Identification Algorithm to Determine the Trajectory of Robots with Singularities

Hossein Sharifi; William C. Black

REVIEW 4 major objections 6 minor 18 references

Identification Algorithm to Determine the Trajectory of Robots with Singularities

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read At a kinematic singularity, a non-redundant robot's feasible escape paths are determined by the sign of a single matrix A derived from the Jacobian.

desk verdict Under the paper's own start-of-path assumption, the proposed sign-of-A rule selects every nonzero path, so the identification algorithm is vacuous; the derivation also has load-bearing math errors. read the letter →

arxiv 1911.06632 v1 pith:GNAJU2XV submitted 2019-08-23 cs.RO

Hossein Sharifi , William C. Black This is my paper · ORCID

classification cs.RO

keywords singularityavoidancetrajectoryidentificationrobotcontrolsingularvaluedecompositionnon-redundantrobotskinematicsingularities

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes an identification algorithm that tells a non-redundant robot which way to move when it loses exactly one degree of freedom at a kinematic singularity. Using the singular value decomposition of the Jacobian, it isolates the singular direction and derives the endpoint acceleration along that direction. When the robot is at the beginning of its path, this acceleration reduces to $\ddot{d} = A \dot{q}_s^2$, so the sign of the scalar $A$ decides whether the robot can escape along the singular direction or against it. Applied to a six-degree-of-freedom robot, the rule gives explicit feasible-path limits and identifies one parameter choice where no escape path exists.

What carries the argument

The driving object is the singular value decomposition of the Jacobian, $J = U \Sigma V^T$, which factors the matrix into rotation-like parts and a diagonal list of singular values. The decomposition splits the endpoint motion into non-singular directions, $\dot{Y} = K \dot{q}$, and the singular direction, $\dot{d} = L \dot{q}$, with $L = 0$ at the singularity. Differentiating $\dot{d}$ with $L = 0$ gives $\ddot{d} = \dot{q}^T \theta L^T \dot{q}$, and partitioning $\dot{q}$ into maintained and free parts turns this into the quadratic form $\ddot{d} = \dot{q}_s^T A \dot{q}_s + B \dot{q}_s + C$. For a non-redundant robot $\dot{q}_s$ is a scalar, and under $\dot{Y}(0)=0$ the identity collapses to $\ddot{d} = A \dot{q}_s^2$. This identity carries the argument: the sign of $A$ selects the admissible trajectories, and the zero set of $A$ marks configurations with no escape path.

What would settle it

Take the six-degree-of-freedom robot at the singular configuration $\theta_3 = \pi/2$ (so $\varepsilon = 1$), keep the non-singular endpoint velocities at zero, and apply a small joint velocity $\dot{q}_s$. The paper predicts the endpoint acceleration $\ddot{d}$ along the singular direction has the sign of $A$; a finite input producing the opposite sign would refute the rule. In the special case $d_4 = a_2$, $\varepsilon = -1$, the paper predicts no feasible path exists, so a robot that accelerates out of the singularity in that configuration would also refute it.

Watch

Extended reading notes

Core claim

The central claim is that in a singular configuration the feasible paths are exactly those whose acceleration along the singular direction has the same sign as $A$, the coefficient obtained from the second-order term of the singular-direction acceleration. After writing the endpoint velocity through the singular value decomposition $J = U \Sigma V^T$ and separating singular from non-singular directions, the paper derives $\ddot{d} = \dot{q}_s^T A \dot{q}_s + B \dot{q}_s + C$. Under the starting-condition assumption $\dot{Y}(0)=0$, the $B$ and $C$ terms vanish, leaving $\ddot{d} = A \dot{q}_s^2$; a positive $A$ moves the endpoint in the singular direction, and a negative $A$ moves it the other way. The algorithm therefore selects paths whose acceleration sign equals the sign of $A$, and when $A=0$ it concludes that no feasible path exists. In the six-degree-of-freedom example, $A = (a_2 \varepsilon)^2 \left(-\frac{d_4 + a_2\varepsilon}{a_2 d_4 \varepsilon}\right)$, so its sign is governed by the scalar $h$, and $A=0$ occurs only when $d_4 = a_2$ and $\varepsilon = -1$.

