2.1. Decoupled dynamics of the 2R spatial serial manipulator

The dynamic decoupling conditions of the 2R spatial serial manipulator (Fig. 1) are reviewed below.

Figure 1. 2R spatial serial manipulator.

The manipulator consists of two orthogonal links, 1 and 2, with revolute joint angles $\theta _1$ and $\theta _2$ . We will distinguish the vectors of the joint angular velocities $\boldsymbol{\dot{\theta }_{1}^{r}}$ and $\boldsymbol{\dot{\theta }_{2}^{r}}$ with $\boldsymbol{\dot{{\theta }_{1}^{r}}=\mathrm{d}\theta _{1}^{r}/\mathrm{d}t}$ , $\boldsymbol{\dot{{\theta }_{2}^{r}}=\mathrm{d}\theta _{2}^{r}/\mathrm{d}t}$ , and the vectors of the absolute angular velocities $\boldsymbol{\dot{\theta }_{1}}$ and $\boldsymbol{\dot{\theta }_{2}}$ with $\boldsymbol{\dot{\theta }_{1}}=\boldsymbol{\dot{\theta }_{1}^{r}}$ and $\boldsymbol{\dot{\theta }_{2}}=\boldsymbol{\dot{\theta }_{1}^{r}+\dot{\theta }_{2}^{r}}$ .

In the study [Reference Arakelian, Geng and Lu23], it was reported that the 2R spatial serial manipulator in Fig. 1 can be dynamically decoupled completely if the following conditions are satisfied:

• the potential energy of the manipulator is constant (or canceled), that is, the gravity balancing is achieved;

• $I_{x_2}=I_{y_2}=I^*$ , where $I_{x_2}$ and $I_{y_2}$ are the axial moments of inertia of link 2 relative to the corresponding coordinate axes of the system associated with link 2.

To ensure the gravity balancing of the manipulator, the center of mass of link 1 should be on the vertical rotation axis of the manipulator (Fig. 1) and the center of mass of link 2 should be at the intersection of the vertical and horizontal rotation axes of the manipulator. As a consequence of such a redistribution of moving masses, the motion equations become linear and decoupled:

(4) \begin{equation} \tau _1 = (I_1 + I^*)\ddot{\theta }_{1}^{r} \end{equation}

(5) \begin{equation} \tau _2 = I_{z_2}\ddot{\theta }_{2}^{r} \end{equation}

The nonlinear dynamic system is thus transformed into a double integrator model and the state space equation of each link can be defined as:

(6) \begin{equation} \dot{\boldsymbol{{x}}} = \boldsymbol{{Ax}} + \boldsymbol{{Bu}} \end{equation}

(7) \begin{equation} \boldsymbol{{y}} = \boldsymbol{{Cx}} \end{equation}

with

(8) \begin{equation} \boldsymbol{{A}} = \begin{bmatrix} 0\;\;\;\; & 1\\ 0\;\;\;\; & 0 \end{bmatrix},\ \boldsymbol{{B}} = \begin{bmatrix} 0 \\ 1/I \end{bmatrix},\ \boldsymbol{{C}} = \begin{bmatrix} 1 & 0\\ \end{bmatrix} \end{equation}

where $I$ is calculated with inertia moment of each link in (4) and (5). $I = (I_1 + I^*)$ for the first link and $I = I_{z_2}$ for the second link.

2.2. Decoupled dynamics of the 3R spatial serial manipulator

As mentioned above, the dynamic decoupling technique can be successfully implemented in the 2R spatial serial manipulators with mass redistribution. However, for the robots with more than 3 DOFs, actuator relocation technique is required in the dynamic decoupling. The supporting structure of the KUKA industrial robot KR 16R1610, which represents an 3R spatial serial manipulator (Fig. 2), is considered as an illustrative example. Its parameters are given in the Table I, which are approximate and round values.

Table I. Length and mass parameters of the industrial robot KR [Reference Vishal and Mohan32].

Figure 2. 3R spatial serial manipulator.

