Pose regulation of a constrained circular object using Echo State Networks

Abstract

For some dexterity robotic tasks, such as turning a constrained dynamic object, rolling motion is a crucial advanced characteristic, which for the human hand is possible from the shape and soft tip in our fingers. This paper presents the manipulation of a rigid constrained dynamic circular object using a robotic finger equipped with a hemispherical and soft tip. Assuming that object angle measurement is not available, a force-position regulator is proposed to guarantee the control of contact force, as well as the object angle throughout the tangential force. To establish the relationship between the rolling angle and the tangential force, the use of Echo State Network (ESN) becomes instrumental. A novel smooth weight adaptation scheme steered by a sliding mode attractor, for readout network is proposed. Stability conditions are obtained to guarantee convergence in finite time using small feedback gains and piecewise continuous functions. Additionally, the local asymptotic stability of the pair robot-object is guaranteed without force sensing, not either any knowledge of the deformation nor penetration measurement, leading surprisingly to a simple blind touch-type controller for such difficult task. Finally, numerical simulations show the learning rule contribution and the behavior of the closed-loop system.

Keywords

Echo State Networks sliding mode control soft fingertip adaptive learning constrained object manipulation

1 Introduction

Based on physiology studies that elucidate how human sensory-motor coordination or motor control is carried out as many distributed processes, roboticists have proposed approaches to reproduce some of these functionalities to increase robot skills. In this way, the robot manipulation has been studied under very hard assumptions, such as exact knowledge of the system and full sensory feedback, which limits its applicability enormously, [1 –3]. To cope with many drawbacks due to model and state-inferred uncertainties, the soft computing and machine learning techniques have been explored, and frequently used.

Object manipulation with curvature is achieved principally through thumb and index fingers that ables the object rotation, wherein index fingertip rolls while the thumb finger acts as the rolling surface for the object. However, notice that to get a satisfactory manipulation, the central nervous system (CNS) should processed information from the skin, muscles, and tendons, who play a crucial sensory-motor role. That is, the object rotation can be carried out using only the perceptual information even if the object is occluded. In this paper, we reproduce this difficult manipulation task using robotic fingers with hemispherical soft tips for a constrained curved object. The difficulty stems from the fact that we assume object angle is not measured, a common situation for we humans but hard to synthesizing for robots. Under this assumption, a novel regulator is proposed where the object angle is estimated by a recurrent echo state (neural) network (ESN), which has not been explored in the literature for this advanced task.

The artificial recurrent neural networks (ARNNs), unlike feedforward neural networks, have feedback connections to influence the output, for a more realistic brain analogy, given that the processing elements are interconnected to propagate inputs throughout the network by activation functions. In this sense, echo state networks (ESNs) are a powerful yet simple paradigm for implementing complex ARNNs [4 –13]. As soon as this paradigm was discovered, many variants were explored and its capacity assessed, [14 –20]. In this work, an alternative training of the ESN output layer is proposed by using (smooth) integral sliding mode control that guarantees finite-time convergence of the learning error, leading to finite-time learning of ESN.

The sliding mode control was developed for control applications [21] and later ported to the neural network field. However its contribution as the driving attractor that sets weights convergence rate has been overlooked. In this regard, sliding mode attractor produces training algorithms that quickly converge, and since the extended error, or sliding surface, also shapes the objective function, there have been proposed for discrete-time and continuous-time systems, [22 –30]. When smooth controllers are used, quasi-sliding modes induce stability only, as shown in these works. However, few approaches prove strictly convergence in finite time at the expense of suffering from chattering, which excites residual dynamics, [31 –33]. In this paper, an integral sliding surface to lead convergence of the ESN in a finite time is proposed, without chattering. Its output computes the necessary tangent force to roll the object a given angle, even without angle measurement.

Skillful manipulation is based, traditionally, on the simultaneous force-position control based on frictionless (infinitesimal) point contact, where neither rolling nor sliding is possible, due to contact occurs at a scalar point, [34]. It simplifies, apparently, the approach at the expense of enormous limitations for dexterous manipulation, [35 –37], because purposely the holonomic constraint that arises at point contact not either models tangential force, essential for rolling an object. In addition, approaches at the actuator level, [38], do not establish a compliant contact to rigid environment due to actively control the apparent impedance of actuation.

