Grasp quality measures: review and performance
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Abstract
The correct grasp of objects is a key aspect for the right fulfillment of a given task. Obtaining a good grasp requires algorithms to automatically determine proper contact points on the object as well as proper hand configurations, especially when dexterous manipulation is desired, and the quantification of a good grasp requires the definition of suitable grasp quality measures. This article reviews the quality measures proposed in the literature to evaluate grasp quality. The quality measures are classified into two groups according to the main aspect they evaluate: location of contact points on the object and hand configuration. The approaches that combine different measures from the two previous groups to obtain a global quality measure are also reviewed, as well as some measures related to human hand studies and grasp performance. Several examples are presented to illustrate and compare the performance of the reviewed measures.
Keywords
Grasping Manipulation Robotic hands Grasp quality1 Introduction
Grasping and manipulation with complex grippers, such as multifingered and/or underactuated hands, is an active research area in robotics. The goal of a grasp is to achieve a desired object constraint in the presence of external disturbances (including the object’s own weight). Robot grasp synthesis is strongly related to the problems of fixture design for industrial parts (Brost and Goldberg 1996; Wang 2000) and design of cabledriven robots (Bruckmann and Pott 2013). Dexterous manipulation involves changing the object’s position with respect to the hand without any external support.

Disturbance resistance: a grasp can resist disturbances in any direction when object immobility is ensured, either by finger positions (form closure) or, up to a certain magnitude, by the forces applied by the fingers (force closure) (Bicchi 1995; Rimon and Burdick 1996). Main problem: determination of contact points on the object boundary.

Dexterity: a grasp is dexterous if the hand can move the object in a compatible way with the task to be performed. When there are no task specifications, a grasp is considered dexterous if the hand is able to move the object in any direction (Shimoga 1996). Main problem: determination of hand configuration.

Equilibrium: a grasp is in equilibrium when the resultant of forces and torques applied on the object (by the fingers and external disturbances) is null (Kerr and Roth 1986; Buss et al. 1996; Liu 1999; Liu et al. 2004a). Main problem: determination and control of the proper contact forces.

Stability: a grasp is stable if any error in the object position caused by a disturbance disappears in time after the disturbance vanishes (Howard and Kumar 1996; Lin et al. 1997; Bruyninckx et al. 1998). Main problem: control of restitution forces when the grasp is moved away from equilibrium.
After this introduction the article is structured as follows. Section 2 summarizes the basic background necessary to formalize the grasp quality measures. Sections 3 and 4 present the quality measures associated with the positions of contact points, and with hand configuration, respectively. Section 5 reviews the approaches that combine different measures from the two previous groups to obtain a global quality measure, and Sect. 6 presents other approaches not included in the previous groups. Finally, Sect. 7 presents the closing discussion.
2 Basic background and nomenclature
2.1 Modeling of contacts, positions, forces and velocities

Punctual contact without friction: the applied force is always normal to the contact boundary.

Punctual contact with friction (hard contact): the applied force has a component normal to the contact boundary and may have another one tangential to it. Several models have been proposed to represent friction (Howe et al. 1988), the most common being Coulomb’s friction cone.

Soft contact: it allows the application of the same forces as the hard contact plus a torque around the direction normal to the contact boundary. This model is valid only for 3D objects (Buss et al. 1996; Xydas and Kao 1999).
A force \(\varvec{F}_i\) applied on the object at a point \(\varvec{p}_i\) generates a torque \(\varvec{\tau }_i=\varvec{p}_i\times \varvec{F}_i\) with respect to the object’s center of mass (CM ). The force and the torque are grouped in a wrench vector \(\varvec{\omega }_i=(\varvec{F}_i , \varvec{\tau }_i/\rho )^T\), with \(\rho \) being a constant that defines the metric of the wrench space. Possible choices for this parameter include the object’s radius of gyration and the largest distance from CM to any point on the object’s surface. A detailed explanation of the implications of such choices can be found in Roa and Suárez (2009a). The dimension of \(\varvec{\omega }\) is \(d=3\) for 2D and \(d=6\) for 3D objects.
The movement of the object is described through the translational velocity \(\varvec{v}\) of CM, and the rotational velocity \(\varvec{w}\) of the object with respect to CM. Both velocities are represented as a twist \(\varvec{\dot{x}}=(\varvec{v},\varvec{w})^T\in \mathbb {R}^d\).
The force \(\varvec{f}_i\) at the fingertip \(i\) is produced by torques \(\varvec{T}_{ij}\), \(j=1,...,m\), applied at each one of the \(m\) joints. In a hand with \(n\) fingers, a vector \(\varvec{T}=\left[ \varvec{T}_{1j}^T \ldots \varvec{T}_{nj}^T \right] ^T\in \mathbb {R}^{nm}\) is defined to group all the torques applied at the hand joints. The velocities in the finger joints, \(\varvec{\dot{\theta }}_{ij}\), are also grouped in a single vector \(\varvec{\dot{\theta }}=\left[ \varvec{\dot{\theta }}_{1j}^T \ldots \varvec{\dot{\theta }}_{nj}^T \right] ^T\in \mathbb {R}^{nm}\).
Forces and velocities at all fingertips can be expressed in a local reference system. Thus, the vector \(\varvec{f}\!=\!\left[ \varvec{f}_{1k}^T \ldots \varvec{f}_{nk}^T \right] ^T\in \mathbb {R}^{nr}\) (\(k=1,...,r\)) groups all the force components applied at the contact points, and the vector \(\varvec{\nu }=\left[ \varvec{\nu }_{1k}^T \ldots \varvec{\nu }_{nk}^T \right] ^T\in \mathbb {R}^{nr}\) contains all the velocity components at the fingertips.
2.2 Relations between forces and velocities
Note that the above analysis relies on a quasistatic approach, as dynamics is not typically considered to play a major role in grasping tasks, although interesting dynamic grasping and manipulation behaviors have been reported (Senoo et al. 2009). Also, it is assumed that every finger has full mobility in its task space, which is not true for defective systems, i.e. systems that have links with limited mobility, such as the palm in a hand that performs a power grasp. For these systems, specific solutions to the problem of distributing perturbation forces to the contact points can be obtained (Bicchi 1994).
3 Quality measures associated with the position of contact points
This first group of quality measures includes those that only take into account the object’s properties (shape, size, weight), friction constraints and form and force closure conditions to quantify grasp quality. These measures are classified into three subgroups: one considering only algebraic properties of the grasp matrix \(G\), another one considering geometric relations in the grasp (assuming in both subgroups that fingers can apply forces without a magnitude limit), and a third subgroup of measures that considers limits in the magnitudes of the finger forces.
3.1 Measures based on algebraic properties of the grasp matrix \(G\)
3.1.1 Minimum singular value of \(G\)
\(Q_{\tiny MSV}\) indicates a physical condition that may be critical in a grasp from a practical point of view. However, it is not invariant under a change in the reference system used to compute torques.
3.1.2 Volume of the ellipsoid in the wrench space
\(Q_{\tiny VEW}\) is invariant under a change in the torque reference system, but it does not provide information about whether some fingers are contributing more than others to the grasp.
3.1.3 Grasp isotropy index
\(Q_{\tiny GII}\) indicates whether the grasp has an equivalent behavior in any direction, which may be useful for general purpose grasps; it also indirectly indicates the same physical condition as \(Q_{\tiny MSV}\).
3.2 Measures based on geometric relations
3.2.1 Shape of the grasp polygon
\(Q_{\tiny SGP}\) has a simple physical interpretation and an easy computation, but it is useful for planar grasps only. The extension to general 3D grasps is not evident, and there may be cases where \(Q_{\tiny SGP}\) leads to unexpected grasps from the practical point of view (for instance grasping an elongated object, like a pencil) because of the object’s geometry.
3.2.2 Area of the grasp polygon
3.2.3 Distance between the centroid of the contact polygon and the object’s center of mass
3.2.4 Orthogonality
3.2.5 Margin of uncertainty in finger positions
The space defined by the \(n\) parameters representing the possible contact points of \(n\) fingers on a 2D object boundary is called grasp space (or contact space), and the subset of the grasp space representing force closure grasps is called force closure space, FCS. For polygonal objects, FCS is the union of a set of convex polyhedra \({CP}_i\), and this is used in several proposals to compute the FCS for polygonal objects and any number of fingers, with or without friction (Liu 2000; Li et al. 2002; Cornellà and Suárez 2005b).
\(Q_{\tiny MUF}\) is quite appropriate to minimize the effect of uncertainty on finger positions during grasp execution, but it is difficult to apply to nonpolygonal 2D or 3D objects due to the complexity and high dimensionality of the resulting grasp space (note that for 3D objects two parameters are needed to fix the position of each finger on the object surface).
3.2.6 Independent contact regions
\(Q_{\tiny ICR}\) has a clear physical interpretation and is particularly useful in the presence of uncertainty in finger positioning. Higher quality also indicates a greater possibility of finding a set of reachable contact points allowing a force closure grasp for a given mechanical hand. As a drawback, it is necessary to compute the set ICRS (i.e. \(B\)), resulting in extra computational cost (Roa and Suárez 2009a).
3.3 Measures considering limitations on the finger forces
The previous subgroups of quality measures are related to the geometric location of contact points, but do not consider any limit in the magnitude of the forces applied by the fingers. Thus, even when the obtained force closure grasps can resist external perturbation wrenches along any direction, nothing is said about the magnitude of the perturbation that can be resisted. This means that in some cases the fingers may have to apply extremely large forces to resist small perturbations. Thus, grasp quality could also consider the module of the perturbation wrench that the grasp can resist when forces applied by fingers are limited. This section includes the quality measures that consider this aspect.
3.3.1 Largestminimum resisted wrench
There are two common constraints on finger forces \(\varvec{f}_i\). The first one is that the module of the force applied by each finger is individually limited, which corresponds to a limited independent power source (or transmission) for each finger. In order to simplify the formalism, and without loss of generality, it is assumed that all finger forces have the same limit and that it is normalized to 1, i.e. \(\left\ \varvec{f}_i\right\ \le 1,\, i=1,...,n\).
There are other proposals of constraints on finger forces, like \(\sum _{i=1}^n\left\ \varvec{f}_i\right\ ^2\le 1\) (Mishra 1995). However, physical interpretations are not as evident as in the previous ones and have not been widely implemented.
The quality measure given by Eq. (24) is interpreted using the metric \(L_2\). In theory, other metrics like \(L_1\) or \(L_\infty \) can be used (Mishra 1995). In practice, these metrics have been used for measuring grasp quality of partial force closure grasps (i.e. grasps which only immobilize the object along certain directions) by considering the sum of the needed forces applied at the existing contact points in order to exert some given unit wrench on the object (Kruger and van der Stappen 2011).
The consideration of the maximum real force that fingers can apply at each contact point is usually not taken into account. However, the real wrenches that fingers with limited torque bounds apply on the object surface, according to (2), can be used for building a set \(\mathcal{P}_{r}\) that includes all the wrench space reachable by the real robot hand (Jeong and Cheong 2012; Zheng and Yamane 2013). Replacing \(\mathcal{P}\) with \(\mathcal{P}_{r}\) still holds for the quality measure defined in (24).
\(Q_{\tiny LRW}\) has a clear and useful physical meaning for general purpose grasps, but depends on the reference system used to compute torques. Selecting the object’s center of mass as the origin of the reference system is coherent with the system dynamics, but as stated above for other measures, in some cases it may be difficult to know the center of mass accurately. Besides, it is necessary to establish a metric in the wrench space to simultaneously consider pure forces and torques, as defined by the factor \(\rho \) introduced in Sect. 2.1 (Roa and Suárez 2009a). \(Q_{\tiny LRW}\) can be normalized with respect to the maximum value that it can reach for a given object, which indicates how far the grasp is from being optimum. However, this requires the computation of the maximum value, implying an additional computational cost. A recent attempt to overcome the dependence of \(Q_{\tiny LRW}\) on the reference frame was proposed by setting the moment origin at the centroid of contact positions so that the grasp wrench sets are frame independent (Zheng and Qian 2009). In that work, instead of dividing the torque component by a factor \(\rho \) it is proposed to multiply the force components by the average distance from the contacts to their centroid, which makes that the grasp wrench sets have the same scale in all wrench directions, and sets the scale factor of the ball in the wrench space directly proportional to the same average distance.
3.3.2 Volume of the Grasp Wrench space (volume of \(\mathcal{P}\))
3.3.3 Decoupling forces and torques
\(Q_{\tiny f}\) and \(Q_{\tiny \tau }\) can be computed in a simpler way by avoiding the definition of a metric of the wrench space, although they are actually two independent measures and the order in which they are considered affects the solution.
3.3.4 Normal components of the forces
3.3.5 Task oriented measure
By considering the set of all possible forces acting on the object surface, an approximation to the most probable perturbations on the object is obtained. In this way, the task polytope \(\mathcal E\) is computed as the convex hull of the wrenches obtained by applying unitary normal forces at each contact point on a discretized object surface; tangential components of perturbation at the contact points are not included for computational reasons (Strandberg and Wahlberg 2006; Jeong and Cheong 2012).
In unstructured environments, estimating the friction coefficient between the hand and object surface is difficult. Therefore, the minimum friction coefficient required to resist perturbations along predefined directions can as well be used as a quality measure (Mantriota 1999). A grasp configuration that minimizes this index is more robust to potential slippage of the object.
3.4 Examples
In order to facilitate their interpretation, the measures presented above were implemented and applied to a simple 2D object, a 4 cm by 2 cm rectangle grasped with 4 frictionless fingers (unless indicated otherwise). The object contour was discretized with 64 points, 11 per each short side and 21 per each long side. For simplicity, it is assumed that a force can be punctually applied in the direction normal to a side of the rectangle, even at the vertices (in practice, a security distance must be considered). As the contacts are frictionless, each finger must lie on a different side of the rectangle, leading to 21*21*11*11=53,361 different grasp combinations, 23,100 of which are force closure grasps. For the FC grasps, different quality measures were computed. Due to the symmetric and discrete nature of the problem, several globally optimal grasps (i.e. same minimum or maximum value for different finger locations) were obtained for a given quality measure. The total number of solutions reported includes symmetric grasps due to symmetries on the finger locations.
Measures based on algebraic properties of the grasp matrix \(G\)

Minimum singular value of \(G\) (\(Q_{\tiny MSV}\)): there are 74 optimal grasps covering different grasping options; Fig. 8a shows one of them.

Volume of the ellipsoid in the wrench space (\(Q_{\tiny VEW}\)): there are two optimal grasps with symmetric locations of the contact points on the object; Fig. 8b shows one of them.

Grasp isotropy index (\(Q_{\tiny GII}\)): there are four optimal grasps achieving the maximum absolute value of the quality measure; Fig. 8c shows one of them.

Shape of the grasp polygon (\(Q_{\tiny SGP}\)): there are two optimal symmetric grasps on the object; Fig. 9a shows one of them.

Area of the grasp polygon (\(Q_{\tiny AGP}\)): there are 400 different optimal grasps, with a variety of positions of the contact points on the object; Fig. 9b shows one of them.

Distance between the centroid of the contact polygon and the object’s center of mass (\(Q_{\tiny DCC}\)): there are 100 optimal grasps that reach the minimum possible value (\(Q=0\)); Fig. 9c shows one of them.

Margin of uncertainty in finger positions (\(Q_{\tiny MUF}\)): Fig. 10a shows the grasp space and force closure space (FCS) for grasps obtained when a contact point has been predefined on the rectangle; in this case, the contact on the left side of the rectangle is fixed in order to obtain a 3dimensional representation that illustrates the concept. The largest hypersphere inscribed in the FCS determines the optimal grasp, as shown in Fig. 10b.

Independent contact regions (\(Q_{\tiny ICR}\)): there are 4,608 optimum grasps that have the same minimum size of one of the ICRs; Fig. 11 shows one example. The figure also shows the ideal grasp according to the uncertainty grasp index for the same independent contact regions, i.e. the contact points are located in the center of their corresponding ICR.

Largest minimum resisted wrench (\(Q_{\tiny LRW}\)): considering a limited common power source for all fingers (\(\sum _{i=1}^n\left\ \varvec{f}_i\right\ \le 1\)) there are two optimal symmetrical grasps; Fig. 12a shows one of them in the wrench space and Fig. 12b shows it on the object.

Volume of the set \(\mathcal{P}\) of possible resultant wrenches on the object (\(Q_{\tiny VOP}\)): there are two optimal symmetrical grasps; Fig. 13a shows one of them in the wrench space and Fig. 13b shows it on the object.

Task oriented measure (\(Q_{\tiny TOM}\)): it is assumed that a task may cause the disturbances shown in Fig. 14a. There are two optimal symmetrical grasps; Fig. 14b shows one of them in the wrench space and Fig. 14c shows it on the object.
Comparison of qualities for optimal grasps according to different criteria
Criterion  \({Q_{\tiny MSV}}^\mathrm{a}\)  \({Q_{\tiny VEW}}^\mathrm{a}\)  \({Q_{\tiny GII}}^\mathrm{a}\)  \({Q_{\tiny SGP}}^\mathrm{b}\)  \({Q_{\tiny AGP}}^\mathrm{a}\)  \({Q_{\tiny DCC}}^\mathrm{b}\)  \({Q_{\tiny MUF}}^\mathrm{a}\)  \({Q_{\tiny ICR}}^\mathrm{a}\)  \( {Q_{\tiny LRW}}^\mathrm{a}\)  \({Q_{\tiny VOP}}^\mathrm{a}\)  \({Q_{\tiny TOM}}^\mathrm{a}\) 

\(Q_{\tiny MSV}\) (Fig. 8a)  1.4142  32.32  0.4975  0.1708  3.2  0  0.5  5  0.0828  1.4667  0.0884 
\(Q_{\tiny VEW}\) (Fig. 8b)  1.4142  40  0.4472  1  0  0  0.9  5  0.3162  2  0.3536 
\(Q_{\tiny GII}\) (Fig. 8c)  1.4142  8  1  0.4647  3.04  0  0.7  5  0.3487  0.9333  0.3333 
\(Q_{\tiny SGP}\) (Fig. 9a)  1.4142  27.2  0.5423  0.0199  2.56  0  0.6  5  0.1655  1.4667  0.1768 
\(Q_{\tiny AGP}\) (Fig. 9b)  0.0858  0.16  0.0260  0.7262  3.98  1.0512  0.9  1  0.0401  0.1333  0.0786 
\(Q_{\tiny DCC}\) (Fig. 9c)  1.4142  28.48  0.53  0.6772  0.8  0  0.9  5  0.3590  1.7333  0.3928 
\(Q_{\tiny MUF}\) (Fig. 10b)  1.1871  6.4  0.7878  0.4163  3.36  0.1  0.7  5  0.254  0.8  0.2357 
\(Q_{\tiny ICR}\) (Fig. 11)  1.3867  11.68  0.7957  0.3932  2.9  0.0707  0.7  5  0.2532  1.0667  0.2525 
\(Q_{\tiny LRW}\) (Fig. 12b)  1.4142  16  0.7071  0.6257  2  0  0.9  5  0.4472  1.3333  0.3333 
\(Q_{\tiny VOP}\) (Fig. 13b)  1.4142  40  0.4472  1  0  0  0.9  5  0.3162  2  0.3536 
\(Q_{\tiny TOM}\) (Fig. 14c)  1.3729  28.48  0.4995  0.8373  0.8  0.2  0.9  5  0.3162  1.7333  0.4419 
4 Quality measures associated with hand configuration
This second group of quality measures includes those that consider hand configuration to estimate the grasp quality. The basic ideas from Sect. 3.1 for quality measures dependent on the properties of the matrix \(G\) can be extended considering the handobject Jacobian \(H\) (Shimoga 1996), taking into account the considerations for the computation of \(H\) presented in Sect. 2.2. In other cases, only hand posture (joint positions) is considered to compute a quality index.
4.1 Measures associated with hand configuration
4.1.1 Distance to singular configurations
4.1.2 Volume of the manipulability ellipsoid
Note that \(Q_{\tiny VME}\) is conceptually equivalent to \(Q_{\tiny VEW}\) given in Eq. (9) but considering the handobject Jacobian \(H\). Therefore, it is also invariant under a change in the reference system, but does not provide information about the finger’s individual contribution.
4.1.3 Uniformity of transformation
As in the previous cases, \(Q_{\tiny UOT}\) is conceptually equivalent to \(Q_{\tiny GII}\) given in Eq. (10). Hence, the same reasonings about the quality properties can be applied.
4.1.4 Positions of the finger joints
4.1.5 Task compatibility
\(Q_{\tiny TCI}\) is specifically oriented to a desired task but, as for all task oriented measures, in practice the task constraints to be considered might be nonconstant and difficult to define.
4.2 Examples

Distance to singular configurations: the optimal gripper configuration is shown in Fig. 16a. Figure 16b illustrates a singular configuration with the minimum singular value equal to zero (the worst possible quality).

Volume of the manipulability ellipsoid: there are 12 optimal gripper configurations (including symmetrical poses); Fig. 17 shows one of them. These configurations allow high manipulability of the object (with respect to infinitesimal movements); however, there are joints close to their range limits.

Uniformity of transformation: there are two optimal gripper configurations, which are the same as those previously obtained using the maximum distance to singular configurations (Fig. 16). The worst quality measure is also obtained in the same singular configurations. Thus, for this particular example the behavior of the two quality measures is similar.

Joint angle deviations: Figure 18a shows the optimal gripper configuration, and Fig. 18b shows a low quality configuration. Note the difference between this optimal configuration, which provides more “comfort” or a larger range of possible hand movements, and the configuration in Fig. 16a, which drives the gripper away from singular configurations.
5 Combinations of quality measures
Grasp quality is measured according to the above criteria, based either on the location of contact points on the object or on the hand configuration. However, the optimal grasp for some particular tasks could be a combination of these criteria; for instance, the selection of optimal contact points on the object surface according to any criteria from Sect. 3, ignoring the actual hand geometry, could lead to contact locations unreachable for the real hand, and vice versa: an optimal hand configuration could generate a weak grasp in the presence of small perturbations. Studies of correlation between quality measures show that in fact using a combination of quality measures allows capturing different aspects of prehension, like geometrical restriction, ability to resist forces, manipulability or comfort (Leon et al. 2012). To evaluate these different aspects, there have been several proposals of quality measures obtained as a combination of those presented in the previous sections, either using them in a serial or in a parallel way.
The serial approach is applied in grasp synthesis by using one of the quality criteria to generate candidate grasps, and the best candidate is chosen among them using another quality measure. For instance, the optimization with respect to the hand configuration using the weighted sum in the task compatibility index given by Eq. (45) generates a preliminary grasp. This grasp is subsequently used as initial one in the search for an optimum grasp under the measure of the largest ball given by Eq. (24) (Hester et al. 1999).
The parallel approach combines different quality measures in a single global index. A simple method uses the algebraic sum of the qualities resulting from each individual criterion (or the inverse of some criteria so that they all must be either maximized or minimized), eventually using suitable weights and normalizations. Simple addition has been used to choose optimum grasps for 2D (Boivin et al. 2004) and 3D objects (Aleotti and Caselli 2010). A variation normalizing the outcome of each criterion, dividing it by the difference between the measures of the best and the worst grasp, has been used to evaluate grasps of 2D objects performed by a 3finger hand (Chinellato et al. 2003). Different combinations can thus be obtained by adding different basic criteria in order to generate indices specifically adapted for different practical applications (Chinellato et al. 2005).
Another approach considers a set of normalized indices and selects as quality output the minimum value among all of the normalized measures. An example of this approach uses normalized quality measures (including uncertainty in finger positions, maximum force transmission ratio, grasp isotropy and stability), assigns weights according to the desired grasp properties, and then selects the grasp with the minimum value out of the normalized and weighted measures (Kim et al. 2004).
Other possibility for combining criteria in a parallel way is to generate ranks of candidate grasps according to different quality measures, and then assign to each grasp a new index obtained as the addition of its place in each one of the original rankings. However, this approach has a high computational cost and has not provided a satisfactory outcome (Chinellato et al. 2003).
6 Other criteria for quality measures
6.1 Relation to human grasp studies
Traditional studies of human grasps have focused on aspects such as the relation between object size and hand aperture (Cuijpers et al. 2004), hand preshaping and fingertip trajectories (Supuk et al. 2005), or force distribution among fingers during object manipulation (Li 2006). Only recently has the application of concepts coming from the robotic world to the analysis of human grasps gained attention. For instance, human experience in grasping has been used to guide a robotic arm and hand to grasp objects, and lately to compare humanguided grasps to grasps obtained with a planner (Balasubramanian et al. 2010). From that work, it was evident that humans prefer to align the palm with the object’s principal axis.
More recent works have collected human grasp data with a sensorized object, and the grasps were later analyzed using different quality measures to evaluate how grasp quality increases with the number of fingers and with the contact area involved in the grasp action, to study the drawbacks of approximating a contact region with simple contact points, and to verify whether subject perception of grasp robustness matches with the prediction of the studied quality measures for both power and precision grasps (Roa et al. 2012).
6.2 Performance based measures
Existing grasp planning approaches rely mainly on quasistatic assumptions, i.e. the object does not move when the contacts are established. Causal correlation between classical quality measures such as \(Q_{LRW}\) and \(Q_{VOP}\) with the actual success in human grasps indicates that a high value of \(Q_{LRW}\) or \(Q_{VOP}\) does not necessarily imply a successful grasp in a real environment (Balasubramanian et al. 2010). The same phenomenon has recently been observed when analyzing grasp databases and comparing them with real grasp executions (Kim et al. 2013). The resulting grasp can be far from the assumed pose at planning time due to uncertainties in real systems, which results in wrong contact information and therefore wrong estimation of grasp quality. However, pose uncertainty can be considered for computing the probability of obtaining a force closure grasp (Weisz and Allen 2012). Incorporation of dynamic simulations into grasp planning systems has recently been proposed to evaluate changes in the relative pose between the hand and the object, and to predict robustness during grasping. Comparisons between simulations and real experiments have been presented for 2D (Zhang et al. 2010) and 3D cases (Kim et al. 2013).
Judging real robotic systems performing grasping actions is more challenging. For this purpose, performancebased measures are proposed to provide a score depending on the success of the system when lifting the object. A simple binary score evaluates whether the robot is able to lift the object and hold it for a predefined amount of time (Saxena et al. 2008), or whether the robot is able to hold the object even after shaking it (Balasubramanian et al. 2010; Morales et al. 2003). More elaborated discrete scoring systems can be created by considering, for instance, resistance to small perturbations directly applied on the object, deliberately trying to break the grasp (Kim et al. 2013).
Performancebased indices measure the success of a grasp after its execution by lifting the object or by applying some small perturbation to it, which allows, for instance, the evaluation of the actual robustness of each grasp to store the results in a database that can be used in future grasp applications. Nevertheless, for real applications one might be interested in predicting the robustness of any grasp before actually executing it, i.e. the object should resist disturbances while being robust to uncertainties in perception and actuation, which can be tackled by using quality measures described in the previous sections.
7 Discussion and conclusions
Grasp quality measures
Group  Subgroup  Quality index  Criterion 

Measures related to the position of the contact points on the object  Based on algebraic properties of \(G\)  Minimum singular value of \(G\)  Maximize 
Volume of the ellipsoid in the wrench space  Maximize  
Grasp isotropy index  Maximize  
Based on geometric relations  Shape of the grasp polygon\(^\mathrm{a}\)  Minimize  
Area of the grasp polygon  Maximize  
Distance between the centroid \(C\) and the center of mass CM  Minimize  
Orthogonality  Minimize  
Margin of uncertainty in finger positions\(^\mathrm{b}\)  Maximize  
Based on independent contact regions  Maximize  
Considering limitations on the finger forces  Largestminimum resisted wrench  Maximize  
Volume of the Grasp Wrench Space  Maximize  
Decoupled forces and torques  Maximize  
Normal components of the contact forces  Minimize  
Coplanarity of the normals\(^\mathrm{a}\)  Minimize  
Task oriented measures  Maximize  
Measures related to hand configuration  Distance to singular configurations  Maximize  
Volume of the manipulability ellipsoid  Maximize  
Uniformity of transformation  Minimize  
Finger joint positions  Minimize  
Similar flexion values  Minimize  
Task compatibility index  Maximize  
Safety margin  Maximize  
Other measures  Biomechanical fatigue  Minimize  
Deviation in object pose  Minimize 
Although some studies compare the optimal grasps obtained according to different criteria for different objects in 2dimensional (Bone and Du 2001; Morales et al. 2002; Borst et al. 2004) and 3dimensional grasps (Miller and Allen 1999), the selection of the best criterion in each real case is not always trivial. Besides, even knowing the criterion to be applied, the complexity of real cases often makes the computational cost of any grasp optimization really high. In order to provide an idea of the behavior of each quality measure, Sects. 3.4 and 4.2 present application examples on simple cases that allow the intuitive interpretation of the measure. In fact, it is not possible to provide a general recommendation for the use of any grasp quality measure, as the quality value depends on several aspects of the grasp. In general, quality measures may consider: (a) locations of the contact points on the object, (b) directions of the forces applied at the contact points, (c) magnitudes of the applied forces at the contact points, and (d) gripper configuration. The consideration of these elements may provide a better idea on the most convenient quality measure for a particular task.
Most of the presented grasp analysis is based on quasistatic considerations. Dynamic manipulability was originally proposed for serial manipulators (Yoshikawa 1985a, 2000), and was formulated for cooperative robots as the ratio between an input torque and the resultant acceleration of the grasped object (Bicchi et al. 1997). The concept has been recently extended to the field of multifingered grasping (Yokokohji et al. 2009).
The commercial availability of hands with integrated tactile sensors and fingertip sensors that can be adapted to specific hands (Silva et al. 2013; Yousef et al. 2011), also provides a new field of application for the presented quality measures, traditionally associated to grasp planning stages. In fact, a fingertip sensor could provide information on the magnitude of the contact force and its point of application, which can be used to estimate the direction of the force being applied on the object. This information is exploited for locally optimizing some quality index by adjusting the grasp force or even the contact location, such that the overall grasp stability during real executions is increased (Dang and Allen 2013; Laaksonen et al. 2012; Bekiroglu et al. 2011).
Some studies have analyzed the change of grasp quality with the location of contacts and the variation of the friction coefficient (Zheng and Qian 2004), and even with the number of contacts (Rosell et al. 2010). It has been suggested that, without other considerations, grasp quality increases slightly for more than a given number of contact points. A large number of contact points is typical in power grasps, but the applicability of quality measures for power grasps has hardly been tackled. One way to quantify the robustness of a power grasp is by considering the minimum virtual work rate required to move the object along a virtual displacement (Zhang et al. 1994). Another metric was proposed to minimize the distance between the object and predefined contact points on the hand, which was used to plan a pregrasp shape that is later used for online grasp planning (Ciocarlie and Allen 2009). Although in theory most of the above measures can be applied to grasps with any number of contact points (Roa et al. 2012), the explicit consideration of the limited forces that some parts of the hand can apply on the object allows the definition of contact robustness, i.e. how far a contact is from violating contact constraints, which is different from grasp robustness, i.e. how far the grasp is from overcoming the object immobilization constraint (Prattichizzo et al. 1997).
Most of the measures presented in this survey were developed for fingertip grasps using fully actuated multifingered hands. The application of the measures to underactuated hands, in particular the measures related to gripper configuration (Sect. 4), requires the development of new theoretical tools. For instance, if finger joints are modeled as elastic elements, the instantaneous kinematics of the hand and object can be predicted by considering a quasistatic equilibrium when the hand is perturbed (Quenouelle and Gosselin 2009; Odhner and Dollar 2011). Such mapping allows the application of the presented manipulability measures. The theoretical framework of parallel robots has been recently proposed as a tool for studying fingertip grasps and dexterous manipulation for underactuated hands (Borras and Dollar 2013). The adaptation of classical manipulability indices (condition number, singular values) to parallel robots has been studied and they do not seem to be consistent for analyzing such robots (Merlet 2006); the adaptation of grasp quality measures for underactuated hands is currently an open area of research (Malvezzi and Prattichizzo 2013).
The grasp quality measures reported in this survey do not consider the effect of compliance. For analyzing compliant grasps a grasp stiffness matrix \(K\) is required; the grasp is stable if the stiffness matrix is positive definite (Howard and Kumar 1996). A measure of grasp stability is based on the eigenvalue decomposition of the generalized matrix \(M^{1}K\), with \(M\) a metric that allows that twists and wrenches lie on the same vector space (Bruyninckx et al. 1998). However, this measure depends on the choice of the metric \(M\). A frameinvariant quality measure can also be developed based on the computation of principal rotational and translational stiffnesses for a grasp with stiffness matrix \(K\) (Lin et al. 2000).
When dealing with wholehand grasps, in general it is not possible to generate forces in all directions. Thus, the concepts of active and passive force closure arise: an external wrench can be counterbalanced if there exist strictly active or passive internal forces (Bicchi and Pratichizzo 2000). Note that in this way, the condition for active force closure is stricter than for pure force closure. A grasp optimization for this case can, for instance, minimize the joint efforts (Ma et al. 2012). Also, when considering hand and contact compliance, specific solutions to the force distribution problem \(\varvec{\omega }=G\varvec{f}\) can be obtained (Bicchi 1994). The implications of compliance in the grasp analysis is receiving a renewed interest due to the evolution of underactuated robotic hands (Prattichizzo et al. 2012).
There are still more open research problems related to the quality measures. First, it is worth mentioning the need for efficient algorithms (both in terms of computational complexity and computational cost) to generate optimal grasps according to different quality criteria. A second aspect is the automatic determination of the relevant quality measures for the problem at hand, either to select the most appropriate one or the most convenient combination. Even when there are already some measures that try to consider the goal of the grasp (i.e. the task to be performed), this is also an aspect that requires further research and more practical proposals. In any case, continuous advances in the development of dexterous grasping devices will require the definition and formalization of new quality measures as well as optimal procedures to apply them.
Footnotes
 1.
Parameter \(d\) is given by the object (2D or 3D). Restricting the analysis to force closure grasps, it is possible to obtain the minimum number of fingers required to guarantee a force closure grasp for a chosen contact model (Mishra et al. 1987; Markenscoff et al. 1990). Using this minimum number of fingers, it is verified that \(d<nr\). Moreover, one of the necessary conditions for force closure is that \(rank(G)=d\) (Murray et al. 1994).
Notes
Acknowledgments
This work was supported by the project SMERobotics, from the European Union Seventh Framework Programme (FP7/2007–2013) under Grant Agreement No. 287787, and by the Spanish Government Projects DPI201015446, DPI201122471 and DPI201340882P.
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