The KR 16R1610 consists of three orthogonal links, 1, 2, and 3, with revolute joint angles $\boldsymbol{\theta _1}$ , $\boldsymbol{\theta _2}$ , and $\boldsymbol{\theta _3}$ . We will distinguish the vectors of the joint angular velocities $\boldsymbol{\dot{\theta }_{1}^{r}}$ , $\boldsymbol{\dot{\theta }_{2}^{r}}$ and $\boldsymbol{\dot{\theta }_{3}^{r}}$ with $\boldsymbol{\dot{{\theta }_{1}^{r}}=\mathrm{d}\theta _{1}^{r}/\mathrm{d}t}$ , $\boldsymbol{\dot{{\theta }_{2}^{r}}=\mathrm{d}\theta _{2}^{r}/\mathrm{d}t}$ , $\boldsymbol{\dot{{\theta }_{3}^{r}}=\mathrm{d}\theta _{3}^{r}/\mathrm{d}t}$ , and the vectors of the absolute angular velocities $\boldsymbol{\dot{\theta }_{1}}$ , $\boldsymbol{\dot{\theta }_{2}}$ , and $\boldsymbol{\dot{\theta }_{3}}$ with $\boldsymbol{\dot{\theta }_{1}}=\boldsymbol{\dot{\theta }_{1}^{r}}$ and $\boldsymbol{\dot{\theta }_{2}}=\boldsymbol{\dot{\theta }_{1}^{r}+\dot{\theta }_{2}^{r}}$ and $\boldsymbol{\dot{\theta }_{3}}=\boldsymbol{\dot{\theta }_{1}^{r}+\dot{\theta }_{2}^{r}+\dot{\theta }_{3}^{r}}$ . Compared to the 2R spatial serial manipulators, the conditions are similar in the 3R manipulators but little more complex. With regard to potential energy, we have

(9) \begin{equation} P_t = P_{t_1}+P_{t_2}+P_{t_3} \end{equation}

with $P_{t_1}=0$ , $P_{t_2} = l_{O_2 C_1}+l_{c_2} \mathrm{sin(\theta _2^r)}$ , and $P_{t_3}=l_{O_2 C_1} +l_{2} \mathrm{sin(\theta _2^r)}+l_{c_3} \mathrm{sin(\theta _2^r+\theta _3^r)}$ , where $l_{O_{2}C_{1}}$ is the distance of the center of mass of link 1 from the joint axis $O_2$ , $l_2$ is the length of link 2, $l_{c_{2}}$ is the distance of the center of mass of link 1 from the joint axis $O_2$ , and $l_{c_3}$ is the distance of the center of mass of link 1 from the joint axis $O_3$ . Now, let us consider that the gravity balancing should be achieved in the manipulator, that is, $P_t = const$ . It can be reached if $l_{c_3} = 0$ and $l_{c_2} = m_3 l_2/m_2$ , considering the location of the center of mass outside $O_2 O_3$ .

Figure 3. Initial structure.

The kinetic energy of the initial structure (Fig. 3) can be written as:

(10) \begin{equation} K = \frac{1}{2}\left(I_{c_1} \dot{\theta }_1^2 + m_2{V_{c_2}}^2 + I_{x_2} \dot{\theta }_{x_2}^2 + I_{y_2} \dot{\theta }_{y_2}^2 + I_{z_2} \dot{\theta }_{z_2}^2 + m_3{V_{c_3}^2} + I_{x_3} \dot{\theta }_{x_3}^2 + I_{y_3} \dot{\theta }_{y_3}^2 + I_{z_3} \dot{\theta }_{z_3}^2\right). \end{equation}

where the vectors of the absolute velocities are expressed as ${\boldsymbol{{V}}}_{\boldsymbol{{c}}_{\textbf{2}}} = {\boldsymbol{{r}}_{\textbf{2}}}\times (\boldsymbol{\dot{\theta }_{1}^{r}} + \boldsymbol{\dot{\theta }_{2}^{r}})$ and $\boldsymbol{V_{c_2}} = \boldsymbol{l_2}\times (\boldsymbol{\dot{\theta }_{1}^{r}} + \boldsymbol{\dot{\theta }_{2}^{r}})$ ; the angles relative to the corresponding coordinate axes of the system are defined as: $\dot{\theta }_{x_2} = \dot{\theta }_{1}^r \mathrm{sin}(\theta _2^r)$ ; $\dot{\theta }_{y_2} = \dot{\theta }_{1}^r \mathrm{cos}(\theta _2^r)$ ; $\dot{\theta }_{z_2} = \dot{\theta }_{2}^r$ ; $\dot{\theta }_{x_3} = \dot{\theta }_{1}^r \mathrm{sin}(\theta _1^r+\theta _2^r)$ ; $\dot{\theta }_{y_3} = \dot{\theta }_{1}^r \mathrm{cos}(\theta _1^r+\theta _2^r)$ ; and $\dot{\theta }_{z_3} = \dot{\theta }_{2}^r+\dot{\theta }_{3}^r$ .

