UR5e Payload Research · Modeling

UR5e Modeling & Dynamics

Reconstructing a consistent six-degree-of-freedom robot and payload model before controller development: coordinate frames, FK/IK, Jacobian conditioning, inverse dynamics, and cross-validation against the Isaac Sim articulation.

UR5eFK / IKJacobianDamped Least SquaresRigid-Body DynamicsPayload CoMIsaac Sim 6.1

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0.000206 mmAnalytical-to-Isaac Tool0 position discrepancy at the checked pose
121 / 121Cartesian-reference IK waypoints converged
15.53Maximum recorded Jacobian condition number along the accepted path
Engineering question: can the geometric, kinematic, and rigid-body definitions used by an analytical controller be reconciled with the independent UR5e articulation before using their outputs to make transport and control decisions?

1. Physical Model and Coordinate Consistency

The study begins with the six-revolute-joint UR5e and its existing robot description. Joint identity, axis direction, joint order, fixed-base configuration, link transformations, and the world-to-base relationship were checked before constructing the later transportation and control experiments. This stage is not another controller benchmark: it establishes whether the mathematical quantities used by a controller describe the same mechanism executed in Isaac Sim.

TBT(q) = TB1(q1) ··· T6T(q6)

The adopted base-to-world rotational convention was diag(−1, −1, 1). Tool0 was held at a fixed orientation for the payload task. Frame consistency matters because a mathematically correct inverse dynamics implementation can still generate physically incorrect joint effort when its joint signs or gravity frame differ from the articulation.

Figure 1 — UR5e Pick-Station Configuration
Figure 1 — UR5e Pick-Station ConfigurationIsaac Sim arrangement used to verify the manipulator, initial station, and payload attachment under the accepted frame conventions.

2. Forward Kinematics and Cross-Validation

A representative joint configuration was evaluated independently through analytical forward kinematics and the Isaac Sim model. The reported Tool0 position discrepancy was 0.000206 mm, supporting agreement of the tested kinematic definitions. This numerical result is not hardware pose accuracy, and one checked configuration does not validate every dynamic parameter.

pTool0 = fp(q),   Δp ≈ Jv(q)Δq
Figure 2 — UR5e Extended Destination Configuration
Figure 2 — UR5e Extended Destination ConfigurationThe place target was selected near 95% of nominal horizontal reach to expose configuration-dependent demands without equating extension with a singularity.

3. Jacobian, Inverse Kinematics and Workspace Feasibility

Differential kinematics connects joint motion and Cartesian transport. Rather than selecting an unrelated inverse-kinematics solution at each point, the accepted path used sequential damped-least-squares IK seeded from the preceding configuration. All 121 reference waypoints converged. The maximum reported Jacobian condition number along this path was approximately 15.53, with a minimum reported singular value of 0.1237 under the adopted numerical convention.

Δq = JT(JJT + λ2I)−1e

Additional checks sampled 481 robot configurations along and between reference waypoints. No mesh intersection was found among the examined candidates. These observations support the sampled path geometry; they do not establish continuous swept-volume collision safety or actuator feasibility.

Figure 3 — Accepted Cartesian Transportation Path
Figure 3 — Accepted Cartesian Transportation PathThe kinematically checked reference establishes the geometric input for later trajectory and inverse-dynamics calculations.
Figure 4 — Cartesian Reference and Joint Coordinates
Figure 4 — Cartesian Reference and Joint CoordinatesShows the relationship between the prescribed workspace motion and the corresponding six-joint reference.

4. Rigid-Body and Attached-Payload Dynamics

The analytical model retains the nonlinear coupled inertia, velocity-dependent, and gravitational terms. A fixed, homogeneous 50 mm cube is attached to Tool0; its center of mass is placed 25 mm along the tool’s local +Z axis. Payload inertia and gravity are mapped through the payload CoM Jacobian, rather than estimated from a single lever arm.

M(q;mp)q̈ + c(q,q̇;mp) + g(q;mp) = τ
Mp,trans = mp Jv,pTJv,p

The complete loaded model already includes the robot and payload contributions. Adding the same payload correction a second time would duplicate inertia and gravity effects. The subsequent study distinguishes total loaded dynamics from the difference between two loaded models.

Figure 5 — Joint-Resolved Reference Torque
Figure 5 — Joint-Resolved Reference TorqueThe joint-torque profiles demonstrate that a prescribed tool motion produces nonuniform effort demands across the six degrees of freedom.
Figure 6 — Predicted Effort Utilization
Figure 6 — Predicted Effort UtilizationReference inverse-dynamics requirements relative to effort limits configured in the simulation asset, not certified motor torque ratings.

5. Why the Physical Model Matters to Later Control

Under the accepted initial 1 kg payload reference, shoulder lift required the largest predicted peak effort, approximately 62.435 Nm. Increasing the transportation time from three to four seconds reduced that peak only to approximately 60.340 Nm: slowing the trajectory lowers inertial demand but does not remove gravity-dependent effort.

Modeling evidenceRecorded resultInterpretation
Analytical/Isaac Tool0 comparison0.000206 mmKinematic agreement at the checked pose
Accepted IK sequence121 / 121 convergedNumerical feasibility of the reference
Maximum Jacobian condition number15.53No detected singularity along the accepted sequence
1 kg peak shoulder-lift torque62.435 NmDominant gravitational and coupled loading

The result of this stage is a traceable physical model and task geometry. It provides the basis for the subsequent controller comparisons and prevents frame, inertia, or attachment errors from being mistaken for feedback-controller limitations.

6. Validation Boundaries

Interpretation: analytical-to-Isaac FK agreement validates the tested geometry, not full actuator dynamics. The rigid attachment excludes grasp/contact behavior; sampled intersection checks are not collision guarantees. Cartesian error reconstructed from joint states is not an independently measured physical tool position.

Project Resources

Source Code & Experiment Records

UR5e analytical modeling, nominal control, ROS 2–PhysX implementation, residual RL training, evaluation routines, and recorded experiment data.

Open Part 3 on GitHub

Technical Reports

The complete integrated investigation and a concise engineering version are provided as two separate PDF files.

Condensed ReportFull Report
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Project by Sooyong Kim · Robotics & Autonomous Systems Portfolio