Modeling and control of planar slippage in object manipulation using robotic soft fingers

Abstract

Slippage occurrence has an important roll in stable and robust object grasping and manipulation. However, in majority of prior research on soft finger manipulation, presence of the slippage between fingers and objects has been ignored. In this paper which is a continuation of our prior work, a revised and more general method for dynamic modeling of planar slippage is presented using the concept of friction limit surface. Friction limit surface is utilized to relate contact sliding motions to contact frictional force and moment in a planar contact. In this method, different states of planar contact are replaced with a second-order differential equation. As an example of the proposed method application, dynamic modeling and slippage analysis of object manipulation on a horizontal plane using a three-link soft finger is studied. Then, a controller is designed to reduce and remove the undesired slippage which occurs between the soft finger and object and simultaneously move the object on a predefined desired path. Numerical simulations reveal the acceptable performance of the proposed method and the designed controller.

Introduction

Human hands are one of the most elaborate organs of the human body from dexterity point of view. They can explore, grasp, and manipulate objects of different shapes and materials. Thus, study on design and development of anthropomorphic and dexterous robotic hands is one of the interesting subjects for research.

Contact modeling is the preliminary step in analysis of object grasping and manipulation. Generally, there are two kinds of contact models; point contact and planar contact. When contact is assumed to be a point or deformation in the contact is negligible, this model can be used. The majority of prior research on grasping and manipulation are based on point contact assumption. Coulomb model is usually used for modeling frictional force in the point contact model. For an extensive literature review on grasping and manipulation analysis based on point contact assumption refer to [1]. However, when the contact area is relatively large or deformation in the contact is significant, e.g., soft contact, the point contact model is not applicable. In planar contact, a frictional moment is exerted at the contact interface along with the normal force and tangential frictional force (Fig. 1). This causes a fewer number of contact interfaces to robustly and stably grasp and manipulate an object using soft fingers compared with rigid fingers. Furthermore, the soft fingertips can conform to the objects’ uneven surfaces and also damp shocks and vibrations to enhance the stability and robustness of grasp. In the planar contact model, friction limit surface is used to relate contact sliding motions to contact frictional force and moment. Goyal et al. [2] first defined the friction limit surface for slippage of a rigid object on a planar surface. Howe and Cutkosky [3] developed a relationship between forces and motions in sliding manipulation. They approximated the friction limit surface to an ellipsoid to simplify control, planning, and simulation of manipulation.

Fig. 1

Normal force, frictional tangential force, and frictional moment in the contact interface of human soft finger

Studies on the robotic hands with soft fingers can be considered in three major groups; soft finger design, soft contact modeling, and soft finger manipulation. The first research group is focused on designing optimized soft fingers [4] and also proper tactile sensors [5]. The second group includes research that focuses on developing a mathematical model for soft contact. Some of the most important soft contact models are presented in [6,7,8,9,10,11,12,13,14,15]. The third group consists of the research focuses on object grasping and manipulation using soft robotic fingers. Arimoto et al. [7, 16] studied grasping and manipulation of a rigid object using soft fingers. To model the softness of fingertips, they assumed a continuous distribution of linear springs located radially within the hemispherical soft fingertip. Kim [8] considered the motion analysis of rigid object manipulation using a pair of soft fingers assuming linear springs and dampers for each contact interface. Inoue and Hirai [9, 17] developed a model for dynamic modeling and orientation control of a rigid object during soft finger grasping and manipulation. To model the softness of fingertips, they assumed a continuous distribution of linear springs positioned perpendicularly to the backplate of the soft fingertip.

Slippage often happens in the contact interface of fingers and objects and control of this slippage is essential for stable and robust grasping and manipulation. However, in the majority of prior studies in this field, it is presumed that the friction coefficients between soft fingers and objects are high enough and slippage never happens during the grasping and manipulation. Hadian et al. [18] studied analysis and control of slippage in object manipulation using a planar rigid-tip finger. Song et al. [19] developed a novel method for prediction and compensation of dynamic slip which occurs between an object and a pair of fingers with hemispherical rigid tips. Engeberg and Meek [20] designed a controller for prosthetic hands to simultaneously prevent slip and minimize the contact force of grasped objects. Kao and Cutkosky [21] modeled the quasi-static sliding manipulation using friction limit surface, however, they did not consider the dynamics of the soft tip. Xu et al. [22] demonstrated a friction model for the contact interface between a soft object and a pair of soft parallel gripper jaws using the Finite Element Method. Ozawa and Tahara [23] investigated past studies on grasping and manipulation of multi-fingered robotic hands from a control viewpoint. Fakhari et al. [24] studied the linear slippage which occurs between a planar soft finger and an object during manipulation. However, analysis and control of planar slippage occurring during manipulating an object using a spatial soft finger has not been considered yet.

Therefore, in this paper which is a continuation of our prior work [25], a revised and more general method for dynamic modeling of all different states of planar slippage is presented using the concept of friction limit surface. In this method, different states of planar contact (i.e., stationary, incipient slip, and slippage) are replaced with a second-order differential equation in which its coefficients are determined based on a table. As an example of the proposed method application, object manipulation using a three-link soft finger is studied first. Then, a controller is designed to reduce and remove the undesired slippage occurring between the soft finger and object and simultaneously move the object on a predefined path despite the slippage occurrence.

