The structural conditions considered in the synthesis process are the following: a) manipulator with three degrees of freedom, b) fully-isotropic parallel manipulator, c) two known elements (fixed and mobile platform), and d) three parallel kinematic chains.

To determine the possible degrees of freedom provided by each kinematic chain in the manipulator, we evaluated the mobility of the kinematic chain using the Chebychev-Grübler-Kutzbach formula [15], along with the contribution of Tsai [15] to the passive degrees of freedom:

where M is the degrees of freedom of the kinematic chain; d is the degrees of freedom of the individual elements without restricting (6 in the spatial scenario and 3 in the plane scenario); n is the number of rigid bodies, elements, or links of the kinematic chain; g is the number of joints; f _i is the number of degrees of freedom per kinematic pair; and f _g is the number of passive degrees of freedom in the kinematic chain, and it may take positive values when degrees of freedom are available (not transferred to the terminal element) or negative values when there are additional restrictions to the movement.

The different configurations of elements and joints that consolidate the kinematic chain can be determined by considering the different assumptions established to each of the parameters of Eq. (1). First, the equations that enable determining the number of elements in the kinematic chain, in function of the joints are defined:

where n is the number of elements that consolidate the parallel manipulator, 2 represents the two platforms (fixed and mobile), and n _cc is the set of elements that consolidate the parallel kinematic chains, which can be calculated in two different ways:

Considering each element as binary regarding the number of joints,

The total number of joints in the manipulator can be expressed as follows:

Substituting Eqs. (2)-(4) into (1) results in M for the scenario of binary elements (5) and ternaries (6):

By knowing the values of M, L, and f _g , the total degrees of freedom provided by the joints in the kinematic chains that consolidate the manipulator can be determined.

The function of the manipulator is to transmit the movement and force of an input link towards an exit link, guaranteeing the transmitted force along the kinematic chains, until the stated element can generate the dynamic equilibrium. Considering that the architecture of the manipulator is completely parallel and only considers one actuator per kinematic chain (considering active prismatic joints, but not the gravitational effects and the friction in the joints), the input transmission index is defined by [16]:

Where λ _i is the input transmission index, γ is the angle of transmission, ψ is the angle of deviation, f _i is the static force, and v _i is the absolute velocity vector.

Fig. 1 shows the inverse transmission and deviation angles, as well as the static force of the coupler:

Figure 1 Source: The Authors.

The angles γ and ψ, in Fig. 1 correspond to two criteria of the manipulator’s input manipulability. The first angle is known as the transmission angle, γ, and is defined as the lower angle between the relative velocity vector, v _C/E , of the coupler element, and the absolute velocity vector, v _C , of the input link, both obtained from the kinematic pair, C (Fig. 1 (a)). The other criterion consists of using the direction of the static force transmitted, F _3/2, from the coupler to the input element and the velocity of the connection point, v _C; _, the angle between them is known as deviation angle, ψ. In the analysed manipulator, the transmission and deviations angles are complementary (Fig. 1 (b)).

According to Tao [18], for systems in which the transmission of movement must be precise and at a high speed, the permissible values for the angle of transmission must be between 45 and 135^o. Sharma [19] postulated that an angle of transmission lower than 20° is unacceptable for a mechanism of high velocity. Martin [20] asserted that mechanisms of general purpose must comply with a minimum angle of transmission higher than 40°. For the configuration studied in this project, the angle of transmission could not attain higher values than 90°; otherwise, the mobile platform would have surpassed the plane where the prismatic joints were located, which was not structurally possible to achieve. Taking as reference the previously mentioned, the input transmission index must not be lower than sin 45° ≈0,7. According to Liu [16] when the input transmission index is higher or equal to 0.7, the manipulator possesses an adequate relation of mobility or force. If within the maximum inscribed workspace, the positions of the kinematic chains of the manipulator are guaranteed, values lower to 0.7 will not be presented in the input transmission index, the mobile platform of the manipulator will have an adequate capacity to transmit movement and force, with a low possibility, and singular positions may result [16].

To determine the parameters, the method considers a cylindrical workspace of radius r₀ and height h. From the cylindrical form obtained, an inscribed hexahedron of a similar form can also be defined. In Fig. 2 the mobile platform is considered to move over the horizontal plane with z=z ₀ measured in regard to the framework of reference (OXYZ).

Figure 2 Source: The Authors.

