Safe Optimal Control using Log Barrier Constrained iLQR

TL;DR

This work addresses constrained trajectory optimization with box constraints on states and controls by integrating a logarithmic barrier interior-point method into iLQR, creating Box-iLQR. The approach reformulates the constrained problem as a barrier-augmented unconstrained subproblem solved via a backward/forward iLQR loop, with an outer barrier-relaxation loop that drives the barrier parameters to zero. Key contributions include a detailed barrier-term formulation with explicit derivatives, a backward pass that accounts for barrier effects, a forward pass with line-search-based progress guarantees, and formal results on inherent regularization, stability, convergence, and constraint-aware feedback. Numerical results on pendulum, cart-pole, and acrobot demonstrate reliable constraint satisfaction, favorable convergence, and characteristic constraint-aware feedback, underscoring the method’s practical potential for safe, constrained optimal control. The work advances constrained iLQR by providing a robust, tunable framework with theoretical guarantees and promising directions for adaptive barrier strategies and MPC-style robustness.

Abstract

This paper presents a constrained iterative Linear Quadratic Regulator (iLQR) framework for nonlinear optimal control problems with box constraints on both states and control inputs. We incorporate logarithmic barrier functions into the stage cost to enforce box constraints (upper and lower bounds on variables), yielding a smooth interior-point formulation that integrates seamlessly with the standard iLQR backward-forward pass. The Hessian contributions from the log barriers are positive definite, preserving and enhancing the positive definiteness of the quadratic approximations in iLQR and providing an intrinsic regularization effect that improves numerical stability and convergence. Moreover, since the negative logarithm is convex, the addition of log barrier terms preserves convexity if the cost is already convex. We further analyze how the barrier-augmented iLQR naturally adapts feedback gains near constraint boundaries. In particular, at convergence, the feedback terms associated with saturated control channels go to zero, recovering a purely feedforward behavior whenever control is saturated. Numerical examples on constrained nonlinear control problems demonstrate that the proposed method reliably respects box constraints and maintains favorable convergence properties.

Figures (10)

• Figure 1: Comparison of unconstrained and constrained control for the pendulum swing-up task. The plots show the resulting (a) control input $u(t)$, (b) pendulum angle $x_1(t) = \theta(t)$, and (c) angular velocity $x_2(t) = \dot{\theta}(t)$ over time. The unconstrained case is shown in blue and the constrained case in red.
• Figure 2: A sequence of snapshots illustrating the pendulum swing-up trajectory. The motion resulting from the unconstrained controller is shown in light blue, while the trajectory under the constrained controller is shown in red.
• Figure 3: Comparison of control inputs $u(t)$ for the cart-pole swing-up. Each subplot contrasts the unconstrained control trajectory (blue) with different constrained scenarios: (\ref{['fig:cartpole_control_comp1']}) control constrained, (\ref{['fig:cartpole_control_comp2']}) state constrained, and (\ref{['fig:cartpole_control_comp3']}) a combination of both. The control constraint boundaries, $-2 \leq u \leq 2$, are shown with dashed black lines in (\ref{['fig:cartpole_control_comp1']}) and (\ref{['fig:cartpole_control_comp3']}).
• Figure 4: A comparative analysis of state evolution for the cart-pole swing-up for the unconstrained and the three constrained conditions.
• Figure 5: Time-lapse visualization of the cart-pole swing-up task, comparing the unconstrained trajectory (light blue) against the trajectory with both state and control constraints (red). The dashed black lines indicate the position constraints on the cart, $-0.2 \leq x_{1} \leq 0.2$.

Theorems & Definitions (13)

• lemma thmcounterlemma
• proof
• remark thmcounterremark
• remark thmcounterremark: Model-Based Formulation
• theorem thmcountertheorem: Convergence to the optima
• proof
• remark thmcounterremark
• theorem thmcountertheorem: Feedback Gain Structure at Control Boundaries
• proof
• remark thmcounterremark