Mech. Sci., 2, 17–26, 2011 www.mech-sci.net/2/17/2011/ doi:10.5194/ms-2-17-2011 © Author(s) 2011. CC Attribution 3.0 License.
Mechanical
Sciences Open Access
A constrained optimization framework for compliant underactuated grasping M. Ciocarlie1 and P. Allen2 1
Willow Garage, Inc., Menlo Park, CA, USA Columbia University, New York, NY, USA
2
Received: 5 March 2010 – Revised: 16 June 2010 – Accepted: 19 August 2010 – Published: 8 February 2011
Abstract. This study focuses on the design and analysis of underactuated robotic hands that use tendons
and compliant joints to enable passive mechanical adaptation during grasping tasks. We use a quasistatic equilibrium formulation to predict the stability of a given grasp. This method is then used as the inner loop of an optimization algorithm that can find a set of actuation mechanism parameters that optimize the stability measure for an entire set of grasps. We discuss two possible approaches to design optimization using this framework, one using exhaustive search over the parameter space, and the other using a simplified gripper construction to cast the problem to a form that is directly solvable using well-established optimization methods. Computations are performed in 3-D, allow arbitrary geometry of the grasped objects and take into account frictional constraints. This paper was presented at the IFToMM/ASME International Workshop on Underactuated Grasping (UG2010), 19 August 2010, Montr´eal, Canada.
1
Introduction
In this study, we present a framework for the analysis and optimization of a class of passively adaptive underactuated robotic hands. In a broad sense, this is the task of replacing elaborate run-time algorithms (often requiring extensive sensor arrays for input and complex actuation mechanisms for execution) with off-line analysis, performed before the hand is even built. We focus on highly underactuated hand models, where the number of joints far exceeds the number of actuators, noting for example that recent studies have shown reliable grasping performance with even a single actuator for multiple fingers and up to 15 joints (Gosselin et al., 2008). In such cases, the only decision available at run-time is the placement of the hand relative to the target object, emphasizing the importance of the design optimization step. As the fingers are closing, the only computation is performed at an implicit level, by the actuation mechanism itself. Two important tools for achieving reliable performance when using passively adaptive hands are grasp analysis and Correspondence to: M. Ciocarlie (
[email protected]) Published by Copernicus Publications.
design optimization. The former traditionally focuses on a single grasp, and attempts to compute a measure of its stability under a given actuation mechanism. The latter aims to compute the parameters of the actuation mechanism itself, so that stability measures are maximized for a broad range of grasps. As the space of hand design parameters is extremely large, especially when taking into account kinematic considerations such as number of joints, number of fingers, finger link and palm shapes, we narrow the scope of this study as follows: – we start from a given kinematic design, with a potentially large number of joints grouped into multiple fingers; – assuming that each joint can be controlled independently, we create a set of stable grasps over a given group of objects. We refer to this set as the optimization pool; – we attempt to derive the design parameters of a a passively adaptive underactuation mechanism that increase the stability of the grasps in the set.
18
M. Ciocarlie and P. Allen: A constrained optimization framework for compliant underactuated grasping
There are multiple ways of achieving passive adaptation with a robotic hand design. One of the earliest examples, the Soft Gripper introduced by Hirose and Umetani (1978), used tendons for both flexion and extension. Ulrich et al. (1988) pioneered the use of a breakaway transmission mechanism which is now used in the Barrett hand (Barrett Technologies, Cambridge, MA). Birglen et al. (2008) presented a detailed and encompassing optimization study for underactuated hands, focusing mainly on four-bar linkages but with applications to other transmission mechanisms as well. Four-bar linkages were also used to construct the MARS hand (Gosselin et al., 1998), which later evolved into the SARAH family of hands (Laliberte et al., 2002). These studies have led to the construction of remarkably efficient grippers and hands. In the process, they have highlighted the fact that optimization of a highly underactuated hand is a complex problem; in other words, simple is hard! Our framework is based on the implementation of passively adaptive underactuation using the mechanics of a tendon-actuated hand combined with compliant, spring-like joints. This is based on the work of Dollar and Howe (2006), who optimized the actuation and compliance forces of a single tendon design. This design was later implemented in the Harvard Hand (Dollar and Howe, 2007), which we also use here as one of our case studies. We found the relative ease of constructing a prototype using this actuation paradigm particularly appealing, and believe it can lower the barriers for experimenting with new hand designs and disseminating research results. However, other actuation methods have their own merits, which must be considered in future iterations. 2
Problem formulation
The starting point for our optimization framework is the quasistatic equilibrium relationship that characterizes a stable grasp. We briefly review this formulation here; for more details we refer the reader to the analysis by Prattichizzo and Trinkle (2008). Consider a grasp with p contacts established between the hand and the target object. For any contact i, the total contact wrench ci must obey two constraints. First, the normal component must be positive (contacts can only push, not pull). Second, friction constraints must be obeyed. A common method is to linearize these constraints, by expressing ci as a linear combination of normal force and possible friction wrenches: ci βi , Fi βi
= Di βi ≥ 0
(1) (2)
where the matrices Di and Fi depend only on the chosen friction model, such as linearized Coulomb friction. The contact wrench is now completely determined by the vector of friction and normal wrench amplitudes βi , which will be computed as part of the grasp analysis algorithm. Mech. Sci., 2, 17–26, 2011
In general, a grasp, as a collection of multiple contacts, is in quasistatic equilibrium if the following conditions are satisfied: – contact forces are balanced by joint forces (hand equilibrium); – resultant object wrench is null (object equilibrium); – contact constraints are met for all contacts in the grasp. We can assemble the complete grasp description as follows: JTc Dβ = τ Gβ = 0 β, Fβ ≥ 0
(3) (4) (5)
where τ is the vector of joint forces, Jc is the Jacobian of the contact locations and G is the grasp map matrix which relates individual contact wrenches to the resultant object wrench. The matrices D and F bring together the individual contact constraint matrices Di and Fi for i = 1... p in block diagonal form. The column vector β contains all contact amplitudes vectors βi in block column form. In order to avoid the trivial solution where all forces in the system are zero, a constraint can be added requiring total actuator forces to sum to a prespecified level. In our framework, we use the term “stable” to refer to grasps that are in quasistatic equilibrium (all of the above constraints are met). It is important to note that, in practice, this is not a sufficient condition for achieving form-closure using the given actuation mechanism. However, it is a necessary prerequisite and, as such, we believe that optimizing a hand to achieve equilibrium under many configurations is a valuable step towards enabling a wide range of grasps. A possible future extension would be to also include a direct measure of grasp quality, according to one of the metrics that have been proposed in the literature (Ferrari and Canny, 1992; Prattichizzo and Trinkle, 2008). So far, this analysis applies to a hand design regardless of its actuation method. To adapt it to the case of underactuated hands, we must look in more detail at the joint force vector τ, which is a result of the actuation mechanism. We use the common tendon-pulley model, as used for example by Kwak et al. (2000), which assumes that a tendon travels through a number of routing points that it can slide through, but which force it to change direction as it follows the kinematic structure. As a result of this change in direction, the routing points are the locations where the tendon applies force to the links of the finger. This model is illustrated in Fig. 1, with the routing points marked with spheres. For clarity, the route shown is on the surface of the links, but in general the tendon can also be tunneled through the inside of the links. www.mech-sci.net/2/17/2011/
M. Ciocarlie and P. Allen: A constrained optimization framework for compliant underactuated grasping 3
19
Grasp analysis
Using this framework, we can analyze the stability of a given grasp by using object equilibrium as an optimization objective. We are interested in detecting finger slip on the object surface, as well as any unbalanced forces in the hand-object system; this analysis can be carried out both as the fingers are closing and after the grasp is complete. For a given hand posture and set of contacts, the goal is to determine the contact forces β and actuation forces δ that balance the system, or, if exact equilibrium is infeasible, result in the smallest magnitude wrench on the object: Figure 1. Illustration of tendon routing points (red spheres) as the
tendon follows a revolute joint (wire frame cylinder).
