

Solving Inverse Kinematics with Gradient Descent

You have to analytically compute all IK solutions. This is basically done with straightforward geometry. Most robotics textbooks with a section on manipulation will have a detailed explanation of ...
user avatar
  • 5,215
3 votes

How to numerically calculate the Jacobian?

Well, although, I didn't well understand what have wrote as a solution to the first part (upper) half of the Jacobian, but AFAIK, the manipulator Jacobian is a $6\times n$ matrix, for that let's say $ ...
user avatar
  • 211
3 votes

Jacobian-based trajectory following

Ugo's answer refers to "Sciavicco-Siciliano" which is a good book I'll quote as well. Chapter 3.6 introduces the so-called analytical Jacobian which is not the same as the so called geometrical ...
user avatar
  • 41
3 votes
Accepted

How to get Max Torque on Robot arm 's Joints (RRR)

Torque is pretty easy to calculate for a single static arm configuration. Torque is just the length of the moment arm * the perpendicular force. And it is easy to decompose the problem into X and Y ...
user avatar
  • 5,215
3 votes

Jacobian of a Robot

Yes, 6x9. Since $$\dot{x} = J \dot{\theta}$$ each column of the Jacobian represents the differential change in one of the six $x$ coordinates with respect to each of the joints.
user avatar
  • 4,326
3 votes

The Jacobian resulted from Screw method is different from analytical one (Example Inside)

You're computing the spatial Jacobian, which relates joint velocities to spatial velocities at the origin. You instead want to compute the body Jacobian, which relates joint velocities to end-effector ...
user avatar
3 votes

How to use Jacobian Method for path following?

You can use some Trajectory Generators to create intermediate points between the points calculated by RRT Algorithm. You can use Moden Robotics Library by Neuroscience and Robotics Lab at Northwestern ...
user avatar
  • 306
2 votes
Accepted

How to get pure end-effector translation through Jacobian?

You don't need the positional IK to solve this problem. All that is required is, that your Jacobian is invertible, i.e. keep away from sinuglar joint configurations. Consider, how the end effector's ...
user avatar
2 votes

Jacobian transpose: How to calculate orientation error?

The math involved in determining orientation error is described in this thread: Jacobian-based trajectory following
user avatar
  • 5,215
2 votes

Can a Jacobian matrix be used to derive joint angles from end-effector linear and rotational velocity (without a filter)?

Joint angles can be determined by looking at the gravity vector of each accelerometer. The paper Low-cost Accelerometers for Robotic Manipulator Perception by Morgan Quigley, Reuben Brewer, Sai P. ...
user avatar
  • 5,215

Only top scored, non community-wiki answers of a minimum length are eligible