Amplify: A Lightweight Library for Reproducible Nonlinear Programming Problems in Robotics

Nelson Rosa

cs.ROSep 23, 2026

Amplify: A Lightweight Library for Reproducible Nonlinear Programming Problems in Robotics

Authors: Nelson Rosa

Abstract

Optimization problems (OPs) are key to solving many challenging research problems in robotics. However, reproducibility still remains a major issue. In this paper, we present Amplify, a lightweight nonlinear programming library aimed at reproducible results of robotic-related trajectory optimization problems. The minimalistic requirements for the 537-line library (80 characters per line) are an Internet connection, familiarity with the AMPL modeling language, and a text editor. Our primary contribution is the formulation of a library where trajectory optimization algorithms are represented directly within the optimization model. Specifically, we implement the algorithms used to compute the dynamics, trajectories, and reference motions as constraints of the OP in a declarative programming paradigm. We outline how our formulation of objectives, decisions variables, and constraints can be implemented in other transcription libraries that want to be lightweight and reproducible. We also compare the Amplify framework with 3 other libraries across examples of benchmark optimization problems across several fields, including bipedal locomotion and grasp planning.

Figures & tables

Library	Languages [Add’l Bindings]	User API	Solvers	Lines of Code [# of Files]
Amplify	AMPL [python, R, C#, C++, Java, Matlab]	Declarative	IPOPT and 25+ others
acado	C++ [Matlab]	Object Oriented	qpOASES and 3+ others	271877 [1968]
acados	C [python, Matlab, Octave]	Object Oriented	qpOASES and 9+ others	787963 [1191]
CasADi’s Opti	C++ [python, Matlab, Octave]	Object Oriented	IPOPT and 21+ others	334875 [1496]
do-mpc	python	Object Oriented	IPOPT	30438 [208]
Drake	C++ [python]	Object Oriented	IPOPT and 12 others	842197 [5172]

TABLE I: A list of trajectory optimization libraries. The languages and API are from the user’s programming perspective. The lines of source code (and files) are taken for their respective online repositories as cloned on Febuary 28, 2026 after running CLOC [ 28 ] .

Panel of Figure 2 — Fig. 1: The modules of the Amplify library. The sets, constraints, and variables listed in each block are defined throughout the paper. They represent a sample of the content defined within a module. Arrows between blocks denotes a dependency. For example, the Bézier module depends on the content defined in the Grid module. In AMPL, decision variables are declared (e.g., $f$ and $\iota$ in the Equations of Motion module). On the other hand, defined variables are variables expressed in terms of other variables (e.g., $M$ in the Composite Rigid Body module depends on $q$ and several other variables).

Panel of Figure 3 — Fig. 2: An arm transports a rectangular object to a ramp. The arm grasping the object is modeled as a physical constraint $y_{p}(q)=0$ . While the object is grasped, the arm has to apply joint torques $u_{1}$ and $u_{2}$ in order to move the object along a given path (the thick black curve). The control law for tracking the trajectory is computed through a virtual constraint $y_{v}(q,t)=0$ .

Panel of Figure 4 — Fig. 3: An articulated rigid-body system.

Sets	Definition	Description
$\mathcal{S}$	specified as input, $\mathbb{R}\setminus\{0\}$	subphases
$\mathcal{P}$	$\left\{\lfloor j\rfloor:j\in\mathcal{S}\right\}$	phases
$\mathcal{S}^{i}$	$\left\{j\in\mathcal{S}:\lfloor j\rfloor=i\right\}$	subphases grouped by phase
$(\mathrm{G}^{i},\leq)$	$\bigcup_{j\in\mathcal{S}^{i}}[a_{j},b_{j}]\subset\mathbb{N}$	grid points grouped by phase
$(\mathrm{G}^{0},\leq)$	$\left[0,n_{G_{\text{max}}}\right]\cap\bigcup_{i\in\mathcal{P}}\mathcal{G}^{i}$	all grid points

TABLE II: Definition of grid sets. Ordered sets are denoted as a pair (see Table III for explanation). Otherwise, the set has no ordering.

Panel of Figure 6 — Fig. 4: An example grid layout for a single-legged hopping robot with $t\in[0,T]$ corresponding to continuous time, $t_{0}$ , $\ldots$ , $t_{n}$ discretized time, and $k\in[0,\ldots,n]$ the grid points. The discretized time points are shown mapped to their corresponding grid points. Discrete times $t_{a-1}$ and $t_{a}$ are switching times such that $t_{a-1}=t_{a}$ and similarly for $t_{c-1}$ and $t_{c}$ . Overall, the hybrid dynamics is modeled with two phases, 1 and 2, and three subphases, 1.1, 2.1, and 1.2. Phase 1 corresponds to the dynamics of the robot in free flight. Phase 2 represents the robot pivoting about its foot on the ground. Subphases 1.1 and 1.2 share the same free-flight dynamics, but one subphase corresponds to flight prior to touchdown and the other after takeoff. The impulse-momentum equations maps the state in one subphase to a state in a different subphase through the coefficient matrices $E^{(a-1)a}$ and $E^{(c-1)c}$ , respectively. Additional constraints can be added pointwise. For example, at grid point $b$ , the ground reaction force can be constrained to be less than a max desired value.

