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arxiv: 1906.09134 · v1 · pith:HAN5K6DOnew · submitted 2019-06-21 · 🧮 math.OC

Symmetry and Motion Primitives in Model Predictive Control

Kathrin Fla{\ss}kamp , Sina Ober-Bl\"obaum , Karl Worthmann This is my paper

Pith reviewed 2026-05-25 18:44 UTC · model grok-4.3

classification 🧮 math.OC
keywords symmetriesmotion primitivesmodel predictive controlasymptotic stabilitymobile robotparallel parkingoptimal controlnonlinear systems
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The pith

Symmetries allow motion primitives to guarantee asymptotic stability in model predictive control closed loops.

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

The paper establishes that symmetries in nonlinear dynamical systems, such as rotational and translational invariances in mechanical systems, permit characterization of solution trajectories and restriction to symmetry-induced dynamics via motion primitives. This restriction is used inside model predictive control to prove asymptotic stability of a desired set point for the closed-loop system. The claim is supported by a detailed mobile robot example with numerical demonstration on parallel parking. When the optimization criterion does not match the symmetry action, explicit guidelines are supplied for deriving stability guarantees anyway. A reader would care because the linkage connects motion planning techniques directly to optimal control while preserving rigorous stability.

Core claim

Restricting the model predictive control optimization to symmetry-induced dynamics through motion primitives establishes asymptotic stability of a desired set point with respect to the MPC closed loop for systems that possess the required symmetries. This is shown numerically for a mobile robot in the parallel parking scenario. When the cost function is inconsistent with the symmetry action, guidelines are provided to derive rigorous stability guarantees based on symmetry exploitation.

What carries the argument

Motion primitives as a quantization of symmetry-induced dynamics inside the model predictive control optimization.

If this is right

  • Asymptotic stability of the set point holds for the MPC closed loop when motion primitives are employed.
  • The approach applies directly to mechanical systems with rotational and translational invariances.
  • Rigorous stability guarantees remain available via provided guidelines even if the cost function breaks symmetry consistency.
  • The restriction to symmetry-induced dynamics functions as a quantization that preserves the ability to reach the target.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry reduction may lower the online computational load of MPC by shrinking the searchable trajectory space.
  • Guidelines for inconsistent costs could be tested on other optimal control formulations beyond MPC.
  • Extension to approximate symmetries in non-ideal mechanical systems would require checking whether the stability margin survives the approximation.

Load-bearing premise

The dynamical system must possess symmetries that allow trajectories to be characterized and restricted via motion primitives while still permitting asymptotic stability to be achieved in the MPC closed loop.

What would settle it

Numerical simulation of the parallel parking scenario in which the MPC closed loop with motion primitives fails to converge asymptotically to the desired set point would falsify the stability result.

Figures

Figures reproduced from arXiv: 1906.09134 by Karl Worthmann, Kathrin Fla{\ss}kamp, Sina Ober-Bl\"obaum.

Figure 1
Figure 1. Figure 1: For u ” pu1 u2q J, u2 ‰ 0, the projection of the trajectory Ť tPr0,δs ϕupt; x 0 q is a segment of a circle. Small arrows indicate the current values of angle x3. To illustrate the symmetry shift, consider ∆x “ p3, 2.5, 1.2q J and the black curve as a starting point. Naively, one may think that the symmetry shift simply equals an addition of g to the state. The two red curves illustrate that this is wrong: … view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the modified terminal constraint for Example 13. 3.3. Optimal Control Problem for the Mobile Robot In Proposition 6 we identified characteristic motions of the mobile robot, i.e. circles and straight lines, which are formally trim primitives. Since any pair pu1, u2q of constant control values generates a trim we now switch perspective and define a class of admis￾sible controls which guarant… view at source ↗
Figure 3
Figure 3. Figure 3: Optimal solution for the mobile robot as defined in Example 23: As it has been shown in Proposition 18, the quadratic part of the optimal control effort is uniformly distributed, i.e. x5 increases linearly, although x4 and u1, u2 do not. 4.2. Energy and Fuel Consumption for Finite Sets of Motion Primitives Let us now focus on the motion primitives setting. That is, we further restrict U in the definition o… view at source ↗
Figure 4
Figure 4. Figure 4: OCP solution for the parallel parking problem of the mobile robot in T “ 8 while minimizing ` “ ||u||2 I ` 1 2 ||u||I for the best sequences of trim primitives obtained from a global search on possible sequences. We restrict to the library of trim primitives as given in [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: OCP solutions for the parallel parking problem of the mobile robot in T “ 8 while minimizing ` “ ||u||2 I ` 1 2 ||u||I for different sequences of trim primitives [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time optimal solution for the parallel parking problem of the mobile robot for different sequences of trim primitives. exists a solution with lower costs (green). Other solutions have much higher costs and would not be chosen in the unconstrained scenario. However, they might become of importance as soon as obstacle avoidance is included in the problem. In [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: MPC solution for the parallel parking problem of the mobile robot in T “ 8 with MPC steps of δ “ 1. After six iterations, the final point is reached. The optimal sequence of trim primitives is replanned at t “ 2 and t “ 3. Additionally, at t “ 2 and t “ 3 a replanning occurs, i.e. another sequence of motion primitives becomes more efficient than the old solution. (Note that a previous solution can always b… view at source ↗
read the original abstract

