Adaptive and sliding mode tracking control for wheeled mobile robots with unknown visual parameters

Abstract

The trajectory tracking control problem of dynamic nonholonomic wheeled mobile robots is considered via visual servoing feedback. A novel visual feedback tracking error model is proposed. Its tracking controller is independent of uncalibrated visual parameters by using new methods. This controller consists of two units: one is an adaptive control for compensation of the uncertainties of dynamic parameters, the other is a variable structure control for the interference suppression. In addition, the torque tracking controller is global and smooth, and the chattering phenomenon is eliminated. The asymptotic convergence of tracking errors to equilibrium point is rigorously proved by the Lyapunov method. Simulation and experiment results are provided to illustrate the performance of the control law.

Keywords

Wheeled mobile robots dynamic tracking visual servoing

Introduction

The control of wheeled mobile robots with nonholonomic constraints has attracted much attention due to the inherent nonlinearity in robot dynamics and the usefulness in many applications. By the theorem of Brockett (1983), a nonholonomic system cannot be stabilized at a single equilibrium point by any continuous, time-invariant, state-feedback controller. However, several other approaches have been proposed (Astolfi, 1996; Tian and Li, 2002; Hu et al., 2004; Samson, 2005; Wang et al., 2015).

In the control of nonholonomic mobile robots, it is usually assumed that the robot states are available and exactly reconstructed using proprioceptive and exteroceptive sensor measurements. Unfortunately, in real-world applications, these assumptions often do not hold due to uncertainties in the kinematic and dynamic models, mechanical limitations and measurement noise. As a consequence, the estimation of the robot states from sensor measurements can be affected by these perturbations. An interesting approach to overcome this position measurement problem is to utilize a vision system to directly obtain the Cartesian position information required by the controller. Since the late 1980s, much effort has been devoted to visual servoing and vision-based manipulations (Fang et al., 2012; Wang et al., 2010). To implement a visual servo controller, an important step is to calibrate the intrinsic and extrinsic parameters of cameras. It is well known that the camera calibration is costly and tedious. To avoid camera calibration, a lot of effort has been made to achieve uncalibrated visual servoing (Chen et al., 2014; Liang et al., 2015).

Recently, the visual tracking control problem of nonholonomic mobile robots has been proposed. In Chen et al. (2006), a visual servoing tracking controller was developed for a monocular camera system mounted on an underactuated wheeled mobile robot subject to nonholonomic motion constraints. The reference (Wang et al., 2010) presented a dynamic feedback tracking controller for a nonholonomic wheeled mobile robot (WMR) kinematic system under the assumption that the unknown visual parameters $α_{1} = α_{2} = α$ .

The methods mentioned above are based on kinematics only and the nonlinear forces in robot dynamics are neglected. However, in practice, it is more realistic to formulate the nonholonomic system control problem at dynamic level, where the torque and force are taken as the control inputs. Liu et al. (2006) presented a new adaptive controller for image-based dynamic control of a robot manipulator using a fixed camera whose intrinsic and extrinsic parameters are not known. Yang et al. (2013) used feedback from an uncalibrated, fixed (ceiling-mounted) camera to develop an adaptive tracking controller for a type (1,1) nonholonomic mobile robot, a novel trajectory tracking torque controller was designed based on Lyapunov’s direct method and backstepping technique. But the method proposed in Yang et al. (2013) cannot be used to deal with the tracking problem discussed in this paper as the typical triangular structure does not meet.

In this paper, we consider the trajectory tracking control problem of a type (2,0) nonholonomic WMR using an uncalibrated, fixed (ceiling-mounted) camera system. The main contributions can be summarized in the following several respects:

A novel uncertain kinematic tracking error model is proposed under the assumption that the visual parameters $α_{1}, α_{2}$ are unknown; the assumption is relaxed compared with Wang et al. (2010). For this kinematic tracking error model, a dynamic feedback tracking controller is presented, which is independent of uncalibrated visual parameters $α_{1}$ and $α_{2}$ .

The trajectory tracking control problem is discussed at an uncertain dynamic level and an adaptive variable structure torque controller is designed via visual servoing feedback. Moreover, the variable structure controller is smooth, global and the common chattering problem is overcome.

The experiment is performed on a real type (2,0) MT-R mobile robot and the results illustrate the effectiveness of the proposed controller.

This paper is organized as follows: In Section Problem formulation, the camera-object visual servoing kinematic and dynamic models of a WMR are introduced. In Section Adaptive controller design, a kinematic tracking controller and an adaptive variable structure torque controller for the WMR are designed. In Section Stability analysis, the asymptotic stability of the closed-loop system is rigorously proved by the Lyapunov method. In Section Simulation and experiment results, the controller’s performance is illustrated through the simulation and experiment results. The last section presents a conclusion and outlines future work.

