On the Design of Underactuated Finger Mechanisms for Robotic Hands

1. Introduction

The mechatronic design of robotic hands is a very complex task, which involves different aspects of mechanics, actuation, and control. In most of cases inspiration is taken by the human hand, which is able to grasp and manipulate objects with different sizes and shapes, but its functionality and versatility are very difficult to mimic. Human hand strength and dexterity involve a complex geometry of cantilevered joints, ligaments, and musculotendinous elements that must be analyzed as a coordinated entity. Furthermore, actuation redundancy of muscles generates forces across joints and tissues, perception ability and intricate mechanics complicate its dynamic and functional analyses.

By considering these factors it is evident that the design of highly adaptable, sensor-based robotic hands is still a quite challenge objective giving in a number of cases devices that are still confined to the research laboratory.

There have been a number of robotic hand implementations that can be found in literature. A selection of leading hand designs reported here is limited in scope, addressing mechanical architecture, not control or sensing schemes. Moreover, because this work is concentrated to finger synthesis and design, the thumb description is excluded, as well as two-fingered constructions, because most of them were designed to work as grippers and would not integrate in the frame of multi-finger configuration.

Significant tendon operated hands are the Stanford/JPL hand and the Utah/MIT hand. In particular, the first one has three 3-DOF fingers, each of them has a double-jointed head knuckle providing 90° of pitch and jaw and another distal knuckle with a range of ±135°. The Utah/MIT dextrous hand has three fingers with 4-DOFs, each digit of this hand has a non anthropomorphic design of the head knuckle excluding circumduction. The inclusion of three fingers minimizes reliance on friction and adds redundant support to manipulations tasks. Each N-DOF finger is controlled by 2-N independent actuators and tension cables. Although these two prototypes exhibit a good overall behaviour, they suffer of limited power transmission capability.

The prototype of the DLR hand possesses special designed actuators and sensors integrated in the hand’s palm and fingers. This prototype has four fingers with 3-DOFs each, a 2-DOFs base joint gives ± 45° of flexion and ±30° of abduction/adduction, and 1-DOF knuckle with 135° of flexion. The distal joint, which is passively driven, is capable of flexing 110°.

A prototype of an anthropomorphic mechanical hand with pneumatic actuation has been developed at Polytechnic of Turin having 4 fingers with 1-DOF each and it is controlled through PWM modulated digital valves.

Following this latter basic idea, several articulated finger mechanisms with only 1-DOF were designed and built at the University of Cassino and some prototypes allowing to carry out suitable grasping tests of different objects were developed.

More recently, the concept of the underactuation was introduced and used for the design of articulated finger mechanisms at the Laval University of Québec.

Underactuation concept deals with the possibility of a mechanical system to be designed having less control inputs than DOFs. Thus, underactuated robotic hands can be considered as a good compromise between manipulation flexibility and reduced complexity for the control and they can be attractive for a large number of application, both industrial and non conventional ones.

2. The underactuation concept

Since the last decades an increasing interest has been focused on the design and control of underactuated mechanical systems, which can be defined as systems whose number of control inputs (i.e. active joints) is smaller than their DOFs. This class of mechanical systems can be found in real life; examples of such systems include, but not limited to, surface vessels, spacecraft, underwater vehicles, helicopters, road vehicles, and robots.

The underactuation property may arise from one of the following reasons:

• the dynamics of the system (e.g. aircrafts, spacecrafts, helicopters, underwater vehicles);

• needs for cost reduction or practical purposes (e.g. satellites);

• actuator failure (e.g. in surface vessel or aircraft).

Furthermore, underactuation can be also imposed artificially to get a complex low-order nonlinear systems for gaining an insight in the control theory and developing new strategies. However, the benefits of underactuation can be extended beyond a simple reduction of mechanical complexity, in particular for devices in which the distribution of wrenches is of fundamental importance. An example is the automobile differential, in which an underactuated mechanism is commonly used to distribute the engine power to two wheels. The differential incorporates an additional DOF to balance the torque delivered to each wheel. The differential fundamentally operates on wheel torques instead of rotations; aided by passive mechanisms, the wheels can rotate along complex relative trajectories, maintaining traction on the ground without closed loop active control.