Slippage analysis in planar contact

Contact forces and moment

Coulomb friction is one of the basic and commonly used dry friction models that determines the tangential frictional force, ${{\mathbf{f}}_t}$, between two rigid objects in point contact as $\left| {{\mathbf{f}}_t} \right| \le \mu N$, where $\mu$ is the friction coefficient between two objects and N is the normal force. However, when two objects come into planar contact (Fig. 2), the friction cone is replaced with a friction limit surface [2, 3]. Friction limit surface relates contact frictional force/moment to contact sliding motions and determines when there is a planar slippage between the objects. By calculating the contact frictional force and moment for all possible translational and rotational slippage that can occur in the contact interface, the friction limit surface is constructed [3, 25]. In Fig. 3, cross-section view of friction limit surface for a rectangular contact interface when pressure distribution is assumed to be uniform is shown as a sample.

Fig. 2

Contact forces and moment of two objects in planar slippage

Fig. 3

Cross section view of friction limit surface for a rectangular contact interface

When the contact frictional forces, (i.e., $f_x$ and $f_y$), and moment, (i.e., $m_z$), as shown in Fig. 2, are placed inside the friction limit surface boundary, there will be no slippage between two objects. By increasing these forces and moment towards the boundary, translational and rotational slippage between two objects begins. This slippage is indeed parallel with the unit vector, $\hat{\mathbf{n}}$, normal to the surface at the point ($f_x$, $f_y$, $m_z$) as shown in Fig. 4. Note that, in pure translational slippage where $m_z=0$, the locus of the contact frictional forces is a circle in the $f_x-f_y$ plane and its radius is $(f_t)_{\text {max}}=\mu N$.

It is proven that an acceptable approximation for friction limit surface, in general, is an ellipsoid [3]. This ellipsoid fits the maximum frictional force, $(f_t)_{\text {max}}$, and also maximum frictional moment, $(m_z)_{\text {max}}$, as shown in Fig. 4. This simplified model is still relatively accurate and it is proper for dynamic modeling and control of object grasping and manipulation. Moreover, it is shown that the skewness in the pressure distribution at the contact interface of a soft hemispherical fingertip due to tangential forces does not have a significant effect on the shape of the friction limit surface [26]. Therefore, the ellipsoidal approximation of friction limit surface (Fig. 4) can be formulated as

$$\begin{aligned} \dfrac{{f_x}^2}{(f_t)_{\text {max}}^2}+\dfrac{{f_y}^2}{(f_t)_{\text {max}}^2}+\dfrac{{m_z}^2}{(m_z)_{\text {max}}^2}=1. \end{aligned}$$

(1)

Fig. 4

Ellipsoidal approximation of friction limit surface

Since the planar slippage in the contact interface of two objects is parallel with the unit vector, $\hat{\mathbf{n}}$, normal to the friction limit surface at the point ($f_x$, $f_y$, $m_z$), the relationship between the translational velocities of sliding object along x and y axes, $(\dot{x},\dot{y})$, angular velocity of sliding object along z axis, $\dot{\theta }$, tangential frictional force, $f_t$, and frictional moment, $m_z$, for ellipsoidal approximation of friction limit surface is

$$\begin{aligned} \frac{{{\sqrt{{\dot{x}}^2+{\dot{y}}^2}}}}{{\dot{\theta } }} = {\lambda ^2}\frac{{{f_t}}}{{{m_z}}}, \end{aligned}$$

(2)

where $\lambda$ is defined as

$$\begin{aligned} {\lambda } = \frac{(m_z)_{\text {max}}}{(f_t)_{\text {max}}}. \end{aligned}$$

(3)

Slippage modeling of planar contact

Based on the features of ellipsoidal approximation of the friction limit surface (i.e., Eqs. 1 and 2) in the slippage state of a planar contact, the relationship between the contact frictional forces, contact frictional moment, linear sliding velocities, and angular sliding velocity is proposed as

$$\begin{aligned} {\begin{bmatrix} {{f_x}}\\ {{f_y}}\\ {{{{m_z}} /\lambda }} \end{bmatrix}}= \frac{{-\mu N}}{{\sqrt{{{\dot{x}}^2} + {{\dot{y}}^2} + {{\left( {\lambda \dot{\theta } } \right) }^2}} }}{\begin{bmatrix} {\dot{x}}\\ {\dot{y}}\\ {\lambda \dot{\theta }} \end{bmatrix}}. \end{aligned}$$

(4)

Thus, the different states of frictional forces and moment between two objects in planar contact can be represented as

$$\begin{aligned} \begin{array}{l r} {f_x}^2 + {f_y}^2 + \dfrac{{{m_z}^2}}{{{\lambda ^2}}} < {\left( {\mu N} \right) ^2} &{} \text {(Stationary)}\\ {f_x}^2 + {f_y}^2 + \dfrac{{{m_z}^2}}{{{\lambda ^2}}} = {\left( {\mu N} \right) ^2} &{} \text {(Incipient Slip)}\\ {\begin{bmatrix} {f_x} \\ {f_y} \\ {m_z} \end{bmatrix}} = \dfrac{{ - \mu N}}{{\sqrt{{{\dot{x}}^2} + {{\dot{y}}^2} + {{\left( {\lambda \dot{\theta } } \right) }^2}} }}{\begin{bmatrix} {\dot{x}} \\ {\dot{y}} \\ {{\lambda ^2}\dot{\theta } } \end{bmatrix}} &{} \text {(Slippage)} \end{array} \end{aligned}$$

