in the grasp matrix in form of an additional virtual contact acting in the inverse direction with a certain magnitude. Then the existence of strict internal forces ...
Task-Oriented Quality Measures for Dextrous Grasping R. Haschke, J. J. Steil, I. Steuwer, and H. Ritter Neuroinformatics Group, Faculty of Technology, Bielefeld University D-33615 Bielefeld, Germany email: {rhaschke,jsteil,helge}@techfak.uni-bielefeld.de Abstract— We propose a new and efficient approach to compute task oriented quality measures for dextrous grasps. Tasks can be specified as a single wrench to be applied, as a rough direction in form of a wrench cone, or as a complex wrench polytope. Based on the linear matrix inequality formalism to treat the friction cone constraints we formulate respective convex optimization problems, whose solutions give the maximal applicable wrench in the task direction together with the needed contact forces. Numerical experiments show that application to complex grasps with many contacts is possible. Index Terms— grasp quality measure, force optimization
I. I NTRODUCTION With the increasing advent of humanoid robots and the concurrent appearance of ever increasingly complex dextrous artifical hands [1], [2], [3], [4], [5] the research on dextrous manipulation gets a source of motivation from the perspective that humanoid robots soon have to be able to carry out every day tasks with common and often irregular shaped objects. This perspective adds new aspects to grasping research like scaling algorithms to five fingered hands, extending specialized methods to flexible objects, and developing grasp quality criteria, which take into account more precisely the given task at hand. From a more classical point of view, grasping is often subdivided into the grasp synthesis problem, which aims at computing contact locations with respect to a given optimality criterion [6], and the grasp analysis, which determines quality measures with respect to a given grasp mainly based on force closure properties and aims at optimizing the respective contact forces [7]. While grasp synthesis still often leads to tedious and numerically costly optimization algorithms [6], [8], grasp analysis has made seminal progress by formalizing friction cone constraints as definiteness criteria of symmetric matrices [9], which allow to employ modern and efficient algorithms for convex optimization [10] or linear matrix inequality (LMI) methods [7]. Below we draw on these results and show that similar methods can be used to evaluate grasp quality with respect to a given direction in which a task wrench has to be applied. In particular, for many tasks only a limited set of wrenches is required in order to manipulate the object. Hence the concept of force closure, which guarantees
resistance of the grasp to any disturbance wrench, i.e. applicability of any object wrench, is too powerful in these cases. Examples are lifting or pushing tasks considered for example in [11], which can be split into two subtasks: (i) acceleration and resistance against a friction force and (ii) deceleration of the object at its final position. Here, in the simplest case it suffices to apply a wrench vector limited to a narrow cone along the movement direction – deceleration may be achieved using gravitation or friction force; other wrenches need not to be applied. Hence, a single point contact with friction would be sufficient for this task. To compute grasp quality measures under such restrictions, appropriate projections of the 6-dimensional wrench space to a lower dimensional space can be used, as proposed in [12]. This approach considers complete linear subspaces and for this reason is not suited for the described task, which needs the restriction to a positively spanned cone. We suggest instead to include the desired task wrench in the grasp matrix in form of an additional virtual contact acting in the inverse direction with a certain magnitude. Then the existence of strict internal forces implies that the task wrench