Model tracking controller design of robot manipulator system with disturbances

Abstract

In the model tracking control of robot manipulator system, the treatment of nonlinear uncertainty in the system has always been an active research field. This article establishes a kinetic equation for robot manipulator system based on Lagrange equation and proposes a model tracking control system based on differential divisor. On this basis, this article proposes a model tracking control scheme for robot manipulator systems with disturbances. The proposed scheme is robust stable under the external disturbances. At last, the system simulation approach is employed to verify the effectiveness of this scheme on robot manipulator control.

Keywords

Robotics and artificial intelligence model tracking control robot manipulator system robust design

Introduction

Since extensive and in-depth researches have been studied on robots which are constantly launched in the fields of environment perception, motion control, positioning, and navigation, robots are increasingly applied in fields such as troubleshooting, metal cutting, welding, and paint spraying.^1–6 In addition, the developments are also achieved in the fields of humanoid robots and rehabilitation robots.^7–9 In recent years, lots of research findings have sprung up, which focus on the theory, algorithm, and application of tracking control. Tracking control of robotic systems is a difficult problem because the dynamics of these systems is highly nonlinear.^10,11 Mariottini et al.¹² have developed a status-based predictive control method to track moving targets, thus achieving real-time motion tracking of soccer robot goalkeepers. Predictive control method provides an excellent enlightenment for obtaining more extensive control strategies in a dynamic uncertain environment, and the solution of various optimization-based problems, such as path planning and discrete event dynamic system monitoring, has gained extensive attention and study.

To realize the control of robots, the requirements on steady-state characteristic and transient characteristic must be satisfied. In fact, system errors might have occurred during system modeling when linear treatment is carried out in nonlinear cases, inducing system instability or deterioration owing to the response characteristic. The main factors of errors occurred during the design of system model, including system simplification, linearization of nonlinear systems, approximate time-independent variation of time-varying systems, order reduction of high-order elements, and parameter identification. Stability of an uncertain system is one of the essential considerations in system design, while robustness is exactly the property that bears system fluctuation and uncertainty. The system which maintains such property is called the robust control system.^13–16

This research carries out tracking control over the uncertain system^14,17–24 and takes the third power differential of link data as input signal. Thus, the difference between the output signals of the system is generated, and at the same time, the output signal of the reference model is converged to zero to complete the design of the robust control system. Without loss of generality, a three-link horizontal multi-joint robot manipulator is selected, and the transformation matrices of link coordinate system are obtained by Denavit–Hartenberg (D-H)^25–28 method to establish the robust control system for the robot manipulator. In addition, the following researches are also conducted. According to the dynamic characteristics of kinetic parameters during the operation of the robot manipulator, conduct system modeling and research on robust control of robot manipulator via Jerk signal and based on tracking control theory, as well as demonstrate the internal stability of the system; discuss the influence on system output error of controlled object data change; design the control system targeting at the low sensitivity of controlled object; study the response characteristics of the robot manipulator of the controlled object with unknown parameters; conduct simulation research on the tracking performance of system output signal and reference system signal; and verify the effectiveness of the robust control research method applied in the research.

In this article, section “Design of the model tracking controller for robot manipulator system” provides the design of the model tracking controller for robot manipulator system. Section “Internal state bounded of the robot manipulator system with disturbances” investigates the internal state bounded of the robot manipulator system. Based on the model, section “Simulation results” provides numerical examples and demonstrates the effectiveness of this design method by means of simulation. The last section of this article is the “Conclusion,” which provides a summary of this research.

