ssik

Reliable enumerative inverse kinematics for 6R and 7R revolute robot arms, including non-Pieper 6R and non-SRS 7R geometries.

The mathematics of inverse kinematics and the numerical behavior of an IK solver are not the same thing. A manipulator may admit an exact algebraic reduction while a particular finite-precision realization loses roots, becomes ill-conditioned, or returns no solution for a pose known to be reachable. ssik is built around the stronger requirement that IK solutions must actually be recoverable and independently verifiable across the robot workspace.

For 6R arms, ssik enumerates the isolated IK branches. For redundant 7R arms, where the solution set is generically a one-dimensional manifold, ssik samples or parameterizes redundancy and enumerates the discrete IK branches conditional on each redundancy value. Every retained candidate is checked by forward-kinematic closure against the original robot model, and runs through the native C++ backend by default (typically 2–100× faster than pure Python, with an automatic pure-Python fallback).

72 arms ship prebuilt, including Universal Robots, Franka, KUKA iiwa, Kinova JACO/Gen3, Flexiv Rizon, Kassow, ABB YuMi, FANUC CRX, and many others. ssik build <your.urdf> specializes the same pipeline to a new robot: it examines the manipulator geometry, selects the simplest structurally valid solver, specializes robot-dependent algebra offline, and — where algebraically equivalent formulations exist — chooses representations for numerical conditioning.

The governing principle is simple:

Solvability is a property of the kinematic equations. Reliability is a property of the solver.

Install

pip install ssik

Python 3.11+. Wheels for Linux x86_64, macOS arm64, macOS x86_64, Windows x86_64. The native C++ backend is bundled in the Linux and macOS wheels (and used by default); Windows and source installs transparently run the identical pure-Python path.

Quickstart

from ssik.prebuilt import franka_panda_ik
import numpy as np

T_target = np.eye(4)
T_target[:3, 3] = [0.5, 0.1, 0.3]

sols = franka_panda_ik.solve(T_target)

sols is a list[Solution]. Each Solution carries:

• q: the joint configuration,
• fk_residual: ‖FK(q) − T_target‖,
• refinement_used: whether numerical polishing was required.

For a 6R arm, the list contains the certified isolated IK branches ssik recovered. For a 7R arm it contains branches obtained across the chosen redundancy samples.

An empty list means no certified solution was returned — which, by itself, is not a mathematical proof that the pose is unreachable. Use explain=True when diagnosing an empty result.

See every branch at once

pip install 'ssik[demo]'
python examples/05_viser_interactive_ik.py

Opens a browser viewer: drag a 3D handle and watch every analytical IK solution render as a live arm in real time. Cycle through the full prebuilt roster, including the non-Pieper 6R and 7R arms EAIK refuses.

Eight arms, every analytical branch

Each loop below is one arm's interactive demo running for ~3 seconds: the live red arm tracks the marker; the faded reds are the other analytical IK branches at the same instant. Captured from examples/05_viser_interactive_ik.py.

UR5: three-parallel 6R (Pieper). EAIK supports this class.

Unitree Z1: three-parallel 6R (UR-class). EAIK supports this class.

Franka Panda: anthropomorphic 7R. EAIK refuses ("only 1–6R").

UFactory xArm6: non-Pieper 6R. EAIK refuses ("6R-Unknown Kinematic Class").

Kinova JACO 2: non-Pieper 6R. EAIK refuses ("6R-Unknown Kinematic Class").

AgileX PiPER: non-Pieper 6R. EAIK refuses ("6R-Unknown Kinematic Class").

KUKA iiwa14: SRS 7R. EAIK refuses ("no 7R DH path").

Flexiv Rizon 4: non-SRS 7R. EAIK refuses ("only 1–6R").

Why ssik exists

General 6R inverse kinematics has been algebraically solvable for decades. Classical work by Raghavan–Roth, Manocha–Canny, and Husty–Pfurner showed how the kinematic equations of a general revolute 6R manipulator reduce to finite polynomial or eigenvalue problems. Geometric approaches such as IK-Geo show how manipulator structure can simplify the same problem dramatically.

The remaining practical problem is numerical recovery. Two algebraically equivalent formulations can behave very differently in floating-point arithmetic. In ssik's Raghavan–Roth implementation, for example, changing which joint is used as the elimination variable on the Kinova JACO 2 changes the conditioning of the quadratic coefficient matrix from approximately

3.75 × 10^16   →   127

while leaving the exact IK problem unchanged. One formulation loses solutions to floating-point error; the other recovers them.

This distinction drives the design of ssik:

kinematic model
      │
      ▼
structural classification ── exploit special geometry when available
      │                       (else general algebraic elimination)
      ▼
numerical representation selection   (choose the best-conditioned equivalent)
      │
      ▼
candidate IK solutions
      │
      ▼
FK certification / optional refinement / recovery
      │
      ▼
certified solutions

Special geometry determines how cheaply and robustly IK is solved, not whether enumerative IK is available at all. The goal is not to possess a derivation that is complete in exact arithmetic — it is to make that derivation survive contact with real robot geometry, finite precision, singular and near-singular configurations, joint limits, and deployment software.

The artifact model

ssik treats IK generation as an offline specialization problem. Each robot becomes a self-contained artifact holding its normalized kinematics, the selected solver, robot-specific constants, and any symbolic or algebraic preprocessing that can be moved off the runtime path:

URDF / robot specification
        │
        ▼
geometry + solver specialization
        │
        ▼
conditioning-aware preprocessing
        │
        ▼
<arm>_ik.py  +  self-contained C++ artifact
        │
        ▼
pure numerical solve at deployment

There is no URDF parsing, urchin, or sympy on the artifact runtime path. This follows the deployment precedent OpenRAVE's IKFast established — do robot-specific symbolic work once, then ship a numerical artifact — extended across a heterogeneous solver hierarchy (geometric closed forms, general 6R algebraic elimination, redundant 7R reductions), and without IKFast's brittleness on non-Pieper geometries. The artifact encodes not just which robot is being solved, but which computational representation of that robot's IK was found to be appropriate.

There are two artifact paths:

Use a prebuilt arm (ssik.prebuilt)

The wheel ships 72 ready-to-import artifacts, grouped by vendor below (expand a vendor to see its arms). Each imports as ssik.prebuilt.<vendor>.<module> (e.g. from ssik.prebuilt.universal_robots import ur5_ik) and the flat from ssik.prebuilt import ur5_ik alias still works. Each was built against a specific URDF (or extracted spec); T_target is the pose of EE_LINK expressed in BASE_LINK:

Universal Robots: ssik.prebuilt.universal_robots (11 arms)

Module	Arm	Class	base_link	ee_link
ur5_ik	Universal Robots UR5	three-parallel 6R	base_link	ee_link
ur3e_ik	Universal Robots UR3e	three-parallel 6R	base_link	tool0
ur5e_ik	Universal Robots UR5e	three-parallel 6R	base_link	tool0
ur10e_ik	Universal Robots UR10e	three-parallel 6R	base_link	tool0
ur16e_ik	Universal Robots UR16e	three-parallel 6R	base_link	tool0
ur20_ik	Universal Robots UR20	three-parallel 6R	base_link	tool0
ur30_ik	Universal Robots UR30	three-parallel 6R	base_link	tool0
ur7e_ik	Universal Robots UR7E	three-parallel 6R	base_link	tool0
ur12e_ik	Universal Robots UR12E	three-parallel 6R	base_link	tool0
ur15_ik	Universal Robots UR15	three-parallel 6R	base_link	tool0
ur18_ik	Universal Robots UR18	three-parallel 6R	base_link	tool0

Unimation: ssik.prebuilt.unimation (1 arm)

Module	Arm	Class	base_link	ee_link
puma560_ik	KUKA Puma 560	Pieper 6R (spherical wrist)	base_link	wrist_3_link

