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Abstract

This paper deals with the design of a variable gain super-twisting algorithm-based controller to solve the high tracking performance, low identification cost, and parameter tuning problem for rapid prototyping manipulators. The key elements of the proposed controller approach are the inverse dynamics-based controller, tracking differentiator (TD), and the variable gain super-twisting algorithm (VGSTA) controller. The inverse dynamic controller relies on part of the dynamic model information, which reduces the identification cost and decouples the manipulator’s torque input. The motion for using VGSTA, apart from its property of providing robustness to the system with respect to the unmodeled part and uncertainties, is also given by its capability of enforcing the controlled objects to the sliding surface in finite time and more explicit tuning parameter, allowing one to tune the system dynamic performance more simply. The proposal has been verified in a 3D-printed rapid prototyping manipulator, the experiment includes the parameter tuning experiment and the controller dynamic performance experiment. The manipulator’s dynamic performance and robustness are identified on the basis of the above experiment.

Keywords

Sliding mode control variable structure control robot manipulators finite-time convergence

Introduction

With the development of the 3D-print tech, the 3D-printed manipulator has wildly applications in rapid prototyping,^1–4 because it allows producing complex and custom shapes in small batches at a low cost, and this manufacturing technique is useful when developing products at the prototype stage.⁵ However, the manipulator controller design is a challenge because of its nonlinear dynamics and load disturbance. Especially for rapid prototyping, to realize the manipulator’s high-performance control, it’s necessary to identify the parameters of the manipulator. However, the process of identification causes certain difficulties and increases the cost of the manipulator development.⁶ To reduce the above identification cost, the PID controller is widely used in robot manipulator control,^5,7,8 but its dynamic performance is limited and parameter tuning is complicated when the manipulator is multi-joint. Combining the above model-based controller and model-free controller problem, it is necessary to design a controller that makes the trade-off between the dynamic model identification cost and controller performance for the 3D-printed rapid prototyping manipulator.

It is well-known that a PID controller is widely used in the robot manipulator, especially for the rapid prototyping manipulator, because it is model-free and can realize the fast deployment of the prototype manipulator. In PID control,⁹ the asymptotic stability of linear PID was proved under the condition that the reference trajectory is constant. However, the PID parameters’ stability condition is not explicit and the parameter tuning process is difficult. Model-based compensation with PD control is an alternative method for PID control, such as the PD controller with gravity compensation.⁹ However, this gravity compensation controller and PID controller cannot render asymptotic stability for tracking tasks.¹⁰ To improve tracking performance, “PD+” controller can realize globally stable tracking control,¹¹ but to realize the “PD+” controller, the full dynamic model is needed which increases the identification cost, and “PD+” controller lacks compensation for the uncertain part and disturbance of the manipulator.¹² proposed a neural-compensate-based PID controller to compensate for the uncertainty or disturbance in the manipulator, although this controller is a model-free controller, it needs a pre-trained model and high real-time performance. The extended states observer (ESO) can realize the uncertain model observation and the ESO-based feedforward system like ADRC was wildly used in the robot manipulator system.^13–15 Although the ESO-based controller can solve the uncertain model, it is hard to prove the system’s stability when it deals with a nonlinear system.¹⁶

The sliding mode control (SMC) is a special mode in the variable structure system (VSS).¹⁷ It’s one of the most useful robust controllers because of its ability to reject disturbance and model uncertainty, with wide applications for handling the uncertainty model, such as power system,^18–21 DC motor control,²² and electric vehicles.²³ However, the SMC has the disadvantage of the chattering²⁴ which is one of the limitations of the controller performance. To address chattering and ensure that the system trajectory stays on the surface, several strategies were developed, In González-Jiménez et al.,²⁵ Ren et al.,²⁶ and Conejo-Benitez et al.,²⁷ the control law can be divided into two parts: the equivalent control part $u_{eq}$ and the SMC part $u_{SMC}$ , the equivalent control part $u_{eq}$ is based on the manipulator dynamic model which needs complex identification to realize. For adaptive SMC,²⁸ although it achieves good dynamic performance and the ability to reduce the chattering, its parameters are hard to tune and lack parameters tuning analysis, that is, the improper switch gain may compromise the controller accuracy. In addition, the current study for the SMC such as barrier function based adaptive SMC and time-delay estimation (TDE) based SMC in Lee et al.,²⁸ Yang et al.,^29,30 focuses on disturbances and reducing chattering, while the parameter tuning analysis and the relationship between the dynamic performance and system parameters are always ignored.

In Dávila et al.,³¹ this paper proposes a second-order SMC to reduce the chattering which is named the variable gain super-twisting algorithm (VGSTA), the VGSTA assumes the uncertain model is a bounded function and divides the system into sliding surface coupled part and differentiable part, providing compensation to above uncertain part. Finally, the stability of the controller can be proved, and this controller has the advantage of strong robustness and the sliding surface will be reached in a finite time.

In this paper, inspired by Dávila et al.,³¹ taking into account the requirement of prototyping manipulator high-performance control, and having the aim of reducing the identification cost, an alternative robust VGSTA-based manipulator controller is proposed. The main contributions of this paper are summarized as follows: (1) a low-identification-cost algorithm is developed for prototyping manipulator with performance between the model-free controller and high high-precision model-based controller (2) the proposed controller has the clear tunning parameter which can adjust the internal parameters to achieve specific physical performance (such as the sliding surface’s reaching time, the system’s tracking error, etc.) (3) the system has the finite time to converge the sliding surface.

The rest of this paper is organized as follows. In Section 2, the problem model of the robot manipulator is introduced and solved by the VGSTA method. Finally, the finite-time convergence of the sliding variable is analyzed by the Lyapunov theorem. The simulation and experiment results are given in Sections 3 and 4 followed by a conclusion in Section 5.