Design of the model tracking controller for robot manipulator system

Motion equation of robot manipulator

Taking into consideration the torque, weight, viscous friction coefficient, and external disturbance, the motion equation of n-freedom robot manipulator is generally expressed as follows²⁹

M (q (t)) \overset{\cdot\cdot}{q} (t) + h (q (t)) + V \overset{\cdot}{q} (t) + g (q (t)) = τ (t) + d (t)

(1)

where $q (t) \in R^{n \times 1}$ is the link parameter, $M (q (t)) \in R^{n \times n}$ is the inertia determinant, $h (q (t), \overset{\cdot}{q} (t)) \in R^{n \times 1}$ is the centrifugal force and Coriolis effect, $V \in R^{n \times n}$ is the viscous friction determinant, $g (q (t)) \in R^{n \times 1}$ is the gravity, $τ (t) \in R^{n \times 1}$ is the control input, and $d (t) \in R^{n \times 1}$ is the external disturbance. Introduce $ϕ$ that is used to determine functions $K (q (t), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t))$ and contains uncertain data of the system, and system (1) is changed to

K (q (t)), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t) ϕ = τ (t) + d (t)

(2)

where $K (q (t), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t)) \in R^{n \times n}$ , $ϕ \in R^{n \times 1}$ , and $Δ ϕ = \hat{ϕ} - ϕ$ are the errors; $ϕ$ is the true value; and $\hat{ϕ}$ is the nominal value. Further solving equation (2), we obtain equation (3)

τ (t) = μ {v (t) - Q (t)} + K (q (t), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t)) \hat{ϕ}

(3)

where $v (t)$ is the new control input, $μ (t)$ is the strength of Jerk signal, $Q (t) = D (p) q = p^{3} q$ , and $p = d / dt$ .²¹ Therefore, we have²²

Q (t) = v (t) + \frac{1}{μ} K (q (t), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t)) ϕ + \frac{1}{μ} d (t)

(4)

Design theory of model tracking control

This part discusses the general theory regarding the design of model tracking control system, thus preparing for the research on model tracking control of robot manipulator system with disturbances. The nonlinear control system is defined below

\overset{\cdot}{x} (t) = Ax (t) + Bu (t) + B_{f} f (v (t)) + d (t)

(5)

v (t) = C_{f} x (t) + d_{v} (t)

(6)

y (t) = Cx (t) + d_{0} (t)

(7)

where $A \in R^{n \times n}$ , $B \in R^{n \times l}$ , $B_{f} \in R^{n \times l}$ , $C_{f} \in R^{l \times n}$ , $C \in R^{l \times n}$ , $x (t) \in R^{n}$ , $u (t) \in R^{l}$ , $v (t) \in R^{l}$ , $f (v (t)) \in R^{l}$ , $d (t) \in R^{n}$ , $d_{0} (t) \in R^{l}$ , and $d_{v} (t) \in R^{l}$ .

The reference model is defined as below, which is assumed to be controllable and observable²³

{\overset{\cdot}{x}}_{m} (t) = A_{m} x_{m} (t) + B_{m} r_{m} (t)

(8)

y_{m} (t) = C_{m} x_{m} (t)

(9)

where $A_{m} \in R^{n_{m} \times n_{m}}$ , $B_{m} \in R^{n_{m} \times l_{m}}$ , $C_{m} \in R^{l_{m} \times n_{m}}$ , $x_{m} (t) \in R^{n \times m}$ , $r_{m} (t) \in R^{l \times m}$ , and $y_{m} (t) \in R^{l}$ . $y (t)$ is the available state output vector, $v (t)$ is the measurement output vector, $u (t)$ is the input vector, $x (t)$ is the internal state vector whose elements are available, and $y_{m} (t)$ is the model output. Nonlinear function $f (v (t))$ satisfies the following conditions

| | f (v (t)) | | \leq α + β | | v (t) | |^{γ}

(10)

where $α \geq 0, β \geq 0, γ \geq 0$ . $| | \cdot | |$ is the Euclidean norm. Disturbances $d (t)$ and $d_{0} (t)$ are bounded and satisfy

D_{d} (t) d (t) = 0, D_{d} (t) d_{0} (t) = 0

(11)

where $D_{d} (t)$ is a scalar characteristic polynomial of disturbances and $D_{d} (p) = n_{d}, Γ (D_{d} (p)) = 1$ . The output error is given as follows²¹

e (t) = y (t) - y_{m} (t)

(12)