We assume that the hand contains a total of d tendons, each with multiple routing points across different links. In this case, joint forces can be expressed as: τ = JTd δ + θk
(6)
where Jd is the Jacobian of the tendon routing points and δ ∈ Rd is the vector of applied tendon forces. θ is a diagonal matrix of joint angle values and k is the vector of joint spring stiffnesses (without loss of generality, we assume 0 is the rest position for all springs). We now have a complete description of the equilibrium state of the grasp, expressed in Eqs. (3) through (6). In practice, one of the relationships comprised in this formulation is used as an optimization objective, rather than a hard constraint, with two important advantages. First, it provides more information for problems where all the constraints are not feasible in their exact form. Second, problems that have a solution in the exact form will often have an infinity of solutions; formulating an optimization objective allows us to choose an “optimal” one. Which of the above constraints is to be used as an optimization objective depends on the nature of the problem; as a result, this formulation is extremely versatile, and can be adapted to a number of practical problems in underactuated grasp analysis: – if the unknown variables include only the contact wrench magnitudes β (and implicitly all the individual contact wrenches ci ), we are computing whether a particular set of actuator forces results in a stable grasp; – if we extend the set of unknowns to also include the vector δ, we are trying to compute the best actuator forces for a grasp characterized by a particular set of contacts; – we can even extend the set of unknowns to include components of JTd or k, in which case we are computing the best hand design parameters for executing a given grasp (or set of grasps). www.mech-sci.net/2/17/2011/
minimize ||Gβ|| = βT GT Gβ subject to : h i JTc D − JTd β δ T = θk δ, β, Fβ ≥ 0
(7) (8)
This is a standard Quadratic Program, with linear constraints. The matrix that defines the quadratic component of the objective function is positive semidefinite by definition, as it is the product of the matrix G and its transpose. Therefore, the optimization problem is convex, so whenever the conditions are feasible, a global minimum can be determined. In this study, we use the Mosek (2010) package to solve all the optimization problems of this form. We can obtain one of three possible results: – the problem is unfeasible; this indicates that contact forces that obey the constraints can not be supported by actuator forces. The fingers will slip on the surface of the object; – the problem is feasible and a non-zero global optimum is found; the contacts are stable but some level of unbalanced force is applied to the object. If this force is not balanced externally (i.e. by interactions between the target object and another surface in the environment), the hand will have to reconfigure itself, also causing the object to move; – the problem is feasible and the global optimum is zero; the contacts are stable and produce a null resulting wrench. The hand-object system is stable in its current configuration. One traditional application of grasp analysis methods takes place at run-time. If the object to be grasped can be modeled (or recognized), this method can be applied to find stable grasps for execution. Another approach uses tactile sensors and proprioception to analyze the grasp currently being executed. However, both of these options require extensive sensing arrays and control options that might not be available in a highly underactuated hand. We believe that a very promising alternative is to use grasp analysis as part of an off-line optimization routine, to improve the overall performance of the hand. We explore this option next. Mech. Sci., 2, 17–26, 2011
20
Hand design optimization
– a “numerical” approach, where, for each possible set of design parameter values, each grasp in the pool is analyzed independently. A global measure of grasp stability is computed by summing (or averaging) the result Mech. Sci., 2, 17–26, 2011
1 0.9 0.8
Stiffness ratio
Using our framework to tackle hand design optimization implies a different approach than the previous section. Grasp analysis focuses on a single grasp at one time, and aims to compute the optimal contact or actuator forces specific to that grasp. In contrast, the study of hand design parameters normally implies solving an optimization problem over a set of grasps. The first step of the hand optimization method is thus to create a batch of grasps that we expect the hand to be able to perform. We refer to this set as the optimization pool. The task of defining the grasp optimization pool, and executing the optimization procedure, is performed in a simulated environment, which seems natural in the context of design analysis performed before the hand is constructed. We use our publicly available GraspIt! simulation engine (Miller and Allen, 2004); all the tools presented here have been integrated in, and are now available as part of GraspIt!. The optimal contact and actuator forces specific to each grasp in the pool are still unknown; now they are joined by a set of unknowns representing actuation parameters which are shared by all the grasps in the pool. A key problem is the decision of which design parameters are the focus of the optimization. We have decided to focus on the parameters of the actuation mechanism, such as tendon route and joint spring stiffness. A second problem, intrinsically related to the first one, regards the method used to perform the optimization. The ideal scenario would intuitively be to assemble a global optimization problem, allowing the direct computation of the optimal design parameters over the entire grasp pool. However, such a global approach is not always possible to implement. Consider for example the problem of optimizing the location of the tendon routing points on their respective links. The effects of the tendon route on the equilibrium condition are encapsulated in the Jacobian of the routing points, Jd . Changing the location of a routing point on a link has a highly non-linear effect on Jd . Furthermore, even if we had a linear relationship between tendon route parameters and the routing point Jacobian, the result must be multiplied by the unknown vector of actuation forces δ. As a result, computing both actuation forces and optimal tendon route parameters results in a higher order equality constraint which can not be handled by the same optimization tools. The general case therefore enables us to quantify a given hand design (by separately computing the quality of each individual grasp in the optimization pool), but not to directly compute a global optimum for the design parameters. We envision two possible solutions to this problem:
0.7 0.6 0.5 0.4 0.3
Fig. 2. 2. MODEL HARVARD Figure Model ofOF theTHE harvard hand andHAND object AND modelsOBJECT used for MODELS USED FOR GRASP POOL GENERATION. grasp pool generation.