(5)

Since in incipient slip state the object is stationary and the contact frictional forces and moment are placed on the friction limit surface boundary, the second relation of Eq. 5 can also be written as

$$\begin{aligned} {\begin{bmatrix} {{f_x}} \\ {{f_y}} \\ {{m_z}} \end{bmatrix}} = \dfrac{{\mu N}}{{\sqrt{{{\bar{f}}_x}^2 + {{\bar{f}}_y}^2 + {{\left( {{{\bar{m}}_z}/\lambda } \right) }^2}} }}{\begin{bmatrix} {{\bar{f}}_x} \\ {{\bar{f}}_y} \\ {{\bar{m}}_z} \end{bmatrix}}, \end{aligned}$$

(6)

where $\bar{f}_x$, $\bar{f}_y$, and $\bar{m}_z$ are the contact forces and moment when ${\dot{\mathbf{X}} }=[\dot{x},\dot{y},\dot{\theta }]^\text {T}=\mathbf{0}$ (i.e., when the contact is assumed to be stationary). Equations 5 and 6 can be combined and rewritten as a single second-order differential equation to model the different states of frictional forces and moment in the planar contact as

$$\begin{aligned} {\beta _1}\ddot{\mathbf{X} } + {\begin{bmatrix} {{\beta _{21}}{{\mathbf {I}}_3}}&\quad{{{{\varvec{\upbeta }}}_{22}}} \end{bmatrix}}{\mathbf{F} } = {\mathbf {0}}, \end{aligned}$$

(7)

where ${\mathbf{F} } = [f_x, f_y, m_z, N]^ \text {T}$, ${\mathbf{I}}_3$ is a $3 \times 3$ identity matrix, and parameters $\beta _1$, $\beta _{21}$, and ${{\varvec{\upbeta }}}_{22}$ can be determined using Table 1 based on the different states of the contact.

Table 1 Parameters $\beta _1$, $\beta _{21}$, and ${{\varvec{\upbeta }}}_{22}$ in different states of planar contact

Manipulating a rigid object using a three-link soft finger

In this section, manipulation of a rigid object on a horizontal surface using a three-link soft finger is studied as an example of application of the method proposed in the previous section. Then, in the next section, a controller is designed to reduce the undesired slippage which occurs during the manipulation of the object. In Fig. 5, top and side views of a soft finger manipulating an object are shown. The soft finger consists of three rigid links and a soft hemisphere attached to the end of the last link to move the object on a horizontal plane. In order to accurately model dynamic equations of the soft finger, dynamics of the soft fingertip is integrated into dynamics of finger linkage.

Fig. 5

Object manipulation using a soft finger, a side view, b top view

Power-law model is one of the time-independent non-linear elastic contact models that has been proposed for deformation in the contact between a soft hemisphere and a rigid plate as shown in Fig. 6. This model has been theoretically and experimentally validated for different types of soft materials [6, 27] and is presented by

$$\begin{aligned} r=cN^\gamma , \end{aligned}$$

(8)

where r is the radius of contact area, N is contact normal force, and c and $\gamma$ are constants which depend on the geometry and material property of the soft fingertip. Due to the accuracy of this model, it is utilized for modeling the elastic behavior of the fingertip in this research. Furthermore, as it is also assumed in the previous research [8, 9, 28], the damping behavior of the hemispherical tip is model by a linear viscous damper, i.e., $C_{\text {eq}}$ in Fig. 7b.

Fig. 6

Contact Interface of a hemispherical soft finger under a normal force

In order to drive the dynamic equations of the soft finger, Lagrange method is used. Since this system includes both viscous damper and external forces, Lagrange equation can be written as

$$\begin{aligned} \frac{\text {d}}{{\text {d}t}}\left( {\frac{{\partial L}}{{\partial {{\dot{q}}_i}}}} \right) - \frac{{\partial L}}{{\partial {q_i}}} + \frac{{\partial F_{\text {st}}}}{{\partial {{\dot{q}}_i}}} = {Q_{i,{\text {nc}}}}, \quad i=1,2,3,4, \end{aligned}$$

(9)

where

$$\begin{aligned} \left\{ {\begin{array}{l} L=T-V: {\text {Lagrangian}}\\ T: {\text {Total Kinetic Energy}}\\ V: {\text {Total Potential Energy}}\\ F_\text {st}=C_\text {eq}{{\dot{d}}^2}/2: \text {Rayleigh Dissipation Function}\\ Q_{i,\text {nc}}: \text {Generalized Force}\\ q_i: \text {Generalized Coordinate} \end{array}} \right. . \end{aligned}$$

(10)

Since dynamics of the soft tip is integrated with dynamics of finger linkage, the vector of soft finger generalized coordinates is defined as ${\mathbf{q}} = \left[ {{q_1}, {q_2}, {q_3}, d} \right] ^\text {T}$, where $q_1$, $q_2$, and $q_3$ are the joint angles and d is the deformation of soft tip (Fig. 5).

