Robot Components

A PR has its inherent modular feature as all limbs of a PR often have the same or similar architecture, constructed by the same type of actuators, joints, and links. Therefore, PRs have excellent reconfigurability, which is identified as a key nature of the future manufacturing system. Figure 7 shows some common building blocks of parallel robots, including revolute and prismatic actuator modules; three types of passive joint modules (without actuators) including the rotary joint, the pivot joint, and the spherical joint; a set of links with various geometrical shapes and dimensions; and a mobile platform. Two exemplar PRs constructed with these components are shown in Fig. 8.

Type Synthesis

Type synthesis, or topological synthesis, is the initial step in PR design, and it is the study of the nature of connection among the members of a mechanism and its mobility. It is concerned primarily with the fundamental relationships among the degrees of freedom, the number of links, the number of joints, and the type of joints used in a mechanism. Type synthesis usually only deals with the general functional characteristics of a mechanism and not with the physical dimensions of the links. This part of the work is difficult because it originates from the creativity of the designer and mostly depends on their intuitiveness, intelligence, and experiences.
In the literature, most of the PMs have identical limbs to simplify the design and fabrication (Gosselin et al. 1996; Huang and Li 2002; Jin et al. 2006a; Tsai 1999, 2001) introduced an enumeration scheme to guide synthesis of PMs with 3–6 DOFs based on Grubler criterion and loop mobility criterion. The Grubler criterion is as follows:

where λ is the motion parameter; for spatial mechanisms, λ = 6; and, for planar and spherical mechanisms, λ = 3. n represents the total number of links (including the fixed link), j the total number of joints, and fi the number of DOF of the ith joint.
Note that the Grubler criterion is valid provided that the constraints imposed by the joints are independent of one another and do not introduce redundant degrees of freedom. The loop mobility criterion is as follows:

where Ck is the connectivity of the kth limb (the DOF associated with all the joints), m the total number of limbs, and l the total number of loops. λ ≥ Ck ≥ F ensure proper mobility and controllability of the end effector. Based on Eqs. 1 and 2, the kinematic architectures of PRs can be enumerated according to their nature of motion and DOF, as shown in Table 1.

Although the enumeration provides a nice guideline of the required connectivity to achieve certain DOF, this approach cannot specify the specific output motion, e. g., 3-DOF translations. For synthesizing PRs with specified DOF, there are two major methods: constraint synthesis approach (Huang and Li 2003; Fang and Tsai 2004; Kong and Gosselin 2004, 2007) based on screw theory and Lie subgroup approach (Herve 1991; Li et al. 2004; Li and Herve´ 2010) based on the algebraic properties of a Lie group of the Euclidean displacement set.
A screw is a mathematical notion which is modeled on a physical screw. It is a quantity associated with a line in the three-dimensional space and a scalar called pitch. According to Chasles’s theorem, any displacement of a rigid body can be effected by a single rotation about a unique axis combined with a unique translation parallel to that axis, so any displacement can be described as a “screw displacement,” where the unique rotation axis, which is also the translation direction, is referred to as the screw axis of the displacement. In this sense, a revolute joint can be represented by a unit screw of zero pitch pointing along the joint axis and a prismatic joint represented by a unit screw of infinite pitch along the direction of the linear joint axis.
Screw theory is a way to study velocities and forces in a three-dimensional space, combining both rotational and translational components. When a screw is used, it is often called a twist. When a screw is used to describe the force system acting on a rigid body, it is called a wrench. If the work done by a wrench $1 acting on a rigid body having a twist $2 motion vanishes, it is said the two screws $1 and $2 are reciprocal. A screw system is a set of linear combination of independent screws. The order of a screw system is equal to the number of basis (independent) screws in the system. A d order screw system is associated with a reciprocal (6-d) screw system. The twist system of a kinematic connection represents all motions permitted by that connection in a given position. The wrench system of the connection represents all constraints that can be transmitted by the connection. The twist system and the wrench system of the connection are uniquely related by reciprocity. The duality and the reciprocality of screws make it a suitable tool for synthesis and analysis of parallel manipulators.
In screw-based method (Huang and Li 2003; Jin et al. 2009), the wrench system of PRs is firstly obtained according to mobility requirement of the end effector. Then the standard base of the mechanism twist system can be formulated and is treated as the standard base of the limb twist system. Subsequently, some other necessary twists are added to obtain the limb twist system. The process of adding other twists mainly depends on the experiences of the designer. Then, all the limbs are connected with the base and the mobile platform according to the geometrical conditions. Lastly, it requires checking if the PMs can work in finite motion as this approach is based on the instantaneous kinematics.
In Lie subset method (Herve 1991), a set of operators is used to describe all possible finite displacements, and each kind of motion is represented by a set of operators. A mechanical bond is represented by the set of allowed displacements of one body with respect to the other. A serial arrangement of kinematic pairs becomes the composition of mechanical bonds, and a parallel arrangement of kinematic pairs is represented by the intersection of mechanical bonds. The first step in type synthesis is to get a mechanical generator of a limb according to the motion characteristics of the end effector. Second is to analyze the mechanical generator to obtain a feasible configuration for a limb. Third is to combine several limbs to form the PMs. And last, checking finite motion on the mechanism is required. Similar to the screw-based method, the second step of this approach very much depends on the experiences of the designer.
So far, a large number of types of PRs have been synthesized, but the advantages of various PR types have not been fully exploited. There is currently still lack of tools and metrics to effectively compare different PR architectures.

