Robotic hand model parameters.

## Abstract

Dynamic and robust control of fingertip grasping is essential in robotic hand manipulation. This study introduces detailed kinematic and dynamic mathematical modeling of a two-fingered robotic hand, which can easily be extended to a multi-fingered robotic hand and its control. The Lagrangian technique is applied as a common procedure to obtain the complete nonlinear dynamic model. Fingertip grasping is considered in developing the detailed model. The computed torque controller of six proportional derivative (PD) controllers, which use the rotation angle of the joint variables, is proposed to linearize the hand model and to establish trajectory tracking. An impedance controller of optimized gains is suggested in the control loop to regulate the impedance of the robotic hand during the interaction with the grasped object in order to provide safe grasping. In this impedance controller, the gains are designed by applying a genetic algorithm to reach minimum contact position and velocity errors. The robustness against disturbances is achieved within the overall control loop. A computer program using MATLAB is used to simulate, monitor, and test the interactional model and the designed controllers.

### Keywords

- modeling robotic hand
- optimized impedance controller
- linearization the dynamics of robotic hand
- fingertip grasping
- Lyapunov stability
- Lagrange technique

## 1. Introduction

A recent research on service robotics has shown that there is an environmental reaction on major applications [1]. The robotic hand is the main interactive part of the environment with the service robotics [2, 3]. Grasping operation [4] has been used explicitly in robotic hand during the interactive period which represents the process of holding objects via two stages: free motion and constrained motion. Free motion is the moving of fingers before detecting the grasped object, and the constrained motion is the state of fingers when the object is detected. The common purpose in grasping is to handle unknown objects. Thus, finding both *a model system* and *a control algorithm* to deal with non-predefined objects is essential in a robotic hand. Regarding modeling, interactional modeling is the demand to perform a specific control algorithm. With interactional modeling, the contact force between the robotic hand and the grasped objects has obtained without needing their properties. In control point of view, the robotic hand holds an object by its fingers and moves the object to another place. In this case, there are different modes of operation which can be listed as before holding the grasped object, holding the grasped object, and moving the robotic hand while holding the grasped object. The aim at each mode is distinct to the others and has to respond to the operation assigned. Thus, the multitude of various goals which have to be achieved with regard to the various operations presents an interesting scenario in the control of a robotic hand.

Generally, it is essential to obtain the modeling of a multi-fingered robotic hand before implementing a specific model-based control algorithm. Arimoto et al. [5] presented a two-fingertip grasping model to control the rolling contact between the fingertips and the grasped object. Arimoto et al. derived the model through the use of the Euler-Lagrange equation with respect to the geometry of the fingertips which is represented by the arc length of a fingertip. Corrales et al. [6] developed a model for a three-fingered robotic hand; in their study the transmission of fingertip contact force from the hand to the grasped object is considered. Boughdiri et al. [7] discussed the issue of modeling a multi-fingered robotic hand as a first step, before designing controllers. They considered free motion only, without the constraints of grasped objects. In a proceeding work, Boughdiri et al. later [8] considered the constraints of grasped objects and the coupling between fingers in grasping tasks.

In the control scenario of a robotic-environment interaction, there is generally also a similar case of a robotic hand-grasped object interaction; force tracking is necessary to obtain appropriate and safe contacts. In force tracking, three main position-force controllers are introduced: stiffness control, hybrid position/force control, and impedance control [9]. Among the above three controllers, impedance control presents an integrated process to constrained manipulators [10]. The impedance controller is introduced by Hogan in 1985 to set the dynamic conduct between the motion of the robot end-effector and the contact force with the environment, instead of regarding the control of motion and force individually [11, 12]. Chen et al. [13] derived impedance controller for an elastic joint-type robotic hand in Cartesian and joint spaces. Chen et al. focused on the compensation of gravity and managing the parameter of uncertainties. Huang et al. [14] proposed the impedance controller for grasping by two fingers to deal with the problem of dropping objects when grasping by one finger. Zhang et al. [15] proposed an adaptive impedance controller with friction compensation to track forces of the multi-fingered robotic hand.

