Locomotion, Humanoids, and Navigation › Locomotion, Humanoids, and Navigation

Imagine three tasks: a quadruped meeting an unfamiliar slope, a humanoid balancing while reaching, and a wheeled robot navigating with a map built earlier. These are illustrative scenarios, not reported trials. They differ in embodiment and horizon, but each requires predicting how actions change the robot's state and surroundings before choosing what to do.

Earlier chapters introduced state, action, partial observability, system identification, and planning. Part V: World-Model Architectures developed learned predictive representations; Part VII: Planning and Agency used them for action selection. Here we ask what those ideas require when a robot makes and breaks contact with terrain while moving toward a goal.

Locomotion and navigation put different pressures on a model. A legged controller must account for changing contact modes and uncertain surface properties; a navigator also needs spatial memory and a way to evaluate routes. Errors can compound over repeated predictions, especially when a planned footfall or route crosses a poorly modeled region. The fixed-base manipulation examples in the previous chapter have different actuation and contact assumptions, but neither domain is categorically easier.

This chapter follows body dynamics and terrain interaction, whole-body control, then spatial memory and goal navigation. We finish with cross-embodiment and sim-to-real transfer. The worked models are deliberately small: they show how contact modes, terrain uncertainty, and map uncertainty enter prediction and planning without claiming to reproduce a full robot stack.

We begin with the mechanics of a floating base so the later control equations have concrete state and force variables. Planning then connects those predictions to actions. Navigation adds a map and longer-horizon decisions; it does not replace the shorter control loop beneath it.

Terrain uncertainty appears at several levels. At the physics level, unknown height or friction changes predicted transitions. At the control level, it can change which candidate motions satisfy contact limits. At the navigation level, it can change the cost or feasibility of a route. A single aggregate prediction score can miss these differences, so we evaluate the quantities each decision uses.

A physical transition model is one component of that planning system:

$$s_{t+1} = f(s_t, a_t)$$

where:

• $s_t$: your robot's physical state at time $t$ (base pose, joint angles, and velocities).
• $a_t$: the control action at time $t$ (joint torques or desired joint positions).
• $f$: the physical transition model mapping current state and action to next state under an assumed body, terrain, and contact mode. Goals, route costs, and spatial memory belong to the task or map model used by the planner; they need not be components of $f$.

A too-smooth terrain estimate and an overestimated friction coefficient might fit familiar trajectories yet fail on a different surface. Aggregate one-step error alone may not identify which component is responsible. Contact-specific tests, terrain-parameter checks, and closed-loop outcomes provide different evidence about the model.