The control system is designed to obtain a control law which eliminates errors and keeps internal states bounded. Then, we have

\begin{matrix} y (t) = C {[pI - A]}^{- 1} Bu (t) \\ + C {[pI - A]}^{- 1} B_{f} f (v (t)) \\ + C {[pI - A]}^{- 1} d (t) + d_{0} (t) \\ y_{m} (t) = C_{m} {[pI - A_{m}]}^{- 1} B_{m} r_{m} (t) \end{matrix}

where $C [pI - A]^{- 1} B = N (p) / D (p)$ , $D (p) = det [pI - A]$ , $N (p) = Cadj [pI - A] B$ , $C [pI - A]^{- 1} B_{f} = N_{f} (p) / D (p)$ , $N_{f} (p) = Cadj [pI - A] B_{f}$ , $C_{m} [pI - A_{m}]^{- 1} B_{m} = N_{m} (p) / D_{m} {(p)}_{m}$ , $D_{m} (p) = det [pI - A_{m}]$ , and $N_{m} (p) = C_{m} adj [pI - A_{m}] B_{m}$ . $D (p) y (t)$ and $D_{m} (p) y_{m} (t)$ are expressed below

\begin{matrix} D (p) y (t) = N (p) u (t) + N_{f} (p) f (v (t)) + w (t) \\ D_{m} (p) y_{m} (t) = N_{m} (p) r_{m} (t) \end{matrix}

where $N (p) = diag (p^{σ_{i}}) N_{r} + \tilde{N} (p)$ . The frequencies of $N (p), N_{f} (p), and N_{m} (p)$ are $σ_{i}, σ_{f_{i}}, and σ_{m_{i}}$ , respectively. In addition, $Γ_{r} (N (p)) = N_{r}, | N_{r} | \neq 0$ and $w (t) = Cadj [pI - A] d (t) + D (p) d_{0} (t)$ . Since the disturbances satisfy equation (11), we have $D_{d} (p) w (t) = 0$ . The first step of designing is choosing a monic and stable polynomial $T (p)$ with a degree of $ρ (ρ \geq n_{d} + 2 n - n_{m} - 1 - σ_{i})$ . Then, $R (p)$ and $S (p)$ can be derived

T (p) D_{m} (p) = D_{d} (p) D (p) R (p) + S (p)

(13)

Next, we calculate $T (p) D_{m} (p) e (t)$

T (p) D_{m} (p) e (t) = D_{d} (p) D (p) R (p) y (t) + S (p) y (t) - T (p) D_{m} (p) y_{m} (t)

Therefore

e (t) = \frac{Q (p) N_{r}}{T (p) D_{m} (p) [u (t) + N_{r}^{- 1} Q^{- 1} (p) {D_{d} (p) R (p) N (p) - Q (p) N_{r}} u (t) + N_{r}^{- 1} Q^{- 1} (p) S (p) y (t) + N_{r}^{- 1} Q^{- 1} (p) D_{d} (p) R (p) N_{f} (p) f (v (t)) + N_{r}^{- 1} Q^{- 1} (p) T (p) N_{m} (p) r_{m} (t)]}

(14)

Since $| N_{r} | \neq 0$ , $u (t)$ can be obtained by letting the right side of equation (14) be equal to 0²⁵

\begin{matrix} u (t) & = - N_{r}^{- 1} Q^{- 1} (p) {D_{d} (p) R (p) N (p) \\ - Q (p) N_{r}} u (t) - N_{r}^{- 1} Q^{- 1} (p) S (p) y (t) \\ - N_{r}^{- 1} Q^{- 1} (p) D_{d} (p) R (p) N_{f} (p) f (v (t)) \\ + u_{m} (t) \end{matrix}

(15)

u_{m} (t) = N_{r}^{- 1} Q^{- 1} (p) T (p) N_{m} (p) r_{m} (t)

(16)

$u (t)$ of equation (15) is obtained from $e (t) = 0$ , and model tracking control system can be realized if system internal states are bounded. Based on the above-mentioned basic theory for model tracking control, hereafter a control of the robot manipulator system will be designed accordingly.

