Kinematic Analysis of Parallel Manipulators by Algebraic Screw Theory

Screw theory refers to the algebra and calculus of six-dimensional entities formed with ordered pairs of three-dimensional vectors related by a reference point, as forces and moments as well as angular and linear velocities concerned with the kinematics and dynamics of a rigid body. Sir Robert Stawell Ball developed the conceptual framework of the theory of screws more than one century ago for applications in kinematics and statics of rigid body mechanics (Ball, The theory of screws: a study in the dynamics of a rigid body. Dublin, Hodges, Foster and Company, 1876). Although screw theory remained only as a promissory mathematical tool for many more years, it has become an important tool in computational geometry, robot mechanics, multibody dynamics, and more recently in the higher-order kinematic analyses of a rigid body. This chapter comprises a brief review of prominent contributions concerned with the development of screw theory.

A parallel manipulator is a mechanical system formed by two linked platforms, namely, the fixed platform and the moving platform. The moving platform is connected to the fixed platform by at least two independent computer-controlled serial chains or limbs working in parallel. Compared with their serial counterparts, parallel manipulators are essentially more accurate and rigid. Furthermore, the possibility to mount the motors near the fixed platform is an attractive feature of robots with parallel kinematic topologies. Then it can be used in a wide range of applications that require precision and higher payload capacity combined with higher speed. This chapter briefly reviews of some well-known parallel manipulators that are currently considered significant contributions.

Jerk, a research field of great interest in quite diverse academic communities, is the time rate of change of acceleration and is related to the rate of change of force, namely an impulse that is considered a hammer blow force. Suh (1971) and Schot (1978) investigated the geometric parameters of the radii of torsion of spatial curves. It is known that there is a direct relationship between jerk and the movements of the human body (Morasso 1981; Flash and Hogan 1985; Uno et al. 2007). Consider, for instance, that Crossman and Goodeve (1983) have shown that when high spatial precision is required, many movements related with the hands present irregularities and multiple velocity peaks. Viviani and coworkers (Viviani and Schneider 1991; Viviani and Flash 1995) proved that there is a correlation among perception, motion planning, and jerk in humans. On the other hand, Sparis and Moroutsos (1984) applied jerk analysis to improve and control the position analysis of planar mechanisms when the mechanism is near a singular configuration. Žefran (1995) presented a coordinate-free formulation to generate shortest paths, and minimum acceleration and jerk properties of rigid bodies undergoing spherical motions, which correspond to the subgroup SO(3) of rotational motions of the Euclidean group SE(3), and general motions of SE(3). Urbinati and Pennestri (1998) developed a computer program for the jerk analysis of a rigid body with respect to an inertial reference frame. Their method is based on the representation of the Euclidean displacements of a rigid body in terms of the Euler parameters and the time derivatives of the corresponding rotation matrices associated with the motion of the rigid body. Gielen et al. (1997) noted that the characteristic pattern of cerebellar ataxia, related with jerk and submovements, is contained in the trajectory of the hand during repeated arm movements. In order to understand how the central nervous system controls the kinematics of rapid finger and hand movement, Novàk et al. (2000) proposed an objective algorithm to identify overlapping submovements, detecting appreciable inflections in the acceleration traces that are evidently related with jerk. To this end, Novàk et al. (2000) studied the motions of subjects turning a knob and concluded that in many trials subjects turned the knob with a single, smooth, and regular motion, as indicated by the angular position and velocity trajectories, but in other cases, subjects produced irregularities in the kinematics, which were considered discrete corrective submovements, detecting appreciable inflections in the acceleration traces. Thus, it is reasonable to assume that jerk is responsible for such irregularities. Goldvasser et al. (2001) investigated high curvature analysis and integrated absolute jerk to differentiate between healthy and cerebellopathy patients performing pointing tasks. Cheng (2002) introduced a procedure to obtain jerk curves that are expressed in explicit parametric forms. Without a doubt, jerk analysis with applications in human motions has been the motive behind a remarkable research field (Panjabi and White 1980; Levin 1995; Willems et al. 1996; Ziddiqui et al. 2006; Ishii et al. 2006; Konz et al. 2006; Howard et al. 2007; Lessard et al. 2007; Uno et al. 2007; Li and Xu 2007; Jones et al. 2008; Zhu et al. 2008; Tsoi and Xie 2010). Furthermore, interesting mechanical applications of jerk include, among many others, the following: (1) the classical design of cam-follower systems; (2) the correct characterization of singularities in closed kinematic chains; (3) the synthesis of spherical and in general spatial linkages; (4) the computation of impact forces caused by rapid changes in acceleration; (5) the improvement of path-planning trajectories in the neighborhood of singular regions. Following that trend, Zahraee and Neff (1996) reported the jerk analysis of some commercial cams, whereas Nguyen and Kim (2007) introduced a method to smooth cam profiles using modified spline curves. Lee and Lin (1999) have shown that in milling machining operations, undesirable chattering effects can be reduced by controlling the continuity of jerks associated with the trajectory of the cutter. Finally, as shown by Erkorkmaz and Altintas (2001) and Macfarlane and Croft (1986b), it is possible to control vibrations within kinematic structures by minimizing jerk. Mandal and Nazcar (2009) designed cam displacement functions based on jerk analysis. To the best knowledge of the authors, the first contribution approaching the jerk analysis of rigid bodies using screw theory is that of Rico et al. (1999).

