Example1_Zhang and Song.html

Implementation of the self-motion parameterization of Example 1

This Maple worksheet follows the notation and computations of the Section IV: EXAMPLE 1: THE ZHANG-SONG PLATFORM in the work

[1] J.Borras, F. Thomas and C. Torras, "On Delta Transforms".

Authors:

Júlia Borràs, Federico Thomas and Carme Torras.

URL:

http://www-iri.upc.es/

Purpose:

This worksheet shows the explicit computations performed in the section IV of the above mentioned work, so the reader can follow the computations by

just executing the present worksheet, to rise to the parameterization shown in equation (25) in [1].

Fig 5-(d) Fig 5-(c) Fig. 5-(b) Fig- 5-(a)

Load the Delta Transform calculus tool, containing the function definitions

Following notation of Fig. 5, we define the local coordinates of the attachments

The cross-ratio conditions, which characterizes architectural singularities, can be written as cr1=cr2, where

We give values at all parameters except one

The last parameter must be so that the condition (22) holds, to obtain an architecturally singular manipulator

Finally, last parameters that must hold

To obtain a set of coherent leg distances, we solve the Inverse Kinematics problem for a given pose of the platform. The local coordinates listed in Table I are

Position vector p=<3,-2,3> and yaw, roll and pitch angles define the pose of the platform, so the Inverse Kinematics solutions will give the length of the corresponding legs

also listed in Table I

First Transform: from Fig. 5-(d) to Fig. 5-(c)

Proceeding backwards all the Delta- transforms defined in Fig.5, we start with the four simultaneous delta-transforms applied to the line-line component. We obtain, from legs d_i, values for legs named l_i.

The system of equations (20) is defined in matrix form by the two following matrices. Matrix A must be rank defective, as we are working with an architecturally singular manipulator

Such system can be linearly solved using

or, if we want to control which of the legs must be the parameter, solving the following system

$(Typesetting:-mprintslash)([solsL := {l[2] = -249+135*2^(1/2)+4*l[4], l[3] = -4*l[4]+350-140*2^(1/2), l[1] = -l[4]+101-35*2^(1/2)}], [{l[2] = -249+135*2^(1/2)+4*l[4], l[3] = -4*l[4]+350-140*2^(1/2), l...$
$(Typesetting:-mprintslash)([solsL := {l[2] = -249+135*2^(1/2)+4*l[4], l[3] = -4*l[4]+350-140*2^(1/2), l[1] = -l[4]+101-35*2^(1/2)}], [{l[2] = -249+135*2^(1/2)+4*l[4], l[3] = -4*l[4]+350-140*2^(1/2), l...$

The obtained parameterization for the legs depending on the fourth leg are (equation (24) in the paper).

Second Transform: from Fig. 5-(c) to Fig. 5-(b)

Now we pass from legs l_i in Fig. 5-(c) to legs named q_i in Fig. 5-(c) by applying two Delta transforms. Legs l_1 and l_3 don't change.

At this step, the present parameterization of legs q_i is

First Transform: from Fig. 5-(b) to Fig. 5-(a)

Now we obtain the final parameterization from legs q_i in Fig. 5-(b) to legs named p_i in Fig. 5-(a). The two legs that remain invariant at this step are p_1 and p_2.

Finally, notice that two of the leg lengths remain invariant during all the past transformations, these are p_5 and p_6, that are equal in Fig. 5-(a) and in Fig. 5-(d)

The legs p_i correspond at a 3-2-1 structure whose base attachments are a1,a6 and a5, and the platform attachments are b2,b5 and b6 (as schemed in Fig. 5-(a)). The forward kinematics of the 3-2-1 structure can be solved using trilaterations. So, we can obtain a parameterization of the self motion by performing three trilaterations (which gives a total of 8 branches) depending on the parameter l_4 that is renamed to x. We write down the parameterization of the 8 branches in file named Zhang_Song_ParametrizationFile.txt.