Proceedings of the 2001 IEEE International Conference on Robotics & Automation Seoul, Korea • May 21-26, 2001 Design of Robot Gripper Jaws Based on Trapezoidal Modules 1 Tao Zhang and Ken Goldberg 2 IEOR Dept., UC Berkeley part as it is acquired. We divide the grasp into three phases: pushing, toppling and fixturing. Initially, one jaw makes contact with the part (the “pushing contact”). The jaw pushes the part along the worksurface until the part makes contact with the “toppling contact” on the second jaw. At this point the part begins to rotate (topple) from its initial orientation to the desired orientation. During toppling, the part is constrained by two contacts and the surface. When the part reaches its final fixtured orientation other jaw surfaces stop its motion. The optimal jaw design guides the part through these phases, avoids premature toppling, jamming, and liftoff, and fixtures the part in form closure with maximal linear contact. ABSTRACT Assembly line reliability can be increased if grippers are carefully designed to capture and align parts. Jaw design depends on the g eometric and mechanical properties of the part as well as its desired orientation. In this paper we propose a class of modular jaws based on rapidly machineable trapezoidal modules. An optimal jaw design is an arrangement of trapezoidal jaw modules that maximizes contacts between the gripper and the part at its desired final orientation over the constraints that: the jaws will capture and rotate the part to its desired orientation and achieve a form-closure grasp. Given the n-sided 2D convex projection of an extruded polygonal part, we d evelop an implemented O(n 5 ) algorithm to find the optimal jaw design. The algorithm combines toppling, jamming, liftoff, accessibility, and form-closure analysis. We verify resulting designs by physical experiments. 2 RELATED WORK There is a substantial body of research on robotic grasping; Bicchi and Kumar provide a concise survey in . A number of papers consider part motion in the horizontal plane and how it can be used to reduce uncertainty. The motion of parts during grasp acquis ition was first analyzed by Mason , who studies push mechanics as a role of passive compliance in grasping and manipulation. Erdmann and Mason  explore the use of motion strategies to reduce uncertainty in the location of objects. They described a systematic algorithm for sensorless manipulation to orient parts using a tilting tray. Brost  applies Mason’s Rule to analyze the mechanics of the parallel-jaw gripper and polygonal parts. He shows that it is possible to align parts using passive push and squeeze mechanics. Goldberg  demonstrates that a modified parallel-jaw gripper could orient polygons up to symmetry by a sequence of normal pushes. Akella et al.  study a minimalist manipulation method to feed planar parts using a one joint robot over a conveyor belt. Several authors address motion of parts in the vertical (gravitational) plane during grasping. Trinkle and Paul  show how to align parts in the gravitational plane by lifting them off work-surface using a planar gripper with two pivoting jaws. The pre-liftoff phase analysis of their paper is related to our toppling analysis. They generated liftability regions corresponding to contact that cause the part to: slide, jam, and break contact with the worksurface. We focus on design of jaw shape and show that parts can be rotated using only translational jaw motion. Abell and Erdmann  study how a planar 1 INTRODUCTION Grippers can be the most design-intensive components of an assembly system . Although grippers are widely used for automated manufacturing, assembly, and packing, the design of gripper jaws is usually ad-hoc and can be a major limiting factor in automated assembly. Causey and Quinn  propose guidelines for the design of grippers in manufacturing. Based upon these criteria, we propose a modular approach where the standard