for each grasp, and the parameter values that yield the The best idealresult scenario are would chosen.intuitively be to assemble a global optimization problem, allowing the direct computation of the optimal design parameters overapproach, the entire grasp – a “global optimization” wherepool. newHowconever,straints such a global approach not always possible(containing to impleare added to theisglobal formulation ment. design parameter values as explicit unknowns) in order to cast itforasexample a solvable problem, the suchlo-as Consider the optimization problem of optimizing a Linear or Quadratic Program. Apart from computacation of the tendon routing points on their respective links. thisroute method alsoequilibrium has the advantage The tional effectsefficiency, of the tendon on the conditionof producing a provable global optimum. Its drawback are encapsulated in the Jacobian of the routing points, Jd .is that the additional constraints Changing theintroduction location of a of routing point on a link has alimits highlyits applicability to a subset of possible hand designs. non-linear effect on Jd . Furthermore, even if we had a linear relationship between tendon route parameters and the routing pointNumerical Jacobian, the result must be multiplied by the unknown 4.1 optimization vector of actuation forces δ. As a result, computing both acWe usedforces the numerical approach investigate howresults grasping tuation and optimal tendontoroute parameters in performance can be improved by which changing design paramea higher order equality constraint can not be handled ters for same the Harvard Hand tools. (Dollar and Howe, 2007). Figure 2 by the optimization shows the GraspIt! model of this hand, which uses a single The general therefore enables us to quantify a given actuator to drivecase eight joints that articulate four fingers. We hand design (by separately computing the quality of each infocused on two design parameters: the actuator torque ratio dividual grasp in the optimization pool), but not to directly between the proximal and distal joints of each finger (circled compute a global optimum for the design parameters. We in the image), and the spring stiffness ratio between the same envision two possible solutions to this problem: joints. These parameters are determinant for the behavior of the as they affect both where, the posture of the hand set before • hand, a “numerical” approach, for each possible of design parameter grasp in the pool is analyzed touching an objectvalues, and theeach forces transmitted after contact is independently. A global measure of grasp stabilitycombinais commade. In particular, we investigated all possible putedranging by summing (or to averaging) the result forfor each tions from 0.2 1.0 (in steps of 0.2) thegrasp, torque and the yield the best for result choratio andparameter from 0.1 values to 1.0 that (in steps of 0.1) theare stiffness sen. ratio. optimization pool consisted 2000new possible grasps •The a “global optimization” approach,ofwhere constraints distributed evenly across 5 object models (glass, flask, toy are added to the global formulation (containing design paplane, mug and phone receiver). For each object, the set rameter values as explicit unknowns) in order to cast it as aof grasps was created by problem, samplingsuch multiple approach directions solvable optimization as a Linear or Quadratic on the faces of the object bounding box and aligning the hand Program. Apart from computational efficiency, this method with the axes of the box. For each torque and stiffness also has the advantage of producing a provable globalcombioptination, wedrawback used the analysis method presented in the previous mum. Its is that the introduction of additional consection all the candidate grasps reported the number straintsfor limits its applicability to a and subset of possible hand ofdesigns. them that are stable throughout their execution. To enable direct comparison across different objects, each set of results was normalized to a scale of 0 to 1 through division by the www.mech-sci.net/2/17/2011/
0.2 0.1 0.2
1
Stiffness ratio
4
M. Ciocarlie and P. Allen: A constrained optimization framework compliant graspingOptimization 4 M.for Ciocarlie and underactuated P. Allen: A Constrained
0.8
0.6
0.4
0.2 0.2
0.4
To
Fig. 3. T THE LIK DARKER GRASPS.
Numeric
We used performa ters for th shows th actuator focused o between in the im joints. T the hand touching made. In tions ran ratio and ratio. The op distribute plane, m grasps w on the fac with the a nation, w section fo
M. Ciocarlie and P. Allen: A Constrained Optimization Framework Grasping M. Ciocarlie and P. Allen: A constrained optimization frameworkfor forCompliant compliant Underactuated underactuated grasping nstrained Optimization Framework for Compliant Underactuated Grasping
of them that are stable throughout their execution. ToGlass enable direct comparison across objects, each set of results Alldifferent objects 1 was normalized to a scale of 0 to 1 through division by the 0.9 maximum number of grasps found for that object. Figure 3 0.8 shows the results for each of the objects, as well as their average over 0.7 the entire set. 0.6 Torque ratio The contour maps reveal which areas of the optimization Plane range offer0.5the best performance; in particular they suggest 0.4 a torque ratio of 0.6 and a stiffness ratio of 0.3. The over0.3 all resemblance between the patterns suggests that the global OBJECT optimum of0.2the average profile is a good compromise, likely to work well 0.1 on all objects. However, the patterns exhibit 0.2 0.4 0.6 0.8 1 Torque ratio enough variation to illustrate the importance of performing Torque ratio this analysis over a large set of models, spanning a wide Phone Mug Flask e a global range of shapes and grasping scenarios. We note that our ion of the ratio is in agreement with the value found in the optorque ol. Howtimization study carried out before the construction of the to impleHarvard Hand prototype (Dollar and Howe, 2006). Our analysis consisted of a total of 20,000 grasps for each g the lo- (400 candidates for each of the 50 combinations of object Torque ratio Torque ratio Torque ratio ive links. force and torque ratios); the typical time spent per object condition Fig. 3.3.THE EFFECT OF HAND DESIGN PARAMETERS ON Figure The effect of hand design parameters on the likelihood was 15 minutes. This performance suggests the possibility of oints, Jd . THE LIKELIHOOD OF OBTAINING Ameans STABLE GRASP. A of obtaining a stable grasp. a darker color a higher number scaling up to significantly larger test sets as, unlike run-time s a highly DARKER COLOR MEANS A HIGHER NUMBER OF STABLE of stable grasps. analysis, GRASPS. off-line optimization can benefit from a time budget d a linear he routing of weeks or months, as well as massively parallel computunknown ing architectures. In addition, it is found possible set 3 maximum number of grasps for to thatextend object.the Figure g both Numerical Optimization ofacanalyzed parameters, including for example link lengths, results in shows the results for each of the objects, as well as their avnumber of links, etc. e handled erage over the entire set. the numerical approach to areas investigate how grasping Theused contour maps that reveal which ofisthe optimization While We our results show this approach feasible (at performance can be improved by changing design offer the best performance; in particular theyparamesuggest least forrange a comparable optimization domain), more advanced for the Harvard Hand and ratio Howe, 2007). Figure 2 y a given a ters ratio of 0.6 and a(Dollar stiffness 0.3. ExamThe overnumericaltorque optimization algorithms can also beofused. the GraspIt! model of this hand, which uses a single f each in- allshows resemblance between theorpatterns suggests that thenuglobal ples include simulated annealing gradient ascent actuator to the driveaverage eight joints that articulate four using fingers.likely We o directly optimum of profile is a good compromise, computation ofdesign the gradient. While such algorithms focused on two parameters: the actuator torque ratio ters.merical We to work well on all objects. However, the patterns exhibit also present disadvantages (like threat in (circled local between the proximal and the distal jointsofofstopping each finger enough variation to illustrate the importance of performing the are image), and the better spring stiffness ratio same optima), in they generally equipped tobetween handle the larger analysis over a large set of models, spanning a wide ble set of this joints. These parameters are determinant for the behavior problems. Another option, discussed in detail inWe thenote nextthat sec-of analyzed range of shapes and grasping scenarios. our the hand, as they affect both the posture of the hand before exhaustive search by with usingthea value different problem y is tion, com- avoids torque ratio is in agreement found in the touching an object and the forces transmitted after contact opis formulation. ch grasp, timization study carried out before the construction of the
21 5
Stiffness ratio
1
0.8
tendon
0.6
pulley
0.4
Stiffness ratio
0.2
0.2
0.4
0.6
0.8
1
links
Stiffness ratio
1
elastic joints
actuator
0.8
0.6
0.4
link i
li
0.2
1
1
0.8
0.8
0.8
0.6
0.4
0.2
0.2
are cho-
Stiffness ratio
1
Stiffness ratio
Stiffness ratio
0.2
0.6
0.4
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0.6
0.4
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
made. In particular, we investigated all possible combina-
1
Harvard Hand from prototype tions ranging 0.2 to(Dollar 1.0 (in and stepsHowe, of 0.2)2006). for the torque Global Optimization Our and analysis a total 000forgrasps for each ratio fromconsisted 0.1 to 1.0of(in stepsof of20 0.1) the stiffness object (400 candidates for each of the 50 combinations of ratio. onstraints Inpaorderforce to illustrate ourratios); global optimization approach tograsps thewas andoptimization torque typical time spent per object esign The poolthe consisted of 2000 possible 15distributed min. Thisevenly performance possibility of scaling problem, we across will build up models athe concrete example, ast ithand as a design 5suggests object (glass, flask, toy to significantly largerreceiver). testmodel sets as, unlike Quadratic mug phone For each run-time object, theanalysis, set of using asupaplane, test bed aand two-finger (which we will refer benefitmultiple from time budget of weeks s method grasps optimization was created bycan approach directions to as a off-line gripper, rather than asampling hand). We awill first describe or months, as well as massively parallel computing obalthe option the faces of the object bounding box and aligning the hand starting model, then discuss the reasons for choosingarchitecthis onal con- tures. with the axes of theit box. For eachtotorque and stiffness combiIn addition, is possible extend the set of analyzed particular design. ible hand parameters, nation, we used the analysis method presented in thenumber previousof including for example link lengths, The basic gripper model is presented in Fig. 4. A single section for all the candidate grasps and reported the number links, etc.