Displacement Modeling

Displacement modeling is to find the relationship between the position (posture) of the end effector and the displacements of the actuators. It is the basis of workspace analysis, dimensional optimization, and motion planning and control. There are two types of displacement models: (1) forward displacement model, which is to find the end-effector pose with given displacements of actuators, and (2) inverse displacement model, in which the displacements of actuators are to be found with given end-effector pose. For a serial manipulator, the forward displacement is straightforward, but the inverse displacement is difficult due to multiple solutions. For the parallel manipulators, the inverse displacement is easy because each limb is much simpler than a serial manipulator. But the forward displacement is generally rather complicated as it requires solving higher-order nonlinear Equations. A geometrical method is commonly used to put down a vector loop equation f(X, q, p) = 0 for each loop while taking the geometrical constraints into account (Altuzarra et al. 2009). Then the passive variables p are to be removed to achieve an equation with only actuators’ inputs q and end effector outputs X. However, the resultant system of equations is nonlinear and often including sine-cosine polynomials. Therefore, it is hard to arrive at closed form solutions; at best a high-order univariate polynomial equation can be obtained and then solved by a numerical method, such as Newton-Raphson’s procedure, but only one solution can be obtained each time (Raghavan 1993;Wang and Chen 1993). Some other researchers studied the forward kinematics of PMs with the help of additional sensors (Notash and Podhorodeski 1995; Chiu and Perng 2001; Baron and Angeles 2000), by which the difficulty is significantly reduced so that the kinematic algorithm is suitable for real-time applications.

Jacobian Analysis

An important limitation of a parallel manipulator is that there may exist singular configurations in its workspace where the manipulator gains or loses one or more DOFs so that the output motion cannot be properly controlled. Jacobian analysis is an important section for checking the motion characteristics of the mechanism. The relationship of the input joint rates and the end-effector output velocity of a PR can be expressed as

where Jx is called direct Jacobian matrix and Jq is called inverse Jacobian matrix. Due to the existence of two Jacobian matrices Jx and Jq, a PR is said to be a singular configuration when either Jx, Jq, or both are singular. According to (Gosselin and Angeles 1990), three different types of singularities are classified, i.e., inverse kinematic singularity (|Jq| = 0), forward kinematic singularity (|Jx| = 0), and combined singularity (|Jx| = |Jq| = 0). The inverse singularity occurs in the manipulator configuration when the moving platform loses one or more degrees of freedom instantaneously. The actuator velocities cannot be determined by the given moving platform velocity. In other words, the moving platform will be constrained by more wrenches instantaneously in the inverse singular configuration. Forward singularity occurs when the moving platform gains one or more degrees of freedom instantaneously. In this configuration, the moving platform can move even though all actuators are locked.
Two common methods are often used to formulate the Jacobian matrix of a PR, i. e., vector loop equation approach and screw theory approach. In the first approach, a loop closure equation is built for each limb, and then the loop closure equation is differentiated to obtain the relationships between the input and output parameters. The input parameters are the rotation velocity or linear velocity of actuators, and the output parameters are the velocity of the end effector. In the second approach, one can always find a screw which is reciprocal to all the twists associated with the passive joints because the instantaneous twist of the moving platform is expressed as a linear combination of all the twists of a limb. Hence, the relationships between the input joint rates and the output velocity can be obtained by the orthogonal product of the screw and twists. Each row of the Jacobian matrix is in fact represented with a twist, and each matrix in essence forms a system of all twists. The singularity will occur when the system is not in its full row rank. Therefore, screw-based line geometry method is usually employed for analyzing singular configurations. For example, the forward singularity of a 3RPR planar PM (Jin et al. 2009; Bonev et al. 2003) occurs when the 3-system of wrenches ( f'1, f'2, f'3) of Jx degenerates. Geometrically, the forward singular configurations occur if and only if the three lines are in parallel or intersect at one common point as shown in Fig. 9. With the three lines in parallel as shown in Fig. 9a, the moving platform will produce an infinitesimal translation along the line direction even though all actuators are locked. With the three lines intersection at a common point as shown in Fig. 9b, the moving platform can produce rotations about that point even though all actuators are locked. Therefore, the end-effector motion path should be purposely designed away from singular postures in motion planning and control.