Kinova: ssik.prebuilt.kinova (5 arms)

Module	Arm	Class	base_link	ee_link
jaco2_ik	Kinova JACO 2	non-Pieper 6R	base_link	ee_link
gen3_ik	Kinova Gen3 7-DOF	approximate-SRS 7R	base_link	end_effector_link
gen3_lite_ik	Kinova Gen3 Lite	non-Pieper 6R	base_link	end_effector_link
j2s6s300_ik	Kinova JACO j2s6s300	Pieper 6R (spherical wrist)	j2s6s300_link_base	j2s6s300_end_effector
j2s7s300_ik	Kinova JACO j2s7s300	approximate-SRS 7R (spherical wrist)	j2s7s300_link_base	j2s7s300_link_7

KUKA: ssik.prebuilt.kuka (4 arms)

Module	Arm	Class	base_link	ee_link
iiwa14_ik	KUKA iiwa LBR 14	SRS 7R	base	iiwa_link_ee_kuka
iiwa7_ik	KUKA iiwa LBR 7	SRS 7R (offset wrist)	iiwa_link_0	iiwa_link_ee
kr6_r900_ik	KUKA KR 6 R900 sixx (Agilus)	Pieper 6R (spherical wrist)	base_link	link_6
kr210_r2700_ik	KUKA KR 210 R2700 (Quantec)	Pieper 6R (spherical wrist)	base_link	link_6

Franka: ssik.prebuilt.franka (2 arms)

Module	Arm	Class	base_link	ee_link
panda_ik	Franka Panda	spherical-shoulder + offset-wrist 7R	panda_link0	panda_link8
fr3_ik	Franka Research 3	spherical-shoulder + offset-wrist 7R (Panda successor)	fr3_link0	fr3_link8

UFactory: ssik.prebuilt.ufactory (2 arms)

Module	Arm	Class	base_link	ee_link
xarm7_ik	UFactory xArm7	approximately-spherical-shoulder 7R	link_base	link7
xarm6_ik	UFactory xArm6	non-Pieper 6R (joint 6 y-offset)	link_base	link_eef

Unitree: ssik.prebuilt.unitree (1 arm)

Module	Arm	Class	base_link	ee_link
z1_ik	Unitree Z1	three-parallel 6R (UR-class)	link00	link06

AgileX: ssik.prebuilt.agilex (1 arm)

Module	Arm	Class	base_link	ee_link
piper_ik	AgileX PiPER	non-Pieper 6R (joints 4 & 6 tilted axis)	base_link	link6

Flexiv: ssik.prebuilt.flexiv (2 arms)

Module	Arm	Class	base_link	ee_link
rizon4_ik	Flexiv Rizon 4	non-SRS 7R	base_link	flange
rizon10_ik	Flexiv Rizon 10	non-SRS 7R (~1.4 m reach)	base_link	flange

Kassow: ssik.prebuilt.kassow (1 arm)

Module	Arm	Class	base_link	ee_link
kr810_ik	Kassow KR810	non-SRS 7R	base	end_effector

FANUC: ssik.prebuilt.fanuc (10 arms)

Module	Arm	Class	base_link	ee_link
crx3ia_ik	FANUC CRX-3iA	non-Pieper 6R (non-spherical wrist)	base_link	tool0
crx5ia_ik	FANUC CRX-5iA	non-Pieper 6R (non-spherical wrist)	base_link	tool0
crx10ia_ik	FANUC CRX-10iA	non-Pieper 6R (non-spherical wrist)	base_link	tool0
crx10ialp_ik	FANUC CRX-10iA/LP	non-Pieper 6R (non-spherical wrist)	base_link	tool0
crx20ial_ik	FANUC CRX-20iA/L	non-Pieper 6R (non-spherical wrist)	base_link	tool0
crx30ia_ik	FANUC CRX-30iA	non-Pieper 6R (non-spherical wrist)	base_link	tool0
crx10ial_ik	FANUC CRX-10iA/L	non-Pieper 6R (non-spherical wrist, 150 mm y-offset)	base_link	tool0
m710ic_ik	FANUC M-710iC/70	Pieper 6R (spherical wrist)	base_link	link_6
lrmate200id_ik	FANUC LR Mate 200iD	Pieper 6R (spherical wrist)	base_link	link_6
r2000ic210l_ik	FANUC R-2000iC/210L	Pieper 6R (spherical wrist)	base_link	link_6

I2RT: ssik.prebuilt.i2rt (2 arms)

Module	Arm	Class	base_link	ee_link
yam_ik	I2RT YAM	non-Pieper 6R	base_link	link_6
big_yam_ik	I2RT big_yam	non-Pieper 6R	base	gripper

Enactic OpenArm: ssik.prebuilt.openarm (2 arms)

Module	Arm	Class	base_link	ee_link
left_ik	Enactic OpenArm v2.0 (left)	SRS 7R (non-Z*Z)	openarm_left_base_link	openarm_left_ee_base_link
right_ik	Enactic OpenArm v2.0 (right)	SRS 7R (non-Z*Z)	openarm_right_base_link	openarm_right_ee_base_link

Galaxea: ssik.prebuilt.galaxea (2 arms)

Module	Arm	Class	base_link	ee_link
r1pro_left_ik	Galaxea R1 Pro (left)	SRS 7R (non-Z*Z)	left_arm_base_link	left_arm_link7
r1pro_right_ik	Galaxea R1 Pro (right)	SRS 7R (non-Z*Z)	right_arm_base_link	right_arm_link7

Standard Bots: ssik.prebuilt.standard_bots (3 arms)

Module	Arm	Class	base_link	ee_link
thor_ik	Standard Bots Thor	three-parallel 6R	base_link	tool0
core_ik	Standard Bots Core	three-parallel 6R	base_link	tool0
spark_ik	Standard Bots Spark	three-parallel 6R	base_link	tool0

Abb: ssik.prebuilt.abb (5 arms)

Module	Arm	Class	base_link	ee_link
yumi_left_ik	ABB YuMi (IRB 14000) left	approximate-SRS 7R	yumi_body	yumi_link_7_l
yumi_right_ik	ABB YuMi (IRB 14000) right	approximate-SRS 7R	yumi_body	yumi_link_7_r
irb120_ik	ABB IRB 120	Pieper 6R (spherical wrist)	base_link	link_6
irb1600_ik	ABB IRB 1600	Pieper 6R (spherical wrist)	base_link	link_6
irb6700_ik	ABB IRB 6700	Pieper 6R (spherical wrist)	base_link	link_6

Yaskawa: ssik.prebuilt.yaskawa (2 arms)

Module	Arm	Class	base_link	ee_link
gp8_ik	Yaskawa GP8	Pieper 6R (spherical wrist)	base_link	link_6_t
hc10_ik	Yaskawa HC10	non-Pieper 6R	base_link	link_6_t

Kawasaki: ssik.prebuilt.kawasaki (1 arm)

Module	Arm	Class	base_link	ee_link
rs007n_ik	Kawasaki RS007N	Pieper 6R (spherical wrist)	base_link	link6

Staubli: ssik.prebuilt.staubli (1 arm)

Module	Arm	Class	base_link	ee_link
rx160_ik	Staubli RX160	Pieper 6R (spherical wrist)	base_link	link_6

Realman: ssik.prebuilt.realman (2 arms)

Module	Arm	Class	base_link	ee_link
rm75_ik	Realman RM75	approximate-SRS 7R	base_link	link_7
gen72_ik	Realman GEN72	approximately-spherical-shoulder 7R	base_link	Link7

Dobot: ssik.prebuilt.dobot (2 arms)

Module	Arm	Class	base_link	ee_link
cr5_ik	Dobot CR5	three-parallel 6R (UR-class)	base_link	Link6
nova5_ik	Dobot Nova5	three-parallel 6R (UR-class)	base_link	Link6

Mitsubishi: ssik.prebuilt.mitsubishi (1 arm)

