Kinematic Singularity
Definition
A kinematic singularity is a configuration at which the end-effector Jacobian
drops rank below the task dimension (Chiaverini 1997). There the
manipulator loses the ability to instantaneously generate end-effector motion along at least one task
direction: a whole subspace of Cartesian velocities becomes infeasible while extra null-space joint
velocities appear. It is intrinsic to the kinematic structure — a singularity of the body Jacobian (Ott
2008), exposed by its SVD (Chiaverini 1997) — and is distinct from an algorithmic singularity — a rank loss of
the constrained secondary-task Jacobian that can occur even when is full rank — and
from a dynamic singularity, which depends on inertia. The notion is regime-
independent (it is purely kinematic), but its consequences differ: for our free-flying base the
fully-actuated 6-DOF base can place the end-effector even when the arm Jacobian is
singular (Bruschi 2025), an escape route unavailable to a fixed-base or purely-reactive free-floating arm.
Key Equations
Symbols per notation.md.
Differential kinematics and the rank condition (Chiaverini Eq. 1–2; is the EE Jacobian, alias here):
\boldsymbol J_E=\boldsymbol U\boldsymbol\Sigma\boldsymbol V^{\top}=\sum_{i=1}^{r}\sigma_i\,\boldsymbol u_i\boldsymbol v_i^{\top}.$$ A kinematic singularity is $\operatorname{rank}\boldsymbol J_E(\boldsymbol q)=r<m$, i.e. the smallest singular value vanishes: $$\sigma_{\min}(\boldsymbol J_E)=0 .$$ Proximity is measured by a scalar; Ott/Yoshikawa use the manipulability volume (notation.md: $w$), which is zero exactly at a singularity: $$w=m_{\mathrm{kin}}(\boldsymbol q)=\sqrt{\det\!\big(\boldsymbol J\boldsymbol J^{\top}\big)} .$$ (In our circumcentroidal model the operative proximity scalar is $\sigma_{\min}(\boldsymbol J_{\nu_e}^\oplus)$ / $\sigma_G$ — see [notation.md](../notation.md).) ## Source Support - [bruschi2025singularity](../sources/bruschi2025singularity.md) — primary for our regime: defines $\boldsymbol J_m(\boldsymbol q)$ losing rank as the kinematic singularity, and shows a free-flying task-priority design tracking the EE *through* arm-singular configurations by moving the actuated base. - [chiaverini1997singularity](../sources/chiaverini1997singularity.md) — canonical definition and SVD analysis (rank $<m$); contrasts kinematic vs algorithmic singularities and the pseudoinverse/damped-least-squares remedies for redundant arms. - [ott2008cartesian](../sources/ott2008cartesian.md) — distinguishes kinematic from *representation* singularities, gives the Yoshikawa manipulability measure, and a singularity-avoidance impedance potential (non-redundant case). ## Related Topics - [singularity_robust_inverse](singularity_robust_inverse.md) — the robust Jacobian inverse used near $\sigma_{\min}\to 0$ to keep joint rates finite. - [algorithmic_singularity](algorithmic_singularity.md) — the *other* rank loss in task-priority control; can occur with full-rank $\boldsymbol J_E$ when tasks conflict. - [manipulability_measure](manipulability_measure.md) — the scalar $w=\sqrt{\det(\boldsymbol J\boldsymbol J^\top)}$ that detects proximity to a kinematic singularity. - [damped_least_squares](damped_least_squares.md) — Chiaverini's variable-damping inverse that trades tracking accuracy for feasible joint velocities inside the singular region. - [kinematic_redundancy](kinematic_redundancy.md) — extra DOF ($n>m$) give null-space freedom but do **not** by themselves remove kinematic singularities. - [dynamic_singularity](dynamic_singularity.md) — inertia-dependent rank loss of the generalized Jacobian; a different phenomenon relevant to the free-floating regime. - [trajectory_tracking](trajectory_tracking.md) — the task whose feasibility degrades at a singularity; Bruschi's contribution is tracking through singular arm configurations. ## Open Questions - [ ] Chiaverini and Ott assume a fixed base; Bruschi shows the actuated base can carry the EE through an arm singularity. What is the right *proximity scalar* for the coupled free-flying system — $\sigma_{\min}(\boldsymbol J_m)$, the circumcentroidal $\sigma_{\min}(\boldsymbol J_{\nu_e}^\oplus)$, or $\sigma_G=\sigma_{\min}(\boldsymbol\Gamma)$? - [ ] Ott's manipulability potential and Chiaverini's damped inverse are derived non-redundant / fixed-base; do their stability arguments survive when the singular direction is instead absorbed by base motion?