Time-optimal path following for non-redundant serial manipulators using an adaptive path-discretization

Kinematic redundancy of manipulators is a well-understood topic, and various methods were developed for the redundancy resolution in order to solve the inverse kinematics problem, at least for serial manipulators. An important question, with high practical relevance, is whether the inverse kinematics solution is cyclic, i.e., whether the redundancy solution leads to a closed path in joint space as a solution of a closed path in task space. This paper investigates the cyclicity property of two widely used redundancy resolution methods, namely the projected gradient method (PGM) and the augmented Jacobian method (AJM), by means of examples. Both methods determine solutions that minimize an objective function, and from an application point of view, the sensitivity of the methods on the initial configuration is crucial. Numerical results are reported for redundant serial robotic arms and for redundant parallel kinematic manipulators. While the AJM is known to be cyclic, it turns out that also the PGM exhibits cyclicity. However, only the PGM converges to the local optimum of the objective function when starting from an initial configuration of the cyclic trajectory.

This paper contains two new works, namely (i) general task space structure of any n d.o.f. manipulator, and (ii) general procedure of determining admissibility of paths in task space for general manipulators. Several examples are presented to illustrate the general theory.

The authors analyse the contour error dynamics in contouring control of dual‐arm robotic manipulators with holonomic constraints. With the modified dynamics of robot, the dynamics of dual‐arm robot with holonomic constraints is symetrically transformed into dynamics error problem by considering the equivalent error method. It becomes the control problem of stabilisation such that it is suitable for robust control approach to improve control performance in terms of contouring control, i.e. contour accuracy with high speed. The proposed method can deal with both analytic and non‐analytic functions of desired path in task space. The experimental results reveal that the proposed method yields excellent performance in comparison to conventional distributed control. As a result, the proposed method can lead to the development of a modified distributed control for high degree of freedom robot manipulators.

The inverse kinematics problem is ill-posed near kinematic singularities leading to extremely large joint velocities. This classical problem is addressed from a different perspective; instead of prescribing the end-effector (EE) motion in terms of a path parameter, the motion is parameterized in terms of the arc length in joint space. This allows tackling the inverse kinematics problem when approaching kinematic singularities. Thereupon a singularity consistent sampling scheme can easily be devised. A sampling of the arc length gives direct control over the joint motion for prescribed EE motion. The arc length parameterization can be used for solving the (time) optimal path following problem, avoiding the ill-conditioning upfront.KeywordsSerial robotsInverse kinematicsSingularitiesPath planningArc length parameterizationNon-equidistant discretization

As Human-Robot Interaction (HRI) develops, robots are expected to learn more complex and demanding interaction skills. Complex HRI tasks are often embodied in robots that are jointly constrained by task space and joint space. In the field of Imitation Learning, scholars have explored the topic of joint constraints from task space and joint space, but there are few relevant studies in Human-Robot Interaction. In this paper, based on the Interaction Primitives framework (a HRI framework), we propose an interaction inference method that first generalizes the robot's movements in two spaces synchronously and then probabilistically fuses the two movements based on Bayesian estimation. This work was validated in the task that the robot follows a human handheld object, and the inference errors (RMSE and MAE) of the method are smaller in both task space and joint space than in IP using only singlespace inference.

This work is concerned with the contouring control problem of robotic manipulators using the method of equivalent errors. The method was previously proposed for multi-axis motion systems with emphasis on the machine tools applications. It is shown in this paper that the method of equivalent errors is equally applicable to the robot manipulators. For robotic systems, a desired path is usually described in terms of the position of the end-effector (task space), but the system dynamics are described in generalized coordinates (joint space). The proposed method utilizes the forward kinematics to transform the desired path in the task space into that in the joint space. As such, the equivalent errors can be easily defined. A two-link robot manipulator contouring a circular path has been numerically studied. It is found that the contouring errors are virtually zero (order of fento meter), verifying the effectiveness of the method.

The design and control of a robot requires the computation of some mathematical models such as the transformation models between the joint space (in which the configuration of the robot is defined) and the task space (in which the location of the end-effector is specified). These transformation models are very important since robots are controlled in the joint space, whereas tasks are defined in the task space. The modeling of robots in a systematic and automatic way requires an adequate method for the description of their structure. Several methods and notations have been proposed. The most popular among these is the Denavit-Hartenberg method. This method is developed for serial structures and presents ambiguities when applied to robots with closed or tree chains. For this reason, we will use the notation of Khalil and Kleinfinger, which gives a unified description for all mechanical articulated systems with a minimum number of parameters.

