Transparent Higher Order Sliding Mode Control for Nonlinear Master-Slave Systems without Velocity Measurement

Abstract

Transparency has been a major objective in bilateral teleoperation systems, even in the absence of time delay induced by the communication channel, since a high degree of transparency would allow humans to drive the remote teleoperator as if he or she were directly interacting with the remote environment, with the remote teleoperator as a physical and sensorial extension of the operator. When fast convergence of position and force tracking errors are ensured by the control system, then complete transparency is obtained, which would ideally guarantee humans to be tightly kinaesthetically coupled. In this paper a model-free Cartesian second order sliding mode (SOSM) PD control scheme for nonlinear master-slave systems is presented. The proposed scheme does not rely on velocity measurements and attains very fast convergence of position trajectories, with bounded tracking of force trajectories, rendering a high degree of transparency with lesser knowledge of the system. The degree of transparency can easily be improved by tuning a feedback gain in the force loop. A unique energy storage function is introduced; such that a similar Cartesian-based controller is implemented in the master and slave sides. The resulting properties of the Cartesian control structure allows the human operator to input directly Cartesian variables, which makes clearer the kinaesthetic coupling, thus the proposed controller becomes a suitable candidate for practical implementation. The performance of the proposed scheme is evaluated in a semi-experimental setup.

1. Introduction

Master-slave systems have been extensively studied since the pioneering works of [1, 2]and [3]; these studies have been motivated by a variety of applications, ranging from space applications to underwater tasks [4] and medical applications [5], as well as teleoperation in cluttered or inaccessible environments, such as dangerous sites or in micro stages [6].

Basically, in bilateral teleoperation the human operator interacts through a robotic master teleoperator to govern the position and contact force of the remote (slave) robot, while the slave robot (teleoperator) reproduces these trajectories onto the remote environment. When the remote environment continuously feeds back some pieces of information, the whole system is coupled and the information flows in both directions to achieve a bilateral teleoperation system, with or without time delay. In this way, scaled slave contact force is perceived continuously by the human operator to kinaesthetically couple him/her to the slave site, so as to improve considerably the closed-loop task performance.

When perfect matching of all trajectories is obtained, it is said that the system is transparent. Today, it still remains an open issue as to how to render a high degree of transparency in nonlinear teleoperation systems, without time delay, when the dynamic model of each teleoperator is not available.

1.1 Background

One of the major objectives in designing bilateral teleoperation control systems is to achieve transparency; that is, a match between the master and slave positions and contact forces [7]. Many teleoperation schemes have been proposed in the last decades, most of them for linear systems, with or without time delay, but all of them seeking stability and a good transparency degree; references [8, 9, 10] and [11] expose in detail some of the most important and popular approaches to bilateral teleoperators.

Position-position (P-P) and position-force (P-F) teleoperation schemes were tested to study their stability and transparency. In [8], the authors demonstrated that this scheme is neither stable for small time delays nor transparent, and that stability is preserved only if a trade-off between controller gains and scale factors versus tracking performance is established. In [12], the force-position (F-P) architecture is applied during the constrained motion, while the position-position (P-P) architecture is applied during the free space motion to attain high transparency. In [13] the effects of controller parameters on transparency of time delayed bilateral teleoperation systems with communication disturbance observer are discussed. The proposed scheme is verified by numerical simulations with linear teleoperators using a Smith predictor at the master side.

Regarding sliding mode control, Buttolo et al. [14] proposed model-based first order sliding mode (FOSM) controllers for both master and slave teleoperators achieving an improvement with respect to position tracking and parametric uncertainty. Although a good level of transparency is achieved, paradoxically the inherent effects of the sliding mode control (introduction of high frequency signals into the system) make it unsuitable to be implemented in a physical system, leading to an eventual instability and, therefore, loss of transparency. Park and Cho [15] proposed to implement an impedance control for the master and a modified FOSM control for the slave, in which the gain that satisfies the sliding condition can be set independently of time delay. Thus, this scheme is robust against constant time delay. Nevertheless, it easily turns unstable when interacting with rigid environments. To deal with this problem, the same authors proposed a sliding mode-based impedance control for the slave in order to reduce impact forces and achieve a sustained stable contact [16]. In [17] the problem of parametric uncertainty is studied, besides which an algorithm for the parameters tuning is provided. However, the signum function, used for the sliding mode, is replaced by a saturation function to reduce chattering, with consequent poor tracking performance, and, therefore, a low degree of transparency. A number of bilateral teleoperation schemes addressing FOSM control, most of them model-based, have been published in recent years: for instance, references [18, 19], and [20], and [21, 22] where some kind of state estimation has been used.

Finally, [23, 24, 25, 26] and [27] propose modified schemes based on higher-order sliding modes to avoid introducing chattering to the system. Formal and stringent stability proofs, taking into account state observers for both master and slave robots, are considered for linear and nonlinear systems. Although the problem of chattering is solved and robustness against constant time delay is accomplished, the system's performance still lacks of transparency.

Regarding velocity observers, a number of state observers techniques have been published for Euler-Lagrange systems, but very few for nonlinear, multi-d.o.f. bilateral teleoperators, such as [21, 24], and [26]-[28].

1.2 The problem

In a teleoperation system, the human drives the master robot in order to command the slave robot to interact with the environment and carry out a task over it. The human then plays the role of generating continuously desired contact force and position trajectory for the slave robot. Several studies [10, 29] reveal that continuous perception of the remote contact force is quite important to increase the quality of the teleoperation. Since we humans evolve in a Cartesian world, a Cartesian-based robot controller would be more intuitive to interact with, and, since each human operator certainly performs differently, it is also important to introduce a transparency that could be tuned at will, in the spatial and time attributes. Then, the formulation of the problem is as follows: “How to guarantee intuitive teleoperation, in a delay free nonlinear teleoperation system?” By intuition it is understood that the teleoperation scheme should be and fulfil the following: i) a Cartesian-based controller for nonlinear teleoperation systems; ii) tuneable degree of transparency; and iii); full access to the state of the system (without resorting on velocity measurements).

1.3 Definition of ideal transparency

A general consensus about transparency is given in [7], defining that the teleoperation system (1)–(2) below is transparent to a human-task interface when the following criteria are accomplished: a) positions x_m and x_s are identical, whatever the environment dynamics is; and b) forces F_h and F_e are identical, whatever the environment dynamics are. When both criteria are fulfilled, the human operator can manoeuvre the system as if he/she were manipulating the remote environment him/herself.

Notice that a) and b) mean nothing but convergence of position and force tracking errors. However, clearly, a modern definition of transparency should include the issues of time delay, visual feedback, as well as the nature of human dynamics. Then, a transparent teleoperation system should consider:

1. Teleoperators are nonlinear Euler Lagrange systems.

2. Time delay should not affect kinaesthetic coupling,

3. Convergence of spatial position and force tracking errors are ensured.

4. Convergence in time of position and force tracking errors are ensured.

While item 1 is introduced to establish a natural perception of a physical system subject to Lagrangian dynamics, item 3 and 4 mimic [7]. Item 2 points out the two main goals and, at the same time, conflicting issues of a teleoperation system under time delay: transparency and stability. Under this definition, item 1 using (1)–(2) is satisfied, while item 2 is fulfilled since no time delay is considered. Finally, the control system proposed here, guarantee item 3 and 4, under a set of certain circumstances.

1.4 Contribution

In this paper, a Cartesian sliding PD control for master-slave bilateral teleoperation systems is proposed in order to achieve the SCD condition given in [30] and [31], and thereby high transparency. The present scheme neither requires the model nor velocity measurements. Some characteristics of the proposed controller are: 1) it is a model-free control, i.e., it does not require any knowledge of the robot dynamics; 2) it is a smooth, but robust control, based on second order sliding modes (SOSM), which means that chattering is no longer a problem; 3) the control system attains exponential convergence, with no velocity measurements; 4) an acceptable force tracking is achieved by properly tuning a feedback gain; 5) it is a very simple inverse Jacobian-based Cartesian control. In this manner, computation of inverse kinematics, and/or Cartesian robot dynamics are not necessary; and 7) formal stability results, in the sense of Lyapunov and variable structure systems, are presented.

Since the velocity observer used here [32] achieves finite time convergence of the estimation errors, there is no need to prove the stability of the nonlinear system (plant-controller plus observer) because the separation principle does not need to be proved [33]. In this manner, master/slave controllers can be designed by separately, independently of the velocity observer, which can be seen as a model-free state observer.

2. Nonlinear dynamics of the master-slave system

The nonlinear dynamic model of a rigid n-link serial joint non-redundant master/slave robot teleoperation system is considered, see Figure 1.

Figure 1.

