A novel iterative learning path-tracking control for nonholonomic mobile robots against initial shifts

Abstract

In this article, we propose a novel discrete-time iterative learning control framework for robust path-tracking problem of nonholonomic mobile robots. The contributions of this iterative learning control framework are threefolds: (1) With the application of a conventional feedback-aided P-type learning algorithm, the tracking error caused by a nonzero initial shift is detected. (2) By the introduction of an initial rectifying term, a novel iterative learning control scheme is proposed to improve the tracking performance. Sufficient conditions of convergence of this approach are given. (3) The convergence of the proposed algorithm for achieving the desired trajectory over a specified interval is proven theoretically. Simulation results validate the effectiveness of the proposed scheme.

Keywords

Iterative learning control nonholonomic mobile robot initial shift path tracking

Introduction

As a hot field of intelligent control, iterative learning control (ILC) has been developed for nearly three decades, since the concept was first published by Arimoto in English in 1984.¹ ILC is a control methodology that performs a given objective system repeatedly to track a reference trajectory over finite time intervals. The prime idea of ILC is to refine the control input to make better operation performance of the system on the next trial by use of updated data on the previous trial. Thus, ILC is applicable to systems with repetitively operated dynamics, such as multiagent systems consensus tracking,² robot arms for manufacturing,³ hard disk drive,⁴ and traffic flow in consecutive sections of a freeway.⁵ As a model-free control, ILC can be applied to systems with strong nonlinearity, unknown dynamics structure, or uncertain parameters. In this regard, ILC is fruitful both in theoretical analysis and practical applications.^6

–10

Widely utilized in civil and military fields, nonholonomic mobile robots (NMR) have gained significantly increasing research interests over the past few decades. The growing demands have brought in the focus on improving the autonomous motion of NMR. One strategy to motion control of mobile robots is to divide control between path planning and path tracking. Utilizing data from perception systems, path planning algorithms can be used to generate a desired path.^11
–13 Given a desired path, this article mainly investigates the tracking problem for NMR to follow the reference path autonomously.

With nonholonomic constraints, wheeled mobile robot is one of the well-known time-varying, coupling and complex nonlinear systems. Thus, the motion control of mobile robot gained increasing attention, and various novel controller design schemes have been developed, such as feedback control,^14,15 backstepping approach,^16,17 sliding mode control,^18
–20 fuzzy control,^21,22 and ILC methodology. In terms of nonlinear feedback control, it is necessarily based on exact mathematical model which is hard to obtain in practice. Combined with adaptive and robust control, backstepping approach can be used to systems that contain unknown parameters and disturbance. However, backstepping approach has the computation expansion problem. As a variable structure control method, sliding mode control is robust to modeling inaccuracy as well as disturbances, whereas its chattering limitation strongly reduces the control performance. Although no need of exact modeling, fuzzy control suffers from the slow response time due to the heavy computation. On the other hand, many ILC schemes have been applied to motion control of the robot systems.^23

–28 For example, iterative learning path-tracking control worked on omnidirectional vehicle robot which, whereas, is not subject to the nonholonomic constraints.^24,25 Oriolo et al.²⁶ presented an ILC method for robust steering of chained-form linear time-varying car-like mobile robot. The seminal work of Kang et al.²³ introduced a robust iterative learning rule with predictive and current learning terms for a two-wheeled mobile robot. Given the discrete kinematic model, Xi et al.²⁸ developed an ILC technique with an adjustable parameter and applied it to a four-wheeled mobile robot. Yu et al.²⁷ designed an ILC law with predictive, current, and past learning items to solve the high-precision trajectory-tracking issue of a wheeled mobile robot.

It should be pointed that most of these proposed schemes are considered with the assumption that the system is reinitialized to the neighborhood of the desired initial condition at each iteration. However, it is practically impossible to set the initial state value of the plant to that of the desired trajectory exactly. Besides, the accumulation of the initial state error may lead to poor tracking performance. In view of this challenge, this article proposes an improved ILC methodology for nonholonomic nonlinear systems in the presence of initial shift. By the term initial shift, we mean the initial condition remains the same for each trial, shifting form the desired initial condition. As an extension of the basic learning strategy used in the study by Kang et al.,²³ an initial rectifying term is added to form learning algorithm. Therefore, the proposed update law adopts feedback-aided P-type ILC to help the tracking error to converge to zero quickly and exploits the initial rectifying strategy to achieve complete tracking with smooth transient. Moreover, conditions on convergence are established. Finally, the effectiveness of our proposed algorithm is demonstrated with simulation studies.

