Modal Kinematic Analysis of a Parallel Kinematic Robot with Low-Stiffness Transmissions

Several industrial robotic applications that require high speed or high stiffness-to-inertia ratios use parallel kinematic robots. In the cases where the critical point of the application is the speed, the compliance of the main mechanical transmissions placed between the actuators and the parallel kinematic structure can be significantly higher than that of the parallel kinematic structure itself. This paper deals with this kind of system, where the overall performance depends on the maximum speed and on the dynamic behavior. Our research proposes a new approach for the investigation of the modes of vibration of the end-effector placed on the robot structure for a system where the transmission’s compliance is not negligible in relation to the flexibility of the parallel kinematic structure. The approach considers the kinematic and dynamic coupling due to the parallel kinematic structure, the system’s mass distribution and the transmission’s stiffness. In the literature, several papers deal with the dynamic vibration analysis of parallel robots. Some of these also consider the transmissions between the motors and the actuated joints. However, these works mainly deal with the modal analysis of the robot’s mechanical structure or the displacement analysis of the transmission’s effects on the positioning error of the end-effector. The discussion of the proposed approach takes into consideration a linear delta robot. The results show that the system’s natural frequencies and the directions of the end-effector’s modal displacements strongly depend on its position in the working space.

A Constraint-Based Approach to Manipulator Kinematics and Singularities

<p>In this thesis, a differential-geometric approach to the kinematics of multibody mechanisms is introduced that enables analysis of singularities of both serial and parallel manipulators in a flexible and complete way. Existing approaches such as those of Gosselin and Angeles [1], Zlatanov et al. [2] and Park and Kim [3] make use of a combination of joint freedoms and constraints and so build in assumptions. In contrast, this new approach is solely constraint-based, avoiding some of the shortcomings of these earlier theories. The proposed representation has two core ingredients. First, it avoids direct reference to the choice of inputs and their associated joint freedoms and instead focuses on a kinematic constraint map (KCM), defined by the constraints imposed by all joints and not requiring consideration of closure conditions arising from closed loops in the design. The KCM is expressed in terms of pose (i.e. position and orientation) variables, which are the coordinates of all the manipulator’s links with respect to a reference frame. The kinematics of a given manipulator can be described by means of this representation, locally and globally. Also, for a family of manipulators defined by a specific architecture, the KCM will tell us how the choice of design parameters (e.g. link lengths) affects these kinematic properties within the family. At a global level, the KCM determines a subset in the space of all pose variables, known as the configuration space (C-space) of the manipulator, whose topology may vary across the set of design parameters. The Jacobian (matrix of first-order partial derivatives) of the KCM may become singular at some specific choices of pose variables. These conditions express a subset called the singular set of the C-space. It is shown that if a family of manipulators, parametrised by a manifold Rd of design parameters, is “well-behaved” then the pose variables can be eliminated from the KCM equations together with the conditions for singularities, to give conditions in terms of design parameters, that define a hypersurface in Rd of manipulators in the class that exhibit C-space singularities. These are referred to as Grashof-type conditions, as they generalise classically known inequalities classifying planar 4-bar mechanisms due to Grashof [4]. Secondly, we develop the theory to incorporate actuator space (A-space) and workspace (W-space), based on a choice of actuated joints or inputs and on the manipulator’s end-effector workspace or outputs. This will facilitate us with a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator’s C-space. This provides new insight into the structure of the forward and inverse kinematics, especially for parallel manipulators. The theory is illustrated by a number of applications, some of which recapitulate classical or known results and some of which are new.</p>