Module	Arm	Class	base_link	ee_link
rv4fr_ik	Mitsubishi RV-4FR	Pieper 6R (spherical wrist)	rv4fr_base	rv4fr_hand_flange

Hyundai: ssik.prebuilt.hyundai (1 arm)

Module	Arm	Class	base_link	ee_link
hh020_ik	Hyundai HH020	Pieper 6R (spherical wrist)	base_link	tool0

Denso: ssik.prebuilt.denso (1 arm)

Module	Arm	Class	base_link	ee_link
vs060_ik	Denso VS-060	Pieper 6R (spherical wrist)	base_link	J6

Doosan: ssik.prebuilt.doosan (2 arms)

Module	Arm	Class	base_link	ee_link
m1013_ik	Doosan M1013	non-Pieper 6R	base_link	link_6
m0609_ik	Doosan M0609	non-Pieper 6R	base_link	link_6

Rokae: ssik.prebuilt.rokae (3 arms)

Module	Arm	Class	base_link	ee_link
xmatepro7_ik	Rokae xMate Pro7	SRS 7R	xMatePro7_base	xMatePro7_link7
xmatecr7_ik	Rokae xMate CR7	non-Pieper 6R	xMateCR7_base	xMateCR7_link6
xmatesr3_ik	Rokae xMate SR3	non-Pieper 6R	xMateSR3_base	xMateSR3_link6

Trossen: ssik.prebuilt.trossen (2 arms)

Module	Arm	Class	base_link	ee_link
viperx300s_ik	Trossen ViperX 300s	Pieper 6R (spherical wrist)	base_link	gripper_link
widowx250s_ik	Trossen WidowX 250s	Pieper 6R (spherical wrist)	wx250s/base_link	wx250s/gripper_link

from ssik.prebuilt import iiwa14_ik
sols = iiwa14_ik.solve(T_target)

Artifacts are organized by vendor, and the flat import above always works as an alias:

import ssik
ssik.list_arms()                             # discover everything, imports nothing
ssik.list_arms(vendor="universal_robots")    # filter by vendor

from ssik.prebuilt.universal_robots import ur5_ik   # vendor path (preferred)
from ssik.prebuilt import ur5_ik                     # flat alias (still supported)

import ssik, import ssik.prebuilt, and import ssik.prebuilt.<vendor> load zero arm artifacts: only importing a specific <arm>_ik module builds anything.

Where each fixture comes from

Each prebuilt's kinematic chain is sourced from a specific upstream URDF (or, for legacy DH arms, the published parameter set), and tests/test_prebuilt_fixture_parity.py asserts module.fk(q) == upstream.fk(q) to machine precision for every arm reachable via robot_descriptions. The full per-arm provenance table lives in the docs: Fixture provenance.

Every prebuilt exposes BASE_LINK, EE_LINK, DOF, and T_HOME (the 4×4 home pose, FK at q = np.zeros(DOF)) as module constants. Use them to verify the baked geometry matches your robot:

from ssik.prebuilt import franka_panda_ik
print(franka_panda_ik.BASE_LINK, "→", franka_panda_ik.EE_LINK, "(", franka_panda_ik.DOF, "DOF)")
# base_link → ee_link ( 7 DOF)
print(franka_panda_ik.T_HOME[:3, 3])
# array([0.088, 0., 0.926])     ← Franka home pose; matches the spec

When a prebuilt is right vs when to ssik build

The prebuilts cover nominal manufacturer geometry with a bare flange. They work when:

• You're using the same URDF source we built against (ros-industrial, manufacturer reference, etc.)
• Your robot's calibration matches the nominal kinematic parameters
• Your end-effector is the flange itself, no gripper, suction cup, or custom tool past it
• Your URDF link names match what we baked (see the table above)

If any of those is false (and especially if you're a 7R arm with anything attached past the flange) build your own:

pip install ssik[urdf]
ssik build <your.urdf> --base <your_base_link> --ee <your_actual_tool_link>
# → <your_arm>_ik.py

ssik build reads your exact URDF, picks the right solver via the same dispatcher we use, and emits a single-file artifact correct for your kinematic chain. That artifact's import / API / public constants are identical to the prebuilts'.

For trajectory tracking and IK-based teleop, the canonical pattern is "give me the IK closest to where the robot is now":

# Robot's current configuration (from joint sensors, last command, etc.).
q_current = np.array([0.0, -0.5, 0.0, 0.7, 0.0, 1.2, 0.0])

# Target pose updates every control tick (VR controller, planner, etc.).
T_target = ...

# max_solutions=1 + q_seed: returns the single solution nearest q_current.
# On 7R jointlock arms the seed drives the lock-outward fast path (~20×
# faster than the full sweep); sub-ms on 6R / SRS arms.
sols = franka_panda_ik.solve(T_target, max_solutions=1, q_seed=q_current)
q_command = sols[0].q if sols else q_current

When a seed is given, two knobs control what "nearest" means:

• seed_metric (default "wrap_linf") ranks by the largest single-joint move, so the arm holds its branch instead of flipping mid-trajectory; "wrap_l2" ranks by summed distance.
• seed_tolerance (radians) is a hard bound: only solutions whose every joint is within the tolerance of the seed are returned. The result may be empty, which is the signal that smooth continuation isn't possible at this pose (replan / accept a jump). Omitted ⇒ best-effort (always returns the nearest if any IK exists).

# "no joint jumps more than 6° from where I am, or tell me it can't":
sols = franka_panda_ik.solve(
    T_target, q_seed=q_current, max_solutions=1, seed_tolerance=np.deg2rad(6)
)
q_command = sols[0].q if sols else replan()   # empty ⇒ discontinuity

Build an artifact for your own arm

For any arm not in the prebuilt set, run ssik build once against the URDF:

ssik build my_arm.urdf --base base_link --ee tool0
# → my_arm_ik.py

Build time depends on solver class:

• <1 s for tier-0 closed-form (UR-class, Pieper, SRS-class 7R)
• ~30 s for non-Pieper 6R (Raghavan–Roth symbolic derivation)
• 7–20 min for non-SRS 7R (cached Husty–Pfurner per lock sample)

Ship the emitted .py alongside your robot stack. Once built, use it exactly like a prebuilt:

import my_arm_ik
sols = my_arm_ik.solve(T_target)

Re-run ssik build after pip install -U ssik if you want the latest solver fixes. Old artifacts keep working. They're frozen against the ssik version that built them. ssik build requires the URDF extras: pip install ssik[urdf].

Development path: Manipulator.from_urdf (not for deployment)

For one-off experiments before committing to a build artifact, ssik also exposes the runtime classifier as a Python class:

import ssik
arm = ssik.Manipulator.from_urdf("my_arm.urdf", base="base_link", ee="tool0")
sols = arm.solve(T_target, max_solutions=1, q_seed=q_current)

Every fresh process re-runs URDF parsing, topology classification, and (for non-Pieper sub-chains) first-call sympy preprocessing, so this path is strictly slower than the build-artifact path in production and requires urchin + sympy on the runtime path (pip install ssik[urdf]). Once dispatch is settled, switch to ssik build.

Contributors extending ssik's own test fixtures (vs deploying for their own arm) use ssik add-arm; see CONTRIBUTING.md.

What solve() returns

A list[Solution]. Each Solution has:

• q: joint-angle vector (length DOF)
• fk_residual: ‖FK(q) − T_target‖_F (Frobenius norm against the original URDF / spec FK)
• refinement_used: "none" or "lm" if Levenberg–Marquardt polish fired

For a generic 6R arm the IK solution set is finite, with at most 16 isolated solutions (8 typical for a Pieper-class arm: 4 shoulder × 2 elbow, wrist deterministic). ssik recovers these discrete branches and rejects any candidate that does not close under the original forward kinematics.