parallel-jaw gripper is augmented with trapezoidal jaw modules. Figure 1 Gripper jaws align the part for assembly: (a) pushing (beginning); (b) pushing (ending); (c) toppling; and (d) fixturing. As illustrated in Figure 1, the part is initially in the resting pose (a). It is necessary to rotate the part to the desired final orientation (d) for assembly. The idea is to design the gripper jaws so as to align the 1 This work was supported in part by Adept Technology, Inc., a 2000 MICRO grant from the State of California, NSF grant CDA-9726389, and NSF Presidential Faculty Fellow Award IRI-9553197. 2 For further information please contact [email protected] 0-7803-6475-9/01/$10.00© 2001 IEEE 1065 Figure 1 illustrates an optimal solution. Without loss of generality, we assume the part rotates counterclockwise. Each jaw consists of a vertical base-plate and a set of trapezoidal modules arranged on the base-plate. To facilitate jaw machining and installation, we consider a class of trapezoidal jaw modules. Each jaw module is determined by the locations of two vertices that make contact with the part. The line segment between these two vertices represents an accessible segment on an edge of the part at its desired orientation. The objective is to find the set of accessible segments with maximum total length. We can define the problem, called OPTIMIZEJAWS, using a nonlinear programming model: polygon can be rotated while stably supported by two frictionless contacts. Rao et al.  give a planar analysis for picking up polyhedral parts using 2 hardpoint contacts with a pivoting bearing, allowing the part to pivot under gravity to rotate into a new configuration. Blind et al.  present a “Pachinko”like device to orient polygonal parts in the vertical plane. It consists of a grid of retractable pins that are programmed to bring the part to a desired orientation as the part falls. Wallack and Canny  develop an algorithm for planning planar grasp configurations using a modular vise. Brown and Brost  turn the vise upside down and invent a modular parallel-jaw gripper. Each jaw consists of a regular grid of precisely positioned holes. By properly locating (inserting) four pins on each grid, the object can be grasped reliably at the desired orientation. Mathematical programming has been employed to analyze grasping properties. Trinkle  introduced a nonlinear programming (NLP) model to predict instantaneous contact force, contact type, and velocity of grasped parts. The model minimizes the power generated by friction and gravity subject to kinematic constraints. Trinkle solves the problem using the primal-dual relationship. And shows that the problem can be reduced to a LP problem if the contacts are frictionless [17,18]. Our work is also motivated by recent research in toppling manipulation. Lynch [11,12] derived sufficient mechanical conditions for toppling parts in term of constraints on contact friction, location, and motion and we built on his analysis. Zhang and Gupta  study how parts can be reoriented as they fall down a series of steps. They derive the critical transition height, which is the minimum step height to topple a part from a given stable orientation to another. Yu et al.  estimate the mass and COM of objects by toppling. In , we introduced the toppling graph to geometrically represent the mechanics of toppling. We extended toppling analysis to parallel-jaw grasping with point contacts in . In the current paper, we consider non-point contacts and solve for optimal jaws based on trapezoidal modules. max: total length of linear contact area between part and gripper at desired orientation 1. pushing is successful; 2. toppling is successful; 3. no jamming occurs; 4. no liftoff occurs; 5. no collisions on part motion trajectory; 6. final grasp is secure. s. t.: Z v5 pushing friction cone z A’ toppling friction cone e4 ψ4 v4 COM ρ A' η A zA X P surface friction cone Figure 2 Notatio