Closed Kinematic Chains Research Articles

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Maximal Loop Method of Automatic Sketching of Planar Closed Kinematic Chains

Automatic sketching of mechanism kinematic chains affects directly the efficiency and reliability of mechanism design. On the basis of a study of the topologic characteristics of planar closed kinematic chains, the maximal loop method is presented, based on contracted link adjacency matrix, to complete automatic sketching of planar closed kinematic chains. Independent loops of kinematic chains are derived by using the breadth-first spanning tree algorithm, based on which the maximal feasible loop is achieved by means of independent loops add-subtract operation. The maximal feasible loop is step by step modified in accordance with the loop relationships so that all independent loops become its normalized non-crossing inner loops. In the process of automatic sketching of kinematic chains, the joints of kinematic chain are located, from outer lane to inner lane, on the vertices of inscribed regular polygon of the concentric circles on the basis of the outer and inner loops’ relationships. As a result, the sketches of the kinematic chains appear to be normal and beautiful. The development process of this algorithm based on the outer loop and the inner loops relationships is demonstrated with the aid of several mechanism examples. The study and the application of the maximal loop method show that it is simple and comprehensive. And it is applicable for the automatic sketching of planar closed kinematic chains with multiple links and multiple degrees of freedom. It can greatly raise the efficiency in mechanism design.

Variational integrators for constrained dynamical systems

AbstractA variational formulation of constrained dynamics is presented in the continuous and in the discrete setting. The existing theory on variational integration of constrained problems is extended by aspects on the initialization of simulations, the discrete Legendre transform and certain postprocessing steps. Furthermore, the discrete null space method which has been introduced in the framework of energy‐momentum conserving integration of constrained systems is adapted to the framework of variational integrators. It eliminates the constraint forces (including the Lagrange multipliers) from the time‐stepping scheme and subsequently reduces its dimension to the minimal possible number. While retaining the structure preserving properties of the specific integrator, the solution of the smaller dimensional system saves computational costs and does not suffer from conditioning problems. The performance of the variational discrete null space method is illustrated by numerical examples dealing with mass point systems, a closed kinematic chain of rigid bodies and flexible multibody dynamics and the solutions are compared to those obtained by an energy‐momentum scheme.

Gait Optimization of Biped Robot during Double Support Phase by Pure Dynamic Synthesis

This paper deals with dynamic optimization of biped locomotion. The main focus of this research is motion optimization of double support phase. The optimization problem is dealt by using Pontryagins Maximum Principal. For motion optimization of double support phase, the closed kinematic chain has been considered to be opened at appropriate joint and the components of ground reaction forces has been applied on the tip of front leg and finally the penalty method has been used to tighten the leg to its prescribed location. The feasible sets of motion are taken into consideration by using inequality constraint to limit the joint motion. Also the components of ground reaction forces on front leg have been introduced as control variables in optimization of double support phase. The proposed technique has the ability to generate optimal free motions without specifying joint trajectories and minimized the performance criterion based on joint actuating torques. The two point boundary value problem has been solved by implementing a shooting method. This technique allows for specifying a few parameters to characterize gait pattern. The optimization process has the ability to generate a motion with a minimum of postural and kinematics data. Unlike previous research which used computational intelligent techniques for biped gait optimization, this study focuses on development of purely dynamic synthesis of biped motion during the double support phase.

Self-Collision Detection in Spatial Closed Chains

A novel methodology for detecting self-collisions in spatial closed kinematic chains is presented. In general these chains generate complex three dimensional motions in which their own links will collide with each other (i.e., a self-collision) without effective motion planning. The self-collision detection is accomplished via a novel algorithm for definitively detecting collisions of right circular, cylindrically shaped, rigid bodies moving in three dimensions. The algorithm uses line geometry and dual number algebra to exploit the geometry of right circular cylindrical objects to facilitate the detection of collisions. In the first stage of the algorithm, cylindrically shaped rigid bodies are modeled by infinite length right circular cylinders. Sufficient and necessary conditions are then used to determine if a pair of infinite length cylinders collide. If the actual finite length rigid bodies collide, then it is necessary that their associate infinite length cylinder models collide, and we proceed to the next stage of the algorithm where the bodies are modeled with finite length cylinders and a definitive necessary and sufficient collision detection algorithm is employed. The result is an efficient approach of detecting collisions of cylindrically shaped bodies moving in three dimensions that has applications in spatial mechanism design and motion planning. A case study examining a spatial 4C mechanism for self-collisions is included.