A Constraint-Based Approach to Manipulator Kinematics and Singularities

<p>In this thesis, a differential-geometric approach to the kinematics of multibody mechanisms is introduced that enables analysis of singularities of both serial and parallel manipulators in a flexible and complete way. Existing approaches such as those of Gosselin and Angeles [1], Zlatanov et al. [2] and Park and Kim [3] make use of a combination of joint freedoms and constraints and so build in assumptions. In contrast, this new approach is solely constraint-based, avoiding some of the shortcomings of these earlier theories. The proposed representation has two core ingredients. First, it avoids direct reference to the choice of inputs and their associated joint freedoms and instead focuses on a kinematic constraint map (KCM), defined by the constraints imposed by all joints and not requiring consideration of closure conditions arising from closed loops in the design. The KCM is expressed in terms of pose (i.e. position and orientation) variables, which are the coordinates of all the manipulator’s links with respect to a reference frame. The kinematics of a given manipulator can be described by means of this representation, locally and globally. Also, for a family of manipulators defined by a specific architecture, the KCM will tell us how the choice of design parameters (e.g. link lengths) affects these kinematic properties within the family. At a global level, the KCM determines a subset in the space of all pose variables, known as the configuration space (C-space) of the manipulator, whose topology may vary across the set of design parameters. The Jacobian (matrix of first-order partial derivatives) of the KCM may become singular at some specific choices of pose variables. These conditions express a subset called the singular set of the C-space. It is shown that if a family of manipulators, parametrised by a manifold Rd of design parameters, is “well-behaved” then the pose variables can be eliminated from the KCM equations together with the conditions for singularities, to give conditions in terms of design parameters, that define a hypersurface in Rd of manipulators in the class that exhibit C-space singularities. These are referred to as Grashof-type conditions, as they generalise classically known inequalities classifying planar 4-bar mechanisms due to Grashof [4]. Secondly, we develop the theory to incorporate actuator space (A-space) and workspace (W-space), based on a choice of actuated joints or inputs and on the manipulator’s end-effector workspace or outputs. This will facilitate us with a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator’s C-space. This provides new insight into the structure of the forward and inverse kinematics, especially for parallel manipulators. The theory is illustrated by a number of applications, some of which recapitulate classical or known results and some of which are new.</p>

Kinematics Analysis and Singularity Avoidance of a Parallel Mechanism with kinematic redundancy

The kinematic redundancy is considered as a way to improve the performance of parallel mechanism. In this paper, the kinematics performance of a three degree-of-freedom parallel mechanism with kinematic redundancy (3-DOF PM-KR) and the influence of redundant part on the PM-KR are analyzed. Firstly, the kinematics model of the PM-KR is established. The inverse solutions, the Jacobian matrix and the workspace of the PM-KR are solved. Secondly, the influence of the redundant redundancy on the PM-KR has been analyzed. Since there exists kinematic redundancy, the PM-KR possesses the fault-tolerant performance. By locking one actuated joint or two actuated joints simultaneously, the fault-tolerant workspace are obtained. When the position of the redundant part is changed, the workspace and singularity will be changed. The results show that the kinematic redundancy can be used to avoid the singularity. Finally, the simulations are performed to prove the theoretical analysis.

Design of Fin-Curvature-Based Feedback Controller for Efficient Swimming

Over the past few decades, biologists and engineers have attempted to elucidate the swimming mechanism of fish and developed a fish-like robot to perform fast swimming in water. Such a robot will have wide applicability in investigations and exploration in the sea. There have been many studies on fish-type robots; however, the propulsion efficiency of the introduced robots is far from that of the actual fish. The main reason is that the robot controller for generating motions is conventionally designed by trial and error, and little attention has been placed on designing a motion controller that matches the body structure of a real fish. In this paper, we present an approach that uses fin-curvature-based feedback to design a motion controller. A swimming robot composed of a body with two actuated joints and a flexible tail fin is developed. After examining the relationship between the swimming speed and tail fin curvature in feedforward (open-loop) system experiments, we propose to reflect the tail fin curvature to the actuation inputs (phase difference between the two cyclic oscillations), which will perform the efficient swimming motion. Further, the results show that implementing the proposed feedback controller in a fish-type robot makes it swim similar to a real fish. In addition, the proposed controller functions to find inappropriate actuation according to the body structure.

