For Review Only. Preprint of a paper from the Industrial Robot, Volume 40, No. 4, pp , 2013

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1 Page of Revised manuscript for submission to : An International Journal July 0 Assisted Design of Linkage-Driven Adaptive Soft Fingers Abstract Purpose Adaptive grippers are versatile end effectors that mechanically adapt their shapes to the objects they seize allowing for soft and delicate grasps while still allowing for strong contact forces if needed and therefore they are well suited for industrial applications. This paper presents a software oriented approach to design optimal architectures of linkage-driven adaptive (often a.k.a underactuated) fingers with three degrees of freedom. Design/methodology/approach The user of the software presented in this paper can design planar underactuated fingers following defined constraints. The software uses an algorithm able to compute the internal and contact forces generated respectively in and by the finger, it is also able of automating the design of non-straight links to eliminate mechanical interferences, and includes results from a topological synthesis to generate all possible architectures. The mechanisms are evaluated for many criteria such as the volume of their workspaces, stability, force isotropy, stiffness of their grasps, and compactness. Findings This article introduces new designs of underactuated fingers for four different usages, and many of these variants are good candidates for a physical realization. One of the interesting results of this work is the recurrence of S variants coupled with torque amplifiers or closely resembling designs using many unrelated performance criteria. Originality/value This paper is the first to the best of the authors' knowledge to investigate the systematic design of underactuated fingers driven by linkages considering not one but dozens of mechanical architectures.. Introduction A simple task such as grasping an apple with a human hand requires many hardware components: up to bones, muscles and joints can be used to complete this task (Tubiana et al, ) which makes it a challenging effector to match. Underactuation can overcome these issues by minimizing the number of actuators and sensors required to drive the finger while maintaining good overall grasping performances (Laliberté and Gosselin, ; Dollar and Howe, 00; Kragten and Herder, 00). The drawback of this technique is that it does not allow manipulation of the seized object: a complex task such as reorienting an object inside the hand by moving one finger after the other is generally impossible. Underactuation in a robotic finger can be achieved using a transmission linkage and passive elements such as mechanical limits and springs (Birglen, 00). Because of the intrinsic compliance of adaptive grippers they are often referred to as soft, e.g. (Hirose and Umetani, ). They also elegantly solve the dilemma of having a gripper soft enough to handle delicate objects but strong enough for industrial applications by combining a rigid transmission linkage with compliant elements. A typical closing sequence of a two-phalanx underactuated finger is illustrated in Fig.. The actuator first rotates the proximal phalanx around the base pivot. Once contact is made with this phalanx, the actuator will overcome the preloading of the spring which was keeping the distal Preprint of a paper from the, Volume 0, No., pp. -, 0

