2. School of Engineering, Qufu Normal University, Rizhao 276826, China
As we all know, time delay in the control input is a pervasive problem in various control applications. Robotic systems with no exception encounter delays which make the control systems more complicated. With the widespread application of communication networks in interconnecting robotic systems and controllers, the flexibility of the robotic systems and the communication distance is increasing, which can lead to the emergence of time delay in the acting force. The presence of time delay in control input is often attributed to deteriorate the effectiveness of associated control systems and is often a cause of instability. Therefore, it is necessary to evaluate the impact of time delay on robotic systems. During the last few decades, the study of control design for the timedelay robotic systems has drawn a great deal of attention and many nice results have been proposed (see [1][7] and the references therein). A control law for teleoperators was presented in [1] which overcame the instability caused by time delay. Using the method of proportionaldifferential (PD) fractional control, Lazarevi in [4] studied the problem of finite time stability for robotic time delay systems. The classical setpoint control problem for rigid robots with inputoutput communication delays was addressed in [5]. The paper demonstrates that if there are transmission delays between the robotic system and the controller, then the use of the scattering variables can stabilize an unstable system with arbitrary unknown constant delays.
In addition to time delay, there always exist a variety of uncertainties in the robotic control systems [8][11]. These uncertainty factors can be divided into two kinds: one is internal uncertainty, such as parametric uncertainties, payload uncertainties, model uncertainty, unmodeled dynamics characteristics and so on; the other is external uncertainty, such as external disturbance, etc. These uncertainties may pose significant impediments to the stabilization problem and potentially degrade the performance of the closed loop system. Therefore, in order to get a better performance of control, these factors cannot be ignored. By using adaptive fuzzy logic, Kim [10] studied output feedback tracking control of robotic systems with model uncertainty. The tracking control problem was concerned in [11] for robotic systems perturbed by timevarying parameters, unmodelled dynamics and external force (and moment) disturbances. Other important research progresses about robotic system control subject to uncertainties can be found in [12][14] and the references therein.
In recent years, Hamiltonian method has been one of effective methods in studying stability and the problem of control for some practical systems, including robotic systems. And many good results have been achieved [15][20]. A key step in using Hamiltonian strategy is to express the system under consideration as a dissipative Hamiltonian system which was firstly put forward by Maschke and Van der Schaft in [21]. The method, in general, can thoroughly use the internal structure properties of real systems during control designs, and the controllers designed by this method are relatively simple in form, easy and effective in operation [22][24]. To obtain the dissipative Hamiltonian system, we first need to express the system as a generalized Hamiltonian system, which is the socalled generalized Hamiltonian realization and then eliminate the nondissipative part of the obtained generalized Hamiltonian realization by a suitable state feedback. Wang and Ge in [24] provided an augmented dissipative Hamiltonian structure for both fully actuated and underactuated uncertain mechanical systems. Besides, [24] has also investigated the energybased robust adaptive control for the uncertain mechanical systems by using the new Hamiltonian formulation. But [24] did not consider the effect when delays inevitably appear in the systems.
Hence, in order to solve the problem of time delay, this paper investigates a robust stabilization problem for a class of uncertain robotic systems with input timevarying delay based on the Hamiltonian method. Firstly, we find a nice delayed Hamiltonian system structure to describe the dynamic action of the robotic systems with timevarying delay. Secondly, we show how to design an adaptive controller based on the obtained Hamiltonian systems which makes the resultant feedback system asymptotically stable. Moreover, the adaptive feedback controller for the uncertain delayed robotic systems is also given. Finally, an illustrative example is presented to show the effectiveness of the method proposed in this paper.
The rest of the paper is organized as follows. Section Ⅱ presents the problem formulation and some preliminaries. In Section Ⅲ, we study the augmented delayed Hamiltonian formulation for uncertain robot manipulators with timevarying delays. The analysis of robust adaptive control of the delayed Hamiltonian system is presented in Section Ⅳ. Section Ⅴ illustrates the obtained results by a twolink robot manipulator example, which is followed by the conclusion in Section Ⅵ.