Problem formulation

In this section, a robust control scheme focusing on compensating the unmodeled part and reducing identification cost was proposed as shown in Figure 1. More specifically, the control scheme consists of three parts: Dynamic inverse part, VGSTA part, and Tracking differentiator part. A Dynamic inverse part aimed at decoupled torque input to realize the basic structure of the VGSTA, based on an inertia matrix $M (q)$ which can be obtained by physical experiment and mass of each joint and rod³²; A VGSTA part, focusing on compensating unmodeled part to realize finite time to the sliding surface; Finally, a Tracking differentiator plays the role of generate the tracking and get its differential value.

Figure 1.

Scheme of the overall VGSTA control for robot manipulators.

Manipulator dynamic model

Consider a perturbed n-joint robotic manipulator system described by the following model:

\begin{matrix} M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + F (\overset{\cdot}{q}) = τ (t) + σ \end{matrix}

(1)

where $q$ , $\overset{\cdot}{q}$ , $\overset{\cdot\cdot}{q} \in R^{n}$ are joint angle, joint angular velocity, and joint acceleration. $M (q) \in R^{n \times n}$ is the manipulator inertia matrix, $C (q, \overset{\cdot}{q}) \in R^{n \times n}$ is the vector of centripetal and Coriolis torques, and $G (q) \in R^{n}$ is the vector of gravitational torques, $F (\overset{\cdot}{q}) \in R^{n}$ is the manipulator joint fraction. $τ (t) \in R^{n}, σ \in R^{n}$ are the vector of applied joint torque and the vector of joint disturbance torque.

Property 1: $M (q)$ is a symmetric and positive definite matrix.

Property 2: There exist positive constants $c_{b}, g_{b}, d$ to satisfy the following bounded condition:

‖ C (q, \overset{\cdot}{q}) \overset{\cdot}{q} ‖ \leq c_{b} ‖ \overset{\cdot}{q} ‖^{2}, ‖ G (q) ‖ \leq g_{b}, ‖ τ (t) ‖ \leq d .

Then, transform (1) into a joint angular acceleration expression:

\begin{matrix} \overset{\cdot\cdot}{q} = M^{- 1} (q) (τ (t) + σ - C (q, \overset{\cdot}{q}) \overset{\cdot}{q} - G (q) - F (\overset{\cdot}{q})) \end{matrix} .

(2)

Transforming (2) to the sliding surface differential form:

\begin{matrix} \overset{\cdot}{S} = M^{- 1} (q) τ (t) + K \overset{\cdot}{e} + \\ M^{- 1} (q) (σ - C (q, \overset{\cdot}{q}) \overset{\cdot}{q} - G (q) - F (\overset{\cdot}{q})) - {\overset{\cdot\cdot}{q}}_{r} \end{matrix}

(3)

where $K = diag (K_{1}, K_{2}, \dots, K_{n})$ denotes a $n \times n$ diagonal matrix, $K_{i} > 0, i = 1, 2, \dots, n$ , $e = q - q_{r}$ is the input-output error, $S = K e + \overset{\cdot}{e}$ $= {(s_{1}, s_{2}, \dots, s_{n})}^{T}$ , $S \in R^{n}$ is the sliding surface vector, and $q_{r} \in R^{n}$ is the joint reference angle.

Inverse dynamic control

To realize the high-performance manipulator trajectory planning and control algorithm implementation, the so-called inverse dynamics approach³³ is used. The inverse dynamics of the robot manipulator can be written, as an auxiliary input and torque output nonlinear equation:

\begin{matrix} τ (t) = M (q) f (t) + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + F (\overset{\cdot}{q}) \end{matrix},

(4)

where $f (t) \in R^{n}$ is the auxiliary input part and $τ \in R^{n}$ is the manipulator input torque. To realize the inverse dynamics control, manipulator dynamic identification is needed which increases the identification cost and development cycle of the rapid prototyping manipulator.

In an effort to decouple the manipulator input torque to realize the VGSTA and reduce the identification cost. The inverse dynamics control of the manipulator in Figure 1 can be simplified to the inertia matrix $M (q)$ product equation:

\begin{matrix} τ (t) = M (q) f (t) \end{matrix},

(5)

where $f (t) \in R^{n}$ is the auxiliary input part, according to Armstrong et al.,³² the inertia matrix $M (q)$ can be obtained directly by the physical experiment and the mass of each joint and rod. Compared with the manipulator dynamic model identification of the online adaptive control identification in Yang et al.³⁴ and off-line identification in Calanca et al.,³⁵ the new simplified inverse dynamics has the advantage of rapid deployment and low cost.

By substituting (5) into (3), the sliding surface equation can be obtained as:

\begin{matrix} \overset{\cdot}{S} = f (t) + K \overset{\cdot}{e} - {\overset{\cdot\cdot}{q}}_{r} \\ M^{- 1} (q) (σ - C (q, \overset{\cdot}{q}) \overset{\cdot}{q} - G (q) - F (\overset{\cdot}{q}) + H), \end{matrix}

(6)

where $H \in R^{n}$ takes into account the modeling uncertainties.

VGSTA controller

The torque input of the joint $i$ can be obtained based on VGSTA in Darvish Falehi²¹:

\begin{matrix} f_{i} = - k_{1 i} (t, q) φ_{1 i} (s_{i}) - \int_{0}^{t} k_{2 i} (t, q) φ_{2 i} (s_{i}) d t \end{matrix}

(7)

\begin{matrix} φ_{1 i} (s_{i}) = {| s_{i} |}^{\frac{1}{2}} sign (s_{i}) + k_{3 i} s_{i} \end{matrix},

(8)

\begin{matrix} φ_{2 i} (s_{i}) = \frac{1}{2} sign (s_{i}) + \frac{3}{2} k_{3 i} {| s_{i} |}^{\frac{1}{2}} sign (s_{i}) + {k_{3 i}}^{2} s_{i} \end{matrix},

(9)

where $k_{1 i}$ and $k_{1 i}$ are constant, and $k_{3 i}$ is the positive constant.

Noting that:

φ_{2 i} (s_{i}) = φ_{1 i} (s_{i}) \frac{d φ_{1 i} (s_{i})}{d s_{i}} .

