Optimal Multingered Grasp Synthesis. J. A. Coelho Jr. R. A. Grupen. Department of Computer Science. approach. 2 Grasp controllers

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1 Optimal Multingere Grasp Synthesis J. A. Coelho Jr. R. A. Grupen Laboratory for Perceptual Robotics Department of Computer Science University of Massachusetts, Amherst, 3 Abstract This paper iscusses how grasp synthesis can be cast as a control composition problem. Two elemental grasp controllers are presente an compose into a general purpose grasp controller through the use of stanar optimal control techniques. The resulting grasp synthesis proceure is scalable on the number of contacts, is not tie to any specic object representation scheme, an can incorporate kinematic constraints into its resolution. grasp synthesis, an is computationally ecient with respect to the number of contacts. It also can be use to incorporate constraints relate to grasp execution (e.g., manipulator kinematics) into the synthesis process. The next section presents the elemental grasp controllers that constitute the basis for the grasp synthesis proceure, followe by the formulation of control composition as an optimal control problem. A simple grasp synthesis problem is then use to illustrate our approach. Introuction Grasp synthesis, as iealize by most researchers, consists in the o-line computation of a grasp con- guration that maximizes a certain grasp metric the author is intereste in [, 6, 7, 9,, ]. Typically, a complete moel [6, 7, ] or a smooth approximation [, 9] of the object geometry is assume. However, the synthesis proceure is computationally expensive [6] or it may involve exhaustive search over the object geometry [7, ]. Some approaches restrict the number of contacts accoring to the object imensionality ( contacts in D an 3D [], contacts in D [6], at least 3 contacts in 3D an contacts in D []), others constrain the object representation (polygons or polyhera are assume in [7, ], an smooth/parametric moels are require in [, 6, 9]). Alternatively, grasp synthesis can be cast as a control composition problem. The unerlying assumption is that a controller for executing wrench closure on general convex geometries can be constructe by composing control moules rawn from a set of basis controllers. This paper escribes how optimal control techniques can be use to solve the control composition problem. The resulting grasp synthesis proceure is not tie to any specic object representation scheme, is applicable to both - an 3-imensional Copyright c994 IEEE. Grasp controllers Grasp synthesis proceures are esigne aroun a grasp metric which is maximize uring the synthesis process. Grasp metrics usually emphasize one or more aspects of the resulting grasp conguration: capacity to resisting external isturbances [9], istance between contacts [], ratio between resulting wrenches an applie forces, or minimization of total forces applie to the object [7], grasp stability [], an others. A grasp conguration is stable if it can resist arbitrary perturbation wrenches. The approach taken observes that homogeneous grasp solutions base on frictionless point contacts moels can exploit whatever real contact friction exists to ilate the contact wrench envelope an therefore improve grasp robustness. Moreover, such grasp geometries can be use to squeeze the object within the \frictionless null space" to scale the size of the friction ilate wrench envelope. This section escribes two controllers, respectively the force closure (FC) controller an the moment closure (MC) controller, esigne to maximize grasp stability. Both controllers are base on estimates of the local wrench closure error surface. The FC controller erives its estimate on the basis of relative contact position alone, while the MC controller employs positions an normals to the contact surfaces. Contacts are assume to be frictionless point contacts an map

2 to forces an moments [; ] with respect to the object frame.. Stable grasp suciency metric In scoring a grasp conguration, it is useful to analyze the associate grasp matrix (also known as the grasp Jacobian). This matrix escribes the transformation from a set of forces applie by contacts on the object surface to a set of object frame wrenches. The grasp matrix W is ene as W = [W W : : :W n ], where W i is a six imensional vector of wrenches applie at the i th contact position. The null space of the frictionless grasp matrix e- nes a basis for a set of equivalent grasp geometries. However, the proceure for computing the null space of a matrix oes not allow for ierentiability an therefore cannot be use as a control surface. The same shortcomings plague other grasp metrics: they are computationally expensive an frequently involve optimization proceures or matrix analysis techniques that o not oer irectional information on which a grasp controller can be base. To overcome those shortcomings, a suciency metric was evise [8], base on the resiual wrench vector ~ = ~ T ~ = ~t? n ^! i! T ~t? n ^! i! ; () where ~ expresses the net