On the Nonlinear H-infinity Optimal Control of Robotic Systems using Two-loop Control Structure and Applications

Abstract

As the robotics field has grown, various robot controllers have been developed for various purposes. Naturally, it is now difficult to define the robust performance in a unified way because the purposes of the controllers are different. To solve this difficulty, a measure of robust performance can be defined by the `deviation' of `actual performance' from the `nominal performance' which is the outcome of the nominal controlled system. To achieve robust performance, this thesis proposes a framework that achieves the robust performance by attenuating the ``deviation'' using optimal control techniques. Once the deviation is attenuated, the real robot behaves like a nominal robot, and thereby, we can apply a controller designed for the nominal robot. This strategy is known as the two-loop control structure.

By invoking the optimality in the control design, we can argue that the controller is the best in a certain sense. In many control problems, H-infinity optimality is of interest because it successfully brings unknown disturbance into the formulation. However, for general nonlinear systems, H-infinity optimal control problems are very difficult to solve. In this thesis, a robust control scheme that brings nonlinear H-infinity is proposed for robotic systems. This scheme is named as nonlinear robust internal-loop compensator (NRIC) framework, in the sense that the NRIC framework is the nonlinear extension of the robust internal-loop compensator (RIC) which is developed for the linear systems.

In the NRIC framework, the `actual performance' is divided into the `nominal performance' and the `deviation' of the actual performance from the nominal performance. By attenuating the `deviation' in the sense of nonlinear H-infinity optimality without affecting the `nominal performance', the `actual performance' recovers the `nominal performance'. Namely, the robust performance is guaranteed by nonlinear H-infinity optimality. Robust stability is guaranteed by proving the extended disturbance (which includes not only the external disturbance, but the model uncertainties) input-to-state stability. According to the framework, a controller that is designed for the nominal plant can achieve robustness (both robust stability and performance) by adding a simple PID-type (or PD-type) auxiliary input to the original control law. Moreover, the NRIC framework preserves the passivity of the original controller. Finally, the performance bound can be predicted, and leads to the gain tuning rules. By virtue of the gain tuning rules, the performance can be tuned using only a single variable. Because the framework can be applied to various controllers, implicit force controllers as well as the motion controllers were implemented in experimental validation.

Noting that the linear systems are special cases of nonlinear systems, the NRIC framework reduces to the standard RIC framework in linear systems. However, one important difference is that the inner-loop and the outer-loop designs can be performed independently because the outer-loop design does not affect the robust stability measure that is derived using the small-gain theorem.

A couple of NRIC-inspired robust control examples will be additionally introduced. As a first example, a robust PD control for flexible joint robots (FJRs) is proposed. Because the joint friction is included in the motor-side dynamics (which is a subsystem of the entire nonlinear robot dynamics), it seems possible to eliminate the friction by applying the DOB technique to the motor-side. However, the DOB-based PD control using typical implementation, in which the measured signal is fed back to the controller, may not satisfy asymptotic stability. On the other hand, when the nominal signal is fed back to the controller instead of the measured signal, global asymptotic stability can be guaranteed. Experimental results using 3 degrees-of-freedom (DOF) FJR show that the robot configuration could not converge to the desired one when the typical implementation is used (i.e., measured signal feedback); the robot fell into a limit cycle around the desired position. In contrast, the robot converged to the desired position when the nominal signal feedback is used.

In the second example, a nonlinear disturbance observer (DOB) is designed for powered upper-limb exoskeleton control. The exoskeleton was developed to carry heavy payloads in industries. Because the exoskeleton is developed for practical purposes, the number of sensors used is limited; encoders as well as F/T sensors that detects user's intention are used; note that the additional force sensors may make the control easy, but will increase the cost (including the potential maintenance cost) and reduce the system robustness. The proposed DOB allows for nonlinear robot dynamics, whereas the ordinary DOBs have not. Passivity ensures that the system is stable with a human operator and environmental interactions in the control loop. The robot interacts with the human operator via F/T sensor and interacts with the environment mainly via end-effectors. Disturbance observation property, which can be proved using two-time scale analysis, ensures that the behavior of the actual system mimics that of a nominal model as the control gain increases. However, because the control gain cannot grow infinitely in practice, the performance limitation according to the achievable control gain is also analyzed. The result of this analysis indicates that the performance of the DOB increases linearly with the achievable gain. Experiments using simple 1 DOF testbed verify the gain tuning rule and passivity property. In addition, The results of experiments in which the powered upper-limb exoskeleton was used to lift and maneuver a payload are presented.

At the last part of the thesis, momentum based DOB design for general robots will be shown. By introducing the generalized momentum, it is possible to utilize full nonlinearities and couplings of the inertia matrix in the DOB design. Moreover, the momentum-based DOB design for the rigid joint robots can be easily extended to the FJR applications by applying it to the link-side dynamics and motor-side dynamics, respectively. As a result, it is possible to estimate the external torque acting on the link-side, and to compensate for the disturbance occurring in the motor-side at the same time. The uniformly ultimately boundedness (UUB) of the closed-loop dynamics can be shown using the Lyapunov-like approaches. This scheme is verified using the numerical simulation.