Neural approximation-based adaptive variable impedance control of robots

Abstract

Variable impedance control improves compliance and robustness in robot-environment interaction through variation of the desired stiffness and the desired damping. This paper proposes neural approximation-based variable impedance controllers for robots in robot-environment interaction. Constraints on variable impedance parameters are given to ensure the exponential stability of the desired first- and second-order variable impedance dynamics. Adaptive neural network controllers are proposed to ensure the achievement of the desired first- and second-order variable impedance dynamics through convergence of variable impedance errors. In the neural networks, deadzone modifications are utilized to enhance robustness by turning off adaptation when auxiliary tracking errors enter the constructed small neighbourhoods of zero. The proposed variable impedance control methods in this paper guarantee the stability and achievement of the desired variable impedance dynamics. Theoretical analysis and simulation results validate the effectiveness of the proposed variable impedance control methods.

Keywords

Robot impedance control neural network adaptive control

Introduction

Impedance control originally proposed by Hogan (1985) has been considered as one of the most powerful compliance control methods for robots. In last decades, many constant-impedance controllers (Cheah and Wang, 2018; He and Dong, 2018; He et al., 2016; Jamwal et al., 2016; Joshua et al., 2015; Jung and Hsia, 1998; Li and Ge, 2014; Li and Liu, 2018; Li et al., 2017; Oh, 1999; Sadeghian et al., 2012; Sharifi et al., 2017; Sun et al., 2019a, 2020) were designed for robots in robot-environment interaction mainly based on the adaptive control approach and the sliding-mode control approach. Compared with constant-impedance controllers, variable impedance control allows variation of the desired stiffness and the desired damping in continuous manner during tasks, and gives more flexibility and more abilities to robots in completion of complex tasks. The benefits of variable impedance have been explored through imitation of human impedance (Ajoudani, 2012; Howard et al., 2013; Yang et al., 2019), reinforcement learning (Buchli et al., 2011) and demonstration (Ficuciello et al., 2015; Park et al., 2019).

An impedance control law for a robot usually aims at achieving the desired second-order linear spring-damping dynamics of the interaction force and the desired-trajectory tracking error. In some applications such as rehabilitation and surgery, the considered robot moves slowly and the desired first-order impedance dynamics can be constructed by neglecting the desired inertial matrix (Calanca et al., 2016; Koivumaki and Mattila, 2017). Compared with the desired second-order impedance dynamics, the desired first-order impedance dynamics has the advantage in easy validation of the performance of impedance errors and control performance. In an impedance control framework, the stability of the desired impedance dynamics should be guaranteed by properly choosing impedance parameters and effective impedance controllers should be designed to achieve the desired impedance dynamics. Otherwise, the compliance of robot-environment interaction would be severely affected. Although some variable impedance controllers (Buchli et al., 2011; Ficuciello et al., 2015; Kronander and Billard, 2016; Park et al., 2019) were designed for robots, the stability properties of the closed-loop control systems are often overlooked. Thus, how to guarantee variable impedance control stability has always been a significant concern in impedance control design.

Choosing proper impedance parameters to guarantee the stability and realizability of the desired impedance dynamics is the first important concern. Almost all constant-impedance controllers (Cheah and Wang, 2018; He and Dong, 2018; He et al., 2016; Jamwal et al., 2016; Jung and Hsia, 1998; Li and Ge, 2014; Li and Liu, 2018; Li et al., 2017, 2018, 2020; Sharifi et al., 2017; Sun et al., 2019a, 2020) were designed under the assumption that the desired inertia, damping and stiffness are positive definite, diagonal, constant matrices which ensures the stability of the desired impedance dynamics. According to the linear system theory, stability of a linear time-varying system at every time cannot guarantee its stability at all time. Thus, the stability guarantee for variable impedance dynamics is much more difficult than the stability guarantee of constant-impedance dynamics. In 2016, Kronander first proposed constraints on variable impedance profiles in order to ensure the asymptotic stability of the desired second-order variable impedance dynamics (Kronander and Billard, 2016). Similar constraints also presented in Dong and Ren (2019) in order to guarantee variable-impedance control stability. However, given bounded interaction force, the constraints in Kronander and Billard (2016) and Dong and Ren (2019) cannot guarantee boundedness of the robot’s position, velocity, or acceleration in the desired variable impedance dynamics. This may lead to drift of the robot’s position, velocity, or acceleration, which results in failure achievement of the desired variable impedance dynamics. Recently, the proposed novel constraints on variable impedance profiles in Sun et al. (2019b) guarantee the exponential stability of the desired impedance dynamics, which ensures boundedness of the robot’s position, velocity and acceleration in the desired impedance dynamics. The stability of the second-order impedance dynamics was guaranteed in Kronander and Billard (2016), Dong and Ren (2019) and Sun et al. (2019b), but how to guarantee the stability the desired first-order variable impedance dynamics is still an open problem.

Designing effective impedance controllers is important for achievement of the desired first- and second-order variable impedance dynamics through the convergence of variable impedance errors to a small neighbourhood of zero. The factors that affect the performances of variable impedance errors mainly come from robots modelling uncertainties and system disturbances. The impedance control robustness can be improved by compensation of modelling uncertainties and system disturbances by their estimators (Liu et al., 2019a, b; Sun et al., 2019; Yu et al., 2019; Zhang et al., 2017). Due to the universal function approximation property of neural networks (NNs), adaptive NNs (Sun et al., 2017) have been demonstrated to be a powerful tool in approximation of structured and unstructured uncertainties in nonlinear systems. To improve control robustness and control stability, adaptive neural approximation-based control has been applied in many robots and has unfolded its powerful abilities in improving control performance.

