Design, Modeling, and Control of Norma: a Slider & Pendulum-Driven Spherical Robot

Hassan Zargarzadeh; M-Mahdi Naddaf-Sh; Saeed Moazami; Srinivas Palanki

REVIEW 3 major objections 5 minor 34 references

Design, Modeling, and Control of Norma: a Slider & Pendulum-Driven Spherical Robot

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper derives the full dynamics of a two-actuator spherical robot and shows, in simulation, that it can roll, turn, and track a smooth path.

desk verdict A novel slider-pendulum spherical robot with a careful EL derivation, but the turning model is imposed kinematically and the validation is self-referential. read the letter →

arxiv 1908.02243 v2 pith:OYMICOBJ submitted 2019-08-06 cs.RO

Saeed Moazami , M-Mahdi Naddaf-Sh , Srinivas Palanki , Hassan Zargarzadeh This is my paper · ORCID

classification cs.RO

keywords SphericalrobotEuler-LagrangedynamicsPendulum-drivenSlideractuationNonholonomicmobilePathtrackingcontrolPIDRollingconekinematics

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

Norma is a spherical robot built around a diagonal shaft fixed inside a ball. A pendulum rotating about the shaft rolls the ball forward and backward, while a slider translating along the shaft shifts the center of gravity sideways, tilting the ball and making it turn. The paper derives the robot's full equations of motion, $M(q)\ddot q + C(q,\dot q)\dot q + G(q)=\tau$, for the four generalized coordinates $q=[\theta,\alpha,\phi,\delta]^T$, using the Euler-Lagrange method under minimal simplifying assumptions. It then couples a pure-pursuit kinematic controller with PID loops that convert desired roll speed and slider position into pendulum torque $\Gamma$ and slider force $\Phi$. In MATLAB simulation the closed-loop system tracks a smooth curved path, which is the evidence offered that the model and controller are sound. If correct, this is a mechanical design that needs only two actuators for both forward motion and steering, with a dynamics model detailed enough to guide control design before hardware is built.

What carries the argument

The load-bearing object is the four-coordinate Euler-Lagrange dynamics written in the standard second-order form, together with the rolling-cone kinematic constraint used for turning. The generalized coordinates are $\theta$ (sphere roll about the transverse shaft), $\alpha$ (pendulum rotation about the shaft), $\phi$ (tilt about the longitudinal axis), and $\delta$ (slider displacement along the shaft). The rolling-cone relation supplies yaw rate from roll rate and tilt angle, while the Lagrangian built from the kinetic and potential energies of the sphere, pendulum, and slider supplies the matrices $M$, $C$, and $G$. Christoffel symbols are used to make the Coriolis/centripetal matrix skew-symmetric with $\dot M-2C$. This structure is what lets the authors separate kinematics, handled by pure pursuit, from dynamics, handled by PID tracking of the desired roll rate and slider position.

What would settle it

Lock the slider at a known off-center position so the ball tilts by $\phi$, roll it forward at a roughly constant roll rate $\dot\theta$ on a flat hard floor, and measure the heading $\psi$ over several full revolutions. Compare $\dot\psi$ with $-\dot\theta\,\sin\phi/(R\cos\phi-R\sin\phi)$; a disagreement larger than the measurement error would refute the rolling-cone model and, with it, the derived dynamics and the simulated tracking results.

Watch

Extended reading notes

Core claim

The central claim is that a spherical robot can obtain both rolling and steering from two internal actuators, and that this behavior is captured by a closed Euler-Lagrange model rather than by a simplified or decoupled approximation. The tilt angle $\phi$ is coupled to yaw rate through a rolling-cone relation $\dot\psi=-\dot\theta\,\sin\phi/(R\cos\phi-R\sin\phi)$, so slider motion that changes $\phi$ produces turning, whereas pendulum torque $\Gamma$ drives $\dot\theta$ and hence forward motion. The derived model has the standard robot form $M(q)\ddot q+C(q,\dot q)\dot q+G(q)=\tau$, and the paper constructs the Coriolis/centripetal matrix using Christoffel symbols so that $\dot M-2C$ is skew-symmetric. A pure-pursuit outer loop specifies desired values for $\dot\theta$ and $\delta$, and PID loops generate $\Gamma$ and $\Phi$; simulations show the robot converging to the desired trajectory with tracking error near zero.

Load-bearing premise

The load-bearing premise is that a tilted sphere rolling on flat ground behaves exactly like a solid cone, without slipping, so each roll changes the heading by an amount fixed by the tilt angle; if that geometric rule is wrong, the yaw rate used to build the dynamics and the controller is wrong too.