Symmetries, e.g. rotational and translational invariances for the class of mechanical systems, allow to characterize solution trajectories of nonlinear dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g. by using the concept of motion primitives, may be considered as a quantization of the system. Symmetry exploitation is well-established in both motion planning and control. However, the linkage between the respective techniques to optimal control is not yet fully explored. In this manuscript, we want to lay the foundation for the usage of symmetries in Model Predictive Control (MPC). To this end, we investigate a mobile robot example in detail where our contribution is twofold: Firstly, we establish asymptotic stability of a desired set point w.r.t. the MPC closed loop, which is also demonstrated numerically by using motion primitives applied to the parallel parking scenario. Secondly, if the optimization criterion is not consistent with the symmetry action, we provide guidelines to rigorously derive stability guarantees based on symmetry exploitation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the incorporation of symmetries (e.g., SE(2) for mobile robots) and motion primitives into Model Predictive Control. It claims to establish asymptotic stability of a desired set point for the MPC closed-loop system when the problem is restricted to symmetry-induced dynamics via motion primitives, demonstrates this numerically on a parallel-parking task, and supplies guidelines for obtaining stability guarantees when the stage cost is not invariant under the symmetry action.

Significance. If the stability result is rigorous, the work would usefully bridge symmetry-based motion planning with MPC, offering a route to reduced computational complexity while retaining formal guarantees for mechanical systems possessing rotational and translational invariances. The numerical parallel-parking example supplies concrete evidence of applicability. No machine-checked proofs or parameter-free derivations are reported.

major comments (2)
  1. [abstract / stability theorem] The central stability claim (abstract) requires that the finite set of motion primitives generates trajectories dense enough in the symmetry group to preserve asymptotic stability to a singleton equilibrium rather than convergence to a discrete lattice. The skeptic note correctly identifies that a non-dense subgroup would violate the claimed property for arbitrary initial conditions; the manuscript must explicitly verify this density condition for the SE(2) primitives employed in the parallel-parking example.
  2. [guidelines section] When the optimization criterion is not symmetry-consistent, the provided guidelines for deriving stability guarantees must be shown to restore a strict Lyapunov function on the full state space (not merely on the orbit). The manuscript should supply the precise modification to the terminal cost or constraint set that achieves this.
minor comments (2)
  1. [introduction] Notation for the symmetry action and the induced dynamics on the quotient space should be introduced earlier and used consistently.
  2. [numerical results] The numerical section should report the specific motion primitives chosen, their number, and the resulting discretization of admissible velocities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight important points for rigorizing the stability claims. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [abstract / stability theorem] The central stability claim (abstract) requires that the finite set of motion primitives generates trajectories dense enough in the symmetry group to preserve asymptotic stability to a singleton equilibrium rather than convergence to a discrete lattice. The skeptic note correctly identifies that a non-dense subgroup would violate the claimed property for arbitrary initial conditions; the manuscript must explicitly verify this density condition for the SE(2) primitives employed in the parallel-parking example.

    Authors: We agree that density of the generated subgroup in SE(2) is essential to guarantee convergence to a singleton rather than a discrete set. The primitives in the parallel-parking example are constructed from a finite set of translations and rotations whose generated subgroup is dense in SE(2) (by standard results on dense subgroups generated by irrational rotations and commensurate translations). To make the argument fully explicit and address the concern, we will add a dedicated paragraph in the revised manuscript verifying this density property for the specific primitives used and confirming that it precludes convergence to a lattice for arbitrary initial conditions. revision: yes

  2. Referee: [guidelines section] When the optimization criterion is not symmetry-consistent, the provided guidelines for deriving stability guarantees must be shown to restore a strict Lyapunov function on the full state space (not merely on the orbit). The manuscript should supply the precise modification to the terminal cost or constraint set that achieves this.

    Authors: We concur that the guidelines must be strengthened to explicitly establish a strict Lyapunov function on the full state space. In the revision we will augment the guidelines section with a precise statement of the required modification: the terminal cost is replaced by V_f(x) + d(x, G·x_0)^2 where d is a distance to the symmetry orbit and G·x_0 denotes the target orbit; this term is shown to dominate the non-invariant part of the stage cost outside any neighborhood of the orbit, thereby restoring strict decrease on the entire state space. A short proof sketch demonstrating the resulting Lyapunov property will be included. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper claims to establish asymptotic stability of an MPC closed loop under symmetry-induced dynamics via motion primitives, with numerical demonstration on parallel parking and guidelines for inconsistent criteria. No equations, definitions, or self-citations in the provided text reduce the stability result to a fitted parameter, self-referential definition, or load-bearing prior result by the same authors. The derivation is presented as an adaptation of standard MPC Lyapunov arguments to symmetries, remaining self-contained against external benchmarks in control theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions about symmetries in mechanical systems and properties of MPC; no free parameters or invented entities are evident from the abstract.

axioms (1)
  • domain assumption The dynamical system possesses symmetries (e.g., rotational and translational invariances for mechanical systems) that allow characterization of solution trajectories.
    Invoked in the abstract to justify use of motion primitives as a quantization of the system.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Towards Velocity Turnpikes in Optimal Control of Mechanical Systems

    math.OC 2019-07 unverdicted novelty 6.0

    Introduces velocity turnpike concepts and shows that optimal solutions and turnpike orbits in one mechanical system example both reduce to optimal trim solutions for every finite horizon.

Reference graph

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