Optimal Hiking in the Wild

• Hiking in the Wild is a study of optimizing navigation through undeveloped, rugged landscapes using optimal control theory and digital elevation models.
• It employs numerical methods, including ENO/WENO schemes and TVD Runge-Kutta integration, to solve Hamilton-Jacobi-Bellman equations for time-optimal paths.
• The approach facilitates practical applications like rescue operations by extracting and clustering optimal trajectories through ODE backtracking.

Hiking in the Wild refers to the act of traversing natural, undeveloped landscapes, often with the goal of optimally moving between two points in steep or rugged terrain. This domain has become technically tractable via the application of optimal control theory, numerical partial differential equations, and high-resolution digital elevation models (DEMs). Central to contemporary research is the formalization of human mobility in wild terrain as a time-optimal control problem, leveraging the Hamilton-Jacobi-Bellman (HJB) framework to synthesize efficient, realistic hiking routes in presence of slopes, obstacles, and environmental constraints (Parkinson et al., 2018).

1. Mathematical Formulation of Optimal Navigation

The determination of optimal hiking paths in wild terrain is modeled as a continuous-time control problem. Let $E(x, y)$ denote the terrain elevation and $a, b \in \mathbb{R}^2$ the initial and terminal locations, respectively. The direction of walking, parameterized by $\theta \in [0, 2\pi)$, acts as the control. Define $s(\theta) = (\cos \theta, \sin \theta)$, yielding the directional slope $S(x, \theta)=\nabla E(x)\cdot s(\theta)$. Human walking speed is encoded by a function $V(S)$ (e.g., $V(S)=1.11\exp[-(100S+2)^2/2345]$), which decays rapidly for steep slopes.

The objective is to find a control strategy $\theta(\cdot)$ that minimizes travel time: $T^* = \inf_{\theta(\cdot)} \int_0^{T} 1\,dt,$ subject to the ODE dynamics

$$\dot{x}(t) = V(\nabla E(x(t))\cdot s(\theta(t)))\, s(\theta(t)),\quad x(0)=a,\, x(T)=b.$$

A value function $u(x, t)$ representing the maximal distance from $a$ to $x$ in time $t$ satisfies a level-set type HJB PDE: $$u_t(x, t) + H(x, \nabla u(x, t)) = 0,\quad u(x, 0) = \|x - a\| - \delta,\, \delta \ll 1,$$ with state Hamiltonian: $$H(x, p) = \sup_{\theta \in [0, 2\pi)} \{ V(\nabla E(x)\cdot s(\theta)) [s(\theta)\cdot p] \}.$$ Alternatively, for minimum travel time $T(x)$, the stationary HJB equation is: $H(x, \nabla T(x)) = 1,\quad T(a)=0.$ At every $x$, the maximizing control $\theta^*(x, p)$ is given by $$\arg\max_{\theta} V(\nabla E(x)\cdot s(\theta))[s(\theta)\cdot p]$$ (Parkinson et al., 2018).

2. Numerical Solution Techniques

The numerical solution pipeline discretizes the problem domain using a uniform Cartesian grid $\{x_{ij}\}$. Elevation data $E(x_{ij})$ is sampled from high-resolution DEMs (e.g., USGS 1/3" data). The spatial gradient $\nabla u$ is approximated via upwind (one-sided) finite differences: $$D^+_x u_{ij} = \frac{u_{i+1, j} - u_{ij}}{\Delta x},\quad D^-_x u_{ij} = \frac{u_{ij} - u_{i-1, j}}{\Delta x},$$ with analogous expressions for $y$.

The Godunov Hamiltonian $\widehat{H}_{ij}$ at grid location $(i,j)$ is formulated as: $$\widehat{H}_{ij} = \max_{u \in [D^-_x u, D^+_x u],\, v \in [D^-_y u, D^+_y u]} H(x_{ij}, (u,v)).$$ Directional control $\theta$ is discretized into $M=32$–$64$ intervals; $V(\nabla E \cdot s(\theta_m))(s(\theta_m) \cdot p)$ is evaluated over these to determine the maximum.

Second-order ENO/WENO schemes provide spatial accuracy, while a 2nd-order TVD Runge–Kutta (RK2) method integrates in time, enforcing the Courant-Friedrichs-Lewy (CFL) condition: $$\Delta t \leq \min\left( \frac{\Delta x}{\max |\partial \widehat{H}/\partial p|},\, \frac{\Delta y}{\max |\partial \widehat{H}/\partial p|} \right).$$ Re-distancing is periodically performed to enforce $|\nabla u| \approx 1$, maintaining numerical stability.