Schematic diagram of the master-slave teleoperation system

${\mathbf{H}}_{\mathrm{m}}\left({\mathbf{q}}_{\mathrm{m}}\right){\stackrel{..}{\mathbf{q}}}_{\mathrm{m}}+{\mathbf{C}}_{\mathrm{m}}\left({\mathbf{q}}_{\mathrm{m}},{\stackrel{.}{\mathbf{q}}}_{\mathrm{m}}\right){\stackrel{.}{\mathbf{q}}}_{\mathrm{m}}+{\mathbf{g}}_{\mathrm{m}}\left({\mathbf{q}}_{\mathrm{m}}\right)={\tau }_{\mathrm{m}}+{\mathbf{J}}_{\mathrm{m}}^{\mathrm{T}}\left({\mathbf{q}}_{\mathrm{m}}\right){\mathbf{F}}_{\mathrm{h}}$

(1)

${\mathbf{H}}_{\mathrm{s}}\left({\mathbf{q}}_{\mathrm{s}}\right){\stackrel{..}{\mathbf{q}}}_{\mathrm{s}}+{\mathbf{C}}_{\mathrm{s}}\left({\mathbf{q}}_{\mathrm{s}},{\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}\right){\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}+{\mathbf{g}}_{\mathrm{s}}\left({\mathbf{q}}_{\mathrm{s}}\right)={\tau }_{\mathrm{s}}-{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}\left({\mathbf{q}}_{\mathrm{s}}\right){\mathbf{F}}_{\mathrm{e}}$

(2)

where the subscript m (s) denotes the master (slave) robot; (q_m,q_s) $\in {\Re }^{2\mathrm{n}}$ , $\left({\stackrel{.}{\mathbf{q}}}_{\mathrm{m}},{\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}\right)\in {\Re }^{2\mathrm{n}}$ are the generalized joint positions and velocity of the master and slave robots, respectively; H_m(q_m) $\in {\Re }^{\mathrm{n}\times \mathrm{n}}$ , H_s(q_s) $\in {\Re }^{\mathrm{n}\times \mathrm{n}}$ denote symmetric positive-definite inertia matrices; C_m(q_m, ${\stackrel{.}{\mathbf{q}}}_{\mathrm{m}}$ ) $\in {\Re }^{\mathrm{n}\times \mathrm{n}}$ , C_s(q_s, ${\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}$ ) $\in {\Re }^{\mathrm{n}\times \mathrm{n}}$ are matrices of Coriolis and centrifugal forces; g_m(q_m) $\in {\Re }^{\mathrm{n}}$ , g_s(q_s) $\in {\Re }^{\mathrm{n}}$ model the torques coming out from the gravitational acceleration, and ${\text{τ}}_{\mathrm{m}}\text{∈}{\mathfrak{R}}^{\mathrm{n}}\text{,\hspace{0.17em}}{\text{τ}}_{\mathrm{s}}\text{∈}{\mathfrak{R}}^{\mathrm{n}}$ represent the controlled torque inputs of the master and the slave manipulator, respectively. Analytical Jacobian matrices of the master and slave manipulator, respectively, are denoted by J_m(q_m) $\in {\Re }^{\mathrm{n}\times \mathrm{n}}$ , J_s(q_s) $\in {\Re }^{\mathrm{n}\times \mathrm{n}}$ ; and finally, F_h $\in {\Re }^{\mathrm{n}}$ is the force that the operator applies to the master manipulator, and F_e $\in {\Re }^{\mathrm{n}}$ represents the force that the slave arm applies to the environment.

2.1 Dealing with Cartesian dynamics approach

Usually, the task is defined in Cartesian space but the teleoperators are defined in the joint space, so either the task (desired trajectory) is transformed into joint space by means of the inverse kinematics mapping or joint robot dynamics (1)–(2) are mapped into Cartesian space dynamics equations of motion as follows, (from here on and without loss of generality, arguments are omitted for simplicity):

${\bar{\mathbf{H}}}_{\mathrm{m}}{\stackrel{..}{\mathbf{x}}}_{\mathrm{m}}+{\bar{\mathbf{C}}}_{\mathrm{m}}{\stackrel{.}{\mathbf{x}}}_{\mathrm{m}}+{\bar{\mathbf{G}}}_{\mathrm{m}}={\mathbf{F}}_{\text{cm}}+{\mathbf{F}}_{\mathrm{h}}$

(3)

${\bar{\mathbf{H}}}_{\mathrm{s}}{\stackrel{..}{\mathbf{x}}}_{\mathrm{s}}+{\bar{\mathbf{C}}}_{\mathrm{s}}{\stackrel{.}{\mathbf{x}}}_{\mathrm{s}}+{\bar{\mathbf{G}}}_{\mathrm{s}}={\mathbf{F}}_{\mathrm{cs}}-{\mathbf{F}}_{\mathrm{e}}$

(4)

with ${\mathbf{F}}_{\text{cm}}={\mathbf{J}}_m^{-T}{\tau }_{\mathrm{m}},{\mathbf{F}}_{\text{cs}}={\mathbf{J}}_{\mathrm{s}}^{-\mathrm{T}}{\tau }_{\mathrm{s}}$ as the Cartesian controllers, ${\bar{\mathbf{H}}}_{\mathrm{m}}={\mathbf{J}}_{\mathrm{m}}^{-T}{\mathbf{H}}_{\mathrm{m}}{\mathbf{J}}_{\mathrm{m}}^{-1}$ , ${\bar{\mathbf{C}}}_{\mathrm{m}}={\mathbf{J}}_{\mathrm{m}}^{-T}{\mathbf{C}}_{\mathrm{m}}{\mathbf{J}}_m^{-1}-{\mathbf{J}}_m^{-T}{\mathbf{H}}_{\mathrm{m}}{\mathbf{J}}_{\mathrm{m}}^{-1}{\stackrel{.}{\mathbf{J}}}_{\mathrm{m}}{\mathbf{J}}_{\mathrm{m}}^{-1}$ and ${\bar{\mathbf{G}}}_{\mathrm{m}}={\mathbf{J}}_{\mathrm{m}}^{-T}{\mathbf{g}}_{\mathrm{m}}$ are expressions for the master robot.; and, ${\bar{\mathbf{H}}}_{\mathrm{s}}={\mathbf{J}}_{\mathrm{s}}^{-T}{\mathbf{H}}_{\mathrm{s}}{\mathbf{J}}_{\mathrm{s}}^{-1},{\bar{\mathbf{C}}}_{\mathrm{s}}={\mathbf{J}}_{\mathrm{s}}^{-T}{\mathbf{C}}_{\mathrm{s}}{\mathbf{J}}_{\mathrm{s}}^{-1}-{\mathbf{J}}_{\mathrm{s}}^{-T}{\mathbf{H}}_{\mathrm{s}}{\mathbf{J}}_{\mathrm{s}}^{-1}{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}{\mathbf{J}}_{\mathrm{s}}^{-1}$ , and ${\bar{\mathbf{G}}}_{\mathrm{s}}={\mathbf{J}}_{\mathrm{s}}^{-T}{\mathbf{g}}_{\mathrm{s}}$ are expressions for the slave robot; where ${\mathbf{x}}_{\mathrm{m}}\in {\Re }^n,\hspace{0.17em}\hspace{0.17em}{\mathbf{x}}_{\mathrm{s}}\in {\Re }^n\hspace{0.17em}$ denote the displacements of the end-effector of the master and slave manipulators, respectively. Note that (1)–(2) are widely preferred since (3)–(4) are far more complex to compute. The problem which persists is then how to compute the joint controllers τ_m, τ_s for (1)–(2) in terms of Cartesian desired trajectories without inverse kinematics mapping. This problem is circumvented by only using the Jacobians.

2.2 Open-loop error dynamics

Since (1)–(2) are linearly parametrizable [34] in terms of a regressor ${\mathbf{Y}}_{\mathrm{i}}={\mathbf{Y}}_{\mathrm{i}}\left({\mathbf{q}}_i,{\stackrel{.}{\mathbf{q}}}_i,{\stackrel{..}{\mathbf{q}}}_i\right)\in {\Re }^{\text{n×p}}$ composed of nonlinear functions, and a vector Θ_i $\in {\Re }^{\mathrm{p}}$ which represents constant parameters (with i = {m, s}). This paramerization Y_iΘ_i can be written in terms of a nominal reference q_mr, q_sr and its derivatives q_mr,q_sr, as follows:

${\mathbf{H}}_{\mathrm{m}}{\stackrel{..}{\mathbf{q}}}_{\text{mr}}+{\mathbf{C}}_{\mathrm{m}}{\stackrel{.}{\mathbf{q}}}_{\text{mr}}+{\mathbf{g}}_{\mathrm{m}}={\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\mathrm{m}}$

(5)

${\mathbf{H}}_{\mathrm{s}}{\stackrel{..}{\mathbf{q}}}_{\text{sr}}+{\mathbf{C}}_{\mathrm{s}}{\stackrel{.}{\mathbf{q}}}_{\text{sr}}+{\mathbf{g}}_{\mathrm{s}}={\mathbf{Y}}_{\mathrm{sr}}{\text{Θ}}_{\mathrm{s}}$

