Node Allocation and Topographical Encoding NATEnet for Inverse Kinematics of a DOF Robot Arm

In order to continuously map desired tool center point TCP movements onto the corre sponding set of joint angles for a DOF robot arm without the need for repeated calculation of the inverse Jacobian we present an adaptive neural network with the following features topographical encoding of the input for smooth interpolation and fast adaptation error driven node allocation for e cent representation of input patterns with respect to the required mapping operation a prestructured network taking advantage of linearities within inverse kinematics Combining these properties a small but e cent network architecture evolves which solves the di erential inverse kinematics problem Introduction Inverse kinematics requires the mapping of a path speci ed for the TCP onto the correspond ing joint angles of the robot arm The nonlinear mapping of the actual position in joint space and the desired movement in workspace onto the movement of joint angles is generally solved using linear algebra involving the Jacobian which represents the local derivatives of the TCP position with respect to joint angles Our approach tackles the problem directly using approximation of the inverse Jacobian without the need to invert matrices In fact the network learns the inverse Jacobian Radial basis functions using topographical encoding of inputs are well known for their rapid adaptation A general drawback for their use is the limitation to low dimensions of the input Several authors tried to overcome this problem using node allocation Some alternative allocation scheme was developed earlier and successfully applied to the inverse kinematics of a redundant four joint robot arm within a planar working area The aim of this paper is to demonstrate the applicability of the proposed scheme to more degrees of freedom Supported by Ministry for Research and Technology BMFT Project SENROB under grant IN A O Error driven node allocation Topographical encoding of input patterns x generates activity ai for neuron i using Gaussian basis functions ai exp r i i with width i and distance ri jjx xijj of the receptive eld center xi to the actual input pattern x Topographical encoding is a distributed representation of the input pattern that ensures smooth interpolation The overall result y of the required mapping is generated using normalized linear summation of the output pattern attached to each neuron y PN i y k i ai x PN i ai x The sum runs over all neurons N Adaptation of the output y towards a desired output y due to an observed error e y y involves a simple delta rule y i ai x e k with learning rate A severe problem is the e cent placement of receptive eld centers xi Without knowledge about the required mapping it is not possible to nd an overall satisfying solution to that problem beforehand One issue is the representation of input pattern due to their statistics There are several methods e g vector quantization data clustering and feature maps to approximate the probability distribution of occuring pattern with the respective density of neurons placed These methods are not very well suited for mappings with uniform distribution of input pattern but varying output An alternate approach is to use the error as criterion whether the already found solution is satisfying or not In our approach a node is allocated everytime the real valued error measureMe y y x is above a threshold Te i e Me y y x Te and the most active neuron is less active than a threshold Ta i e maxNi fai x g Ta If both conditions are ful lled simultaneously a node is allocated with center xN x output y N y k and width N min N i jjxN xijj and every width of the already allocated neurons is reset to i min jjxN xijj i allows for adjustable interpolation properties Chosing close to zero approximates nearest neighbor table lookup with xed outputs within each Voronoi cell Choosing much larger than one takes global properties of the mapping for each node into account but gradient learning becomes more di cult since there are more parameters to adjust simultaneously Earlier approaches never altered the width i after allocation of the nodes resulting in a hierarchical order of decreasing width for the allocated nodes In our approach the local density of neurons and the receptive eld width are strongly coupled resulting in a democracy of importance for all nodes If the allocation conditions are not ful lled a usual adaptation step using the delta rule is done The threshold for the error Te depends on the number of learning steps s i e Te s e exp s ef decaying exponentiallly to a nal desired error ef The threshold for the activity Ta is kept xed during learning Using this scheme areas with large errors are covered rst and it is possible to suspend allocation of new neurons after a satisfying behaviour is reached or all the available resources came into use Prestructured network for inverse kinematics Using robot arms requires a mapping of a desired path for the TCP in work space x t to a path given in joint angles t The variables describing the path are the position and orientation of the TCP Di erential inverse kinematics uses the desired velocity xpos and angular velocity x of the TCP x t xpos t x t and the initial con guration of the robot to generate the desired path The general solution for the di erential inverse kinematics problem for the non redundant case is J x where J is the inverse of the Jacobian J