A Passivity-based Nonlinear Admittance Control with Application to Powered Upper-limb Control under Unknown Environmental Interactions

This paper presents an admittance controller based on the passivity theory for a powered upper-limb exoskeleton robot which is governed by the nonlinear equation of motion. Passivity allows us to include a human operator and environmental interaction in the control loop. The robot interacts with the human operator via F/T sensor and interacts with the environment mainly via end-effectors. Although the environmental interaction cannot be detected by any sensors (hence unknown), passivity allows us to have natural interaction. An analysis shows that the behavior of the actual system mimics that of a nominal model as the control gain goes to infinity, which implies that the proposed approach is an admittance controller. 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 in the sense of infinite norm increases linearly with the control gain. In the experiments, the proposed properties were verified using 1 degree-of-freedom testbench, and an actual powered upper-limb exoskeleton was used to lift and maneuver the unknown payload.Comment: Accepted in IEEE/ASME Transactions on Mechatronics (T-MECH

The converse of the passivity and small-gain theorems for input-output maps

We prove the following converse of the passivity theorem. Consider a causal system given by a sum of a linear time-invariant and a passive linear time-varying input-output map. Then, in order to guarantee stability (in the sense of finite L2-gain) of the feedback interconnection of the system with an arbitrary nonlinear output strictly passive system, the given system must itself be output strictly passive. The proof is based on the S-procedure lossless theorem. We discuss the importance of this result for the control of systems interacting with an output strictly passive, but otherwise completely unknown, environment. Similarly, we prove the necessity of the small-gain condition for closed-loop stability of certain time-varying systems, extending the well-known necessity result in linear robust control.Comment: 15 pages, 3 figure

Generalized Scattering-Based Stabilization of Nonlinear Interconnected Systems

The research presented in this thesis is aimed at development of new methods and techniques for stability analysis and stabilization of interconnections of nonlinear systems, in particular, in the presence of communication delays. Based on the conic systems\u27 formalism, we extend the notion of conicity for the non-planar case where the dimension of the cone\u27s central subspace may be greater than one. One of the advantages of the notion of non-planar conicity is that any dissipative system with a quadratic supply rate can be represented as a non-planar conic system; specifically, its central subspace and radius can be calculated using an algorithm developed in this thesis. For a feedback interconnection of two non-planar conic systems, a graph separation condition for finite-gain L2-stability is established in terms of central subspaces and radii of the subsystems\u27 non-planar cones. Subsequently, a generalized version of the scattering transformation is developed which is applicable to non-planar conic systems. The transformation allows for rendering the dynamics of a non-planar conic system into a prescribed cone with compatible dimensions; the corresponding design algorithm is presented. The ability of the generalized scattering transformation to change the parameters of a system\u27s cone can be used for stabilization of interconnections of non-planar conic systems. For interconnections without communication delays, stabilization is achieved through the design of a scattering transformation that guarantees the fulfilment of the graph separation stability condition. For interconnected systems with communication delays, scattering transformations are designed on both sides of communication channel in a way that guarantees the overall stability through fulfilment of the small gain stability condition. Application to stabilization of bilateral teleoperators with multiple heterogeneous communication delays is briefly discussed. Next, the coupled stability problem is addressed based on the proposed scattering based stabilization techniques. The coupled stability problem is one of the most fundamental problems in robotics. It requires to guarantee stability of a controlled manipulator in contact with an environment whose dynamics are unknown, or at least not known precisely. We present a scattering-based design procedure that guarantees coupled stability while at the same time does not affect the robot\u27s trajectory tracking performance in free space. A detailed design example is presented that demonstrates the capabilities of the scattering-based design approach, as well as its advantages in comparison with more conventional passivity-based approaches. Finally, the generalized scattering-based technique is applied to the problem of stabilization of complex interconnections of dissipative systems with quadratic supply rates in the presence of multiple heterogeneous constant time delays. Our approach is to design local scattering transformations that guarantee the fulfilment of a multi-dimensional small-gain stability condition for the interconnected system. A numerical example is presented that illustrates the capabilities of the proposed design method

Stability results for constrained dynamical systems

Differential-Algebraic Equations (DAE) provide an appropriate framework to model and analyse dynamic systems with constraints. This framework facilitates modelling of the system behaviour through natural physical variables of the system, while preserving the topological constraints of the system. The main purpose of this dissertation is to investigate stability properties of two important classes of DAEs. We consider some special cases of Linear Time Invariant (LTI) DAEs with control inputs and outputs, and also a special class of Linear switched DAEs. In the first part of the thesis, we consider LTI systems, where we focus on two properties: passivity and a generalization of passivity and small gain theorems called mixed property. These properties play an important role in the control design of large-scale interconnected systems. An important bottleneck for a design based on the aforementioned properties is their verification. Hence we intend to develop easily verifiable conditions to check passivity and mixedness of Single Input Single Output (SISO) and Multiple Input Multiple Output (MIMO) DAEs. For linear switched DAEs, we focus on the Lyapunov stability and this problem forms the basis for the second part of the thesis. In this part, we try to find conditions under which there exists a common Lyapunov function for all modes of the switched system, thus guaranteeing exponential stability of the switched system. These results are primarily developed for continuous-time systems. However, simulation and control design of a dynamic system requires a discrete-time representation of the system that we are interested in. Thus, it is critical to establish whether discrete-time systems, inherit fundamental properties of the continuous-time systems from which they are derived. Hence, the third part of our thesis is dedicated to the problems of preserving passivity, mixedness and Lyapunov stability under discretization. In this part, we examine several existing discretization methods and find conditions under which they preserve the stability properties discussed in the thesis

Stability results for constrained dynamical systems

Differential-Algebraic Equations (DAE) provide an appropriate framework to model and analyse dynamic systems with constraints. This framework facilitates modelling of the system behaviour through natural physical variables of the system, while preserving the topological constraints of the system. The main purpose of this dissertation is to investigate stability properties of two important classes of DAEs. We consider some special cases of Linear Time Invariant (LTI) DAEs with control inputs and outputs, and also a special class of Linear switched DAEs. In the first part of the thesis, we consider LTI systems, where we focus on two properties: passivity and a generalization of passivity and small gain theorems called mixed property. These properties play an important role in the control design of large-scale interconnected systems. An important bottleneck for a design based on the aforementioned properties is their verification. Hence we intend to develop easily verifiable conditions to check passivity and mixedness of Single Input Single Output (SISO) and Multiple Input Multiple Output (MIMO) DAEs. For linear switched DAEs, we focus on the Lyapunov stability and this problem forms the basis for the second part of the thesis. In this part, we try to find conditions under which there exists a common Lyapunov function for all modes of the switched system, thus guaranteeing exponential stability of the switched system. These results are primarily developed for continuous-time systems. However, simulation and control design of a dynamic system requires a discrete-time representation of the system that we are interested in. Thus, it is critical to establish whether discrete-time systems, inherit fundamental properties of the continuous-time systems from which they are derived. Hence, the third part of our thesis is dedicated to the problems of preserving passivity, mixedness and Lyapunov stability under discretization. In this part, we examine several existing discretization methods and find conditions under which they preserve the stability properties discussed in the thesis