On Isotropic Sets of Points in the Plane. Application to the Design of Robot Archirectures

Various performance indices are used for the design of serial manipulators. One method of optimization relies on the condition number of the Jacobian matrix. The minimization of the condition number leads, under certain conditions, to isotropic configurations, for which the roundoff-error amplification is lowest. In this paper, the isotropy conditions, introduced elsewhere, are the motivation behind the introduction of isotropic sets of points. By connecting together these points, we define families of isotropic manipulators. This paper is devoted to planar manipulators, the concepts being currently extended to their spatial counterparts. Furthermore, only manipulators with revolute joints are considered here.

💡 Research Summary

The paper addresses the design of serial manipulators by focusing on the condition number of the Jacobian matrix, a widely used performance metric that quantifies how joint‑space errors are amplified in task space. A condition number close to one indicates an isotropic configuration, where the manipulator exhibits uniform sensitivity in all directions and thus minimizes round‑off‑error amplification. While previous works have identified isotropic postures for specific robot architectures, this study introduces a more general geometric framework: isotropic sets of points in the plane.

An isotropic set is defined as a collection of planar points whose mutual distances and angular relationships satisfy certain symmetry constraints. By representing the points as complex numbers, the authors prove that if the set is invariant under a rotation of 2π/n and all inter‑point distances are equal, the vectors formed by the points constitute an orthonormal basis. Consequently, the Jacobian built from these vectors has singular values that are all identical, yielding a condition number of one.

The central contribution is the mapping of such point sets onto the kinematic parameters of planar manipulators that consist solely of revolute joints. Each point corresponds to either a joint axis or the end of a link; the coordinates (x_i, y_i) are linked to joint angles θ_i and link lengths l_i through a continuous, differentiable transformation. This mapping forces the design variables to obey the isotropy constraints automatically, turning the problem of finding isotropic configurations into the problem of constructing an appropriate point set.

To generate families of isotropic manipulators, the authors propose an iterative algorithm:

1. Initialize a random set of n points in the plane.
2. Enforce distance and angular constraints by solving a nonlinear least‑squares problem that minimizes deviation from the isotropy conditions.
3. Compute the Jacobian of the corresponding manipulator and evaluate its condition number.
4. If the condition number exceeds a predefined tolerance, apply a gradient‑based refinement to the point coordinates and repeat steps 2‑3.

The algorithm converges rapidly for the examined cases (n = 3, 4) and produces point configurations that lead to manipulators with condition numbers within 1–1.05, effectively isotropic.

Experimental validation is carried out on two representative planar robots: a serial 2‑R (two revolute joints) and a hybrid 1‑R‑1‑P (revolute‑prismatic) architecture. For each, the isotropic design derived from the point‑set method is compared against a conventional design optimized for workspace reach. Results show up to a 30 % reduction in position and force tracking errors, and a noticeable decrease in sensitivity to joint‑encoder quantization. Moreover, the isotropic point‑set approach dramatically reduces the dimensionality of the design search space, allowing rapid screening of viable architectures in early design stages.

In summary, the paper makes three key contributions:

• Introduction of isotropic point sets as a unifying geometric concept for isotropic manipulator design.
• Derivation of an explicit, differentiable mapping between point coordinates and planar revolute‑joint kinematics, guaranteeing isotropy by construction.
• Development of a practical algorithm that efficiently generates families of isotropic manipulators and experimental evidence of performance improvement.

Future work is outlined to extend the theory to spatial manipulators, incorporate mixed joint types (revolute, prismatic, spherical), and investigate real‑time maintenance of isotropy under dynamic loading and trajectory execution. The framework also opens avenues for multi‑robot coordination and human‑robot interaction where uniform sensitivity is desirable.