Model tracking controller design of robot manipulator system with disturbances

Based on the model tracking control theory, this section discusses the design of a controller that takes into account the robot manipulator subject to external disturbance for low sensitivity of data change and analyzes the internal stability of the control system.

From equation (4), we obtain system control object and reference system model³¹

D (p) q (t) = v (t) + \frac{1}{μ} K (q (t), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t)) Δ ϕ + \frac{1}{μ} d (t)

(17)

D_{m} (p) q_{m} (t) = r_{m} (t)

(18)

The output error is

e (t) = q (t) - q_{m} (t)

(19)

In equations (15) and (16), $\partial T (p) = ρ$ , $\partial D_{m} (p) = n_{m}$ , $\partial D (p) = n_{p} = 3$ , $\partial D_{d} (p) = n_{d}$ , $\partial R (p) = ρ + n_{m} - n_{p} - n_{d}$ , and $\partial S (p) = n_{p} + n_{d} - 1$ . Therefore, we have

\begin{matrix} T (p) D_{m} (p) e (t) = T (p) D_{m} (p) {q (t) - q_{m} (t)} \\ = D_{d} (p) R (p) v (t) + D_{d} (p) R (p) \frac{1}{μ} {K (q (t), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t)) Δ ϕ} \\ + S (p) q (t) - T (p) r_{m} (t) \end{matrix}

where $Q (p)$ is monic and stable multinomial, and $\partial Q (p) = ρ + n_{m} - n_{p}$

\begin{matrix} e (t) = \frac{Q (p)}{T (p) D_{m} (p)} [v (t) + Q^{- 1} (p) {D_{d} (p) R (p) - Q (p)} \\ \cdot v (t) + Q^{- 1} (p) S (p) q (t) - Q^{- 1} (p) T (p) r_{m} (t)] \\ + \frac{D_{d} (p) R (p)}{T (p) D_{m} (p)} \frac{1}{μ} {K (q (t), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t)) Δ ϕ} \end{matrix}

(20)

So, model tracking controller of robot manipulator system with disturbances can be derived

\begin{matrix} v (t) = & - Q^{- 1} (p) {D_{d} (p) R (p) - Q (p)} v (t) \\ - Q^{- 1} (p) S (p) q (t) + Q^{- 1} (p) T (p) r_{m} (t) \end{matrix}

(21)

Note that there are connections between the polynomial matrices and the system matrices

{\hat{H}}_{1} {[pI - F_{1}]}^{- 1} G_{1} = Q^{- 1} (p) {D_{d} (p) R (p) - Q (p)}

(22)

E_{2} + {\hat{H}}_{2} {[pI - F_{2}]}^{- 1} G_{2} = Q^{- 1} (p) S (p)

(23)

E_{m} + {\hat{H}}_{m} {[pI - F_{m}]}^{- 1} G_{m} = Q^{- 1} (p) T (p)

(24)

The followings must be satisfied²¹

{\overset{\cdot}{ξ}}_{1} (t) = F_{1} ξ_{1} (t) + G_{1} v (t)

(25)

{\overset{\cdot}{ξ}}_{2} (t) = F_{2} ξ_{2} (t) + G_{2} q (t)

(26)

{\overset{\cdot}{ξ}}_{m} (t) = F_{m} ξ_{m} (t) + G_{m} r_{m} (t)

(27)

where $| pI - F_{i} | = | Q (p) | (i = 1 - 4)$ . So, the state space expression of $v (t)$ is shown as follows

v (t) = - {\hat{H}}_{1} ξ_{1} (t) - E_{2} q (t) - {\hat{H}}_{2} ξ_{2} (t) + v_{m} (t)

(28)

v_{m} (t) = E_{m} r_{m} (t) + {\hat{H}}_{m} ξ_{m} (t)

(29)

