Elastostatic modeling and shape optimization of a 6-dof haptic interface device

This paper deals with the shape optimization of a six degree-of-freedom haptic interface device. This six-dof epicyclicparallel manipulator has all actuators located on the ground. A regular dexterous workspace is introduced to represent the mobility of user’s hand. Throughout this workspace, the deviation of the mobile platform is bounded to provide a better feeling to the user and the masses in motion are minimized to increase the transparency of the haptic device. The stiffness model is written using a virtual joint method and compared with the results obtained with the finite element analysis to be validated. Finally, the shape of the links are optimized in order to minimize the masses in motion while guaranteeing a given stiffness throughout the regular workspace of the mechanism. INTRODUCTION There is a large number of parallel manipulators, which have been reported over the past three decades. They can be divided into three-dof, four-dof, five-dof and six-dof parallel manipulators [1–4]. In three-dof parallel manipulators, there are three-dof translational parallel manipulators [5–7], three-dof rotational parallel manipulators [8, 9] and three-dof parallel manipulators with mixed (translational and rotational) motion capabilities [10, 11]. Among the four-dof parallel manipulators, examples include the four-dof four-URU parallel mechanism [12] and the Schönflies-motion generator [13]. There are also fivedof parallel manipulators, such as the 3T2R parallel manipulators [14–16]. For six-dof manipulators, the number of all possible architectures can be extremely large [17]. Six-legged six-dof parallel manipulators, such as the Gough-Stewart platform, have high stiffness and accuracy but suffer from a small workspace and limb interference. Three-legged six-dof manipulators were introduced to overcome this workspace limitation and do not suffer from the same limb interference as their six-legged counterparts [18]. However, to achieve six-dof with only three legs requires actuators to be mounted on the moving limbs, thus increasing the mass and inertia of the moving parts. Three-legged manipulators have been reported, such as [19–24]. However, very few of them have all actuators allocated on the ground, [21, 24] being some examples. A differential drive system is used in [21] to allow for all actuators to be mounted on the base whereas [24] use a gimbal mechanism. This paper deals with the shape optimization of a six-dof three-legged parallel manipulator with all actuators mounted to the base, called the Monash Epicyclic-Parallel Manipulator (MEPaM) [25, 26]. The design is achieved by taking advantages of two-dof planetary belt systems. By mounting actuators on the base, the mass and inertia of the moving links is greatly reduced resulting in a lightweight six-dof parallel manipulator. The shape of the links are optimized in order to minimize the masses in motion while guaranteeing a given stiffness throughout a regular workspace of the mechanism. The displacement of the mobile platform due to external loading is limited to be compatible with the user’s perception. The paper is organized as follows. First, a prototype of the MEPaM is presented. Then, its elastostatic model is obtained by using the Virtual Joint method (VJM) introduced in [27, 28] and validated with the structural analysis workbench of CATIA commercial software. Finally, the shape optimization of the manipulator is performed and the results are discussed. MECHANISM UNDER STUDY A prototype of the MEPaM is shown in Fig. 1. The moving platform has six degree-of-freedom with two actuated joints in each leg. The revolute joints are driven by belt connected to the actuators mounted to the base of each leg. Links 2 and 3 are connected with a cylindrical joint and Link 3 is connected to the moving platform through passive universal joint. Figure 1. A prototype of the MEPaM


