The aim of the article is to expand the functionality of the copy control of an anthropomorphic manipulator using an exoskeleton master device. To achieve the goal, the article proposed a mathematical method for mapping the con guration space of the master manipulator into the con guration space of the slave manipulator. The developed mathematical method is based on the requirements of maximizing the working space of the slave manipulator, the uniqueness of the correspondence of the position of the slave manipulator to the position of the master manipulator and the human-like movements.
Copyright c 2019 by the paper's authors. Copying permitted for private and academic purposes. the operator's hand based on the rotation angles of the driver [Tebueva2018, Teb2018, Pet2018] allows the capture of the operator's hands movements directly.
The developed exoskeleton complexes [Kutlubaev2016, Kut2016] for control of anthropomorphic manipulators (AM) in the majority represent the lever mechanism consisting of three rigid links (shoulder, elbow, hand) and the gripper identical to a hand of the person. In working condition, the exoskeleton is put on the operator, while the lever system is located parallel to the human hand. Registration of the rotation angles of the exoskeleton is carried out with the help of built-in encoders, the information from which is transmitted to the actuator or simulation program. At the same time, the centers of the end of the links and the rotation angles of the exoskeleton and the operator's hands do not coincide, since they are in di erent positions in space. Di erent anthropometric parameters between the human hand and the manipulator create errors in the formation of control laws that a ect on the accuracy of the target operations of the actuator.
The existing approach to the control of anthropomorphic manipulator [Tebueva2018, Teb2018, Pet2018]] using a copy-type master device involves the use of generalized coordinates of the operator's hand as control signals, rather than the master device. The scheme of the generalized coordinates calculation of the operator's hand is shown in gure 1.
The most complex computational problem is the calculation of generalized coordinates of the operator's hand on the basis of data on the position of the operator's hand joints, for which the methods of solving direct and inverse problems of kinematics are used [Tebueva2018, Teb2018].
The solution of the direct kinematics problem is necessary to convert the information about the position of the manipulator from its own coordinate system to the working (absolute) coordinate system to determine the coordinates of the links of the manipulator. The solution of the inverse kinematics problem is designed to calculate the spatial con guration of the manipulator by the position of its links [Tebueva2018, Teb2018]. This solution requires a description of the overall characteristics of the manipulator in a form convenient for their analysis and recording the equations of coordinate transformation. Of the existing approaches to the description of the dimensional characteristics of the manipulator, the main ones are their expression in the form of a system of linear or matrix equations [Jaz2007].
The basis for solving the problems of kinematics is the choice of parameters that uniquely determine the orientation of a solid in space. For this purpose, there are a number of kinematic parameters: guiding cosines and Euler angles [Tebueva2018, Teb2018, Per2019, Mar2014], Rodrigue-Hamilton parameters (quaternions) [Rad2012, Isa2018, Gou2012, Gor2016], Cayley-Klein parameters [Oni2006, Jaz2007] and the DenavitHartenberg representation [Sic2009].
The possibility of using a geometric approach to solving the inverse problem of kinematics [Sha2018] and the high prevalence of using the Denavit-Hartenberg apparatus provides advantages when it is used in describing the kinematics of the master device and calculating the generalized coordinates of the operator's hand.
When calculating the rotation angles of the operator's hand, the task of determining the coordinates of the elbow joint of the operator's hand arises, since there is an in nite number of positions of this joint. The solution of this problem is proposed in the work [Pet2018]. Another important factor is the di erence between the lengths of the manipulator links from the lengths of the operator's arm parts, as well as the di erent "shoulder width" of the operator and the anthropomorphic robot. This leads to various kinds of problems in the transmission of the rotation angles of the operator's hand as the rotation angles of the Slave. For example, even with the coincidence of the links lengths, with a di erent distance between the shoulders, a situation is possible when two Masters joined their palms, but Slaves did not. In this case, the connection of the palms of the two Slaves becomes impossible. Creating anthropomorphic manipulators with adjustable length links is possible, but it is a complex task. A simpler task is to map the con guration space of the Master to the con guration space of the Slave. This article proposes a method that solves this problem. iT j { transformation matrix from j-th to i-th coordinate system; i 1Ai { homogeneous matrix of complex transformation for adjacent coordinate systems; Tz; ( ) { is a homogeneous matrix of elementary rotation around the z-axis by the angle ; Tz;d (d) { homogeneous matrix of elementary shift along the z-axis by distance d; Tx:a (a) { homogeneous matrix of elementary shift along the x-axis by distance a; Tx; ( ) { is a homogeneous matrix of elementary rotation around the x-axis by an angle ; a; d; ; { Denavit-Hartenberg parameters describing the kinematic structure of the manipulator. Oi { is the beginning of the i-th coordinate system. The matrix of the uniform transformation Ti from thei-th coordinate system to the global coordinate system associated with the humeral articulation can be found using the formula:
Solution of the direct kinematics problem, i.e. determining the position of points Oi in the Cartesian coordinate system for a given value of has the form:
Ti = T 00T i; i > 0; T0 = T x:a ( 90 ) T z; ( 90 ) :
The left upper submatrix Ri of size 3x3 of the matrix Ti describes the rotation of the i-th coordinate system relative to the global coordinate system. This rotation can also be described using Euler angles. Euler angles i; i; i can be calculated by the rotation matrix Ri as follows:
Ri 2 Ri { is the rotation matrix of the i-th coordinate system relative to the global coordinate system; ni,si,ai,pi {Ri matrix elements; AT AN 2 { arctangent function, taking into account the angle's quadrant.
