6

A coordinate transformation of a point P from Frame 1 to Frame 0 is given by: $$ \mathbf{p}^0=\mathbf{o}^0_1+\mathbf{R}^0_1\mathbf{p}^1. $$ Differentiating with respect to time gives: $$ \dot{\mathbf{p}}^0=\dot{\mathbf{o}}^0_1+\mathbf{R}^0_1\dot{\mathbf{p}}^1+\dot{\mathbf{R}}^0_1\mathbf{p}^1. $$ Considering that $\dot{\mathbf{p}}^1=0$ as $\mathbf{p}^1$ is ...


5

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 Jacobians for under-actuated manipulators (as is your case) can be found in my book Robotics, Vision & Control" section 8.4.1. For information about ...


5

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 (Siciliano etal., Spong etal., Corke) it relates joint velocities to end-effector velocity (translational and rotational) expressed in either the world or end-...


4

About why screw axes: According to Kevin Lynch in his video of Twists, "just like the time-derivative of a rotation matrix is not equivalent to the angular velocity, the time-derivative of a transformation matrix is not equivalent to the rigid-body velocity" (linear and angular). Also he mentions that, instead, "any rigid-body velocity is ...


4

The author expects a background that includes a course in physics or mechanics where this equation is taught. When that is the case, this equation gives you instantaneous velocity of a particle (point) moving on a circular path. The $\times$ in $\dot{p} = \omega \times p$ is the cross product. (This may already be obvious to you, it's hard to tell from the ...


4

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 frame rigidly attached to the end effector but currently coincident with the origin frame. It may help to think of the rigid body as extending to cover the whole ...


4

Actually, the 6x1 vector is sometimes better referred to as the coordinates of the twist. The twist itself is a 4x4 matrix, element of $SE(3)$, found by $$ \begin{align} A &= \begin{bmatrix} \widehat{\omega} & v \\ 0 & 0 \end{bmatrix} \\ v &\triangleq -\omega \times q \end{align} $$ where $\omega$ is the unit vector pointing along the axis of ...


4

I will try to make it as simple as possible. Imagine you have a SCREW, when you WRENCH it, it TWIST forward or backward. From your wiki link The components of the screw define the Plücker coordinates of a line in space and the magnitudes of the vector along the line and moment about this line. It means that any system can be described as those ...


3

The body velocity $V_{b}$ is the velocity of the frame with respect to the world, as seen from the frame's perspective. Its rotational component $\omega_{b}$ contains the rotation rates around the world-fixed axes instantaneously pointing in the frame's forward, lateral, and dorsal directions (local $x$, $y$, and $z$), and its translational component $v_{b}$ ...


3

Lynch and Park's Modern Robotics book uses the product of exponentials formula and screw axes to describe manipulators, and they have a well-documented library available in Python, MATLAB, and Mathematica. Plus there is a community-released C++ port using CMake/Eigen. Book is available on this site (for free): http://modernrobotics.org/ Original library is ...


3

I will try not to skip too many steps. Assuming a Global coordinate frame at the base and the arm is fully extended along the Y-axis of the base frame. Since SCARA has four joints, we will create four 6D spatial vectors (screws) ${ξ}_{i}$ with respect to a global coordinate frame. Keep in mind that all spatial vectors are described with respect to the ...


3

Hint: First, write the transformation matrix as $$ T = \begin{bmatrix} R &p\\0_{1\times3} &1 \end{bmatrix}. $$ Now we use the relations $\omega_a = R\omega_b$ and $q_a = Rq_b + p$. Then since $v_a = -\omega_a \times q_a$, we can derive $v_a$ in terms of all the known quantities. In fact, two twists representing the same screw motion described in ...


3

Actuators Forces Do I get this right: you have a theoretical model of a rigid multibody system and would like to perform rigid body dynamics computations. You have implemented the model and now would like to compute how the model behaves when driven by an actuator. However what is an actuator for you? Is it simply a force acting at that joint? Is it a DC ...


2

If you haven't come across the Rigid Body Dynamics Library (RBDL) you might want to look at how they implement it, and/or contact the author Martin Felis.