Workspace Analysis

Workspace analysis is a challenging problem for PRs. The solution of this problem is critical in design and motion planning of the robot. As the complete workspace of a 6-DOF PR is embedded in a 6-dimensional space, there is no simple way to visualize it in a human-readable way. Therefore, the workspace of PR is usually visualized in various 3D subspaces (Merlet 2000), such as the constant-orientation workspace, the reachable orientation workspace, and the reachable positional workspace. As the major drawback of PRs is their limited workspace, workspace volume becomes an important performance index of PRs (Monsarrat and Gosselin 2003; Merlet et al. 1998; Gosselin 1990; Kumar 1992; Jin et al. 2011). Some researchers made use of geometrical methods to calculate the workspace of PMs (Merlet et al. 1998; Gosselin 1990). Other researchers studied the design for the desired workspace by optimization (Monsarrat and Gosselin 2003; Jin et al. 2011; Lou et al. 2004).
In geometrical method, the reachable positional workspace of each limb is firstly obtained based on the geometrical relationships of joints and links of the limb. The workspace boundaries are then obtained by the intersection of the reachable workspace generated by each limb. With numerical method, a large workspace envelope through the extreme reach of each limb is firstly assumed, and then the workspace envelope will be parametrically and proportionally divided into a number of finite elements of the same volume. When the number of the finite elements is sufficiently large, each element can be kinematically represented by a point located inside the element. And inverse kinematics will then be used to check if the point is inside or outside the workspace by checking all constraints including joint angles and actuators’ displacements. And the workspace volume is converted into a simple summation of all volume elements in the workspace.

Dimensional Synthesis

A PR topology can only give its best performance when its geometrical parameters are properly assigned. Dimensional synthesis is used to find the optimal dimensions of every link (L) for achieving the desired performance through certain cost functions f(L), subject to a number of constraints. So the problem is often formulated as an optimization question as follows:

Min : f(L) subject to constraints.

However, constructing the optimization model and solving the optimization problem are rather challenging because of the nonlinear function and often conflicted design criteria (e.g., use of stiffness and weight, workspace, and/or accuracy) (Cardou et al. 2010; Merlet 2006). A number of performance metrics have been introduced for designing PRs, and most of them are related to dexterity/conditioning or workspace volume of the end effector (Gosselin and Angeles 1991; Castelli et al. 2008; Jin et al. 2006b; Carbone et al. 2007). Gosselin and Angeles proposed the global conditioning index (GCI) for measuring the global dexterity of manipulators over their entire workspace in design optimization (Gosselin and Angeles 1991). Liu and Gao (2000) studied optimum design of 3-DOF spherical parallel manipulators with respect to the conditioning and stiffness indices. Chablat and Wenger (2003) studied architecture optimization of the Orthoglide with prescribed kinetostatic performances in a prescribed workspace. Huang et al. (2005a) introduced dimension optimization for the TriVariant PR with two performance indices, i.e., global and comprehensive conditioning indices. Li and Xu (2006) introduced a new optimization approach which utilized both the global dexterity index and space utility ratio for design a 3-PUU translational parallel mechanism. Pierrot et al. (2009) introduced optimal design of a 4-DOF parallel manipulator with the cost of links as an objective while keeping machine cycle time and dexterity as optimization constraints. Liu et al. (2007) designed a hybrid PR with large workspace/limb-stroke ratio, and a global conditioning index based on the minimum singular value of Jacobian is defined for dimension optimization. Ottaviano and Ceccarelli (2002) used specified workspace volume as the design objective for synthesizing the design parameters of the CaPaMan. Altuzarra et al. (2011) studied dimensional synthesis using Pareto optimization with three design objectives, including workspace volume, dexterity, and energy consumption. The current trend is to incorporate practical dynamic performance indices in design optimization as dynamic performance is the key advantage of PRs comparing to serial robots (Huang et al. 2013). Apart from the objective function-based optimal design, another method based on the performance chart is emerging recently (Liu and Wang 2007; Liu 2006). In this method, the performance maps of various measures are drawn in the entire or desired workspace of PRs, so that the performance can be evaluated through visual analytics so as to achieve the global optimization. But the approach is limited by the number of parameters.