Editorial extensions

If this is right

A physical Norma robot would need only the pendulum torque and the slider force as inputs to execute both forward motion and turns on flat ground.
Because the model is in $M(q)\ddot q+C(q,\dot q)\dot q+G(q)=\tau$ form, standard robot-control techniques such as computed torque, gain scheduling, or adaptive parameter estimation can be applied directly to it.
The pure-pursuit plus PID architecture gives a decoupled design procedure: choose desired roll speed and slider position from path error, then tune two independent PID loops for the actuators.
The simulations imply that, with the stated masses, lengths, and PID gains, tracking error converges to a small neighborhood of zero on a smooth path.
A working two-input design would make spherical robots simpler to build than multi-actuator gimbal or flywheel designs, since steering and propulsion come from the same internal mass shifts.

Reading between the lines

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

A direct hardware test of the cone kinematic relation is the fastest way to decide whether the model transfers to a real robot: lock the slider at fixed $\phi$, roll the ball at known $\dot\theta$, and compare measured heading change with $\dot\psi=-\dot\theta\,\sin\phi/(R\cos\phi-R\sin\phi)$.
The point-mass assumptions for the slider and pendulum bob, plus a massless rod, are convenient for the Lagrangian but will be violated in a physical build; adding those inertias or showing that they are negligible is a natural extension the paper does not address.
If the ideal no-slip cone model were replaced by a contact model with a finite contact patch and slip, the same control law could be re-evaluated to show how much of the simulated tracking performance depends on the no-slip assumption.
The control demonstration is limited to smooth trajectories; extending the scheme to paths with corners or to 3D terrain would likely require a modified kinematic law for $\dot\psi$.

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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

Colleague, here's my take on Moazami et al. The Norma mechanism is a genuine new combination — a slider and a pendulum on the same diagonal shaft — and the Euler-Lagrange derivation is careful and complete as far as it goes. The inertia matrix is symmetric, the Coriolis construction uses the standard Christoffel trick, and the Appendix gives the full expansion. That is real work, and someone wanting to build a slider-pendulum SR could start from these equations.

The soft spot is load-bearing. The turning yaw rate ψ̇ is introduced through a rolling-cone analogy in Section III-A, but ψ never appears as a generalized coordinate. The dynamics in (46)–(47) involve only θ, α, φ, δ; there is no ψ̇ in the shell angular velocity (8), in the kinetic energies, or in M, C, G. The world trajectory is then recovered by integrating the assumed algebraic relation (6). So the 'complete' model does not predict turning; it imposes turning kinematically. For a sphere rolling without slip on a plane, the vertical spin component is unconstrained by the no-slip condition, so Eq. (6) is not a general kinematic law. It may be a legitimate planning constraint if enforced by control, but it is not a dynamic consequence. As written, the paper overstates the completeness of its model.

Second, the validation is circular. The model is verified by simulating the same model; there is no experiment, no independent data, no comparison against a simpler benchmark, and no released code. The abstract I was given claims validation 'against experimental results,' but the full text has no experiments. That needs to be corrected either way.

Minor issues: the pure-pursuit plus PID control is standard and fine; the gains are hand-tuned, which is normal. The citation pattern looks honest, and the self-citation in the conclusions is to a companion kinematics paper, which is fine.

Bottom line: the mechanism is worth knowing about, and the derivation is a useful starting point, but the turning model is a serious gap. The authors could fix it by adding ψ (or an equivalent spin coordinate) to the generalized coordinates and re-deriving the kinetic energy, or by clearly reframing the paper as modeling the robot under a prescribed nonholonomic rolling constraint. As submitted, I would not trust the simulated turning trajectories as predictions.

Referee Report

3 major / 5 minor

Summary. The paper presents the design, dynamics modeling, and control of Norma, a spherical robot actuated by a pendulum and a slider on a diagonal shaft. The authors use the Euler-Lagrange method to derive an explicit 4-DOF model M(q)q̈ + C(q,q̇)q̇ + G(q) = τ with generalized coordinates q = [θ, α, φ, δ], then propose a pure-pursuit kinematic controller together with PID dynamic controllers for the pendulum torque and slider force. The reported contribution is that this 2-actuator mechanism can perform rolling and turning maneuvers, and that MATLAB simulations demonstrate accurate path tracking. The Appendix gives element-wise expressions for M, C, and G, and the Coriolis matrix is constructed via Christoffel symbols so that Ṁ − 2C is skew-symmetric.