Notations:
Consider the following
$ M(q)\ddot q+C(q, \dot q)\dot q+G(q) = \tau $  (1) 
where
Assumption 1: The unknown part in
$ G(q)=\Lambda(qq^{0})+{\Lambda_1}(q)\theta $  (2) 
where
Moreover, the above equations exhibit certain fundamental properties due to their Lagrangian dynamic structure.
Property 1: The inertia matrix
$ mI\leq M(q)\leq MI. $  (3) 
We consider the existence of timevarying delays in the input signals applied to the robot joints. Let
$ 0\leq d(t)\leq h $  (4) 
and
$ 0\leq\dot d(t)\leq \mu <1 $  (5) 
where
$ M(q)\ddot q+C(q, \dot q)\dot q+G(q)=\tau(td(t)). $  (6) 
Our aim in this paper is to investigate the robust adaptive control problem of system (6) in the presence of input delay such that all signals in the closedloop system are bounded and all bounded trajectories converge to the largest invariant set where
Under Assumption 1, we consider a Hamilton function for system (6) as follows:
$ H(q, p, \hat{\theta})=K(q, p)+P_g(q)+ \frac{1}{2}(\hat{\theta}\theta)^{ T}\Gamma_0(\hat{\theta}\theta) $  (7) 
where
$ K(q, p):= \frac{1}{2}p^{ T} M^{1}(q)p = \frac{1}{2}\dot q^{ T} M(q)\dot q $  (8) 
is the system's kinetic energy,
$ P_g(q):=\frac{1}{2}(qq^0)^{ T}\Lambda(qq^0) $  (9) 
is the socalled virtual potential energy,
Obviously, we have
$ \frac{\partial H(q, p, \hat{\theta}) }{\partial p}=M^{1}(q)p=\dot q $  (10) 
which means
$ p=M(q)\dot q. $  (11) 
Let
Property 2:
We conclude this section by recalling some auxiliary results to be used in Hamiltonian realization of system (6) and the robust controller designed in this paper.
Lemma 1 [24]: Assume that
$ \frac{\partial \left(\alpha^{ T} A(x)\beta\right)}{\partial x}=\left(I_n \otimes \alpha^{ T}\right)\left(\Gamma_n \frac{\partial A(x)}{\partial x}\right)\beta $ 
where
Lemma 2 [27]: For any positivedefinite matrix
$ \begin{align*} &h\int_{td(t)}^t w^{ T}(s)Z w(s)ds \\ &\qquad \geq\left[\int_{td(t)}^tw(s)ds\right]^{ T} Z\left[\int_{td(t)}^tw(s)ds\right], \quad t\geq0. \end{align*} $ 
Lemma 3 [28]: The linear matrix inequality (LMI)
$ \begin{align} \left[ \begin{array}{cc} \Lambda_1(x) & \Lambda_2(x) \\ \ast & \Lambda_3(x) \\ \end{array} \right]>0 \end{align} $  (12) 
is equivalent to
Remark 1: The derivative of the timevarying delay
According to Lemma 1 and (7), we get
$ \frac{\partial H(q, p, \hat{\theta})}{\partial q}=\frac{1}{2}\left(I_n\otimes p^{ T}\right)\left(\Gamma_n\frac{\partial M^{1}(q)}{\partial q}\right)p+\Lambda\left(qq^0\right). $  (13) 
From system (6) and (11), the derivative of
$ \begin{align} \dot p &=\dot M(q)\dot q+M(q)\ddot q \\ &=\dot M(q)\dot qC(q, \dot q)\dot qG(q)+\tau_t. \end{align} $  (14) 
In order to get a nice Hamiltonian structure for system (6), we design a prefeedback law as follows:
