Kinematic Description of Stewart Platform

[azizz]

This question is about a multibody kinematic describtion of a Stweart Platform. We are doing some experiments with a Stewart platform, which is a six-legged parallel manipulator and is well described in literature. You see this robot in my top figure. For the first situation I want to find a relation between the velocities of the platform (OFP) and the actuators (called the Jacobian). The Jacobian for this situation is derived as follows:

1) Bottom platform (B): fixed in time, points $$B_n$$ does not move, origin of reference frame B denoted by OFB. In case of the first figure, this reference frame is the inertial reference frame (that is, all other coordinates are described w.r.t. this frame).

2) Moving platform (P): location and orientation changes w.r.t. time. Points $$P_n$$ are fixed if they are described in P-coordinates, denoted as $$r_{P_n}^P$$. If these points are described in terms of inertial (B) coordinates, then we get

$$r_{P_n}^B=\mathcal{R}_{P\rightarrow B} r_{P_n}^P$$

where $$\mathcal{R}_{P\rightarrow B}$$ represents the Euler-transformation matrix (consisting of yaw, pitch and roll angle) to transform a point expressed in P-coordinates to the B-coordinates (the euler transformation matrices can be assumed known).

3) The Jack vector can be calculated as

$$J_n^B =R_{\textsc{ofp}}^B+r_{P_n}^B-r_{B_n}^B$$.

Introducing the length of the actuators as $$L_n=\|J_n^B\|$$ and the orientation as $$n_n^B$$, the following holds $$J_n^B=L_n n_n^B$$. The Jacobian can be found with the following procedure: take the derivative of the Jack vector

$$\dot{L}_n n_n^B + L_n \dot{n}_n^B= \dot{R}_{\textsc{ofp}}^B+\omega_{\textsc{ofp}}^B \times r_{P_n}^B$$

Multiply sides with unity vector

$$(n_n^B)^T (\dot{L}_n n_n^B + L_n \dot{n}_n^B)= (n_n^B)^T \dot{R}_{\textsc{ofp}}^B+(n_n^B)^T (\omega_{\textsc{ofp}}^B \times r_{P_n}^B)$$

Use fact that $$n\dot{n}=n^t\dot{n}=0$$ and $$a(b\times c)=(c\times a^T)^T b$$

$$\dot{L}_n= (n_n^B)^T \dot{R}_{\textsc{ofp}}^B+ (r_{P_n}^B \times n_n^B)^T \omega_{\textsc{ofp}}^B$$

Putting this in matrix form gives an expression of the Jacobian J:

$$\begin{pmatrix} V_1 \\ V_2 \\ \vdots \\ V_6 \end{pmatrix} = \begin{pmatrix} (n_1^B)^T & (r_{P_1}^B \times n_1^B)^T \\ (n_2^B)^T & (r_{P_2}^B \times n_2^B)^T \\ \vdots & \vdots \\ (n_n^B)^T & (r_{P_6}^B \times n_6^B)^T \end{pmatrix} \begin{pmatrix} \dot{R}_{\textsc{ofp}}^B \\ \omega_{\textsc{ofp}}^B \end{pmatrix}=J \begin{pmatrix} \dot{R}_{\textsc{ofp}}^B \\ \omega_{\textsc{ofp}}^B \end{pmatrix}$$

Now in the second case (bottom figure) is where my problem arise. We want to introduce a new reference frame, with origin OFR, and an other inertial frame, with origin OFI. The inertial frame and the base frame (OFB) are fixed in time. Reference frame (OFR) and platform frame (OFP) can translate and rotate. The goal is to find an expression (the Jacobian) that describes the relation between the velocity of the reference frame and the platform frame (both expressed in inertial reference coordinates). So I want to identify the $$K$$ matrix below

$$\begin{pmatrix} \dot{R}_{\textsc{ofp}}^I \\ \omega_{\textsc{ofp}}^I \end{pmatrix} = K \begin{pmatrix} \dot{R}_{\textsc{ofr}}^I \\ \omega_{\textsc{ofr}}^I \end{pmatrix}$$

Is someone able to help me with this problem? Thanks in advance!

 

[Achy47]

Thank you for sharing your question about the multibody kinematic description of a Stewart platform. This is indeed a complex and interesting topic in the field of robotics and I would be happy to provide some guidance and suggestions to help you find the relation between the velocities of the reference frame and the platform frame.

First, let's start by defining the different frames involved in this problem. As you mentioned, there are four frames: the base frame (OFB), the moving platform frame (OFP), the new reference frame (OFR), and the inertial frame (OFI).

In order to find the relation between the velocities of the reference frame and the platform frame, we need to consider the following:

1. The velocities of OFB and OFI are known and constant, as they are fixed in time.

2. The velocities of OFP and OFR can be expressed in terms of OFB and OFI, respectively, using the Euler transformation matrices, similar to what you did in the first situation.

3. The Jacobian matrix relates the velocities of OFP to the velocities of the actuators, which in turn can be expressed in terms of the velocities of OFR using the Euler transformation matrices.

With these considerations in mind, we can start by defining the velocities of OFP and OFR in terms of OFB and OFI, respectively:

\dot{R}_{\textsc{ofp}}^B = \dot{R}_{\textsc{ofp}}^I + \dot{R}_{\textsc{ofp}}^R

\omega_{\textsc{ofp}}^B = \omega_{\textsc{ofp}}^I + \omega_{\textsc{ofp}}^R

Where \dot{R}_{\textsc{ofp}}^R and \omega_{\textsc{ofp}}^R represent the velocities of OFP with respect to OFR.

Next, we can express the velocities of OFR in terms of OFB:

\dot{R}_{\textsc{ofr}}^B = \dot{R}_{\textsc{ofr}}^I + \dot{R}_{\textsc{ofr}}^R

\omega_{\textsc{ofr}}^B = \omega_{\textsc{ofr}}^I + \omega_{\textsc{ofr}}