(6)

where the regressor ${\mathbf{Y}}_{\text{ir}}={\mathbf{Y}}_{\text{ir}}\left({\mathbf{q}}_i,{\stackrel{.}{\mathbf{q}}}_i,{\stackrel{.}{\mathbf{q}}}_{\text{ir}},{\stackrel{..}{\mathbf{q}}}_{\text{ir}}\right)\in {\Re }^{\text{n×p}}$ . Adding and subtracting (5)–(6) to (1)–(2) yields the open-loop error dynamics in terms of the extended error coordinates S_m, S_sr as follows:

${\mathbf{H}}_{\mathrm{m}}{\stackrel{.}{\mathbf{S}}}_{\text{mr}}+{\mathbf{C}}_{\mathrm{m}}{\mathbf{S}}_{\text{mr}}={\tau }_{\mathrm{m}}+{\mathbf{J}}_{\mathrm{m}}^{\mathrm{T}}{\mathbf{F}}_{\mathrm{h}}-{\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\mathrm{m}}$

(7)

${\mathbf{H}}_{\mathrm{s}}{\stackrel{.}{\mathbf{S}}}_{\text{sr}}+{\mathbf{C}}_{\mathrm{s}}{\mathbf{S}}_{\text{sr}}={\tau }_{\mathrm{s}}-{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}{\mathbf{F}}_e-{\mathbf{Y}}_{\mathrm{sr}}{\text{Θ}}_{\mathrm{s}}$

(8)

where the error coordinates S_m, S_sr are defined as follows:

${\mathbf{S}}_{\text{mr}}={\stackrel{.}{\mathbf{q}}}_{\mathrm{m}}-{\stackrel{.}{\mathbf{q}}}_{\text{mr}\hspace{0.17em}}$

(9)

${\mathbf{S}}_{\text{sr}}={\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}-{\stackrel{.}{\mathbf{q}}}_{\text{sr}}$

(10)

The key up to this point is to design joint nominal references ${\stackrel{.}{\mathbf{q}}}_{\text{mr\hspace{0.17em}}},\hspace{0.17em}{\stackrel{.}{\mathbf{q}}}_{\text{sr}}$ in terms of Cartesian nominal references.

3. Cartesian error manifold

According to robot forward kinematics ${\mathbf{x}}_i=ℱ\left({\mathbf{q}}_i\right)$ , and the well-posed differential kinematics:

${\stackrel{.}{\mathbf{q}}}_{\mathrm{m}}={\mathbf{J}}_{\mathrm{m}}^{-1}{\stackrel{.}{\mathbf{x}}}_{\mathrm{m}}$

(11)

${\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}={\mathbf{J}}_{\mathrm{s}}^{-1}{\stackrel{.}{\mathbf{x}}}_{s\hspace{0.17em}}$

(12)

for J _m ⁻¹, J_s^{− 1} as the inverse analytical Jacobian of the master and slave, respectively; a natural choice to relate joint nominal references ${\stackrel{.}{\mathbf{q}}}_{mr\hspace{0.17em}},\hspace{0.17em}{\stackrel{.}{\mathbf{q}}}_{sr}$ in (9)–(10), to Cartesian nominal references is based on (11)–(12), as follows:

${\stackrel{.}{\mathbf{q}}}_{\text{mr}}={\mathbf{J}}_{\mathrm{m}}^{-1}{\stackrel{.}{\mathbf{x}}}_{\text{mr}}$

(13)

${\stackrel{.}{\mathbf{q}}}_{\text{sr}}={\mathbf{J}}_{\mathrm{s}}^{-1}{\stackrel{.}{\mathbf{x}}}_{\text{sr}}$

(14)

Then using (11) to (14), equations (9)–(10) become:

${\mathbf{S}}_{\text{mr}}={\mathbf{J}}_{\mathrm{m}}^{-1}\left({\stackrel{.}{\mathbf{x}}}_{\mathrm{m}}-{\stackrel{.}{\mathbf{x}}}_{\text{mr}}\right)\triangleq {\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}$

(15)

${\mathbf{S}}_{\mathrm{sr}}={\mathbf{J}}_{\mathrm{s}}^{-1}\left({\stackrel{.}{\mathbf{x}}}_{\mathrm{s}}-{\stackrel{.}{\mathbf{x}}}_{\text{sr}}\right)\triangleq {\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\mathrm{sr}}$

(16)

where ${\bar{\mathbf{S}}}_{mr}={\stackrel{.}{\mathbf{x}}}_{\mathrm{m}}-{\stackrel{.}{\mathbf{x}}}_{mr},\hspace{0.17em}{\bar{\mathbf{S}}}_{sr}={\stackrel{.}{\mathbf{x}}}_{\mathrm{s}}-{\stackrel{.}{\mathbf{x}}}_{sr}$ . Time derivatives of (15)–(16) are the following:

${\stackrel{.}{\mathbf{S}}}_{\text{mr}}={\stackrel{.}{\mathbf{J}}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}+{\mathbf{J}}_{\mathrm{m}}^{-1}{\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{mr}}$

(17)

${\stackrel{.}{\mathbf{S}}}_{\text{sr}}={\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{sr}}+{\mathbf{J}}_{\mathrm{s}}^{-1}{\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{sr}}$

(18)

Now, consider the following Cartesian nominal references ${\stackrel{.}{\mathbf{x}}}_{mr},\hspace{0.17em}{\stackrel{.}{\mathbf{x}}}_{sr}$ :

${\stackrel{.}{\mathbf{x}}}_{\text{mr}}={\stackrel{.}{\mathbf{x}}}_{\text{md}}-\alpha \left({\mathbf{x}}_{\mathrm{m}}-{\mathbf{x}}_{\text{md}}\right)+{\mathbf{S}}_{\text{dm}}-{\text{γ}}_{\mathrm{m}}\underset{{\mathrm{t}}_0}{\overset{\mathrm{t}}{{\displaystyle \int }}}\text{sign}\left({\mathbf{S}}_{\mathbf{q}\mathrm{m}}\right)\text{dt}\hspace{0.17em}\hspace{0.17em}$

(19)

${\stackrel{.}{\mathbf{x}}}_{\text{sr}}={\stackrel{.}{\mathbf{x}}}_{\mathrm{sd}}-\alpha \left({\mathbf{x}}_{\mathrm{s}}-{\mathbf{x}}_{\mathrm{sd}}\right)+{\mathbf{S}}_{\mathrm{ds}}-{\text{γ}}_{\mathrm{s}}\underset{{\mathrm{t}}_0}{\overset{\mathrm{t}}{{\displaystyle \int }}}\text{sign}\left({\mathbf{S}}_{\mathrm{qs}}\right)\text{dt}\hspace{0.17em}\hspace{0.17em}$

(20)

where x_md =x_s/k_p, x_sd=k_px_m, and subscript d denotes the desired trajectories. Constant k is the position scaling factor, feedback gains α_i, γ_i are diagonal positive definite nxn matrices (remember that subscript i = {m,s}), the function $\text{sign}\left({{\displaystyle \prod }}^{\text{}}\right)={\left[\text{sgn}\left({\text{π}}_1\right),\cdots ,\text{sgn}\left({\text{π}}_{\mathrm{n}}\right)\right]}^{\mathrm{T}}$ stands for the input-wise discontinuous signum function of $\prod \in {\Re }^{\mathrm{n}}$ and:

$\begin{array}{l}{\mathbf{S}}_{\mathbf{q}\mathrm{m}}={\mathbf{S}}_{\mathrm{m}}-{\mathbf{S}}_{\mathrm{dm}},{\mathbf{S}}_{\mathbf{q}\mathrm{s}}={\mathbf{S}}_{\mathrm{s}}-{\mathbf{S}}_{\text{ds}}\\ {\mathbf{S}}_{\mathrm{m}}={\stackrel{.}{\tilde{\mathbf{x}}}}_{\mathrm{m}}+{\alpha }_{\mathrm{m}}{\tilde{\mathbf{x}}}_{\mathrm{m}},{\mathbf{S}}_{\mathrm{s}}={\stackrel{.}{\tilde{\mathbf{x}}}}_{\mathrm{s}}+{\alpha }_{\mathrm{s}}{\tilde{\mathbf{x}}}_{\mathrm{s}}\\ {\mathbf{S}}_{\text{dm}}={\mathbf{S}}_{\mathrm{m}}\left({\mathrm{t}}_0\right)e^{-{\mathrm{\kappa }}_{\mathrm{m}}t},{\mathbf{S}}_{\mathrm{ds}}={\mathbf{S}}_{\mathrm{s}}\left(t_0\right)e^{-{\kappa }_{\mathrm{s}}t}\end{array}$