$ \begin{align} \begin{cases} \tau_t=\Lambda_1\hat{\theta}_tK_{D1}\dot q_t+u_t\\ \dot{\hat{\theta}}=\Gamma_0^{1}\Lambda_1^{ T}\dot q_t\\ \end{cases} \end{align} $  (15) 
where
Substituting (15) into (6) and considering (10)(14), we obtain
$ \begin{align} \left( \begin{array}{c} \dot q \\ \dot p \\ \dot{\hat{\theta}} \\ \end{array} \right)=& \left[\begin{array}{ccc} 0 & I_n & 0 \\ I_n & K_c(q, p) & 0\\ 0 & 0 & 0 \\ \end{array} \right] \left[ \begin{array}{c} \frac{\partial{H(q, p, \hat{\theta})}}{\partial q} \\ \frac{ \partial{H(q, p, \hat{\theta})}}{\partial p} \\ \frac{ \partial{H(q, p, \hat{\theta})}}{\partial\hat{\theta}}\end{array}\right] \\ & + \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & K_{D1} & \Lambda_1\Gamma_0^{1}\\ 0 & \Gamma_0^{1}\Lambda_1^{ T} & 0 \\ \end{array} \right) \\ & \times \left[ \begin{array}{c} \partial{H(q_t, p_t, \hat{\theta}_t)}{q} \\ \partial{H(q_t, p_t, \hat{\theta}_t)}{p} \\ \partial{H(q_t, p_t, \hat{\theta}_t)}{\hat{\theta}} \end{array}\right] + \left( \begin{array}{c} 0 \\ I_n\\ 0 \\ \end{array} \right)u_t \end{align} $  (16) 
where
$ \begin{align*} K_c(q, p)=&\ \dot M(q)C(q, \dot q)+\frac{1}{2}(I_n \otimes P^{ T})\\ & \times (\Gamma_n\frac{\partial M^{1}(q)}{\partial q})M(q). \end{align*} $ 
Since
$ \begin{align} \frac{\partial}{\partial x}\left[A^{1}(x)\right]=\Gamma_n\left(I_n\otimes A^{1}(x)\right)\left(\Gamma_n\frac{\partial A(x)}{\partial x}\right) A^{1}(x) \end{align} $ 
holds, we can prove that
Thus, system (6) can be transformed into the following delayed portcontrolled Hamiltonian system
$ \begin{align} \begin{cases} \dot X=[J(X)R(X)]\frac{\partial{H(X)}}{\partial X} +[J_1(X)\\ \qquad R_1(X)]\frac{\partial{H(X_t)}}{\partial X}+g u_t\\[2mm] X(t)= \phi(t), \quad t\in[h, 0]\end{cases} \end{align} $ 
where
$ \begin{align*} J(X)=\left( \begin{array}{ccc} 0 & I_n & 0 \\ I_n &0 & 0\\ 0 & 0 & 0 \\ \end{array} \right)\\ J_1(X)=\left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & \Lambda_1(q)\Gamma_0^{1}\\ 0 &\Gamma_0^{1}\Lambda_1^{ T}(q)& 0 \\ \end{array} \right] \\ R(X)= \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 &K_c(q, p) & 0\\ 0 & 0 & 0 \\ \end{array} \right] \\ R_1(X)= \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & K_{D1} & 0\\ 0 & 0 & 0 \\ \end{array} \right), g= \left( \begin{array}{c} 0 \\ I_n\\ 0 \\ \end{array} \right). \end{align*} $ 
Obviously,
Summarizing the above, we have the following theorem.
Theorem 1: Consider the robotic system (6) with input delay and Assumption 1. With the Hamilton function (7) and the adaptive prefeedback law (15), system (6) can be transformed into a delayed Hamiltonian system described as (17).
Ⅳ. ADAPTIVE CONTROLIn this section, we study the adaptive control problem of the uncertain robotic system (6) with input delay by using the equivalent delayed Hamiltonian formulation (17).