n. As shown in Figure 2, the part sits on a flat work-surface at a stable initial orientation. We define the World frame, W, to be a Cartesian coordinate system originating at pivot point P with X-axis on the surface pointing right, Z-axis vertical to the surface pointing up. The COM is a distance ρ from the origin and angle η from the +X direction at its initial orientation. The pushing contact, A’, is a distance zA’ from the surface; the toppling contact, A, is a distance zA from the surface. Starting from the pivot, we consider each edge of the part in counter-clockwise order, namely e1 , e2 , …, en . The edge ei , with vertices vi at (xi , zi ) and v(i+1) at (x(i+1), z(i+1)), is in direction ψi from the X-axis. The surface friction cone half-angle is αs = tan -1 µs , and the toppling (pushing) friction cone half-angle is αt = tan-1 µt . Let Ft , Fp , and Fs denote the contact force at A, A’, and the surface, respectively, and ft , fp , and fs denote the direction of the corresponding contact force. We also denote the direction at the left edge of 3 DESIGN PROBLEM DEFINITION We assume the part can be treated as a rigid extrusion of a polygon; both the part and the jaws are rigid; part geometry and location of the COM and the jaws are known; part motion is sufficiently slow to apply quasi-static analysis. We consider the following design problem. The input is the n-sided convex projection of an extruded polygonal part, its COM, its initial and desired orientations, ε vertex clearance radius, µt , µs : friction coefficients between gripper-part and part-surface, respectively. The output is a pair of gripper jaws, each specified by a list of trapezoidal modules. 1066 Since cos ω≥ 0 and µ s Mg ≥ 0 , then: the toppling (/ pushing / surface) friction cone as ftl (/fpl / fsl ) and the right edge as ftr (/fpr / fsr ). Let fg denote the vertical line through the part’s COM. Let a Ib denote the intersection of line a and line b, where a and b can be anyone of ftl , fpl , fsl , ftr , fpr , fsr , fg ft , fp , and fs . Let θ denote the rotation angle of the part; initially θ =0 and finally θ =θd . Let θt denote the rotation angle where the COM is right above P; therefore, θt = π /2 -η. We partition edge ei by wi that is the distance from vertex vi . Therefore, a point on ei can be expressed as (xi + wi cos ψi , zi + wi sin ψi ). We say an edge ek is visible if it can be seen from +X direction; invisible, otherwise. Therefore, ek is visible if 0 < ψk +θ < π; ek is invisible if π < ψk +θ < 2π. Notice that A can only make contact with visible edges and A’ with invisible edges. Fp 1 + µ s tan ω ≥ 0 π 1 ± αt ) ≥ − 2 µs 3π or ψ i ≥ 2 ⇒ tan(ψ i + . #6 Therefore, given the pushing contact makes contact with edge ei , to guarantee no jamming equation #6 must be satisfied. Z P1 Fp X 4 ALGORITHM We develop a numerical algorithm to find the optimal jaw design. The algorithm is based on an efficient enumeration of feasible designs that exploits part geometry and a graphical force analysis. The feasible region for the optimization problem is a curve plane defined by (zA , zA’, L), where L is a function of (zA , zA’) and represents the total contact length between the part and the gripper at the desired final orientation. We approximate the maximal (optimal) point of this plane by sampling the plane. P2 P3 Figure 3 Condition to prevent toppling. To prevent the part from toppling during the pushing phase, we begin by constructing a Lynchlike triangle with vertices P1 , P2 , and P3 as shown in Figure 4. P1 (xp1 , zp1 ) is g Isl , P2 (xp2 , zp2 ) is P, and P3 (xp3 , zp3 ) is g Isr . From these definitions we have: xp1 = s, #7 #8 zp1 = (x2 -s)/µs , xp2 = 0, #9 zp2 = x2 , # 10 # 11 xp3 = s, # 12 zp3 = - (x2 -s)/µs , # 13 where s = ρ cosη. To guarantee the pushing contact does not roll the part clockwise, every force in the pushing friction cone must not make a positive moment (i.e., the contact force passes