Redundant Control of a Planar Snake Robot with Prismatic Joints

This paper presents a control method of a planar snake robot with prismatic joints. The kinematic model is derived considering velocity constraints caused by passive wheels. The proposed control method based on the model allows the robot to track a target trajectory by appropriately changing its link length using prismatic joints. The degrees of freedom of prismatic joints are represented as kinematic redundancy in the model and are used in realizing subtasks such as singularity avoidance and obstacle avoidance. In addition, the link length is below its limit when introducing a sigmoid function into the kinematic model. Simulations are carried out to demonstrate the effectiveness of the proposed method and show a novel motion that avoids singular configurations through changes in link lengths.

     

Analytical development of dynamic equations of motion for a three-dimensional flexible link manipulator with revolute and prismatic joints

In this paper, a mathematical model capable of handling a three-dimensional (3D) flexible n-degree of freedom manipulator having both revolute and prismatic joints is considered. This model is used to study the longitudinal, transversal, and torsional vibration characteristics of the robot manipulator and obtain kinematic and dynamic equations of motion. The presence of prismatic joints makes the mathematical derivation complex. In this paper, for the first time, prismatic joints as well as revolute joints have been considered in the structure of a 3D flexible n-degree of freedom manipulator. The kinematic and dynamic equations of motion representing longitudinal, transversal, and torsional vibration characteristics have been solved in parametric form with no discretization. In this investigation, in order to obtain an analytical solution of the vibrational equations, a novel approach is presented using the perturbation method. By solving the equations of motion, it is shown that mode shapes of the link with prismatic joints can be modeled as the equivalent clamped beam at each time instant. As an example, this method is applied to a three degrees of freedom robot with revolute and prismatic joints. The obtained equations are solved using the perturbation method and the results are used to simulate vibrational behavior of the manipulator.     

A constrained assumed modes method for dynamics of a flexible planar serial robot with prismatic joints

In the present study, for the first time, flexible multibody dynamics for a three-link serial robot with two flexible links having active prismatic joints is presented using an approximate analytical method. Transverse vibrations of flexible links/beams with prismatic joints have complicated differential equations. This complexity is mostly due to axial motion of the links. In this study, first, vibration analysis of a flexible link sliding through an active prismatic joint having translational motion is considered. A rigid-body coordinate system is used, which aids in obtaining a new and rather simple form of the kinematic differential equation without the loss of generality. Next, the analysis is extended to include dynamic forces for a three-link planar serial robot called PPP (Prismatic, Prismatic, Prismatic), in which all joints are prismatic and active. The robot has a rigid first link but flexible second and third links. To model the prismatic joint, time-variant constraints are written, and a motion equation in a form of virtual displacement and virtual work of forces/moments is obtained. Finally, an approximate analytical method called the “constrained assumed modes method” is presented for solving the motion equations. For a numerical case study, approximate analytical results are compared with finite element results, which show that the two solutions closely follow each other.     

Nonlinear Control of a Robot Manipulator with a Nonholonomic Jerk Constraint

We study the control of a prismatic‐prismatic‐revolute (PPR) robot manipulator subject to a nonholonomic jerk constraint, i.e., a third‐order nonintegrable design constraint. The mathematical model is obtained using the method of Lagrange multipliers. The control inputs are two forces and a torque applied to the prismatic joints and the revolute joint, respectively. The control objective is to control the robot end‐effector movement while keeping the transverse jerk component as zero. The main result of the paper is the construction of a feedback control algorithm that transfers the manipulator from any initial equilibrium configuration to the zero equilibrium configuration in finite time. The effectiveness of the algorithm is illustrated through a simulation example.     

Joints Position Estimation of a Redundant Scara Robot by Means of the Unscented Kalman Filter and Inertial Sensors

A method is presented for estimating the position of rotational and prismatic joints of a novel SCARA‐type fault‐tolerant redundant manipulator using inertial sensors (accelerometers and gyroscopes). The estimation is based on the integration of the different sensors by means of the modified AUKF algorithm. The results of the evaluation of this integration scheme are compared with the CMRGD and DCMR methods, which are used for the estimation of the positions of rotational joints based on inertial sensors, showing a clear advantage of the proposed integration method over existing methods for estimating the joint angles, in addition to allowing the calculation of the position of prismatic joints.     

