Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots

In plain words

Legged robots solve many small optimisation problems every second to decide how to move: where to step, how hard to push on each joint, and how to stay within limits such as friction and motor torque. A common type of problem for this is called a quadratic program (QP). Several software libraries solve QPs, and they work in very different ways.

This paper is a guide to those libraries. It explains the main ways a QP solver can work, sorts the popular solvers into those families, and collects published timing comparisons. Its practical message is that no single solver wins everywhere. A solver that suits a long, structured planning problem can be a poor fit for a short, dense, high-frequency control loop. The paper also points out that energy use on battery-powered robots matters as much as speed.

TL;DR

• A review of convex QP solvers for real-time legged-robot control, covering four algorithm families and code generation §2.
• It runs no new experiments of its own. Benchmark claims come from prior work and from its own figure §3.
• Guidance: a structure-exploiting interior-point solver (HPIPM) for long-horizon MPC; dense active-set solvers or ProxQP for high-rate whole-body control §4.
• Its selection tables are qualitative judgements by the authors, not measurements tab. 1 tab. 2.
• Use the paper as a map of solver families, then check the cited benchmarks before choosing.

Problem

Model Predictive Control (MPC) and Whole-Body Control (WBC) on legged robots both reduce to QPs that must be solved within tight time and energy budgets §1. MPC yields sparse QPs with optimal-control structure, often over long horizons. WBC yields small-to-medium dense QPs solved at high rates, which the paper gives as 100 to 1000 Hz §1. Solver speed, memory, numerical robustness and warm-starting decide whether the control loop stays feasible §1.

The paper's concern is that most earlier comparisons report solve time and accuracy, but not energy per solve §1. It proposes the metric Solve Frequency per Watt (SFPW) for this, attributing it to prior work §1.

Assumptions: the paper treats the QP as convex, with a positive semi-definite quadratic term §2. It does not treat nonconvex or mixed-integer problems.

Method

The paper is a review, so this section restates its formulations and its solver taxonomy. Notation: n decision variables, n_e equality constraints, n_i inequality constraints §2.

1. Standard convex QP. The paper writes the target form as \min \tfrac12 x^\top H x + q^\top x subject to Ax = b and Cx \le u §2. It uses q for the linear term so that g(\cdot) keeps its meaning as the nonlinear inequality function §2.

2. SQP subproblem. Sequential Quadratic Programming (SQP) solves a nonlinear program by repeatedly solving a QP built around the current iterate x_k §2. The mapping is H = B_k, q = \nabla f(x_k), A = J_h(x_k), b = -h(x_k), C = J_g(x_k), u = -g(x_k), with decision variable x \equiv d §2. B_k approximates the Hessian of the Lagrangian §2. The paper does not say how B_k is formed.

# SQP subproblem as a standard convex QP  [§2]
# Input:  B_k (n, n); grad_f (n,); jac_h (n_e, n); h (n_e,);
#         jac_g (n_i, n); g (n_i,)
# Output: H, q, A, b, C, u for min 1/2 x'Hx + q'x, Ax = b, Cx <= u

def sqp_to_qp(B_k, grad_f, jac_h, h, jac_g, g):
    # Quadratic term: approximate Hessian of the Lagrangian  [§2]
    H = B_k
    # Linear term: objective gradient at the current iterate  [§2]
    q = grad_f
    # Equality rows: constraint Jacobian, negated residual  [§2]
    A = jac_h
    b = -h
    # Inequality rows: constraint Jacobian, negated residual  [§2]
    C = jac_g
    u = -g
    return H, q, A, b, C, u

3. Equality-only case. With only equalities, the paper solves the KKT linear system for x^* and the multipliers \lambda^* §2, eq. 1:

\begin{bmatrix} H & A^\top \\ A & 0 \end{bmatrix} \begin{bmatrix} x^* \\ \lambda^* \end{bmatrix} = \begin{bmatrix} -q \\ b \end{bmatrix} §2, eq. 1

4. Active-set methods. These guess the set of tight inequalities \mathcal{A} and solve an equality-constrained subproblem on that set [§2, eq. 2]. They warm-start well when the active set changes slowly, and their timing can jitter when it switches §2 tab. 1. Solvers named: qpOASES, Eiquadprog, qpmad, DAQP §2.

