By Leondes, Cornelius T

Meant for engineers, electric engineers and keep watch over engineers, this ebook offers insurance of the newest advances in robot structures, from the applying of neural networks to robotics

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T h e corresponding network is illustrated in ( c ) , and the three dimensional view of the collision penalty function is shown in ( d ) . Note that the shape of the collision penalty function can be changed by adjusting the parameter Τ in the activation function. 4. B. Path Planning of A Point Object A collision-free path planning problem is equivalent to the optimiza- tion problem with two constraints. One is to avoid the collision between the object and obstacles, and the other is to minimize the length of the path.

T h e computation of the necessary torques requires consideration of such parameters as inertia and damping. In view of the fact that the arm segments are coupled, the inertia matrix is not diagonal and the dynamical equations are highly nonlinear. Gravitational, Coriolis and centripetal forces must also b e considered. The general form of the dynamics equation can be expressed a citecrai: τ = Μ{θ)θ + τΏ(θ, θ) + τ3{θ) + τ}{θ, θ) (30) 31 APPLICATIONS OF NEURAL NETWORK TO ROBOTICS where r is the torque vector, M ( 0 ) is the mass matrix, τν(θ,θ) a vector of centrifugal and Coriolis terms, τ9(θ) is is a vector of gravity terms and Tf(6,6) is a vector of friction terms.

51 APPLICATIONS OF NEURAL NETWORK TO ROBOTICS {N} goal configuration initial configuration Figure 27: Representation of a via configuration of the object along the path. For the determination of the degree of collision between the poly- hedral object and the obstacles, we select a set of test points from the o b j e c t . T h e collision penalty of the object is defined as the sum of the collision penalties at all the test points. These test points are fixed with reference to the assigned object frame, such that their actual positions with respect to the world frame can be determined by the position and orientation of the corresponding object frame as follows: Qi where Pf st = st 3 yf**, zf ] ' *3 * 3 tj = RxPf st + (46) Pi represents the position vector of the jth test point with respect to the object frame, Qij = [Xiji Yi,j, Zij] repre- sents the position vector of the jth test point at the ith via point with respect to the world frame, Pi(xi,yi, Z{) is the position of the origin of the object frame at the ith via point, and Ri represents the rotational matrix corresponding to the orientation of the object frame at the ith via point.