Sampling-Based Planning and Model Predictive Control › Limitations and Impact

The limitations of sampling-based MPC fall into four buckets: sample complexity, model exploitation, constraint handling, and real-time discipline. Each of these is an obstacle rather than a footnote, and each corresponds to an active area of research. It is also worth keeping the four notions of model quality from the introduction in mind here, because different failure modes are failures of different qualities, and conflating them leads to fixing the wrong thing.

Sample complexity is fundamental. The difficulty of covering action-sequence space grows with action dimension and horizon. Our 2D example uses 800 samples over 15 CEM iterations at $H=25$ and produces a useful plan for one task and seed; it does not establish a general sample requirement. Applying the same independent, diagonal-Gaussian proposal to a 20-dimensional humanoid over $H=100$ would search a 2,000-dimensional sequence space and is unlikely to cover useful modes efficiently. Temporally correlated proposals, spline or skill parameterizations, and learned low-dimensional action representations restrict search to a smaller structured family. They reduce effective dimension only when that family still contains adequate plans.

Model exploitation is the second hazard. Optimization searches for low predicted cost and may select out-of-distribution actions where model error is systematically optimistic. A deterministic conditional-mean predictor also omits outcome dispersion unless uncertainty is modeled separately; that can make risky sequences look safer than they are. Possible mitigations include calibrated uncertainty penalties, ensembles or probabilistic models, conservative objectives, and shorter horizons. Each changes the planning objective or compute cost, and none substitutes for closed-loop evaluation on the planner-induced distribution.

Constraint handling depends on the method. The finite penalties used here trade task cost against violation and cannot guarantee feasibility. Rejection, projection, constrained optimization, robust tubes, invariant sets, barrier-based filters, and supervisory shutdown mechanisms provide different levels of assurance under different assumptions. A safety-critical design must state which layer owns the guarantee and validate that layer against timing, estimation, and model errors.

Real-time discipline is the fourth bucket. A controller is real-time only relative to a stated deadline on its target hardware. Average notebook timing is insufficient; the design needs worst-case or high-quantile latency, an overrun policy, and end-to-end accounting for sensing and actuation. Sampling and iteration budgets are therefore part of the controller specification, not merely performance-tuning details.

MPC is established in process control and widely studied in robotics and autonomous systems. Sampling-based inner solvers extend receding-horizon control to models and objectives for which derivatives or structured solvers are unavailable or inconvenient. Their minimal computational interface is a forward rollout, a scalar evaluation, an action proposal, and a time budget. Practical deployment adds less minimal requirements: state estimation, constraint handling, uncertainty treatment, and a deadline policy.

Looking ahead, differentiable planning (Part VII, Chapter 2) uses model gradients to optimize actions, while belief-space planning (Part VII, Chapter 3) represents uncertainty in the state used for planning. Part VIII then separates several decision-centric lineages rather than forcing them into one loop: PETS and PlaNet use online trajectory optimization, MuZero performs tree search, TD-MPC combines trajectory optimization with learned values, and Dreamer commonly executes a policy trained in imagination without online MPC.