International Journal of Automation and Computing  2018, Vol. 15 Issue (6): 728-735   PDF    
Adaptive Tracking Control of Mobile Manipulators with affine Constraints and Under-actuated Joint
Wei Sun1, Wen-Xing Yuan1, Yu-Qiang Wu2     
1 School of Mathematics Science, Liaocheng University, Liaocheng 252000, China;
2 Institute of Automation, Qufu Normal University, Qufu 273165, China
Abstract: Adaptive motion/force tracking control is considered for a class of mobile manipulators with affine constraints and under-actuated joints in the presence of uncertainties in this paper.Dynamic equation of mobile manipulator is transformed into a controllable form based on dynamic coupling technique.In view of the asymptotic tracking idea and adaptive theory, adaptive controllers are proposed to achieve the desired control objective. Detailed simulation results confirm the validity of the control strategy.
Key words: Tracking control     affine constraints     mobile manipulators     under-actuated joints     dynamic coupling    
1 Introduction

Due to the higher performances possessed by mobile manipulator, such as much larger work space, the better kinematic flexibility beyond that of the traditional one, considerable efforts[1-4] have been made to guarantee stability and robustness for it. However, control design for this mechanical system is still a challenging problem owing to complex and strongly coupled dynamics of the mobile platform and the robotic arm.

The motion and force tracking control for mobile manipulators have been systematically investigated in literatures[5-9] by state-feedback, output-feedback and neural network, etc. However, most researches have been done to investigate mobile manipulators with the full-actuated joints. The under-actuation in any joint of mobile manipulators may occur in the full-actuated manipulators, and the effective control of under-actuated robotic system could enhance the fault-tolerance if the actuator fails. For these reasons, great efforts[10-12] have been made to design controllers for the under-actuated mobile manipulators.

A new mobile manipulator shown in Fig. 1, is made up of a multi-link manipulator with under-actuated joints and a boat, and is subjected to affine constraints[13-16]. The traditional control methods are hardly applicable to such mechanical systems. Therefore, investigating tracking control problems for such mobile manipulator has theoretical and practical meanings. Considering the mentioned problems, this paper considers the tracking control of the affine constraint mobile manipulators with under-actuated joints, and addresses mathematical modeling, algorithm design and theory analysis for practical mechanical systems. Using the technique of dynamic coupling, dynamic equation of mobile manipulator is transformed into a controllable form. By constructing an appropriate upper bound parameter, adaptive control design becomes much easier, and only one parameter updating law is needed. Hence, the dynamic order of adaptive controller is reduced to be minimum.

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Fig. 1. Manipulators mounted on a boat

2 System description 2.1 Dynamic model

Consider an $n$-DOF mobile manipulator shown in Fig. 1. It consists of a mobile boat and a multi-link manipulator. According to the kinematic analysis of the boat on a running river[5], the affine constraints can be written as

$ \begin{align} J_a(q_b)\dot q_b=A(q_b) \end{align} $ (1)

where $q_b\in {\bf R}^3=[q_{b_1}, q_{b_2}, q_{b_3}]^{\rm T}$ is the coordinates of the boat, $J_a(q_b)=[\cos q_{b_3}, -\sin q_{b_3}, 0]$ and $A(q_b)=C(q_{b_2})\cos q_{b_3}$. The affine constraint forces are given by

$ \begin{align} f_a=J_a^{\rm T}(q_b) l_a \end{align} $ (2)

where $l_a\in {\bf R}^{m}$ is a Lagrangian multiplier corresponding to $m$ affine constraints.

The manipulator is a series-chain multi-link manipulator, $q_a\in {\bf R}^{n_a}$ and $q_u\in {\bf R}^{n_u}$ are the coordinates of the active and under-actuated joints of the manipulator, respectively. For convenience, let $q = [q_b^{\rm T}, q_a^{\rm T}, q_u^{\rm T}]^{\rm T}\in {\bf R}^n$ be the vector of generalized coordinates of the whole system.

According to Euler-lagrangian formulation, after considering the affine constraints, dynamic equations of the mobile manipulator are described by

$ \begin{align} M(q)\ddot q+C(q, \dot q)\dot q+G(q)+F=B(q)\tau+f \end{align} $ (3)

with

$ \begin{align*} &M(q)=\left[ \begin{array}{ccc} M_b&M_{ba}&M_{bu} \\ M_{ba}&M_a&M_{au} \\ M_{bu}&M_{au}&M_{u} \\ \end{array} \right], ~B(q)\tau=\left[ \begin{array}{c} B_b \tau_b\\ \tau_a \\ 0\\ \end{array} \right]\\ &C(q, \dot q)=\left[ \begin{array}{ccc} C_b&C_{ba}&C_{bu} \\ C_{ba}&C_a&C_{au} \\ C_{bu}&C_{au}&C_{u} \\ \end{array} \right], ~ f=\left[ \begin{array}{c} f_a \\ 0 \\ 0\\ \end{array} \right] \end{align*} $

where $M_b$, $M_a$ and $M_u$ describe the inertia matrices for the mobile boat and the active links and the passive links, respectively. $M_{ba}$, $M_{bu}$ $M_{au}$ are the coupling inertia matrices of the boat, the active links and the passive links, respectively. $C_b$, $C_a$ and $C_u$ denote the Centripetal and Coriolis torques for the boat, the active links and the passive links. $C_{ba}$, $C_{bu}$ and $C_{au}$ are the coupling Centripetal and Coriolis torques of the boat, the active links and the passive links, respectively. $G(q)=[G_b, G_a, G_u]^{\rm T}$ is the gravitational torque vector. $B_b$, as input transformation matrix of the boat, is assumed to be known because it is a function of fixed geometry of the system. $\tau_b$ and $\tau_a$ are the control input vectors for the mobile boat and the active links, $F=[F_b, F_a, F_u]^{\rm T}$ denotes the external forces.

