Adaptive fuzzy-based motion generation and control of mobile under-actuated manipulators

doi:10.1016/j.engappai.2013.12.013

Engineering Applications of Artificial Intelligence

Volume 30, April 2014, Pages 86-95

https://doi.org/10.1016/j.engappai.2013.12.013 Get rights and content

Abstract

In this paper, adaptive fuzzy-based motion generation and control are investigated for nonholonomic mobile manipulators with an under-actuated dyanmics model, in the presence of parametric and functional uncertainties. It is well known that the constraints of this kind of system consist of kinematic constraints for the mobile platform and dynamic constraints for the under-actuated manipulator with a passive joint. Through using dynamic coupling property of nonholonomic mobile manipulators, we can decouple the dynamics into a fully actuated subsystem and an unactuated subsystem. Then adaptive control is employed for the fully actuated subsystem using fuzzy logic approximation. Since the non-actuated subsystem cannot be directly manipulated by torque inputs but can be indirectly affected by the motion of the actuated subsystem, the reference trajectory of the actuated subsystem is planned by the fuzzy logic system based motion generator. Rigorous theoretic analysis has been established to show that the proposed trajectory generation and control are able to achieve dynamic stability, motion tracking and optimized dynamics. Simulation studies have further validated the efficiency of the developed scheme.

Introduction

Mobile manipulators with the full-actuated joints have been extensively investigated (Lin and Goldenberg, 2001, Li et al., 2007, Li and Chen, 2008). Mobile under-actuated manipulators first appeared in Li et al. (2006), by simultaneously integrating both kinematic constraints and dynamic constraints. Moreover, they are also different from mobile wheeled inverted pendulums (Li and Luo, 2009) or pendulums (Zhang and Tarn, 2002), whose dynamic balances in the vertical plane are achieved due to the gravity.

In robotics (Hsu, 2013, Du et al., 2013), nonholonomic constraints are formulated as non-integrable differential equations containing time-derivatives of generalized coordinates (velocity, acceleration, etc.) (Oya et al., 2003). Due to Brockett's theorem (Brockett, 1983), it is well known that nonholonomic systems with restricted mobility cannot be stabilized to a desired configuration nor posture-via differentiable, or even continuous, pure state feedback. In general, such nonholonomic constraints include (i) only kinematic constraints which geometrically restrict the direction of mobility, i.e., wheeled mobile robots (Su and Stepanenko, 1994); (ii) only dynamic constraints due to dynamic balance at passive degrees of freedom where no force or torque is applied, i.e., the manipulators with passive link (Arai and Tanie, 1998, De Luca and Oriolo, 2002, Bergerman et al., 1995); (iii) not only kinematic constraints but also dynamic constraints (Li et al., 2006), such as mobile underactuated manipulators investigated in this paper. The common features of these systems are governed by under-actuated configuration, i.e., the number of control inputs is less than the number of degrees of freedom, which makes it difficult to apply the conventional robotics approaches for controlling Euler–Lagrange systems. For a mobile under-actuated manipulator, there exist not only kinematic constraints but also dynamic constraints. For the nonholonomic constraints, there exists a first-order non-integrable differential equation in a Pfaffian form, $A (q) \dot{q} = 0$ , where A(q) denotes nonholonomic constraints, and q and $\dot{q}$ are the generalized coordinate vector and the velocity vector, respectively. For dynamic constraints, there exist a class of dynamic constraints formulated as a second-order differential equation, for example, $M (q) \ddot{q} + V (q, \dot{q}) = 0$ with the inertia matrix M(q) and the Centripetal and Coriolis matrix $V (q, \dot{q})$ , which includes the generalized acceleration $\ddot{q}$ and cannot be transformed into a Pfaffian form. These constraints are called second-order nonholonomic constraints. The zero torques of the under-actuated joints result in second-order nonholonomic constraints (Arai and Tanie, 1998, De Luca and Oriolo, 2002, Spong, 1995, Bergerman et al., 1995).

