2. Problem formulation

A parallel manipulator consists of two platforms connected by n independent kinematic chains, as illustrated in the scheme in Fig. 1. A fixed reference frame $\{S_0\}$ can be defined on one of the platforms (“base”) of the robot, and a mobile reference frame $\{S_H\}$ describes the pose of the other platform (“end-effector”). In serial architectures, a single kinematic chain determines the pose of the end-effector, which can be described with a homogeneous transformation $T_{0H} \in SE(3)$ from $\{S_0\}$ to $\{S_H\}$ [Reference Ceccarelli1]. This transformation defines $R_{0H} \in SO(3)$ as the rotation and $t_{0H} \in R^3$ as the translation from $\{S_0\}$ to $\{S_H\}$ , with each joint of the chain controlled by an actuator. Conversely, in parallel manipulators each limb includes both active and passive joints. Because of the presence of idle joints, the pose of a limb cannot be defined by the limb’s motor only but depends on the other limbs.

Figure 1. Elements of a parallel manipulators: (a) Kinematic scheme with base, end-effector, and n independent kinematic chains; (b) workspace $\Lambda$ of a parallel manipulator as intersection of the limb workspaces $\Lambda_i$ .

Table I. Nomenclature.

As it is difficult to decouple the pose of each limb from the others, the inverse and forward kinematic problems are usually defined through a system of loop-closure equations, which is tailored to the specific architecture of the parallel manipulator under examination [Reference Ceccarelli1, Reference Tsai2]. However, when the kinematic chain of the $i{\rm th}$ limb is known in both topology and constraints, the locus of all the poses $\Lambda_i$ that describe a possible location of the end-effector (“limb workspace”) can be defined by considering that chain only and neglecting the constraints imposed by the others. From there, the workspace of the parallel manipulator $\Lambda$ can be obtained from the intersection of the limb workspaces as

(1) \begin{equation} \Lambda = \bigcap\limits_{i=1}\limits^n\Lambda_i,\end{equation}

similarly to the generation of the workspace of n-R serial manipulators [Reference Ceccarelli6].

3. Workspace volume generation

In order to obtain the workspace of each limb of a parallel manipulator, conventional techniques [Reference Ceccarelli6] can be used as long as the kinematic chain of the limb has a serial architecture. Each kinematic chain consists of a succession of rigid links connected by joints such as revolute (R), prismatic (P), universal (U), and spherical (S) joints. In the notation used in this paper, an active joint (i.e., actuated by a motor) is underlined (e.g., X). As outlined in the previous section, the workspace of each limb, given its architecture, can be obtained as a combination of rotations and translations along its links. Each R joint corresponds to a rotation around a single axis, while U joints and S joints result in consecutive rotations around two and three normal axes, respectively; a P joint is defined as a translation along a single axis. In this section, we use the mobility of each joint to determine the limb workspace with examples describing the limb architectures of different popular 3-DoF and 6-DoF fully parallel manipulators with identical limbs. In particular, as illustrated in Fig. 2, we consider the following chains as examples: UPS, PUS, UPU, Delta robot.

Figure 2. Kinematic schemes of common limb architectures with their motion variables: (a) UPS kinematic chain [Reference Russo and Ceccarelli22]; (b) PUS kinematic chain [Reference Russo and Dong23]; (c) UPU kinematic chain [Reference Baigunchekov and Kaiyrov24]; (d) Delta robot kinematic chain [Reference Tsai and Joshi25].

3.1. UPS and UPU kinematic chains

Arguably, the most common kinematic chain, the UPS limb, in Fig. 2(a), is used in Gough-Stewart platforms among other architectures, with many applications from flight simulators [Reference Tsai2] to high-accuracy machining [Reference Russo and Dong23]. The workspace of a UPS limb can be obtained by modeling the universal (U) and spherical (S) joints as a succession of two and three revolute (R) joints, respectively. Thus, following the procedure in ref. [Reference Guo, Zhao, Li, Liu, Deng and Tian4] and with reference to the general scheme in Fig. 1, the workspace of this joint can be described by

(2) \begin{equation} \Lambda_i = Rot_x(\vartheta_{i,1}) \cdot Rot_y(\vartheta_{i,2}) \cdot Tr_z(l_{i}) \cdot Rot_x(\vartheta_{i,3}) \cdot Rot_y(\vartheta_{i,4}) \cdot Rot_z(\vartheta_{i,5}),\end{equation}

where $Rot_k$ describes a rotation around the k-axis, $Tr_k$ describes a translation along the k-axis, $\vartheta_{i,j}$ are the motion variables for the passive joints of the chain, and $l_i$ is the length of the limb, as controlled by the linear actuator. Each rotational degree of freedom has been numbered sequentially from base to mobile platform, so that indices 1 and 2 correspond to the two rotational degrees of freedom of the universal joint and indices 3, 4, and 5 refer to the three rotational degrees of freedom of the spherical joint.