2.2 State transformation

For the affine constraints (1), according to our previous results [13, 14], there exists a known full-rank matrix $S(q_b)\in {\bf R}^{n_b\times (n_b-m)}$ satisfying

$ \begin{align} \dot{q_b}=S(q_b)\dot z+\eta(q_b) \end{align} $ (4)

where $z$ corresponds to the internal state variable of $q_b$, $S(q_b)$ and $\eta(q_b)$ satisfying $J_a(q_b)S(q_b)= 0$ and $J_a(q_b)\eta(q_b)= A(q_b)$, respectively. As mentioned by Wang et al.[17], the internal states $z(q)$ and $\dot z(q)$ possess practical physical meanings and $z(q)$ can be considered as $(n-m)$ "output equations" of original system. Substituting (4) into (3) gives

$ \begin{align} MH\ddot\xi+(M\dot H+CH)\dot \xi+G+F=\tau+J^{\rm T}l_a \end{align} $ (5)

with

$ \begin{align} &\xi=\left[ \begin{array}{c} z \\ q_a \\ q_u \\ \end{array} \right], ~ H=\left[ \begin{array}{ccc} S&0&0 \\ 0&I&0 \\ 0&0&I \\ \end{array} \right], ~F=\left[ \begin{array}{c} F_b \\ F_a \\ F_u \\ \end{array} \right]\notag\\ &G=\left[ \begin{array}{c} M_b\dot \eta +C_b\eta+G_b \\ M_{ba}\dot \eta+C_{ba}\eta+G_a \\ M_{bu}\dot\eta+C_{bu}\eta+G_u \\ \end{array} \right], ~ \tau=\left[ \begin{array}{c} B_b\tau_b \\ \tau_a \\ 0 \\ \end{array} \right]. \end{align} $ (6)

If considering the control input $\tau$ in the form

$ \begin{align} \tau=\hat{\tau} + J^{\rm T}\tau_{0} \end{align} $ (7)

and pre-multiplying $H^{\rm T}(q)$ on both sides of (5), and noting $J(q)H(q)=0$, one can obtain

$ \begin{align} \bar M\ddot\xi+\bar C\dot \xi+\bar G+\bar F=\bar \tau \end{align} $ (8)

where $\bar M=H^{\rm T}MH$ is symmetric and positive definite, $\bar C=H^{\rm T}(M\dot H+CH)$, $\bar G=H^{\rm T}G$, $\bar F=H^{\rm T}F$, $\bar \tau=H^{\rm T}\hat\tau$, and the force multipliers can be obtained by (5)

$ \begin{align} l_a=J^{*}\left( (M\dot H+CH)\dot \xi+ G+F-\hat\tau\right)-\tau_{0} \end{align} $ (9)

where $J^{*}=\left( J{M}^{-1} J^{\rm T}\right)^{-1} J M^{-1}$.

2.3 Dynamic coupling

The state variable of mobile manipulators is partitioned in quantities related to the active joints, the passive joints, and the remaining joints as $\xi_1$, $\xi_3$, and $\xi_2$, respectively, such that the dimension of $\xi_1$ and $\xi_3$ are equal. According to these partitions, we have the partition structure for (8) as

$ \begin{align} &\bar M(q)=\left[ \begin{array}{ccc} M_{11}&M_{12}&M_{13} \\ M_{21}&M_{22}&M_{23} \\ M_{31}&M_{32}&M_{33} \\ \end{array} \right]\notag\\ &\bar C(q, \dot q)\xi=\left[ \begin{array}{c} C_{1} \\ C_{2} \\ C_{3} \\ \end{array} \right]=\left[ \begin{array}{c} C_{11}\dot \xi_1 + C_{12}\dot \xi_2 + C_{13} \dot \xi_3 \\ C_{21} \dot \xi_1 + C_{22}\dot \xi_2+ C_{23}\dot \xi_3 \\ C_{31} \dot \xi_1 + C_{32} \dot \xi_2 + C_{33}\dot \xi_3 \\ \end{array} \right]\notag\\ &\bar G(q)=\left[ \begin{array}{c} G_1 \\ G_2 \\ G_3 \\ \end{array} \right], \quad F=\left[ \begin{array}{c} F_1\\ F_2 \\ F_3\\ \end{array} \right], \quad \tau=\left[ \begin{array}{c} \tau_1\\ \tau_2 \\ 0\\ \end{array} \right]. \end{align} $ (10)

In order to make $\xi_3$ controllable, we assume that matrices $M_{13}$ and $M_{31}$ are not equal to 0 and $M^{-1}_{11}$ exists. Considering the new partition in (10), after some manipulations, one obtains the following dynamic equations

