Hippo: High-performance Interior-Point and Projection-based Solver for Generic Constrained Trajectory Optimization

As robot capabilities increase, quadratic programming (QP)-based controllers must account for a similarly increasing number of constraints to ensure safe, reliable operation. Yet, with each added constraint, this introduces more chances of momentary conflict: in which case, a QP solver that returns an "infeasible" status leaves the controller with nothing to execute. To address this, we introduce ElastiQP, a modified dual active-set QP solver that relaxes every inequality constraint with an exact, per-constraint l1 penalty while keeping equality constraints (dynamics) hard. Notably, ElastiQP does so by folding the slack variables into the solver analytically, maintaining a constant size of the condensed linear system. On a suite of robot control benchmarks, ElastiQP achieves microsecond-level performance, matching or outperforming leading modern solvers on feasible problems. On infeasible problems, ElastiQP handles these gracefully, confining violations to strictly the conflicting inequality terms, returning a usable solution up to 40x faster than the best alternative solvers. ElastiQP is available as an open-source C++ header-only library, with Python and JAX interfaces, at https://github.com/StanfordASL/elastiqp.

Contact-implicit trajectory optimization avoids prescribing contact sequences, but yields mathematical programs with complementarity constraints (MPCCs) whose degeneracy challenges conventional primal--dual solvers. Existing contact-specific methods improve robustness to this degeneracy but do not leverage a stagewise optimal-control factorization and primal--dual consistency, while structure-exploiting optimal-control solvers are not designed for complementarity constraints. We show that these capabilities can be combined in a single primal--dual method. ContactIPM identifies complementary inequality pairs, embeds them through a barrier-coupled elastic interior relaxation, eliminates slack and dual variables stagewise, and solves the reduced Newton system using a Riccati recursion. A fixed multi-phase MPCC recovery schedule provides four continuation and restart attempts from naive initializations, while termination is gated by the unrelaxed physical complementarity residual. We compare ContactIPM with two contact-specific MPCC solvers, CRISP and IMPACT, using matched benchmark conditions and common post-solve acceptance criteria. On four fixed CRISP benchmark cases, ContactIPM is $2.17$--$8.87\times$ faster over 20 paired timing repetitions per case and achieves higher success on the Push Box and Push-T robustness suites. Against IMPACT, ContactIPM is $2.96\times$ faster on Push T and $4.91\times$ faster on Cart Transport, but $4.46\times$ slower on Push Box. In 50 closed-loop Push Box rollouts spanning model mismatch, measurement noise, initial-pose errors, and state resets,

Many robotics problems, including trajectory optimization, inverse kinematics, and contact-rich motion planning, reduce to nonlinear programs (NLPs). Mature NLP solvers such as IPOPT can solve these problems, offering hard constraint satisfaction, optimality guarantees, and favorable scaling with problem dimension. These solvers underpin gradient-based methods in robotics, yet remain CPU-bound and solve only one problem at a time, preventing their integration into GPU-batched learning pipelines. On the other hand, sampling-based approaches such as reinforcement learning, model predictive path integral, and imitation learning have become the core of modern robotics research due to their ability to leverage GPU-batched simulators. These simulators can generate orders of magnitude more dynamics rollouts per second than was previously possible. If a GPU-batched NLP solver existed, it would unlock similar speedups in the number of constrained, locally optimal solutions generated per second. This regime of solving many problems concurrently versus solving a single problem at a time is a key requirement for integrating NLP solvers in modern GPU-batched robotics frameworks. To this end, we introduce jaxipm, the first GPU-batched NLP solver, based on IPOPT, and implemented in JAX. We accomplish this by redesigning IPOPT's algorithm to eliminate control flow with heterogeneous iteration fusion, and by minimizing GPU idle time with iteration level batching. We evaluate jaxipm on a variety of quadrotor nonlinear model predictive control benchmarks, including reference tracking in the presence of obstacles, multi-quadrotor navigation without collision, and navigation in a cluttered environment. We demonstrate up to a $32.85\times$ increase in throughput over IPOPT. Our complete open-source codebase is available at https://github.com/johnviljoen/jaxipm.