around the triangle in a clockwise fashion.) respect to the P1 P2 P3 triangle . Therefore, fpl must pass below the highest vertex of the triangle. Let 1 wi denote the edge contact on ei where fpl passes exactly through point P1 . We find 1 wi by projecting a line from P1 at the angle of fpl until it intersects the edge of the part: 1 w i = ( zi - (x2 -s)/µt - (xi - s) tan βil ) /(cos ψi tan βil # 14 sin ψi ), where βil = ψi + π/2 + αt . The contacts for fpl passing through P2 and P3 are given by: 2 w i =( zi -x2 -xi tan βil ) /(cos ψi tan βil - sin ψi ), # 15 3 w i = ( zi + (x2 -s)/µt - (xi - s) tan βil ) /(cos ψi tan βil # 16 sin ψi ). To guarantee the pushing contact on an edge ei cannot roll the part clockwise, the following inequality must be satisfied: # 17 i zA’ < DHi = zi + w Hi sinψi , 4.1 Location of Contacts The contacts could switch edges as the part rolls. Therefore, we have to check which edge the contacts keep touch with at the different part rotation angle θ. We first consider the location of A’. It keeps touch with a invisible edge ek at θ if xk sinθ + zk cosθ > zA’ > xk+1 sinθ + zk+1 cosθ. Therefore, we can find the corresponding xA’ by: z − x k sin θ − z k cosθ . # 1 x = x cosθ − z sin θ + A ' A' k k tan(ψ k + θ ) The location of the toppling contact (xA , zA ) can be derived in a similar way . 4.2 Pushing Constraints To guarantee the linear translation of the part on the surface during pushing, neither toppling nor jamming could occur. To eliminate the pushing contact that may result in jamming, Fp should overcome the friction between the part and the surface. By static analysis, ω= ψi + π/2 ± αt #2 #3 Mg – Fp sin ω = N Fp cos ω≥ µs N #4 Therefore, we have: cosω + µ sin ω ≥ µ s Mg Fp #5 1067 ψi ≥ π − αt + αs w that is the wHi = 1 i π − αt + αs ≥ ψi 2 wi minimum of 1 wi , 2 wi, and 3 wi . Similarly, to guarantee the pushing contact does not roll the part counter-clockwise, the following inequality must be satisfied: # 18 i zA’ > DLi = zi + w Li sinψi , βir = ψi + π/2 αt and where ψi ≥ 2π + αt − αs that is the maximum of 2 wi w = 4.5 Liftoff Constraints As the part rotates, it may be lifted off from the surface. We now find the condition that insures the contact between the part and the surface. In rolling phase, we only need to consider the cases where: ψi > π + αs - αt if xA’ < 0; or ψi > π + αs + αt if xA’ > 0 because there is no toppling area otherwise. In these cases, the part keeps contact with the surface if and only if Ft makes negative moment about the line segment between g Ip and p Is . Therefore, the toppling constraints include the nonliftoff conditions when θ <θt . When θ >θt , we assume the part is just about to be lifted off from the surface if a jamming area exists. Therefore, we have: # 19 Fp cos αi + Ft cos αj = 0, where Li 3 wi 2 π + αt − αs ≥ ψi 1 w i , 2 w i, and 3 w i . In summary, to guarantee linear translation of the part without toppling and jamming, inequality #6, #17, and #18 must be satisfied if the pushing contact makes contact with edge ei . where αi = β il β ir 4.3 Toppling Constraints During toppling, the part rotates about P. The system of forces on the part: Fp , Ft , Fs , and the part’s weight Mg, must generate a negative moment on the part with respective to P. The contact forces lie on the edge of the corresponding friction cones. Since P slides to the right, the contact force from the surface is on the direction of fsl . Ft is on the direction of ftl if xA > 0; ftr if xA < 0. Fp is on the direction of fpr if xA’ > 0; fpl if xA’ < 0 . Given the part stays on the surface and the height of A’, the toppling function , Hj (θ), is the minimum height at θ that A in contact with edge ej must be in order to roll the part instantaneously, where θ = 0 ~θd during toppling. Consider an area in W defined by linear edges. Let a toppling area denote an area such that toppling is guaranteed if Ft makes a negative moment about every point in the area. When θ <θt , there only exists a toppling area. We derive the toppling function depending on the direction of Fp , i.e., xA’ > 0 or xA’ < 0 (see  for details). if x A' < 0 and αj = β jl if x A > 0 . if x A' > 0 β jr if x A < 0 To guarantee non-liftoff, forces must satisfy: Fp sin αi + Ft sin αj – G < 0, # 20 Therefore, tan αi - tan αj < 0. Notice that when both Fp and Ft point down, i.e. π < αi < 2π and π < αj < 2π, the inequality is always true; when both Fp and Ft point up, i.e. 2π < αi < 3π and 0 < αj < π, the inequality is always false. 