We define three sliding states $x_s$, $y_s$, and $\theta _s$ as

$$\begin{aligned} \begin{array}{l} {{\dot{x}}_s} = V_c^x - {{\dot{x}}_o}\\ {{\dot{y}}_s} = V_c^y - {{\dot{y}}_o}\\ {{\dot{\theta } }_s} = \omega _c^z - {{\dot{\theta } }_o} \end{array}. \end{aligned}$$

(11)

In these equations, $V_c^x$ and $V_c^y$ are the linear velocities of point c (Fig. 5), along x and y axes respectively, $\omega _c^z$ is the angular velocity of the contact interface about z axis, $x_o$ and $y_o$ represent coordinate of the object center of mass, and $\theta _o$ is the rotation of the object (Fig. 5).

Fig. 7

a Contact forces and moment applied on the object (side and top view), b soft tip model and the contact forces and moment between the soft finger and the object

Hence, the dynamic equations of the soft finger manipulating a rigid object can be derived as

$$\begin{aligned}&{\mathbf{M}}\left( {\mathbf{q}} \right) \ddot{\mathbf{q}} + {\mathbf{h}}\left( {{\mathbf{q}},{\dot{\mathbf{q}}}} \right) = \mathbf{B}{\varvec{\uptau }} + {\mathbf{J}}{\left( {\mathbf{q}} \right) ^\text {T}}{{\mathbf{F}}_c}\\&{{\mathbf{M}}_o}{{\ddot{\mathbf{q}}}_o} + {{\mathbf{h}}_o} = {{\mathbf{F}}_g} - {{\mathbf{B}}_c}{{\mathbf{F}}_c}\\&{\mathbf{J}}\left( {\mathbf{q}} \right) \ddot{\mathbf{q}} - {\mathbf{B}}\left( {{{\ddot{\mathbf{q}}}_o} + {{\ddot{\mathbf{q}}}_s}} \right) = - {\dot{\mathbf{J}}}\left( {{\mathbf{q}},{\dot{\mathbf{q}}}} \right) {\dot{\mathbf{q}}}\\&{\beta _{s,1}}{{\ddot{\mathbf{q}}}_s} + {{\mathbf{B}}_s}\left( {{{{\dot{\mathbf{q}}}}_s}} \right) {{\mathbf{F}}_c} = {\mathbf{0}}\\&{\beta _{o,1}}{{\ddot{\mathbf{q}}}_o} + {{\mathbf{B}}_o}\left( {{{{\dot{\mathbf{q}}}}_o}} \right) {{\mathbf{F}}_g} = {\mathbf{0}} \end{aligned},$$

(12)

where $\mathbf{F}_c = \left[ f_{c,x},f_{c,y},M_c,N_c \right] ^\text {T}$ represents the contact forces and moment between the soft tip and the object and $\mathbf{F}_g = \left[ f_{g,x},f_{g,y},M_g,N_g \right] ^\text {T}$ represents the contact forces and moment between the object and the ground as shown in Fig. 7, respectively. Note that the first relation is the soft finger dynamics, the second relation is the object dynamics, the third relation represents the contact constraint and also sliding states (Eq. 11). Two last relations demonstrate the dynamics of frictional forces and moments in the both contact interfaces, i.e., between soft tip and the object and also between the object and ground, derived using the proposed method in the previous section (Eq. 7). In Eq. 12, $\mathbf{M}_{\text{4}} \times 4$ is the soft finger inertia matrix, $\mathbf{h}_4 \times 1$ is vector of centrifugal, Coriolis, and gravity terms of the soft finger, ${{\varvec{\uptau }}} = \left[ \tau _1,\tau _2,\tau _3 \right] ^\text {T}$ is vector of soft finger joint torques (Fig. 5), $\mathbf{B}_4 \times 3$ is a linear mapping between dynamics of the finger linkage (including the soft tip) and the finger joint torques, ${\mathbf{B}}_{c, {4 \times 4}}$ is a matrix used to include moments of the forces $f_{c,x}$ and $f_{c,y}$ about the object center of mass in the object dynamics, $\mathbf{J}_ {4 \times 4}$ is soft finger Jacobian matrix, ${\mathbf{M}}_{o,{4 \times 3}}$ is object inertial matrix, ${\mathbf{h}}_{o, {4 \times 1}}$ is object gravity vector, ${\mathbf{q}}_o = \left[ {{x_o},{y_o},{\theta _o}} \right] ^\text {T}$ is a vector that represents position of the object center of gravity and orientation of the object about its center of gravity, ${\mathbf{q}}_s = \left[ {{x_s},{y_s},{\theta _s}} \right] ^\text {T}$ is vector of sliding states, ${{{\mathbf{B}}_s} = \left[ {{\beta _{s,21}}{{\mathbf{I}}_{3}},{{{\varvec{\upbeta }}}_{s,22}}} \right] }$, and ${{{\mathbf{B}}_o} = \left[ {{\beta _{o,21}}{{\mathbf{I}}_{3}},{{{\varvec{\upbeta }}}_{o,22}}} \right] }$. The parameters $\beta _{s,1}$, $\beta _{s,21}$, ${{{\varvec{\upbeta }}}_{s,22}}$, $\beta _{o,1}$, $\beta _{o,21}$, and ${{{\varvec{\upbeta }}}_{o,22}}$ can be determined using Table 1 based on the different states of each contact and knowing the friction coefficient between soft tip and object, $\mu _c$, and between object and ground, $\mu _g$. Moreover, parameter $\lambda$ (Eq. 3) which appears in ${{{\varvec{\upbeta }}}_{s,22}}$ as $\lambda _c$ for a circular contact interface and in ${{{\varvec{\upbeta }}}_{o,22}}$ as ${\lambda _g}$ for a rectangular contact interface is derived as