2 Page of Revised manuscript for submission to periodical : An International Journal May, 00 phalanx aligned with the proximal one and begins the rotation of this second phalanx until it makes itself contact with the object. This shape adaptation can also be used in case of a collision with an object thus resulting in an overall softness of the gripper. "Take in Figure " While underactuation can also be achieved using tendons and pulleys, e.g. in (Hirose and Umetani, ; Carrozza et al, 00, Dollar and Howe, 00), focus is however placed on linkages in this paper. Compared to tendons, linkages have the advantage of being able to generate greater forces and can have links with a curved shape (often necessary to avoid mechanical interferences for instance) without any complex routing. They are however usually less compact though. The design of an underactuated finger with three degrees of freedom (DOF) driven by linkages is not trivial. For example, many possibilities exist for the connection pattern between the links of such a finger. In (Birglen, 00) was proved that there exists 0 of these connection patterns (called architectures) for a three DOF finger with revolute joints and a reasonable number of links. These architectures are illustrated in Fig.. The transmission linkage of each architecture is composed of either (class S, one architecture) or joints (classes A, B, C, D, E: a total of architectures). Since each of these joints can be either prismatic or revolute, the overall number of architectures is, ( ), still with a relatively limited number of links. Furthermore, the actuator can usually be located in several valid locations between two specific links for each of these architectures. In this paper, a variant refers to an architecture combined with an actuator located in a particular joint. Thus, one architecture can have several variants depending on which joint is actuated. The naming convention of the variants discussed in this paper (e.g. A-) is illustrated in Fig., while the digit after the dash indicates the actuated joint, also illustrated in the latter Figure. "Take in Figure (-column width)" A first question now arises: which variant is the most well-suited for a particular task? Since a very large number of variants exists and each of them is defined by up to geometrical parameters, the authors decided to write a computer software to help dealing with this task. This approach is inspired by the results obtained by many authors with serial and parallel robots (Zhuang et al, 00; Mayer St-Onge and Gosselin, 000; Macho et al, 00) as well as the famous GraspIt! simulation tool (Miller and Allen, 00) and generalizes the work presented in (Wu et al, 00). In our case, the search for the best variant unsurprisingly involves an optimization procedure and the evaluation of performance criteria. The criteria presented in this paper are taken from the literature and mostly involve the grasping of a few typical objects. The associated contact forces are then related to the stiffness of the grasp, its stability, the workspace and compactness of the mechanism while their relative magnitudes are also studied. With such a large number of possible underactuated fingers and without any a priori knowledge, the only valid design methodology is to consider all possible variants. Each one must be analyzed separately, optimized for the task at hand and finally, all of these optimal variants must be compared in order to select the best overall design. This represents a huge amount of work and lead the authors to write a computer program to handle as much as possible of this methodology. Knowing that simulations will be run with multiple objects for a large number of variants, a second question must be answered: how should the software be organized for these computations to be performed in a reasonable amount of time? This is not as trivial as it seems because one cannot rely on assuming that a fast computer will be used to deal with the huge amount of computation required. The answer to this question has been found to be the usage of vectorization in the program. This technique is defined as the process of converting a program performing one operation at the time to one working with many sets of data in parallel (simultaneously). Vectorization is however somewhat restrictive regarding the order in which the operations are performed: misplaced loops and conditional statements related to specific elements within the vectorized process must generally be avoided or the parallelism will be lost and computation times will increase. Therefore, Preprint of a paper from the, Volume 0, No., pp. -, 0

3 Page of Revised manuscript for submission to periodical : An International Journal May, 00 vectorization requires some similarities between the data so that each computation can be performed on all the data at the same time. As will be presented, these restrictions have heavily influenced the structure and the strategy used in the developed software. Because it is readily available in academia, the software developed by the authors to design underactuated fingers has been implemented with Matlab. A wireframe architecture is defined for each underactuated finger to represent the kinematics of the mechanism. This architecture is obtained from elementary classical mechanisms (referred to as classes), such as Watt's of Stephenson's linkages. Each joint is assumed to be either revolute or prismatic and an actuated joint is chosen. From the three-dof classes, 0 architectures are derived, and for each architecture, the actuator can be located in at least three joints. Considering all these choices leads to an astounding, ( ) different fingers.. Performance Criteria The performance of an underactuated finger cannot be judged solely upon inspection: the best finger for a defined task is the result of an optimization with performance criteria (Kragten and Herder, 00). Along with more traditional criteria related to the size of the workspace and characteristics of the contact forces (distribution, magnitudes, positiveness), compactness is also introduced in this paper with a novel approach.. Force isotropy Force isotropy can be measured by the normalized standard deviation of the norms associated to the contact forces (Doria and Birglen, 00). This criterion improves when the contact forces on each phalanx are similar, namely in situations where the grasping pressure is equally distributed amongst the phalanges. This could also arguably be another interpretation of the softness of a gripper. It is especially useful when grasping fragile objects since it penalizes unbalanced grasp where one contact force is significantly larger than the others. With -phalanx fingers (constituted by phalanges) grasping objects (indexed here by ) through the contact forces each assumed to be perpendicular to the phalanx, the overall average contact force on an object is: An approach where the performance index associated with the force isotropy would simply be the standard deviation of the norms of the contact forces would benefit the variants where the contact forces are lower for an actuator with a certain torque. To avoid this unpractical phenomenon, the standard deviation of the contact forces for each object is normalized with the average contact force on this object. The performance index associated with force isotropy then becomes: which is ideally zero.. Workspace The workspace of an underactuated finger can be seen as a function of the limits of the three variables defining its pose, e.g. the angles between the phalanges. Since the limits of a planar (D) workspace are a function of the three variables defining the pose, it is possible to evaluate the volume of the workspace with an analytical or numerical method. However, these methods are unsuitable for vectorization or for a generic approach usable on multiple architectures. To address these issues each of the three variables defining the pose of the finger (cf. Fig. ) is taken as a coordinate in a D reference frame. A volume is thus computed from the cloud of discrete points Preprint of a paper from the, Volume 0, No., pp. -, 0 () ()