Let us consider the dynamic decoupling of the manipulator by combining two different approaches: the mass redistribution and the actuator relocation [Reference Arakelian and Xu24]. For this purpose, let us modify the design of the manipulator by placing the third actuator on the axis $O_2$ and coupling it with link 3 by a transmission. This transmission can be a parallelogram, a belt transmission, a gear transmission, or other type of motion generation.

Let us now consider the same manipulator with relocated actuator (Fig. 4). In this case, the kinetic energy can be expressed as follows:

(11) \begin{equation} K_e = \frac{1}{2}(I_{c_1} \dot{\varphi }_{z_1}^2 + m_2{V_{c_2}^2} + I_{x_2} \dot{\varphi }_{x_2}^2 + I_{y_2} \dot{\varphi }_{y_2}^2 + I_{z_2} \dot{\varphi }_{z_2}^2 + m_3{V_{c_3}^2} + I_{x_3} \dot{\varphi }_{x_3}^2 + I_{y_3} \dot{\varphi }_{y_3}^2 + I_{z_3} \dot{\varphi }_{z_3}^2). \end{equation}

where $\boldsymbol{\varphi _i}$ (where i = 1, 2, and 3) is the generalized angles defined as: $\boldsymbol{\varphi _1 = \dot{\theta }_{1}^r}$ ; $\boldsymbol{\varphi _2 = \dot{\theta }_{1}^r+\dot{\theta }_{2}^r}$ ; $\boldsymbol{\varphi _3 = \dot{\theta }_{1}^r+\dot{\theta }_{2}^r+\dot{\theta }_{3}^r}$ . Then, the components of the angles relative to the corresponding coordinate axes of the system are defined as follows: $\dot{\varphi }_{x_1}=\dot{\varphi }_{y_1} = 0$ , $\dot{\varphi }_{z_1} = \dot{\theta }_{1}^r$ ; $\dot{\varphi }_{x_2} = \dot{\theta }_{1}^r \mathrm{sin}(\theta _2^r)$ , $\dot{\varphi }_{y_2} = \dot{\theta }_{1}^r \mathrm{cos}(\theta _2^r)$ and $\dot{\varphi }_{z_2} = \dot{\theta }_{2}^r$ ; $\dot{\varphi }_{x_3} = \dot{\theta }_{1}^r \mathrm{sin}(\theta _2^r+\theta _3^r)$ , $\dot{\varphi }_{y_3} = \dot{\theta }_{1}^r \mathrm{cos}(\theta _2^r+\theta _3^r)$ and $\dot{\varphi }_{z_3} = \dot{\theta }_{2}^r+\dot{\theta }_{3}^r$ ; $\dot{\varphi }_{z_1}$ , $\dot{\varphi }_{z_2}$ and $\dot{\varphi }_{z_3}$ are the actuator velocities.

Figure 4. With relocated actuators.

Then, the following condition of the mass redistribution of the links is ensured:

(12) \begin{equation} I_{x_2} = I_{y_2} + m_3 l_{2}^{2} + m_2 r_{2}^{2} = I_{2}^{*} \end{equation}

(13) \begin{equation} I_{x_3} = I_{y_3} = I_{3}^{*} \end{equation}

Thus, the Lagrangian can be represented as follows:

(14) \begin{equation} L = \frac{1}{2}[I_{z_1}(\dot{\varphi }_{z_1})^2 + I_{2}^{*}(\dot{\varphi }_{z_1})^2 + (I_{z_2}+ m_3 l_{2}^{2} + m_2 r_{2}^{2})(\dot{\varphi }_{z_2})^2 + I_{3}^{*} (\dot{\varphi }_{1})^2 + I_{z_3}(\dot{\varphi }_{z_3})^2] \end{equation}

As a consequence of such a redistribution of moving masses, the motion equations become linear and decoupled:

(15) \begin{equation} \tau _1 = (I_{z_1}+I_{2}^{*}+I_{3}^{*})\ddot{\varphi }_{z_1} \end{equation}

(16) \begin{equation} \tau _2 = (I_{z_2}+ m_3 l_{2}^{2} + m_2 r_{2}^{2}) \ddot{\varphi }_{z_2} \end{equation}

(17) \begin{equation} \tau _3 = I_{z_3} \ddot{\varphi }_{z_3} \end{equation}

The nonlinear dynamic system is thus transformed into a double integrator model as obtained in the 2R spatial serial manipulator (6)–(8).

