Exact Fractional Order Impedance Rendering for Highly Flexible and Multi-Jointed Robots Using Time-Delay Estimation

Fractional-order (FO) control, which employs non-integer orders of differintegration, has been adopted for its advantages in generalized controller design or reduced modeling complexity. For compliance control, FO impedance is employed as a generalization of conventional integer-order (IO) impedance, where forces are generated based on IO derivatives such as position, velocity, and acceleration. However, due to model uncertainties and friction, accurately rendering FO impedance on flexible joint robots (FJRs) remains challenging. Therefore, this letter proposes a model-free and robust controller that leverages the time-delay estimation (TDE) technique to estimate nonlinear and unknown terms using historical data. Furthermore, detailed tuning procedures and stability conditions are provided. The proposed controller is evaluated on a single-joint FJR, where it is compared with other FJR impedance controllers, and its robustness against disturbances and model perturbations is examined. Additionally, FO and IO impedance behaviors are compared using the proposed controller. The proposed controller is implemented on the DLR C-Runner, a legged FJR with six degrees of freedom (DoF), to validate the controller's performance in more complex systems.