$ M_{11}\ddot \xi_1+M_{12}\ddot \xi_2+M_{13}\ddot \xi_3+C_1+G_1+F_1=\tau_1 $ (11)
$ \Gamma_{11}\ddot \xi_2+\Gamma_{12}\ddot \xi_3+\Upsilon_1+\Pi_1=\tau_2-M_{21}M_{11}^{-1}\tau_1 $ (12)
$ \Gamma_{21}\ddot \xi_2+\Gamma_{22}\ddot \xi_3+\Upsilon_2+\Pi_2=-M_{31}M_{11}^{-1}\tau_1 $ (13)

where

$ \begin{align*} &\Gamma_{11}\!=\!M_{22}\!-\!M_{21}M_{11}^{-1}M_{12}\\ &\Gamma_{12}\!=\!M_{23}\!-\!M_{21}M_{11}^{-1}M_{13}\\ &\Gamma_{21}\!=\!M_{32}\!-\!M_{31}M_{11}^{-1}M_{12}\\ &\Gamma_{22}\!=\!M_{33}\!-\!M_{31}M_{11}^{-1}M_{13}\\ &\Upsilon_1\!=\!(C_{22}\!-\!M_{21}M_{11}^{-1}C_{12})\dot\xi_2+(C_{22}-M_{21}M_{11}^{-1}C_{13})\dot\xi_3\\ &\Upsilon_2\!=\!(C_{32}\!-\!M_{31}M_{11}^{-1}C_{12})\dot\xi_2+(C_{33}-M_{31}M_{11}^{-1}C_{13})\dot\xi_3\\ &\Pi_1\!=\!(C_{21}\!-\!M_{21}M_{11}^{-1}C_{11})\dot\xi_1+G_2+F_2-M_{21}M_{11}^{-1}(G_1+F_1)\\ &\Pi_2\!=\!(C_{31}\!-\!M_{31}M_{11}^{-1}C_{11})\dot\xi_1+G_3+F_3-M_{31}M_{11}^{-1}(G_1+F_1). \end{align*} $

Let $y=[\xi_3^{\rm T}, \xi_2^{\rm T}]^{\rm T}$, we can rewrite (12) and (13) as

$ \begin{align} M_1(\xi)\ddot y+C_1(\xi, \dot \xi)\dot y +G_1+F_1=B_1u \end{align} $ (14)

where

$ \begin{align*} &C_1=\left[ \begin{array}{cc} C_{33}-M_{31}M_{11}^{-1}C_{13}&C_{32}-M_{31}M_{11}^{-1}C_{12} \\ C_{23}-M_{21}M_{11}^{-1}C_{13}&C_{22}-M_{21}M_{11}^{-1}C_{12} \\ \end{array} \right]\\ &M_1=\left[ \begin{array}{cc} \Gamma_{22}&\Gamma_{21} \\ \Gamma_{12}&\Gamma_{11} \\ \end{array} \right], G_1=\left[ \begin{array}{c} \Upsilon_2 \\ \Upsilon_1 \\ \end{array} \right], F_1=\left[ \begin{array}{c} \Pi_2 \\ \Pi_1 \\ \end{array} \right]\\ &B_1=\left[ \begin{array}{cc} -M_{31}M_{11}^{-1}&0 \\ -M_{21}M_{11}^{-1}&I \\ \end{array} \right], u=\left[ \begin{array}{c} \tau_1 \\ \tau_2 \\ \end{array} \right]. \end{align*} $

It should be noted that dynamic (14) possesses considerable properties which are listed as follows:

Property 1. The matrix $M_1$ is symmetric and positive definite.

Property 2. One can decompose $C_1=\hat{C}_1+\tilde{C}_1$ such that the matrix $\dot M_1-2\hat{C}_1$ is skew symmetric.

In practice, by virtue of structural complexity of the mobile manipulator and pay-load variation from task to task, the inertia parameters of the system are often unknown. Hence, we suppose matrix functions $M_1$, $C_1$, $G_1$ and $F_1$ in system (14) are unknown due to uncertain parameters. $B_1$ is known because input transformation matrix $B_b$ and $S(q_b)$ are all known.

3 Control design and stability analysis 3.1 $\xi_2$ and $\xi_3$ subsystems control

The control objective is specified as: the designed controllers ensure that the tracking errors $e_y=y-y_d=[\xi_3^{\rm T}-\xi_{3d}^{\rm T}, \xi_2-\xi_{2d}^{\rm T}]^{\rm T}$ remain within a small neighborhood of 0, i.e.,

$ \begin{align*} \|\xi_i-\xi_{id}\|\leq\varepsilon_i \end{align*} $

where $\|\cdot\|$ denotes the Euclidean norm, $i=2, 3$, $\varepsilon_i$ are arbitrarily small constants, $l-l_d$ and $\xi$ are bounded.

Lemma 1[17]. There exist time varying positive functions $\delta(t)$ and $\iota(t)$ converging to 0 as $t\to \infty$, and they satisfy

$ \begin{align*} \int_0^\infty\delta(t){\rm d}t=\nu_1 < +\infty, ~~\int_0^\infty\iota(t){\rm d}t=\nu_2 < +\infty \end{align*} $

where $\nu_1$ and $\nu_2$ are known non-negative constants.

Lemma 2[7]. $\forall x\geq0$ and $\forall \alpha \geq1$, one has ${\rm ln}\big(\cosh(x)\big)+\alpha\geq x$.