2.3. Motion generation via bang-bang profile

As discussed in ref. [Reference Arakelian, Geng and Lu23], to generate joint angular motion in the dynamically decoupled spatial serial manipulators, it is preferable to apply the “bang-bang profile” (Fig. 5), which enables reduction of the maximal input torque values.

Bang-bang motion profile is defined as:

(18) \begin{equation} \theta _d(t)=\left \{ \begin{array}{rcl} &\theta _d(t_0)+2\left(\dfrac{t}{t_f}\right)^2\theta _d(t_f), &\;{t_0\lt t\lt t_f/2}\\[12pt] &\theta _d(t_0)+\left [-1+4\left(\dfrac{t}{t_f}\right)-2\left(\dfrac{t}{t_f}\right)^2\right ]\theta _d(t_f), &\;{t_f/2\lt t\lt t_f}\\ \end{array} \right. \end{equation}

Figure 5. “Bang-bang” profile used for generation of motion in the dynamically decoupled 2R spatial serial manipulator.

3.1. Performance indices

For the controller design, performance indices are introduced to quantify and evaluate system performance. Two performance indices were considered in our case: the integral of the square of the error (ISE) and the maximum input torque. The ISE of the $i$ th link in the decoupled manipulator is defined as:

(19) \begin{equation} \mathrm{ISE_i} = \int _{t_0}^{t_f} e_i^2 \mathrm{d}t,\ i = 1,\cdots,n \end{equation}

where $e_i = \theta _i-\theta _{id}$ is the tracking error of the first link. The ISE discriminates between excessively over-damped and excessively under-damped systems. Tracking accuracy can be evaluated using this criterion. Another criterion introduced was the maximum input torque (MT), which represents input energy. The actuator capacity requirement also depends on the maximum input torque. Besides, the energy efficiency is affected by Joule effect, which describes the process where the energy of an electric current is converted into heat as it flows through a resistance. In electric motors, neglecting the friction, most part of the energy consumption is due to the loss by Joule effect. To evaluate the loss, a criterion is introduced as follows for the $i$ th link in the decoupled manipulator:

(20) \begin{equation} \mathrm{W_i} = \int _{t_0}^{t_f} \tau _i ^2 \mathrm{d}t,\ i = 1,\cdots,n \end{equation}

where $\tau _i$ is the actuator torque for the $i$ th link. For a DC motor, considering that the torque supplied by motor is proportional to the armature current, the criterion (20) is proportional to the energy consumption produced by Joule effect. It characterizes the energy that must be produced by the battery to allow the desired motion. Besides, the heat generated by Joule effect degrades the reliability of electrical systems components. For example, the results in ref. [Reference Jiao, Jermsittiparsert, Krasnopevtsev, Yousif and Salmani33] has revealed that the useful lifetimes of solder joints have been significantly decreased due to the Joule heating effect at high current densities. Therefore, the effect is undesirable and must be considered in controller design. Various straightforward control techniques are applied to track the bang-bang profile, and the influence on these performance indices will be investigated.

3.2. The performance of lead compensation

The primary function of the lead compensator is to reshape the frequency-response curve to provide a phase lead angle sufficient to offset the excessive lag phase associated with the components of the fixed system. The frequency-domain expression of the decoupled system is presented as follows:

(21) \begin{equation} G(\mathrm{s}) = \frac{I}{\mathrm{s}^{2}} \end{equation}

where the letter s, a complex variable, defines the Laplace transform operator; as mentioned above, $I$ can be obtained by the dynamic decoupling. For example, to each linear system (15)–(17) in the decoupled 3R manipulator, $I = (I_{z_1} + I_{2}^{*} + I_{3}^{*})$ for the first link, $I = I_{z_2}+ m_3 l_{2}^{2} + m_2 r_{2}^{2}$ for the second link, and $I = I_{z_3}$ for the third link. As all the dynamic systems are transformed into the similar structure (6)–(8) of double integrators, we will study the control performance about the first link with the biggest inertial moment, which has the lowest dynamic respond speed. $I=0.4\ \mathrm{kg m^2}$ for the first link in the decoupled 2R manipulator, and $ I = 57.60\ \mathrm{kg m^2}$ for the first link in the decoupled 3R manipulator. Since our new linear system, obtained by dynamic decoupling, was a double integrator model, a lead compensator could be used to stabilize the system by increasing the phase margin. A lead compensator in the following form will be used:

(22) \begin{equation} G_{c}(\mathrm{s}) = K_{c}\frac{T\mathrm{s}+1}{\alpha T\mathrm{s}+1} \end{equation}

where $K_{c}$ , $T$ , and $\alpha$ are the coefficients determined with the maximum phase lead angle and the cutoff frequency. Note however that the system’s dynamic characteristics need to be modified by increasing the cutoff frequency, which increases the dynamic response speed to track the bang-bang profile. The gain crossover frequency of the decoupled system is therefore increased to $89.44\ \mathrm{rad/s}$ for the first link. This implies an increase in the speed of response. Then, we assumed that the necessary maximum phase lead angle $\phi _m$ is $70^{\circ }$ , and therefore the coefficients in the lead compensator, could be determined. The Bode plot of the compensated system is presented in the Fig. 6.

As observed in the Bode plot, the maximum phase lead margin was increased to $70^{\circ }$ at the modified gain crossover frequency $89.44\ \mathrm{rad/s}$ , which improved the asymptotic stability. To see the tracking performance and the input torque, the simulation was carried out in Matlab. The value of the time step was set at $1 \times 10^{-5}\ \mathrm{s}$ . The simulation results are presented in the Figs. 7–10.

It is obvious that the systems (2R and 3R) with the modified gain crossover frequency and $70^{\circ }$ phase lead angle is capable of tracking the desired bang-bang profile trajectory. For example, in the decoupled 3R manipulator, the ISE is $8.29\times 10^{-2}$ $\mathrm{rad^2 s}$ and the maximum torque is $4.78\times 10^{2}$ Nm. The energy consumption produced by Joule effect is $1.3252\times 10^{5}$ $\mathrm{rad^2 s}$ $\mathrm{N^2 m^2 s}$ . In addition, the Tables II and III with a variety of phase lead angles were created to investigate the influence of different phase lead angles on three criteria in both 2R and 3R decoupled manipulators.

Table II. Results of different phase lead angles in the decoupled 2R spatial serial manipulator.

Figure 6. Frequency domain response with $70^{\circ }$ phase lead compensator.

Figure 7. Trajectory tracking of lead compensator in the decoupled 2R spatial serial manipulator.

Figure 8. Torque generation of lead compensator in the decoupled 2R spatial serial manipulator.

Figure 9. Trajectory tracking of lead compensator in the decoupled 3R spatial serial manipulator.

Table III. Results of different phase lead angles in the decoupled 3R spatial serial manipulator.

Figure 10. Torque generation of lead compensator in the decoupled 3R spatial serial manipulator.

The table shows that the ISE is diminished, and the maximum torque increased when the phase lead angle is increased. In application, a large phase lead angle is required to lower the actuator capacity and energy consumption. However, for a system requiring high tracking accuracy, a small phase lead angle is needed. Obviously, there is a trade-off between the tracking accuracy and input energy efficiency, which also means that the required performance can be achieved by adjusting the phase lead angle, which facilitates the design of the robot manipulator controller. To achieve a more satisfactory control performance, full-state feedback method is considered in the following parts.

3.3. The performance of linear quadratic regulator

Since the double integrator model system is unstable, the control technique should increase the stability of the system. Therefore, the speed of response and tracking error requirement should be improved to track the bang-bang profile. Linear quadratic regulator (LQR) is a powerful control technique, which stabilizes and operates a linear dynamic system by minimizing a given cost function. In order to implement a LQR controller, a linear time-invariant (LTI) dynamic system must be available, which has already been obtained by the dynamic decoupling, and the system must be completely controllable. We assumed that all state variables were measurable and also available for feedback in our system. The controllability matrix is given by:

(23) \begin{equation} \textbf{C} = \begin{bmatrix} \boldsymbol{{B}} & \boldsymbol{{AB}} & \boldsymbol{{A}}^{\textbf{2}}\textrm{B}&\ldots &{\boldsymbol{{A}}^{\boldsymbol{{n}}-1}\boldsymbol{{B}}} \end{bmatrix}= \begin{bmatrix} 0 & 1/I\\ 1/I & 0 \end{bmatrix} \end{equation}

Since the control matrix C of the state space equation obtained has full row rank 2, this decoupled dynamic system is controllable. LQR algorithme aims to minimize the quadratic cost function as follows:

(24) \begin{equation} \int _{0}^{\infty } (\boldsymbol{{x}}^{\top }\boldsymbol{{Q}}\boldsymbol{{x}}+\boldsymbol{{u}}^{\top }\boldsymbol{{R}}\boldsymbol{{u}}) {\textbf{d}\boldsymbol{{x}}} \end{equation}

where $Q$ and $R$ are the weights related to the system state error (angle and angular velocity) and the control input (motor torque), respectively. The appropriate selection of $Q$ and $R$ determines the control performance. The optimal control law to minimize the quadratic cost function is given by:

(25) \begin{equation} {\boldsymbol{{u}} = -\boldsymbol{{Kx}}} \end{equation}

where

(26) \begin{equation} \boldsymbol{{K}} = \boldsymbol{{R}}^{-\textbf{1}} \boldsymbol{{B}}^{\boldsymbol\top } \boldsymbol{{P}} \end{equation}

and $P$ is found by solving the continuous time algebraic Riccati equation:

(27) \begin{equation} {\boldsymbol{{A}}^{\boldsymbol\top } \boldsymbol{{P}} + \boldsymbol{{PA}} - \boldsymbol{{P}} \boldsymbol{{B}} \boldsymbol{{R}}^{-\textbf{1}} \boldsymbol{{B}}^{\boldsymbol\top } \boldsymbol{{P}} + \boldsymbol{{Q}} = \textbf{0}} \end{equation}

The design steps may be stated as follows:

1. 1. Solve equation, the reduced matrix Riccati equation, for the matrix $P$

2. 2. Substitute this matrix $P$ into equation. The resulting matrix $K$ is the optimal one.

The simulation was carried out in Matlab software with $I = 0.4 \ \mathrm{kg m^2}$ (the first link with lower frequency) and the time step set at $1 \times 10^{-5}$ s. By using the steps mentioned above, the optimal control gains $K_1$ and $K_2$ were calculated as $K_1 = 3464.10$ and $K_2 = 634.64$ . The values of $\boldsymbol{{Q}}$ and $\boldsymbol{{R}}$ were initialized as follows for the control of the first link:

(28) \begin{equation} \boldsymbol{{Q}} = \begin{bmatrix} 600,000 & 0\\ 0 & 20,000 \end{bmatrix}, \boldsymbol{{R}} = \begin{bmatrix} 0.05 & 0\\ 0 & 0.05 \end{bmatrix} \end{equation}

The initial and final values of the rotating angles were as follows: $\theta _i = 0^{\circ }$ and $\theta _f = 90^{\circ }$ . The tracking performance and the input torque of the decoupled 2R manipulator were obtained in the Figs. 11 and 12.

The simulation has also been carried out in the decoupled 3R manipulator ( $I = 57.60 \ \mathrm{kg m^2}$ ) with $K_1 = 109,540$ and $K_2 = 40,160$ . The Figs. 13 and 14 present the simulation results.

Figure 11. Trajectory tracking of LQR in the decoupled 2R manipulator.

Figure 12. Torque generation of LQR in the decoupled 2R manipulator.

Figure 13. Trajectory tracking of LQR in the decoupled 3R manipulator.

Figure 14. Torque generation of LQR in the decoupled 3R manipulator.

Figure 15. Joint friction curve.

From the simulation results, it was observed that in the decoupled dynamic system, LQR results in excellent control performance by the bang-bang profile. For example, in the 3R decoupled manipulator, the ISE is $0.2144\ \mathrm{rad^2 s}$ and the maximum torque is $3.64\times 10^{5}$ Nm. The energy consumption produced by Joule effect is $1.3109\times 10^{5}$ $\mathrm{N^2 m^2 s}$ . From the data obtained above, it can be seen that LQR controller performs better than lead compensator in terms of energy consumption. However, these linear controllers did not take the parameter uncertainty and external unknown disturbances into account in the control process, which may destabilize the decoupled dynamic system.