In this paper, we propose two variable impedance controllers of robots to realize the desired variable impedance dynamics. The main contributions of this paper include:

Novel constraints on impedance profiles are proposed to ensure the exponential stability of the desired first-order variable impedance dynamics which guarantees the boundedness of the robot’s position and velocity in the desired first-order variable impedance dynamics.

Adaptive NN controllers with dead-zone modifications are proposed for robots to realize the desired first- and second-order variable impedance dynamics.

Compared with the existing constant-impedance controllers (Cheah and Wang, 2018; He and Dong, 2018; He et al., 2016; Jamwal et al., 2016; Joshua et al., 2015; Jung and Hsia, 1998; Li and Ge, 2014; Li and Liu, 2018; Li et al., 2017; Oh, 1999; Sadeghian et al., 2012; Sharifi et al., 2017), this paper proposes adaptive NN approximation-based variable impedance controllers for robots realization of the desired first-order and second-order variable impedance dynamics, respectively. Compared with the impedance controllers (Buchli et al., 2011; Dong and Ren, 2019; Ficuciello et al., 2015; Kronander and Billard, 2016; Park et al., 2019; Sun et al., 2019b) aiming at achievement of the desired second-order variable impedance dynamics, this paper guarantees the exponential stability of the desired first-order variable impedance dynamics. The variable impedance controller in Sun et al. (2019b) was designed based on approximated dynamic inversion and high control gains may be required in implementation of the controller. Compared with the controller (Sun et al., 2019b), modelling uncertainties and low-frequency disturbances are approximated and compensated by NNs, which effectively decreases control gains.

Problem formulation

Problem statement

The Euler-Lagrangian dynamics of the considered robot can be described as

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + F (\overset{\cdot}{q}) + d = τ + τ_{e},

(1)

where $q = [q_{1}, q_{2}, \dots, q_{n}]^{T} \in R^{n}$ is the joint angle vector, $M (q) \in R^{n \times n}$ denotes the inertia matrix, $C (q, \overset{\cdot}{q}) \overset{\cdot}{q} \in R^{n}$ represents the centripetal and Coriolis torque, $G (q) \in R^{n}$ is the gravity vector, $F (\overset{\cdot}{q}) \in R^{n}$ denotes the friction force, $d$ denotes the low-frequency disturbance term, $τ \in R^{n}$ is the control torque, and $τ_{e} \in R^{n}$ is the robot-environment interaction force.

Property 1. The matrix $M (q)$ is symmetric and positive definite. There exist positive constants $σ_{1}$ and $σ_{2}$ such that $σ_{1} I \leq M (q) \leq σ_{2} I$ .

Property 2. The matrix $\overset{\cdot}{M} (q) - 2 C (q, \overset{\cdot}{q})$ is skew symmetric.

Assumption 1. The desired trajectory $q_{d}$ and its time derivatives ${\overset{\cdot}{q}}_{d}$ and ${\overset{\cdot\cdot}{q}}_{d}$ are bounded.

Assumption 2. $M (q), C (q, \overset{\cdot}{q}), G (q), F (\overset{\cdot}{q}), d$ are robot modelling uncertainties.

Assumption 3. The low-frequency disturbance $d$ is a slowly-varying signal.

The objective of this paper is to design NN approximation-based variable impedance controllers to achieve the following desired first-order variable impedance dynamics

- τ_{e} = B_{d 0} (t) ({\overset{\cdot}{q}}_{d} - \overset{\cdot}{q}) + K_{d 0} (t) (q_{d} - q)

(2)

where $B_{d 0} (t)$ and $K_{d 0} (t)$ are the desired variable damping matrix, stiffness matrix, respectively, and to achieve the following desired second-order variable impedance dynamics

- τ_{e} = M_{d} ({\overset{\cdot\cdot}{q}}_{d} - \overset{\cdot\cdot}{q}) + B_{d} (t) ({\overset{\cdot}{q}}_{d} - \overset{\cdot}{q}) + K_{d} (t) (q_{d} - q)

(3)

where $M_{d}$ , $B_{d} (t)$ and $K_{d} (t)$ are the desired inertia matrix, damping matrix and the desired stiffness matrix, respectively.

The desired variable impedance dynamics in equations (2) and (3) can be achieved by obtaining the convergence of variable impedance errors $E_{0}$ and $e$ to zero or a small neighbourhood of zero, where $E_{0}$ and $e$ are defined as

E_{0} = B_{d 0} (t) ({\overset{\cdot}{q}}_{d} - \overset{\cdot}{q}) + K_{d 0} (t) (q_{d} - q) + τ_{e},

(4)

e = M_{d} ({\overset{\cdot\cdot}{q}}_{d} - \overset{\cdot\cdot}{q}) + B_{d} (t) ({\overset{\cdot}{q}}_{d} - \overset{\cdot}{q}) + K_{d} (t) (q_{d} - q) + τ_{e} .