4 Page of Revised manuscript for submission to periodical : An International Journal May, 00 inside this space which correspond to all the attainable poses. This simple method gives satisfactory results with respect to computing speed. The workspace is discretized with a resolution r (a -phalanx finger is thus evaluated for samples), the variable is the number of points where the mechanism can be physically assembled. A suitable performance index would then be Therefore, the performance index () is improved when its value is maximal (ideally one).. Compactness The compactness of an adaptive finger is measured in this paper by the ratio of the finger width by the sum of the lengths of its phalanges. The finger width is defined by computing the area of the convex polygon enclosing the finger while grasping an object divided by the vertical position of its highest joint (cf. Fig. ). The performance index proposed here is: where is the computed finger width, is the length of the phalanx, and is the number of objects considered. The smaller is, the more compact the finger (the better). "Take in Figure ". Grasp Stiffness The total force-in a particular direction-created on a grasped object by all the contact forces generated by the phalanges of an underactuated finger can be defined as the ratio between a small displacement in this particular direction of the object and the related actuator displacement times the actuator torque (using the principle of virtual work). To obtain an optimal grasp, conditions are then imposed: first, this ratio should be positive for object displacements opposed to the palm ( displacements) and towards the finger displacements, cf. Fig. ); second, its first derivative along should also be positive. The first condition ensures that the actuator is pushing the grasped object towards the palm of the hand and the opposing finger (Laliberté and Gosselin, ) while the second condition ensures that the object stays centered with two opposing fingers. It should be noted that this condition is usually demanding and can be discarded if a non-backdrivable actuator is used (Bégoc et al, 00). The performance index, which improves when increasing, is then defined as: where is a binary condition ensuring that the object if pulled away from the gripper will be brought back, is the horizontal ratio for the object o and is the vertical ratio still for the object o. It should be noted that this criterion is bounded in the program to prevent the optimization procedure from generating a mechanism very stiff for only one object.. Grasp stability As previously defined in (Birglen et al, 00), an adaptive finger is stable if all its contact forces are positive. However, adaptive fingers do not always have all their contact forces positive in every contact situation so the ease of implementing this criterion has the downside of being usually extremely severe for three DOF fingers. Still, this technique is also implemented here. To evaluate the stability of a variant, an object is grasped Preprint of a paper from the, Volume 0, No., pp. -, 0 () ()