In contrast, recent control schemes for manipulation of planar objects [39], have studied different contact models using soft tip wherein area contact arises, [40 , 44], leading to more skill in manipulation. Also, a deformable soft tip dissipates impulsive contact forces that allows stable transition from free and constrained motion regimes for safe interaction, [42 –46]. The significant advantage is also that the hemispherical shape of the tip induces the rolling at contact, which leads to repositioning the contact point by the rolling motion, without losing interaction. However, there remains to solve how much tangent force is necessary for a given desired rolling.

1.1 Our proposal

The control design is built upon the intricate relationship between the normal force subspace, arises along the maximum radial deformation, and the orthogonal velocity subspace, exactly wherein tangent force evolves. This fundamental observation leads to consider that to manipulate a circular object by a robotic finger with soft-tip, the ESN must be capable of outputting the necessary rolling motion under the following conditions: the smaller (larger) orientation error, the smaller (larger) rolling motion is needed, consequently the smaller(larger) tangent force is required. However, since there does not exists a direct relationship that models such behavior, the challenge is to design a ESN that quickly provides that, noticing that we cannot use controllers developed for rigid point contact because no explicit model of tangent force arises. To combine the stable time-invariant linear systems of the ESN architecture as a stable filter module, where the string of integrators is equivalent to the pure delay element, its reservoir is designed taking the states as inputs to the read-out network. Thus, the sliding mode-based learning in the readout network stands for a complementary tool to control or approximate arbitrary dynamical systems, similarly to [47 –49], but never explored for complex robotic tasks. In this way, a regulator that yields local asymptotic convergence is proposed, and simulations results show the performance of the closed-loop system.

1.2 Organization

This paper is organized as follows. Section 2 introduces the kinematics and dynamics relationships of a constrained object, and Section 3 shows the general structure of ESN and its learning process in the adaptive readout network based on sliding mode attractors. In Section 4 a controller is proposed without any sensing of contact forces, deformation nor contact area, with stability analysis. Simulation results are presented in Section 5 to illustrate the numerical performance under various operational conditions, finally concluding remarks are addressed in Section 6.

2 Robotic Finger with Deformable Fingertip

2.1 Kinematics

Consider a three degree of freedom (DOF) robotic finger with a hemispherical, and soft, tip in contact to a circular constrained rigid object, as depicted in Fig. 1. Let Σ _r be the fixed inertial frame, Σ _b is the center position of the base of the deformable fingertip, and Σ ₀ is the object center of mass (CoM) that coincides with its rotational axis. Let q _i and l _i be the generalized positions of the robotic finger and the link length i, respectively for i = 1, 2, 3. Finally, r > 0 stands for the fingertip radius with center at Σ _b, R > 0 is the radii of the circular object while θ is the object angle measured with respect to the usual right hand side convention, for positive angular displacement.

Fig.1

Robotic finger with a soft tip and circular object pivoting around Σ ₀.

2.2 Deformation Model

A contact area when a soft fingertip establishes contact with circular object, whose deformation varies according to the applied force. Due to maximal force arises at the point of maximum radial deformation Δx, [44, 50], then Δx can be computed by localizing the center of the contact area with respect to the frame Σ ₀, $\begin{matrix} Δ x = r + R - R_{0} \\ R_{0} = (x_{0} - x_{b}) cos (α) + (y_{0} - y_{b}) sin (α) \end{matrix}$ (1) where $α = {tan}^{- 1} (\frac{y_{o} - y_{b}}{x_{o} - x_{b}})$ is the angle between the force vector and the horizontal, see Fig. 2.

Fig.2

Kinematic relations between fingertip and the circular object.