(5)

RBF NN approximation

Let $Ω_{x} \in R^{N}$ be a compact set. The RBF NN approximation of a function $f (x) : Ω_{x}^{N} \to R^{n}$ can be expressed as follows:

\hat{f} (x) = W^{T} Φ (x)

(6)

where $W \in R^{N \times n}$ is the adjustable NN weight and $Φ (x) = [1, ϕ_{1} (x), ϕ_{2} (x), \dots, ϕ_{N - 1} (x)]^{T} \in R^{N}$ is an regression matrix, where $ϕ_{i} (.)$ being the Gaussian RBF functions defined as

ϕ_{i} (x) = \exp (- \frac{| | x - ξ_{i} {| |}^{2}}{2 b_{i}^{2}}), i = 1, 2, \dots, N - 1

(7)

with $ξ_{i}, b_{i}$ being the centre and width of the $i$ th neuron, respectively.

The universal approximation property for NNs (He et al., 2016) shows that for any continuous function $f (x) : Ω_{x} \to R^{n}$ , there exists an optimal NN approximation on $W \in Ω_{w} = {[w_{1}, \dots, w_{N}] : | | w_{i} | | \leq c_{wi}, i = 1, 2, \dots, N}$ with $c_{wi}$ being predefined constants, such that

f (x) = W^{* T} Φ (x) + ε^{*} (x)

(8)

where $ε^{*} (x)$ is the optimal approximation error which satisfies $| | ε^{*} (x) | | \leq ε_{x}$ with $ε_{x}$ being a positive constant, and $W^{*}$ is the optimal approximation weight given by

W^{*} = \underset{W \in Ω_{w}}{\arg \min} {sup_{x \in Ω_{x}} | | f (x) - \hat{f} | |} .

(9)

First-order variable impedance control design

Stability of the desired first-order variable impedance dynamics

Assumption 4. There exist positive constants $α_{i}, i = 1, 2, 3, 4$ , such that the diagonal matrices $B_{d 0} (t)$ and $K_{d 0} (t)$ satisfy $α_{1} I \leq B_{d 0} (t) \leq α_{2} I$ and $α_{3} I \leq K_{d 0} (t) \leq α_{4} I$ with I being the $2 \times 2$ sized identity matrix.

Proposition 1. Suppose Assumption 1 and Assumption 4 are satisfied. The desired trajectory tracking error $q_{d} - q$ in (2) will converge to zero when the interaction force $τ_{e} = 0$ , and the robot position and velocity in (2) are bounded when the interaction force $τ_{e}$ is bounded.

Proof. The dynamics in (2) can be expressed as

{\overset{\cdot}{q}}_{d} - \overset{\cdot}{q} = - B_{d 0}^{- 1} (t) K_{d 0} (t) (q_{d} - q) - B_{d 0}^{- 1} (t) τ_{e} .

(10)

Since Assumption 4 holds, the matrix $B_{d 0}^{- 1} (t) K_{d 0} (t)$ is a diagonal matrix and satisfies $- α_{1} α_{4} I \leq - B_{d 0}^{- 1} (t) K_{d 0} (t) \leq - α_{2} α_{3} I$ , which shows that the state transition matrix for the dynamics (10) is uniformly exponentially stable. Therefore, the desired trajectory tracking error $q_{d} - q$ in (2) will exponentially converge to zero when the interaction force $τ_{e} = 0$ . Since $B_{d 0}^{- 1} (t) \leq α_{1} I$ , the vector $B_{d 0}^{- 1} (t) τ_{e}$ is bounded if $τ_{e}$ is bounded. Then, the robot position and velocity in (2) are bounded when the interaction force $τ_{e}$ is bounded.

Adaptive NN impedance control design

Assumption 5. $B_{d 0} (t)$ , $K_{d 0} (t) \in C^{2}$ and their twice time derivatives are bounded.

Let the auxiliary signal $q_{r 0} \in R^{n}$ be generated by

B_{d 0} (t) {\overset{\cdot}{q}}_{r 0} = - K_{d 0} (t) q_{r 0} + [B_{d 0} (t) {\overset{\cdot}{q}}_{d} + K_{d 0} (t) q_{d} + τ_{e}] .

(11)

Then, the impedance error $E_{0}$ can be expressed as

\begin{matrix} E_{0} = B_{d 0} (t) ({\overset{\cdot}{q}}_{r} - \overset{\cdot}{q}) + K_{d 0} (t) (q_{r} - q) \\ = B_{d 0} (t) {\overset{\cdot}{e}}_{0} + K_{d 0} (t) e_{0} \end{matrix}

(12)

with $e_{0} = q_{r 0} - q$ . From equation (12), one can conclude that if $e_{0}$ and ${\overset{\cdot}{e}}_{0}$ converge to a small neighbourhood of zero, then the variable impedance error $E_{0}$ will converge to a small neighbourhood of zero.

Define the auxiliary error $r = [r_{1}, r_{2}, \dots, r_{n}]^{T} = {\overset{\cdot}{e}}_{0} + k_{0} e_{0}$ with $k_{0}$ being a positive design parameter. Then, based on the dynamics (1), one can obtain

M (q) \overset{\cdot}{r} = - C (q, \overset{\cdot}{q}) r + f (q, \overset{\cdot}{q}, {\overset{\cdot\cdot}{q}}_{r} + k_{0} {\overset{\cdot}{e}}_{0}, r) - τ - τ_{e}

(13)

where $f (x) = M (q) ({\overset{\cdot\cdot}{q}}_{r} + k_{0} {\overset{\cdot}{e}}_{0}) + G (q) + C (q, \overset{\cdot}{q}) (\overset{\cdot}{q} + r) + d (\overset{\cdot}{q})$ is an uncertain nonlinear function with $x = [1, q^{T}, {\overset{\cdot}{q}}^{T}, {\overset{\cdot\cdot}{q}}_{r}^{T} + k_{0} {\overset{\cdot}{e}}_{0}^{T}, r^{T}]^{T}$ .