Global positioning of robot manipulators with mixed revolute and prismatic joints

The existing controllers for robot manipulators with uncertain gravitational force can globally stabilize only robot manipulators with revolute joints. The main obstacles to the global stabilization of robot manipulators with mixed revolute and prismatic joints are unboundedness of the inertia matrix and the Jacobian of the gravity vector. In this note, a class of globally stable controllers for robot manipulators with mixed revolute and prismatic joints is proposed. The global asymptotic stabilization is achieved by adding a nonlinear proportional and derivative term to the linear proportional-integral-derivative (PID) controller. By using Lyapunov's direct method, the explicit conditions on the controller parameters to ensure global asymptotic stability are obtained.     

Frictional contact analysis of spatial prismatic joints in multibody systems

The contact analysis of spatial prismatic joints remains a hard problem due to its complex nature. In this paper, a methodology for the frictional contact analysis of rigid multibody systems with spatial prismatic joints is presented, which is free of calculating the relative motion between the slider and guide, and is particularly suitable to the case of clearances being tiny. Under the assumption of the slider and guide being rigid, we prove that all types of contacts in the joint can be converted to point-to-point contacts. At each of the candidate points, two gap functions are introduced. However, in the proposed method, not the values of these gap functions but the relations between them are essential. In view of the non-colliding contacts being predominant when clearances of joints are tiny, we formulate the contact forces in terms of resultant frictional forces in the joint, resulting in a linear complementarity problem. By the proposed method, details about the contacts including the impact instants can be obtained, although impacts are not taken into consideration explicitly, as indicated by the numerical examples in this paper.     

双臂机械手干涉区的分析

本文在考虑构件形状和尺寸的条件下,提出了当双臂机械手在三维空间实现各自预定轨迹时判别干涉区的方法.以构件的简化模型建立起确定干涉区的模型和准则.本文所提出的方法适用于具有转动副和移动副的一般双臂机械手的干涉区分析.最后给出了一个数字例.     

Dynamics and Coupling Actuation of Elastic Underactuated Manipulators

This paper investigates the constraint and coupling characteristics of underactuated manipulators by proposing an elastic model of the manipulator and examining the second order constraint equation. A dynamic model and a coupling constraint equation are developed from a Jacobian matrix and the Newton‐Euler formulation. The inertia matrix and the Christoffel tensor are analyzed and decomposed into the part concerning actuated joints and the part concerning passive joints. This decomposition is further extended to the dynamic coupling equation and generates an actuation coupling matrix and a dynamic coupling tensor. Two new dynamic coupling indices are hence identified. One is related to an actuation input and the other is related to centrifugal and Coriolis forces. The former reveals the dynamic coupling between the input and the acceleration of passive joints and gives the actuation effect on the passive joints. The latter reveals the dynamic coupling between the centrifugal and Coriolis forces and the acceleration of passive joints and provides the centrifugal and Coriolis effect on the acceleration of passive joints. The study reveals the coupling characteristics of an underactuated manipulator. This is then demonstrated in a three‐link manipulator and extended to a serial manipulator with passive prismatic joint. © 2003 Wiley Periodicals, Inc.     

Improving the absolute positioning accuracy of robot manipulators

The absolute positioning accuracy of robot manipulator can be increased substantially by updating the nominal link parameters in the control software. This paper presents a general method to estimate the link parameter errors for any serial link manipulator (i.e., n links, and any combination of revolute and prismatic joints). The parameters are estimated through a linear kinematic model which relates the link bias errors to the end-effector positioning error. Only end-effector measurements are required instead of individual link measurements to implement this method.     

New closed-form solution for kinematic parameter identification ofa binocular head using point measurements

This paper proposes a new closed-form solution for identifying the kinematic parameters of an active binocular head having four revolute joints and two prismatic joints by using three-dimensional (3-D) point (position) measurements of a calibration point. Since this binocular head is composed of off-the-shelf components, its kinematic parameters are unknown. Therefore, we can not directly apply those existing nonlinear optimization methods. Even if we want to use the nonlinear optimization methods, a closed-form solution can be first applied to obtain accurate enough initial values. Hence, this paper considers only methods that provide closed-form solutions, i.e., those requiring no initial estimates. Notice that most existing closed-form solutions require pose (i.e., both position and orientation) measurements. However, as far as we know, there is no inexpensive technique which can provide accurate pose measurements. Therefore, existing closed-form solutions based on pose measurements can not give us the required accuracy. As a result, we have developed a new method that does not require orientation measurements and can use only the position measurements of a calibration point to obtain highly accurate estimates of kinematic parameters using closed-form solutions. The proposed method is based on the complete and parametrically continuous (CPC) kinematic model, and can be applied to any kind of kinematic parameter identification problems with or without multiple end-effecters, providing that the links are rigid, the joints are either revolute or prismatic and no closed-loop kinematic chain is included.     