However, the synthesis of impedance control design continues to exist as a major challenge in the performance advancement of robotic hand in order to provide safe and robust grasping. Besides this, the greatest problems in robotic hand-grasped object interaction are the existence of nonlinearity in the dynamic system, dealing with different grasped objects, disturbances, and interference. In the previous studies, the aforementioned problems have not taken in consideration. The aim of this paper is to treat these problems using interactional modeling and the design gains of impedance control. The action of the suggested control expression is to provide a fingertip interactional robust control algorithm that takes into account the dynamics of the robotic hand under unknown contact conditions. To accomplish this, we propose first to linearize the dynamics on the level of position-tracking controller through six PD controllers. When executed, the calculated gains of the designed PD controllers result in zero steady errors without overshoot. Second, through an impedance controller, the interactional fingertips’ force is regulated by the robotic hand’s impedance. The gains of the impedance are designed based on genetic algorithm. As a result, the introduced control expression enables the position and velocity of fingertips to be controlled simultaneously without changing over control subspace. This control expression avoids selecting between different controllers which causes indirect control and consumption of high energy. The operation and the robustness of the presented control expression are approved by MATLAB/Simulink in several tests.

The rest of the chapter is organized as follows. In Section 2, the mathematical representation of kinematics and dynamics of the robotic hand with environment interaction is derived and explained. In Section 3, the overall controllers of position tracking and impedance control are designed. In addition, the suggested procedure of obtaining optimum values of impedance gains using genetic algorithm is explained. In Section 4, following the joint path and fingertip position is tested. Then, the robustness of the system is examined. Finally, the conclusions of this study are presented in Section 5.

## 2. Mathematical model

The models of a robotic hand system are the mathematical equations that describe all the contact forces and motion of the fingers at any time [16]. These models are essential for grasping and manipulation operations which include kinematics, dynamics, and contacts between the fingers and the environment. A multi-fingered robotic hand with an existing grasped object has a complex structure. Thus, deriving the mathematical representation is not a simple task. In this section, the mathematical representation for a two-fingered robotic hand with an environment is developed. Changing the orientation of the robotic hand with respect to the inertial (earth) frame during manipulation operation is considered in this analysis.

### 2.1 Reference frames

The system is a two-fingered robotic hand that can adjust fingertip grasp forces by the torque of actuators in the joints. Each finger includes three joints (MCP, PIP, and DIP) and three rigid phalanges (proximal, intermediate, and distal). The motion of the finger actuates through the torque in the joints. The variables

The result of orienting the robotic hand with respect to the earth coordinates is expressed by defining the three rotation angles roll, pitch, and yaw (**RPY**). In which, roll is (

where

### 2.2 Kinematic analysis

Forward kinematics is implemented here to express the relationship between the Cartesian spaces of the fingertip contact points, which are represented by coordinates (

and the forward kinematic of finger 2 is

where

where

and

The robotic hand transforms the generated torque by each joint to the wrenches exerted on the grasped object through the following Jacobian matrix:

### 2.3 Contact interaction

A contact interaction, which uses contact behavior between rigid fingertips and a flexible grasped object, is proposed in this study (see Figure 2). The grasped object is assumed as a linear spring of stiffness

where

### 2.4 Dynamics of robotic hand

The motion equations are derived to describe the dynamics of the robotic hand when interacting with the grasped object; the explained fingertip forces in Eqs. (10)–(12) are involved in the dynamic equation. When ignoring friction, each finger in free motion can be described by the following general motion equation:

where

where

to derive Eqs. (15) and (16), the Lagrange equations of motion are used. For each finger manipulator, the equation of motion is given by [18]

where

the position of the mass center of intermediate phalanx with respect to MP joint is represented by

and the position of the mass center of distal phalanx with respect to MP joint is represented by

where

and to obtain the kinetic energy, the above formulas (

where

The procedure for calculating

by using the kinetic energy in Eq. (18) and the potential energy in Eqs. (25)–(27), it results that Eqs. (19)–(21) after being rearranged in a matrix form result in

The formula of matrices’ elements in Eq. (28) is listed in Appendix A. From Eqs. (15) and (16), the overall dynamics of robotic hand can be constituted as

with the following matrix forms;

Thus, the six motion equations are presented by rearranging Eq. (29) to result in

The desired finger configuration in joint space can be obtained by solving the above equation, which will be explained later in Section 3.1 for designing the position-tracking controller.

