Adaptive learning tracking for robot manipulators with varying trial lengths

Abstract

This paper proposes adaptive iterative learning control schemes for robot manipulator systems with iteration-varying lengths. To prove the asymptotical convergence of the joint position tracking error along the iteration axis, this paper develops a new composite energy function based on the newly introduced auxiliary variables for the analysis. Moreover, the traditional assumption of identical initialization condition is relaxed to be arbitrarily varying and then an initial rectifying mechanism is introduced to tackle initial shift problem of robotic systems. Illustrative simulations on a two degree-of-freedom robot manipulator are provided to verify the theoretical results.

Introduction

Learning is a basic skill of human beings. That is, people keep trying and learning from failures, which makes them behave better and better in daily work and lives. Inspired by this learning concept, some scholars speculated on whether industrial machines could own certain learning ability to serve human better. In fact, this basic cognition directly motivates the introduction and developments of iterative learning control (ILC). This novel learning control method is very effective in dealing with repetitive industrial processes, where the operation intervals for each batch are finite and fixed, and the learning objectives are identical for all iterations [1], [2], [3], [4]. The iterative learning controller is simple in structure, and actually is a data-driven mode requiring little knowledge of system dynamics. Therefore, ILC has been widely used in modeling various industrial systems, such as robotic system [5], [6], [7], [8], [9], [10]. Through experiments while by learning and correction, the iterative learning controller can gradually improve its tracking performance and achieve accurate tracking as the iteration number increases.

In order to ensure the desired control or system information can be learned over iterations, a fundamental requirement for ILC is that the operation lengths should be iteration-invariant. However, this precondition is often violated for many practical applications as a result of unknown uncertainties or unpredictable factors. For example, while applying ILC in humanoid and biped walking robots, the gaits are divided into phases whose duration is different in every batch [11]. Another example can be found in analyzing the lab-scale gantry crane, where the iteration will terminate once the output exceeds the specified boundary, and hence the trial length is varying [12]. A third example can be seen in [13], where the authors investigated the functional electrical stimulation for drop foot treatment, and they found that a trial might end early due to safety considerations. All these application examples motivate us to consider the varying-length problem in ILC design and analysis.

There were several primary researches trying to resolve this issue. In [14], Seel et al. considered the linear system and first completed the proof of the monotonic convergence. It should be emphasized that no specific formulation of varying length was imposed in this paper. Li et al. proposed the random model of the iteration-varying trial length problem based on the introduced Bernoulli variable whose distribution is known to the controller in [15], [16], [17], where an iteration-averaging operator was provided to compensate for the missing information caused by the early termination of the batch. Afterwards, to derive strong convergence of the conventional P-type ILC scheme, Shen et al. [18], [19] provided new analysis techniques where the iteration-averaging operator was not involved. Additionally, [20] applied the iteration-moving-average operator and proposed two new update laws for linear systems. However, in these papers, the considered systems are all linear or globally Lipschitz nonlinear. In our recent works [21], [22], we develop adaptive ILC laws for continuous-time parametric nonlinear systems, where the nonlinear functions do not satisfy the globally Lipschitz continuous condition and the trial lengths vary randomly. It has been known that the multi-degree-of-freedom robot manipulator described by Lagrange equation is a very complex nonlinear system. Therefore, these results obtained in these studies may not be directly applicable to such systems while the kernel concept can be borrowed.

Moreover, another challenge with ILC on robotic systems is the well-known identical initialization conditions (i.i.c.), which matches the repetition requirement of ILC and has been widely used in most ILC literature [23], [24], [25], [26]. Specifically, i.i.c. requires that the system states be accurately reset to the desired values at the start of each iteration. This assumption is so strict that it may be difficult to realize due to various reasons. In practical applications, the initial state of the control system is usually not equal to the expected value or even iteration-varying. An alternative solution for this problem is to introduce some initial rectifying mechanism, whose essence is that the initial tracking error is introduced into the traditional ILC law for a small time interval [0, T₁) with T₁ can be arbitrarily small, to eliminate the influence of the initial error for the rest part [T₁, T]. In [27], Sun et al. proposed this mechanism when studying the initial problem for a class of continuous-time nonlinear systems with time-delay. Subsequently, the rectifying mechanism was extended to the nonlinear systems with general high-order relative degree [28] and the SISO discrete-time nonlinear systems with high-order relative degree [29]. However, the initial states in [27], [28], [29] are usually fixed. Motivated by this, Li et al. [30] investigated a continuous-time affine multi-variable system and the initial conditions were relaxed to non-fixed case. Moreover, Jin also provided interesting results on the relaxation of initial state condition for learning control with output and state constraints [31], [32], [33], [34], where the initial state can be arbitrarily set. These results have inspired us to relax the identical initial condition for extending the possible application environments in this paper.

In particular, we consider the ILC problem for robot manipulator systems with random trial lengths. The manipulator system is of high-order multi-input-multi-output form, which cannot be covered by the existing results. To handle this system, we employ the back-stepping adaptive scheme, which can be further extended to more complicated nonlinear systems. To derive the asymptotical convergence, we define a new composite energy function (CEF), in which a virtual tracking error is introduced for the untrodden part of each iteration. To address the different initial states, we modify the original reference trajectory to guarantee the convergence property beyond a small initial time interval. The major contributions of this paper are summarized as follows:

• We consider the adaptive learning control of robot manipulator systems, in which the operation length is randomly varying from trial to trial.

• We apply the initial rectifying mechanism similar to [31], [32], [33], [34] and modify the reference trajectory to guarantee a precise tracking to the original objective beyond a small initial interval.

• A novel CEF is defined with the help of newly introduced auxiliary variables to facilitate the convergence analysis.

The paper is organized as follows. In Section 2, the general formulation of the robot manipulator system and the iteration-varying trial length problem are given. Section 3 discusses the controller design and convergence analysis. The relaxation of the i.i.c. is addressed in Section 4. Illustrative simulations are given in Section 5. In the end, Section 6 concludes this paper.

Notations: $\mathbf{R}$ denotes the set of real numbers, ${\mathbf{R}}^n$ is the space of all n-dimensional vectors, and $\mathbf{N}$ is the set of positive integers. $\mathbb{E}[·],$ $\mathbb{P}[·],$ $\mathbb{F}[·]$ denote the mathematical expectation of a random variable, the probability of an event, and the probability distribution function of a random variable, respectively.