P’ represents the closer position of the center of the mobile platform with respect to the action line of the prismatic pair Q, while P” is the farther position. P’H’Q’ and P”H”Q” correspond to the kinematic chain orientations when the center of the mobile platform coincides with the borders of the cylinder. In Fig. 3, the orientations that can test the kinematic chain are illustrated according to the extreme positions the mobile platform may reach.

Figure 3 Source: The Authors.

From Figs. 3 (b) and (c) the following relations are obtained between the dimensional parameters of the manipulator:

With the aim of establishing a technical condition and structural reliability between the maximum and minimum deviation angles, these are related to the maximum opening angle (in practical terms, the spherical joints have a permissible interval of values for the opening angle between 15 and 30°) that enables the spherical joint Ci, of the form

If the deviation angles are specified, the dimensional parameters L and R-r can be determined, using Eq. (9) and (10), in the form

Under the same considerations, in Figs. 3 (b) and (c), the range of movement can be determined in the prismatic joints, and it corresponds to the defined height for the manipulator’s workspace through the equations.

Let 𝝉=[𝜏₁,𝜏₂,𝜏₃,⋯,𝜏_𝑛]^𝑇 be the vector of motor forces or torques that act on the manipulator and Δq= [Δq₁, Δq₂, Δq₃,…, Δq_n]^T the vector of joint displacements. According to Tsai [15], the vectors t and Δq can be related by the equation

In which 𝜒=diag [𝑘₁,𝑘₂,𝑘₃,⋯,𝑘_𝑛, ] is a diagonal n×n matrix of the stiffness coefficients for each of the kinematic chains.

The resultant force of the mobile platform is related to the vector of motor forces or torques that act on the manipulator following the equation

In which J ^t ,is the transposed Jacobian matrix.

Joint displacements are related to linear and angular displacements (Δx= [Δx, Δy, Δz, Δ𝜙, Δ𝜃,Δ𝜓]^T of the mobile platform following the Jacobian Matrix [15], as:

Substituting Eqs. (21) and (23) into (22) we obtain

In which 𝐾=𝐽^t𝜒 𝐽, is the manipulator’s stiffness matrix.

For the manipulator studied, the three kinematic chains were considered symmetrical and completely rigid; hence, we assumed that the only source of flexibility was concentrated in the active kinematic pair governed by the actuator [15]. Here, the stiffness coefficients were considered equal (𝑘_a=𝑘_b=𝑘_c=𝑘_act); thus, the stiffness matrix can be rewritten as follows:

Using Eq. (25) in conjunction with Eq. (20), the stiffness matrix is obtained:

Because Eq. (25) depends on the Jacobian matrix, the stiffness also depends on the position in which the manipulator is; hence, the results obtained are instantaneous. The instantaneous stiffness associated with the mobile platform for each axis is associated with the components of the main diagonal of the stiffness matrix, as shown below:

In which

The previous results indicate that the stiffness along the z-axis depends only on the stiffness coefficient of the active kinematic pairs governed by the actuator for a given workspace.

The total stiffness (K _STT ) is determined as the trace of the stiffness matrix:

The stiffness of the active prismatic kinematic pair depends on many factors such as the mechanical components, power of the electric motors, and control system. The authors in [29,30] only considered the effect of the Jacobian Matrix in the stiffness analysis, which is why they assumed 𝑘_act=1 kN/m. For most developed stiffness analyzes, Zhang [1] used a value of 𝑘_act=1 GN/m, whereas Pashkevich [31] proposed a value for linear actuators as 𝑘_act=100 MN/m. For the stiffness analysis developed in this study, the stiffness value for the linear actuators was considered as 𝑘_act=100 MN/m, which represented an intermediate value for initial design purposes.

The dexterity analysis is directly related to the condition index (CI) which is the reciprocal of the conditioning number Eq. (31) of the Jacobian matrix [16,33].

where k is the condition number, which is determined by the multiplication of any of the rules of the Jacobian matrix and its inverse, that is

Another approach to calculating the condition number is in terms of the stiffness matrix and its eigenvalues:

In which

The condition index can have a value between 0 and 1, which enables the skill, manipulator’s isotropy level (the ability to transmit movement and static charge in all directions), and the remoteness of the manipulator of singular positions to be determined [33]. When CI= 0, the robot is in a singular position, at which it is considered to be in a state out of control [17]. When CI= 1, the manipulator is in an isotropic configuration.