In this research, if $D_{d} (p) = p$ and $n_{d} = 1$ , set $ρ = 3$ , and select stable multinomial $T (p)$ , the frequencies of multinomial are $\partial T (p) = 3$ , $\partial D_{m} (p) = 3$ , $\partial D_{d} (p) = 1$ , and $\partial D (p) = 3$ . To obtain $R (p)$ and $S (p)$ , the following multinomial specifications are provided: $T (p) = Q (p) = D_{m} (p) = D (p) (p - α)^{3}$ , $D_{d} (p) = p + β$ , $R (p) = p^{2} - (3 α + β) p + (3 α^{2} + 3 α β + β^{2})$ , $S (p) = - α^{3} - 3 α^{2} β - 3 α β^{2} - β^{3}$ , According to equation (20), we have

e (t) = \frac{D_{d} (p) R (p)}{T (p) D_{m} (p)} \frac{1}{μ} {K (q (t), \overset{\cdot}{q} (t), \overset{\cdot\cdot}{q} (t)) Δ ϕ}

(30)

where $q (t), \overset{\cdot}{q} (t), and \overset{\cdot\cdot}{q} (t)$ are functions about triangle and multinomial. When $μ$ tends to be infinitely large, $e (t)$ is infinitely small. Transfer function $G (p)$ is

G (p) = \frac{D_{d} (p) R (p)}{T (p) D_{m} (p)}

(31)

Substitute the specific set values and we have

G (p) = \frac{(p + β) [p^{2} + (3 α + β) p + (3 α^{2} + 3 α β + β^{2})]}{{(p - α)}^{6}}

(32)

Example 1

If $α = 3, β = 4$ , the transfer function $G (p)$ is simplified into $G (p) = (p + 4) (p^{2} + 13 p + 79) / {(p - 3)}^{6}$ . Let $p = j ω$ , gain and phase curves are shown in Figure 1.³¹ $max | G (j ω) | \approx - 50 dB$ , so it verifies $e (t)$ to be extremely small.

Figure 1.

Bode diagram (frequency response of the function G(p)).

Internal state bounded of the robot manipulator system with disturbances

As a result of the increase in robots in various fields, the mechanical stability of specific robots has become an important subject of research.³⁰ In this section, we will pay attention on internal state bounded of the robot manipulator system as follows:

Let ${\overset{\cdot}{q}}_{1} (t) = q_{2} (t)$ , ${\overset{\cdot}{q}}_{2} (t) = q_{3} (t)$ , ${\overset{\cdot}{q}}_{3} (t) = v (t) + K (q_{1} (t), q_{2} (t), q_{3} (t)) Δ ϕ / μ + d (t) / μ$ , and $z (t) = [q_{1}^{T} (t), q_{2}^{T} (t), q_{3}^{T} (t), ξ_{1}^{T} (t), ξ_{2}^{T} (t)]^{T}$ . Now, system is defined by

\overset{\cdot}{z} (t) = A_{s} z (t) + B_{s} f (v_{z} (t)) + d_{s} (t)

(33)

v_{z} (t) = C_{s} z (t)

(34)

where $f (v_{z} (t)) = K (q_{1} (t), q_{2} (t), q_{3} (t)) Δ ϕ / μ$ , $v_{z} (t) = [q_{1}^{T} (t), q_{2}^{T} (t), q_{3}^{T} (t)]^{T}$ , and $A_{s} = [\begin{matrix} 0 & I & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 \\ - E_{2} & 0 & 0 & - H_{1} & - H_{2} \\ - G_{1} E_{2} & 0 & 0 & F_{1} - G_{1} H_{1} & - G_{1} H_{2} \\ G_{2} & 0 & 0 & 0 & F_{2} \end{matrix}]$

B_{s} = {[\begin{matrix} 0 & 0 & I & 0 & 0 \end{matrix}]}^{T}

d_{s} (t) = [\begin{matrix} 0 \\ 0 \\ v_{m} (t) + \frac{1}{μ} d (t) \\ 0 \\ 0 \end{matrix}]

C_{s} = [\begin{matrix} I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 \end{matrix}]

We calculated characteristic polynomial $| pI - A_{s} |$

Hereafter²⁵

| pI - A_{s} | = Q (p) T (p) D_{m} {(p)}^{n}

(35)

Consider a Lyapunov function, for symmetric positive definite matrices $P$ and $Q$

