AUTHOR ONE et al

*Mien Van, School of electronics, electrical and computer science, Queen’s University of Belfast, Belfast BT7 1NN.

# Fixed-time Integral Sliding Mode Control for Admittance Control of a Robot
Manipulator

Yuzhu Sun  Mien Van  Stephen McIlvanna  Seán McLoone  Dariusz Ceglarek  School
of electronics, electrical and computer science, Queen’s University of
Belfast, Belfast, UK  School of electronics, electrical and computer science,
Queen’s University of Belfast, Belfast, UK  School of electronics, electrical
and computer science, Queen’s University of Belfast, Belfast, UK  School of
electronics, electrical and computer science, Queen’s University of Belfast,
Belfast, UK  Warwick Manufacturing Group, University of Warwick, Coventry, UK
<EMAIL_ADDRESS>Sun Y  Van M  McIlvanna S  McLoone S  Ceglarek D

(26 April 2016; 6 June 2016; 6 June 2016)

###### Abstract

[Summary]This paper proposes a novel fixed-time integral sliding mode
controller for admittance control to enhance physical human-robot
collaboration. The proposed method combines the benefits of compliance to
external forces of admittance control and high robustness to uncertainties of
integral sliding mode control (ISMC), such that the system can collaborate
with a human partner in an uncertain environment effectively. Firstly, a
fixed-time sliding surface is applied in the ISMC to make the tracking error
of the system converge within a fixed-time regardless of the initial
condition. Then, a fixed-time backstepping controller (BSP) is integrated into
the ISMC as the nominal controller to realize global fixed-time convergence.
Furthermore, to overcome the singularity problem, a non-singular fixed-time
sliding surface is designed and integrated into the controller, which is
useful for practical application. Finally, the proposed controller is
validated for a two-link robot manipulator with uncertainties and external
human forces. The results show that the proposed controller is superior in the
sense of both tracking error and convergence time, and at the same time, can
comply with human motion in a shared workspace.

, , , , and (2022), Fixed-time Integral Sliding Mode Control for Admittance
Control of a Robot Manipulator, Manuscript submitted for publication..

###### keywords:

Integral sliding mode control, fixed-time convergence, admittance control,
robot manipulator, physical human-robot collaboration

††articletype: Article Type

##  1 Introduction

Recent developments in the biomedical, social and industrial robotics have led
to an increasing interest in the field of physical human-robot collaboration
(pHRC).1, 2 Robots that can physically collaborate with human partners enable
the cognitive strengths of human and the high-precision and repeatability of
robots to be combined. Unlike fully automated industrial production lines, to
effectively work with the human partner in a shared workspace involving
activities such as object handovers or co-carrying, it is imperative that the
cooperative robots are capable of simultaneously complying with human forces
at the contact point rather than rejecting them as external disturbances.3
Therefore, impedance/admittance control based compliance control strategies
have been attracting significant interest for this application. These include
the adaptive admittance control,4 adaptive impedance control,5, 6 and robust
impedance control.7, 8

Figure 1: Structure of the proposed FTISMC Admittance Control Scheme.

Consider an industrial robot that is required to work with a human partner in
an uncertain environment. The controller of the robot should be capable of
tracking a certain trajectory with external uncertainties and perturbations,
and at the same time, be compliant to human forces without the loss of
stability. A considerable literature exists on techniques for improving the
tracking performance of robot manipulators in an uncertain environment. For
many years, model-free control strategies such as PID have been extensively
employed in the industry.9 Sliding mode control (SMC) based methods have also
emerged and been widely studied in the field of robotics to provide robustness
against the effects of parametric uncertainties and external disturbances.10
The SMC scheme involves (i) the reaching phase, in which the system converges
from the initial state to the sliding surface, and (ii) the sliding phase, in
which the system reaches the sliding surface and tries to stay on the
equilibrium point. However, the effectiveness of SMC is limited by the
reaching phase which is sensitive to disturbances. Therefore, to overcome the
limitation of SMC, integral sliding mode control (ISMC) has been developed.
ISMC eliminates the reaching phase by finding a suitable initial position to
reject disturbances from the beginning, which is helpful for practical
applications.10

Furthermore, to obtain the property of fast tracking-error convergence, a
further extension of ISMC, called fixed-time integral sliding mode control
(FTISMC),11, 12, 13 which incorporates the benefits of ISMC and fixed-time
sliding mode control (FxTSMC)14 has been proposed. Compared with traditional
finite-time control strategies,15, 16 the fixed-time controller does not
depend on the initial condition of the system with the result that the
convergence time can be determined a priori based on the design parameters.16
In the work of Huang and Wang,11 a FTISMC is proposed that stabilizes a highly
complex nonlinear system within a fixed time and effectively restrains the
irregular and nonlinear vibrations of the system. In the work of Li et al.12 a
new integral high order sliding mode controller is proposed for high-order
nonlinear systems with fixed-time convergence. In addition, the singularity
problem always comes up in the sense of fixed-time stability when calculating
the derivative of sliding variables. To address this issue a number of
approaches have been developed to avoid or eliminate the singularity. 17, 18,
9, 19, 20 In the work of Feng et al.17 a global non-singular terminal sliding
mode controller is presented for second-order systems to overcome the
singularity problem. In the work of Zuo,19 a new sliding surface is proposed
to circumvent the singularity problem. In the work of Van and Ceglarek,20 a
new fault-tolerant control scheme based on a nonsingular fixed-time sliding
mode controller is proposed for robot manipulators, which guarantees a global
fixed-time convergence.

So far there is little literature that combines admittance control and FTISMC
for physical human-robot collaboration. Motivated by the above discussion this
paper proposes a new ISMC control scheme for admittance control of a robot
manipulator, as depicted in Fig. 1. The stability and tracking error of the
closed-loop system are converged within a fixed time. Furthermore, for
practical application, a nonsingular fixed-time integral sliding mode
controller is designed. The proposed method is then simulated on a two-link
robot manipulator and compared with other state-of-the-art control strategies.
The simulation results show that the proposed controller is superior in the
sense of both tracking error and convergence time. The contributions and
innovations of the proposed approach can be highlighted in a comparison with
other approaches as follows:

  1. 1.