All expressions in this subsection are also valid for Slave.
In this case, the following requirements for copying control should be met: the movements similarity of the Master and Slave; the coincidence of the characteristic positions of the Master and Slave; maximization of the Slave working space involved; avoid self-collision of the Slave manipulator links. 2.3
Proposed solution Consider the radius vector re, connecting the origin and center of the manipulator e ector. jrej reaches the maximum value of rm with full straightening of the manipulator (Figure 3), zero - with the coincidence of the e ector position with the origin, and intermediate in all other cases.
We introduce the value of kr { the coe cient of directness of the manipulator:
To maximize the involved working area of the operating space, it is necessary for kr and k0r reach their maximum and minimum values simultaneously. Otherwise, this means that a bent executive arm will correspond to a fully extended master manipulator and vice versa. In the rst case, it will be impossible to rectify the executive manipulator. In the second case, further straightening of the master manipulator will not have any kr = jrej :
rm e ect on the executive. To solve this problem, it is proposed to observe the following relationship in the process of copying control:
In addition to the module r0e it is also necessary to know its direction in order to determine the position of the e ector of the actuator arm. To describe the vector r0e it is convenient to use a spherical coordinate system that uses its module r0e , azimuth angle a and zenith angle a to de ne the end of the radius vector (Figure 4).
At rst glance it may seem that the following equations can be used for copy control: however, this approach has some problems. Figure 5 shows a top view of an anthropomorphic robot. When r0e hits in area 1, it is indeed possible to assume that a0 = a. The problem is connected with the connection of the \palms" of two robot manipulators in the sagittal plane. In general, the operator's shoulder width and the robot's \shoulder width" are di erent. If the distance between the shoulders of the robot is greater than the distance between the shoulders of the operator, then when a0 = a the robot will not be able to connect the palms as an operator (Figure 5). If the distance between the shoulders of the robot is less than that between the shoulders of the operator, the robot will bring the palms together before the person connects them, which will lead to poor control.
k0r = kr: r0e = rm0 jrrmej : k0r = kr; a0 = a; a0 = a;
To solve this problem, it is proposed to use the following relationships: ci { empirically determined coe cients of the polynomial based on ergonomic estimates.
The above relations completely de ne r0e but do not allow to nd the vector of the generalized coordinates of the executive manipulator 0 .
The rst four degrees of mobility AM, counting from the base, are designed to move the gripper, while the other three ensure its orientation in space. The orientation of the gripper in space is determined by the triple Euler angles; we denote them as h; h; h. For comfortable manipulation of objects using a copy control, the following relationships must be observed: h0 = h0 = h; h; h0 = h: O06 =
the rst matrix is a homogeneous transformation matrix from the coordinate system associated with the seventh link; the second matrix describes the coordinates of the wrist joint O06 in the seventh coordinate system; a7 { is the Denavit-Hartenberg parameter of the 7th link.
Due to the kinematic redundancy of the anthropomorphic manipulator (Figure 6), to solve the inverse problem of kinematics and calculate the desired vector 0 , there is not enough knowledge of the Cartesian coordinates of the r0e e ector and the wrist joint O06 of the Slave. The Cartesian coordinates O03 of the Slave elbow joint are also required. The kinematic redundancy of AM lies in the fact that the given position O06 of the wrist joint corresponds in general to an in nite set of possible positions O03 of the elbow joint.
It is proposed to choose one of the possible positions on the basis of the Slave's human-like requirement. 0 To enhance human-like, let the orientation in space of the plane formed by the Slave shoulder joint O1, the ulnar joint O03 and the wrist joint O06 coincides with the similar plane formed by the joints of the Master As a coordinate system associated with this plane, a coordinate system associated with the elbow joint can be taken. In this case, we can assume that:
R03=R3:
O01 =
Based on the expressions obtained, the rotation angles of the Slave can be calculated using the method given in [Pet2018]. To test the e ectiveness of the proposed method, a numerical simulation of the process of copying control and mapping the Master con guration space to the Slave con guration space was performed. An example of the correspondence between the position of the Master and the Slave is shown in Figure 7. Figure 7a shows the position of the Master and the Slave in a single coordinate system, and in Figure 7b in two coordinate systems scaled for clarity and superimposed on each other. As can be seen from the simulation results, when applying the proposed method, there is a clear similarity between the position of the Master and the Slave.
The proposed method of mapping the con guration space of the Master con guration space to the Slave conguration space allows copy control to be performed even with signi cant di erences in the lengths of the links of the Master and the Slave. This feature allows to e ectively use promising exoskeleton Master devices. The proposed method is based on the requirements of safety and human-like movements, maximizing the use of the AM workspace. It is planned to introduce the proposed method into existing copy management systems. 5
The research was carried out within the framework of the implementation of a research project on the development of a software and hardware control system based on solving the inverse problem of dynamics and kinematics within the framework of FCNIR 2014-2020 (unique identi er RFMEFI57517X0166) with the nancial support of the Ministry of Science and Higher Education of the Russian Federation.