4.6 Toppling Graph Our analysis involves the graphical construction of a set of functions. All of these functions are piecewise sinusoidal and dependent on θ. They map from part orientation to height: S1 →ℜ ℜ +, where S1 is the set of planar orientations. The functions include vertex functions, toppling functions, and jamming functions. The toppling graph , which consists of these functions, helps us to identify the range of the contact permits toppling. 4.7 Trajectory Analysis Once the part has been toppled to θd , the jaws must stop the part’s rotation and securely hold it. We want to maximize the contact between the part and the jaws for the most robust performance. During toppling, the part is constrained by two contacts and the surface. We first consider the motion trajectory of the visible edges. We take the toppling contact as a fixed point; therefore, the part motion is constrained by A and the surface (see Figure 4). The motion trajectory of the invisible edges can be obtained by the same method taking A’ as a fixed point; therefore, the part motion is constrained by A’ and the surface. The part performs both rotation and linear translation during toppling. We decompose the part motion into pure rotation and pure linear translation. The part first rotates about P to semi-position, and then translates to actual-position. Let (θ xj , θ zj ) and (θ x’j , θ z’j ) denote the actual-position and the semiposition of vertex vj after the part is toppled by θ, 4.4 Jamming Constraints The part continues to rotate after it has reached θt , and jamming may occur due to friction. Given the part stays on the surface and the height of A’, the jamming function , Jj (θ), is the minimum height at θ that A in contact with edge ej must be in order to guarantee no jamming. Let a jamming area denote an area such that no jamming is guaranteed if Ft does not make a positive moment about every point in the area. We consider the jamming function by investigating the location of A’. We divide quadrant II of W into four zones by fg and fsl . Therefore, A’ must lie in one of these zones during toppling. We first check which zone A’ belongs to at θ. Then we check if there exists a jamming area or a toppling area. Finally, we derive the jamming function if a jamming area exists and the toppling function if a toppling area exists (see  for details). 1068 Qjk(θ) = x Q jk (θ, z A ) Q (θ, z ) A z jk = respectively. Let (d xj , d zj ) and (d x’j , d z’j ) denote the actual-position and the semi-position of vertex vj after the part is toppled to its desired orientation, respectively. Let θxt and d xt denote the distance between the actual-position and the semi-position of any point after the part is toppled by θ and θd , respectively. No obstacle (any portion of the jaws) can block the motion trajectory of the part during toppling. We developed quasi-vertex functions to represent the motion trajectory of vertices. Given zA , the quasivertex function Qjk(θ, zA ) indicates the location of vk in Fj as the part rotates. ( θ x' k −d x' j +θ xt − d xt ) cos(ψ j −1 + θd ) +( θ z 'k − d z ' j ) sin(ψj −1 + θd ) −( θ x' k −d x' j +θ xt − d xt ) sin(ψj −1 + θd ) +( θ z 'k − d z ' j ) cos(ψj −1 + θd ) # 24 The line segment, connecting two corresponding points of Qjk(θ, zA ) and Qj(k+1)(θ, zA ) at the same rotation angle, represents the configuration of edge ek in Fj . We