Recursive Kinematics and Inverse Dynamics for Parallel Manipulators

We examine here the modular and recursive formulation of the inverse dynamics of parallel architecture manipulators. The concept of the decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism of the robots of interest to develop an inverse dynamics algorithm which is both recursive and modular. INTRODUCTION The modular and recursive inverse dynamics formulation for parallel architecture manipulators is the subject of this paper. The modular formulation of mathematical models is attractive because existing sub-models may be assembled to create different topologies, e.g. cooperative robotic systems. Recursive algorithms are desirable from the viewpoint of simplicity and uniformity of computation, despite the ever-increasing complexity of mechanisms. Additionally we note that, prior to the dynamics computation stage, a forward or inverse kinematics stage is ∗Address all correspondence to this author. often required. Hence, the development of efficient recursive dynamics algorithms also necessitates the careful investigation of recursive kinematics algorithms. However, the development of modular and recursive kinematics and dynamics algorithms for parallel manipulator architectures remains a challenging research problem. The literature on mathematical modelling of manipulators has a rich history spanning several decades. We will summarize some critical aspects presently while referring the interested reader to any number of books on the subject [1–5], for details. Methods for formulation of equations of motion (EOM) fall into two main categories: a) Euler-Lagrange and b) Newton-Euler formulations. Euler-Lagrange methods, some times referred to as analytic, are commonly used in the robotics community to obtain the equations of motion (EOM) of robotic manipulators. Typically, such approaches use the joint-based relative coordinates as the configuration space. For serial chain manipulators, these form a minimal coordinate description and permit a direct mapping to actuator coordinates. Newton-Euler (NE) methods, also referred to as synthetic, on the other hand, typically favor 1 Copyright c © 2003 by ASME the use of Cartesian variables as configuration-space variables, and develop recursive formulations from the free-body diagram (FBD) of each single body. The uncoupled governing equations are then assembled to obtain the model of the entire system. While efficient formulations exist for serial-chain and treestructured multi-body systems, the adaptation of these methods to the simulation of closed-chain linkages and parallel manipulators is relatively more difficult. Such systems possess one or more kinematic loops, requiring the introduction of algebraic constraints, typically nonlinear, into the formulation. Considerable work has been reported in the literature on the specialization of the above methods to formulate the EOM of constrained mechanical systems, while including both holonomic and nonholonomic constraints. Parallel mechanisms and manipulators form a special class of constrained mechanical systems where the multiple kinematic loops give rise to systems of nonlinear holonomic constraints; in the ensuing discussion we will focus on the development of EOM of such systems. Non-Recursive Newton-Lagrange Formulations The dynamics of constrained mechanical systems with closed loops using the Newton-Lagrange approach is traditionally obtained by cutting the closed loops to obtain various open loops and tree-structured systems, and then writing a system of ordinary differential equations (ODEs) for the corresponding chains in their corresponding generalized coordinates [6]. The solution to these equations are required to satisfy additional algebraic equations guaranteing the closure of the cut-open loops. A Lagrange multiplier term is introduced to represent the forces in the direction of the constraint violation. The resulting formulation, often referred to as a descriptor form, yields a usually simpler, albeit larger, system of index-3 differential algebraic equations (DAEs). Typical methods used to solve the forward and inverse dynamics problems for such constrained systems cover a broad spectrum, namely, Direct elimination where the surplus variables are eliminated directly, using the position-level algebraic constraints to explicitly reduce index-3 DAE to an ODE in a minimal set of generalized coordinates (conversion into Lagrange’s equations of the second kind) [7]; Explicit Lagrange-multiplier computation together with the unknown accelerations computed from the augmented index-1 differential algebraic equation (DAE) formed by appending the differentiated acceleration level constraints to the system equations [1, 