Industrial roboticsManufacturing & robotics

UR5 6-axis manipulator

A spatial 6-DoF UR5 arm as one multibody model — closed-form inverse kinematics driving a Cartesian end-effector path through per-joint PD control with gravity feedforward.

DjiniousLabModelling & simulation

The UR5 6-axis robotic arm rendered in 3D at a representative mid-reach pose, showing all six revolute joints along the DH kinematic chain

revolute joints: 6Rrevolute joints
spatial workspace: 6 DoFspatial workspace
IK round-trip (rad): ~1e-16IK round-trip (rad)
Cartesian tracking (peak): 0.485 cmCartesian tracking (peak)
Cartesian tracking (steady): 0.41 cmCartesian tracking (steady)
joint tracking error: <0.0013 radjoint tracking error

One robot arm. One model. Joint space to Cartesian.

The UR5 — a 6-revolute industrial arm — modelled end-to-end as a multibody DAE. Closed-form inverse kinematics feeds a per-joint PD controller with gravity feedforward that holds a Cartesian end-effector path in simulation. Kinematics, inertia, and control, all on one living digital replica.

Every subsystem, on one canvas.

Nothing is stubbed — the kinematic chain, link inertia, and control all carry their own physics.

Six revolute joints

Standard UR5 DH parameters chain six revolute joints from base to wrist. Rotor inertia (Jr) is available as an opt-in parameter but held at 0 for the 6R chain — Jr > 0 triggers an MTK index-reduction pathology that freezes chains with N ≥ 5 joints (ADR-0072).

Full-inertia links

Link masses and diagonal principal inertias come from the published UR5 URDF. Off-diagonal inertia terms are neglected; the multibody DAE captures the dominant inertial forces without full tensor coupling.

Closed-form IK

A decoupled-wrist analytic solver closes the inverse-kinematics loop in one branch. Round-trip error (FK ∘ IK) is ~1e-16 rad — at floating-point machine precision.

PD + gravity feedforward control

Per-joint PD plus gravity feedforward: in_tau → −q. Each joint's control law is independent, with gains tuned to the arm's natural frequency.

FK end-effector observer

Forward kinematics propagates joint angles to a Cartesian end-effector position that is compared to the reference path, providing the tracking error signal.

Multibody DAE composition

Revolute, FixedTranslation, FixedRotation, and Link primitives from the pure-DJL multibody kit compose the DH chain. The engine lowers this to a symbolic DAE.

Built from the DH chain, not hand-assembled matrices.

The UR5 model is assembled from the same pure-DJL multibody kit that powers the Djinborn quadcopter — Revolute, FixedTranslation, FixedRotation, and Link primitives wired according to the standard DH parameters. Kinematics and inertial forces emerge from the composition; there are no hand-coded Jacobians or mass matrices. The engine init-policy is set to NoInit (ADR-0071) so the analytic IK can pre-condition joint angles before the DAE integrator starts — a requirement for the decoupled-wrist closed-form solver to operate cleanly in the loop.

Pure-DJL multibody kit (Revolute/FixedTranslation/FixedRotation/Link) — no Rust or Julia changes. — ADR-0067

Closed-form decoupled-wrist IK, one branch, non-singular poses. — ADR-0072

NoInit engine policy lets analytic IK pre-condition the DAE state. — ADR-0071

The UR5 full-system model opened as a node graph in the DjiniousLab canvas, showing the entire closed-loop composition wired from multibody-kit primitives — The ur5_full_system composition opened as a canvas in DjiniousLab — the whole closed loop as one elaborated graph: the Cartesian path feeds the closed-form inverse kinematics, through per-joint reference filters into PD + gravity-feedforward control, driving the 6R DH multibody plant, with a forward-kinematics observer tapping the tip. One model the worker solves end-to-end.

The UR5 6-axis arm at a representative pose, rendered from the DjiniousLab 3D viewer — The UR5 at a representative mid-reach pose — six revolute joints, URDF link masses and diagonal inertias, closed-form IK in the loop, and per-joint PD control with gravity feedforward, all simulated as one composition.

Four gates. Real numbers.

Simulation verified against four gates on the feat/multibody-ur5-arm branch. Scope: non-singular poses, one IK branch. No physical robot — these are digital-twin simulation results.

Gate

G-B1: gravity-comp pose hold: PASS
G-B2: joint tracking error: PASS
G-B3a: IK round-trip (FK∘IK): PASS
G-B3b: Cartesian tracking (peak): PASS
G-B3b: Cartesian tracking (steady): PASS

Result

G-B1: gravity-comp pose hold: joint angles stable, no drift
G-B2: joint tracking error: <0.0013 rad per joint
G-B3a: IK round-trip (FK∘IK): ~1e-16 rad (machine precision)
G-B3b: Cartesian tracking (peak): 0.485 cm
G-B3b: Cartesian tracking (steady): 0.41 cm

The bug that looked like a physics gap.

When the 6R chain first ran, the DAE appeared under-determined — the solver couldn't close. The first hypothesis was missing rotor inertia (an easy fix: add Jr per joint). The model ran, but tracking remained poor. The real cause was a dangling terminal frame: the sixth joint's output Frame connector had no downstream body to react against, so the constraint equations were degenerate. Identifying that — and not the inertia — as the root cause required reading the residual structure rather than guessing. The fix was one line. Framing the misdiagnosis honestly is part of the record: engineering rigour means following the residual, not the first plausible story.