5 Page of Revised manuscript for submission to periodical : An International Journal May, 00 providing up to Therefore, one has contact forces. If all these contact forces are positive, the mechanism is deemed stable. where H the Heaviside function, is the contact force on the object, are grasped. The performance increases with the index.. Overall performance Finally, all the previously discussed criteria are combined into a global performance index defined by () and is a vector containing the associated weights of the criteria. Since some criteria improve when their values go up and others when their values go down, the signs of the associated values in the weighting vector are defined to allow a minimization of the global performance index.. Internal procedures. Interference computations The first step in the software is to determinate the angles of the phalanges when an object is grasped. To do this, the program assumes a typical closing motion: first, the proximal phalanx is rotated by an angle (c.f. Fig. ) until one of the phalanges comes in contact with the object. Three cases are evaluated: this contact can be with the proximal, intermediate or distal phalanx. Once a contact is found, the algorithm is repeated with the next phalanx until all angles are known.. Dimension generation, resolution order and reassembly The mechanism generation is a fundamental step since it allows the optimization procedure to define the lengths of the links. This step might be assumed to be trivial by trying the lengths directly generated by the optimization procedure. However, this sometimes leads to links often too long or too short (Figs. a) and c)). "Take in Figure " Therefore, another approach has been implemented. Assuming two points and separated by a distance d to be connected at point by two links of lengths and and the vector from the origin of the coordinate system of the finger to point with, the unit vector is from to and vector is perpendicular to vector. The lengths and are bounded using and where and define the location of point, i.e. () The distances lengths and are then obtained from and as will be detailed in the next Section. Once the phalanges are positioned on the object from the previous step (Section.), the remaining joints are positioned one by one following a predetermined resolution order. Assuming two joints at points and with known position with, and joint with an unknown position, but distant of and respectively from joints and (Fig. ). The position of joint is found by calculating the length d between and and the angle between vectors and : Preprint of a paper from the, Volume 0, No., pp. -, 0 () ()

6 Page of Revised manuscript for submission to periodical : An International Journal May, 00 Angle Then, one has: Vector is then defined as: is finally However, some architectures cannot be solved with the technique discussed here, e.g. C, with a linkage similar to a -RPR planar parallel mechanism. Since the approach to solve these mechanisms requires numerical methods which are unsuited for vectorization, a few ( out of 0) architectures have been discarded.. Force Analysis Once all the links of a finger are positioned, it is possible to compute the internal and generated contact forces. If a rigid body such as this finger is in static equilibrium, it is possible to write the equations describing this equilibrium: () where is a loading vector accounting for the external contact forces, the springs keeping the phalanges aligned, etc. and is a vector containing the unknown internal forces. The matrix describes the geometry of the mechanism. Every link is modeled as a line connecting two endpoints, namely point to point with where and are the vectors from the origin of the coordinate frame to respectively points and At each one of these endpoints, an unknown force is applied: for and for. Since each link is in static equilibrium, one has: () This equation accounts for three lines in the matrix When an external contact acts on the link (i.e., is a phalanx) eq. () must be modified to: () where is the angle of this contact force, is the Euclidean norm of and is the location of the contact force along the phalanx with respect to its base joint (where the moments are computed). It is assumed that the external contact force on the phalanx is perpendicular to this phalanx. Finally, when all these equations are concatenated in matrix the latter is mostly made of ones and zeros but has large dimensions. Indeed, a three-dof finger might require up to a matrix elements). As an example, the sparse matrix of the A- finger described in Section.. is illustrated in Fig.. "Take in Figure ". Automated design of the shape of the links In some cases, mechanisms generated from the optimization procedure must use non-straight links to avoid mechanical interferences. If two links of the transmission mechanism intersect it is possible to overcome this interference by having each link moving in a different plane. However, if this situation happens between a Preprint of a paper from the, Volume 0, No., pp. -, 0 (0) () () ()