INTRODUCTION
There is a large number of parallel manipulators, which have been reported over the past three decades.They can be divided into three-dof, four-dof, five-dof and six-dof parallel manipulators [1][2][3][4].In three-dof parallel manipulators, there are three-dof translational parallel manipulators [5][6][7], three-dof rotational parallel manipulators [8,9] and three-dof parallel manipulators with mixed (translational and rotational) motion capabilities [10,11].Among the four-dof parallel manipulators, examples include the four-dof four-URU parallel mechanism [12] and the Schönflies-motion generator [13].There are also fivedof parallel manipulators, such as the 3T2R parallel manipulators [14][15][16].For six-dof manipulators, the number of all possible architectures can be extremely large [17].Six-legged six-dof parallel manipulators, such as the Gough-Stewart platform, have high stiffness and accuracy but suffer from a small workspace and limb interference.Three-legged six-dof manipulators were introduced to overcome this workspace limitation and do not suffer from the same limb interference as their six-legged counterparts [18].However, to achieve six-dof with only three legs requires actuators to be mounted on the moving limbs, thus increasing the mass and inertia of the moving parts.Three-legged manipulators have been reported, such as [19][20][21][22][23][24].However, very few of them have all actuators allocated on the ground, [21,24] being some examples.A differential drive system is used in [21] to allow for all actuators to be mounted on the base whereas [24] use a gimbal mechanism.This paper deals with the shape optimization of a six-dof three-legged parallel manipulator with all actuators mounted to the base, called the Monash Epicyclic-Parallel Manipulator (MEPaM) [25,26].The design is achieved by taking advantages of two-dof planetary belt systems.By mounting actuators on the base, the mass and inertia of the moving links is greatly reduced resulting in a lightweight six-dof parallel manipulator.The shape of the links are optimized in order to minimize the masses in motion while guaranteeing a given stiffness throughout a regular workspace of the mechanism.The displacement of the mobile platform due to external loading is limited to be compatible with the user's perception.
The paper is organized as follows.First, a prototype of the MEPaM is presented.Then, its elastostatic model is obtained by using the Virtual Joint method (VJM) introduced in [27,28] and validated with the structural analysis workbench of CATIA commercial software.Finally, the shape optimization of the manipulator is performed and the results are discussed.

MECHANISM UNDER STUDY
A prototype of the MEPaM is shown in Fig. 1.The moving platform has six degree-of-freedom with two actuated joints in each leg.The revolute joints are driven by belt connected to the actuators mounted to the base of each leg.Links 2 and 3 are connected with a cylindrical joint and Link 3 is connected to the moving platform through passive universal joint.In the scope of this paper, the planetary belt systems are sup-posed to be replaced by direct actuation systems and the ball bearings located at the cylindrical joints are not considered in the model.The simplified model of the mechanism is shown in Fig. 2. F 0 and F P denote the base frame and the moving-platform frame, respectively.The geometric parameters of the MEPaM are given in Table 1.

ELASTOSTATIC MODELING OF THE MEPaM Stiffness model with a virtual joint method
The virtual joint method used to derive the Cartesian stiffness matrix of the manipulator under study and the notations used in this section are explained in [28].The control loop stiffness k i (a1, a2) associated with the two actuators of the ith leg and the links' deflections are considered while defining the stiffness model of the MEPaM.The former are defined as follows: The actuated joint stiffness is supposed to be high enough in order to focus mainly on the effect of the links's flexibility on the overall manipulator stiffness.The first and the second links of each leg are supposed to be axi-symmetrical about the x-axis of their local frame.Hence, the inverse of their stiffness matrix is expressed as: Likewise, the third link of each leg is supposed to be axisymmetrical about the z-axis of its local frame.Hence, the in-verse of its stiffness matrix is expressed as: The links are supposed to be made up of stainless steel.Therefore, the Young's modulus E is equal to 210 × 10 6 MPa and the Poisson's ratio is equal to 0.3.I ix , I iy , and I iz are the area moment of inertia and J i is the torsion moment of the ith link.From Eqs. ( 1) to (3), the aggregated spring stiffness matrix associated with the ith leg of the MEPaM takes the form: Hence, the Cartesian stiffness matrix defining the motion-toforce mapping associated with the ith leg of the MEPaM is obtained by solving the following equation: where The inverse of the first left-hand side matrix gives the Cartesian stiffness of the kinematic chain, K i C where the 6 × 6 matrix is extracted from the top left hand corner of the inverse matrix.
Once the Cartesian stiffness matrices K i C , i = 1, . . ., 3, for the three kinematic chains are computed, the Cartesian stiffness matrix of the MEPaM can be found by simple addition, namely, It is noteworthy that this model only stands for unloaded modes and assumes a linearization of displacement around the equilibrium position.A more advanced model is described in [29], but is more time consuming as the external loads are performed iteratively.

CAD and FEA modeling
A Computer Aided Design (CAD) model of the MEPaM manipulator was built under CATIA commercial software.For the Finite Element Analysis (FEA), the components of the MEPaM are designed individually and the manipulator is assembled as shown in Fig. 3. Rigid virtual joints, represented by some red lines in Fig. 3, are used to connect all the components.The benefit of using rigid virtual joints connection is that the users can define the joint stiffness in their desired directions.
Then, the actuator stiffness is defined by connection properties between the rigid virtual joints.A slider connection property is introduced for the cylindrical joint that connects link 2 to link 3.This connection property enables link 3 to slide freely inside the connection hole of link 2.

