Convex-Objective Path Parameterization

Optimal motion,
solved in milliseconds.

The professional engine for time- and energy-optimal trajectory parameterization — with third-order jerk and torque-rate limits, general convex objectives, and real-time speed.

Request Pro access Explore the open-source core ↗
31× faster3rd-order vs. traditional LP
10.6 msper 1000-interval, 7-DOF solve
3rd-orderjerk + torque-rate aware
∞ objectivesany convex cost
How it works

Three inputs, one solver, a ready-to-track trajectory.

Path, constraints, and objective feed into the COPP solver core. It returns a complete, densely-sampled timed trajectory — positions, velocities and accelerations for every axis, ready for your controller.

INPUT · PATH Geometric path q(s) INPUT · CONSTRAINTS vel · acc · torque · jerk INPUT · OBJECTIVE min time / convex J COPP SOLVER RDDP · RA · SOCP OUTPUT · TIMED TRAJECTORY dense q(t), q̇(t), q̈(t) — executed by the robot robot executes q(t)
INPUT · PATH
Geometric path q(s)
↓
INPUT · CONSTRAINTS
vel · acc · torque · jerk
↓
INPUT · OBJECTIVE
min time / convex J
↓
COPP SOLVER
RDDP · RA · SOCP
↓
OUTPUT · TIMED TRAJECTORY
dense q(t), q̇(t), q̈(t) — executed by the robot

Third-order constraints

Universal jerk and torque-rate limits — the bang-bang control rides exactly between its bounds, giving smooth motion with less wear and no structural-mode excitation.

Any convex objective

Beyond time-optimal: minimize thermal energy, torque variation, or any convex cost. The bowl shape guarantees a single global optimum — one engine, many goals.

Real-time speed

Proprietary RDDP solvers reach millisecond solve times on 1000-interval problems — fast enough for online replanning.

Pro vs. Community

An order of magnitude faster, at the same optimum.

100 random 7-DOF spline paths, 1000 intervals each. Like-for-like: 2nd-order vs 2nd-order, 3rd-order vs 3rd-order.

2nd-order · vel / acc / torque
computation time · lower is better
COPP2-RDDP PRO 5.4 ms
COPP2-SOCP OSS 150 ms
TOPP2-RA OSS 0.6 ms

≈28× faster than COPP2-SOCP · same traversal time ≈ 40.9 s

3rd-order · + jerk / torque-rate
computation time · lower is better
TOPP3-RA PRO 10.6 ms
TOPP3-LP OSS 327 ms
TOPP3-SOCP OSS 290 ms

≈31× faster than traditional LP · traversal time ≈ 41.5 s

CapabilityCommunityPro
Time-optimal & convex objectives✓✓
3rd-order jerk / torque-rate limits✓✓
Ultra-fast RDDP / RA solvers—✓
Long-horizon numerical stabilitylimited✓
Commercial license & supportMIT, community✓
Get started

Two ways in.

COMMUNITY · MIT

Start with open source

Globally-optimal TOPP2-RA, COPP2-SOCP, TOPP3-LP/SOCP and COPP3-SOCP solvers, in Rust and C. Read the docs, run the examples, benchmark it yourself.

github.com/TOPP-THU/copp ↗
PRO · COMMERCIAL

Talk to us about Pro

RDDP/RA solvers, long-horizon stability, commercial licensing and integration support for production robotics and CNC. Let's discuss your application.

hello@copp.pro