7 Page of Revised manuscript for submission to periodical : An International Journal May, 00 phalanx and a link of the transmission, this technique cannot be applied since the phalanges separate the mechanism area from the object. This can be solved using the technique described below. Assuming three points where with. By definition, and are the endpoints of the same link, and is higher than (needed to determine if is at the left or at the right). is the position of a phalanx joint, and is currently inspected for interference. One must ensure that: () This equation is presented in the D form to improve readability. It is evaluated for all the links of the mechanism and with all the phalanx joints. If this equation is not true at least once, an interference is detected and the shape of the link between and must be altered. Once all non-straight links have been identified, the instantaneous positions of all the different points causing interferences must be located in the reference frame of the link. This is done by using an affine transformation where is the angle of the link m with the horizontal axis when grasping case the object o. Thus, one has: with and the rotation matrix is defined as: Therefore, the inverse of the affine transformation matrix is: Figs. a) and b) illustrate examples of interferences leading to the design of non-straight links. Fig. shows the non-straight link capable of avoiding the interference in Fig. b). "Take in Figure " "Take in Figure " "Take in Figure " Once all the points causing the interferences are defined in the reference frame of the link, a convex polygon of minimal area including all these points is generated. This polygon is divided in two parts according to a line passing through the two endpoints of the link. One of the two branches will be in the object area, while the other will circumvent the area. Using eqs. () and (), it is possible to determine the correct branch which eliminates mechanical interferences. This defines the non-straight shape of the link under scrutiny. The process of the definition of the link shape is illustrated in Fig.. Point is the relative position of when object o is grasped. The illustrated link is positioned as if the finger is grasping object as in Fig.. The grasp with object for this finger is shown in Fig. b) and represents the problematic situation. The final shape of the link is constituted by the line segments of the polygon joining to.. Design inputs The user inputs related to the design of underactuated fingers strongly influenced the structure of the software by their effects on the execution speed of the software. Three different input types have been found to be Preprint of a paper from the, Volume 0, No., pp. -, 0 () () (0) ()

8 Page of Revised manuscript for submission to periodical : An International Journal May, 00 common amongst designer of underactuated fingers namely: ) external situations, ) chosen (wireframe) architecture, and ) design instructions.. External situations Ideally, the measure of the performance of a finger should be obtained with criteria that are not linked to arbitrarily chosen grasped objects. However, most of the criteria discussed earlier involve objects, and not intrinsic properties of the finger. Therefore, arbitrary choices in the selection of the objects have to be made: in this paper, the user is assumed to define in the software the shape and position of a few typical objects to be seized. The external situations encompass these objects for which the finger will be designed. These objects are assumed to be in a position fixed in space and with a shape well defined as well as constant (i.e. not deformable). With the software developed by the authors, the angles of the phalanges are a function of the grasped objects and the lengths of the phalanges. Wireframe architecture The wireframe architecture of the finger is the geometrical arrangement of how the joints of the finger are connected together. As discussed in Section., the connections between the points generate a specific sparse pattern in the matrix representing the mechanism. For a different pose or different dimensions of the same architecture, the sparse pattern remains identical but the numerical values of the elements of change. Since the values of the forces are often required in the computation of the performance indices, it would be required to build this matrix at each step of the optimization algorithm and for each mechanism. However, it has been decided that only one variant (an architecture with a defined actuation joint) will be evaluated at a time. This decision makes possible the efficient parallel construction of the matrix : even if each Cartesian position of a joint is different for each set of geometric parameters under scrutiny, the matrix sparse structure representing the joints connection is the same.. Automated design Once the dimensions of the mechanism have been generated, it is possible to perform predefined automated design steps. Even if this type of input is not represented by numbers, it is considered as an input nonetheless since it allows the implementation of common design practices. In our case, the automated design steps allows for instance to overcome mechanical interferences by designing non-straight links. This step directly influences some of the performance criteria such as compactness. Since it allows correcting major issues at a very early stage, the assisted design in the software provides much help with minimal efforts.. Results Three typical cases of weighting values for the optimization criteria have been selected to represent various practical challenges. The first one aims at creating a finger with isotropic (equal) contact forces as often as possible. Such finger would be useful in the food processing industry to grasp various fragile products such as fruits or vegetables. The second set of weighting values stands for industrial grippers where powerful and stiff grasps are required. The third one aims at creating an anthropomorphic finger by maximizing the compactness of the mechanism so it could fit inside a prosthetic glove. The weights of the global performance index has been set accordingly to each case and the program ran an optimization based on genetic algorithms to find the best design. Genetic algorithms have been selected to this task since they are reported in the literature to work well with these mechanisms and in our case, no a priori solution was known. The genetic algorithm used a population of,000 individuals with 0 elites for 00 generations. The optimal geometric parameters obtained are these illustrated in Figs. and onwards. In these figures, the green triangles are either the distal phalanx of a finger or a ternary link, actuation is indicated by a large red circle while the seized object is in yellow. Preprint of a paper from the, Volume 0, No., pp. -, 0