(10)

Considering the torque input $f = {(f_{1}, f_{2}, \dots, f_{n})}^{T} \in R^{n}$ as the joint torque input:

\begin{matrix} \overset{\cdot}{S} = f + K \overset{\cdot}{e} - {\overset{\cdot\cdot}{q}}_{r} + \\ M^{- 1} (q) (σ - C (q, \overset{\cdot}{q}) \overset{\cdot}{q} - G (q) - F (\overset{\cdot}{q}) + H) . \end{matrix}

(11)

Assuming the right side of the input torque $τ_{0} (t)$ in (6) can divide into two bounded parts:

\begin{matrix} g_{1} (s, q, t) + g_{2} (s, q, t) \end{matrix},

(12)

where $g_{1} (s, q, t) \in R^{n}$ is about $g_{1 i} \in R$ element column vector, $g_{2} (s, q, t) \in R^{n}$ is about $g_{2 i} \in R$ element column vector and $g_{1 i} (s_{i}, q, t), g_{2 i} (s_{i}, q, t)$ are followed above bounded condition:

\begin{matrix} | g_{1 i} (s_{i}, q, t) | = | \frac{g_{1 i} (s_{i}, q, t)}{φ_{1} (s_{i})} φ_{1} (s_{i}) | = | α_{1 i} | | φ_{1} (s_{i}) | \\ \leq ρ_{1 i} (t, q) | φ_{1} (s_{i}) | \end{matrix}

(13)

| \frac{d g_{2 i} (s, q, t)}{dt} | = | α_{2 i} | | α_{2 i} φ_{2} (s_{i}) | \leq ρ_{2 i} (t, q) | φ_{2} (s_{i}) |

(14)

where $ρ_{1 i} (t, q) \in R$ and $ρ_{2 i} (t, q) \in R$ are positive value, both of them are continuous functions or constants in this system.

To sum up, the (6) can be transformed into:

\begin{matrix} \overset{\cdot}{S} = - k_{1} φ_{1} (S) + Z + g_{1} \end{matrix}

(15)

\begin{matrix} \overset{\cdot}{Z} = - k_{2} φ_{2} (S) + \frac{d g_{2}}{d t} \end{matrix},

(16)

where $Z = {[\begin{matrix} Z_{1} & Z_{2} & \dots & Z_{n} \end{matrix}]}^{T} \in R^{n}$ is about $Z_{i}$ element column vector, $k_{1} \in R^{n \times n}$ is about $k_{1 i}$ element diagonal matrix, $k_{2} \in R^{n \times n}$ is about $k_{2 i}$ element diagonal matrix, $φ_{1} (S) \in R^{n}$ is about $φ_{1 i}$ element column vector, and $φ_{2} (S) \in R^{n}$ is about $φ_{2 i}$ element column vector.

The sliding surface s_i in this system can reach the s_i = 0 in finite time T when gains k_1ik_2i meet the following requirements:

\begin{matrix} k_{1 i} = δ_{i} + \frac{1}{β_{i}} [\frac{1}{4 ϵ_{i}} {(2 ϵ_{i} ρ_{1 i} + ρ_{2 i})}^{2} + 2 ϵ_{i} ρ_{2 i} \\ + ϵ_{i} + (2 ϵ_{i} + ρ_{1 i}) (β_{i} + 4 {ϵ_{i}}^{2})] \end{matrix}

(17)

\begin{matrix} k_{2 i} = β_{i} + 4 {ϵ_{i}}^{2} + 2 ϵ_{i} k_{1 i} \end{matrix},

(18)

where $β_{i} > 0, ϵ_{i} > 0, δ_{i} > 0$ are positive constants. And the reaching time of the sliding surface T can be estimated:

\begin{matrix} T = \frac{2}{γ_{2}} \ln (\frac{γ_{2}}{γ_{1}} {V_{i}}^{\frac{1}{2}} (0) + 1) \end{matrix},

(19)

\begin{matrix} γ_{1} = \frac{ϵ_{i} λ_{m i n}^{\frac{1}{2}} {P_{i}}}{λ_{m a x} {P_{i}}}, γ_{2} = \frac{2 ϵ_{i} k_{3 i}}{λ_{m a x} {P_{i}}} \end{matrix},

where $P_{i} = (\begin{matrix} β_{i} + 4 ϵ_{i}^{2} & - 2 ϵ_{i} \\ - 2 ϵ_{i} & 1 \end{matrix})$ , $ε_{i}^{T} = (\begin{matrix} φ_{1 i} (s) & Z_{i} \end{matrix})$ , $V_{i} = {ε_{i}}^{T} P_{i} ε_{i}$ , $λ_{\max} {Δ}$ , and $λ_{\min} {Δ}$ are about the maximum and minimum eigenvalue in the matrix $Δ$ .

Stability proof

Choose the function (20) as the Lyapunov function:

\begin{matrix} V = ε^{T} P ε \end{matrix}

(20)

where $P = Blkdiag (P_{1}, P_{2}, \dots, P_{n})$ denotes a $2 n \times 2 n$ block diagonal matrix, $ε^{T} = (ε_{1}^{T}, ε_{2}^{T}, \dots, ε_{n}^{T}) \in R^{2 n}$ , $P_{i} = (\begin{matrix} β_{i} + 4 ϵ_{i}^{2} & - 2 ϵ_{i} \\ - 2 ϵ_{i} & 1 \end{matrix})$ , $ε_{i}^{T} = (\begin{matrix} φ_{1 i} (s) & Z_{i} \end{matrix})$ , $i = 1, 2, \dots, n$ . The Lyapunov function $V$ in (20) is positive definite and it’s a continuously differentiable function except on the point $s = 0$ .