Design the following neural approximation-based first-order variable impedance controller

τ = {\bar{k}}_{0} r + \hat{f} - τ_{e}

(14)

where ${\bar{k}}_{0}$ is a positive design parameter, $Γ_{i}$ is a positive definite, diagonal matrix, and $\hat{f} = W^{T} Φ (x)$ is the RBF NN approximator with the NN weight $W = [w_{1}, w_{2} \dots, w_{n}] \in R^{N \times n}$ being updated by the following dead-zone modification

{\overset{\cdot}{w}}_{i} = {\begin{matrix} Γ_{i} Φ (x) r_{i}, & for | | r | | > ε \\ 0, & otherwise \end{matrix}

(15)

with $ε$ being a small positive design parameter.

Substituting the controller in (14) into the dynamics in (13), yields

M (q) \overset{\cdot}{r} = - C (q, \overset{\cdot}{q}) r + {\tilde{W}}^{T} Φ (x) + ε^{*} (x) - {\bar{k}}_{0} r .

(16)

Theorem 1. Suppose Assumptions 1–5 be satisfied. Consider the robot with the dynamics in (1) driven by the proposed adaptive NN controller in (14). If the control parameters ${\bar{k}}_{0}$ and $ε$ are chosen to satisfy $({\bar{k}}_{0} - 0.5) ε^{2} > 0.5 ε_{x}^{2}$ , then, the signals in the closed-loop control system are bounded and the error $r$ are ultimately bounded by $ε$ . Thus, the variable impedance error will converge to a small neighbourhood of zero by properly choosing control parameters and the desired first-order variable impedance dynamics in (2) can be achieved.

Proof. Consider the following continuously differentiable positive-definite function

V_{0} = \frac{1}{2} r^{T} M (q) r + \frac{1}{2} \sum_{i = 1}^{n} {\tilde{w}}_{i}^{T} Γ_{i}^{- 1} {\tilde{w}}_{i}

(17)

with $\tilde{W} = [{\tilde{w}}_{1}, \dots, {\tilde{w}}_{n}] = W^{*} - W$ .

Differentiating $V_{0}$ with respect to time $t$ and substituting equation (17), yields

\begin{matrix} {\overset{\cdot}{V}}_{0} = \frac{1}{2} r^{T} (\overset{\cdot}{M} - 2 C (q, \overset{\cdot}{q}) r + r^{T} (- {\bar{k}}_{0} r + {\tilde{W}}^{T} Φ (x) \\ + ε^{*} (x)) - \sum_{i = 1}^{N} {\tilde{w}}_{i}^{T} Γ_{i}^{- 1} {\overset{\cdot}{w}}_{i} . \end{matrix}

(18)

Since the matrix $\overset{\cdot}{M} - 2 C (q, \overset{\cdot}{q})$ is skew symmetric, the first term of the right hand of equation (19) is equal to zero. Then,

\begin{matrix} {\overset{\cdot}{V}}_{0} = - {\bar{k}}_{0} r^{T} r + r^{T} ε^{*} (x) - \sum_{i = 1}^{N} {\tilde{w}}_{i}^{T} Γ_{i}^{- 1} ({\overset{\cdot}{w}}_{i} - Γ_{i} Φ (x) r_{i}) \\ \leq - ({\bar{k}}_{0} - 0.5) r^{T} r + 0.5 ε_{x}^{2} - \sum_{i = 1}^{N} {\tilde{w}}_{i}^{T} Γ_{i}^{- 1} ({\overset{\cdot}{w}}_{i} - Γ_{i} Φ (x) r_{i}) . \end{matrix}

(19)

The dead-zone weight update law in (15) guarantees that $- \sum_{i = 1}^{N} {\tilde{w}}_{i}^{T} Γ_{i}^{- 1} ({\overset{\cdot}{w}}_{i} - Γ_{i} ϕ (x) r_{i}) = 0$ if $| | r | | > ε$ . Thus, for any time at which $| | r | | > ε$ , one has that ${\overset{\cdot}{V}}_{0} \leq - δ$ with $δ = ({\bar{k}}_{0} - 0.5) ε^{2} - 0.5 ε_{x}^{2} > 0$ , from which we can see that the error $r$ enters the deadzone ${r : | | r | | \leq ε}$ in finite time.

The deadzone projection in (15) is designed to turn off adaptation when $| | r | | \leq ε$ . As described in Figure 1, for $j = 1, 2, \dots$ , let times with odd subscripts $t_{2 j + 1}$ denote times at which $| | r (t_{2 j + 1}) | | = ε$ and the error is entering the deadzone, while times with even subscripts $t_{2 j}$ denote times at which $| | r (t_{2 j}) | | = ε$ and the error is leaving the deadzone (see Figure 1). Then, one obtains

$V_{0} (t_{2 j - 1}) = V_{0} (t_{2 j})$ with $V_{0} (t) \leq V_{0} (t_{2 j - 1})$ for $t \in [t_{2 j - 1}, t_{2 j}]$ ;

${\overset{\cdot}{V}}_{0} (t) \leq - δ$ for $t \in [t_{2 j}, t_{2 j + 1}]$ , from which we further obtain $t_{2 j + 1} - t_{2 j} \leq \frac{1}{δ} (V_{0} (t_{2 j}) - V_{0} (t_{2 j + 1}))$ .

Figure 1.

Time indices on error convergence.