tendon provides flexion forces for both fingers, which are coactuated using a pulley mechanism, similar to the one used in the Harvard Hand. Note that the pulley allows one finger to continue flexing even if the other finger is blocked by contact www.mech-sci.net/2/17/2011/ with the object. Extension forces are provided by spring-like joints. In practice, these joints can be constructed using a compliant, rubber-like material; this design enables
r joint i
li+1
li și
r d
Fig. 4. OVERVIEW AND DETAILED JOINTfor DESCRIPTION Figure 4. Overview and detailed joint description two-fingered gripper used as optimization case study. FOR TWO-FINGERED GRIPPER USED AS OPTIMIZATION CASE STUDY.
While our results show that this approach is feasible (at leastjoints for a comparable optimization domain), moreare advanced distal to flex even when proximal joints stopped. optimization algorithms can also ExamWenumerical assume that the kinematic behavior is be thatused. of ideal revples include simulated annealing or gradient ascent using nuolute joints, with the center of rotation placed halfway bemerical computation of the gradient. While such algorithms tween the connected links. The tendon itself follows a route also present disadvantages (like the threat of stopping in local in the flexion-extension plane of the gripper. The essentially optima), they are generally better equipped to handle larger two-dimensional design prevents theinlinks this problems. Another option, discussed detailfrom in theleaving next secplane the application forces. problem However, tion,without avoids exhaustive search of by external using a different theformulation. tendon route inside this plane is not specified, and is one of the targets of the optimizations. Figure 4 also shows in detail the design parameters of the 4.2 Global optimization gripper. The tendon route is determined by the location of theInentry exit points each link; moreapproach specifically, orderand to illustrate our for global optimization to thethe hand design build up a concrete parameter thatproblem, we use iswe thewill distance between the example, tendon enas point a test and bed athe two-finger model (whichthe welink will and referthe tryusing or exit connection between to as a gripper, rather than a hand). We will first describe joint. We also make the simplifying assumption that, for a the i,starting model, then discuss reasonslink for choosing joint the exit point from the the proximal and the this entry particular design. point in the distal link have the same value for this parameThe basic gripper model is presented in the Fig.joint 4. Aissingle ter, which we call li . The current value of θi . r is tendon provides flexion forces for both fingers, which are cothe joint radius (shared by all the joints), while the length of actuated using a pulley mechanism, similar to the one used theinlinks is denoted by d. the Harvard Hand. Note that the pulley allows one finThetoreason forflexing usingeven this ifdesign andfinger formulation that ger continue the other is blockedisby they yieldwith a compact and, Extension more importantly, relationcontact the object. forces arelinear provided by ship betweenjoints. the construction andbe theconstructed joint forces spring-like In practice, parameters these joints can applied the rubber-like tendon. Ifmaterial; we consider the parameter using athrough compliant, this design enables distal joints to flex even when proximal joints are stopped. vector p = [l0 l1 l2 l3 l4 l5 r d], we obtain a relationship of assume that the kinematic behavior is that of ideal revtheWe form: olute joints, with the center of rotation placed halfway between the+connected τ= δ(Bp a) + θk links. The tendon itself follows a route(9) in the flexion-extension plane of the gripper. The essentially
where the matrix B ∈ R8x8 and the vector a ∈ R8 depend only on the joint values θ0 ...θ5 .Mech. A sketch the derivation Sci., for 2, 17–26, 2011 of these matrices is presented in the Appendix. Furthermore, since we are using a single tendon, we can normalize its value
22
M. Ciocarlie and P. Allen: A constrained optimization framework for compliant underactuated grasping 6 M. Ciocarlie and P. Allen: A Constrained Optimization
two-dimensional design prevents the links from leaving this plane without the application of external forces. However, the tendon route inside this plane is not specified, and is one of the targets of the optimizations. Figure 4 also shows in detail the design parameters of the gripper. The tendon route is determined by the location of the entry and exit points for each link; more specifically, the parameter that we use is the distance between the tendon entry or exit point and the connection between the link and the joint. We also make the simplifying assumption that, for a joint i, the exit point from the proximal link and the entry point in the distal link have the same value for this parameter, which we call li . The current value of the joint is θi . r is the joint radius (shared by all the joints), while the length of the links is denoted by d. The reason for using this design and formulation is that they yield a compact and, more importantly, linear relationship between the construction parameters and the joint forces applied through the tendon. If we consider the parameter vector p = [l0 l1 l2 l3 l4 l5 r d], we obtain a relationship of the form: τ = δ(B p+ a) + θk
(9) 8x8
8
where the matrix B ∈ R and the vector a ∈ R depend only on the joint values θ0 ...θ5 . A sketch for the derivation of these matrices is presented in the Appendix. Furthermore, since we are using a single tendon, we can normalize its value without loss of generality to δ = 1N. The joint force relationship, and by extension the grasp equilibrium conditions, are now linear in all of the unknowns. Having established the general characteristics of the gripper, the next step was to generate a pool of grasps over which to optimize its performance. We created a kinematic model of the gripper for the GraspIt! environment, assuming each joint could be controlled independently. Then, using the interaction tools in the simulator, we manually specified a number of grasps over a set of 3-D models of common household objects. The set comprised 70 grasps distributed across 15 objects; the process is illustrated in Fig. 5. Each grasp was defined by the set of gripper joint angles, the location of the contacts on each link, and the contact surface normals, resulting in a purely “geometric” description of a grasp, with no reference to the actuation mechanism. Most of the grasps in the pool used different postures for the two fingers of the gripper. We added to the set the “transpose” of each grasp, obtained by rotating the gripper by 180◦ around the wrist roll axis, reversing the roles of the left and right finger. The complete optimization pool thus comprised 140 grasps. The inclusion of the transposed grasps also ensured that the final optimized parameters, presented in the next section, where symmetrical, with identical results for both fingers. A key restriction during the creation of the optimization pool was that all the grasps therein were required to have form-closure. GraspIt! integrates a number of analysis tools Mech. Sci., 2, 17–26, 2011
model d T Jcj D
G j
β , F
Fig. 5. 5.GRIPPER MODEL FOR GRASPIT! ANDofEXAMPLES Figure Gripper model for graspit! and examples grasps from OFoptimization GRASPS FROM the pool.THE OPTIMIZATION POOL.