Some examples found in Robotics can be considered as underactuated systems such as: legged robots, underwater and flying robots, and grasping and manipulation robots.

In particular, underactuated robotic hands are the intermediate solution between robotic hands for manipulation, which have the advantages of being versatile, guarantee a stable grasp, but they are expensive, complex to control and with many actuators; and robotic grippers, whose advantages are simplified control, few actuators, but they have the drawbacks of being task specific, and perform an unstable grasp.

In an underactuated mechanism actuators are replaced by passive elastic elements (e.g. springs) or limit switches. These elements are small, lightweight and allow a reduction in the number of actuators. They may be considered as passive elements that increase the adaptability of the mechanism to shape of the grasped object, but can not and should not be handled by the control system.

The correct choice of arrangement and the functional characteristics of the elastic or mechanical limit (mechanical stop) ensures the proper execution of the grasping sequence. In a generic sequence for the grasping action, with an object with regular shape and in a fixed position, one can clearly distinguish the different phases, as shown in Fig. 1.

In Fig.1a the finger is in its initial configuration and no external forces are acting. In Fig.1b the proximal phalanx is in contact with the object. In the Fig.1c the middle phalanx, after a relative rotation respect to the proximal phalanx, starts the contact with the object. In this configuration, the first two phalanges can not move, because of the object itself. In Fig.1d, finally, the finger has completed the adaptation to the object, and all the three phalanges are in contact with it. A similar sequence can be described for an irregularly shaped object, as shown in Fig.2, in which it is worth to note the adaptation of the finger to the irregular object shape.

An underactuated mechanism allows the grasping of objects in a more natural and more similar to the movement obtained by the human hand. The geometric configuration of the finger is automatically determined by external constraints related with the shape of the object and does not require coordinated activities of several phalanges. It is important to note that the sequences shown in Fig. 1 and fig. 2 can be obtained with a continuous motion given by a single actuator.

Few underactuated finger mechanisms for robotic hands have been proposed in the literature. Some of them are based on linkages, while others are based on tendon-actuated mechanisms. Tendon systems are generally limited to rather small grasping forces and they lead to friction and elasticity. Hence, for applications in which large grasping forces are

Figure 1.

A sequence for grasping a regularly shaped object: a) starting phase; b) first phalange is in its final configuration; c) second phalange is in its final configuration; d) third phalange is in its final configuration.

Figure 2.

A sequence for grasping an irregularly shaped object: a) starting phase; b) first phalange is in its final configuration; c) second phalange is in its final configuration; d) third phalange is in its final configuration.

required, linkage mechanisms are usually preferred and this Chapter is focused to the study of the latter type of mechanisms.

An example of underactuation based on cable transmission is shown in Fig.3a, it consists of a cable system, which properly tensioned, act in such a way as to close the fingers and grasp the object.

The underactuation based on link transmission, or linkages, consists of a mechanism with multiple DOFs in which an appropriate use of passive joints enables to completely envelop the object, so as to ensure a stable grasp. An example of this system is shown in Fig.3.b. This type solution for robotic hands has been developed for industrial or space applications with the aim to increase functionality without overly complicating the complexity of the mechanism, and ensuring a good adaptability to the object in grasp.

Figure 3.

Examples of underactuation systems: a) tendon-actuated mechanism; b) linkage mechanism; c) differential mechanism; d) hybrid mechanism.