$$\begin{aligned} \begin{aligned} \lambda _c&=\dfrac{3}{4}\dfrac{{\Gamma (3/k)}^2}{\Gamma (2/k)\Gamma (4/k)}r\\ {\lambda _g}&= \dfrac{1}{{ab}}\int _{ - b/2}^{b/2} {\int _{ - a/2}^{a/2} {\sqrt{{x^2} + {y^2}} } \text {d}x} \text {d}y \end{aligned}, \end{aligned}$$

(13)

where r is the radius of contact area, $\Gamma (\cdot )$ is the Gamma function, k depends on the shape of the pressure distribution profile in the contact interface of the soft tip [6], and a and b are length and width of the object, respectively. Vector ${{\mathbf{h}}_o}$ and matrices ${\mathbf{J}}\left( {\mathbf{q}} \right)$, ${{\mathbf{M}}_o}$, ${\mathbf{B}}$, and ${{\mathbf{B}}_c}$ are presented in Appendix.

Slippage control in planar manipulation of the object

Human hands can grasp and manipulate different objects without having knowledge about the weight of objects and the friction coefficient between the fingertips and objects. Indeed, human hands can first sense the incipient slip which occurs in the contact interface of the fingertip by detecting micro-vibrations using the tactile mechanoreceptors and then control the grasp force unconsciously to have a stable grasp and robust manipulation without damaging the objects [29].

In this section, a controller is proposed for the soft finger to move the object on a predefined desired path in a horizontal plane and simultaneously reduce and remove the slippage occurring between the soft fingertip and the object by increasing the normal contact force. The controller is designed based on hybrid position/force control concept and independent of the object parameters, i.e., object mass, $m_o$, and friction coefficient of object/ground contact, $\mu _g$.

The dynamic equation of the soft finger (i.e., the first relation of Eq. 12) can be rewritten as

$$\begin{aligned}&{\mathbf{M}_r}{\ddot{\mathbf{q}}}_r + \mathbf{h}_r = {\varvec{\uptau }} + {\mathbf{J}_c}^{\text T}{\mathbf{F}_c}\\&{N_c} = \left( R^2-d^2\right) ^{\dfrac{1}{2\gamma }}\left( \dfrac{1}{c} \right) ^{\dfrac{1}{\gamma }} - C_{eq}\dot{d} \end{aligned},$$

(14)

where ${\mathbf{M}_r}=\mathbf{M}(1{:}3,1{:}3)$ is inertia matrix of the finger linkage, ${\mathbf{h}_r}=\mathbf{h}(1{:}3)$ is vector of centrifugal, Coriolis, and gravity terms of the finger linkage, ${\mathbf{J}_c}=\mathbf{J}(1{:}4,1{:}3)$ is Jacobian matrix of soft finger contact point c, and $\mathbf{q}_r = \left[ {{q_1}, {q_2}, {q_3}} \right] ^\text {T}$ is the vector of soft finger joint angles. Note that $\mathbf{M}(1{:}3,1{:}3)$ represents a $3 \times 3$ matrix constructed by the first three rows and columns of $\mathbf{M}$, $\mathbf{h}(1{:}3)$ represents a $3 \times 1$ vector constructed by the first three components of $\mathbf{h}$, and $\mathbf{J}(1{:}4,1{:}3)$ represents a $4 \times 3$ matrix constructed by the first four rows and the first three columns of $\mathbf{J}$.

The relation between angular velocity of the soft finger joints and velocity of the center of soft finger hemisphere, $o_{tip}$, with respect to xyz-coordinate system (Fig. 5), can be written by a $3 \times 3$ Jacobian matrix, $\mathbf{J}_t$, as

$$\begin{aligned} {\dot{\mathbf{X}}}_t = {\mathbf{J}_t}{\dot{\mathbf{q}}}_r, \end{aligned}$$

(15)

where ${{\mathbf{X}}_t} = {\left[ {{x_t},{y_t},{z_t}} \right] ^{\text {T}}}$ is the position of the point $o_{\text {tip}}$. By differentiating Eq. 15, ${\ddot{\mathbf{q}}_r}$ can be calculated as

$$\begin{aligned} {\ddot{\mathbf{q}}}_r = {\mathbf{J}}_{t}^{-1}\left( {{\ddot{\mathbf{X}}}_t} - {{{\dot{\mathbf{J}}}}_t}{\dot{\mathbf{q}}}_r \right) , \end{aligned}$$