According to the (10), (15), and (16), the derivative of $ε_{i}$ can be obtained:

\begin{matrix} {\overset{\cdot}{ε}}_{i} = (\begin{matrix} \frac{d φ_{1 i} (s)}{d s_{i}} [- k_{1 i} φ_{1 i} (s) + Z_{i} + g_{1 i}] \\ - k_{2 i} φ_{2 i} (s) + \frac{d g_{2 i}}{dt} \end{matrix}) \\ = \frac{d φ_{1 i} (s)}{d s_{i}} (\begin{matrix} - (k_{1 i} - α_{1 i}) & 1 \\ - (k_{2 i} - α_{2 i}) & 0 \end{matrix}) ε_{i} = \frac{d φ_{1 i} (s)}{d s_{i}} A_{i} ε_{i} \end{matrix}

(21)

\begin{matrix} \overset{\cdot}{ε} = (\begin{matrix} {\overset{\cdot}{ε}}_{1} \\ {\overset{\cdot}{ε}}_{2} \\ ⋮ \\ {\overset{\cdot}{ε}}_{n} \end{matrix}) = ψ A ε \end{matrix}

(22)

where $ψ_{i} = diag (\frac{d φ_{1 i} (s)}{d s_{i}}, \frac{d φ_{1 i} (s)}{d s_{i}})$ , $ψ_{i} \in R^{2 \times 2}$ , $ψ = Blkdiag (ψ_{1}, ψ_{2}, \dots, ψ_{n})$ denotes a $2 n \times 2 n$ block diagonal matrix, $A = Blkdiag (A_{1}, A_{2}, \dots, A_{n})$ denotes a $2 n \times 2 n$ block diagonal matrix.

To calculate the derivative of the Lyapunov function:

\begin{matrix} \overset{\cdot}{V} = ε^{T} (A^{T} ψ^{T} P + P ψ A) ε \\ = \sum_{i = 1}^{n} \frac{d φ_{1 i} (s)}{d s_{i}} {ε_{i}}^{T} (A_{i}^{T} P_{i} + P_{i} A_{i}) ε_{i} \\ = - \sum_{i = 1}^{n} \frac{d φ_{1 i} (s)}{d s_{i}} {ε_{i}}^{T} Q_{i} ε_{i} . \end{matrix}

(23)

With the bounded parameter $| α_{1} | < ρ_{1 i}$ , $| α_{2} | < ρ_{2 i}$ , and substitute $k_{1 i}$ and $k_{2 i}$ into the matrix $Q_{i} - 2 ϵ_{i} I$ , we have

\begin{matrix} Q_{i} - 2 ϵ_{i} I = (\begin{matrix} Δ & 2 ϵ_{i} α_{1 i} - α_{2 i} \\ 2 ϵ_{i} α_{1 i} - α_{2 i} & 2 ϵ_{i} \end{matrix}) \end{matrix}

(24)

where $Δ = 2 β_{i} k_{1 i} - (β_{i} + 4 {ϵ_{i}}^{2}) (4 ϵ_{i} + 2 α_{1 i}) + 4 ϵ_{i} α_{2 i} - 2 ϵ_{i}$ and the matrix $Q_{i} - 2 ϵ_{i} I$ is always positive when the parameter $| α_{1 i} | < ρ_{1 i}, | α_{2 i} | < ρ_{2 i}$ (For more detail please see Appendix 2), and the (25) can be obtained:

\begin{matrix} \overset{\cdot}{V} = - \sum_{i = 1}^{n} \frac{d φ_{1 i} (s)}{d s_{i}} {ε_{i}}^{T} Q_{i} ε_{i} \leq - \sum_{i = 1}^{n} 2 ϵ_{i} \frac{d φ_{1 i} (s)}{d s_{i}} {ε_{i}}^{T} ε_{i} \\ = - \sum_{i = 1}^{n} 2 ϵ_{i} (\frac{1}{2 {| s_{i} |}^{\frac{1}{2}}} + k_{3 i}) \frac{d φ_{1 i} (s)}{d s_{i}} {ε_{i}}^{T} ε_{i} \leq 0 . \end{matrix}

(25)

According to the Lyapunov stability theorem, the sliding surface $s$ in the system (11) is asymptotically stable.

Convergence proof

Since $P$ in (20) is the positive definite matrix, then we have

\begin{matrix} λ_{\min} {P} ε^{T} ε \leq ε^{T} P ε < λ_{\max} {P} ε^{T} ε \end{matrix}

(26)

and based on the definition of $ε_{i}$ and $ε$ in (20), we have

\begin{matrix} {| s_{i} |}^{\frac{1}{2}} < {| | ε_{i} | |}_{2} < {| | ε | |}_{2} < {(\frac{ε^{Τ} P ε}{λ_{m i n} {P}})}^{\frac{1}{2}} = {(\frac{V}{λ_{m i n} {P}})}^{\frac{1}{2}} \end{matrix} .

(27)

To sum up:

\begin{matrix} \overset{\cdot}{V} \leq - 2 ϵ_{\min} (\frac{1}{2 {(\frac{V}{λ_{\min} {P}})}^{\frac{1}{2}}} + {(k_{3})}_{\min}) \sum_{i = 1}^{n} {ε_{i}}^{T} ε_{i} \\ \leq - 2 ϵ_{\min} (\frac{1}{2 {(\frac{V}{λ_{\min} {P}})}^{\frac{1}{2}}} + {(k_{3})}_{\min}) \frac{ε^{T} P ε}{λ_{\max} {P}} \\ = - γ_{1} V^{\frac{1}{2}} - γ_{2} V \end{matrix}

(28)

where $γ_{1} = \frac{ϵ_{\min} λ_{\min}^{\frac{1}{2}} {P}}{λ_{\max} {P}}, γ_{2} = \frac{2 ϵ_{\min} {(k_{3})}_{\min}}{λ_{\max} {P}}$ , $k_{3}$ is about $k_{3 i}$ element column vector, and the symbol $(Δ)_{\min}$ is the minimal element in the vector $Δ \in R^{n}$ .