Stiffness of the MEPaM throughout its regular workspace
The VJM and FEA models are simulated with three individual forces and three individual moments applied on the moving platform for six different locations within the workspace.The magnitudes of the corresponding force and moments are equal to 10 N and 0.5 Nm, respectively.The 2-norm of the pointdisplacement screw, δp 2 , of the moving-platform and the 2norm of its orientation vector, δφ 2 , are defined as: Thanks to the symmetry of the MEPaM, a cylindrical regular workspace is defined and shown in Fig. 4. Its diameter is equal to 170 mm and the Z-coordinate of the geometric center of the moving platform varies between 50 mm and 250 mm in the base frame.This regular workspace turns to be similar to the workspace of the Virtuose 6D Desktop from Haption [30].

Comparison between VJM model and FEA model
The VJM model described is computed with MATLAB computational software.In general, the difference between the results obtained with the VJM model and those obtained with the CAD model are below 10%.At the center of the working plane, it appears that the difference between the two models in terms of rotation angle is relatively high.The same phenomenon appears for the linear displacement of the moving platform when some torques are applied on the latter.To explain the phenomenon, it is necessary to understand the moving platform's displacement corresponding to the stiffness of the leg of the manipulator.Due to the leg configuration, all the legs of the manipulator have different stiffness in different directions.The stiffness differences amongst the legs resulted the moving platform to translate a small distance and to rotate in a small angle when forces or torques are applied.All the legs of the manipulator are connected to the moving platform through a passive universal joint.The passive universal joint allows the moving platform to rotate freely.The differences in leg stiffness and passive universal joint affect the translation and rotation of the moving platform.Accordingly, the results obtained with the VJM model and those obtained with the CAD model differ slightly for some poses of the moving-platform because of the different modelings of the passive universal joints in the two models.Nevertheless, we can claim that the VJM model of the MEPaM provides a good approximation of its real stiffness throughout the regular workspace.

OPTIMIZATION PROBLEM Objective function
The stiffness of the MEPaM depends on its configuration and the dimensions of its components.The relation between the MEPaM stiffness and its configuration is characterized by Eq. ( 6).This section aims at defining the optimum shape of the MEPaM's components in order to minimize the mass of the components in motion while satisfying a given stiffness throughout the regular workspace of the manipulator.Therefore, the objective function of the optimization problem is the mass m MEPaM of the MEPaM and is defined as follows: where vector x contains the decision variables of the optimization problem at hand.Vector q is composed of the parameters of the optimization problem at hand, namely, the Young's modulus E, the Poisson's ratio µ and the density ρ of the material and the link lengths.

Decision variables
Figure 11 represents the possible cross section types for the links of the MEPaM.On the one hand, the first two links of each leg are supposed to be composed of a rectangular hollow, a Ishaped or a H-shaped cross section type.On the other hand, the cross section of the third link is assumed to be circular hollow to realize the cylindrical joint between link 2 and link 3.As shown in Fig. 11, the rectangular hollow, I-shaped or a H-shaped cross section types are characterized by design variables b, h, s and t.The circular hollow cross section is parameterized by its inner and outer radii r in and r out , respectively.Moreover, the architecture of the MEPaM is supposed to be symmetrical, namely, its three legs are the same.
As a result, the decision variable vector x takes the form: where b i , h i , s i , t i characterize the dimensions of the cross section of the ith link, i = 1, 2, while k i denotes its cross section type, namely,  • k i = 1 stands for a rectangular hollow cross section type; • k i = 2 stands for a I-shaped cross section type; • k i = 3 stands for a H-shaped cross section type.
r in and r out are the inner and outer radii of the circular hollow section of the third link.