Compared to conventional admittance control,2 the proposed approach integrates
an admittance with a FTISMC to guarantee a fixed-time convergence, while
compliant with external forces. Therefore, the proposed controller provides
faster transient response, lower tracking errors and better disturbance
rejection capacity.

  2. 2.

Compared to the existing FTISMC,11, 12, 13 which uses the tracking errors and
the velocity of the tracking errors to reconstruct the integral sliding
surface, the proposed controller develops a new nonsingular fixed-time sliding
surface and uses the dynamics model of the system, which is reconstructed
based on the nonsingular fixed-time sliding surface, to reconstruct a new
integral sliding surface. Therefore, the proposed approach is able to avail of
the full advantages of both ISMC and FxTSMC.

The paper is organized as follows. The mathematical model describing the
motion of a robot manipulator and the problem formulation are presented in
Section 2. The proposed fixed-time ISMC scheme for admittance control is
developed in Section 3. Simulation results demonstrating the effectiveness of
the proposed controller are presented in Section 4. Finally, section 5
discusses the conclusions and directions for future works.

##  2 Problem formulation and preliminaries

###  2.1 Problem Formulation

We consider a robot manipulator whose joint space dynamics equation can be
written in the form:

$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G\left(q\right)+F\left(\dot{q}\right)=\tau_{c}+\tau_{e}$
(1)

where $q$, $\dot{q}$, $\ddot{q}$ are the joint positions, velocities and
accelerations of the robot manipulator, respectively.
$q=\left[q_{1},q_{2},...,q_{N}\right]^{T}$ and $N$ is the number of joints.
$M\left(q\right)$ is the $N\times N$ mass matrix, $C\left(q,\dot{q}\right)$ is
the $N\times N$ Coriolis and centrifugal forces matrix, $G\left(q\right)$ is
the $N\times 1$ gravity vector, and $F\left(\dot{q}\right)$ is the $N\times 1$
vector of frictional forces. $\tau_{c}$ denotes the control torque acting at
the joints, and $\tau_{e}$ is the external torque from the interaction with
the human partner. $F\left(\dot{q}\right)$ and $\tau_{e}$ together form the
time-varying lumped uncertainties of the system. Fig. 2 above shows the
structure of a classical two-link robot manipulator interacting physically
with a human.

Figure 2: Two-Link robot manipulator.

Since it is easier to formulate the trajectory in Cartesian space, we transfer
the joint space dynamics of the robot manipulator (1) into Cartesian space:

$M_{x}\ddot{x}+C_{x}\dot{x}+G_{x}+F_{x}=f_{c}+f_{e}$ (2)

where $x=\left[x_{1},x_{2},...,x_{N}\right]$ is the position of the joints in
Cartesian space, $M_{x}$, $C_{x}$, $G_{x}$ and $F_{x}$ are coefficient
matrices in Cartesian space, $f_{c}$ is the control force, and $f_{e}$ is the
external force. The joint space velocities and Cartesian space velocities are
related as:

$\dot{x}=J\left(q\right)\dot{q}$ (3)

where $J\left(q\right)$ is the Jacobian of the robot manipulator. Inserting
(3) into (1), the coefficient matrices in Cartesian space can be written as:

$M_{x}=J^{-T}\left(q\right)M\left(q\right)J^{-1}\left(q\right)$
$C_{x}=J^{-T}\left(q\right)\left(C\left(q,\dot{q}\right)-M\left(q\right)J^{-1}\left(q\right)\dot{J}\left(q\right)\right)J^{-1}\left(q\right)$
$G_{x}=J^{-T}\left(q\right)G\left(q\right),\
F_{x}=J^{-T}\left(q\right)F\left(\dot{q}\right)$
$f_{c}=J^{-T}\left(q\right)\tau_{c},\ \ f_{e}=J^{-T}\left(q\right)\tau_{e}$

Letting $\eta_{1}=x$ and $\eta_{2}=\dot{x}$, the Cartesian space model of the
robot can be represented as:

$\left\\{\begin{array}[]{l}\dot{\eta}_{1}=\eta_{2}\\\
\dot{\eta}_{2}=M_{x}^{-1}\left(-C_{x}\dot{x}-G_{x}-F_{x}+f_{e}+f_{c}\right)\\\
\end{array}\right.$ (4)

For ease of control design, let $\varXi=M_{x}^{-1}$,
$\varGamma\left(x,\dot{x}\right)=M_{x}^{-1}\left(-C_{x}-G_{x}\right)$,
$\varPsi\left(x,\dot{x},t\right)=M_{x}^{-1}\left(-F_{x}+f_{e}\right)$, and
$u=f_{c}$, such that the Cartesian space model can be further written as:

$\left\\{\begin{array}[]{l}\dot{\eta}_{1}=\eta_{2}\\\ \dot{\eta}_{2}=\varXi
u+\varGamma\left(x,\dot{x}\right)+\varPsi\left(x,\dot{x},t\right)\\\
\end{array}\right.$ (5)

_Assumption 1_ : There exists a positive constants $\rho$, such that the
unknown lumped uncertainties term is bounded by:

$\varPsi\left(x,\dot{x},t\right)=M_{x}^{-1}\left(-F_{x}+f_{e}\right)\leq\rho$
(6)

where $\rho$ is a positive constant.
$\varPsi\left(x,\dot{x},t\right)=[\varPsi_{i}\left(x,\dot{x},t\right),...,\varPsi_{N}\left(x,\dot{x},t\right)]^{T}$.
This assumption means that the lumped uncertainties such as joint errors,
friction and external human forces are bounded, which is usually satisfied in
real application scenarios.

The control objective of this paper is to develop a fixed-time ISMC scheme to
make the joint position $x$ track a certain trajectory $x_{r}$ which is
generated from admittance control. Therefore, the robot manipulator can comply
with human forces rather than reject them as external disturbances.
Furthermore, we require that the tracking error converges within an
arbitrarily small interval of time, regardless of the initial conditions.

###  2.2 Preliminaries

Consider a time-varying differential equation:

$\dot{x}=f\left(x,t\right)$ (7)

where $x\in R^{n}$, and $f:\ R^{n}\rightarrow R^{n}$ is a continuous nonlinear
function. Assume that the origin of (7) is the stable equilibrium point and
$f\left(0\right)=0$.