To this end, the following milder assumptions should be imposed on the unknown matrix functions of system (14):

Assumption 1. There exist some unknown finite positive constants $\sigma_i > 0(i=1, \cdots, 8)$, such that

$ \begin{align*} &\|M_1- M_{\Delta}\|\leq\sigma_1\\ &\|\hat{C}_1-\hat{C}_{\Delta}\|\leq\sigma_2+\sigma_3\|\dot \xi\|\\ &\|\tilde{C}_1-\tilde{C}_{\Delta}\|\leq\sigma_4+\sigma_5\|\dot\xi\|\\ &\| G_1-G_{\Delta}\|\leq\sigma_6+\sigma_7\|\dot\xi\|\\ &\| F_1-F_{\Delta}\|\leq \sigma_8 \end{align*} $

where $M_{\Delta}$, $\hat{C}_{\Delta}$, $\tilde{C}_{\Delta}$, $G_{\Delta}$ and $F_{\Delta}$, as nominal parameter matrices, are known exactly.

Assumption 2. The desired reference trajectory $\xi_d$ is assumed to be bounded and uniformly continuous, and has bounded and uniformly continuous derivatives up to the second order. The desired $l_d(t)$ is bounded and uniformly continuous.

In order to reduce the dynamic order of the designed controller or the number of adaptive updating laws, we choose $\Theta=\max\{\sigma_i\}(i=1, \cdots, 8)$. Next, the following filtered tracking errors are introduced:

$ \begin{align} \begin{cases} &\dot y_r= \dot y_d-K_{y}e_{y}\\ &r~= \dot e_{y}+K_{y}e_{y}\\ &e_{l} = l_h-l_{hd} \end{cases} \end{align} $ (15)

where $K_{y}={\rm diag}\{K_{y j}\}$ is positive diagonal matrix.

Consider the following adaptive control laws given by

$ \begin{align} \begin{cases} B_1u =-Kr+M_{\Delta}\ddot y_r+\tilde{C}_{\Delta}r+C_{\Delta}\dot y_r+G_{\Delta}+F_{\Delta}-\\ \qquad \dfrac{r\hat \Theta(t)\Gamma}{\|r\|\hat \Theta(t)\Gamma+\iota(t)}\Big(\ln\big(\cosh(\hat \Theta(t)\Gamma)\big)+\alpha\Big)\\ \tau_0\, =-l_{hd}+Pe_{\lambda} \end{cases} \end{align} $ (16)

with

$ \begin{align} \dot{\hat \Theta}(t)=-\delta(t)\hat \Theta(t)+\gamma\|r\|\Gamma, ~~~\hat \Theta(0)>0 \end{align} $ (17)

where $K$ is positive definite, $\alpha$ is an arbitrary positive constant and must satisfy $\alpha\geq1$, $\hat\Theta(t)\in{\bf R}$ is the parameter estimation of $\Theta$ and $\Gamma=\|\ddot y_r\|+\|r\|+\|\dot \xi \|\cdot\|r\|+\|\dot y_r\|+\|\dot \xi \|\cdot\|\dot y_r\|+\|\dot \xi \|+2$, $\delta(t)$ and $\iota(t)$ are defined in Lemma 1, without loss of generality, let $\iota(t)=\delta(t)=\frac{1}{(1+t)^2}$.

Lemma 3. For the $\xi_2$-subsystem (12) and $\xi_3$-subsystem (13), considering controllers (16) and (27), then the output tracking errors $\xi_2-\xi_{2d}$ and $\xi_3-\xi_{3d}$ asymptotically converge to 0 and $e_{l}$, $\tau$ and all the other signals in the closed-loop system are bounded.

Proof. In view of (15), system dynamics (14) can be rewritten as

$ \begin{align} M_1\dot r=B_1u -M_1\ddot y_r-C_1r-C_1\dot y_r-G_1-F_1. \end{align} $ (18)

Choose a continuously differentiable, positive definite and radially unbounded function

$ \begin{align} V_1=\frac{1}{2}r^{\rm T}M_1r+\frac{1}{2\gamma}\tilde{\Theta}^2 \end{align} $ (19)

where $\tilde\Theta(t)=\Theta-\hat \Theta(t)$ represents the parameter estimation error. Taking the time derivative of $V_1$ and substituting (16) and (17) into it, one has

$ \begin{align} &\dot V_1 =r^{\rm T}M_1\dot r+\frac{1}{2}r^{\rm T}\dot M_1r-\frac{1}{\gamma}\tilde{\Theta}\dot{\hat \Theta}\leq\notag\\ &\qquad -r^{\rm T}Kr+\hat \Theta\|r\|\Gamma-\frac{\|r\|^2\hat \Theta^2(t)\Gamma^2}{\|r\|\hat \Theta(t)\Gamma+\iota(t)}+\notag\\ &\qquad \tilde{\Theta}\|r\|\Gamma+\tilde{\Theta}\Big(\frac{\delta(t)}{\gamma}\hat \Theta-\|r\|\Gamma\Big)\leq\notag\\ &\qquad -r^{\rm T}Kr+\iota(t)-\frac{\delta(t)}{\gamma}\left(\hat\Theta(t)-\frac{1}{2}\Theta\right)^2+\frac{\Theta^2\delta(t)}{4\gamma}\leq\notag\\ &\qquad -\lambda_{\min}(K)\|r\|^2+\iota(t)+\frac{\Theta^2\delta(t)}{4\gamma}. \end{align} $ (20)