Problem formulation

System configuration

A mobile robot measured by using a fixed camera is shown in Figure 1. It is assumed that a pinhole camera is fixed to the ceiling, the type (2,0) mobile robot is under the camera, the camera plane and the robot plane are parallel. There are three coordinate frames, namely the inertial frame X–Y–Z, the camera frame x–y–z and the image frame $u - o_{1} - v$ . Assume that the $x - y$ plane of the camera frame is identical with the u–v plane of the image frame. Here C is the crossing point between the optical axis of the camera and the X–Y plane. Its coordinate relative to the X–Y plane is $(c_{x}, c_{y})$ . The coordinate of the original point of the camera frame with respect to the image frame is denoted by $(O_{c 1}, O_{c 2})$ . $(x, y)$ is the coordinate of the mass centre P of the robot with respective to the X–Y plane. Suppose that $(x_{m}, y_{m})$ is the coordinate of $(x, y)$ relative to the image frame. A pinhole camera model yields (Yang et al., 2013)

[\begin{matrix} x_{m} \\ y_{m} \end{matrix}] = [\begin{matrix} α_{1} & 0 \\ 0 & α_{2} \end{matrix}] R [[\begin{matrix} x \\ y \end{matrix}] - [\begin{matrix} c_{x} \\ c_{y} \end{matrix}]] + [\begin{matrix} O_{c 1} \\ O_{c 2} \end{matrix}]

(1)

where $α_{1}$ , $α_{2}$ are constants, which are dependent on the depth information, focal length, scalar factors along the u axis and the v axis, respectively. In (1),

R = [\begin{matrix} \cos θ_{0} & \sin θ_{0} \\ - \sin θ_{0} & \cos θ_{0} \end{matrix}],

where $θ_{0}$ denotes the angle between u axis and X axis with a positive anticlockwise orientation.

Kinematic and dynamic models

As shown in Figure 1, the two rear wheels of the robot are controlled independently by motors, and a front castor wheel prevents the robot from tipping over as it moves on a plane. Both wheels have the same radius denoted by r, and $2 L$ is the distance between two wheels. Assume that the geometric centre point and the mass centre point P of the robot are identical. The pose of the robot in the inertial coordinate frame ${O, X, Y}$ is defined as $q = (x, y, θ)^{T}$ , where $(x, y)$ is the coordinate of position P, and $θ$ is the orientation angle of robot between the robot frame ${P, X_{1}, X_{2}}$ and the inertial frame ${O, X, Y}$ with a positive anticlockwise direction. The kinematic of the robot can be modelled by the following differential equations (Campion et al., 1996)

\overset{\cdot}{q} = S (q) v (t)

(2)

where $v (t) = [ν_{1}, ω]^{T}$ , $ν_{1}$ is the forward velocity while $ω$ is the angular velocity of the robot, $S (q)$ is expressed as follows

S (q) = [\begin{matrix} \cos θ & 0 \\ \sin θ & 0 \\ 0 & 1 \end{matrix}]

(3)

According to the Euler-Lagrangian formulation, the dynamic model of the mobile robot can be described as follows (Fierro and Lewis, 1995)

M (q) \overset{\cdot\cdot}{q} + C (q, \dot{q}) \dot{q} + τ_{d} = B (q) τ + A^{T} (q) λ

(4)

where $M (q) \in R^{3 \times 3}$ is a positive definite symmetric inertial matrix, $C (q, \overset{\cdot}{q}) \in R^{3 \times 3}$ represents centripetal and Coriolis torque, $B (q) \in R^{3 \times 2}$ is an input transformation matrix, and rank $B = 2$ , $A (q) \in R^{1 \times 3}$ is the matrix associated with the constraints and $λ \in R^{1}$ is a Lagrange multiplier which expresses the constraint force, $τ_{d} \in R^{3}$ denotes bounded unknown disturbances including unstructured unmodelled dynamics, $τ \in R^{2}$ is the torque vector applied to the right and left wheels.

Differentiating both sides of (2), substituting it into (4) and pre-multiplying both sides by $S^{T} (q)$ , one obtains

M_{1} (q) \overset{\cdot}{v} + C_{1} (q, \overset{\cdot}{q}) v + {\bar{τ}}_{d} = B_{1} (q) τ

(5)

where $M_{1} (q) = S^{T} MS$ , $C_{1} (q, \overset{\cdot}{q}) = S^{T} (M \overset{\cdot}{S} + CS)$ , ${\bar{τ}}_{d} = S^{T} τ_{d}$ , $B_{1} (q) = S^{T} B (q)$ . System (5) is more appropriate for controller design as the constraint $λ$ has been eliminated from dynamic equation (4).

Assumption 1. $∥ {\bar{τ}}_{d} ∥$ is bounded by a known scalar; i.e., $∥ {\bar{τ}}_{d} ∥ \leq d_{B}$ , where $d_{B}$ is a known constant.

Remark 1. In order to facilitate the design of the controller in the following section, we generally assume that the bound of disturbance is known and its boundary is available to the designer. In theory, the boundary of disturbance may be obtained by the actual measured signal minus the ideal signal.

In system (2), generally, $(x, y)$ can be obtained from the encoders of motors and other sensors such as ultrasonic sensors, infrared sensors, etc. However, for complex environments, it is difficult to implement this strategy. Instead, we will take advantage of the vision information to deal with this challenge.

In this paper, a monocular camera is used to measure the position $(x, y)$ and determine the desired target. A key strategy is that the error between the mass centre point of the robot and its desired point in the image frame can be used in the closed-loop feedback control. As for the angle $θ$ , it can be obtained easily from the angle sensor such as compass. Therefore, $θ$ is still included in the error model.