can be applied and force optimization with respect to the magnitude even yields an elegant way to find the maximally applicable force in the task wrench direction by means of solving a certain LMI-problem similar to [7]. The remainder of the paper is organized as follows. In Section II, we give the basic notation following the standard screw theory formalism [13] and comment on the existing task oriented measures. Section III formulates our approach for single task directions and gives a numerical example for a complex lifting task with many contacts. In Section IV we generalize our concept to force optimization with respect to a task cone and section V treats the even more complex case of task polytopes. Finally Section VI adds some discussion. II. F ORCE
CLOSURE AND TASK ORIENTED MEASURES
We consider an object grasped by a multifingered robotic hand using k point contacts with the finger links or the palm. Each point contact may assume one of the following considered contact models: frictionless point contact (FPC), point contact with friction (PCWF), or soft finger contact with elliptic approximation (SFCE). Depending on the contact model there exists respectively a one-, three- or four-
dimensional local contact force xi ∈ Rmi for each contact. Employing a matrix Gi ∈ R6×mi these can be transformed into a resultant object wrench F = Gi xi .PSfrag If we replacements combine all contact forces into a vector x = [xT1 , . . . , xTk ]T and all matrices Gi into the grasp matrix G = [G1 , . . . , Gk ], the transformation from applied contact forces x to the net resultant object wrench reads X F = Gx x ∈ Rm , m = mi . (1)
The contact force x is constrained to the joint friction cone FC = {x ∈ Rm | xi ∈ FC i , i = 1, . . . , k}
(2)
FC i = {xi ∈ Rmi | 0 ≤ kxi,t kw ≤ xi,n } ,
(3)
where FC i is the local friction cone constraint which can be written formally as i.e. the tangential force components xi,t are appropriately bounded by the normal component. The weighted norms kxi,t kw are defined for the different contact models as follows: FPC: PCWF: SFCE:
kxi,t k2w = 0 1 kxi,t k2w = 2 x2i,1 + x2i,2 µi 1 1 2 kxi,t kw = 2 x2i,1 + x2i,2 + 2 x2i,4 , µi γi
(4) (5) (6)
where xi,1 and xi,2 are orthogonal friction force components in the tangential contact plane, xi,4 is the moment along the contact normal, µi is the usual Coulomb friction coefficient, and γi is the torsional friction coefficient. Helmke et al. [10] observed that the local cone constraints (3) are equivalent to the positive definiteness of the following symmetric matrices: FPC: Pi = xi,n 0 µi xi,3 + xi,1 xi,2 PCWF: Pi = 0 xi,2 µi xi,3 − xi,1 1 1 xi,3 + µ1i xi,1 µi xi,2 − j γi xi,4 0, SFCE: Pi = 1 1 xi,3 − µ1i xi,1 µi xi,2 + j γi xi,4 where j is the imaginary unit. The joint friction cone (2) can be expressed by combining all friction conditions into a single block diagonal matrix: P (x) = diag(P1 , . . . , Pk ) 0 .
(7)
A grasp (G, FC) is called force closure if and only if it can resist any external disturbance wrench without violating the friction cone constraints (2). This was shown to be equivalent to the following conditions [13]: 1) G is surjective, i.e. rank(G) = 6, 2) there exist strict internal forces xI , i.e. P (xI ) 0 and GxI = 0. In order to estimate the quality of a given force closure grasp, several quality metrics were proposed. References [14], [15] use the magnitude of the worst-case disturbance wrench that can be resisted by the grasp. Due to the linear relationship (1) the magnitude of a resultant object wrench
O PSfrag replacements α (a) worst-case Fig. 1.
W
O (b) task ellipsoid
W
Grasp quality measures sketched in grasp wrench space W.
linearly scales with the magnitude of the contact forces. Hence, the magnitude of the contact forces are typically bounded to unity with respect to some norm of their normal components, e.g. L1 or L∞ [14]. Hence, the compact and convex set of admissible object wrenches of a grasp G is given by W(G) = {Gx | x ∈ FC and kxk ≤ 1} .
(8)
µBall = sup{α ≥ 0 | Bα ⊂ W} .
