Yunjiang Lou

Singularities of parallel manipulators: a geometric treatment

A parallel manipulator is naturally associated with a set of constraint functions defined by its closure constraints. The differential forms arising from these constraint functions completely characterize the geometric properties of the manipulator. In this paper, using the language of differential forms, we provide a thorough geometric study on the various types of singularities of a parallel manipulator, their relations with the kinematic parameters and the configuration spaces of the manipulator, and the role redundant actuation plays in reshaping the singularities and improving the performance of the manipulator. First, we analyze configuration space singularities by constructing a Morse function on some appropriately defined spaces. By varying key parameters of the manipulator, we obtain homotopic classes of the configuration spaces. This allows us to gain insight on configuration space singularities and understand how to choose design parameters for the manipulator. Second, we define parametrization singularities which include actuator and end-effector singularities (or other equivalent definitions) as their special cases. This definition naturally contains the closure constraints in addition to the coordinates of the actuators and the end-effector and can be used to search a complete set of actuator or end-effector singularities including some singularities that may be missed by the usual kinematics methods. We give an intrinsic classification of parametrization singularities and define their topological orders. While a nondegenerate singularity poses no problems in general, a degenerate singularity can sometimes be a source of danger and should be avoided if possible.

Randomized Optimal Design of Parallel Manipulators

This work intends to deal with the optimal kinematic synthesis problem of parallel manipulators under a unified framework. Observing that regular (e.g., hyper-rectangular) workspaces are desirable for most machines, we propose the concept of effective regular workspace, which reflects simultaneously requirements on the workspace shape and quality. The effectiveness of a workspace is characterized by the dexterity of the mechanism over every point in the workspace. Other performance indices, such as manipulability and stiffness, provide alternatives of dexterity characterization of workspace effectiveness. An optimal design problem, including constraints on actuated/passive joint limits and link interference, is then formulated to find the manipulator geometry that maximizes the effective regular workspace. This problem is a constrained nonlinear optimization problem without explicitly analytical expression. Traditional gradient based approaches may have difficulty in searching the global optimum. The controlled random search technique, as reported robust and reliable, is used to obtain an numerical solution. The design procedure is demonstrated through examples of a Delta robot and a Gough-Stewart platform.

Optimal design of parallel manipulators for maximum effective regular workspace

Kinematic design of parallel manipulators is addressed in this paper. By observation that regular (e.g., hyper-rectangular) workspaces are desirable for most machines, we propose the concept of effective regular workspace, which reflects both requirements on the workspace shape and quality. Dexterity index is utilized to characterize the effectiveness of the workspace. The optimal design problem is then formulated to find a manipulator geometry that maximizes the effective regular workspace. Since the optimal design problem is a constrained nonlinear optimization problem without explicit analytical expressions, the controlled random search (CRS) technique, which was reported robust and reliable, is applied to numerically solve the problem. The commonly-used Stewart-Gough platform is employed as an example to demonstrate the design procedure.

A general approach for optimal kinematic design of parallel manipulators

This paper deals with the problem of optimal geometry design of parallel manipulators. In order to reduce the main drawbacks of parallel manipulators, relatively small workspace and more singularities, two requirements, workspace and condition number, are considered. The design problem is thus formulated to find a parallel mechanism such that its Cartesian workspace contains a prescribed workspaces with good condition numbers in it. By observing that those requirements can be locally cast into Linear Matrix Inequalities (LMIs), we formulate the design problem locally as a convex optimization problem subject to LMIs with a max-det function as its objective function. Hence, at each node of discretized space of design parameters, there is an LMI-based convex optimization problem. A two-level algorithm can be applied to solve for a set of optimal design parameters: (1) Discretize the space of design parameters into a set of discrete nodes; (2) At each node the Newton algorithm is applied to solve the max-det optimization problem. By comparing all the locally optimal costs, we can obtain a corresponding set of globally optimal design parameters correspondingly. Simulation results verify the effectiveness of the proposed approach.

Task Polar Coordinate Frame-Based Contouring Control of Biaxial Systems

Contouring control is crucial in high-speed and high-precision manufacturing. In this paper, a novel task polar coordinate frame (TPCF), moving along the desired contour, is proposed to naturally calculate and control the estimated contouring error by the circular approximation, a second-order approximation. The dynamics in the world Cartesian coordinate frame is transformed into radial and angular dynamics in the local polar coordinate frame. By the feedback linearization technique and an input feedforward compensation, the closed-loop dynamics are decoupled in terms of the estimated contouring error and the angular error, respectively. Proportional-plus-derivative controllers can be assigned to stabilize the individual axis dynamics in the TPCF. By tuning the control parameters, different strengthening on estimated contouring error and angular error can be imposed explicitly and directly. Various experiments on an XY-stage biaxial system with typical contours, a circle and a figure-“8,” were conducted. Comparative studies are carried out for the TPCF- and traditional Frenet frame-based controls. The contouring errors were drastically reduced by the proposed approach, particularly in high-speed and large-curvature contouring cases.