for establishing the form-closure building the without loss of generality to δ = 1Nproperty . The jointbyforce relationGrasp Wrench Space, as described by Ferrari and Canny ship, and by extension the grasp equilibrium conditions, are (1992). Thisinformulation is equivalent to the ability of a set now linear all of the unknowns. of contacts to apply a null resulting wrench on the object Having established the general characteristics of the gripwhile satisfying contact friction constraints, but disregarding per, the next step was to generate a pool of grasps over which any kinematic or actuation constraints. to optimize its performance. We created a kinematic model grasp in our optimization pool, assuming we can apply ofFor the each gripper for the GraspIt! environment, each the equilibrium formulation using the actuation mechanism joint could be controlled independently. Then, using the inmodel described in this section: teraction tools in earlier the simulator, we manually specified a num ber j Tof grasps over a j j j set of j 3Dj models of common household Jc D β = B p+ a + θ k (10) objects. The set comprised 70 grasps distributed across 15 j j G βthe process = 0 is illustrated in Fig. 5. Each grasp was (11) objects; j j by j the set of gripper joint angles, the location of the defined β, Fβ ≥ 0 (12) contacts on each link, and the contact surface normals, rewhere the superscript denote the number sultingwe in use a purely ”geometric”j to description of index a grasp, with of particular grasp from the optimization pool that we are nothe reference to the actuation mechanism. referring to.the The unknowns are the grasp contact forces for β j, Most of grasps in the pool used different postures the parameter p and the vector spring the hand two fingers of thevector gripper. We added to theof setjoint the ”transstiffnesses k. Note that p and k do not have a superscript ◦ pose” of each grasp, obtained by rotating the gripper by 180as they are shared between all the grasps in the pool. around the wrist roll axis, reversing the roles of the left and To finger. obtain The a global optimization problem, wecomprised assemble right complete optimization pool thus these relationships in block over the grasps entire also poolenof 140 grasps. The inclusion of form the transposed j grasps. Thethe matrices for individual grasps presented (Jcj )T D j , B θ j, sured that final optimized parameters, in ,the j G andsection, F j are assembled in block diagonal form inresults the matrinext where symmetrical, with identical for ˜ T ˜ and F, ˜ G ˜ ˜ θ, ˜ respectively. The vectors β j and a j ces D, B, bothJcfingers. are assembled in block columns in the vectors β˜ and a˜ . FiA key restriction during the creation of the optimization nally, the joint equilibrium condition (10) assembled for all pool was that all the grasps therein were required to have the grasps in the pool becomes the optimization objective: form-closure. GraspIt! integrates a number of analysis tools ˜ property for establishing the form-closure by building the h i β Grasp Wrench Space, as described by Ferrari ˜ T ˜ ˜ ˜ minimize Jc D − B − θ p − a˜ subject to: and Canny formulation is equivalent to the ability of a set (1992). This k of contacts to apply a null resulting wrench on the object ˜ β˜ = contact while satisfying friction constraints, but disregarding G 0 (13) any ˜kinematic or actuation constraints. ˜ ˜ β, Fβ ≥ 0 (14) For each grasp in our optimization pool, we can apply pmin ≤ p ≤ pmax (15) the equilibrium formulation using the actuation mechanism kmin ≤ k ≤ kmax (16) www.mech-sci.net/2/17/2011/
where w of the pa referring the hand stiffness they are To ob these re grasps. θ j , Gj matrices and aj a ˜ . Final a all the g
minimiz
˜ β,
pmin ≤
kmin ≤
The min rameters ical con section. We n gram th straints having f nism, w ingly. A optimum
Constru
The fina tion of a This req could be limit of was insi Using be part o (link len can be k generati
The minimum and maximum values for the construction parameters p and k can be set to reflect constraints in the physical construction of the gripper, as we will show in the next section. We note that the result is again a convex Quadratic Program that, by construction, always accepts a solution: constraints (13) and (14) are equivalent to each individual grasp having form-closure independently of the actuation mechanism, which we ensured by building our grasp pool accordingly. As a result, the problem can be solved and a global optimum can be computed. 4.3
Construction of an optimized gripper
The final step of using our framework was physical construction of a gripper according to the results of the optimization. This required setting limits for the optimized parameters that could be implemented in practice. In particular, we used a limit of −5 mm ≤ li ≤ 5 mm ∀ i to ensure that the tendon route was inside the physical volume of each link. Using a fixed grasp pool affects the parameters that can be part of the analysis: if the values of r (joint radius) and d (link length) can change as part of the optimization, the result can be kinematically different than the model used for the generation of the grasp pool. Taking this aspect into account, we performed two optimizations. For the first one (referred to as Optimization 1), we fixed the values to r = 5 mm and d = 20 mm. For the second one (Optimization 2) we set the limits 3 mm ≤ r ≤ 10 mm and 15 mm ≤ d ≤ 22 mm. We also added the constraint r + d = 25 mm, to ensure that the overall length of the fingers would not change. The joint stiffness levels require additional discussion, as there are two cases to consider. First, during the early stages of the grasp, spring forces and tendon forces play equal parts in the process. Once the fingers are closed however, tendon forces can be increased arbitrarily, while spring forces do not change if the grasp is stable and no joint movement occurs. In the limit, tendon forces dominate to the point that spring forces become negligible. An ideal grasp would be stable in both of these phases. For each of the two optimizations, we used the following convention. Tendon route values were derived by solving the optimization problem without joint springs, in an attempt to ensure grasp stability in the limit. The resulting values were then plugged back into the optimization, in order to compute spring stiffness values, using as limits 1.0 Nmmrad−1 ≤ ki ≤ 2.0 Nmmrad−1 . The dual nature of joint stiffness vs. tendon force optimization presents interesting possibilities and requires more detailed exploration than presented here. In particular, we envision a case where joint springs are used to determine the postures that the hand achieves in “free motion”, thus affecting pre-grasp behavior, while tendon routes are optimized for stable grasps in the limit. The transition between these two phases will present additional challenges to address. We bewww.mech-sci.net/2/17/2011/
Avg. joint unbalanced torque (Nmm)
M. Ciocarlie and P. Allen: A constrained optimization framework compliant grasping Optimization Framework 23 M.for Ciocarlie andunderactuated P. Allen: A Constrained for C Ad-hoc
Optim. 1
Optim. 2
1 0.8 0.6 0.4 0.2 0 0
20
40
60
80
100
120
Grasp number (ordered in increasing quality value)
Figure 6. Unbalanced joint forces over the set ofTHE grasps the Fig. 6. UNBALANCED JOINT FORCES OVER SETin OF optimization pool.OPTIMIZATION POOL. GRASPS IN THE
Table GRIPPER OPTIMIZATION. Table1. 1.RESULTS Results of OF gripper designDESIGN optimization.