Denote $N$ as the numbers of times that the error $r$ enters the deadzone, where $1 \leq N \leq \infty$ . Then, the total time outside the deadzone satisfies

\begin{matrix} \sum_{j = 0}^{N} (t_{2 j + 1}) \leq \frac{1}{δ} \sum_{j = 0}^{N} (V_{0} (t_{2 j}) - V_{0} (t_{2 j + 1})) \\ \leq \frac{1}{δ} (V_{0} (t_{0}) - V_{1} (t_{2 N + 1})) \\ \leq \frac{1}{δ} V_{0} (t_{0}) < \infty . \end{matrix}

(20)

Therefore, the total time outside the deadzone is finite and the error $r$ are ultimately bounded by $ε$ . Since $V_{0} (t) \leq V_{0} (t_{0}), \forall t > t_{0}$ , the signals in the closed-loop control system are bounded.

Solving the differential equation ${\overset{\cdot}{e}}_{0} + k_{0} e_{0} = r$ yields

e_{0} (t) = e^{- k_{0} t} e_{0} (0) + \int_{0}^{t} e^{- k_{0} (t - τ)} r (τ) d τ .

(21)

If $| | r (t) | | \leq ε$ , then $| | e_{0} (t) | | \leq e^{- k_{0} t} (| | e_{0} (0) | | - ε / k_{0}) + ε / k_{0}$ . Since $r$ is bounded by the small constant $ε$ ultimately, and ${\overset{\cdot}{e}}_{0} = r - k_{0} e_{0}$ will converge to a small neighbourhood of zero. Therefore, we can conclude that the variable impedance error will converge to a small neighbourhood of zero and the desired first-order variable impedance dynamics in (2) can be achieved.

Second-order variable impedance control design

Stability of the second-order variable impedance dynamics

Assumption 6. $M_{d}$ , $B_{d} (t)$ and $K_{d} (t)$ are positive definite, diagonal matrices.

Denote ${\bar{τ}}_{e} = [{\bar{τ}}_{e 1}, {\bar{τ}}_{e 2}, \dots, {\bar{τ}}_{en}]^{T} = M_{d}^{- 1} τ_{e}$ , $α = [α_{1}, α_{2}, \dots, α_{n}]^{T}$ , ${\bar{B}}_{d} (t) = diag {{\bar{b}}_{ii} (t)} \overset{Δ}{=} M_{d}^{- 1} B_{d} (t)$ and ${\bar{K}}_{d} (t) = diag {{\bar{k}}_{ii} (t)} \overset{Δ}{=} M_{d}^{- 1} K_{d} (t)$ . Then, the dynamics in (3) can be expressed as

[\begin{matrix} \overset{\cdot}{α} \\ \overset{\cdot\cdot}{α} \end{matrix}] = A (t) [\begin{matrix} α \\ \overset{\cdot}{α} \end{matrix}] + [\begin{matrix} 0 \\ {\bar{τ}}_{e} \end{matrix}]

(22)

with $α = q_{d} - q$ and

A (t) = [\begin{matrix} 0 & I \\ - {\bar{K}}_{d} (t) & - {\bar{B}}_{d} (t) \end{matrix}] .

(23)

Since Assumption 5 holds, the dynamics in (22) is equivalent to

\begin{matrix} [\begin{matrix} {\overset{\cdot}{α}}_{i} \\ {\overset{\cdot\cdot}{α}}_{i} \end{matrix}] = [\begin{matrix} 0 & 1 \\ - {\bar{k}}_{ii} (t) & - {\bar{b}}_{ii} (t) \end{matrix}] [\begin{matrix} α_{i} \\ {\overset{\cdot}{α}}_{i} \end{matrix}] + [\begin{matrix} 0 \\ {\bar{τ}}_{ei} \end{matrix}], \\ i = 1, 2, \dots, n, \end{matrix}

(24)

where the state transition matrices of the dynamics (24) are defined as $Φ_{i} (t, t_{0}) i = 1, 2, \dots, n$ .

Assumption 7. There exists a positive constant $β_{1}$ , such that ${\bar{k}}_{ii} (t) \geq β_{1}, i = 1, 2, \dots, n$ .

Assumption 8. There exist positive constants $β_{2}$ and $β_{3}$ , such that the functions $K_{i} (t) = {\bar{b}}_{ii} (t) + 0.5 {\overset{\cdot}{\bar{k}}}_{ii} (t) / {\bar{k}}_{ii} (t) i = 1, 2, \dots, n$ satisfy $β_{2} \leq K_{i} (t) / \sqrt{{\bar{k}}_{ii} (t)} \leq β_{3}$ .

Proposition 2. (Sun et al., 2019b) Suppose Assumption 1 and Assumptions 6–8 hold. Then, (a) The desired variable impedance dynamics in (3) is exponential stable if the interaction force is zero; (b) The robot’s position, velocity and acceleration in the desired impedance dynamics are all bounded if the interaction force is bounded.

Remark 1. Assumption 8 is satisfied if and only if $(β_{2} - {\bar{b}}_{ii} (t)) {\bar{k}}_{ii}^{3 / 2} (t) \leq {\overset{\cdot}{\bar{k}}}_{ii} (t) \leq (β_{3} - {\bar{b}}_{ii} (t)) {\bar{k}}_{ii}^{3 / 2} (t)$ which gives bounds for variation rate of ${\bar{k}}_{ii} (t)$ . The assumption states that $k_{ii} (t)$ cannot vary too fast and is general for robots in practice. For example, $M_{d} = 1 I$ , $B_{d} (t) = (20 - 6 \cos (t)) I$ , $K_{d} (t) = (50 + 20 \sin (t)) I$ . For the variable impedance profiles, there exist $β_{3} = 150 / (7 \sqrt{30})$ and $β_{2} = 2 / \sqrt{70}$ , such that Assumption 8 is satisfied.