9 Page of Revised manuscript for submission to periodical : An International Journal May, 00. Case : priority on isotropy In this case, two facts have arisen from the results of the optimization: first, the overall best performing finger, A- (illustrated in Fig. a) is extremely close to an S finger. The sole difference is a part of the transmission linkage at its base that increases the actuation torque. Secondly, one can note a striking similarity between some of the best variants, which seems to indicate that the topology found is well suited for isotropic grasps. As seen in Fig. a), the variant A- is somewhat compact and has an acceptable standard deviation for the contact forces ( ). Once compared to the second best performing variant, namely D- (Fig. b)), the latter is wider but has a smaller standard deviation ( ). "Take in Figure " By looking closely at the base of the A- variant (Fig. 0), it is possible to see a mechanism acting as a torque amplifier compared to the very similar S architecture. One can see that a rotation of of the actuator (red circle) produces a rotation of at the lower joint of the finger connected to the ground. This mechanism increases the mechanical advantage of the actuation is known as a crossed double-rocker (McCarthy, 000), and unlike a crank-rocker, the actuated link can not perform a complete rotation. Small changes in the lengths of the links of the double-rocker mechanism could easily turn it into a crank-rocker while keeping the mechanical advantage. This discontinuous mechanism was unfortunately obtained for two reasons. First, the workspace criterion maximizes the number of attainable positions in the workspace regardless of their continuity (Fig. ). Second, the performance of the grasping of the finger is only evaluated for few poses, and none of these poses lied in the discontinuity of the workspace. If this would have been the case, it would not have been possible to assemble this finger and thus, it would have been rejected. Hence, the designer willing to use the A- variant should modify the part of the mechanism illustrated in Fig. 0 into a crankrocker. "Take in Figure 0" "Take in Figure ". Case : priority on rigidity Out of the four best fingers found for this case, three points can be noticed: first, the second phalanx is always a binary link. Second, the best finger, A- is again the best variant but with different geometric parameters. Third, several variants of the architecture D show a lot of potential for this case as in addition to good results, most of the links can be located under the base of the first phalanx, i.e. inside the palm of the finger. The variant A- has again outperformed the other designs in this case and it also obtained great compactness. As illustrated in Fig., the transmission mechanism of this variant is again similar to an S finger with a torque amplifier (note the ratio of rotations between the red circle joint and the lower joint connected to the ground). "Take in Figure " The second up to the fourth best performing fingers for this case were all D variants, where D- had the best performance and D- ranked fourth. These fingers have great stiffness and average compactness compared to the variant A-. By looking at these designs (Figs. a) to c)) it is possible to notice they have much in common: most of the transmission linkage and especially all their actuated joints are located under the base of the first phalanx, and they all use a long C-shaped link reaching to the distal phalanx. Since most of the transmission mechanism is located in an area where it is less prone to damage and can be protected, these fingers might indeed be well-suited for applications where robustness is a requirement. "Take in Figure " Preprint of a paper from the, Volume 0, No., pp. -, 0

