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

Haizhou Zhao; Ludovic Righetti; Majid Khadiv

cs.ROMar 1, 2026

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

Authors: Haizhou Zhao, Ludovic Righetti, Majid Khadiv

Organizations: Tandon School of Engineering, New York University, USA · Munich Institute of Robotics and Machine Intelligence (MIRMI), Technical University of Munich (TUM), Germany

Abstract

Trajectory optimization is the core of modern model-based robotic control and motion planning. Existing trajectory optimizers, based on sequential quadratic programming (SQP) or differential dynamic programming (DDP), are often limited by their slow computation efficiency, low modeling flexibility, and poor convergence for complex tasks requiring hard constraints. In this paper, we introduce Hippo, a solver that can handle inequality constraints using the interior-point method (IPM) with an adaptive barrier update strategy and hard equality constraints via projection or IPM. Through extensive numerical benchmarks, we show that Hippo is a robust and efficient alternative to existing state-of-the-art solvers for difficult robotic trajectory optimization problems requiring high-quality solutions, such as locomotion and manipulation.

Figures & tables

Solver	Parallelized	Eq. ii ii ii Eq./Ineq. - Equality/Inequality constraint handling.	Ineq.	Dynamics
Hippo (ours)	yes	proj. iii iii iii projection-based handling. /IPM	IPM	impl.
fatrop [ 26 ]	no	proj.	IPM	expl.
acados [ 7 ]	no	IPM	IPM	impl.
aligator [ 4 , 6 ]	yes	ALM	ALM	impl.
mim_solver [ 5 ]	yes	ALM	ALM	expl.

TABLE I: Solver Attributes

Solver	Inner Loop iv iv iv ’lin’ means only updating zero-order sensitivities. ’quad’ means com- puting new Newton steps of the LQ subproblems.	Param Update Scheme
Hippo /fatrop	iterative refinement (lin)	per outer loop
acados	hpipm qp loops (quad)	reset per outer loop
aligator(p) v v v the serial version of aligator does not have inner loop w.r.t. SQP steps	iterative refinement (lin)	if primal res. $>$ tol
mim_solver	ADMM step (lin)	per fixed num. updates

TABLE II: Comparison of Solvers and Their Characteristics

Solver	Iters	Inner Iters	Inner Tol.	Param vi vi vi barrier parameter for IPM and penalty parameter for ALM/ADMM.
Hippo	100	2	1e-10 (abs)	1
acados	100	50	1e-3 (abs)	1
fatrop	100	5	1e-8 (abs)	1
aligator(s)	400	0	varying	0.1
mim_solver(1)/(2)	400	5(15)/20(60) vii vii vii (..) the number of fixed inner steps to update the penalty parameters.	1e-3 (abs)	0.1

TABLE III: Solver Parameters for UR5 Reaching Benchmark

Fig. 1: UR5 random SE3 reaching benchmark results. Left plot is the number of sucess v.s. the wall time. Right plot shows the number of QP iterations distribution of the tests.

	Hippo	Hippo(i)	fatrop	acados	aligator	mim_solver
#	43	65	56	36/81 viii viii viii Only 36 out of the 81 solutions have acceptable slack values ( $\leq$ 1e-3).	61	(1):67;(2):64
t ix ix ix ‘#‘: number of solved problems; ‘ t ‘: average timing per QP (ms)	0.33	0.32	0.86	0.13	1.02	(1):0.61;(2):1.1

TABLE IV: Statistics for UR5 example

Solver	SQP Iters	Inner Iters	Inner Tol.	Param x x x barrier parameter for IPM and penalty parameter for ALM/ADMM
aligator(p)	200	5	varying	0.1
mim_solver	200	100(50) xi xi xi (..) the number of fixed steps to update the penalty parameters	1e-3 (abs)	0.1

TABLE V: ALM Solver Parameters Go2 Locomotion Benchmark

Fig. 2: Go2 2-step locomotion benchmark results. 100 random trials are tested for each setting. Missing solver data for a setting indicates that the solver failed all tests for that setting.

	RS xii xii xii R: reduced; S: simple; F: full; H: hard. ’#’ : number of solved problems. ’ t ’: average timing (ms) per QP.		FS		RH		FH
trot	#	t	#	t	#	t	#	t
Hippo	100	2.35	100	2.75	96	2.55	100	2.67
Hippo(i)	100	2.93	100	3.13	99	3.01	100	2.95
fatrop	100	24.9	100	25.0	100	24.9	100	23.9
acados	100	2.86	100	2.99	89	2.60	100	2.81
aligator	96	8.68	0	-	81	9.95	0	-

TABLE VI: Statistics for Go2 example

Trajectory Optimization

Trajectory optimization computes motion or control sequences that minimize costs while satisfying system dynamics, safety limits, and task constraints.

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Departments of Mechanical Engineering and Aeronautics & Astronautics, Stanford University, USA · Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, USA

Robotic Control Constrained Optimization Quadratic Programming

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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.

Abstract

Trajectory Optimization

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