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Pillar 4 • Physical AI & Embodied Robotics

Robotics & Physical AI Systems Lab

Explore the fundamental physics, computational geometry, and multi-tier software architectures powering embodied artificial intelligence: from forward and inverse kinematics solvers to actuator torque density sizing, multi-rate control loops, and sim-to-real domain randomization.

Planar Kinematics & Coordinate Transformations

Step through Forward Kinematics (computing end-effector coordinates from joint angles) and numerical Inverse Kinematics via Jacobian Transpose gradient descent. Click or drag anywhere on the workspace to command the arm to track target points in real time.

Interactive Tip: In IK mode, click or drag on the canvas to relocate the blue crosshair target.
End-Effector (X, Y) 0.42m, 0.38m Cartesian 2D Frame
Reach Radius 0.85 m Max Extension Limit
Tracking Error 0.00 mm Euclidean Delta
Workspace State Nominal Manipulability Metric

Actuator Torque Density, Gearbox Physics & Reflected Inertia

Calculate dynamic joint torque, motor phase currents, thermal dissipation ($I^2 R$), and reflected rotor inertia ($J_{rotor} \cdot N^2$) across Quasi-Direct Drive (QDD), Planetary, Cycloidal, and Strain Wave (Harmonic) architectures.

Total Joint Torque (τload) 27.3 Nm Static Gravity + Inertia
Motor Shaft Torque (τm) 3.71 Nm Reflected to Rotor
Required Phase Current 30.9 A I = τm / Kt
Ohmic Heat Dissipation 114.8 W Ploss = I² · R (0.12 Ω)
Reflected Rotor Inertia at Joint (Jrotor · N²) 0.0064 kg·m²

With a 8:1 gear ratio, reflected rotor inertia scales by N² = 64. Low gear ratios preserve backdrivability, allowing the humanoid leg to absorb unexpected ground impacts without stripping gear teeth.

Actuator Torque Density (Nm / kg) 28.4 Nm/kg
Standard Drone Motor (5 Nm/kg) Unitree G1 (25 Nm/kg) Optimus Gen-2 (38 Nm/kg) State-of-the-Art (50+ Nm/kg)

Multi-Rate Control Hierarchy: VLA to Motor Actuation

Physical robots cannot run single monolithic neural control loops. Cognitive Vision-Language-Action (VLA) models evaluate deep vision transformers at 1–5 Hz, generating chunked trajectories. Mid-level planners interpolate at 20–50 Hz, while Field-Oriented Control (FOC) joint loops run at 200–1000 Hz.

Tier 1: High-Level VLA Policy (2 Hz) Active • Next chunk ready
Tier 2: Trajectory Spline Interpolation (30 Hz) Interpolating Waypoints
Tier 3: Low-Level Joint FOC / PD Servo (500 Hz) Real-Time Microcontroller Loop
Buffer Starvation Safety Intercept: Watchdog Guard OK (0 Underruns)

Sim-to-Real Domain Randomization Sandbox

Examine how learned physical neural policies perform when deployed from idealized simulations (MuJoCo/Isaac Sim) into the real world. Adjust friction variation, unmodeled mass perturbations, and communication latency to evaluate trajectory divergence.

Root-Mean-Square Error 0.034 m Nominal vs Real Trajectory
Overshoot / Phase Lag 6.2 % Damping Deviation
Sim-to-Real Viability Robust Policy Stability Envelope

Physical AI & Robotics Engineering Curriculum