(21)

where ${\tilde{\mathbf{x}}}_{\mathrm{s}}={\mathbf{x}}_{\mathrm{s}}-{\mathbf{x}}_{\text{sd}}$ denote position tracking error. Constants k_m, k_s>0, and S_m(t₀), S_s(t₀) stand for S_m, S_s at t = t₀. Notice that ${\stackrel{.}{\mathbf{x}}}_{\text{mr}},{\stackrel{.}{\mathbf{x}}}_{\text{sr}}$ are continuous, and S_qm(t₀) = S_qs(t ₀)=0 for any initial condition at t = t₀. Therefore ${\bar{\mathbf{S}}}_{\text{mr}},{\bar{\mathbf{S}}}_{\text{sr}}$ in (15)–(16) becomes:

${\bar{\mathbf{S}}}_{\text{mr}}={\mathbf{S}}_{\text{qm}}+{\text{γ}}_{\mathrm{m}}\underset{t_0}{\overset{t}{{\displaystyle \int }}}\text{sign}\left({\mathbf{S}}_{\text{qm}}\right)dt$

(22)

${\bar{\mathbf{S}}}_{\text{sr}}={\mathbf{S}}_{\text{qs}}+{\text{γ}}_{\mathrm{s}}\underset{t_0}{\overset{t}{{\displaystyle \int }}}\text{sign}\left({\mathbf{S}}_{\text{qs}}\right)dt$

(23)

It is important to mention that both stations exchange their respective position, velocity, and force (see Figure 1 for instance), the master sends towards the slave its measured position, estimated velocity, and human force, and the slave sends backward its measured position, estimated velocity, and environment force. Notice that computation of inverse kinematics has been circumvented since desired trajectories have been defined in the Cartesian space all the time. The inverted Jacobian matrix plays the role of transforming the velocity and extended errors $\left({\bar{\mathbf{S}}}_{\text{mr}},{\overline{\hspace{0.17em}\mathbf{S}}}_{\text{sr}}\right)$ from the Cartesian space into joint space.

4. Model-free exact differentiator as a velocity observer

In 2003, [32] proposed an arbitrary-order real-time differentiator, that allows a robust exact differentiation up to any given order l, provided the next ((l + 1)th derivative is bounded by a known constant. These features allow wide application of the differentiator in nonlinear control theory. In this work, the Cartesian velocity signal of the master robot is obtained by means of the exact differentiator, preserving the stability of the system and achieving excellent performance in comparison to the usual Euler-differentiator. In our case l = 1, which means that the measured position is differentiated once. The following differentiator arises:

$\begin{array}{l}\hspace{0.17em}{\stackrel{.}{z}}_{0j}=-{\lambda }_{0j}{\left|z_{0j}-{\mathbf{x}}_j\right|}^{1/2}sign\left(z_{0j}-{\mathbf{x}}_j\right)\\ {\stackrel{.}{z}}_{1j}=-{\lambda }_{1j}sign\left(z_{1j}-{\stackrel{.}{z}}_{0j}\right)\end{array}$

(24)

where subscript j = {x, y, z} denotes the Cartesian position component. For instance, let x _z be the end-effector Cartesian master position in z-axis. From (24), it is evident that a single exact differentiator for each Cartesian position component (x, y.z) has to be implemented in the master robot in order to achieve the estimated Cartesian velocity. End-effector position is obtained by means of the forward kinematics. Constants Δ_0j, Δ_1j>0 are tuned carefully according to [32]. Since the exact differentiator attains finite time convergence of the estimated states to the real ones, it is not necessary for the separation principle to be proved [33], provided the differentiator dynamics is chosen fast enough so that the states estimates are available for the controller before the system's trajectories leave certain region of stability. In this fashion, master/slave controllers can be designed by separately, independently of the velocity observer.

The main result is given in the following section.

5. Cartesian sliding PD controller

Consider the following controllers:

${\tau }_{\mathrm{m}}=-{\mathbf{K}}_{\mathrm{m}}{\mathbf{S}}_{\text{mr}}-{\mathbf{J}}_{\mathrm{m}}^{\mathrm{T}}{\mathbf{F}}_{\mathrm{h}}+{\mathbf{J}}_{\mathrm{m}}^{\mathrm{T}}{\mathbf{K}}_{\text{fm}}{\tilde{\mathbf{F}}}_{\mathrm{m}}$

(25)

${\tau }_{\mathrm{s}}=-{\mathbf{K}}_{\mathrm{s}}{\mathbf{S}}_{\text{sr}}+{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}{\mathbf{F}}_e-{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}{\mathbf{K}}_{\text{fs}}{\tilde{\mathbf{F}}}_{\mathrm{s}}$

(26)

where K_m, K_s, K_fm and K_fs are n × n positive definite diagonal feedback matrices. Force tracking errors are ${\tilde{\mathbf{F}}}_{\mathrm{m}}={\mathbf{F}}_{\mathrm{h}}-{\mathbf{F}}_{\text{hd}}$ , with F_hd=Fe/k_f, and ${\tilde{\mathbf{F}}}_{\mathrm{s}}={\mathbf{F}}_{\mathrm{e}}-{\mathbf{F}}_{\text{ed}}$ , with F_ed=k_fF_h. Constant k_f>0 is the force-scaling factor. It is worth noticing that (25)–(26) do not require any knowledge of the teleoperator dynamics, but only gain matrices, position, velocity, and force tracking errors. However, in order to perform the stability proof, some manipulation of the regressor must be performed. Substituting (25)–(26) into (7)–(8) gives rise to the following closed-loop error dynamics:

${\mathbf{H}}_{\mathrm{m}}{\stackrel{.}{\mathbf{S}}}_{\text{mr}}=-\left({\mathbf{C}}_{\mathrm{m}}+{\mathbf{K}}_{\mathrm{m}}\right){\mathbf{S}}_{\text{mr}}-{\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\mathrm{m}}+{\mathbf{J}}_{\mathrm{m}}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{fm}}{\tilde{\mathbf{F}}}_{\mathrm{m}}$

(27)

${\mathbf{H}}_{\mathrm{s}}{\stackrel{.}{\mathbf{S}}}_{\text{sr}}=-\left({\mathbf{C}}_{\mathrm{s}}+{\mathbf{K}}_{\mathrm{s}}\right){\mathbf{S}}_{\text{sr}}-{\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\mathrm{s}}-{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}{\mathbf{K}}_{fs}{\tilde{\mathbf{F}}}_{\mathrm{s}}$

(28)

equations (27)–(28) can be rewritten in terms of (15)–(16) and (17)–(18) as follows:

$\begin{array}{l}{\mathbf{H}}_{\mathrm{m}}\left({\mathbf{J}}_m^{-1}{\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{mr}}\right)=-\left({\mathbf{C}}_{\mathrm{m}}+{\mathbf{K}}_{\mathrm{m}}\right){\mathrm{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}+{\mathbf{J}}_{\mathrm{m}}^{\mathrm{T}}{\mathbf{K}}_{\text{fm}}{\tilde{\mathbf{F}}}_{\mathrm{m}}\\ -{\mathbf{H}}_{\mathrm{m}}{\stackrel{.}{\mathbf{J}}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}-{\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\mathrm{m}}\end{array}$

(29)

$\begin{array}{l}{\mathbf{H}}_{\mathrm{s}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{sr}}\right)=-\left({\mathbf{C}}_{\mathrm{s}}+{\mathbf{K}}_{\mathrm{s}}\right){\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{sr}}+{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}{\mathbf{K}}_{\text{fs}}{\tilde{\mathbf{F}}}_{\mathrm{s}}\\ -{\mathbf{H}}_{\mathrm{s}}{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{sr}}-{\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\mathrm{s}}\end{array}$

(30)

Since ${\bar{\mathbf{S}}}_{\text{sr}}=-{\mathrm{k}}_{\mathrm{p}}{\bar{\mathbf{S}}}_{\text{mr}}$ , thus substituting it into (30) yields:

$\begin{array}{l}-{\mathbf{H}}_{\mathrm{s}}{\mathbf{J}}_{\mathrm{s}}^{-1}{\mathrm{k}}_{\mathrm{p}}{\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{mr}}=\left({\mathbf{C}}_{\mathrm{s}}+{\mathbf{K}}_{\mathrm{s}}\right){\mathbf{J}}_{\mathrm{s}}^{-1}{\mathrm{k}}_{\mathrm{p}}{\bar{\mathbf{S}}}_{\text{mr}}-{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}{\mathbf{K}}_{\text{fs}}{\tilde{\mathbf{F}}}_{\mathrm{s}}\\ +{\mathbf{H}}_{\mathrm{s}}{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}{\mathrm{k}}_{\mathrm{p}}{\bar{\mathbf{S}}}_{\text{mr}}-{\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\text{s\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\end{array}$

Then multiplying by K_fm K_f⁻_s¹ k⁻¹ _f gives rise to:

$\begin{array}{r}\beta {\mathbf{H}}_{\mathrm{s}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{mr}}\right)=-\text{β}\left({\mathbf{C}}_{\mathrm{s}}+{\mathbf{K}}_{\mathrm{s}}\right){\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}+\left({\mathbf{K}}_{\text{fm}}{\mathrm{k}}_{\mathrm{f}}^{-1}\right){\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}\\ -\beta {\mathbf{H}}_{\mathrm{s}}{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}+\left({\mathbf{K}}_{\text{fm}}{\mathbf{K}}_{\text{fs}}^{-1}{\mathrm{k}}_f^{-1}\right){\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\text{s\hspace{0.17em}}}\hspace{0.17em}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=}-\beta \left({\mathbf{C}}_{\mathrm{s}}+{\mathbf{K}}_{\mathrm{s}}\right){\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}-{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}{\mathbf{K}}_{\text{fm}}{\tilde{\mathbf{F}}}_{\mathrm{m}}\\ -\beta {\mathbf{H}}_{\mathrm{s}}{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}+\left({\mathbf{K}}_{\text{fm}}{\mathbf{K}}_{\text{fs}}^{-1}{\mathrm{k}}_{\mathrm{f}}^{-1}\right){\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\text{s\hspace{0.17em}}}\end{array}$

(31)

where $\beta ={\mathrm{k}}_{\mathrm{p}}{\mathbf{K}}_{\text{fm}}{\mathbf{K}}_{\text{fs}}^{-1}{\mathrm{k}}_f^{-1}$ . Notice that due to the system being dynamically similar, a symmetric control has been considered to attain high transparency, as suggested in [31]. Then, taking into account (29) and (31), the stability analysis is given below.