_Definition 1_9: The system (7) is finite time stable if the origin is
Lyapunov stable and any solution $x\left(t\right)$ starting from $x_{0}$
satisfies $\lim_{t\rightarrow T\left(x_{0}\right)}x\left(t\right)\rightarrow
0$ and $x\left(t\right)=0,\forall t>T\left(x_{0}\right)$, where
$T\left(x_{0}\right)$ represents the settling time of the system.

_Definition 2_9: The system (7) is fixed-time stable if it is globally finite
time stable and its settling time $T\left(x_{0}\right)$ satisfies
$T\left(x_{0}\right)<T_{\max}$, where $T_{\max}$ is a positive number.

_Lemma 1_ 16: The following inequality holds for any real number that
satisfies $x_{1},x_{2},...,x_{n}>0$ and $0<k<1$:

$\sum_{i=1}^{n}{x_{i}^{k+1}\geq\left(\sum_{i=1}^{n}{x_{i}^{2}}\right)}^{\frac{k+1}{2}}$
(8)

_Lemma 2_ 16: The following inequality holds for any real number that
satisfies $x_{1},x_{2},...,x_{n}>0$ and $k>1$:

$\sum_{i=1}^{n}{x_{i}^{k}\geq n^{1-k}\left(\sum_{i=1}^{n}{x_{i}}\right)}^{k}$
(9)

_Lemma 3_ 16: Consider a scalar differential system below:

$\dot{y}=\lambda_{1}\left[y\right]^{\alpha}+\lambda_{2}\left[y\right]^{\beta},\
y\left(0\right)=y_{0}$ (10)

where $\lambda_{1}$, $\lambda_{2}$, $\alpha$, $\beta$ are all positive real
number satisfying $\lambda_{1},\lambda_{2}>0$ , $0<\alpha<1$ and $\beta>1$.
Then the system (10) is fixed-time stable and the convergence time is
independent with respect to the initial state of the system and upper bounded
by the design parameters as:

$T\left(x_{0}\right)<T_{\max}=\frac{1}{\lambda_{1}\left(1-\alpha\right)}+\frac{1}{\lambda_{2}\left(\beta-1\right)}$
(11)

##  3 Control design and stability analysis

###  3.1 Admittance Trajectory Shaping

To comply with external human forces, the system at the contact point can be
modeled as a mass-spring-damper system (shown in Fig. 3). The objective of
modeling the system with virtual mass, spring and damper is to make sure the
interaction forces are elastic and never vibrate at the contact point.

Figure 3:  The mass-spring-damper system.

Let $\xi_{i}=x_{r_{i}}-x_{d_{i}},i=1,2,...,N$, where the $x_{r_{i}}$ is the
reference trajectory generated from admittance control and the $x_{d_{i}}$ is
the initial desired trajectory of the robot manipulator. The dynamics for a
robot manipulator rendering an impedance can be written as:

$k_{m_{i}}\ddot{\xi}_{i}+k_{b_{i}}\dot{\xi}_{i}+k_{k_{i}}\xi_{i}=f_{e_{i}}$
(12)

where $k_{m_{i}}$, $k_{b_{i}}$ and $k_{k_{i}}$ are mass, spring and damping
coefficient. $f_{e}$ is the external human force. By integrating the impedance
equation (12), we obtain $x_{r_{i}}$ which will be tracked by the controller.

###  3.2 Design of Fixed-time ISMC

Let the tracking error be written as:

$e=x-x_{r}$ (13)

where $x_{r}$ is the reference trajectory generated from the admittance
control, and $x$ is the real trajectory of robot. The derivative of tracking
error $e$ is $\dot{e}=\dot{x}-\dot{x}_{r}$. To obtain a fixed-time integral
sliding mode controller, the error is reconstructed based on Lemma 3:

$s=\dot{e}+k_{1}\left[e\right]^{\alpha}+k_{2}\left[e\right]^{\beta}$ (14)

where $k_{1}$, $k_{2}$, $\alpha$, $\beta$ are all positive constants
satisfying $k_{1},k_{2}>0$ , $0<\alpha<1$ and $\beta>1$. When the sliding
surface $s$ converges to zero, (14) can be rewritten as:

$\dot{e}=-k_{1}\left[e\right]^{\alpha}-k_{2}\left[e\right]^{\beta}$ (15)

According to Lemma 3, the convergence time of (15) is bounded by:

$T_{s1}\leq\frac{1}{k_{1}\left(1-\alpha\right)}+\frac{1}{k_{2}\left(\beta-1\right)}$
(16)

Differentiating (14) with respect to time and using (5), we have:

$\displaystyle\dot{s}=\ddot{x}-\ddot{x}_{r}+k_{1}\alpha\left[e\right]^{\alpha-1}\dot{e}+k_{2}\beta\left[e\right]^{\beta-1}\dot{e}=\varXi
u+\varGamma\left(x,\dot{x}\right)+\varPsi\left(x,\dot{x},t\right)-\ddot{x}_{r}+k_{1}\alpha\left[e\right]^{\alpha-1}\dot{e}+k_{2}\beta\left[e\right]^{\beta-1}\dot{e}$
(17)

Based on (14), the integral sliding surface is selected as:

$\displaystyle\sigma_{1}\left(t\right)=s\left(t\right)-s\left(0\right)-\int\limits_{0}^{t}{\left(\varXi
u_{0}+\varGamma\left(x,\dot{x}\right)-\ddot{x}_{r}+k_{1}\alpha\left[e\right]^{\alpha-1}\dot{e}+k_{2}\beta\left[e\right]^{\beta-1}\dot{e}\right)dt}$
(18)

where term $s\left(0\right)$ represents the value of the sliding surface when
$t=0$, which is added to eliminate the initial error. $u_{0}$ is the output of
the nominal controller, which is designed for the non-disturbance system by
another method (e.g., PID, CTC, etc.) and will be presented in the Section
3.4. The derivative of the integral sliding surface can be further written as:

$\displaystyle\dot{\sigma_{1}}\left(t\right)=\left(\varXi
u+\varGamma\left(x,\dot{x}\right)+\varPsi\left(x,\dot{x},t\right)-\ddot{x}_{r}+k_{1}\alpha\left[e\right]^{\alpha-1}\dot{e}+k_{2}\beta\left[e\right]^{\beta-1}\dot{e}\right)-\left(\varXi
u_{0}+\varGamma\left(x,\dot{x}\right)-\ddot{x}_{r}+k_{1}\alpha\left[e\right]^{\alpha-1}\dot{e}+k_{2}\beta\left[e\right]^{\beta-1}\dot{e}\right)$
(19)

Therefore, we have:

$\dot{\sigma_{1}}\left(t\right)=\varXi\left(u-u_{0}\right)+\varPsi\left(x,\dot{x},t\right)$

Letting $u=u_{0}+u_{s1}$, where $u_{s1}$ is designed to compensate for
disturbances based on $u_{0}$, we obtain:

$\dot{\sigma_{1}}\left(t\right)=\varXi u_{s1}+\varPsi\left(x,\dot{x},t\right)$
(20)

From (20), the proposed compensating controller $u_{s1}$ is designed as:

$u_{s1}=\varXi^{-1}\left(-\left(\rho+\varepsilon\right)sign\left(\sigma_{1}\right)-k_{3}\left[\sigma_{1}\right]^{p}-k_{4}\left[\sigma_{1}\right]^{q}\right)$
(21)

where the $k_{3},k_{4}>0$, $0<p<1$ and $q>1$. Inserting the compensating
controller $u_{s1}$ into (20), yields:

$\dot{\sigma_{1}}\left(t\right)=\varPsi\left(x,\dot{x},t\right)-\left(\rho+\varepsilon\right)sign\left(\sigma_{1}\right)-k_{3}\left[\sigma_{1}\right]^{p}-k_{4}\left[\sigma_{1}\right]^{q}$
(22)

and

$\dot{\sigma_{1_{i}}}\left(t\right)=\varPsi_{i}\left(x,\dot{x},t\right)-\left(\rho+\varepsilon\right)sign\left(\sigma_{1_{i}}\right)-k_{3}\left[\sigma_{1_{i}}\right]^{p}-k_{4}\left[\sigma_{1_{i}}\right]^{q}$
(23)

Consider a Lyapunov function candidate:

$V_{1}=\frac{1}{2}\sum_{i=1}^{N}{\sigma_{1_{i}}^{2}}$ (24)

Differentiating the Lyapunov function (24), we have:

$\displaystyle\dot{V_{1}}$
$\displaystyle=\sum_{i=1}^{N}{\sigma_{1_{i}}\dot{\sigma_{1_{i}}}}$ (25)
$\displaystyle=\sum_{i=1}^{N}{\sigma_{1_{i}}\left(\varPsi_{i}\left(x,\dot{x},t\right)-\left(\rho+\varepsilon\right)sign\left(\sigma_{1_{i}}\right)-k_{3}\left[\sigma_{1_{i}}\right]^{p}-k_{4}\left[\sigma_{1_{i}}\right]^{q}\right)}$
$\displaystyle=\sum_{i=1}^{N}{\left(\sigma_{1_{i}}\varPsi_{i}\left(x,\dot{x},t\right)-\left(\rho+\varepsilon\right)\left|\sigma_{1_{i}}\right|-k_{3}\left[\sigma_{1_{i}}\right]^{p+1}-k_{4}\left[\sigma_{1_{i}}\right]^{q+1}\right)}$

By using Assumption 1, we have:

$\displaystyle\dot{V_{1}}\leq\sum_{i=1}^{N}{\left(-k_{3}\left[\sigma_{1_{i}}\right]^{p+1}-k_{4}\left[\sigma_{1_{i}}\right]^{q+1}\right)}$
(26)

By using Lemma 1 and Lemma 2, we have:

$\displaystyle\dot{V_{1}}$ $\displaystyle\leq-
k_{3}\sum_{i=1}^{N}{\left|\sigma_{1_{i}}\right|^{p+1}-}k_{4}\sum_{i=1}^{N}{\left|\sigma_{1_{i}}\right|^{q+1}}$
(27) $\displaystyle\leq-
k_{3}\sum_{i=1}^{N}{\left(\left|\sigma_{1_{i}}\right|^{2}\right)^{\frac{p+1}{2}}}-k_{4}\sum_{i=1}^{N}{\left(\left|\sigma_{1_{i}}\right|^{2}\right)^{\frac{q+1}{2}}}$
$\displaystyle\leq-
k_{3}\left(\sum_{i=1}^{N}{\left|\sigma_{1_{i}}\right|^{2}}\right)^{\frac{p+1}{2}}-k_{4}N^{\frac{1-q}{2}}\left(\sum_{i=1}^{N}{\left|\sigma_{1_{i}}\right|^{2}}\right)^{\frac{q+1}{2}}$
$\displaystyle=-k_{3}2^{\frac{p+1}{2}}\left(V_{1}\right)^{\frac{p+1}{2}}-k_{4}2^{\frac{q+1}{2}}N^{\frac{1-q}{2}}\left(V_{1}\right)^{\frac{q+1}{2}}$

Based on the Lemma 3, the above system is fixed-time stable and the
convergence time is upper bounded as:

$T_{r1}\leq\frac{1}{2^{\frac{p+1}{2}}k_{3}\left(1-\frac{p+1}{2}\right)}+\frac{1}{2^{\frac{q+1}{2}}k_{4}N^{\frac{1-q}{2}}\left(\frac{q+1}{2}-1\right)}$
(28)

_Theorem 1_. The system (5) with controller $u_{s}$ is fixed-time stable and
the settling time is bounded by:

$T\leq T_{s1}+T_{r1}+T_{n}$ (29)

where $T_{s1}$ and $T_{r1}$ represent the convergence time of the sliding
variable $s$ and the system (5) with $u_{s1}$ as the controller, respectively.
$T_{n}$ represents the convergence time of the system (5) with $u_{0}$ as the
nominal controller, which will be determined in Section 3.4.

_Proof_. It is clear that the convergence time of the entire system is smaller
than the sum of the convergence times of each component (i.e., the convergence
times of the sliding surface, compensating controller and nominal controller).
This completes the proof.