wrench over j n contacts, ~t is a user-specie wrench closure bias, an ~! i is the wrench vector resulting from the i th interaction force. The elements t j [?; ] of ~t an the elements w ij [?; ] of ~w i are qualitative in the sense that they o not reect engineering units of force an torque, but express the relative ability of a contact conguration to transmit forces an torques through the object's surface. Minima in correspon to linear combinations of these qualitative wrenches which map into the grasp matrix's null space. By moving the contacts in irections that minimize, the grasp controller approaches the esire null grasp conguration. One plausible approach is to compute the graient of with respect to the contact coorinates i an move the contacts accoringly: i = ~t? n For the null grasp task, ~t = ~.! T? ~ t? n ^! i i i P i ^! i =? ~t? n =? n ~T G i i ^! i! T!i i n =? n GT i ~; () where G i is the wrench graient with respect to the contact coorinate i. In vectorial notation ~ =? n 6 4 G T G T. G T n ~ =? n GT ~: (3) Equation 3 evaluates to ~ in one of two conitions: (a) ~ = ~ an (b) when G T ~ = ~. The rst conition is the esire convergence criterion an can be use to suppress external perturbations by scaling grasp wrenches to span the task. The secon conition is equivalent to a local minimum of. This class of minima results when a local tangent to the wrench surface, G, is orthogonal to the resiual vector, ~. Both grasp controllers are base on Equation 3: what istinguishes them is exactly how ^! i varies with the contact coorinate, i.. Force closure controller The force closure (FC) controller is use to navigate frictionless point contacts on the continuous Gaussian sphere. This simplie system uses only relative contact position to eliminate the force closure component of the resiual (Equation ). If an are the angular coorinates on the Gaussian sphere, then the corresponing wrench omain moel of the object ~W (; ) = [F x F y F z M x M y M z ] becomes w = f x =?cos()cos() w 4 = m x = w = f y =?sin()cos() w 5 = m y = w 3 = f z =?sin() w 6 = m z = The suciency metric for the FC controller yiels: FC = " " " t fx + n t fy + n t fz + n cos( i )cos( i )# + sin( i )cos( i )# + sin( i )# (4)

3 The graient of Equation 4 with respect to k yiels: z FC k = (A k + B k) = sin( k + k ) (5) where A k, B k, an k are constants inepenent of k : A k = n B k =? n tfy + n tfx + n i6=k (sin( i )cos( i )) A cos(k ) i6=k (cos( i )cos( i )) A cos(k ) ^r φ r o φ o θo N ^ r θ y k = tan? (A k =B k ) Equation 5 emonstrates that the suciency metric employing the force closure moel generates a unimoal error function..3 Moment closure controller Reasoning in terms of surface curvature, a scale version of the FC controller can capture the local shape of the suciency metric for all positive an negative curvatures: the scaling constant is proportional to r, where r is the raius of curvature. However, as r! the FC graient vanishes, since forces remain constant on planar surface facets. The moment closure (MC) controller minimizes the moment resiual by ajusting each contact position on the local surface facet. We use the perpenicular of a plane, (r ; ; ), to parameterize the contact plane. Figure illustrates the parameterization an the geometry from which the wrench graient is erive. The forces transmitte through any planar face are constant at all contact positions on that face. The moments applie to the object, ~m = ~r f, ~ vary linearly with surface coorinate an pass through zero where the perpenicular passes through the plane. The resulting wrench moel is: w = f x =?cos( )cos( ) w = f y =?sin( )cos( ) w 3 = f z =?sin( ) w 4 = m x =?r sin( )cos( ) + r sin( ) w 5 = m y =?r sin( )sin( )? r cos( ) w 6 = m z = r cos( ) (6) where ( ^r ; r^ ) are the surface coorinates of the contact in the plane. Note that this approximation yiels x Figure : The Moment Closure Moel erive from a Parametric Object Plane constant contact forces which implies that control is erive from the moment iagrams exclusively; thus the esignation \moment closure". The moment closure suciency metric can be written as follows: MC = i wi MC = "t wi? n wi MC ; (7) i w i # Figure (a) shows the plot of both the force closure metric (Equation 4, ashe curve) an the plot for the suciency metric (Equation, soli curve), for the square shape, contacts. Those plots are obtaine by xing the black contact at the origin =, an moving the white contact counterclockwise over the object surface. Figure (b) shows the complete object moel; in the same gure, the suciency metric's local minima are marke over the object surface with triangles, an the white contact is positione over the metric's global minimum. Figure (c) shows the corresponing constant curvature unit Gaussian sphere, an its global (an only) minimum. Figure illustrates how the FC controller may contribute to grasp synthesis: it is a global approximation of every manifol geometry with no local minima. This heuristic control surface selects combinations of faces that minimize the force closure resiual.