For mobile under-actuated manipulators, the development of intelligent control approach is important since the dynamics uncertainties cause unknown coupling among the joints. The mobile under-actuated manipulator is a strong dynamic coupled nonlinear system, and there exists strong dynamic coupling between the mobile platform and the manipulator, which is beyond the fixed manipulators. In the previous works, the adaptive model reference control of robots has been extensively investigated. In Klancar and Skrjanc (2007), a model-predictive trajectory-tracking control was applied to a mobile robot, where linearized tracking-error dynamics are used to predict future system behavior and a control law is derived from a quadratic cost function with the system tracking error and the control effort. In Li et al. (2007), a model reference adaptive control design was proposed for mobile manipulators. In Takubo et al. (2002), the cooperation between mobile manipulator and human was developed using an impedance model. But it is noted that they require the precise knowledge of the robot structure, which is often difficult to obtain in practice. To solve this problem, several methods that require little dynamics information are proposed in Ge et al. (1998) and Li et al. (2012a).

Recently, fuzzy logic control has found extensive applications for complex and ill-defined systems, especially in the presence of incomplete knowledge of the plant or the situation where precise control action is unavailable. Based on the universal approximation theorem (Zhao and Gao, 2012), stable fuzzy schemes have been developed for unknown single-input and single-output (SISO) nonlinear systems (Li et al., 2012b, Xia et al., 2011, Liu et al., 2009), and multiple-input and multiple-output (MIMO) nonlinear systems (Chang and Chen, 2000, Liu et al., 2010, Liu et al., 2011), and achieve stable performance criteria.

Pioneered by Hogan (1985), impedance control has been embedded in extensive research on interaction control, which does not attempt to track the motion or force trajectory but rather to regulate the mechanical impedance specified by a target model. Based on our previous work (Li et al., 2006; Yang et al., 2013) and considering unknown dynamics, in this paper, we first break down nonlinear mobile underactuated manipulators dynamics into the fully actuated dynamic subsystem and the zero dynamic subsystem. Aiming at minimizing optimal motion tracking errors and state variable accelerations, we employ the linear quadratic regulation (LQR) optimization technique to obtain an optimal reference model. The property of the optimized mass-spring-damper reference model guarantee smooth velocities. In addition, due to the under-actuated mechanism of mobile underactuated manipulators, the zero dynamics cannot be independently controlled. In our previous works (Li et al., 2010), the vehicle forward velocity dynamics are regarded as zero dynamics. Zero dynamics theories have been used to analyze the stability. As the vehicle forward velocity dynamics are directly affected by the under-actuated joint dynamics, in this paper, we manipulate the forward velocity by using the reference trajectory of the under-actuated joint angle as a virtual “controller”. A fuzzy logic system based reference trajectory generator has been designed for the zero dynamics, such that the vehicle forward velocity can be indirectly manipulated to follow the desired forward velocity.

The main contributions of this paper lie in:

(i)
a reference model for the actuated subsystems of the mobile underactuated manipulators system using the LQR optimization approach is employed for motion tracking and improves smoothness of the tracking performance;
(ii)
sliding mode has been utilized to design the adaptive fuzzy control in order to make the controlled dynamics to match the reference model dynamics in finite time;
(iii)
instead of leaving the unactuated dynamics uncontrolled, the reference trajectory for the under-actuated joint angle is designed to indirectly affect the forward velocity such that the desired tracking can be achieved.

Section snippets

Preliminaries

For a linear time-invariant (LTI) system $\dot{x} (t) = Ax (t)$ , the system is stable if and only if there exists a symmetric positive definite solution P_L to the Lyapunov equation $A^{T} P_{L} + P_{L} A = - Q_{L}$ , where Q_L is an arbitrary symmetric positive-definite matrix.

