Analytical Inverse Kinematic Solution for "Moz1" NonSRS 7-DOF Robot arm with novel arm angle

This paper presents an analytical solution to the inverse kinematic problem(IKP) for the seven degree-of-freedom (7-DOF) Moz1 Robot Arm with offsets on wrist. We provide closed-form solutions with the novel arm angle . it allow fully self-motion and solve the problem of algorithmic singularities within the workspace. It also provides information on how the redundancy is resolved in a new arm angle representation where traditional SEW angle faied to be defined and how singularities are handled. The solution is simple, fast and exact, providing full solution space (i.e. all 16 solutions) per pose.

💡 Research Summary

The paper addresses the inverse kinematics problem (IKP) for the seven‑degree‑of‑freedom (7‑DOF) Moz1 robot arm, which belongs to the non‑SRS (non‑spherical‑wrist) family. Unlike classic 7‑R anthropomorphic manipulators that have intersecting shoulder‑elbow‑wrist axes, Moz1 features linear offsets in the elbow and wrist joints, making the widely used SEW (Shoulder‑Elbow‑Wrist) angle inapplicable because the shoulder and wrist points cannot remain static during self‑motion.

To overcome this limitation, the authors introduce a novel “arm angle” ψ. ψ is defined using three key points: the shoulder center S, the elbow center E, and the seventh‑joint axis center C. The reference plane is spanned by the vector SC and the negative z‑axis of joint 7 (−z₇); the arm plane is spanned by SC and SE. ψ is the dihedral angle between these two planes about the SC axis. Because S, C, and the SC line stay fixed while the robot moves in its null‑space, ψ provides a continuous redundancy parameter even when SEW cannot be defined.

The analytical derivation proceeds in two stages. First, a special configuration is considered where the wrist center C lies on the base z‑axis directly above the shoulder and the seventh joint axis z₇ lies in the base xz‑plane. In this configuration the end‑effector pose 0T₇ can be expressed by three scalar parameters: d_sc = ‖SC‖, the angle q between SC and −z₇, and the rotation α between the reference and arm planes. The pose matrix (Eq. 2) is written in terms of these parameters and ψ.

Using geometric constraints, the authors derive two key equations linking ψ, q₆, and q₇ (Eqs. 5 and 10). Substituting the expressions for the wrist‑center point and the elbow point leads to a quartic polynomial in the auxiliary variable t₆ = a_wr + d_sc cos q (Eq. 13). Solving this quartic yields up to four real solutions for q₆; the sign of q₆ is resolved via an auxiliary condition (Eq. 12). Once q₆ is known, q₈ = q₇ − α is obtained from Eq. 11, and finally q₇ = q₈ + α.

The next step solves for q₄ and q₅. By examining the transformation from frame 6 to frame 5 (Eq. 14) and squaring the resulting relations, a closed‑form expression for cos q₄ is obtained (Eq. 15), giving two possible values. Substituting back yields q₅ via an atan2 expression (Eq. 18).

Finally, the base joints q₁, q₂, q₃ are extracted by equating the product of the first three joint rotation matrices (Eq. 19) with the known overall rotation 0R₇ (Eq. 20). This leads to two solutions for q₂ (Eq. 21) and corresponding q₁, q₃ obtained via atan2 formulas (Eqs. 22‑23). The combination of the binary choices (q₄/q₅, q₂, and the two q₆ solutions) yields the full set of 16 possible IK solutions for any reachable pose.

For arbitrary poses, the authors show that any end‑effector configuration can be reduced to the special case by a “sphere rotation”: the first three joints (the spherical shoulder) are used to rotate the robot until the wrist center aligns with the base z‑axis, after which the same analytical steps apply. The required parameters d_sc, q, and α are extracted directly from the given 0T₇ using vector algebra (section V).

Singularity analysis is performed using a reciprocity‑based method. Kinematic singularities occur when q₄ = 0 or π, when q₂ = ±π/2 together with q₃ = 0 or π, and when q₆ reaches the limit acos(−a_wr/d_ew). Algorithmic singularities arise when the denominator in the quartic formulation vanishes, specifically when q₂ = ±π/2 and the expression (d_ew + a_wr cos q₆) sin q₄ − a_wr cos q₄ sin q₅ sin q₆ = 0. These conditions are explicitly listed, allowing a controller to detect and avoid degenerate configurations.

Implementation details emphasize real‑time safety. The C++ code relies only on the Eigen library, guaranteeing deterministic execution time because the number of arithmetic operations is fixed and independent of the pose. Unlike numerical solvers, the analytical method does not require an initial guess and always returns a mathematically exact solution when one exists; even for poses outside the workspace it produces a geometrically reasonable fallback.

The authors claim that the presented arm‑angle based analytical IK can be adapted to other non‑SRS manipulators such as the Franka Emika Panda, opening avenues for fast, singularity‑aware redundancy resolution in modern collaborative robots.

In summary, the paper delivers a complete closed‑form IK solution for the Moz1 7‑DOF arm, introduces a novel redundancy parameter that works where SEW fails, provides a systematic way to enumerate all 16 solutions, analyses singularities rigorously, and supplies a real‑time‑compatible implementation, thereby advancing both theoretical understanding and practical control of redundant, offset‑rich robotic manipulators.