4.1. The performance of model-reference adaptive control

The parameter uncertainty or variation occurs in many practical problems. For instance, robot manipulators carry out their tasks such as “pick and place” with inaccurate inertial parameters. It may cause inaccuracy or instability for control systems. In our case, thanks to the dynamic decoupling, the 2R and 3R spatial manipulator models have been simplified as a double integrator. However, it is very strict to assume that the inertial parameters of the dynamically decoupled manipulator are the exactly same as we know in the design process. To resolve the problem, model-reference adaptive control (MARC) has been proposed in ref. [Reference Slotine and Li37], which can be adopted for the decoupled dynamic system. An adaptation mechanism in MARC aims to adjust the parameters in the control law so that the perfect tracking performance can be achieved [Reference Slotine and Li37]. The MRAC law is presented as follows:

(31) \begin{equation} u = \hat{I}(\ddot{\theta }_d-2\lambda \dot{\hat{\theta }}-\lambda ^2\hat{\theta }) \end{equation}

where tracking error $\hat{\theta } = \theta -\theta _d$ ; $\lambda$ is the positive constant chosen to reflect the performance specification. $\hat{I}$ is defined as the estimate of inertial. The following update law is used to adjust the parameter:

(32) \begin{equation} \dot{\hat{I}} = -\gamma v s \end{equation}

where $\gamma$ is a position constant called the adaptation gain; $v = \ddot{\theta }_d-2\lambda \dot{\hat{\theta }}-\lambda ^2\hat{\theta }$ . The combined tracking error $s$ is defined as:

(33) \begin{equation} s = \dot{\hat{\theta }}+\lambda \hat{\theta } \end{equation}

The equations above present an adaptation mechanism based on system signals. The proof of the global asymptotic stability is demonstrated in ref. [Reference Slotine and Li37] with the Lyapunov theory. The interested reader may refer to ref. [Reference Slotine and Li37] for further details. The simulation is carried out in Matlab with the following parameter. For the decoupled 2R manipulator: $\gamma _1=2$ ; $\lambda _1=50$ ; and $\hat{I}_1(0) = 0.4$ $\mathrm{kgm^2}$ . The real inertial value is $0.5$ $\mathrm{kgm^2}$ . For the decoupled 3R manipulator: $\gamma _2=200$ ; $\lambda _2=50$ ; and $\hat{I}_2(0) = 57.60$ $\mathrm{kgm^2}$ . The real inertial value is $72$ $\mathrm{kgm^2}$ . The time step is set as $1 \times 10^{-5}$ s.

As shown in the Figs. 16 and 17, there is no significant difference between the desired trajectory and the trajectory generated by the MRAC. At the beginning of the control process in the Figs. 18 and 19, an increasing torque is generated due to the wrong inertial information but then the torque generation becomes almost the same as the bang-bang profile by adjusting the inertial value during the process. The control performance indices were calculated from the simulations:

Figure 16. Trajectory tracking of MRAC in the decoupled 2R manipulator.

Figure 17. Trajectory tracking of MRAC in the decoupled 3R manipulator.

Figure 18. Torque generation of MRAC in the decoupled 2R manipulator.

As can be seen from the Table IV, MRAC consumes less energy than LQR controller in the decoupled dynamic system but the maximum torque is increased a little. Besides, the tracking accuracy of MRAC is much better than LQR controller. It should be noted that MARC could maintain a high tracking accuracy in the presence of parameter uncertainty, which is attractive for the design of controller.

4.2. The performance of modified twisting controller

A lot of nonlinear controllers have been proposed in the literature to overcome the external disturbances [Reference Slotine and Li37]. In the friction model (30), it should be noticed that there always exist the discontinuous terms $\mathrm{sign}(\dot{\theta })$ . Attributed to these discontinuous terms of friction, the continuous control algorithms are unable to stabilize the dynamic system. Thus, the discontinuous controllers are brought into sight to resolve the stability problem. Among them, the twisting algorithms are well recognized for the robustness properties and finite time stability. The controllers, which have switching terms, are capable of forcing the dynamic system to the zero dynamics of error in spite of the external disturbances. However, in the twisting controller, the chattering occurs in the dynamics system due to the important switching gain and results in low reliability of mechanical systems. Yury Orlov has proposed a modification of the twisting controller to resolve the problem, which aims to avoid the undesired chattering phenomenon appearing in the closed loop when driven by an important switching input. The main advantages of this control are to ensure a convergence in finite time and to be continuous. The effectiveness of the proposed method was demonstrated for a double integrator model [Reference Orlov, Aoustin and Chevallereau38]. The feedback law is stated as follows:

(34) \begin{equation} u = -\mu \left | e_2 \right |^\epsilon \mathrm{sign}(e_2)-v\left | e_1 \right |^{\dfrac{\epsilon }{2-\epsilon }}\mathrm{sign}(e_1) \end{equation}

where $e_1 = x_1-x_{1d}$ and $e_2 = x_2-x_{2d}$ . Parameters $v\gt \mu \gt 0$ and $\epsilon \in [0,1)$ are proposed to globally stabilize the double integrator. Suppose that the external disturbance friction satisfies the growth condition:

Table IV. Results of two controllers in the presence of parameter uncertainty.

Figure 19. Torque generation of MRAC in the decoupled 3R manipulator.

Figure 20. Trajectory tracking of modified twisting controller in the decoupled 2R manipulator.

Figure 21. Trajectory tracking of modified twisting controller in the decoupled 3R manipulator.

(35) \begin{equation} \left | \tau _f(\dot{\theta }) \right | \leq \mu _0 \left | e_2 \right |^\epsilon \end{equation}

where $\mu _0$ is the upper bound of the external disturbance. According to ref. [[Reference Orlov, Aoustin and Chevallereau38], Theorem 1], for any disturbance $w$ satisfying the growing condition, the continuous closed-loop system is globally asymptotically stable if $\mu _0 \leq \mu$ . The proof is given in ref. [Reference Orlov, Aoustin and Chevallereau38] by applying the Lyapunov function and invariance principle. The simulation is carried out in Matlab with the following parameters of controller: $\mu _1 = 1000$ , $v_1 = 1500$ , and $\epsilon _1=0.8$ for the 2R decoupled maniplator, and $\mu _2 = 140,000$ , $v_2 = 230,000$ , and $\epsilon _2=0.8$ for the 2R decoupled maniplator. In terms of friction, the parameters in the decoupled 2R manipulator are set to be: $F_c = 3.40 \times 10^{-2}\ \mathrm{Nm}$ ; $F_s = 4.60 \times 10^{-2}\ \mathrm{Nm}$ ; $F_v = 3.68 \times 10^{-4}\ \mathrm{Nm/(rad/s)}$ ; $\dot{\theta }_s=10.68\ \mathrm{rad/s}$ ; and $\beta = 1.93$ . In the decoupled 3R manipulators, the friction is supposed to be 30 times of the friction in the decoupled 2R manipulator. The errors of the moment of inertia in the previous section are also considered in the decoupled manipulator. In the simulation, the time step is set as $1 \times 10^{-5}$ s.

As shown in the Figs. 20 and 21, no significant tacking differences were found after the rise time between the desired trajectory and the trajectory generated by the modified twisting controller. Even if there exists the discontinuous terms in the friction model (described by sign functions without any approximation in the simulation), it can be observed from the Figs. 22 and 23 that the torque generated by the modified twisting controller is close to the ideal bang-bang profile except the discontinuous points. It has been shown in the Figs. 24 and 25 that the maximum difference after the rise time between the actuator torque and the ideal torque is $0.0626$ and $7.1040\ \mathrm{Nm}$ (both below $2 \%$ of the ideal torque) for the decoupled 2R and 3R spatial serial manipulators. The result shows that an appropriate choice of $\varepsilon$ in the modified twisting controller attenuates the chattering effects. Then, the control performance indices could be calculated from the simulation:

Table V. Results of different controllers in the presence of joint friction.

Figure 22. Torque generation of modified twisting controller in the decoupled 2R manipulator.

Figure 23. Torque generation of modified twisting controller in the decoupled 3R manipulator.

As can be seen from the Table V, the best control performance is attained with the modified twisting controller, which consumes less energy and maintains high tracking accuracy. The simulation results demonstrate that the modified twisting controller is effective to stabilize the decoupled dynamic system in the presence of parameter uncertainty and external disturbances.

Figure 24. Difference between the actuator torque and the ideal torque in the decoupled 2R manipulator.

Figure 25. Difference between the actuator torque and the ideal torque in the decoupled 3R manipulator.