Theorem Consider the master-slave dynamics (1)–(2) in closed-loop with the controllers (25)–(26). Then, the closed-loop dynamics (29) and (31) guarantee exponential tracking provided that γ _m , γ _s in (22)–(23) and K_m, K_s are large enough for a small initial condition error.

Proof. The stability proof is organized into three parts.

Part I. Boundedness of closed-loop trajectories. Consider the time derivative of the following Lyapunov's candidate function:

$V=\underset{V_1}{\underbrace{\frac{1}{2}{\left({\mathbf{J}}_m^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{H}}_{\mathrm{m}}\left({\mathbf{J}}_m^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}}+\underset{V_2}{\underbrace{\frac{1}{2}\beta {\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{H}}_{\mathrm{s}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}}$

(32)

along the solutions of the system. For the sake of simplicity, (32) is divided into two parts V = V₁ + V₂, and so its time derivative is as follows:

$\begin{array}{l}\hspace{0.17em}{\stackrel{.}{V}}_1={\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{H}}_{\mathrm{m}}\left(\frac{d}{dt}\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\right)\\ +\frac{1}{2}{\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\stackrel{.}{\mathbf{H}}}_{\mathrm{m}}\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\\ ={\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{H}}_{\mathrm{m}}\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{mr}}+{\stackrel{.}{\mathbf{J}}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\\ +\frac{1}{2}{\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\stackrel{.}{\mathbf{H}}}_{\mathrm{m}}\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\end{array}$

(33)

and

$\begin{array}{l}{\stackrel{.}{V}}_2=\beta {\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{H}}_{\mathrm{s}}\left(\frac{d}{dt}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\right)\\ +\frac{1}{2}\beta {\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\stackrel{.}{\mathbf{H}}}_{\mathrm{s}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\\ ={\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{H}}_{\mathrm{s}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{mr}}+{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\\ +\frac{1}{2}\beta {\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\stackrel{.}{\mathbf{H}}}_{\mathrm{s}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\end{array}$

(34)

Now, substituting the closed-loop dynamics (29) and (31) into (33) and (34), respectively; applying the well-known skew-symmetry property $({\mathbf{X}}^{\mathrm{T}}\left(\stackrel{.}{\mathbf{H}}-2\mathbf{C}\right)\mathbf{X}\equiv 0,\forall \mathbf{X}\hspace{0.17em}\in \hspace{0.17em}{\Re }^n,\mathbf{X}\ne 0)$ , and eliminating common terms, yields:

$\begin{array}{c}{\stackrel{.}{V}}_1=-{\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{m}}\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)+{\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{J}}_{\mathrm{m}}^{\mathrm{T}}{\mathbf{K}}_{\text{fm}}{\tilde{\mathbf{F}}}_{\mathrm{m}}\\ -{\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\mathrm{m}}\end{array}$

(35)

$\begin{array}{l}{\stackrel{.}{V}}_2=-\beta {\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{s}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\\ -{\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}{\mathbf{K}}_{\text{fm}}{\tilde{\mathbf{F}}}_{\mathrm{m}}\\ -{\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}\left({\mathbf{K}}_{\text{fm}}{\mathbf{K}}_{\text{fs}}^{-1}{\mathrm{k}}_{\mathrm{f}}^{-1}\right){\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\mathrm{s}}\end{array}$

(36)

Thus, adding (35) and (36) and getting rid of common terms, gives rise to:

$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\stackrel{.}{V}=\hspace{0.17em}{\stackrel{.}{V}}_1+\hspace{0.17em}{\stackrel{.}{V}}_2\\ =-{\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{m}}\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)-\hspace{0.17em}{\left({\mathbf{J}}_{\mathrm{m}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\mathrm{m}}\\ -\beta {\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{s}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)\\ -{\left({\mathbf{J}}_{\mathrm{s}}^{-1}{\bar{\mathbf{S}}}_{\text{mr}}\right)}^{\mathrm{T}}\left({\mathbf{K}}_{\text{fm}}{\mathbf{K}}_{\text{fs}}^{-1}{\mathrm{k}}_{\mathrm{f}}^{-1}\right){\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\mathrm{s}}\end{array}$

(37)

Expression (37) can be rewritten in joint space as follows:

$\begin{array}{l}\stackrel{.}{V}={\mathbf{S}}_{\text{mr}}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{m}}{\bar{\mathbf{S}}}_{\text{mr}}-{\mathbf{S}}_{\text{mr}}^{\mathrm{T}}{\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\mathrm{m}}-\beta {\text{\hspace{0.17em}k}}_{\mathrm{p}}^{-2}{\mathbf{S}}_{\text{sr}}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{s}}{\mathbf{S}}_{\text{sr}}\\ \hspace{0.17em}-{\mathrm{k}}_{\mathrm{p}}^{-1}{\mathrm{k}}_{\mathrm{f}}^{-1}{\mathbf{K}}_{\text{fm}}{\mathbf{K}}_{\text{fs}}^{-1}{\mathbf{S}}_{\text{sr}}^{\mathrm{T}}{\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\mathrm{s}}\end{array}$

(38)

where expression ${\bar{\mathbf{S}}}_{\text{mr}}={\text{\hspace{0.17em}k}}_{\mathrm{p}}^{-1}{\bar{\mathbf{S}}}_{\text{sr}}$ has been used, as well as (15)–(16). It can be proven that Y_mrΘ_m, Y_srΘ_s in (5)–(6) are upper bounded by state-dependent functions η_m(t), η_s(t) as shown for the slave case in the following. Consider (20):

$\Vert {\stackrel{.}{\mathbf{x}}}_{\text{sr}}\Vert \le {\rho }_1+{\lambda }_{\mathrm{M}}\left({\alpha }_{\mathrm{s}}\right)\Vert {\tilde{\mathbf{x}}}_{\mathrm{s}}\Vert +\Vert {\sigma }_{\mathrm{s}}\Vert$

where ${\text{ρ}}_1\ge {\mathrm{k}}_{\mathrm{p}}\hspace{0.17em}\Vert {\stackrel{.}{\mathbf{x}}}_{\text{md}}\Vert ,{\text{σ}}_{\mathrm{s}}={\text{γ}}_{\mathrm{s}}\underset{\mathrm{t}0}{\overset{\mathrm{t}}{{\displaystyle \int }}}\text{sign\hspace{0.17em}}\left({\mathbf{S}}_{\text{qs}}\right)dt$ (in [35] it is proven that σ _s is bounded). In this case, differentiating (20), one obtains:

$\Vert {\stackrel{..}{\mathbf{x}}}_{\text{sr}}\Vert \le {\alpha }_{\mathrm{s}}\Vert {\stackrel{.}{\tilde{\mathbf{x}}}}_{\mathrm{s}}\Vert$

Hence, q_sr= J_s⁻¹x_sr is bounded since it is assumed that J_s⁻¹ is well posed within the workspace free of singular configurations. Then, (14) yields:

$\Vert {\stackrel{.}{\mathbf{q}}}_{\text{sr}}\Vert \le {\text{λ}}_{\mathrm{M}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}\right)\left[{\text{\hspace{0.17em}ρ}}_1+{\text{λ}}_{\mathrm{M}}\left({\alpha }_{\mathrm{s}}\right)\Vert {\tilde{\mathbf{x}}}_{\mathrm{s}}\Vert +\Vert {\sigma }_{\mathrm{s}}\Vert \right]$

(39)

where Λ_M represents the maximum eigenvalue of J _s ⁻¹. As a consequence, apparently, ${\stackrel{..}{\mathbf{q}}}_{\text{sr}}={\mathbf{J}}_{\mathrm{s}}^{-1}{\stackrel{..}{\mathbf{x}}}_{\text{sr}}+{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}{\stackrel{.}{\mathbf{x}}}_{\text{sr}}$ is bounded from above:

$\Vert {\stackrel{..}{\mathbf{q}}}_{\text{sr}}\Vert \le {\text{λ}}_{\mathrm{M}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}\right){\alpha }_{\mathrm{s}}\Vert {\stackrel{.}{\tilde{\mathbf{x}}}}_{\mathrm{s}}\Vert +{\text{\hspace{0.17em}λ}}_{\mathrm{M}}\left({\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}\right)\left[{\text{\hspace{0.17em}ρ}}_1+{\alpha }_{\mathrm{s}}\Vert {\tilde{\mathbf{x}}}_{\mathrm{s}}\Vert +\Vert {\sigma }_{\mathrm{s}}\Vert \right]$

but it is not because ${\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}$ is ill posed when q_s=0, that is, each entry of the time derivative inverse Jacobian matrix ${\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}^{-1}$ is divided by the velocity. To overcome the latter, define ${\stackrel{.}{\mathbf{x}}}_{\text{sr}}={\mathbf{J}}_s{\stackrel{.}{\mathbf{q}}}_{\text{sr}}$ , whose time derivative is ${\stackrel{..}{\mathbf{x}}}_{\text{sr}}={\mathbf{J}}_s{\stackrel{..}{\mathbf{q}}}_{\text{sr}}+{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}{\stackrel{.}{\mathbf{q}}}_{\text{sr}}$ , and solve for ${\stackrel{..}{\mathbf{q}}}_{\text{sr}}$ to obtain:

$\begin{array}{l}{\stackrel{..}{\mathbf{q}}}_{\text{sr}}={\mathbf{J}}_{\mathrm{s}}^{-1}\left({\stackrel{..}{\mathbf{x}}}_{\text{sr}}-{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}{\stackrel{.}{\mathbf{q}}}_{\text{sr}}\right)\\ \le {\text{λ}\hspace{0.17em}}_{\mathrm{M}}\left({\mathbf{J}}_{\mathrm{s}}^{-1}\right)\left(\Vert {\stackrel{..}{\mathbf{x}}}_{\text{sr}}\Vert +{\text{λ}\hspace{0.17em}}_{\mathrm{M}}\left({\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}\right)\Vert {\stackrel{.}{\mathbf{q}}}_{\text{sr}}\Vert \right)\end{array}$

(40)

Then, from (6) the norm of Y_srΘ_s is the following:

$\begin{array}{l}\Vert {\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\mathrm{s}}\Vert \le \Vert {\mathbf{H}}_{\mathrm{s}}\Vert \Vert {\stackrel{..}{\mathbf{q}}}_{\mathrm{r}}\Vert +\Vert C_{\mathrm{s}}\Vert \Vert {\stackrel{.}{\mathbf{q}}}_{\mathrm{r}}\Vert +\Vert g_{\mathrm{s}}\Vert \\ \le {\text{λ}}_{\text{M\hspace{0.17em}}}\left({\mathbf{H}}_{\mathrm{s}}\right)\Vert {\stackrel{..}{\mathbf{q}}}_{\mathrm{r}}\Vert +{\text{ρ}}_{\mathrm{s}2}\Vert {\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}\Vert \Vert {\stackrel{.}{\mathbf{q}}}_{\text{sr}}\Vert +{\text{ρ}}_{\mathrm{s}3}\\ \le {\text{\hspace{0.17em}η}}_{\mathrm{s}}\left(t\right)\end{array}$

(41)

where ${\text{η}}_{\mathrm{s}}\left(\mathrm{t}\right)\doteq {\text{η}}_{\mathrm{s}}\left(t,{\tilde{\mathbf{x}}}_{\mathrm{s}},{\stackrel{.}{\tilde{\mathbf{x}}}}_{\mathrm{s}},{\sigma }_{\mathrm{s}},{\text{ρ}}_{\mathrm{s}2},{\text{ρ}}_{\mathrm{s}3}\right)$ , and some well-known properties of the dynamic model have been used. Hence, η_s(t) is called state-dependent function. Likewise, for the master one obtains the following:

$\Vert {\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\text{m\hspace{0.17em}}}\Vert \le {\text{η}}_{\mathrm{m}}\left(t\right)$

(42)

where ${\text{η}}_{\mathrm{m}}\left(t\right)\doteq {\text{η}}_{\mathrm{m}}\left(t,{\tilde{\mathbf{x}}}_{\mathrm{m}},{\stackrel{.}{\tilde{\mathbf{x}}}}_{\mathrm{m}},{\sigma }_{\mathrm{m}},{\text{ρ}}_{\mathrm{m}2},{\text{ρ}}_{\mathrm{m}3}\right)$ . Thus, (38) becomes:

$\begin{array}{l}\stackrel{.}{V}\le {\text{λ}}_{\mathrm{m}}\left({\mathbf{K}}_{\mathrm{m}}\right){\Vert {\mathbf{S}}_{\text{mr}}\Vert }^2+\Vert {\mathbf{S}}_{\text{mr}}\Vert {\text{η}}_{\mathrm{m}}\left(t\right)\\ -{\text{\hspace{0.17em}δλ}}_{\mathrm{m}}\left({\mathbf{K}}_{\mathrm{s}}\right){\Vert {\mathbf{S}}_{\text{sr}}\Vert }^2+\text{\hspace{0.17em}δ}\Vert {\mathbf{S}}_{\text{sr}}\Vert {\text{η}}_{\mathrm{s}}\left(t\right)\end{array}$

(43)

with $\text{δ}=\left|{\text{\hspace{0.17em}k}}_{\mathrm{p}}^{-1}\right|\left|{\text{\hspace{0.17em}k}}_{\mathrm{f}}^{-1}\right|{\text{λ}}_{\mathrm{m}}\left({\mathbf{K}}_{\text{fm}}\right){\text{\hspace{0.17em}λ}}_{\mathrm{m}}({\mathbf{K}}_{\text{fs}}^{-1})$ which is always a positive number. Then, if K_m and K_s are large enough and the initial errors are small enough, one concludes the negative definiteness of (43) outside of the very small ball ${ϵ}_0=\left\{{\mathbf{S}}_{\text{mr}},{\mathbf{S}}_{\text{sr}}|\stackrel{.}{V}\ge 0\right\}$ centered at the origin $\stackrel{.}{V}\left({\mathbf{S}}_{\text{mr}},{\mathbf{S}}_{\text{sr}}\right)=\left(0,0\right)$ , such that the bounded closed-loop signals are obtained:

${\mathbf{S}}_{\text{mr}},{\mathbf{S}}_{\text{sr}}\in {\mathcal{L}}_{\infty }\Rightarrow {\bar{\mathbf{S}}}_{\text{mr}},{\bar{\mathbf{S}}}_{\text{sr}}\in {\mathcal{L}}_{\infty }$

(44)

According to (44), $\Vert {\mathbf{S}}_{\text{mr}}\Vert \le {ϵ}_1,\Vert {\mathbf{S}}_{\text{sr}}\Vert \le {ϵ}_2$ , are upper bounded by given constants (₁, ₂) >(0, 0), respectively. Since ${\bar{\mathbf{S}}}_{\text{mr}},{\overline{\hspace{0.17em}\mathbf{S}}}_{\text{sr}}$ are bounded then $\left({\mathbf{S}}_{\text{qm}},{\mathbf{S}}_{\text{qs}}\right)\in {ℒ}_{\infty }$ and $\left({\text{σ}}_{\mathrm{m}},{\text{\hspace{0.17em}σ}}_{\mathrm{s}}\right)\in {ℒ}_{\infty }$ are bounded, too. Likewise, by virtue of (H_m, H_s), (C_m, C_s), (g_m, g_s) are upper bounded and since the trajectory generated by the human operator is bounded (it is assumed the human operator is passive, then he/she generates bounded trajectories for the slave robot), and the feedback gains are bounded, then $\left({\stackrel{.}{\mathbf{x}}}_{\text{mr}},{\stackrel{.}{\mathbf{x}}}_{\text{sr}}\right)\in {ℒ}_{\infty }$ and, therefore, $\left({\stackrel{.}{\mathbf{q}}}_{\text{mr}},{\stackrel{.}{\mathbf{q}}}_{\text{sr}}\right)\in {ℒ}_{\infty }$ which implies that $\left({\mathbf{Y}}_{\text{mr}}{\text{Θ}}_{\mathrm{m}},{\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\mathrm{s}}\right)\in {ℒ}_{\infty }$ are also bounded such as shown in (41)–(42). Thus, from (27)–(28), one can conclude that $\Vert {\stackrel{.}{\mathbf{S}}}_{\text{mr}}\Vert \le {\text{\hspace{0.17em}ζ}}_{\mathrm{m}}\left(t\right)$ are bounded by state-dependent functions ${\text{ζ}}_{\mathrm{m}}(*),{\text{\hspace{0.17em}ζ}}_{\mathrm{s}}(*)$ , provided that the human operator and the environment are passive systems, and forces F_h, F_e are bounded such that $\stackrel{.}{V}\in {ℒ}_{\infty }$ . So far, it has only been proved that all closed-loop signals remain bounded. Now it will be shown that for a given γ_m, γs a sliding mode is induced for all time on S_qm=0, S_qs =0, at master and slave sites.