Significance. If the derived model and kinematic relations were correct, the paper would be a useful reference for simulation-based control design of pendulum-and-slider spherical robots. The first-principles derivation is detailed, the inertia matrix is symmetric by construction, and the explicit Appendix A is a valuable checkable artifact. However, the turning kinematics rests on an unproven cone analogy, the yaw coordinate is absent from the dynamics, and the 'verification' is a self-simulation of the same model with no independent experimental or model-based benchmark. These issues directly affect the paper's central claim of a complete model that enables turning control, so the current manuscript does not establish that claim.

major comments (3)

[Section III-A, Eqs. (3)-(6)] Equation (6) does not follow algebraically from Eqs. (3)-(5). Substituting c_r = R C_φ and ρ = R C_φ − R S_φ into c_r θ̇ = ρ ψ̇ gives ψ̇ = C_φ/(C_φ − S_φ) θ̇, not the printed ψ̇ = −S_φ/(R(C_φ − S_φ)) θ̇. Beyond the algebra, the imaginary cone model is not justified for a sphere: for a sphere rolling without slip on a horizontal plane, the no-slip condition constrains only the horizontal velocity at the contact point, and the yaw rate should be the vertical component of the sphere's angular velocity, which from Eq. (8) is ψ̇ = −S_φ θ̇ with no dependence on R. Because Eq. (12) and all of the Section V trajectories inherit this relation, the simulated turning behavior is not grounded in the dynamics of a rolling sphere.
[Section III-B and Appendix A] The yaw angle ψ is not a generalized coordinate. The Lagrangian in Eq. (45) and the explicit M, C, and G elements in Appendix A depend only on θ, α, φ, δ and their derivatives; the angular velocities in Eqs. (8), (29), and (36) contain no ψ̇ term; and the generalized force vector in Eq. (47) has zero component for yaw. Consequently there is no equation of motion governing ψ, and the turning motion in the simulations is imposed by the algebraic relation of Section III-A rather than produced by the dynamics. The paper's claim of a complete dynamic model that enables turning control is therefore not supported as written.
[Section V, Simulations] The claimed verification of the mathematical model is circular: the simulation plant is the same Euler-Lagrange model (46)-(47) whose accuracy is being claimed, and the paper provides no independent experimental data, no comparison against a separately derived model, and no benchmark from the spherical-robot literature. The abstract's phrase 'against experimental results' is not supported by any experimental section in the manuscript. The claims should either be reduced to 'closed-loop simulation of the derived model' or an independent validation should be added.

minor comments (5)

[Throughout] The equations are heavily garbled, with missing hats on unit vectors, missing dot notation, and corrupted trigonometric subscripts (for example in Eqs. (6), (8), and (45)); the manuscript needs a careful typesetting pass before it can be checked reliably.
[Section IV-B and Table 1] The text does not specify which PID block in Fig. 5 corresponds to PID1, PID2, and PID3 in Table 1, and the gains are said to be selected by trial-and-error without any stability or robustness analysis.
[Eqs. (50)-(51)] The expressions for the desired slider displacement and rolling velocity are ambiguous: the normalized error gain ||e||/(||e||+k3) appears in different forms, and the units of k1, k2, and k3 are not discussed.
[Section V, Eq. (52)] The desired trajectory specification is corrupted ('0.01 0.02, 2 ,1.5 S S') and should be written explicitly so that the simulation can be reproduced.
[References] The paper repeatedly cites the authors' own preprint [4] but does not compare the proposed model with existing pendulum-driven spherical-robot models from references [12]-[16], [21]-[27]; such a comparison would help position the contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Euler-Lagrange derivation is self-contained; the turning-cone relation is a stated modeling assumption, not a fitted, self-cited, or definitionally forced prediction.

