Miomir Vukobratović

Kinematics and trajectory synthesis of manipulation robots

1 Kinematic Equations.- 1.1. Introduction.- 1.2. Definitions.- 1.3. Manipulator hand position.- 1.3.1. Rodrigues formula approach.- 1.3.2. Homogeneous transformations.- 1.3.3. Spherical coordinates.- 1.3.4. Cylindrical coordinates.- 1.4. Manipulator hand orientation.- 1.4.1. Euler angles.- 1.4.2. Euler parameters.- 1.5. Manipulator hand velocities.- 1.5.1. Recursive and nonrecursive relations for linear and angular velocities.- 1.5.2. The Jacobian matrices.- 1.6. Summary.- 2 Computer-aided Generation of Kinematic Equations in Symbolic Form.- 2.1. Introduction.- 2.2. Symbolic kinematic equations.- 2.2.1. Backward and forward recursive relations.- 2.2.2. Kinematic equations for the UMS-3B manipulator.- 2.2.3. Backward recursive symbolic relations.- 2.2.4. Forward recursive symbolic relations.- 2.2.5. Treatment of revolute joints with parallel joints axes.- 2.3. The Jacobian matrix with respect to the hand coordinate frame.- 2.3.1. The Jacobian for the UMS-3B manipulator.- 2.3.2. Recursive symbolic relations for the Jacobian with respect to the hand coordinate frame.- 2.3.3. The Jacobian columns corresponding to parallel joints.- 2.4. The Jacobian matrix with respect to the base coordinate frame.- 2.4.1. The Jacobian for the UMS-3B manipulator.- 2.4.2. Recursive symbolic relations for the Jacobian with respect to the base coordinate frame.- 2.4.3. The Jacobian columns corresponding to parallel joints.- 2.5. Program implementation, numerical aspects and examples.- 2.5.1. Block-diagram of the program for the symbolic model generation.- 2.5.2. Examples.- 2.5.3. Numerical aspects.- 2.6. Summary.- Appendix I Direct Kinematic Problem for the Arthropoid Manipulator.- Appendix II The Jacobian with Respect to the Hand Coordinate Frame for the Arthropoid Manipulator.- Appendix III The Jacobion with Respect to the Base Coordinate Frame for the Arthropoid Manipulator.- 3 Inverse Kinematic Problem.- 3.1. Introduction.- 3.2. Analytical solutions.- 3.3. Numerical solutions.- 3.4. Summary.- 4 Kinematic Approach to Motion Generation.- 4.1. Introduction.- 4.2. Manipulation task.- 4.3. Trajectory planning.- 4.4. Motion between positions.- 4.4.1. Joint-interpolated motion.- 4.4.2. External coordinates motion.- 4.5. Procedurally defined motion.- 4.6. Summary.- 5 Dynamic Approach to Motion Generation.- 5.1. Introduction.- 5.2. Manipulation system dynamic model.- 5.3. An overview of methods for dynamic motion synthesis.- 5.4. Determination of the energy optimal velocity distribution using dynamic programming.- 5.5. Quasioptimal nominal trajectory synthesis using decentralized system model.- 5.6. Summary.- 6 Motion Generation for Redundant Manipulators.- 6.1. Introduction.- 6.2. Kinematic methods for redundant manipulator motion generation.- 6.3. Energy optimal motion synthesis.- 6.4. Obstacle avoidance using redundant manipulators.- 6.5. An algorithm for redundant manipulator motion synthesis in the presence of obstacles.- 6.6. Summary.- References.