The passive joints can be characterized by mechanical limits as $\vartheta_{i,j}\in[\vartheta_{i,j,\min};\;\vartheta_{i,j,\max}]$ , while the limb’s actuator makes the limb’s length range between its extremities as $l_i\in[l_{i,\min};\;l_{i,\max}]$ . Therefore, the position workspace of this limb is a family of spheres S (or spherical sectors in case of angular limits), centered on the base extremity of the limb $A_i$ and with radius $l_i$ . This family is defined as

(3) \begin{equation} S\;:\;(x-x_{Ai})^2+(y-y_{Ai})^2+(z-z_{Ai})^2=l_i^2,\end{equation}

and its workspace can be obtained as the union of these spheres

(4) \begin{equation} \Lambda_i = \bigcup\limits_{l_i} S.\end{equation}

When no limits are considered for the rotational joints, the volume of this workspace can be determined as

(5) \begin{equation} V_{\Lambda_i} = \frac{4\pi(l_{i,\max}^3-l_{i,\min}^3)}{3}.\end{equation}

A more general formulation for the volume can be obtained by assuming a symmetric limit for the universal joint ( $\vartheta_{i,1,\min}=\vartheta_{i,2,\min};\vartheta_{i,1,\max}=\vartheta_{i,2,\max}$ ). Thus, the volume of the workspace can be evaluated from the volume of a spherical sector with a cone angle equal to the angular range of the joint ( $\vartheta_{i,1,\max}-\vartheta_{i,1,\min}$ ), as

(6) \begin{equation} V_{\Lambda_i} = \frac{2\pi(l_{i,\max}^3-l_{i,\min}^3)}{3}\left(1-\cos{\frac{\vartheta_{i,1,\max}-\vartheta_{i,1,\min}}{2}}\right).\end{equation}

Thus, by assuming a constraint on the linear actuator that limits the limb length $l_i$ to range between $l_{\min}$ and $l_{\max}$ , the workspace of the robot can be defined as the volume comprised between two spheres, both with center in $A_i$ and radii of $l_{\min}$ and $l_{\max}$ , respectively.

When considering the reachable position workspace only, by fixing one or more degrees of freedom in the chain, this formulation can be used to model similar kinematic chains that start with a universal joint and a prismatic joint. A third joint allows rotations only (e.g., UPR and UPS architectures). For example, the UPU chain is used as a limb of lower-mobility manipulator structures [Reference Aref and Taghirad19] because of the additional constraint. Its design is shown in Fig. 2(c), while its position workspace is equivalent to the one of a UPS leg as explained in Eq. (2), by imposing $\vartheta_{i,5}=0$ .

3.2. PUS kinematic chain

Obtained from a kinematic inversion of the Gough-Stewart design, this kinematic chain [Reference Perreault, Cardou, Gosselin and Otis14, Reference Aref and Taghirad19], which is shown in Fig. 2(b), is usually characterized by a lower inertia and better performance than the previous one as the motor is not embedded in a mobile part of the robot but has a larger footprint due to the linear actuators on the base. The limb workspace for this architecture can be defined as

(7) \begin{equation} \Lambda_i = Tr_z(l_{i}) \cdot Rot_x(\vartheta_{i,1}) \cdot Rot_y(\vartheta_{i,2}) \cdot Tr_z(d_{i}) \cdot Rot_x(\vartheta_{i,3}) \cdot Rot_y(\vartheta_{i,4}) \cdot Rot_z(\vartheta_{i,5}),\end{equation}

where $d_i$ is a geometrical parameter that describes the length of the rigid link that connects the double revolute (RR) and spherical (S) joints of the kinematic chain. The position workspace of this limb is again a family of spheres S (or spherical sectors in case of angular limits), centered on the base extremity of the limb $A_i$ translated by $l_i$ along the z-axis and with radius $d_i$ . This family is defined as

(8) \begin{equation} S:\;(x-x_{Ai})^2+(y-y_{Ai})^2+(z-z_{Ai}-l_i)^2=d_i^2,\end{equation}

and its workspace can be obtained as the union of these spheres as in Eq. (4), with a resulting volume of

(9) \begin{equation} V_{\Lambda_i} = \frac{4\pi d_{i}^3}{3} + \pi d_i^2 (l_{i,\max}-l_{i,\min})\end{equation}

when not considering angular limitations.

4. Workspace analysis

As explained in the previous section, the limb workspaces of fully parallel manipulators with at least two rotational joints (i.e., a wide majority of the existing parallel mechanisms) can be usually geometrically defined by a family of spheres, or spherical sectors, with the actuation variable as family parameter, and the overall workspace of the manipulator can be found as the intersection of these limb workspaces.