Kinematic and dynamic models with monocular camera

Differentiating (1) with respect to time and using (2), in the image frame, a camera-object visual servoing kinematic model can be obtained

[\begin{matrix} {\overset{\cdot}{x}}_{m} \\ {\overset{\cdot}{y}}_{m} \\ \overset{\cdot}{θ} \end{matrix}] = [\begin{matrix} ν_{1} α_{1} \cos (θ - θ_{0}) \\ ν_{1} α_{2} \sin (θ - θ_{0}) \\ ω \end{matrix}]

(6)

According to (1), the state transformation from the inertial frame to the image frame can be described by

q_{m} = Fq + H,

(7)

where $q_{m} = (x_{m}, y_{m}, θ)^{T}$ , F and H are constant matrices, with

F = [\begin{matrix} α_{1} \cos θ_{0} & α_{1} \sin θ_{0} & 0 \\ - α_{2} \sin θ_{0} & α_{2} \cos θ_{0} & 0 \\ 0 & 0 & 1 \end{matrix}], H = [\begin{matrix} - c_{x} α_{1} \cos θ_{0} - c_{y} α_{1} \sin θ_{0} + O_{c 1} \\ c_{x} α_{2} \sin θ_{0} - c_{y} α_{2} \cos θ_{0} + O_{c 2} \\ 0 \end{matrix}]

According to the transformation (7), the dynamic subsystem (5) can be written as

M_{2} (q_{m}) \overset{\cdot}{v} + C_{2} (q_{m}, {\overset{\cdot}{q}}_{m}) v + {\bar{τ}}_{d} = B_{2} (q_{m}) τ

(8)

where

\begin{matrix} M_{2} (q_{m}) = M_{1} (q {) |}_{q = F^{- 1} (q_{m} - H)} C_{2} (q_{m}, {\overset{\cdot}{q}}_{m}) = C_{1} (q, \overset{\cdot}{q} {) |}_{q = F^{- 1} (q_{m} - H)} B_{2} (q_{m}) = B_{1} (q {) |}_{q = F^{- 1} (q_{m} - H)} \end{matrix}

According to Campion et al. (1996), we know that the coefficients $M, C, B$ of the dynamic equation (4) are only dependent on the angle $θ$ and have nothing to do with x and y. However, the transformation (1) is independent of the angle $θ$ . Again with the nonsingularity of S, the transformed system (8) has the same properties as (4) or (5), and $B_{2}$ is a known invertible matrix. Due to the dynamic characteristics of (4), some interesting properties of the dynamic model (8) are listed below.

Property 1. $M_{2}$ is a positive definite symmetric matrix.

Property 2. ${\overset{\cdot}{M}}_{2} - 2 C_{2}$ is skew-symmetric.

Property 3. For any differentiable vector $ξ$

\begin{matrix} M_{2} (q_{m}) \overset{\cdot}{ξ} + C_{2} (q_{m}, {\overset{\cdot}{q}}_{m}) ξ = Y (q_{m}, {\overset{\cdot}{q}}_{m}, ξ, \overset{\cdot}{ξ}) p \end{matrix}

(9)

where the regressor matrix $Y (q_{m}, {\overset{\cdot}{q}}_{m}, ξ, \overset{\cdot}{ξ})$ is a known matrix of $q_{m}, {\overset{\cdot}{q}}_{m}, ξ$ , and $\overset{\cdot}{ξ}$ , p is an inertia parameter vector of the robot system.

Adaptive controller design

Controller design of the kinematic model

In this paper, we consider the tracking problem when $α_{1}$ and $α_{2}$ are unknown and $θ_{0}$ is known.

Assumption 2. ${\underline{α}}_{1} \leq α_{1} \leq {\bar{α}}_{1}$ , ${\underline{α}}_{2} \leq α_{2} \leq {\bar{α}}_{2}$ , ${\underline{α}}_{1}$ , ${\bar{α}}_{1}$ , ${\underline{α}}_{2}$ and ${\bar{α}}_{2}$ are positive known constants.

In order to discuss the tracking problem of the system (6), we give a desired trajectory $q_{r} (t) = (x_{r} (t), y_{r} (t), θ_{r} (t))^{T}$ , generated by a reference robot whose equation of motion is

{\begin{matrix} {\overset{\cdot}{x}}_{r} & = & ν_{r} α_{1} \cos (θ_{r} - θ_{0}) \\ {\overset{\cdot}{y}}_{r} & = & ν_{r} α_{2} \sin (θ_{r} - θ_{0}) \\ {\overset{\cdot}{θ}}_{r} & = & ω_{r} \end{matrix}

(10)

where $(x_{r}, y_{r})$ is the desired path of the mass centre $(x, y)$ in the image frame, $θ_{r}$ is the desired direction. $ν_{r}$ and $ω_{r}$ are the forward velocity and angular velocity of the reference mobile robot, respectively.

Assumption 3. $ν_{r}, ω_{r}$ and their derivatives are bounded.

The dynamic tracking problem discussed in this paper is defined as finding a feedback control law $τ$ such that

lim_{t \to \infty} (q_{m} - q_{r}) = 0

when the inertia parameter vector of the robot and the camera parameters are unknown.

If $α_{1}, α_{2}$ are known, system (6) can be reduced to a common nonholonomic chained-form system through state and input transformations, its tracking problem can be addressed by lots of methods (Dong, 2012; Jiang and Nijmeijer, 1999). However, when $α_{1}$ and $α_{2}$ are unknown, the existing methods cannot be applied because the state and input transformations can not be implemented. Next we consider the controller design for system (6) with unknown $α_{1}$ and $α_{2}$ .