A differential mechanism, shown in Fig. 3c, is a device, usually but not necessarily used for gears, capable of transmitting torque and rotation through three shafts, almost always used in one of two ways: in one way, it receives one input and provides two outputs, this is found in most automobiles, and in the other way, it combines two inputs to create an output that is the sum, difference, or average, of the inputs. These differential mechanisms have unique features like the ability to control many DOFs with a single actuator, mechanical stops or elastic limits. The differential gear, commonly used in cars, distributes the torque from the engine on two-wheel drive according to the torque acting on the wheels. Applying this solution to robotic hands, the actuation can be distributed to the joints according to the reaction forces acting to each phalanx during its operation.

Hybrid solutions have been also developed and make use of planetary gears and linkages, together with mechanical stops or elastic elements. An example is shown in Fig. 3 d.

3. Design of underactuated finger mechanism

An anthropomorphic robotic finger usually consists of 2-3 hinge-like joints that articulates the phalanges. In addition to the pitch enabled by a pivoting joint, the head knuckle, sometimes also provides yaw movement. Usually, the condyloid nature of the human metacarpal-phalangeal joint is often separated into two rotary joints or, as in the case under-study, simplified as just one revolute joint.

Maintaining size and shape of the robot hand consistent to the human counterpart is to facilitate automatic grasp and sensible use of conventional tools designed for human finger placement. This holds true for many manipulative applications, especially in prosthesis and tele-manipulation where accuracy of a human hand model enables more intuitive control to the slave. Regarding to the actuation system in most of cases adopted solutions do not attempt to mimic human capabilities, but assume some of the pertinent characteristics of the force generation, since complex functionality of tendons and muscles that have to be replaced and somehow simplified by linear or revolute actuators and rotary joints.

The design of a finger mechanism proposed here uses the concept of underactuation applied to mechanical hands. Specifically, underactuation allows the use of n – m actuators to control n-DOFs, where m passive elastic elements replace actuators, as shown in Fig. 4. Thus, the concept of underactuation is used to design a suitable finger mechanism for mechanical hands, which can automatically envelop objects with different sizes and shapes through simple stable grasping sequences, and do not require an active coordination of the phalanges. Referring to Figs. 4 and 5, the underactuated finger mechanism of Ca.U.M.Ha. (Cassino-Underactuated-Multifinger-Hand) is composed by three links m_j for j = 1, 2, 3, which correspond to the proximal, median and distal phalanges, respectively. Dimensions of the simplified sketch reported in Fig.4 have been chosen according to the overall characteristics of the human finger given in Table 1. In particular, in Fig. 4, θ_iM are the maximal angles of rotation, and torsion springs are denoted by S₁ and S₂. In the kinematic scheme of Fig.5, two four-bar linkages A, B, C, D and B, E, F, G are connected in series through the rigid body B, C, G, for transmitting the motion to the median and distal phalanges, respectively, where the rigid body A, D, P represents the distal phalange. Likewise to the human finger, links m_j ( j = 1, 2, 3) are provided of suitable mechanical stoppers in order to avoid the hyper-extension and hyper-flexion of the finger mechanism. Both revolute joints in A and B are provided of torsion springs in order to obtain a statically determined system in each configuration of the finger mechanism.

Figure 4.

Simplified sketch of underactuated finger mechanism.

Phalanx	Length	Angle
m1	l1 = 43 mm	( 1M = 83°
m2	l2 = 25 mm	( 2M = 105°
m3	l3 = 23 mm	( 3M = 78°

Table 1.

Characteristics of an index human finger.

Figure 5.

Kinematic sketch of the underactuated finger mechanism.

3.1. Optimal kinematic synthesis

The optimal dimensional synthesis of the function-generating linkage shown in Fig. 5, which is used as transmission system from the pneumatic cylinder to the three phalanxes of the proposed underactuated finger mechanism, is formulated by using the Freudenstein’s equations and the transmission defect, as index of merit of the force transmission. The three linkages connected in series are synthesized as in the following by starting from the four-bar linkage, which moves the third phalanx.