Considering a feedback controller
Theorem 2: For a given scalar
$ \begin{align} \Xi=\left(\begin{array}{cccc} \Gamma_{1, 1} & I_{2n+l} & \Gamma_{1, 3} & \Gamma_{1, 4}\\ \ast & 2I_{2n+l} & I_{2n+l} & \Gamma_{2, 4}\\ \ast & \ast & I_{2n+l} & 0 \\ \ast & \ast & 0 & I_{2n+l} \end{array} \right) <0 \end{align} $  (18) 
where
$ \begin{align*} &\Gamma_{1, 1}=2R2R_12gKg^{ T}\\ &\Gamma_{1, 3}=R_1J_1+gKg^{ T}\\ &\Gamma_{1, 4}=h\nu_2(JR)^{ T}\\ &\Gamma_{2, 4}=h\nu_2(J_1R_1gKg^{ T})^{ T} \end{align*} $ 
then the delayed Hamiltonian system (17) under the output feedback control
Proof: Substituting
$ \begin{align} \dot X=&\ [J(X)R(X)]\frac{\partial{H(X)}}{\partial X} \\ & +[J_1(X)R_1(X)gKg^{ T}]\frac{\partial{H(X_t)}}{\partial X}. \end{align} $  (19) 
For the sake of simplicity, we denote
Construct a Lyapunov functional candidate as follows:
$ V(X_t)=V_1+V_2 $  (20) 
where
$ \begin{align} &V_1 = 2H(X) \\ &V_2= h\int_{h}^0\int_{t+\alpha}^t [\nabla^{ T} H(X(s))]'[\nabla H(X(s))]'dsd\alpha. \end{align} $ 
Since the Hamilton function
$ \epsilon_1(\X_t\)\leq V(X_t)\leq\varepsilon(\X_t\_\mathcal{C}) $  (21) 
where
According to the NewtonLeibniz formula, we have
$ \nabla H(x)\nabla H(x_t)={\int^{~t}_{td(t)}}[\nabla H(x(s))]'ds. $  (22) 
Then system (19) can be rewritten as
$ \begin{align} \dot X=&\ \left(J+J_1RR_1gKg^{ T}\right)\nabla H(X) \\ & \left(J_1R_1gKg^{ T}\right){\int^{~t}_{td(t)}}[\nabla H(X(s))]'ds. \end{align} $  (23) 
Computing the derivatives of
$ \begin{align} \dot V_1=&\ 2\nabla^{ T} H(X)(J+J_1RR_1gKg^{ T})\nabla H(X) \\ & 2\nabla^{ T} H(X)(J_1R_1gKg^{ T}) \\ & \times{\int^{~t}_{td(t)}}[\nabla H(X(s))]'ds \end{align} $  (24) 
$ \begin{align} \dot V_2\leq &\ h^2\dot X^{ T}\times Hess^{ T}(H(X))\times Hess(H(X))\times\dot X \\ & h{\int^{~t}_{td(t)}}[\nabla^{ T} H(X(\beta))]'[\nabla H(X(\beta))]'d\beta. \end{align} $  (25) 
According to Property 2, by replacing
$ \begin{align} \dot V_2\leq&\ h^2\nu_2^2\Big[(JR)\nabla H(X) +(J_1R_1gKg^{ T}) \\ &\times \nabla H(X_t)\Big]^{ T} \Big[(JR)\nabla H(X) +(J_1R_1 \\ &gKg^{ T})\nabla H(X_t)\Big] {\int^{~t}_{td(t)}}[\nabla^{ T} H(X(\beta))]'d\beta \\ &\times{\int^{~t}_{td(t)}}[\nabla H(X(\beta))]'d\beta. \end{align} $  (26) 
Then the following inequality about
$ \begin{align} \dot V(X_t)=&\ \dot V_1+\dot V_2 \\ \leq &\ \nabla^{ T} H(X)\Big[2R2gKg^{ T}2R_1+h^2\nu_2^2 \\ &\times(JR)^{ T}(JR)\Big]\nabla H(X) +2\nabla^{ T} H(X) \\ &\times\Big[h^2\nu_2^2(JR)^{ T} (J_1R_1gKg^{ T})\Big] \\ &\times\nabla H(X_t)2\nabla^{ T} H(X)(J_1R_1gKg^{ T}) \\ &\times{\int^{~t}_{td(t)}}[\nabla H(X(s))]'{d}s +\nabla^{ T} H(X_t)\Big[h^2\nu_2^2 \\ &\times(J_1R_1gKg^{ T})^{ T} (J_1R_1gKg^{ T})\Big] \\ &\times\nabla H(X_t) {\int^{~t}_{td(t)}}[\nabla^{ T} H(X(\beta))]'d\beta \\ &\times{\int^{~t}_{td(t)}}[\nabla H(X(\beta))]'d\beta. \end{align} $  (27) 