The remainder of this article is organized as follows: In the second section, we present the mobile robot model and the formulation of tracking problem. With established tracking properties, a new scheme is proposed to design an appropriate ILC update law in the third section. Convergence is also analyzed. Simulation results are given in the fourth section, and the fifth section concludes this work.

Robot model and problem formulation

In this article, the following notations and definitions are introduced. The equation $| | x | | = {(x^{T} x)}^{1 / 2}$ denotes the vector norm for $x \in R^{n}$ in the n-dimensional Euclidean space. For an $(p \times m)$ -dimensional matrix with real elements $C \in R^{p \times m}$ , its corresponding matrix norm is denoted as $| | C | | = \sqrt{λ_{max} (C^{T} C)}, where λ_{max} (\cdot)$ represents the maximum eigenvalue for symmetric matrix. Let N refers to the positive integers ${0, 1, 2, \dots, n}$ . The α-norm of the positive real function $q : N \to R$ is defined as, with α ≥ 1

| | q (\cdot) | |_{α} = sup_{k \in N} q (k) {(\frac{1}{α})}^{k}

which is equivalent to the ∞-norm defined as $| | q (\cdot) | |_{\infty} = {sup}_{k \in N} q (k)$ by noting $| | q (k) | |_{α} \leq | | q (k) | |_{\infty} \leq α^{k} | | q (k) | |_{α}$ . Therefore, the sufficient convergence condition of our algorithm is derived by using the α-norm.

Consider a two-wheeled mobile robot located on a global Cartesian coordinate system O − xy as shown in Figure 1. The robot which is represented as P(k) possesses three degrees of freedom. The representation of its positioning can be denoted as ${[x_{p} (k), y_{p} (k), θ_{p} (k)]}^{T},$ where k is a discrete-time index, and the heading direction θ is taken counterclockwise from the x-axis. As shown in Figure 1, the coordinates of reference posture P_d(k) are defined as ${[x_{d} (k), y_{d} (k), θ_{d} (k)]}^{T}$ . The motion of the mobile robot is controlled by its linear velocity $υ_{p} (k)$ and rotational velocity $ω_{p} (k)$ . Then in the discrete-time domain, the kinematic model of the mobile robot can be defined as

[\begin{matrix} x_{p} (k + 1) \\ y_{p} (k + 1) \\ θ_{p} (k + 1) \end{matrix}] = [\begin{matrix} x_{p} (k) \\ y_{p} (k) \\ θ_{p} (k) \end{matrix}] + Δ T [\begin{matrix} cos θ_{p} (k) & 0 \\ sin θ_{p} (k) & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} υ_{p} (k) \\ ω_{p} (k) \end{matrix}]

Figure 1.

A two-wheeled nonholonomic mobile robot.

where ΔT is the sampling time. With regard to nonholonomic constraints, we assume that there is no sliding between the tire and the road.

As Figure 2 illustrated, the mobile robot dynamics inevitable exists some random uncertainties and disturbances, so we assume these influences as system state disturbances. Let $q (k) = {[x_{p} (k), y_{p} (k), θ_{p} (k)]}^{T} and u (k) = {[υ_{p} (k), ω_{p} (k)]}^{T}$ , then equation (1) can be converted into

q_{i} (k + 1) = q_{i} (k) + B (q_{i} (k), k) u_{i} (k) + β_{i} (k)

y_{i} (k) = q_{i} (k) + γ_{i} (k)

Figure 2.

The kinematic path-tracking control scheme using ILC for nonholonomic mobile robot. ILC: iterative learning control.

where i denotes the number of trial cycle and k is the time index ranging from 0 to n to complete a cycle. For $k \in N, q_{i} (k) \in R^{3}, u_{i} (k) \in R^{2}, y_{i} (k) \in R^{3}, β_{i} (k) \in R^{3}, and γ_{i} (k) \in R^{2}$ denote the system states, control inputs, outputs, state disturbance, and measurement noise at the ith iteration, respectively, and

B (q_{i} (k), k) = Δ T [\begin{matrix} cos θ_{p} (k) & 0 \\ sin θ_{p} (k) & 0 \\ 0 & 1 \end{matrix}]

Remark 1

In this article, we use different fonts to distinguish the mobile robot output trajectory y with its vertical coordinate y.