Figure 3. Resulting workspace volumes of common kinematic chains: (a) sphere (with a spherical void) from an unconstrained SP or UP kinematic chain; (b) cylinder with sphere caps from an unconstrained PS or PU chain.

A first estimation of the workspace of the robot can be obtained from the external shape of the limb workspace, which is characterized by a strong dependence on the first joints in the limb’s kinematic chain, as shown in the examples in Fig. 3. Three types of kinematic chains can be identified, according to their outer workspace boundary shape:

• • Type A: Limbs starting with joints that allow two or more rotations. In general, limb chains starting with two or more rotational degrees of freedom will have a spherical surface as external volume boundary, with subsequent linear and rotational degrees of freedom modifying the sphere’s radius, as in Fig. 3(a).

• • Type B: Limbs with a prismatic joint as first or second joint of their kinematic chains. Chains with a linear degree of freedom as first or second degree of freedom of the kinematic chain will result in a cylindrical outer surface, with hemispheres on the top and bottom faces (Fig. 3(b)).

• • Type C: Limbs beginning with multiple prismatic joints with non-parallel axes. Consecutive linear degrees of freedom are rarely used in parallel mechanisms; however, in case two or more prismatic joints (with normal or anyway distinct axes of motion) are at the beginning of a limb chain, the resulting workspace is characterized by a parallelepipedal shape (with rounded corners and edges in case of subsequent rotational degrees of freedom).

Applying limits to the linear and rotational joints constraints these volumes further, depending on their surface type:

• • Type A: Linear constraints to the prismatic joints result into a maximum (and minimum) radius for the spherical surface swept by the end-effector. Angular constraints reduce the sphere to a spherical sector.

• • Type B: Linear constraints on the first prismatic joint define the height of the cylindrical volume. Linear constraints on subsequent prismatic joints modify the maximum (and minimum) radius of the cylinder, as well as the one of the hemispheres. Angular constraints on rotational degrees of freedom change the hemispheres to spherical caps and affect the height of the resulting domes.

• • Type C: Linear constraints on the prismatic joints define the side lengths of the parallelepiped. Angular constraints on subsequent rotational joints, together with the length of the related links, define fillet radius and incidence angle of the rounded corners and edges.

By identifying the type of each kinematic chain and considering joint limits, the workspace volume can be easily visualized and represented. Furthermore, these types can be used to classify parallel manipulators according to their limb characteristics, in order to evaluate the suitability of a given architecture for a desired workspace according to the workspace’s features. For example, when working on tasks with a principal direction, Type B limbs with parallel linear actuators as first joint of each limb might be favored, while other tasks on axial-symmetric or near-planar workspaces might be reached better by Type A limbs or Type B limbs with concurrent linear actuators as first joint of each limb. Hybrid manipulators with limbs of different types can be also considered for specific tasks.

Another key information when analyzing the workspace of a parallel manipulator is given by the volume that cannot be reached by the manipulator, either to avoid the collision with an obstacle or for any other kind of hazard. A limitation of the proposed procedure is given by internal obstacles, such as self-collision among limbs, which cannot be implemented directly with the proposed procedure without solving the kinematics of the mechanism. This is a consequence of the proposed two-step procedure, which computes the limb workspaces independently and then generates the overall volume as an intersection. In this way, no information on limb placement for each point of the workspace can be carried over into the intersection operation.

Conversely, external obstacles can be easily implemented in the proposed formulation with a volume subtraction, which can be expressed from (1) as a difference with set algebra as:

(10) \begin{equation} \Lambda = \bigcap\limits_{i=1}\limits^n\Lambda_i-\bigcap\limits_{k=1}\limits^m\Omega_k,\end{equation}

where $\Omega_j$ represents the volume occupied by the $k{\rm th}$ obstacle and m is the number of obstacles.

5. An example: 3-SPR mechanism

The procedure introduced in the previous section is here validated with a numerical example on a 3-SPR parallel manipulator, in Fig. 4(a). The design and kinematics of this mechanism are detailed in ref. [Reference Russo, Ceccarelli and Takeda27], where the reachable position workspace of the mechanism is computed by using forward kinematics through a discretization of the range of motion of each linear actuator. In this section, a geometrical formulation of the workspace is used, and we compare its results and efficiency (measured as computational time to solution) with the discrete formulation used in ref. [Reference Russo, Ceccarelli and Takeda27]. Its limb workspace, in Fig. 4(b), can be seen as a subset of the UPU case reported in Section 3, as the 3-SPR mechanism is an overconstrained variant of a conventional 3-UPU manipulator which requires the end-effector to collapse into a single point to enable motion [Reference Russo, Ceccarelli and Takeda27].