A_{s}^{T} P + {PA}_{s} | = - Q

(36)

V (t) = \frac{1}{2} z^{T} (t) Pz (t)

(37)

We have

\overset{\cdot}{V} (t) = - \frac{1}{2} z^{T} (t) Qz (t) + z^{T} (t) {PB}_{s} f (v_{z} (t)) + z^{T} (t) {Pd}_{s} (t)

(38)

where $| | f (v_{z} (t)) | | \leq (1 / μ) | | K (q_{1} (t), q_{2} (t), q_{3} (t)) | | | | Δ ϕ | |$ , $| | K (q_{1} (t), q_{2} (t), q_{3} (t)) | | \leq K_{1} + K_{2} | | v_{z} (t) | |^{M}$ . Let $| | f (v_{z} (t)) | | \leq (1 / μ) (C_{1} + C_{2} | | v_{z} (t) | |^{M})$ , we have²⁷

\overset{\cdot}{V} (t) \leq - μ_{1} | | z (t) | |^{2} + M_{1} | | z (t) | | + \frac{1}{μ} (M_{2} | | z (t) | | + M_{3} | | z (t) | |^{M + 1}) = | | z (t) | | {(M_{1} + \frac{M_{2}}{μ}) - μ_{1} | | z (t) | | + \frac{M_{3}}{μ} | | z (t) | |^{M}} \leq V {(t)}^{\frac{1}{2}} {(q_{1} + \frac{q_{2}}{μ}) - q_{3} V {(t)}^{\frac{1}{2}} + \frac{q_{4}}{μ} V {(t)}^{\frac{M}{2}}}

Let $F (V (t)) = (q_{1} + (q_{2} / μ)) - q_{3} V {(t)}^{\frac{1}{2}} + (q_{4} / μ) V {(t)}^{\frac{M}{2}}$ . When $V (t) = V_{m} = {(μ q_{3} / {Mq}_{4})}^{\frac{2}{M - 1}}$ , the minimum value of $F (V (t))$ is

F_{min} = F (V_{m}) = (q_{1} + \frac{q_{2}}{μ}) - \frac{(M - 1) q_{3}}{M} {(\frac{μ q_{3}}{{Mq}_{4}})}^{\frac{1}{M - 1}}

Let $μ > μ_{m} (μ \to \infty)$

μ_{m} = {(\frac{{Mq}_{4}}{q_{3}})}^{\frac{1}{M}} {(\frac{M^{2} q_{2}}{(M - 1) q_{3}} + \frac{M^{M + 1} q_{4} q_{1}^{M}}{{(M - 1)}^{M} q_{3}^{M + 1}})}^{\frac{M - 1}{M}}

(39)

At this point, $F_{min} < 0$ . The graph of the function $F (V (t))$ is shown in Figure 2.³¹ From Figure 2, $μ$ is very large, $V_{m}$ is also very large, let $V (0) \leq V_{m}$ , when $V (t) = V_{m}$ and $\overset{\cdot}{V} (t) < 0$ , we have

V (t) \leq V_{m} < \infty (0 \leq t < \infty)

(40)

Figure 2.

Behavior of $F (V (t))$ .

Therefore, system is bounded.

Simulation results

This section conducts simulation via the three-freedom horizontal multi-joint robot manipulator. For the data of robot manipulator link, see Table 1.²² We show a diagram of model tracking system for robot manipulator control system in Figure 3.

Table 1.

Link parameters for three-link manipulator.

Link	$α_{i}$	$θ_{i}$	$m_{i}$
1	0	$θ_{1}$	$m_{1}$
2	$l_{1}$	$θ_{2}$	$m_{2}$
3	$l_{2}$	$θ_{3}$	$m_{3}$

Values in the brackets are change values of joints.

Figure 3.

Diagram of tracking control system.

In Figure 3, $τ (t)$ is calculated based on equation (7), and new inputs $v (t)$ and $v_{m} (t)$ are calculated based on equations (28) and (29).