Obviously, we arrive at $\dot V_1 \leq -o V +\iota(t)+\epsilon\delta(t)$. $\iota(t)$ and $\delta(t)$ are bounded time varying positive functions converging to 0 as $t\to \infty$ and $o$ is an appropriate constant. Therefore, there exist $t>T$ and $\iota(t)+\frac{\Theta^2\delta(t)}{4\gamma} < \varepsilon$, when $\|r\|\geq \sqrt{\frac{\varepsilon}{\lambda_{\min}(K)}}$, $\dot V_1\leq0$. Hence, it implies that $r$ converges to a set containing the origin as $t\to \infty$.

On the other hand, integrating both sides of (20) gives

$ \begin{align*} V_1(t)\leq V_1(0)-\int_0^t\lambda_{\min}(K)\|r(\varsigma)\|^2 {\rm d}\varsigma+\frac{\Theta^2}{4\gamma}\nu_1+\nu_2 < \infty. \end{align*} $

Hence, $V_1$ is bounded, which implies $r\in L_{\infty}^{n-k-m}$ and $\hat\Theta(t)\in L_{\infty}^{n-k-m}$. $\int_0^tr^{\rm T}Kr{\rm d}r=V_1(0)-V_1(t)+\frac{\Theta^2}{4\gamma}\nu_1+\nu_2\in L_{\infty}$ can be used to show that $r\in L_2^{n-k-m}$. $r=\dot e_{\xi}+K_{\xi}e_{\xi}$ means $\dot e_{\xi}, e_{\xi}\in L_{\infty}^{n-k-m}\bigcap L_2^{n-k-m}$. $\xi$, $\dot \xi$, $\dot\xi_r$, $\ddot \xi_r\in L_{\infty}^{n-k-m}$ can be further concluded from (15). Therefore, all the signals on the right-hand side of (17) are bounded. one can deduce that $\dot r$ and $\ddot \xi$ are bounded. Therefore, by the well-known Barbalat Lemma[18], we immediately get $\lim_{t\to\infty}r=0$ and $\lim_{t\to\infty}e_{\xi}=0$. Consequently, one has $\lim_{t\to\infty}\dot e_{\xi}=0$.

Finally, substituting the control (16) into dynamic (19) yields

$ \begin{align} (I+P)e_{l} = J^{*}\left( (M\dot H+CH)\dot \xi+ G+F-\hat\tau\right). \end{align} $ (21)

Since all the signals on the right-hand side of (21) are bounded, $(I+P)e_{l}$ is also bounded. Hence, the size of $e_{l}$ can be adjusted by choosing the proper gain matrix $P$.

3.2 Stability analysis of $\xi_1$ subsystem

Finally, for system (11) $-$ (13) under control laws (16) $-$ (17), apparently, the $\xi_1$-subsystem (11) can be rewritten as

$ \begin{align} \dot\chi=g(\phi, \chi, \varphi) \end{align} $ (22)

where $\chi=[\xi_1^{\rm T}, \dot \xi_1^{\rm T}]^{\rm T}$, $\phi=[r^{\rm T}, \dot r^{\rm T}]^{\rm T}$, $\varphi=[\tau_1^{\rm T}, \tau_2^{\rm T}]^{\rm T}$. To analyse the stability of $\xi_1$-subsystems, we need the following assumption:

Assumption 3. There exist Lipschitz positive constants $L_i$, $i=1, \cdots, 4$ such that

$ \|C_{1}+G_1+F_1\|\leq L_1\|\omega\|+L_2 $ (23)
$ \|\Upsilon_2+\Pi_2\|\leq L_3\|\omega\|+L_4\label{mm34}. $ (24)

Moreover, from the stability analysis of $\xi_2$ and $\xi_3$ subsystems, $\omega$ converges to a small neighborhood of $\omega_d=[\xi_{2d}^{\rm T}, \dot\xi_{2d}^{\rm T}, \xi_{3d}^{\rm T}, \dot\xi_{3d}^{\rm T}]^{\rm T}$, i.e., $\|\omega-\omega_d\|\leq\epsilon_1$, it is easy to obtain $\|\omega\|\leq\|\omega_d\|+\epsilon_1$, and similarly, $\|[\ddot\xi_2^{\rm T}, \ddot\xi_3^{\rm T}]^{\rm T}\| \leq\|[\ddot\xi_{2d}^{\rm T}, \ddot\xi_{3d}^{\rm T}]^{\rm T}\|+\epsilon_2$, where $\epsilon_1$ and $\epsilon_2$ are small bounded errors.

Lemma 4. The $\xi_1$-subsystem (11) is stable.