No Need for a Body Model: Positive Velocity Feedback for the Control of an 18-DOF Robot Walker

In a multilegged walking robot several legs usually have ground contact and thereby form a closed kinematic chain. The control of such a system is generally assumed to require the explicit calculation of the body kinematics. Such a computation requires knowledge concerning all relevant joint angles as well as the segment lengths. Here, we propose a biologically inspired solution that does not need such a body model. This is done by using implicit communication through the body mechanics (embodiment) and a local positive velocity feedback strategy (LPVF) on the single joint level. In this control scheme the locally measured joint velocity of an elastic joint is fed into the same joint during the next time step to maintain the movement. At the same time, an additional part of this joint controller observes the mechanical joint power to confine the positive feedback. This solution does not depend on changes of the geometry, e.g. length of individual segments, and allows for a simple solution of negotiation of curves. The principle is tested in a dynamics simulation on a six-legged walker and, for the first time, also on a real robot.

A General-Purpose 7 DOF Haptic Device: Applications Toward Robot-Assisted Surgery

A 7 DOF haptic device has been designed and developed with applications towards robot-assisted minimally invasive surgery. The device consists of four degrees of force feedback (X, Y, Z, and grasping) capability and seven degrees of position feedback capability. It has a closed kinematic chain with two halves (user interface and spatial mechanism) that connect together via a universal joint. The user interface contains four degrees of position feedback, namely, the roll, pitch, yaw, and linear motion of the hand and forearm. In addition, a grasping mechanism with two thimbles mounted at the end of the user interface provides force feedback to the fingers of the user. The spatial mechanism provides force feedback to the user interface through a universal joint located at the grasping mechanism. This paper presents the design and development of this haptic device. In addition, a kinematic and workspace analysis of the device has been completed to compute the position of the slave robot and end-effector tool. Friction estimation has been presented to enable a higher transparency of the haptic device. Finally, a simulation of needle insertion into soft tissue was developed to test the device.

A unifying framework for robot control with redundant DOFs

Recently, Udwadia (Proc. R. Soc. Lond. A 2003:1783–1800, 2003) suggested to derive tracking controllers for mechanical systems with redundant degrees-of-freedom (DOFs) using a generalization of Gauss’ principle of least constraint. This method allows reformulating control problems as a special class of optimal controllers. In this paper, we take this line of reasoning one step further and demonstrate that several well-known and also novel nonlinear robot control laws can be derived from this generic methodology. We show experimental verifications on a Sarcos Master Arm robot for some of the derived controllers. The suggested approach offers a promising unification and simplification of nonlinear control law design for robots obeying rigid body dynamics equations, both with or without external constraints, with over-actuation or underactuation, as well as open-chain and closed-chain kinematics.

Comparison of the Effects of Open and Closed Kinematic Chain and Different Target Position on the Knee Joint Position Sense

Comparison of the Effects of Open and Closed Kinematic Chain and Different Target Position on the Knee Joint Position Sense

Application of joint coordinates and homogeneous transformations to modeling of vehicle dynamics

The paper presents a method of modeling dynamics of multibody systems with open and closed kinematic chains. The joint coordinates and homogeneous transformations are applied in order to formulate the equations of motion of a rigid body. In this method, constraint equations are introduced only in the case when closed subchains are considered or when the joint reactions have to be calculated. This allows the number of generalized coordinates in the system to be reduced in comparison to the case when absolute coordinates are applied. It is shown how the method can be applied to modeling of vehicle dynamics. The calculation results are compared with those obtained when the ADAMS/Car package is used. Experimental verification has been performed and is reported in the paper, as well. In both cases, a good correspondence of results has been achieved. Final remarks concerning advantages and disadvantages of the method are formulated at the end of the paper.

Impulse‐based dynamic simulation in linear time

AbstractThis paper describes an impulse‐based dynamic simulation method for articulated bodies which has a linear time complexity. Existing linear‐time methods are either based on a reduced‐coordinate formulation or on Lagrange multipliers. The impulse‐based simulation has advantages over these well‐known methods. Unlike reduced‐coordinate methods, it handles nonholonomic constraints like velocity‐dependent ones and is very easy to implement. In contrast to Lagrange multiplier methods the impulse‐based approach has no drift problem and an additional stabilisation is not necessary. The presented method computes a simulation step in O(n) time for acyclic multi‐body systems containing equality constraints. Closed kinematic chains can be handled by dividing the model into different acyclic parts. Each of these parts is solved independently from each other. The dependencies between the single parts are solved by an iterative method. In the same way inequality constraints can be integrated in the simulation process in order to handle collisions and permanent contacts with dynamic and static friction. Copyright © 2007 John Wiley & Sons, Ltd.