######  Remark 3.1.

The value for $\rho$ of the controller in (21) was selected based on
Assumption 1. In practice, we can use experiments to determine the bounded
value of the uncertainty $\rho$. Another method which can be applied to
approximate the value of parameter of $\rho$ is to use an adaptive technique;
for example, we can use an adaptive second-order super-twisting method 20 to
estimate the adaptive gain and eliminate the chattering at the same time.

######  Remark 3.2.

The use of the ‘sign’ function in the controller (21) will generate a
chattering. To eliminate the chattering, a boundary layer method or second-
order super-twisting method can be employed 20.

###  3.3 Nonsingular Problem

From (17), when $e=0$ and $\dot{e}\neq 0$ the term
$k_{1}\alpha\left[e\right]^{\alpha-1}\dot{e}$ results in a singular problem.
To avoid the singularity, the error can be reconstructed as:

$s=e+\frac{1}{k_{5}^{m}}\left[\dot{e}+k_{6}\left[e\right]^{n}\right]^{\frac{1}{m}}$
(30)

where $k_{5}$, $k_{6}$, $m$, $n$ are all positive constants satisfying
$k_{5},k_{6}>0$ , $0<m<1$ and $n>1$. When the sliding surface variable
converges to zero, (30) is equal to (14). Therefore, according to the Lemma 3,
the convergence time of (30) is bounded by:

$T_{s2}\leq\frac{1}{k_{5}\left(1-m\right)}+\frac{1}{k_{6}\left(n-1\right)}$
(31)

Differentiating (30) with respect to time, we have:

$\dot{s}=\dot{e}+\frac{1}{mk_{5}^{m}}\left[\dot{e}+k_{6}\left[e\right]^{n}\right]^{\frac{1}{m}-1}\left(\ddot{e}+k_{6}n\left[e\right]^{n-1}\dot{e}\right)$
(32)

The integral sliding surface is selected as:

$\displaystyle\sigma_{2}\left(t\right)=s\left(t\right)-s\left(0\right)-\int_{0}^{t}{\left(\dot{e}+\frac{1}{mk_{5}^{m}}\left[\dot{e}+k_{6}\left[e\right]^{n}\right]^{\frac{1}{m}-1}\left(\ddot{e}+k_{6}n\left[e\right]^{n-1}\right)\dot{e}\right)dt}$
(33)

According to the (5), we have:

$\ddot{e}=\ddot{x}-\ddot{x_{r}}=\varXi
u+\varGamma\left(x,\dot{x}\right)-\ddot{x_{r}}$ (34)

Inserting (34) into (33) and letting:

$\displaystyle
T_{1}\left(e,\dot{e}\right)=\frac{1}{mk_{5}^{m}}\left[\dot{e}+k_{6}\left[e\right]^{n}\right]^{\frac{1}{m}-1}$
(35) $\displaystyle T_{2}\left(e\right)=k_{6}n\left[e\right]^{n-1}\dot{e}$

The derivative of integral sliding surface can be written as:

$\displaystyle\dot{\sigma_{2}}\left(t\right)=\left(\dot{e}+T_{1}\left(e,\dot{e}\right)\left(\varXi
u+\varGamma\left(x,\dot{x}\right)+\varPsi\left(x,\dot{x},t\right)-\ddot{x}_{r}+T_{2}\left(e\right)\right)\right)-\left(\dot{e}+T_{1}\left(e,\dot{e}\right)\left(\varXi
u_{0}+\varGamma\left(x,\dot{x}\right)-\ddot{x}_{r}+T_{2}\left(e\right)\right)\right)$
(36)

Setting $u=u_{0}+u_{s2}$, we have:

$\dot{\sigma_{2}}\left(t\right)=T_{1}\left(e,\dot{e}\right)\left(\varXi
u_{s2}+\varPsi\left(x,\dot{x},t\right)\right)$ (37)

Again, the controller $u_{s2}$ can be designed as:

$u_{s2}=\varXi^{-1}\left(-\left(\rho+\varepsilon\right)sign\left(\sigma_{2}\right)-k_{3}\left[\sigma_{2}\right]^{p}-k_{4}\left[\sigma_{2}\right]^{q}\right)$
(38)

Inserting the compensating controller into (37), we have:

$\dot{\sigma_{2}}\left(t\right)=T_{1}\left(e,\dot{e}\right)\left(\varPsi\left(x,\dot{x},t\right)-\left(\rho+\varepsilon\right)sign\left(\sigma_{2}\right)-k_{3}\left[\sigma_{2}\right]^{p}-k_{4}\left[\sigma_{2}\right]^{q}\right)$
(39)

Considering a Lyapunov function candidate:

$V_{2}=\frac{1}{2}\sum_{i=1}^{N}{\sigma_{2_{i}}^{2}}$ (40)

Differentiating the Lyapunov function (24), we have:

$\displaystyle\dot{V_{2}}$
$\displaystyle=\sum_{i=1}^{N}{\sigma_{2_{i}}\dot{\sigma_{2_{i}}}}$ (41)
$\displaystyle=\sum_{i=1}^{N}{\sigma_{2_{i}}T_{1}\left(e,\dot{e}\right)\left(\varPsi_{i}\left(x,\dot{x},t\right)-\left(\rho+\varepsilon\right)sign\left(\sigma_{2_{i}}\right)-k_{3}\left[\sigma_{2_{i}}\right]^{p}-k_{4}\left[\sigma_{2_{i}}\right]^{q}\right)}$
$\displaystyle=T_{1}\left(e,\dot{e}\right)\sum_{i=1}^{N}{\left(\sigma_{2_{i}}\varPsi_{i}\left(x,\dot{x},t\right)-\left(\rho+\varepsilon\right)\left|\sigma_{2_{i}}\right|-k_{3}\left[\sigma_{2_{i}}\right]^{p+1}-k_{4}\left[\sigma_{2_{i}}\right]^{q+1}\right)}$
$\displaystyle\leq
T_{1}\left(e,\dot{e}\right)\sum_{i=1}^{N}{\left(-k_{3}\left[\sigma_{2_{i}}\right]^{p+1}-k_{4}\left[\sigma_{2_{i}}\right]^{q+1}\right)}$
$\displaystyle\leq
T_{1}\left(e,\dot{e}\right)\left(-k_{3}2^{\frac{p+1}{2}}\left(V_{2}\right)^{\frac{p+1}{2}}-k_{4}2^{\frac{q+1}{2}}N^{\frac{1-q}{2}}\left(V_{2}\right)^{\frac{q+1}{2}}\right)$

_Theorem 2_. The system (5) with controller $u_{s2}$ is fixed-time stable and
its settling time is bounded by:

$T\leq T_{s2}+T_{r2}+T_{n}+\epsilon\left(\tau\right)$ (42)

where $T_{s2}$ and $T_{r2}$ represent the convergence time of sliding variable
$s$ and the system (5) with $u_{s2}$ as the controller, respectively. $T_{n}$
represent the convergence time of the system (5) with $u_{0}$ as the nominal
controller.