Ad-hoc Ad-hoc Optimization 1 Optimization21 Optimization Optimization 2
lo lo 5.0 5.0 5.0 5.0 5.0 5.0
l1 l1 5.0 5.0 5.0 5.0 4.64 4.64
l2 l2 5.0 5.0 1.72 1.72 1.02 1.02
ko ko 1.0 1.0 1.0 1.0 1.0 1.0
k1 k1 1.0 1.0 1.0 1.0 1.0 1.0
k2 k2 1.0 1.0 2.0 2.0 2.0 2.0
r r 5.0 5.0 5.0 5.0 7.89 7.89
d d 20.0 20.0 20.0 20.0 17.11 17.11
we performed two optimizations. For the first one (referred to as that Optimization 1), we framework fixed the values to r=5mm lieve the optimization presented here canand be d=20mm. Forthis thedirection. second one (Optimization 2) we set the a first step in limits ≤ of r ≤the 10mm and 15mm d ≤ 22mm. We1.also The3mm results optimizations are≤shown in Table We added the constraint to ensure the as overall only show the valuesr+d=25mm, for one of the fingers,that since, menlength the fingers would tioned of before, the results fornot thechange. other finger are symmetrical. The joint stiffness levels of require additional discussion, as For a quantitative analysis the computed optimal configuthere arewe twocompared cases to consider. First,anduring early stages rations, them against ad-hoctheparameter set, −1 forces play equal parts of theligrasp, spring and tendon with = 5 mm and kforces ∀ i. The comparison crii = 1 Nmmrad in the process. Onceofthe fingers arejoint closed however, tendon terion was the level unbalanced forces for each grasp forces increased forces do not in the can limitbecase. The arbitrarily, results are while shownspring in Fig. 6. We nochange the optimized grasp is stable and no joint movement occurs. tice thatif the configurations provide significantly In the stable limit, grasps tendon across forces the dominate to the pool. point In that spring more optimization addition, forces become negligible. An ideal wouldradius be stable allowing a change in the link lengthgrasp and joint proin bothadditional of these phases. of theare two optimizations, vides stability,For buteach the gains diminished comwe used convention. Tendonare route values pared to the thefollowing case where these parameters fixed. Thewere total derived by formulating solving the optimization problem without probjoint time spent and solving each optimization springs, anthan attempt to ensure stabilitydesktop in the limit. lem was in less a minute, usinggrasp a commodity comThe resulting values the were thenpossibility plugged back into the optiputer. This suggests future of scaling to much mization, in optimization order to compute larger grasp pools.spring stiffness values, using −1 −1 as limits 1.0Nmmrad ≤ ki ≤ gripper 2.0Nmmrad . results of We constructed a prototype using the The dual nature of joint stiffness vs. tendon optiOptimization 1. The links were built using aforce Stratasys mization presents interesting possibilities and requires FDM rapid prototyping machine, and assembled usingmore elasdetailed than of presented here.Each In particular, we tic jointsexploration cut from a sheet hard rubber. link contained envision case with where springs arepoints used tosetdetermine a tendonaroute thejoint entry and exit accordingthe to postures that the hand achieves “freeofmotion”, the optimization results. The in width the stripthus of affectrubber ing behavior, while tendon routes are optimized waspre-grasp varied for each joint to provide the desired stiffnessfor rastable grasps in the limit. The transition between these two tios. As this prototype is intended as a proof-of-concept for phases will present additional challenges to address.noWe bethe kinematic configuration and design parameters, motor lieve that the optimization presented can be or sensors were installed; framework instead, actuation washere performed amanually. first step in this direction. Mech. Sci., 2, 17–26, 2011
Fig. 7. EXECU
The r only sh tioned b For a qu rations, with li = terion w in the li tice tha more sta allowin vides ad pared to time spe lem was puter. T larger g We c Optimiz FDM ra tic joint a tendon the opti was var tios. As the kine or senso manuall We fo range o tioning ability i of two g and lead contrast an asym success the spec per. All and the
248for Compliant Underactuated M. and P. Allen: Allen: A A Constrained constrained optimization Framework framework for Compliant compliant underactuated grasping M.Ciocarlie CiocarlieGrasping and P. Underactuated Grasping on Framework 7 Optimization
tle time or effort spent positioning the gripper relative to the target.
5
120
)
SET OF
ON. d 20.0 20.0 17.11
referred mm and set the We also overall
sion, as y stages al parts tendon s do not occurs. t spring e stable zations, es were ut joint e limit. he optis, using
ce opties more ular, we mine the s affectized for ese two We becan be
Discussion and Conclusions
A qualitative analysis of the optimization results can start from the observation that the prototype gripper is capable of executing both fingertip grasps (of varying finger spans) and enveloping grasps (of both regular and irregular shapes). Intuitively, fingertip grasps require relatively low torques on the 7.distal joints, so (centered that fingertip forcesASYMMETRICAL) are inexecuted opposition, Fig. TWO GRASPS (CENTERED AND Figure 7. Two grasps and asymmetrical) with rather than oriented towards the palm. Conversely, larger EXECUTED WITH THE PROTOTYPE GRIPPER. the prototype gripper. torques on the distal joints benefit enveloping grasps; as a result, the optimization process was required to combine two The found results of the are shown in Table 1. We somewhat opposing goals. Thegripper results indicate thatofthe soluWe that theoptimizations prototype is capable a wide only theenables values forand one of of thegrasps, fingers, as mention show indeed both kinds butsince, the distal joint range of grasping tasks does not require precise positioned before, for theobject. other are symmetrical. is both stifferthe and lesstarget powerful thanfinger thepassive proximal ones. In tioning relative toresults the Its adaptation For a quantitative analysis of the computed optimal configufact, our optimization framework achieves thecharacterisexecution ability is exemplified in Fig. 7, which showsthis rations, we compared them an ad-hoc parameter set, tictwo by grasps. “saturating” ofagainst the hand parameters, which take of The many first one starts from a centered position −1 with l =5mm and k =1Nmmrad ∀ i. The comparison crieither the minimum or maximum value allowed. i i similar joint values for both fingers. In and leads to relatively terion thesecond level unbalanced joint each In was thisthe sense, theofresult of the optimization could begrasp intercontrast, grasp requires theforces jointsfor to conform to in the limit case. The results are shown in Fig. 6. We nopreted as meaning that the addition of a third link to the gripan asymmetrical, irregular shape. Both grasps were executed tice the optimized configurations provide significantly perthat provides little benefit. The resulting gripper comes close successfully. Figure 8 attempts to provide an illustration of more stable grasps across the optimization pool. In addition, to a model with two links per finger, a design also the spectrum of grasps that can be carried out with confirmed this gripallowing change instudies the grasps link joint successfully radius in the of length Dollar Howe (2006).proWe per. Alloptimization ofa the presented were and executed vides additional stability, but the gains are diminished combelieve that this is the type of analysis that our framework is and the object was securely lifted off the table, with very litpared to the case where these parameters are fixed. The total natively suited for: in future iterations, we can directly comtle time or effort spent positioning the gripper relative to the time andmodels, solving and eachcompute optimization probparespent two- formulating