Real-time dynamics of manipulation robots

1 Survey of computer-aided robot modelling methods.- 1.1. Introduction.- 1.2. Methods based on Lagranges equation.- 1.2.1. Uicker-Kahns method v.- 1.2.2. Algorithms by Mahil, Megahed and Renaud.- 1.2.3. Algorithms by Waters and Hollerbach.- 1.2.4. Vukobratovi?-Potkonjaks recursive method.- 1.3. Methods based on Newton-Eulers equations.- 1.3.1. Vukobratovi?-Stepanenkos method.- 1.3.2. Huston-Kanes method.- 1.3.3. A new recursive method.- 1.4. Methods based on Appels equations.- 1.5. Symbolic methods.- 1.6. Numeric-symbolic method.- 2 Computer-aided method for closed-form dynamic robot model construction.- 2.1. Introduction.- 2.2. Model based on Newton-Eulers equations.- 2.3. Closed-form dynamic model.- 2.4. Properties of dynamic model matrices: symmetry, positive definiteness and antisymmetry.- 2.5. Closed-form linearized model.- 2.6. Closed-form sensitivity model.- Appendix 2.1.- Appendix 2.2.- 3 Computer-aided generation of numeric-symbolic robot model.- 3.1. Introduction.- 3.2. Numeric-symbolic representation of variables.- 3.3. Algebra of polynomial matrices.- 3.4. Optimization of polynomial matrices.- 3.5. Nonlinear model.- 3.6. Linearized and sensitivity model.- 3.7. Approximate models.- Appendix 3.1.- 4 Model optimization and real-time program-code generation.- 4.1. Introduction.- 4.2. Optimal computation of polynomial matrices.- 4.3. Real-time program-code generation.- Appendix 4.1.- 5 Examples.- 5.1. Introduction.- 5.2. A cylindrical robot.- 5.3. An arthropoid robot.- 5.4. An anthropomorphic robot.- 5.5. Microcomputer implementation of analytical robot models - - time-memory requirements.- Appendix 5.1.- References.

Humanoid Robotic System with and without Elasticity Elements Walking on an Immobile/Mobile Platform

This work is concerned with the modeling and analysis of a complex humanoid robotic system walking on an immobile/mobile platform. For this purpose, a software package was synthesized which allows one to select configuration of both the humanoid and the platform. Each joint of the biped and platform can be defined by the user via the motor state (active or locked) and gear type (rigid or elastic). The user can also form very diverse configurations of the humanoid and platform. The software package forms a mathematical model. By selecting system’s parameters the simulation allows user to analyze dynamic behavior of the biped of selected configuration, walking on either an immobile or mobile platform of selected configuration. In the moment when the biped steps on the platform, the latter, by its dynamics, acts on the biped dynamics, and the biped on the other hand, by its characteristics, influences dynamics of the platform motion. These two complex contacting systems form a more complex system, whose mathematical model has to encompass all the elements of coupling between the humanoid joints and platform joints. The phenomenon of coupling is analyzed first on a humanoid robotic system with all rigid elements, which is in contact with the platform mechanism having also all rigid elements. It has been shown that coupling is more influenced when elasticity elements are included into the configuration. Insufficient knowledge of coupling characteristics may present a serious disturbance to the system in the robotic task realization. The deviation of the ZMP (Zero-Moment Point) from the reference trajectory is presented, which implies the need for the synthesis of new control structures for stabilizing biped motion on the immobile/mobile platform. The reference trajectory may be defined in very different ways and from several aspects. Reference trajectory of each joint can be defined so to encompass or not encompass elastic deformations. The control structure for the biped walking on the platform should be defined so that it satisfies the requirement for the ZMP to be within the given boundaries in every sampling instant, which guarantees dynamic balance of the locomotion mechanism in the real regime. The control is defined as CR (Centralized Reference control, calculated from the reference state), plus LO (control via local feedbacks of motor motion with respect to position and velocity). In the case of the biped motion on a mobile platform CR control is defined separately under the real conditions of unknown characteristics of coupling between the two complex systems, as well as unknown elasticity properties. The analysis of simulation results of the humanoid robot motion on a mobile platform gives evidence for all the complexity of this system and shows how much system parameters (choice of trajectory, configuration, geometry, elasticity characteristics, motor, etc.) influence stabilization of its humanoid motion.

BIOLOGICAL PRINCIPLES OF CONTROL SELECTION FOR A HUMANOID ROBOT'S DYNAMIC BALANCE PRESERVATION

This paper presents a contribution to the study of control law structures and to the selection of relevant sensory information for humanoid robots in situations where dynamic balance is jeopardized. In the example considered, the system first experiences a large disturbance, and then by an appropriate control action resumes a normal posture of standing on one leg. In order to examine the control laws used by humans, an experiment was performed in which a human subject was subjected to perturbations and the ensuing reactions were recorded to obtain complete information about the subjects motion and ground reaction force. Then, a humanoid model was advanced with characteristics matching those of the experimental human subject. The whole experiment was simulated so as to achieve a simulated motion that was similar to that of the human test subject. The analysis of the control laws applied, and the behavior of selected ground reference points (ZMP, CMP and CM projection on the ground surface), provided valuable insight into balance strategies that humanoid robots might employ to better mimic the kinetics and kinematics of humans compensating for balance disturbances.