_Proof_. From (39), we can see the fixed-time stability depends both on the
term
$\left(-k_{3}2^{\frac{p+1}{2}}\left(V_{2}\right)^{\frac{p+1}{2}}-k_{4}2^{\frac{q+1}{2}}N^{\frac{1-q}{2}}\left(V_{2}\right)^{\frac{q+1}{2}}\right)$
and the term $T_{1}\left(e,\dot{e}\right)$. According to the work of Van and
Ceglarek 20, for the case of $\dot{e}+k_{6}\left[e\right]^{n}\neq 0$, we
divide the state space into two different areas
$\varOmega_{1}=\left\\{\left(e,\dot{e}\right)\left|T\left(e,\dot{e}\right)\geq
1\right.\right\\}$ and
$\varOmega_{2}=\left\\{\left(e,\dot{e}\right)\left|T\left(e,\dot{e}\right)<1\right.\right\\}$.

i. When the system states are in the area of $\varOmega_{1}$, we have:

$\displaystyle\dot{V_{2}}\leq-
k_{3}2^{\frac{p+1}{2}}\left(V_{2}\right)^{\frac{p+1}{2}}-k_{4}2^{\frac{q+1}{2}}N^{\frac{1-q}{2}}\left(V_{2}\right)^{\frac{q+1}{2}}$
(43)

$V_{2}=0$ implies the sliding surface $s=0$. Therefore, according to the Lemma
3, the system states will reach the sliding surface $s=0$ within a fixed time,
which is upper bounded as:

$T_{r2}\leq\frac{1}{2^{\frac{p+1}{2}}k_{3}\left(1-\frac{p+1}{2}\right)}+\frac{1}{2^{\frac{q+1}{2}}k_{4}N^{\frac{1-q}{2}}\left(\frac{q+1}{2}-1\right)}$
(44)

ii. When the system states are in the area of $\varOmega_{2}$, according to
(41), the sliding surface $s=0$ is still an attractor. In addition, for the
case of $\dot{e}+k_{6}\left[e\right]^{n}=0$, for a given $\tau$, there exists
a positive constant $\epsilon\left(\tau\right)$ such that the sliding surface
$s_{2}=0$ can be reached from anywhere in the phase plane within a fixed time
$t_{r}<T_{r2}+\epsilon\left(\tau\right)$. 21 Therefore, the total setting time
$T$ is bounded as (42). This completes the proof.

######  Remark 3.3.

Compared to the design of the integral sliding surfaces of the existing FTISMC
11, 12, 13, the proposed controller uses the sliding surface in (33), which
takes the dynamics of the system account in the design of the integral sliding
surface. This approach provides two advantages: (1) it allows us to design the
nominal controller $u_{0}$ and the reaching controller $u_{s}$ separately.
This facilitates the design procedure and preserves the advantages of both the
nominal controller and ISMC.

###  3.4 Design of the Nominal Controller

The nominal controller is designed to stabilize the system neglecting the
lumped uncertainties. To stabilize the whole system within a fixed time, we
choose fixed-time backstepping control 22 as the nominal controller. The
dynamics of the robot manipulator without lumped uncertainties can be written
as:

$\left\\{\begin{array}[]{l}\dot{\eta}_{1}=\eta_{2}\\\ \dot{\eta}_{2}=\varXi
u+\varGamma\left(x,\dot{x}\right)\\\ \end{array}\right.$ (45)

Letting $s_{1}=\eta_{1}-x_{r}$, then, $\dot{s}_{1}=\eta_{2}-\dot{x_{r}}$, and
we can design the stabilizing function as:

$\alpha_{s}=-\left(\lambda_{1}s_{1}+\lambda_{2}s_{1}^{\alpha}+\lambda_{3}s_{1}^{\beta}\right)+\dot{x}_{r}$
(46)

where $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\alpha$, $\beta$ are all
positive constants satisfying $0<\alpha<1$ and $\beta>1$. Thus, $\dot{s}_{1}$
can be written as:

$\dot{s}_{1}=-\left(\lambda_{1}s_{1}+\lambda_{2}s_{1}^{\alpha}+\lambda_{3}s_{1}^{\beta}\right)$
(47)

Selecting the Lyapunov function candidate as:

$V_{3}=\frac{1}{2}s_{1}^{T}s_{1}$ (48)

The derivative of the Lyapunov function $V_{3}$ is:

$\displaystyle\dot{V}_{3}$ $\displaystyle=s_{1}^{T}\dot{s}_{1}$ (49)
$\displaystyle=-s_{1}^{T}\left(\lambda_{1}s_{1}+\lambda_{2}s_{1}^{\alpha}+\lambda_{3}s_{1}^{\beta}\right)$
$\displaystyle=-\lambda_{1}s_{1}^{T}s_{1}-\lambda_{2}\left(s_{1}^{T}s_{1}\right)^{\frac{\alpha+1}{2}}-\lambda_{3}\left(s_{1}^{T}s_{1}\right)^{\frac{\beta+1}{2}}$
$\displaystyle\leq-\lambda_{2}\left(s_{1}^{T}s_{1}\right)^{\frac{\alpha+1}{2}}-\lambda_{3}\left(s_{1}^{T}s_{1}\right)^{\frac{\beta+1}{2}}$
$\displaystyle\leq-2^{\frac{\alpha+1}{2}}\lambda_{2}\left(\frac{1}{2}s_{1}^{T}s_{1}\right)^{\frac{\alpha+1}{2}}-2^{\frac{\beta+1}{2}}\lambda_{3}\left(\frac{1}{2}s_{1}^{T}s_{1}\right)^{\frac{\beta+1}{2}}$
$\displaystyle=-2^{\frac{\alpha+1}{2}}\lambda_{2}\left(V_{3}\right)^{\frac{\alpha+1}{2}}-2^{\frac{\beta+1}{2}}\lambda_{3}\left(V_{3}\right)^{\frac{\beta+1}{2}}$