and three-link a numerical target. lem was less a minute, using aby commodity desktop commeasure of than the benefit provided the additional link. The puter. This suggests the future possibility of scaling much relatively simple two-fingered design that we usedtohere al5larger Discussion and conclusions grasp optimization pools. of the design choices (which lows an intuitive understanding makes it well suited for initialgripper testing using and proof-of-concept We constructed a prototype the results of A qualitative analysis of the were optimization results can start implementations). However, for more emOptimization 1. The links builtcomplex using amodels, Stratasys from the observation that the prototype gripper is capable pirical analysis becomes unfeasible, and quantitative tools, FDM rapid prototyping machine, and assembled using elasof executing bothpresented grasps (of varying finger spans) as cut the from one can prove more valuable. ticsuch joints afingertip sheet ofhere, hard rubber. Each link contained and enveloping grasps both regular andstudies irregular a tendon route with the(of entry and exit points set according to We also believe that future hand design willshapes). consist Intuitively, fingertip require relatively low torques on the results. The width of theoptimization strip of rubber of aoptimization combination of grasps exhaustive search and probthe distal joints, so joint that fingertip forces are opposition, was varied for each to provide the desired stiffness ralems for which more efficient algorithms are in available. The rather than oriented towards theand palm. Conversely, larger tios. As prototype isdesigns, intended as implicitly a proof-of-concept forof space ofthis possible hand the domain torques on the distal jointsand benefit enveloping grasps; as a the kinematic design parameters, noVirtually motor parameters toconfiguration be optimized, is practically limitless. result, the optimization process was required tocompromise combine two orany sensors were installed; instead, actuation performed hand design ever proposed involves somewas of somewhat opposing goals. The results indicate that the solumanually. ad-hoc decisions vs. informed, optimized parameter choices. tion indeed kinds of grasps, butasthe distal joint InWe our case,enables we have discussed aspects such tendon routes found that theboth prototype gripper is capable of a wide is both stiffer and less powerful than the proximal ones. and joint stiffness.tasks However, by not moving up in the scale at range of grasping and does require precise posi-In fact, our optimization framework achieves this characteriswhich we are analyzing the hand, we can uncoveradaptation many more tioning relative to the target object. Its passive tic by “saturating” many of hand which take design which wethe assumed as given: and ability isdecisions, exemplified in Fig. 7, whichparameters, shows thenumber execution either the minimum or maximum value allowed. links, Someposition of these ofconfiguration two grasps. of The first kinematic one starts chains, from a etc. centered Inleads this sense, theimpossible result of the optimization could be interwill likely to encapsulate a solvable opand to prove relatively similar joint values for in both fingers. In preted as meaning that the addition of a third link to the griptimization thus some contribution numerical contrast, theproblem, second grasp requires the joints from to conform to per provides little benefit. The resulting gripper comes close will irregular be unavoidable. anapproaches asymmetrical, shape. Both grasps were executed to a An model with two links per finger, a on-line design confirmed interesting aspect concerns algorithms that successfully. Figure 8 attempts to the provide an also illustration of Dollar and Howe (2006). We in the optimization studies of arespectrum used to control the that handcan during grasping Traditionthe of grasps be carried outtasks. with this gripbelieve that thispresented is the type of analysis that after our framework is ally,All these algorithms have been designed the hand was per. of the grasps were executed successfully natively suited for: in future iterations, we can directly comconstructed, carefully tuned to extract the best performance and the object was securely lifted off the table, with very litpare two- and three-link models, and compute a numerical Mech. Sci., 2, 17–26, 2011
Fig. 8. 8. GRASPS EXECUTED THE PROTOTYPE GRIPFigure Grasps executed with WITH the prototype gripper. PER.
measure of the benefit provided the additional link. The from a given mechanical design. by Off-line hand optimization relatively two-fingered thatmechanism we used here alenables thesimple opposite approach: design the hand is delows antointuitive understanding of the design choices (which signed suit a particular algorithm. The same applies to makes well suited initial testing proof-of-concept sensor it arrays: we canfor build a hand thatand is optimized for the implementations). for more complex models, emtypes of grasps thatHowever, we can expect to perform based on data pirical becomes quantitative tools, from a analysis certain sensor. In unfeasible, this way, theand hand is intrinsically such as thetoone presented here, can prove equipped handle the shortcomings of themore inputvaluable. data. Overall,We it seems naturalthat to ask: what comes first, the hand or the also believe future hand design studies will consist algorithm? of a combination of exhaustive search and optimization probAsfor robots with the efficient ability toalgorithms operate inare unstructured lems which more available. setThe tings are constantly evolving, thatthe research space of possible hand designs,we andbelieve implicitly domainonof adaptive and designs has the potential to ulparameters to underactuated be optimized, is practically limitless. Virtually timately us with inexpensive and easy-to-build, yetof any hand provide design ever proposed involves some compromise effective robotic hands for a variety of applications inchoices. human ad-hoc decisions vs. informed, optimized parameter environments. In our case, we have discussed aspects such as tendon routes and joint stiffness. However, by moving up in the scale at which we are analyzing the hand, we can uncover many more Appendix A Derivation of Gripper Joint Torque design decisions, which we assumed as given: number and configuration of links, kinematicforchains, etc. Somebetween of these In order to sketch the derivation the relationship will likely prove impossible to encapsulate in a solvable opthe tendon route parameters and the resulting joint torques, timization problem, thus some contribution from numerical we start by focusing on how tendon entry and exit points on approaches be unavoidable. link i affect will the torque applied at joint j. Using the notation An interesting algorithms that shown in Fig. A1,aspect we useconcerns joint i asthe ouron-line reference coordinate are used to control the hand during grasping tasks. Traditionframe, and assume that the translation from joint j to joint i ij T ally, algorithms have been designed after the hand was is tijthese = [tij x ty ] . constructed, carefully tuned to extract thechanges best performance In general, for any point where a tendon direction, from a given mechanical design. Off-line hand optimization such as the link entry point in the figure, the force applied to enables opposite approach: the tendon hand mechanism is dethe link the is the resultant of the total force applied in signed suit aand particular algorithm. The same applies both thetoinitial the changed direction, or f = fin + foutto . sensor arrays: we can build a hand that is optimized for the We note that ||f || = ||f || = δ. However, since we norin out types of grasps that we can expect to perform based on data www.mech-sci.net/2/17/2011/