8]; Lagrange-multiplier approximation/Penalty formulation, where the Lagrange multipliers are estimated using a compliance-based force-law, based on the extent of constraint violation and assumed penalty factors [2, 9]; Projection of dynamics onto the tangent space of the constraint manifold, where the constraint-reaction dynamics equations are taken into the orthogonal and tangent subspaces of the vector space of the system generalized velocities. A family of choices exist for the projection, as surveyed by Garcı́a de Jalón and Bayo [2] and Shabana [5]. Recursive Newton-Euler Formulations Many variants of fast and readily-implementable recursive algorithms have been formulated within the last two decades, principally within the robotics community to overcome the limitations posed by the complexity of the dynamics equations based on classic Lagrange approaches [6]. The first researchers to develop O(N) algorithms for inverse dynamics for robotics used a Newton-Euler formulation of the problem. Stepanenko and Vukobratovic [10] developed a recursive NE method for human limb dynamics, while Orin et al. [11] improved the efficiency of the recursive method by referring forces and moments to local link-coordinates for the real-time control of a leg of a walking machine. Luh et al. [12] developed a very efficient Recursive Newton-Euler algorithm (RNEA) by referencing most quantities to link-coordinates; this is the most often cited. Further gains have been made in the efficiency over the years, as reported, for example, by Balafoutis et al. [13] and Goldenberg and He [14]. In multi-loop mechanisms, the generalized coordinates (joint variables) are no longer independent, since they are subject to the typically non-linear loop-closure constraints. The most common method for dealing with kinematics is to cut the loop, introduce Lagrange multipliers to substitute for the cut joints and use a recursive scheme for the open-chain system to obtain a recursive algorithm. Decoupled Natural Orthogonal Complement The natural orthogonal complement (NOC), first introduced in Angeles and Lee [15], belongs to the class of projection methods for dynamics evaluation. Saha [16] showed a method for splitting the NOC of a serial chain into two matrices, one diagonal and one lower block-diagonal, thus introducing the decoupled natural orthogonal complement (DeNOC). By doing so, Saha was able to exploit the recursive nature of the DeNOC and apply the concept to model simple serial manipulators. Further, although recursive kinematics algorithms for serial chains have had a long history [10–12], a recursive algorithm for the forward kinematics of closed-chain systems first appeared in Saha and Schiehlen [17]. In this work, Saha and Schiehlen [17] showed that the NOC of a parallel manipulator may be split into three parts–one full, one block-diagonal and one lower-triangular, and proposed a recursive minimal-order forward dynamics algorithm for parallel manipulators. Examples of up to two degree-offreedom planar manipulators were included and various physical 2 Copyright c © 2003 by ASME Figure 1. TWO BODIES CONNECTED BY A KINEMATIC PAIR interpretations were reported. BACKGROUND Twists, Wrenches and Equations of Motion In this section, some definitions and concepts associated with the formulation of the kinematics and dynamics of articulated rigid body systems coupled by lower kinematic pairs will be briefly reviewed. See [18] and [19] for further details. Figure 1 shows two rigid links connected by a kinematic pair. The mass center of ith link is at Ci while that of link i − 1 is at Ci−1. The axis of the ith pair is represented by a unit vector ei. We attach a frame Fi with origin Oi and axes xi, yi and zi, to link i − 1, such that zi is along ei. The global inertial reference frame F0 with axes x, y and z is attached to the base of the manipulator, and unless otherwise specified, all quantities will be represented in this global frame in the balance of the paper. Further, we define, the three-dimensional position vectors di from the Oi to the mass center of link i and ri−1 from the mass center of link i− 1 to Oi. The six dimensional twist and wrench vector associated with link i, at its mass center Ci, are now defined as ti = [ ωi vi ] ; wi = [ ni fi ] (1) where ω, v, n, and f are three dimensional angular velocity, linear velocity, moment and force vectors, respectively, associated with link i and represented about Ci. The Newton-Euler equations for link i are ni = Iiω̇i + İiωi = Iiω̇i + ωi × Iiωi fi = miv̇i where Ii is the 3 × 3 inertia tensor about Ci and mi is the mass of link i. The above set, in matrix form, may be written as wi = [ Ii O O mi1 ] } {{ } Mi [ ω̇i v̇i ] + [ Ωi O O O ] } {{ } Wi [ Ii O O mi1 ] [ ωi vi ] (2)