Abstract
This paper highlights some of the degeneracies encountered in the inverse kinematics problem (IKP) of a general 6R (revolute jointed) serial robot. It is known that algebraic eliminations introduce certain mathematical degeneracies, encountering which an algorithm may fail. The degeneracy conditions are often overlooked or remains unreported when an algorithm is proposed. Awareness of these conditions becomes important when these algorithms form important cogs of bigger objectives like design, path-planning or control. Kinematic problems often lead to matrices whose entries are polynomials in a single variable. These can, in turn, be treated as polynomials whose coefficients are matrices. Degeneracy study of such systems can be attempted akin to their scalar counterparts, by investigating the conditions that result in the determinant of coefficient matrix associated with the leading term evaluating to zero. The same conditions have been analysed for the IKP of a general 6R, and some of the degenerate cases presented in this paper.
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Notes
- 1.
The DH parameters [2] are used as per convention with \(a_i, d_i\) representing the link length and link offset, respectively, \(\alpha _i\) is the twist angle and \(\theta _i\) is the joint angle. It may be noted that in this paper \(\sin \theta _i \equiv s_i\) and \(\cos \theta _i \equiv c_i\).
- 2.
Following [5], \(\mathbf {P}=\begin{pmatrix} \mathbf {1} &{}\mathbf {0} \\ \mathbf {0} &{}\mathbf {A} \end{pmatrix}, \quad \mathbf {Q}=\begin{pmatrix} \mathbf {0} &{}\mathbf {1} \\ -\mathbf {B} &{} -\mathbf {C} \end{pmatrix}\), where \(\mathbf {1} \in \mathbb {R}^{12 \times 12}\) is the identity matrix.
- 3.
\(\boldsymbol{\delta }\equiv [a_i,d_i,\alpha _i]^\top ,~i=1,\ldots ,6\), \(\boldsymbol{\rho } \equiv [\mathbf {l},\mathbf {m}, \mathbf {n}, p_x, p_y, p_z]^\top \) .
- 4.
The idea of studying degenerate surfaces was used in [10] to study the degeneracy conditions in the algorithm of the forward kinematic problem of the Stewart platform manipulator.
- 5.
The discrete cases presented in this paper are meant as pointers that need to be care of while adopting elimination based solution techniques and are not exhaustive.
- 6.
The failure could have been termed intrinsic if \(u_{11}\) identically evaluated to zero for \(\alpha _1 \ne 0,\pi \). It may also be argued that intrinsic failures would invariably change the cardinality of the solution space as explained in [8].
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Das, A., Nag, A., Saha, S.K. (2022). Degenerate Cases in the Inverse Kinematics Problem of a General 6R Serial Robot. In: Altuzarra, O., Kecskeméthy, A. (eds) Advances in Robot Kinematics 2022. ARK 2022. Springer Proceedings in Advanced Robotics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-031-08140-8_45
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