Neural approximation-based variable impedance control design

Let the auxiliary signal $γ \in R^{n}$ be generated by

\overset{\cdot\cdot}{γ} + M_{d}^{- 1} B_{d} (t) \overset{\cdot}{γ} + M_{d}^{- 1} K_{d} (t) γ = M_{d}^{- 1} τ_{e} .

(25)

Then, the impedance error $e$ can be presented as

\begin{matrix} e = M_{d} ({\overset{\cdot\cdot}{q}}_{d} + \overset{\cdot\cdot}{γ} - \overset{\cdot\cdot}{q}) + B_{d} (t) ({\overset{\cdot}{q}}_{d} + \overset{\cdot}{γ} - \overset{\cdot}{q}) \\ + K_{d} (t) (q_{d} + γ - q) \\ = M_{d} ({\overset{\cdot\cdot}{q}}_{r} - \overset{\cdot\cdot}{q}) + B_{d} (t) ({\overset{\cdot}{q}}_{r} - \overset{\cdot}{q}) + K_{d} (t) (q_{r} - q) \\ = M_{d} {\overset{\cdot\cdot}{e}}_{1} + B_{d} (t) {\overset{\cdot}{e}}_{1} + K_{d} (t) e_{1} \end{matrix}

(26)

where $q_{r}$ and $e$ are defined as $q_{r} = q_{d} + γ$ and $e_{1} = q_{r} - q$ . From equation (26), the impedance error can be considered as the low-pass filtered signal of $e_{1}$ .

This section aims at designing an adaptive NN control law to obtain the convergence $e_{1}$ , ${\overset{\cdot}{e}}_{1}$ to a small neighbourhood of zero, which indicates the realization of the desired variable second-order impedance dynamics in the low-frequency range.

Remark 2. From Proposition 2, we can obtain the boundedness of $γ$ and its first-order and second-order time derivatives. From Assumption 1, we can further obtain the boundedness of $q_{r}$ and its first- and second-order time derivatives.

Define the auxiliary error $e_{2}$ as

e_{2} = {\overset{\cdot}{e}}_{1} + k_{1} e_{1}

(27)

with $k_{1}$ being a positive design parameter. Then, the convergence of $e_{1}$ and ${\overset{\cdot}{e}}_{1}$ to a small neighbourhood of zero can be obtained from the convergence of $e_{2}$ to a small neighbourhood of zero.

Taking the time derivative of $e_{2}$ and substituting equation (1) into it, yields

M (q) {\overset{\cdot}{e}}_{2} = - C (q, \overset{\cdot}{q}) e_{2} + f_{1} (X_{1}) - τ - τ_{e} .

(28)

where $f_{1} (X_{1}) = M (q) ({\overset{\cdot\cdot}{q}}_{r} + k_{1} {\overset{\cdot}{e}}_{1}) + C (q, \overset{\cdot}{q}) (e_{2} + \overset{\cdot}{q}) + G (q) + d (\overset{\cdot}{q})$ with $X_{1} = [1, q^{T}, {\overset{\cdot}{q}}^{T}, {\overset{\cdot\cdot}{q}}_{r}^{T} + k_{1} {\overset{\cdot}{e}}_{1}^{T}, e_{2}^{T}]^{T}$ .

Design the neural approximation-based second-order variable impedance control law as

τ = - τ_{e} + {\bar{k}}_{1} e_{2} + {\hat{f}}_{1}

(29)

where ${\bar{k}}_{1}$ is a positive control gain, ${\hat{f}}_{1} = W_{1}^{T} Φ_{1} (X_{1})$ is the RBF NN approximator with $W_{1} = [w_{11}, \dots, w_{1 n}]^{T}$ and $Φ_{1}$ being the NN weight and the basis function vector, respectively, and the NN weight $W_{1} = [w_{11}, w_{12} \dots, w_{1 n}] \in R^{N \times n}$ is updated by the following dead-zone modification

{\overset{\cdot}{w}}_{1 i} = {\begin{matrix} Γ_{1 i} ϕ_{1} (X_{1}) e_{2 i}^{T}, & for | | e_{2} | | > ε_{1} \\ 0, & otherwise \end{matrix}

(30)

with $ε_{1}$ being a small positive design parameter and $Γ_{1 i}$ being positive definite matrices.

Substituting the control law in (29) into the dynamics in (28), one can obtain

M (q) {\overset{\cdot}{e}}_{2} = - C (q, \overset{\cdot}{q}) e_{2} + {\tilde{f}}_{1} - {\bar{k}}_{1} e_{2}

(31)

where ${\tilde{f}}_{1} = f_{1} - {\hat{f}}_{1}$ .

Theorem 2. Suppose Assumptions 1–3 and Assumptions 6–8 be satisfied. Consider the robot with the Euler-Lagrangian dynamics in equation (1) driven by the variable impedance controller in (29). If the control parameters ${\bar{k}}_{1}$ and $ε_{1}$ are chosen to satisfy $({\bar{k}}_{1} - 0.5) ε^{2} > 0.5 ε_{x}^{2}$ , then, the signals in the closed-loop control system are bounded and the error $e_{2}$ are ultimately bounded by $ε$

Proof. The proof of this theorem is similar to the proof of Theorem 1 and is omitted here for simplicity.