Chapter 3 - PID Control in Task Space

A cuspidal robot can move from one inverse kinematics (IK) solution to another without crossing a singularity. Multiple industrial robots are cuspidal. They tend to have a beautiful mechanical design, but they pose path-planning challenges. A task-space path may have a valid IK solution for each point along the path, but a continuous joint-space path may depend on the choice of the IK solution or even be infeasible. This article presents new analysis, path planning, and optimization methods to enhance the utility of cuspidal robots. We first demonstrate an efficient method to identify cuspidal robots and show, for the first time, that the ABB GoFa and certain robots with three parallel joint axes are cuspidal. We then propose a new path-planning method for cuspidal robots by finding all IK solutions for each point along a task-space path and constructing a graph to connect each vertex corresponding to an IK solution. Graph edges have a weight based on the optimization metric, such as minimizing joint velocity. The optimal feasible path is the shortest path in the graph. This method can find nonsingular paths as well as smooth paths that pass through singularities. Finally, we incorporate this path-planning method into a path optimization algorithm. Given a fixed workspace toolpath, we optimize the offset of the toolpath in the robot base frame while ensuring continuous joint motion. Code examples are available in a publicly accessible repository.

Chapter 1 - Terminology and general definitions

Earlier work [1] shows that using nonlinear feedback in the joint space, global practical stability and uniform attractivity of the closed loop robot system can be obtained. Furthermore, the closed loop system maintains robustness with respect to bounded parameter uncertainty, and remains stable after collision with the environment, thus returning to the desired trajectory rapidly. However, most robot tasks are usually specified in task space. This paper extents the earlier work to task space feedback. It is shown that using task space feedback the same results can be achieved, however, it is more effective to handle the trajectory planning and controller design directly in terms of task space. Numerical simulations are presented for illustration.

Control problems of motion of multi-body mechanical systems under constraints and/or with redundancy in system’s degrees-of-freedom (DOF) are treated from the standpoint of Riemannian geometry. A multi-joint reaching problem with excess DOF is tackled and it is shown that a task space PD feedback with damping shaping in joints maneuvers the endpoint of the robot arm to reach a given target in the sense of exponentially asymptotic convergence. An artificial potential inducing the position feedback in task space can be regarded as a Morse-Bott function introduced in Riemannian geometry, from which the Lagrange stability theorem can be directly extended to this redundant case. The speed of convergence of both the orbit of the endpoint in task space and the trajectory of joint vector in joint space can be adjusted by damping shaping and adequately choosing a single stiffness parameter. In the case that the endpoint is constrained on a hypersurface in E 3, the original Lagrange dynamics expressed in an implicit form by introducing a Lagrange multiplier is decomposed into two partial dynamics with the aid of decomposition of the tangent space into the image of the endpoint Jacobian matrix and the kernel orthogonally complemented to the image. The stability problem of point-to-point endpoint movement on the constraint surface is reduced to the former case without constraint.

In this paper, the possibility of bilateral delta robot with micro-macro control to enhance human ability and transfer experience of operator is considered. Human's hand movement usually occur in three dimension space with high speed and small impendence from hand's mass which is not capable with conventional serial robot that have slow speed and heavy moving part. As a result, parallel delta robot with properties of high speed and light moving part is chosen to suit human's hand behavioral. Micro-macro bilateral or scaling bilateral robot is a robot that scale the position or force from master robot to smaller slave robot. Convectional micro-macro in joint space or motor space bilateral system is not enough to control the task space of the delta robot due to non-linearity of delta robot kinematic. As a result, a micro-macro task space for delta robot in xyz plane by transform joint space into task space using inverse and forward kinematic of delta robot is proposed. Delta robot have rigid body properties which is necessary for detail work and can be considered appropriate micro-macro bilateral manipulator. However, direct kinematics of delta robot have around 40 solutions due to the complexity of pair of spherical joint that may present difficulty in design and manufacturing. Disturbance force is estimated by disturbance observer and scaled by conventional micro-macro method. The experiment and simulation is carry out to verify the proposed method.

TASK SPACE CONTROL OF THE DELTA PARALLEL ROBOT

This paper presents the inverse kinematic analysis of a planar bio-inspired soft robot in a discrete task space using artificial intelligence (AI). The soft robot manipulator comprises a stack of hydraulically amplified dielectric elastomer actuators, which exhibit snap-through instability, resulting in a discrete robot task space. The model simulates the manipulator with a series of adjustable-length rods connected with revolute joints and satisfies the piecewise constant curvature (PCC) requirement for each robot segment. The discrete task space of the robot was estimated using the Monte Carlo method. The inverse kinematic analysis was carried out through genetic algorithm optimization with an embedded forward kinematic model to minimize the error for both the position and the orientation of the robot's end-effector. We analyze the fully controlled robot at each discrete segment, and we compare the results with the reduced complexity model obtained by grouping the actuator in sets of different sizes. The algorithm showed excellent results and can potentially be used in the analysis of similar soft manipulators with discrete task spaces.