Let
$ \begin{align} \xi(t)=\left[ \begin{array}{ccc} \nabla H(X) & \nabla H(X_t) & {\int^{~t}_{td(t)}}[\nabla H(X(\beta))]'d\beta \\ \end{array} \right]^{ T} \end{align} $  (28) 
and
$ \begin{align} \Theta=\left(\begin{array}{cccc} \Theta_{1, 1} & \Theta_{1, 2} & \Theta_{1, 3}\\ \Theta_{2, 1}& \Theta_{2, 2} & 0\\ \Theta_{3, 1}& 0 & I_{2n+l} \\ \end{array} \right) \end{align} $  (29) 
with
$ \begin{align*} &\Theta_{1, 1}=2R2R_12gKg^{ T}\\ &~~~~~~~~~+h^{2}\nu_2^2(JR)^{ T}(JR)\\ &\Theta_{1, 2}=h^{2}\nu_2^2(JR)(J_1R_1gKg^{ T})\\ &\Theta_{1, 3}=J_1+R_1gKg^{ T}\\ &\Theta_{2, 1}=h^{2}\nu_2^2(J_1R_1gKg^{ T})^{ T}(JR)^{ T}\\ &\Theta_{2, 2}=h^{2}\nu_2^2(J_1R_1gKg^{ T})^{ T} (J_1R_1gKg^{ T})\\ &\Theta_{3, 1}=J^{ T}_1+R_1gKg^{ T} \end{align*} $ 
further obtain
$ \dot V(X_t)\leq\xi^{ T}(t)\Theta\xi(t). $  (30) 
Due to
$ \begin{align} \dot V(X_t)\leq& \rho\{\int^{~t}_{td(t)}}[\nabla^{ T} H(X(\beta))]'d\beta \\ & \times{\int^{~t}_{td(t)}}[\nabla H(X(\beta))]'d\beta\ \\ \leq & \rho\nu_1^2\X\^2 \end{align} $  (31) 
where
By LyapunovKrasovskii stability theorem [34], we can conclude that system (17) under the output feedback control law
From Theorem 1, since (17) is an equivalence transformation of uncertain delayed robotic system (6) under the prefeedback law (15), the controller
Corollary 1: Consider the uncertain delayed robotic system (6). Suppose Assumption 1 holds. For a given scalar
$ \begin{align} \begin{cases} \tau_t=\Lambda_1\hat{\theta}_tK_{D1}\dot q_tKg^{ T}\nabla H(X_t)\\ \dot{\hat{\theta}}=\Gamma_0^{1}\Lambda_1^{ T}\dot q_t\end{cases} \end{align} $  (32) 
can asymptotically stabilize system (6), where
Remark 2: It is necessary to point out that the result obtained in this paper is different from that in [34]. Apart from the differences of the delays, there exist essential differences between the two controllers in [34] and this paper. In [34], the control signals under consideration include two parts: the local control signal and the remote one. However, we only need the information
In this section, we give an example to show: 1) how to transform the robot manipulator with time delay into delayed Hamiltonian system; and 2) how to design the adaptive controller for the delayed robot systems under Hamiltonian system framework. A planar twolink manipulator with two nodes in the vertical plane is considered as shown in Fig. 1, where we assume that the mass
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Fig. 1 Planar twolink manipulator with payload. 