Remark 3. The convergence of $e_{1}$ and ${\overset{\cdot}{e}}_{1}$ to a small neighbourhood of zero in Theorem 2 ensures the convergence of the impedance error to a small neighbourhood of zero and the achievement of the desired second-order variable impedance dynamics in the low-frequency range (Li and Liu, 2018). The reasons are stated as follows: (a) Since the error $e_{2}$ converges to a small neighbourhood of zero and the neural approximation error ${\tilde{f}}_{1}$ is small, from the dynamics in (31), one can obtain that the signal ${\overset{\cdot}{e}}_{2}$ and then ${\overset{\cdot\cdot}{e}}_{1}$ is ultimately bounded by a small positive constant. (b) In rehabilitation, surgery and some other applications, the desired impedance dynamics are designed in low-frequency range in which the acceleration in the desired impedance dynamics varies slowly. Moreover, the stiffness and damping signals are much larger than the stiffness signal in the desired impedance dynamics. Thus, the error $e_{1}$ and its time derivative ${\overset{\cdot}{e}}_{1}$ play a dominant role in the low-pass filtered form of $e_{1}$ in (26).

Simulation results

To illustrate the effectiveness of the proposed variable impedance control laws, simulations are taken on a five-bar parallel robot with the structure described by Figure 2, where links 1 and 3 are parallel, links 2 and 4 are parallel and have same length, and the point $P$ is the end effector point. The matrices $M (q)$ , $C (q, \overset{\cdot}{q})$ in equation (1) are presented as

\begin{matrix} M (q) = [\begin{matrix} d_{11} (q) & d_{12} (q) \\ d_{21} (q) & d_{22} (q) \end{matrix}], C (q, \overset{\cdot}{q}) = [\begin{matrix} 0 & h {\overset{\cdot}{q}}_{2} \\ - h {\overset{\cdot}{q}}_{1} & 0 \end{matrix}], \\ F (\overset{\cdot}{q}) = diag (d_{1}, d_{2}) \overset{\cdot}{q} \end{matrix}

(32)

where

\begin{matrix} d_{11} (q) = m_{1} l_{c 1}^{2} + m_{3} l_{c 3}^{2} + m_{4} l_{1}^{2} + I_{1} + I_{3}, \\ d_{12} (q) = d_{21} (q) = (m_{3} l_{2} l_{c 3} + m_{4} l_{1} l_{c 4}) \cos (q_{2} - q_{1}), \\ d_{22} (q) = m_{2} l_{c 2}^{2} + m_{3} l_{2}^{2} + m_{4} l_{c 4}^{2} + I_{2} + I_{4}, \\ h = - (m_{3} l_{2} l_{c 3} + m_{4} l_{1} l_{c 4}) \sin (q_{2} - q_{1}) \end{matrix}

(33)

with $l_{1}$ , $l_{2}, l_{3}$ and $l_{4}$ being lengths of the joints links, $l_{c 1}$ , $l_{c 2}, l_{c 3}$ and $l_{c 4}$ being distances between the joints and respective centre of masses, $I_{1}, I_{2}, I_{3}$ and $I_{4}$ being inertial moments of the respective links, and $m_{1}, m_{2}, m_{3}$ and $m_{4}$ being masses of the respective links. The robot model defined by equation (1) has the following parameters: $l_{1} = 0.2 m, l_{2} = l_{4} = 0.4 m, l_{3} = 0.5 m, l_{c 1} = 0.1 m, l_{c 2} = l_{c 4} = 0.2 m, l_{c 3} = 0.25 m, I_{1} = 0.1 kg . m^{2}, I_{2} = I_{4} = 0.2 kg . m^{2}, I_{3} = 0.3 kg . m^{2}, m_{1} = 0.5 kg, m_{2} = m_{4} = 1 kg, m_{3} = 1.5 kg, d_{1} = d_{2} = 0.4$ , and $τ_{d} = 0.5 [\sin (0.5 t), \cos (0.5 t)]^{T}$ ,

Figure 2.

The structure of the considered parallel robot.

The robot-environment interaction force at the end-effector is defined as $τ_{e} = J^{T} (q) f_{e} \in R^{2}$ with the Jacobian matrix $J (q)$ being defined as

J (q) = [\begin{matrix} - l_{3} \sin q_{1} & - l_{2} \sin q_{2} \\ l_{3} \cos q_{1} & l_{2} \cos q_{2} \end{matrix}]

(34)

and the interaction force $f_{e} = [f_{e 1}, f_{e 2}]^{T}$ is chosen as $f_{e 1} = 10 \sin (2 t), f_{e 2} = 10 \cos (2 t)$ . The robot initial position and velocity are chosen as zero vectors, respectively. The desired trajectory $q_{d} = [q_{d 1}, q_{d 2}]^{T}$ in the desired impedance dynamics is chosen as $q_{d 1} = 1 + 0.3 \cos (π t / 3)$ , $q_{d 2} = 1 + 0.3 \sin (π t / 3)$ . Simulations are carried out in MATLAB software, where the solver is fixed-step ode 4, the step size is 0.01 s, other settings are kept at their defaults, and the measurement noises for $q$ and $\overset{\cdot}{q}$ are generated by normrnd with the mean value 0 and the standard deviation 0.02. Simulations are taken on Case 1 and Case 2 to verify the effectiveness of the proposed first-order and second-order variable impedance controllers, respectively.