In this article, we consider the case that there exists an initial shift between the initial condition and the desired one and the resetting might not be exact. Therefore, our control objective is to propose a robust ILC algorithm, so that the NMR output trajectory $y (k), 0 \leq k \leq n$ , follows the given desired trajectory $y_{d} (k)$ as closely as possible in the presence of the initial shifts. Furthermore, the NMR kinematic systems (2) and (3) satisfy the properties and bounds stated as follows.

Assumption 1

Suppose the uncertainties $β (k) = 0 and γ (k) = 0$ are satisfied for k ∈ N. For any realizable desired trajectory which is given as a bounded sequences $y_{d} (k), k \in N$ , and a bounded state sequence $q_{d} (k)$ , there exists a unique control input $u_{d} (k)$ that can drive the NMR to track $y_{d} (k)$ exactly, that is

q_{d} (k + 1) = q_{d} (k) + B (q_{d} (k), k) u_{d} (k)

y_{d} (k) = q_{d} (k)

Assumption 2

The matrix function B(q,k) is globally uniformly Lipschitz with respect to q on N, that is

| | B (q_{1}, k) - B (q_{2}, k) | | \leq c_{B} | | q_{1} - q_{2} | |

for positive Lipschitz constant c_B.

Assumption 3

The matrix $B (q, k)$ is defined in $R^{3} \times N$ with bounds as $| | B (q, k) | | \leq b_{B}$ , where b_B is a positive constant.

Assumption 4

For $i \geq 0, k \in N$ , the disturbances and noises are bounded as $| | β_{i} (k) | | \leq b_{β}, | | γ_{i} (k) | | \leq b_{γ}, where b_{β} and b_{γ}$ are positive constants.

Assumption 5

All operations start at a biased initial state $q_{i} (0)$ . Besides, the resetting may not be exact that the initial state $q_{i} (0)$ is reset within a neighbor of q₀, that is, $| | q_{0} - q_{i} (0) | | \leq c_{q_{0}}$ for i ≥ 0 and a positive constant $c_{q_{0}}$ .

Controller design and convergence analysis

A conventional learning algorithm for NMR tracking problem takes the form of²³

u_{i + 1} (k) = u_{i} (k) + L_{1} (k) e_{i} (k + 1) + L_{2} (k) e_{i + 1} (k)

where $e_{i} (k) = y_{d} (k) - y_{i} (k)$ is the tracking error, $L_{1} (k) and L_{2} (k)$ are bounded learning gains and should be chosen such that

| | I - L_{1} (k) B (q_{i}, k) | | \leq ρ < 1

Now, we provide the following theorem without proof.

Theorem 1

If assumptions (A1) to (A5) are satisfied for NMR systems (2) and (3), and the updating law (7) be applied. Assume (8) holds for all $(q_{i} (k), k) \in R^{3} \times N$ , in the absence of state disturbances and output noises, the system output $y_{i} (k)$ converges uniformly to $y^{*} (k) on [0, n \cdot Δ T] as k \to \infty$ , where

y^{*} (k) = y_{d} (k) + (q_{0} - q_{d} (0))

In the presence of disturbances and noises, the system output will converge to $y^{*} (k)$ with bounds that are functions of $b_{β} and b_{γ}$ .

The above theorem indicates that the output trajectory converged to $y^{*} (k)$ which deviates from $y_{d} (k) by (q_{0} - q_{d} (0)), 0 \leq k \leq n$ . Thus, the trajectory tracking of the updating law (7) can never be achieved if the initial shift exists. To settle the deviated convergence shown in Theorem 1, a rectifying term is introduced to form the following iterative learning rule

u_{i + 1} (k) = u_{i} (k) + L_{1} (k) e_{i} (k + 1) + L_{2} (k) e_{i + 1} (k) + L_{1} e^{- Q k} θ_{h} (k) L_{3} (k) e_{i} (0) + L_{2} e^{- Q k} θ_{h} (k) L_{3} (k) e_{i + 1} (0)

where $θ_{h} : [0, n]$ is defined by

θ_{h} (k) = {\begin{array}{l} \frac{2}{Δ T * h} (1 - \frac{k}{h}), & (0 \leq k \leq h) \\ 0, & (h < k \leq n) \end{array}