Ordered Set	Definition	Description
$(\mathrm{L}_{0},\leq)$	$\left(0,1,\ldots,n_{q}\right)$	link from root to leaves
$(\mathrm{L}_{1},\leq)$	$\left(1,2,\ldots,n_{q}\right)$	links excluding root
$(\mathrm{L}_{-1},\geq)$	$\left(n_{q},n_{q}-1,\ldots,0\right)$	links from leaves to root
$(\mathrm{P}_{i},\leq)$	specified as input	the parent of link $i\in\mathrm{L}_{0}$
$(\mathrm{C}_{i},\leq)$	$\left(c:i\in\mathrm{P}_{c}\right)$	the children of link $i\in\mathrm{L}_{0}$
$(\mathrm{S}_{i},\leq)$	$\left(i\right)\cup\bigcup_{c\in\mathrm{C}_{i}}\mathrm{S}_{c}$	the subtree rooted at link $i\in\mathrm{L}_{-1}$

TABLE III: Connectivity sets of a kinematic tree as indexed ordered sets. The notation

(S,r)

represents the set’s name

S

and its ordering relation

r

Set	Definition	Description
$\mathcal{R}$	$\left\{1,\ldots,n_{r}\right\}$	constraint IDs
$\mathcal{R}^{i}_{r}$	$\{(k_{1},b_{1}),(k_{2},b_{2}),\ldots\}$	constraint frames by row
$\mathcal{P}_{\text{con}}^{i}$	specified as input	physical constraint IDs
$\mathcal{V}_{\text{con}}^{i}$	specified as input	virtual constraint IDs
$\mathcal{U}_{\text{con}}^{i}$	specified as input	control input IDs
$\mathcal{M}_{\text{con}}^{i}$	$\mathcal{P}_{\text{con}}^{i}\cup\mathcal{V}_{\text{con}}^{i}$	motion constraints

TABLE IV: Sets of active constraints during phase

i\in\mathcal{P}

and row

r\in\mathcal{J}_{\text{con}}^{i}

J^{i}

\mathcal{R}

is independent of phase and

\mathrm{A}_{\text{con}}^{i}

is an ordered set.

Demo	Library	RK Method	Objective	Wall Time (s)	Iterates	# Vars.	# Eq. Cons.	# of Ineq. Constr.
Acrobot	OptimTraj	TRAP	38.96	0.65 $\pm$ 0.50	55	102	76	102
Acrobot	Amplify	TRAP	69.88	0.23 $\pm$ 0.03	14	132	116	18
Cart-pole (force)	OptimTraj	TRAP	229.29	0.43 $\pm$ 0.02	38	152	116	152
Cart-pole (force)	Amplify	TRAP	229.29	0.23 $\pm$ 0.02	12	202	176	0
Cart-Pole (time)	OptimTraj	TRAP	1.15	0.46 $\pm$ 0.05	158	52	36	52
Cart-Pole (time)	Amplify	TRAP	1.15	0.22 $\pm$ 0.02	30	80	74	0

TABLE V: Comparison Between OptimTraj + fmincon v24.1 and Amplify + Ipopt 3.12.13 across 10 runs. Better performance is highlighted in gray. TRAP and HS correspond to the trapezoidal and Hermite-Simpson direct collocation methods, respectively.

Library	Objective	Wall Time (s)	Iterates	Eval. Time (s)	# Vars.	# Eq. Cons.	# Ineq. Cons.	Note
TROPIC	4.12	2.69 $\pm$ 0.41	101	1.15 $\pm$ 0.17	1404	6	1642	TROPIC defaults with $\epsilon$ constraints
TROPIC	6.09	18.97 $\pm$ 0.44	793	9.20 $\pm$ 0.21	1404	1390	258	TROPIC defaults with equality constraints
TROPIC	5.92	207.48 $\pm$ 2.46	1141	69.76 $\pm$ 0.80	1404	1390	258	Ipopt defaults with equality constraints
Amplify	5.73	31.95 $\pm$ 0.63	206	11.90 $\pm$ 0.29	2345	2331	206	Ipopt defaults with equality constraints

TABLE VI: Comparison between TROPIC + Ipopt 3.12.3 + spat_v2 and Amplify + Ipopt 3.12.13 across 10 runs of TROPIC’s five-link biped NLP. The RK method is Hermite Simpson. Better performance is highlighted in gray.