Case 1: The desired first-order variable impedance dynamics. In this case, the desired variable impedance profiles are chosen as $B_{d} (t) = (10 - 3 \cos (t)) I$ and $K_{d} (t) = (25 + 10 \sin (t)) I$ which satisfy Assumptions 4–5 and guarantee the stability of the desired first-order variable impedance dynamics. For the proposed adaptive NN impedance controller, choose the following control parameters: $k_{0} = 2$ , ${\bar{k}}_{0} = 4$ , $Γ_{i} = 20 I$ , $ε = 0.005$ . The NN has $N = 37$ neurons, where the neuron centres are chosen as $[- 4.5 + 1.8 σ_{1}, - 4.5 + 1.8 σ_{2}, - 4.5 + 0.9 σ_{1}, - 4.5 + 0.9 σ_{2}, - 4.5 + 1.35 σ_{1}, - 4.5 + 1.35 σ_{2}, - 4.5 + 0.9 σ_{1}, - 4.5 + 0.9 σ_{2}]^{T}$ for $σ_{1} = 0, 1, \dots, 5, σ_{2} = 0, 1, \dots, 5$ , neuron widths are chosen as $b_{1} = b_{2} = \dots = b_{36} = 4.5$ , and the initial value of the NN weight is chosen as the zero vector.

The simulation results in Case 1 are presented in Figures 3–4. From the performance of the impedance error $E_{0} = [E_{0} (1), E_{0} (2)]^{T}$ and the errors $e_{0} = [e_{0} (1), e_{0} (2)]^{T}$ and ${\overset{\cdot}{e}}_{0} = [{\overset{\cdot}{e}}_{0} (1), {\overset{\cdot}{e}}_{0} (2)]^{T}$ in Figure 3, we can see that the impedance error and the tracking error $e_{0}$ and its time derivative converge to a small neighbourhood of zero after 4 seconds. From the performance of the NN approximation in Figure 3, the designed NN approximates the uncertain function $f (x)$ well with the approximation error $\tilde{f} = [{\tilde{f}}_{1} (1), {\tilde{f}}_{2}]^{T}$ being bounded by 0.1 after 4 seconds. By comparison of Figure 3 and Figure 4, we can see favourable control robustness to measurement noises. Thus, we can conclude that the proposed adaptive neural variable impedance controller guarantees the convergence of the errors $e_{0}$ and ${\overset{\cdot}{e}}_{0}$ to a small neighbourhood of zero, which ensures the favourable performance of the impedance error and the achievement of the desired first-order variable impedance dynamics.

Case 2: The desired second-order variable impedance dynamics. The desired variable impedance profiles are chosen as $M_{d} = 0.5 I, B_{d} (t) = 10 - 3 \cos (t) I, K_{d} (t) = 25 + 10 \sin (t) I$ which satisfy Assumptions 6–8 and guarantee the stability of the desired second-order variable impedance dynamics. In this case, the control parameters are chosen as $k_{1} = 2$ , $k_{2} = 4$ , $Γ_{1 i} = 20 I$ , $ε_{1} = 0.005$ , the NN is designed as the same as in Case 1, and the simulation results are given in Figures 5–6. From the performance of the impedance error $e = [e (1), e (2)]^{T}$ and the errors $e_{1} = [e_{1} (1), e_{1} (2)]^{T}$ and ${\overset{\cdot}{e}}_{1} = [{\overset{\cdot}{e}}_{1} (1), {\overset{\cdot}{e}}_{1} (2)]^{T}$ , we can see that the impedance error and the tracking error $e_{1}$ and its time derivative converge to a small neighbourhood of zero after 4 seconds. From the performance of the NN approximation, the designed NN approximates the uncertain function $f_{1} (x)$ well with the approximation error ${\tilde{f}}_{1} = [{\tilde{f}}_{1} (1), {\tilde{f}}_{1} (2)]^{T}$ being bounded by 0.1 after 4 seconds. By comparison of Figure 5 and Figure 6, we can see favourable control robustness to measurement noises. Thus, we can conclude that the proposed adaptive neural variable impedance controller guarantees the convergence of the errors $e_{1}$ and ${\overset{\cdot}{e}}_{1}$ to a small neighbourhood of zero, which ensures the performance of the impedance error and the achievement of the desired second-order variable impedance dynamics. Therefore, the effectiveness of the stability-guaranteed variable impedance controller can be concluded.

Figure 3.

Simulation results by the proposed controller in (14) in Case 1 without measurement noise.

Figure 4.

Simulation results by the proposed controller in (14) in Case 1 with measurement noise.

Figure 5.

Simulation results by the proposed controller in (29) in Case 2 without measurement noise.

Figure 6.

Simulation results by the proposed controller in (29) in Case 2 with measurement noise.

Conclusions

Novel constraints on variable impedance profiles have been given to guarantee the exponential stability of the desired first- and second-order variable impedance dynamics, which ensures the boundedness of the robot’s position, velocity and acceleration in the desired impedance dynamics. Two adaptive neural impedance control strategies have been developed for robots to realize the desired first- and second-order variable impedance dynamics, respectively. By theoretical analysis and simulation results, we have validated the convergence of impedance errors to a small neighbourhood of zero and the effectiveness of the proposed variable impedance controllers. This paper contributes to stability-guaranteed variable impedance control design.

Footnotes

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: This paper was supported in part by the the National Key Research and Development Project under Grant 2019YFB1312503.

ORCID iD

Tairen Sun

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