2.3 Rolling Constraint

Clearly the hemispherical shape of the fingertip allows rolling which implies simultaneous translation of Σ _b, a characteristic that is is considered a robotic ability. In this case, ee Fig. 1, the 2D contact area is depicted by the light blue arc length1. Assuming initial contact of fingertip with the object, Δx > 0, thus the velocity rolling constraint becomes, [44], $\dot{Φ} = R \dot{θ} + r \dot{θ_{p}} = 0$ (2) where θ _p = π + α - q^Te, e = [1 , 1] ^T. Noting that (2) is integrable in 2D for the pinned object, the vector-valued solution S ∈ R, called kinematic rolling constraint, is defined as $Φ = R θ + r θ_{p} + C_{θ_{p 0}} = 0$ (3) where C _{θ
_p0} stands for integration constant at initial condition.

2.4 Dynamic Model

Using the unconstrained Lagrangian defined as $L = K_{\dot{q}} + K_{\dot{θ}} - P_{g} (q) - P_{c} (Δ x)$ where $K_{\dot{q}}$ and $K_{\dot{θ}}$ represent the kinetic energies of the rigid robotic finger, and the object, respectively while the potential energies P _g (q) and P _c (Δx) stands for the potential energy of the Earth gravitational field, and the elastic potential energy from deformation of the fingertip, respectively. The dynamical model of the robotic finger, with a deformable and hemispherical fingertip, that evolves onto holonomic constraint (3) is defined as $H (q) \ddot{q} + C (q, \dot{q}) \dot{q} + g (q) + f \frac{\partial Δ x}{\partial q} - λ \frac{\partial Φ}{\partial q} = τ$ (4) $I \ddot{θ} + f \frac{\partial Δ x}{\partial θ} - λ \frac{\partial Φ}{\partial θ} = 0$ (5) where H (q) ∈ R ^3×3 and scalar I stand for the inertial matrix and the inertia angular moment of the robot and the object, respectively, $C (q, \dot{q}) \in R^{3 \times 3}$ represents the Coriolis matrix that gives rise to centrifugal and centripetal forces, $g (q) = \frac{\partial P_{g} (q)}{\partial q} \in R^{3}$ models the torques due to Earth gravitational field, τ ∈ R ³ stands for the robot control input and $f = f (Δ x) = \frac{\partial P_{c} (Δ x)}{\partial Δ x} = k Δ x^{2}$ as the contact force, for k = 2πE the fingertip stiffness. Notice that (4)–(5) represent a highly non-linear under-actuated system given that the object dynamics coupled with (3). It has been subject of intensive research in robotics, whose robotic problem is to control de state of the underactuated object dynamics (5), by controlling (4), despite unmeasurable (θ, f, λ).

System (4)–(5) can be written in a compact vector-matrix notation as follows $H \ddot{z} + C \dot{z} + G + B Λ = Π$ (6) where $\begin{matrix} H & = & diag (H (q), I), is the inertial matrix \\ C & = & diag (C (q, \dot{q}), 0), is the Coriolis matrix \\ G & = & [g (q), 0]^{T}, is the gravity vector \\ Π & = & [τ, 0]^{T}, is the input vector \\ z & = & [q^{T}, θ]^{T}, is the state \\ Λ & = & [f, λ]^{T}, is the contact forces \\ B & = & [\begin{matrix} \frac{\partial Δ x}{\partial q} & \frac{\partial Φ}{\partial q} \\ \frac{\partial Δ x}{\partial θ} & \frac{\partial Φ}{\partial θ} \end{matrix}] = [\begin{matrix} J^{T} (q) r_{X} & - r (J_{S} + e) \\ 0 & R \end{matrix}], \end{matrix}$ where B maps contact forces into torques, J (q) ∈ R ^3×2 stands for the non-square Jacobian matrix of the manipulator at (x _e, y _e) with r _X = [sin(α) , cos(α)] ^T and $J_{S} = \frac{1}{D} J^{T} (q) r_{W} \in R^{3}$ yields the tangential direction of λ with r _W = [y ₀ - y _b, x ₀ - x _b] ^T and D = (y ₀ - y _b) ² + (x ₀ - x _b) ².

2.5 Useful Structural Properties

P1: Antisymmetric property. Matrix $(\dot{H} (q) - 2 C (q, \dot{q}))$ is anti-symmetric, [44].