Denote $e_{1} = x_{m} - x_{r}, e_{2} = y_{m} - y_{r}, e_{3} = θ - θ_{r}$ . By using (6) and (10), we have

\begin{matrix} {\overset{\cdot}{e}}_{1} = ν_{1} α_{1} \cos (θ - θ_{0}) - ν_{r} α_{1} \cos (θ_{r} - θ_{0}) \\ = [ν_{1} α_{1} \cos (θ - θ_{0}) - ν_{r} α_{1} \cos (θ - θ_{0})] \\ + [ν_{r} α_{1} \cos (θ - θ_{0}) - ν_{r} α_{1} \cos (θ_{r} - θ_{0})] \\ = (ν_{1} - ν_{r}) α_{1} \cos (θ - θ_{0}) - 2 ν_{r} α_{1} \sin \frac{e_{3}}{2} \sin (\frac{θ + θ_{r}}{2} - θ_{0}) \\ {\overset{\cdot}{e}}_{2} = ν_{1} α_{2} \sin (θ - θ_{0}) - ν_{r} α_{2} \sin (θ_{r} - θ_{0}) \\ = [ν_{1} α_{2} \sin (θ - θ_{0}) - ν_{r} α_{2} \sin (θ - θ_{0})] \\ + [ν_{r} α_{2} \sin (θ - θ_{0}) - ν_{r} α_{2} \sin (θ_{r} - θ_{0})] \\ = (ν_{1} - ν_{r}) α_{2} \sin (θ - θ_{0}) + 2 ν_{r} α_{2} \sin \frac{e_{3}}{2} \cos (\frac{θ + θ_{r}}{2} - θ_{0}) \\ {\overset{\cdot}{e}}_{3} = ω - ω_{r} \end{matrix}

Then, a new visual feedback kinematics tracking error model can be obtained

{\begin{matrix} {\overset{\cdot}{e}}_{1} = (ν_{1} - ν_{r}) α_{1} \cos (θ - θ_{0}) - 2 ν_{r} α_{1} \sin \frac{e_{3}}{2} \sin (\frac{θ + θ_{r}}{2} - θ_{0}) \\ {\overset{\cdot}{e}}_{2} = (ν_{1} - ν_{r}) α_{2} \sin (θ - θ_{0}) + 2 ν_{r} α_{2} \sin \frac{e_{3}}{2} \cos (\frac{θ + θ_{r}}{2} - θ_{0}) \\ {\overset{\cdot}{e}}_{3} = ω - ω_{r} \end{matrix}

(11)

As a basis of the full dynamic system study, let us first neglect the part of dynamics and consider the kinematic tracking problem of the system (6) only.

Based on (11) and the subsequent closed-loop error system development, a dynamic feedback tracking controller that achieves the error states $e_{1}, e_{2}, e_{3}$ to zero under the assumption

lim_{t \to + \infty} inf_{σ \in [t, + \infty)} | ν_{r} (σ) | = ε

(12)

is given by

{\begin{matrix} \dot{h} = - k_{4} h - e_{1} \cos (θ - θ_{0}) - e_{2} \sin (θ - θ_{0}) \\ v_{d} = [\begin{matrix} ν_{1 d} \\ ω_{d} \end{matrix}] = \\ {\begin{matrix} ν_{r} + h, \\ ω_{r} - k_{3} e_{3} + 2 ν_{r} e_{1} \frac{\sin \frac{e_{3}}{2}}{e_{3}} \sin (\frac{θ + θ_{r}}{2} - θ_{0}) - 2 ν_{r} e_{2} \frac{\sin \frac{e_{3}}{2}}{e_{3}} \cos (\frac{θ + θ_{r}}{2} - θ_{0}), \end{matrix} \end{matrix}

(13)

where $ε$ is a small positive constant, $k_{3}$ and $k_{4}$ denote positive constant control gains. Note that, $\frac{\sin \frac{e_{3}}{2}}{\frac{e_{3}}{2}} = \int_{0}^{1} \cos (s \frac{e_{3}}{2}) ds$ is a smooth function in $\frac{e_{3}}{2}$ .

Remark 2. The tracking controller (13) is independent of uncalibrated visual parameters $α_{1}$ and $α_{2}$ , which is different to other controllers (Jiang and Nijmeijer, 1999; Dong, 2012; Yang et al., 2013).

If only the kinematic model of the mobile robot (2) with velocity input (13) is considered, and assuming ‘perfect velocity tracking’, then the kinematic model is asymptotically stable with respect to the reference trajectory. But the perfect velocity tracking assumption does not hold in practice.

To design the more realistic control law $τ$ and generate the desired velocity $v_{d}$ for the complete dynamic systems (6) and (8), the auxiliary velocity tracking error signal, denoted by $η (t) \in R^{2}$ , is defined as follows

η = [η_{1}, η_{2}]^{T} = v - v_{d} = [ν_{1} - ν_{1 d}, ω - ω_{d}]^{T}

(14)

After substituting (13) and (14) into (11), we have $ν_{1} - ν_{r} = ν_{1 d} + η_{1} - ν_{r} = h + η_{1}$ , $ω - ω_{r} = ω_{d} - ω_{r} + η_{2}$ . Furthermore, the following closed-loop error system is obtained