3.1.1. Synthesis of the four bar linkage A, B, C,D

By considering the four-bar linkage A, B, C, D in Fig. 5, one has to refer to Fig.6 and the Freudenstein’s equations can be expressed in the form

$$R_1\cos {\epsilon }_i-R_2\cos {\rho }_i+R_3=\cos ({\epsilon }_i-{\rho }_i)i=1,2,3$$E1

with

$$R_1=l_2/a;R_2=l_2/c;R_3=(a_{}^2-b^2+c_{}^2+l{}_2^2)/2ac$$E2

where l₂ is the length of the second phalanx, a, b and c are the lengths of the links AD, DC and CB respectively, and ε_i and ρ_i for i = 1, 2, 3 are the input and output angles of the four-bar linkage ABCD.

Equations (1) can be solved when three positions 1), 2) and 3) of both links BC and AD are given through the pairs of angles (ε_i, ρ_i) for i = 1, 2, 3. According to a suitable mechanical design of the finger, (zoomed view reported in Fig.7) some design parameters are assumed, such as α = 50° for the link AD, γ = 40° and β ₁ = 25° for the link BC, the pairs of angles (ε₁ = 115°, ρ₁ = 130°) and (ε₃ = 140°, ρ₃ = 208°) are obtained for the starting 1) and final 3) configurations respectively. Angle ρ₃ is given by the sum of ρ₁ and θ_3M. Since only two of the three pairs of angles required by the Freudenstein’s equations are assigned as design specification of the function-generating four-bar linkage ABCD, an optimization procedure in terms of force transmission has been developed by assuming (ε₂, ρ₂) as starting values of the optimization, which correspond to both middle positions between 1) and 3) of links BC and AD respectively.

The transmission quality of the four-bar linkage is defined as the integral of the square of the cosine of the transmission angle. The complement of this quantity is defined “transmission defect” by taking the form

$$z\text{'}=\frac{1}{{\epsilon }_3-{\epsilon }_1}{\displaystyle {\int }_{\epsilon {}_1}^{\epsilon {}_3}{\cos }^2}\mu {}_1\text{d}\epsilon$$E3

where the transmission angle μ ₁ is expressed as

$${\mu }_1\text{=}{\cos }^{-1}\left(\frac{l_2^2+c^2-a^2-b^2-2l_{2}c\cos (\pi -\epsilon )}{2ab}\right)$$E4

The optimal values of the pair of angles (ε₂, ρ₂) are obtained through the optimization of the transmission defect z’. In particular, the outcome of the computation has given (ε₂ = 132.5°, ρ₂ = 180.1°) and consequently, a = 22.6 mm, b = 58.3 mm and c = 70.9 mm have been obtained from the Eq (1) and Eq. (2).

It is worth to note that, as reported from Fig.8a to Fig.8c, these plots give many design solutions, the choice can be related to the specific application and design requirements. In the case under-study parameters ε₂ and ρ₂ have been obtained in order to have the maximum of the mean values for the transmission angle. The transmission angle µ₁ versus the input angle ε for the synthesized mechanism is shown in Fig.8 d.

Figure 8, shows a parametric study of the a, b, c, parameters as function of ε₂ and η₂. The colour scale represents the relative link length. For each plot the circle represents the choice that has been made for ε₂ and ρ₂, by assuming the length a = 23 mm, for the case under-study.

Figure 6.

Sketch for the kinematic synthesis of the four bar linkage ABCD, shown in Fig. 5.

Figure 7.

Mechanical design of a particular used to define the angle α and the link length a of A, B, C, D, in Fig. 6.

Figure 8.

Map of the link length versus the angles ε₂ and ρ₂.; a) link AD, b) link DC; c) link BC, d) Transmission angle μ₁ versus angle ε for the moving link c.