Part II. Existence of SOSM. Multiplying (16) by Js it results ${\mathbf{J}}_{\mathrm{s}}{\mathbf{S}}_{\text{sr}}={\bar{\mathbf{S}}}_{\text{sr}}$ whose time derivative yields:

$\begin{array}{l}{\mathbf{J}}_{\mathrm{s}}{\stackrel{.}{\mathbf{S}}}_{\text{sr}}+{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}{\mathbf{S}}_{\text{sr}}={\stackrel{.}{\stackrel{̄}{\mathbf{S}}}}_{\text{sr}}\\ ={\stackrel{.}{\mathbf{S}}}_{\text{qs}}+{\text{γ}}_{\mathrm{s}}\text{\hspace{0.17em}sign}\left({\mathbf{S}}_{\text{qs}}\right)\end{array}$

Solving for ${\stackrel{.}{\mathbf{S}}}_{\text{qs}}$ , and multiplying by S_q^T_s, the sliding mode condition is attained:

$\begin{array}{l}{\mathbf{S}}_{\text{qs}}^{\mathrm{T}}{\stackrel{.}{\mathbf{S}}}_{\text{qs}}=-{\mathbf{S}}_{\text{qs}}^{\mathrm{T}}{\text{γ}}_{\mathrm{s}}\text{\hspace{0.17em}sign}\left({\mathbf{S}}_{\text{qs}}\right)+{\mathbf{S}}_{\text{qs}}^{\mathrm{T}}\left({\mathbf{J}}_{\mathrm{s}}{\stackrel{.}{\mathbf{S}}}_{\text{sr}}+{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}{\mathbf{S}}_{\text{sr}}\right)\\ \le -{\text{λ}}_{\mathrm{m}}\left({\text{γ}}_{\mathrm{s}}\right)\left|{\mathbf{S}}_{\text{qs}}^{\mathrm{T}}\right|+\left|{\mathbf{S}}_{\text{qs}}^{\mathrm{T}}\right|\left|\left({\mathbf{J}}_{\mathrm{s}}{\stackrel{.}{\mathbf{S}}}_{\text{sr}}+{\stackrel{.}{\mathbf{J}}}_{\mathrm{s}}{\mathbf{S}}_{\text{sr}}\right)\right|\\ \le -{\text{\hspace{0.17em}μ}}_{\mathrm{s}}\left|{\mathbf{S}}_{\text{qs}}\right|\end{array}$

(45)

with ${\text{μ}}_{\mathrm{s}}=\left[{\text{λ}}_{\text{min}}\left({\text{γ}}_{\mathrm{s}}\right)-{ϵ}_3\right],{ϵ}_3=\left|\frac{d}{dt}\left({\mathbf{J}}_{\mathrm{s}}{\mathbf{S}}_{\text{sr}}\right)\right|$ with Λ_min denoting the minimum eigenvalue of γ _s. Thus, the sliding mode condition is accomplished if Λ_min(γ_s)>₃, such that μ_s>0 of (45) guarantees the sliding mode at ${\mathbf{S}}_{\text{qs}}=0$ . Notice that for any initial condition S_qs(t₀)=0, then t_r =0, which implies that the sliding mode is enforced for all time [36]. This result means that at t = t₀ and for any initial condition, the closed-loop dynamics is governed by a homogeneous first order differential equation, invariant to any physical parameter:

${\mathbf{S}}_{\mathrm{s}}={\mathbf{S}}_{\text{ds}}\forall t\Rightarrow {\mathbf{S}}_{\mathrm{s}}={\mathbf{S}}_{\mathrm{s}}\left(t_0\right){\mathrm{e}}^{-{\text{κ}}_{\mathrm{s}}t}$

(46)

which vanishes exponentially fast.

Part III. Tracking performance.

1. Position tracking performance. Solving (46), one obtains:

   ${\mathbf{S}}_{\mathrm{s}}={\mathbf{S}}_{\text{ds}}\forall t\Rightarrow {\stackrel{.}{\tilde{\mathbf{x}}}}_{\mathrm{s}}=-{\alpha }_{\mathrm{s}}{\tilde{\mathbf{x}}}_{\mathrm{s}}+{\mathbf{S}}_{\mathrm{s}}\left(t_0\right){\mathrm{e}}^{-{\text{κ}}_{\mathrm{s}}t}$

   (47)
   which implies that the position tracking error tends toward zero exponentially fast, that is, ${\mathbf{x}}_{\mathrm{s}}\to {\mathrm{k}}_{\mathrm{p}}{\mathbf{x}}_{\mathrm{m}},{\stackrel{.}{\mathbf{x}}}_{\mathrm{s}}\to {\mathrm{k}}_{\mathrm{p}}{\stackrel{.}{\mathbf{x}}}_{\mathrm{m}}$ . A similar procedure leads to prove that a sliding mode exists for the master, which, in turn, guarantees that the position tracking error converges exponentially fast, i.e., ${\mathbf{x}}_{\mathrm{m}}\to {\mathbf{x}}_{\mathrm{s}}/{\mathrm{k}}_{\mathrm{p}},{\stackrel{.}{\mathbf{x}}}_{\mathrm{m}}\to {\stackrel{.}{\mathbf{x}}}_{\mathrm{s}}/{\mathrm{k}}_{\mathrm{p}}$ . Hence, a strong kinaesthetic coupling between the human operator and the environment is achieved.

2. Force racking performance. According to the boundedness of H_s, C_s, K_s, S _sr , Θ _sr , and ${\mathbf{Y}}_{sr}{\text{Θ}}_{\mathrm{s}}$ , considering the slave closed-loop dynamics (28), and solving for the force error, one has the following:

   $\begin{array}{l}{\tilde{\mathbf{F}}}_{\mathrm{s}}={\mathbf{K}}_{\text{fs}}^{-1}{\mathbf{J}}_{\mathrm{s}}^{-\mathrm{T}}\left[{\mathbf{H}}_{\mathrm{s}}{\stackrel{.}{\mathbf{S}}}_{\text{sr}}+\left(C_{\mathrm{s}}+{\mathbf{K}}_{\mathrm{s}}\right){\mathbf{S}}_{\text{sr}}+{\mathbf{Y}}_{\text{sr}}{\text{Θ}}_{\mathrm{s}}\right]\\ \le {\text{λ}}_{\mathrm{M}}\left({\mathbf{K}}_{\text{fs}}^{-1}\right){\text{λ}}_{\mathrm{M}}\left({\mathbf{J}}_{\mathrm{s}}^{-\mathrm{T}}\right)[{\text{λ}}_{\mathrm{M}}\left({\mathbf{H}}_{\mathrm{s}}\right){\text{ζ}}_{\mathrm{s}}\left(t\right)+\\ +\left({\text{ρ}}_{\mathrm{s}1}{\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}+\hspace{0.17em}{\text{λ}}_{\mathrm{M}}\left({\mathbf{K}}_{\mathrm{s}}\right)\right){\in }_2+{\text{\hspace{0.17em}η}}_{\mathrm{s}}\left(t\right)]\end{array}$

   (48)

From (48), it is evident that the force tracking error remains bounded. In the steady state, the error can be reduced to a reasonable small value by increasing K_fs. A similar procedure can be performed for the master and is therefore omitted.

6. Discussions

6.1 Well-posed inverse Jacobian

Apparently there might be a problem with J_m^{− 1}, J_s-¹. However, it has been proven that J_m, J_s are well posed for all t, because ${\mathbf{x}}_{\mathrm{s}}\to {\mathbf{x}}_{\text{sd}}$ and ${\mathbf{x}}_{\mathrm{m}}\to {\mathbf{x}}_{\text{md}}$ with no overshooting. Assuming that reasonably desired trajectories (generated by the human operator) have been designed such that the slave trajectories belong to its workspace free of any singular configuration Ω_i= $\left\{{\mathbf{q}}_{\mathrm{i}}|\text{rank}\left({\mathbf{J}}_{\mathrm{i}}\right)=n,\forall {\mathbf{q}}_{\mathrm{i}}\in {\Re }^{\mathrm{n}}\right\}$ within Ω_i and therefore J_i⁻¹ is well-posed ∀t (for subscript i=m denoting master or i =s denoting slave).

6.2 Exact differentiator robustness: Noisy signals vs. accuracy

In the absence of input noises the outputs (state estimates) can be considered as exact direct measurements [32]. In the presence of measurement noises the differentiation accuracy inevitably deteriorates rapidly with the growth of the differentiation order. Nevertheless, this problem may be alleviated by reducing the sampling period.