and $L_{1} (k), L_{2} (k), and L_{3} (k)$ are bounded learning gain matrices such that $| | L_{1} (k) | | \leq b_{L_{1}}, | | L_{2} (k) | | \leq b_{L_{2}}, and | | L_{3} (k) | | \leq b_{L_{3}}, where b_{L_{1}}, b_{L_{2}}, b_{L_{3}} > 0$ . The learning gain Q is a positive constant that satisfies $| | e^{- Q k} | | \leq b_{Q}, (b_{Q} > 0)$ . As equation (11) shown, θ_h is bounded as $| | θ_{h} | | \leq b_{θ_{h}} . u_{i} (k) + L_{1} e_{i} (k + 1)$ is the widely used predictive learning term, $L_{2} (k) e_{i + 1} (k)$ is the current learning term which serves as a feedback controller that ensures the uniform boundedness of the tracking errors, and the last two terms are initial error rectifying terms that boost system outputs converge to the desired trajectory in the sense that $y_{i} (k) \to y_{d} (k), h \leq k \leq n$ , as the learning trials increases.

Theorem 2

Consider assumptions (A1) to (A5) hold for systems (2) and (3) and the ILC updating law (10) is applied. If (8) is satisfied, given a bounded output sequence $y_{d} (k), k \in N$ then in the absence of state disturbances, output noises and $c_{q_{0}}, y_{i} (k)$ converges to $y_{d}^{*} (k) on [0, n \cdot Δ T] as k \to \infty, where y_{d}^{*} (k)$ is given by

y_{d}^{*} (k) = {\begin{array}{l} y_{d} (k) + e^{- Q k} L_{3} (k) \sum_{j = h}^{k} θ_{h} (j) (y_{d} (0) - y_{i} (0)), & (0 \leq k \leq h) \\ y_{d} (k), & (h < k < n) \end{array}

On the other hand, in the presence of disturbances and noise, $| | y_{i} (k) - y_{d}^{*} (k) | |$ will be ultimately bounded as a function of $b_{β}, b_{γ}, and c_{q_{0}}$ .

Proof

Clearly, for an initial state $q_{0}$ and a realizable trajectory $y_{d}^{*} (k)$ , assumption (A1) ensures that there exists a unique control input $u_{d}^{*} (k)$ that will drive the output of the system located to $y_{d}^{*} (k)$ . Namely

q_{d}^{*} (k + 1) = q_{d}^{*} (k) + B (q_{d}^{*} (k), k) u_{d}^{*} (k)

y_{d}^{*} (k) = q_{d}^{*} (k)

where $q_{d}^{*} (0) = q_{0}$ . Then denote $Δ u_{i}^{*} (k) = u_{d}^{*} (k) - u_{i} (k) and Δ q_{i}^{*} (k) = q_{d}^{*} (k) - q_{i} (k)$ . Subtracting (13) from (5) yields

Δ q_{i}^{*} (k + 1) = q_{d}^{*} (k + 1) - q_{i} (k + 1) = [q_{d}^{*} (k) + B (q_{d}^{*} (k), k) u_{d}^{*} (k)] - [q_{i} (k) + B (q_{i} (k), k) u_{i} (k) + β_{i} (k)] = Δ q_{i}^{*} (k) + B (q_{d}^{*} (k), k) u_{d}^{*} (k) - B (q_{i} (k), k) {u_{i} (k) - u_{d}^{*} (k) + u_{d}^{*} (k)} - β_{i} (k) = Δ q_{i}^{*} (k) + [B (q_{d}^{*} (k), k) - B (q_{i} (k), k) u_{i} (k)] u_{d}^{*} (k) + B (q_{i} (k), k) Δ u_{i}^{*} (k) - β_{i} (k)

Taking the function norm of (15), we have

| | Δ q_{i}^{*} (k + 1) | | \leq | | Δ q_{i}^{*} (k) | | + c_{B} b_{u_{d}^{*}} | | Δ q_{i}^{*} (k) | | + b_{B} | | Δ u_{i}^{*} (k) | | + b_{β}

Let $h_{2} = 1 + c_{B} b_{u_{d}^{*}}, where b_{u_{d}^{*}} = max_{1 \leq k \leq n} | | u_{d}^{*} (k) | |$ , then (16) can be converted into the following form

| | Δ q_{i}^{*} (k + 1) | | \leq h_{2} | | Δ q_{i}^{*} (k) | | + b_{B} | | Δ u_{i}^{*} (k) | | + b_{β}

from which we arrive at the following equation

| | Δ q_{i}^{*} (k) | | \leq \sum_{j = 0}^{k - 1} h_{2}^{k - 1 - j} [b_{B} | | Δ u_{i}^{*} (j) | | + b_{β}] + h_{2}^{k} c_{q_{0}}

