Modelling and Control of The Barrett Hand for Grasping

Md , , , School of Engineering and Materials Science.

Queen Mary, University of London, London, UK. E mail: , , , .

Abstract— Grasping of an object by robot hand is always challenging due to the uncertainties such as robot non-linearities, unstructured objects with unknown location and stiffness, unknown contact types. To address these challenges, the fundamental task is to achieve the mathematical model of the robot hand, collect information about the object and model the contact between the hand and the object. In this paper, mathematical model of a three fingered Barrett Hand is developed using the Lagrangian method. Joint and cartesian space motions are simulated on the basis of computed torque algorithm. A contact model is then developed for hand-object coupled system assuming that the object properties are well known. Contact model is also simulated under different environment stiffness. Simulation studies verify the joint and cartesian space motions along with the estimated contact force. Implementation of the control for optimizing the contact force will be presented soon.

Keywords- Robot Hand; Modeling; Grasping; Control; SimMechanics; Interaction etc.

I. INTRODUCTION

From the ancient age, human hands have made an enormous contribution to the progress and achievements of human civilization. Extensive effort has been made over the last two decades to construct robot hand mimicking the mechanism and dynamics of the human hand. A robotic manipulator with a human like hand as its end-effector will be exceedingly effective in all its current and future applications.

In fact, robot manipulators were first designed to emulate human limbs by roughly mapping the human hand and arm structure. In the beginning, the gripper was used as an end-effector. It was only effective in performing picking and placing objects of a particular shape. The limitation of the gripper included the lack of dexterity due to the absence of several degrees of freedom, inability to pick multi shaped object and insufficient space to mount enough sensors [4]. The multifingered hand has been introduced to eliminate the limitation of conventional grippers and to increase the efficiency of a manipulator to perform better in grasping and manipulation tasks. Salisbury [2] was the first person to successfully design a human like robot hand, an advanced manipulator end-effector, for the purpose of grasping and manipulation research. Since then, many robot hands have been designed and used to perform tasks in industries, hazardous and remote places, power plants, space. The DLR

hand, the UTAH MIT hand and the Barrett hand are some hands popularly used in industries and as research platforms to study grasping and manipulation [9, 10, 11].

Most of the hands can perform a few tasks in the different environments, but cannot offer a large range of task completion. It is due to the lack of dexterity in the hand, modeling error and uncertainties, unstructured shape of the objects, incorrect contact force in terms of robot-object compliance, unknown contact friction and inaccuracy of sensor information [5]. As such, no robot hand exists today, that can offer similar efficiency as human hands in performing tasks in unknown environments. This paper studies the modeling and control of the robot hand to improve the efficiency in executing tasks.

II. PREVIOUS WORK AND GRASPING ISSUES Grasp studies concern to increase the ability of a robot

hand to grasp objects of different shapes and stiffness with dexterity, by ensuring that no slippage occurs between the object and the hand during contact. Asada and Hanafusa [12] have carried out a substantial investigation analyzing multifingered hand. They determined the essential conditions for the grasping and manipulating capabilities of a multifingered hand. Kerr [4] formulated the force required to squeeze an object and introduced the concept of internal forces in grasping. Yashima and Yamaguchi [13] discussed the theory behind the whole finger manipulation (WFM). In whole finger manipulation, all links of a finger make contact with the object which brings more complexity in hand-object dynamics. The kinematics, dynamics and the control of WFM are detailed in their work. Sastry and Li [14] have formulated the mathematics of grasp planning and proposed nonlinear control laws considering point contact. Seraji [7] derived the force tracking characteristics of the hand-object system and proposed adaptive based impedance control to maintain contact force under unknown object properties.

Based on the previous studies, a number of issues such as hand modeling, hand-object contact modeling and control are required to be addressed to improve the efficiency and dexterity of the multifingered hand for object grasping. The performance of the hand in grasping depends on the accurate information of the object’s properties, hand modeling, contact modeling between the object and the hand, grasp planning on the basis of the object location and contact point between the object and the hand, contact force

2013 UKSim 15th International Conference on Computer Modelling and Simulation

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estimation and control of the hand according to hand-object dynamics [6]. Considering all these factors, the grasping of the objects with a multifingered hand is presented in the following steps:

� Object Location: Determine the object location by Tactile or Visual Sensor. Object recognition, visual tracking and 3-D pose estimation are the current advanced technology used in identifying objects in environment [15].

� Grasp Planning: Plan the grasp on the basis of the sensor data. At this stage, it is required to decide the number of contact points to grasp the object. Grasp planning itself is considered as an independent research area in the field of grasping.

� Contact Modelling: The contact model is necessary to determine the contact locations in the hand and object space, estimate the contact force and avoid any slippage due to friction while the hand makes contact with the object. The object and the hand materials are also considered in contact modelling. At present, force sensors are mounted on fingertip in industries to measure the contact force [16].

� Grasp control: The essential part is the control of the hand based on grasp planning, contact modelling and hand and object dynamics. Control for grasping is complicated when the hand and object model is not known apriori. Type of contact defines the control difficulty of the hand which depends on the shape of fingertip and object at the contact point. The control task is accomplished in 2 steps: First, the hand reaches the desired object location and subsequently grasps the object and secondly it requires an optimum force to be applied on the object, depending on the object’s stiffness, to grasp it.