Letting $s_{2}=\eta_{2}-\alpha_{s}$, we have:

$\dot{s}_{2}=\dot{\eta}_{2}-\dot{\alpha}_{s}=\varXi
u_{0}-\varGamma\left(x,\dot{x}\right)-\dot{\alpha}_{s}$ (50)

Selecting the Lyapunov function candidate as:

$V_{4}=\frac{1}{2}s_{2}^{T}s_{2}$ (51)

To obtain the property of fixed-time convergence, we design the nominal
controller as:

$u_{0}=\varXi^{-1}\left(-\varGamma\left(x,\dot{x}\right)+\dot{\alpha}_{s}-\lambda_{1}s_{2}-\lambda_{2}s_{2}^{\alpha}-\lambda_{3}s_{2}^{\beta}\right)$
(52)

Inserting (52) into (50), we have:

$\dot{s}_{2}=-\lambda_{1}s_{2}-\lambda_{2}s_{2}^{\alpha}-\lambda_{3}s_{2}^{\beta}$
(53)

The derivative of the Lyapunov function $V_{2}$ is:

$\displaystyle\dot{V}_{4}$ $\displaystyle=s_{2}^{T}\dot{s}_{2}$ (54)
$\displaystyle=s_{2}^{T}\left(-\lambda_{1}s_{2}-\lambda_{2}s_{2}^{\alpha}-\lambda_{3}s_{2}^{\beta}\right)$
$\displaystyle\leq-2^{\frac{\alpha+1}{2}}\lambda_{2}\left(V_{4}\right)^{\frac{\alpha+1}{2}}-2^{\frac{\beta+1}{2}}\lambda_{3}\left(V_{4}\right)^{\frac{\beta+1}{2}}$

Choosing the Lyapunov function candidate of system as:

$V_{n}=V_{3}+V_{4}$ (55)

The derivative of Lyapunov function $V$ is:

$\displaystyle\dot{V_{n}}$
$\displaystyle\leq-2^{\frac{\alpha+1}{2}}\lambda_{2}\left(V_{3}\right)^{\frac{\alpha+1}{2}}-2^{\frac{\beta+1}{2}}\lambda_{3}\left(V_{3}\right)^{\frac{\beta+1}{2}}-2^{\frac{\alpha+1}{2}}\lambda_{2}\left(V_{4}\right)^{\frac{\alpha+1}{2}}-2^{\frac{\beta+1}{2}}\lambda_{3}\left(V_{4}\right)^{\frac{\beta+1}{2}}$
(56)
$\displaystyle\leq-2^{\frac{\alpha+1}{2}}\lambda_{2}\left(V_{n}\right)^{\frac{\alpha+1}{2}}-2^{\frac{\beta+1}{2}}\lambda_{3}\left(V_{n}\right)^{\frac{\beta+1}{2}}$

According to the Lemma 3, when applying the proposed nominal controller
$u_{0}$, the system without lumped uncertainties is fixed-time stable and the
convergence time is bounded as:

$T_{n}\leq\frac{2}{\lambda_{2}2^{\frac{\alpha+1}{2}}\left(1-\alpha\right)}+\frac{2}{\lambda_{3}2^{\frac{\beta+1}{2}}\left(\beta-1\right)}$
(57)

######  Remark 3.4.

It is noted that the nominal controller can be designed using other
controllers, e.g, PID, CTC and Backstepping. However, in this paper, we use
the fixed-time backstepping controller to achieve global fixed-time
convergence for the system.

##  4 Simulation and results

In the simulation, we consider a two-link robot in the horizontal plane 23.
The dynamics of the robot are described as:

$\displaystyle\tau_{1}$
$\displaystyle=m_{2}l_{2}^{2}\left(\ddot{q}_{1}+\ddot{q}_{2}\right)+m_{2}l_{1}l_{2}c_{2}\left(2\ddot{q}_{1}+\ddot{q}_{2}\right)+\left(m_{1}+m_{2}\right)l_{1}^{2}\ddot{q}_{1}-m_{2}l_{1}l_{2}s_{2}\dot{q}_{2}^{2}-2m_{2}l_{1}l_{2}s_{2}\dot{q}_{1}\dot{q}_{2}+m_{2}l_{2}gc_{12}+\left(m_{1}+m_{2}\right)l_{1}gc_{1}$
(58) $\displaystyle\tau_{2}$
$\displaystyle=m_{2}l_{1}l_{2}c_{2}\ddot{q}_{1}+m_{2}l_{1}l_{2}s_{2}\dot{q}_{1}^{2}+m_{2}l_{2}gc_{12}+m_{2}l_{2}^{2}\left(\ddot{q}_{1}+\ddot{q}_{2}\right)$

where $c_{i}=\cos\left(q_{i}\right)$, $c_{ij}=\cos\left(q_{i}+q_{j}\right)$,
$s_{i}=\sin\left(q_{i}\right)$, and $s_{ij}=\sin\left(q_{i}+q_{j}\right)$,
$i,j=1,2$. The value of masses are $m_{1}=1.5kg$, and $m_{2}=1.0kg$. The
length of the links are $l_{1}=l_{2}=0.3m$. Based on the (1), the mass matrix
$M\left(q\right)$, Coriolis and centrifugal forces matrix
$C\left(q,\dot{q}\right)$ and the gravity matrix $G\left(q\right)$ are given
as:

$M\left(q\right)=\left[\begin{matrix}m_{2}l_{2}^{2}+2m_{2}l_{1}l_{2}c_{2}+\left(m_{1}+m_{2}\right)l_{1}^{2}&m_{2}l_{2}^{2}+m_{2}l_{1}l_{2}c_{2}\\\
m_{2}l_{2}^{2}+m_{2}l_{1}l_{2}c_{2}&m_{2}l_{2}^{2}\\\ \end{matrix}\right]$
(59)
$C\left(q,\dot{q}\right)=\left[\begin{matrix}-2m_{2}l_{1}l_{2}s_{2}\dot{q}_{2}&-m_{2}l_{1}l_{2}s_{2}\\\
m_{2}l_{1}l_{2}s_{2}\dot{q}_{1}&0\\\ \end{matrix}\right]$ (60)
$G\left(q\right)=\left[\begin{array}[]{c}m_{2}l_{2}gc_{12}+\left(m_{1}+m_{2}\right)l_{1}gc_{1}\\\
m_{2}l_{2}gc_{12}\\\ \end{array}\right]$ (61)