Proof. One can choose the following Lyapunov function candidate as

$ \begin{align} V_2=V_1+\ln(\cosh(\dot\xi_1)). \end{align} $ (25)

Differentiating (25) and substituting (13) into it will give

$ \begin{align*} &\dot V_2=\dot V_1+\tanh(\dot\xi_1)\ddot\xi_1=\\ &\qquad \dot V_1+\tanh(\dot\xi_1)\Big(-M^{-1}_{31}(\Gamma_{21}\ddot \xi_2+\Gamma_{22}\ddot \xi_3+\Upsilon_2+\Pi_2)-\\ &\qquad M^{-1}_{11}(M_{12}\ddot \xi_2+M_{13}\ddot \xi_3+C_1+G_1+F_1)\Big)=\\ &\qquad \dot V_1+\tanh(\dot\xi_1)(-M^{-1}_{31}(\Upsilon_2+\Pi_2)-\tanh(\dot\xi_1)M^{-1}_{11}C_1-\\ &\qquad\left[ \begin{array}{c} \tanh(\dot\xi_1)M^{-1}_{31}\Gamma_{21}+\tanh(\dot\xi_1)M^{-1}_{11}M_{12} \\ \tanh(\dot\xi_1)M^{-1}_{31}\Gamma_{22}+\tanh(\dot\xi_1)M^{-1}_{11}M_{13}\\ \end{array} \right]^{\rm T}\left[ \begin{array}{c} \ddot \xi_2 \\ \ddot \xi_3 \\ \end{array} \right]. \end{align*} $

$\|\tanh(\dot\xi_1)\|\leq1$ and the boundedness of $M_{12}$, $M_{13}$, $M_{11}^{-1}$ and $M_{31}^{-1}$ mean that there exist bounded constants $\varpi_1$, $i=1, 2, 3$, such that

$ \begin{align*} &\left\|\left[ \begin{array}{c} \tanh(\dot\xi_1)M^{-1}_{31}\Gamma_{21}+\tanh(\dot\xi_1)M^{-1}_{11}M_{12} \\ \tanh(\dot\xi_1)M^{-1}_{31}\Gamma_{22}+\tanh(\dot\xi_1)M^{-1}_{11}M_{13}\\ \end{array} \right]^{\rm T}\right\|\leq\varpi_1\\ &\|M^{-1}_{11}\|\leq\varpi_2, ~ \|M^{-1}_{31}\|\leq\varpi_3. \end{align*} $

With these in mind and using Assumption 3, we finally get

$ \begin{align} &\dot V_2 \leq-\lambda_{\min}(K)\|r\|^2+\iota(t)+\frac{\Theta^2\delta(t)}{4\gamma}+\varpi_3(L_3(\|\omega_d\|+\epsilon_1)+\notag\\ &\qquad L_4)+\varpi_2(L_1(\|\omega_d\|+\epsilon_1)+L_2)+\notag\\ &\qquad \varpi_1(\|[\ddot\xi_{2d}^{\rm T}, \ddot\xi_{3d}^{\rm T}]^{\rm T}\|+\epsilon_2). \end{align} $ (26)

Let $p=\varpi_3(L_3(\|\omega_d\|+\epsilon_1)+L_4)+\varpi_2(L_1(\|\omega_d\|+\epsilon_1)+L_2)+ \varpi_1(\|[\ddot\xi_{2d}^{\rm T}, \ddot\xi_{3d}^{\rm T}]^{\rm T}\|+\epsilon_2)$, when $\|r\|\geq\sqrt{\frac{2(\varepsilon+p)}{(\lambda_{\min}(K))}}$, one has $\dot V_2\leq0$. Furthermore, $r$ can be arbitrarily small by choosing a proper $K$. Therefore, the $\xi_1$-subsystem (11) is stable.

Theorem 1. Consider the mobile manipulators described by (3) with affine constraints (1). Using the adaptive control laws (16) and (17), the following are guaranteed:

1) The output tracking errors $e_{y}$ and $\dot e_{y}$ of $\xi_2$ and $\xi_3$ subsystems converge to 0 as $t\to \infty$.

2) The $\xi_1$-subsystem (11) is stable and $e_{l}$, $\tau$ in the closed-loop system are bounded for all $t\geq0$.

4 Simulation example

In this section, computer simulation is conducted to examine the performance of the tracking controller for the mobile manipulator as shown in Fig. 2. According to the Euler-Lagrangian equations, the following standard form can be obtained.

$ \begin{align*} &q_b=[q_{b_1}, \, q_{b_2}, \, q_{b_3}]^{\rm T}, ~[q_{u}, \, q_{a}]^{\rm T}=[\theta_1, \, \theta_2]^{\rm T}\\ &M_b=\left[ \begin{array}{ccc} m_{b12}& 0&m_{12}d\cos q_{b_3} \\ 0&m_{b12} &-m_{12}d\sin q_{b_3} \\ m_{12}d\cos q_{b_3} &-m_{12}d\sin q_{b_3}&I_{b12}+m_{12}d^2 \\ \end{array} \right]\\ &M_m=\left[ \begin{array}{cc} I_{12}&0 \\ 0&I_2 \\ \end{array} \right], M_{mb}=\left[ \begin{array}{ccc} 0&0&I_{12} \\ 0&0&0 \\ \end{array} \right], M_{bm}=M_{mb}^{\rm T} \end{align*} $
$ \begin{align*} &C_b=\left[ \begin{array}{ccc} 0&0 &-m_{12}\dot q_{b_3} d\sin q_{b_3} \\ 0&0 &-m_{12}\dot q_{b_3} d\cos q_{b_3} \\ 0&0&0 \\ \end{array} \right], C_{mb}=0, C_{bm}^{\rm T}=C_{mb}\\ &G_b=0, G_m=[0, \, m_2gl_2\sin\theta_2]^{\rm T}, F_b=[f_{b_1}, \, f_{b_2}, \, f_{b_3}]^{\rm T}\\ &B_b=\left[ \begin{array}{cc} -\cos q_{b_3} &-\cos q_{b_3} \\ \sin q_{b_3} &-\sin q_{b_3}\\ -\frac{l}{2}&\frac{l}{2} \\ \end{array} \right], ~~\tau_b=\left[ \begin{array}{c} \tau_r \\ \tau_l \\ \end{array} \right], ~~\tau_a=\tau_1. \end{align*} $
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Fig. 2. 2-DOF manipulator mounted on boat

