Variable Geometry Truss Manipulator Research Articles

Towards Design Optimization of Compliant Mechanisms: A Hybrid Pseudo-Rigid-Body Model-Finite Element Method Approach and an Accurate Empirical Compliance Equation for Circular Flexure Hinges.

Innovative designs such as morphing wings and terrain adaptive landing systems are examples of biomimicry and innovations inspired by nature, which are actively being investigated by aerospace designers. Morphing wing designs based on Variable Geometry Truss Manipulators (VGTMs) and articulated helicopter robotic landing gear (RLG) have drawn a great deal of attention from industry. Compliant mechanisms have become increasingly popular due to their advantages over conventional rigid-body systems, and the research team led by the second author at Toronto Metropolitan University (TMU) has set their long-term goal to be exploiting these systems in the above aerospace applications. To gain a deeper insight into the design and optimization of compliant mechanisms and their potential application as alternatives to VGTM and RLG systems, this study conducted a thorough analysis of the design of flexible hinges, and single-, four-, and multi-bar configurations as a part of more complex, flexible mechanisms. The investigation highlighted the flexibility and compliance of mechanisms incorporating circular flexure hinges (CFHs), showcasing their capacity to withstand forces and moments. Despite a discrepancy between the results obtained from previously published Pseudo-Rigid-Body Model (PRBM) equations and FEM-based analyses, the mechanisms exhibited predictable linear behavior and acceptable fatigue testing results, affirming their suitability for diverse applications. While including additional linkages perpendicular to the applied force direction in a compliant mechanism with N vertical linkages led to improved factors of safety, the associated increase in system weight necessitates careful consideration. It is shown herein that, in this case, adding one vertical bar increased the safety factor by 100N percent. The present study also addressed solutions for the precise modeling of CFHs through the derivation of an empirical polynomial torsional stiffness/compliance equation related to geometric dimensions and material properties. The effectiveness of the presented empirical polynomial compliance equation was validated against FEA results, revealing a generally accurate prediction with an average error of 1.74%. It is expected that the present investigation will open new avenues to higher precision in the design of CFHs, ensuring reliability and efficiency in various practical applications, and enhancing the optimization design of compliant mechanisms comprised of such hinges. A specific focus was put on ABS plastic and aluminum alloy 7075, as they are the materials of choice for non-load-bearing and load-bearing structural components, respectively.

Two Actuation Methods for a Complete Morphing System Composed of a VGTM and a Compliant Parallel Mechanism

Abstract In this paper, a complete morphing system consisting of a variable geometry truss manipulator (VGTM) is presented that is fully covered by a flexible panel skin. Two approaches are studied for the morphing control. The first one is to have the VGTM act as a driving mechanism and the flexible panels as a passive system. In this case, the VGTM is composed of active members and passive lockable members. It is shown that the morphing system can reach the desired shapes through intermediate steps. The second method is to have the flexible panels act as drivers and the VGTM as a passive supporting structure. In this case, the VGTM is only composed of passive lockable members. The morphing system can also achieve the desired poses through several steps. The control strategies of the two methods are discussed along with kinematic analysis, a comparison study is conducted to show their pros and cons, two prototypes are fabricated, and experiments are carried out to verify the feasibility of two actuation methods.

Design and Analysis of a Multisegment Shape Morphing Mechanism

Abstract This article presents a novel design of a multisegment shape morphing mechanism that combines a lockable reconfigurable variable geometry truss manipulator (VGTM) with an active parallel compliant mechanism. The structure of the VGTM is in a parallel-serial structure, and its hyper-redundant degree-of-freedom (DOF) can be fully controlled by using two active flexible panels and some lockable joints. This mechanism is suitable for aerospace applications that require light and compact structure with high load-carrying ability as well as achieve multiple DOFs for large-scale shape deformation. To make shape morphing process simple and efficient, the mobility and topological configuration of the mechanism are analyzed first. Then, a control strategy combining the approximate motion mode and the exact motion mode is proposed. The kinematic models for different motion modes are established and solved analytically. It has been found that, under the exact motion mode, two approaches could be realized for the pose control under external loads for each segment. The one with the shorter moving path is selected in this article. Finally, a prototype was constructed to demonstrate the feasibility of this structure and to verify the proposed kinematic model.

