# Adaptive Robust Admittance Control of Robots Using Duality Principle-Based Impedance Selection
###### Abstract  
The performance of robot admittance  
environmental stiffness and damping have been conducted based on the signal processing method, recursive least-squares method, and Lyapunov-based adaptation [12; 13; 14], without estimators being contained in the prescribed sets.  
Robot modeling uncertainties and disturbances are the main difficulties for robots in realizing the desired impedance dynamics. How to improve control robustness [15] for robot impedance/admittance control is an important issue to be solved. Impedance control is achieved with a force-controlled outer loop and a position-controlled inner loop and is better suited for interaction with stiff environments [16; 17], whereas admittance control is achieved with a position-controlled outer loop and a force-controlled inner loop and is better suited for interaction with soft environments. Therefore, the ways to tackle modeling uncertainties and disturbances are different for impedance control and admittance control. Impedance control robustness is improved mainly by enhancing robot modeling accuracy, whereas admittance control robustness is enhanced through position control based on an adaptive control approach, a sliding-mode control approach, iterative learning, neural networks, and fuzzy logics [18; 19; 20; 21; 22]. However, adaptive control requires robot dynamic parameterization, sliding-mode control has an inherent chattering problem, iterative learning is mainly applied in systems with periodicity characteristics, and the complexities of neural networks and fuzzy logics limit their applications in real-time control. Through time-delay estimation (TDE) [23; 24; 25], time-delay control can compensate for complex robot dynamics without a robot dynamics model. In addition, time-delay control is simple in structure and numerically efficient and has been applied in robot manipulators and exoskeletons. However, discontinuous terms in robotic models, especially the Coulomb friction and static friction, introduce non-negligible time-delay estimation errors and affect control stability and robustness. As sliding-mode control (SMC) is well known for its strong robustness [26; 27], it is promising to integrate TDE and SMC to improve control robustness [28]; however, the inherent possible chattering problem of SMC limits its applications.  
In this study, we propose an adaptive robust admittance control strategy to improve robot-environment interaction. The environmental impedance is estimated using an adaptive law with projection modification to guarantee the impedance estimator constrained in a predefined set. Then, the desired robot stiffness and damping are determined based on the duality principle. To obtain the desired impedance dynamics, an adaptive robust admittance controller based on TDE is designed for the robot to track the admittance trajectory with the prescribed performance. In the adaptive robust controller, the possible chattering problem is highly alleviated due to the use of the saturation function instead of the conventional SMC. Simulations are conducted to show the effectiveness of the environmental impedance estimation and the proposed adaptive robust admittance control.  
Compared with the environmental impedance estimation in [12; 13; 14], the projection modification-based adaptive impedance learning law can guarantee that the estimated environmental stiffness and damping are constrained in prescribed sets, which is of significant value for the determination of the range of the desired robot impedance. Compared with the TDE-based SMC in [28] for position regulation, the designed adaptive robust admittance controller can make robots realize the desired impedance dynamics without serious chattering and the requirement of the bound of the TDE error.