On the other hand, from (10) we get

\begin{array}{l} Δ u_{i + 1}^{*} (k) = u_{d}^{*} (k) - u_{i + 1} (k) = Δ u_{i}^{*} (k) + u_{i} (k) - u_{i + 1} (k) \\ = Δ u_{i}^{*} (k) - L_{1} (k) e_{i} (k + 1) - L_{2} (k) e_{i + 1} (k) \\ - (L_{1} (k) + L_{2} (k)) e^{- Q k} θ_{h} (k) L_{3} (k) e_{i} (0) \\ = Δ u_{i}^{*} (k) - L_{1} (k) [y_{d} (k + 1) - y_{i} (k + 1) \\ + e^{- Q k} θ_{h} (k) L_{3} (k) (y_{d} (0) - y_{i} (0)] - L_{2} (k) [y_{d} (k) \\ - y_{i + 1} (k) + e^{- Q k} θ_{h} (k) L_{3} (k) (y_{d} (0) - y_{i + 1} (0)] \\ = Δ u_{i}^{*} (k) - L_{1} (k) [y_{d}^{*} (k + 1) - y_{i} (k + 1)] \\ - L_{2} (k) [y_{d}^{*} (k) - y_{i + 1} (k)] \\ = Δ u_{i}^{*} (k) - L_{1} (k) [q_{d}^{*} (k + 1) - q_{i} (k + 1) \\ - γ_{i} (k + 1)] - L_{2} (k) [q_{d}^{*} (k) - q_{i + 1} (k) - γ_{i + 1} (k)] \\ = Δ u_{i}^{*} (k) - L_{1} (k) [Δ q_{i}^{*} (k) + B (q_{d}^{*} (k), k) u_{d}^{*} (k) \\ - B (q_{i} (k), k) u_{i} (k) - β_{i} (k) - γ_{i} (k + 1)] \\ - L_{2} (k) [Δ q_{i + 1}^{*} (k) - γ_{i + 1} (k)] \\ = Δ u_{i}^{*} (k) - L_{1} (k) Δ q_{i}^{*} (k) \\ - L_{1} (k) [B (q_{d}^{*} (k), k) u_{d}^{*} (k) - B (q_{i} (k), k) {u_{i} (k) \\ - u_{d}^{*} (k) + u_{d}^{*} (k)}] + L_{1} (k) [β_{i} (k) + γ_{i} (k + 1)] \\ - L_{2} (k) [Δ q_{i + 1}^{*} (k) - γ_{i + 1} (k)] \\ = Δ u_{i}^{*} (k) - L_{1} (k) Δ q_{i}^{*} (k) \\ - L_{1} (k) B (q_{i} (k), k) Δ u_{i}^{*} (k) - L_{1} (k) [B (q_{d}^{*} (k), k) \\ - B (q_{i} (k), k)] u_{d}^{*} (k) + L_{1} (k) [β_{i} (k) + γ_{i} (k + 1)] \\ - L_{2} (k) Δ q_{i + 1}^{*} (k) + L_{2} (k) γ_{i + 1} (k) \end{array}

Applying assumptions 2–5 to (19), it follows that

| | Δ u_{i + 1}^{*} (k) | | \leq | | I - L_{1} (k) B (q_{i} (k), k) | | | | Δ u_{i}^{*} (k) | | + b_{L_{1}} | | Δ q_{i}^{*} (k) | | + b_{L_{1}} c_{B} b_{u_{d}^{*}} | | Δ q_{i}^{*} (k) | | + b_{L_{2}} | | Δ q_{i + 1}^{*} (k) | | + b_{L_{1}} (b_{β} + b_{γ}) + b_{L_{2}} b_{r}