For a 7R arm the situation is different: the solution set is generically a one-dimensional self-motion manifold, so there is no finite set of "all 7R IK solutions." ssik parameterizes or samples that redundancy and enumerates the discrete algebraic branches associated with each sample. A result count of 128 therefore means, for example, 16 redundancy samples × 8 conditional branches — not that the arm has only 128 IK solutions. This distinction is intentional: 6R enumeration is over isolated solutions; 7R enumeration is conditional on a redundancy parameterization.

By default solve() runs respect_limits=True: out-of-URDF-limit branches are dropped (with a q ± 2π rescue pass first), then duplicates are merged. On 7R jointlock arms the limits filter runs during the lock-sweep, so max_solutions=1 short-circuits on the first in-limits candidate. Pass respect_limits=False for the raw geometric set. Seed ranking, seed tolerances, and max_solutions then select among the recovered branches for trajectory continuation or control.

The allow_refinement=True opt-in runs LM polish per algebraic candidate at a few hundred microseconds per branch, useful when an algebraic candidate lands just above fk_atol near a kinematic singularity.

Joints that can turn more than once

A joint whose limits span more than a full turn — the UR family's [−2π, 2π], and 46 of the 72 prebuilt arms have at least one — reaches the same pose at several different joint coordinates. With limits [−2π, 2π], −10° and +350° are the same place and different configurations: different distance from where the robot is now, different room to keep turning, different chance of hitting the limit mid-trajectory.

Since v6.0, solve() returns all of them:

sols = ur5e_ik.solve(T)            # 256 = 8 geometric branches x 2^5 wide joints
sols = ur5e_ik.solve(T, enumerate_windings=False)   # 8, one per geometric branch

These are finite-limit lifts of the same geometric branch, not additional IK branches, and the distinction is kept in the diagnostics rather than blurred:

sols, diag = arm.solve(T, explain=True)
diag.geometric_branches        # 8   -- real IK branches
diag.winding_representatives   # 256 -- their in-limit coordinates

Ranking accounts for it. A finite joint's configuration space is an interval, not a circle: it cannot rotate through its limit, so its distance to a seed is the ordinary coordinate difference, and +3 really is 6 radians from −3. Only continuous joints (limits=None), which can always take the short way, are measured modulo 2π. So a seed at +350° gets the +350° representative back rather than an equivalent one a full turn away.

Returning 32× more configurations costs proportionally more, but a capped solve never pays for what it discards: solve(T, q_seed=q, max_solutions=1) — the tracking idiom — returns the globally nearest configuration for the same cost as the un-enumerated solve, and is identical to enumerating, ranking and truncating.

The redundant arm's other answers: self-motion charts

A 7R arm holding a 6-DOF pose is not at a point in configuration space, it is on a curve: the self-motion manifold, the one-parameter family of postures with the same end-effector pose. solve() samples that curve and hands back points. Since v6.1 you can ask for the curve itself.

manifold = arm.self_motion(T_target)   # every branch at this pose
chart = manifold.charts[0]             # one continuous branch q(t)

chart.q(t)          # the posture at t
chart.tangent(t)    # the self-motion direction and rate; J @ direction == 0
chart.domain        # the intervals of t where the branch exists
chart.in_limits()   # the sub-arcs of those that stay inside the joint box
chart.sample(20)    # ~20 postures along the branch, spaced evenly in arc length
                    # (returned per continuous segment, as (t, q) arrays)

The closed-form solvers already compute this internally, as a branch label plus a redundancy coordinate, and used to discard both at the Solution boundary. The tangent comes from implicit differentiation of FK(q(t)) = T, so J·q' = 0 holds by construction rather than by choosing a step size, and a fold returns NaN instead of a fabricated direction.

Beyond one branch, the manifold knows its own structure: sheets() glues charts that meet at folds, so you get the components a controller can actually reach by self-motion; gap(a, b) and escape(a, b) measure how close two sheets come and the task-space twist that closes the distance fastest; locate(q) inverts back to a chart and parameter. Along a path, solve_path(poses) tracks every branch by label, reporting folds, unreachable poses, and on a closed loop the monodromy permutation.

Available for seven_r.spherical_shoulder (Panda, FR3), seven_r.srs (iiwa and other exactly-concurrent SRS arms), and ikgeo.three_parallel (UR-class 6R, where the charts are the isolated solutions with geometric labels). Other families raise NotImplementedError. What a label promises differs per family and is written down in the ssik.chart module docstring, because the joint-lock and polished solvers cannot offer the same guarantee.

On the native backend a family builds in roughly 20 µs and q(t) costs a few µs, so this fits inside a 1 kHz loop. See docs/api.md for the full surface.

Diagnosing an empty result: explain=True

If solve() returns [], you can attribute the failure with explain=True instead of guessing:

import ssik
arm = ssik.Manipulator.from_urdf("my_arm.urdf", base="base_link", ee="tool0")
sols, diag = arm.solve(T_target, explain=True)
if not sols:
    print(diag.summary())
    # solver: ikgeo.three_parallel (tier 0)
    # dispatch: Three consecutive parallel axes at joints (1, 2, 3) ...
    #   -> 0 raw candidates: pose appears unreachable
    #      (or outside this solver's analytical envelope)

The Diagnostic record distinguishes:

• Unreachable (raw_candidates == 0): pose is outside the solver's analytical envelope
• All-filtered (raw_candidates > 0, final_count == 0): try respect_limits=False for the raw geometric set
• Capped (dropped_by_max_solutions > 0): pass a larger max_solutions

Available on ssik.Manipulator.solve today; per-prebuilt explain mode tracked in #265.

Tuning knobs

TolerancePolicy: six thresholds, one object

solve() accepts an optional policy= kwarg. The default ssik.DEFAULT_TOLERANCE_POLICY works for every shipped fixture; reach for a custom policy when a real arm's URDF has structural near-degeneracies (axes that almost but not exactly meet) or when you want tighter / looser FK closure than the defaults provide.

from ssik import TolerancePolicy, DEFAULT_TOLERANCE_POLICY

policy = TolerancePolicy(
    axis_parallel=1e-8,         # ||a × b||: when two axes are "parallel"
    axis_intersect=1e-8,        # perpendicular distance: when two lines "meet"
    subproblem_feasibility=1e-9,# is_ls boundary inside SP1-SP6
    subproblem_numerical=1e-5,  # FK-closure filter on algebraic candidates
    subproblem_degeneracy=1e-12,# rank-drop threshold; below this, return []
    subproblem_dedup=1e-3,      # angle-space tolerance for collapsing duplicates
)
sols = my_arm_ik.solve(T_target, policy=policy)

The fields are named for why they exist so log messages can say "SP6 sign branch rejected: closure 1.2e-4 > subproblem_numerical 1e-5" instead of citing magic numbers.

How to read fk_residual, and how to tighten it

fk_residual is ‖FK(q) − T_target‖_F: a Frobenius norm of a 4×4 SE(3) matrix mixing rotation (radians, dimensionless when small) and translation (meters). For a typical 1 m-reach arm:

fk_residual	Position-error scale	Note
1e-3	1 mm	visible to the naked eye
1e-4	0.1 mm	typical robot repeatability (manufacturer spec)
1e-5 (default)	10 µm	sub-repeatability; fine for control
1e-9	1 nm	math / analysis territory
1e-13	0.1 pm	float64 epsilon

The default subproblem_numerical = 1e-5 is intentionally pragmatic, already two orders below what any physical robot can mechanically repeat, but cheap enough that all prebuilts hit it without LM polish. Most control / planning users want exactly this default.