Internal Redundancy – the Way to Improve Robot Dynamics and Control Performances

The research follows the concept of variable geometry. The concept was originally introduced for general mechanical system in order to improve its dynamic behaviour [14]. Here, we apply the concept to robots. It can be shown that enhancement of robot dynamic performances is achieved. In this paper variation of geometry is considered equally as the motion in robot joints. Thus, a new set of degrees of freedom is introduced. This leads to redundancy – different internal motions are possible for the given external motion of the end-effector. However, there is an important difference from the usual notion of redundancy. Here, the additional joints do not influence the external motion and accordingly cannot improve the end-effector ability for maneuvering. For this reason the new notion is defined the internal redundancy.Although variation of geometry is treated equally with the motion in robot joints, there is still a difference in the aim of these new degrees of freedom. They should contribute to overcoming the limits of robot actuators, achieving better static compensation, etc. One might say that internal redundancy improves the robot dynamic capabilities. In this paper the mathematical formulation of kinematics and dynamics of robots with internal redundancy is carried out. A case study is presented in order to support the main idea.

Software Package for Computer-Aided Synthesis of Control for Manipulation Robots

Abstract A software package for interactive, computer-aided synthesis of control for manipulation robots is described in the paper. The package enables the synthesis of the simplest control law which can accommodate all requirements of a particular robotic task. The choice of microprocessor structure for robot control is also provided by the package. The package uses stability analysis of the nonlinear model of the robot to synthesize decentralized control and introduce global feedback loops. The package is designed so as to allow users experience in the control synthesis to be utilized. An example of control synthesis for a six-degree-of-freedom industrial robot by applying the package is presented.

Dynamics of Biped Locomotion

The motion of living organisms by means of legs, especially the locomotion of bipeds, has always been a challenging problem to scientists of different vocations: biologists, physiologists, medicine specialists, mathematicians, and engineers. In spite of their efforts, however, this problem has not been solved yet in a satisfactory way.

Roads to Regional Conflict Resolution via Academic and National Goals of Co-operation

Abstract Experience of the impact of gradually increasing scientific co-operation during most difficult times suggest that academia can do much more for enhancing a paradigm shift towards rethinking new value patterns and revision of national goals, which as a rule have been formulated a century ago. A system analysis of the causes and of the effects is clearly in favour to rethinking to academia a novel role in improving the bilateral mutual relationships via mutually beneficial, multilateral, scientific co-operation. It is an influential way for creating new spiritual forces of fairness and justice among and within long confronted communities, groups, societies, nations. A South-European case of an ever growing and flourishing academic co-operation involving a multinational group of control scientists and engineers in a dramatic period ofmost difficult time suggests this approach is feasible and useful. It remains only to find efficient means how to influence the decision making centres of powers at national level and in the region into a process ofre-examination and re-thinking the existing outdated national goals and society value patterns

Humanoid Robot Motion in Unstructured Environment - Generation of Various Gait Patterns from a Single Nominal

Humanoid robotics has been in the focus of scientific community for decades. For a long time already, robots have not been present only in the industrial plants, at a time their traditional work space, but have been increasingly more engaged in the close living and working environment of humans. This fact inevitably leads to the need of “working coexistence” of man and robot and sharing their common working environment. The fact that no significant rearrangement of the environment of humans could be expected as a consequence of the presence of their mechatronic partners, robots will have to further “adapt” to the environment previously dedicated only to men. Besides, it is expected that the robots cooperating with humans will have an operation efficiency matching that of humans. The working and living environment, adapted to humans, imposes on robots mechanical-control structure at least two types of tasks related to its motion: motion in a specific environment with the obstacles of th e type of staircases, thresholds, multi-level floors, etc., and the motion within a very dynamic scene in which the trajectory of system’s motion can be only globally planned and the actual trajectory determined on the basis of the instantaneous situation.

Dynamical Method for the Evaluation and Choice of Industrial Manipulators

One of the basic reasons for studying the dynamics of active mechanisms applicable to robotics is that it is desirable to be thoroughly acquainted with the dynamical properties of robots-manipulators during their design.