Load-bearing premise

The whole sign rule rests on assuming the robot is at the very beginning of its path when the singularity occurs; if it is already moving, the acceleration along the singular direction is no longer decided by the sign of $A$ alone.

Editorial extensions

If this is right

A controller can evaluate the sign of $A$ at a singular configuration and steer the endpoint along the direction that sign indicates.
If $A=0$, the robot should avoid the configuration entirely, since no acceleration can move it out of the singularity under the paper's assumptions.
For the six-degree-of-freedom robot, the first three degrees of freedom separate from the rest, and the sign of $h$ gives the boundary directions the robot must respect.
The same singular-value-decomposition procedure can be applied to the remaining degrees of freedom to find their feasible paths as well.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The sign rule is local: if a singularity occurs mid-path with $\dot{Y}(0) \neq 0$, the full quadratic form must be checked and the sign of $A$ alone would not decide feasibility.
For redundant robots, $\dot{q}_s$ is a vector rather than a scalar, so the natural extension would be a definiteness condition on the matrix $A$ instead of a single sign.
The special case $d_4 = a_2$, $\varepsilon = -1$ could be used as a geometric design constraint, telling engineers to avoid link lengths that make the singularity inescapable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: the paper's selection rule doesn't select anything. Under the paper's own start-of-path assumption (Section III, ˙Y(0)=0), B=C=0, so Eq. (16) gives ¨d = A ˙q_s^2. For a non-redundant robot, ˙q_s^2 ≥ 0, so the sign of ¨d always equals sign(A). That means every nonzero candidate path satisfies the proposed criterion, and the direction of ˙q_s—the only thing you could choose—drops out. The stress-test note you passed along is right, and the paper's own example confirms it: in Eq. (29), B=C=0 and h is fixed by robot parameters, so no trajectory is actually identified. The word "identification" is doing no work.

What's worth a second glance: the paper does set up the singular-direction acceleration as a quadratic form in the null-space velocity, and the decomposition into A, B, C (Eqs. 13–14) is a defensible Taylor-expansion view of second-order effects. That's a legitimate idea, though not new—null-space and SVD analysis of singularities goes back at least to Nakamura and Hanafusa [7]. The paper is also explicit about the start-of-path limitation, which is more than some singularity papers do.

The soft spots are serious. The derivation has internal inconsistencies: Section II says all singular values vanish in a singularity, then uses only the trailing ones; Eq. (9) has a dangling L^T that makes the expression dimensionally wrong (the time derivative of L should contract with ˙q, not with L); and the example in Section IV introduces an undefined parameter a4 and then equates a singular value to a joint-angle condition (σ3 = kπ + π/2). These aren't cosmetic; they're in the chain that leads to the sign-of-A rule. On top of that, there's no simulation or experiment, just an algebraic example, and the conclusion claims effectiveness without evidence.

The citation pattern is not the main problem; the self-citation [10] is used only for the standard Jacobian relation, and [7] is directly relevant, though the paper doesn't clearly differentiate itself from it.

Who is this for? Maybe someone cataloguing flawed singular-robustness proposals. I wouldn't bring it to reading group or cite it. A serious editor should desk-reject; sending this to referees would waste their time.

Best,

[You]

Referee Report

4 major / 6 minor

Summary. The paper proposes an 'identification algorithm' for selecting a feasible robot trajectory when a non-redundant, single-rank robot enters a singular configuration. The derivation starts from the SVD of the Jacobian, defines a singular-direction velocity ˙d = u_m^T ˙X, and expresses the singular-direction acceleration as a quadratic form in the remaining joint-velocity scalar ˙q_s (Eq. 13). Under the assumption that the singularity occurs at the start of motion (˙Y(0)=0), the linear and constant terms vanish, giving ¨d = A ˙q_s^2 (Eq. 16). The paper claims that the sign of the coefficient A determines whether motion in the singular direction is feasible, and applies the criterion to a six-DOF manipulator, deriving an expression for A in Eq. (29).