To get machine precision (RL training, differentiable IK, sample-based planning, math validation), tighten the one field that gates FK closure and opt into LM polish:

from dataclasses import replace
from ssik import DEFAULT_TOLERANCE_POLICY
from ssik.prebuilt.franka import panda_ik

tight = replace(DEFAULT_TOLERANCE_POLICY, subproblem_numerical=1e-9)  # 4 orders tighter
sols = panda_ik.solve(T_target, policy=tight, allow_refinement=True)
# every returned IK FK-closes ~3e-10 (~0.3 nm position error)

The allow_refinement=True flag engages Levenberg-Marquardt polish on candidates that don't meet subproblem_numerical. On the jointlock 7R arms (Franka, Rizon 4, Kassow KR810) this lifts worst-case FK from ~5×10⁻⁶ (default) to ~3×10⁻¹⁰ (tight + LM). Cost: a few hundred microseconds per polished candidate. Sub-repeatability arms (UR5, Puma 560, JACO 2, iiwa14, Gen3) already hit machine precision at the default policy and don't need the opt-in.

Per-arm worst-case behaviour under both policies is documented in docs/arm_coverage.md.

ssik.postprocess: composable filters

solve() returns the geometric IK set. For application-specific filtering, five helpers in ssik.postprocess compose into the typical "robot-aware IK" pipeline:

from ssik.postprocess import (
    respect_limits, wrap_to_limits, nearest_to_seed, within_seed_tolerance, take_first,
)

sols = my_arm_ik.solve(T_target, respect_limits=False)       # raw geometric set
sols = wrap_to_limits(sols, my_arm_ik._KB)                   # try q ± 2π to bring in
sols = respect_limits(sols, my_arm_ik._KB)                   # drop anything still outside
sols = within_seed_tolerance(sols, q_current, np.deg2rad(6)) # drop big-jump branches (may empty)
sols = nearest_to_seed(sols, q_current, metric="wrap_linf")  # rank by max-joint-move
sols = take_first(sols, k=4)                                 # top-k after ranking

By default solve() already runs wrap_to_limits + respect_limits (and, when q_seed/seed_tolerance/seed_metric are passed, the seed filter + ranking); the standalone helpers exist for callers who want a different order, a different metric, or to add their own filters (collision-aware filtering, dexterity scoring) between the layers.

Native (C++) backend — the default

Every one of the 72 prebuilt arms runs a bundled C++ implementation of the full solve() contract by default — typically 2–100× faster than the pure-Python path (median ~16×; see docs/native_benchmark.md for the full per-arm table). Nothing to opt into:

sols = ur5_ik.solve(T_target)                                # native by default
sols = ur5_ik.solve(T_target, native=False)                  # identical algorithm, pure Python

• Same answers. Native reproduces the Python result's solution set. Without a seed the order and the near-singular representative may differ (numpy vs Eigen), and redundant-7R arms may sample the self-motion manifold differently; with a seed the nearest solution is stable. Parity is gated in CI against the Python solve() across every arm and option (limits / seed / max / tolerance).
• Automatic fallback. Where the native extension isn't bundled (Windows wheels, source installs), solve() transparently runs the identical pure-Python path — it never fails for unavailability. Pass native=False to force it explicitly (e.g. for bit-reproducible results across machines).

Using ssik from C++ (no Python)

The same solvers are also emitted as zero-runtime-Python, header-only artifacts — cpp/gen/<arm>_ik.hpp — for MoveIt / real-time / any-C++ use where the Python library isn't an option. Each header bakes one robot's geometry and exposes solve(T) returning every IK branch, depending only on Eigen:

#include "iiwa14_ik.hpp"
namespace arm = ssik::iiwa14_ik;
auto sols = arm::solve(T);          // std::vector<ssik::Solution<arm::DOF>>, all in-limits IK

find_package(ssik_cpp REQUIRED)
target_link_libraries(my_app PRIVATE ssik::ssik_cpp)   # every <arm>_ik.hpp on the include path

Full guide (install/find_package, the bare-c++ path, the Solution/ArtifactParams types, solve_batch, generating any arm): cpp/README.md. Runnable examples: cpp/examples/solve_arm.cpp and the standalone downstream project cpp/examples/consumer/ (built in CI). A MoveIt KinematicsBase plugin generator is tracked in #493.

Out of scope: collision filtering (use FCL or similar at the application layer) and continuous-trajectory smoothness (typically a separate planner concern).

Reliability and FK certification

Every candidate an internal solver produces is checked against the original forward kinematics. For target T and candidate q, ssik evaluates ‖FK(q) − T‖ before exposing the solution to the caller — a common correctness check independent of how the candidate was generated. A candidate may originate from a geometric decomposition, Raghavan–Roth elimination, Husty–Pfurner elimination, a redundancy reduction, or numerical polishing; the final question is always the same: does this configuration actually reproduce the requested pose?

This separates two properties that are often conflated:

• Soundness — every returned configuration actually solves the requested IK problem to tolerance.
• Recovery / completeness — the solver does not silently lose valid branches or fail on reachable configurations.

FK closure directly checks soundness. Recovery requires stronger testing: independent cross-solver agreement, randomized reachable-pose sweeps, adversarial singularity probes, representation diversity, and branch-count checks. That is why ssik's tests emphasize reachable-pose recovery and worst-case behavior over average FK error or average runtime.

Conditioning-aware algebraic IK

For general non-Pieper 6R arms, an exact algebraic derivation does not uniquely determine its numerical realization — equivalent elimination choices can yield polynomial-eigenvalue problems with radically different conditioning. ssik therefore treats representation as part of the solver. For algebraically equivalent representations r1, r2, …,

exact_solution_set(r1) = exact_solution_set(r2)

does not imply

numerical_recovery(r1) = numerical_recovery(r2).

AE-3 exploits this: it evaluates alternative Raghavan–Roth elimination variables and specializes the artifact to a better-conditioned choice (the JACO 2 3.75e16 → 127 result above). This turns conditioning from a post-hoc debugging statistic into an input to solver construction. The same principle drives structural dispatch, alternative algebraic formulations, refinement, rescue, and FK certification — all serving one goal: expand the part of the reachable workspace on which the solver reliably recovers valid IK.

Some configurations where a naive closed form quietly fails, and how ssik handles them:

Configuration	Failure mode	ssik
Rank-deficient RR ridge (reachable, measure-zero)	analytical path returns []	reach-gated T-perturbation rescue
180° twist, α = π (JACO 2 joint 2)	tan(α/2)→∞, roots lost in float64	chart cap + LM polish
Symmetric-DH locked-7R	RR incomplete / long hang	Husty–Pfurner Study-quaternion dispatch
Ill-conditioned 80×80 pencil	Eigen QZ non-converges (~38%)	monic-companion reduction
Offset wrist (iiwa7, ±6 cm)	canonical path mislocates the wrist	route to the general path
Anti-parallel joint trio (Standard Bots)	signed-sum collapse → FK-wrong	three-parallel sign guards

How it compares

There are three different questions an IK system can answer: (1) can the kinematic equations be solved? (2) can a finite-precision implementation reliably recover the solution set? (3) how much computation does that recovery cost? ssik is primarily concerned with the second.

Numerical IK (MINK, TRAC-IK, KDL-LMA) solves a local optimization from a seed and returns one converged configuration — often exactly the right interface for servoing. ssik instead exposes multiple kinematically valid branches, separating kinematic feasibility from which feasible configuration is best for the task (a planner can enumerate branches and choose by clearance, limits, manipulability, or distance from the current pose).

EAIK (Ostermeier 2024) automatically recognizes several geometric manipulator families and derives efficient subproblem-decomposition solvers for them; on those families it is extremely fast and accurate. The table below compares the current EAIK implementation on the supplied fixtures as-is against ssik. A refuses row means EAIK did not produce a valid solution for that fixture through this benchmark path — not a claim that the robot could never be handled via joint-locking or remodeling. On geometries EAIK recognizes, its specialized C++ is generally faster; ssik's emphasis is retaining enumerative IK as the geometry becomes less structurally convenient, always checking returned solutions against the original FK. The ssik column is solve() on the native backend (the default) over the geometric solution set, which is the like-for-like comparison against what EAIK returns. On an arm with a joint that can turn more than once, ssik's default additionally lifts each branch to its in-limit coordinates (above) and costs proportionally more for the larger set it returns; enumerate_windings=False is the benchmarked behaviour.