3.1.2. Synthesis of the four-bar linkage B, E, F, G

The same method has been applied to the synthesis of the function-generating four-bar linkage BEFG. In fact, referring to Fig.9, the Freudenstein’s equations can be expressed in the form

$$R_1\cos {\psi }_i-R_2\cos {\phi }_i+R_3=\cos ({\psi }_i-{\phi }_i)i=1,2,3$$E5

with

$$R_1=l_1/d;R_2=l_1/f;R_3=(d_{}^2-e^2+f_{}^2+l{}_1^2)/2df$$E6

where l₁ is the length of the first phalanx, d, e and f are lengths of the links BG, GF and FE respectively, and _i and φ_i for i = 1, 2, 3 are the input and output angles of the four-bar linkage BEFG.

Likewise to the four-bar linkage ABCD, Eq. (5) can be solved when three positions 1), 2) and 3) of both links EF and BG are given through the pairs of angles ( _i, φ_i) for i = 1, 2, 3. In particular, according to a suitable mechanical design of the finger, the design parameters γ = 40°, β ₂ = 30° and = 10° are assumed empirically. Consequently, the pairs of angles ( ₁ = 80°, φ₁ = 60°) and ( ₃ = 140°, φ₃ = 190°) are obtained for the starting 1) and final 3) positions of both links EF and BG.

Since only two of the three pairs of angles required by the Freudenstein’s equations are assigned as design specification of the function-generating four-bar linkage BEFG, an optimization procedure in terms of force transmission has been carried out by assuming (₂, φ₂) as starting values of the optimization the middle positions between 1) and 3) of links EF and BG respectively. The transmission defect z ’ of the function-generating four-bar linkage BEFG takes the form

$$z\text{'}=\frac{1}{{\psi }_3-{\psi }_1}{\displaystyle {\int }_{\psi {}_1}^{{\psi }_{{}_3}}{\cos }^2}{\mu }_2\text{d}\psi$$E7

where the transmission angle ₂ is expressed as

$${\mu }_2\text{=}{\cos }^{-1}\left(\frac{l{}_1^2+f^2-d^2-e^2-2l_{1}f\cos (\pi -\psi )}{2de}\right)$$E8

The optimal values of the pair of angles (₂, φ₂) are obtained and the output of the computation gives ( ₂ = 115.5°, φ₂ = 133.7°). Consequently, d = 53.4 mm, e = 96.3 mm and f = 104.9 mm have been obtained from Eq. (5) and Eq. (6). Figure 10, shows a parametric study of the d, e, f, parameters as a function of ₂ and φ ₂. The colour scale represents the relative link length. For each plot the circle represents the choice that has been made for ₂ and φ₂, for the case under study. The diagram of the transmission angle µ₂ versus the input angle of the moving link EF of the synthesized mechanism BEFG is shown in Fig. 10 d.

Figure 9.

Sketch for the kinematic synthesis of the four-bar linkage BEFG.

Figure 10.

Map of the link length versus the angles ψ₂ and φ₂; a) link BG, b) link GF, c) link EF, d) transmission angle μ₂ versus angle ψ of the moving link EF.

3.1.3. Synthesis of the slider-crank mechanism EHI

Likewise to both four-bar linkages ABCD and BEFG, the offset slider-crank mechanism EHI of Fig. 11 is synthesized by using the Freudenstein’s equations, which takes the form

$$R_1(s{}_1-x_i)\cos {\lambda }_i+R_2\sin {\lambda }_i-R_3={(s{}_1-x_i)}^{2}i=1,2,3$$E9

with

$$\begin{array}{l}{R}_1=2g;\\ R_2=2g{o}_f;\\ R_3=g_{}^2+o_f^2-h^2\end{array}$$E10

where o_f is the offset, g and h are the lengths of the links EH and HI respectively, and x _i and λ _i for i = 1, 2, 3 are the input displacement of the piston and the output rotation angle of the link EH of the slider-crank mechanism EHI. Eq. (9) can be solved when three positions 1), 2) and 3) of both piston and link EH are given through the pairs of parameters (x _i, _i) for i = 1, 2, 3. In particular, according to a suitable mechanical design of the finger, the design parameters (x₁ = 0 mm, ₁ = 37°) and (x₃ = 75 mm, ₃ = 180°) are assumed empirically for the starting 1) and final 3) positions of both piston and link EH. The optimization procedure in terms of force transmission has been carried out by assuming as starting values of the optimization the middle position between 1) and 3) of the piston and link EH respectively. The transmission defect z ’of the function-generating slider-crank mechanism EHI takes the form