Feedback passivity‐based control of discrete nonlinear systems with time‐delay for variable geometry truss manipulator

AbstractIn this paper, the feedback passivity‐based control of nonlinear discrete time‐delay systems for variable geometry truss manipulators is investigated. To determine an appropriate communication channel in the sense of feedback passivation, we first model the dynamics of the variable geometry truss manipulator as a generalized discrete nonlinear system with time‐delay. Then we further prove that when the infinite norm of estimated error is bounded, as long as there is a controller enables the closed‐loop system to be input‐strictly passive, there must be a deterministic equivalent controller to ensure that the system is stochastically quasi passive. After that, on the basis of the conclusion obtained, a more conclusive corollary is addressed for linear plants. Though passivity is a stricter condition than stability, feedback passivation does not impose more restrictions on estimate errors, and therefore does not require more communication channel information than mean square stability. Finally, we simplify the variable geometry truss dynamics to a linear plant to simulate to verify the validity of our method, and also compared the experimental results with the methods in the existing literature.

Inverse kinematics of asymmetric octahedral variable geometry truss manipulator with obstacle avoidance through inexact interior point optimization

This article studies the inverse kinematics for asymmetric octahedral variable geometry truss manipulator with obstacle avoidance. The inverse kinematics problem is cast as a nonconvex optimization that having quadratic objective function subject to quadratic constraints. This article uses an inexact interior point optimization to solve it, which is developed on the basis of the imprecise algorithm Ipopt. According to the particularity of our actual optimization problem, each iteration undergoes specific modifications so as to minimize the memory consumption as well as computation time. Utilizing the sparse and binary characteristics of the coefficient matrix, respectively, the algorithm allocates the computation to the finite sparse matrix vector multiplication and changes the storage form, which greatly reduces the memory space. Based on the unique rules of inverse kinematics, the iteration direction of the algorithm becomes more clear. With the aid of mechanical constraints inherent in the manipulator, the algorithm omits the feasibility recovery part that embedded in the solver Ipopt. All these make us save the operation time greatly while utilizing Ipopt algorithm. To demonstrate the effectiveness of the proposed approach, the scheme was applied to obstacle avoidance inverse kinematics of variable geometry truss manipulator with three modules.

Trajectory Tracking of a Planer Parallel Manipulator by Using Computed Force Control Method

Despite small workspace, parallel manipulators have some advantages over their serial counterparts in terms of higher speed, acceleration, rigidity, accuracy, manufacturing cost and payload. Accordingly, this type of manipulators can be used in many applications such as in high-speed machine tools, tuning machine for feeding, sensitive cutting, assembly and packaging. This paper presents a special type of planar parallel manipulator with three degrees of freedom. It is constructed as a variable geometry truss generally known planar Stewart platform. The reachable and orientation workspaces are obtained for this manipulator. The inverse kinematic analysis is solved for the trajectory tracking according to the redundancy and joint limit avoidance. Then, the dynamics model of the manipulator is established by using Virtual Work method. The simulations are performed to follow the given planar trajectories by using the dynamic equations of the variable geometry truss manipulator and computed force control method. In computed force control method, the feedback gain matrices for PD control are tuned with fixed matrices by trail end error and variable ones by means of optimization with genetic algorithm.

Forward kinematics analysis for a class of asymmetrical parallel manipulators

This article focuses on the forward kinematic analysis of a class of asymmetrical parallel manipulators by the proposed elimination approach. To solve the key forward kinematic constraint equations with transcendental parameters of the manipulator, an improved elimination algorithm is presented. First, by analyzing the geometry structure of the manipulator, we find the inherent triangular-topology relations of the manipulator. Further, by utilizing the parameter transformation of angular, the key transcendental equations of forward kinematic analysis are formulated into compact polynomial ones. In this context, comparing with the screw approach by Gallardo-Alvarado suggested that the computation efficiency of our proposed approach is superior. Finally, an example of the asymmetrical variable geometry truss manipulator illustrates the effectiveness of the proposed approach.