Based on inequality in (28), the reaching time of the sliding surface $S = 0$ can be estimated by the differential equation:

\begin{matrix} \overset{\cdot}{v} = - γ_{1} v^{\frac{1}{2}} - γ_{2} v \end{matrix}

(29)

and assume the $v (0) = V (0)$ , the solution of the (26) is given by:

\begin{matrix} v = e^{- γ_{2} t} {[v {(0)}^{\frac{1}{2}} + \frac{γ_{1}}{γ_{2}} (1 - e^{- \frac{γ_{2}}{2} t})]}^{2} \geq V \\ = ε^{T} P ε \geq 0 \end{matrix}

(30)

Let $v = 0$ , the reaching time $T$ of the sliding surface $S = 0$ can be estimated by the (19), and the convergence of the equation is proved.

To sum up, a single joint $i$ convergence time can be proved by setting the Lyapunov function $V_{i} = {ε_{i}}^{T} P_{i} ε_{i}$ , according to the (21), (25), and (27), the differentia of the Lyapunov function can be obtained:

\begin{matrix} {\overset{\cdot}{V}}_{i} = - \frac{d φ_{1 i} (s)}{d s_{i}} {ε_{i}}^{T} Q_{i} ε_{i} \\ \leq - 2 ϵ_{i} (\frac{1}{2 {| s_{i} |}^{\frac{1}{2}}} + k_{3 i}) \frac{d φ_{1 i} (s)}{d s_{i}} {ε_{i}}^{T} ε_{i} \\ < - 2 ϵ_{i} (\frac{1}{2 {(\frac{V_{i}}{λ_{\min} {P}})}^{\frac{1}{2}}} + k_{3 i}) \frac{{ε_{i}}^{T} P_{i} ε_{i}}{λ_{\max} {P}} . \end{matrix}

(31)

The reaching time of the single joint $i$ can be estimated by the differential equation:

\begin{matrix} \overset{\cdot}{v} = - γ_{1} v^{\frac{1}{2}} - γ_{2} v \end{matrix}

(32)

where $γ_{1} = \frac{ϵ_{i} λ_{\min}^{\frac{1}{2}} {P_{i}}}{λ_{\max} {P_{i}}}, γ_{2} = \frac{2 ϵ_{i} k_{3 i}}{λ_{\max} {P_{i}}}$ . And assume the $v (0) = V (0)$ , the solution of (26) is given by:

v = e^{- γ_{2} t} {[v {(0)}^{\frac{1}{2}} + \frac{γ_{1}}{γ_{2}} (1 - e^{- \frac{γ_{2}}{2} t})]}^{2} \geq V_{i} .

(33)

To sum up, the single joint sliding surface can converge to zero in finite time and the reaching time in (19) is proved by letting (33) equal to zero.

Tracking differentiator

To calculate the sliding surface $S = K e + \overset{\cdot}{e}$ , a tracking differentiator (TD) is needed. The TD is a useful differentiator and can make the reference trajectory “softer” to improve the system’s transient performance. In Han,³⁶ this paper proposed a discrete fast-tracking differentiator :

\begin{matrix} {\begin{matrix} f h = f h a n (x_{1} (k) - q_{r e f}, x_{2} (k), r, h) \\ x_{1} (k + 1) = x_{1} (k) + h x_{2} (k) \\ x_{2} (k + 1) = x_{2} (k) + h f h \end{matrix} \end{matrix},

(34)

\begin{matrix} f h a n (x_{1}, x_{2}, r, h) = {\begin{matrix} d = r h \\ d_{0} = h d \\ y = x_{1} + h x_{2} \\ a_{0} = \sqrt{d^{2} + 8 r | y |} \\ a = {\begin{matrix} x_{2} + \frac{(a_{0} - d)}{2} s i g n (y), | y | > d_{0} \\ x_{2} + \frac{y}{h}, | y | \leq d_{0} \end{matrix} \\ f h a n = {\begin{matrix} r s i g n (a), | a | > d \\ r \frac{a}{d}, | a | \leq d \end{matrix} \end{matrix} \end{matrix},

(35)

where $q_{ref}, x_{1}, x_{2}$ are reference trajectory, tracking value of the reference trajectory, and tracking Differential value. H is the integral step, and r is the state tracking rate.

Simulation

In this section, a 3DOF robot manipulator simulation experiment based on MATLAB/Robotics Toolbox is performed to evaluate the performance of the proposed VGSTA-based controller, and the reference trajectories are given by:

\begin{matrix} q_{r} = [q_{0}, q_{1}, q_{2}]^{T} \\ = [120 \sin (2 π t), 120 \sin (2 π t), 120 \sin (2 π t)]^{T} . \end{matrix}

To verify the dynamic performance of the VGSTA controller, The present controller will be compared with the nonsingular terminal sliding mode controller (NTSMC)³⁷ which needs the full kinetic model, and model-free controller (PID). As shown in Figure 2: the VGSTA controller can be well-handled multi-joint manipulator control with the partial kinetics model, its transient performance and tracking performance are between the full kinetic model-based controller (NTSMC) and the model-free controller (PID), which can be well-handled the fast-prototyping manipulator with the certain dynamic performance requirement.

Figure 2.

Tracking error simulation: (a) tracking error of joint q₀, (b) tracking error of joint q₁, and (c) tracking error of joint q₂.