$$z\text{'}=\frac{1}{x_3-x_1}{\displaystyle {\int }_{x_1}^{x_3}{\cos }^2}{\mu }_3\text{d}x$$E11

where the transmission angle µ ₃ is expressed as

$${\mu }_3={\cos }^{-1}\left(\frac{{(s_1-x)}^2+o{}_f^2-g^2-h^2}{2gh}\right)$$E12

The optimal values of the pair of parameters (x₂, ₂) are obtained, and the outcome of the computation has given (x₂ = 47.5 mm, ₂ = 126.9°). Consequently, o_f = 43.4 mm, g = 35.7 mm and h = 74.7 mm have been obtained from the Eq. (9) and Eq. (10).

Figures 12a, 12b and 12c, show a parametric study of the parameters g, h and o_f, as a function of λ₂ and s₂. The colour scale represents the relative link length and for each plot the marked circle represents the choice that has been made for values ₂ and s_2. The diagram of the transmission angle µ₃ versus the input displacement x of the moving piston of the synthesized slider-crank mechanism EHI is shown in Fig. 12d.

Figure 11.

Kinematic scheme of the offset slider-crank mechanism EHI.

Figure 12.

Map of the link length versus angles λ₂ and s₂; a) link EI_, b) link HI_, c) link EH_, d) transmission angle μ₃ versus distance x of the moving link.

3.2. Mechanical design

Figure 13 shows a drawing front view of the designed underactuated finger mechanism. In particular, EHI indicates the slider-crank mechanism, ABCD indicates the first four-bar linkage, and DEFG indicates the second four-bar linkage. In order to obtain the underactuated finger mechanism, two torsion springs (S₁ and S₂) have been used at joints A and B and indicated with 1 and 2, respectively.

Aluminium has been selected for its characteristics of lightness and low-cost. It has the disadvantage of low hardness, therefore for the manufacturing of the revolute joints, ferrules have been considered. In particular, in Fig. 13, it is possible to note that the finger mechanism, which allows the finger motion, is always on the upper side of the phalanges. This is to avoid mechanical interference between the object in grasp and the links’ mechanism. Furthermore, the finger is asymmetric, this is due to the fact that is necessary to have a suitable side to mount the torsion spring.

Figure 13.

Mechanical design of the underactuated finger.

Each phalange has a flat surface to interact with the object to be grasped. This is to further consider force sensors to develop a suitable force control of the robotic hand prototype, as reported in Section 4.

The common operation of the four underactuated fingers gives an additional auto-adaptability of the Ca.U.M.Ha. robotic hand, because each finger can reach a different closure configuration according to the shape and size of the object to grasp. This behaviour is due to the uniform distribution of the air pressure inside the pneumatic tank and pushing chambers, as it will be described below.

3.3. Actuation and control

The lay-out of the electro-pneumatic circuit of the proposed closed-loop pressure control system is sketched in Fig.14, where the pressure P_OUT in the rigid tank is controlled by means of two PWM modulated pneumatic digital valves V₁ and V₂, which are connected in supply, at the supply pressure P_S, and in exhaust, at the atmospheric pressure P_A, respectively.

Thus, both valves V₁ and V₂ approximate the behaviour of a three-way flow proportional valve, which allows the pressure regulation in the tank. These valves are controlled through the voltage control signals V_{PWM 1} and V_{PWM 2,} which are modulated in PWM at 24 V, as it is required by the valves V₁ and V₂. These signals are given by a specific electronic board supplied at 24 V, which allows the generation of both signals V_{PWM 1} and V_{PWM 2} and the amplification at 24 V from the input signal V_PWM that lies within the range of [– 5; + 5 ] V. The PWM modulated control signal V_PWM is generated via software because of a suitable Lab-View program.