Assume there exist delays in the input signals,
$ M(q)\ddot q+C(q, \dot q)\dot q+G(q) = \tau_t $  (33) 
where
$ \begin{align*} &M(q)=\left[\begin{array}{cc} \eta + 2\overline{m}_3 \cos q_2 & \overline{m}_2 + \overline{m}_3 \cos q_2 \\ \overline{m}_2 + \overline{m}_3 \cos q_2 & \overline{m}_2 \\ \end{array} \right]\\ &C(q, \dot q)=\left[\begin{array}{cc} \overline{m}_3\dot q_2 \sin q_2 &\overline{m}_3(\dot q_1+\dot q_2) \sin q_2 \\ \overline{m}_3 \dot q_1 \sin q_2 & 0 \\ \end{array} \right]\\ &G(q)=\left[\begin{array}{c} \overline{m}_4 g \cos q_1+\overline{m}_5 g \cos (q_1+q_2) \\ \overline{m}_5 g \cos (q_1+q_2) \\ \end{array} \right]\\ &\eta =\overline{m}_1 + \overline{m}_2\\ & \overline{m}_1=m_1l^2_{c1} + m_2l^2_1 + I_1 + m_pl^2_1\\ & \overline{m}_2=m_2l^2_{c2} + I_2 + m_pl^2_2\\ & \overline{m}_3=m_2l_1l_{c2} + m_pl_1l_2\\ & \overline{m}_4=(m_1l_{c2} + m_2l_1)(q_1q^{0}_1)+ m_pl_1\\ & \overline{m}_5=m_2l_{c2}(q_2q^{0}_2) + m_pl_2. \end{align*} $ 
Because the payload's mass
Let
$ G(q)=\Lambda(qq^{0})+\Lambda_1(q)\theta $  (34) 
where
$ \begin{align} \Lambda_1(q)=&\ \left[\begin{array}{c} l_1g \cos(q_1) + l_2g \cos(q_1 + q_2) \\ l_2g \cos(q_1 + q_2)\\ \end{array} \right] \\ :=&\ \left[\begin{array}{c} \Lambda_{11}(q)\\ \Lambda_{12}(q)\\ \end{array} \right]. \end{align} $ 
We choose
$ \begin{align} \Lambda=\left(\begin{array}{cc} m_1l_{c2} + m_2l_1 & 0 \\ 0 & m_2l_{c2}\\ \end{array}\right). \end{align} $ 
Consider
$ \begin{align*} H(q, p, \hat{\theta})&=K(q, p) + P_g(q) +\frac{1}{2}(\hat{\theta}\theta)^{ T}\Gamma_0(\hat{\theta}\theta)\\ &=\frac{1}{2}p^{ T} M^{1}(q)p + \frac{1}{2}(qq^0)^{ T} \Lambda (qq^0)\\ &\quad +\frac{\Gamma_0}{2}(\hat{\theta}\theta)^2 \end{align*} $ 
as the Hamilton function and
$ \dot{\hat{\theta}}=\Gamma_0^{1}\Lambda_1^{ T}\dot q_t. $  (35) 
The prefeedback law can be designed as follows:
$ \begin{align} \begin{cases} \tau_t=\Lambda_1\hat{\theta}_tK_{D1}\dot q_t+u_t\\ \dot{\hat{\theta}}=\Gamma_0^{1}\Lambda_1^{ T}\dot q_t \end{cases} \end{align} $  (36) 
where
According to Theorem 1, by the Hamilton function (35) and the prefeedback law (36), system (33) can be transformed into the following delayed Hamiltonian system
$ \begin{align} \begin{cases} \dot X=[J(X)R(X)]\nabla H(X) +[J_1(X)\\ \qquadR_1(X)]\nabla H(X_t)+ g_c u_t\\ X(t)= \phi(t), ~~~t\in[h, 0] \end{cases} \end{align} $  (37) 
where
$ \begin{align*} &J(X)= \left( \begin{array}{ccccc} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\\ \end{array} \right) \\ &J_1(X) = \left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & J_{35} \\ 0 & 0 & 0 & 0 & \frac{m_2l_{c2}}{\lambda_0} \\ 0 & 0 & J_{53} & \frac{m_2l_{c2}}{\lambda_0} & 0 \\ \end{array} \right)\\ &R_1(X)=\left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & k_{d11} & 0 & 0 \\ 0 & 0 & 0 & k_{d22} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\ & g_c = \left (\begin{array}{ccccc} 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ \end{array} \right)^{ T}. \end{align*} $ 
Obviously,
It is easy to verify that the Hamilton function in system (37) satisfies Assumption 1. In addition, according to Properties 1 and 2, the Hessian matrix of
In the following, we illustrate whether the output feedback controller
$ \begin{align*} K=\left(\begin{array}{cc} k_{11}&k_{12} \\ k_{21}&k_{22}\\ \end{array} \right) \end{align*} $ 
where
$ \begin{align*} &k_{11}=h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4}2k_{d11}\\ &k_{12}=2h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4}\\ &k_{21}=2h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4}\\ &k_{22}=h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4}2k_{d22}. \end{align*} $ 