Let $h_{1} = b_{L_{1}} (1 + c_{B} b_{u_{d}^{*}}) = b_{L_{1}} h_{2} and b_{1} = b_{L_{1}} (b_{β} + b_{γ}) + b_{L_{2}} b_{r}$ . Then (20) can be rewritten as

\begin{array}{l} | | Δ u_{i + 1}^{*} (k) | | \leq ρ | | Δ u_{i}^{*} (k) | | + h_{1} | | Δ q_{i}^{*} (k) | | + b_{L_{2}} | | Δ q_{i + 1}^{*} (k) | | + b_{1} \\ \leq ρ | | Δ u_{i}^{*} (k) | | + h_{1} [\sum_{j = 0}^{k - 1} h_{2}^{k - 1 - j} (b_{B} | | Δ u_{i}^{*} (j) | | + b_{β}) + h_{2}^{k} c_{q_{0}}] + b_{1} \\ + b_{L_{2}} [\sum_{j = 0}^{k - 1} h_{2}^{k - 1 - j} (b_{B} | | Δ u_{i + 1}^{*} (j) | | + b_{β}) + h_{2}^{k} c_{q_{0}}] \leq ρ | | Δ u_{i}^{*} (k) | | \\ + h_{1} \sum_{j = 0}^{k - 1} h_{2}^{k - 1 - j} (b_{B} | | Δ u_{i}^{*} (j) | | + b_{β}) + b_{1} + (h_{1} + b_{L_{2}}) h_{2}^{k} c_{q_{0}} \\ + b_{L_{2}} \sum_{j = 0}^{k - 1} h_{2}^{k - 1 - j} (b_{B} | | Δ u_{i + 1}^{*} (j) | | + b_{β}) \end{array}

Multiply (21) by ${(1 / α)}^{k}$ to compute the α-norm, we get

| | Δ u_{i + 1}^{*} (k) | | \leq ρ | | Δ u_{i}^{*} (k) ∥ {(\frac{1}{α})}^{k} + b_{1} {(\frac{1}{α})}^{k} + (h_{1} + b_{L_{2}}) c_{q_{0}} {(\frac{h_{2}}{α})}^{k} + (\frac{h_{1}}{α}) \sum_{j = 0}^{k - 1} {(\frac{h_{2}}{α})}^{k - 1 - j} \times [b_{B} | | Δ u_{i}^{*} (j) | | {(\frac{1}{α})}^{j} + b_{β} {(\frac{1}{α})}^{j}] + (\frac{b_{L_{2}}}{α}) \sum_{j = 0}^{k - 1} {(\frac{h_{2}}{α})}^{k - 1 - j} \times [b_{B} | | Δ u_{i + 1}^{*} (j) | | {(\frac{1}{α})}^{j} + b_{β} {(\frac{1}{α})}^{j}]

Taking $α > max [1, h_{2}]$ , we have

\begin{array}{l} | | Δ u_{i + 1}^{*} | |_{α} \leq ρ | | Δ u_{i}^{*} | |_{α} + b_{1} + (h_{1} + b_{L_{2}}) c_{q_{0}} \\ + (b_{B} | | Δ u_{i}^{*} | |_{α} + b_{β}) (\frac{h_{1} [1 - {(h_{2} / α)}^{n}]}{α - h_{2}}) \\ + (b_{B} | | Δ u_{i + 1}^{*} | |_{α} + b_{β}) (\frac{b_{L_{2}} [1 - {(h_{2} / α)}^{n}]}{α - h_{2}}) \\ = (ρ + b_{B} h_{1} \frac{1 - {(h_{2} / α)}^{n}}{α - h_{2}}) | | Δ u_{i}^{*} | |_{α} + b_{1} \\ + b_{B} b_{L_{2}} \frac{1 - {(h_{2} / α)}^{n}}{α - h_{2}} | | Δ u_{i + 1}^{*} | |_{α} \\ + \frac{b_{β} (h_{1} + b_{L_{2}}) [1 - {(h_{2} / α)}^{n}]}{α - h_{2}} + (h_{1} + b_{L_{2}}) c_{q_{0}} \end{array}

Let $α > max [1, h_{2}, h_{2} + b_{B} b_{L_{2}}]$ , then equation (23) can be written as

| | Δ u_{i + 1}^{*} | |_{α} \leq \hat{ρ} | | Δ u_{i}^{*} | |_{α} + ∊

where

\hat{ρ} = \frac{ρ + b_{B} h_{1} (1 - {(h_{2} / α)}^{n} / α - h_{2})}{1 - b_{B} b_{L_{2}} (1 - {(h_{2} / α)}^{n} / α - h_{2})}

and

ε = \frac{b_{1} + (h_{1} + b_{L_{2}}) (c_{q_{0}} + b_{β} (1 - {(h_{2} / α)}^{n} / α - h_{2}))}{1 - b_{B} b_{L_{2}} (1 - {(h_{2} / α)}^{n} / α - h_{2})}