5. Interior-point methods. A logarithmic barrier keeps iterates strictly inside Cx < u. The paper writes the barrier problem with parameter \mu > 0 §2, eq. 3:

\min_x \tfrac12 x^\top H x + q^\top x - \mu \sum_{i=1}^{n_i} \log\!\bigl(u_i - (Cx)_i\bigr) \quad \text{s.t.} \quad Ax = b §2, eq. 3

Named solvers: HPIPM (structure-exploiting), qpSWIFT, PIQP §2.

6. Operator splitting (ADMM). Inequalities are turned into equalities with a slack s \ge 0, so that Cx + s = u §2, eq. 4. The paper gives only this reformulation, not the ADMM iterations §2. OSQP is named as the representative §2.

\min_{x,\, s \ge 0} \tfrac12 x^\top H x + q^\top x \quad \text{s.t.} \quad Ax = b,\; Cx + s = u §2, eq. 4

7. Augmented Lagrangian and proximal methods. The paper gives a Powell–Hestenes–Rockafellar style penalised Lagrangian with multipliers \lambda \in \mathbb{R}^{n_e}, \mu \in \mathbb{R}^{n_i} and penalties \rho_e, \rho_i > 0 §2, eq. 5. Its dual updates are §2, eq. 6 and §2, eq. 7:

\lambda^{k+1} = \lambda^k + \rho_e\,(Ax^{k+1} - b) \qquad \mu^{k+1} = \max\!\bigl(0,\; \mu^k + \rho_i\,(Cx^{k+1} - u)\bigr) §2, eq. 6 §2, eq. 7

ProxQP is the representative solver in this family §2.

8. Code generation and conic solvers. For fixed-structure problems where only parameters change, code generators (CVXGEN, CVXPYgen, FORCESPRO) emit specialised C code §2. When conic constraints or certificates are needed, Clarabel is proposed as a general route that includes QPs as a special case §2.

9. Overview tables. tab. 1 gives, per family and per representative solver, flags for warm-start (WS), high accuracy (Acc), early termination (Early) and practical real-time suitability (RT), with a "best scenario" column. The caption calls RT "scenario-dependent" tab. 1. tab. 2 lists dense/sparse matrix support, warm-start, hot-start, early termination, dual-gap reporting and an "I-H" column tab. 2.

Table 1 tab. 1. Columns: WS, Acc, Early, RT. ✓ = yes, ✗ = no, N/A = not applicable.

tab. 1 Source: the paper's unified overview. The "best scenario" column is condensed from the table's own text.

Table 2 tab. 2. Columns as printed. ✓ = yes, ✗ = no.

tab. 2 Hot-S, Dual-gap and I-H are not defined in the paper.

The selection guidance in §3 and §4 can be condensed to this sketch. It is the paper's own advice, not an algorithm the paper states §3 §4.

# Candidate solvers by problem shape  [§3] [§4]
# Input:  shape, one of 'ocp', 'dense_wbc', 'large_sparse', 'fixed',
#         'conic', 'ill_conditioned', 'condensed_mpc'
# Output: list of candidate solver names (suggestions, not a ranking)

def candidate_solvers(shape):
    # Sparse optimal-control structure, long horizon  [§4]
    if shape == 'ocp':
        return ['HPIPM']
    # Small dense whole-body control problems  [§3] [§4]
    if shape == 'dense_wbc':
        return ['qpmad', 'Eiquadprog', 'qpOASES', 'ProxQP']
    # Large generic sparse QPs, moderate accuracy  [§3]
    if shape == 'large_sparse':
        return ['OSQP']
    # Fully condensed dense MPC, fixed structure  [§3]
    if shape == 'condensed_mpc':
        return ['DAQP']
    # Fixed structure with strong tooling needs  [§3]
    if shape == 'fixed':
        return ['FORCESPRO', 'CVXGEN', 'CVXPYgen']
    # Conic constraints or certificates  [§3]
    if shape == 'conic':
        return ['Clarabel']
    # Ill-conditioned sparse QPs  [§3]
    if shape == 'ill_conditioned':
        return ['PIQP', 'ProxQP']
    return []

Hyperparameters

The paper specifies no solver settings for its own experiments beyond the problem size and simulation length of its one figure. Everything else is not stated.

