A Relialble Approach to Compute the Forward Kinematics of Robot with Uncertain Geometric Parameters

rd kinematics of robot with uncertain geometric parameters (1) Determinate computational model of robot Fig. 1 D-H convention for robot link coordinate system The robot kinematic model is based on the Denavit-Hartenberg (DH) convention. The relative translation and rotation between link coordinate frame i-1 and i can be described by a homogenous transformation matrix, which is a function of four kinematic parameters , , and as shown in Fig. 1. The homogenous transformation Ai is given in Eq. (1) (1) Using the homogenous transformation matrix the relationship of the end-effector frame with respect to the robot base frame can be represented as in Eq. (2): (2) (2) The robot kinematic model using parameters with interval uncertainty When the kinematic parameters θi, di, αi, ai have no fixed value but having the values falling in the intervals [θi], [di], [αi], [ai] randomly, expanding the Eq. (2) with the intervals, we get, (3) with solution of the interval computational model of robot with uncertain geometric parameters (1) Brief review of some definitions and properties in interval mathematics [7-8] For two interval number and , ( , is the set of real compact intervals), the interval arithmetic was defined as follows. , , and (for ). If , then the interval degenerates to a real number a, i.e. . In this way, interval mathematics can be considered as a generation of real numbers mathematics. However, only some of the algebraic laws, valid for real numbers, remain valid for intervals. The other laws hold only in a weaker form. For example, a non-degenerate interval has no inversion with respect to addition or multiplication. Even the distributive law has to be replaced by the so-called subdistributivity (4) Let be given by a mathematical expression , which is composed by finitely many elementary operations and standard functions . The following inclusion monotone holds. for (5) where,f([x]) is an interval also and which stands for an interval arithmetic evaluation of f over .As x∈[x] , the relation (6) can be obtained. (6) whence (7) Where R(f,[x]) denotes the range of f over . (2) A new approach to evaluate interval functions Overestimation is a major drawback in interval computation. Based on the inclusion monotone relation (7) and the physical/real means expressed by the interval function, a new approach to evaluate interval functions was proposed in this work. Relation (7) is the fundamental property on which nearly all applications of interval arithmetic are based. It shows that it is possible to compute lower and upper bounds for the range over an interval by using only the bounds of the given interval without any further assumption. Obviously, the true value of is existing and unique. means the range of over . One of the original idea to introduce the interval function is to evaluate the range of the value of the function when the variable changes in the range of in a statement of interval way. However, because only some of the algebraic laws which valid for real numbers hold only in a weaker form for interval numbers, the computational results of depends on the calculating order severely, and they are often larger than the value of . A number of literatures took efforts on finding the skills to obtain the better results of . And some valuable rules were found. For example, (1) If each variable , , occurs at most once in , then ; (2) To make the most of the subdistributivity, i.e., to execute the addition and subtraction operation first, to execute multiplication and division operations then. For instance, the better result of the polynomial can be obtained through computing its reformed form . However, the similar results to improv