{\begin{matrix} {\overset{\cdot}{e}}_{1} = (h + η_{1}) α_{1} \cos (θ - θ_{0}) - 2 ν_{r} α_{1} \sin \frac{e_{3}}{2} \sin (\frac{θ + θ_{r}}{2} - θ_{0}) \\ {\overset{\cdot}{e}}_{2} = (h + η_{1}) α_{2} \sin (θ - θ_{0}) + 2 ν_{r} α_{2} \sin \frac{e_{3}}{2} \cos (\frac{θ + θ_{r}}{2} - θ_{0}) \\ {\overset{\cdot}{e}}_{3} = - k_{3} e_{3} + 2 ν_{r} e_{1} \frac{\sin \frac{e_{3}}{2}}{e_{3}} \sin (\frac{θ + θ_{r}}{2} - θ_{0}) \\ - 2 ν_{r} e_{2} \frac{\sin \frac{e_{3}}{2}}{e_{3}} \cos (\frac{θ + θ_{r}}{2} - θ_{0}) + η_{2} \\ \overset{\cdot}{h} = - k_{4} h - e_{1} \cos (θ - θ_{0}) - e_{2} \sin (θ - θ_{0}) . \end{matrix}

(15)

Controller design of the dynamic system

In this subsection, we utilize the dynamic model given by (8) to design an adaptive variable structure torque controller that regulates the kinematic tracking error signals defined in (11) to zero when the camera parameters and robot dynamic parameters are all unknown.

Differentiating (14) and using the formula (8), the robot dynamics using the velocity tracking error can be rewritten as

M_{2} (q_{m}) \overset{\cdot}{η} + C_{2} (q_{m}, {\overset{\cdot}{q}}_{m}) η + Y_{c} p + {\bar{τ}}_{d} = B_{2} (q_{m}) τ

(16)

where $Y_{c} p$ is defined as

Y_{c} p = M_{2} {\overset{\cdot}{v}}_{d} + C_{2} v_{d}

(17)

and $Y_{c} (\cdot) \in R^{2 \times 3}$ denotes the known desired regression matrix, p is defined in (9).

In order to design the final torque tracking controller, firstly, we choose a positive integrable function $δ (t)$ on $[0, + \infty)$ such that

$\int_{0}^{+ \infty} δ (σ) d σ < + \infty$

Remark 3. The examples of time-varying function $δ (t)$ satisfies (18), for instance, can be chosen as exponential function $e^{- t}$ , $e^{- (1 + t)}$ ; power function $\frac{1}{t^{n}}$ , $\frac{1}{{(1 + t)}^{n}}$ , $n > 1$ .

Based on the subsequent stability proof and the regulation of $η (t)$ , we design the torque control input $τ (t)$ as follows

τ (t) = B_{2}^{- 1} \bar{τ}

(19)

where $\bar{τ} \in R^{2}$ is an auxiliary control signal designed as shown in the following

\bar{τ} = Y_{c} \hat{p} - u_{s} - K_{d} η - [\begin{matrix} e_{1} \cos (θ - θ_{0}) + e_{2} \sin (θ - θ_{0}) \\ e_{3} \end{matrix}]

(20)

and

u_{s} = d_{B} \tanh \frac{k_{5} d_{B} η}{δ (t)}

(21)

where $K_{d} = diag (k_{d 1}, k_{d 2})$ , $k_{d 1}$ and $k_{d 2}$ are positive constant control gains, $k_{5} \in R^{1}$ is positive small constant, $δ (t)$ is bounded time-varying positive scalar defined in (18). $\hat{p}$ is the estimation of p, the parameter update law for $\hat{p}$ is designed as follows

\overset{\cdot}{\hat{p}} = - Γ Y_{c}^{T} η

(22)

where $Γ = diag (Γ_{1}, Γ_{2})$ is a positive definite gain matrix.

Remark 4. The time-varying function $δ (t)$ defined in (18) can be explained as a boundary layer of variable structure controller. The difference from the common variable structure controller is that this boundary layer is time-varying.

After utilizing (16), (20), and the definition of $B_{2}$ , we can obtain the following expression of the closed-loop error system for $η (t)$

\begin{matrix} M_{2} (q_{m}) \overset{\cdot}{η} = - C_{2} (q_{m}, {\overset{\cdot}{q}}_{m}) η - Y_{c} \tilde{p} - {\bar{τ}}_{d} - u_{s} - K_{d} η - [\begin{matrix} e_{1} \cos (θ - θ_{0}) + e_{2} \sin (θ - θ_{0}) \\ e_{3} \end{matrix}] \end{matrix}

(23)

where $\tilde{p}$ is the parameter error signal

\tilde{p} = p - \hat{p}

Stability analysis

Before describing the main theorem, the following useful Lemma can be given first:

Lemma 1 (Yang et al., 2013). If continuous functions $f (t)$ and $g (t)$ are boundedness, and

lim_{t \to + \infty} inf_{σ \in [t, + \infty)} | f (σ) | = ϵ

(24)

where $ϵ$ is a small positive constant. Then

lim_{t \to + \infty} f (t) g (t) = 0

implies

lim_{t \to + \infty} g (t) = 0

In the following, we present the main result of our paper.

Theorem 1. Under Assumptions 1–3, the control inputs given in (13), (18)–(22) can guarantee the asymptotic convergence of the states of the closed loop systems defined by (15) and (23) in the sense that

lim_{t \to + \infty} (e_{1}, e_{2}, e_{3}, η_{1}, η_{2}) = 0

provided that

lim_{t \to + \infty} inf_{σ \in [t, + \infty)} | ν_{r} (σ) | = ε

where $ε$ is a small positive constant.

