This paper describes spatial operators in robot dynamics, emphasizing their physical interpretation, while avoiding lengthy mathematical derivations. The spatial operators are rooted in the function space approach to the estimation theory developed in the decades that followed the introduction of the Kalman filter. In the mid 1980's, the authors recognized the analogy between Kalman filtering and robot dynamics, and began to use this approach on a wide range of multi-body systems of increasing complexity. This paper reviews the spatial operator approach to robot dynamics, and outlines current applications to the modeling, simulation and control of space robot dynamics and large molecular structures.

A diagonal equation for robot dynamics is developed by combining mass matrix factorization results with classical Lagrangian mechanics. Diagonalization implies that at each fixed time instant the equation at each joint is decoupled from all of the other joint equations. The equation involves two important variables: a vector of total joint rotational rates and a corresponding vector of working joint moments. The nonlinear Coriolis term depends on the joint angles and the rates. The total joint rates are related to the relative joint-angle rates by a linear spatial operator. The total rate at a given joint k reflects the total rotational velocity about the joint, and includes the combined effects from all the links between joint k and the base. Similarly, the working moments are related to the applied moments by the spatial operator . The working moment at a given joint is that part of the applied moment which does actual mechanical work. The diagonal equations are obtained by using the mass matrix factorization in the system Lagrangian.>

The freedom of choice in designating a base-body for free-flying manipulators gives these manipulators a base-invariance symmetry that is not encountered in fixed-base manipulators. This paper analyzes the relationship between this natural symmetry and the dynamical equations for free-flying manipulators. The base-invariance symmetry is used to develop a new formulation of the manipulator dynamics in which two independent O(N) recursions proceeding in opposite directions are summed together to obtain the complete free-flying manipulator dynamics. Computation of the operational space inertia for the links in the manipulator is also discussed.>

A diagonal equation /spl nu//spl dot/+C(/spl theta/, /spl nu/)=/spl epsiv/ for robot dynamics is developed by combining mass matrix factorization results with classical Lagrangian mechanics. The nonlinear Coriolis term C(/spl theta/, /spl nu/) depends on the joint angles /spl theta/ and the rates /spl nu/ and does no work. The total joint rates /spl nu/=m*(/spl theta/)/spl theta//spl dot/ are related to the relative joint-angle rates /spl theta//spl dot/ by a linear spatial operator m*(/spl theta/) mechanized by a base-to-tip spatially recursive algorithm.>

The dynamics and kinematics of manipulators that have fewer actuators than degrees of freedom are studied. These underactuated manipulators arise in a number of important applications such as free-flying space robots, hyperredundant manipulators, manipulators with structural flexibility, etc. In the analysis such underactuated manipulators are decomposed into component active and passive arms. This decomposition allows techniques previously developed for regular (fully actuated) manipulators to be applied to underactuated systems. Spatial operator identities are used to develop closed-form expressions for the generalized accelerations for the system. These expressions form the basis for a recursive O(N) dynamics algorithm. The structure of this algorithm is a hybrid of known forward and inverse dynamics algorithms for regular manipulators. Expressions and computational algorithms are also developed for the generalized and disturbance Jacobians for underactuated manipulators. The application of the results in the paper to space manipulators is also described.>

Techniques from the spatial operator algebra are used to obtain closed-form operator expressions for two types of linearized dynamics models: the linearized inverse and forward dynamics models. Spatially recursive algorithms of O(n) and O(n/sup 2/) complexity for the computation of the perturbation vector and coefficient matrices for the linearized inverse dynamics model (LIDM) are developed. Operator factorization and inversion identities are used to develop corresponding closed-form expressions for the linearized forward dynamics model (LFDM). Once again, these are used to develop algorithms of O(n) and O(n/sup 2/) complexity for the computation of the perturbation vector and the coefficient matrices. The algorithms for the LFDM do not require the explicit computation of the mass matrix nor its numerical inversion and are also of lower complexity than the conventional O(n/sup 3/) algorithms.>

This paper advances two linear operator factorizations of the manipulator mass matrix. Embedded in the factorizations are many of the techniques that are regarded as very efficient computational solutions to inverse and forward dynamics problems. The operator factorizations provide a high-level architectural understanding of the mass matrix and its inverse, which is not visible in the detailed algorithms. They also lead to an approach to the development of computer programs or the organisation of complexity in robot dynamics.>

The kinematics and dynamics of underactuated manipulators are discussed. It is shown that, due to the presence of passive hinges, the dynamics of the underactuated manipulators are considerably more complex than those of fully actuated manipulators. Spatial operator algebra techniques are used to develop expressions for the dynamical equations of motion, the mass matrix, the generalized Jacobian, and the Jacobian that describes the coupling between the motions of the active and passive hinges. An efficient and recursive inverse dynamics algorithm, the complexity of which depends only linearly on the number of degrees of freedom, is also described. The structure of this algorithm is a hybrid combination of well-known inverse and forward dynamics algorithms for fully actuated manipulators.>

The spatial operator algebra methodology is used to develop efficient spatially recursive algorithms for the dynamics of flexible link manipulators. The operator-based formulation closely parallels that for rigid link manipulators. The computational complexity of the forward dynamics algorithm is linear in the number of links and cubic in the number of modes.>

A spatial operator algebra approach to modeling and analysis of multibody robotic systems is discussed. It is shown that this modeling and analysis method can be used to develop O(n) recursive algorithms that compute the operational space mass matrix and the operational space coriolis/centrifugal and gravity terms of an n-link serial chain manipulator. It is also shown that, taken together, these algorithms enable an O(n) recursive implementation of operational space control.>

The Jacobian matrix, J, relates joint velocity to Cartesian velocity for an N-link serial manipulator. An operator factorization and inversion of /sup J/* /sup J/ is shown to result in an order (N) spatially recursive filtering and smoothing algorithm that solves the inverse Jacobian problems of finding the joint angle velocities given the end-effector velocity or finding the joint angle accelerations given the end-effector acceleration. It is shown that, with a proper model, these inverse Jacobian problems are equivalent to solving the forward dynamics problem for the same model. The recursive algorithm developed by G. Rodriguez (1987) to solve the forward dynamics problem is applied directly to solve these inverse Jacobian problems.>

This study extends to flexible multibody manipulators the recent results of the author (1989) on the use of spatially recursive filtering and smoothing techniques for robot arm dynamics. The configuration analyzed is that of a mechanical system of flexible bodies joined together by articulated joints. The inverse and forward dynamics problems are solved using the techniques of spatially recursive Kalman filtering and smoothing. The algorithms are easily developed using a set of identities associated with mass matrix factorization and inversion. The identities are easily derived using a spatial operator algebra developed by the author.>

It is known that the difference equations of Kalman filtering and smoothing recursively factor and invert the covariance of the output of a linear state-space system driven by a white-noise process. It is shown that similar recursive techniques factor and invert the inertia matrix of a multibody robot system. The random field models are based on the assumption that all of the inertial (D'Alembert) forces in the system are represented by a spatially distributed white-noise model. They are easier to describe than the models based on classical mechanics, which typically require extensive derivation and manipulation of equations of motion for complex mechanical systems. With the spatially random models, more primitive locally specified computations result in a global collective system behavior equivalent to that obtained with the deterministic models. The primary goal in investigating robot dynamics from the point of view of random field estimation is to provide a concise analytical foundation for solving robot control and motion planning problems.>

The spatial operator algebra developed by the authors for modeling control and trajectory design of manipulation is discussed, with emphasis on the analytical formulation of the operator algebra and its simple implementation in the Ada programming language. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of the manipulator. Inversion of operators can be efficiently obtained using techniques of recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanization of specific algorithm is greatly simplified.>

Recursive forward dynamics algorithms are developed and presented for an arbitrary number of robot arms moving a commonly held object. The multiarm forward dynamics solutions provide a thorough physical and mathematical understanding of the way several arms behave in response to a set of applied joint moments. The forward dynamics algorithms also provide the necessary analytical foundation for conducting analysis and simulation studies. The multiarm algorithms are based on the filtering and smoothing approach advanced by the author for single-arm dynamics (IEEE Trans. Robots Autom., Dec. 1987, no.6, Dec. 1987 pp.624-39), and they can be built up modularly from the single-arm algorithms. The algorithms compute recursively the joint angle-accelerations, the contact forces, and the task-object accelerations. Algorithms are also developed to evaluate in closed form the linear transformations from the active joint moments to the joint angle accelerations, to the task object accelerations, and to the task-object contact forces. A possible computing architecture is presented as a precursor to a more complete investigation of the computational performance of the dynamics algorithms.>

This paper describes the architecture and Ada language implementation of a process-level run-time control subystem for the Jet Propulsion Laboratory (JPL) telerobot system. The concept of run-time control in a combined robot-teleoperation environment is examined and the telerobot system at JPL is described. An Ada language implementation of the JPL Telerobot Run-Time Controller (RTC) is described by highlighting the functional behavior of the subsystem, defining the internal modules, and providing a functional flow time sequence of internal module activity.

The inverse and forward dynamics problems for multilink serial manipulators are solved by using recursive techniques from linear filtering and smoothing theory. The pivotal step is to cast the system dynamics and kinematics as a two-point boundary-value problem. Solution of this problem leads to filtering and smoothing techniques similar to the equations of Kalman filtering and Bryson-Frazier fixed time-interval smoothing. The solutions prescribe an inward filtering recursion to compute a sequence of constraint moments and forces followed by an outward recursion to determine a corresponding sequence of angular and linear accelerations. An inward recursion refers to a sequential technique that starts at the tip of the terminal link and proceeds inwardly through all of the links until it reaches the base. Similarly, an outward recursion starts at the base and propagates out toward the tip. The recursive solutions are O(N), in the sense that the number of required computations only grows linearly with the number of links. A technique is provided to compute the relative angular accelerations at all of the joints from the applied external joint moments (and vice versa). It also provides an approach to evaluate recursively the composite multilink system inertia matrix and its inverse. The main contribution is to establish the equivalence between the filtering and smoothing techniques arising in state estimation theory and the methods of recursive robot dynamics. The filtering and smoothing architecture is very easy to understand and implement. This provides for a better understanding of robot dynamics. While the focus is not on exploring computational efficiency, some initial results in that direction are obtained. This is done by comparing performance with other recursive methods for a planar chain example. The analytical foundation is laid for the potential use of filtering and smoothing techniques in robot dynamics and control.