where $ m_{b12}=m_b+m_1+m_2$, $m_{12}=m_1+m_2$, $I_{b12}=I_b+I_1+I_2$, $I_{12}=I_1+I_2$. The parameters used in the simulation are $m_b=500\, {\rm kg}$, $m_1=50\, {\rm kg}$, $l=l_1=l_2=4\, {\rm m}$, $d=6\, {\rm m}$ $I_b=50\, {\rm kg\cdot m^2}$, $I_1=10\, {\rm kg\cdot m^2}$, $f_{b_i}=\bar b\sin t$ and $f_{m_i}=\bar m \cos t$ with unknown $\bar b$ and $\bar m$, $C(q_{b_2})=q_{b_2}$. Because of the second operating arm with varied pay-load, one assumes that $m_2$ and $I_2$ are unknown. The system is subject to the affine constraint:

$ \begin{align*} \dot q_{b_1}\cos q_{b_3} -\dot q_{b_2}\sin q_{b_3} =C(q_{b_2})\cos q_{b_3}. \end{align*} $

We select

$ \begin{align*} &S(q_b)=\left[ \begin{array}{ccc} \tan q_{b_3}&0 \\ 1&0 \\ 0&1 \\ \end{array} \right], ~~ ~\eta(q_b)=\left[ \begin{array}{ccc} q_{b_2} \\ 0 \\ 0 \\ \end{array} \right]. \end{align*} $

Considering the above transformation, the whole system dynamics are converted as

$ \begin{align*} &\left[ \begin{array}{cccc} m_{11}&0&0&0 \\ 0&m_{22}&m_{23}&0 \\ 0&m_{32} &m_{33}&0 \\ 0&0&0 &m_{44}\\ \end{array} \right]\ddot \zeta +\left[ \begin{array}{cccc} c_{11}&c_{12}&0&0 \\ c_{21}&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ \end{array} \right] \dot \zeta+\\ \\ &\quad \left[ \begin{array}{c} g_1 \\ g_2 \\ 0 \\ g_4 \\ \end{array} \right] + \left[ \begin{array}{c} f_1 \\ f_2\\ f_3 \\ f_4 \\ \end{array} \right] = \left[ \begin{array}{c} u_1 \\ u_2 \\ 0 \\ u_3 \\ \end{array} \right]. \end{align*} $

where $\zeta=[z_1, z_2, q_{u}, q_{a}]^{\rm T}$, $m_{11}=\frac{550+m_2}{\cos^2 q_{b_3}}$, $m_{22}=1860+36m_2+I_2$, $m_{23}=10+I_2$ $m_{32}=10+I_2$, $m_{33}=10+I_2$, $m_{44}=I_2$, $c_{11}=\frac{(550+m_2)\sin q_{b_3}\dot q_{b_3}}{\cos^3 q_{b_3}}$, $c_{12}=-\frac{6(50+m_2)}{\cos^2 q_{b_3}}\dot q_{b_3}$, $c_{21}=\frac{6(50+m_2)}{\cos q_{b_3}}\dot q_{b_3}$, $g_{1}=(550+m_2)\tan q_{b_3}\dot z_1$, $g_{2}=6(50+m_2)\cos q_{b_3}\dot z_1$, $g_{4}=4m_2g\sin\theta_2 $, $f_{1}= \tan q_{b_3}f_{b_1}+f_{b_2}$, $f_{2}= f_{b_3}$, $f_{3}=f_{m_1}$, $f_{4}=f_{m_2}$, $u_1=-2\sin q_{b_3}\tau_r$, $u_2=-2\tau_r+2\tau_l$, $u_3=\tau_1$.

Then the above dynamic equation can be further transformed into the following dynamics:

$ \begin{align*} &\left[ \begin{array}{cccc} m_{33}-\frac{m_{32}m_{23}}{m_{22}}&0&0 \\ 0&m_{11}&0 \\ 0&0&m_{44} \\ \end{array} \right]\ddot{\bar \zeta}+\\ & \qquad\left[ \begin{array}{cccc} 0 &-\frac{m_{32}c_{11}}{m_{22}}&0 \\ 0&c_{11}&0 \\ 0&0&0 \\ \end{array} \right] \dot{\bar \zeta}+ \\ \\ &\qquad\left[ \begin{array}{c} \frac{-m_{32}g_2}{m_{22}}\\ g_1+c_{12}\dot z_2\\ g_4 \\ \end{array} \right]+ \left[ \begin{array}{c} f_3-\frac{m_{32}f_2}{m_{22}} \\ f_1\\ f_4 \\ \end{array} \right] =\\ &\qquad \left[ \begin{array}{cccc} \frac{-m_{32}}{m_{22}}&0&0 \\ 0&1&0 \\ 0&0&1 \\ \end{array} \right]\bar u.\\ \\ & \ddot z_2=\frac{1}{m_{22}}(u_2-c_{21}\dot z_1-g_2-f_2-m_{23}\ddot q_{u}) \end{align*} $