Proof. To prove Theorem 1, we consider a Lyapunov function candidate

\begin{matrix} V = \frac{1}{2 α_{1}} e_{1}^{2} + \frac{1}{2 α_{2}} e_{2}^{2} + \frac{1}{2} e_{3}^{2} + \frac{1}{2} h^{2} + \frac{1}{2} η^{T} M_{2} η + \frac{1}{2} {\tilde{p}}^{T} Γ^{- 1} \tilde{p} \end{matrix}

(25)

After taking the time derivative of (25) and making the appropriate substitutions from (15), (22), (23) and by Property 2, we can conclude that

\begin{matrix} \overset{\cdot}{V} = \frac{1}{α_{1}} e_{1} {\overset{\cdot}{e}}_{1} + \frac{1}{α_{2}} e_{2} {\overset{\cdot}{e}}_{2} + e_{3} {\overset{\cdot}{e}}_{3} + h \overset{\cdot}{h} + \frac{1}{2} η^{T} {\overset{\cdot}{M}}_{2} η \\ + η^{T} M_{2} \overset{\cdot}{η} + {\tilde{p}}^{T} Γ^{- 1} \overset{\cdot}{\tilde{p}} \\ = - k_{4} h^{2} - k_{3} e_{3}^{2} + e_{1} η_{1} \cos (θ - θ_{0}) + e_{2} η_{1} \sin (θ - θ_{0}) \\ + e_{3} η_{2} - k_{d 1} η_{1}^{2} - k_{d 2} η_{2}^{2} \\ + {\tilde{p}}^{T} Γ^{- 1} \overset{\cdot}{\tilde{p}} - η^{T} Y_{c} \tilde{p} - η^{T} {\bar{τ}}_{d} - η^{T} u_{s} - η^{T} \\ [\begin{matrix} e_{1} \cos (θ - θ_{0}) + e_{2} \sin (θ - θ_{0}) \\ e_{3} \end{matrix}] \\ = - k_{4} h^{2} - k_{3} e_{3}^{2} - k_{d 1} η_{1}^{2} - k_{d 2} η_{2}^{2} - η^{T} {\bar{τ}}_{d} - η^{T} u_{s} \end{matrix}

(26)

From a review of the literature (Polycarpou, 1996), one may find that

l ∥ x ∥ - x^{T} l \tanh \frac{lx}{ϕ} \leq k_{5} \sum_{i = 1}^{n} ϕ_{i}

(27)

where $x \in R^{n}, l \in R, ϕ_{i} \in R^{+}$ , $k_{5}$ is a constant that satisfies $k_{5} = e^{- (k_{5} + 1)}$ , and the following notation is used

\tanh \frac{lx}{ϕ} = {[\tanh \frac{l x_{1}}{ϕ_{1}}, \dots, \tanh \frac{l x_{n}}{ϕ_{n}}]}^{T}

By applying (27) to (26), one obtains

- η^{T} {\bar{τ}}_{d} - η^{T} u_{s} \leq ∥ η^{T} ∥ d_{B} - η^{T} d_{B} \tanh \frac{k_{5} d_{B} η}{δ (t)} \leq k_{5} \frac{δ (t)}{k_{5}} + k_{5} \frac{δ (t)}{k_{5}} = 2 δ (t)

(28)

therefore

\begin{matrix} \overset{\cdot}{V} \leq - k_{4} h^{2} - k_{3} e_{3}^{2} - k_{d 1} η_{1}^{2} - k_{d 2} η_{2}^{2} + 2 δ (t) \end{matrix}

(29)

So, when we choose the smooth variable structure control law $u_{s}$ in (21), the inequality (29) is always valid.

From (29), we have

\overset{\cdot}{V} \leq 2 δ (t)

(30)

By integrating both sides of (30), it is seen that

V (t) \leq V (0) + 2 \int_{0}^{t} δ (σ) d σ

(31)

According to the formula (18), we obtain that $V (t)$ is bounded, thus, $h, e_{1}, e_{2}, e_{3}$ , $η_{1}, η_{2}, \tilde{p}$ are all bounded, and $\hat{p}$ is bounded as well. In view of (15), (23) and Assumption 3 ( $ν_{r}, ω_{r}$ are all bounded), we obtain that, ${\overset{\cdot}{e}}_{1}, {\overset{\cdot}{e}}_{2}, {\overset{\cdot}{e}}_{3}, \overset{\cdot}{h}, {\overset{\cdot}{η}}_{1}, {\overset{\cdot}{η}}_{2}$ are all bounded. Hence, $e_{3}, h, η_{1}, η_{2}$ are uniformly continuous.

The inequality (29) implies that

k_{4} h^{2} + k_{3} e_{3}^{2} + k_{d 1} η_{1}^{2} + k_{d 2} η_{2}^{2} \leq - \overset{\cdot}{V} (t) + 2 δ (t)

(32)

By integrating both sides of (32), again with the boundedness of $V (t)$ and the formula (18), we have $e_{3}, h, η_{1}, η_{2} \in L_{2}$ . By using the Barbalat’s lemma (Khalil, 2002), we can obtain

lim_{t \to \infty} e_{3}, h, η_{1}, η_{2} = 0

(33)

Taking the time derivative of ${\overset{\cdot}{e}}_{3}$ , yields

\begin{matrix} {\overset{\cdot\cdot}{e}}_{3} = - k_{3} {\overset{\cdot}{e}}_{3} + (ν_{r} e_{1} \sin (\frac{θ + θ_{r}}{2} - θ_{0}))' \frac{\sin \frac{e_{3}}{2}}{\frac{e_{3}}{2}} \\ + ν_{r} e_{1} \sin (\frac{θ + θ_{r}}{2} - θ_{0}) (\frac{\sin \frac{e_{3}}{2}}{\frac{e_{3}}{2}})' \\ - (ν_{r} e_{2} \cos (\frac{θ + θ_{r}}{2} - θ_{0}))' \frac{\sin \frac{e_{3}}{2}}{\frac{e_{3}}{2}} - ν_{r} e_{2} \\ \cos (\frac{θ + θ_{r}}{2} - θ_{0}) (\frac{\sin \frac{e_{3}}{2}}{\frac{e_{3}}{2}})' + {\overset{\cdot}{η}}_{2} \end{matrix}

