Identifying contact formations in the presence of uncertainty

The efficiency of the automatic execution of complex assembly tasks can be enhanced by the identification of the contact state. In this paper we derive a new method for testing a hypothesized contact state using force sensing in the presence of sensing and control uncertainty. The hypothesized contact state is represented as a collection of elementary contacts. The feasibility of the elementary contacts is tested by solving a linear program. No knowledge of the contact pressure distribution or of the contact forces is required, so our method can be used even when the contact forces are statically indeterminate. We give a geometric interpretation of the contact identification problem using the theory of polyhedral convex cones. If more than one contact state is feasible, we use the geometric interpretation to determine the likelihood of elZch feasible contact formation. 1 Introd uction In this paper, we present a linear programming approach to the force/moment testing of the different contact topologies. This apf)roach yields two benefits over the previous techniques: no estimate of the pressure distribution in the contact interface is required, and the contact formation can be identified even when the contact forces are statically indeterminate. The work presented here extends that of Xiao [1], on contact identification, Desai [21, who introduced the concept of contact formation and sensor-based recognition of contact formations, Hirai and Asada [3], who used Monte Carlo techniques to develop contact formation classifiers from a CAD model, and of Taylor [4] and Brooks [5] on spatial and force uncertainty. Consider a movable polyhedral object W, which is held by a robot, contacting polyhedron obstacles 6'. The forces and moments acting on the movable object can be measured by a force sensor. The exact contact formation is not known. The problem to be solved can be stated as: given a set of measured forces and moments, a set of hypothetical contact formations, and an uncertainty model, find the contact formation. We make the following assumptions: 1. The bodies in the system are rigid polyhedra, 2. Friction forces obey Coloumb's Law, 3. Force/moment sensing errors are bounded, 4. No uncertainty exists in the model of the environment, 5. The position and orientation of the robot are specified. The first four assumptions are commonly made in analyses of robotic manipulation systems and experimental results support their validity. The last assumption does not exclude uncertainty in the location of robot. The uncertainty in the location of the robot is handled by the first phase of the contact formation identification algorithm: the hypothesis phase. In the hypothesis phase, we use Xiao's algorithm to obtain a discrete set V.. of possible locations of the robot and a list of possible contact formations corresponding to each element of V... Each element of V, along with the list of possible contact formations is then passed to the testing phase of the contact identification algorithm. 2 Contact Models For simplicity we consider contacts between two polyhedra. Let W denote the movable object and e denote the polyhedral environment. The contacts can be described in terms of three elementary types of contacts [6] Type A between a face of Wand a vertex of e. Type B between a vertex of W and a face of e. Type C between an edge of W and an edge of e. 3 Uncertainty Models The most important sources of uncertainty are: uncertainty in the force/moment sensor, uncertainty in the positions of the contact points with respect to the force sensing frame, the uncertainty in the direction of the contact normal with respect to the force sensing *This research was supported by the National Science Foundation (grant IRI-9304734), the Texas Advanced Research Prc>gram (999903-078), and the Texas Advanced Technology Prc>gram (999903-095). 59 0-8186-7108-4/95 $4.00 ~ 1995 IEEE frame, and the uncertainty in the position and orientation of the force sending frame with respect to a world frame. All but the first source of uncertainty are handled by the first phase of the contact identification algorithm. Let S be the force sensor internal coordinate frame. Let C be a coordinate frame attached to the object's center of gravity and aligned with the world frame W. For the rest of this paper we use "generalized force" to imply force and moment. Let gs denote the actual generalized force applied at frame S, and let t.s denote the measured value for the generalized force. Also, let c5g.s be the vector of uncertainty in the force sensing given by: where g= [gz-,g,,-, ]T with g+ defined similarly. Equation (11) represents a rectangular solid in generalized force space centered around the nominal generalized forces at C gC that bounds the actual generalized force gc at C. 4 Model Description When a frictionless rigid body is in contact with its environment, the effect of the contact forces and the external forces must balance. Let <i denote the velocity of the center of gravity of the object in the world frame and 8 denote the joint rates of the polyhedral obstacles. Let Cn denote the vector of contact wrench intensities. The following equations represent the kinematic and equilibrium constraints that must be satisfied if the object is stationary or moving quasistatically while maintaining the contact formation [8]: og.s = [ 6gSa 6gs. 6gs. 6gso 6gsj! 6gS.,]T (1) Thus: is -6g.s < gs < is + 6g.s. (2) The above inequality applies element by element. Let the homogeneous transform from S to C be given by -Rll R12 R13 XlR21 R22 R23 Yl R31 R32 R33 Zl 0 0 0 1 T. . Wn q+JnO = 0 (12) W nCn = -gob; (13)