(16)

and by substituting Eq. 16 into the first relation of Eq. 14, ${{\varvec{\uptau }}}$ can be derived as

$$\begin{aligned} {{\varvec{\uptau }}} = {{\mathbf{M}}_r}{\mathbf{J}}_{t}^{-1}\left( {{{\ddot{\mathbf{X}}}_t} - {{{\dot{\mathbf{J}}}}_t}{{{\dot{\mathbf{q}}}}_r}} \right) + {{\mathbf{h}}_r} - {\mathbf{J}}_c^{\text {T}}{{\mathbf{F}}_c}. \end{aligned}$$

(17)

The corresponding feedback linearization control for Eq. 17 is given by

$$\begin{aligned} {{{\varvec{\uptau }}}_c} = {{\mathbf{M}}_r}{\mathbf{J}}_{t}^{-1}\left( {{\mathbf{a}} - {{{\dot{\mathbf{J}}}}_t}{{{\dot{\mathbf{q}}}}_r}} \right) + {{\mathbf{h}}_r} - {\mathbf{J}}_c^{\text {T}}{{\mathbf{F}}_c}, \end{aligned}$$

(18)

where ${{{\varvec{\uptau }}}_c}$ is the control torques vector of the soft finger joints and $\mathbf{a}$ is the control input vector used to represent the position and force control strategies. By substituting Eq. 18 into Eq. 17, the close-loop equation can be written as

$$\begin{aligned} {\ddot{\mathbf{X}}_t} = {\mathbf{a}}. \end{aligned}$$

(19)

This equation represents that the task space motion has been globally linearized and decoupled. Since z-axis is always perpendicular to the motion plane of the soft finger contact interface, the position and force controllers can be designed independently. That is, the position controller is designed for the task space variables which represent the tangent motion (i.e., x and y) and the force controller is designed for the task space variable which represents the normal motion (i.e., z). Therefore, vector $\mathbf{a}$ can be written as

$$\begin{aligned} {\mathbf{a}} =\begin{bmatrix} {\mathbf{a}_T}\\ {a_N} \end{bmatrix}, \end{aligned}$$

(20)

where

$$\begin{aligned} \begin{aligned} {{\mathbf{a}}_T}&= {{\ddot{\mathbf{X}}}^{\text {des}}}_{t,T} + {{\mathbf{K}}_{T,v}}{{\dot{\mathbf{e}}}_T} + {{\mathbf{K}}_{T,p}}{{\mathbf{e}}_T}\\ a_N&= {{\ddot{X}}^{\text {des}}}_{t,N} + {K_{N,v}}{\dot{e}_N} + {K_{N,p}}{e_N} \end{aligned}, \end{aligned}$$

(21)

and $\mathbf{X}^{\text {des}}_{t,T}$ is the object desired trajectory, ${{\mathbf{K}}_{T,v}}$ and ${{\mathbf{K}}_{T,p}}$ are diagonal positive definite $2 \times 2$ matrices, ${K_{N,v}}$ and ${K_{N,p}}$ are positive constant parameters, $e_N=X^{\text {des}}_{t,N}-X_{t,N}$, $X_{t,N}={z_t}$, and ${{\dot{X}}^{\text {des}}}_{t,N}={{\ddot{X}}^{\text {des}}}_{t,N}=0$. In order to move the object on a desired path even after the slippage between fingertip and object occurs, we define ${\mathbf{e}}_T$ as ${\mathbf{e}}_T=\mathbf{X}^{\text {des}}_{t,T}-\mathbf{X}_{o,T}$ where $\mathbf{X}_{o,T}=\left[ {x_o},{y_o} \right] ^{\text {T}}$.

A method for reducing and removing the slippage between the fingertip and object is increasing the normal contact force from an initial value to when the slippage decreases. This method is similar to what human hands do when manipulating an object on a horizontal plane. Hence, the desired normal contact force is proposed as

$$\begin{aligned} N_c^{\text {des}}=N_c^{\text {ini}}+K_s\Phi \left( \delta _s,t \right) , \end{aligned}$$

(22)

where $N_c^{\text {ini}}$ is the initial normal contact force, $K_s$ is a constant positive parameter, $\delta _s$ is a slippage parameter defined as

$$\begin{aligned} \delta _s=\left( {{\dot{x}}_s}^2+{{\dot{y}}_s}^2+\lambda _c^2{{\dot{\theta }}_s}^2 \right) ^{1/2}, \end{aligned}$$

(23)

and the function $\Phi \left( \delta _s,t \right)$ determine the maximum value of $\delta _s$ between time 0 and t. Therefore, $X^{\text {des}}_{t,N}$ can be calculated as

$$\begin{aligned} X^{\text {des}}_{t,N}=z_c+d^{\text {des}}, \end{aligned}$$

(24)

where $z_c$ is position of center of contact area, c, along z-axis and $d^{\text {des}}$ represents the deformation of soft tip when the normal contact force is $N_c^{\text {des}}$ and based on Eq. 8, it can be written as

$$\begin{aligned} d^{\text {des}}=\left( R^2-c^2\left( N_c^{\text {des}}\right) ^{2\gamma } \right) ^{1/2}. \end{aligned}$$

(25)