The hyper-jerk, also known as snap or jounce, is the time rate of the jerk and is more than an academic pursuit; for example, Wohlhart (2010) showed that any planar parallel manipulator at a singular configuration could be architecturally mobile according to hyper-jerk analysis. In this chapter the hyper-jerk analysis of rigid bodies is approached using screw theory. A few decades ago the theory of screws introduced by Ball (1900) seemed to be an “old-fashioned” mathematical tool limited to elementary kinematic analyses, such as the displacement and velocity analyses of rigid bodies. Although the robustness of the definition of reduced acceleration state of a rigid body was introduced by Brand (1947), the representation in screw form of the acceleration analysis of kinematic chains was formally published half a century later by Rico and Duffy (1996). Before that contribution, most kinematicians commonly accepted that screw theory could not handle higher-order analyses of mechanisms, which today we recognize to be a wrong assumption. The escape from singular configurations of serial manipulators, the characterization of singularities of closed chains, as well as the acceleration analysis of parallel manipulators have been the most recurrent subjects since the formulation introduced by Rico and Duffy (1996) was proved successfully; see, for instance, Rico et al. (1995), Rico and Gallardo (1996), Gallardo-Alvarado et al. (2007, 2010), Gallardo et al. (2008). Furthermore, the elucidation of the acceleration analysis in screw form opened the possibility to extend it to the jerk analysis (Rico et al. 1999; Gallardo-Alvarado and Rico-Martínez 2001), which was applied in the higher-order kinematic analyses of parallel manipulators (Gallardo-Alvarado 2003, 2012; Gallardo-Alvarado and Camarillo-Gómez 2011). In that concern, as shown by Lipkin (2005), screw theory is not limited by the order of the kinematic analysis. Without a doubt, the higher-order analysis is a subject of growing interest covering topics like path-planning control (Kyriakopoulos and Saridis 1994), improvement of maneuvers of mobile robots (Oyadiji et al. 2005), high-speed machining (Zhang et al. 2011), jerk dynamics (Maccari 2011), smooth path planning for conventional vehicles (Villagra et al. 2012), optimal time-jerk algorithms for trajectory planning (Gasparetto et al. 2012), and so forth.

The first robot manipulator selected to exemplify the application of the kinematic analysis method proposed in this book is a partially decoupled five-degree-of-freedom parallel manipulator generator of the 3R2T motion.

A parallel wrist is a robot manipulator where one point embedded in the moving platform is unable to perform translational motions with respect to the fixed platform. This chapter presents the kinematic analysis of a two-degree-of-freedom parallel wrist.

This chapter reports on the kinematics of a symmetric three-legged six-degree-of-freedom parallel manipulator equipped with a combination of rotary and linear actuators. The introduction of this class of robot manipulators has been a recursive option to avoid the complexity of the forward displacement analysis of the general hexapod.

The combination of rotary and linear actuators in a parallel manipulator is an interesting option to overcome some of its limitations, such as its well-known limited workspace and recurrent problem of singular configurations. This chapter uses screw theory to address the kinematics of a six-degree-of-freedom parallel manipulator provided with a combined scheme of actuation.

This chapter uses screw theory to address the kinematics of the most celebrated six-legged six-degree-of-freedom parallel manipulator. A novel strategy to solve the forward displacement analysis of the robot is explained in detail in this chapter.

In this chapter the kinematics of the original flight simulator introduced by Stewart (A platform with six degrees of freedom. Proc Inst Mech Eng 180(1):371–386, 1965–1966; and A platform with six degrees of freedom: a new form of mechanical linkage which enables a platform to move simultaneously in all six degrees of freedom developed by Elliott-Automation. Aircr Eng Aerosp Technol 38(4):30–35, 1966) is widely discussed. Curiously, most kinematicians believe that Stewart was the inventor of the hexapod. In that regard, while the kinematics of Gough’s tyre testing machine (Gough, Contribution to discussion of papers on research in automobile stability. Proc Autom Div Inst Mech Eng 392–394, 1957; Gough and Whitehall, Universal tyre testing machine. In G. Eley (Ed.), Proceedings 9th International Automobile Technical Congress, discussion pp. 250ff; Fédération Internationale des Sociétés d’Ingénieurs des Techniques de lÁutomobile (FISITA). IMechE 1, London, pp. 117–137, 1962) has been extensively investigated, the kinematics of the Stewart platform has been practically overlooked.

Parallel manipulators with topologies based on parallelograms have proved to be the most efficient robot manipulators, and their applications have therefore had a great impact in an industrial context. This revolution of modern robotics began with the Delta robot, and in this chapter its kinematics is revisited according to screw theory.

This chapter provides full solutions to selected exercises. The figures in this chapter are intentionally not numbered.

There are several well-known methods to compute the rotation matrix. However, such methods require a little experience dealing with the handling of two related reference frames through the introduction of a wide class of orientation angles, such as Euler angles, Tait–Bryan angles (roll, pitch, and yaw), XYZ angles, and so on. This appendix provides a simple method for computing the rotation matrix. The method is based on the fact that the coordinates of three points of a rigid body are sufficient to determine the pose of the rigid body as observed from another body or reference frame.

In this book, Maple was chosen as the software package to develop a few computer codes because of its affordability and higher characteristics to implement symbolic computation.