4 .75 MC error FC error y x 3 Control composition as an optimal control problem Error Degrees (a) (b) y x (c) A small an somewhat constraine version of the grasp synthesis problem was chosen to illustrate our approach, namely the null grasp task of planar, convex objects with contacts. There is no theoretical impeiment in generalizing this technique to multicontact grasp of 3D objects. For the general control composition problem, the only requirements are continuity of the metric being optimize, an ierentiability of the constituent controllers. The problem Figure : (a) Suciency metric error plot, for the complete object moel (soli curve) an for the smoothe object moel (ashe curve). (b) The triangles mark the position of the suciency metric local minima on the object surface. (c) FC object moel. In this case, the global minima for both metrics coincie on = 8 o. (a) β θ (b) β.4 Controller properties The FC controller is base on contact positions exclusively; the contacts navigate the surface of the continuous Gaussian sphere, an the resulting controller can be mae globally convex [4]. The MC controller consiers both contact position an normal; the resulting control surface is piecewise convex. Each controller encoes a moel of how local contact geometry transforms contact forces into contact wrenches, but no explicit moel of the object itself is ever require { only local sensory information. Moreover, it is possible to assign an inepenent asynchronous controller to each contact. As a consequence, the complexity of the composite controller grows linearly with the number of contacts. In [4, 5], knowlege-base heuristic rules, learning techniques, an pre-imaging control are use in the composition of the FC an MC controllers. These techniques o not require a complete moel of the object; control actions can be compute irectly from sensor reaings of contact positions an normals. Although this is probably the most aequate paraigm for the control of real-worl complex systems, one can o better if complete or very goo moels about the problem is available, using optimal control techniques. Portraying the composition of controllers as an optimal control problem illustrates a general solution strategy that can be use in other omains to solve complex control problems. Figure 3: (a) Initial conguration (b) Final conguration. aresse in this paper consists of activating the FC an MC controllers, so as to minimize the number of control steps require to take the system from a initial to a nal (unknown) conguration (see Figure 3). For simplicity, we will consier that just one contact moves over the object surface, while the other stays xe at its initial position. Also, we will assume that the activation for the MC controller is constraine to be (? ), where [; ] is the activation for the FC controller. 3. Problem formulation Maximize Subject to _(t) =? () = : V = [; ] Z T FC()? MC () t + (? ) MC() In the formulation above, MC is the wrench closure error (Equation 7). For two contact grasps of square geometries, MC () can be written as MC () = E + ( tan? ) ; 6

5 where E is a constant corresponing to the force error incurre in positioning the moving contact in the various faces of the square. For the face corresponing to =, E = ; for faces corresponing to = = an = 3=, E = :5, an for the face corresponing to =, E =. Similarly, FC () is the force closure error, expresse as FC () = :5 ( + cos(? )); being the angle for the xe contact. In the evelopment that follows, the rst an secon erivatives of MC () an FC () with respect to will be require: MC = tan? 4 cos MC 8 + (tan? ) sin = 6 cos 4 FC =?:5 sin(? ) FC =?:5 cos(? ) The Hamiltonian function for the problem can now be expresse as H(t; ; ; ) =? MC () + (t) t : The costate variable is the Lagrangian multiplier require by the metho [, 3]. The Maximum Principle conitions are H(t; ; ; ) H(t; ; ; ) 8t [; T ] (8) _(t) = H =? _(t) =? H FC() = MC + (t)( FC (T ) = + (? ) WC() (9) + (? ) MC ) () Equation 8 means that the resulting Hamiltonian H is to be maximize over time, an (t) is the activation policy that maximizes it. Notice that selecting such that H = oes not cover all cases, because of the constraine range of values can assume it might be the case that correspons to one value in the limits of the close interval [; ]. Activation This is the case for the problem we are intereste in; for this example, H oes not epen on : H =?