Numerical simulation and discussion

In this section, performance of the controller designed for manipulating a rigid object on a predefined desired path in the horizontal plane using a three-link soft finger (Fig. 5) is numerically evaluated. In order to solve the dynamic equations of system (i.e., Eq. 12), they can be rewritten in form $\mathbf {b=Ax}$ as

$$\begin{aligned} {\begin{bmatrix} {\mathbf{B}{\varvec{\uptau }} - \mathbf{h}}\\ { - {{\mathbf{h}}_o}}\\ { - {\dot{\mathbf{J}}\dot{\mathbf{q}}}}\\ {\mathbf{0}}\\ {\mathbf{0}} \end{bmatrix}} = {\begin{bmatrix} {\mathbf{M}}&{\mathbf{0}}&{\mathbf{0}}&{ - {{\mathbf{J}}^\text {T}}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{{{\mathbf{M}}_o}}&{{{\mathbf{B}}_c}}&{ - {\mathbf{I}}_4}\\ {\mathbf{J}}&{ - {\mathbf{B}}}&{ - {\mathbf{B}}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{0}}&{{\beta _{s,1}}{\mathbf{I}}_3}&{\mathbf{0}}&{{\mathbf{B}}_s}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{{\beta _{o,1}}{\mathbf{I}}_3}&{\mathbf{0}}&{{\mathbf{B}}_o} \end{bmatrix}} {\begin{bmatrix} {\ddot{\mathbf{q}}}\\ {{{\ddot{\mathbf{q}}}_s}}\\ {{{\ddot{\mathbf{q}}}_o}}\\ {{{\mathbf{F}}_c}}\\ {{{\mathbf{F}}_g}} \end{bmatrix}}. \end{aligned}$$

(26)

Block diagram of the closed-loop system is depicted in Fig. 8. In Table 2, the values of the parameters utilized to simulate the system performance are presented. The mass of the soft tip is assumed to be included in the mass of the third link. Note that soft materials generally have a larger friction coefficient compared to rigid materials [30].

Fig. 8

Block diagram of closed-loop system

Table 2 Simulation parameters

The predefined trajectory for the position of the object is assumed to be elliptical as

$$\begin{aligned} \left\{ {\begin{array}{l} {{x^{{\text {des}}}} = A\sqrt{1 - {{\left( y^{\text {des}}/B\right) }^2}} }\\ {{y^{{\text {des}}}} = {a_0} + {a_1}t + {a_2}{t^2} + {a_3}{t^3}} \end{array}} \right. , \end{aligned}$$

(27)

where the coefficients are selected as ${A=0.3200}$, $B=-0.0905$, ${a_0=-0.0813}$, ${a_1=0}$, ${a_2=0.1219}$, and $a_3=-0.0406$. Moreover, the control parameters are chosen to be ${\mathbf {K}}_{T,v}=10{\mathbf{I}}_2$, ${\mathbf {K}}_{T,p}=100{\mathbf{I}}_2$, $K_{N,v}=20$, $K_{N,p}=100$, and $K_s=20$.

In Fig. 9, the slippage parameter $\delta _s$ defined in Eq. 23 and in Fig. 10, the normal force at the contact interface of the soft tip and object are presented. Since the selected initial normal contact force between soft tip and object, i.e., $N_c^{\text {ini}}=0.05$N, is not sufficient to move the object on the plane, planar slippage begins at the soft tip contact interface. Therefore, the controller increases the normal contact force to reduce the slippage simultaneously. In Fig. 11, the predefined desired path and object path are shown. This figure represents that although the slippage occurs at the finger contact interface, the controller is able to move the object on the predefined desired path. In Figs. 12 and 13, the frictional moment and force at the contact interface of soft tip and object are presented (refer to Fig. 7b). Since soft fingers can sustain frictional moment along with tangential frictional force and normal force at the contact interface, they are able to remove the rotational slippage of the object during planar manipulation compared with rigid fingers. Control torques of the finger joints and deformation of the soft tip during manipulation are shown in Figs. 14 and 15.

Fig. 9

Slippage parameter

Fig. 10

Normal force at the contact interface of the soft tip and object

Fig. 11

Comparison of the object path and desired path

Fig. 12

Frictional moment at the contact interface of the soft tip and object

Fig. 13

Tangential frictional forces at the contact interface of the soft tip and object

Fig. 14

Control torques of the finger joints

Fig. 15

Deformation of the soft tip during manipulation

Uncertainty in friction coefficient

In order to evaluate performance of the controller when the surface condition is changed, we increase the friction coefficient between the object and ground ($\mu _o$) by 100% and 200% in a specific region of the object path (rough surface) as

$$\begin{aligned} {\mu _{\text {o}}} = \left\{ \begin{array}{ll} {\mu _{\text {o1}}} &{} \quad {y_{\text {o}}} \le {y_{\text {g1}}},\,\,{y_{\text {o}}} \ge {y_{\text {g2}}}\\ {\mu _{\text {o2}}} &{} \quad {y_{\text {g1}}}< {y_{\text {o}}} < {y_{\text {g2}}} \end{array} \right. , \end{aligned}$$

