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azizz

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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 [tex]B_n[/tex] 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 [tex]P_n[/tex] are fixed if they are described in P-coordinates, denoted as [tex]r_{P_n}^P [/tex]. If these points are described in terms of inertial (B) coordinates, then we get

[tex]r_{P_n}^B=\mathcal{R}_{P\rightarrow B} r_{P_n}^P[/tex]

where [tex]\mathcal{R}_{P\rightarrow B}[/tex] 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

[tex]J_n^B =R_{\textsc{ofp}}^B+r_{P_n}^B-r_{B_n}^B[/tex].

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

[tex]\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 [/tex]

Multiply sides with unity vector

[tex](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)[/tex]

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

[tex]\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 [/tex]

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

[tex]\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}[/tex]

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 [tex]K[/tex] matrix below

[tex] \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} [/tex]

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

1) Bottom platform (B): fixed in time, points [tex]B_n[/tex] 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 [tex]P_n[/tex] are fixed if they are described in P-coordinates, denoted as [tex]r_{P_n}^P [/tex]. If these points are described in terms of inertial (B) coordinates, then we get

[tex]r_{P_n}^B=\mathcal{R}_{P\rightarrow B} r_{P_n}^P[/tex]

where [tex]\mathcal{R}_{P\rightarrow B}[/tex] 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

[tex]J_n^B =R_{\textsc{ofp}}^B+r_{P_n}^B-r_{B_n}^B[/tex].

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

[tex]\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 [/tex]

Multiply sides with unity vector

[tex](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)[/tex]

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

[tex]\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 [/tex]

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

[tex]\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}[/tex]

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 [tex]K[/tex] matrix below

[tex] \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} [/tex]

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