3. Path Extraction via Back-Tracking

After propagating the HJB PDE until the zero-level set $\{u=0\}$ reaches the terminal state $b$ (or, in the stationary case, after solving for $T(x)$), the optimal trajectory is retrieved by integrating the following ODE system backward in time: $$\dot{x}(s) = -\nabla_p H(x(s), p(s)),\quad x(0) = b$$

$$\dot{p}(s) = \nabla_x H(x(s), p(s)),\quad p(0) = \nabla u(b, t^*)$$

In common applications, only the state equation is evolved with $ẋ = V(\nabla E \cdot s(\theta^*))\, s(\theta^*)$ and $$\theta^*(s) = \arg\max_{\theta} V(\nabla E(x(s))\cdot s(\theta))[s(\theta)\cdot p(s)]$$; $p(s)$ is periodically reset as $p(s)\leftarrow \nabla u(x(s))$. Integration employs a 2nd-order Runge–Kutta method, or a stiff ODE solver for irregular $E$.

4. Practical Implementation Workflow

A standardized pipeline for synthesizing optimal hiking paths from DEM data incorporates the following sequential steps:

1. Acquire DEM from authoritative sources (e.g., USGS 1/3" data), preprocess using GIS tools (QGIS, TopoToolbox), and resample to a uniform Cartesian grid in MATLAB.
2. Compute $\nabla E$ via central differences on the grid.
3. Specify or calibrate the walking speed profile $V(S)$. The prototype $V(S) = 1.11\exp[-(100S+2)^2/2345]$ ensures $V(0)\approx1$ m/s and $V\to 0$ as $S$ increases.
4. Optionally, penalize regions of extreme omnidirectional slope $S_\mathrm{max} = \|\nabla E\|_\infty$ by modifying the Hamiltonian, e.g., multiplying by $$P(S_\mathrm{max}) = \frac{1}{2}[1- \tanh(S_\mathrm{max} - 1)]$$.
5. Discretize the domain and initialize the level-set function. For uncertain initial regions $A$, reverse-propagate from $b$ and record arrival times $t^*(a)$ for $a\in A$.
6. March $u_t + \widehat{H} = 0$ until $b$ (or $A$) are reached, performing re-distancing every 20–50 iterations.
7. Backtrack the ODE from each desired start point to extract the optimal path.

The following table summarizes key steps:

Step	Input/Method	Output
DEM acquisition/preprocessing	USGS/QGIS/TopoToolbox + resampling	Gridded elevation data
Gradient computation	Central finite differences	$\nabla E$
HJB PDE solving	Upwind ENO/WENO + RK2 time stepping	Value function $u(x,t)$
Path extraction	Backward ODE integration	Spatial trajectory

5. Case Study: Yosemite and Empirical Observations

The method was demonstrated in the El Capitan region of Yosemite National Park (38–39° N, 119–120° W) using 1/3" DEM resolution (Parkinson et al., 2018). Optimal paths from summit area $a$ to destination $b$ systematically avoid the near-vertical cliff, instead descending through east or west gullies. When the initial location $a$ is uncertain within a region $A$ near the summit, reverse-propagation from $b$ was performed, sampling $N=100$ start points drawn from $A$. Each trajectory was backtracked using the ODE scheme.

Clustering analysis via a chord-metric

$d(P, Q) = \int_0^{0.8}\|P(t) - Q(t)\|\,dt$

and k-means (with $k=2$) revealed that 24% of paths followed the eastern funnel and 76% the western. This partitioning supports resource allocation in patrol or rescue scenarios, as trajectory clusters reflect the geometry of the accessible landscape.

Observed limitations are primarily related to DEM noise and discretization artifacts, which degrade the accuracy of $\nabla E$. Smoothing or higher-order interpolation mitigates these effects. Trajectories generated by this method do not inherently account for trails, undergrowth, bodies of water, or private property; such features must be encoded as high-cost (effectively impassable) barriers in the Hamiltonian. Extremely steep regions can be demarcated impassable by setting $V=0$. Computational complexity scales as $O(N_x N_y N_\theta \times \text{timesteps})$, with $M\approx 32$–$64$ angular directions yielding satisfactory resolution.

6. Implications and Extensions

The outlined HJB pipeline offers a comprehensive, numerically robust approach for planning human hiking paths in wild terrain. The versatility of this methodology enables adaptation to different speed profiles, constraint sets, and environmental inputs. Clustering of resulting path ensembles reveals emergent “funnels” imposed by terrain, facilitating decision-making for law enforcement or rescue operations. A plausible implication is that such optimal-control frameworks can be generalized to multi-agent or adversarial navigation, or integrated with stochastic models to accommodate environmental uncertainty.

Explicit limitations include the algorithm’s dependence on DEM fidelity, inability to directly model behavioral avoidance of non-topographic obstacles without additional cost terms, and computational expense for large domains or high angular discretization. Nevertheless, the approach rigorously encapsulates the interplay between human locomotion physiology, terrain geometry, and optimal path synthesis in the context of hiking in the wild, providing a reproducible pipeline from digital elevation data to actionable navigation paths (Parkinson et al., 2018).