P2: Passivity. Considering the dot product of output $\dot{z}$ and input u, its integral leads to the open-loop energy as follows $\int_{0}^{t} ({\dot{z}}^{T} u) d τ = E (t) - E (0) \geq - E (0),$ (7) in virtue that $\frac{d}{dt} Φ = {\dot{q}}^{T} \frac{\partial Φ}{\partial q} + \dot{θ} \frac{\partial Φ}{\partial θ} = 0$ , with E (t) = K + P. Thus, system (6) is passive on the so-called primary constrained manifold given as $M_{3} = {z ∣ Φ = 0}$ . Notice that if joint or contact affine friction in $\dot{z}$ were considered, it would imply dissipativity. For robots in contact tasks, the hemispherical soft tip yields two interaction forces that are fundamentals for dexterity tasks. A restitution force f due to deformation arises in the normal direction while the rolling mechanism, between the object and hemispherical soft tip, induce λ that is not a dissipative force. The joint work of these forces is essential for complex applications in safe robotics, impact control and physical human-robot interaction.

P3: Equilibrium of force. Notice that the monotonously increasing f = kΔx ² implies that given a desired constant force f _d > 0 corresponds necessarily to $f_{d} (Δ x_{d}) = k Δ x_{d}^{2}$ and δx = x - x _d = Δx - Δx _d, then if force f (Δx) → f _d (Δx _d) for Δx → Δx _d, implying that x → x _d. The control of this relationship, between force and deformation, is basic to obtain to manipulate a rigid constrained dynamic circular object without force sensing through measurement or deformation estimation, however so far it is unclear how to compute f _d when for an unmeasurable θ. In the next section, we propose to exploit the ESN to solve it.

3 Echo-State Networks

The ESN is a type of ARNN that consist of: (i) an artificial recurrent neural network with random and not trainable weights, that acts as a reservoir, on which external excitations impinge and produce a broad class of dynamical echoes, and (ii) the output layer produces a linear combination of those echoes. This architecture can be seen as a combination of a large and fixed ARNN, and a simple-to-train linear output stage, which allows to produce a broad and rich dynamics while keeping the training requirements to a minimum [52, 53]. A typical architecture of ESN is composed of three layers: the input layer and hidden layer called a dynamical reservoir, and output layer known as adaptive readout network, see Fig. 3. As is pointed in [6], let us assume that reservoir of the ESN is based on continuous-time dynamics of leaky-integrator, that is $\dot{x} = \frac{1}{c} (- a x + f (W_{in} u + W_{re} x))$ (8) where u ∈ R ^m is the input to dynamic reservoir, x ∈ R ⁿ is internal reservoir state, f (*) is a nonlinear activation vector, a > 0 is the leaking rate, c > 0 is a time constant global to the ESN while W_in ∈ R ^n×m and W_re ∈ R ^n×n depicts the weight input matrix and the internal connection weight matrix, respectively. Finally, the readout network, is made up by the states of reservoir and the adaptive weights, that is $y = g (W_{out}^{T} x)$ (9) where y ∈ R ^l is the ESN output vector, g (*) is linear or non-linear activation function, and W_out ∈ R ^n×l is the time-varying weights matrix. To difference other approaches where the training of memoryless readout network is based on Mean Square Error or Recursive Least Square, principally, [54 –58]; in this paper an on-line adaptation is proposed using sliding mode control to guarantee the convergence of learning error to zero in finite time, [21, 59].

Fig.3

General scheme of an echo state network (ESN).