Results

Author-reported, and mostly relayed from other papers. The paper runs no new benchmark with reported numbers. Setups as stated in the text.

Figure description (not reproduced here): fig. 1 plots solve times over a 100-step simulation for selected solvers at two tolerances, with a ✗ marking a solver that did not reliably solve some instances at the required tolerance fig. 1. The text does not state the data source or hardware for this figure.

Paper vs code

No official code.

The paper names no code repository in the abstract, the body or the bibliography. Its own benchmark tools (qpbenchmark, qpmad_benchmark) belong to third-party projects and are cited, not implemented here. GitHub searches for "Van Nam Dinh" and for "QP solver legged robots" returned no matching repository, so no repo was cloned.

Pitfalls

• Review, not a new benchmark. Most performance claims are relayed from [stark2025benchmarking] or [qpmad_benchmark], or from the paper's own qualitative judgements tab. 1. Check the original sources before relying on a number.
• Figure 1 cannot be reproduced from the text. Solver versions, tolerances, hardware and the test set source are not stated fig. 1.
• SFPW is not defined. The paper names "Solve Frequency per Watt" but gives no formula and no measurement method §1 §3.
• Overloaded symbol. \mu is the barrier parameter in the IPM section §2, eq. 3 and the augmented-Lagrangian multiplier for inequalities in §2, eq. 5. Keep them apart in code.
• SQP Hessian unspecified. B_k is described only as an approximation of the Lagrangian Hessian §2. The paper does not say whether it is exact, quasi-Newton or regularised.
• Convexity assumption. The subproblem is convex only when B_k is positive semi-definite §2. The qpmad benchmark uses positive-definite QPs only §3. Check that the solver supports semi-definite H before using it.
• ADMM and AL are not algorithms here. The paper gives the penalised form and the dual updates, but no iteration loop, no \rho adaptation rule and no termination test §2, eqs. 4–7.
• Undefined table columns. "Hot-S", "Dual-gap" and "I-H" in tab. 2 are not explained tab. 2. "Matrix" entries also differ from tab. 1 for qpOASES: "Dense / Sparse" in tab. 2 against "small/medium dense" in tab. 1.
• Citation mismatch. The claim that qpmad suits embedded pipelines cites a field-robotics paper on point cloud registration §3 [qpmad_app]. That reference does not support the claim and should not be used as evidence.
• Timing claims lack units and setups. "Sub-millisecond" §3 and "up to twofold" §3 are given without hardware, sizes or variance.
• RT and Acc columns are judgements. tab. 1 states these are scenario-dependent, not measured.
• Version. This card follows arXiv v2 (updated 2025-12-12), as fetched.
• Equation numbers. Numbers are counted from the source TeX (LNCS class, one global counter): the KKT system is eq. 1, the active-set problem eq. 2, the barrier problem eq. 3, the slack form eq. 4, the augmented Lagrangian eq. 5, and the two rows of the dual update eqs. 6 and 7. The three displays at the start of §2 (the SQP and convex QP forms) are unnumbered and are cited by section only.

Lineage

• Builds on: [stark2025benchmarking] (arXiv:2502.01329), the MPC and WBC benchmark whose results the paper summarises §3. Also qpbenchmark and qpmad_benchmark, cited for benchmark evidence §3.
• Compared against: the solver set itself. qpOASES, OSQP, HPIPM, ProxQP, qpmad, Eiquadprog, DAQP, PIQP, qpSWIFT, FORCESPRO, CVXGEN, CVXPYgen, Clarabel, Gurobi, MOSEK, SCS, quadprog tab. 1 tab. 2.
• Superseded by: not known from the content of this wiki.