The table is measured automatically by scripts/regen_bench.py (both libraries over the same 200 random reachable poses per arm, mean ± 95% CI via 1000-resample bootstrap) and stored in the manifest, so it refreshes when an arm is added — no hand-maintained numbers. FK residual is the Frobenius norm ‖FK(q) − T‖. Each library is fed the same manufacturer fixture as-is (no manual joint-locking).

Universal Robots: ssik.prebuilt.universal_robots (11 arms)

Arm (class)	EAIK	ssik
UR5 (Pieper 6R, three-parallel)	4 ± 0 µs / FK 2e-15 / 2-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR3e (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 2-6 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR5e (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 4-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR10e (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 2-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR16e (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 4-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR20 (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 4-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR30 (Pieper 6R, three-parallel)	4 ± 0 µs / FK 2e-15 / 2-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR7E (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 4-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR12E (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 2-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols
UR15 (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 4-8 sols	20 ± 0 µs / FK 8e-11 / 2-8 sols
UR18 (Pieper 6R, three-parallel)	4 ± 0 µs / FK 1e-15 / 2-8 sols	20 ± 0 µs / FK 1e-8 / 2-8 sols

Unimation: ssik.prebuilt.unimation (1 arm)

Arm (class)	EAIK	ssik
Puma 560 (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 8e-12 / 8 sols	10 ± 0 µs / FK 9e-9 / 8 sols

Kinova: ssik.prebuilt.kinova (5 arms)

Arm (class)	EAIK	ssik
JACO 2 (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	350 ± 10 µs / FK 3e-9 / 2-12 sols
Gen3 (approximate-SRS 7R, 12 mm offset)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	2.26 ± 1.15 ms / FK 1e-12 / 14-95 sols
Gen3 Lite (non-Pieper 6R)	refuses ("Intersection point can't be calculated for two parallel axes")	350 ± 10 µs / FK 7e-9 / 1-12 sols
JACO j2s6s300 (Pieper 6R, spherical wrist)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	10 ± 0 µs / FK 6e-8 / 6-8 sols
JACO j2s7s300 (approximate-SRS 7R, 1.6 mm offset)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	2.68 ± 0.16 ms / FK 1e-12 / 18-80 sols

KUKA: ssik.prebuilt.kuka (4 arms)

Arm (class)	EAIK	ssik
iiwa14 (SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	210 ± 0 µs / FK 4e-14 / 128 sols
iiwa7 (SRS 7R, offset wrist)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	200 ± 0 µs / FK 2e-13 / 128 sols
KR 6 R900 (Pieper 6R, spherical wrist)	4 ± 1 µs / FK 9e-12 / 4 sols	20 ± 10 µs / FK 9e-9 / 4 sols
KR 210 R2700 (Pieper 6R, spherical wrist)	8 ± 5 µs / FK 1e-15 / 4 sols	20 ± 0 µs / FK 4e-9 / 4 sols

Franka: ssik.prebuilt.franka (2 arms)

Arm (class)	EAIK	ssik
Franka Panda (spherical-shoulder 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	130 ± 0 µs / FK 6e-12 / 32-132 sols
FR3 (spherical-shoulder 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	120 ± 0 µs / FK 6e-12 / 32-132 sols

UFactory: ssik.prebuilt.ufactory (2 arms)

Arm (class)	EAIK	ssik
xArm7 (approx spherical-shoulder 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	360 ± 10 µs / FK 1e-10 / 82-108 sols
xArm6 (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	450 ± 10 µs / FK 4e-9 / 8-16 sols

Unitree: ssik.prebuilt.unitree (1 arm)

Arm (class)	EAIK	ssik
Z1 (Pieper 6R, three-parallel)	4 ± 0 µs / FK 2e-15 / 4-8 sols	20 ± 0 µs / FK 5e-15 / 4-8 sols

AgileX: ssik.prebuilt.agilex (1 arm)

Arm (class)	EAIK	ssik
PiPER (non-Pieper 6R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	410 ± 10 µs / FK 5e-6 / 1-10 sols

Flexiv: ssik.prebuilt.flexiv (2 arms)

Arm (class)	EAIK	ssik
Rizon 4 (non-SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	1.72 ± 0.22 ms / FK 3e-10 / 4-60 sols
Rizon 10 (non-SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	1.10 ± 0.02 ms / FK 2e-9 / 6-64 sols

Kassow: ssik.prebuilt.kassow (1 arm)

Arm (class)	EAIK	ssik
Kassow KR810 (non-SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	12.02 ± 1.31 ms / FK 1e-7 / 5-49 sols

FANUC: ssik.prebuilt.fanuc (10 arms)

Arm (class)	EAIK	ssik
CRX-3iA (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	370 ± 0 µs / FK 2e-9 / 8-12 sols
CRX-5iA (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	380 ± 0 µs / FK 4e-8 / 8-12 sols
CRX-10iA (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	370 ± 0 µs / FK 2e-9 / 8-12 sols
CRX-10iA/LP (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	380 ± 0 µs / FK 1e-9 / 4-12 sols
CRX-20iA/L (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	370 ± 0 µs / FK 1e-9 / 4-12 sols
CRX-30iA (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	380 ± 0 µs / FK 4e-7 / 8-12 sols
CRX-10iA/L (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	380 ± 0 µs / FK 3e-9 / 4-12 sols
M-710iC (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 8e-12 / 4-8 sols	10 ± 0 µs / FK 9e-9 / 4-8 sols
LR Mate 200iD (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 4e-12 / 8 sols	10 ± 0 µs / FK 3e-9 / 8 sols
R-2000iC/210L (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 8e-12 / 4-8 sols	10 ± 0 µs / FK 3e-9 / 4-8 sols

I2RT: ssik.prebuilt.i2rt (2 arms)

Arm (class)	EAIK	ssik
YAM (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	460 ± 30 µs / FK 8e-7 / 8 sols
big_yam (non-Pieper 6R)	refuses ("Intersection point can't be calculated for two parallel axes")	400 ± 0 µs / FK 2e-8 / 8 sols

Enactic OpenArm: ssik.prebuilt.openarm (2 arms)

Arm (class)	EAIK	ssik
OpenArm L (SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	190 ± 0 µs / FK 3e-14 / 128 sols
OpenArm R (SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	190 ± 0 µs / FK 3e-15 / 128 sols

Galaxea: ssik.prebuilt.galaxea (2 arms)

Arm (class)	EAIK	ssik
R1 Pro L (SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	190 ± 0 µs / FK 4e-15 / 128 sols
R1 Pro R (SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	190 ± 0 µs / FK 4e-15 / 128 sols

Standard Bots: ssik.prebuilt.standard_bots (3 arms)

Arm (class)	EAIK	ssik
Thor (Pieper 6R, three-parallel)	refuses ("classifies as 6R-THREE_INNER_PARALLEL but returns FK-incorrect solutions (max FK 3e+00)")	10 ± 0 µs / FK 8e-9 / 1-4 sols
Core (Pieper 6R, three-parallel)	4 ± 0 µs / FK 9e-16 / 2-6 sols	10 ± 0 µs / FK 1e-8 / 1-4 sols
Spark (Pieper 6R, three-parallel)	refuses ("classifies as 6R-THREE_INNER_PARALLEL but returns FK-incorrect solutions (max FK 3e+00)")	10 ± 0 µs / FK 8e-8 / 1-4 sols

Abb: ssik.prebuilt.abb (5 arms)