Significance. If correct, the sign-of-A criterion would be a simple closed-form condition for identifying singularity-avoiding paths at the instant of singularity. The paper presents a self-contained symbolic derivation and a worked example, and the assumption structure is explicit. However, the central selection rule is vacuous under the paper's own assumptions: for a scalar ˙q_s, ¨d = A ˙q_s^2 makes the sign of the acceleration equal to sign(A) for every nonzero joint velocity, so the proposed condition does not single out any trajectory. In addition, the derivation contains several dimensionally inconsistent steps. The useful idea of examining the singular-direction acceleration is standard (see, e.g., Nakamura and Hanafusa [7]), but the claimed identification algorithm as written does not perform identification.

major comments (4)

[Section III, Eq. (16) and the paragraph after it] The proposed selection rule is vacuous. With L=0 and ˙Y(0)=0, the acceleration in the singular direction is ¨d = A ˙q_s^2. For a non-redundant robot, ˙q_s is a scalar, so ˙q_s^2 ≥ 0; every nonzero ˙q_s gives sign(¨d)=sign(A). The condition "we will pick the paths in which the acceleration sign in a singular situation is equal to matrix A sign" is therefore satisfied by all nonzero trajectories and selects none. The sign of ˙q_s, which is the only quantity distinguishing the direction of motion, enters quadratically and is not determined by the criterion.
[Section II, near Eqs. (5)–(6)] The text states "In a singularity case, σ1, σ2, ..., σm = 0." This is inconsistent with the rank-r structure used immediately before and after, where only the trailing singular values σ_{r+1}, ..., σ_m vanish. If all singular values were zero the Jacobian would be the zero matrix and no motion would be possible. The manuscript should state the rank-one (or rank-r) deficiency consistently.
[Section III, Eqs. (9)–(14)] The derivation of the quadratic form is dimensionally and algebraically unsupported. In Eq. (9), L is a row vector (1×n), θ in Eq. (10) is an n×n matrix of partial derivatives, and the product ˙q^T θ L^T ˙q does not yield a scalar with the dimensions implied. The definitions in Eq. (14) mix row and column vectors without correct transposes; for example, A = N θ L N requires N to be both column and row. Consequently, Eqs. (13)–(16) do not follow from the preceding definitions.
[Section IV, Eqs. (22)–(24) and (29)] The example is not internally consistent. The expression for σ1 in Eq. (22) is missing the square-root operation and the condition "σ3 = kπ + π/2" equates a singular value with an angle, suggesting a confusion with the joint angle θ3. Eq. (23) introduces a4, which is not defined in Table I. The matrices θ×L and K in Eqs. (24)–(25) are asserted without derivation, and the relationship of Eq. (29) to the general formula for A is not shown. These gaps make the example unverifiable.

minor comments (6)

[Abstract] "Base on" should be "Based on"; "Singularity is robot controls" should be "Singularity in robot controls."
[Section II, Eq. (7)] After stating that the last columns of U (u_{r+1} to u_m) are zero, defining ˙d via u_m^T makes ˙d identically zero in a singularity. Clarify whether u_m denotes a left singular vector in the null space or a row of U^T, and how multiple singular directions are handled for rank deficiency r < m−1.
[Section III, Eqs. (8) and (12)] The dimensions of ˙Y, K, L, ˙q_p, and ˙q_s are not stated consistently. Specify all matrix dimensions.
[Section III, after Eq. (16)] The assumption ˙Y(0)=0 restricts the result to a single time instant. The paper should state explicitly that the feasibility criterion is local and provide conditions under which it remains valid along an escape trajectory.
[Section IV, conclusion] The conclusion states that the strategy is "very efficient and helpful," but no simulation or experimental results are presented; the example is analytic only.
[References] Several references ([11], [12], [16]) do not appear to be related to robot singularity or identification, and [13] duplicates [6]. The related-work discussion would be stronger with a more focused citation list.