(9)
Using an appropriate weighting of the contact forces’ normal components in the norm kxk, it is possible to apply specific upper bounds to each contact, which take into account applicable joint torques. Notice, that the set W always includes the origin – possibly at its boundary, in which case the grasp is not force closure. The measures employed in [14], [15] relate to the radius α of the largest ball Bα around the origin, which fits into the grasp wrench space W(G)1 (see fig. 1a):
Ref. [16] replaced the uniform ball – which weights all wrench directions equally – by a task ellipsoid, and defined a task-oriented quality measure based on the size of the largest task ellipsoid fitting into W again (see fig. 1b). Although this measure allows a weighting of applicable object wrenches according to their relevance for a specific task, to the best of our knowledge there doesn’t exist an efficient algorithm to compute this measure. III. TASK -O RIENTED G RASP E VALUATION We propose a measure, which builds upon an efficient algorithm for the force optimization problem. Given a grasp (G, FC) and an external disturbance wrench Fext , the grasp force optimization problem amounts to finding minimal contact forces x ∈ FC, which resist the disturbance. As shown in [7], the friction cone constraint (7) can be cast into P a Linear Matrix Inequality (LMI) of the form P (x) = m i=1 Ai xi 0, and the force optimization can be formulated as a determinant maximization (maxdet) problem with LMI constraints , which can be solved efficiently with the interior point algorithm [17]. Dual to this minimization of contact forces for a given external wrench, we try to maximize the magnitude α of a given normalized task wrench Fˆt given admissible and bounded contact forces: x ∈ FC and kxk ≤ 1 (see fig. 2): µ(Fˆt ) = sup{α ≥ 0 | αFˆt ∈ W} .
(10)
1 Actually, they used a discretized version of the admissible wrench space W obtained by approximating of each local friction cone by a smaller polyhedral convex cone.
PSfrag replacements
˜ z ∈ Rm yields an equivalent LMI:
αFˆ
P˜ (z) = P (V z) =
O
m ˜ X i=1
W Fig. 2. The proposed grasp quality measure µ(Fˆt ) maximizes the magnitude α of a given task wrench Fˆt .
This measure can be used to evaluate the grasp for the specific task of applying a wrench along the single direction Fˆt , and specifies the magnitude of the largest applicable wrench along this direction. If a wrench along the task direction Fˆt is not applicable at all, the quality measure µ(Fˆt ) equals zero, thus indicating an infeasible task for the considered grasp. A. Computation The task wrench optimization is in the form of a maxdet problem: maximize Ψ(x, α) = α − ε log det P −1 (x) subject to Gx − αFˆt = 0 α≥0 P (x) 0 and kxk ≤ 1
(11)
The objective function Ψ(x, α) maximizes the magnitude α of the task wrench, while simultaneously keeping the contact forces away from the friction cone boundaries according to the second term log det P −1 (x). A pure linear optimization with convex constraints yields an optimal solution on the boundaries of the friction cones, resulting in a grasp which is not robust to infinitely small disturbances. P The term log det P −1 (x) = − log(λi (P )) decreases to negative infinity, if any friction force approaches the cone boundary. The factor ε ≥ 0 allows a weighting of both contributions to the objective function. In order to cast the optimization problem (11) into a simpler form, we rewrite the linear equality constraint as x ˆ = G 0 · x0 = 0 . (12) G −F α According to this equation, we can interpret (Fˆ , α) as an additional virtual contact acting with wrench Fˆ in the objects’ coordinate frame and having the additional virtual friction cone constraint α ≥ 0. Appropriately, the contact force vector is extended by an additional component: 0 x0 = [xt , α]t ∈ Rm , m0 = m + 1. The constraint (12) restricts search space to the null space of the admissible G0 = G −Fˆ . If a set of m ˜ basis vectors of this null 0 ˜ space is concatenated to form a matrix V ∈ Rm ×m each 0 internal force xI can be expressed as x0I = V z
˜ z ∈ Rm .