Experiment

In this section, the parameter tuning experiment is proposed which includes tracking performance tuning and dynamic performance tuning. And the effectiveness of the proposed controller in (7)–(9) is verified by the tracking experiment and comparison with the PID controller in PID control⁹ and the nonsingular terminal sliding mode controller (NTSMC) in Feng et al.³⁷ The inertia matrix $M (q)$ for the VGSTA controller in (5) is obtained by the offline identification method which needs each joint mass and rod mass in Armstrong et al.³²

Figure 3(a) and (b) show the 3DOF manipulator in this experiment is made by 3D-print tech. Its three joints are actuated by brushless DC servo motors which communicate with the controller (STM32F427IIH6) via the CAN bus. As shown in Figure 3(b), the manipulator parameter $M_{i}$ , $i = 1, 2, 3$ , $m_{j}$ , $j = 1, 2$ , $l_{p}$ , $p = 1, 2, 3$ denotes the i-th joint mass, j-th rod mass, and p-th rod length, respectively. Furthermore, consider the 3DOF manipulator kinetics model in this experiment can be expressed as:

\begin{matrix} M (q) & = [\begin{matrix} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} & M_{23} \\ M_{31} & M_{32} & M_{33} \end{matrix}], C (q, \overset{\cdot}{q}) = [\begin{matrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{matrix}] \\ G (q) & = [\begin{matrix} g_{1} \\ g_{2} \\ g_{3} \end{matrix}], \end{matrix}

where $M (q)$ is the inertia matrix, $C (q, \overset{\cdot}{q})$ is the vector of centripetal and Coriolis torques, and G(q) is the vector of gravitational torques. The manipulator dynamic model parameter can be seen in Appendix 1.

Figure 3.

3D-printed manipulator: (a) experiment manipulator and (b) schematic diagram of the experimental manipulator.

Control objective & parameter tuning strategy

For the VGSTA controller in the system (10), we seek to tune suitable parameters to achieve the following objectives:

Steady performance requirement

Regulate the joint angle $q$ to their destinations $q_{ref}$ , respectively, that is:

lim_{t \to \infty} q = q_{r}

for actual turning, the steady performance requirement can be reduced to:

lim_{t \to \infty} | q - q_{r} | \leq b_{0} .

To achieve the above steady performance requirement, the parameter tuning process can be divided into the following parts:

Phase error tuning

The phase error (such as Figure 4) in the system (1) mainly comes from the TD in (34), and (35), the tracking error can be reduced by increasing the tracking rate parameter $r$ in (34).

Figure 4.

Before phase difference tuning.

Stability tuning

Stability tuning is necessary for the steady performance requirement due to the well-tuned VGSTA controller can enforce the system phase plane into the sliding surface, making the error gradually converge to zero. According to (24) and (25), the sliding surface stability can be determined by the parameter $ρ_{1}$ , $ρ_{2}$ , and the parameter $ρ_{1}$ , $ρ_{2}$ can be increased until the system can follow the reference trajectory when the system can’t follow the reference trajectory.

Sliding surface tuning

According to the sliding surface $S = K e + \overset{\cdot}{e}$ , the system error convergence time can be reduced by increasing the parameter $K$ .

Overshoot performance requirement

Regulate the system output error overshoot $σ_{p}$ under a certain range $σ_{0}$ , respectively, that is:

| σ_{p} | < σ_{0} .

To achieve the above steady performance requirement, the parameter tuning process can be divided into the following parts:

Convergence time tuning

The reaching time of the sliding surface during the transient time is one of the causes of overshoot. When the overshoot problem occurs, according to (19), the convergence time can be reduced by increasing the parameter $k_{3}$ until the performance meets the requirement.

Transient reference trajectory tuning

The TD can not only calculate the differential of the reference trajectory $q_{ref}$ , but also generate the tracking trajectory $q_{tra}$ for the VGSTA controller input. The TD can produce the transient process of the VGSTA controller input, and the transient process length can be prolonged with the parameter $r$ in (34) reduced. To meet the overshoot performance requirement, the parameter $r$ can be reduced to prolong the transient process and reduce the system overshoot.

Sliding surface tuning

According to the differential (19), reducing the sliding surface gain $k$ to reduce the Lyapunov function $V_{i}$ can effectively reduce the convergence time and the overshoot.

Parameter tuning experiment

In this contrast experiment, the trajectory signal:

\begin{matrix} q_{r} = [q_{0}, q_{1}, q_{2}]^{T} \\ = [70 \sin (2 π t) + 170, 40 \sin (2 π t) + 200, 50 \sin (2 π t - π / 8) + 80]^{T} \end{matrix}

is selected as the reference trajectory signal. And the VGSTA turning process as flows:

Select the discrete cycle as step h, and estimate the rate $r$ as a small value. Select the suitable parameter $β$ , $ϵ$ to determine the suitable eigenvalue for the matrix $P$ in (20).

Estimate the $ρ_{1}, ρ_{2}, δ$ as a small value, select a suitable parameter $k$ in the sliding surface $s = K e + \overset{\cdot}{e}$ , and set other parameters $k_{3}$ as a small value.

Gradually increase the $ρ_{1}$ and $ρ_{2}$ until the system can follow the reference trajectory $q_{ref}$ , after the system can be stable, the tracking differential rate parameter $r$ can be increased if the system exists the phase difference (Figure 4). After the above tuning process (Figure 5), if the system exists the tracking error, the tracking error can be decreased by increased the parameter $k$ until the tracking error meet the user’s requirement (Figure 6).

After the above dynamic performance tuning, it’s necessary to tune the transiting process. As we can see the overshoot problem in Figure 7. According to the convergence time tuning mentioned above, Increasing the parameter $k_{3}$ and decreasing the sliding surface parameter $k$ can reduce the reaching time of the sliding surface and solve the overshoot problem. After the above parameter tuning process, the well-tuned dynamic controller can be obtained in Figure 8

Figure 5.

After phase difference tuning.

Figure 6.

After tracking error tuning.

Figure 7.

Before overshoot tuning.

Figure 8.

After overshoot tuning.

Dynamic performance experiment

The contrast tracking results of each joint’s angle are presented in Figures 9 –11. The result investigated that the VGSTAs’ transient performance and tracking accuracy are between the model-free controller (PID) and high identification cost controller (NTSMC), and the further comparison of the above performance is shown in Table 1:

Figure 9.

Position tracking results of the VGSTA: (a) tracking error of joint q₀, (b) tracking error of joint q₁, and (c) tracking error of joint q₂.

Figure 10.

Position tracking results of the PID: (a) tracking error of joint q₀ (b)tracking error of joint q₁ and (c) tracking error of joint q₂.

Figure 11.

