Vertical Planetary Landing on Sloped Terrain Using Optical Flow Divergence Estimates

TL;DR

The paper tackles autonomous landings of small, resource-constrained spacecraft on sloped terrain by leveraging two local optical flow divergences to separately regulate thrust and attitude. It formulates a nonlinear control law within Incremental Nonlinear Dynamic Inversion (INDI) to track flow-divergence observables and to align attitude with incline, including a dedicated touchdown strategy for slopes. Numerical 2D simulations demonstrate exponential decay of height and vertical speed on horizontal terrain and robust slope alignment on inclined surfaces, with superior tracking and drift mitigation compared to PID controls. The work offers a lightweight, robust landing approach suitable for future planetary missions with limited sensing and processing resources.

Abstract

Autonomous landing on sloped terrain poses significant challenges for small, lightweight spacecraft, such as rotorcraft and landers. These vehicles have limited processing capability and payload capacity, which makes advanced deep learning methods and heavy sensors impractical. Flying insects, such as bees, achieve remarkable landings with minimal neural and sensory resources, relying heavily on optical flow. By regulating flow divergence, a measure of vertical velocity divided by height, they perform smooth landings in which velocity and height decay exponentially together. However, adapting this bio-inspired strategy for spacecraft landings on sloped terrain presents two key challenges: global flow-divergence estimates obscure terrain inclination, and the nonlinear nature of divergence-based control can lead to instability when using conventional controllers. This paper proposes a nonlinear control strategy that leverages two distinct local flow divergence estimates to regulate both thrust and attitude during vertical landings. The control law is formulated based on Incremental Nonlinear Dynamic Inversion to handle the nonlinear flow divergence. The thrust control ensures a smooth vertical descent by keeping a constant average of the local flow divergence estimates, while the attitude control aligns the vehicle with the inclined surface at touchdown by exploiting their difference. The approach is evaluated in numerical simulations using a simplified 2D spacecraft model across varying slopes and divergence setpoints. Results show that regulating the average divergence yields stable landings with exponential decay of velocity and height, and using the divergence difference enables effective alignment with inclined terrain. Overall, the method offers a robust, low-resource landing strategy that enhances the feasibility of autonomous planetary missions with small spacecraft.

Figures (4)

• Figure 1: Simplified 2D spacecraft model during landing on an inclined surface.
• Figure 2: Landing experiments on horizontal terrain with flow divergence setpoints of $\vartheta_{1}^{*}$ = -0.1, $\vartheta_{2}^{*}$ = -0.2, $\vartheta_{3}^{*}$ = -0.3 rad/s. Plots (left to right) show the height $h$, vertical velocity $\dot{h}$, observed flow divergence $y_1$, and control input $u_1$ of the vehicle. The INDI controller achieves accurate tracking of the flow divergence setpoints, resulting in the expected exponential decay of height and velocity, with initial acceleration followed by deceleration toward touchdown.
• Figure 3: Landing experiments on inclined terrain with slopes of $10^\circ$, $20^\circ$, and $30^\circ$ (left to right). Plots show the vehicle's height $h$, roll angle $\phi$, flow divergence difference $y_2$, and control inputs $u_1$ and $u_2$. The results demonstrate exponential decay of height, activation of roll alignment near touchdown as $y_2$ is regulated to zero, and coordinated thrust and attitude control for stable landings across all slopes.
• Figure 4: Landing trajectories on inclined terrain with slopes of (a) $10^\circ$, (b) $20^\circ$, and (c) $30^\circ$ using the proposed method, and a final touchdown without drift compensation (rightmost). (a-c) Vehicle motion is shown in the $(Z,Y)$ plane with snapshots overlaid at fixed time intervals. The sequence highlights the initial acceleration (large spacing between snapshots), gradual deceleration (smaller spacing), and final touchdown with the roll angle aligned to the slope. The rightmost case without drift compensation shows the vehicle drifting far away from the intended landing site.