In this paper, a three fingered multifingered hand called Barrett Hand is selected for investigating the fundamental tasks of grasping which involve: hand and contact model, hand position control in both the joint space and the task space and estimation of the contact force in the task space. There are some assumptions made to simplify the modeling and control problems associated with the tasks. It is assumed that the hand fingers are rigid. Frictions in each of the joints at motor shaft are ignored. Contact location is assumed to be known and to be without friction. Trajectories are planned in both the joint space and task space to implement the control based on the model of the hand.

III. GENERAL PROBLEM FORMULATION

The primary task in grasping involves the interaction between the object and the hand. For simplicity, consider a fixed object with known coordinates and a single finger of the robot hand. The task of grasping can be broken down into two distinct problems. First problem is to command the hand to reach the object location. With a given trajectory, the joint position of the finger is controlled to reach the desired location. As soon as the finger hits the object, reaction force is generated, which needs to be dealt with. The reaction force is needed to be limited to a certain

optimum value so that it does not cause any slippage or damage of the object while maintaining contact. It is assumed that, the object does not produce any motion i.e.

. In both cases, the goal is to achieve the desired location and desired force

such that,

(1)

where, are the fingertip position and orientation vector, are the finger force and moment vector. and are called the position and the force error respectively. The problem of limiting the reaction force problem defined above assumes that applied reference force must be tracked during contact.

IV. MULTIFINGERED HAND KINEMATICS

A. Barrett Hand Description The BH8-series is a three-fingered

hand with the optimum flexibility that is required to grasp objects of different shapes and sizes [9]. This hand can be used with any existing industrial manipulators available in the industry. The hand consists of three fingers placed on the palm . Three fingers are identical in terms of length. Each finger has two revolute joints and and consists of three links . Initial joint angle of the first and the second joints are and respectively. and can rotate around the entire palm with a spread angle of ( ) which gives similar agility to human thumbs. is fixed on the base and considered on planar axis only due to the absence of .

B. Forward Kinematics Forward kinematics is required to calculate the end

position of the hand finger. Consider the finger to determine the forward kinematics where

and . The Cartesian fingertip position of finger can be determined as below,

(2)

(3) Equation (2) and (3) is used to determine the forward kinematics of the finger and . In this paper, Spread angle is not considered in kinematics as hand is analyzed on planar axis with only finger .

C. Inverse Kinematics Inverse kinematics is used to calculate the angle required

to reach the desired position. For current cartesian position of ( , the joint angles and of the finger can be calculated using simple algebra and trigonometry The joint angles are found as,

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(4)

(5) where, . Finding the inverse

kinematics of the other two fingers is similar to (4) and (5).

V. BARRETT HAND MODEL

A. Physical model The Solidworks based CAD model of the Barrett hand is

collected from [9] and converted as a physical model in SimMechanics, a toolbox of MATLAB & Simulink, used for modeling and simulation of rigid machine and analyzing the motion [8]. Any CAD design of a mechanical system can be converted into a physical model in SimMechanics. In SimMechanics hand model, representative of each body part (Position and Orientation) refers to the DH transformation of the hand [1]. The DH parameters of the Barrett Hand are shown in Table I. The SimMechanics model allows defining joints according to DOF of the robot hand. Motion or torque can be applied as input to all joints of the hand. Sensors are available to measure the hand position in joint and workspace. The SimMechanics model of the Barrett hand is shown in Fig. 1.

Figure 1. Kinematic description of the Barrett Hand [9].

B. Mathematical Model The general mathematical model of an open chain manipulator can be derived using well known energy based Lagrangian method [5]. Lagrangian can be determined as the difference of the kinetic ( ) and the potential energy ( ) of the system. Differentiating the Lagrangian determines

the equation of motion of the system in joint space of the manipulator as,

(6)

is considered as DOF. is the total of all torques acting on the system in terms of rotational motion .

TABLE I. DH PARAMETERS OF EACH BARRETT FINGER

Frame Link length

Twist Angle

Joint Offset

Joint angle

{1} r 90 0

{2} 0 0

{3} 0 0

a. r = [-1,1,0] for Finger ; i=1,2,3…n.

C. Joint and Cartesian Space Model Following (6), manipulator dynamics of an open chain

manipulator joint space can be written as, (7)

where, is non-singular symmetric inertia matrix, is the vector including centrifugal, coriolis terms, is the gravitational vector terms.

are the joint position, velocity and acceleration of the manipulator links respectively. The system input is defined as torque which is a vector. The task space dynamics of the manipulator can be represented as follows,

(8)

Equation (5) is derived from (4) using the relation between fingertip and joint velocities as follows,

(9)

Where is the Jacobian of the manipulator end-effector [6]. Therefore, represents the end-effector cartesian position, velocity and acceleration respectively. is the generalized input force which is related to joint torque as . This relation is used to transform the input force

to the joint input torque is called the Jacobian Transpose method [5]. The dynamic model of the Barrett Hand is derived using Lagrangian. Each finger of the hand is similar to open chain manipulator dynamics in (7). All fingers have similar kinematics except the spread angle between and . The Barrett finger dynamics is shown below:

(10)

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and are the input torque of first and second joint of the finger . The dynamics of the finger and are similar to

, excluding the spread angle on plane (Fig. 1). The planar plane is considered for the hand simulation and the spread angle is not considered in (10).