Position tracking results of the NTSMC: (a) tracking error of joint q₀, (b)tracking error of joint q₁, and (c) tracking error of joint q₂.

Table 1.

The average tracking error and reach time of different algorithm.

	${\bar{e}}_{0}, {\bar{e}}_{1}, {\bar{e}}_{2}$ (°)	$t_{0}, t_{1}, t_{2}$ (s)
VGSTA	1.74,1.82,1.28	0.157,0.189,0.181
PID	2.44,2.41,1.80	0.234,0.270,0.247
NTSMC	1.43,1.61,1.21	0.142,0.140,0.161

Where ${\bar{e}}_{i}$ is the average static error of the joint $i$ , and the $t_{i}$ is the reach time of the joint $i$ reference trajectory.

Compared with the model-free controller, the VGSTA controller can track reference trajectory more accurately and fast response. It also reduces the cost of the identification compared with the mode-based controller (NTSMC) by making trade-off in some dynamic performance.

Discussion of the experiment result

To sum up, in this experiment part, the VGSTA-based controller has the explicit parameter to tuning its dynamic performance, having the advantage of the identification cost, and can realize the fast deployment of the rapid prototyping 3D-printed manipulator in a certain dynamic performance.

However, manipulator parameters used in the experiment are estimated by joint and rod mass, and unknown disturbance which will lead the parameter errors in the manipulator model. Moreover, there is still a lot of room for improvement in the design of linear sliding surfaces, such as terminal sliding surface.

Thus, in our future work, our work will focus on handling the uncertainty of the model error and introduce the other sliding surface to improve our manipulator performance.

Conclusion

In this paper, a VGSTA-based controller is proposed for the rapid deployment of the prototyping manipulator. This controller can reduce the identification cost and realize rapid deployment by the simplified inverse dynamic part and the VGSTA part to compensate for the unmodeled dynamic model. The inverse dynamic part can be obtained easily through the manipulator’s physical parameter, compared with the PID and NTSMC which are widely used in the rapid prototyping manipulator, the new controller has the advantage of high dynamic performance and the tuning parameter has a certain physical meaning which can easily to tune the dynamic performance to meet the user’s requirements. The above conclusion has been validated in a 3D-printed manipulator experiment, and its rapid deployment performance has been identified on the basis of experimental tests. In our future studies, the adaptive law^29,30 and fuzzy logic systems approximation technique³⁸ will be further combined with the VGSTA controller to handle the uncertainty of the offline identification and further improve the robustness of the VGSTA.

Footnotes

Appendix 1

Manipulator dynamic model parameter:

\begin{matrix} M_{11} = \frac{l_{1}^{3} m_{1}}{3 l_{2}}; M_{12} = \frac{4 m_{1} l_{1}^{3} - 6 l_{2} m_{1} \sin (θ_{1} + 2 θ_{2}) l_{1}^{2}}{12 l_{2}}; \\ M_{13} = 0; M_{21} = \frac{1}{6 l_{2}} (2 m_{1} l_{1}^{3} - 3 l_{2} m_{1} \sin (θ_{1} + 2 θ_{2}) l_{1}^{2}); \\ M_{22} = \frac{1}{6 l_{2}} (6 M_{1} l_{2}^{3} + 2 l_{1}^{3} m_{1} + 2_{2}^{3} m_{2} + 6 l_{1} l_{2}^{2} m_{1} \\ - 6 l_{1}^{2} l_{2} m_{1} \sin (θ_{1} + 2 θ_{2})); \\ M_{23} = 0; \\ M_{33} = \frac{1}{6 l_{2}} (6 J_{2} l_{2} + l_{1}^{3} m_{1} + 6 M_{1} l_{2}^{3} \cos^{2} (θ_{2}) + 3 l_{1} l_{2}^{2} m_{1} \\ + 2 l_{2}^{3} m_{2} \cos^{2} (θ_{2}) + l_{1}^{3} m_{1} \cos (2 θ_{1} + 2 θ_{2}) \\ + 3 l_{1}^{2} l_{2} m_{1} \cos (θ_{1}) + 3 l_{1} l_{2}^{2} m_{1} \cos (2 θ_{2}) \\ + 3 l_{1}^{2} l_{2} m_{1} \cos (θ_{1} + 2 θ_{2})); \end{matrix}