3.1 Adaptive readout network based on sliding mode learning

Motivated by the seminal work of [32], we propose that an adaptive readout network is an adaline-based neural network that preserves the stability of an universal approximator of smooth functions in finite time. For adaptive readout network is assumed, without loss of generality, that input and output signal vectors are bounded time-varying, i.e. ∥x ∥ ₂ < V _x and ∥y ∥ ₂ < V _y where V _x and V _y are positive constants. Also, we will assume that the derivatives of its inputs and outputs are bounded according to $∥ \dot{x} ∥_{2} < V_{\dot{x}}$ and $∥ \dot{y} ∥_{2} < V_{\dot{y}}$ where $V_{\dot{x}}$ and $V_{\dot{y}}$ are positive constants. Now, in order to train the neural network we will define the learning error as: $e (t) = y - y_{d}$ (10) where $y = g (W_{out}^{T} x)$ with g (*) a linear activation function for l = 1 while y_d stands for the reference bounded time-varying scalar signal, that is, ∥y_d (t) ∥ ₂ < V _yd, $∥ {\dot{y}}_{d} (t) ∥_{2} < V_{\dot{y} d}$ with V _yd and $V_{\dot{y} d}$ positive constants. Notice that (10) plays the role of cost function that indicates if the desired function is learned, if e = 0 the learning process has been achieved.

Using the continuous-time sliding mode control concepts [21, 59], we propose an extended learning error by adding a time-varying sliding surface that integrates the sign of the error over time, and introduces stronger error corrections that counterweight the chattering noise. That is, time-varying sliding surface is defined as $Q = e (t) + k_{N} \int_{0}^{t} sign (e (δ)) d δ$ (11) with $k_{N} \in R^{+}$ is a positive scalar that modulates the integral effect and $sign (e (t)) = {\begin{matrix} + 1 & if e (t) > 0 \\ 0 & if e (t) = 0 \\ - 1 & if e (t) < 0 \end{matrix}$ (12) For simplicity in the rest of the paper we assume that e (t) = e, and y_d (t) = y_d.

Definition [31]: A sliding motion is said to exist on a sliding surface σ = e = 0, after t _h, if the condition $σ \dot{σ} < 0$ is satisfied for all t in some nontrivial semi-open subinterval of time of the form [t, t _h) ⊂ (- ∞ , t _h) .

All the previous definitions allow us to state the following proposition:

Proposition 1: Second Order Sliding Mode-based Learning. Consider that adaptive parameters of the readout network, W_out, are updated according to ${\dot{W}}_{out} = - (\frac{x}{x^{T} x}) (k_{1} Q)$ (13) where k ₁ is a positive constant feedback gain. Then, for any initial condition e (0), the learning error e converges to zero in finite time t _h once the sliding mode condition is satisfied. Thus, the error is sustained over the sliding surface for all t > t _h

Proof. Part 1. Consider a Lyapunov candidate function given by $V_{ESN} = \frac{1}{2} Q^{T} Q$ (14) Taking the derivative of (14), we have that

$\begin{matrix} {\dot{V}}_{ESN} = Q^{T} \dot{Q} = Q^{T} [\dot{e} + k_{N} sign e] \\ = - Q^{T} k_{1} Q + Q^{T} (W_{out}^{T} \dot{x} - {\dot{y}}_{d} + k_{N} sign e) \\ \leq - k_{1} | | Q | | + (V_{W} V_{\dot{x}} + V_{\dot{y}} + V_{kN}) ∥ Q ∥ \end{matrix}$ (15) In this way, there exists a large enough feedback gain k ₁ such that Q → ∈ ₀ where ∈ ₀ is a hyperball centered in Q = 0 with radii r ₀ > 0. That is, Q → ∈ ₀ as t→ ∞. Thus, Q is upper bounded by a constant ∈ ₀. Since $\dot{x}$ , $\dot{y}$ , and ${\dot{y}}_{d}$ are bounded and ${\dot{W}}_{out}$ is a function of x, and Q bounded, too; we have that $\dot{Q}$ is also bounded. This boundedness in the L ₂ sense, leads to the existence of constants ∈ ₀ > 0, ∈ ₁ > 0 such that ∥Q ∥ ≤ ∈ ₀ and $∥ \dot{Q} ∥ \leq \in_{1}$ . This does not yet proves that learning process has been achieved. To prove that e converges, we need to prove the existence of sliding mode condition $e^{T} \dot{e} \leq η | e |$ , for some positive scalar η.