Study on operational space control of a redundant robot with un-actuated joints: experiments under actuation failure scenarios

This paper analyzes operational space dynamics for redundant robots with un-actuated joints and reveals their highly nonlinear dynamic impacts on operational space control (OSC) tasks. Unlike conventional OSC approaches that partly address the under-actuated system by introducing rigid grasping or contact constraints, we deal with the problem even without such physical constraints which have been overlooked, yet it includes a wide range of applications such as free-floating robots and manipulators with passive joints or unwanted actuation failure. In addition, as an intuitive application example of the drawn result, an OSC is formulated as an optimization problem to alleviate the dynamics disturbance stemmed from the un-actuated joints and to satisfy other inequality constraints. The dynamic analysis and the proposed control method are verified by a number of numerical simulations as well as physical experiments with a 7-degrees-of-freedom robotic arm. In particular, we consider joint actuation failure scenarios that can be occurred at certain joints of a torque-controlled robot and practical case studies are performed with an actual redundant robot arm.

Direct Position Analysis of a Particular Translational 3-URU Manipulator

Direct position analysis (DPA) of parallel manipulators (PMs) is in general difficult to solve. Over on PMs' topology, DPA complexity depends on the choice of the actuated joints. From an analytic point of view, the system of algebraic equations that one must solve to implement PMs' DPA is usually expressible in an apparently simple form, but such a form does not allow an analytic solution and even the problem formalization is relevant in PMs' DPAs. The ample literature on the DPA of Stewart platforms well document this point. This paper addresses the DPA of a particular translational PM of 3-URU type, which has the actuators on the frame while the actuated joints are not adjacent to the frame. The problem formulation brings to a closure-equation system consisting of three irrational equations in three unknowns. Such a system is transformed into an algebraic system of four quadratic equations in four unknowns that yields a univariate irrational equation in one of the four unknowns and three explicit expressions of the remaining three unknowns. Then, an algorithm is proposed which is able to find only the real solutions of the DPA. The proposed solution technique can be applied to other DPAs reducible to a similar system of irrational equations and, as far as this author is aware, is novel. Keywords: Kinematics, Position Analysis, Parallel Manipulators, Lower Mobility, Translational Manipulators

Simulation of Upward Jump Control for One-Legged Robot Based on QP Optimization

An optimization framework for upward jumping motion based on quadratic programming (QP) is proposed in this paper, which can simultaneously consider constraints such as the zero moment point (ZMP), limitation of angular accelerations, and anti-slippage. Our approach comprises two parts: the trajectory generation and real-time control. In the trajectory generation for the launch phase, we discretize the continuous trajectories and assume that the accelerations between the two sampling intervals are constant and transcribe the problem into a nonlinear optimization problem. In the real-time control of the stance phase, the over-constrained control objectives such as the tracking of the center of moment (CoM), angle, and angular momentum, and constraints such as the anti-slippage, ZMP, and limitation of joint acceleration are unified within a framework based on QP optimization. Input angles of the actuated joints are thus obtained through a simple iteration. The simulation result reveals that a successful upward jump to a height of 16.4 cm was achieved, which confirms that the controller fully satisfies all constraints and achieves the control objectives.

A Novel 3-URU Architecture with Actuators on the Base: Kinematics and Singularity Analysis

Translational parallel manipulators (TPMs) with DELTA-like architectures are the most known and affirmed ones, even though many other TPM architectures have been proposed and studied in the literature. In a recent patent application, this author has presented a TPM with three equal limbs of Universal-Revolute-Universal (URU) type, with only one actuated joint per limb, which has overall size and characteristics similar to DELTA robots. The presented translational 3-URU architecture is different from other 3-URUs, proposed in the literature, since it has the actuators on the frame (base) even though the actuated joints are not on the base, and it features a particular geometry. Choosing the geometry and the actuated joints highly affects 3-URU’s behavior. Moreover, putting the actuators on the base allows a substantial reduction of the mobile masses, thus promising good dynamic performances, and makes the remaining part of the limb a simple chain constituted by only passive R-pairs. The paper addresses the kinematics and the singularity analysis of this novel TPM and proves the effectiveness of the new design choices. The results presented here form the technical basis for the above-mentioned patent application.