Lemma 2.1

Anderson, 1990

Given a linear system with completely stabilizable pair $[A, B]$ , $\dot{x} = Ax + Bu$ , $x (t_{0}) = x_{0}$ , $x \in R^{n}, u \in R^{n}$ , the optimal control $u^{⁎} (t)$ , $t > 0$ , that minimizes the following performance index $J = \int_{t_{0}}^{t_{f}} (x^{T} Qx + u^{T} Ru) dt, R = R^{T} > 0, Q = Q^{T} \geq 0$ , is given by $u^{⁎} = - R^{- 1} B^{T} Px$ , where P is the

Dynamics

Consider an n DOF fixed manipulator mounted on a two-wheeled driven mobile platform, which can be subdivided into the active, passive and vehicles joints, the dynamics can be described as $M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + d (t) = B (q) τ + f$ where $M (q) = [\begin{matrix} M_{v} & M_{va} & M_{vp} \\ M_{av} & M_{a} & M_{ap} \\ M_{pv} & M_{pa} & M_{p} \end{matrix}], G (q) = [\begin{matrix} G_{v} \\ G_{a} \\ G_{p} \end{matrix}], C (q, \dot{q}) = [\begin{matrix} C_{v} & C_{va} & C_{vp} \\ C_{av} & C_{a} & C_{ap} \\ C_{pv} & C_{pa} & C_{p} \end{matrix}],$ $J^{T} = {[J_{v}^{T} 0 0]}^{T}$ , $d (t) = [d_{v}^{T} d_{a}^{T} d_{p}^{T}]$ , $B (q) τ = {[τ_{v}^{T} τ_{a}^{T} 0]}^{T}$ , $f = J^{T} λ$ , q are the generalized coordinates for the mobile manipulators with $q = {[q_{v}^{T}, q_{a}^{T}, q_{p}^{T}]}^{T} \in R^{n}$ , q_a are the coordinates of the active joints

Functional universal approximation using FLSs

Consider an n-inputs, single-output fuzzy logic system (Liu et al., 2010) with the product-inference rule, singleton fuzzifier, center average defuzzifier, and Gaussian membership function given by m fuzzy IF–THEN rules $R^{(j)} : IF x_{1} is A_{1}^{j} and \dots and x_{n} is A_{n}^{j} THEN y is W^{j}, j = 1, \dots, m$ , where R^j denotes the jth rule, $1 \leq j \leq m$ , ${(x_{1}, x_{2}, \dots, x_{n})}^{T} \in U \subset R^{n}$ and $y \in R$ are the linguistic variables associated with the inputs and output of the fuzzy logic system, respectively, A_i^j and W^j denote the fuzzy sets in U and R. The fuzzy

Controller structure

Define the trajectories ξ_r and ${\dot{ξ}}_{r}$ as ${\dot{ξ}}_{r} = {\dot{ξ}}_{d} - Λ e$ ${\ddot{ξ}}_{r} = {\ddot{ξ}}_{d} - Λ \dot{e}$

Since $\dot{ξ} = {\dot{ξ}}_{r} + r$ , $\ddot{ξ} = {\ddot{ξ}}_{r} + \dot{r}$ , considering the decomposition (19), Eq. (18) becomes $M_{1} \dot{r} + {\tilde{C}}_{1} r = - M_{1} {\ddot{ξ}}_{r} - C_{1} {\dot{ξ}}_{r} - {\hat{C}}_{1} r - D_{1} + B_{1} Λ_{1} U_{1}$ where $r = {[r_{3}^{T}, r_{2}^{T}]}^{T}$ , ${\ddot{ξ}}_{r} = {[{\ddot{ζ}}_{3 r}^{T}, {\ddot{ζ}}_{2 r}^{T}]}^{T}$ .

