Workspace Coordinates Research Articles

Robust task-space control of robot manipulators using differential equations for uncertainty estimation

SUMMARYMost control algorithms for rigid-link electrically driven robots are given in joint coordinates. However, since the task to be accomplished is expressed in Cartesian coordinates, inverse kinematics has to be computed in order to implement the control law. Alternatively, one can develop the necessary theory directly in workspace coordinates. This has the disadvantage of a more complex robot model. In this paper, a robust control scheme is given to achieve exact Cartesian tracking without the knowledge of the manipulator kinematics and dynamics, actuator dynamics and nor computing inverse kinematics. The control design procedure is based on a new form of universal approximation theory and using Stone–Weierstrass theorem, to mitigate structured and unstructured uncertainties associated with external disturbances and actuated manipulator dynamics. It has been assumed that the lumped uncertainty can be modeled by linear differential equations. As the method is Model-Free, a broad range of manipulators can be controlled. Numerical case studies are developed for an industrial robot manipulator.

Real-Time Optimization for Eye-in-Hand Visual Search

This paper presents a multistage, real-time optimization algorithm to perform visual searches. A cost function relates the success of the visual search at any given pose in the workspace. This function is measurable at every camera pose, but it is not necessarily known as a function of changing pose, and its gradient with respect to the workspace or joint space coordinates is not known. In the first stage, a camera is constrained on a fixed sphere centered on the robot and searches for a general target direction by multidirectional direct search. In the second stage, the camera is constrained on a series of spheres centered on refined estimates of the search target's position. A series of extremum seeking control (ESC) algorithms are executed to search the best pose to observe the target. Since there is no existing method to detect when ESC has converged, we propose a novel frequency domain criteria to decide when to terminate the current ESC and launch another on the next candidate sphere. We prove that, under certain conditions, ESC will converge to known neighborhood of the optimal point in finite time. Simulations and experiments are presented to illustrate the effectiveness of the proposed algorithm.

Velocity and Force Observers for the Control of Robot Manipulators

In this paper, position/force tracking control for rigid robot manipulators interacting with its environment is considered. It is assumed that only joint angles are available for feedback, so that velocity and force observers are designed. The principle of orthogonalization is employed for this particular purpose and some of its main properties are fully exploited to guarantee local asymptotical stability. Only the force observer requires the dynamic model of the robot manipulator for implementation, and the scheme is developed directly in workspace coordinates, so that no inverse kinematics is required. The proposed approach is tested experimentally and compared with a well–known algorithm.

The role of the mass matrix in the analysis of mechanical systems

This paper proposes a simple, intuitive method of analyzing and understanding the mass matrix of a mechanical system. With very little calculation, it is possible to determine four important values that describe the inertia and inertial coupling of a mechanism: the locked effective inertia, the force coupling, the free effective inertia, and the acceleration coupling. These values can be determined for a system formulated in terms of joint coordinates, but when used with systems formulated in terms of workspace coordinates, this work provides a unified method of determining important, well-established concepts such as effective mass (particularly important in the field of human-robot interaction) and dynamic isotropy (an important consideration for robotic manipulators and haptic devices). Two examples are provided which highlight the use of this method to analyze robotic manipulators.

Global Manipulation Planning in Robot Joint Space With Task Constraints

We explore global randomized joint-space path planning for articulated robots that are subjected to task-space constraints. This paper describes a representation of constrained motion for joint-space planners and develops two simple and efficient methods for constrained sampling of joint configurations: tangent-space sampling (TS) and first-order retraction (FR). FR is formally proven to provide global sampling for linear task-space transformations. Constrained joint-space planning is important for many real-world problems, which involves redundant manipulators. On the one hand, tasks are designated in workspace coordinates: to rotate doors about fixed axes, to slide drawers along fixed trajectories, or to hold objects level during transport. On the other hand, joint-space planning gives alternative paths that use redundant degrees of freedom (DOFs) to avoid obstacles or satisfy additional goals while performing a task. We demonstrate that our methods are faster and more invariant to parameter choices than the techniques that exist.

Cartesian control of robots without dynamic model and observer design

Most control algorithms for rigid robots are given in joint coordinates. However, since the task to be accomplished is expressed in Cartesian coordinates, inverse kinematics has to be computed in order to implement the control law. Alternatively, one can develop the necessary theory directly in workspace coordinates. This has the disadvantage of a more complex robot model. In this paper, a control-observer scheme is given to achieve exact Cartesian tracking without the knowledge of the manipulator dynamics nor computing inverse kinematics. Also, only joint measurements are used.