Figure 4. Modeling of a 3-SPR parallel manipulator: (a) Kinematic diagram of the mechanism with main geometrical parameters; (b) limb workspace evaluated with the proposed geometrical method.

In this example, joint limits are considered for prismatic and spherical joints. The minimum and maximum stroke of the linear actuator is defined as an inequality constraint that limits the limb length $l_i$ to range between $l_{\min}$ and $l_{\max}$ . Thus, the workspace of the robot can be defined as the volume comprised between two spheres, both with center in $A_i$ and radii of $l_{\min}$ and $l_{\max}$ , respectively. The angular limit for the base spherical joint is given as $[-\pi/2; +\pi/2]$ in each axis and is implemented as shown in Eq. (6). By considering limb volumes $\Lambda_i$ as per Eq. (5), the overall workspace of the manipulator $\Lambda$ can be evaluated as per Eq. (1). The corresponding workspace volume $V_{\Lambda}$ can be evaluated analytically from the formulation of the volume of intersection of three spheres presented in ref. [Reference Chkhartishvili28], as

(11) \begin{equation} V_\Lambda = \bigcap\limits_{i=1}\limits^3 V_{S_{i,\max}}-\bigcap\limits_{i=1}\limits^3 V_{S_{i,\min}},\end{equation}

where $V_{S_{i,\max}}$ is the volume of $S:(x-x_{Ai})^2+(y-y_{Ai})^2+(z-z_{Ai})^2=l_{\max}^2$ , and $V_{S_{i,\min}}$ is the volume of $S:(x-x_{Ai})^2+(y-y_{Ai})^2+(z-z_{Ai})^2=l_{\min}^2$ . For the formulation in ref. [Reference Chkhartishvili28], the geometry described in ref. [Reference Russo, Ceccarelli and Takeda27] is considered for the manipulators, with each base joint at a vertex of an equilateral triangle with side d, and $l_{\min} > d$ . Thus, the volume can be explicitly expressed as

(12a) \begin{equation} V_{\Lambda,\max} = \frac{d \sqrt{3l_{\max}^4-d^4}}{6}-\frac{d(12l_{\max}^2-d^2)}{4}\arccos{\frac{d\sqrt{3}}{\sqrt{4l_{\max}^2-d^2}}}+2l^3\arccos{\frac{2l_{\max}^2-d^2}{-4l_{\max}^2+d^2}}\end{equation}

(12b) \begin{equation} V_{\Lambda,\min} = \frac{d \sqrt{3l_{\min}^4-d^4}}{6}-\frac{d(12l_{\min}^2-d^2)}{4}\arccos{\frac{d\sqrt{3}}{\sqrt{4l_{\min}^2-d^2}}}+2l^3\arccos{\frac{2l_{\min}^2-d^2}{-4l_{\min}^2+d^2}}\end{equation}

(12c) \begin{equation} V_{\Lambda} = V_{\Lambda,\max}-V_{\Lambda,\min}.\end{equation}

Both the proposed geometrical method and a discretized approach [Reference Russo, Ceccarelli and Takeda27] have been used to compute the workspace volume and shape of a 3-SPR manipulator with $l_{i,\min}=200$ , $l_{i,\max}=300$ , $d=100$ . The calculation has been implemented in MATLAB on a Windows 10 computer with a 2.60 GHz quad-core CPU (Intel i7-6700HQ). The results of the geometrical approach are reported in Fig. 5, in which the boundaries of the three limb workspaces (in red, yellow, and blue) are shown with the boundary of their intersection (in black), whereas the discrete workspace computed as a point cloud of 106 reachable locations is shown in Fig. 6.

Figure 5. Workspace $\Lambda$ of a 3-SPR parallel manipulator evaluated by using the proposed geometrical approach as intersection of the limb workspaces $\Lambda_i$ : (a) Side view of the three spherical limb workspaces, outlined in blue, red and yellow, and the overall workspace of the manipulator given by their intersection, outlined in black; (b) top view of a close-up on the intersection of the limb workspaces, outlined in black.

Figure 6. Workspace $\Lambda$ of a 3-SPR parallel manipulator computed as a point cloud of reachable locations by discretizing pose parameters as per [Reference Liu, Wang, Oh and Kim26]: (a) side view; (b) top view.

The geometrical method provides an exact solution in less than 0.02 s, while the numerical approach returns an approximated result after 1.20 s. The two computed workspaces perfectly overlap. The volume obtained with the algebraic geometric approach is the geometrically exact one, as per Eq. (9) as reported also in ref. [Reference Russo, Ceccarelli and Takeda27], whereas the value computed with the discretized approach comes from a voxelization of the workspace and therefore includes an approximation error. Thus, the proposed method proves to be more efficient and better than the numerical one when determining the workspace volume of a parallel manipulator with given joint constraints.