(13)
Rewriting the friction cone constraints P (x, α) = diag(P1 , . . . , Pk , α) 0 in terms of the new free variable
A˜i zl 0 ,
(14)
because the LMI structure is preserved under affine transformations [7]. Now the optimization problem (11) can be written as: ˜ maximize Ψ(z) = vαt · z − ε log det P˜ −1 (z) (15) subject to P˜ (z) 0 kV zk ≤ 1
t where vαt = em ˜ · V is the last row of V which includes contributions to the wrench magnitude α = vαt · z. An external wrench Fext , which has to be resisted by the grasp, modifies the equality constraint in (11) to Gx − αFˆt = −Fext and can be eliminated by substituting x0 with + x00 + V z, where x00 = −G0 Fext is a particular solution of the constraint equation obtained using the pseudo inverse of G0 . Effectively the whole grasp wrench space W is shifted along Fext . Because this approach is well studied in literature [6], [7] we ignore external wrenches to simplify matters. We solve the determinant maximization problem (15) applying the maxdet optimization package [18], which in turn employs an interior point convex programming algorithm. In order to find a valid initial force z which is needed for this algorithm, we utilize the method proposed in [7], which transforms the LMI feasibility problem into a semidefinite programming problem which can be solved using the sdpsol package [19]. It is important to note, that a solution to the optimization problem not only yields the maximal magnitude α of the task wrench Fˆt , but corresponding contact forces x as well. In contrast to other grasp quality measures this becomes possible, because we maximize the magnitude of a single task wrench direction, and directly use the contact forces x – or more precisely the free variables z spanning the space of internal forces – as free variables for the optimization process.
B. Numerical Results from Physics-Based Simulation We employ the real-time physics-based simulation engine Vortex [20] in order to prepare and evaluate grasps before their final execution with a three-fingered dextrous robot hand [3]. The simulation package allows for accurate object motion and interaction based on Newtonian physics, and we have added features to provide static and dynamic friction for contacts, which is crucial for successful grasp simulation. Although contacts are simulated on the basis of point contacts and thus are necessarily coarse, they provide full force feedback, which is not available with our real world tactile fingertip and palm sensors. Our heuristic grasping algorithm coherently closes all fingers in a cage-like fashion around the object until no further finger movements are possible due to contacts with the object. At this time the grasp is considered as complete and is evaluated according to the quality measure. Quantitative evaluations of the measure which allow for a detailed comparison between different grasps require a
Z X Y
(b) maximally applicable forces around the unit ball
(a) three-finger grasp of a cube
Fig. 3. Three-finger grasp of a baufix cube with visualized friction cones at contact points (a). The grid of maximally applicable translational forces around the unit ball (b) approximates the corresponding subspace of the grasp wrench space W.
carefully chosen norm for the contact forces (kxk), which takes applicable joint torques into account. But, precise upper boundaries are usually not available and depend on the hand Jacobian, i.e. the current hand posture. In order to simplify matters, we have chosen an uniform weighting of all normal contact forces, which still allows for a qualitative comparison of different grasps. Fig. 3 shows the evaluation of a three-fingered simulated grasp, gripping a baufix cube (a), studied in our GRAVIS robotics system [21]. Due to the approximation of the fingertips and the toy cube with meshes, each finger tip has a different number of contacts with the cube. Altogether the shown grasp has eight contacts. In order to give an impression of the capabilities of this grasp, we have sampled possible task wrench directions Fˆt around the unit ball in steps of five degrees giving a total of 2592 sampling points. The resulting polytope (fig. 3b) approximates the projection of the grasp wrench space W to the 3-dimensional subspace of translational forces. Because the two fingers to the right and the single finger to the left both have four contacts, the magnitudes of applicable forces along the positive and negative x-axis are nearly identical and twice as large as appropriate forces along orthogonal movement directions, which have to rely on friction only. IV. F ORCE O PTIMIZATION
TASK C ONE ˆ If we can interpret the task wrench (F , α) as an additional virtual frictionless contact (FPC), we may ask what effect a virtual contact with friction (PCWF) or a soft finger contact (SFCE) would have. In order to answer this question, we have to consider the contribution Gi of a contact i to the grasp matrix G in more detail. Following the notation in [13], this contribution reads
or torsional components, which in turn are subject to the friction cone constraints (4-6). Carrying over the principle of virtual frictionless contacts – whose wrenches directly act at the objects frame O – to virtual contacts with friction (PCWF) we have to define a major task direction Fˆz corresponding to the normal direction of real contacts, and perpendicular tangential directions Fˆx , Fˆy , which together span a three-dimensional subspace of the whole wrench space. This subspace is confined to a task wrench cone if the friction cone constraints (5) are applied to the virtual contact forces (see fig. 4). In contrast to wrench cones of real contacts, the cones of virtual contacts may contain any mixture of translational and rotational force components. Especially it is possible to define wrench cones containing pure translational or pure rotational forces in order to express translational or rotational movement tasks. Similar to (11) we obtain the following optimization problem: maximize Ψ(x, αx , αy , αz ) =αz − ε log det P −1 (x) αx subject to Gx − [Fˆx , Fˆy , Fˆz ] αy = 0 αz q α2x + α2y ≤ αz tan γ
P (x) 0 and kxk ≤ 1 .
WITHIN A
Accordingly, the task wrench F = [Fˆx , Fˆy , Fˆz ] · [αx , αy , αz ]T can move freely within the task cone with aperture angle γ (fig. 4a), while we maximize the magni-
g
^
Gi = AdTT −1 Bci ,
(cone constraint)
Fy
(16) O PSfrag replacements
PSfragF replacements ^
O W where Bci is a contact force basis depending on the contact W F model and AdTT −1 is a matrix transforming wrenches from (b) Maximization on the surface of (a) Definition of the task cone oci the grasp wrench space. for fuzzy task wrenches. local contact coordinate frames Ci to the object coordinate frame O. Hence, columns of Gi describe orthogonal Fig. 4. Optimization within a task cone, defined by the major axis Fˆz and wrenches acting at the objects frame O which correspond the aperture angle γ. In comparison to simple task wrench optimization, the search space extends to a whole surface segment of W. to specific contact force components, i.e. normal, tangential oci
z
^
x
(a) Friction cones.
(b) Optimization along single direction.
(c) Optimization within task cone.
Fig. 5. Grasping a ball with a three-fingered simple gripper, whereas each finger produces a single contact with the ball as visualized by friction cones in (a). The maximization of a virtual contact with friction searches for the largest possible force within a whole cone (c). In contrast, the optimization of a frictionless virtual contact maximizes the wrench magnitude along a single wrench direction (b).
tude of its major component Fˆz . Consequently the search space for the optimal task wrench extends to the intersection of the task cone segment with the surface of the grasp wrench space W (fig. 4b), i.e. not only the wrench magnitude is maximized, but also its direction within the task cone. Fig. 5 illustrates this behavior with grasping a ball with a three-fingered simple gripper. The contact normals of all three contacts are located in the equatorial plane of the ball, such that the largest possible wrench of translational forces is parallel to this plane and runs along the symmetry axis of the gripper. Comparing the optimization result along an appropriately chosen single task wrench direction (fig. 5b) with the result obtained from an optimization within a task cone (fig. 5c), it can be seen that both approaches yield the same optimal task wrench – although the axis of the task cone differs from the optimal wrench direction. It is important, to note that wrench optimization within the task cone does not guarantee applicability of any wrenches in the cone, even if they are small in magnitude. Rather, the largest possible wrench is found, which is applicable by the grasp and is located in the task cone.