Circularity Check

2 steps flagged · score 8.0 of 10

The path-selection rule is vacuous: Eq. (16) makes the sign of the singular-direction acceleration equal to sign(A) for every nonzero scalar qdot_s, so the algorithm's "pick paths with acceleration sign equal to A sign" selects all admissible paths and identifies no trajectory.

self definitional [Section III, between Eqs. (15) and (16), and the paragraph immediately after Eq. (16).]
"In this paper, we assume that robot is in the beginning of its path and the singularity occurs. Therefore, ˙Y (0) = 0. Based on Eq. 14, matrices B and C will be zero too. Therefore, we will have the Equation below. If A in Eq. 16 is non-zero, the acceleration sign will be the same as A matrix sign. This means that the end-point will be the same sign as A. If A matrix is positive, the end-point will be in the same direction of the singularity, and if it is negative, it will be in the adverse singularity direction."

With L=0 and Ydot(0)=0, Eq. (16) is ddot_d = A qdot_s^2, and qdot_s is a scalar for a non-redundant robot. Since qdot_s^2 >= 0, every path with qdot_s != 0 has sign(ddot_d) = sign(A) automatically; the sign of qdot_s, the only remaining free choice, enters quadratically and is invisible to the criterion. The proposed selection rule "pick the paths in which the acceleration sign is equal to matrix A sign" is therefore equivalent, by construction, to "pick the paths with nonzero qdot_s". It does not discriminate among candidate trajectories and does not determine a direction of motion, so the central identification claim reduces to its input assumption rather than producing an independently identifying condition.
self definitional [Section IV, paragraph after Eq. (29), around Eq. (30).]
"Considering that A is always non-zero, and the acceleration in a singular direction follows the sign of A matrix, the possible paths are the ones that the acceleration sign is the same as h parameter sign in Eq. 30."

In the worked example Eq. (29) gives B=C=0 and A = (a2*eps)^2 * (-(d4+a2*eps)/(a2*d4*eps)), a fixed expression in robot parameters. Substituting into Eq. (16), ddot_d = A qdot_s^2, so sign(ddot_d) = sign(A) for every nonzero qdot_s. The statement that possible paths are those whose acceleration sign equals the sign of h is therefore a tautological restatement of Eq. (16) in the example; the criterion imposes no restriction on qdot_s and cannot select a feasible path.

full rationale

The central derivation of Eq. (16) from the stated assumptions is internally consistent, and the paper does not fit parameters to data. The only self-citation [10] supports the standard forward-kinematics relation and is not load-bearing. However, the paper's load-bearing claim is the selection rule after Eq. (16): feasible paths are those whose singular-direction acceleration has the same sign as A. Under the paper's own assumptions, qdot_s is a scalar and B=C=0, so Eq. (16) gives ddot_d = A qdot_s^2; since qdot_s^2 is nonnegative, every nonzero path automatically satisfies sign(ddot_d)=sign(A). The rule therefore selects no particular path and provides no direction choice; it is a tautological restatement of Eq. (16) rather than an identification of a feasible trajectory. The worked example repeats this vacuity: with B=C=0 and A fixed by robot parameters, "possible paths" are again just all paths with nonzero qdot_s. This makes the central identification claim reduce by construction to its input assumptions, warranting score 8. The zero-A case (ddot=0 -> no feasible path) is a derived special case but does not rescue the general selection algorithm.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted parameters: all quantities are symbolic robot dimensions (a2, d4) and standard DH parameters. The key axioms are the start-of-path assumption and the rank-reduction structure of the Jacobian at a singularity. No new physical entities are postulated.

assumptions (3)

domain assumption The robot is at the beginning of its path when the singularity occurs, so the initial non-singular velocity is zero ( ˙Y(0) = 0 ).
Section III, paragraph after Eq. (15): this assumption zeroes matrices B and C and reduces the singular-direction acceleration to a single quadratic term. Without it, the sign-based selection rule does not follow.
domain assumption At a singularity, the Jacobian rank drops to m-1 and only the trailing singular values are zero, so the singular direction is the last column of U.
Section II, Eqs. (4)-(7): the paper first states that all sigma values vanish but then uses the standard result that only the last columns of U are affected, implying only trailing singular values are zero.
standard math Standard SVD properties and the differential kinematics relation ˙X = J ˙q hold.
Section II, Eq. (1): the paper builds on the standard Jacobian velocity mapping and its singular value decomposition.