The feed-back signal V_F/B is given by the pressure transducer Tp with static gain K_T = 1 V/bar, which is installed on the rigid tank directly.

Thus, a typical PID compensation of the error between the input electric signal V_SET and the feed-back electric signal V_F/B is carried out through a PC controller, which is provided of the electronic board PCI 6052-E and a terminal block SCB-68.

Figure 14.

Scheme for the pressure control of the robotic hand prototype finger.

3.3.1. Experimental test-bed

The closed-loop pressure control system and a test bed of Fig.15 have been designed and built according to the scheme of Fig.14. In particular, this test-bed is mainly composed by: 1) and 2), two 2/2 (two-way/two-position) pneumatic digital valves of type SMC VQ21A1-5Y0-C6-F-Q; 3) a tank of type Festo with a volume of 0.4 lt; 4) a pressure transducer of type GS Sensor XPM5-10G, connected to an electronic board of type PCI 6052-E with the terminal block SCB-68, which is connected to the PC in order to generate the control signal V_PWM ; 5) a specific electronic board to split and amplify at 24 V the control signals V_{PWM 1} and V_{PWM 2}.

The electronic circuit of Fig.15b splits and amplifies the modulated electric signal V_PWM that comes from the PWM driver into the signals V_{PWM 1} and V_{PWM 2}, which control the digital valves V₁ and V₂ respectively. This circuit is composed by a photodiode FD, three equal electric resistors R₁, a MOSFET M and a diode D. In fact, the working range of the electronic board NI DAQ AT MIO-16E-2 is amplified from [–5 / +5 ] V to the working range [ 0 / + 24] V of the digital valves because of the electric supply at 24 V DC. Moreover, this signal is decomposed and sent alternatively to V₁ and V₂ because of the effects of the MOSFET M.

A suitable software in the form of virtual instrument has been conceived and implemented by using the Lab-View software, as shown in Fig.16. This solutions gives the possibility of using the electronic board NI DAQ PCI-6052-E for driving the PWM modulated pneumatic digital valves and acquiring both voltage signals V_SET and V_F/B of the proposed closed-loop pressure control system. Thus, the program can be considered as composed by three main blocks, where the first is for acquiring analogical signals through a suitable scan-rate, the second gives the PID compensation of the pressure error and the third one is for generating the PWM signal.

Figure 15.

Test-bed a) of the proposed closed-loop pressure control system and b) a scheme of the electronic circuit.

Figure 16.

Lab-View program for controlling the pressure in the tank through PWM modulated pneumatic digital valves.

3.3.2. Experimental results

The static and dynamic performances of the proposed closed-loop pressure control system have been analyzed by using the test-bed of Fig.15. Some experimental results in the time domain are reported in Fig.17 in order to show the effects of the proportional gain Kp of the PID compensator. In particular, the reference and output pressure signals P_SET and P_OUT are compared by increasing the values of the proportional gain Kp from 0.3 to 2.4, as shown in Figs.17a to 17d, respectively. Taking into account that the pressure transducer Tp is characterized by a static gain K_T = 1 V/bar, the pressure diagrams of P_SET and P_OUT show the same shape and values of the correspondent voltage diagrams V_SET and V_F/B, respectively. Moreover, the diagram of Fig.17c shows a good behaviour at high values of Kp, even if some instability of the system may appear, as shown in Fig.17d for Kp = 2.4. The experimental closed-loop frequency response of the proposed pressure control system has been carried out by using a Gain-Phase-Analyzer of type SI 1253. The Bode diagrams of Fig.18a and 18bhave been obtained for the periods of the PWM modulation, T = 50 ms and T = 100 ms, respectively.