Arm (class)	EAIK	ssik
YuMi L (approximate-SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	2.66 ± 0.19 ms / FK 1e-12 / 42-89 sols
YuMi R (approximate-SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	2.56 ± 0.09 ms / FK 1e-12 / 39-94 sols
IRB 120 (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 3e-12 / 8 sols	10 ± 0 µs / FK 9e-9 / 8 sols
IRB 1600 (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 5e-12 / 4-8 sols	10 ± 0 µs / FK 3e-9 / 4-8 sols
IRB 6700 (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 8e-12 / 4-8 sols	10 ± 0 µs / FK 3e-9 / 4-8 sols

Yaskawa: ssik.prebuilt.yaskawa (2 arms)

Arm (class)	EAIK	ssik
GP8 (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 8e-12 / 8 sols	10 ± 0 µs / FK 9e-9 / 8 sols
HC10 (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	410 ± 10 µs / FK 2e-8 / 4-16 sols

Kawasaki: ssik.prebuilt.kawasaki (1 arm)

Arm (class)	EAIK	ssik
RS007N (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 4e-12 / 8 sols	10 ± 0 µs / FK 3e-9 / 4-8 sols

Staubli: ssik.prebuilt.staubli (1 arm)

Arm (class)	EAIK	ssik
RX160 (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 8e-12 / 4-8 sols	20 ± 0 µs / FK 9e-9 / 2-8 sols

Realman: ssik.prebuilt.realman (2 arms)

Arm (class)	EAIK	ssik
RM75 (approximate-SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	600 ± 10 µs / FK 1e-12 / 128 sols
GEN72 (approximately-spherical-shoulder 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	300 ± 0 µs / FK 1e-10 / 50-72 sols

Dobot: ssik.prebuilt.dobot (2 arms)

Arm (class)	EAIK	ssik
CR5 (three-parallel 6R)	4 ± 0 µs / FK 2e-15 / 2-4 sols	10 ± 0 µs / FK 3e-11 / 1-4 sols
Nova5 (three-parallel 6R)	4 ± 0 µs / FK 1e-15 / 2-4 sols	10 ± 0 µs / FK 7e-8 / 1-4 sols

Mitsubishi: ssik.prebuilt.mitsubishi (1 arm)

Arm (class)	EAIK	ssik
RV-4FR (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 8e-12 / 8 sols	10 ± 0 µs / FK 9e-9 / 8 sols

Hyundai: ssik.prebuilt.hyundai (1 arm)

Arm (class)	EAIK	ssik
HH020 (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 2e-14 / 4-8 sols	10 ± 0 µs / FK 9e-8 / 4-8 sols

Denso: ssik.prebuilt.denso (1 arm)

Arm (class)	EAIK	ssik
VS-060 (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 8e-12 / 8 sols	10 ± 0 µs / FK 3e-9 / 4-8 sols

Doosan: ssik.prebuilt.doosan (2 arms)

Arm (class)	EAIK	ssik
M1013 (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	410 ± 10 µs / FK 8e-6 / 2-9 sols
M0609 (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	500 ± 90 µs / FK 1e-5 / 2-9 sols

Rokae: ssik.prebuilt.rokae (3 arms)

Arm (class)	EAIK	ssik
xMate Pro7 (SRS 7R)	refuses ("Currently, only 1-6R robots are solvable with EAIK")	230 ± 10 µs / FK 1e-12 / 128 sols
xMate CR7 (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	400 ± 10 µs / FK 8e-9 / 4-12 sols
xMate SR3 (non-Pieper 6R)	refuses ("6R-Unknown Kinematic Class")	340 ± 10 µs / FK 2e-9 / 2-12 sols

Trossen: ssik.prebuilt.trossen (2 arms)

Arm (class)	EAIK	ssik
ViperX 300s (Pieper 6R, spherical wrist)	3 ± 0 µs / FK 9e-16 / 8 sols	10 ± 0 µs / FK 3e-9 / 8 sols
WidowX 250s (Pieper 6R, spherical wrist)	4 ± 0 µs / FK 1e-15 / 8 sols	10 ± 0 µs / FK 9e-9 / 8 sols

The sols column is the branch-count range across reachable poses: constant for Pieper-class 6R (Puma → 8), variable for non-Pieper 6R (spurious roots of the degree-8 Sylvester resultant fall complex at some poses), and the redundancy-sample × algebraic-branch product for 7R (iiwa14: 16 swivel samples × 8 = 128).

The tradeoff is not "ssik is always faster." It is:

more exploitable geometric structure  →  simpler, faster solver
less exploitable geometric structure  →  more general algebraic machinery,
                                          higher cost, enumerative semantics kept

What the benchmark should be read as

Mean runtime alone does not characterize an IK solver. For reachable targets generated as q ~ joint distribution; T = FK(q), the target is known to have a solution — so a solver that returns nothing there has suffered a recovery failure, regardless of whether an exact derivation exists. A solver that is very fast on 99% of the workspace but develops numerical holes in the remaining 1% can be less useful than a slower one with predictable recovery. ssik's evaluation therefore emphasizes reachable-pose recovery rate, branch recovery for nonredundant arms, worst-case and tail FK residual, behavior near singularities, median and tail latency, and cross-solver agreement — not average runtime alone. (Refusal strings are EAIK's own, captured verbatim; a numerical-IK comparison against MINK is tracked in #236.)

Under the hood

The mathematical ingredients have a long lineage: geometric subproblem decomposition, Raghavan–Roth and Manocha–Canny general-6R elimination, Singh–Kreutz redundancy parameterization, and Husty–Pfurner general 6R kinematics. ssik does not claim these classical derivations as new.

The implementation problem is that algebraic solvability does not guarantee numerical recoverability. A textbook derivation must still make choices about representation, elimination order, linearization, tolerances, singular cases, reconstruction, and floating-point recovery — and those choices decide whether solutions that exist in exact arithmetic are actually returned by a deployed solver. ssik organizes them into a common hierarchy:

1. normalize the robot into a common kinematic representation;
2. identify structural conditions that permit simpler geometric solvers;
3. fall back to general algebraic IK when special geometry is absent;
4. choose among algebraically equivalent formulations for numerical conditioning;
5. reconstruct and independently validate candidate joint configurations;
6. refine or retry numerically difficult candidates when appropriate;
7. apply application-level constraints such as joint limits and seed continuity.

The JACO 2 conditioning result shows why: two exact formulations of the same IK equations can differ from cond ≈ 3.75e16 to cond ≈ 127. The equations are equally solvable; the resulting numerical solvers are not equally reliable. Cython hot loops cover the leaf primitives on the pure-Python path (POE forward kinematics, LM polish, analytical Jacobian); the native C++ backend covers the full solve.

How a solver is picked

dispatch() searches from specialized to general representations: a solver is eligible only when its structural assumptions hold, and among eligible solvers ssik prefers the one that avoids unnecessary algebraic complexity. When no convenient Pieper-style geometry exists, dispatch falls through to general algebraic machinery rather than interpreting the geometry as analytically unsolvable. The same classifier runs whether you load a URDF with Manipulator.from_urdf or bake an artifact with ssik build.

flowchart TD
    START(["T_target<br/>POE-normalized chain"]) --> DOF{"6R or 7R?"}

    %% 7R: concurrent-shoulder closed-form by family, else jointlock
    DOF -->|7R| SH{"shoulder axes<br/>concurrent?<br/>within drift"}
    SH -->|yes| WR{"wrist axes<br/>concurrent?"}
    WR -->|"yes · SRS"| A0["seven_r.srs<br/>+ srs_polished for drift<br/>KUKA iiwa · Kinova Gen3"]:::cf
    WR -->|"no · offset wrist"| A1["seven_r.spherical_shoulder<br/>+ polished for drift<br/>Franka / FR3 · xArm7"]:::cf
    SH -->|no| JL["jointlock.seven_r<br/>lock 1 joint · sweep 16 · inner 6R"]:::fb
    JL --> BUILT{"artifact built?"}
    BUILT -->|"yes · ssik build"| CRR["cached Raghavan–Roth<br/>~17 ms · Rizon · Kassow"]:::rr
    BUILT -->|"no · from_urdf"| HP["Husty–Pfurner backstop<br/>symmetric-DH safe · slower"]:::fb

