COPP for MATLAB

COPP solves optimal path-parameterization problems. A geometric path is written as

where \(q\) is robot configuration and \(s\) is a scalar path coordinate. The solver does not change the geometric path. It computes a time law

so the robot traverses the path while respecting velocity, acceleration, torque, jerk, and user-supplied constraints. In short, COPP turns a geometry-space path into a feasible and optimized time schedule.

The MATLAB package is named copp. Public code should call APIs through that package namespace.

Solver profiles

Solvers return compact path-speed profiles rather than time-sampled robot trajectories. In second-order solvers the main output is \(a\), the squared path speed at station nodes. Third-order solvers return copp.Profile3rd, which stores the node profiles needed by the third-order interpolation helpers.

Use the interpolation functions to convert solver output into time-domain data:

TOPP and COPP

TOPP means time-optimal path parameterization. A TOPP solver seeks the shortest feasible traversal time under a selected constraint model.

COPP means convex-objective path parameterization. COPP solvers keep the same path-domain constraints, but optimize convex objective terms such as time, linear penalties, thermal energy, or total variation of torque.

The MATLAB binding exposes both second-order and third-order families:

Package layout

The MATLAB API mirrors the Rust and Python structure while using MATLAB idioms:

MATLAB public station indices are 1-based. Matrices use MATLAB's natural column-major convention: a \(\mathrm{dim} \times N\) matrix stores one station/sample per column.

The MATLAB binding is a workflow facade, not a one-to-one C ABI mirror. It prioritizes the public Path, Robot, solver, objective, and interpolation workflows. Raw native pointers, owned C buffers, status-code plumbing, and low-level debug/introspection calls are intentionally hidden unless they make a user-facing workflow clearer.

Shape glossary

The binding deliberately uses one convention across all MATLAB entry points:

A quick mental check: if a matrix is about robot coordinates, rows are axes and columns are stations. If a matrix is about raw inequalities, rows are inequality rows and columns are stations.

First complete workflow

This example solves a compact TOPP2-RA problem. The path is \(q(s)=s\), sampled on seven stations. The robot starts and ends at rest, with unit velocity and acceleration bounds.

n = 7;
s = linspace(0.0, 1.0, n).';

robot = copp.Robot(1, Capacity=n);
cleanup = onCleanup(@() robot.release());
robot.append_s(s);
% q, dq, and ddq are dim x N matrices. Here dim=1 and N=n.
robot.set_q_2nd(s.', ones(1, n), zeros(1, n));
robot.add_velocity_limits(1, -1);
robot.add_acceleration_limits(1, -1);

problem = copp.solver.topp2_ra.Problem( ...
    robot, ...
    idx_s_interval=[1, n], ...
    a_boundary=[0, 0]);
a = copp.solver.topp2_ra.solve(problem);
[t_final, t_s] = copp.interpolation.s_to_t_topp2(s, a);

fprintf("a shape: %d x %d, t_s shape: %d x %d\n", ...
    size(a, 1), size(a, 2), size(t_s, 1), size(t_s, 2));
summary = table(s, a, t_s, 'VariableNames', {'s', 'a', 't_s'});
disp(summary)
fprintf("Final traversal time: %.6f seconds\n", t_final);

Reading the rest of the documentation

A useful learning order is: