We address the motion planning problem for a robotic system whose configuration manifold contains a group of rotations. Our approach is applied to a free-floating space robot composed of a three-dimensional base (a spacecraft) and an anthropomorphic onboard manipulator. The robot is actuated by the torques exerted at the joints of the onboard manipulator. A coordinate-free representation of rotations is utilized. The Lagrangian formalism is employed in order to derive a dynamics model of the robot that takes the form of a control system defined on the group of rotations and the joint space of the onboard manipulator. Using the conservation of angular momentum of the robot, a Jacobian motion planning algorithm is designed relying on the Endogenous Configuration Space Approach. The performance of the algorithm is verified by computer simulations.

This paper discusses the nonholonomic robotic systems whose motion constraints assume the Pfaffian form, and the equations of motion are represented by driftless control systems with outputs. By reference to the end point map of such a control system, we define the system's Jacobian and study Jacobian motion-planning algorithms. A new Lagrange-type Jacobian inverse, referred to as the General Lagrangian Jacobian Inverse (GLJI), is designed as the solution of an optimal control problem with a Lagrange-type objective function. Singularities of GLJI are examined. A special choice of the objective function illustrates features of GLJI. A new motion-planning algorithm based on GLJI is proposed. Theoretical arguments are illustrated with a motion-planning problem of a space robot.

The paper presents a task-priority motion planning algorithm for a balancing robot, derived from the endogenous configuration space approach. The motion planning problem is composed of three sub-tasks. Firstly, the robot has to reach a desirable point in task space (proper motion planning). Secondly, the upward position of the robot torso should be maintained during the motion. Thirdly, the motion should save the control energy. Performance of our algorithm is presented in simulations.

This paper addresses a constrained motion planning problem for mobile manipulators. The constraints are included into the system model by means of a sort of penalty function, and then processed in accordance with the endogenous configuration space approach. Main novelty of this paper lies in deriving a constrained Jacobian motion planning algorithm with the following features: inequality constraints are included into an extended kinematics model using a smooth approximation of the plus function, the model is then regularized against singularities, and the resulting imbalance in error equations is handled as a perturbation of an exponentially stable linear dynamic system. The operation of the constrained motion planning algorithm is illustrated by a motion planning problem of a mobile manipulator with bounds imposed on a platform variable. Performance of the algorithm is tested by computer simulations.

Extended Jacobian inverse kinematics algorithms for redundant robotic manipulators are defined by combining the manipulator's kinematics with an augmenting kinematics map in such a way that the combination becomes a local diffeomorphism of the augmented taskspace. A specific choice of the augmentation relies on the optimal approximation by the extended Jacobian of the Jacobian pseudoinverse (the Moore-Penrose inverse of the Jacobian). In this paper, we propose a novel formulation of the approximation problem, rooted conceptually in the Riemannian geometry. The resulting optimality conditions assume the form of a Poisson equation involving the Laplace-Beltrami operator. Two computational examples illustrate the theory.

Based on the concept of endogenous configuration space we derive an extended Jacobian inverse kinematics algorithm for doubly nonholonomic mobile manipulators, i.e. mobile manipulators built of a nonholonomic manipulator and a nonholonomic mobile platform. A key step in the derivation consists in de fining a local diffeomorphism between the endogenous configuration space and the suitably augmented (extended) taskspace of the mobile manipulator. This has been achieved by decomposing the Fourier series representation of control functions driving the mobile manipulator into a finite dimensional part isomorphic to the taskspace, and a complementary in finite dimensional part augmenting the taskspace. By definition, our algorithm has the property of repeatability. Theoretical considerations are illustrated with computer simulations accomplished for a 3 d.o.f. planar nonholonomic manipulator mounted on a kinematic car-type platform.

On the basis of a geometric characterization of repeatability we present a repeatable extended Jacobian inverse kinematics algorithm for mobile manipulators. The algorithm's dynamics have linear invariant subspaces in the configuration space. A standard Ritz approximation of platform controls results in a band-limited version of this algorithm. Computer simulations involving an RTR manipulator mounted on a kinematic car-type mobile platform are used in order to illustrate repeatability and performance of the algorithm.

We study doubly nonholonomic mobile manipulators composed of a nonholonomic mobile platform and a nonholonomic manipulator fixed to the platform. The kinematics of such mobile manipulators are represented by a pair of driftless control systems driven by platform and manipulator controls, and sharing an output function that describes positions and orientations of the end effector. The approach assumed in this paper is based on the concept of endogenous configuration defined as a pair of control functions of the platform, and of the aboard manipulator. For doubly nonholonomic mobile manipulators we define local performance measures and examine the inverse kinematic problem. Our results consist in a definition of the dexterity ellipsoid, a characterization of dextrous and isotropic endogenous configurations, and a derivation of a Jacobian pseudoinverse inverse kinematics algorithm. Computer simulations involving a 3 d.o.f. planar nonholonomic manipulator mounted on a kinematic car type platform illustrate our approach.

The continuous inverse kinematic problem amounts to computing a configuration of a mobile manipulator capable of producing a prescribed trajectory in the taskspace. By adopting a control system representation of kinematics of the mobile manipulator this problem is made equivalent to the problem of exact or asymptotic output tracking that may be solved by the dynamic extension algorithm. We study the continuous inverse kinematic problem for a mobile manipulator comprising a 3 DOF RTR manipulator mounted on a kinematic car type mobile platform. Our main result is a continuous asymptotic inverse kinematics algorithm employing dynamic state feedback. The performance of this inversion algorithm has been examined in computer simulations.

In this paper the problem of global trajectory tracking for the kinematic model of a unicycle-type mobile robot is considered. It is assumed that some of the tracking error coordinates are not measurable. Using a cascaded systems approach we develop full order and reduced order observers, and introduce an observer based controller resulting in K-exponential convergence of the tracking error. Simulations are provided to illustrate the results.

We consider the kinematics of a mobile manipulator that consists of a nonholonomic (or holonomic) mobile platform and a holonomic manipulator mounted atop of the platform. At any fixed time instant and with fixed initial posture of the platform these kinematics are viewed as a map from a Hilbert space of controls and manipulator joint positions into a task-space of the mobile manipulator, and called the instantaneous kinematics. A derivative of this map is referred to as the instantaneous analytic Jacobian of the mobile manipulator. We derive an instantaneous dexterity matrix and a dexterity measure of the mobile manipulator. The dexterity matrix is used in order to introduce an instantaneous dexterity ellipsoid of the mobile manipulator. Concepts are illustrated with examples of mobile manipulators equipped with both holonomic and nonholonomic mobile platforms.

We present a normal form approach to solving the singular inverse kinematic problem for non-redundant robot kinematics. A standing assumption is made that singular configurations of the kinematics have corank 1 and that the kinematics are equivalent to the quadratic normal form. The approach has been illustrated with a simulation case study of 2R kinematics. A comparison with the singularity-robust inverse technique and the null-space based approach has demonstrated advantages of the normal form approach.

We derive a rank condition under which the redundant robot kinematics around a corank 1 singular configuration can be given a quadratic normal form. This normal form is further exploited to introduce new sufficient conditions for local avoidability and unavoidability of singular configurations.

In this article, a new method of solving the singular inverse kinematic problem for nonredundant robotic manipulators is proposed, based on normal forms of the manipulator kinematics. A complete solution to the inverse problem is presented under assumption that the kinematics are equivalent to the quadratic normal form. Comparisons with other existing methods, including the singularity-robust inverse, are made that reveal advantages of the normal form approach. Experimental results concerned with singular tracking in the AdeptOne manipulator are reported.

We study singular kinematics of robot manipulators around internal configurations at which the manipulator Jacobian drops rank by 1. Conditions are presented under which such kinematics can be transformed to a simple quadratic normal form. The conditions are proved using standard methods of singularity theory, and illustrated with an example.>

The calibration of manipulator kinematics subject to both geometric and nongeometric variations is addressed. A general definition of calibration is introduced, based on the concept of the group action. A specific calibration method is proposed, called calibration by diffeomorphisms. For calibration by diffeomorphisms, the existence of calibrating transformations has been shown to be equivalent to the structural stability of the nominal kinematics. In the case of parametric variations of kinematics of either geometric or nongeometric origin, the calibration by diffeomorphisms has yielded approximate affine calibrating transformations defined as solutions to a homological equation. For structurally stable nominal kinematics, the homological equation is solvable and provides calibrating transformations defined globally, applicable to both nonsingular and singular kinematics. The affine calibrating transformations have been illustrated with examples including general 2R position kinematics. >