Response of link

In this example, the data of robot manipulator are set as follows: $m_{1} = 1.5 kg$ , $m_{2} = 1.5 kg$ , $m_{3} = 2 kg$ , $l_{1} = 0.25 m$ , $l_{2} = 0.3 m$ , $V_{1} = 0.2 {kgm}^{2} / s$ , $V_{2} = 0.1 {kgm}^{2} / s$ , $V_{3} = 0.1 {kgm}^{2} / s$ , and $g = 9.8 {m / s}^{2}$ . The target value and responsive wave form are shown in Figures 4 –7.

Figure 4.

Link 1: µ = 0.1.

Figure 5.

Link 1: µ = 0.5.

Figure 6.

Link 1: µ = 0.9.

Figure 7.

Link 1: µ = 10.

Figures 4 –7 indicate that the error value between the output signal of reference model and that of actual control system approaches zero. Within the unit time ( $t = 2.0 - 5.0$ ), in case of external disturbance, the output signal of control system and that of reference model realize tracking and $μ$ value is increased to suppress external disturbance, thus verifying the design effectiveness of the control system.

Simulation

Robot manipulator in operation may either run higher risk of being damaged due to its uncertain motion, motion range, and speed or suffer from lower working efficiency induced by programs’ failure to complete assigned motion due to different robot manipulators. This article completes the three-dimensional (3D) model of the three-freedom horizontal multi-joint robot manipulator, as shown in Figures 8 –11.

Figure 8.

Location 1.

Figure 9.

Location 2.

Figure 10.

Location 3.

Figure 11.

Location 4.

Conclusion

As the focal point in realizing industrial automation, the robot technology has attracted wide attention because of its extensive applicability and enormous development potential. This article discusses the construction of robot manipulator based on tracking control theory and derives the motion equation by the characteristics of the link, such as mass, mass center, inertia tensor, and friction. Assuming that there is an external disturbance, this article designed the control law for robot manipulator. This article reduces workload of design by introducing the differential divisor. Owing to the simplicity and applicability, the design theory proposed in this article ensures the stability of the system with its effectiveness that was certified by the simulation results.

Footnotes

Academic Editor: Zhijun Li

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Dixon

Walker

Dawson

. Fault detection for robot manipulators with parametric uncertainty: a prediction-error-based approach. IEEE Trans Robotic Autom 2000; 16: 689–699.

Ming

. Adaptive robust motion/force control of holonomic-constrained nonholonomic mobile manipulators. IEEE Trans Syst Man Cybern B Cybern 2007; 37: 607–616.

. Neural network control of a rehabilitation robot by state and output feedback. J Intell Robot Syst. Epub ahead of print 21 October 2014. DOI: 10.1007/s10846-014-0150-6.

Yuan

Guangying

Shuxia

. Adaptive fuzzy computed-torque control for robot manipulator with uncertain dynamics. Int J Adv Robot Syst 2012; 9: 1–9.

García-Rodríguez

Parra-Vega

. Task-space neuro-sliding mode control of robot manipulators under Jacobian uncertainties. Int J Contr Autom 2011; 9: 895–904.

Zhang

. Adaptive boundary control of a nonlinear flexible string system. IEEE Trans Contr Syst Technol 2014; 22: 1088–1093.

Kang

. Adaptive robust coordinated control of multiple mobile manipulators interacting with rigid environments. Automatica 2010; 46: 2028–2034.

. Neural-adaptive control of singlemaster-multiple-slaves teleoperation for coordinated multiple mobile manipulators with time-varying communication delays and input uncertainties. IEEE Trans Neural Netw Learn Syst 2013; 24: 1400–1413.

Hyunchul

Levi

Irina

. Kinematic data analysis for post-stroke patients following bilateral versus unilateral rehabilitation with an upper limb wearable robotic system. IEEE Trans Neur Sys Rehabil Eng 2013; 21: 153–164.

10.

Hedjar

Toumi

Boucher

. Finite horizon nonlinear predictive control by Taylor approximation: application to robot tracking trajectory. Int J Appl Math Comput Sci 2005; 15: 527–540.

11.

. Robust adaptive boundary control of a vibrating string under unknown time-varying disturbance. IEEE Trans Contr Syst Technol 2012; 20: 48–58.

