Energy Optimal Control of a 3R Spatial Robotic Arm via Iterative Nonlinear LQR

Energy consumption is a growing concern in robotic manipulation, particularly for industrial arms that execute repetitive, high-speed trajectories. This paper addresses the energy-optimal motion planning and control of a three-degree-of-freedom (3R) spatial robotic arm with a swivelling waist, shoulder, and elbow joints arranged in a body–shoulder–elbow topology. A complete, closed-form Euler–Lagrange dynamical model, comprising the configuration-dependent mass matrix M (q), the Coriolis/centrifugal matrix C(q, q̇), and the gravity vector G(q) were derived from first principles and verified through the passivity condition. This derivation, to the best of the authors’ knowledge, has not been reported in prior literature for this specific topology. The energy-minimisation problem is formulated as a nonlinear optimal control problem (OCP) and solved using the iterative Linear Quadratic Regulator (iLQR), which iteratively linearises the nonlinear dynamics and applies a backward Riccati recursion followed by a forward rollout with line search. A classical proportional–integral–derivative (PID) controller tracking a cubic polynomial reference trajectory serves as the comparative baseline. Simulation results demonstrate that the iLQR controller produces qualitatively different torque profiles that exploit dynamic coupling to reduce total actuator energy relative to PID control, while achieving accurate rest-to-rest manoeuvres in three-dimensional space.