(28)

where ${\mu _{\text {o1}}}=0.1$, ${y_{\text {g1}}}=-0.02\,\text {m}$, and ${y_{\text {g2}}}=0.02\,\text {m}$. Simulation results for $N_c^{\text {ini}}=0.5\,\text {N}$ and three values of ${\mu _{\text {o2}}}$, i.e., 0.1, 0.2, and 0.3, are given in Figs. 16, 17 and 18. As shown in Fig. 16, when the slippage first begins between the finger and object, the controller quickly reduces and cancels the slippage by increasing the normal force (Fig. 17) and moves the object on the predefined desired path. When the object reaches the region with ${\mu _{\text {o2}}}=0.2$, the slippage between the finger and object does not occur again because the increased normal force is large enough to move the object without slippage. However, when the object reaches the region with a larger friction coefficient, i.e., ${\mu _{\text {o2}}}=0.3$, the slippage occurs again and the controller increases the normal force to cancel the slippage as much as possible and move the object on the predefined desired path. As shown in Fig. 18, despite reoccurrence of slippage, the object can properly follow the desired path with a deviation less than 4% for when ${\mu _{\text {o2}}}=0.3$. Hence, the controller has an acceptable performance in moving objects on the surfaces with variable friction coefficients. Moreover, the performance of the controller has been also evaluated when there is up to 10% uncertainty in the length and mass parameters of the finger and object. The results confirmed the satisfactory robustness of the controller to the parametric uncertainty.

Fig. 16

Slippage parameter

Fig. 17

Normal force at the contact interface of the soft tip and object

Fig. 18

Comparison of the object path and desired path

It is worth mentioning that we proposed our dynamic model and controller based on the ellipsoidal approximation of the friction limit surface, which is a generalization of the Coulomb.s friction law. Although this approximation has been validated experimentally [3], due to some other frictional factors, e.g., pre-sliding displacement, stiction effect, Stribeck effect, viscous effect, and frictional lag, this approximation may not be valid for some cases. Hence, in spite of the acceptable performance of the controller in presence of uncertainties, for the situations that the ellipsoidal approximation is not accurate, the controller may have different behavior in the real world from the simulation results and this is a possible limitation of our model.

Conclusions

Contact modeling is one of the first steps in the analysis of grasping and manipulation. When the contact interface is relatively large or deformation in the contact is not negligible, e.g., soft contact, a frictional moment, in addition to tangential frictional force and normal force, can be sustained by the contact interface. Therefore, the friction limit surface is used instead of friction cone to relate contact frictional force/moment to contact sliding motions. In this study, a novel method for dynamic modeling of planar slippage was proposed using the concept of friction limit surface. In this method, a single differential equation was presented to model the different states of frictional force and moment in the planar contact. This method was utilized in the analysis of manipulating a rigid object on a horizontal plane using a three-link soft finger. A controller was designed to reduce and remove the undesired slippage which occurs between the soft finger and object and simultaneously move the object on a predefined path. Numerical simulations revealed that the presented controller has an acceptable performance not only in reducing and removing the undesired finger slippage, but also in moving the object on a predefined desired path despite the slippage occurrence. In the next step, the proposed methods will be utilized for slippage analysis and control in object grasping and manipulation using soft multi-fingered robotic hands.

Appendix: Vectors and matrices

The vectors and matrices introduced in Eq. 12 are obtained as

$$\begin{aligned}&{\begin{array}{l} {\mathbf{J}}\left( {\mathbf{q}} \right) = {\begin{bmatrix} { - s1J_1}&{}{c1J_2}&{}{c1J_3}&{}0\\ {c1J_1}&{}{s1J_2}&{}{s1J_3}&{}0\\ 1&{}0&{}0&{}0\\ 0&{}{ - {l_3}s23 - {l_2}s2}&{}{ - {l_3}s23}&{}{ - 1} \end{bmatrix}}\\ J_1={{l_3}s23 + {l_2}s2}\\ J_2={{l_3}c23 + {l_2}c2 - d}\\ J_3={{l_3}c23 - d} \end{array},} \end{aligned}$$

(29)

$$\begin{aligned}&{{{\mathbf{M}}_o} = {\begin{bmatrix} {{m_o}}&0&0\\ 0&{{m_o}}&0\\ 0&0&{{I_o}}\\ 0&0&0 \end{bmatrix}}, \qquad {{\mathbf{h}}_o} = {\begin{bmatrix} 0\\ 0\\ 0\\ {{m_o}g} \end{bmatrix}},} \end{aligned}$$

(30)

$$\begin{aligned}&{{\mathbf{B}} = {\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ 0&0&0 \end{bmatrix}}, {{\mathbf{B}}_c} = {\begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ {{y_o} - {y_c}}&{{x_c} - {x_o}}&1&0\\ 0&0&0&1 \end{bmatrix}},} \end{aligned}$$

(31)

where $l_1$, $l_2$, and $l_3$ are lengths of the finger links (Fig. 5), $m_o$ is the object mass, $I_o$ is moment of inertia of the object about its rotation axis, g is the gravitational acceleration, $x_o$ and $y_o$ represent coordinate of the object center of mass, and $x_c$ and $y_c$ are the position of center of contact area, c. Furthermore, $s\text {I}$, $c\text {I}$, $s\text {IJ}$, and $c\text {IJ}$ represent $\sin \left( {{\theta _\text {I}}} \right)$, $\cos \left( {{\theta _\text {I}}} \right)$, $\sin \left( {{\theta _\text {I}+\theta _\text {J}}} \right)$, and $\cos \left( {{\theta _\text {I}+\theta _\text {J}}} \right)$, respectively.