6.3 How to tune the controller

The stability proof suggests that arbitrary small gains γ_m, γ_s and small α_m, α_s can be set as a starting point. Increase feedback gains K_m, K_s until acceptable boundedness of S_m, S_sr and ${\stackrel{.}{\mathbf{S}}}_{\text{mr}},{\stackrel{.}{\mathbf{S}}}_{\text{sr}}$ appears. Then increase gradually γ_m, γ _s until the sliding mode for both manipulators arises. Finally, increase α_m, α_s to achieve a better position tracking performance. Notice that γ _m, γ_s are not a high gain result since larger γ _m, γ _s do not mean a larger domain of stability. On the other hand, the larger K_fm, K_fs the smaller the force tracking error, however, a trade-off between position tracking force tracking boundedness and force scaling must be taken into account to ensure a desired system's stability performance.

6.4 Cartesian and joint positions

In order to implement the proposed controllers, only Cartesian trajectories x_m, x_s and ${\stackrel{.}{\mathbf{x}}}_{\mathrm{m}},{\stackrel{.}{\mathbf{x}}}_{\mathrm{s}}$ and joint position readings q_m, q_s are required, but neither acceleration measurements nor ${\stackrel{.}{\mathbf{q}}}_{\mathrm{m}},{\stackrel{.}{\mathbf{q}}}_{\mathrm{s}}$ . Phantom and Catalyst-5 Cartesian positions are obtained by means of its driver and forward kinematics, respectively. Cartesian velocities of Phantom and Catalyst-5 are estimated by a differentiation method based on higher-order sliding modes [32], and by means of the differential kinematics, respectively.

6.5 Robustness

The system has inherent robustness typical of variable structure systems, since the invariance property is attained for all time through (45) whose convergence is governed solely by (47).

6.6 Smooth controller

Higher-order sliding modes, in this case SOSM, have emerged to solve the problem of chattering, which is induced by first order sliding modes (FOSM). In addition to preserve the advantages of FOSM, the scheme SOSM totally removes the chattering effect of FOSM and provides for even higher accuracy [37]. In our case, SOSM is induced and chattering is circumvented by integrating the signum function of S_qm and S_qs.

6.7 How to compute master and slave control signal

Taking into account equations (15) and (16), notice that ${\bar{\mathbf{S}}}_{\text{mr}}={\stackrel{.}{\mathbf{x}}}_{\mathrm{m}}-{\stackrel{.}{\mathbf{x}}}_{mr},\hspace{0.17em}{\bar{\mathbf{S}}}_{sr}={\stackrel{.}{\mathbf{x}}}_{\mathrm{s}}-{\stackrel{.}{\mathbf{x}}}_{sr}$ are expressions in Cartesian space that are mapped into joint space by the inverse of their respective Jacobian. Thus, S_mr, S_mr in (25) and (26) are already in joint space units. The resulting control signal units are ton meter [Nm]. Regarding the haptic interface, the digital value expressed in[Nm]units is internally converted by driver functions into electrical current that powers the actuators. Regarding the virtual robot (Catalyst-5), its torque control value is directly written into the dynamics Equation (2), then solved online to compute joint acceleration, which is in turn integrated once and twice to compute joint velocity and position, respectively.

7. Experiments and results

7.1 Semi-experimental setup

In this section, the performance of the proposed control scheme is evaluated by means of experiments carried out in a semi-experimental setup, in the absence of time delay. The semi-experimental setup (shown in Figure 2) is fitted with a Phantom Premium 1.0 haptic interface that is used as the master robot and a NANO17 six dof force/torque sensor that is attached to the Phantom's end-effector to measure the human operator force. The complete system, haptic interface and virtual environment, run in a PC under the Windows XP operating system. The virtual slave robot mimics the industrial robot Catalyst-5, and its dynamics is solved on line by a well-known numerical integrator. The combination of Visual C++, CHAI3D and OpenGL libraries allow the user to interact with a high fidelity and realistic virtual environment in real time.

Figure 2.

Semi-experimental setup

7.2 Results

The slave robot (virtual robot) is teleoperated in such a way its end-effector follows a predefined trajectory as it is shown in Figure 3. The sequence is as follows. The master and slave robot's initial conditions are different and away from the virtual plane. The operator approaches to the plane to make contact at the red point shown in Figure 3 (right). Then the operator requests the slave robot's end-effector to follow the circumference while exerting a sustained level of force onto the plane. After the completion of one circumference the operator follows a segment of a straight line, stops exerting force, and again starts exerting a sustained force onto the plane while moving along the last straight line segment. The experiments last approximately 42 seconds.

Figure 3.

Definition of the task: standard trajectory to be followed by the operator

1. Feedback gains

Feedback gains for the master and slave controllers are set as follows:

K_m= diag (0.01, 0.04, 0.04), Ks= diag (2.0, 5.5, 2.0), K_fm=K_fs=0.01 I_3times3,α_m= diag (0.8, 3.3, 2.3), α_s= diag (1.8, 4.0, 2.4), γ_m= diag (0.01, 0.018, 0.01), γ_s= diag (0.8, 1.5, 1.7), κ_m=6.0, κ_m=9.0.

2. Scaling factors

Scaling factors are set as follows: k_p=2.7, k_f=1.0.

3. Environment parameters

The virtual plane (environment) is modelled as a spring whose stiffness coefficient K_ez = 360 N/m. The virtual plane is located at z_e = 0.1 m in z - axis, and is parallel to the x-y plane of the slave robot base frame; thus, the environment force is F_e = K _ez Δz, with Δz = z-z_e as the deformation of the virtual plane (see Figure 4 for reference).

Figure 4.

Location of the virtual plane with respect to the slave robot base frame

4. Slave robot parameters

Parameters are those of the Catalyst-5 industrial robot. Only the first three dof are modelled and controlled; pitch and roll joints are fixed.

Table 1.

Main parameters of the slave robot

Link	Mass	Length
Link 1	9.0	0.2504
Link 2	2.0	0.2504
Link 3	1.0	0.3914
(units)	kg	m

5. Exact differentiator parameters

Differentiator parameters are set as follows: Λ_0j = 10, Λ_{1 j} =5, where j=x,y,z.

Figure 5 depicts master and slave Cartesian trajectories in 3D (three dimensional) view. Notice that the slave Cartesian trajectory is greater than the master trajectory; this is because of the scaling factor k_p=2.7.

Figure 5.

Cartesian task: free and constrained motion. Slave position is scaled by a factor k_p=2.7.

Figure 6 shows position tracking errors in x, y, z coordinates, at the slave side. Notice that a very small position tracking error in the z coordinate arises at the contact stages, that is, when the slave robot exerts a sustainable force onto the virtual plane. Furthermore, notice in Figure 7 that the human force is practically equal to that of the environment force, which implies a high level of transparency.

Figure 6.

Position tracking errors in Cartesian coordinates at the slave side. Notice that master position has been conveniently scaled to be compared against the slave position according to ${\tilde{\mathbf{x}}}_{\mathrm{s}}={\mathbf{x}}_{\mathrm{s}}-k_p{\mathbf{x}}_{\mathrm{m}}$ .

Figure 7.

Force tracking performance. The operator exerts a force profile along the z-axis. Force scaling factor k_f is set to 1.0.

According to Figure 3, there are two contact stages during the experiment: the first one, evident from 10 to 30 seconds; and the second one, from approximately 30 to 38 seconds, which can be seen in Figures 6 and 7.

In Figures 8 and 9, it is possible to see the sliding mode surfaces response in Cartesian coordinates, for the master and the slave robots, respectively. Notice that both S_qm and S_qs equal zero for all time, even at the contact stages.

Figure 8.

At the master controller, the sliding mode condition (S_qm=S_m-S_dm=0) is enforced since the beginning of the experiment

Figure 9.

At the slave controller, the sliding mode condition is enforced for all time, that is, (S_qs=S_s-S_ds=0) since the beginning

Finally, Figures 10 and 11 depicts, respectively, applied torques to master and slave robots.

Figure 10.

Torque control input applied to each master joint actuator. Notice that the control signal is free of chattering.

Figure 11.

Torque control input applied to each slave joint actuator. Notice that the control signal is free of chattering.

8. Conclusions

A simple yet straightforward approach based on the second order sliding mode control for nonlinear bilateral teleoperators has been synthesized. The proposed scheme accomplishes the transparency criteria without velocity measurements, nor robot dynamics knowledge, nor computing Cartesian robot dynamics. Exponential fast convergence of position and velocity, without overshooting, is ensured; in fact, since the sliding mode exists for all time it is possible to achieve a fast response.

Force tracking error exhibits a strong performance, similar to other approaches using impedance control reported in the international literature.

Since the velocity observer (exact differentiator) achieves finite time convergence of the estimate errors, the separation principle proof can be omitted, provided the observer is carefully tuned so that it exhibits a faster response than the master dynamics. Semi-experimental results considering nonlinear systems reveal the good performance of the proposed scheme.