Part 2. The time derivative of (11) produces $\dot{Q} = \dot{e} + k_{N} sign e$ . Then, we have that $\begin{matrix} e^{T} \dot{e} & = & e^{T} [- k_{N} sign e + \dot{Q}] \\ \leq & - k_{N} | e | + ɛ_{1} | e | \\ \leq & - η | e | \forall e \neq 0, \end{matrix}$ where η = k _N - ∈ ₁ with $∥ \dot{Q} ∥ \leq ɛ_{1}$ . Thus, if η > 0 is guaranteed the sliding mode existence at time $t_{h} \leq \frac{| e (0) |}{η}$ . Now we are in conditions to present the controller design to achieve the force and orientation control of the circular object.

4 Controller synthesis

4.1 Force-orientation controller

We aim at controlling the orientation of the circular object, under the assumption that object angle is not available. To that end, let the following control law be, see Fig. 5, $τ = - K_{d} \dot{q} + g (q) + J_{z}^{T} f_{d} - J_{s} λ_{d}$ (16) where K _d is a positive feedback gain, f _d > 0 stands for the desired force, and λ _d represents the desired tangential force. To proceed, consider the following:

i) orientation error Δθ = θ - θ _d, where θ _d is the desired object orientation and θ the object angle, which clearly depends on the applied tangential force λ, then it seems reasonable to argue that the desired tangential force be $λ_{d} = Δ \hat{θ} = \hat{θ} - θ_{d}$ , where $\hat{θ}$ will be estimated online by ESN, since θ is not available. The rationale follows: for a smaller or larger orientation error, a smaller or larger rolling motion is needed to reduce or augment this error, while the object is motionless when rolling motion is zero (that is, when λ = 0);

ii) $Δ \hat{θ}$ means the total expected object orientation which leads from $\hat{θ} (0) = {tan}^{- 1} (\frac{y_{b} (0)}{x_{b} (0)})$ where x _b (0) and y _b (0) represent the initial center position of the base of the deformable fingertip, which are known. Thus, object angle θ is not necessary for the proposed approach only but only for modeling purposes.

Fig.4

Adaptive readout network for the output-i, i = 1 … l.

Fig.5

Block diagram of the closed-loop system.

Proposition 2. Consider the dynamical model of the robotic finger with a soft tip in contact with circular object (6), in closed loop with control law (16) and the adaptive law (13). For small errors on initial conditions, closed-loop system trajectories converge to the basic manifold $M_{1} = {(z, \dot{z} = 0) ∣ Φ = 0, Δ f = 0, Δ λ = 0}$ . Thus, the local asymptotic convergence of Δf → 0, and Δλ → 0 ⇒Δθ → 0 is guaranteed.

Proof. Substituting (16) into (6), we obtain the closed-loop error equation $H \ddot{z} + C \dot{z} + B Λ_{cl} = F + F_{0}$ (17) where Λ _cl = [f - f _d, λ - λ _d] ^T= [Δf, Δλ] ^T ∈ R ³ stand for the force error vector, $F = [- K_{v} \dot{q}, - R λ_{d}]^{T}$ , and F₀ will be useful only for analysis purposes. Using property P1 for input F₀ = 0 and output $\dot{z}$ , the power balance yields $〈 F_{0}, \dot{z} 〉 = {\dot{V}}_{1} (z, Δ f) + {\dot{q}}^{T} K_{v} \dot{q} + \dot{θ} R λ_{d}$ , that is, ${\dot{V}}_{1} (z, Δ f) = - {\dot{q}}^{T} K_{v} \dot{q} - \dot{θ} R λ_{d}$ (18) in virtue that $\dot{Φ} = 0$ . In this way, the closed-loop storage energy function is defined as V ₁ (z, Δf) = K + P _Δf > 0 with $P_{Δ f} = \int_{0}^{δ x} {f (Δ x_{d} + ɛ) - f (Δ x_{d})} d ɛ > 0$ . However, V ₁ (z, Δf) does not qualify as a Lyapunov function in the 8-dimensional space $(z, \dot{z})$ , hence stability in the sense of Lyapunov cannot be concluded but certainty V ₁ (z, Δf) is useful to analyze the convergence properties of the autonomous system (17). Applying the maximum invariance set to (18) for origin z = 0, it leads to $\dot{z} = (\dot{q}, \dot{θ}) = (0, 0)$ as an isolated annihilator, that is ${\dot{V}}_{1} (z) = 0$ , implying that $\ddot{z} = 0$ . Thus, there exists a maximum connected set containing $(z, \dot{z}, \ddot{z}) = (0, 0, 0)$ . That is, (17) becomes $B Λ_{cl} = 0$ (19) which stands for the residual error dynamics. Now, let z (t ₀) ∈ Ω ₀ and z_d ∈ Ω ₀ with $Ω_{0} = {z | Δ x = \sqrt{\frac{f}{k}}, z \in M_{3}}$ a non-empty connected set, and B a non-degenerate matrix which means that matrix J (q) is full column rank which is true when q _i ≠ aπ for a ∈ Z, with i = 1, 2, 3. Then, for small errors on initial conditions there exists a ball B (z_d, η) with η > 0, such that the trajectories of the closed loop system remain within $B (z_{d}, η) \cap M_{1}$ for all t ∈ [0, + ∞ [. Thus, a locally asymptotically convergence is guaranteed such that Λ _cl → 0implies (Δf, Δλ) → (0, 0).