( FC? MC ): Therefore, the time optimal policy for this particular problem implies switching from to an back. This optimal pattern can be easily compute by integrating Equations 9 an above over time:. Initialize ;. Compute H H ; if make = ; 3. Compute t+ = _ t t; 4. Compute t+ = _ t t; > ; select = ; otherwise, 5. Check for convergence on ; return to () above if not converge. 4 Results In all results, we use () = 5, an t = :. The metho is not sensitive to the values chosen for these parameters; an ample range of values can be use. However, selecting a low initial value for may result in solutions corresponing to local minima. The xe contact is statione at =?:46365 raians Time MC active FC active Figure 4: Time optimal activation pattern uring the grasp. = correspons to full activation of the FC controller, while = correspons to full activation of the MC controller. Figure 4 shows the optimal activation pattern for as a function of time. Figure 4 also isplays the corresponing areas on the object surface where the controllers were selecte: gray areas correspon to where

6 in the object the MC controller was selecte ( = ), an white areas label the those areas in which the FC controller receive full activation ( = ). Contact Angle Time Figure 5: Evolution of with time. Figure 5 shows how varies with time. Starting from () = :, the contact progresses steaily to the optimal solution (T ) = 3:65 raians. Firstorer iscontinuities are present whenever the moving contact changes faces on the square. 5 Conclusion As long as the metric chosen to be optimize oesn't epen on the intermeiate activations (t), the time optimal activation policy will always be a bang-bang type solution, no matter the object being graspe. Another interesting aspect is the fact that this approach also yiels an eective grasp controller: just by evaluating H one can ecie on the next activation. What remains to be etermine is how robust this controller is with respect to object geometries, an uncertain actuation an sensory information. Because this approach essentially involves integration over time, it is not clear how to revise the control history incrementally: presumably, a object moel is being continuously or perioically upate, an sensory ata may yiel iscontinuities in the state information. How to accommoate such changes in the control scheme is yet to be etermine. Acknowlegements This work is supporte in part by NSF CDA- 8957, IRI-9697, IRI-989, an CNPq 7/9.6 (National Research Council, Brazil). References [] Bryson, A., an Ho, Y.-C. Applie Optimal Control: Optimization, Estimation, an Control. Hemisphere Publishing Corporation, New York, NY, 975. [] Chen, I., an Burick, J. Fining antipoal point grasps on irregularly shape objects. In Proc. 99 IEEE Int. Conf. Robotics Automat. (Nice, FRANCE, May 99), vol. 3, pp. 78{83. [3] Chiang, A. Elements of Dynamic Optimization. McGraw-Hill, Inc., New York, NY, 99. [4] Coelho Jr., J. Eective multingere grasp synthesis. Master's thesis, UMass, Department of Computer Science, Amherst, MA, Sept [5] Coelho Jr., J., an Grupen, R. Control preimaging for multingere grasp synthesis. In Proc. 994 IEEE Int. Conf. Robotics Automat. (San Diego, CA, May 994). [6] Faverjon, B., an Ponce, J. On computing two- nger force-closure graps of curve objects. In Proc. 99 IEEE Int. Conf. Robotics Automat. (Sacramento, CA, May 99), vol., pp. 44{ 49. [7] Ferrari, C., an Canny, J. Planning optimal grasps. In Proc. 99 IEEE Int. Conf. Robotics Automat. (Nice, FRANCE, May 99), vol. 3, pp. 9{95. [8] Grupen, R., Coelho Jr, J., an Souccar, K. Online grasp estimator: A partione state space approach. Tech. Rep. COINS Technical Report 9-75, COINS Department, University of Massachusetts, Oct. 99. [9] Guo, G., Gruver, W., an Jin, K. Grasp planning for multingere robot hans. In Proc. 99 IEEE Int. Conf. Robotics Automat. (Nice, FRANCE, May 99), vol. 3, pp. 84{89. [] Jameson, J., an Leifer, L. Automatic grasping: An optimization approach. IEEE Trans. Syst., Man, Cybern. SMC-7, 5 (Sept./Oct. 987). [] Nguyen, V. Constructing stable grasps. The Int. J. Robotics Res. 8, (989), 6{37.