D. Contact Dynamics Model Contact dynamics modeling is essential in grasping to estimate the collision force between the object and the robot hand. The contact model between the modeled finger and a fixed rectangular shaped object is shown in Fig. 2. The initial and current fingertip position of the arm are and respectively. Other parameters are: the object position ( ), object stiffness ( ) and damping coefficient ( ). and is modified to observe the reaction force under different values. According to the model, the collision will only occur when horizontal fingertip position,

and the reaction force then can be calculated as,

(11)

where, is a conditional variable defined as,

(12)

in equation (11) is called the interaction or contact

force exerted by the object. From (7), the manipulator model considering contact dynamics can be rewritten as,

(13)

Equation (13) is called the interactional dynamics of the

manipulator in joint space. Interactional torque can be calculated from force-torque relationship using Jacobian transpose as .

Figure 2. Contact model between a 2 DoF manipulator and fixed object.

VI. HAND CONTROL (JOINT AND WORK SPACE)

Refer to the problem defined in Section III, control of the hand can be divided into two categories: 1. Control the position of the finger to reach the object location. 2. Control the force of the finger while making contact with the object. In this paper, position control in joint and cartesian space is considered. For joint position control, hand fingers follow command to track any given position trajectory to reach the object. The dynamics of each finger in (7) is rearranged in (14). (14)

Now define the desired trajectory for tracking. To

ensure the tracking, the joint position error can be shown as, where is called the tracking

error. Now, the torque input can be chosen as,

(15)

where, is the new control input. The new input can be chosen in several ways. In this case, the

Proportional–Derivative (PD) control law is applied as new control input where, . and are

diagonal matrices. Apply (15) in (14) then simplifies to,

(16)

which solves the position control problem given in (1) as long as and are positive definite. The above control strategy is used to simulate the position of each finger for any given trajectory. The desired trajectory equation has the following exponential form, (17)

where, is the demanded step command, is a positive constant. For cartesian position control, To reach a specific object location by the hand, inverse kinematics is required. Equation (4) and (5) are used to calculate the joint angles from desired cartesian position . Then, control torque

is applied to the joints to move the arm to the correct location. To reach the correct location is important to grasp the object in grasping case. Wrong kinematics results in failure of the hand to grasp an object even in known object environment.

VII. SIMULATION RESULTS

For each finger of the Barrett Hand simulation, the mass, link and other parameters are: link lengths,

=.05m, link mass, , time step constant, , base

height of the hand, The desired angle,

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, and the initial position of joint two and three are and respectively. The control gains a found as

In Fig. 3 and Fig. 4, the simulation results are presented for all three fingers of the Barrett hand. All desired commands for the simulation are available in Table II. Trajectory tracking in joint space of the fingers , and are found satisfactory. Fig. 5 shows the simulation of the finger to reach a desired location in the workspace. Desired cartesian trajectory is given as

and .

TABLE II. DESIRED JOINT ANGLE OF THE BARRETT HAND

Finger Desired Position (deg)

Desired Position (deg)

25 35

10 25

15 30

The contact model based on (11), (12) and (13) is simulated in cartesian space. In Fig. 6, the simulated contact force is shown. Fig. 7 shows the maximum contact force in zoomed view which is around 175N. This force is large enough to damage the joint motor or object which is not suitable for grasping. The optimization of this force is possible based on the derived contact model. Future works will include the implementation of the adaptive control to optimize the reaction force during contact.

VIII. CONCLUSION

This paper solved the position control problem of a multifingered robot hand. The kinematics and modeling process of the Barrett hand is described and simulated. The control is designed to track the finger in both joint and cartesian space. The contact between object and fingertip is modeled and the reaction force response is observed. The mathematical models are simulated and the results proved the authenticity against the physical model. Investigation on optimizing the reaction force is under process. Future works include unknown object environment, consideration of adaptive control law to estimate the accurate force required for grasping.

Figure 3. Second Joint Position (deg) of Barrett fingers.

Figure 4. Third joint position (deg) of Barrett fingers.

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Figure 5. Fingertip position (m) of in cartesian space.

Figure 6. Contact force (N) of the finger .

Figure 7. Zoomed contact force (N) of the finger .

Figure 8. Hand position in free space (Left) and contact with object (Right)

(SimMechanics)

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[14] Z. Li, S. Sastry, “Grasping and Coordinated Manipulation by a Multifingered Robot Hand”, The International Journal of Robotics Research, Vol. 8(4) (1989) pp. 33-50.

[15] P. K. Alen, A. T. Miller, P. Y. Oh and B. S. Leibowitz. “Using tactile and visual sensing with a robotic hand”, Symposium proceedings of IEEE, Vol. 1,1997, pp. 20-25.

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