Constraints
The constraints of the optimization problem at hand are related to the feasibility of the cross sections and the maximum point-and angular displacements of the moving-platform allowed for a given wrench applied on the latter.
As a matter of fact, the maximum point-displacement of the moving-platform allowed for a force of 10 N applied on the latter is equal to ±1 mm.Likewise, the maximum angular displacement of the moving-platform allowed for a moment of 0.5 Nm applied on the latter is equal to ±0.0349 rad (±2 • ).
Twelve points P j , j = 1, . . ., 12, are selected through the regular workspace of the MEPaM, as shown in Fig. 12, to check whether the constraints related to the maximum point and angular displacements of the moving platform are satisfied or not.The Cartesian coordinates p j of points P j , j = 1, . . ., 12 are the following: As a consequence, the following constraints should be satisfied: δp ymin ≤ δp jy ≤ δp ymax (12) δp zmin ≤ δp jz ≤ δp zmax (13) where δp jx , δp jy and δp jz are point-displacements of the moving-platform at point P j along the x-, yand z-axes of the base frame F 0 , respectively.Likewise, δϕ jx , δϕ jy and δϕ jz are the angular displacements of the moving platform at point P j about the x-, yand z-axes, respectively.Besides, In order the cross sections of the first two links of each leg to be feasible, the following constraints should be satisfied: In order the cross sections of the third link of each leg to be feasible, the following constraint should be satisfied:

Formulation of the optimization problem
From Eqs.( 9) to (22), the optimization problem to be solved takes the form: where vectors x lb and x ub contain the lower and upper bounds of the decision variables, namely, x lb = 1 3 3 0.5 0.5 1 3 3 0.5 0.5 0.1 2.5 T x ub = 3 100 100 50 50 3 100 100 50 50 10 10 T Vector q is defined as follows: where ρ, E and µ are the density, Young's modulus and Poisson's ratio of the material.l 1 and l 2 are the lengths of links 1 and 2, respectively.r b and r p are the base and moving-platform radii, respectively.

Results of the optimisation problem
The optimization problem (23) was solved for the nine pairs of cross section types for links 1 and 2, namely, for the nine sets (k 1 , k 2 ) by means of the fmincon Matlab function.Table 2 shows the solutions corresponding to the nine pairs of cross section types.Pair 1-1 has the lowest mass (highlighted in blue in Table 2) and correspond to a MEPaM with both links 1 and 2 having a rectangular hollow cross section.A CAD model is built and shown in Fig. 13.Pair 2-2 has the highest mass (highlighted in red in Table 2) among the nine pairs of cross section types.From the results obtained, it is noteworthy that none of the Ishaped cross sections and H-shaped cross sections converge into rectangular cross sections.

CONCLUSIONS
This paper dealt with the shape optimization of a six degreeof-freedom haptic interface device, named MEPaM.A regular dexterous workspace was introduced to represent the mobility of user's hand.Throughout this workspace, the deviation of the mobile platform was bounded to provide a better feeling to the user and the masses in motion were minimized to increase the transparency of the haptic device.The stiffness model was written using a virtual joint method and compared with the results obtained with a finite element analysis model.
The shape of the links of the MEPaM were optimized in order to minimize the masses in motion while guaranteeing a given stiffness throughout the regular workspace of the mechanism.Three cross section types were considered for the first two links of each leg of the MEPaM, namely, the rectangular hollow, the I-shaped and the H-shaped cross section types.The third link of each leg had a circular hollow cross section.Therefore, an optimization problem was formulated in order to find the optimal shape and dimensions of each link of the MEPaM that minimize its mass in motion while guaranteeing a given stiffness throughout its regular workspace.The virtual joint method was used to obtain the elasto-static model of the MEPaM with regard to the decision variables and the design parameters of the optimization problem at hand and the manipulator configuration.
Finally, the MEPaM with rectangular hollow cross sections for the first two links of its legs turns to be the optimal solution, namely, the solution that minimizes the mass in motion of the manipulator while satisfying the constraints related to the feasibility of the cross sections and the manipulator's stiffness.

Figure 1 Figure 2 .
Figure 1.A prototype of the MEPaM 0.1450 m Platform r p = 0.0275 m

Figure 3 .
Figure 3. FEA assembly model of the MEPaM under CATIA

Figures 5 Figure 4 .Figure 5 .Figure 6 .Figure 7 .
Figures 5 to 10 illustrate the isocontours of the linear and angular displacements of the moving-platform throughout the cylindrical regular workspace of the MEPaM for different wrenches applied on the moving-platform, namely, F x = 10 N,

Figure 8 .Figure 9 .
Figure 8. Linear and angular displacement of the moving-platform throughout the cylindrical regular workspace for M x = 0.5 N.m

Figure 10 .
Figure 10.Linear and angular displacement of the moving-platform throughout the cylindrical regular workspace for M z = 0.5 N.m

Figure 11 .
Figure 11.Possible cross-section types for the links

Figure 12 .
Figure 12.Location of the sampling points within the regular workspace