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Cite this review

Pith. "Pith review of Identification Algorithm to Determine the Trajectory of Robots with Singularities." pith.science (2026). https://pith.science/paper/GNAJU2XV

@misc{pith2026191106632,
  author       = {Pith},
  title        = {Pith review of: Identification Algorithm to Determine the Trajectory of Robots with Singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GNAJU2XV}},
  note         = {Machine review of arXiv:1911.06632}
}

read the original abstract

Singularity in robot controls is an important problem. By identifying an appropriate trajectory for the robots, the singular situations can be avoided. In this paper an identification algorithm is proposed to control the robot such that it can change its direction to avoid the singularity situation. Base on the singular value decomposition, the proposed algorithm is developed for the non-redundant, single-rank robots. The proposed method is employed on a robot with six degrees of freedom, in order to identify its feasible trajectory. Keywords: Singularity; Trajectory identification; Robot control; Identification algorithm; Singular value decomposition.

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Works this paper leans on

18 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

”Inverse kinematic solutions with singularity robustness for robot manipulator control.” Journal of dynamic systems, measurement, and control 108, no

Nakamura, Yoshihiko, and Hideo Hanafusa. ”Inverse kinematic solutions with singularity robustness for robot manipulator control.” Journal of dynamic systems, measurement, and control 108, no. 3 (1986): 163-171

work page 1986

[2] [10]

5, Issue 08, pp

Hossein Shariﬁ, ”Singular Identiﬁcation of a Constrained Rigid Robot”, International Research Journal of Engineering and Technology (IRJET), V ol. 5, Issue 08, pp. 941-946, 2018

work page 2018

[3] [1]

Modeling, Identiﬁcation & Control of Robots

Khalil, W., and Dombre, E. Modeling, Identiﬁcation & Control of Robots. London ; Sterling, V A: Kogan Page Science, 2004. Kogan Page Science Paper Edition

work page 2004

[4] [2]

Cheah, Chien Chern., and Li, Xiang. Author. Task-Space Sensory Feedback Control of Robot Manipulators. 1st Ed. 2015.. ed. 2015. Intelligent Systems, Control and Automation: Science and Engineering

work page 2015

[5] [3]

Advances in Robot Kinematics

Lenarcic, J., and Khatib, Oussama, Editor. Advances in Robot Kinematics. 2014

work page 2014

[6] [4]

”Singularity Analysis of a Plane-Symmetry 3-RPS Parallel Robot Based on Translational/Rotational Jacobian Matrices.” Applied Mechanics and Materials 121-126 (2011): 1590-594

Shi, Yan, Hong Xin Yue, Yi Lu, and Lian He Guo. ”Singularity Analysis of a Plane-Symmetry 3-RPS Parallel Robot Based on Translational/Rotational Jacobian Matrices.” Applied Mechanics and Materials 121-126 (2011): 1590-594

work page 2011

[7] [5]

Structural Synthesis of Parallel Robots

Gogu, Grigore. Structural Synthesis of Parallel Robots. Dordrecht ; [London]: Springer, 2008. Solid Mechanics and Its Applications; v. 149, 159, 173, 183

work page 2008

[8] [8]

”Classiﬁcation of serial robot-manipulators with nonremovable singularities.” Journal of Mechanical Design 118, no

Karger, A. ”Classiﬁcation of serial robot-manipulators with nonremovable singularities.” Journal of Mechanical Design 118, no. 2 (1996): 202-208

work page 1996