Control co-design jointly optimizes a robot's physical design and its controller. Existing techniques are limited in the number of model parameters and the complexity of the physical models they can handle. In this paper, we propose a gradient-based bilevel formulation that replaces the dynamics models of earlier techniques with a differentiable neural surrogate and differentiates through the trajectory optimizer using a structure-exploiting approach. We validate the framework on fixed-wing gliders, using NeuralFoil as the aerodynamic surrogate. The tasks are maximum-distance gliding, minimum-distance landing, and perching. Experiments on 2D airfoils and 3D wings show that our method runs about five times faster than a state-of-the-art evolutionary baseline. It also outperforms pipelines that optimize aerodynamic proxy metrics.
Figures & tables
Fig. 1 : Co-design . We jointly optimize the airfoil shape (left, colored by iteration) and the controls. The shape determines the trajectories (right) through a neural surrogate, and the gradient flows back by implicit differentiation. Without the validity constraint (bottom), the optimizer drives the shape to where the surrogate is unreliable.
Fig. 2 : Overview. The neural surrogate is evaluated on the flight envelope once per design update and replaced by a polynomial, which the trajectory optimizations, one per task and launch state, use instead of the network. The design gradient and the validity constraint are detailed in Secs. III-B and III-C .
Fig. 3 : Glider model. (a) Dots mark the centers of mass. Lengths on the left are measured from O . (b) Model parameters.
Fig. 4 : Tasks and controls. Optimized airfoils (left), trajectories (middle), and mission cost over the outer iterations (right) for the three tasks, colored by run progress (dashed: initial design).
Method
Gliding
Landing
Perching
min
Initial design
− 24.3 ± 0.5
– (0/3)
– (2/3)
16
Seq. max L/D
− 40.8 ± 12.8
– (2/3)
– (1/3)
93
Seq. max CL,max
− 25.5 ± 4.9
– (2/3)
25.49 ± 14.79
83
Seq. max CD
− 37.3 ± 2.0
– (2/3)
– (2/3)
81
GA
− 33.0 ± 7.2
244 ± 7
8.63 ± 0.19
246
CMA-ES
− 39.3 ± 9.5
– (2/3)
9.13 ± 0.57
265
TABLE I : Comparison with the baselines. Mission cost Φ (lower is better), re-scored with XFoil. Mean ± std over I1–I3 if feasible on all three, else the number of feasible inits ( t/c≥0.05 , c≥0.6 at visited operating points). Bold : best; min: median wall-clock.
Fig. 5 : Pareto front between landing cost and glide distance. The line interpolates all the non-dominated designs among all those evaluated by the sweep. Gray represents dominated designs.
Fig. 6 : 3D wing, perching task. Time per design update versus number of design parameters. Only the time to compute the wing model grows. The insets show wings with 4 and 7 sections.
Fig. 7 : Time per design update versus surrogate size. Ours stays constant; with the network inside the OCP or the simultaneous NLP, the cost grows.
Fig. 8 : Without the confidence constraint, the optimizer exploits the model. Short landing. Left: model cost and mean confidence per iteration. Right: model error against XFoil for each final design (gray: XFoil fails; blue: flights). Bottom: final airfoils.
Robotics and Biology Laboratory, Technische Universität Berlin · Science of Intelligence (SCIoI), Cluster of Excellence, Berlin · Robotics Institute Germany (RIG)