The bounded disturbance term is given as:

$F\left(q\right)=\left[\begin{array}[]{c}2c_{1}s_{2}+5c_{1}^{2}\\\
-2c_{1}s_{2}-5c_{1}^{2}\\\ \end{array}\right]$ (62)

The Jacobian of the robot manipulator is given as:

$J\left(q\right)=\left[\begin{matrix}-l_{1}s_{1}-l_{2}s_{12}&-l_{2}s_{12}\\\
l_{1}c_{1}+l_{2}c_{12}&l_{2}c_{12}\\\ \end{matrix}\right]$ (63)

Assume the desired trajectory 3 of the two joints is:

$\displaystyle x_{d1}\left(t\right)$ $\displaystyle=0.14\cos\left(0.5t\right)$
(64) $\displaystyle x_{d2}\left(t\right)$
$\displaystyle=0.14\sin\left(0.5t\right)$

Figure 4: Desired/reference trajectory.

Figure 5: External human forces.

The external human forces, as shown in Fig. 5, are given by 3:

$fe_{i}\left(t\right)\left\\{\begin{array}[]{l}0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t<10\ or\ t\geq 21\\\ a_{i}\left(1-\cos\pi
t\right)\ \ \ \ \ \ \ \ \ 10\leq t<11\\\ 2a_{i}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ 11\leq t<20\\\ a_{i}\left(1+\cos\pi t\right)\ \ \ \ \ \
\ \ \ 20\leq t<21\\\ \end{array}\right.$ (65)

The external human forces are applied when $t=10s$ and removed at $t=21s$. By
applying admittance control, the reference trajectory of the two joints
$x_{r1}$ and $x_{r2}$ can be derived by integrating equation (12) twice. Fig.
5 shows both the desired trajectory and the reference trajectory. We can see
the two trajectories are the same before the external forces are applied. When
$t>10s$, the reference trajectory, which complies with thr external human
forces, is different from the desired trajectory. When $t>21s$, the external
human forces are removed from the contact point and the two trajectories
coincide again.

Figure 6: Trajectories obtained with the different controllers.

Figure 7: Tracking error for joint 1.

Figure 8: Tracking error for joint 2.

Figure 9: Control input for joint 1.

Figure 10: Control input for joint 2.

To verify the effectiveness of the proposed controller, we compare it with
PID, computed torque control (CTC), traditional ISMC with PID as the nominal
controller (ISMC PID), traditional ISMC with CTC as the nominal controller
(ISMC CTC), and traditional ISMC with BSP as the nominal controller (ISMC
BSP). Assume the initial position of the end-effector is
$q\left(0\right)=\left[0.5236,2.0944\right]^{T}$, and the initial value of the
task variable is $x\left(0\right)=\left[0,0\right]^{T}$. The design parameters
of the admittance control are $k_{m_{i}}=20$, $k_{b_{i}}=20$ and
$k_{k_{i}}=100$, $i=1,2$. The PID parameters are $k_{p}=300$, $k_{d}=400$,
$k_{i}=10$. The parameters of BSP are $\lambda_{1}=3$, $\lambda_{2}=20$,
$\lambda_{3}=50$, $\alpha=\frac{5}{7}$, $\beta=\frac{5}{3}$. These parameters
are selected based on a trial-and-error procedure and by experience. The
design parameters of the proposed controller are $k_{1}=k_{3}=20$,
$k_{2}=k_{4}=50$, $m=p=\frac{5}{7}$, $n=q=\frac{5}{3}$.

The trajectories of the benchmark controllers and the proposed fixed-time ISMC
controller with BSP as the nominal controller are shown in Fig. 6, and the
corresponding Root Mean Square Error (RMSE) of tracking errors are reported in
Table 1. We can see that the PID controller has the poorest tracking
performance. The performances of CTC is a little better than PID. The
performance of ISMC PID and ISMC CTC are better than pure PID and CTC,
respectively, which means the tracking performance of the controller is
improved by adding the integral sliding surface. The ISMC BSP and proposed
fixed-time ISMC controller have the best tracking performance. They almost
achieve perfectly tracking of the reference trajectory, which makes it
difficult to compare them in Fig. 6. However, as can be seen in Fig. 8 and
Fig. 8, it is clear that the proposed controller has both the smallest
tracking error and shortest convergence time. Note that the proposed
controller is global fixed-time convergent because both the integral sliding
surface, reaching controller and the nominal controller can converge within a
fixed time. In addition, from Fig. 10 and Fig. 10, which show the control
inputs of the two joints, it can be seen that the proposed solution provides
the smoothest and most efficient control input.

Table 1: The RMSE performance of each controller. Joint |  PID |  CTC |  ISMC+PID |  ISMC+CTC |  ISMC+BSP |  FTISMC+BSP
---|---|---|---|---|---|---
1 |  0.1211 |  0.0070 |  0.0446 |  0.0064 |  $2.0901\times 10^{-4}$ |  $5.7841\times 10^{-5}$
2 |  0.0845 |  0.0314 |  0.0644 |  0.0220 |  0.0016 |  $4.4889\times 10^{-4}$

##  5 Conclusions

In this paper, a fixed-time ISMC controller based on admittance control has
been proposed for robot manipulators such that the robot can comply with the
external human forces rather than reject them as disturbances. Furthermore,
the convergence time of the system can be predefined regardless of the initial
conditions. The simulation results show that the proposed controller can track
the reference trajectory precisely with faster convergence of the tracking
error. In future work, a fixed-time force observer of external human forces
will be discussed and integrated into the system. Safety constraints of the
system will also be considered. The whole system will be tested on a robotics
platform such as the Baxter robot to further validate the controller’s
effectiveness and evaluate real world performance.

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