Thus, the diagrams of the pressure signals P_SET and P_OUT versus time, which have been acquired through the Lab-View Data-Acquisition-System, are shown in continuous and dash-dot lines, respectively. In particular, Fig.19a and 19bshow both frequency responses of Fig.18a and 18bin the time domain for a P_SET sinusoidal pressure signal with frequency f = 0.1 Hz, average value Av = 3 bar rel and amplitude A = 2 bar rel. Likewise to the diagrams of Fig.20and still referring to the Bode diagrams of Fig.18, the frequency responses in the time domain for a P_SET with frequency f = 1.5 Hz are shown respectively in Fig.20a and 20b for T = 50 ms and T = 100 ms.

Figure 17.

Effects of the proportional gain: a) Kp = 0.3; b) Kp = 0.9; c) Kp = 1.8; d) Kp = 2.4.

Figure 18.

Closed-loop frequency responses of the proposed pressure control system for different periods of the PWM modulation; a) T = 50 ms; b) T = 100 ms.

Figure 19.

Frequency responses in the time domain for a sinusoidal P_SET with f = 0.1 Hz, Av = 3 bar rel and A = 2 bar rel: a) T = 50 ms; b) T = 100 ms.

Figure 20.

Frequency responses in the time domain for a sinusoidal P_SET with f = 1.5 Hz, Av = 3 V and A = 2 V: a) T = 50 ms; b) T = 100 ms.

4. The CaUMHa underactuated robotic hand: overall design

According to the mechatronic design proposed and described in Sections II and III, a prototype of Ca.U.M.Ha. robotic hand has been built and tested by using the experimental test-bed of Fig.21, which shows: 1) Ca.U.M.Ha. robotic hand prototype; 2) pneumatic cylinder; 3) PWM modulated pneumatic digital valves; 4) 3/2 pneumatic digital valve; 5) 5/2 pneumatic digital valve; 6) external block SCB-68; 7) electronic board to convert the signal V_PWM to V_{PWM 1} and V_{PWM 2}; 8) electronic board to control the thumb of the robotic hand.

The mechanical parts of Ca.U.M.Ha., i.e. underactuated fingers along with their linkage systems, palm and thumb, have been manufactured in aluminum, while the tank is made by steel.

Figure 21.

Prototype and experimental test-bed of the Ca.U.M.Ha. robotic hand, 1) Ca.U.M.Ha. robotic hand; 2) double-acting pneumatic cylinder; 3) two PWM modulated pneumatic digital valves; 4) 3/2 pneumatic digital valve; 5) 5/2 pneumatic digital valve; 6) terminal block SCB-68; 7) electronic board to split and amplify at 24 V the control signals V_{PWM 1} and V_{PWM 2}; 8) electronic board to split and amplify at 24 V both signals to control the thumb of the robotic hand.

5. Conclusions

In this Chapter the mechatronic design has been reported for the Ca.U.M.Ha. (Cassino-Underactuated-Multifinger-Hand) robotic hand. In particular, the underactuation concept is addressed by reporting several examples and kinematic synthesis and the mechatronic design have been developed for a finger mechanism of the robotic hand. As a result the Ca.U.M.Ha. robotic hand shows a robust and efficient design, which gives good flexibility and versatility in the grasping operation at low-cost. The kinematic synthesis and optimization of the underactuated finger mechanism of Ca.U.M.Ha. have been formulated and implemented. In particular, two function-generating four-bar linkages and one offset slider-crank mechanism have been synthesized by using the Freudenstein’ equations and optimizing the force transmission, which can be considered as a critical issue because of the large rotation angles of the phalanxes. A closed-loop pressure control system through PWM modulated pneumatic digital valves has been designed and experimentally tested in order to determine and analyze its static and dynamic performances. The proposed and tested closed-loop control system is applied to the Ca.U.M.Ha. robotic hand in order to control the actuating force of the pneumatic cylinders of the articulated fingers. Consequently, a force control of the grasping force has been developed and tested according to a robust and low-cost design of the robotic hand.