Choose α large enough so that $\hat{ρ} < 1$ , then (24) can be rewritten as

lim_{i \to \infty} | | Δ u_{i}^{*} | |_{α} \leq \frac{ε}{1 - \hat{ρ}}

Similarly, multiplying both sides of (18) by ${(1 / α)}^{k}$ and applying inequality (25), we obtain

lim_{i \to \infty} | | Δ q_{i}^{*} | |_{α} \leq b_{B} \frac{1 - {(h_{2} / α)}^{n}}{α - h_{2}} \frac{ε}{1 - \hat{ρ}} + b_{β} \frac{1 - {(h_{2} / α)}^{n}}{α - h_{2}} + c_{q_{0}}

Next, subtracting (14) from (3) yields

Δ y_{i}^{*} (k) = y_{d}^{*} (k) - y_{i} (k) = q_{d}^{*} (k) - [q_{i} (k) + γ_{i} (k)] = Δ q_{i}^{*} (k) - γ_{i} (k)

Multiplying both sides of (27) by ${(1 / α)}^{k}$ and applying inequality (18) gives

lim_{i \to \infty} | | Δ y_{i}^{*} | |_{α} \leq b_{B} \frac{1 - {(h_{2} / α)}^{n}}{α - h_{2}} \frac{ε}{1 - \hat{ρ}} + b_{β} \frac{1 - {(h_{2} / α)}^{n}}{α - h_{2}} + b_{γ} + c_{q_{0}}

Hence, from the inequalities (25), (26), and (28), we can conclude that $| | Δ u_{i}^{*} | |_{α}, | | Δ q_{i}^{*} | |_{α}, and | | Δ y_{i}^{*} | |_{α}$ all converge to zero in the absence of disturbances and with perfect repeatability, that is, $b_{β} = 0, b_{γ} = 0, and c_{q_{0}} = 0$ . This completes the proof.□

As $b_{β}, b_{γ}, and c_{q_{0}}$ tend to zero, theorem 2 shows that the application of ILC updating law (10) ensures the convergence of the system outputs to the desired trajectory $y_{d}^{*} (k) for k \in [h, n]$ . In the time interval $[0, h]$ , the added initial rectifying term generates a smooth transition from initial biased position to the desired trajectory. It implies that the introduction of our proposed ILC scheme helps systems with initial shifts to achieve the complete tracking performance with specified transient.

Simulation results

In this section, simulation studies are presented to verify the obtained theoretical results via MATLAB Simulink platform. We have run three simulations with different kinds of trajectories as the desired paths. For comparison, tracking performance of learning algorithm (7) are also demonstrated in all simulations. Define the absolute value of maximum tracking errors in i-th iteration as $x_{e} = max_{0 \leq k \leq n} | x_{d} (k) - x_{i} (k) |, y_{e} = max_{0 \leq k \leq n} | y_{d} (k) - y_{i} (k) |, and θ_{e} = max_{1 \leq k \leq n} | θ_{d} (k) - θ_{i} (k) |$ .

Case 1: circular trajectory

In this case, we mainly discuss a circular expected trajectory whose initial condition is $q_{d} (0) = {[1, 0, π / 2]}^{T}$ , and 2000 samples are predefined for $x_{d} (k), y_{d} (k), and θ_{d} (k)$ with sample time $Δ T = 0.001 s$ . The initial state at each iteration is reset as $q_{i} (0) = q_{0} = {[0.9, 0.1, 14 π / 30]}^{T}$ . The gain matrices $L_{1} (k), L_{2} (k), and L_{3} (k)$ in control laws (10) and (7) are set to

L_{1} (k) = L_{2} (k) = 0.1 [\begin{matrix} cos θ_{p} (k) & sin θ_{p} (k) & 0 \\ 0 & 0 & 1 \end{matrix}], L_{3} (k) = L_{1}^{- 1} [\begin{matrix} 0.1 & 0.1 & 0.01 \\ 0.01 & 0.01 & 0.01 \end{matrix}]

Figures 3 to 5 show simulation results for 100 iterations using the learning scheme (10) with Q = 0.003 and the conventional updating law (7).

Figure 3.

Simulation results of the change of mobile robot trajectory in the iterative process by using the ILC law (10). ILC: iterative learning control.

Figure 4.

Simulation results of the change of mobile robot trajectory in the iterative process by using the ILC law (7). ILC: iterative learning control.

Figure 5.

The convergence performance of the learning algorithm (10).

