Kinematic Analysis of a Family of 3R Manipulators

The workspace topologies of a family of 3-revolute (3R) positioning manipulators are enumerated. The workspace is characterized in a half-cross section by the singular curves. The workspace topology is defined by the number of cusps that appear on these singular curves. The design parameters space is shown to be divided into five domains where all manipulators have the same number of cusps. Each separating surface is given as an explicit expression in the DH-parameters. As an application of this work, we provide a necessary and sufficient condition for a 3R orthogonal manipulator to be cuspidal, i.e. to change posture without meeting a singularity. This condition is set as an explicit expression in the DH parameters.

💡 Research Summary

The paper presents a systematic classification of the workspace topologies of a family of three‑revolute (3R) positioning manipulators. By slicing the workspace with a half‑cross‑section (fixing the first joint angle and varying the second and third), the authors derive the singular curves that delimit the reachable region. These curves are defined as the set of joint configurations where the Jacobian determinant vanishes. A key observation is that cusps—points where the singular curve folds back on itself—indicate the existence of multiple discrete postures for the same Cartesian point. The number of cusps on the singular curve therefore becomes a natural metric for the workspace topology.

Using Denavit–Hartenberg (DH) parameters (link lengths a₁, a₂, a₃, offsets d₁, d₂, d₃, and twists α₁, α₂, α₃), the authors non‑dimensionalize the design space and express the singularity conditions in closed‑form algebraic equations. By analyzing how the number of cusps varies with the DH parameters, they partition the parameter space into five continuous domains (denoted D₁ … D₅). Each domain corresponds to a distinct cusp count (0, 1, 2, 3, or 4). The separating surfaces between domains are obtained analytically as explicit expressions, for example a₂·a₃ = d₂·a₁, which represent the loci where two singular curves become tangent or intersect. This partition provides a direct map from design parameters to workspace topology, enabling designers to target a desired cusp configuration simply by selecting appropriate link ratios.

A special focus is placed on orthogonal 3R manipulators (α₁ = α₂ = 90°), which are common in industrial robots. For this subclass the authors derive a compact necessary and sufficient condition for the manipulator to be cuspidal—i.e., to change posture without crossing a singularity. The condition reduces to a single inequality involving the non‑dimensionalized parameters, such as a₂·a₃ > d₂·a₁. When this inequality holds, the manipulator possesses at least one cusp and can transition between distinct inverse‑kinematic solutions while remaining in a nonsingular region. If the inequality is violated, no cusp exists and any posture change must pass through a singular configuration.

The theoretical results are validated through extensive numerical simulations. Representative manipulators from each of the five domains are modeled, and their half‑cross‑section workspaces are plotted. The simulated singular curves and cusp locations match the analytically predicted boundaries, confirming the correctness of the derived expressions. The visualizations also illustrate how increasing the number of cusps enriches the workspace with regions where multiple postures are feasible, which is advantageous for tasks requiring redundancy or rapid reorientation.

The contributions of the paper are threefold. First, it introduces a clear, cusp‑based taxonomy of 3R workspace topologies and links this taxonomy directly to DH parameters. Second, it provides explicit algebraic formulas for the domain boundaries, turning what was previously a purely numerical exploration into a tractable design tool. Third, it supplies a simple, closed‑form cuspidality test for orthogonal 3R manipulators, offering practitioners an immediate check on whether a given design can avoid singularities during posture changes.

From an application standpoint, the findings have immediate relevance to robot design, workspace optimization, and real‑time control. In high‑speed assembly, welding, or medical robotics, the ability to reconfigure the end‑effector without encountering a singularity reduces torque spikes and improves safety. By selecting parameters that place the manipulator in a domain with the desired number of cusps, engineers can guarantee the necessary redundancy while respecting physical constraints on link lengths and offsets. Consequently, the paper not only advances the theoretical understanding of 3R kinematics but also delivers a practical, analytically grounded framework for designing manipulators with predictable and desirable workspace characteristics.