full rationale

The central derivation chain is direct: the paper defines generalized coordinates q=[theta, alpha, phi, delta], builds kinematic velocity expressions for the shell, pendulum, and slider (Eqs. 8, 9, 22-36), assembles kinetic and potential energies (Eqs. 20, 37-45), and applies the Euler-Lagrange equation (Eq. 13) to obtain M, C, and G in Eq. 46. The Appendix A entries are obtained by differentiating that same Lagrangian; no parameter is fitted to data and then renamed as a prediction. The turning relation psi_dot = -S_phi theta_dot / (R(C_phi - S_phi)) in Eq. 6 is explicitly introduced as a rolling-cone modeling assumption and is used for kinematic trajectory integration (Eqs. 8 and 12); it is not derived from the dynamics, but it is also not an output of the dynamics. Thus, even if that kinematic assumption is physically questionable or incomplete, the issue is modeling fidelity rather than circularity. The PID gains in Table 1 are trial-and-error tuned, but they are controller gains, not model parameters, and the simulation-based verification is a validation limitation rather than a circular derivation. The only self-citation, reference [4], concerns future 3D-terrain kinematics of Norma and is not load-bearing for the dynamics or control claims presented here. No step reduces to its own input by construction, and no load-bearing claim depends on an unverified self-citation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central model derivation relies on standard rigid-body mechanics and the stated idealizations (no-slip, point masses, rigid shaft). No new physical entities, forces, or conserved quantities are introduced. The only hand-chosen quantities are the controller gains, which affect the control demonstration but not the model derivation. The rolling-cone turning model is an ad hoc geometric postulate specific to this paper and is the most fragile nonstandard assumption.

free parameters (4)

k1 = 0.1
Kinematics controller gain in Eq. (50); chosen by hand via trial-and-error.
k2 = -0.1
Kinematics controller gain in Eq. (51); chosen by hand via trial-and-error.
k3 = 5
Normalized error gain in Eqs. (50)-(51); chosen by hand via trial-and-error.
PID gains (Table 1) = PID1: KP=1,KI=0,KD=0; PID2: KP=10,KI=7,KD=3; PID3: KP=15,KI=3,KD=10
Dynamics controller gains tuned by trial-and-error; required for the simulated tracking performance but not fitted to physical data.

assumptions (5)

domain assumption The sphere rolls without slipping over a flat horizontal surface.
Assumption 1 in Section 2; underlies the rolling-cone kinematics in Eqs. (3)-(6) and the velocity relations in Eqs. (8)-(9).
domain assumption The slider and pendulum bob are point masses, and the pendulum rod is massless.
Assumption 2 in Section 2; used to compute potential and kinetic energies in Eqs. (20), (21), (42), and (44).
ad hoc to paper Turning motion is modeled by an imaginary rolling cone with apex angle 2φ and the equal-arc-length no-slip rule.
Section III-A, Eqs. (3)-(6); this geometric model converts tilt angle φ into yaw rate ψ̇ and is not validated against experiments or alternative kinematic models.
domain assumption The shaft is rigidly fixed to the spherical shell, so they are treated as a single rigid body.
Section 2 model description; used to combine shell and shaft inertia in Eqs. (38)-(40).
standard math Euler-Lagrange equations with generalized coordinates q = [θ, α, φ, δ] describe the robot dynamics.
Eq. (13); standard Lagrangian mechanics applied to the chosen coordinates, assuming ideal constraints of the mechanism.

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Cite this review

Pith. "Pith review of Design, Modeling, and Control of Norma: a Slider & Pendulum-Driven Spherical Robot." pith.science (2026). https://pith.science/paper/OYMICOBJ

@misc{pith2026190802243,
  author       = {Pith},
  title        = {Pith review of: Design, Modeling, and Control of Norma: a Slider & Pendulum-Driven Spherical Robot},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OYMICOBJ}},
  note         = {Machine review of arXiv:1908.02243}
}

read the original abstract

This paper discusses the design, modeling, and control of Norma, a novel 2 DOF mobile spherical robot (SR). The propelling mechanism of this robot consists of two actuators: a slider, and a rotational pendulum located on the SR's diagonal shaft. The slider can translate along the shaft and shift the robot's center of gravity towards the robot's sides. The pendulum rotates around the shaft to propel the SR to roll forward and backward. These two actuators enable the SR to perform both rolling and turning maneuvers as a nonholonomic robot. The advantage of the proposed mechanical design lies in its convenience of physical implementation, agility, and accurate mathematical model. The Euler Lagrange approach is utilized to derive the dynamics of the proposed mechanical structure using minimum simplifications possible. Further, a path tracking control scheme is introduced for a smooth trajectory. Finally, simulations are carried out in MATLAB to verify the accuracy of the mathematical model and the effectiveness of the controller against experimental results.

Figures

Figures reproduced from arXiv: 1908.02243 by the authors.

**Figure 2.** The reference frames used in the spherical robot kinematics model. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Rolling cone model of the spherical robot’s turning maneuver. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: The schematic of PID controller structure for the spherical robot. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Robot’s desired and simulation trajectories in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: SR trajectory in X andY directions versus time [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: Control actions governed by the controllers. [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Robot’s Tilting angle [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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

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