Rapid learning with parametrized self-organizing maps

The construction of computer vision and robot control algorithms from training data is a challenging application for artificial neural networks. However, many practical applications require an approach that is workable with a small number of data examples. In this contribution, we describe results on the use of ‘Parametrized Self-organizing Maps’ (‘PSOMs’) with this goal in mind. We report results that demonstrate that a small number of labeled training images is sufficient to construct PSOMs to identify the position of finger tips in images of 3D-hand shapes to within an accuracy of only a few pixel locations and we present a framework of hierarchical PSOMs that allows rapid ‘one-shot-learning’ after acquiring a number of ‘basis mappings’ during a previous ‘investment learning stage’. We demonstrate the potential of this approach with the task of constructing the position-dependent mapping from camera coordinates to the work space coordinates of a Puma robot.

Control of human arm movements in two dimensions: paths and joint control in avoiding simple linear obstacles.

In order to examine path planning and the control of redundant degrees of freedom in the human arm, the movements of the shoulder, elbow and wrist were recorded as subjects moved a pointer to a target and avoided a simple obstacle. With respect to joint control, the results show that the extra degree of freedom provided by the wrist is incorporated into target movements in a systematic manner for both large and small obstacles; it is not used only when there is no geometrical alternative. For the wrist, two strategies are apparent, depending upon the length of the obstacle. Wrist extension predominates for shorter obstacles, while flexion or extension and flexion predominate for longer obstacles. These wrist movements shorten the effective length of the distal segments (lower arm plus hand and pointer) and thus reduce the excursion required at the proximal joints. In part, they correspond to assuming the most comfortable arm configuration at each point in the new path necessitated by the obstacle and can be described by static cost functions. However, wrist extension is also used to move the hand and pointer away from the obstacle as shoulder and elbow movements carry the wrist itself towards the obstacle. Wrist flexion is also used to move the pointer tip rapidly past the obstacle. These components, which cannot be explained by static cost functions alone, confirm for the human arm the hypothesized use of redundant degrees of freedom in obstacle avoidance. With respect to path planning, the results show that the minimum distance between pointer and obstacle remains fairly constant over a large range of obstacle lengths; this relative invariance is interpreted to support the hypothesis that workspace coordinates are important for movement planning. However, minimum distance and several other path parameters do depend significantly on the orientation and location of the movement in the workspace. This inhomogeneity implies that movement planning does not occur exclusively in workspace coordinates; it suggests an influence of joint space criteria. In frontal movements, for example, the systematic decline in the minimum distance with increasing obstacle length is interpreted as a compromise reducing the amount of extra joint movement and the discomfort of arm configurations.

Solution of the direct and inverse kinematic problems by a common algorithm based on the mean of multiple computations

For the control of the movement of a multijoint manipulator a “mental model” which represents the geometrical properties of the arm may prove helpful. Using this model the direct and the inverse kinematic problem could be solved. Here we propose such a model which is based on a recurrent network. It is realized for the example of a three-joint manipulator working in a two-dimensional plane, i.e., for a manipulator with one extra degree of freedom. The system computes the complete set of variables, in our example the three joint angles and the two work-space coordinates of the endpoint of the manipulator. The system finds a stable state and a geometrically correct solution even if only a part of these state variables is given. Thus, the direct and the inverse kinematic problem as well as any mixed problem, including the underconstrained case, can be solved by the network.

An algorithmic approach to coordinate transformation for robotic manipulators

Coordinate transformation is one of the most important issues in robotic manipulator control. Robot tasks are naturally specified in work space coordinates, usually a Cartesian frame, while control...

Coordinate Transformation: A Solution Algorithm for One Class of Robots

One of the most important features of an advanced control system for articulated robots is the capability of transforming the work space coordinates, which naturally characterize any robot task, into the corresponding joint coordinates, on which control actions are developed. For each task, the coordinate transformation problem consists in calculating one trajectory in the joint space which corresponds to the end effector trajectory, usually given in the Cartesian space. While simple kinematical structures allow for closed-form solutions, there is a class of robots for which this is not true. Typical articulated robot structures have three revolute joints at the end effector; the geometric parameters of these joints actually determine the spatial configuration of the last axes of motion. The large majority of today's nonredundant structures have three intersecting axes at the end effector, and closed form solutions do exist in this case. If the axes intersect two-by-two, as in some rather common arm design, an exact solution seems not to exist. A quite different solution algorithm is established, as compared to the trigonometric approach widely adopted so far, which yields solutions in case of two-by-two intersecting axes. The convergence of the algorithm along any trajectory is proved. Effectiveness of the proposed technique can be argued by the fact that it only makes use of direct kinematics, thus resulting in a contained computational burden. A robot prototype of the kind described above is taken as a reference in order to discuss digital implementation and develop numerical examples.