Dynamics and control of variable geometry truss manipulator

Variable geometry truss manipulator (VGTM) has potential to work in the future space applications, of which a dynamic model is important to dynamic analysis and control of the system. In this paper, an approach is presented to model the dynamic equations of a VGTM by independent variables, which consists of two double-octahedral truss units and a 3-revolute-prismatic-spherical (3-RPS) parallel manipulator. In this approach, the kinematic recursive relations of two adjacent bodies and geometric constrains are used to deduce the kinematic equations of the VGTM, and Jourdain’s velocity variation principle is adopted to establish the dynamic equations of the system. The validity of the proposed dynamic model is verified by comparison of numerical simulations with the software ADAMS. Besides, an active controller for trajectory tracking of the system is designed by the computed torque method. The effectiveness of the controller is numerically proved.

Inverse kinematics for the variable geometry truss manipulator via a Lagrangian dual method

This article studies the inverse kinematics problem of the variable geometry truss manipulator. The problem is cast as an optimization process which can be divided into two steps. Firstly, according to the information about the location of the end effector and fixed base, an optimal center curve and the corresponding distribution of the intermediate platforms along this center line are generated. This procedure is implemented by solving a non-convex optimization problem that has a quadratic objective function subject to quadratic constraints. Then, in accordance with the distribution of the intermediate platforms along the optimal center curve, all lengths of the actuators are calculated via the inverse kinematics of each variable geometry truss module. Hence, the approach that we present is an optimization procedure that attempts to generate the optimal intermediate platform distribution along the optimal central curve, while the performance index and kinematic constraints are satisfied. By using the Lagrangian duality theory, a closed-form optimal solution of the original optimization is given. The numerical simulation substantiates the effectiveness of the introduced approach.

Statically redundant parallel robots

Presented in this paper is a method for the development of variable geometry truss manipulators (VGTMs) with enhanced static characteristics. The concept of statically redundant parallel robots is introduced and used to design the VGTM module through under-actuation. Under this design, each module is over-constrained, composed of both active and passive limbs. To enhance the stiffness, the passive limbs are designed using lockable prismatic joints. It is shown that through proper sequencing of the locking and unlocking of these passive limbs in correspondence to actuated motion, system rigidity can be maintained in the course of reconfiguration. In this paper, the proposed method is first explained using a planar system, then it is shown how to apply this method to the design of the variable geometry wing-box for an aircraft morphing wing.

Fast Trajectory Planning for VGT Manipulator via Convex Optimization

Variable-geometry truss (VGT) can be used as joints in a large, lightweight, high load-bearing manipulator for many industrial applications. This paper introduces the modelling process of a multisection three-degree-freedom double octahedral VGT manipulator and proposes a new point-to-point trajectory planning algorithm for a VGT manipulator with a nonlinear convex optimization approach. The trajectory planning problem is converted to a quadratic convex optimization problem, and if some parameters are appropriately chosen the proposed algorithm is proved to be of global convergence and to have a super-linear convergence rate. The effectiveness of the proposed algorithm is illustrated by numerical simulation.

Analytical model method for dynamics of N-celled tetrahedron–tetrahedron variable geometry truss manipulators

In this paper some fundamental formulae are derived for a tetrahedron-based variable geometry truss manipulator which is composed of a series of tetrahedrons stacked upon each other such that one link in each cell is made variable in length. Analytical model for dynamics of the manipulator is established, and expressions in numeric–symbolic form of model matrices are derived. An example is given for illustration.

Inverse kinematic analysis for triple-octahedron variable-geometry truss manipulators

In this paper, a new triple-octahedron variable-geometry truss manipulator is presented. Its inverse kinematic solutions in closed form are studied. An input-output displacement equation in one output variable is derived. The solution procedure is given in detail. A numerical example is illustrated.

Velocity analysis of parallel manipulators by truss transformations

A method of velocity analysis of general parallel manipulators is presented. The method consists of transforming the parallel manipulator to an equivalent variable-geometry truss (VGT), and then generating the set of linear velocity equations that govern the transformed manipulator. VGTs are a special class of parallel manipulators. Their actuation is provided by extensible links terminating in spheric joints. A method for velocity analysis of VGT manipulators will be briefly reviewed. This method employs a connectivity chart to generate a system of linear algebraic equations for the manipulator velocity. In addition to the spheric joints and two-force member links that make up truss actuators, general parallel manipulators commonly contain revolute, prismatic, and cylindric joints and multi-force links, links that are more than two-force members. Further, a number of actuation schemes are possible. This work shows how such general parallel manipulators can be conceptually transformed into equivalent VGTs and thereby allow a straightforward velocity analysis approach. While this approach is applicable to general parallel manipulators, the method is most effective when applied to parallel manipulators with truss-like structures.