M. Ciocarlie P. Allen:AA Constrained constrained optimization framework for compliant 25 M. Ciocarlie and and P. Allen: Optimization Framework forunderactuated Compliantgrasping Underactuated Grasping
# " #! −sin(θi /2) sinθi × + −cos(θi /2) cosθi References "
x fout
y și /2 fin t ij
a
li r și
joint i A1. Torque computation for tendon entry point. Fig. A1.Figure Torque computation for tendon entry point
(18)
Through a similar computation, using the notation from T.,torque and Gosselin, C.: Underact Figs. 4 andBirglen, A1, we L., can Laliberte, compute the applied at the Hands, tendon exit point from Springer link i as: Tracts in Adv. Robotics, 2008. i j A. Dollar, Design of U " and Howe, R.: #" Joint #Coupling t x cosθ sinθ l ij i i i+1 in: 30th Mech. and Robotics Conf., 2006 × τexit = Grippers, + ij −sinθi cosθi r+d t y Dollar, A. and Howe, R.: Simple, Robust Autonom " Unstructured #Environments, " #! in: IEEE Intl. Con in sin(θi + θi+1 /2) −sinθi × and Automation,+2007. (19) cos(θi + θi+1 /2) −cosθi
Ferrari, C. and Canny, J.: Planning optimal grasps,
If li , li+1 , the tendon must also change direction someConf. on resulting Roboticstorque and Automation, 1992. where inside link i. The is simply:
Gosselin, C., Laliberte, T., and Degoulange, T.: U
j τichange = li+1 − li robotic hand, in: Video Proceedings(20) of the IEEE
Robotics and Automation, 1998. the total All of these contributions are added to obtain from a certain sensor. In this way, the hand is intrinsically Gosselin, C.,j Pelletier, F., and Laliberte, T.: An Ant torque applied on joint due to tendon routing points on handleto theδshortcomings of the inputitdata. Overmalize equipped tendon toforce = 1 we can omit from the comwith link i. Finally,Underactuated the computationRobotic above is Hand repeated for 15 all Dofs de- and a Si all, it seems natural to ask: what comes first, the hand or the putations. We then obtain the torque applied around a given sired combinations of i and j. By explicitly computing cross in: IEEE Intl. Conf. on Robotics and Automation, algorithm? T products asHirose, u × v = [v − v ][u u ] we obtain the respecjoint by cross-product with the joint moment arm. Using this y x x y S. and Umetani, Y.: The development of so As robots with the ability to operate in unstructured settive entries in the matrix B and the vector a, which are then notation, thearetorque around jointwej applied at research the tendon tings constantly evolving, believe that on entry the versatile robot hand, Mechanism and Machin assembled in the linear relationship adaptive and underactuated designs has the potential to ulpoint in link i is: 351–358, 1978. timately provide us with inexpensive and easy-to-build, yet τtendon = B(θ) p+ a(θ) (21) Kwak, S. D., Blankevoort, L., and Ateshian, G.: A ij effective ij robotic hands for a variety of applications in human τentry = (t + a) × (f + f ) (A1) in out formulation quasi-static multibody models o which can then integrated infor the 3D complete grasp formulation environments. " # ! joints, Computer Meth. in Biomech. and Biomed. E presented in the paper. ij
tx cosθi sinθi li 2000. + = × Acknowledgements. This work was partially supported by the ij Appendix −sinθ cosθ r i i ty Laliberte, T., under Birglen, and Gosselin, C. M.: Und National Science Foundation GrantL., no. 0904514. robotic grasping hands, Machine Intelligence & Ro of gripper joint torque Derivation Edited by: J. L. Herder 4, 1–11, 2002. −sin(θi /2) sinθi × + (A2) Reviewed by: two anonymous referees Miller, A. and Allen, P. K.: GraspIt!: a versatile In order to−cos(θ sketch thei /2) derivation for the irelationship between cosθ the tendon route parameters and the resulting joint torques,
robotic grasping, IEEE Robotics and Automation
References we start focusingcomputation, on how tendon entry andthe exit points on from Through a by similar using notation 110–122, 2004. link i affect the torque applied at joint j. Using the notation ApS Denmark, http://www.mosek.com Figs. 4shown and inA1, we can compute the torque applied at Birglen, the L., Mosek: Laliberte, Mosek T., and Gosselin, C.: Underactuated Robotic Fig. A1, we use joint i as our reference coordinate Hands, Spr. Tra. Adv. Robot., 2008. Prattichizzo, D. and Trinkle, J. C.: Grasping, in: Sp tendon frame, exit point fromthat link as: and assume theitranslation from joint j to joint i Dollar, A. and Howe, R.: Joint Coupling Design of Underactuated i j i j j book of Robotics, pp. 671–700, Springer, 2008. is t i" = [t x # ty ]T . Grippers, in: 30th Mech. and Robotics Conf., Philadelphia, USA, ! ij Ulrich, N., Paul, R., and Bajcsy, R.: A medium-comp In general, point where a tendon changes direction, tx for any 10–13, 2006. cosθ li+1 ij i sinθi + entry point τexit =such asijthe link × to in the figure, the force applied Dollar, A. and ant Howe, R.:effector, Simple, Robust Autonomous Grasping end in: IEEE Intl. Conf. on Robotics −sinθofi the cosθ r +force d applied in i tendon tyis the resultant the link total in Unstructured Environments, in: IEEE Intl. Conf. on Robotics tion, 1988. and Automation, Rome, Italy, 4693–4700, 2007. both the initial and the changed or f = f in + f out . direction, Ferrari, C. and Canny, J.: Planning optimal grasps, in: IEEE Intl. We note sin(θ that || f ini ||+ =θ || fi+1 = δ. However, since iwe normalize out ||/2) −sinθ Conf. on Robotics and Automation, Nice, France, 2290–2295, × + (A3) tendon force to δ = 1 we can omit it from the computations. cos(θ + θ /2) −cosθ 1992. i i+1 i We then obtain the torque applied around a given joint by Gosselin, C., Laliberte, T., and Degoulange, T.: Underactuated some-robotic hand, in: Video Proceedings of the IEEE Intl. Conf. on Robotics and Automation, Leuven, Belgium, 1998. Gosselin, C., Pelletier, F., and Laliberte, T.: An Anthropomorphic Underactuated Robotic Hand with 15 Dofs and a Single Actuaj ij ij τientry (17) (A4) tor, in: IEEE Intl. Conf. on Robotics and Automation, Pasadena, τchange = li+1= −(tli + a) × ( f in + f out ) i j " USA, 749–754, 2008. #" # t x cosθi sinθi li Hirose, S. and Umetani, Y.: The development of soft gripper for the = i j + × All of these contributions added −sinθiarecosθ r to obtain the total i versatile robot hand, Mech. Mach. Theory, 13, 351–358, 1978. ty
cross-product with the joint moment arm. Using this notaIf li tion, 6= li+1 the tendon mustj applied also change direction the, torque around joint at the tendon entry point in link link i is: where inside i. The resulting torque is simply:
torque applied on joint j due to tendon routing points on link i. Finally, the computation above is repeated for all desired www.mech-sci.net/2/17/2011/ combinations of i and j. By explicitly computing cross products as u × v = [vy − vx ][ux uy ]T we obtain the respective
Mech. Sci., 2, 17–26, 2011
26
M. Ciocarlie and P. Allen: A constrained optimization framework for compliant underactuated grasping
Kwak, S. D., Blankevoort, L., and Ateshian, G.: A mathematical formulation for 3D quasi-static multibody models of diarthrodial joints, Comput. Method Biomech., 3, 41–64, 2000. Laliberte, T., Birglen, L., and Gosselin, C. M.: Underactuation in robotic grasping hands, Machine Intelligence & Robotic Control, 4, 1–11, 2002. Miller, A. and Allen, P. K.: GraspIt!: a versatile simulator for robotic grasping, IEEE Robot. Autom. Mag., 11, 110–122, 2004.
Mech. Sci., 2, 17–26, 2011
Mosek: Mosek ApS Denmark, http://www.mosek.com, last access: February 2010. Prattichizzo, D. and Trinkle, J. C.: Grasping, in: Springer Handbook of Robotics, Springer, 671–700, 2008. Ulrich, N., Paul, R., and Bajcsy, R.: A medium-complexity compliant end effector, in: IEEE Intl. Conf. on Robotics and Automation, Leuven, Belgium, 434–439, 1988.
www.mech-sci.net/2/17/2011/