Type Synthesis, Kinematic Analysis, and Optimal Design of a Novel Class of Schönflies-Motion Parallel Manipulators

A novel class of spatial four degree-of-freedom Schönflies-motion parallel manipulators with four identical subchains is presented. Their features are that each serial subchain undergoes the pure Schönflies motion without redundant joints. The parallel mechanisms possess the simplest topology and are suitable for pick-and-place operations. Kinematic analysis of the 4-PRPaR parallel manipulator, including its inverse and forward kinematics, singularity, and workspace, is discussed in detail. The analysis shows that the moving platform and the base must be in dissimilar dimension for good manipulability performance. The optimal design of the parallel manipulator is formulated as a multiobjective optimization problem. A novel performance index characterizing the approximation of the generated workspace to the prescribed regular workspace, the regular workspace share, is proposed to serve as one of the design objectives. The other objective is the global condition index, which measures the manipulability. The multiobjective optimization problem provides multiple optimal solutions for choice. Simulation verifies that the designed parallel manipulator can approximate the prescribed regular workspace with good condition index.

Optimization Algorithms for Kinematically Optimal Design of Parallel Manipulators

Optimal design is an inevitable step for parallel manipulators. The formulated optimal design problems are generally constrained, nonlinear, multimodal, and even without closed-form analytical expressions. Numerical optimization algorithms are thus applied to solve the problems. However, the optimization algorithms are usually chosen ad arbitrium. This paper aims to provide a guideline to choose algorithms for optimal design problems. Typical algorithms, the sequential quadratic programming (SQP) with multiple initial points, the controlled random search (CRS), the genetic algorithm (GA), the differential evolution (DE), and the particle swarm optimization (PSO), are investigated in detail for their convergence performances by using two canonical design examples, the Delta robot and the Gough-Stewart platform. It is shown that SQP with multiple initial points can be efficient for simple design problems, while DE and PSO perform effectively and steadily for all design problems. CRS can be used to generate good initial points since it exhibits excellent convergence evolution in the starting period.

Improved and modified geometric formulation of POE based kinematic calibration of serial robots

We propose in this paper an improved geometric formulation of POE (Product Of Exponential) based kinematic calibration of serial robots, which is based on the work of [1]. We use both joint offset-free formulation and adjoint transformation errors of joint screws, and apply it to the calibration of an elbow manipulator. Our formulation explains why the original POE calibration always fails with the existence of joint offset errors; the adjoint formulation of joint screw errors eliminates joint screw constraints that was imposed in the original iterated least square calibration algorithm. The second contribution of this paper is the proposal of a modified POE formulation which adopts point measurement data instead of frame measurement data of the end-effector, which can be more realistic and convenient for practical implementation. Simulation results show that the proposed method is plausible and effective. An experiment is under preparation to verify the effectiveness of the proposed calibration method on an elbow manipulator built by Googol Technology.

A Novel 3-DoF Purely Translational Parallel Mechanism

A novel 3-DoF purely translational parallel mechanism, the Orthotripod, is proposed. It is a variant of the tripod based parallel machine and has a similar architecture to the Orthoglide. In order to reduce the number of passive joints and remove the effect of ease of abrasion of revolute joints, spherical joints are applied in the parallelogram. A mathematic mobility analysis shows the mechanism is indeed 3-DoF purely translational. We optimally design the Orthotripod and the tripod based parallel machine by maximizing the well-conditioned workspace. The optimized Orthotripod possesses a nearly ball-shaped workspace and has much better kinematic performance than the optimized tripod based parallel mechanism. The proposed mechanism is adaptable for machine tool applications

Optimal design of parallel manipulators via LMI approach

This paper deals with the problem of optimal design of parallel manipulators which are singularityless, of high stiffness and manipulability and the most economic. By observing that those requirements can be cast into Linear Matrix Inequalities (LMIs), we formulate the design problem as a convex optimization problem subject to LMIs with either a linear function or a max-det function as its objective function. The variables x associated with LMIs are nonlinear functions of some key kinematic parameters /spl alpha/. If the dimension of x, t, is equal to the number of kinematic parameters, l/sub 0/, a two-level algorithm can be applied to solve for a set of optimal kinematic parameters: (1) Applying the interior point algorithm for solving of x; (2) Applying Newton method to a set of nonlinear algebraic equations for solving of /spl alpha/. If the dimension of x is greater than the number of kinematic parameters (i.e., x are not linearly independent), we consider the constrained semi-definite programming problems and the constrained max-det problems by taking account of an additional set of nonlinear constraints. We propose a simplified constrained gradient algorithm for solving of x in such cases, /spl alpha/ derives from x using Newton method. Simulation results verify the effectiveness of the proposed algorithms.