Dynamics of closed linked variable geometry truss manipulators

This paper presents dynamical characteristics of a variable geometry truss (VGT) manipulator system which consists of a two-dimensional statically determinate truss for space applications. Formulation takes into account geometrical effects of the closed-link constraints, variable length mechanisms, rotational degrees of freedom at the joints and internal control forces; and is developed by Kane's dynamical equations. The conventional formulae of tree-like truss structures are extended to the procedure, which is adapted to statically determine truss structures using a Jacobian matrix derived by geometrical considerations. Inverse kinetics of the variable geometry truss have been formulated considering the minimum norm solution to the equations of motion. A two-dimensional variable geometry truss manipulator is simulated numerically as an example of activities on Space Station. Results show the effects of internal control forces on the attitude of the manipulator system in a space environment, and the characteristics of the inverse kinetics of the manipulator are investigated.

Forward displacement analysis of the decahedral variable geometry truss manipulator

A forward displacement analysis in closed form is performed for the decahedral variable geometry truss manipulator. Based on the simplified geometry model, a system of the constraint equations are formulated. With elimination, this system of equations can be reduced to a 48th degree polynomial equation in one output variable.

The Inverse Kinematic Solution of the Tetrahedron Based Variable-Geometry Truss Manipulator

The Tetrahedron Based Variable Geometry Truss Manipulator is comprised of a series of tetrahedrons stacked upon one another in a spiraling manner. When one link of each “cell” is made variable in length, the system can be effectively used as a remote manipulator. The manipulator is kinematically equivalent to a serial link manipulator consisting only of revolute joints where the angle between successive axes is 120°. In this paper, a closed-form analytic solution for the six-celled (6R series) variable geometry truss manipulator is obtained. It is shown that a sixteenth order polynomial in one of the joint angles is found which leads to at most sixteen solutions.

Closed-Form Inverse Kinematic Analysis of Variable-Geometry Truss Manipulators

A variety of applications for variable-geometry truss manipulators (VGTMs) have been demonstrated or proposed in the literature. Most of these applications require solution to the inverse kinematic problem, yet only a few isolated examples of closed-form solution methods have been presented to date. This paper provides an overview to the general problem of inverse kinematic analysis of variable-geometry truss manipulators and presents new closed-form solution techniques for problems of practical importance.

Shape Synthesis and Optimization Using Intrinsic Geometry

The present paper outlines a method of shape synthesis using intrinsic geometry to be used for two-dimensional shape optimization problems. It is observed that the shape of a curve can be defined in terms of intrinsic parameters such as the curvature as a function of the arc lenght. The method of shape synthesis, proposed here, consists of selecting a shape model, defining a set of shape design variables and then evaluating Cartesian coordinates of a curve. A shape model is conceived as a set of continuous piecewise linear segments of the curvature; each segment defined as a function of the arc length. The shape design variables are the values of curvature and/or arc lengths at some of the end-points of the linear segments. The proposed method of shape synthesis and optimization is general in nature. It has been shown how the proposed method can be used to find the optimal shape of a planar Variable Geometry Truss (VGT) manipulator for a prespecified position and orientation of the end-effector.

Inverse kinematics of variable‐geometry truss manipulators

AbstractA new class of robotic arm consists of a periodic sequence of truss substructures, each of which has several variable‐length members. Such variable‐geometry truss manipulators (VGTMs) are inherently highly redundant and promise a significant increase in dexterity over conventional anthropomorphic manipulators. This dexterity may be exploited for both obstacle avoidance and controlled deployment in complex workspaces. The inverse kinematics problem for such unorthodox manipulators, however, becomes complex because of the large number of degrees of freedom, and conventional solutions to the inverse kinematics problem become inefficient because of the high degree of redundancy. This paper presents a solution to this problem based on a spline‐like reference curve for the manipulator's shape. Such an approach has a number of advantages: (1) direct, intuitive manipulation of shape; (2) reduced calculation time; and (3) direct control over the effective degree of redundancy of the manipulator. Furthermore, although the algorithm has been developed primarily for variable‐geometry‐truss manipulators, it is general enough for application to other manipulator designs.