Thus, the stabilization controller can be expressed as
$ \begin{align} u_t=& \left( \begin{array}{cc} k_{11} & k_{12} \\ k_{21} & k_{22}\\ \end{array} \right) \left( \begin{array}{ccccc} 0&0&1&0&0 \\ 0&0&0&1&0 \\ \end{array} \right) \\ & \ \left( \begin{array}{c} \frac{\partial H(X_t)}{\partial q_1} \\ \frac{\partial H(X_t)}{\partial q_2} \\ \dot q_{1t} \\ \dot q_{2t} \\ \Gamma_0(\hat{\theta}(td(t))\theta)\end{array}\right) \\[2mm] =&  \left(\begin{array}{c} u_1 \\ u_2\\ \end{array} \right) \end{align} $  (38) 
where
$ \begin{align*} u_1=&\ (h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4}2k_{d11})\dot q_{1t}\\ & + 2h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4}\dot q_{2t}\\ u_2=&\ 2h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4}\dot q_{1t}\\ & +(h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4}2k_{d22})\dot q_{2t}. \end{align*} $ 
Therefore, an adaptive controller of system (33)
$ \begin{align} \begin{cases} \tau_t=\Lambda_1\hat{\theta}_tK_{D1}\dot q_t+u_t\\ \dot{\hat{\theta}}=\Gamma_0^{1}\Lambda_1^{ T}\dot q_t \end{cases} \end{align} $  (39) 
can be expressed as follows:
$ \begin{align*} \begin{cases} \tau_{1t}= [l_1g \cos(q_1) + l_2g \cos(q_1 + q_2)]\hat{\theta}_t\\ ~~~~~~~+(k_{d11}h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4})\dot q_{1t}\\ ~~~~~~~(2h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4})\dot q_{2t} \\ \tau_{2t} = l_2g \cos(q_1 + q_2)\hat{\theta}_t\\ ~~~~~~~(2h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4})\dot q_{1t}\\ ~~~~~~~(k_{d22}h^2(\lambda+M^{2}\\dot q\^2)^2\lambda^{2}_0m^{4})\dot q_{2t}\\ \dot{\hat{\theta}}=\lambda_0^{1}[(m_1l_{c2} + m_2l_1)\dot q_1+m_2l_{c2}\dot q_2] \end{cases} \end{align*} $ 
where
In order to show the effectiveness of controller (39), simulation is carried out for system (33) whose physical parameters are the same as those in [35]. The target point
A timevarying delay is considered as
Figs. 25 are the responses of the system with the timevarying delay
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Fig. 2 Responses of joint angle position 
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Fig. 3 Control 
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Fig. 4 Estimate 
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Fig. 5 New control 
The above simulation results demonstrate that the energybased robust adaptive controller (39) is effective for the control of position and velocity, and for dealing with both unknown parameters and time delays. Furthermore, fast convergence of the controller
In this paper, the stabilization of uncertain robotic systems in the presence of input delay has been investigated. A new delayed Hamiltonian formulation has been proposed for the uncertain robotic systems under consideration. Based on the obtained delayed Hamiltonian formulation, the stabilization problem has been investigated by using LyapunovKrasovskii technique. The output feedback control law, by which the asymptotic stability of the obtained time delay Hamiltonian system is guaranteed, is determined by LMI constraints. Simulation has shown the effectiveness of the controller in handling time delays and unknown parameters in delayed robot manipulator. In the future work, the stabilization and trajectory tracking control problems will be considered for the general robotic systems and teleoperation systems under input/output delays. In this topic, the crucial difficulty may lie in the realization of the suitable delayed Hamiltonian formulation of the system under consideration.
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