\begin{array}{l} C_{11} = 0; \\ C_{12} = - \frac{1}{2} ({l_{1}}^{2} m_{1} \cos (θ_{1} + 2 θ_{2})) ω_{2}; \\ C_{13} = \frac{1}{12 l_{2}} (2 l_{1}^{3} m_{1} \sin (2 θ_{1} +2 θ_{2}) + 3 l_{1}^{2} l_{2} m_{1} \sin (θ_{1}) + 3 l_{1}^{2} l_{2} m_{1} \sin (θ_{1} + 2 θ_{2})) ω_{3}; \\ C_{21} = - \frac{1}{2} (l_{1}^{2} m_{1} \cos (θ_{1} + 2 θ_{2})) ω_{1}; \\ C_{22} = - l_{1}^{2} m_{1} \cos (θ_{1} + 2 θ_{2}) ω_{1} - l_{1}^{2} m_{1} \cos (θ_{1} + 2 θ_{2}) ω_{2}; \\ C_{23} = \frac{1}{6 l_{2}} (3 M_{1} l_{2}^{3} \sin (2 θ_{2}) + l_{2}^{3} m_{2} \sin (2 θ_{2}) + l_{1}^{3} m_{1} \sin (2 θ_{1} + 2 θ_{2}) \\ + 3 l_{1} l_{2}^{2} m_{1} \sin (2 θ_{2}) + 3 l_{1}^{2} l_{2} m_{1} \sin (θ_{1} + 2 θ_{2})) ω_{1}; \\ C_{31} = \frac{1}{2} (l_{1}^{2} m_{1} \cos (θ_{1} + 2 θ_{2})) ω_{2}; \\ C_{32} = (\frac{l_{1}^{2} m_{1} \cos (θ_{1} + 2 θ_{2})}{2}) ω_{2} + (- M_{1} l_{2}^{2} \sin (2 θ_{2}) - \frac{l_{2}^{2} m_{2} \sin (2 θ_{2})}{3} \\ - l_{1}^{2} m_{1} \sin (θ_{1} + 2 θ_{2}) - l_{1} l_{2} m_{1} \sin (2 θ_{2}) - \frac{l_{1}^{3} m_{1} \sin (2 θ_{1} + 2 θ_{2})}{3 l_{2}} ω_{3}; \\ C_{33} = (- \frac{1}{2} l_{1}^{2} m_{1} \sin (θ_{1} + 2 θ_{2}) - \frac{1}{2} l_{1}^{2} m_{1} \sin (θ_{1}) - \frac{1}{3 l_{2}} l_{1}^{3} m_{1} \sin (2 θ_{1} + 2 θ_{2})) ω_{1} \\ + \frac{1}{12 l_{2}} (l_{1} m_{1} (2 l_{1}^{2} \sin (2 θ_{1} + 2 θ_{2}) + 3 l_{1} l_{2} \sin (θ_{1}) + 3 l_{1} l_{2} \sin (θ_{1} + 2 θ_{2}))) ω_{3}; \\ g_{1} = \frac{1}{2} g l_{1} m_{1} \cos (θ_{1} + θ_{2}); \\ g_{2} = \frac{1}{6 l_{2}} (6 M_{1} {gl}_{2}^{2} \cos (θ_{2}) + 9 {gl}_{2}^{2} m_{1} \cos (θ_{2}) \\ + 3 g l_{1} l_{2} m_{1} \cos (θ_{1} + θ_{2})); \\ g_{3} = \frac{1}{2} g l_{1} m_{1} \cos (θ_{1} + θ_{2}); \end{array}

Appendix 2

The positive definiteness of the matrix $Q_{i} - 2 ϵ_{i} I$ :

The positive definite matrix $Q_{i} - 2 ϵ_{i} I$ in (24) can be proved by

(A1)

Δ > 0

(A2)

det (Q_{i} - 2 ϵ_{i} I) > 0

where $det (α)$ denotes the determinant of the square matrix $α$ . Then, the inequality (A1) can be directly proved by applying (17) into $Δ$

(A3)

\begin{matrix} Δ = 2 β_{i} k_{1 i} - (β_{i} + 4 {ϵ_{i}}^{2}) (4 ϵ_{i} + 2 α_{1 i}) + 4 ϵ_{i} α_{2 i} - 2 ϵ_{i} \\ = 2 β_{i} δ_{i} + \frac{1}{2 ϵ_{i}} {(2 ϵ_{i} ρ_{1 i} + ρ_{2 i})}^{2} + 4 ϵ_{i} (ρ_{2 i} + α_{2 i}) \\ + 2 (β_{i} + 4 {ϵ_{i}}^{2}) [ρ_{1 i} - α_{1 i}] \end{matrix}

Then, based on inequality in (13), (14) and positive constant parameters $ϵ_{i}$ , $β_{i}$ and $δ_{i}$ , we can obtain that equation (A3) is always positive.

By applying define of $k_{1 i}$ into (A2), the expansion of the $det (Q_{i} - 2 ϵ_{i} I)$ can be written as

(A4)

\begin{matrix} det (Q_{i} - 2 ϵ_{i} I) = {(2 ϵ_{i} ρ_{1 i} + ρ_{2 i})}^{2} - {(2 ϵ_{i} α_{1 i} + α_{2 i})}^{2} \\ + 8 ϵ_{i}^{2} (ρ_{2 i} + α_{2 i}) + 4 ϵ_{i} β_{i} δ_{i} \\ + 4 ϵ_{i} (β_{i} + 4 {ϵ_{i}}^{2}) [ρ_{1 i} - α_{1 i}] \\ = 4 ϵ_{i}^{2} (ρ_{1 i}^{2} - α_{1 i}^{2}) + (4 ϵ_{i} + 8 ϵ_{i}^{2}) (ρ_{2 i} + α_{2 i}) \\ + ρ_{2 i}^{2} - α_{2 i i}^{2} + 4 ϵ_{i} (ρ_{1 i} ρ_{2 i} - α_{1 i} α_{2 i}) \\ + 4 ϵ_{i} (β_{i} + 4 {ϵ_{i}}^{2}) [ρ_{1 i} - α_{1 i}] + 4 ϵ_{i} β_{i} δ_{i} \end{matrix}

Analogously, based on inequality in (13) and (14) and positive constant parameters $ϵ_{i}$ , $β_{i}$ , and $δ_{i}$ , the determinant of the matrix $Q_{i} - 2 ϵ_{i} I$ is always positive.

To sum up, (A1) and (A2) are always held. Thus, the matrix $Q_{i} - 2 ϵ_{i} I$ is always the positive definite matrix.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Jiangsu Science and Technology Plan Program (BK20220263), Suzhou basic research pilot project(SSD2023018) CAS Project for Young Scientists in Basic Research (YSBR-067) National Key R&D Program of China (2021YFF0700503, 2022YFC2404201).

ORCID iD

Sujian Wu

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Robust trajectory tracking of a 3D-printed rapid prototyping manipulator through variable gain super-twisting algorithm

Abstract

Keywords

Introduction

Problem formulation

Manipulator dynamic model

Inverse dynamic control

VGSTA controller

Stability proof

Convergence proof

Tracking differentiator

Simulation

Experiment

Control objective & parameter tuning strategy

Steady performance requirement

Phase error tuning

Stability tuning

Sliding surface tuning

Overshoot performance requirement

Convergence time tuning

Transient reference trajectory tuning

Sliding surface tuning

Parameter tuning experiment

Dynamic performance experiment

Discussion of the experiment result

Conclusion

Footnotes

Appendix 1

Appendix 2

Declaration of conflicting interests

Funding

ORCID iD

Data availability statement

References