Let $M_{0}$ , $C_{0}$ , ${\hat{C}}_{0}$ , $D_{0}$ , and $B_{1}^{0}$ be nominal parameter vectors which give the corresponding nominal function $M_{0} {\ddot{ξ}}_{r} + C_{0} {\dot{ξ}}_{r} + {\hat{C}}_{0} r + D_{0}$ and ${(B_{1}^{0})}^{- 1}$ , respectively. The unknown continuous function $M_{1} {\ddot{ξ}}_{r} + C_{1} {\dot{ξ}}_{r} + {\hat{C}}_{1} r + D_{1}$ in (44) can be approximated by FLSs to any arbitrary accuracy as $M_{1} {\ddot{ξ}}_{r} + C_{1} {\dot{ξ}}_{r} + {\hat{C}}_{1} r + D_{1} =$

Motion generation for indirectly controlling ζ₁ subsystem

We see that r=0 and subsequently w=0 for $t > t_{z}$ have been guaranteed by the adaptive controller developed in the previous section. From the definition of w in (33), the solution of K_d and C_d in (31), (32), it is clear that $\ddot{e} = 0$ for $t > t_{f}$ , i.e., $\ddot{ξ} = {\ddot{ξ}}_{d}$ . The ζ₁-subsystem (21) can be rewritten as ${\dot{φ}}_{1} = φ_{2} {\dot{φ}}_{2} = f (φ, ξ_{d}, {\dot{ξ}}_{d}, μ, ν)$ where it is assumed that ξ has converged to ξ_d, $φ = {[φ_{1}^{T}, φ_{2}^{T}]}^{T} = {[ζ_{1}^{T}, {\dot{ζ}}_{1}^{T}]}^{T}$ , $μ = {\ddot{ζ}}_{3 d}$ , $ν = {\ddot{ζ}}_{2 d}$ .

Consider the desired position and velocity of the first generalized coordinate ζ₁ as $φ_{d} = [ζ_{1 d}^{T}, ζ$

Simulation

The overall control system scheme combining both adaptive controller and fuzzy-based motion generation is shown in Fig. 3. For the given ζ_2d and ζ_3d, we can obtain optimal reference model (23). Using the filter (37), we can construct the controller (47). Through the augmented motion subsystem (18), we can produce the actual ζ₂ and ζ₃. For the given ζ_1d, we use trajectory generation (72) to produce the desired ζ_2d, the error between the actual ζ₁ and the desired ζ_1d is used to update the fuzzy

Conclusion

In this paper, adaptive fuzzy-based motion generation and control are investigated for nonholonomic mobile manipulators with an under-actuated joint, in the presence of parametric and functional uncertainties. Although in the previous works, nonlinear approximator such as NN and SVM was used to emulate the system dynamics and the unactuated subsystem is usually treated as zero dynamics, in this work, adaptive control has been employed for the fully actuated subsystem using fuzzy logic

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^☆: This work is supported in part by the Foundation of Key Laboratory of Autonomous Systems and Networked Control (Chinese Ministry of Education) Grant nos. 2012A04, 2013A04, the National Natural Science Foundation of China Grants 61174045, U1201244, the Fundamental Research Funds for the Central Universities under Grant 2013ZG0035, the Program for New Century Excellent Talents in University under Grant NCET-12-0195, and the Ph.D. Programs Foundation of Ministry of Education of China under Grant 20130172110026.

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Adaptive fuzzy-based motion generation and control of mobile under-actuated manipulators☆

Abstract

Introduction

Section snippets

Preliminaries

Anderson, 1990

Dynamics

Functional universal approximation using FLSs

Controller structure

Motion generation for indirectly controlling ζ1 subsystem

Simulation

Conclusion

Eng. Appl. Artif. Intell.

Robot. Auton. Syst.

Eng. Appl. Artif. Intell.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Nonholonomic control of a three-DOF planar underactuted manipulator

IEEE Trans. Robot. Autom.

Optimal Control

A dynamic coupling index for underactuated manipulators

J. Robot. Syst.

Asymptotic stability and feedback stabilization

Robust tracking designs for both holonomic and nonholonomic constrained mechanical systemsadaptive fuzzy approach

IEEE Trans. Fuzzy Syst.