## 3. Controller design

The derived mathematical representation is applied using MATLAB/Simulink to implement the proposed controllers and the tests of this study. The parameters of the model, which are used in Simulink, are shown in Table 1. The overall controllers of the robotic hand are depicted as a complete block diagram in Figure 3. In the following sections, the design of the controllers is described in detailed.

Parameter | Value | Parameter | Value |
---|---|---|---|

10 mm | |||

7083 g.mm^{2} | 400 g.mm^{2} | ||

100 mm | |||

50 mm | |||

70 mm | |||

1125 g.mm^{2} |

### 3.1 Position tracking

Recently, a variant of the PD controller explicitly has been considered in the control of nonlinear system where the aim is to make the system follow a specific trajectory [19, 20, 21, 22]. The joint angle controller shown in Figure 3 is the inner feedforward loop which presents the effect of the input torque on the error in Eq. (30). The detailed block diagram of this controller is shown in Figure 4. The tracking error is formed by subtracting the joint angle vector (

The objective is to minimize this error at any time, by differentiating Eq. (31) twice, solving for

where

where

The system can be stabilized in Eq. (34) by a proper design of PD controller to track the error

where

The design of gains is selected for the closed-loop time constant (

where

The stability of robustness analysis by Lyapunov’s direct method is implemented here to verify the ability of applying the proposed PD controller in real situations where unmodeled disturbances are included in the actual robotic hand. The effect of unmodeled disturbances in robotic manipulator can be represented by replacing the PD controller gain matrices with matrices of nonlinear gains [23]. Considering the nonlinear matrices gains, the robotic hand input in Eq. (34) becomes

where

The stability of Eq. (32) together with Eq. (36) is assumed if there exists

The stability analysis is implemented by proposing the following Lyapunov function:

Considering the smallest eigenvalues

thus

where

Then, Eq. (38) becomes

The first two terms of Lyapunov function are positive. Regarding the third and the fourth terms, according to Eqs. (39) and (40), we have

thus

Equations (44) and (45) result in

and

Hence, the Lyapunov function presented in Eq. (38) is globally positive definitive function.

Taking the derivative of Eq. (38) with respect to time, we get

Finally, using the rule of Leibnitz for representation, the integrals in differentiation form and Substituting Eq. (37) in (50) yield

for all positive values of gain matrices

In this way, the system is globally exponential stabile.

### 3.2 Impedance controller

The PD impedance controller has been widely applied in modern researches for controlling the interaction with the environment [24, 25]. The fundamental principle of the implemented impedance control in this study is that the fingers should track a motion trajectory, and adjust the mechanical impedance of the robotic hand. The mechanical impedance of the robotic hand is defined in terms of velocity and position as

By virtue of the above two equations, the fingertip contact force

where

where

where

and

Hence, the contact forces are regulated in joint space, and the robotic hand can adapt the collision of the fingers with the grasped objects.