(34)

Based on the fact that

(\frac{\sin \frac{e_{3}}{2}}{\frac{e_{3}}{2}})' = \frac{{\overset{\cdot}{e}}_{3}}{2} (\frac{\frac{e_{3}}{2} \cos \frac{e_{3}}{2} - \sin \frac{e_{3}}{2}}{{(\frac{e_{3}}{2})}^{2}})

(35)

and

lim_{e_{3} \to 0} \frac{\frac{e_{3}}{2} \cos \frac{e_{3}}{2} - \sin \frac{e_{3}}{2}}{{(\frac{e_{3}}{2})}^{2}} = 0

we have $(\frac{\sin \frac{e_{3}}{2}}{\frac{e_{3}}{2}})'$ is bounded.

Again with $lim_{e_{3} \to 0} \frac{\sin \frac{e_{3}}{2}}{\frac{e_{3}}{2}} = 1$ and the boundedness of $e_{1}, {\overset{\cdot}{e}}_{1}, e_{2}, {\overset{\cdot}{e}}_{2}, {\overset{\cdot}{e}}_{3}, ν_{r}, {\overset{\cdot}{ν}}_{r}$ , ${\overset{\cdot}{η}}_{2}$ , we can conclude that ${\overset{\cdot\cdot}{e}}_{3}$ is bounded. Then, ${\overset{\cdot}{e}}_{3}$ is uniformly continuous. By using Barbalat’s Lemma (Khalil, 2002), we have $lim_{t \to \infty} {\overset{\cdot}{e}}_{3} = 0$ . Noting that the third equation of system (15), we can obtain

lim_{t \to \infty} ν_{r} (e_{1} \sin (\frac{θ + θ_{r}}{2} - θ_{0}) - e_{2} \cos (\frac{θ + θ_{r}}{2} - θ_{0})) = 0

(36)

Due to the assumption on $ν_{r}$ in Theorem 1 and the conclusion of Lemma 1, it yields

lim_{t \to \infty} (e_{1} \sin (\frac{θ + θ_{r}}{2} - θ_{0}) - e_{2} \cos (\frac{θ + θ_{r}}{2} - θ_{0})) = 0

(37)

Furthermore, by using the fact that $e_{3} = θ - θ_{r}$ tends to zero, we have

lim_{t \to \infty} (e_{1} \sin (θ - θ_{0}) - e_{2} \cos (θ - θ_{0})) = 0

(38)

Applying the extended Barbalat’s Lemma (Khalil, 2002) to the fourth equation of system (15), yields

lim_{t \to \infty} (e_{1} \cos (θ - θ_{0}) + e_{2} \sin (θ - θ_{0})) = 0

The result can be used in conjunction with (38); we have

lim_{t \to \infty} [\begin{matrix} \sin (θ - θ_{0}) & - \cos (θ - θ_{0}) \\ \cos (θ - θ_{0}) & \sin (θ - θ_{0}) \end{matrix}] \cdot [\begin{matrix} e_{1} \\ e_{2} \end{matrix}] = 0

(39)

According to the nonsingularity of the matrix

[\begin{matrix} \sin (θ - θ_{0}) & - \cos (θ - θ_{0}) \\ \cos (θ - θ_{0}) & \sin (θ - θ_{0}) \end{matrix}]

(40)

we can obtain the conclusion

lim_{t \to \infty} e_{1}, e_{2} = 0

Therefore, we have proved that $e_{1}, e_{2}, e_{3}, η_{1}, η_{2}$ tend to zero and $\hat{p}$ is bounded.

Remark 5. The proposed adaptive robust algorithm can guarantee the asymptotical stability of the system. The bounded disturbance can be completely suppressed by using a smoothed sliding mode control law (21) taking into account the boundary layer (18), the chattering problem is overcome.

Simulation and experiment results

Simulation results

For the mobile robot system considered here, according to (Fierro and Lewis, 1995), the dynamic equation parameters of (4) can be given

M (q) = [\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I \end{matrix}], C (q, \overset{\cdot}{q}) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

B (q) = \frac{1}{r} [\begin{matrix} \cos θ & \cos θ \\ \sin θ & \sin θ & - L \end{matrix}], A^{T} (q) = [\begin{matrix} \sin θ \\ - \cos θ \\ 0 \end{matrix}]

where m is the mass of the mobile robot and I being its inertia moment around the vertical axis at point P.

According to the transformation of the section on problem formulation, we have

M_{2} (q_{m}) = [\begin{matrix} m & 0 \\ 0 & I \end{matrix}], C_{2} (q_{m}, {\dot{q}}_{m}) = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}], B_{2} (q_{m}) = \frac{1}{r} [\begin{matrix} 1 & 1 \\ L & - L \end{matrix}]

The robot regressor $Y_{c}$ and the inertia parameter vector p can be selected as $Y_{c} = diag ({\overset{\cdot}{ν}}_{1 d}, {\overset{\cdot}{ω}}_{d})$ , $p = diag (m, I)$ .