The Hamel Representation: A Diagonalized Poincaré Form

The Poincaré equations, also known as Lagrange’s equations in quasicoordinates, are revisited with special attention focused on a diagonal form. The diagonal form stems from a special choice of generalized speeds that were first introduced by Hamel (Hamel, G., 1967, Theorctische Mechanik, Springer-Verlag, Berlin, Secs. 235 and 236) nearly a century ago. The form has been largely ignored because the generalized speeds create so-called Hamel coefficients that appear in the governing equations and are based on the partial derivative of a mass-matrix factorization. Consequently, closed-form expressions for the Hamel coefficients can be difficult to obtain. In this paper, a newly developed operator overloading technique is used within a simulation code to automatically generate the Hamel coefficients through an exact partial differentiation together with a numerical evaluation. This allows the diagonal form of Poincaré’s equations to be numerically integrated for system simulation. The diagonal form and the techniques used to generate the Hamel coefficients are applicable to general systems, including systems with closed kinematic chains. Because of Hamel’s original influence, these special Poincaré equations are called the Hamel representations and their usefulness in dynamic simulation and control is investigated.

Decentralized Control of Elastic Limbs in Closed Kinematic Chains

The generation of movements in closed kinematic chains as opposed to open kinematic chains is a challenging task because all participating joints have to be moved in a highly coordinated manner in order to avoid destructive tensions in the limb. In this paper we present a new decentral joint controller which uses low level interactions between a moving joint and its environment consisting of neighboring joints, the body and the surroundings the agent is placed in. This local joint controller is based on a Local Positive Velocity Feedback (LPVF) mechanism which exploits the elastic properties of the joint. The control strategy is inspired by biological findings in the walking system of stick insects. We will show that a closed kinematic chain consisting of several LPVF controlled joints, though lacking a central controller, can solve tasks which need a high level of inter-joint coordination. As an example, a planar manipulator turning a crank is presented. In a further step, the LPVF algorithm is extended to switched LPVF in order to improve the mechanical power conversion. The extended capabilities are proven in a second experiment in which a 3DoF test leg generates powerful stance movements.

Virtual kinematic chains to solve the underwater vehicle-manipulator systems redundancy

This paper addresses the kinematics of the Underwater Vehicle-Manipulator Systems (UVMSs). Due the adittional degrees of freedom (dofs) provided by the vehicle, such systems are kinematically redundant, i.e. they possess more dofs than those required to execute a given task, and need to be solved using some redundancy technique. We present an approach based on introducing kinematic constraints. The approach uses the screw representation of movement and is based on the so-called Davies method used to solve the kinematics of closed kinematic chains. We describe the vehicle-manipulator system as an open-loop chain and present a virtual kinematic chain concept that allows closing this open chain. Applying the Davies method to this resulting closed chain, the UVMS direct kinematic is solved. The inverse kinematics is obtained using the same approach, by introducing extra constraints derived from energy savings requirements. The proposed approach is compared to another redundancy resolution method to illustrate the ability of the proposed strategy.

Adaptive control of holonomic constrained systems: a feedforward fuzzy approximation-based approach

This paper proposes a novel adaptive fuzzy control scheme for the motion/force tracking control of holonomic constrained systems with poorly understood models and disturbances. Some disadvantages of traditional adaptive fuzzy controllers are removed here. In comparison to typical state-feedback fuzzy approximation, the uncertainties are compensated based on a feedforward fuzzy approximation (FFA), which takes desired commands as the premise variables of fuzzy rules. In detail, a unified control model is introduced for representing well-known holonomic systems with an environmental constraint or a set of closed kinematic chains. Then, the FFA-based fuzzy system, adaptation mechanism, and auxiliary-compensating control are derived to ensure robust motion and force tracking in a global manner. Furthermore, a feasible solution for the derived linear matrix inequality guarantees the attenuation of both disturbances and fuzzy parameter errors in an L/sub 2/-gain sense. Finally, two applications are carried out on: 1) a two-link constrained robot and 2) two planar robots transporting a common object. Numerical simulation results show the expected performance.

Animation planning for virtual characters cooperation

This paper presents an approach to automatically compute animations for virtual (human-like and robot) characters cooperating to move bulky objects in cluttered environments. The main challenge is to deal with 3D collision avoidance while preserving the believability of the agent's behaviors. To accomplish the coordinated task, a geometric and kinematic decoupling of the system is proposed. This decomposition enables us to plan a collision-free path for a reduced system, then to animate locomotion and grasping behaviors independently, and finally to automatically tune the animation to avoid residual collisions. These three steps are applied consecutively to synthesize an animation. The different techniques used, such as probabilistic path planning, locomotion controllers, inverse kinematics and path planning for closed kinematic chains are explained, and the way to integrate them into a single scheme is described.

