by Jacob Goodman
NTNU Norwegian University of Science and Technology
Modern engineering systems are expected to operate in environments that are complex, uncertain, and full of restrictions. Robots must avoid obstacles, aerial vehicles must remain within safe operating envelopes, and mechanical systems often contain configurations that are mathematically admissible but physically undesirable. The central difficulty is not only steering the system, but ensuring that its motion remains feasible at all times. Many widely used control approaches treat constraints as external additions to the dynamics. Optimization-based methods, such as model predictive control, enforce constraints through repeated numerical solves. Feedback-based approaches rely on barrier functions or penalties. While powerful, these methods can become computationally demanding, sensitive to modeling errors, and difficult to reconcile with the intrinsic geometric structure of mechanical systems [3].
The MSCA project GNACS (Geometric and Numerical Analysis of Control Systems) attacks this problem from a different perspective: instead of only shaping the control inputs, we shape the geometry of the system itself.
From Geometry to Control
Many controlled systems evolve on curved configuration spaces rather than flat Euclidean ones. Examples include rigid body orientation, configurations of robotic manipulators, and vehicles with rolling constraints. In practice, not all mathematically possible states are physically meaningful or desirable. A helicopter model may allow upside-down orientations even if the real vehicle cannot safely operate there. A robot’s configuration space may include regions corresponding to collisions. In other situations, certain regions are technically allowed but undesirable because they correspond to high energy use, risk, or poor performance.
The key idea of GNACS is to modify this geometry so that undesirable states become dynamically disfavored. Instead of explicitly forbidding regions, we reshape the “kinetic landscape” of the system so that entering those regions requires substantially more effort. Safe or desirable regions, by contrast, remain dynamically natural. This idea is illustrated schematically in Figure 1. We then use control inputs to align the real system with this modified, geometry-shaped behavior. This geometric shaping acts as a form of preconditioning for control. Additional control design may be needed, but the system dynamics are already biased toward avoiding unsafe regions, remaining within feasible operating zones, and favoring low-cost or high-performance behaviors.
Figure 1: The original surface (a plane) is modified to include large hills around certain regions. When simulating a mechanical system on such a geometry, solutions will naturally avoid these hilly (high cost) regions. Control is then used to make the agent behave as if it were moving on the deformed surface; see [1].
Geometry as a Modeling Device
The geometric modification has another powerful interpretation. Changing the geometry can be seen as deforming the underlying space itself. This allows us to use simple model spaces to represent complex environments. A surface that is flat in the original model can behave as if it were bumpy or textured after geometric shaping. In this sense, geometry becomes a modeling layer for environmental effects.
We can even introduce randomness through geometry: by making the geometric modification depend on stochastic ingredients, we inject structured uncertainty into the dynamics without destroying the underlying geometric framework (see Figure 2). This contrasts with many standard approaches in control, where disturbances are introduced directly into the equations of motion, often obscuring or breaking important dynamical and geometric structure. In this way, GNACS aims to provide a principled setting for testing the robustness of control laws and numerical methods under uncertainty.
Figure 2: Simulating the solutions to a simple planar dynamical system over a bumpy surface. Noise is injected into the geometry of the surface, rather than directly into the dynamics, providing a structured model of motion over a rough environment [4].
Curvature Control for Numerical Methods
As dynamical systems and control progressively lean into their geometric foundations, so too do many numerical methods. The areas of Riemannian optimization and geometric integration attempt to adapt algorithms traditionally used for optimization and simulation on flat spaces to spaces with curvature. In some cases, taking advantage of geometry even acts a way to provide stability and long-term accuracy in simulations. However, a challenge that has emerged in these efforts is that algorithmic performance often depends strongly on the curvature of the space [2]. In some cases, algorithms only apply in regions of negative sectional curvature. In other cases, negative sectional curvature acts a hard barrier to how much an algorithm can be accelerated [5].
Counterintuitive as it may seem, curvature is determined by the choice of metric rather than the underlying topological space. From a topological perspective, many objects we call ‘spheres’ look nothing like the round sphere we visualize, yet they are equivalent in the sense that matters to topology. For numerical analysts, the tools they use often depend strongly on a choice of metric, while the underlying problems they aim to solve (minimizing an objective function, simulating the solutions of a differential equation, etc.) often do not. This separation creates an opportunity: when the problem itself is metric-independent, the metric can be chosen or modified to improve algorithmic behavior without altering the structure of the original task, or even to extend the application of an algorithm to regimes where it would fail to be applicable with the standard choice in metric.
[1] J.A. Acosta, A. Bloch, D. Martín de Diego Completeness of Riemannian Metrics: An Application to the Control of Constrained Mechanical Systems, IEEE Transactions on Automatic Control, 2025.
[2] M. Ghirardelli, B. Owren, E. Celledoni. Conditional stability of the Euler method on Riemannian manifolds, Preprint, 2025. Available at:
[3] A.J. Krener and H. J. Sussmann, Lie brackets and local controllability of nonlinear systems, SIAM Journal on Control, 11(3), 443–472, 1973.
[4] J.A. Lázaro-Camí and J.P. Ortega, Stochastic Hamiltonian dynamical systems, Reports on Mathematical Physics, 61(1), 65–122, 2008.
[5] H. Zhang and S. Sra, First-order methods for geodesically convex optimization, Proceedings of the Conference on Learning Theory (COLT), 2016.