## Abstract

This paper presents a method to solve the forward kinematic problem of parallel robots, obtaining all real solutions. The method is illustrated with the 3UPS-PU parallel robot, and consists in eliminating one unknown to obtain an equation that constrains the admissible values of the remaining unknowns. This constraint defines a curve in the plane of the remaining unknowns, which contains all real solutions. The proposed method samples and scans this constraint curve in order to find all real solutions, using only analytical operations and k-dimensional trees. This method is able to find all real solutions and performs at least one order of magnitude faster than other numerical methods, while avoiding the complicated and lengthy symbolic expressions that traditional elimination procedures involve. The proposed method is also extended to the complex domain and another parallel robot (the 3RRR robot).

### Keywords

- Forward kinematics
- K-d trees
- Parallel robots
- Algebraic elimination
- Curve tracing

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## Appendices

### Appendix A

This appendix derives the expressions of all coefficients \(\omega _i\) of the quartic polynomial of Eq. (22). The next demonstration can also be used for obtaining all coefficients \(\omega _i\) of the quartic polynomial of Eq. (29), by swapping every \(\alpha \) and \(\beta \) appearing in this appendix. First, let us rewrite Eq. (21) as follows, with the denominator already omitted:

where:

Since \(\mathbf {U}\), \(\mathbf {V}\), \(\mathbf {W}\) are \(2\times 2\) matrices, the expansion of the left-hand side of Eq. (36) yields the quartic polynomial of Eq. (22), i.e.:

The next objective is to obtain the expressions of the coefficients \(\omega _i\) as simple functions of the matrices \(\mathbf {U}\), \(\mathbf {V}\), and \(\mathbf {W}\). Note that \(\omega _0\) can be directly obtained by substituting \(t_\beta =0\) into the previous equation, which yields:

To obtain \(\omega _1\), we first differentiate Eq. (38) once with respect to \(t_\beta \):

where Jacobi’s formula for the derivative of a determinant has been used when differentiating the left-hand side of Eq. (38):

where “\(\hbox {adj}(\mathbf {Q})\)” is the adjugate of \(\mathbf {Q}\), which satisfies: \(\mathbf {Q} \cdot \hbox {adj}(\mathbf {Q})=\det (\mathbf {Q}) \cdot \mathbf {I}\) (\(\mathbf {I}\) is the identity matrix with the same size as \(\mathbf {Q}\)). Inserting \(t_\beta =0\) into (40) yields:

Following the previous process, one might continue differentiating Eq. (40) repeatedly and substituting \(t_\beta =0\) to obtain the remaining coefficients (\(\omega _2\), \(\omega _3\), \(\omega _4\)). However, this task becomes too tedious and error-prone due to the many terms generated by repeatedly differentiating the determinant. Alternatively, the remaining coefficients can be obtained more easily as explained next.

First, it can be checked that reversing the order of the matrix coefficients of the polynomial appearing inside the determinant on the left-hand side of Eq. (38) (i.e., swapping \(\mathbf {U}\) and \(\mathbf {W}\)) reverses the order of the coefficients \(\omega _i\) on the right-hand side of the same equation, obtaining:

Accordingly, one can compute \(\omega _4\) and \(\omega _3\) by swapping \(\mathbf {U}\) and \(\mathbf {W}\) in the formulas already derived for \(\omega _0\) and \(\omega _1\), respectively, obtaining:

Finally, after all \(\omega _i\) except \(\omega _2\) are known, the central coefficient \(\omega _2\) can be obtained by substituting \(t_\beta =1\) into Eq. (38) and solving for \(\omega _2\):

### Appendix B

This appendix shows how to solve \(\beta \) (respectively \(\alpha \)) from Eq. (34) when \(\alpha \) (respectively \(\beta \)) is given. If \(\alpha \) is given, Eq. (34) can be rewritten as follows:

where the coefficients *C*, *S*, *I* have the following expressions:

Equation (46) can be converted to a quadratic polynomial using the substitution of Eq. (19), which gives: \((I-C)t_\beta ^2 + (2S)t_\beta + (I+C)=0\). This quadratic equation is easily solved.

Similarly, if \(\beta \) is given, Eq. (34) can also be rewritten as “\(C\cos \alpha + S\sin \alpha + I=0\)”, where the coefficients *C*, *S*, *I* are:

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Peidró, A., Payá, L., Cebollada, S., Román, V., Reinoso, Ó. (2022). Solution of the Forward Kinematics of Parallel Robots Based on Constraint Curves. In: Gusikhin, O., Madani, K., Zaytoon, J. (eds) Informatics in Control, Automation and Robotics. ICINCO 2020. Lecture Notes in Electrical Engineering, vol 793. Springer, Cham. https://doi.org/10.1007/978-3-030-92442-3_20

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