The paper deals with sti ness modeling of 3RRR parallel spherical manipulator. This manipulator is proposed to be used as a rehabilitation device for wrist trauma. The desired elastostatic model is obtained using a matrix structural analysis approach (MSA) which divides robot representation into two sets of constraints describing elasticity of the links and connection between them. The model operates with matrices of size 84x84 for each leg. As a result, the closed-form solution for the Cartesian sti ness matrix for this type of robot is derived.
with compliant relation between these elements [
A spherical 3-DOF parallel manipulator with revolute actuators for wrist rehabilitation consists of 3
serial kinematic chains and an end-e ector platform [
(a) Robot prototype in CAD (b) Kinematics description of parallel spherical manipulator for i-th leg
The kinematic structure of the robot leg could be described using DH notation (Table 1). Where 1 and 2 are the rst and the second bend angle of links, i is an active joint rotation angle, 1i is the rst passive joint rotation angle, 2i is the second passive joint rotation angle and i = 2(iN1) where N is a number of legs, i = 1; 2:::N is an angle of leg connection on the lower platform. 3
As a complex parallel structure, the use of the MSA technique is reasonable for the spherical parallel
manipulator sti ness modeling. The fundamentals of these technique in general form and all theoretical
basis could be found in [
In Fig. 2, u1, u2 and u3 show torque vector while u4, u5 and u6 show the force vector directions, thus using Castigliano's theorem, the compliance matrix of the curved link takes the form of: the elements of this matrix are:
R C11 = 2
R C22 = 2
C35 = s1 + GIx s2 + GIx s2 EIy s1 EIy ; ;
C12 =
C26 = s6R2 EIz ;
R
C44 = 2A
R C55 = 2A s1 + s2 E G s1 + s2 E G + s2R3 2EIz
; with: s1 = s2 =
L + sin L cos L L sin L cos L The rst step in MSA modeling is deriving the MSA model of the robot. Here, two cases are possible. In the rst method ( g.3), the robot is split into four parts: the robot platform and three legs. This allows taking into account a complex platform elasticity of the robot if needed. In the second case, robot split only into three leg, where each leg has part of the end-e ector platform as the last link ( g. 3a). For simplicity, let's assume that end-e ector platform is rigid, so we can easily implement second method.
Since the spherical wrist robot is symmetrical, the legs of the robot are the same, so equations only for one leg are presented. According to node numbering in g. 3a, links 2-3 and 4-5 are exible and their sti ness is described by equations presented earlier, link 6-e is rigid, joints 3-4 and 5-6 are passive and joint 1-2 is active and elastic.
The equations notation is the following: ti and tj are the de ections at the link ends,Wi and Wj are the link end wrenches, i and j are the node indices, and Ki1;1j , Ki1;2j , Ki2;1j , Ki2;2j are 6x6 sti ness matrices.
(a) The MSA model of i-th leg
(b) The MSA model for full robot The node 1 is connected to the rigid base and described by the following constraint equation: Flexible links 2-3 and 4-5 constraints on de ection and loading could be described as: 2 W2 3 2664 000I6666 6666 000I6666 6666 000I6666 6666 000I6666 6666 00KK662211;;213366 00KK662221;;223366 00KK664411;;125566 00KK664412;;225566 3775 66666666664 WWWttt453234 77777777757 = 2664 0000 3775 t5 Rigid platform presented as a rigid link 6-e: Rij is ijth element of rotation matrix in joint. p calculated the same way as e
The external loading denotes by the following equation: 06 6
= Wext where where [d6;e] is denotes the 3 3 skew-symmetric matrix derived from the vector d6;e describes the link geometry and is directed from the 6th to the e-th node.
Active elastic joint 1-2 is described by the following equation: 3 2 W1 3 where Kact is a sti ness of the actuator.
The passive joints 3-4 and 5-6: Now we can aggregate the equations (2), (3), (4), (5), (6) and (7) to the following type: A B C D 4 06 6 06 6 06 6 06 6 06 6 06 6 06 6 06 6
r1;2 05 6 06 6 06 6 Ka e1;2 01 6 05 6 r3;4 05 6 05 6 01 6 01 6 01 6 01 6 05 6 05 6 05 6 05 6 01 6 01 6 01 6 01 6 Then the resulting sti ness matrix KC of the whole manipulator could be found as:
It should be noted, that rank of the KCi will not be full, because of the passive joints, but after assembly step, where sti ness matrices of all legs are combined KC , rank of this matrix will be full.
In this work, closed-form equations for the Cartesian sti ness matrix of the spherical parallel manipulator are presented. The desired models were obtained using the enhanced matrix structural analysis (MSA) approach that is able to analyze the under-actuated and over-constrained structures with numerous passive joints. In order to obtain sti ness model, the robot was split into 3 legs, each of them supporting part of the end-e ector platform. The total robot sti ness matrix was obtained in two steps: calculating sti ness for each leg and aggregating legs in total robot structure. The algorithm deals with matrices of size 84x84 for each leg and computationally light.
In future work, the sti ness model will be used for design optimization. This work was supported by RFBR grant 18-38-20186