Part II. Convergence of Δθ. So far, the convergence of force errors have been proved, now let us prove it for orientation error. Consider the sliding surface defined as $Q_{Δ θ} = Δ θ + k_{N} \int_{0}^{t} sign (Δ θ) dt$ , using Proposition 1 it is easy to show the sliding mode condition for Δθ in finite time t _h, thus, Δθ remains on Q _Δθ such that $λ_{d} = Δ θ \to 0 \hat{θ} \to θ_{d}$ , which proves this Proposition.

Table 1

Physical parameters of the robot

(l₁, l₂, l₃)	(0.05, 0.04, 0.03) [m]
(l_c1, l_c2, l_c3)	(0.025, 0.02, 0.015) [m]
(m₁, m₂, m₃)	(0.05, 0.03, 0.02) [kg]
(I₁₁, I₂₂, I₃₃)	(14.167, 6.25, 3) ×10^-6 [kgm²]

5 Simulation Results

Consider a circular object pivoting at its axis, see Fig. 1 whose task is convergece to a constant θ _d. A simulator is programmed using a stiff numerical solver with a Constrained Stabilization Method (CSM) [60]. The physical parameters of the robot manipulator are shown in Table I, where l _i, m _i and I _i are the length, mass and the inertia moment of each link, respectively. Additionally, the mass, moment of inertia and radius of the object are defined as M = 0.1 [Kg], I = 5.67 × 10^-6 [kgm ²] and R = 0.03 [m], respectively. Finally, k = 500 [N/m ²] and r = 0.02 [m] is the stiffness of the material and the radius of the soft fingertip, respectively.

5.1 Simulation conditions

The task goal is to roll the constrained circular object to a desired angle θ _d by applying a desired force f _d with a 3 degrees of freedom robotic fingertip. Consider the soft-fingertip in stable contact to the object, that is, Δx (0) >0impliesf (0) >0 and consistent as a DAE-2 formulation at t = t ₀. Finger state is measurable but not object angle, which is estimated as $\hat{θ} (0) = {tan}^{- 1} (y_{b} (0) / x_{b} (0))$ . The input vector and the internal reservoir state are defined as $u = [q, \dot{q}]^{T} \in R^{8}$ and $x \in R^{30}$ , respectively. The values of weight input matrix and internal connection matrix are defined randomly between [-1, 1]. According to (11), the sliding surface used in this experiment is defined as $Q = Δ θ + k_{N} \int_{0}^{t} sign (Δ θ) dt$ where $Δ θ = \hat{θ} - θ_{d}$ and $\hat{θ}$ is the output of ESN. Finally, the update rule of W_out is defined by (13).