Library	RK Method	Objective	Wall Time (s)	Iterates	Eval. Time (s)	# Vars.	# Eq. Cons.	# Ineq. Cons.
Horizon	COL3	-	-	-	-	-	-	-
Amplify	COL3	13250.85	199.74 $\pm$ 2.11	172	152.02 $\pm$ 1.20	14854	14228	480
Horizon	RK1	-	-	-	-	-	-	-
Amplify	RK1	13087.32	217.56 $\pm$ 4.39	201	168.97 $\pm$ 3.84	3706	3080	480
Horizon	RK4	13127.37	43.20 $\pm$ 0.85	330	9.12 $\pm$ 0.26	3400	2787	1180
Amplify	RK4	13343.08	159.83 $\pm$ 2.95	136	119.61 $\pm$ 2.46	14836	14210	480

TABLE VII: Comparison Between Horizon + Ipopt 3.12.3 and Amplify + Ipopt 3.12.13 on an 18-DOF Spot quadruped model across 10 runs. Gray highlights better performance, dashes denote solver failed to converge.

Panel of Figure 12 — Fig. 5: (a) A time-optimal motion for a differential-drive kinematic car (blue dot) and its heading (orange cone) from its start (red circle) to its goal (green circle) configuration. The optimal motion is to turn left, drive forward, turn right, and then drive backwards. Only 4 of the 9 solvers tested converged to a time-optimal solution of $t\approx$ 1.96\text{,}\mathrm{s}$$ . (b) and (c) the control inputs applied by the solvers. The black dots are the grid points where the inputs are applied.

Panel of Figure 13 — Fig. 6: Select solver results of the best minimal fingered form-closure grasp. The optimal solution, e.g., (a) and (d), will always use 4 frictionless point-contact fingers each placed on a side of the block as depicted by the object’s normals (black arrows). The problem is formulated as a mixed-integer program. Grasp points are shown in black. RAPOSa used Knitro as an underlying solver, which is why it found the same solution as Knitro.

Demo	RK	Elapsed Time (s)	Elapsed Time (s)	Ratio of Means	Amplify	Presolve	Amplify	Presolve
		Local	NEOS		# of Vars.		# of Eq. and Ineq. Constr.
acrobot	TRAP	$0.217\pm 0.037$	$93.448\pm 74.709$	$430.237$	8779	132	8824	134
cart-pole (force)	TRAP	$0.234\pm 0.039$	$43.404\pm 56.936$	$185.170$	13169	202	13234	176
cart-pole (time)	TRAP	$0.198\pm 0.022$	$75.756\pm 86.706$	$381.833$	4388	80	4413	74
moving block	TRAP	$0.174\pm 0.014$	$20.149\pm 15.103$	$116.133$	6689	116	6722	88
five-link biped	HS	$0.486\pm 0.054$	$66.036\pm 49.696$	$135.931$	99109	266	98877	214

TABLE VIII: Wall time and problem size comparisons. Wall time comparison is between local and NEOS computations across 10 runs. Local timing is the duration from when the command script starts until the solver returns control back to the script. NEOS wall time includes query for submission template, waiting in queue, and running on a compute node. The Ratio of Means is the ratio of NEOS’s average wall time to the average local wall time. The problem size comparison lists the number of variables or constraints created by Amplify’s and the user’s initial formulation and the reduction after running AMPL’s presolver on the NLP.

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OpenSCvx: An Open-Source Modular and Extensible Nonlinear Trajectory Planning Package

Christopher R. Hayner, Griffin J. Norris, Fabio Spada, Samet Uzun+3

Dept. of Aeronautics and Astronautics, University of Washington, Seattle, WA, USA. · General Robotics, Seattle, WA, USA. · Dept. of Mechanical Engineering, University of California, Berkeley, Berkley CA, USA.+2

STL Motion Planning Trajectory Optimization Motion Planning Constrained Motion Planning

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Trajectory optimization computes dynamically feasible motions that enable autonomous systems to accomplish complex tasks while satisfying operational and environmental constraints. This tutorial presents OpenSCvx, an open-source Python framework that bridges the gap between high-level problem specification and efficient numerical optimization. Rather than requiring users to derive solver-specific mathematical formulations, OpenSCvx provides a symbolic modeling interface that automatically constructs and solves trajectory optimization problems from modular descriptions of objectives, dynamics, and constraints. Beyond simplifying problem formulation, OpenSCvx supports (i) continuous-time constraint modeling, (ii) temporal and logical specifications, (iii) automatic vectorization for scalable and batched optimization, and (iv) a modular architecture that enables new algorithms, models, and solver backends to be incorporated with minimal effort. These capabilities allow researchers and practitioners to rapidly prototype, solve, and extend state-of-the-art trajectory optimization methods.

Abstract

Robotic Optimization

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