    %% 6R: Pieper-class closed-form, else Raghavan–Roth
    DOF -->|6R| P3{"3 parallel axes<br/>at joints 1·2·3?"}
    P3 -->|yes| B0["ikgeo.three_parallel<br/>UR3 / UR5 / UR10"]:::cf
    P3 -->|no| WM{"spherical wrist?<br/>axes 3·4·5 meet"}
    WM -->|yes| B1["ikgeo.spherical_*<br/>shoulder specialisation picks the variant<br/>Puma · Fanuc · IRB120 · xArm6"]:::cf
    WM -->|no| B4["ikgeo.general_6r<br/>Raghavan–Roth + AE-3<br/>JACO 2 · Piper"]:::rr

    classDef cf fill:#d3f9d8,stroke:#2f9e44,color:#0b2e13;
    classDef rr fill:#dbe4ff,stroke:#4263eb,color:#0b1a40;
    classDef fb fill:#ffe8cc,stroke:#e8590c,color:#3d1900;

Every solver returns algebraic candidates that pass through one shared tail: an optional Levenberg–Marquardt polish, an empty-result rescue, then limit / seed / truncate finalisation.

flowchart LR
    C["algebraic IK<br/>candidates"] --> R{"allow_refinement<br/>or *_polished solver?"}
    R -->|yes| LM["lm_refine<br/>LM on spatial Jacobian<br/>to FK tolerance"]:::post
    R -->|no| E{"empty<br/>result?"}
    LM --> E
    E -->|"yes · allow_rescue"| RS["T-perturbation<br/>rescue + LM polish"]:::post
    E -->|no| F["finalize_solutions<br/>limits → seed-sort → truncate"]:::post
    RS --> F
    F --> OUT(["list of Solution"])

    classDef post fill:#e7f5ff,stroke:#1c7ed6,color:#08324f;

The tree folds a few details for readability:

• Exact vs _polished. The _polished 7R solvers cover arms whose shoulder or wrist axes only nearly meet (Kinova Gen3's 12 mm / 0.4 mm drift, xArm7's near-concurrent wrist): the exact recipe seeds candidates, then LM polish recovers machine precision against the true FK. Exact solvers require true concurrence; the split is a drift threshold (≤ 40 mm for the SRS family).
• The three 6R spherical-wrist variants. ikgeo.spherical_* is one of spherical_two_parallel (axes 1 ∥ 2: Puma / Fanuc / KUKA KR), spherical_two_intersecting (‖p₁‖ ≈ 0, shared shoulder origin: ABB IRB120 / xArm6), or plain spherical (generic). All are closed-form; the shoulder geometry picks the tightest-conditioned one.
• Tier-1 search solvers. two_parallel / two_intersecting are importable but never auto-dispatched: Raghavan–Roth handles the same chains 50–200× faster.
• When lm_refine runs. _polished solvers (and the T-perturbation rescue) run it unconditionally as part of their algorithm; every other solver runs it only under allow_refinement=True, and only on candidates that miss the FK tolerance.

Testing the distinction between solvability and recovery

A solver can be mathematically general and still fail numerically, so the test suite asks a stronger question than whether each algorithm implements its derivation. For reachable poses, ssik checks that solutions are actually recovered; for returned candidates, it checks independent FK closure (≤ 1e-10 on retained IK). On shared geometries it uses N-way cross-solver agreement, while adversarial and randomized tests (500+ Hypothesis-fuzzed poses per fixture) probe conditioning, singularities, reconstruction, joint limits, and branch loss, and an explicit speed bench must clear a regression gate.

The discipline follows one invariant:

No silent wrong answers.

A failure to recover a reachable pose, the loss of a valid branch, or an FK-inconsistent candidate is treated as a solver failure — never hidden behind an average-error metric. Negative-result investigations (a Cython estimate that missed by 2–5×, a codegen-bake on a part that was 0.3% of runtime) are published as closed issues with profile data so the next contributor doesn't repeat the path.

Documentation

Full docs site: https://personalrobotics.github.io/ssik/

• Quickstart: install, prebuilts, trajectory tracking, explain mode
• Setting up your robot: URDF readiness, --base/--ee selection, tool baking, verification
• Arm coverage: per-arm fixtures, speeds, FK floors
• Architecture: solver tier catalog, dispatch flow, algorithmic lineage
• API reference: Manipulator, Solution, Diagnostic, TolerancePolicy
• Semver policy: what's public, what counts as breaking
• CONTRIBUTING.md: repo layout, dev setup, testing discipline

Related libraries

ssik sits within a long line of analytical, algebraic, geometric, and numerical IK systems. These packages make different tradeoffs; the distinctions below are about solver semantics and current implementations, not a claim that one method dominates.

• IK-Geo (Elias–Wen 2022/2025): a unified geometric-subproblem formulation for revolute IK. It covers any 6R manipulator in principle — robots with enough intersecting/parallel-axis structure get closed forms, less-structured commercial arms use 1D search, fully general 6R uses 2D search (the search forms can also be polynomialized). ssik shares IK-Geo's aggressive geometry exploitation, but uses general finite algebraic elimination as a first-class fallback. The ik-geo PyPI wheel currently pins pyo3==0.20.3 (incompatible with Python 3.13).
• EAIK (Ostermeier, Külz, Althoff): automatically recognizes supported kinematic structure and builds analytical IK via subproblem decomposition. Its current implementation covers a set of nonredundant families and handles redundant chains by locking a joint when the resulting subchain is supported. Directly benchmarked above on the supplied fixtures.
• IKFast (Diankov/OpenRAVE): the influential analytical-IK codegen system that established the offline-symbolic → deployed-numerical-artifact pattern ssik also follows. Works well on the families it was tuned for (Pieper-class 6R, spherical-wrist 7R via joint lock); its sympy pipeline fails on modern sympy for non-Pieper geometries (mpmath.polyroots NoConvergence, Matrix.inv/det stalls). LGPL.
• Raghavan–Roth / Manocha–Canny / Husty–Pfurner: classical general-6R algebraic methods establishing that a lack of Pieper structure does not imply a lack of a finite IK method. These are part of ssik's foundation; ssik's concern is their reliable finite-precision realization on contemporary geometries.
• MINK (Zakka): MuJoCo-native optimization-based numerical IK. Takes a seed, searches locally to a single configuration — applicable to any geometry and natural for control, but different semantics from enumeration. FK closure tracks the convergence tolerance (typically 1e-3–1e-6).
• TRAC-IK and KDL: mature numerical IK centered on seeded, one-branch-per-solve solution finding — the right interface when one nearby solution is what you want. (pytracik's arm64 wheel is currently broken; the ROS-native binding works.)

The relevant tradeoff is therefore not simply analytical versus numerical. It is among structural specialization, generality, solution-set semantics, numerical reliability, and computational cost.

License

BSD-3-Clause. The library incorporates clean-room reimplementations of algorithms from BSD-3-licensed IK-Geo (Elias–Wen 2022/2025) and from the academic publications of Raghavan–Roth (1990), Manocha–Canny (1994), Singh–Kreutz (1989), and Husty–Pfurner (2007). Algorithmic lineage is documented in module docstrings.

Citation

If you use ssik in academic work, please cite it. Machine-readable metadata is in CITATION.cff; GitHub renders that as a "Cite this repository" button on the repo sidebar.

@software{ssik,
  author    = {Srinivasa, Siddhartha},
  title     = {ssik: reliable enumerative inverse kinematics for 6R and 7R revolute arms},
  url       = {https://github.com/personalrobotics/ssik},
  doi       = {10.5281/zenodo.20278005},
  year      = {2026},
  publisher = {Zenodo},
}