The simulation is implemented for the controllers defined in (13) and (20). The corresponding closed loop systems are written by (15) and (23). Take $α_{1} = 1$ , $α_{2} = 2$ , $θ_{0} = 0$ rad, $m = 10$ kg, $I = 5 kg \cdot m^{2}$ , $L = 1$ m, $r = 0.1$ m. The desired velocities are chosen as $ν_{r} = 1$ m/s, $ω_{r} = 0.3$ rad/s. The reference trajectory starts from $[2, 0, 0.5]$ , and the actual robot initial posture is taken as $[1, 2, - 1.5]$ . The parameters of the controller are chosen as $k_{3} = 2, k_{4} = 2, k_{5} = 0.2785$ , $k_{d 1} = k_{d 2} = 20$ , $d_{B} = 2$ , and the gain $Γ = diag (Γ_{1}, Γ_{2}) = diag (1, 1)$ . The external disturbance vector is taken as ${\bar{τ}}_{d} = (\sin (t), \cos (t))^{T}$ , and the boundary layer is $δ (t) = \frac{1}{{(1 + t)}^{3}}$ .

The simulation is implemented and the results are shown in Figures 2 –7, respectively. From Figures 2 –4, we can see that the posture errors $e_{1}, e_{2}, e_{3}$ and velocity errors $η_{1}, η_{2}$ tend to zero asymptotically. From Figure 5, the estimate of the parameter $\hat{p}$ is bounded. Figure 6 demonstrates the good tracking performance of actual robot with respect to the desired trajectory. As shown in Figure 7, the control inputs $τ_{1}$ and $τ_{2}$ are tend to zero as well, and the common chattering problem is overcome. The effectiveness of the proposed controller is verified by the simulation results.

In the simulation, as long as the controller parameters $k_{3}, k_{4}, Γ_{1}, Γ_{2}, k_{d 1}, k_{d 2}$ are chosen positive, $k_{5} = 0.2785$ , the errors are convergent. Good positive gains $k_{3}, k_{4}$ will be chosen in the range 0.1 ~ 20, the positive gains $Γ_{1}, Γ_{2}, k_{d 1}, k_{d 2}$ can be chosen in the range 1–100, outside this range, the speed of convergence will be slower and the simulation time will be longer. When one of $k_{3}, k_{4}, Γ_{1}, Γ_{2}, k_{d 1}, k_{d 2}$ is chosen negative, the convergence of the errors could not be guaranteed. Generally, there is little effect caused by the change of initial value of the tracking error.

Experiment results

In this section, an experiment is proposed to illustrate the theoretical results. As is displayed by Figure 8, the experimental setup is composed by 1) a pinhole camera fixed to the ceiling, and 2) a type (2,0) MT-R mobile robot. Two pieces of red rectangular marks are pasted on the top of the mobile robot. The larger mark is denoted by Goal B, and is used to describe the position of the mobile robot. It is supposed that the centre of Goal B can represent the mass centre of the mobile robot. The smaller rectangular mark is used to calculate the orientation of the mobile robot, and can be denoted by Goal C.

In the experiment, the reference forward velocity and reference angular velocity are denoted by $ν_{r}$ and $ω_{r}$ , respectively; they are obtained by using the coordinates of goal B and goal C, respectively. The bounds of $ν_{r}$ and $ω_{r}$ are based on the maximum velocity of the mobile robot and the processing time of the upper computer, and it mainly limited to the processing time of the upper computer. The forward velocity and angular velocity control input of the actual robot are given by the controller (13) and conducted appropriate saturated processing.

The initial kinematic errors are taken as $e_{1} = 28.75, e_{2} = 35.18, e_{3} = 2.67$ , the controller parameters are chosen as $k_{3} = k_{4} = 2$ , $ν_{r} = 0.1$ m/s, $ω_{r} = 0.02$ rad/s, the diameter of two standard wheels is $0.2$ m, the distance between two motorized standard wheels is 0.4 m, and the constant torque is 25.5 Nm.

Experimental results are shown in Figures 9 –12. As can be seen from Figures 9 and 10, with the proposed controller, the position of the mobile robot is capable of converging to the desired trajectory after a transient process of 700 frames. In Figure 11, a peak of the angle error appeared between 1250 and 1350 frames. This is because when the robot moved into the third quadrant, the change of position as well as the lighting effect resulted in some missing and discontinued images. When the robot across this area, its orientation gradually coincides with the desired trajectory. It can be illustrated in Figure 12 that, the tracking error of position is smaller than five pixels, and the tracking error of orientation is smaller than $5^{\circ}$ , indicating that performances of the proposed trajectory tracking controller are satisfactory.

Conclusions

A new adaptive variable structure torque controller has been proposed for WMR with unknown camera parameters, dynamic inertial parameters, and bounded disturbances. The global asymptotic tracking of the actual robot with respect to the reference robot has been achieved by using the designed control input. The closed-loop system stability and estimated error boundedness were proved by Lyapunov stability theory. Simulation and experiment results were presented to illustrate the performance of the proposed controller. Due to the limitation of laboratory equipment, the experiment of kinematic tracking control was implemented only, the dynamic system experiment results will be given in the future.

Footnotes

Conflict of Interest Statement

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: This paper was supported by the Natural Science Foundation (grant numbers 61503205 and 61374040), the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (grant number ICT1501), and the Ningbo Science and Technology Plan Project (grant number 2014C50052).

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