The robot start motionless, with an initial angular position of q (0) = [1.2, - 0.6, - 0.6] ^T. Two sets of simulations are presented:

Simulation 1: θ _d = -4.87 [rad], f _d = 2 [N], and k _N = 0.1

Simulation 2: θ _d = 4.8744 [rad], f _d = 1.5 [N], diag (K _d) =2.0 * I _3×3 k ₁ = 1.7 and k _N = 0.1

Simulation 1: it is shows that contact force reaches the desired force in approximately 6 [s], Fig. 6. At the same time, in Fig. 7 is shown the tangential force behavior which apparently converge to zero rapidly. However, zooming in on the plot between 5.5 - 8.3 [s], we can notice that although the λ is small and very close to zero, the convergence to zero is reached in 8 [s] approximately. This means that λ trajectory is very close to the sliding surface, however, until the convergence of the contact force is guaranteed, the sliding condition is satisfied and Δθ → 0. This can be verified in Fig. 8 where the convergence of Δθ is reached in approximately 6 [s] while the object angle reaches the value of -0.04 [rad]. In Fig. 9 is shown how are driving the readout network weights by the sliding mode control, that is, once the sliding mode condition is satisfied the weights remain bounded on the sliding surface. Finally, in Fig. 10 is shown the rolling motion contribution to move the object through tangential force, changing position of x _b and y _b, while the contact force guarantees to grasp firmly the object during the task.

Fig.6

Radial deformation of the soft tip and contact force convergence to the desired value.

Fig.7

Tangential force performance.

Fig.8

Convergence of pose error Δθ, in intrisic relationship to desired tangential force.

Fig.9

The internal reservoir states and weights of the readout network.

Fig.10

Simulation 1. Trajectories of the center position of the deformable soft fingertip during the task.

Simulation 2: See Figures 11 –14. A little bit deformation with a smooth convergence to the desired force is observed in Fig. 11. As in previous simulation, the tangential force tends to the zero rapidly, however the sliding surface is reached in 5 [s], Fig. 12. At the same time that sliding mode condition is satisfied, the convergence of Δθ is guaranteed, see Fig. 12. Finally, the performance of the internal weights and the readout network is presented in Fig. 14.

Fig.11

Simulation 2. Deformation of the soft tip and the contact force.

Fig.12

Simulation 2. Performance of the tangential force.

Fig.13

Simulation 2. Smooth convergence to the desired object angle, θ _d, once the sliding mode condition is satisfied.

Fig.14

Simulation 2. The internal reservoir states and weights of the readout network.

Comparative Performance: Figures 15 –17 shows the comparison of the proposed approach with the traditional learning algorithm gradient descent. The convergence of Δθ for both algorithms is shown in Fig. 15. As we can see, the performance of pose error with the proposed approach is more smooth and converge in a finite time. To show that once the learning error reaches the sliding surface Q = 0 remain in it, Fig. 16 show how λ → λ _d = Δθ as is expected by the proposed algorithm while by gradient descent the convergence to zero is not guaranteed. And last but not least, in Fig. 17 we can notice the bounded-weights in adaptive readout network as it is expected by the proof, part 1, of Proposition 1. On the other hand, since gradient descent algorithm cannot guaranteed the boundedness of weights globally, we can see that there appears a undesirable dead zone in the beginning signaling the well-known problems of local minima.

Fig.15

Comparative simulation: Performance of the pose error, Δθ.

Fig.16

Comparative convergence: Tangential force convergence (zooming of last seconds is shown).

Fig.17

Comparative performance: Weights in the adaptive readout network.

6 Conclusions

This work presents the key contribution of ESN for angle manipulation of circular object through a robotic finger with hemispherical soft tip subject to unmeasurable object angle measurement. Stability conditions guarantee the local convergence of the closed-loop system based on online adaptive learning of the ESN weights. Its structure and proposed learning, based on sliding mode control, has been studied and highlighted the usefulness in this complex robotic problem by exploiting precise, smooth and fast online weight adaptation to estimate the intrinsic relationship of the nonlinear under-actuated robotic task.

Footnotes

for the 3D case, an ellipsoid area is conformed at contact, which is useful for a less conservative motion planning because there arises an enlarged convex hull at contact, []

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