Contact Conditions for Cylindrical,Prismatic, and Screw Joints in Flexible Multibody Systems

This paper focuses on the modeling of the contact conditionsassociated with cylindrical, prismatic, and screw joints in flexiblemultibody systems. In the classical formulation these joints aredeveloped for rigid bodies, and kinematic constraints are enforcedbetween the kinematic variables of the two bodies. These constraintsexpress the conditions for relative translation and rotation of the twobodies along and about a body-fixed axis, and imply the relative slidingand rotation of the two bodies which remain in constant contact witheach other. However, these kinematic constraints no longer implyrelative sliding with contact when one of the bodies is flexible. Toremedy this situation, a sliding joint and a sliding screwjoint are proposed that involves kinematic constraints at theinstantaneous point of contact between the sliding bodies. For slidingscrew joints, additional constraints are added on the relative rotationof the contacting bodies. Various numerical examples are presented thatdemonstrate the dramatically different behavior of cylindrical,prismatic, or screw joints and of the proposed sliding and sliding screwjoints in the presence of elastic bodies, and the usefulness of theseconstraint elements in the modeling of complex mechanical systems.     

A linear complementarity approach to the articulated multifingered friction gripper

In this article, the problem of grasping by an articulated multifingered gripper is investigated. The fingers' joints are idealized by angular and linear springs for the revolute and prismatic joints, respectively. The method, which was developed here, considers the cases of both hard and soft fingers. The unilateral frictional contact problem for the gripper-object system is formulated as a linear complementarity problem. Numerical examples illustrating the theory are given.     

Forward displacement analysis and uncertainty configurations of parallel manipulators with a redundant branch

The effect of adding a redundant branch in terms of reduction of the number of assembly modes and elimination of potential uncertainty configuration types is investigated for a class of parallel manipulators. Considered is a broad class that includes all three-branch manipulators where each branch is comprised of a serial arrangement of three main-arm joints supporting a common payload platform through a passive spherical branch end joint-group. The addition of a redundant branch effectively yields a four-branch manipulator class. Considered in particular is a 3–4 form of the manipulator where two branch ends meet at one point on the mobile platform. Symmetric main-arm joint sensing and actuation (two sensed/acutated main-arm joints per branch) is utilized. Synthetic geometry is used to study the number of assembly configurations of the resulting 3–4 four-branch parallel manipulators. It is presented that the number of assembly modes of three-branch parallel manipulators with passive spherical branch end joints can be reduced by utilizing a redundant branch. It is shown that there exist up to eight and up to four assembly modes when all unsensed joints are revolute and when all unsensed joints are prismatic, respectively. Combinations of unsensed prismatic and revolute joints are also investigated. It is determined that there are up to eight and up to four assembly modes when the unsensed main-arm joint of one of the concurrent branches is prismatic and when the unsensed joints of both concurrent branches are prismatic joints, respectively. Resolving the potential assembly modes require only the consideration of, at highest, second-order single-variable polynomials. In addition, kinematic design considerations allowing reduction of feasible assembly modes are discussed. The investigation of potential uncertainty configuration types is based on examining degeneracies of the screw systems formed by wrenches associated with the forces that the actuated-joints can apply. All linear dependency cases that could potentially cause uncertainties for the class of four-branch manipulators are identified. It is shown that while significantly reducing potential uncertainty configuration types, the addition of a redundant branch number cannot eliminate all potential dependency (uncertainty) cases completely. For the remaining potential uncertainty configuration types, the characteristics of the corresponding unconstrained instantaneous degrees of freedom are discussed. © 1996 John Wiley & Sons, Inc.     