Case 2: cardioid-like trajectory

In this case, we mainly discuss a cardioid-like desired trajectory which is comprised of three semicircles. Starting from (1,0), the cardioid-like trajectory first rotates counterclockwise along a unit circle centered at the origin (0,0), then it rotates counterclockwise along a semicircle with center coordinates (1,0) and radius 2. Rotate counterclockwise continuously along a semicircle of radius 1 centered at (2,0), the cardioid-like trajectory finally terminates at (1,0). This desired trajectory is predefined for $k = 1, \dots, 3000$ with sample time $Δ T = 0.001 s$ . The initial state at each iteration is reset as $q_{i} (0) = q_{0} = {[0.9, 0, π / 2]}^{T}$ . The gain matrices $L_{1} (k), L_{2} (k), and L_{3} (k)$ in control laws (10) and (7) are set to

L_{1} (k) = L_{2} (k) = 0.1 [\begin{matrix} cos θ_{p} (k) & sin θ_{p} (k) & 0 \\ 0 & 0 & 1 \end{matrix}], L_{3} (k) = L_{1}^{- 1} [\begin{matrix} 0.4 & 0.1 & 0 \\ 0.04 & 0 & 1 \end{matrix}]

Figures 6 to 8 show simulation results for 200 iterations using the learning scheme (10) with Q = 0.001 and the conventional updating law (7).

Figure 6.

Simulation results of the change of mobile robot trajectory in the iterative process by using the ILC law (10). ILC: iterative learning control.

Figure 7.

Simulation results of the change of mobile robot trajectory in the iterative process by using the ILC law (7). ILC: iterative learning control.

Figure 8.

The convergence performance of the learning algorithm (10).

Case 3: spiral-like trajectory

In this case, we mainly discuss a spiral-like desired trajectory which is comprised of seven semicircles with different radiuses. Starting from (1,0), the spiral-like trajectory first rotates counterclockwise along a unit circle centered at the origin (0,0), then it rotates counterclockwise along a semicircle of radius 1.5 centered at (0.5, 0). Rotate counterclockwise continuously along a semicircle of radius 2 centered at the origin, the trajectory continues to rotate counterclockwise along a semicircle with center coordinates (0.5, 0) and radius 2.5. Such like this, the desired trajectory rotates counterclockwise along semicircles whose center coordinates change between (0,0) and (0.5, 0), and the radius of them grow at the pace of 0.5. Finally, the desired trajectory terminates at point (−4, 0). This desired trajectory is predefined for $k = 1, \dots, 6000$ with sample time $Δ T = 0.001 s$ . The initial state at each iteration is reset as $q_{i} (0) = q_{0} = {[0.8, 0, π / 2]}^{T}$ . The gain matrices $L_{1} (k), L_{2} (k), and L_{3} (k)$ in control laws (10) and (7) are set to

L_{1} (k) = L_{2} (k) = 0.1 [\begin{matrix} cos θ_{p} (k) & sin θ_{p} (k) & 0 \\ 0 & 0 & 1 \end{matrix}], L_{3} (k) = L_{1}^{- 1} [\begin{matrix} 0.9 & 0.9 & 0.1 \\ 0.08 & 0.8 & 0.08 \end{matrix}]

Figures 9 to 11 show simulation results for 100 iterations using the learning scheme (10) with Q = 0.003 and the conventional updating law (7).

Figure 9.

Simulation results of the change of mobile robot trajectory in the iterative process by using the ILC law (10). ILC: iterative learning control.

Figure 10.

Simulation results of the change of mobile robot trajectory in the iterative process by using the ILC law (7). ILC: iterative learning control.

Figure 11.

The convergence performance of the learning algorithm (10).

Conclusions

In this article, a novel practical ILC updating law is proposed to improve the path-tracking accuracy for NMR. Its main feature is its ability to track the desired trajectory with biased initial condition. Given the kinematic model of NMR, the proposed updating law exploits feedback-aided P-type learning terms to enhance stability as well as robustness characteristics. Furthermore, the initial rectifying term is introduced to guarantee the uniform convergence of the output trajectories to the desired one with a smooth transient. Numerical simulations are given to validate the obtained theoretical results.

Footnotes

Acknowledgements

The authors would like to thank the editors and anonymous referees for their constructive comments and suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research is supported by the National Natural Science Foundation of China (61375084 and 61374101) and the Natural Science Foundation of Shandong Province (ZR2015QZ08).

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