denote it by the quasi-edge function Ejk(θ, zA ). Figure 5 illustrates Q43 (θ, zA ) and Q44 (θ, zA ), and E43 (θ, zA ) for the sample part given zA’ = 0.5cm and zA = 3.65cm. x (dxj, dzj) 80 Q43 z 80 60 e j-1 40 Q44 -80 -60 -40 -20 40 20 -80 -40 -80 (a) (dxj-1, dzj-1) (b) Figure 5 Quasi-vertex functions (a) and quasi-edge functions (b). We obtain an accessibility graph of Fj by combining Qjk(θ, zA )’s for all the visible vertices vk’s into one graph. We say a portion of an edge is an accessible segment if the gripper can contact this portion at the desired final orientation of the part without blocking its motion trajectory. Therefore, the portion of edge e( j-1) is accessible if there is no intersection between the quasi-edge functions and this portion of X-axis in the accessibility graph of Fj . Figure 5 illustrates the accessibility graph of F4 based on the sample part given zA’ = 0.5cm and zA = 3.65cm. Figure 4 Frame Fj notation. We define a frame of reference Fj at the desired orientation of the part. As shown in Figure 5, Fj originates at vj . The Z-axis of Fj is the interior normal of edge e( j-1), and the X-axis is on edge e( j-1) obeying the right-hand rule. We obtain Qjk(θ, zA ) by transforming the motion trajectory of vk in W to Fj . We know that d xt = d xA - d x’A , where d xA = xA since A is fixed. A makes contact with edge em if d z’m < zA < d z’m+1 . Therefore, we can find d x’A , and then we obtain: 70 ( z A − d z ' m )(d x ' m +1 − d x ' m ) - d x’m . # 21 d xt = xA d z ' m+ 1 − d z ' m So the transformation matrix Fj TW is given by: Fj TW = [WTFj ]-1 = cos(ψ j −1 +θ d ) 0 − sin(ψ j −1 + θ d ) 0 0 sin(ψ j −1 + θ d ) −(d x' j + d xt ) cos(ψ j −1 + θ d )− d z ' j sin(ψ 1 0 0 0 cos(ψ j −1 + θ d ) + (d x' j + d xt ) sin(ψ j −1 +θ d )−d z ' j cos(ψ 0 0 1 50 40 j −1 j −1 xk cosθ − z k sin θ+θ xt , 0 x k sin θ + z k cosθ 1 where θ xt = xA - Q43 +θ d ) + θ d ) # 22 We next describe the motion trajectory of vertex vk in W numerically. At each sampling θ, we compute the position of vk: VW = Q45 60 30 20 10 Q44 -100 -80 -60 -40 0 -20 0 20 -10 Figure 6 The accessibility graph of F4 . # 23 Starting from any point of Qjk(θ, zA )’s in the accessibility graph of Fj with z Qjk(θ, zA ) < 0, we search the trajectory of ek and the neighbor edges recursively to find all the intersections between the corresponding quasi-edge functions and X-axis. We obtain the accessible segment on edge e( j-1) by eliminating those intersections. Repeating this ( z A −θ z ' l )(θ x' l + 1 −θ x ' l ) - θ x’l if θ z’l θ z ' l +1 − θ z ' l < zA < θ z’l+1 . Thus, quasi-vertex function Qjk(θ, zA ) for vertex vk can be shown to be: 1069 analysis on all the edges of the part, we get the set of accessible segments for the pair of (zA , zA’). Additionally, we require the jaws to achieve a form-closure grasp on the part. It is well known that to check the form-closure grasps is equivalent to solving a system of equations . We find the optimal design by ranking the form-closure check survivors based upon the total length of contact. Figure 7 demo nstrates the optimal design with maximal L for the sample part where zA’ = 0.5cm and zA = 22.6cm.         Figure 7 An example: the optimal jaw design.  Given the n-sided convex projection of an extruded polygonal part, our algorithm takes O(n 2 ) to find a pair of feasible (zA , zA’). For each (zA , zA’), we apply O(n 3 ) time trajectory analysis to find the set of the accessible segments. Therefore, the running time is O(n 5 ).     5. IMPLEMENTATION RESULTS We conducted a physical experiment using an AdeptOne industrial robot and a parallel-jaw gripper with jaws designed using the methodology described in this paper. The part and the jaws were machined from aluminum. The corresponding friction cone half angles are αt = 5° ± 2°, and αp = 5° ± 2°. As illustrated in Figure 1, the part begins at stable orientation (a). 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