The application of Lyapunov function technique determines the conditions of stability of impedance controller [26]. In the forward kinematic model, the equations of fingers are

On the other hand, the errors are given as

where

where

then, we get

The error dynamics in Eqs. (61) and (62) is asymptotically stable if the following candidate Lyapunov function

satisfies the following conditions:

The derivative of Eq. (70) is

Equations (61) and (62) can be rearranged as below

By substituting Eqs. (72) and (73) in Eq. (71) and applying the relations in Eqs. (57)–(60), it yields

Hence, the robotic hand system is asymptotically stable under the following boundaries:

or

### 3.3 Gains design using genetic algorithm

Genetic algorithm is a well-known optimization procedure for complex problems which ascertains optimum values for different systems [27]. The genetic algorithm is written to find the optimum impedance gains. The code takes the calculated data from the robotic hand Simulink model to compute impedance’s gains. However, the detailed genetic algorithm that is applied in the design of the gains can be defined by the following proceedings:

(1) *Generate a number of solutions*. This represents the random number of population which includes the gains

(2) *Fitness function*. During the running of the program, the fitness is evaluated for each individual in the population. In this study, the objective of the fitness function is to minimize the errors

where

(3) *Create the next generation* [28]. At each step, genetic algorithm implements the following three rules to create a new generation of the recent population:

Individuals called parents are selected by the selection rules, and then the population of the next generation is obtained by contributing the parents.

Two parents are combined to generate children by crossover rules for the next generation.

Children are formed by mutation rules. The process of forming children is dependent on applying random changes on the individual parents.

(4) *Setting the genetic algorithm*. The following parameters are used for setting the proposed genetic algorithm in this study: the number of generations = 80, the size of population = 30, and the range of gains are

## 4. Simulation results

The obtained interactional model was applied in the simulation to verify the designed controllers through three tests as follows:

Test 1: Following a specific joint path.

Test 2: Following fingertip position.

Test 3: Robustness of the controllers.

The Simulink model is implemented using MATLAB as shown in Figure 5.

### 4.1 Following a specific joint path

This test is focused on position-tracking performance of joints. We assumed the two fingers are in initial position (

The simulation result explained in Figure 6 shows that the designed position-tracking controller can follow a specific path with zero steady error and without overshoot. A unit step input function of amplitude

### 4.2 Following fingertip position

The second test is implemented to monitor the operation of the designed impedance controller in task space. We assumed the initial position of the fingertip 1 is (135, 100, −25 mm) and it has to reach the position (115, 153.7, −25 mm). The position of the grasped object is assumed at

For comparison purposes, another study is implemented without considering the impedance controller in the overall control loop, i.e.,

### 4.3 System robustness

The robustness of the designed controllers is checked in this section. This test is implemented within the same parameters of the desired position and object location in the test of Section 4.2. The disturbances on the robotic hand cause change of the fingertip contact forces. Here, we assumed that the disturbances result in a contact force with a white Gaussian noise [29, 30, 31, 32] shown in Figure 11. The Gaussian noise model is generated using block AWGN channel in Simulink. The position of the AWGN channel is shown in Figure 5. The parameters of white Gaussian noise are set as shown in Figure 12. The designed controllers have achieved the functionality of rejecting the disturbances and kept the motion of the finger as shown in Figure 13.

## 5. Conclusion

Grasping by two fingertips plays a distinctive role in its performance related to robotic hand-environment interaction. A mathematical model including the dynamic equation has been derived based on the Lagrange technique. The coupling between two fingers and the fingertip contacts with unknown grasped objects has been presented in this model. The performance of the derived model is confirmed when a designed based model controller is implemented in Simulink. A computed torque controller of six PD controllers has been implemented to control the rotation of joints. The impedance controller is applied to operate as the outer loop of the overall control system. This controller allowed the regulation of the fingertip contact forces without overshoot response, which is essential for safe grasping, especially in holding fragile objects. Besides this, the impedance gains have been obtained using a genetic algorithm with a new behavior of improving the response of the position and velocity errors. The applied genetic algorithm method has minimized the position error and the velocity error while keeping the operation of the overall control system. The robot finger has been ordered to follow a specific joint path and fingertip position grasping; the results of simulations have achieved a typical accomplishment of trajectory tracking. Finally, the overall controllers have shown a perfect rejection of disturbances in terms of fingertip contact force. As a future work, it is recommended to apply the proposed controllers of this study to mobile manipulator for grasping and manipulation objects.

Let

and

Then

The partial derivatives of

The matrices’ elements in Eq. (28) are