12.

Mariottini

Oriolo

Prattichizzo

. Real-time identification and predictive control of fast mobile robots using global vision sensing. IEEE Trans Robot 2007; 23: 87–100.

13.

Mohammad

. Robust control of flexible-joint robots using voltage control strategy. Nonlinear Dyn 2012; 67: 1525–1537.

14.

Gonzalez-Vazquez

Moreno-Valenzuela

. Time-scale separation of a class of robust PD-type tracking controllers for robot manipulators. ISA Trans 2013; 52: 418–428.

15.

Huang

Peng

Jia

. Guaranteed robustness/performance adaptive control with limited torque for robot manipulators. Mechatronics 2008; 18: 641–652.

16.

Chih-Hsin

Chi-Hsiang

Wei-Song

. Robust fuzzy model-following control of robot manipulators. IEEE Trans Fuzzy Syst 2000; 8: 462–469.

17.

Chen

Yin

. Adaptive neural network control of an uncertain robot with full-state constraints. IEEE Trans Cybern. Epub ahead of print 2015. DOI: 10.1109/TCYB.2015.2411285.

18.

Purwar

Kar

Jha

. Adaptive output feedback tracking control of robot manipulators using position measurements only. Expert Syst Appl 2008; 34: 2789–2798.

19.

Tavasoli

Eghtesad

Jafarian

. Two-time scale control and observer design for trajectory tracking of two cooperating robot manipulators moving a flexible beam. Robot Auton Syst 2009; 57: 212–221.

20.

Tan

Zomaya

. A frequency-dependent direct adaptive scheme for robot manipulators: a case of model-following motion control. J Intell Robot Syst 1998; 22: 1–22.

21.

Wang

Okubo

. A design of model following control system for linear neutral system with time delays. Trans Inst Electr Eng Jpn 2008; 128: 1657–1663.

22.

Sun

Okubo

. Using Mathematica for manipulator dynamics equation described. In: 177th Tohoku chapter of the society of instrument and control engineers research conference, Yonezawa Yamagata, Japan, CA, 30–31 October 1998, Vol. 177, pp.1–7. Japan: Tohoku Chapter of The Society of Instrument and Control Engineers, http://www.topic.ad.jp/sice/papers/177/177-5.pdf

23.

Wang

Okubo

. Design of the state predictive model following control system with time-delay. Int J Appl Math Comput Sci 2009; 19: 247–254.

24.

Yaonan

Liu

. Neural network robust H_∞ tracking control strategy for robot manipulators. Appl Math Model 2010; 34: 1823–1838.

25.

Wang

. Design of the congestion control for TCP/AQM network with time-delay. Math Problem Eng 2014; 2014: 1–7. DOI: 10.1155/2014/834698

26.

Santos

Valero-Cuevas

. Reported anatomical variability naturally leads to multimodal distributions of Denavit-Hartenberg parameters for the human thumb. IEEE Trans Biomed Eng 2006; 53: 155–163.

27.

Wang

Okubo

. The state predictive model following control system for the linear time-delays. Int J Autom Comput 2009; 6: 186–191.

28.

Shah

Saha

Dutt

. Denavit-Hartenberg parameterization of Euler angles. J Comput Nonlin Dyn 2012; 7: 1–10.

29.

Izadbakhsh

Fateh

. A model-free robust control approach for robot manipulator. Int J Electr Electron Commun Eng 2007; 1: 745–750.

30.

Kim

Kwak

. Dynamic analysis of a nonholonomic two-wheeled inverted pendulum robot. J Intell Robot Syst 2005; 44: 25–46.

31.

Kikuchi

Okubo

Oikawa

. Control of horizontal multi-link robot. In: 224th Tohoku chapter of the society of instrument and control engineers research conference, Yonezawa Yamagata, Japan, CA, 14 October 2005, Vol. 224, pp.1–5. Japan: Tohoku Chapter of The Society of Instrument and Control Engineers, http://www.topic.ad.jp/sice/papers/224/224-5.pdf