10 Page 0 of Revised manuscript for submission to periodical : An International Journal May, 00. Case : priority on compactness Compact fingers are the most likely to be used in prostheses, since they can be covered by a cosmetic glove. Two types of architectures have shown interesting features: first, the architectures similar to S with torque amplifiers still perform well; second, the two topologies found in case appear again amongst the best designs. Once again, an architecture close to S but with torque amplifiers, namely A-, is the best overall. This mechanism is illustrated in Fig. a), along with the seventh (S-, Fig. b)), and the eighth best (A-, Fig. c)). The latter two have the advantage over the first one of having their actuator located closer to the ground which is usually desired from a practical point of view. "Take in Figure " Even if the optimization weights for this case are very different than for the isotropic case, the topologies presented in the case appear again. One of them, B- is surprisingly close to the previous design of case, as seen in Fig. b). C- (Fig. a)) ranked fourth, while B- ranked sixth. "Take in Figure ". Prototype Following the topological optimization described before a new prototype of underactuated finger was designed as illustrated in Fig.. It was used to demonstrate the effectiveness of the software to assist the designer of soft adaptive fingers and the prototype itself was used to grasp several objects with various shapes and stiffnesses ranging from metallic weights, boxes to fruits (see Fig. ). It is worth mentioning again that all these grasps are achieved without any sensors or control scheme besides powering the motor driving the finger and yet stable and soft envelopments of the objects are obtained. "Take in Figure -column width". Conclusion This paper is the first to the best of the authors' knowledge to investigate the systematic design of underactuated fingers driven by linkages considering not one but dozens of mechanical architectures. The computations leading to the topologies presented here were obtained with a genetic algorithm which optimizes each variant. Furthermore, the developed software is capable of complex tasks such as eliminating mechanical interferences. The mechanisms are evaluated for many criteria such as the volume of their workspaces, stability, force isotropy, stiffness of their grasps and compactness. This article has introduced new designs of underactuated fingers for four different usages, and many of these variants are good candidates for a physical realization. However, some details could still be improved. First, it has been observed that the ratio of the displacement of the third phalanx with respect to the displacement of the actuator when the third phalanx is about to touch the object is often small. This issue could cause problems when trying to close this phalanx due to the presence of the springs in the transmission linkage. Second, it has also been observed in many situations that the optimal topology for an architecture has an actuator located in an unpractical location, namely far from the base of the finger. Third, it might be useful to evaluate the variance of each criterion to generate weighting coefficients which truly gives equal importance to each of them. This could avoid the situations encountered in case where a finger with an overall better performance outperformed fingers with greater stiffness. Preprint of a paper from the, Volume 0, No., pp. -, 0 0

