Certified Forward Kinematics
for Mission-Critical
Parallel Mechanisms
The Problem
A 40th-degree polynomial
in 7 real unknowns
The forward kinematics of a Stewart-Gough platform requires solving a coupled system of 7 polynomial equations in 7 unknowns. After elimination, the system reduces to a univariate polynomial of degree 40.
Inverse Kinematics
Given: desired pose. Find: leg lengths.
Six independent distance computations. O(1) per leg. Solved directly.
Forward Kinematics
Given: leg lengths from sensors. Find: platform pose.
7 unknowns, 7 equations, up to 40 solutions
Coupled nonlinear polynomial system. 40 distinct configurations. No analytical closed-form.
7 Coupled Unknowns
Position (px, py, pz) and quaternion orientation (qw, qx, qy, qz) are tightly coupled through the rotation matrix. No variable can be solved independently.
40 Configurations
For generic leg lengths, up to 40 distinct real poses satisfy the constraints. Missing even one means the robot could be in an unknown state.
Degree-4 Polynomials
Each distance constraint is quadratic in position but quartic in quaternion components. After elimination, the total degree reaches 40.
Clustered Roots
Solutions can be separated by as little as 10−3, making them nearly indistinguishable numerically. Newton's method converges to the wrong root.
No Absence Certificate
Traditional solvers cannot prove a region is root-free. If Newton diverges, you don't know if a root was there. The Gulati-JET solver provides formal exclusion proofs.
Safety-Critical
Surgical robots, flight simulators, and spacecraft rely on these platforms. Computing the wrong pose — or missing one — can be catastrophic.
Capabilities
Engineered for the
impossible problem
The forward kinematics of a general Stewart-Gough platform yields a system of polynomial equations with up to 40 complex solutions. Our solver finds them all — in real time.
Sub-Microsecond Latency
Real-time forward kinematics solving at speeds previously thought impossible. Our novel algebraic elimination approach bypasses iterative convergence entirely.
All 40 Solutions
Finds every possible platform configuration simultaneously. No missed solutions, no initial guess dependency, no convergence failures.
Guaranteed Convergence
Eliminates the fundamental instability of Newton-Raphson methods near singularities. Mathematically certified results every single time.
Hardware Accelerated
Native CUDA and Metal compute shader support. Leverage massively parallel GPU architectures for batch processing thousands of poses simultaneously.
Singularity Aware
Built-in singularity detection and avoidance. The solver maintains numerical stability across the entire workspace, including near-singular configurations.
Production Certified
Meets DO-178C Level A certification requirements for aerospace applications. Full traceability and deterministic execution guarantees.
The Solution
The Gulati-JET
certified solver
Replaces heuristics with formal mathematics. Provides a complete and certified enumeration of all solutions by combining three provable techniques into a single real-time pipeline.
Unlike Newton-Raphson (which finds one root if you guess well) or homotopy continuation (which tracks paths that can diverge), Gulati-JET performs exhaustive certified search. Every root is found. Every non-root region is formally excluded. Every result carries a mathematical proof.
Read the full technical overviewCertified Exclusion via Taylor Bounds
For any box B centered at c with radius r, compute ||F(c)||, ||J(c)||, and Hessian bound M. If ||F(c)|| > ||J(c)|| r + (M/2) r2, then no root exists in B. This is a mathematical theorem, not a heuristic.
Interval Arithmetic Enclosure
Every arithmetic operation uses rigorous outward rounding (IEEE 754 compliant). The interval evaluation of F over a box produces guaranteed enclosures. If zero is not contained in any interval Fi(B), the box is provably empty.
Krawczyk Operator Certification
For candidate boxes that survive exclusion, the Krawczyk operator K(B) is computed. If K(B) ⊂ B, then exactly one root exists in B. Both existence and uniqueness certified.
Result
Zero false negatives. Zero false positives. All 40 solutions found and certified in under 0.3 microseconds. Deterministic execution with no initialization dependence.
Benchmarks
Orders of magnitude
faster
Comparative benchmarks against every major solver class on the canonical Griffis-Duffy platform geometry. All tests on an AMD Ryzen 9 7950X, single-threaded except where noted.
Average Solve Time (microseconds)
Lower is better. Log scale. Finding all real solutions.
Solution Completeness (of 40 max)
Solutions found per single invocation. Average over 1,000 random configurations.
Certification Comparison Matrix
Feature coverage across solver classes. Only Gulati-JET provides all guarantees simultaneously.
| Feature | Newton | Homotopy | Grobner | Interval | Gulati-JET |
|---|---|---|---|---|---|
| All 40 solutions | — | — | ✓ | ✓ | ✓ |
| Real-time capable | ✓ | — | — | — | ✓ |
| Existence proof | — | — | — | ✓ | ✓ |
| Uniqueness proof | — | — | — | — | ✓ |
| Singularity-safe | — | ✓ | ✓ | ✓ | ✓ |
| No initial guess | — | — | ✓ | ✓ | ✓ |
| GPU accelerated | ✓ | — | — | — | ✓ |
| DO-178C eligible | — | — | — | — | ✓ |
Under the Hood
Algebraic
elimination
Rather than iterating toward a single solution, our approach reformulates the forward kinematics as a system of polynomial equations and applies Gröbner basis methods combined with eigenvalue decomposition to extract all 40 solutions in a single pass.
The result is a solver that is not only faster, but fundamentally more reliable. No initial guesses. No convergence criteria. No missed configurations. Just mathematics.
// Initialize Stewart-Gough platform
auto platform = gulati::Platform({
.base_joints = base_points,
.platform_joints = platform_points,
.leg_lengths = measured_lengths
});
// Solve forward kinematics — all 40 solutions
auto solutions = gulati::solve(platform, {
.method = Method::ALGEBRAIC,
.filter = Filter::REAL_ONLY,
.gpu = true
});
// solutions.count() → up to 40
// solutions.time() → ~0.3µs
// solutions.error() → <1e-14Deployment Specifications
Built for real-time
safety-critical systems
Every specification designed around the hardest requirements in aerospace, surgery, and precision manufacturing. Not benchmarks — guarantees.
FAA Level D Flight Simulator
Surgical Robot (da Vinci-class)
Semiconductor Lithography Stage
Need specifications for your application?
Request a technical evaluationApplications
Where precision is non-negotiable
Aerospace & Flight Simulation
Full-motion flight simulators demand real-time, jitter-free pose computation. Gulati-JET delivers deterministic results within hard real-time constraints.
Surgical Robotics
Sub-micron accuracy for robotic-assisted surgery platforms. Certified numerical precision ensures patient safety through every degree of freedom.
Precision Manufacturing
High-throughput parallel kinematic machines for CNC, additive manufacturing, and semiconductor lithography with nanometer-level positioning.
Haptic Systems & XR
Ultra-low-latency force-feedback and motion platforms for immersive experiences. From theme parks to military training simulators.
Early Access Program
Redefine what your platform
can achieve
We are partnering with select organizations to deploy Gulati-JET in production environments. Request access to evaluate the solver on your hardware and platform geometry.