2

Worked example $\hspace{2.5em}$ $\vec{q}$ = $[q_{1}\hspace{1em}q_{2}]^{T}$ $\hspace{1.5em}$ [Generalized coordinate] $\hspace{2.5em}$ $\vec{J}$ = $\frac{\partial \vec{r}_{OA}(\vec{q})}{\partial\vec{q}}$ = $\begin{bmatrix} \frac{\partial \vec{r}_{1}}{\partial\vec{q}_{1}} & ... & \frac{\partial \vec{r}_{1}}{\partial\vec{q}_{n}} \\ ... & & .....


2

First note that $p(0)$ travels along an arc of the circle of radius $r = \Vert p \Vert \sin(\phi)$ centered at a point on the axis of $\omega$; and the velocity $\dot{p}$ is perpendicular to the arc of the circle; and (from the definition) $\omega = \dot{\theta} u$, where $u$ is a unit vector perpendicular to the plane of rotation. Now we try to relate $\...


2

You want to use the product of exponentials to calculate the transformation of $\zeta_1$ and $\zeta_2$ for $\theta_1$ and $\theta_2$. To be more clear, using your notation of $g_{12}$: \begin{equation} g_{12} = e^{\hat{\zeta_1} \theta_1} \cdot e^{\hat{\zeta_2} \theta_2} \end{equation} Where $\hat{\zeta_i}$ is the skew-symmetric matrix representation of ...


2

Adding to Peter Corke's answer, there's also a Coursera course by Kevin Lynch which uses the Modern Robotics book as a reference and explains how to derive the screw based Jacobian. The Jacobian can be either with respect to the "space frame" (frame attached to base of the manipulator) or the "body frame" (frame attached to the end-effector). Here's a ...


2

You still haven't posted the (full) code that gives the results you've presented; when I run your snippet I don't the results you posted. Instead, I get: [xdfb] = FDfb(sphere, xfb, [], [], []) xdfb = 0 -2.5000 0 0 0 -1.0000 -1.0000 0 0 0 0 5.0000 -14.8066 Here xdfb is ...


2

Due to the way that frames are defined in the Modern Robotics book (and in this type of vector-field mechanics in general, such as those of Featherstone), both the spatial frame and the body frames are defined as stationary inertial frames. This requires a bit of a different conceptual understanding than the more traditional moving frames that have been &...


1

1) There are many ways to express velocities. All of them are mathematical constructs to describe the same motion. They can have some minor advantages/disadvantages depending on the applications. The one main disadvantage of an Euler angle based approach is the gimbal lock problem. An advantage of Euler angle based approaches is, that they can be more ...


1

If I'm understanding you correctly, you're attempting to put the position of the joint in for the translational velocity component of the screw axis. What you actually want to put in there is the velocity of a point rigidly attached to the rotating part of the joint, but currently located at the origin of the base frame, $$ v = \begin{bmatrix} \omega_{x} \\ \...


1

maybe need some transformation from centers of mass to the joint frame? Isn't that what $A_i$ is? I don't have the book with me, but from your excerpt: Let $A_i$ be the screw axis of joint $i$ defined in $\{i\}$ where it says at the top, frames $\{1\}$ to $\{N\}$ [are attached] to the centers of mass of links $\{1\}$ to $\{N\}$


1

There are in fact two types of Jacobians, a geometric Jacobian and an analytical Jacobian. The intro to chapter 3 in the book: Robotics: Modelling, Planning and Control by Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani, Giuseppe Oriolo says it well: ...differential kinematics is the relationship between the joint velocities and the corresponding end-...


1

Try looking for terms like robot calibration, robot kinematic calibration or kinematic calibration. Have a look at Chapter 6 of [1]. [1] Springer Handbook of Robotics, Eds: B.Siciliano, O.Khatib, 2016


1

According to the notation in Corke's book, it looks like that top-right 3x3 block that you're looking for is the translation from end effector frame E to base frame 0 represented in so(3). It looks like the trouble here comes from viewing the velocity in different frames. Lynch's spatial twist (defined on page 99) differs from Corke's spatial velocity (...


1

Summarizing from Murray, Li, and Sastry (chapters 3 and 5) there are 3 related things: Twist: An element of se(3) (which is a bit like the derivative of an element of SE(3), which is the set of translations + rotations) Screw: A translation+rotation (i.e. and element of SE(3)) Wrench: Generalized force (combination of linear force and torque)


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