11 Page of Revised manuscript for submission to periodical : An International Journal May, 00 REFERENCES Bégoc, V., Durand, C., Krut, S, Dombre, E. and Pierrot, F. On the Form-Closure Capability of Robotic Underactuated Hands th International Conference on Control, Automation, Robotics and Vision, ICARCV'0, page, 00. Birglen, L., Type Synthesis of Linkage-Driven Self-Adaptive Fingers ASME Journal of Mechanisms and Robotics, Vol., No., 00. Birglen, L., Laliberté, T., and Gosselin, C., Underactuated Robotic Hands, Springer Special Tracts in Advanced Robotics, New-York, 00. Carrozza, M. C., Suppo, C., Sebastiani, F., Massa, B., Vecchi, F., Lazzarini, R., Cutkosky, M. R., and Dario P, The spring hand: Development of a selfadaptive prosthesis for restoring natural grasping, Autonomous Robots, Vol., pp. -, 00. Dollar, A.M., and Howe, R. D., A robust compliant grasper via shape deposition manufacturing IEEE/ASME Transactions on Mechatronics, Vol., No., pp. -, April 00. Dollar, A.M. and Howe, R.D. Simple, Robust Autonomous Grasping in Unstructured Environments, Proceedings 00 IEEE International Conference on Robotics and Automation, pp. -00, April 00. Doria, M., and Birglen, L., Design of an Underactuated Compliant Gripper for Surgery Using Nitinol ASME Journal of Medical Devices, Vol., No., 00. Hirose, S., and Umetani, Y., The development of soft gripper for the versatile robot hand Mechanism and Machine Theory, Vol., No., pp. -,. Kragten, G. A., and Herder, J. L., The ability of underactuated hands to grasp and hold objects Mechanism and Machine Theory, Vol., No., pp. 0-, March 00. Laliberté, T., and Gosselin, C. Simulation and design of underactuated mechanical hands Mechanism and Machine Theory, Vol., No., pp. -,. Larouche L.A., and Birglen, L., Design and first experiments with a new underactuated finger, Proceedings of the st International Workshop on Underactuated Grasping, UG00, Montréal, 00. Macho, E., Altuzarra, O., Amezua, E., and Hernandez, A. Obtaining configuration space and singularity maps for parallel manipulators Mechanism and Machine Theory, Vol., No., pp. 0 -, 00. Mayer St-Onge, B., and Gosselin, C., Singularity Analysis and Representation of the General Gough-Stewart Platform, International Journal of Robotics Research, Vol., No., pp. -, 000. McCarthy, J.M., Geometric Design of Linkages, Springer-Verlag, New-York, 000. Miller,A.T., and Allen, P.K., Graspit! a versatile simulator for robotic grasping, IEEE Robotics Automation Magazine, Vol., No., pp. 0-, 00. Tubiana, R., Thomine, J.M., and Mackin, E., Examination of the Hand and Wrist, Taylor & Francis, nd edition,. Wu, L.C., Carbone, G., and Ceccarelli, M. Designing an underactuated mechanism for a active dof finger operation, Mechanism and Machine Theory, Vol., No., pp. -, 00. Zhuang, H.J., Bauer, F., Khan, W.A., and Angeles, J., The development and implementation of a robot visualisation system for windows Proceedings of the CCToMM Symposium on Mechanisms, Machines, and Mechatronics, St-Hubert, QC, Canada, May -June st, 00. Preprint of a paper from the, Volume 0, No., pp. -, 0

12 Page of Revised manuscript for submission to periodical : An International Journal May, 00 Figure. Closing sequence of a two-dof finger (Birglen et al, 00). Figure. Architectures of three-phalanx adaptive fingers with less than nine revolute joints, possible locations of actuator(s) indicated by numerals. Preprint of a paper from the, Volume 0, No., pp. -, 0

13 Page of Revised manuscript for submission to periodical : An International Journal May, 00 Figure. Phalanx joints for an S finger and definitions required with the compactness criterion. Figure. Possible cases when assembling two links: a) too short, b) valid, c) too long. Figure. Sparse matrix A of a finger A-. Preprint of a paper from the, Volume 0, No., pp. -, 0

14 Page of Revised manuscript for submission to periodical : An International Journal May, 00 Figure. Interference between links with two fingers grasping an object: a) must go around, b) must go around. Figure. Different positions of point various objects. Figure.The case illustrated in Fig. b) with link interference. with the architecture in Fig. b) for Preprint of a paper from the, Volume 0, No., pp. -, 0 modified to avoid

15 Page of Revised manuscript for submission to periodical : An International Journal May, 00 Figure. Variants a) A- and b) D-. Figure 0. a) Torque amplifier and b) illustration. Figure. Discontinuous workspace for the optimally isotropic A- finger. Preprint of a paper from the, Volume 0, No., pp. -, 0

16 Page of Revised manuscript for submission to periodical : An International Journal May, 00 Figure.A A- variant capable of stiff grasps, a) geometry, b) links around the actuator, c) torque amplification illustration. Figure. Fingers shown during their closing motions: a) D-, b) D-, c) D-. Figure.Compact variants similar to S, a) A-, b) S-, c) A-. Preprint of a paper from the, Volume 0, No., pp. -, 0

17 Page of Revised manuscript for submission to periodical : An International Journal May, 00 Figure. Variants optimized for maximal compactness: a) C-, b) B-. Figure. Enveloping grasp of a metal cylinder, a fruit (peach) and a plastic box. Preprint of a paper from the, Volume 0, No., pp. -, 0