Open AccessTo build mathematical models of actuating mechanisms (AM) of robots with linear kinematic structure, J. Denavit and R.S. Hartenberg (D-H) proposed to use a special coordinate system (CS). It uses a matrix of homogeneous transformation and provides clear and unambiguous rules to build mathematical models of robot actuating mechanisms. Developers widely use this approach because of its clarity and binding to AM design parameters. However, its use to describe robots the AM of which has a spatial tree-like kinematic structure revealed the certain hardship.
The paper offers a method for building a modified D-H coordinate system. Its using allows to create mathematical models of robot AM, having spatial kinematic structure. The method is based on using both the graphs theory and the known D-H coordinate system jointly.
The paper defines concepts of main and auxiliary coordinate systems and the order of their location on the tree-like AM. The values of the modified parameters of D-H are obtained. There its known parameters are complemented by parameters f(i), which determine the father number of the link i, and ns(i), that shows, which son in succession is the link i for the link f(i). An algorithm to form the matrix of reachability and the graph of reachability of the tree-like AM is proposed.
It is proved that the use of the modified D-H coordinate system allows us to record in a block-matrix form both kinematic and dynamic equations for all links of the tree-like AM of robot. The use of these equations together with D'Alembert principle allowed us to write the equations of the tree-like AM dynamics of robot in the form traditional for recording the AM dynamics equations with open kinematic structure.
Via examples of mathematically described kinematics and dynamics of tree-like AM of specific robots (robot-dog, robot-crab, anthropomorphic walking robot, etc.) the paper shows efficiency of modified D-H coordinate system.
Obtained research results of the medical robot AM have shown that the proposed model can be successfully used to describe robots with linear open kinematic structure, which is a special case of tree-like kinematic one.