with $\bar\zeta=[q_u, z_1, q_{a}]^{\rm T}$, $\bar u=[u_2, u_1, u_3, ]^{\rm T}$. Given the desired trajectory $y_d=[q_{ud}, \, z_{1d}, \, q_{ad}]^{\rm T}=[\frac{\pi}{4}, \, \sin t+\cos t, \, \frac{\pi}{4}(1-0.5\sin t)]^{\rm T}$, $l_{hd}=10\, {\rm N}$. The control objective is to determine an adaptive controller so that the trajectory $y=[q_{u}, \, z_{1}, \, q_{a}]^{\rm T}$ and $\dot y$ follow $y_d$ and $\dot y_d$, respectively, and $z_2$-subsystem is stable and $l$ is bounded. We get the actual controller

$ \begin{align*} &B_1u = \left[ \begin{array}{cccc} m^*&0&0 \\ 0&m_{11}&0 \\ 0&0&m_{44} \\ \end{array} \right]\ddot y_r +\left[ \begin{array}{cccc} 0&\frac{-m_{32}c_{11}}{m_{22}}&0 \\ 0&c_{11}&0 \\ 0&0&0 \\ \end{array} \right]\dot y_r+\\ &\quad \quad \left[ \begin{array}{c} \frac{-m_{32}g_2}{m_{22}}\\ g_1+c_{12}\dot z_2\\ g_4 \\ \end{array} \right]+ \frac{r\hat \Theta(t)\Gamma}{\|r\|\hat \Theta(t)\Gamma+\iota(t)}\left(\ln(*)+2\right)-r \end{align*} $

with

$ \begin{align*} \begin{cases} \dot{\hat \Theta}=-\delta(t)\hat \Theta(t)+\|r\|\Gamma, ~\hat \Theta(0)=2\\ \tau_0=-10+P(l_h-10) \end{cases} \end{align*} $

where $m^*=\frac{m_{33}m_{22}-m_{32}m_{23}}{m_{22}}$, $*=(\cosh(\hat \Theta(t)\Gamma)\big)+2)$, $r=[\dot q_u+q_u-\frac{\pi}{4}, \, \dot z_1+z_1-2\cos t, \, \dot q_{a}+q_{a}+\frac{\pi}{8}\cos t+\frac{\pi}{8}\sin t-\frac{\pi}{4}]^{\rm T}$, $\ddot y_r=[-\dot q_u, \, -\dot z_1-2\sin t, \, -\dot q_{a}+\frac{\pi}{8}\sin t-\frac{\pi}{8}\cos t]^{\rm T}$, $\dot y_r=[-q_u+\frac{\pi}{4}, \, - z_1+2\cos t, \, -q_{a}-\frac{\pi}{8}\sin t-\frac{\pi}{8}\cos t+\frac{\pi}{4}]^{\rm T}$, $\iota(t)=\delta(t)=\frac{1}{(1+t)^2}$ and $\Gamma=\|\ddot y_r\|+\|r\|+\|\dot \xi \|\cdot\|r\|+\|\dot y_r\|+\|\dot \xi \|\cdot\|\dot y_r\|+\|\dot \xi \|+2$.

The position state performances of $q_{u}$, $z_{1}$ and $q_{a}$ are illustrated in Figs. 3-5 and the velocity tracking results of $\dot q_{u}$, $\dot z_{1}$ and $\dot q_{a}$ are presented in Figs. 6-8. The stability of $z_2$-subsystem is shown in Fig. 9. The force tracking error of $l_a-l_{ad}$ becomes arbitrarily small and the parameter updating laws $\hat \Theta $ are bounded as shown in Figs. 10 and 11. The input torques are all bounded as shown in Fig. 12.

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Fig. 3. Trajectories of $q_{u}$, $q_{ud}$

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Fig. 4. Trajectories of $z_{1}$, $z_{1d}$

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Fig. 5. Trajectories of $q_{a}$, $q_{ad}$

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Fig. 6. Trajectories of ${\dot q}_{u}$

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Fig. 7. Trajectories of $\dot z_{1}$, ${\dot z}_{1d}$

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Fig. 8. Trajectories of ${\dot q}_{a}$, ${\dot q}_{ad}$

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Fig. 9. Trajectories of $z_2$, ${\dot z}_2$

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Fig. 10. Trajectory of $\lambda_a-\lambda_{ad}$

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Fig. 11. Trajectory of $\hat\Theta$

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Fig. 12. Trajectories of inputs

5 Conclusions

This paper discusses tracking control of the uncertain affine constraints mobile manipulator with under-actuated joints, and mainly discusses mathematical modeling, algorithm design and theory analysis for it. An adaptive tracking controller is proposed. Practical simulation is presented to illustrate the effectiveness of the control strategy.

Acknowledgements

This work was supported by National Natural Science Foundation of China (Nos. 61273091 and 61403228), Project of Taishan Scholar of Shandong Province of China, and Ph. D. Programs Foundation of Ministry of Education of China (No. 20123705110002).

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