A Biologically Inspired Active Compliant Joint Using Local Positive Velocity Feedback (LPVF)

Starting from studies which revealed that positive feedback is found in the control system for walking in arthropods, we have constructed a new positive feedback driven joint that can be used for solving compliant motion tasks. We propose two different joint constructions each of which shows passive compliance. Based on these joints we introduce three different local positive velocity feedback (LPVF) controllers and discuss their properties in the context of motion generation in closed kinematic chains. The third circuit named undelayed dLPVF is used for the control of a compliant planar manipulator which turns a crank. Our concept is of highly decentralized nature and follows the idea of embodiment. In our case this means that a process which is controlled by LPVF controllers reveals its nature when the controllers interact with this process.

Control of Closed Kinematic Chains Using A Singularly Perturbed Dynamics Model

In this paper, we propose a novel approach to the control of closed kinematic chains (CKCs). This method is based on a recently developed singularly perturbed model for CKCs. Conventionally, the dynamics of CKCs are described by differential-algebraic equations (DAEs). Our approach transfers the control of the original DAE system to the control of an artificially created singularly perturbed system in which the slow dynamics corresponds to the original DAE when the perturbation parameter tends to zero. Compared to control schemes that rely on solving nonlinear algebraic constraint equations, the proposed method uses an ordinary differential equation (ODE) solver to obtain the dependent coordinates, hence, eliminates the need for Newton-type iterations and is amenable to real-time implementation. The composite Lyapunov function method is used to show that the closed-loop system, when controlled by typical open kinematic chain schemes, achieves asymptotic trajectory tracking. Simulations and experimental results on a parallel robot, the Rice planar Delta robot, are also presented to illustrate the efficacy of our method.

How to Reach the Dynamic Limits of Parallel Robots? An Autonomous Control Approach

Based on closed kinematic chains, parallel robots obtain favorable dynamic properties as well as high stiffness. Hence, their application can significantly enlarge the productivity of automated production processes. A control concept for tapping the high potential concerning low cycle times and high path-tracking accuracy is presented. The proposed approach adapts autonomously to changing dynamic parameters as varying payload. The autonomous behavior is achieved by combining an adaptive control approach with an adaptive, time-optimal trajectory planning concept and an online-trajectory adaption mechanism. Extensive experimental results prove the performance of the proposed approach. Note to Practitioners -Many applications in the field of production automation (material handling, assembly, etc.) require high operating speeds and accelerations. During the past years, parallel robots proved to be an efficient and suitable supplement to serial robots. Unfortunately, the promising possibilities of parallel robots often cannot yield profit because their dynamic potential is still not fully exploited. The payload/robot mass ratio of parallel structures is even higher compared to serial robots, where the influence of the payload on the impedance of the robot is negligible. By use of direct drives the influence of a variable payload cannot be ignored. A modified adaptive control concept, which adapts autonomously to changing dynamic parameters-as varying payload due to diversity of assembly processes-guarantees high tracking accuracy and therefore better process quality as well as accurate estimates of changing dynamic parameters and therefore better process quality. In addition the productivity of the process can be enlarged, if the full drive power can be used at each point on the path. Thus, a new adaptive time-optimal trajectory planning algorithm is used to exploit the dynamic potential of the direct drives and consequently to shorten the cycle times. The aim of time-optimal trajectory planning, as it is commonly understood, is the determination of the maximum velocity profile along a given path that complies with all given dynamic and kinematic robot constraints like limited drive forces/torques, limited path and/or drive velocities and limited path jerk. Combining the adaptive control scheme and the adaptive, time-optimal trajectory planning algorithm with an online trajectory adaption mechanism, a control concept is realized, which autonomously adapts to changing dynamic robot behavior. Using this new approach, the advantages of parallel robots-as well as serial robots with direct drives-can better be utilized. This is a necessary prerequisite for a larger extension of PKMs for industrial applications.

Design and analysis of kinematically redundant parallel manipulators with configurable platforms

Redundancy can, in general, improve the ability and performance of parallel manipulators by implementing the redundant degrees of freedom to optimize a secondary objective function. Almost all published researches in the area of parallel manipulators redundancy were focused on the design and analysis of redundant parallel manipulators with rigid (nonconfigurable) platforms and on grasping hands to be attached to the platforms. Conventional grippers usually are not appropriate to grasp irregular or large objects. Very few studies focused on the idea of using a configurable platform as a grasping device. This paper highlights the idea of using configurable platforms in both planar and spatial redundant parallel manipulators, and generalizes their analysis. The configurable platform is actually a closed kinematic chain of mobility equal to the degree of redundancy of the manipulator. The additional redundant degrees of freedom are used in reconfiguring the shape of the platform itself. Several designs of kinematically redundant planar and spatial parallel manipulators with configurable platform are presented. Such designs can be used as a grasping device especially for irregular or large objects or even as a micro-positioning device after grasping the object. Screw algebra is used to develop a general framework that can be adapted to analyze the kinematics of any general-geometry planar or spatial kinematically redundant parallel manipulator with configurable platform.