magnitudes αi of the task wrenches Fˆi simultaneously according to an objective function like X Ψ(x, α1 , · · · , αn ) = max w i αi (17) with weighting factors wi is not possible, because this results in the optimization of a single effective wrench P ˜i Fˆi , but does not ensure applicability direction Fˆeff = i w of all wrenches within the polytope (compare fig. 6b). To make this point even clearer, imagine a grasp wrench space W which is totally flat, i.e. does not allow applicability of wrenches orthogonal to its principal axis. Instead of a simultaneous optimization of all task wrenches, a serial approach is required, optimizing for every considered task wrench Fˆi separately. Finally all ˆ ˆ obtained maximal wrench magnitudes alpha i = µ(Fi ) can be combined to form an overall quality measure for the grasp w.r.t. the described task: µ(Fˆ1 , . . . , Fˆn ) =
n X
wi α ˆi ,
(18)
i=1
where a weighting of the different wrenches according to their relevance can be achieved with the factors wi > 0 again. It is important to notice, that all wrenches within V. G RASP Q UALITY W. R . T. A TASK P OLYTOPE the positively spanned subspace X We have to point out that both proposed quality measures C = { replacements αi ; ·; α ˆ i Fˆi | 0 ≤ αi ≤ 1} (19) PSfrag replacements PSfrag ˆ – optimization w.r.t. a task wrench Ft or w.r.t. a task cone – do not guarantee robustness of the grasp to small Ft disturbance wrenches which differ from the optimal task Fˆ1 Feff wrench. As an example consider fig. 6a showing a very Fˆ1 Ft ˆ flat grasp wrench space W, which allows application / F2 W resistance to a very restricted set of wrenches. While Feff W Fˆ2 ˆ the task oriented measure µ(Ft ) yields a large value for (b) Multiple task wrenches Fˆi . (a) Optimization within a cone. ˆ wrenches αFt along the principal axis, the corresponding Fig. 6. The task oriented measure µ(Fˆt ) does not guarantee robustness grasp is not robust to orthogonal disturbance wrenches. of the grasp, neither if a task cone is considered (a) nor if several Typically tasks require application of arbitrary wrenches task wrenches are considered simultaneously (b). In the latter case the within a narrow wrench polytope as indicated by the shaded optimization yields an effective wrench Feff which is linearly combined from the task wrenches Fˆi . region in fig. 6b. The naive approach of optimizing all
replacements
PSfrag replacements Fˆ2
O Fˆ3
Fˆ1
R EFERENCES Fˆ2 Fˆ3
W
(a) Emphasis on robustness.
Fˆ1
W
(b) Emphasis on magnitude.
Fig. 7. A separate optimization of multiple task wrenches Fˆi and a subsequent weighting the achieved magnitudes αi allows to consider whole task wrench polytopes.
spanned by the optimal task wrenches α ˆ i Fˆi are applicable by the given grasp due to the convexity of the grasp wrench space W. Depending on the selection of the set of wrenches spanning the task polytope, emphasis can be given to robustness (fig. 7a) or to a maximal wrench magnitude (fig. 7b). Actually, this approach allows to approximate the task ellipsoid proposed in [16] with a task polytope employing task wrenches Fˆi along the principal axes of the task ellipsoid. VI. C ONCLUSION We give an effective means to compute grasp quality measures for tasks given as vectors, cones, or polytopes in wrench space. Our modified usage of the LMI-based force optimization approach originally developed in [7] allows to maximize the force which is applicable along task wrench directions and at the same time to obtain the respective contact forces. This distinguishes our approach from other quality measures, which are solely based on geometric considerations with respect to the stability polytope in wrench space and usually assume that a force closure grasp is given. In contrast, a given grasp does not have to be force closure in order to apply our quality measure, which allows to consider tasks like pushing, lifting, or rotating objects, which will be typical for the next generations of humanoid robots to come. In contrast to common approaches which approximate friction cones by convex polyhedral cones causing an exponential increase of the computation time with the number of contacts [14], [15], the usage of LMIs and quadratical programming allows the consideration of true friction cones. Although we use non-optimized general purpose software, it takes only about 1 CPU second on a Pentium III–1GHz to solve the LMI optimization problem for the grasp shown in fig. 3 having eight contacts. A further extension of the formalism could also take into account the kinematic constraints of the hand. Our current work aims at including the task oriented grasp evaluation in the context of an evolving integrated architecture for imitation learning of grasps in a situated learning scenario [21], [22].
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