A new family of hybrid 4‐DOF parallel mechanisms with two platforms and its application to a footpad device

This paper proposes a new family of 4‐degrees‐of‐freedom (DOF) parallel mechanisms with two platforms and its application to a footpad device that can simulate the spatial motions of the human foot. The new mechanism consists of front and rear platforms, and three limbs. Two limbs with 6‐DOF serial joints (P ‐S‐P‐P) are attached to each platform and are perpendicular to the base plate, while the middle limb is attached to the revolute joint that connects the front and rear platforms. The middle limb is driven by the 2‐DOF driving mechanism that is equivalent to active serial prismatic and revolute joints (P_e ‐R_e ), or prismatic and prismatic joints (P_e ‐P_e ) with two base‐fixed prismatic actuators. Since the middle limb perpendicular to the base plate has 3‐DOF serial joints (P_e ‐R_e ‐R or P_e ‐P_e ‐R), two new 4‐DOF parallel mechanisms with two platforms can generate pitch motion of each platform, and roll and heave motions (1T‐3R) or pitch motion of each platform and two translational motions (2T‐2R) at both platforms, according to the type of the 2‐DOF driving mechanism. Kinematic analyses of the 1T‐3R mechanism were performed, including inverse and forward kinematics and velocity analysis. Based on the 1T‐3R mechanism, a footpad device was designed to generate foot trajectories for natural walking. © 2005 Wiley Periodicals, Inc.     

Modeling of flexible-link manipulators with prismatic joints

The axially translating flexible link in flexible manipulators with a prismatic joint can be modeled using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separation of variables cannot be applied to solve this partial differential equation. In this paper, we present a nondimensional form of the Euler-Bernoulli beam equation using the concept of group velocity and present conditions under which separation of variables and assumed modes method can be used. The use of clamped-mass boundary conditions lead to a time-dependent frequency equation for the translating flexible beam. We present a novel method to solve this time-dependent frequency equation by using a differential form of the frequency equation. We then present a systematic modeling procedure for spatial multi-link flexible manipulators having both revolute and prismatic joints. The assumed mode/Lagrangian formulation of dynamics is employed to derive closed form equations of motion. We show, using a model-based control law, that the closed-loop dynamic response of modal variables become unstable during retraction of a flexible link, compared to the stable dynamic response during extension of the link. Numerical simulation results are presented for a flexible spatial RRP configuration robot arm. We show that the numerical results compare favorably with those obtained by using a finite element-based model.     

PPR型平面欠驱动机械臂的点位控制

研究了PPR型平面欠驱动机械臂(第1个关节和第2个关节是移动关节且是受控的,第3个关节为被动的转动关节)在水平面运动的点位控制问题.首先,通过输入和坐标变换方法,系统的动力学方程被变换成二阶链式形式.其次,提出用反步法推导出保证系统指数渐近稳定的控制器.仿真结果表明,机械臂能够稳定地从任意初始位置运动到任意给定的位置,从而证明了控制器设计的有效性.     

Unified topology and joint types optimization of general planar linkage mechanisms

We propose a gradient-based topology optimization method to synthesize a planar linkage mechanism consisting of links and revolute/prismatic joints, which converts an input motion to a desired output motion at its end effector. Earlier gradient-based topology optimization methods were mainly applicable to the synthesis of linkage mechanisms connected by revolute joints only. The proposed method simultaneously determines not only the topology of planar linkage mechanisms but also the required revolute and/or prismatic joint types. For the synthesis, the design domain is discretized into rectangular rigid blocks that are connected to each other by the newly proposed revolute and prismatic joint elements, the joint states of which vary depending on the corresponding design variables. The new concept of joint elements is materialized thorough an elaborately configured set of zero-length springs whose stiffnesses vary as the functions of the design variables. Therefore, any connectivity state among unconnected, rigidly connected, revolute joint, and prismatic joint states can be represented by properly adjusting the stiffnesses. After presenting our modeling, formulation, and sensitivity analysis, the developed method is tested with verification examples. Then the developed method is extended to be able include additional shape design variables and applied to solve a realistic problem of synthesizing a finger rehabilitation robotic device. We expect the developed method to play a critical role in synthesizing a wide class of general linkage mechanisms.     

A method for computing the Hessian tensor of loop closing conditions in multibody systems

This paper presents a method for computing the Hessian tensor of loop closing conditions employing the relative coordinate systems for the multibody system dynamics. Based on the efficient Jacobian computation method proposed by Orin et al., the derivatives of Jacobian matrix are evaluated for all combinations of revolute and prismatic joints. It is shown that all cases can be summarized in one simple formula and the Hessian tensor can be computed easily from the elements of the Jacobian matrix.